From 7f9eb1690c506f9ca42a3026e8bdd21816dba7ba Mon Sep 17 00:00:00 2001 From: Alex-Zughaid <117576511+Alex-Zughaid@users.noreply.github.com> Date: Thu, 30 Jul 2026 12:06:59 +0100 Subject: [PATCH 1/3] monthly updates --- .github/workflows/monthly-updates.yml | 126 + web2/.gitignore | 6 + web2/data/monthly-updates/2026-06.json | 13102 +++++++++++++++++++++ web2/data/monthly-updates/README.md | 30 + web2/public/monthly-updates/2026-06.pdf | Bin 0 -> 934876 bytes web2/scripts/generate-monthly-updates.js | 1166 ++ web2/scripts/lean-decls.js | 403 + web2/src/app/monthly-updates/page.tsx | 150 + web2/src/lib/site.ts | 1 + web2/src/lib/yaml.ts | 84 +- 10 files changed, 15067 insertions(+), 1 deletion(-) create mode 100644 .github/workflows/monthly-updates.yml create mode 100644 web2/data/monthly-updates/2026-06.json create mode 100644 web2/data/monthly-updates/README.md create mode 100644 web2/public/monthly-updates/2026-06.pdf create mode 100644 web2/scripts/generate-monthly-updates.js create mode 100644 web2/scripts/lean-decls.js create mode 100644 web2/src/app/monthly-updates/page.tsx diff --git a/.github/workflows/monthly-updates.yml b/.github/workflows/monthly-updates.yml new file mode 100644 index 0000000..0dcc27e --- /dev/null +++ b/.github/workflows/monthly-updates.yml @@ -0,0 +1,126 @@ +name: Generate monthly update + +on: + schedule: + # Runs at 03:15 UTC on the 1st of every month. Uses the UTC clock so the + # "previous month" that the script picks matches the calendar month that + # just ended. + - cron: "15 3 1 * *" + workflow_dispatch: + inputs: + month: + description: "Optional YYYY-MM to generate (default: previous month)" + required: false + type: string + backfill: + description: "Optional number of previous months to backfill" + required: false + type: string + force: + description: "Overwrite existing JSON files" + required: false + type: boolean + default: false + +permissions: + contents: write + +# Two runs committing to the same branch would race on push. +concurrency: + group: monthly-updates + cancel-in-progress: false + +jobs: + generate: + runs-on: ubuntu-latest + steps: + - name: Checkout + uses: actions/checkout@v4 + with: + persist-credentials: true + + - name: Set up Node.js + uses: actions/setup-node@v4 + with: + node-version: "20" + + - name: Install Tectonic (LaTeX engine) + uses: wtfjoke/setup-tectonic@v3 + with: + github-token: ${{ secrets.GITHUB_TOKEN }} + + # The generator reads whole Lean files at both month-boundary commits, so + # it needs a clone of physlib. Caching it turns a full clone into a small + # incremental fetch on subsequent runs. + - name: Cache physlib clone + uses: actions/cache@v4 + with: + path: web2/.cache/physlib.git + key: physlib-clone-${{ github.run_id }} + restore-keys: physlib-clone- + + # Inputs go through env, never string-interpolated into the script: a + # `${{ inputs.month }}` spliced straight into `run:` lets a dispatch value + # like `2026-01; curl … | sh` execute as shell. + - name: Run generator + working-directory: web2 + env: + GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }} + MONTH: ${{ inputs.month }} + BACKFILL: ${{ inputs.backfill }} + FORCE: ${{ inputs.force }} + run: | + ARGS=() + if [ -n "$MONTH" ]; then + ARGS+=(--month "$MONTH") + fi + if [ -n "$BACKFILL" ]; then + ARGS+=(--backfill "$BACKFILL") + fi + if [ "$FORCE" = "true" ]; then + ARGS+=(--force) + fi + node scripts/generate-monthly-updates.js "${ARGS[@]}" + + # Only the JSON and the PDF are published. The .tex intermediate lives in + # web2/.monthly-updates-build/ and is gitignored. + - name: Commit & push new report(s) + id: commit + env: + MONTH: ${{ inputs.month }} + run: | + git config user.name "github-actions[bot]" + git config user.email "41898282+github-actions[bot]@users.noreply.github.com" + git add web2/data/monthly-updates web2/public/monthly-updates + if git diff --cached --quiet; then + echo "No new monthly reports to commit." + echo "committed=false" >> "$GITHUB_OUTPUT" + exit 0 + fi + MSG="chore(monthly-updates): auto-generated report" + if [ -n "$MONTH" ]; then + MSG="$MSG for $MONTH" + fi + git commit -m "$MSG" + # main may have moved while TeX was installing. + git pull --rebase --autostash origin "${GITHUB_REF_NAME}" + git push origin "HEAD:${GITHUB_REF_NAME}" + echo "committed=true" >> "$GITHUB_OUTPUT" + + # The site deploys via Gabrielebattimelli/Physlib-Website (Vercel), which + # mirror-to-personal.yml normally keeps in sync on push. That workflow + # will NOT fire for the commit above: GitHub deliberately does not trigger + # workflows for pushes made with the default GITHUB_TOKEN, to avoid + # recursion. Without this step the report lands on main and never reaches + # the live site. So mirror it here, using the same secret and target as + # mirror-to-personal.yml. + - name: Mirror to personal repo (Vercel) + if: steps.commit.outputs.committed == 'true' + env: + MIRROR_PAT: ${{ secrets.MIRROR_PAT }} + run: | + git config --unset-all http.https://github.com/.extraheader || true + git config --global credential.helper store + echo "https://x-access-token:${MIRROR_PAT}@github.com" > ~/.git-credentials + git remote add mirror https://github.com/Gabrielebattimelli/Physlib-Website.git + git push mirror "HEAD:refs/heads/${GITHUB_REF_NAME}" diff --git a/web2/.gitignore b/web2/.gitignore index 5ef6a52..b20ef95 100644 --- a/web2/.gitignore +++ b/web2/.gitignore @@ -39,3 +39,9 @@ yarn-error.log* # typescript *.tsbuildinfo next-env.d.ts + +# monthly-updates generator intermediates. Only data/monthly-updates/*.json and +# public/monthly-updates/*.pdf are committed. +/.monthly-updates-build/ +/.cache/ +/build.log diff --git a/web2/data/monthly-updates/2026-06.json b/web2/data/monthly-updates/2026-06.json new file mode 100644 index 0000000..2614fa6 --- /dev/null +++ b/web2/data/monthly-updates/2026-06.json @@ -0,0 +1,13102 @@ +{ + "slug": "2026-06", + "label": "Jun 2026", + "year": 2026, + "month": 6, + "startDate": "2026-06-01T00:00:00.000Z", + "endDate": "2026-07-01T00:00:00.000Z", + "repo": "leanprover-community/physlib", + "branch": "upstream/master", + "baseSha": "d44dd713876a1cf7817747d9846cfd362353057f", + "headSha": "05930fe0159096a15956f518d5917bc72a69175d", + "compareUrl": "https://github.com/leanprover-community/physlib/compare/d44dd713876a1cf7817747d9846cfd362353057f...05930fe0159096a15956f518d5917bc72a69175d", + "commitsUrl": 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"file": "Physlib/ClassicalMechanics/FreeParticle/Basic.lean", + "kind": "def", + "name": "ClassicalMechanics.FreeParticle.linearMomentum", + "docstring": null, + "snippet": "def linearMomentum (s : FreeParticle) (q : Trajectory) (t : Time) : ℝ", + "line": 109 + }, + { + "file": "Physlib/ClassicalMechanics/FreeParticle/Basic.lean", + "kind": "lemma", + "name": "ClassicalMechanics.FreeParticle.linearMomentum_conserved_of_velocity_const", + "docstring": "If a free-particle trajectory has constant velocity, then its linear momentum is constant.", + "snippet": "lemma linearMomentum_conserved_of_velocity_const (s : FreeParticle) (q : Trajectory)\n (h : ∃ v₀, ∀ t, s.velocity q t = v₀) :\n ∃ p, ∀ t, s.linearMomentum q t = p", + "line": 179 + }, + { + "file": "Physlib/ClassicalMechanics/FreeParticle/Basic.lean", + "kind": "theorem", + "name": "ClassicalMechanics.FreeParticle.linearMomentum_conserved", + "docstring": "A free particle satisfying the equation of motion conserves linear momentum.\n\nNewton's second law implies that the acceleration vanishes, so the velocity is constant.\nSince the particle mass is fixed, the linear momentum is constant in time.", + "snippet": "theorem linearMomentum_conserved (s : FreeParticle) (q : Trajectory)\n (h : ∀ t, s.NewtonsSecondLaw q t) (hcont : ContDiff ℝ 2 q) :\n ∃ p, ∀ t, s.linearMomentum q t = p", + "line": 193 + } + ] + }, + { + "filename": "Physlib/ClassicalMechanics/HarmonicOscillator/ConfigurationSpace.lean", + "entries": [ + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/ConfigurationSpace.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.ConfigurationSpace.coe_apply", + "docstring": null, + "snippet": "lemma coe_apply (x : ConfigurationSpace) (i : Fin 1) : x i = x.val i", + "line": 87 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/ConfigurationSpace.lean", + "kind": "instance", + "name": "ClassicalMechanics.HarmonicOscillator.ConfigurationSpace.«instance@8aef359a»", + "docstring": "The topology on configuration space, induced by the chosen coordinate\n`ConfigurationSpace.val` into `EuclideanSpace ℝ (Fin 1)`.", + "snippet": "instance : TopologicalSpace ConfigurationSpace", + "line": 103 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/ConfigurationSpace.lean", + "kind": "def", + "name": "ClassicalMechanics.HarmonicOscillator.ConfigurationSpace.valEquiv", + "docstring": "The coordinate equivalence between configuration space and its `EuclideanSpace ℝ (Fin 1)`\nmodel, given by `ConfigurationSpace.val` with its wrapper inverse.", + "snippet": "def valEquiv : ConfigurationSpace ≃ EuclideanSpace ℝ (Fin 1) where", + "line": 108 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/ConfigurationSpace.lean", + "kind": "instance", + "name": "ClassicalMechanics.HarmonicOscillator.ConfigurationSpace.«instance@f9bea586»", + "docstring": "Configuration space is Hausdorff, transported from `EuclideanSpace ℝ (Fin 1)` across the\ncoordinate homeomorphism.", + "snippet": "instance : T2Space ConfigurationSpace", + "line": 127 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/ConfigurationSpace.lean", + "kind": "instance", + "name": "ClassicalMechanics.HarmonicOscillator.ConfigurationSpace.«instance@572511a8»", + "docstring": "Configuration space is second countable, transported from `EuclideanSpace ℝ (Fin 1)`\nacross the coordinate homeomorphism.", + "snippet": "instance : SecondCountableTopology ConfigurationSpace", + "line": 131 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/ConfigurationSpace.lean", + "kind": "instance", + "name": "ClassicalMechanics.HarmonicOscillator.ConfigurationSpace.«instance@9e9cd011»", + "docstring": "The structure of a charted space on `ConfigurationSpace`, modeled on its\n`EuclideanSpace ℝ (Fin 1)` coordinate via the single global chart `valHomeomorphism`.", + "snippet": "instance : ChartedSpace (EuclideanSpace ℝ (Fin 1)) ConfigurationSpace where", + "line": 144 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/ConfigurationSpace.lean", + "kind": "instance", + "name": "ClassicalMechanics.HarmonicOscillator.ConfigurationSpace.«instance@66ad2370»", + "docstring": "The structure of a smooth (indeed analytic) manifold on `ConfigurationSpace`. With a\nsingle global chart, the only coordinate change is the chart's self-transition, which is\nanalytic.", + "snippet": "instance : IsManifold 𝓘(ℝ, EuclideanSpace ℝ (Fin 1)) ω ConfigurationSpace where", + "line": 158 + } + ] + }, + { + "filename": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "entries": [ + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "def", + "name": "ClassicalMechanics.HarmonicOscillator.InitialConditionsFromTwoPositions.toInitialConditions", + "docstring": "Convert two-position initial conditions to standard initial conditions at `t = 0`.\n\n Obtained by solving the 2×2 linear system from the trajectory formula at `t₁` and `t₂`.\n See `toInitialConditions_trajectory_at_t₁` and `toInitialConditions_trajectory_at_t₂` in\n section D.2 for the correctness proofs (valid under `sin (S.ω * (t₂ - t₁)) ≠ 0`).", + "snippet": "noncomputable def toInitialConditions (S : HarmonicOscillator)\n (IC : InitialConditionsFromTwoPositions) : InitialConditions where", + "line": 249 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "def", + "name": "ClassicalMechanics.HarmonicOscillator.InitialConditionsFromTwoVelocities.toInitialConditions", + "docstring": "Convert two-velocity initial conditions to standard initial conditions at `t = 0`.\n\n Obtained by solving the 2×2 linear system from the velocity formula at `t₁` and `t₂`.\n See `toInitialConditions_velocity_at_t₁` and `toInitialConditions_velocity_at_t₂` in\n section D.3 for the correctness proofs (valid under `sin (S.ω * (t₂ - t₁)) ≠ 0`).", + "snippet": "noncomputable def toInitialConditions (S : HarmonicOscillator)\n (IC : InitialConditionsFromTwoVelocities) : InitialConditions where", + "line": 303 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.InitialConditionsFromTwoPositions.toInitialConditions_trajectory_at_t₁", + "docstring": "The trajectory from `toInitialConditions` passes through `x_t₁` at time `t₁`,\n provided `sin (S.ω * (t₂ - t₁)) ≠ 0`.", + "snippet": "lemma toInitialConditions_trajectory_at_t₁ (S : HarmonicOscillator)\n (IC : InitialConditionsFromTwoPositions)\n (hΔ : sin (S.ω * (IC.t₂ - IC.t₁)) ≠ 0) :\n (IC.toInitialConditions S).trajectory S IC.t₁ = IC.x_t₁", + "line": 680 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.InitialConditionsFromTwoPositions.toInitialConditions_trajectory_at_t₂", + "docstring": "The trajectory from `toInitialConditions` passes through `x_t₂` at time `t₂`,\n provided `sin (S.ω * (t₂ - t₁)) ≠ 0`.", + "snippet": "lemma toInitialConditions_trajectory_at_t₂ (S : HarmonicOscillator)\n (IC : InitialConditionsFromTwoPositions)\n (hΔ : sin (S.ω * (IC.t₂ - IC.t₁)) ≠ 0) :\n (IC.toInitialConditions S).trajectory S IC.t₂ = IC.x_t₂", + "line": 692 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.InitialConditionsFromTwoVelocities.toInitialConditions_velocity_at_t₁", + "docstring": "The trajectory from `toInitialConditions` has velocity `v_t₁` at time `t₁`,\n provided `sin (S.ω * (t₂ - t₁)) ≠ 0`.", + "snippet": "lemma toInitialConditions_velocity_at_t₁ (S : HarmonicOscillator)\n (IC : InitialConditionsFromTwoVelocities)\n (hΔ : sin (S.ω * (IC.t₂ - IC.t₁)) ≠ 0) :\n ∂ₜ ((IC.toInitialConditions S).trajectory S) IC.t₁ = IC.v_t₁", + "line": 718 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.InitialConditionsFromTwoVelocities.toInitialConditions_velocity_at_t₂", + "docstring": "The trajectory from `toInitialConditions` has velocity `v_t₂` at time `t₂`,\n provided `sin (S.ω * (t₂ - t₁)) ≠ 0`.", + "snippet": "lemma toInitialConditions_velocity_at_t₂ (S : HarmonicOscillator)\n (IC : InitialConditionsFromTwoVelocities)\n (hΔ : sin (S.ω * (IC.t₂ - IC.t₁)) ≠ 0) :\n ∂ₜ ((IC.toInitialConditions S).trajectory S) IC.t₂ = IC.v_t₂", + "line": 731 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "def", + "name": "ClassicalMechanics.HarmonicOscillator.AmplitudePhase.toInitialConditions", + "docstring": "Convert amplitude–phase initial conditions to standard initial conditions at `t = 0`, via\n `x₀ = A cos φ` and `v₀ = A ω sin φ`.\n\n See `toInitialConditions_trajectory_eq_cos` and `toInitialConditions_velocity_eq_sin` in\n section E.3 for the correctness proofs.", + "snippet": "noncomputable def toInitialConditions (S : HarmonicOscillator) (IC : AmplitudePhase) :\n InitialConditions where", + "line": 804 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.AmplitudePhase.toInitialConditions_trajectory_eq_cos", + "docstring": "The trajectory of amplitude–phase initial conditions is the cosine normal form\n `x(t) = A cos (ω t - φ)`.", + "snippet": "lemma toInitialConditions_trajectory_eq_cos (S : HarmonicOscillator) (IC : AmplitudePhase)\n (t : Time) :\n (IC.toInitialConditions S).trajectory S t\n = EuclideanSpace.single 0 (IC.A * cos (S.ω * t - IC.φ))", + "line": 821 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.AmplitudePhase.toInitialConditions_velocity_eq_sin", + "docstring": "The velocity of the amplitude–phase trajectory is `v(t) = -A ω sin (ω t - φ)`.", + "snippet": "lemma toInitialConditions_velocity_eq_sin (S : HarmonicOscillator) (IC : AmplitudePhase)\n (t : Time) :\n ∂ₜ ((IC.toInitialConditions S).trajectory S) t\n = EuclideanSpace.single 0 (-(IC.A * S.ω * sin (S.ω * t.val - IC.φ)))", + "line": 832 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "def", + "name": "ClassicalMechanics.HarmonicOscillator.AmplitudePhase.fromInitialConditions", + "docstring": "Recover amplitude–phase data from standard initial conditions, as the polar coordinates of the\n phase vector `(x₀, v₀ / ω)` embedded as `z = x₀ + (v₀ / ω) i`: the amplitude is `‖z‖` and the\n phase is `Complex.arg z`.\n\n See `toInitialConditions_fromInitialConditions` for the right-inverse identity.", + "snippet": "noncomputable def fromInitialConditions (S : HarmonicOscillator) (IC : InitialConditions) :\n AmplitudePhase where", + "line": 862 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.AmplitudePhase.toInitialConditions_fromInitialConditions", + "docstring": "`fromInitialConditions` is a right inverse of `toInitialConditions`: converting initial\n conditions to amplitude–phase form and back recovers them exactly.", + "snippet": "lemma toInitialConditions_fromInitialConditions (S : HarmonicOscillator)\n (IC : InitialConditions) :\n (fromInitialConditions S IC).toInitialConditions S = IC", + "line": 869 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.InitialConditions.trajectory_eq_cos", + "docstring": "Every trajectory of the harmonic oscillator is a single shifted cosine after converting its\n initial conditions to amplitude-phase form.", + "snippet": "lemma trajectory_eq_cos (IC : InitialConditions) (t : Time) :\n IC.trajectory S t =\n EuclideanSpace.single 0 ((AmplitudePhase.fromInitialConditions S IC).A *\n cos (S.ω * t - (AmplitudePhase.fromInitialConditions S IC).φ))", + "line": 929 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.InitialConditions.trajectory_velocity_eq_sin", + "docstring": "The velocity of every trajectory is the corresponding shifted sine in amplitude-phase form.", + "snippet": "lemma trajectory_velocity_eq_sin (IC : InitialConditions) (t : Time) :\n ∂ₜ (IC.trajectory S) t =\n EuclideanSpace.single 0 (-((AmplitudePhase.fromInitialConditions S IC).A * S.ω *\n sin (S.ω * t.val - (AmplitudePhase.fromInitialConditions S IC).φ)))", + "line": 939 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.InitialConditions.trajectory_velocity_eq_zero_iff_sin_eq_zero", + "docstring": "For nonzero amplitude, the velocity vanishes exactly when the sine factor in\n amplitude-phase form vanishes.", + "snippet": "lemma trajectory_velocity_eq_zero_iff_sin_eq_zero (IC : InitialConditions)\n (hA : (AmplitudePhase.fromInitialConditions S IC).A ≠ 0) (t : Time) :\n ∂ₜ (IC.trajectory S) t = 0 ↔\n sin (S.ω * t.val - (AmplitudePhase.fromInitialConditions S IC).φ) = 0", + "line": 960 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.InitialConditions.trajectory_velocity_eq_zero_iff_exists_int", + "docstring": "For nonzero amplitude, the velocity is zero exactly at phase times `φ + nπ`.", + "snippet": "lemma trajectory_velocity_eq_zero_iff_exists_int (IC : InitialConditions)\n (hA : (AmplitudePhase.fromInitialConditions S IC).A ≠ 0) (t : Time) :\n ∂ₜ (IC.trajectory S) t = 0 ↔\n ∃ n : ℤ,\n (t : ℝ) =\n ((AmplitudePhase.fromInitialConditions S IC).φ + n * π) / S.ω", + "line": 986 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.InitialConditions.trajectory_velocity_eq_zero_iff_norm_eq_amplitude", + "docstring": "The velocity vanishes exactly when the trajectory has norm equal to the amplitude.", + "snippet": "lemma trajectory_velocity_eq_zero_iff_norm_eq_amplitude (IC : InitialConditions)\n (t : Time) :\n ∂ₜ (IC.trajectory S) t = 0 ↔\n ‖IC.trajectory S t‖ = (AmplitudePhase.fromInitialConditions S IC).A", + "line": 1028 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.InitialConditions.trajectory_eq_zero_iff_cos_eq_zero", + "docstring": "For nonzero amplitude, the trajectory passes through zero exactly when the cosine factor in\n amplitude-phase form vanishes.", + "snippet": "lemma trajectory_eq_zero_iff_cos_eq_zero (IC : InitialConditions)\n (hA : (AmplitudePhase.fromInitialConditions S IC).A ≠ 0) (t : Time) :\n IC.trajectory S t = 0 ↔\n cos (S.ω * t.val - (AmplitudePhase.fromInitialConditions S IC).φ) = 0", + "line": 1082 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.InitialConditions.trajectory_eq_zero_iff_exists_int", + "docstring": "For nonzero amplitude, the trajectory passes through zero exactly at phase times\n `φ + (2n + 1)π / 2`.", + "snippet": "lemma trajectory_eq_zero_iff_exists_int (IC : InitialConditions)\n (hA : (AmplitudePhase.fromInitialConditions S IC).A ≠ 0) (t : Time) :\n IC.trajectory S t = 0 ↔\n ∃ n : ℤ,\n (t : ℝ) =\n ((AmplitudePhase.fromInitialConditions S IC).φ + (2 * n + 1) * π / 2) / S.ω", + "line": 1103 + }, + { + "file": "Physlib/ClassicalMechanics/HarmonicOscillator/Solution.lean", + "kind": "lemma", + "name": "ClassicalMechanics.HarmonicOscillator.return_time", + "docstring": "Assuming that the initial coordinate and velocity are not simultaneously zero,\nthe time stamps when the harmonic oscillator returns to its initial coordinate and velocity is\na multiple of its period", + "snippet": "lemma return_time (IC : InitialConditions) (non_trivial : IC.x₀ ≠ 0 ∨ IC.v₀ ≠ 0)\n (t : Time) (ht : IC.trajectory S t = IC.x₀ ∧ ∂ₜ (IC.trajectory S) t = IC.v₀) :\n ∃ n : ℤ, (n : ℝ) * (T S) = t", + "line": 1202 + } + ] + }, + { + "filename": "Physlib/ClassicalMechanics/Lagrangian/TotalDerivativeEquivalence.lean", + "entries": [ + { + "file": "Physlib/ClassicalMechanics/Lagrangian/TotalDerivativeEquivalence.lean", + "kind": "lemma", + "name": "ClassicalMechanics.Lagrangian.isTotalTimeDerivative_explicit", + "docstring": "Explicit reformulation (by the chain rule):\n δL(t, q, dₜ q) = ∂F/∂t(t, q) + ⟨∇ᵣF(t, q), dₜ q⟩\n\n or\n\n δL(t, q, dₜ q) = fderiv ℝ F (t, q) (1, dₜ q)", + "snippet": "lemma isTotalTimeDerivative_explicit {δL : Time → X → X → ℝ} :\n IsTotalTimeDerivative δL ↔ (∃ (F : Time → X → ℝ) (_ : ContDiff ℝ ∞ ↿F),\n ∀ t q v, δL t q v = fderiv ℝ ↿F (t, q) ((1 : Time), v))", + "line": 94 + } + ] + }, + { + "filename": "Physlib/Electromagnetism/Distributional/Basic.lean", + "entries": [ + { + "file": "Physlib/Electromagnetism/Distributional/Basic.lean", + "kind": "def", + "name": "Electromagnetism.DistElectromagneticPotential.ofStaticScalarPotentialFunction", + "docstring": "The creation of an electromagnetic potential from a static scalar-potential function.\n\nIf the function is distribution-bounded, it is first promoted to a distribution using\n`Space.distOfFunction`. Otherwise the constructor returns zero.", + "snippet": "noncomputable def ofStaticScalarPotentialFunction {d} (c : SpeedOfLight)\n (φ : Space d → ℝ) : DistElectromagneticPotential d", + "line": 99 + }, + { + "file": "Physlib/Electromagnetism/Distributional/Basic.lean", + "kind": "lemma", + "name": "Electromagnetism.DistElectromagneticPotential.ofStaticScalarPotentialFunction_of_isDistBounded", + "docstring": null, + "snippet": "@[simp]\nlemma ofStaticScalarPotentialFunction_of_isDistBounded {d} (c : SpeedOfLight)\n (φ : Space d → ℝ) (hφ : Space.IsDistBounded φ) :\n ofStaticScalarPotentialFunction c φ =\n ofStaticScalarPotential c (Space.distOfFunction φ hφ)", + "line": 106 + }, + { + "file": "Physlib/Electromagnetism/Distributional/Basic.lean", + "kind": "lemma", + "name": "Electromagnetism.DistElectromagneticPotential.ofStaticScalarPotentialFunction_eq_zero_of_not_isDistBounded", + "docstring": null, + "snippet": "@[simp]\nlemma ofStaticScalarPotentialFunction_eq_zero_of_not_isDistBounded {d}\n (c : SpeedOfLight) (φ : Space d → ℝ) (hφ : ¬ Space.IsDistBounded φ) :\n ofStaticScalarPotentialFunction c φ = 0", + "line": 114 + }, + { + "file": "Physlib/Electromagnetism/Distributional/Basic.lean", + "kind": "def", + "name": "Electromagnetism.DistElectromagneticPotential.ofVectorPotential", + "docstring": "The creation of an electromagnetic potential from a vector potential.", + "snippet": "noncomputable def ofVectorPotential {d} (c : SpeedOfLight) :\n ((Time × Space d) →d[ℝ] EuclideanSpace ℝ (Fin d)) →ₗ[ℝ]\n DistElectromagneticPotential d where", + "line": 121 + }, + { + "file": "Physlib/Electromagnetism/Distributional/Basic.lean", + "kind": "def", + "name": "Electromagnetism.DistElectromagneticPotential.ofStaticVectorPotential", + "docstring": "The creation of an electromagnetic potential from a static vector potential.", + "snippet": "noncomputable def ofStaticVectorPotential {d} (c : SpeedOfLight) :\n ((Space d) →d[ℝ] EuclideanSpace ℝ (Fin d)) →ₗ[ℝ] DistElectromagneticPotential d", + "line": 133 + }, + { + "file": "Physlib/Electromagnetism/Distributional/Basic.lean", + "kind": "lemma", + "name": "Electromagnetism.DistElectromagneticPotential.distTensorDeriv_eq_sum_sum", + "docstring": null, + "snippet": "lemma distTensorDeriv_eq_sum_sum {d} (A : DistElectromagneticPotential d)\n (ε : 𝓢(SpaceTime d, ℝ)) :\n distTensorDeriv A ε =∑ μ, ∑ ν, (SpaceTime.distDeriv μ A ε ν) •\n Lorentz.CoVector.basis μ ⊗ₜ[ℝ] Lorentz.Vector.basis ν", + "line": 146 + }, + { + "file": "Physlib/Electromagnetism/Distributional/Basic.lean", + "kind": "lemma", + "name": "Electromagnetism.DistElectromagneticPotential.distTensorDeriv_basis_repr_apply", + "docstring": null, + "snippet": "@[simp]\nlemma distTensorDeriv_basis_repr_apply {d} {μν : (Fin 1 ⊕ Fin d) × (Fin 1 ⊕ Fin d)}\n (A : DistElectromagneticPotential d)\n (ε : 𝓢(SpaceTime d, ℝ)) :\n (Lorentz.CoVector.basis.tensorProduct Lorentz.Vector.basis).repr (distTensorDeriv A ε) μν =\n distDeriv μν.1 A ε μν.2", + "line": 167 + }, + { + "file": "Physlib/Electromagnetism/Distributional/Basic.lean", + "kind": "lemma", + "name": "Electromagnetism.DistElectromagneticPotential.toTensor_distTensorDeriv_basis_repr_apply", + "docstring": null, + "snippet": "lemma toTensor_distTensorDeriv_basis_repr_apply {d} (A : DistElectromagneticPotential d)\n (ε : 𝓢(SpaceTime d, ℝ)) (b : ComponentIdx (S := realLorentzTensor d)\n (Fin.append ![Color.down] ![Color.up])) :\n (Tensor.basis _).repr (Tensorial.toTensor (distTensorDeriv A ε)) b =\n distDeriv (b 0) A ε (b 1)", + "line": 186 + } + ] + }, + { + "filename": "Physlib/Electromagnetism/Kinematics/EMPotential.lean", + "entries": [ + { + "file": "Physlib/Electromagnetism/Kinematics/EMPotential.lean", + "kind": "instance", + "name": "Electromagnetism.ElectromagneticPotential.«instance@94a1323b»", + "docstring": null, + "snippet": "instance {d} : Zero (ElectromagneticPotential d) where", + "line": 97 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/EMPotential.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.zero_val", + "docstring": null, + "snippet": "@[simp]\nlemma zero_val {d} : (0 : ElectromagneticPotential d).val = 0", + "line": 101 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/EMPotential.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.zero_apply", + "docstring": null, + "snippet": "@[simp]\nlemma zero_apply {d} (x : SpaceTime d) : (0 : ElectromagneticPotential d) x = 0", + "line": 104 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/EMPotential.lean", + "kind": "instance", + "name": "Electromagnetism.ElectromagneticPotential.«instance@3f7e7f60»", + "docstring": null, + "snippet": "instance {d} : Neg (ElectromagneticPotential d) where", + "line": 117 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/EMPotential.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.neg_val", + "docstring": null, + "snippet": "@[simp]\nlemma neg_val {d} (A : ElectromagneticPotential d) :\n (- A).val = - A.val", + "line": 121 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/EMPotential.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.neg_apply", + "docstring": null, + "snippet": "@[simp]\nlemma neg_apply {d} (A : ElectromagneticPotential d) (x : SpaceTime d) :\n (- A) x = - A x", + "line": 125 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/EMPotential.lean", + "kind": "instance", + "name": "Electromagnetism.ElectromagneticPotential.«instance@6c8e2750»", + "docstring": null, + "snippet": "instance {d} : Sub (ElectromagneticPotential d) where", + "line": 128 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/EMPotential.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.sub_val", + "docstring": null, + "snippet": "@[simp]\nlemma sub_val {d} (A B : ElectromagneticPotential d) :\n (A - B).val = A.val - B.val", + "line": 132 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/EMPotential.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.sub_apply", + "docstring": null, + "snippet": "@[simp]\nlemma sub_apply {d} (A B : ElectromagneticPotential d) (x : SpaceTime d) :\n (A - B) x = A x - B x", + "line": 136 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/EMPotential.lean", + "kind": "instance", + "name": "Electromagnetism.ElectromagneticPotential.«instance@2040b1c1»", + "docstring": null, + "snippet": "instance {d} : AddCommGroup (ElectromagneticPotential d) where", + "line": 139 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/EMPotential.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.action_apply", + "docstring": null, + "snippet": "@[simp]\nlemma action_apply {d} (Λ : LorentzGroup d) (A : ElectromagneticPotential d)\n (x : SpaceTime d) :\n (Λ • A) x = Λ • A (Λ⁻¹ • x)", + "line": 265 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/EMPotential.lean", + "kind": "instance", + "name": "Electromagnetism.ElectromagneticPotential.«instance@f3d05234»", + "docstring": null, + "snippet": "noncomputable instance {d} :\n DistribMulAction (LorentzGroup d) (ElectromagneticPotential d) where", + "line": 277 + } + ] + }, + { + "filename": "Physlib/Electromagnetism/Kinematics/FieldStrength.lean", + "entries": [ + { + "file": "Physlib/Electromagnetism/Kinematics/FieldStrength.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.toFieldStrength_eval_eq_basis_repr", + "docstring": "Evaluating both tensor indices of the field strength gives the coefficient in the\nstandard tensor-product basis.", + "snippet": "lemma toFieldStrength_eval_eq_basis_repr {d} (A : ElectromagneticPotential d)\n (x : SpaceTime d) (μ ν : Fin 1 ⊕ Fin d) :\n toField {A.toFieldStrength x | [μ] [ν]}ᵀ =\n (Lorentz.CoVector.basis.tensorProduct Lorentz.Vector.basis).repr\n (A.toFieldStrength x) (μ, ν)", + "line": 378 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/FieldStrength.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.toFieldStrength_eval_apply", + "docstring": "The evaluated components of the field strength tensor in terms of derivatives of the\nelectromagnetic potential.", + "snippet": "lemma toFieldStrength_eval_apply {d} (A : ElectromagneticPotential d)\n (x : SpaceTime d) (μ ν : Fin 1 ⊕ Fin d) :\n toField {A.toFieldStrength x | [μ] [ν]}ᵀ =\n ∑ κ, (η μ κ * ∂_ κ A x ν - η ν κ * ∂_ κ A x μ)", + "line": 393 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/FieldStrength.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.toFieldStrength_eval_apply_eq_single", + "docstring": "The evaluated components of the field strength tensor after using diagonal form of the\nMinkowski metric.", + "snippet": "lemma toFieldStrength_eval_apply_eq_single {d} (A : ElectromagneticPotential d)\n (x : SpaceTime d) (μ ν : Fin 1 ⊕ Fin d) :\n toField {A.toFieldStrength x | [μ] [ν]}ᵀ =\n η μ μ * ∂_ μ A x ν - η ν ν * ∂_ ν A x μ", + "line": 402 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/FieldStrength.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.toFieldStrength_eval_eq_fieldStrengthMatrix", + "docstring": "Index evaluation of the field strength tensor agrees with the corresponding component of\nthe field strength matrix.", + "snippet": "lemma toFieldStrength_eval_eq_fieldStrengthMatrix {d} (A : ElectromagneticPotential d)\n (x : SpaceTime d) (μ ν : Fin 1 ⊕ Fin d) :\n toField {A.toFieldStrength x | [μ] [ν]}ᵀ = A.fieldStrengthMatrix x (μ, ν)", + "line": 430 + } + ] + }, + { + "filename": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "entries": [ + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "def", + "name": "Electromagnetism.ElectromagneticPotential.ofGradient", + "docstring": "The pure-gauge electromagnetic potential `A^μ = ∂^μ χ = η^{μν} ∂_ν χ` built from a\n gauge function `χ`. The metric-contracted form `∑ κ, η μ κ * ∂_ κ χ` is the definition;\n the diagonal form (which equals `η μ μ * ∂_ μ χ` since `η` is diagonal) is derived in\n `ofGradient_apply`.", + "snippet": "noncomputable def ofGradient {d} (χ : SpaceTime d → ℝ) : ElectromagneticPotential d where", + "line": 96 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.ofGradient_apply_sum", + "docstring": "Unfolding of the summed definition of `ofGradient`.", + "snippet": "lemma ofGradient_apply_sum {d} (χ : SpaceTime d → ℝ) (x : SpaceTime d) (μ : Fin 1 ⊕ Fin d) :\n ofGradient χ x μ = ∑ κ, η μ κ * ∂_ κ χ x", + "line": 100 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.ofGradient_apply", + "docstring": "Evaluation of `ofGradient` in the diagonal form; the off-diagonal entries of `η` vanish so\n only the `κ = μ` term survives.", + "snippet": "lemma ofGradient_apply {d} (χ : SpaceTime d → ℝ) (x : SpaceTime d) (μ : Fin 1 ⊕ Fin d) :\n ofGradient χ x μ = η μ μ * ∂_ μ χ x", + "line": 105 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.ofGradient_zero", + "docstring": "The pure-gauge potential built from the zero gauge function has all components zero.", + "snippet": "lemma ofGradient_zero {d} (x : SpaceTime d) (μ : Fin 1 ⊕ Fin d) :\n ofGradient (0 : SpaceTime d → ℝ) x μ = 0", + "line": 115 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.ofGradient_add", + "docstring": "`ofGradient` is additive in the gauge function (when both summands are differentiable).", + "snippet": "lemma ofGradient_add {d} {χ₁ χ₂ : SpaceTime d → ℝ}\n (hχ₁ : Differentiable ℝ χ₁) (hχ₂ : Differentiable ℝ χ₂) :\n ofGradient (χ₁ + χ₂) = ofGradient χ₁ + ofGradient χ₂", + "line": 121 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.differentiable_ofGradient", + "docstring": "The pure-gauge potential is differentiable when `χ` is `C^2`.", + "snippet": "lemma differentiable_ofGradient {d} {χ : SpaceTime d → ℝ} (hχ : ContDiff ℝ 2 χ) :\n Differentiable ℝ (ofGradient χ)", + "line": 137 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.contDiff_ofGradient", + "docstring": "The pure-gauge potential is `C^n` when `χ` is `C^{n+1}`.", + "snippet": "lemma contDiff_ofGradient {n} {d} {χ : SpaceTime d → ℝ} (hχ : ContDiff ℝ (n + 1) χ) :\n ContDiff ℝ n (ofGradient χ)", + "line": 146 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.toFieldStrength_ofGradient", + "docstring": "A pure-gauge potential has vanishing field strength.", + "snippet": "lemma toFieldStrength_ofGradient {d} {χ : SpaceTime d → ℝ} (hχ : ContDiff ℝ 2 χ)\n (x : SpaceTime d) : (ofGradient χ).toFieldStrength x = 0", + "line": 162 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.ofGradient_equivariant", + "docstring": "`ofGradient` intertwines the Lorentz action on potentials with composition by Λ⁻¹ on the\n gauge function: `Λ • ofGradient χ = ofGradient (χ ∘ (Λ⁻¹ • ·))`.", + "snippet": "lemma ofGradient_equivariant {d} (χ : SpaceTime d → ℝ) (hχ : Differentiable ℝ χ)\n (Λ : LorentzGroup d) :\n Λ • ofGradient χ = ofGradient (χ ∘ (Λ⁻¹ • ·))", + "line": 199 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "def", + "name": "Electromagnetism.ElectromagneticPotential.gaugeTransform", + "docstring": "The gauge transformation `A^μ ↦ A^μ + ∂^μ χ` of an electromagnetic potential.", + "snippet": "noncomputable def gaugeTransform {d} (χ : SpaceTime d → ℝ) (A : ElectromagneticPotential d) :\n ElectromagneticPotential d", + "line": 249 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.gaugeTransform_apply", + "docstring": "Evaluation of `gaugeTransform`.", + "snippet": "lemma gaugeTransform_apply {d} (χ : SpaceTime d → ℝ) (A : ElectromagneticPotential d)\n (x : SpaceTime d) : gaugeTransform χ A x = A x + ofGradient χ x", + "line": 253 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.toFieldStrength_gaugeTransform", + "docstring": "The field strength tensor is invariant under gauge transformations.", + "snippet": "lemma toFieldStrength_gaugeTransform {d} (A : ElectromagneticPotential d)\n (χ : SpaceTime d → ℝ) (hA : Differentiable ℝ A) (hχ : ContDiff ℝ 2 χ) (x : SpaceTime d) :\n (gaugeTransform χ A).toFieldStrength x = A.toFieldStrength x", + "line": 267 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.fieldStrengthMatrix_gaugeTransform", + "docstring": "The field strength matrix is invariant under gauge transformations.", + "snippet": "lemma fieldStrengthMatrix_gaugeTransform {d} (A : ElectromagneticPotential d)\n (χ : SpaceTime d → ℝ) (hA : Differentiable ℝ A) (hχ : ContDiff ℝ 2 χ) (x : SpaceTime d) :\n (gaugeTransform χ A).fieldStrengthMatrix x = A.fieldStrengthMatrix x", + "line": 274 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.gaugeTransform_zero", + "docstring": "Shifting by the zero gauge function is the identity.", + "snippet": "lemma gaugeTransform_zero {d} (A : ElectromagneticPotential d) :\n gaugeTransform (0 : SpaceTime d → ℝ) A = A", + "line": 293 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.gaugeTransform_gaugeTransform", + "docstring": "Two successive gauge shifts compose: shifting by χ₂ then χ₁ equals shifting by χ₁ + χ₂.\n This upgrades one-step F-invariance to invariance along any finite chain of gauge shifts.", + "snippet": "lemma gaugeTransform_gaugeTransform {d} (A : ElectromagneticPotential d)\n (χ₁ χ₂ : SpaceTime d → ℝ) (hχ₁ : Differentiable ℝ χ₁) (hχ₂ : Differentiable ℝ χ₂) :\n gaugeTransform χ₁ (gaugeTransform χ₂ A) = gaugeTransform (χ₁ + χ₂) A", + "line": 301 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.gaugeTransform_equivariant", + "docstring": "Gauge transformations commute with Lorentz transformations: applying Λ and then performing\n a gauge transformation by `χ` equals performing a gauge transformation by `χ ∘ (Λ⁻¹ • ·)`\n and then applying Λ.", + "snippet": "lemma gaugeTransform_equivariant {d} (A : ElectromagneticPotential d)\n (χ : SpaceTime d → ℝ) (hχ : Differentiable ℝ χ) (Λ : LorentzGroup d) :\n Λ • gaugeTransform χ A = gaugeTransform (χ ∘ (Λ⁻¹ • ·)) (Λ • A)", + "line": 322 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.fieldStrengthMatrix_bareGradient_inl_inr", + "docstring": "The `(inl 0, inr i)` component of the field strength matrix of the bare-gradient potential\n `B^μ := ∂_μ χ` for `χ(x) = x⁰·xⁱ` equals `2`. This witnesses that the bare covariant gradient\n does not produce a gauge-invariant field strength, so the raised-index contraction\n `η^{μν} ∂_ν χ` in `ofGradient` is necessary (see the module overview).", + "snippet": "lemma fieldStrengthMatrix_bareGradient_inl_inr {d : ℕ} (i : Fin d)\n (x : SpaceTime d) :\n let χ : SpaceTime d → ℝ", + "line": 347 + }, + { + "file": "Physlib/Electromagnetism/Kinematics/GaugeTransformation.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.toFieldStrength_bareGradient_ne_zero", + "docstring": "The field strength of the bare-gradient potential `B^μ := ∂_μ χ` for\n `χ(x) = x⁰·xⁱ` is nonzero (follows from `fieldStrengthMatrix_bareGradient_inl_inr`).", + "snippet": "lemma toFieldStrength_bareGradient_ne_zero {d : ℕ} (i : Fin d)\n (x : SpaceTime d) :\n let χ : SpaceTime d → ℝ", + "line": 395 + } + ] + }, + { + "filename": "Physlib/Electromagnetism/Kinematics/MagneticField.lean", + "entries": [ + { + "file": "Physlib/Electromagnetism/Kinematics/MagneticField.lean", + "kind": "lemma", + "name": "Electromagnetism.ElectromagneticPotential.magneticField_coord_eq_fieldStrengthMatrix", + "docstring": null, + "snippet": "lemma magneticField_coord_eq_fieldStrengthMatrix {i : Fin 3} {c : SpeedOfLight}\n (A : ElectromagneticPotential) (t : Time)\n (x : Space) (hA : Differentiable ℝ A) :\n A.magneticField c t x i =\n - A.fieldStrengthMatrix ((toTimeAndSpace c).symm (t, x)) (Sum.inr (i+1), Sum.inr (i+2))", + "line": 90 + } + ] + }, + { + "filename": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "entries": [ + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "def", + "name": "Physlib.Wirtinger.weightedDirDeriv", + "docstring": "The base-point field `p ↦ (1/2)(d_{b₁} f + c·d_{b₂} f)`, a weighted combination of the\nreal Fréchet derivative of `f` along two directions `b₁`, `b₂`. The directional Wirtinger\noperators are its two specializations: `dWirtingerDir f v` at `c = -i`, `(b₁, b₂) = (v, i·v)`,\nand `dWirtingerAntiDir f v` at `c = i`. Keeping `b₁`, `b₂` free lets §G differentiate it a\nsecond time once and reuse the bridge for both operators.", + "snippet": "def weightedDirDeriv (f : V → ℂ) (c : ℂ) (b₁ b₂ : V) : V → ℂ", + "line": 178 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "def", + "name": "Physlib.Wirtinger.dWirtingerDir", + "docstring": "The holomorphic directional Wirtinger derivative `∂_v f = (1/2)(d_v f − i·d_{i·v} f)`\nof `f : V → ℂ` along the direction vector `v : V`, the `weightedDirDeriv` at `c = -i`.", + "snippet": "def dWirtingerDir (f : V → ℂ) (v u : V) : ℂ", + "line": 183 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "def", + "name": "Physlib.Wirtinger.dWirtingerAntiDir", + "docstring": "The anti-holomorphic directional Wirtinger derivative\n`∂̄_v f = (1/2)(d_v f + i·d_{i·v} f)` of `f : V → ℂ` along the direction vector `v : V`,\nthe `weightedDirDeriv` at `c = i`.", + "snippet": "def dWirtingerAntiDir (f : V → ℂ) (v u : V) : ℂ", + "line": 189 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerDir_apply", + "docstring": "Definitional unfolding of `dWirtingerDir` to the explicit Wirtinger combination, used to\nexpand the outer operator of a composition without touching the inner one.", + "snippet": "lemma dWirtingerDir_apply (g : V → ℂ) (v u : V) :\n dWirtingerDir g v u\n = (1 / 2 : ℂ) * (fderiv ℝ g u v - Complex.I * fderiv ℝ g u (Complex.I • v))", + "line": 194 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiDir_apply", + "docstring": "Definitional unfolding of `dWirtingerAntiDir` to the explicit Wirtinger combination.", + "snippet": "lemma dWirtingerAntiDir_apply (g : V → ℂ) (v u : V) :\n dWirtingerAntiDir g v u\n = (1 / 2 : ℂ) * (fderiv ℝ g u v + Complex.I * fderiv ℝ g u (Complex.I • v))", + "line": 200 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerDir_add", + "docstring": "Additivity of `dWirtingerDir`, `∂_v(g + h) = ∂_v g + ∂_v h`.", + "snippet": "lemma dWirtingerDir_add {g h : V → ℂ} (hg : DifferentiableAt ℝ g u)\n (hh : DifferentiableAt ℝ h u) (v : V) :\n dWirtingerDir (g + h) v u = dWirtingerDir g v u + dWirtingerDir h v u", + "line": 238 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiDir_add", + "docstring": "Additivity of `dWirtingerAntiDir`, `∂̄_v(g + h) = ∂̄_v g + ∂̄_v h`.", + "snippet": "lemma dWirtingerAntiDir_add {g h : V → ℂ} (hg : DifferentiableAt ℝ g u)\n (hh : DifferentiableAt ℝ h u) (v : V) :\n dWirtingerAntiDir (g + h) v u = dWirtingerAntiDir g v u + dWirtingerAntiDir h v u", + "line": 244 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerDir_smul", + "docstring": "Compatibility of `dWirtingerDir` with complex scalar multiplication,\n`∂_v(c·g) = c·∂_v g`.", + "snippet": "lemma dWirtingerDir_smul (c : ℂ) {g : V → ℂ} (hg : DifferentiableAt ℝ g u) (v : V) :\n dWirtingerDir (c • g) v u = c • dWirtingerDir g v u", + "line": 251 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiDir_smul", + "docstring": "Compatibility of `dWirtingerAntiDir` with complex scalar multiplication,\n`∂̄_v(c·g) = c·∂̄_v g`.", + "snippet": "lemma dWirtingerAntiDir_smul (c : ℂ) {g : V → ℂ} (hg : DifferentiableAt ℝ g u) (v : V) :\n dWirtingerAntiDir (c • g) v u = c • dWirtingerAntiDir g v u", + "line": 258 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.fderiv_mul_apply", + "docstring": "The real Fréchet derivative of a product, evaluated at a tangent `v`.", + "snippet": "private lemma fderiv_mul_apply {g h : V → ℂ} (hg : DifferentiableAt ℝ g u)\n (hh : DifferentiableAt ℝ h u) (v : V) :\n fderiv ℝ (g * h) u v = g u * fderiv ℝ h u v + h u * fderiv ℝ g u v", + "line": 265 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerDir_mul", + "docstring": "The Wirtinger Leibniz rule for `dWirtingerDir`,\n`∂_v(g·h) = ∂_v g·h + g·∂_v h`.", + "snippet": "lemma dWirtingerDir_mul {g h : V → ℂ} (hg : DifferentiableAt ℝ g u)\n (hh : DifferentiableAt ℝ h u) (v : V) :\n dWirtingerDir (g * h) v u = dWirtingerDir g v u * h u + g u * dWirtingerDir h v u", + "line": 272 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiDir_mul", + "docstring": "The Wirtinger Leibniz rule for `dWirtingerAntiDir`,\n`∂̄_v(g·h) = ∂̄_v g·h + g·∂̄_v h`.", + "snippet": "lemma dWirtingerAntiDir_mul {g h : V → ℂ} (hg : DifferentiableAt ℝ g u)\n (hh : DifferentiableAt ℝ h u) (v : V) :\n dWirtingerAntiDir (g * h) v u =\n dWirtingerAntiDir g v u * h u + g u * dWirtingerAntiDir h v u", + "line": 279 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerDir_fun_sum", + "docstring": "Finite-sum rule for `dWirtingerDir`, `∂_v(∑ₐ Fₐ) = ∑ₐ ∂_v Fₐ`.", + "snippet": "lemma dWirtingerDir_fun_sum {α : Type*} {s : Finset α} {F : α → V → ℂ}\n (hF : ∀ a ∈ s, DifferentiableAt ℝ (F a) u) (v : V) :\n dWirtingerDir (fun p => ∑ a ∈ s, F a p) v u = ∑ a ∈ s, dWirtingerDir (F a) v u", + "line": 286 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiDir_fun_sum", + "docstring": "Finite-sum rule for `dWirtingerAntiDir`, `∂̄_v(∑ₐ Fₐ) = ∑ₐ ∂̄_v Fₐ`.", + "snippet": "lemma dWirtingerAntiDir_fun_sum {α : Type*} {s : Finset α} {F : α → V → ℂ}\n (hF : ∀ a ∈ s, DifferentiableAt ℝ (F a) u) (v : V) :\n dWirtingerAntiDir (fun p => ∑ a ∈ s, F a p) v u = ∑ a ∈ s, dWirtingerAntiDir (F a) v u", + "line": 293 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.fderiv_star_eq", + "docstring": "Differentiation commutes with conjugation: the real Fréchet derivative of the\npointwise conjugate `p ↦ star (f p)` is `conjCLE` (conjugation on `ℂ`) composed with\n`fderiv ℝ f u`; in physicists' notation, `d f̄ = conj(d f)`. Conjugation is `ℝ`-linear,\nso it slides through the real derivative unchanged, whereas it does *not* commute with\nthe holomorphic Wirtinger derivative `∂_v`. The `star` conjugates the *output* `f p`,\nso this is not a derivative in a conjugate variable.\n\nThis is the analytic core of the conjugation lemmas below\n(`dWirtingerDir_star_comp` and its dual): distributed over the Wirtinger split of\n`fderiv ℝ f u` (`realLinear_apply_eq_wirtinger`, §D), the outer `conjCLE` conjugates the two\ncoefficients and swaps the holomorphic and anti-holomorphic parts.", + "snippet": "lemma fderiv_star_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]\n {f : E → ℂ} {u : E} (hf : DifferentiableAt ℝ f u) :\n fderiv ℝ (fun p : E => star (f p)) u =\n Complex.conjCLE.toContinuousLinearMap.comp (fderiv ℝ f u)", + "line": 332 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerDir_star_comp", + "docstring": "Conjugating the function swaps the operators up to an outer conjugation:\n`∂_v f̄ = conj (∂̄_v f)`.", + "snippet": "lemma dWirtingerDir_star_comp (hf : DifferentiableAt ℝ f u) (v : V) :\n dWirtingerDir (fun p => star (f p)) v u = star (dWirtingerAntiDir f v u)", + "line": 342 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiDir_star_comp", + "docstring": "Conjugating the function swaps the operators up to an outer conjugation:\n`∂̄_v f̄ = conj (∂_v f)`. Dual of `dWirtingerDir_star_comp`.", + "snippet": "lemma dWirtingerAntiDir_star_comp (hf : DifferentiableAt ℝ f u) (v : V) :\n dWirtingerAntiDir (fun p => star (f p)) v u = star (dWirtingerDir f v u)", + "line": 353 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.realLinear_apply_eq_wirtinger", + "docstring": "Split a real-linear map `ℂ → ℂ` into its Wirtinger components. Any\nreal-linear `L : ℂ →L[ℝ] ℂ` splits into a holomorphic and an anti-holomorphic part\nwith the Wirtinger coefficients `a = ½(L 1 - i * L i)`, `b = ½(L 1 + i * L i)` as\nweights:\n\n `L w = a * w + b * star w`.\n\nThis is purely algebraic: `L` is an arbitrary real-linear map, no derivative\ninvolved. Its use is the Wirtinger chain rule (`dWirtingerDir_comp` below), where the\nweights of the outer differential `L = fderiv ℝ g (f u)` are the coefficients `∂g/∂f`,\n`∂g/∂f̄`.", + "snippet": "lemma realLinear_apply_eq_wirtinger (L : ℂ →L[ℝ] ℂ) (w : ℂ) :\n L w =\n ((1 / 2 : ℂ) * (L 1 - Complex.I * L Complex.I)) * w\n + ((1 / 2 : ℂ) * (L 1 + Complex.I * L Complex.I)) * star w", + "line": 391 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerDir_comp", + "docstring": "The two-term Wirtinger chain rule for `dWirtingerDir`, outer `g : ℂ → ℂ` and inner\n`f : V → ℂ`:\n\n `∂_v(g∘f) = (∂g/∂f)·∂_v f + (∂g/∂f̄)·∂_v f̄`.\n\n`realLinear_apply_eq_wirtinger` splits the chain rule's outer `ℝ`-linear factor into the\n`∂g/∂f`, `∂g/∂f̄` coefficients, each multiplying its inner directional derivative `∂_v f`,\n`∂_v f̄`.", + "snippet": "lemma dWirtingerDir_comp {g : ℂ → ℂ} (hg : DifferentiableAt ℝ g (f u))\n (hf : DifferentiableAt ℝ f u) (v : V) :\n dWirtingerDir (fun p => g (f p)) v u =\n dWirtingerDir g 1 (f u) * dWirtingerDir f v u\n + dWirtingerAntiDir g 1 (f u) * dWirtingerDir (fun p => star (f p)) v u", + "line": 416 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiDir_comp", + "docstring": "The two-term Wirtinger chain rule for `dWirtingerAntiDir`, the anti-holomorphic dual of\n`dWirtingerDir_comp`:\n\n `∂̄_v(g∘f) = (∂g/∂f)·∂̄_v f + (∂g/∂f̄)·∂̄_v f̄`.\n\nSame outer `∂g/∂f`, `∂g/∂f̄` coefficients, now each multiplying its anti-holomorphic inner\nderivative `∂̄_v f`, `∂̄_v f̄`; same proof as `dWirtingerDir_comp`.", + "snippet": "lemma dWirtingerAntiDir_comp {g : ℂ → ℂ} (hg : DifferentiableAt ℝ g (f u))\n (hf : DifferentiableAt ℝ f u) (v : V) :\n dWirtingerAntiDir (fun p => g (f p)) v u =\n dWirtingerDir g 1 (f u) * dWirtingerAntiDir f v u\n + dWirtingerAntiDir g 1 (f u) * dWirtingerAntiDir (fun p => star (f p)) v u", + "line": 437 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.fderiv_comp_clm_apply", + "docstring": "Chain rule for an inner continuous linear map `L`. Because the derivative of a linear\nmap is the map itself, the real Fréchet derivative of `g ∘ L` at `u`, applied to `x`, equals\nthe derivative of `g` at `L u` applied to `L x`.", + "snippet": "private lemma fderiv_comp_clm_apply {g : V' → ℂ} {L : V →L[ℝ] V'} {u : V}\n (hg : DifferentiableAt ℝ g (L u)) (x : V) :\n fderiv ℝ (fun p => g (L p)) u x = fderiv ℝ g (L u) (L x)", + "line": 499 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerDir_comp_conjLinear", + "docstring": "Domain conjugation swaps the operators: precomposing with a conjugate-`ℂ`-linear `L`\nturns the holomorphic derivative of `g ∘ L` at `u` into the anti-holomorphic derivative of\n`g` at the mapped point `L u`, in the mapped direction `L v`:\n`∂_v(g ∘ L)` at `u` equals `∂̄_{L v} g` at `L u`.", + "snippet": "lemma dWirtingerDir_comp_conjLinear {g : V' → ℂ} {L : V →L[ℝ] V'} {u : V}\n (hL : ∀ x : V, L (Complex.I • x) = -(Complex.I • L x))\n (hg : DifferentiableAt ℝ g (L u)) (v : V) :\n dWirtingerDir (fun p => g (L p)) v u = dWirtingerAntiDir g (L v) (L u)", + "line": 510 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiDir_comp_conjLinear", + "docstring": "Dual of `dWirtingerDir_comp_conjLinear`: the anti-holomorphic derivative of `g ∘ L` at\n`u` is the holomorphic derivative of `g` at the mapped point `L u`, in the mapped direction\n`L v`: `∂̄_v(g ∘ L)` at `u` equals `∂_{L v} g` at `L u`.", + "snippet": "lemma dWirtingerAntiDir_comp_conjLinear {g : V' → ℂ} {L : V →L[ℝ] V'} {u : V}\n (hL : ∀ x : V, L (Complex.I • x) = -(Complex.I • L x))\n (hg : DifferentiableAt ℝ g (L u)) (v : V) :\n dWirtingerAntiDir (fun p => g (L p)) v u = dWirtingerDir g (L v) (L u)", + "line": 521 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerDir_eq_of_clinear", + "docstring": "Holomorphic collapse: along a direction where the real derivative is `ℂ`-linear, the\nholomorphic derivative is the full real derivative, `∂_v f = d_v f`.", + "snippet": "lemma dWirtingerDir_eq_of_clinear {v : V}\n (h : fderiv ℝ f u (Complex.I • v) = Complex.I • fderiv ℝ f u v) :\n dWirtingerDir f v u = fderiv ℝ f u v", + "line": 558 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiDir_eq_zero_of_clinear", + "docstring": "Holomorphic collapse: the anti-holomorphic derivative vanishes along a direction\nof `ℂ`-linearity, `∂̄_v f = 0`.", + "snippet": "lemma dWirtingerAntiDir_eq_zero_of_clinear {v : V}\n (h : fderiv ℝ f u (Complex.I • v) = Complex.I • fderiv ℝ f u v) :\n dWirtingerAntiDir f v u = 0", + "line": 565 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerDir_eq_zero_of_antilinear", + "docstring": "Anti-holomorphic collapse: a direction of conjugate-`ℂ`-linearity kills the\nholomorphic derivative, `∂_v f = 0`.", + "snippet": "lemma dWirtingerDir_eq_zero_of_antilinear {v : V}\n (h : fderiv ℝ f u (Complex.I • v) = -(Complex.I • fderiv ℝ f u v)) :\n dWirtingerDir f v u = 0", + "line": 572 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiDir_eq_of_antilinear", + "docstring": "Anti-holomorphic collapse: the anti-holomorphic derivative is the full real\nderivative along a direction of conjugate-`ℂ`-linearity, `∂̄_v f = d_v f`.", + "snippet": "lemma dWirtingerAntiDir_eq_of_antilinear {v : V}\n (h : fderiv ℝ f u (Complex.I • v) = -(Complex.I • fderiv ℝ f u v)) :\n dWirtingerAntiDir f v u = fderiv ℝ f u v", + "line": 579 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.hasFDerivAt_fderiv_apply", + "docstring": "The field `p ↦ d_b f` is the evaluation map `· b` composed with `fderiv ℝ f`,\nso when `fderiv ℝ f` is differentiable its derivative is `fderiv ℝ (fderiv ℝ f) u`\npost-composed with that evaluation.", + "snippet": "private lemma hasFDerivAt_fderiv_apply (hf' : DifferentiableAt ℝ (fderiv ℝ f) u)\n (b : V) :\n HasFDerivAt (fun p => fderiv ℝ f p b)\n ((ContinuousLinearMap.apply ℝ ℂ b).comp (fderiv ℝ (fderiv ℝ f) u)) u", + "line": 635 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.hasFDerivAt_weightedDirDeriv", + "docstring": "The `weightedDirDeriv` is differentiable wherever `fderiv ℝ f` is.", + "snippet": "private lemma hasFDerivAt_weightedDirDeriv (hf' : DifferentiableAt ℝ (fderiv ℝ f) u)\n (c : ℂ) (b₁ b₂ : V) :\n HasFDerivAt (weightedDirDeriv f c b₁ b₂)\n ((1 / 2 : ℂ) • ((ContinuousLinearMap.apply ℝ ℂ b₁).comp (fderiv ℝ (fderiv ℝ f) u)\n + c • (ContinuousLinearMap.apply ℝ ℂ b₂).comp (fderiv ℝ (fderiv ℝ f) u))) u", + "line": 643 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.fderiv_weightedDirDeriv", + "docstring": "The bridge: differentiating a `weightedDirDeriv` along a third direction `a`\nlands on the second real Fréchet derivative `fderiv ℝ (fderiv ℝ f) u a b` in the\ntwo slots.", + "snippet": "private lemma fderiv_weightedDirDeriv (hf' : DifferentiableAt ℝ (fderiv ℝ f) u)\n (c : ℂ) (b₁ b₂ a : V) :\n fderiv ℝ (weightedDirDeriv f c b₁ b₂) u a\n = (1 / 2 : ℂ) * (fderiv ℝ (fderiv ℝ f) u a b₁\n + c * fderiv ℝ (fderiv ℝ f) u a b₂)", + "line": 655 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiDir_eq_weightedDirDeriv", + "docstring": "A directional derivative is a `weightedDirDeriv`: anti-holomorphic with `c = i`.", + "snippet": "private lemma dWirtingerAntiDir_eq_weightedDirDeriv (w : V) :\n (fun p => dWirtingerAntiDir f w p) = weightedDirDeriv f Complex.I w (Complex.I • w)", + "line": 666 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerDir_eq_weightedDirDeriv", + "docstring": "A directional derivative is a `weightedDirDeriv`: holomorphic with `c = -i`.", + "snippet": "private lemma dWirtingerDir_eq_weightedDirDeriv (v : V) :\n (fun p => dWirtingerDir f v p) = weightedDirDeriv f (-Complex.I) v (Complex.I • v)", + "line": 671 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.fderiv_dWirtingerAntiDir", + "docstring": "Differentiating the anti-holomorphic directional derivative lands on the second\nreal Fréchet derivative in the two slots.", + "snippet": "private lemma fderiv_dWirtingerAntiDir (hf' : DifferentiableAt ℝ (fderiv ℝ f) u)\n (w a : V) :\n fderiv ℝ (fun p => dWirtingerAntiDir f w p) u a\n = (1 / 2 : ℂ) * (fderiv ℝ (fderiv ℝ f) u a w\n + Complex.I * fderiv ℝ (fderiv ℝ f) u a (Complex.I • w))", + "line": 677 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.fderiv_dWirtingerDir", + "docstring": "Differentiating the holomorphic directional derivative lands on the second real\nFréchet derivative in the two slots.", + "snippet": "private lemma fderiv_dWirtingerDir (hf' : DifferentiableAt ℝ (fderiv ℝ f) u)\n (v a : V) :\n fderiv ℝ (fun p => dWirtingerDir f v p) u a\n = (1 / 2 : ℂ) * (fderiv ℝ (fderiv ℝ f) u a v\n - Complex.I * fderiv ℝ (fderiv ℝ f) u a (Complex.I • v))", + "line": 686 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.differentiableAt_dWirtingerDir", + "docstring": "On a `C²` field the holomorphic directional derivative is itself\nreal-differentiable.", + "snippet": "lemma differentiableAt_dWirtingerDir (hf2 : ContDiffAt ℝ 2 f u) (v : V) :\n DifferentiableAt ℝ (fun p => dWirtingerDir f v p) u", + "line": 723 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.differentiableAt_dWirtingerAntiDir", + "docstring": "On a `C²` field the anti-holomorphic directional derivative is itself\nreal-differentiable.", + "snippet": "lemma differentiableAt_dWirtingerAntiDir (hf2 : ContDiffAt ℝ 2 f u) (w : V) :\n DifferentiableAt ℝ (fun p => dWirtingerAntiDir f w p) u", + "line": 732 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerDir_congr_of_eventuallyEq", + "docstring": "The holomorphic directional derivative depends only on the field near the point:\nfields agreeing on a neighbourhood have equal derivative.", + "snippet": "lemma dWirtingerDir_congr_of_eventuallyEq {f₁ f₂ : V → ℂ} {u : V}\n (h : f₁ =ᶠ[nhds u] f₂) (v : V) :\n dWirtingerDir f₁ v u = dWirtingerDir f₂ v u", + "line": 741 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiDir_congr_of_eventuallyEq", + "docstring": "The anti-holomorphic directional derivative depends only on the field near the\npoint; dual of `dWirtingerDir_congr_of_eventuallyEq`.", + "snippet": "lemma dWirtingerAntiDir_congr_of_eventuallyEq {f₁ f₂ : V → ℂ} {u : V}\n (h : f₁ =ᶠ[nhds u] f₂) (v : V) :\n dWirtingerAntiDir f₁ v u = dWirtingerAntiDir f₂ v u", + "line": 748 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Basic.lean", + "kind": "theorem", + "name": "Physlib.Wirtinger.dWirtingerDir_dWirtingerAntiDir_comm", + "docstring": "**Schwarz's theorem** for the directional Wirtinger operators: on a `C²` field `f` the\nholomorphic and anti-holomorphic directional derivatives commute in any two directions,\n`∂_v ∂̄_w f = ∂̄_w ∂_v f`.\n\nThe commutation adds no analytic input. By the §G bridge each order expands into a real-linear\ncombination of the second real Fréchet derivative `fderiv ℝ (fderiv ℝ f) u` on the four\ndirections `v`, `i·v`, `w`, `i·w`. The two orders give the same combination up to transposing\nthe two slots of that second derivative, and `ContDiffAt.isSymmSndFDerivAt`, the symmetry of\nordinary mixed second partials, equates them.", + "snippet": "theorem dWirtingerDir_dWirtingerAntiDir_comm (hf2 : ContDiffAt ℝ 2 f u) (v w : V) :\n dWirtingerDir (fun p => dWirtingerAntiDir f w p) v u\n = dWirtingerAntiDir (fun p => dWirtingerDir f v p) w u", + "line": 768 + } + ] + }, + { + "filename": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "entries": [ + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "def", + "name": "Physlib.Wirtinger.dWirtingerCoord", + "docstring": "Holomorphic Wirtinger derivative along the I-th coordinate of `ι → ℂ`.", + "snippet": "def dWirtingerCoord (f : (ι → ℂ) → ℂ) (I : ι) : (ι → ℂ) → ℂ", + "line": 167 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "def", + "name": "Physlib.Wirtinger.dWirtingerAntiCoord", + "docstring": "Anti-holomorphic Wirtinger derivative along the I-th coordinate of `ι → ℂ`.", + "snippet": "def dWirtingerAntiCoord (f : (ι → ℂ) → ℂ) (I : ι) : (ι → ℂ) → ℂ", + "line": 171 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerCoord_apply", + "docstring": "Real-Fréchet form of `dWirtingerCoord`:\n`dWirtingerCoord f I u = (1/2)(∂_x − i · ∂_y) f`,\nthe derivatives along the slot-I real and imaginary coordinate directions.\nUnconditional — the directional definition makes it definitional\n(`Complex.I • Pi.single I 1 = Pi.single I Complex.I`); no differentiability\nhypothesis is needed.", + "snippet": "lemma dWirtingerCoord_apply {f : (ι → ℂ) → ℂ}\n {u : (ι → ℂ)} (I : ι) :\n dWirtingerCoord f I u = (1 / 2 : ℂ) * (fderiv ℝ f u (Pi.single I 1)\n - Complex.I * fderiv ℝ f u (Pi.single I Complex.I))", + "line": 180 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiCoord_apply", + "docstring": "Real-Fréchet form of `dWirtingerAntiCoord`:\n`∂̄_I f = (1/2)(∂_x + i · ∂_y) f`,\nmirror of `dWirtingerCoord_apply` with the sign flip on the imaginary-direction term.\nUnconditional, as for `dWirtingerCoord_apply`.", + "snippet": "lemma dWirtingerAntiCoord_apply {f : (ι → ℂ) → ℂ}\n {u : (ι → ℂ)} (I : ι) :\n dWirtingerAntiCoord f I u = (1 / 2 : ℂ) * (fderiv ℝ f u (Pi.single I 1)\n + Complex.I * fderiv ℝ f u (Pi.single I Complex.I))", + "line": 190 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.clinear_of_holomorphic", + "docstring": "The real derivative of a holomorphic `f : E → ℂ` is `ℂ`-linear along every direction —\nthe hypothesis the foundation collapse `dWirtingerDir_eq_of_clinear` consumes. A holomorphic\n`f` has `fderiv ℝ f u = (fderiv ℂ f u).restrictScalars ℝ`\n(`DifferentiableAt.fderiv_restrictScalars`), so its real derivative agrees with the `ℂ`-linear\ncomplex derivative on every direction. Used at `E = ι → ℂ` for the coordinate collapse and at\n`E = ℂ` for the outer-`g` chain rule (§D).", + "snippet": "private lemma clinear_of_holomorphic {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]\n [NormedSpace ℂ E] [IsScalarTower ℝ ℂ E] {f : E → ℂ} {u : E}\n (hf : DifferentiableAt ℂ f u) (d : E) :\n fderiv ℝ f u (Complex.I • d) = Complex.I • fderiv ℝ f u d", + "line": 226 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerCoord_congr_of_eventuallyEq_apply", + "docstring": "`dWirtingerCoord` is local: functions agreeing on a neighbourhood of `u` have equal\nholomorphic Wirtinger derivative at `u` (`f₁ =ᶠ[nhds u] f₂ ⟹ ∂_I f₁ u = ∂_I f₂ u`).", + "snippet": "lemma dWirtingerCoord_congr_of_eventuallyEq_apply {f₁ f₂ : (ι → ℂ) → ℂ}\n {u : (ι → ℂ)} (h : f₁ =ᶠ[nhds u] f₂) (I : ι) :\n dWirtingerCoord f₁ I u = dWirtingerCoord f₂ I u", + "line": 239 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerCoord_neg_apply", + "docstring": "Pointwise negation rule for the holomorphic coordinate derivative at `u`:\n`∂_I (−f) = −∂_I f`. Used with `dWirtingerCoord_add_apply` to assemble the\ncoordinate-difference rule `dWirtingerCoord_coordDiff` (§C).", + "snippet": "lemma dWirtingerCoord_neg_apply {u : (ι → ℂ)} (I : ι) :\n dWirtingerCoord (fun v => -(f v)) I u = -(dWirtingerCoord f I u)", + "line": 252 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.fderiv_coordProj", + "docstring": "The real Fréchet derivative of the J-th coordinate projection `v ↦ v J`. Consumed by the\nKronecker coordinate-value lemmas `dWirtingerCoord_coordProj` / `dWirtingerAntiCoord_coordProj`,\nwhich feed it the slot-I real and imaginary directions.", + "snippet": "private lemma fderiv_coordProj (J : ι) (u d : (ι → ℂ)) :\n fderiv ℝ (fun v : (ι → ℂ) => v J) u d = d J", + "line": 260 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerCoord_add_apply", + "docstring": "Pointwise additivity of the holomorphic coordinate derivative at `u`:\n`∂_I (f + g) = ∂_I f + ∂_I g`. Used to assemble the coordinate-difference rule\n`dWirtingerCoord_coordDiff` (§C).", + "snippet": "lemma dWirtingerCoord_add_apply {u : (ι → ℂ)}\n (hf : DifferentiableAt ℝ f u) (hg : DifferentiableAt ℝ g u) (I : ι) :\n dWirtingerCoord (f + g) I u = dWirtingerCoord f I u + dWirtingerCoord g I u", + "line": 283 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerCoord_smul_apply", + "docstring": "Pointwise compatibility with complex scalar multiplication at `u`:\n`∂_I (c • f) = c • ∂_I f`.", + "snippet": "lemma dWirtingerCoord_smul_apply {u : (ι → ℂ)}\n (c : ℂ) (hf : DifferentiableAt ℝ f u) (I : ι) :\n dWirtingerCoord (c • f) I u = c • dWirtingerCoord f I u", + "line": 290 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerCoord_mul_apply", + "docstring": "Pointwise Leibniz rule for the holomorphic coordinate derivative at `u`:\n`∂_I (f · g) = ∂_I f · g + f · ∂_I g`.", + "snippet": "lemma dWirtingerCoord_mul_apply {u : (ι → ℂ)}\n (hf : DifferentiableAt ℝ f u) (hg : DifferentiableAt ℝ g u) (I : ι) :\n dWirtingerCoord (f * g) I u =\n dWirtingerCoord f I u * g u + f u * dWirtingerCoord g I u", + "line": 297 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerCoord_fun_sum_apply", + "docstring": "Pointwise finite-sum rule for holomorphic coordinate derivatives at `u`:\n`∂_I (∑ a ∈ s, F a) = ∑ a ∈ s, ∂_I (F a)`.", + "snippet": "lemma dWirtingerCoord_fun_sum_apply {α : Type*} {s : Finset α}\n {F : α → (ι → ℂ) → ℂ} {u : (ι → ℂ)}\n (hF : ∀ a ∈ s, DifferentiableAt ℝ (F a) u) (I : ι) :\n dWirtingerCoord (fun v => ∑ a ∈ s, F a v) I u =\n ∑ a ∈ s, dWirtingerCoord (F a) I u", + "line": 305 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerCoord_eq_complex_fderiv_apply", + "docstring": "For a holomorphic function, `dWirtingerCoord` is the complex Fréchet derivative in\nthe corresponding coordinate direction: `∂_I f = fderiv ℂ f u (Pi.single I 1)`.", + "snippet": "lemma dWirtingerCoord_eq_complex_fderiv_apply {u : (ι → ℂ)}\n (hf : DifferentiableAt ℂ f u) (I : ι) :\n dWirtingerCoord f I u = fderiv ℂ f u (Pi.single I 1)", + "line": 314 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "def", + "name": "Physlib.Wirtinger.conjCLM", + "docstring": "Pointwise conjugation bundled as an ℝ-linear CLM (conjugate-ℂ-linear): the I-th\ncomponent is conjugation of the I-th coordinate `Complex.conjCLE ∘ proj I`. Its underlying\nfunction is the `star` of `ι → ℂ` (`conjCLM_apply`). Bundling `star` as a `→L[ℝ]` is what\nlets the anti-holomorphic lemmas below feed it to the foundation `dWirtingerDir_comp_conjLinear`,\nwhich requires its domain map as a continuous linear map.", + "snippet": "private def conjCLM : (ι → ℂ) →L[ℝ] (ι → ℂ)", + "line": 326 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.conjCLM_smul_I", + "docstring": "`conjCLM` is conjugate-ℂ-linear: `conj (i·d) = -(i · conj d)`. The `hL` hypothesis the\nfoundation `dWirtingerDir_comp_conjLinear` / `dWirtingerAntiDir_comp_conjLinear` consume.", + "snippet": "private lemma conjCLM_smul_I (d : ι → ℂ) :\n conjCLM (Complex.I • d) = -(Complex.I • conjCLM (ι := ι) d)", + "line": 336 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerCoord_eq_zero_of_antiHolomorphic_apply", + "docstring": "An *anti-holomorphic* function `v ↦ g (star v)` (a holomorphic `g` precomposed\nwith conjugation) depends on the coordinates only through their conjugates `z̄`. Its\nholomorphic derivative therefore vanishes, `∂_I (g ∘ star) = 0`, since `∂_I`\ndifferentiates w.r.t. `z` and treats `z̄` as constant. Dual to `∂̄_I f = 0` for\nholomorphic `f`.", + "snippet": "lemma dWirtingerCoord_eq_zero_of_antiHolomorphic_apply {u : (ι → ℂ)}\n (hg : DifferentiableAt ℂ g (star u)) (I : ι) :\n dWirtingerCoord (fun v : (ι → ℂ) => g (star v)) I u = 0", + "line": 348 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiCoord_congr_of_eventuallyEq_apply", + "docstring": "`dWirtingerAntiCoord` is local (`f₁ =ᶠ[nhds u] f₂ ⟹ ∂̄_I f₁ u = ∂̄_I f₂ u`).", + "snippet": "lemma dWirtingerAntiCoord_congr_of_eventuallyEq_apply {f₁ f₂ : (ι → ℂ) → ℂ}\n {u : (ι → ℂ)} (h : f₁ =ᶠ[nhds u] f₂) (I : ι) :\n dWirtingerAntiCoord f₁ I u = dWirtingerAntiCoord f₂ I u", + "line": 385 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiCoord_neg_apply", + "docstring": "Pointwise negation rule for the anti-holomorphic coordinate derivative at `u`:\n`∂̄_I (−f) = −∂̄_I f`. Used with `dWirtingerAntiCoord_add_apply` to assemble the\ncoordinate-difference rule `dWirtingerAntiCoord_coordDiff` (§C).", + "snippet": "lemma dWirtingerAntiCoord_neg_apply {u : (ι → ℂ)} (I : ι) :\n dWirtingerAntiCoord (fun v => -(f v)) I u = -(dWirtingerAntiCoord f I u)", + "line": 398 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerCoord_conjCoord", + "docstring": "`∂_I z̄^J = 0`. The conjugate of `dWirtingerAntiCoord_coordProj` (`z̄^J = star z^J`),\nread off through `dWirtingerDir_star_comp` rather than recomputed. Used to assemble the\ncoordinate-difference rule `dWirtingerCoord_coordDiff` (§C).", + "snippet": "lemma dWirtingerCoord_conjCoord (I J : ι) :\n dWirtingerCoord (fun u : (ι → ℂ) => star (u J)) I = 0", + "line": 418 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiCoord_conjCoord", + "docstring": "`∂̄_I z̄^J = δ_IJ`. The conjugate of `dWirtingerCoord_coordProj`, read off through\n`dWirtingerAntiDir_star_comp` rather than recomputed. Used to assemble the\ncoordinate-difference rule `dWirtingerAntiCoord_coordDiff` (§C).", + "snippet": "lemma dWirtingerAntiCoord_conjCoord (I J : ι) :\n dWirtingerAntiCoord (fun u : (ι → ℂ) => star (u J)) I =\n fun _ => if I = J then 1 else 0", + "line": 432 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiCoord_add_apply", + "docstring": "Pointwise additivity of the anti-holomorphic coordinate derivative at `u`:\n`∂̄_I (f + g) = ∂̄_I f + ∂̄_I g`. Used to assemble the coordinate-difference rule\n`dWirtingerAntiCoord_coordDiff` (§C).", + "snippet": "lemma dWirtingerAntiCoord_add_apply {u : (ι → ℂ)}\n (hf : DifferentiableAt ℝ f u) (hg : DifferentiableAt ℝ g u) (I : ι) :\n dWirtingerAntiCoord (f + g) I u = dWirtingerAntiCoord f I u + dWirtingerAntiCoord g I u", + "line": 448 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiCoord_smul_apply", + "docstring": "Pointwise compatibility with complex scalar multiplication at `u`:\n`∂̄_I (c • f) = c • ∂̄_I f`.", + "snippet": "lemma dWirtingerAntiCoord_smul_apply {u : (ι → ℂ)}\n (c : ℂ) (hf : DifferentiableAt ℝ f u) (I : ι) :\n dWirtingerAntiCoord (c • f) I u = c • dWirtingerAntiCoord f I u", + "line": 455 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiCoord_mul_apply", + "docstring": "Pointwise Leibniz rule for the anti-holomorphic coordinate derivative at `u`:\n`∂̄_I (f · g) = ∂̄_I f · g + f · ∂̄_I g`.", + "snippet": "lemma dWirtingerAntiCoord_mul_apply {u : (ι → ℂ)}\n (hf : DifferentiableAt ℝ f u) (hg : DifferentiableAt ℝ g u) (I : ι) :\n dWirtingerAntiCoord (f * g) I u =\n dWirtingerAntiCoord f I u * g u + f u * dWirtingerAntiCoord g I u", + "line": 462 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiCoord_fun_sum_apply", + "docstring": "Pointwise finite-sum rule for anti-holomorphic coordinate derivatives at `u`:\n`∂̄_I (∑ a ∈ s, F a) = ∑ a ∈ s, ∂̄_I (F a)`.", + "snippet": "lemma dWirtingerAntiCoord_fun_sum_apply {α : Type*} {s : Finset α}\n {F : α → (ι → ℂ) → ℂ} {u : (ι → ℂ)}\n (hF : ∀ a ∈ s, DifferentiableAt ℝ (F a) u) (I : ι) :\n dWirtingerAntiCoord (fun v => ∑ a ∈ s, F a v) I u =\n ∑ a ∈ s, dWirtingerAntiCoord (F a) I u", + "line": 470 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiCoord_eq_complex_fderiv_apply", + "docstring": "For an anti-holomorphic function the anti-holomorphic Wirtinger derivative equals\nthe complex Fréchet derivative of `g` at `star u` along the slot-I real\ncoordinate direction, `∂̄_I (g ∘ star) = fderiv ℂ g (star u) (Pi.single I 1)`.\nDual of `dWirtingerCoord_eq_complex_fderiv_apply`.", + "snippet": "lemma dWirtingerAntiCoord_eq_complex_fderiv_apply {u : (ι → ℂ)}\n (hg : DifferentiableAt ℂ g (star u)) (I : ι) :\n dWirtingerAntiCoord (fun v : (ι → ℂ) => g (star v)) I u =\n fderiv ℂ g (star u) (Pi.single I 1)", + "line": 481 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiCoord_eq_zero_of_holomorphic_apply", + "docstring": "Holomorphic functions have zero anti-holomorphic coordinate derivative: `∂̄_I f = 0`.", + "snippet": "lemma dWirtingerAntiCoord_eq_zero_of_holomorphic_apply {u : (ι → ℂ)}\n (hf : DifferentiableAt ℂ f u) (I : ι) :\n dWirtingerAntiCoord f I u = 0", + "line": 497 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.coordDiff_eq_add_neg", + "docstring": "The coordinate difference `z^J − z̄^J` as a sum of the holomorphic coordinate\nand the negated conjugate coordinate — the form on which the `dWirtingerCoord` /\n`dWirtingerAntiCoord` additivity rules apply.", + "snippet": "private lemma coordDiff_eq_add_neg (J : ι) :\n (fun v : (ι → ℂ) => v J - star (v J))\n = (fun v => v J) + (fun v => -(star (v J)))", + "line": 517 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerCoord_coordDiff", + "docstring": "`∂_I (z^J − z̄^J) = δ_IJ`, from `∂_I z^J = δ_IJ` and\n`∂_I z̄^J = 0`.", + "snippet": "lemma dWirtingerCoord_coordDiff (I J : ι) :\n dWirtingerCoord (fun v : (ι → ℂ) => v J - star (v J)) I\n = fun _ => if I = J then 1 else 0", + "line": 524 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiCoord_coordDiff", + "docstring": "`∂̄_I (z^J − z̄^J) = −δ_IJ`, from `∂̄_I z^J = 0` and\n`∂̄_I z̄^J = δ_IJ`.", + "snippet": "lemma dWirtingerAntiCoord_coordDiff (I J : ι) :\n dWirtingerAntiCoord (fun v : (ι → ℂ) => v J - star (v J)) I\n = fun _ => if I = J then -1 else 0", + "line": 534 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerCoord_comp_apply", + "docstring": "The two-term coordinate chain rule for a real-differentiable outer `g`, pointwise at `u`:\n`∂_I (g ∘ f) = (∂g/∂f) · ∂_I f + (∂g/∂f̄) · ∂_I f̄`,\nwith coefficients `∂g/∂f = dWirtingerDir g 1 (f u)` and `∂g/∂f̄ = dWirtingerAntiDir g 1 (f u)`.\nThe `d = Pi.single I 1` case of the foundation `dWirtingerDir_comp`. The single-term\nholomorphic specialization `dWirtingerCoord_comp_holomorphic_apply` is proved from this.", + "snippet": "lemma dWirtingerCoord_comp_apply {u : (ι → ℂ)}\n (hg : DifferentiableAt ℝ g (f u)) (hf : DifferentiableAt ℝ f u) (I : ι) :\n dWirtingerCoord (fun v => g (f v)) I u =\n dWirtingerDir g 1 (f u) * dWirtingerCoord f I u\n + dWirtingerAntiDir g 1 (f u) *\n dWirtingerCoord (fun v : (ι → ℂ) => star (f v)) I u", + "line": 580 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerCoord_comp_holomorphic_apply", + "docstring": "The single-term coordinate chain rule for a holomorphic outer `g`, pointwise at `u`:\n`∂_I (g ∘ f) = deriv g (f u) · ∂_I f`. From the two-term `dWirtingerCoord_comp_apply`: for\nholomorphic `g` the anti-holomorphic coefficient `dWirtingerAntiDir g 1 (f u)` vanishes and\n`dWirtingerDir g 1 (f u)` collapses to `deriv g (f u)`, both off the `ℂ`-linearity\n`clinear_of_holomorphic` (§B).", + "snippet": "lemma dWirtingerCoord_comp_holomorphic_apply {u : (ι → ℂ)}\n (hg : DifferentiableAt ℂ g (f u)) (hf : DifferentiableAt ℝ f u) (I : ι) :\n dWirtingerCoord (fun v => g (f v)) I u =\n deriv g (f u) * dWirtingerCoord f I u", + "line": 593 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerCoord_star_comp_apply", + "docstring": "Conjugating the inner field swaps the two operators: `∂_I f̄ = conj (∂̄_I f)`. The\n`d = Pi.single I 1` case of the foundation `dWirtingerDir_star_comp`.", + "snippet": "lemma dWirtingerCoord_star_comp_apply {u : (ι → ℂ)}\n (hf : DifferentiableAt ℝ f u) (I : ι) :\n dWirtingerCoord (fun v : (ι → ℂ) => star (f v)) I u =\n star (dWirtingerAntiCoord f I u)", + "line": 605 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiCoord_comp_apply", + "docstring": "The two-term coordinate chain rule for a real-differentiable outer `g`, anti-holomorphic\nversion, pointwise at `u`: `∂̄_I (g ∘ f) = (∂g/∂f) · ∂̄_I f + (∂g/∂f̄) · ∂̄_I f̄`. Same outer\ncoefficients as `dWirtingerCoord_comp_apply`, now multiplying the anti-holomorphic inner\nderivatives. The `d = Pi.single I 1` case of the foundation `dWirtingerAntiDir_comp`. The\nsingle-term holomorphic specialization `dWirtingerAntiCoord_comp_holomorphic_apply` is proved\nfrom this.", + "snippet": "lemma dWirtingerAntiCoord_comp_apply {u : (ι → ℂ)}\n (hg : DifferentiableAt ℝ g (f u)) (hf : DifferentiableAt ℝ f u) (I : ι) :\n dWirtingerAntiCoord (fun v => g (f v)) I u =\n dWirtingerDir g 1 (f u) * dWirtingerAntiCoord f I u\n + dWirtingerAntiDir g 1 (f u) *\n dWirtingerAntiCoord (fun v : (ι → ℂ) => star (f v)) I u", + "line": 617 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiCoord_comp_holomorphic_apply", + "docstring": "The single-term coordinate chain rule for a holomorphic outer `g`, anti-holomorphic\nversion, pointwise at `u`: `∂̄_I (g ∘ f) = deriv g (f u) · ∂̄_I f`. As in\n`dWirtingerCoord_comp_holomorphic_apply`, the `∂g/∂f̄` channel vanishes and `∂g/∂f`\ncollapses to `deriv g (f u)`.", + "snippet": "lemma dWirtingerAntiCoord_comp_holomorphic_apply {u : (ι → ℂ)}\n (hg : DifferentiableAt ℂ g (f u)) (hf : DifferentiableAt ℝ f u) (I : ι) :\n dWirtingerAntiCoord (fun v => g (f v)) I u =\n deriv g (f u) * dWirtingerAntiCoord f I u", + "line": 629 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.dWirtingerAntiCoord_star_comp_apply", + "docstring": "Conjugating the inner field swaps the two operators: `∂̄_I f̄ = conj (∂_I f)`. Dual of\n`dWirtingerCoord_star_comp_apply`, the `d = Pi.single I 1` case of the foundation\n`dWirtingerAntiDir_star_comp`.", + "snippet": "lemma dWirtingerAntiCoord_star_comp_apply {u : (ι → ℂ)}\n (hf : DifferentiableAt ℝ f u) (I : ι) :\n dWirtingerAntiCoord (fun v : (ι → ℂ) => star (f v)) I u =\n star (dWirtingerCoord f I u)", + "line": 642 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.differentiableAt_dWirtingerCoord", + "docstring": "On a `C²` function the holomorphic coordinate Wirtinger derivative is itself\nreal-differentiable (`DifferentiableAt ℝ (∂_I f) u`) — the `d = Pi.single I 1` case of\n`differentiableAt_dWirtingerDir`.", + "snippet": "lemma differentiableAt_dWirtingerCoord (hf2 : ContDiffAt ℝ 2 f u) (I : ι) :\n DifferentiableAt ℝ (fun v => dWirtingerCoord f I v) u", + "line": 672 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "lemma", + "name": "Physlib.Wirtinger.differentiableAt_dWirtingerAntiCoord", + "docstring": "On a `C²` function the anti-holomorphic coordinate Wirtinger derivative is itself\nreal-differentiable (`DifferentiableAt ℝ (∂̄_J f) u`) — the `d = Pi.single J 1` case of\n`differentiableAt_dWirtingerAntiDir`.", + "snippet": "lemma differentiableAt_dWirtingerAntiCoord (hf2 : ContDiffAt ℝ 2 f u) (J : ι) :\n DifferentiableAt ℝ (fun v => dWirtingerAntiCoord f J v) u", + "line": 679 + }, + { + "file": "Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean", + "kind": "theorem", + "name": "Physlib.Wirtinger.dWirtingerCoord_dWirtingerAntiCoord_comm", + "docstring": "**Schwarz's theorem** for the coordinate Wirtinger operators: on a `C²` function the\nholomorphic and anti-holomorphic derivatives commute, `∂_I ∂̄_J f = ∂̄_J ∂_I f`. The\n`d = Pi.single I 1`, `e = Pi.single J 1` case of the general\n`dWirtingerDir_dWirtingerAntiDir_comm`.", + "snippet": "theorem dWirtingerCoord_dWirtingerAntiCoord_comm (hf2 : ContDiffAt ℝ 2 f u) (I J : ι) :\n dWirtingerCoord (fun v => dWirtingerAntiCoord f J v) I u\n = dWirtingerAntiCoord (fun v => dWirtingerCoord f I v) J u", + "line": 687 + } + ] + }, + { + "filename": "Physlib/Mathematics/ConjModule.lean", + "entries": [ + { + "file": "Physlib/Mathematics/ConjModule.lean", + "kind": "def", + "name": "ConjModule", + "docstring": "The conjugate module of `M`: the same additive group with the scalar action twisted by\nconjugation, `r • v = star r • v`. A type synonym so the twisted action stays off `M`.", + "snippet": "def ConjModule (M : Type*)", + "line": 45 + }, + { + "file": "Physlib/Mathematics/ConjModule.lean", + "kind": "instance", + "name": "ConjModule.«instance@88e8497f»", + "docstring": null, + "snippet": "instance : AddCommGroup (ConjModule M)", + "line": 49 + }, + { + "file": "Physlib/Mathematics/ConjModule.lean", + "kind": "instance", + "name": "ConjModule.instModule", + "docstring": "The twisted action `r • v = star r • v`, obtained by restricting scalars along the\nconjugation ring endomorphism `starRingEnd k`.", + "snippet": "instance instModule : Module k (ConjModule M)", + "line": 53 + }, + { + "file": "Physlib/Mathematics/ConjModule.lean", + "kind": "def", + "name": "conjEquiv", + "docstring": "The canonical conjugate-linear equivalence `M ≃ₛₗ[starRingEnd k] ConjModule M`, the identity on\nthe underlying additive group.", + "snippet": "def conjEquiv : M ≃ₛₗ[starRingEnd k] ConjModule M where", + "line": 60 + }, + { + "file": "Physlib/Mathematics/ConjModule.lean", + "kind": "def", + "name": "ConjModule.involution", + "docstring": "Conjugating twice returns the original module: the `k`-linear isomorphism\n`ConjModule (ConjModule M) ≃ₗ[k] M`. It is `k`-linear, not merely semilinear, because\n`starRingEnd k` composed with itself is the identity.", + "snippet": "def involution : ConjModule (ConjModule M) ≃ₗ[k] M", + "line": 73 + }, + { + "file": "Physlib/Mathematics/ConjModule.lean", + "kind": "def", + "name": "ConjModule.starFinsupp", + "docstring": "Coordinate-wise conjugation on `ι →₀ k`, a conjugate-linear self-equivalence.", + "snippet": "noncomputable def starFinsupp : (ι →₀ k) ≃ₛₗ[starRingEnd k] (ι →₀ k) where", + "line": 79 + }, + { + "file": "Physlib/Mathematics/ConjModule.lean", + "kind": "def", + "name": "Basis.conj", + "docstring": "A basis of `M` transported to a basis of `ConjModule M`: the same basis vectors, with\ncoordinates conjugated (`(Basis.conj b).repr v = star ∘ b.repr v`).", + "snippet": "noncomputable def _root_.Basis.conj (b : Basis ι k M) : Basis ι k (ConjModule M)", + "line": 92 + } + ] + }, + { + "filename": "Physlib/Mathematics/CrossProductMatrix.lean", + "entries": [ + { + "file": "Physlib/Mathematics/CrossProductMatrix.lean", + "kind": "def", + "name": "Matrix.crossProductMatrix", + "docstring": "The hat map `[ω]ₓ`: the skew-symmetric `3 × 3` matrix acting as the cross product with `ω`,\ni.e. `[ω]ₓ *ᵥ v = ω ⨯₃ v`.", + "snippet": "def crossProductMatrix (ω : Fin 3 → ℝ) : Matrix (Fin 3) (Fin 3) ℝ", + "line": 28 + }, + { + "file": "Physlib/Mathematics/CrossProductMatrix.lean", + "kind": "lemma", + "name": "Matrix.crossProductMatrix_mulVec", + "docstring": "The hat matrix acts on a vector as the cross product.", + "snippet": "@[simp]\nlemma crossProductMatrix_mulVec (ω v : Fin 3 → ℝ) :\n crossProductMatrix ω *ᵥ v = ω ⨯₃ v", + "line": 33 + }, + { + "file": "Physlib/Mathematics/CrossProductMatrix.lean", + "kind": "lemma", + "name": "Matrix.crossProductMatrix_transpose", + "docstring": "The hat map produces skew-symmetric matrices.", + "snippet": "@[simp]\nlemma crossProductMatrix_transpose (ω : Fin 3 → ℝ) :\n (crossProductMatrix ω)ᵀ = - crossProductMatrix ω", + "line": 41 + }, + { + "file": "Physlib/Mathematics/CrossProductMatrix.lean", + "kind": "def", + "name": "Matrix.crossProductVee", + "docstring": "The vee map: reads the vector off a `3 × 3` matrix. It is a left inverse of the hat map.", + "snippet": "def crossProductVee (A : Matrix (Fin 3) (Fin 3) ℝ) : Fin 3 → ℝ", + "line": 47 + }, + { + "file": "Physlib/Mathematics/CrossProductMatrix.lean", + "kind": "lemma", + "name": "Matrix.crossProductVee_crossProductMatrix", + "docstring": "The vee map is a left inverse of the hat map.", + "snippet": "@[simp]\nlemma crossProductVee_crossProductMatrix (ω : Fin 3 → ℝ) :\n crossProductVee (crossProductMatrix ω) = ω", + "line": 51 + } + ] + }, + { + "filename": "Physlib/Mathematics/KroneckerDelta.lean", + "entries": [ + { + "file": "Physlib/Mathematics/KroneckerDelta.lean", + "kind": "def", + "name": "KroneckerDelta.generalizedKroneckerDelta", + "docstring": "Generalized Kronecker delta:\n`δ^{μ₁...μₙ}_{ν₁...νₙ} = det (δ[μᵢ, νⱼ])`.\n\nThis is defined for any finite type `α` with decidable equality.", + "snippet": "def generalizedKroneckerDelta {α ι : Type} [DecidableEq α]\n [DecidableEq ι] [Fintype ι]\n (μ : ι → α) (ν : ι → α) : ℤ", + "line": 151 + }, + { + "file": "Physlib/Mathematics/KroneckerDelta.lean", + "kind": "lemma", + "name": "KroneckerDelta.generalizedKroneckerDelta_swap", + "docstring": "Swapping two of the upper indices of the generalized Kronecker delta negates it.\nThis is one row transposition of the underlying determinant.", + "snippet": "lemma generalizedKroneckerDelta_swap {α ι : Type} [DecidableEq α] [DecidableEq ι] [Fintype ι]\n (μ ν : ι → α) {i j : ι} (hij : i ≠ j) :\n generalizedKroneckerDelta (μ ∘ Equiv.swap i j) ν = - generalizedKroneckerDelta μ ν", + "line": 158 + } + ] + }, + { + "filename": "Physlib/Mathematics/LinearPMap.lean", + "entries": [ + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.sub_le_zero", + "docstring": null, + "snippet": "lemma sub_le_zero : f - f ≤ 0", + "line": 64 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.neg_add_le_zero", + "docstring": null, + "snippet": "lemma neg_add_le_zero : -f + f ≤ 0", + "line": 66 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.le_iff_neg_le_neg", + "docstring": null, + "snippet": "lemma le_iff_neg_le_neg : g₁ ≤ g₂ ↔ -g₁ ≤ -g₂", + "line": 68 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.le_neg_iff_neg_le", + "docstring": null, + "snippet": "lemma le_neg_iff_neg_le : g₁ ≤ -g₂ ↔ -g₁ ≤ g₂", + "line": 71 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.add_sub_le_cancel", + "docstring": null, + "snippet": "lemma add_sub_le_cancel : f₁ + (f₂ - f₁) ≤ f₂", + "line": 73 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.add_sub_le_cancel_left", + "docstring": null, + "snippet": "lemma add_sub_le_cancel_left : f₁ + f₂ - f₁ ≤ f₂", + "line": 76 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.add_sub_le_cancel_right", + "docstring": null, + "snippet": "lemma add_sub_le_cancel_right : f₁ + f₂ - f₂ ≤ f₁", + "line": 78 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.add_add_sub_le_cancel", + "docstring": null, + "snippet": "lemma add_add_sub_le_cancel : f₁ + f₂ + (f₃ - f₂) ≤ f₁ + f₃", + "line": 80 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.add_sub_sub_le_cancel", + "docstring": null, + "snippet": "lemma add_sub_sub_le_cancel : f₁ + f₂ - (f₁ - f₃) ≤ f₂ + f₃", + "line": 83 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.sub_sub_sub_le_cancel_right", + "docstring": null, + "snippet": "lemma sub_sub_sub_le_cancel_right : f₁ - f₂ - (f₃ - f₂) ≤ f₁ - f₃", + "line": 86 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.sub_sub_sub_le_cancel_left", + "docstring": null, + "snippet": "lemma sub_sub_sub_le_cancel_left : f₁ - f₂ - (f₁ - f₃) ≤ f₃ - f₂", + "line": 90 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.sub_le_of_le_add", + "docstring": null, + "snippet": "lemma sub_le_of_le_add (h : g ≤ g₁ + g₂) : g - g₂ ≤ g₁", + "line": 93 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.sub_add_le_cancel", + "docstring": null, + "snippet": "lemma sub_add_le_cancel : f₁ - f₂ + f₂ ≤ f₁", + "line": 99 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.add_le_of_le_sub", + "docstring": null, + "snippet": "lemma add_le_of_le_sub (h : g ≤ g₁ - g₂) : g + g₂ ≤ g₁", + "line": 102 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.add_left_le_of_le", + "docstring": null, + "snippet": "lemma add_left_le_of_le (h : g₁ ≤ g₂) : f + g₁ ≤ f + g₂", + "line": 105 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.add_right_le_of_le", + "docstring": null, + "snippet": "lemma add_right_le_of_le (h : g₁ ≤ g₂) : g₁ + f ≤ g₂ + f", + "line": 112 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.sub_right_le_of_le", + "docstring": null, + "snippet": "lemma sub_right_le_of_le (h : g₁ ≤ g₂) : g₁ - f ≤ g₂ - f", + "line": 115 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.sub_left_le_of_le", + "docstring": null, + "snippet": "lemma sub_left_le_of_le (h : g₁ ≤ g₂) : f - g₁ ≤ f - g₂", + "line": 118 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.zero_smul_le", + "docstring": null, + "snippet": "lemma zero_smul_le (f : E →ₗ.[𝕜] F) : (0 : 𝕜) • f ≤ 0", + "line": 131 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.zero_smul_eq", + "docstring": null, + "snippet": "@[simp]\nlemma zero_smul_eq {f : E →ₗ.[𝕜] F} (h : f.domain = ⊤) : (0 : 𝕜) • f = 0", + "line": 134 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "def", + "name": "LinearPMap.sum", + "docstring": "A finite sum of partial linear maps.\n\n `sum f` and `∑ a, f a` are equal, but not by definition.\n With `sum f` both `domain` and `toFun` are made explicit.", + "snippet": "def sum : E →ₗ.[R] F where", + "line": 151 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.sum_domain", + "docstring": null, + "snippet": "lemma sum_domain : (sum f).domain = ⨅ a, (f a).domain", + "line": 155 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.sum_domain_le", + "docstring": null, + "snippet": "lemma sum_domain_le (a : α) : (sum f).domain ≤ (f a).domain", + "line": 157 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.sum_apply", + "docstring": null, + "snippet": "@[simp]\nlemma sum_apply (ψ : (sum f).domain) : sum f ψ = ∑ a, f a ⟨ψ, sum_domain_le f a ψ.2⟩", + "line": 160 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "def", + "name": "LinearPMap.compRestricted", + "docstring": "`g ∘ᵣ f` is the composition of `g` with `f` restricted to a domain consisting of exactly those\n `x : f.domain` for which `f x ∈ g.domain`.", + "snippet": "def compRestricted : E →ₗ.[R] G", + "line": 177 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.compRestricted_domain_le", + "docstring": null, + "snippet": "lemma compRestricted_domain_le : (g ∘ᵣ f).domain ≤ f.domain", + "line": 189 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.compRestricted_domain", + "docstring": null, + "snippet": "lemma compRestricted_domain : (g ∘ᵣ f).domain = (g.domain.comap f.toFun).map f.domain.subtype", + "line": 191 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.mem_compRestricted_domain_iff", + "docstring": null, + "snippet": "lemma mem_compRestricted_domain_iff {x : E} :\n x ∈ (v ∘ᵣ u).domain ↔ ∃ h : x ∈ u.domain, u ⟨x, h⟩ ∈ v.domain", + "line": 200 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.mem_compRestricted_domain_iff'", + "docstring": null, + "snippet": "lemma mem_compRestricted_domain_iff' {x : E} :\n x ∈ (v ∘ᵣ u).domain ↔ ∃ y : u.domain, x = y ∧ ∃ y' : v.domain, u y = y'", + "line": 204 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.mem_domain_of_mem_compRestricted_domain", + "docstring": null, + "snippet": "lemma mem_domain_of_mem_compRestricted_domain (x : (v ∘ᵣ u).domain) : u ⟨x, x.2.2⟩ ∈ v.domain", + "line": 208 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.compRestricted_apply", + "docstring": null, + "snippet": "@[simp]\nlemma compRestricted_apply (x : (v ∘ᵣ u).domain) :\n (v ∘ᵣ u) x = v ⟨u ⟨x, x.2.2⟩, mem_domain_of_mem_compRestricted_domain x⟩", + "line": 212 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.compRestricted_zero", + "docstring": "The zero map is right-absorbing.", + "snippet": "@[simp]\nlemma compRestricted_zero : g ∘ᵣ (0 : E →ₗ.[R] F) = 0", + "line": 217 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.compRestricted_assoc", + "docstring": null, + "snippet": "lemma compRestricted_assoc {H : Type*} [AddCommGroup H] [Module R H]\n (f₁ : G →ₗ.[R] H) (f₂ : F →ₗ.[R] G) (f₃ : E →ₗ.[R] F) :\n (f₁ ∘ᵣ f₂) ∘ᵣ f₃ = f₁ ∘ᵣ f₂ ∘ᵣ f₃", + "line": 222 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.compRestricted_eq_comp", + "docstring": "`compRestricted` is the same as `comp` when the range of `u` is contained in `v.domain`.", + "snippet": "lemma compRestricted_eq_comp (h : ∀ x : u.domain, u x ∈ v.domain) :\n v ∘ᵣ u = v.comp u h", + "line": 231 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.comp_le_compRestricted", + "docstring": "`compRestricted` is maximal amongst compositions of `v` with domain restrictions of `u`.", + "snippet": "lemma comp_le_compRestricted\n {S : Submodule R E} (h : ∀ x : (u.domRestrict S).domain, u ⟨x, x.2.2⟩ ∈ v.domain) :\n v.comp (u.domRestrict S) h ≤ v ∘ᵣ u", + "line": 239 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.compRestricted_mono_left", + "docstring": null, + "snippet": "lemma compRestricted_mono_left {g g' : F →ₗ.[R] G} (h : g ≤ g') (f : E →ₗ.[R] F) :\n g ∘ᵣ f ≤ g' ∘ᵣ f", + "line": 244 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.compRestricted_mono_right", + "docstring": null, + "snippet": "lemma compRestricted_mono_right (g : F →ₗ.[R] G) {f f' : E →ₗ.[R] F} (h : f ≤ f') :\n g ∘ᵣ f ≤ g ∘ᵣ f'", + "line": 254 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.neg_compRestricted", + "docstring": null, + "snippet": "@[simp]\nlemma neg_compRestricted : (-g) ∘ᵣ f = -g ∘ᵣ f", + "line": 264 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.compRestricted_neg", + "docstring": null, + "snippet": "@[simp]\nlemma compRestricted_neg : g ∘ᵣ (-f) = -g ∘ᵣ f", + "line": 267 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.add_compRestricted", + "docstring": null, + "snippet": "lemma add_compRestricted : (g₁ + g₂) ∘ᵣ f = g₁ ∘ᵣ f + g₂ ∘ᵣ f", + "line": 273 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.sub_compRestricted", + "docstring": null, + "snippet": "lemma sub_compRestricted : (g₁ - g₂) ∘ᵣ f = g₁ ∘ᵣ f - g₂ ∘ᵣ f", + "line": 279 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.compRestricted_add_ge", + "docstring": null, + "snippet": "lemma compRestricted_add_ge : g ∘ᵣ f₁ + g ∘ᵣ f₂ ≤ g ∘ᵣ (f₁ + f₂)", + "line": 282 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.compRestricted_sub_ge", + "docstring": null, + "snippet": "lemma compRestricted_sub_ge : g ∘ᵣ f₁ - g ∘ᵣ f₂ ≤ g ∘ᵣ (f₁ - f₂)", + "line": 293 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.compRestricted_smul", + "docstring": null, + "snippet": "lemma compRestricted_smul {S : Type*} [DivisionRing S]\n [Module S E] [Module S F] [Module S G] [SMulCommClass S S F] [SMulCommClass S S G]\n {c : S} (hc : c ≠ 0) (g : F →ₗ.[S] G) (f : E →ₗ.[S] F) :\n g ∘ᵣ (c • f) = c • (g ∘ᵣ f)", + "line": 297 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.smul_compRestricted", + "docstring": null, + "snippet": "@[simp]\nlemma smul_compRestricted {M : Type*} [Monoid M] [DistribMulAction M G] [SMulCommClass R M G]\n (c : M) (g : F →ₗ.[R] G) (f : E →ₗ.[R] F) :\n (c • g) ∘ᵣ f = c • (g ∘ᵣ f)", + "line": 307 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "instance", + "name": "LinearPMap.instMonoid", + "docstring": null, + "snippet": "instance instMonoid : Monoid (E →ₗ.[R] E) where", + "line": 325 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.mul_def", + "docstring": null, + "snippet": "lemma mul_def (f₁ f₂ : E →ₗ.[R] E) : f₁ * f₂ = f₁ ∘ᵣ f₂", + "line": 340 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.one_domain", + "docstring": null, + "snippet": "@[simp]\nlemma one_domain : (1 : E →ₗ.[R] E).domain = ⊤", + "line": 343 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.one_toFun", + "docstring": null, + "snippet": "@[simp]\nlemma one_toFun : (1 : E →ₗ.[R] E).toFun = topEquiv.toLinearMap", + "line": 346 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.one_coe", + "docstring": null, + "snippet": "@[simp]\nlemma one_coe : (1 : E →ₗ.[R] E).toFun' = ⇑topEquiv.toLinearMap", + "line": 349 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.inverse_ker", + "docstring": null, + "snippet": "lemma inverse_ker : f.inverse.toFun.ker = ⊥", + "line": 362 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.inverse_inverse", + "docstring": null, + "snippet": "lemma inverse_inverse : f.inverse.inverse = f", + "line": 367 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.inverse_compRestricted_eq", + "docstring": null, + "snippet": "lemma inverse_compRestricted_eq : f.inverse ∘ᵣ f = domRestrict 1 f.domain", + "line": 374 + }, + { + "file": "Physlib/Mathematics/LinearPMap.lean", + "kind": "lemma", + "name": "LinearPMap.compRestricted_inverse_eq", + "docstring": null, + "snippet": "lemma compRestricted_inverse_eq : f ∘ᵣ f.inverse = domRestrict 1 f.inverse.domain", + "line": 379 + } + ] + }, + { + "filename": "Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/Basic.lean", + "entries": [ + { + "file": "Physlib/Particles/BeyondTheStandardModel/RHN/AnomalyCancellation/Basic.lean", + "kind": "lemma", + "name": "SMνCharges.sum_one", + "docstring": null, + "snippet": "lemma sum_one [AddCommMonoid M] (f : Fin (SMνSpecies 1).numberCharges → M) :\n ∑ i, f i = f ⟨0, by simp⟩", + "line": 35 + } + ] + }, + { + "filename": "Physlib/Particles/StandardModel/AnomalyCancellation/Basic.lean", + "entries": [ + { + "file": "Physlib/Particles/StandardModel/AnomalyCancellation/Basic.lean", + "kind": "lemma", + "name": "SMCharges.sum_SMSpecies_numberCharges_one", + "docstring": null, + "snippet": "lemma sum_SMSpecies_numberCharges_one {M} [AddCommMonoid M]\n (f : Fin (SMSpecies 1).numberCharges → M) :\n ∑ i, f i = f ⟨0, by simp⟩", + "line": 34 + }, + { + "file": "Physlib/Particles/StandardModel/AnomalyCancellation/Basic.lean", + "kind": "lemma", + "name": "SMCharges.toSpecies_apply_eq", + "docstring": null, + "snippet": "lemma toSpecies_apply_eq (i : Fin 5) (S : (SMCharges n).Charges) :\n toSpecies i S = fun j => toSpeciesEquiv S i j", + "line": 55 + } + ] + }, + { + "filename": "Physlib/Particles/StandardModel/AnomalyCancellation/NoGrav/One/LinearParameterization.lean", + "entries": [ + { + "file": "Physlib/Particles/StandardModel/AnomalyCancellation/NoGrav/One/LinearParameterization.lean", + "kind": "lemma", + "name": "SM.SMNoGrav.One.linearParameters.toSpecies_apply_asCharges", + "docstring": null, + "snippet": "lemma toSpecies_apply_asCharges (S : linearParameters) (i : Fin 5) :\n toSpecies i S.asCharges = fun _ => S.asCharges i", + "line": 69 + } + ] + }, + { + "filename": "Physlib/Particles/StandardModel/Basic.lean", + "entries": [ + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.gaugeGroupℤ₆UnitaryOfRoot", + "docstring": "The unitary complex number associated to a sixth root of unity.", + "snippet": "noncomputable def gaugeGroupℤ₆UnitaryOfRoot (α : rootsOfUnity 6 ℂ) : unitary ℂ", + "line": 125 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₆UnitaryOfRoot_coe", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₆UnitaryOfRoot_coe (α : rootsOfUnity 6 ℂ) :\n (gaugeGroupℤ₆UnitaryOfRoot α : ℂ) = ((α : ℂˣ) : ℂ)", + "line": 135 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.gaugeGroupℤ₆SU3OfRoot", + "docstring": "The `SU(3)` scalar matrix associated to a sixth root of unity.", + "snippet": "noncomputable def gaugeGroupℤ₆SU3OfRoot (α : rootsOfUnity 6 ℂ) :\n specialUnitaryGroup (Fin 3) ℂ", + "line": 139 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.gaugeGroupℤ₆SU2OfRoot", + "docstring": "The `SU(2)` scalar matrix associated to a sixth root of unity.", + "snippet": "noncomputable def gaugeGroupℤ₆SU2OfRoot (α : rootsOfUnity 6 ℂ) :\n specialUnitaryGroup (Fin 2) ℂ", + "line": 163 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.gaugeGroupℤ₆OfRoot", + "docstring": "The element of `GaugeGroupI` associated to a sixth root of unity.", + "snippet": "noncomputable def gaugeGroupℤ₆OfRoot (α : rootsOfUnity 6 ℂ) : GaugeGroupI", + "line": 193 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₆OfRoot_toSU3", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₆OfRoot_toSU3 (α : rootsOfUnity 6 ℂ) :\n GaugeGroupI.toSU3 (gaugeGroupℤ₆OfRoot α) = gaugeGroupℤ₆SU3OfRoot α", + "line": 197 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₆OfRoot_toSU2", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₆OfRoot_toSU2 (α : rootsOfUnity 6 ℂ) :\n GaugeGroupI.toSU2 (gaugeGroupℤ₆OfRoot α) = gaugeGroupℤ₆SU2OfRoot α", + "line": 201 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₆OfRoot_toU1", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₆OfRoot_toU1 (α : rootsOfUnity 6 ℂ) :\n GaugeGroupI.toU1 (gaugeGroupℤ₆OfRoot α) = gaugeGroupℤ₆UnitaryOfRoot α", + "line": 205 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₆OfRoot_mem_center", + "docstring": null, + "snippet": "lemma gaugeGroupℤ₆OfRoot_mem_center (α : rootsOfUnity 6 ℂ) :\n gaugeGroupℤ₆OfRoot α ∈ Subgroup.center GaugeGroupI", + "line": 208 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.gaugeGroupℤ₆Hom", + "docstring": "The homomorphism from sixth roots of unity to `GaugeGroupI`.", + "snippet": "noncomputable def gaugeGroupℤ₆Hom : rootsOfUnity 6 ℂ →* GaugeGroupI where", + "line": 226 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₆Hom_apply", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₆Hom_apply (α : rootsOfUnity 6 ℂ) :\n gaugeGroupℤ₆Hom α = gaugeGroupℤ₆OfRoot α", + "line": 258 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₆Hom_toSU3", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₆Hom_toSU3 (α : rootsOfUnity 6 ℂ) :\n GaugeGroupI.toSU3 (gaugeGroupℤ₆Hom α) = gaugeGroupℤ₆SU3OfRoot α", + "line": 262 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₆Hom_toSU2", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₆Hom_toSU2 (α : rootsOfUnity 6 ℂ) :\n GaugeGroupI.toSU2 (gaugeGroupℤ₆Hom α) = gaugeGroupℤ₆SU2OfRoot α", + "line": 266 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₆Hom_toU1", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₆Hom_toU1 (α : rootsOfUnity 6 ℂ) :\n GaugeGroupI.toU1 (gaugeGroupℤ₆Hom α) = gaugeGroupℤ₆UnitaryOfRoot α", + "line": 270 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₆OfRoot_mem", + "docstring": null, + "snippet": "lemma gaugeGroupℤ₆OfRoot_mem (α : rootsOfUnity 6 ℂ) :\n gaugeGroupℤ₆OfRoot α ∈ gaugeGroupℤ₆SubGroup", + "line": 282 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.mem_gaugeGroupℤ₆SubGroup_iff", + "docstring": null, + "snippet": "lemma mem_gaugeGroupℤ₆SubGroup_iff (g : GaugeGroupI) :\n g ∈ gaugeGroupℤ₆SubGroup ↔ ∃ α : rootsOfUnity 6 ℂ, gaugeGroupℤ₆OfRoot α = g", + "line": 286 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₆SubGroup_le_center", + "docstring": null, + "snippet": "lemma gaugeGroupℤ₆SubGroup_le_center :\n gaugeGroupℤ₆SubGroup ≤ Subgroup.center GaugeGroupI", + "line": 290 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "instance", + "name": "StandardModel.gaugeGroupℤ₆SubGroup_normal", + "docstring": null, + "snippet": "instance gaugeGroupℤ₆SubGroup_normal : gaugeGroupℤ₆SubGroup.Normal where", + "line": 297 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "instance", + "name": "StandardModel.«instance@25db9a02»", + "docstring": null, + "snippet": "noncomputable instance : Group GaugeGroupℤ₆", + "line": 315 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.GaugeGroupℤ₆.mk", + "docstring": "The quotient map from `GaugeGroupI` to `GaugeGroupℤ₆`.", + "snippet": "noncomputable def mk : GaugeGroupI →* GaugeGroupℤ₆", + "line": 321 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.GaugeGroupℤ₆.mk_gaugeGroupℤ₆OfRoot", + "docstring": null, + "snippet": "@[simp]\nlemma mk_gaugeGroupℤ₆OfRoot (α : rootsOfUnity 6 ℂ) :\n mk (gaugeGroupℤ₆OfRoot α) = 1", + "line": 325 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.gaugeGroupℤ₂RootToℤ₆Root", + "docstring": "The inclusion of second roots of unity into sixth roots of unity.", + "snippet": "noncomputable def gaugeGroupℤ₂RootToℤ₆Root : rootsOfUnity 2 ℂ →* rootsOfUnity 6 ℂ", + "line": 340 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.gaugeGroupℤ₂OfRoot", + "docstring": "The element of `GaugeGroupI` associated to a second root of unity.", + "snippet": "noncomputable def gaugeGroupℤ₂OfRoot (α : rootsOfUnity 2 ℂ) : GaugeGroupI", + "line": 344 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₂OfRoot_toSU3", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₂OfRoot_toSU3 (α : rootsOfUnity 2 ℂ) :\n GaugeGroupI.toSU3 (gaugeGroupℤ₂OfRoot α) =\n gaugeGroupℤ₆SU3OfRoot (gaugeGroupℤ₂RootToℤ₆Root α)", + "line": 348 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₂OfRoot_toSU2", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₂OfRoot_toSU2 (α : rootsOfUnity 2 ℂ) :\n GaugeGroupI.toSU2 (gaugeGroupℤ₂OfRoot α) =\n gaugeGroupℤ₆SU2OfRoot (gaugeGroupℤ₂RootToℤ₆Root α)", + "line": 353 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₂OfRoot_toU1", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₂OfRoot_toU1 (α : rootsOfUnity 2 ℂ) :\n GaugeGroupI.toU1 (gaugeGroupℤ₂OfRoot α) =\n gaugeGroupℤ₆UnitaryOfRoot (gaugeGroupℤ₂RootToℤ₆Root α)", + "line": 358 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₂OfRoot_mem_center", + "docstring": null, + "snippet": "lemma gaugeGroupℤ₂OfRoot_mem_center (α : rootsOfUnity 2 ℂ) :\n gaugeGroupℤ₂OfRoot α ∈ Subgroup.center GaugeGroupI", + "line": 362 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.gaugeGroupℤ₂Hom", + "docstring": "The homomorphism from second roots of unity to `GaugeGroupI`.", + "snippet": "noncomputable def gaugeGroupℤ₂Hom : rootsOfUnity 2 ℂ →* GaugeGroupI", + "line": 367 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₂Hom_apply", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₂Hom_apply (α : rootsOfUnity 2 ℂ) :\n gaugeGroupℤ₂Hom α = gaugeGroupℤ₂OfRoot α", + "line": 371 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₂Hom_toSU3", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₂Hom_toSU3 (α : rootsOfUnity 2 ℂ) :\n GaugeGroupI.toSU3 (gaugeGroupℤ₂Hom α) =\n gaugeGroupℤ₆SU3OfRoot (gaugeGroupℤ₂RootToℤ₆Root α)", + "line": 375 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₂Hom_toSU2", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₂Hom_toSU2 (α : rootsOfUnity 2 ℂ) :\n GaugeGroupI.toSU2 (gaugeGroupℤ₂Hom α) =\n gaugeGroupℤ₆SU2OfRoot (gaugeGroupℤ₂RootToℤ₆Root α)", + "line": 380 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₂Hom_toU1", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₂Hom_toU1 (α : rootsOfUnity 2 ℂ) :\n GaugeGroupI.toU1 (gaugeGroupℤ₂Hom α) =\n gaugeGroupℤ₆UnitaryOfRoot (gaugeGroupℤ₂RootToℤ₆Root α)", + "line": 385 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₂OfRoot_mem", + "docstring": null, + "snippet": "lemma gaugeGroupℤ₂OfRoot_mem (α : rootsOfUnity 2 ℂ) :\n gaugeGroupℤ₂OfRoot α ∈ gaugeGroupℤ₂SubGroup", + "line": 398 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.mem_gaugeGroupℤ₂SubGroup_iff", + "docstring": null, + "snippet": "lemma mem_gaugeGroupℤ₂SubGroup_iff (g : GaugeGroupI) :\n g ∈ gaugeGroupℤ₂SubGroup ↔ ∃ α : rootsOfUnity 2 ℂ, gaugeGroupℤ₂OfRoot α = g", + "line": 402 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₂SubGroup_le_gaugeGroupℤ₆SubGroup", + "docstring": null, + "snippet": "lemma gaugeGroupℤ₂SubGroup_le_gaugeGroupℤ₆SubGroup :\n gaugeGroupℤ₂SubGroup ≤ gaugeGroupℤ₆SubGroup", + "line": 406 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₂SubGroup_le_center", + "docstring": null, + "snippet": "lemma gaugeGroupℤ₂SubGroup_le_center :\n gaugeGroupℤ₂SubGroup ≤ Subgroup.center GaugeGroupI", + "line": 413 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "instance", + "name": "StandardModel.gaugeGroupℤ₂SubGroup_normal", + "docstring": null, + "snippet": "instance gaugeGroupℤ₂SubGroup_normal : gaugeGroupℤ₂SubGroup.Normal where", + "line": 420 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "instance", + "name": "StandardModel.«instance@6ca2807b»", + "docstring": null, + "snippet": "noncomputable instance : Group GaugeGroupℤ₂", + "line": 438 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.GaugeGroupℤ₂.mk", + "docstring": "The quotient map from `GaugeGroupI` to `GaugeGroupℤ₂`.", + "snippet": "noncomputable def mk : GaugeGroupI →* GaugeGroupℤ₂", + "line": 444 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.GaugeGroupℤ₂.mk_gaugeGroupℤ₂OfRoot", + "docstring": null, + "snippet": "@[simp]\nlemma mk_gaugeGroupℤ₂OfRoot (α : rootsOfUnity 2 ℂ) :\n mk (gaugeGroupℤ₂OfRoot α) = 1", + "line": 448 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.gaugeGroupℤ₃RootToℤ₆Root", + "docstring": "The inclusion of third roots of unity into sixth roots of unity.", + "snippet": "noncomputable def gaugeGroupℤ₃RootToℤ₆Root : rootsOfUnity 3 ℂ →* rootsOfUnity 6 ℂ", + "line": 463 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.gaugeGroupℤ₃OfRoot", + "docstring": "The element of `GaugeGroupI` associated to a third root of unity.", + "snippet": "noncomputable def gaugeGroupℤ₃OfRoot (α : rootsOfUnity 3 ℂ) : GaugeGroupI", + "line": 467 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₃OfRoot_toSU3", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₃OfRoot_toSU3 (α : rootsOfUnity 3 ℂ) :\n GaugeGroupI.toSU3 (gaugeGroupℤ₃OfRoot α) =\n gaugeGroupℤ₆SU3OfRoot (gaugeGroupℤ₃RootToℤ₆Root α)", + "line": 471 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₃OfRoot_toSU2", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₃OfRoot_toSU2 (α : rootsOfUnity 3 ℂ) :\n GaugeGroupI.toSU2 (gaugeGroupℤ₃OfRoot α) =\n gaugeGroupℤ₆SU2OfRoot (gaugeGroupℤ₃RootToℤ₆Root α)", + "line": 476 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₃OfRoot_toU1", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₃OfRoot_toU1 (α : rootsOfUnity 3 ℂ) :\n GaugeGroupI.toU1 (gaugeGroupℤ₃OfRoot α) =\n gaugeGroupℤ₆UnitaryOfRoot (gaugeGroupℤ₃RootToℤ₆Root α)", + "line": 481 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₃OfRoot_mem_center", + "docstring": null, + "snippet": "lemma gaugeGroupℤ₃OfRoot_mem_center (α : rootsOfUnity 3 ℂ) :\n gaugeGroupℤ₃OfRoot α ∈ Subgroup.center GaugeGroupI", + "line": 485 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.gaugeGroupℤ₃Hom", + "docstring": "The homomorphism from third roots of unity to `GaugeGroupI`.", + "snippet": "noncomputable def gaugeGroupℤ₃Hom : rootsOfUnity 3 ℂ →* GaugeGroupI", + "line": 490 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₃Hom_apply", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₃Hom_apply (α : rootsOfUnity 3 ℂ) :\n gaugeGroupℤ₃Hom α = gaugeGroupℤ₃OfRoot α", + "line": 494 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₃Hom_toSU3", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₃Hom_toSU3 (α : rootsOfUnity 3 ℂ) :\n GaugeGroupI.toSU3 (gaugeGroupℤ₃Hom α) =\n gaugeGroupℤ₆SU3OfRoot (gaugeGroupℤ₃RootToℤ₆Root α)", + "line": 498 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₃Hom_toSU2", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₃Hom_toSU2 (α : rootsOfUnity 3 ℂ) :\n GaugeGroupI.toSU2 (gaugeGroupℤ₃Hom α) =\n gaugeGroupℤ₆SU2OfRoot (gaugeGroupℤ₃RootToℤ₆Root α)", + "line": 503 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₃Hom_toU1", + "docstring": null, + "snippet": "@[simp]\nlemma gaugeGroupℤ₃Hom_toU1 (α : rootsOfUnity 3 ℂ) :\n GaugeGroupI.toU1 (gaugeGroupℤ₃Hom α) =\n gaugeGroupℤ₆UnitaryOfRoot (gaugeGroupℤ₃RootToℤ₆Root α)", + "line": 508 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₃OfRoot_mem", + "docstring": null, + "snippet": "lemma gaugeGroupℤ₃OfRoot_mem (α : rootsOfUnity 3 ℂ) :\n gaugeGroupℤ₃OfRoot α ∈ gaugeGroupℤ₃SubGroup", + "line": 521 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.mem_gaugeGroupℤ₃SubGroup_iff", + "docstring": null, + "snippet": "lemma mem_gaugeGroupℤ₃SubGroup_iff (g : GaugeGroupI) :\n g ∈ gaugeGroupℤ₃SubGroup ↔ ∃ α : rootsOfUnity 3 ℂ, gaugeGroupℤ₃OfRoot α = g", + "line": 525 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₃SubGroup_le_gaugeGroupℤ₆SubGroup", + "docstring": null, + "snippet": "lemma gaugeGroupℤ₃SubGroup_le_gaugeGroupℤ₆SubGroup :\n gaugeGroupℤ₃SubGroup ≤ gaugeGroupℤ₆SubGroup", + "line": 529 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.gaugeGroupℤ₃SubGroup_le_center", + "docstring": null, + "snippet": "lemma gaugeGroupℤ₃SubGroup_le_center :\n gaugeGroupℤ₃SubGroup ≤ Subgroup.center GaugeGroupI", + "line": 536 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "instance", + "name": "StandardModel.gaugeGroupℤ₃SubGroup_normal", + "docstring": null, + "snippet": "instance gaugeGroupℤ₃SubGroup_normal : gaugeGroupℤ₃SubGroup.Normal where", + "line": 543 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "instance", + "name": "StandardModel.«instance@2ccf6670»", + "docstring": null, + "snippet": "noncomputable instance : Group GaugeGroupℤ₃", + "line": 561 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.GaugeGroupℤ₃.mk", + "docstring": "The quotient map from `GaugeGroupI` to `GaugeGroupℤ₃`.", + "snippet": "noncomputable def mk : GaugeGroupI →* GaugeGroupℤ₃", + "line": 567 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.GaugeGroupℤ₃.mk_gaugeGroupℤ₃OfRoot", + "docstring": null, + "snippet": "@[simp]\nlemma mk_gaugeGroupℤ₃OfRoot (α : rootsOfUnity 3 ℂ) :\n mk (gaugeGroupℤ₃OfRoot α) = 1", + "line": 571 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "instance", + "name": "StandardModel.«instance@e87cb8a7»", + "docstring": null, + "snippet": "noncomputable instance (q : GaugeGroupQuot) : Group (GaugeGroup q)", + "line": 612 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.GaugeGroupQuot.subgroup", + "docstring": "The central subgroup of `GaugeGroupI` quotiented by a gauge-group quotient choice.", + "snippet": "noncomputable def subgroup : GaugeGroupQuot → Subgroup GaugeGroupI\n | .ℤ₆ => gaugeGroupℤ₆SubGroup\n | .ℤ₂ => gaugeGroupℤ₂SubGroup\n | .ℤ₃ => gaugeGroupℤ₃SubGroup\n | .I => ⊥", + "line": 618 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.GaugeGroupQuot.subgroup_le_center", + "docstring": "The subgroup attached to a gauge-group quotient choice lies in the center of `GaugeGroupI`.", + "snippet": "lemma subgroup_le_center (q : GaugeGroupQuot) :\n subgroup q ≤ Subgroup.center GaugeGroupI", + "line": 625 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "instance", + "name": "StandardModel.GaugeGroupQuot.subgroup_normal", + "docstring": "The subgroup attached to a gauge-group quotient choice is normal in `GaugeGroupI`.", + "snippet": "instance subgroup_normal (q : GaugeGroupQuot) : (subgroup q).Normal", + "line": 637 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "def", + "name": "StandardModel.GaugeGroupQuot.quotientMap", + "docstring": "The quotient map from `GaugeGroupI` to the gauge group selected by a quotient choice.", + "snippet": "noncomputable def quotientMap (q : GaugeGroupQuot) : GaugeGroupI →* GaugeGroup q", + "line": 646 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.GaugeGroupQuot.quotientMap_I_apply", + "docstring": null, + "snippet": "@[simp]\nlemma quotientMap_I_apply (g : GaugeGroupI) :\n quotientMap .I g = g", + "line": 654 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.GaugeGroupQuot.quotientMap_ℤ₆_gaugeGroupℤ₆OfRoot", + "docstring": null, + "snippet": "@[simp]\nlemma quotientMap_ℤ₆_gaugeGroupℤ₆OfRoot (α : rootsOfUnity 6 ℂ) :\n quotientMap .ℤ₆ (gaugeGroupℤ₆OfRoot α) = 1", + "line": 658 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.GaugeGroupQuot.quotientMap_ℤ₂_gaugeGroupℤ₂OfRoot", + "docstring": null, + "snippet": "@[simp]\nlemma quotientMap_ℤ₂_gaugeGroupℤ₂OfRoot (α : rootsOfUnity 2 ℂ) :\n quotientMap .ℤ₂ (gaugeGroupℤ₂OfRoot α) = 1", + "line": 663 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.GaugeGroupQuot.quotientMap_ℤ₃_gaugeGroupℤ₃OfRoot", + "docstring": null, + "snippet": "@[simp]\nlemma quotientMap_ℤ₃_gaugeGroupℤ₃OfRoot (α : rootsOfUnity 3 ℂ) :\n quotientMap .ℤ₃ (gaugeGroupℤ₃OfRoot α) = 1", + "line": 668 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.GaugeGroupQuot.mem_subgroup_iff_quotientMap_eq_one", + "docstring": "The kernel of the quotient map is the subgroup selected by the quotient choice.", + "snippet": "lemma mem_subgroup_iff_quotientMap_eq_one (q : GaugeGroupQuot) (g : GaugeGroupI) :\n g ∈ subgroup q ↔ quotientMap q g = 1", + "line": 673 + }, + { + "file": "Physlib/Particles/StandardModel/Basic.lean", + "kind": "lemma", + "name": "StandardModel.GaugeGroupQuot.quotientMap_eq_iff", + "docstring": "Two representatives have the same image under the selected quotient map exactly when their\nquotient lies in the subgroup selected by the quotient choice.", + "snippet": "lemma quotientMap_eq_iff (q : GaugeGroupQuot) (g h : GaugeGroupI) :\n quotientMap q g = quotientMap q h ↔ g / h ∈ subgroup q", + "line": 690 + } + ] + }, + { + "filename": "Physlib/Particles/StandardModel/HiggsBoson/Basic.lean", + "entries": [ + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/Basic.lean", + "kind": "instance", + "name": "StandardModel.HiggsVec.«instance@d58bd5cc»", + "docstring": null, + "snippet": "instance : SMulCommClass ℂ GaugeGroupI HiggsVec where", + "line": 199 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/Basic.lean", + "kind": "instance", + "name": "StandardModel.HiggsVec.«instance@34481c4b»", + "docstring": null, + "snippet": "instance : SMulCommClass ℝ GaugeGroupI HiggsVec where", + "line": 204 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/Basic.lean", + "kind": "def", + "name": "StandardModel.HiggsVec.toRealScalars", + "docstring": "The underlying real values of the Higgs vector.", + "snippet": "def toRealScalars : HiggsVec →ₗ[ℝ] (Fin 4 → ℝ) where", + "line": 439 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/Basic.lean", + "kind": "lemma", + "name": "StandardModel.HiggsVec.toRealScalars_smul_real", + "docstring": null, + "snippet": "lemma toRealScalars_smul_real (a : ℝ) (φ : HiggsVec) :\n toRealScalars (a • φ) = a • toRealScalars φ", + "line": 452 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/Basic.lean", + "kind": "lemma", + "name": "StandardModel.HiggsVec.ofReal_toRealScalars", + "docstring": null, + "snippet": "lemma ofReal_toRealScalars (a : ℝ) :\n toRealScalars (ofReal a) = !₄[Real.sqrt a, 0, 0, 0]", + "line": 456 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/Basic.lean", + "kind": "lemma", + "name": "StandardModel.HiggsVec.ofReal_toRealScalars_norm", + "docstring": null, + "snippet": "lemma ofReal_toRealScalars_norm (φ : HiggsVec) :\n toRealScalars (ofReal (‖φ‖ ^ 2)) = !₄[‖φ‖, 0, 0, 0]", + "line": 462 + } + ] + }, + { + "filename": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "entries": [ + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "abbrev", + "name": "StandardModel.HiggsField.EffectivePotential", + "docstring": "A general potential of the Higgs field.", + "snippet": "abbrev EffectivePotential : Type", + "line": 34 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "def", + "name": "StandardModel.HiggsField.EffectivePotential.IsInvariant", + "docstring": "The proposition that the general potential is invariant under\n the global action of the gauge group.", + "snippet": "def IsInvariant (V : EffectivePotential) : Prop", + "line": 46 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "lemma", + "name": "StandardModel.HiggsField.EffectivePotential.IsInvariant.eq_on_orbits", + "docstring": "An invariant potential is equal on gauge orbits.", + "snippet": "lemma eq_on_orbits {φ1 φ2 : HiggsVec} {V : EffectivePotential} (h : IsInvariant V)\n (hφ : φ1 ∈ MulAction.orbit GaugeGroupI φ2) :\n V φ1 = V φ2", + "line": 52 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "lemma", + "name": "StandardModel.HiggsField.EffectivePotential.IsInvariant.eq_of_norm_eq", + "docstring": "An invariant potential is equal on Higgs vectors with identical norms.", + "snippet": "lemma eq_of_norm_eq {φ1 φ2 : HiggsVec} {V : EffectivePotential} (h : IsInvariant V)\n (hφ : ‖φ1‖ = ‖φ2‖) :\n V φ1 = V φ2", + "line": 60 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "lemma", + "name": "StandardModel.HiggsField.EffectivePotential.IsInvariant.factors_through_norm", + "docstring": null, + "snippet": "lemma factors_through_norm {V : EffectivePotential} (h : IsInvariant V) :\n ∃ (f : ℝ → ℝ), V = f ∘ norm", + "line": 64 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "def", + "name": "StandardModel.HiggsField.EffectivePotential.HasMaxMassDimLE", + "docstring": "The proposition that the potential `V` has a maximum mass dimension\n less then or equal to `n` - also implying it is a polynomial.", + "snippet": "def HasMaxMassDimLE (V : EffectivePotential) (n : ℕ) : Prop", + "line": 83 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "def", + "name": "StandardModel.HiggsField.EffectivePotential.polynomial", + "docstring": "The polynomial associated to a potential `V` with a maximum mass dimension\n less than or equal to `n`.", + "snippet": "def polynomial (V : EffectivePotential) {n : ℕ} (h : HasMaxMassDimLE V n) :\n MvPolynomial (Fin 4) ℝ", + "line": 89 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "lemma", + "name": "StandardModel.HiggsField.EffectivePotential.polynomial_totalDegree", + "docstring": null, + "snippet": "lemma polynomial_totalDegree {V : EffectivePotential} {n : ℕ} (h : HasMaxMassDimLE V n) :\n (polynomial V h).totalDegree ≤ n", + "line": 92 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "lemma", + "name": "StandardModel.HiggsField.EffectivePotential.apply_eq_polynomial", + "docstring": null, + "snippet": "lemma apply_eq_polynomial {V : EffectivePotential} {n : ℕ} (h : HasMaxMassDimLE V n)\n (φ : HiggsVec) : V φ = (polynomial V h).eval φ.toRealScalars", + "line": 95 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "def", + "name": "StandardModel.HiggsField.EffectivePotential.termOfMassDim", + "docstring": "The part of a potential at a given mass-dimension.", + "snippet": "def termOfMassDim (V : EffectivePotential) {n : ℕ} (h : HasMaxMassDimLE V n) (m : ℕ) :\n HiggsVec → ℝ", + "line": 105 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "lemma", + "name": "StandardModel.HiggsField.EffectivePotential.termOfMassDim_eq_zero_of_max_lt", + "docstring": null, + "snippet": "lemma termOfMassDim_eq_zero_of_max_lt {V : EffectivePotential} {n : ℕ} (h : HasMaxMassDimLE V n)\n {m : ℕ} (hm : n < m) (φ : HiggsVec) :\n termOfMassDim V h m φ = 0", + "line": 108 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "lemma", + "name": "StandardModel.HiggsField.EffectivePotential.termOfMassDim_homogeneity", + "docstring": null, + "snippet": "lemma termOfMassDim_homogeneity {V : EffectivePotential} {n : ℕ} (h : HasMaxMassDimLE V n) (m : ℕ)\n (φ : HiggsVec) (t : ℝ) : termOfMassDim V h m (t • φ) = t ^ m * termOfMassDim V h m φ", + "line": 117 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "lemma", + "name": "StandardModel.HiggsField.EffectivePotential.apply_eq_sum_termOfMassDim", + "docstring": null, + "snippet": "lemma apply_eq_sum_termOfMassDim {V : EffectivePotential} {n : ℕ} (h : HasMaxMassDimLE V n)\n (φ : HiggsVec) :\n V φ = ∑ m ∈ Finset.range (n + 1), termOfMassDim V h m φ", + "line": 130 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "lemma", + "name": "StandardModel.HiggsField.EffectivePotential.apply_smul_eq_sum_termOfMassDim", + "docstring": null, + "snippet": "lemma apply_smul_eq_sum_termOfMassDim {V : EffectivePotential} {n : ℕ} (h : HasMaxMassDimLE V n)\n (φ : HiggsVec) (t : ℝ) :\n V (t • φ) = ∑ m ∈ Finset.range (n + 1), t ^ m * termOfMassDim V h m φ", + "line": 145 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "lemma", + "name": "StandardModel.HiggsField.EffectivePotential.termOfMassDim_isInvariant", + "docstring": null, + "snippet": "lemma termOfMassDim_isInvariant {V : EffectivePotential} {n : ℕ} (h : HasMaxMassDimLE V n)\n (m : ℕ) (hV : IsInvariant V) : IsInvariant (termOfMassDim V h m)", + "line": 153 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "lemma", + "name": "StandardModel.HiggsField.EffectivePotential.termOfMassDim_eq_mul_norm", + "docstring": null, + "snippet": "lemma termOfMassDim_eq_mul_norm {V : EffectivePotential} {n : ℕ}\n (h : HasMaxMassDimLE V n) (m : ℕ) (hV : IsInvariant V) (φ : HiggsVec) :\n ∃ c, termOfMassDim V h m φ = c * ‖φ‖ ^ m", + "line": 179 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "lemma", + "name": "StandardModel.HiggsField.EffectivePotential.termOfMassDim_zero_of_odd", + "docstring": null, + "snippet": "lemma termOfMassDim_zero_of_odd {V : EffectivePotential} {n : ℕ} (h : HasMaxMassDimLE V n) (m : ℕ)\n (hV : IsInvariant V) (φ : HiggsVec) (hodd : Odd m) :\n termOfMassDim V h m φ = 0", + "line": 187 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "lemma", + "name": "StandardModel.HiggsField.EffectivePotential.apply_eq_sum_even_termOfMassDim", + "docstring": null, + "snippet": "lemma apply_eq_sum_even_termOfMassDim {V : EffectivePotential} {n : ℕ} (h : HasMaxMassDimLE V n)\n (hV : IsInvariant V) (φ : HiggsVec) :\n V φ = ∑ m ∈ Finset.range (n / 2 + 1), termOfMassDim V h (2 * m) φ", + "line": 203 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "lemma", + "name": "StandardModel.HiggsField.EffectivePotential.apply_eq_sum_even_termOfMassDim_fin", + "docstring": null, + "snippet": "lemma apply_eq_sum_even_termOfMassDim_fin {V : EffectivePotential} {n : ℕ} (h : HasMaxMassDimLE V n)\n (hV : IsInvariant V) (φ : HiggsVec) :\n V φ = ∑ m : Fin (n/2 + 1), termOfMassDim V h (2 * m) φ", + "line": 228 + }, + { + "file": "Physlib/Particles/StandardModel/HiggsBoson/EffectivePotential.lean", + "kind": "lemma", + "name": "StandardModel.HiggsField.EffectivePotential.apply_eq_sum_norm_pow", + "docstring": "The potential is equal to the sum of norms to even powers.", + "snippet": "lemma apply_eq_sum_norm_pow {V : EffectivePotential} {n : ℕ} (h : HasMaxMassDimLE V n)\n (hV : IsInvariant V) (φ : HiggsVec) :\n ∃ c : Fin (n/2 + 1) → ℝ, V φ = ∑ m, c m • ‖φ‖ ^ (2 * m.1)", + "line": 234 + } + ] + }, + { + "filename": "Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/Basic.lean", + "entries": [ + { + "file": "Physlib/Particles/SuperSymmetry/MSSMNu/AnomalyCancellation/Basic.lean", + "kind": "lemma", + "name": "MSSMCharges.sum_MSSMSpecies_numberCharges_eq_expand", + "docstring": null, + "snippet": "lemma sum_MSSMSpecies_numberCharges_eq_expand [AddCommMonoid M]\n (f : Fin MSSMSpecies.numberCharges → M) :\n ∑ i, f i = f ⟨0, by simp⟩ + f ⟨1, by simp⟩ + f ⟨2, by simp⟩", + "line": 32 + } + ] + }, + { + "filename": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "entries": [ + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "abbrev", + "name": "SUSY.N1.ChiralScalarConfiguration", + "docstring": "The chiral scalar configuration: a complex value for each chiral label. This is\nthe sector's only field data. Declared as an `abbrev` so that unification sees\nthrough it to `ChiralIndexingType → ℂ` and applies Mathlib's function-space calculus\nlemmas directly.", + "snippet": "abbrev ChiralScalarConfiguration (ChiralIndexingType : Type*)", + "line": 102 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "inductive", + "name": "SUSY.N1.ChiralColor", + "docstring": "The four colours carried by a chiral-sector index: holomorphy (`chiral` versus\n`anti`, a scalar versus its complex conjugate) crossed with variance (`up` versus\n`down`, contravariant versus covariant). Carrying both axes here lets the single index\ntype `ι` label the scalars.", + "snippet": "inductive ChiralColor | chiralUp | chiralDown | antiUp | antiDown", + "line": 115 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "def", + "name": "SUSY.N1.ChiralColor.tau", + "docstring": "The dual colour: flips variance and preserves holomorphy. Two indices may contract\nexactly when their colours are `τ`-related, so `V^I` pairs only with `V_I` (same\nholomorphy, opposite variance) and never with a conjugate index.", + "snippet": "def tau : ChiralColor → ChiralColor\n | chiralUp => chiralDown\n | chiralDown => chiralUp\n | antiUp => antiDown\n | antiDown => antiUp", + "line": 123 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "def", + "name": "SUSY.N1.ChiralColor.bar", + "docstring": "The conjugate colour: flips holomorphy (`chiral`↔`anti`) and preserves variance. Complex\nconjugation sends an index to its conjugate carrier, so `bar` swaps `chiral*` with `anti*`.\nDistinct from the variance dual `tau`; the two commute (`bar_tau`).", + "snippet": "def bar : ChiralColor → ChiralColor\n | chiralUp => antiUp\n | antiUp => chiralUp\n | chiralDown => antiDown\n | antiDown => chiralDown", + "line": 132 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "abbrev", + "name": "SUSY.N1.chiralModule", + "docstring": "The carrier module of each colour, distinct for all four: the holomorphic vectors `ι → ℂ` and\ntheir dual `Module.Dual ℂ (ι → ℂ)` on the chiral side, and the conjugate module `ConjModule …` of\neach (where `i` acts as `−i`) on the anti side. Variance is the vector/dual axis, holomorphy the\nconjugate-module axis; both are genuine carrier data, not labels tracked separately.", + "snippet": "abbrev chiralModule : ChiralColor → Type\n | .chiralUp => ι → ℂ\n | .chiralDown => Module.Dual ℂ (ι → ℂ)\n | .antiUp => ConjModule (ι → ℂ)\n | .antiDown => ConjModule (Module.Dual ℂ (ι → ℂ))", + "line": 163 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "instance", + "name": "SUSY.N1.instAddCommGroupChiralModule", + "docstring": null, + "snippet": "instance instAddCommGroupChiralModule : ∀ c, AddCommGroup (chiralModule (ι := ι) c)\n | .chiralUp | .chiralDown | .antiUp | .antiDown => inferInstance", + "line": 169 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "instance", + "name": "SUSY.N1.instModuleChiralModule", + "docstring": null, + "snippet": "noncomputable instance instModuleChiralModule : ∀ c, Module ℂ (chiralModule (ι := ι) c)\n | .chiralUp | .chiralDown | .antiUp | .antiDown => inferInstance", + "line": 172 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "def", + "name": "SUSY.N1.chiralRep", + "docstring": "The representation on each colour, taken trivial over the trivial group `Unit`: the chiral\nscalars carry no charge in this sector.", + "snippet": "def chiralRep : (c : ChiralColor) → Representation ℂ Unit (chiralModule (ι := ι) c)", + "line": 177 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "def", + "name": "SUSY.N1.piBasis", + "docstring": "The standard basis of the vector carrier `ι → ℂ` (the indicator functions); the basis of\n`chiralUp` and the reference basis for the δ pairing.", + "snippet": "def piBasis : Basis ι ℂ (ι → ℂ)", + "line": 182 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "def", + "name": "SUSY.N1.chiralBasis", + "docstring": "The basis of each colour's carrier, all indexed by `ι`: `piBasis` on the holomorphic vectors,\nits dual `piBasis.dualBasis` on the holomorphic covectors, and the `Basis.conj` of each on the\nanti-holomorphic side (coordinates `star`-ed).", + "snippet": "noncomputable def chiralBasis : (c : ChiralColor) → Basis ι ℂ (chiralModule (ι := ι) c)\n | .chiralUp => piBasis\n | .chiralDown => piBasis.dualBasis\n | .antiUp => Basis.conj piBasis\n | .antiDown => Basis.conj piBasis.dualBasis", + "line": 187 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "def", + "name": "SUSY.N1.deltaCap", + "docstring": "The δ cap `∑_I b_I ⊗ b_I`: the rank-2 tensor in `M ⊗ M` with two upper indices, whose\ncomponents in the basis `b` are `δⁱʲ`. It is an element of `M ⊗ M` (the inverse-metric \"cap\"),\nnot a linear map, and serves as the `metric` field of the species (whose two slots share a\ncolour).", + "snippet": "def deltaCap (b : Basis ι ℂ M) : M ⊗[ℂ] M", + "line": 215 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "def", + "name": "SUSY.N1.deltaBil₂", + "docstring": "The δ pairing between two based modules sharing the index `ι`: the dot product of coordinate\nvectors `(x, y) ↦ ∑_I (b x)_I (b' y)_I`. Built from Mathlib's `Basis.toDual b` (the canonical δ map\n`M → Module.Dual M`, sending `b` to its dual basis) precomposed on the second slot with the basis\ntransport `b' ≃ b`.", + "snippet": "def deltaBil₂ (b : Basis ι ℂ M) (b' : Basis ι ℂ N) : M →ₗ[ℂ] N →ₗ[ℂ] ℂ", + "line": 223 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "lemma", + "name": "SUSY.N1.deltaBil₂_apply", + "docstring": "`deltaBil₂ b b' x y = ∑_I (b x)_I (b' y)_I`.", + "snippet": "lemma deltaBil₂_apply (b : Basis ι ℂ M) (b' : Basis ι ℂ N) (x : M) (y : N) :\n deltaBil₂ b b' x y = ∑ I, b.equivFun x I * b'.equivFun y I", + "line": 227 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "def", + "name": "SUSY.N1.deltaContr₂", + "docstring": "The two-module δ contraction `M ⊗ N → ℂ`.", + "snippet": "def deltaContr₂ (b : Basis ι ℂ M) (b' : Basis ι ℂ N) : M ⊗[ℂ] N →ₗ[ℂ] ℂ", + "line": 236 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "lemma", + "name": "SUSY.N1.deltaContr₂_tmul", + "docstring": "`deltaContr₂ b b' (x ⊗ₜ y) = ∑_I (b x)_I (b' y)_I`.", + "snippet": "lemma deltaContr₂_tmul (b : Basis ι ℂ M) (b' : Basis ι ℂ N) (x : M) (y : N) :\n deltaContr₂ b b' (x ⊗ₜ[ℂ] y) = ∑ I, b.equivFun x I * b'.equivFun y I", + "line": 240 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "lemma", + "name": "SUSY.N1.deltaContr₂_tmul_basis", + "docstring": "`deltaContr₂ b b' (x ⊗ₜ b' J) = x_J`: pairing with the second basis reads off a coordinate.", + "snippet": "lemma deltaContr₂_tmul_basis (b : Basis ι ℂ M) (b' : Basis ι ℂ N) (x : M) (J : ι) :\n deltaContr₂ b b' (x ⊗ₜ[ℂ] b' J) = b.equivFun x J", + "line": 245 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "lemma", + "name": "SUSY.N1.deltaContr₂_basis_basis", + "docstring": "`deltaContr₂ b b' (b I ⊗ₜ b' J) = δ_{IJ}`: the two bases are δ-dual.", + "snippet": "lemma deltaContr₂_basis_basis (b : Basis ι ℂ M) (b' : Basis ι ℂ N) (I J : ι) :\n deltaContr₂ b b' (b I ⊗ₜ[ℂ] b' J) = if I = J then 1 else 0", + "line": 250 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "lemma", + "name": "SUSY.N1.deltaContr₂_comm", + "docstring": "`deltaContr₂ b b' (x ⊗ₜ y) = deltaContr₂ b' b (y ⊗ₜ x)`: swapping slots swaps the two bases.", + "snippet": "lemma deltaContr₂_comm (b : Basis ι ℂ M) (b' : Basis ι ℂ N) (x : M) (y : N) :\n deltaContr₂ b b' (x ⊗ₜ[ℂ] y) = deltaContr₂ b' b (y ⊗ₜ[ℂ] x)", + "line": 255 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "def", + "name": "SUSY.N1.deltaCap₂", + "docstring": "The two-module δ cap `∑_I b_I ⊗ b'_I ∈ M ⊗ N`.", + "snippet": "def deltaCap₂ (b : Basis ι ℂ M) (b' : Basis ι ℂ N) : M ⊗[ℂ] N", + "line": 261 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "lemma", + "name": "SUSY.N1.deltaCap₂_comm", + "docstring": "`comm (deltaCap₂ b b') = deltaCap₂ b' b`: swapping the two factors swaps the two bases.", + "snippet": "lemma deltaCap₂_comm (b : Basis ι ℂ M) (b' : Basis ι ℂ N) :\n TensorProduct.comm ℂ M N (deltaCap₂ b b') = deltaCap₂ b' b", + "line": 265 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "lemma", + "name": "SUSY.N1.deltaUnit₂_symm", + "docstring": "The `unit_symm` law (two-module, `toSpanSingleton` form): `deltaCap₂ b' b` is the swap of\n`deltaCap₂ b b'`.", + "snippet": "lemma deltaUnit₂_symm (b : Basis ι ℂ M) (b' : Basis ι ℂ N) :\n LinearMap.toSpanSingleton ℂ _ (deltaCap₂ b' b) 1 =\n LinearMap.lTensor N (LinearEquiv.refl ℂ M).toLinearMap\n (TensorProduct.comm ℂ M N (LinearMap.toSpanSingleton ℂ _ (deltaCap₂ b b') 1))", + "line": 273 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "lemma", + "name": "SUSY.N1.deltaContr₂_unit", + "docstring": "The snake identity (two-module, `contr_unit` law): contracting `x ∈ M` into the `M`-leg of\n`deltaCap₂ b' b ∈ N ⊗ M` returns `x`.", + "snippet": "lemma deltaContr₂_unit (b : Basis ι ℂ M) (b' : Basis ι ℂ N) (x : M) :\n (TensorProduct.lid ℂ M) ((deltaContr₂ b b').rTensor M\n ((TensorProduct.assoc ℂ M N M).symm\n (x ⊗ₜ[ℂ] LinearMap.toSpanSingleton ℂ _ (deltaCap₂ b' b) 1))) = x", + "line": 283 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "lemma", + "name": "SUSY.N1.deltaContr₂_metric", + "docstring": "The `contr_metric` law (two-module): contracting the inner `M`/`N` legs of\n`deltaCap b ⊗ deltaCap b'` yields `deltaCap₂ b' b`.", + "snippet": "lemma deltaContr₂_metric (b : Basis ι ℂ M) (b' : Basis ι ℂ N) :\n (TensorProduct.comm ℂ M N ((TensorProduct.lid ℂ N).lTensor M\n (((deltaContr₂ b b').rTensor N).lTensor M\n (((TensorProduct.assoc ℂ M N N).symm.toLinearMap.lTensor M)\n ((TensorProduct.assoc ℂ M M (N ⊗[ℂ] N))\n (LinearMap.toSpanSingleton ℂ _ (deltaCap b) 1 ⊗ₜ[ℂ]\n LinearMap.toSpanSingleton ℂ _ (deltaCap b') 1)))))) =\n LinearMap.toSpanSingleton ℂ _ (deltaCap₂ b' b) 1", + "line": 296 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "def", + "name": "SUSY.N1.chiralTensor", + "docstring": "The chiral-index tensor species, bundled with its conjugation. Its four colours\n`chiral`/`anti` × `up`/`down` carry the four distinct carriers of §C. `contr c` is the two-module δ\npairing of a colour against its variance dual `τ c` (`V c ⊗ V (τ c) → ℂ`); `unit c` is the δ cap\nacross those two carriers; `metric c` is the single-colour δ cap `∑_I b_I ⊗ b_I`. Each\n`TensorSpecies` coherence law reduces, by case analysis on the colour, to the corresponding abstract\ntwo-module δ lemma of §D. The conjugation flips holomorphy (`ChiralColor.bar`) while preserving\nvariance; every basis is indexed by `ι`, so `barIdx_eq` is `rfl`, and `conj_contrComm` is\n`star δ = δ`. Instantiating `ConjTensorSpecies` this way gives the chiral sector both\nthe framework's generic tensor API and its conjugation API (`conjT` and its laws) on one object.", + "snippet": "def chiralTensor : ConjTensorSpecies ℂ ChiralColor Unit (chiralModule (ι := ι)) (fun _ => ι)\n (chiralRep (ι := ι)) (chiralBasis (ι := ι)) where", + "line": 328 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "def", + "name": "SUSY.N1.conjScalar", + "docstring": "Conjugation of a scalar tensor, normalized back to the scalar colour list `![]`.", + "snippet": "def conjScalar (t : (chiralTensor (ι := ι)).Tensor ![]) :\n (chiralTensor (ι := ι)).Tensor ![]", + "line": 399 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "def", + "name": "SUSY.N1.conjChiralCovector", + "docstring": "Conjugation of a holomorphic covector, normalized to the anti-holomorphic covector colour\nlist `![antiDown]`.", + "snippet": "def conjChiralCovector\n (t : (chiralTensor (ι := ι)).Tensor ![chiralDown]) :\n (chiralTensor (ι := ι)).Tensor ![antiDown]", + "line": 406 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "lemma", + "name": "SUSY.N1.toField_conjScalar", + "docstring": "For scalar tensors, `toField` of the normalized tensor conjugate is the complex conjugate of\n`toField`.", + "snippet": "lemma toField_conjScalar (t : (chiralTensor (ι := ι)).Tensor ![]) :\n (conjScalar t).toField = star t.toField", + "line": 414 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "lemma", + "name": "SUSY.N1.repr_conjChiralCovector", + "docstring": "Component formula for the holomorphic covector conjugate: the `![I]` basis component of\n`conjChiralCovector t` is the complex conjugate of the `![I]` component of `t`.", + "snippet": "lemma repr_conjChiralCovector\n (t : (chiralTensor (ι := ι)).Tensor ![chiralDown]) (I : ι) :\n (basis (S := (chiralTensor (ι := ι)).toTensorSpecies) ![antiDown]).repr\n (conjChiralCovector t) ![I] =\n star ((basis (S := (chiralTensor (ι := ι)).toTensorSpecies) ![chiralDown]).repr t ![I])", + "line": 426 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "lemma", + "name": "SUSY.N1.conjChiralCovector_add", + "docstring": "Conjugation of a holomorphic covector is additive.", + "snippet": "@[simp]\nlemma conjChiralCovector_add\n (t₁ t₂ : (chiralTensor (ι := ι)).Tensor ![chiralDown]) :\n conjChiralCovector (t₁ + t₂) = conjChiralCovector t₁ + conjChiralCovector t₂", + "line": 444 + }, + { + "file": "Physlib/Particles/SuperSymmetry/N1/Basic.lean", + "kind": "lemma", + "name": "SUSY.N1.conjChiralCovector_smul", + "docstring": "Conjugation of a holomorphic covector is conjugate-linear: a scalar `r` pulls out as\n`star r`.", + "snippet": "@[simp]\nlemma conjChiralCovector_smul (r : ℂ)\n (t : (chiralTensor (ι := ι)).Tensor ![chiralDown]) :\n conjChiralCovector (r • t) = star r • conjChiralCovector t", + "line": 453 + } + ] + }, + { + "filename": "Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/AllowsTerm.lean", + "entries": [ + { + "file": "Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/AllowsTerm.lean", + "kind": "instance", + "name": "SuperSymmetry.SU5.ChargeSpectrum.«instance@7b86df89»", + "docstring": null, + "snippet": "instance (x : ChargeSpectrum 𝓩) (q5 : 𝓩) (T : PotentialTerm) :\n Decidable (AllowsTermQ5 x q5 T)", + "line": 488 + }, + { + "file": "Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/AllowsTerm.lean", + "kind": "instance", + "name": "SuperSymmetry.SU5.ChargeSpectrum.«instance@35144a57»", + "docstring": null, + "snippet": "instance (x : ChargeSpectrum 𝓩) (q10 : 𝓩) (T : PotentialTerm) :\n Decidable (AllowsTermQ10 x q10 T)", + "line": 786 + } + ] + }, + { + "filename": "Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/PhenoClosed.lean", + "entries": [ + { + "file": "Physlib/Particles/SuperSymmetry/SU5/ChargeSpectrum/PhenoClosed.lean", + "kind": "instance", + "name": "SuperSymmetry.SU5.ChargeSpectrum.«instance@1aa8df08»", + "docstring": null, + "snippet": "instance {S5 S10 : Finset 𝓩} {charges : Multiset (ChargeSpectrum 𝓩)} :\n Decidable (ContainsPhenoCompletionsOfMinimallyAllows S5 S10 charges)", + "line": 193 + } + ] + }, + { + "filename": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "entries": [ + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "abbrev", + "name": "LinearPMap.resolvent", + "docstring": "The resolvent, `(T - z • 1)⁻¹`.", + "snippet": "abbrev resolvent (T : H →ₗ.[ℂ] H) (z : ℂ) : H →ₗ.[ℂ] H", + "line": 105 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "def", + "name": "LinearPMap.IsLowerBound", + "docstring": "`IsLowerBound T z c` is the property that `c * ‖x‖ ≤ ‖T x - z • x‖` for all `x : T.domain`.", + "snippet": "def IsLowerBound (T : H →ₗ.[ℂ] H) (z : ℂ) (c : ℝ) : Prop", + "line": 115 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.isLowerBound_neg", + "docstring": null, + "snippet": "lemma isLowerBound_neg {T : H →ₗ.[ℂ] H} {z : ℂ} {c : ℝ} (h : IsLowerBound T z c) :\n IsLowerBound (-T) (-z) c", + "line": 117 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.isLowerBound_of_right_le", + "docstring": null, + "snippet": "lemma isLowerBound_of_right_le\n {T : H →ₗ.[ℂ] H} {z : ℂ} {c₁ c₂ : ℝ} (hle : c₁ ≤ c₂) (h : IsLowerBound T z c₂) :\n IsLowerBound T z c₁", + "line": 122 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.isLowerBound_of_left_le", + "docstring": null, + "snippet": "lemma isLowerBound_of_left_le\n {T₁ T₂ : H →ₗ.[ℂ] H} (hle : T₁ ≤ T₂) {z : ℂ} {c : ℝ} (h : IsLowerBound T₂ z c) :\n IsLowerBound T₁ z c", + "line": 127 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.isLowerBound_closure", + "docstring": null, + "snippet": "lemma isLowerBound_closure\n {T : H →ₗ.[ℂ] H} {z : ℂ} {c : ℝ} (h : IsLowerBound T z c) : IsLowerBound T.closure z c", + "line": 132 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "def", + "name": "LinearPMap.regularityDomain", + "docstring": "The regular points for `T`.\n\n `z : ℂ` is a regular point for `T` iff there exists a constant `c > 0` such that\n `c * ‖x‖ ≤ ‖(T - z • 1) x‖` for all `x ∈ T.domain`.", + "snippet": "def regularityDomain (T : H →ₗ.[ℂ] H) : Set ℂ", + "line": 150 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.regularityDomain_neg", + "docstring": null, + "snippet": "@[simp]\nlemma regularityDomain_neg (T : H →ₗ.[ℂ] H) : (-T).regularityDomain = -T.regularityDomain", + "line": 153 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.regularityDomain_smul", + "docstring": null, + "snippet": "@[simp]\nlemma regularityDomain_smul (T : H →ₗ.[ℂ] H) {w : ℂ} (hw : w ≠ 0) :\n (w • T).regularityDomain = w • T.regularityDomain", + "line": 160 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.regularityDomain_antitone", + "docstring": "`T ≤ T'` implies `T'.regularityDomain ⊆ T.regularityDomain`.", + "snippet": "lemma regularityDomain_antitone : Antitone (regularityDomain (H := H))", + "line": 180 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.mem_regularityDomain_iff", + "docstring": "`z` is a regular point for `T` iff `T - z • 1` has\n a continuous (equivalently, bounded) inverse.", + "snippet": "lemma mem_regularityDomain_iff {T : H →ₗ.[ℂ] H} {z : ℂ} :\n z ∈ T.regularityDomain ↔ (T - z • 1).toFun.ker = ⊥ ∧ Continuous (𝑅 T z)", + "line": 185 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.ball_subset_regularityDomain", + "docstring": "The regularity domain of `T` contains open balls with radii controlled by the lower bounds.", + "snippet": "lemma ball_subset_regularityDomain\n {T : H →ₗ.[ℂ] H} {z : ℂ} {c : ℝ} (h : IsLowerBound T z c) : ball z c ⊆ T.regularityDomain", + "line": 213 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.regularityDomain_isOpen", + "docstring": "The regularity domain is an open set.", + "snippet": "lemma regularityDomain_isOpen (T : H →ₗ.[ℂ] H) : IsOpen T.regularityDomain", + "line": 224 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.regularityDomain_closure", + "docstring": "`T` and `T.closure` have the same regularity domain.", + "snippet": "lemma regularityDomain_closure (T : H →ₗ.[ℂ] H) :\n T.closure.regularityDomain = T.regularityDomain", + "line": 228 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosable.closure_range_sub_eq_range_closure_sub", + "docstring": null, + "snippet": "lemma IsClosable.closure_range_sub_eq_range_closure_sub [CompleteSpace H]\n {T : H →ₗ.[ℂ] H} (hT : T.IsClosable) {z : ℂ} (hz : z ∈ T.regularityDomain) :\n (T - z • 1).toFun.range.closure = (T.closure - z • 1).toFun.range", + "line": 233 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosed.sub_range_isClosed", + "docstring": null, + "snippet": "lemma IsClosed.sub_range_isClosed [CompleteSpace H]\n {T : H →ₗ.[ℂ] H} (hT : T.IsClosed) {z : ℂ} (hz : z ∈ T.regularityDomain) :\n _root_.IsClosed ((T - z • 1).toFun.range : Set H)", + "line": 274 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.IsUnbounded.orthogonal_closure_sub_range", + "docstring": "`(T.closure - z • 1).rangeᗮ = (T† - conj z • 1).ker`", + "snippet": "lemma IsUnbounded.orthogonal_closure_sub_range [CompleteSpace H]\n {T : H →ₗ.[ℂ] H} (hT : T.IsUnbounded) (z : ℂ) :\n (T.closure - z • 1).toFun.rangeᗮ\n = (T† - conj z • 1).toFun.ker.map (T† - conj z • 1).domain.subtype", + "line": 281 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.IsUnbounded.orthogonal_adjoint_sub_ker", + "docstring": "`(T† - conj z • 1).kerᗮ = (T.closure - z • 1).range`", + "snippet": "lemma IsUnbounded.orthogonal_adjoint_sub_ker [CompleteSpace H]\n {T : H →ₗ.[ℂ] H} (hT : T.IsUnbounded) {z : ℂ} (hz : z ∈ T.regularityDomain) :\n ((T† - conj z • 1).toFun.ker.map (T† - conj z • 1).domain.subtype)ᗮ\n = (T.closure - z • 1).toFun.range", + "line": 315 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "def", + "name": "LinearPMap.deficiencySubspace", + "docstring": "For a partial linear map `T` and any complex number `z`, the closed submodule which\n is orthogonal to the range of `T - z • 1`.\n\n `T.defectNumber z` is defined as the rank of this subspace.", + "snippet": "def deficiencySubspace (T : H →ₗ.[ℂ] H) (z : ℂ) : ClosedSubmodule ℂ H", + "line": 332 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.deficiencySubspace_coe", + "docstring": null, + "snippet": "@[simp]\nlemma deficiencySubspace_coe (T : H →ₗ.[ℂ] H) (z : ℂ) :\n T.deficiencySubspace z = (T - z • 1).toFun.rangeᗮ", + "line": 336 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "def", + "name": "LinearPMap.defectNumber", + "docstring": "The rank of `T.deficiencySubspace z = (T - z • 1).rangeᗮ`.", + "snippet": "def defectNumber (T : H →ₗ.[ℂ] H) (z : ℂ) : Cardinal", + "line": 340 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.defectNumber_eq", + "docstring": null, + "snippet": "lemma defectNumber_eq (T : H →ₗ.[ℂ] H) (z : ℂ) :\n T.defectNumber z = Module.rank ℂ (T.deficiencySubspace z)", + "line": 342 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosed.defectNumber_eq_zero_iff", + "docstring": null, + "snippet": "lemma IsClosed.defectNumber_eq_zero_iff [CompleteSpace H]\n {T : H →ₗ.[ℂ] H} (hT : T.IsClosed) {z : ℂ} (hz : z ∈ T.regularityDomain) :\n T.defectNumber z = 0 ↔ (T - z • 1).toFun.range = ⊤", + "line": 345 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.defectNumber_closure", + "docstring": "`T` and `T.closure` have the same defect number at points in their regularity domain.", + "snippet": "lemma defectNumber_closure [CompleteSpace H]\n {T : H →ₗ.[ℂ] H} {z : ℂ} (hz : z ∈ T.regularityDomain) :\n T.closure.defectNumber z = T.defectNumber z", + "line": 353 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "Submodule.inf_ne_bot_of_rank_lt", + "docstring": null, + "snippet": "lemma _root_.Submodule.inf_ne_bot_of_rank_lt\n {E F : Submodule ℂ H} [E.HasOrthogonalProjection] (h_rank : Module.rank ℂ E < Module.rank ℂ F) :\n Eᗮ ⊓ F ≠ ⊥", + "line": 361 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosed.exists_inner_eq_zero_of_defectNumber_lt", + "docstring": null, + "snippet": "lemma IsClosed.exists_inner_eq_zero_of_defectNumber_lt [CompleteSpace H]\n {T : H →ₗ.[ℂ] H} (hT : T.IsClosed)\n {z₁ z₂ : ℂ} (hz₁ : z₁ ∈ T.regularityDomain) (h : T.defectNumber z₁ < T.defectNumber z₂) :\n ∃ x : T.domain, x ≠ 0 ∧ ⟪T x - z₁ • x, T x - z₂ • x⟫_ℂ = 0", + "line": 374 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosable.defectNumber_eq_of_mem_ball", + "docstring": null, + "snippet": "lemma IsClosable.defectNumber_eq_of_mem_ball [CompleteSpace H] {T : H →ₗ.[ℂ] H} (hT : T.IsClosable)\n {z₁ z₂ : ℂ} {c : ℝ} (h : IsLowerBound T z₁ c) (h_ball : z₂ ∈ ball z₁ c) :\n T.defectNumber z₁ = T.defectNumber z₂", + "line": 388 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosable.defectNumber_const", + "docstring": "The defect number is constant on each connected component of the regularity domain.", + "snippet": "lemma IsClosable.defectNumber_const [CompleteSpace H]\n {T : H →ₗ.[ℂ] H} (hT : T.IsClosable)\n {z₁ z₂ : ℂ} (hz : z₂ ∈ connectedComponentIn T.regularityDomain z₁) :\n T.defectNumber z₁ = T.defectNumber z₂", + "line": 417 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "def", + "name": "LinearPMap.numericalRange", + "docstring": "The set `{⟪x, T x⟫_ℂ | x ∈ T.domain ∧ ‖x‖ = 1} ⊆ ℂ`.", + "snippet": "def numericalRange (T : H →ₗ.[ℂ] H) : Set ℂ", + "line": 482 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.numericalRange_eq", + "docstring": null, + "snippet": "lemma numericalRange_eq (T : H →ₗ.[ℂ] H) : Θ T = (fun x ↦ ⟪↑x, T x⟫_ℂ) '' {x | ‖x‖ = 1}", + "line": 487 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.mem_numericalRange", + "docstring": null, + "snippet": "lemma mem_numericalRange {T : H →ₗ.[ℂ] H} {x : T.domain} (hx : x ≠ 0) :\n (‖x‖ ^ 2)⁻¹ * ⟪↑x, T x⟫_ℂ ∈ Θ T", + "line": 489 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.numericalRange_nonempty", + "docstring": null, + "snippet": "lemma numericalRange_nonempty {T : H →ₗ.[ℂ] H} (hT : T.domain ≠ ⊥) : (Θ T).Nonempty", + "line": 495 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.numericalRange_neg", + "docstring": null, + "snippet": "@[simp]\nlemma numericalRange_neg (T : H →ₗ.[ℂ] H) : Θ (-T) = -Θ T", + "line": 501 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.numericalRange_smul", + "docstring": null, + "snippet": "@[simp]\nlemma numericalRange_smul (T : H →ₗ.[ℂ] H) (c : ℂ) : Θ (c • T) = c • Θ T", + "line": 506 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.numericalRange_sub_const", + "docstring": null, + "snippet": "lemma numericalRange_sub_const (T : H →ₗ.[ℂ] H) (c : ℂ) : Θ (T - c • 1) = Θ T - {c}", + "line": 510 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.compl_closure_numericalRange_subset_regularityDomain", + "docstring": "The regularity domain contains the exterior of the numerical range.", + "snippet": "lemma compl_closure_numericalRange_subset_regularityDomain (T : H →ₗ.[ℂ] H) :\n (_root_.closure (Θ T))ᶜ ⊆ T.regularityDomain", + "line": 522 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.exists_phase_add_im_eq_zero", + "docstring": null, + "snippet": "private lemma exists_phase_add_im_eq_zero (z₁ z₂ : ℂ) :\n ∃ θ : ℝ, (exp (I * θ) * z₁ + exp (-I * θ) * z₂).im = 0", + "line": 556 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "theorem", + "name": "LinearPMap.numericalRange_convex", + "docstring": "The Toeplitz-Hausdorff theorem.", + "snippet": "theorem numericalRange_convex (T : H →ₗ.[ℂ] H) : Convex ℝ (Θ T)", + "line": 568 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "def", + "name": "LinearPMap.resolventSet", + "docstring": "The resolvent set, `ρ`, of a partial linear map.\n\n A complex number `z` is in `ρ T` iff the linear map `T - z • 1` from `T.domain` to `H`\n is a bijection with continuous (equivalently, bounded) inverse.", + "snippet": "def resolventSet (T : H →ₗ.[ℂ] H) : Set ℂ", + "line": 654 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.resolventSet_eq", + "docstring": null, + "snippet": "lemma resolventSet_eq (T : H →ₗ.[ℂ] H) :\n ρ T = {z | (T - z • 1).toFun.ker = ⊥ ∧ (T - z • 1).toFun.range = ⊤ ∧ Continuous (𝑅 T z)}", + "line": 660 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.mem_resolventSet_iff", + "docstring": null, + "snippet": "lemma mem_resolventSet_iff {T : H →ₗ.[ℂ] H} {z : ℂ} :\n z ∈ ρ T ↔ (T - z • 1).toFun.ker = ⊥ ∧ (T - z • 1).toFun.range = ⊤ ∧ Continuous (𝑅 T z)", + "line": 664 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.resolventSet_eq_empty", + "docstring": "If an operator is not closed then its resolvent set is empty.", + "snippet": "lemma resolventSet_eq_empty [CompleteSpace H] {T : H →ₗ.[ℂ] H} (h : ¬T.IsClosed) : ρ T = ∅", + "line": 669 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.resolventSet_subset_regularityDomain", + "docstring": null, + "snippet": "lemma resolventSet_subset_regularityDomain (T : H →ₗ.[ℂ] H) : ρ T ⊆ T.regularityDomain", + "line": 681 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosed.resolventSet_eq", + "docstring": "For a closed operator the continuity of the resolvent is redundant\n in the definition of the resolvent set.", + "snippet": "lemma IsClosed.resolventSet_eq [CompleteSpace H] {T : H →ₗ.[ℂ] H} (hT : T.IsClosed) :\n ρ T = {z : ℂ | (T - z • 1).toFun.ker = ⊥ ∧ (T - z • 1).toFun.range = ⊤}", + "line": 686 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosed.resolventSet_eq'", + "docstring": "For a closed operator the resolvent set consists of those regular points for which\n the defect number is zero.", + "snippet": "lemma IsClosed.resolventSet_eq' [CompleteSpace H] {T : H →ₗ.[ℂ] H} (hT : T.IsClosed) :\n ρ T = T.regularityDomain ∩ T.defectNumber ⁻¹' {0}", + "line": 698 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.resolventSet_isOpen", + "docstring": "The resolvent set is an open subset of ℂ.", + "snippet": "lemma resolventSet_isOpen [CompleteSpace H] (T : H →ₗ.[ℂ] H) : IsOpen (ρ T)", + "line": 710 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "def", + "name": "LinearPMap.spectrum", + "docstring": "The spectrum, `σ`, of a partial linear map.\n\n `σ T` is the complement of `ρ T`. A complex number `z` is in `σ T` iff the linear map `T - z • 1`\n from `T.domain` to `H` fails to be bijective or `(T - z • 1)⁻¹` is not continuous\n (equivalently, is not bounded).", + "snippet": "def spectrum (T : H →ₗ.[ℂ] H) : Set ℂ", + "line": 731 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.spectrum_eq", + "docstring": null, + "snippet": "lemma spectrum_eq (T : H →ₗ.[ℂ] H) : σ T = (ρ T)ᶜ", + "line": 736 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.mem_spectrum_iff", + "docstring": null, + "snippet": "lemma mem_spectrum_iff {T : H →ₗ.[ℂ] H} {z : ℂ} :\n z ∈ σ T ↔ (T - z • 1).toFun.ker ≠ ⊥ ∨ (T - z • 1).toFun.range ≠ ⊤ ∨ ¬Continuous (𝑅 T z)", + "line": 738 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.spectrum_eq_univ", + "docstring": "If an operator is not closed then its spectrum is all of ℂ.", + "snippet": "lemma spectrum_eq_univ [CompleteSpace H] {T : H →ₗ.[ℂ] H} (h : ¬T.IsClosed) : σ T = univ", + "line": 744 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.spectrum_isClosed", + "docstring": "The spectrum is a closed subset of ℂ.", + "snippet": "lemma spectrum_isClosed [CompleteSpace H] (T : H →ₗ.[ℂ] H) : _root_.IsClosed (σ T)", + "line": 748 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "def", + "name": "LinearPMap.pointSpectrum", + "docstring": "The point spectrum, `σᵖ`, of a partial linear map.\n\n A complex number `z` is in `σᵖ T` iff `T - z • 1` is not injective.", + "snippet": "def pointSpectrum (T : H →ₗ.[ℂ] H) : Set ℂ", + "line": 758 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.pointSpectrum_eq", + "docstring": null, + "snippet": "lemma pointSpectrum_eq (T : H →ₗ.[ℂ] H) : σᵖ T = {z | (T - z • 1).toFun.ker ≠ ⊥}", + "line": 763 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.mem_pointSpectrum_iff", + "docstring": null, + "snippet": "lemma mem_pointSpectrum_iff {T : H →ₗ.[ℂ] H} {z : ℂ} : z ∈ σᵖ T ↔ (T - z • 1).toFun.ker ≠ ⊥", + "line": 765 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.pointSpectrum_subset_spectrum", + "docstring": null, + "snippet": "lemma pointSpectrum_subset_spectrum (T : H →ₗ.[ℂ] H) : σᵖ T ⊆ σ T", + "line": 768 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "def", + "name": "LinearPMap.residualSpectrum", + "docstring": "The residual spectrum, `σʳ`, of a partial linear map.\n\n A complex number `z` is in `σʳ T` iff `T - z • 1` is injective but not surjective\n and `(T - z • 1)⁻¹` is continuous (equivalently, bounded).", + "snippet": "def residualSpectrum (T : H →ₗ.[ℂ] H) : Set ℂ", + "line": 779 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.residualSpectrum_eq", + "docstring": null, + "snippet": "lemma residualSpectrum_eq (T : H →ₗ.[ℂ] H) :\n σʳ T = {z | (T - z • 1).toFun.ker = ⊥ ∧ (T - z • 1).toFun.range ≠ ⊤ ∧ Continuous (𝑅 T z)}", + "line": 785 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.mem_residualSpectrum_iff", + "docstring": null, + "snippet": "lemma mem_residualSpectrum_iff {T : H →ₗ.[ℂ] H} {z : ℂ} :\n z ∈ σʳ T ↔ (T - z • 1).toFun.ker = ⊥ ∧ (T - z • 1).toFun.range ≠ ⊤ ∧ Continuous (𝑅 T z)", + "line": 789 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.residualSpectrum_subset_spectrum", + "docstring": null, + "snippet": "lemma residualSpectrum_subset_spectrum (T : H →ₗ.[ℂ] H) : σʳ T ⊆ σ T", + "line": 793 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.residualSpectrum_subset_regularityDomain", + "docstring": null, + "snippet": "lemma residualSpectrum_subset_regularityDomain (T : H →ₗ.[ℂ] H) : σʳ T ⊆ T.regularityDomain", + "line": 796 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "def", + "name": "LinearPMap.continuousSpectrum", + "docstring": "The continuous spectrum, `σᶜ`, of a partial linear map.\n\n A complex number `z` is in `σᶜ T` iff the range of `T - z • 1` is not closed.", + "snippet": "def continuousSpectrum (T : H →ₗ.[ℂ] H) : Set ℂ", + "line": 806 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.continuousSpectrum_eq", + "docstring": null, + "snippet": "lemma continuousSpectrum_eq (T : H →ₗ.[ℂ] H) :\n σᶜ T = {z | ¬_root_.IsClosed ((T - z • 1).toFun.range : Set H)}", + "line": 812 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.mem_continuousSpectrum_iff", + "docstring": null, + "snippet": "lemma mem_continuousSpectrum_iff {T : H →ₗ.[ℂ] H} {z : ℂ} :\n z ∈ σᶜ T ↔ ¬_root_.IsClosed ((T - z • 1).toFun.range : Set H)", + "line": 815 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.continuousSpectrum_subset_spectrum", + "docstring": null, + "snippet": "lemma continuousSpectrum_subset_spectrum (T : H →ₗ.[ℂ] H) : σᶜ T ⊆ σ T", + "line": 818 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosed.spectrum_eq", + "docstring": null, + "snippet": "lemma IsClosed.spectrum_eq [CompleteSpace H] {T : H →ₗ.[ℂ] H} (hT : T.IsClosed) :\n σ T = σᵖ T ∪ σʳ T ∪ σᶜ T", + "line": 825 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.pointSpectrum_inter_residualSpectrum", + "docstring": null, + "snippet": "lemma pointSpectrum_inter_residualSpectrum (T : H →ₗ.[ℂ] H) : σᵖ T ∩ σʳ T = ∅", + "line": 843 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.resolvent_sub", + "docstring": null, + "snippet": "lemma resolvent_sub\n {T₁ T₂ : H →ₗ.[ℂ] H} (hT : T₂.domain ≤ T₁.domain) {z : ℂ} (hz₁ : z ∈ ρ T₁) (hz₂ : z ∈ ρ T₂) :\n 𝑅 T₁ z - 𝑅 T₂ z = 𝑅 T₁ z * (T₂ - T₁) * 𝑅 T₂ z", + "line": 852 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Basic.lean", + "kind": "lemma", + "name": "LinearPMap.resolvent_sub'", + "docstring": null, + "snippet": "lemma resolvent_sub' {T : H →ₗ.[ℂ] H} (z₁ z₂ : ℂ) (hz₁ : z₁ ∈ ρ T) (hz₂ : z₂ ∈ ρ T) :\n 𝑅 T z₁ - 𝑅 T z₂ = (z₁ - z₂) • (𝑅 T z₁ * 𝑅 T z₂)", + "line": 882 + } + ] + }, + { + "filename": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SelfAdjoint.lean", + "entries": [ + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SelfAdjoint.lean", + "kind": "lemma", + "name": "LinearPMap.IsSelfAdjoint.resolventSet_eq_regularityDomain", + "docstring": null, + "snippet": "lemma resolventSet_eq_regularityDomain : ρ T = T.regularityDomain", + "line": 49 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SelfAdjoint.lean", + "kind": "lemma", + "name": "LinearPMap.IsSelfAdjoint.mem_resolventSet_of_range_eq_top", + "docstring": "`(T - z • 1).range = ⊤` is a sufficient condition for `z ∈ ρ T`\n (and it is a necessary condition by definition of `ρ`).", + "snippet": "lemma mem_resolventSet_of_range_eq_top {z : ℂ} (h : (T - z • 1).toFun.range = ⊤) : z ∈ ρ T", + "line": 66 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SelfAdjoint.lean", + "kind": "lemma", + "name": "LinearPMap.IsSelfAdjoint.spectrum_real", + "docstring": "The spectrum of a self-adjoint operator is real.", + "snippet": "lemma spectrum_real : σ T ⊆ range ofReal", + "line": 84 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SelfAdjoint.lean", + "kind": "lemma", + "name": "LinearPMap.IsSelfAdjoint.residualSpectrum_eq_empty", + "docstring": "The residual spectrum of a self-adjoint operator is empty.", + "snippet": "lemma residualSpectrum_eq_empty : σʳ T = ∅", + "line": 89 + } + ] + }, + { + "filename": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SpectralMeasure.lean", + "entries": [ + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SpectralMeasure.lean", + "kind": "instance", + "name": "«instance@38b2a7ba»", + "docstring": null, + "snippet": "instance (H : Type*) [SeminormedAddCommGroup H] [InnerProductSpace ℂ H] :\n IsAddTorsionFree (H →L[ℂ] H) where", + "line": 47 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SpectralMeasure.lean", + "kind": "structure", + "name": "SpectralMeasure", + "docstring": "A _spectral measure_ on a measurable space `α` is a σ-additive function `Set α → H →L[ℂ] H`\n such that each set is mapped to a star projection on `H`, the empty set and non-measurable sets\n are mapped to zero, and `univ` is mapped to the identity.", + "snippet": "structure SpectralMeasure\n (α : Type*) [MeasurableSpace α]\n (H : Type*) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]\n extends VectorMeasure α (H →L[ℂ] H) where\n isStarProjection' : ∀ A, IsStarProjection (measureOf' A)\n univ' : measureOf' univ = 1", + "line": 61 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SpectralMeasure.lean", + "kind": "instance", + "name": "SpectralMeasure.instCoeVectorMeasure", + "docstring": null, + "snippet": "instance instCoeVectorMeasure : Coe (SpectralMeasure α H) (VectorMeasure α (H →L[ℂ] H))", + "line": 76 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SpectralMeasure.lean", + "kind": "instance", + "name": "SpectralMeasure.instCoeFun", + "docstring": null, + "snippet": "instance instCoeFun : CoeFun (SpectralMeasure α H) fun _ ↦ Set α → H →L[ℂ] H", + "line": 79 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SpectralMeasure.lean", + "kind": "lemma", + "name": "SpectralMeasure.isStarProjection", + "docstring": null, + "snippet": "lemma isStarProjection (A : Set α) : IsStarProjection (μS A)", + "line": 82 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SpectralMeasure.lean", + "kind": "lemma", + "name": "SpectralMeasure.univ", + "docstring": null, + "snippet": "@[simp]\nlemma univ : μS univ = 1", + "line": 85 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SpectralMeasure.lean", + "kind": "lemma", + "name": "SpectralMeasure.comp_self", + "docstring": null, + "snippet": "@[simp]\nlemma comp_self (A : Set α) : μS A ∘L μS A = μS A", + "line": 92 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SpectralMeasure.lean", + "kind": "lemma", + "name": "SpectralMeasure.comp_of_disjoint", + "docstring": null, + "snippet": "lemma comp_of_disjoint\n {A B : Set α} (h : Disjoint A B) (hA : MeasurableSet A) (hB : MeasurableSet B) :\n μS A ∘L μS B = 0", + "line": 94 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SpectralMeasure.lean", + "kind": "lemma", + "name": "SpectralMeasure.comp_eq_of_inter", + "docstring": null, + "snippet": "lemma comp_eq_of_inter {A B : Set α} (hA : MeasurableSet A) (hB : MeasurableSet B) :\n μS A ∘L μS B = μS (A ∩ B)", + "line": 104 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/SpectralMeasure.lean", + "kind": "lemma", + "name": "SpectralMeasure.commute", + "docstring": null, + "snippet": "lemma commute (A B : Set α) : Commute (μS A) (μS B)", + "line": 113 + } + ] + }, + { + "filename": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "entries": [ + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "kind": "def", + "name": "LinearPMap.IsSymmetric.realNumericalRange", + "docstring": "The projection of the numerical range onto the real axis.", + "snippet": "def realNumericalRange (T : H →ₗ.[ℂ] H) : Set ℝ", + "line": 56 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "kind": "lemma", + "name": "LinearPMap.IsSymmetric.realNumericalRange_eq", + "docstring": null, + "snippet": "lemma realNumericalRange_eq (T : H →ₗ.[ℂ] H) : Θᵣₑ T = re '' Θ T", + "line": 61 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "kind": "lemma", + "name": "LinearPMap.IsSymmetric.realNumericalRange_neg", + "docstring": null, + "snippet": "@[simp]\nlemma realNumericalRange_neg (T : H →ₗ.[ℂ] H) : Θᵣₑ (-T) = -Θᵣₑ T", + "line": 64 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "kind": "lemma", + "name": "LinearPMap.IsSymmetric.im_eq_zero_of_mem_numericalRange", + "docstring": "The numerical range of a symmetric operator is contained in the real axis.", + "snippet": "lemma im_eq_zero_of_mem_numericalRange {z : ℂ} (hz : z ∈ Θ T) : z.im = 0", + "line": 72 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "kind": "lemma", + "name": "LinearPMap.IsSymmetric.numericalRange_subset", + "docstring": "The numerical range of a symmetric operator is contained in the real axis.", + "snippet": "lemma numericalRange_subset : Θ T ⊆ range ofReal", + "line": 78 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "kind": "lemma", + "name": "LinearPMap.IsSymmetric.numericalRange_eq", + "docstring": "The numerical range of a symmetric operator is equal to its projection onto the real axis.", + "snippet": "lemma numericalRange_eq : Θ T = ofReal '' Θᵣₑ T", + "line": 82 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "kind": "lemma", + "name": "LinearPMap.IsSymmetric.closure_numericalRange_subset", + "docstring": null, + "snippet": "lemma closure_numericalRange_subset : _root_.closure (Θ T) ⊆ range ofReal", + "line": 87 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "kind": "lemma", + "name": "LinearPMap.IsSymmetric.mem_regularityDomain_of_im_ne_zero", + "docstring": "The regularity domain of a symmetric operator contains all complex numbers with non-zero\n imaginary part.", + "snippet": "lemma mem_regularityDomain_of_im_ne_zero {z : ℂ} (hz : z.im ≠ 0) : z ∈ T.regularityDomain", + "line": 97 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "kind": "lemma", + "name": "LinearPMap.IsSymmetric.compl_ofReal_subset_regularityDomain", + "docstring": "The regularity domain of a symmetric operator contains all complex numbers with non-zero\n imaginary part.", + "snippet": "lemma compl_ofReal_subset_regularityDomain : (range ofReal)ᶜ ⊆ T.regularityDomain", + "line": 105 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "kind": "lemma", + "name": "LinearPMap.IsSymmetric.Iio_subset_regularityDomain", + "docstring": "If `m` is a lower bound on the numerical range then the regularity domain contains `(-∞,m)`.", + "snippet": "lemma Iio_subset_regularityDomain {m : ℝ} (h : m ∈ lowerBounds (Θᵣₑ T)) :\n ofReal '' Iio m ⊆ T.regularityDomain", + "line": 109 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "kind": "lemma", + "name": "LinearPMap.IsSymmetric.Ioi_subset_regularityDomain", + "docstring": "If `m` is an upper bound on the numerical range then the regularity domain contains `(m,∞)`.", + "snippet": "lemma Ioi_subset_regularityDomain {m : ℝ} (h : m ∈ upperBounds (Θᵣₑ T)) :\n ofReal '' Ioi m ⊆ T.regularityDomain", + "line": 129 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "kind": "lemma", + "name": "LinearPMap.IsSymmetric.regularityDomain_isConnected_iff", + "docstring": "The regularity domain of a symmetric operator is connected iff it contains a real number.", + "snippet": "lemma regularityDomain_isConnected_iff :\n IsConnected T.regularityDomain ↔ (ofReal ⁻¹' T.regularityDomain).Nonempty", + "line": 138 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "kind": "lemma", + "name": "LinearPMap.IsSymmetric.regularityDomain_isConnected_of_bddBelow", + "docstring": "The regularity domain of a lower semibounded symmetric operator is connected.", + "snippet": "lemma regularityDomain_isConnected_of_bddBelow (h : BddBelow (Θᵣₑ T)) :\n IsConnected T.regularityDomain", + "line": 186 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "kind": "lemma", + "name": "LinearPMap.IsSymmetric.regularityDomain_isConnected_of_bddAbove", + "docstring": "The regularity domain of an upper semibounded symmetric operator is connected.", + "snippet": "lemma regularityDomain_isConnected_of_bddAbove (h : BddAbove (Θᵣₑ T)) :\n IsConnected T.regularityDomain", + "line": 193 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/SpectralTheory/Symmetric.lean", + "kind": "lemma", + "name": "LinearPMap.IsSymmetric.pointSpectrum_real", + "docstring": "Eigenvalues of a symmetric unbounded operator are real.", + "snippet": "lemma pointSpectrum_real : σᵖ T ⊆ range ofReal", + "line": 204 + } + ] + }, + { + "filename": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "entries": [ + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.HasDenseDomain.orthogonal_range", + "docstring": "`U.rangeᗮ = U†.ker`\n\n c.f. `LinearMap.orthogonal_range` and `ContinuousLinearMap.orthogonal_range`", + "snippet": "lemma HasDenseDomain.orthogonal_range [CompleteSpace H] (h : U.HasDenseDomain) :\n U.toFun.rangeᗮ = U†.toFun.ker.map U†.domain.subtype", + "line": 191 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.HasDenseDomain.orthogonal_adjoint_ker", + "docstring": "`U†.kerᗮ = U.range.closure`", + "snippet": "lemma HasDenseDomain.orthogonal_adjoint_ker [CompleteSpace H] [CompleteSpace H']\n (h : U.HasDenseDomain) :\n (U†.toFun.ker.map U†.domain.subtype)ᗮ = U.toFun.range.closure", + "line": 203 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosed.closure_eq", + "docstring": null, + "snippet": "lemma IsClosed.closure_eq (h : U.IsClosed) : U.closure = U", + "line": 212 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.adjoint_zero", + "docstring": null, + "snippet": "@[simp]\nlemma adjoint_zero [CompleteSpace H] : (0 : H →ₗ.[ℂ] H')† = 0", + "line": 313 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.adjoint_sub_le_sub_adjoint", + "docstring": null, + "snippet": "lemma adjoint_sub_le_sub_adjoint [CompleteSpace H]\n (U₁ U₂ : H →ₗ.[ℂ] H') (h₁₂ : (U₁ - U₂).HasDenseDomain) : U₁† - U₂† ≤ (U₁ - U₂)†", + "line": 367 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearMap.continuous_iff_bounded", + "docstring": "`f : E →ₗ[𝕜] F` is continuous iff there exists `M > 0` s.t. `‖f x‖ ≤ M * ‖x‖` for all `x : E`.\n\n This is a (convenient) immediate consequence of\n `IsBoundedLinearMap.isLinearMap_and_continuous_iff_isBoundedLinearMap`.", + "snippet": "lemma _root_.LinearMap.continuous_iff_bounded {𝕜 E F : Type*} [NontriviallyNormedField 𝕜]\n [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [SeminormedAddCommGroup F] [NormedSpace 𝕜 F]\n {f : E →ₗ[𝕜] F} : Continuous f ↔ ∃ M, 0 < M ∧ ∀ x : E, ‖f x‖ ≤ M * ‖x‖", + "line": 403 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.isClosable_of_continuous", + "docstring": "Continuous operators are closable.", + "snippet": "lemma isClosable_of_continuous (h : Continuous U) : U.IsClosable", + "line": 411 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.closure_domain_eq_domain_closure_of_continuous", + "docstring": "A strengthening of `closure_domain_le_domain_closure` for continuous operators.", + "snippet": "lemma closure_domain_eq_domain_closure_of_continuous [CompleteSpace H'] (h : Continuous U) :\n U.closure.domain = U.domain.closure", + "line": 426 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.isClosed_iff_isClosed_domain_of_continuous", + "docstring": "A continuous operator is closed iff its domain is closed.", + "snippet": "lemma isClosed_iff_isClosed_domain_of_continuous [CompleteSpace H'] (h : Continuous U) :\n U.IsClosed ↔ _root_.IsClosed (U.domain : Set H)", + "line": 454 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosed.isClosed_toFun_graph", + "docstring": null, + "snippet": "lemma IsClosed.isClosed_toFun_graph (hU : U.IsClosed) :\n _root_.IsClosed (U.toFun.graph : Set (U.domain × H'))", + "line": 463 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosed.continuous_of_isClosed_domain", + "docstring": "The **closed graph theorem** for partial linear maps:\n a closed operator with closed domain is continuous.\n\n This follows immediately from `LinearMap.continuous_of_isClosed_graph`\n and the completeness of `H` and `H'`.", + "snippet": "lemma IsClosed.continuous_of_isClosed_domain [CompleteSpace H] [CompleteSpace H']\n (hU : U.IsClosed) (h : _root_.IsClosed (U.domain : Set H)) :\n Continuous U", + "line": 480 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosable.add_continuous", + "docstring": "Closability is preserved upon adding a continuous operator.", + "snippet": "lemma IsClosable.add_continuous\n (h₁ : U₁.IsClosable) (h₂ : Continuous U₂) (h : U₁.domain ≤ U₂.domain) :\n (U₁ + U₂).IsClosable", + "line": 488 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosable.sub_continuous", + "docstring": "Closability is preserved upon subtracting a continuous operator.", + "snippet": "lemma IsClosable.sub_continuous\n (h₁ : U₁.IsClosable) (h₂ : Continuous U₂) (h : U₁.domain ≤ U₂.domain) : (U₁ - U₂).IsClosable", + "line": 507 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosed.add_continuous", + "docstring": "Closedness is preserved upon adding a continuous operator.", + "snippet": "lemma IsClosed.add_continuous [CompleteSpace H']\n (h₁ : U₁.IsClosed) (h₂ : Continuous U₂) (h : U₁.domain ≤ U₂.domain) : (U₁ + U₂).IsClosed", + "line": 512 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.IsClosed.sub_continuous", + "docstring": "Closedness is preserved upon subtracting a continuous operator.", + "snippet": "lemma IsClosed.sub_continuous [CompleteSpace H']\n (h₁ : U₁.IsClosed) (h₂ : Continuous U₂) (h : U₁.domain ≤ U₂.domain) : (U₁ - U₂).IsClosed", + "line": 548 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.IsUnbounded.orthogonal_adjoint_range", + "docstring": "`U†.rangeᗮ = U.closure.ker`", + "snippet": "lemma IsUnbounded.orthogonal_adjoint_range [CompleteSpace H] [CompleteSpace H']\n (h : U.IsUnbounded) : U†.toFun.rangeᗮ = U.closure.toFun.ker.map U.closure.domain.subtype", + "line": 818 + }, + { + "file": "Physlib/QuantumMechanics/DDimensions/Operators/Unbounded.lean", + "kind": "lemma", + "name": "LinearPMap.IsUnbounded.orthogonal_closure_ker", + "docstring": "`U.closure.kerᗮ = U†.range`", + "snippet": "lemma IsUnbounded.orthogonal_closure_ker [CompleteSpace H] [CompleteSpace H'] (h : U.IsUnbounded) :\n (U.closure.toFun.ker.map U.closure.domain.subtype)ᗮ = U†.toFun.range.closure", + "line": 823 + } + ] + }, + { + "filename": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "entries": [ + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "def", + "name": "QuantumMechanics.FiniteHilbertSpace.equivEuclidean", + "docstring": "The equivalence between `FiniteHilbertSpace d` and `EuclideanSpace ℂ d`\n given by `val`.", + "snippet": "def equivEuclidean : FiniteHilbertSpace d ≃ EuclideanSpace ℂ d where", + "line": 77 + }, + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "instance", + "name": "QuantumMechanics.FiniteHilbertSpace.«instance@89dd6b4e»", + "docstring": null, + "snippet": "noncomputable instance : AddCommGroup (FiniteHilbertSpace d)", + "line": 92 + }, + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "instance", + "name": "QuantumMechanics.FiniteHilbertSpace.«instance@ba152d22»", + "docstring": null, + "snippet": "noncomputable instance : Module ℂ (FiniteHilbertSpace d)", + "line": 94 + }, + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "lemma", + "name": "QuantumMechanics.FiniteHilbertSpace.val_add", + "docstring": null, + "snippet": "@[simp]\nlemma val_add (ψ φ : FiniteHilbertSpace d) : (ψ + φ).val = ψ.val + φ.val", + "line": 97 + }, + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "lemma", + "name": "QuantumMechanics.FiniteHilbertSpace.val_smul", + "docstring": null, + "snippet": "@[simp]\nlemma val_smul (c : ℂ) (ψ : FiniteHilbertSpace d) : (c • ψ).val = c • ψ.val", + "line": 100 + }, + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "lemma", + "name": "QuantumMechanics.FiniteHilbertSpace.val_zero", + "docstring": null, + "snippet": "@[simp]\nlemma val_zero : (0 : FiniteHilbertSpace d).val = 0", + "line": 103 + }, + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "def", + "name": "QuantumMechanics.FiniteHilbertSpace.linearEquivEuclidean", + "docstring": "The equivalence between `FiniteHilbertSpace d` and `EuclideanSpace ℂ d`\n as a `ℂ`-linear equivalence, upgrading `equivEuclidean`.", + "snippet": "noncomputable def linearEquivEuclidean : FiniteHilbertSpace d ≃ₗ[ℂ] EuclideanSpace ℂ d", + "line": 107 + }, + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "instance", + "name": "QuantumMechanics.FiniteHilbertSpace.«instance@a3aa9745»", + "docstring": null, + "snippet": "noncomputable instance : NormedAddCommGroup (FiniteHilbertSpace d)", + "line": 122 + }, + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "lemma", + "name": "QuantumMechanics.FiniteHilbertSpace.norm_eq_val", + "docstring": null, + "snippet": "@[simp]\nlemma norm_eq_val (ψ : FiniteHilbertSpace d) : ‖ψ‖ = ‖ψ.val‖", + "line": 126 + }, + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "instance", + "name": "QuantumMechanics.FiniteHilbertSpace.«instance@d6d3ba74»", + "docstring": null, + "snippet": "noncomputable instance : InnerProductSpace ℂ (FiniteHilbertSpace d)", + "line": 128 + }, + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "lemma", + "name": "QuantumMechanics.FiniteHilbertSpace.inner_eq_val", + "docstring": null, + "snippet": "@[simp]\nlemma inner_eq_val (ψ φ : FiniteHilbertSpace d) : inner ℂ ψ φ = inner ℂ ψ.val φ.val", + "line": 132 + }, + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "instance", + "name": "QuantumMechanics.FiniteHilbertSpace.«instance@df6e240e»", + "docstring": null, + "snippet": "instance : FiniteDimensional ℂ (FiniteHilbertSpace d)", + "line": 134 + }, + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "instance", + "name": "QuantumMechanics.FiniteHilbertSpace.«instance@25da29e8»", + "docstring": null, + "snippet": "instance : CompleteSpace (FiniteHilbertSpace d)", + "line": 137 + }, + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "def", + "name": "QuantumMechanics.FiniteHilbertSpace.isometryEquivEuclidean", + "docstring": "The equivalence between `FiniteHilbertSpace d` and `EuclideanSpace ℂ d`\n as a linear isometry equivalence, upgrading `linearEquivEuclidean`.", + "snippet": "noncomputable def isometryEquivEuclidean : FiniteHilbertSpace d ≃ₗᵢ[ℂ] EuclideanSpace ℂ d where", + "line": 141 + }, + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "def", + "name": "QuantumMechanics.FiniteHilbertSpace.basisFun", + "docstring": "The standard orthonormal basis of `FiniteHilbertSpace d`, indexed by `d`.", + "snippet": "noncomputable def basisFun (d : Type*) [Fintype d] [DecidableEq d] :\n OrthonormalBasis d ℂ (FiniteHilbertSpace d)", + "line": 152 + }, + { + "file": "Physlib/QuantumMechanics/FiniteTarget/HilbertSpace.lean", + "kind": "lemma", + "name": "QuantumMechanics.FiniteHilbertSpace.basisFun_apply", + "docstring": null, + "snippet": "lemma basisFun_apply (i : d) : basisFun d i = ⟨EuclideanSpace.single i 1⟩", + "line": 156 + } + ] + }, + { + "filename": "Physlib/Relativity/LorentzGroup/Basic.lean", + "entries": [ + { + "file": "Physlib/Relativity/LorentzGroup/Basic.lean", + "kind": "lemma", + "name": "LorentzGroup.continuous_self_mul_dual", + "docstring": null, + "snippet": "lemma continuous_self_mul_dual {d : ℕ} :\n Continuous (fun Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ => Λ * dual Λ)", + "line": 155 + }, + { + "file": "Physlib/Relativity/LorentzGroup/Basic.lean", + "kind": "lemma", + "name": "LorentzGroup.isClosed", + "docstring": "The Lorentz group is closed in the ambient matrix space.", + "snippet": "lemma isClosed (d : ℕ) :\n IsClosed (LorentzGroup d : Set (Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ))", + "line": 161 + }, + { + "file": "Physlib/Relativity/LorentzGroup/Basic.lean", + "kind": "lemma", + "name": "LorentzGroup.isClosedEmbedding_val", + "docstring": "The coercion from the Lorentz group to its ambient matrix space is a closed embedding.", + "snippet": "lemma isClosedEmbedding_val {d : ℕ} :\n Topology.IsClosedEmbedding\n (Subtype.val : LorentzGroup d → Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)", + "line": 167 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/Basic.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/Basic.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.Pure.update_eq_function_update", + "docstring": null, + "snippet": "lemma update_eq_function_update {n : ℕ} {c : Fin n → C} [inst : DecidableEq (Fin n)]\n (p : Pure S c) (i : Fin n)\n (x : V (c i)) : update p i x = Function.update p i x", + "line": 133 + }, + { + "file": "Physlib/Relativity/Tensors/Basic.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.Pure.update_diff", + "docstring": null, + "snippet": "lemma update_diff {n : ℕ} {c : Fin n → C} [inst : DecidableEq (Fin n)] (p : Pure S c) (i j : Fin n)\n (x : V (c i)) (hij : i ≠ j) : (update p i x) j = p j", + "line": 142 + }, + { + "file": "Physlib/Relativity/Tensors/Basic.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.componentMap_eq_repr", + "docstring": "`componentMap` is the basis representation `(basis c).repr` definitionally; this bridges the two\nnotations wherever a component computation meets a `repr` rewrite.", + "snippet": "lemma componentMap_eq_repr {n : ℕ} (c : Fin n → C) (t : S.Tensor c)\n (ψ : ComponentIdx (S := S) c) : componentMap c t ψ = (basis c).repr t ψ", + "line": 415 + }, + { + "file": "Physlib/Relativity/Tensors/Basic.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.componentMap_ext", + "docstring": "Two tensors with the same colour sequence are equal when all their components agree.", + "snippet": "lemma componentMap_ext {n : ℕ} {c : Fin n → C} {t₁ t₂ : S.Tensor c}\n (h : ∀ b, componentMap c t₁ b = componentMap c t₂ b) : t₁ = t₂", + "line": 419 + }, + { + "file": "Physlib/Relativity/Tensors/Basic.lean", + "kind": "instance", + "name": "TensorSpecies.Tensor.«instance@180d6a48»", + "docstring": null, + "snippet": "instance {k : Type} [Field k] {C : Type} {G : Type} [Group G]\n {V : C → Type} [∀ c, AddCommGroup (V c)] [∀ c, Module k (V c)]\n {basisIdx : C → Type} [∀ c, Fintype (basisIdx c)] [∀ c, DecidableEq (basisIdx c)]\n {rep : (c : C) → Representation k G (V c)} {b : (c : C) → Basis (basisIdx c) k (V c)}\n (S : TensorSpecies k C G V basisIdx rep b)\n {c : Fin n → C} : FiniteDimensional k (S.Tensor c)", + "line": 432 + }, + { + "file": "Physlib/Relativity/Tensors/Basic.lean", + "kind": "instance", + "name": "TensorSpecies.Tensor.«instance@357a4078»", + "docstring": null, + "snippet": "noncomputable instance {k : Type} [RCLike k] {C : Type} {G : Type} [Group G]\n {V : C → Type} [∀ c, AddCommGroup (V c)] [∀ c, Module k (V c)]\n {basisIdx : C → Type} [∀ c, Fintype (basisIdx c)] [∀ c, DecidableEq (basisIdx c)]\n {rep : (c : C) → Representation k G (V c)} {b : (c : C) → Basis (basisIdx c) k (V c)}\n (S : TensorSpecies k C G V basisIdx rep b)\n {c : Fin n → C} : TopologicalSpace (S.Tensor c)", + "line": 440 + }, + { + "file": "Physlib/Relativity/Tensors/Basic.lean", + "kind": "instance", + "name": "TensorSpecies.Tensor.«instance@958788b8»", + "docstring": null, + "snippet": "instance {k : Type} [RCLike k] {C : Type} {G : Type} [Group G]\n {V : C → Type} [∀ c, AddCommGroup (V c)] [∀ c, Module k (V c)]\n {basisIdx : C → Type} [∀ c, Fintype (basisIdx c)] [∀ c, DecidableEq (basisIdx c)]\n {rep : (c : C) → Representation k G (V c)} {b : (c : C) → Basis (basisIdx c) k (V c)}\n (S : TensorSpecies k C G V basisIdx rep b)\n {c : Fin n → C} : IsTopologicalAddGroup (S.Tensor c)", + "line": 448 + }, + { + "file": "Physlib/Relativity/Tensors/Basic.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.Pure.rep_cast", + "docstring": null, + "snippet": "lemma rep_cast {g : G} {c c1 : C} (p : V c) (h : c = c1) :\n LinearEquiv.cast (R := k) h (rep c g p) = rep c1 g (LinearEquiv.cast (R := k) h p)", + "line": 476 + }, + { + "file": "Physlib/Relativity/Tensors/Basic.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.Pure.actionP_cast", + "docstring": null, + "snippet": "lemma actionP_cast {g : G} {p : Pure S c} (h : c = c1) :\n LinearEquiv.cast (R := k) h (g • p) = g • LinearEquiv.cast (R := k) h p", + "line": 481 + }, + { + "file": "Physlib/Relativity/Tensors/Basic.lean", + "kind": "instance", + "name": "TensorSpecies.Tensor.«instance@713c1a5c»", + "docstring": null, + "snippet": "noncomputable instance : SMul G (S.Tensor c) where", + "line": 502 + }, + { + "file": "Physlib/Relativity/Tensors/Basic.lean", + "kind": "instance", + "name": "TensorSpecies.Tensor.«instance@62e8adcb»", + "docstring": null, + "snippet": "noncomputable instance : DistribMulAction G (S.Tensor c) where", + "line": 543 + }, + { + "file": "Physlib/Relativity/Tensors/Basic.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.Pure.permP_equivariant", + "docstring": null, + "snippet": "lemma Pure.permP_equivariant {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}\n {σ : Fin m → Fin n} (h : IsReindexing c c1 σ) (g : G) (p : Pure S c) :\n Pure.permP σ h (g • p) = g • Pure.permP σ h p", + "line": 578 + }, + { + "file": "Physlib/Relativity/Tensors/Basic.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.toField_injective", + "docstring": null, + "snippet": "lemma toField_injective {c : Fin 0 → C} :\n Function.Injective (toField : S.Tensor c → k)", + "line": 781 + }, + { + "file": "Physlib/Relativity/Tensors/Basic.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.toField_basis", + "docstring": null, + "snippet": "lemma toField_basis {c : Fin 0 → C} (b : ComponentIdx (S := S) c) :\n toField (basis c b) = 1", + "line": 807 + }, + { + "file": "Physlib/Relativity/Tensors/Basic.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.eq_smul_toField", + "docstring": null, + "snippet": "lemma eq_smul_toField {c : Fin 0 → C} (t : Tensor S c) :\n t = toField t • (basis c (@default (ComponentIdx (S := S) c) Unique.instInhabited))", + "line": 835 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComplexTensor/Basic.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Basic.lean", + "kind": "abbrev", + "name": "complexLorentzTensor.modules", + "docstring": "The modules associated with each of the different types of complex Lorentz vector space.", + "snippet": "abbrev modules : Color → Type\n | Color.upL => Fermion.LeftHandedWeyl\n | Color.downL => Fermion.DualLeftHandedWeyl\n | Color.upR => Fermion.RightHandedWeyl\n | Color.downR => Fermion.DualRightHandedWeyl\n | Color.up => Lorentz.ContrℂModule\n | Color.down => Lorentz.CoℂModule", + "line": 93 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Basic.lean", + "kind": "instance", + "name": "complexLorentzTensor.modulesAddCommGroup", + "docstring": null, + "snippet": "instance modulesAddCommGroup : ∀ c, AddCommGroup (modules c)\n | Color.upL => inferInstance\n | Color.downL => inferInstance\n | Color.upR => inferInstance\n | Color.downR => inferInstance\n | Color.up => inferInstance\n | Color.down => inferInstance", + "line": 101 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Basic.lean", + "kind": "instance", + "name": "complexLorentzTensor.modulesModule", + "docstring": null, + "snippet": "noncomputable instance modulesModule : ∀ c, Module ℂ (modules c)\n | Color.upL => inferInstance\n | Color.downL => inferInstance\n | Color.upR => inferInstance\n | Color.downR => inferInstance\n | Color.up => inferInstance\n | Color.down => inferInstance", + "line": 109 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Basic.lean", + "kind": "abbrev", + "name": "complexLorentzTensor.basis", + "docstring": "The basis associated with each of the different types of complex Lorentz vector space.", + "snippet": "abbrev basis (c : Color) : Module.Basis (Fin (repDim c)) ℂ (modules c)", + "line": 232 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Basic.lean", + "kind": "abbrev", + "name": "complexLorentzTensor.rep", + "docstring": "The reps associated with each of the different types of complex Lorentz vector space.", + "snippet": "abbrev rep (c : Color) : Representation ℂ SL(2, ℂ) (modules c)", + "line": 242 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.contrPCoeff_basis", + "docstring": null, + "snippet": "lemma contrPCoeff_basis {n : ℕ} {c : Fin n → complexLorentzTensor.Color} (i j : Fin n)\n (hij : i ≠ j ∧ (complexLorentzTensor.τ (c i) = c j))\n (b : ComponentIdx (S := complexLorentzTensor) c) :\n Pure.contrPCoeff i j hij (Pure.basisVector c b) = if b i =\n Fin.cast (by simp [← hij.2, repDim_tau]) (b j)\n then 1 else 0", + "line": 273 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean", + "kind": "abbrev", + "name": "complexLorentzTensor.dualLeftMetric", + "docstring": "The metric `εₐₐ` as a complex Lorentz tensor.", + "snippet": "abbrev dualLeftMetric : ℂT[.downL, .downL]", + "line": 45 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean", + "kind": "abbrev", + "name": "complexLorentzTensor.dualRightMetric", + "docstring": "The metric `ε_{dot a}_{dot a}` as a complex Lorentz tensor.", + "snippet": "abbrev dualRightMetric : ℂT[.downR, .downR]", + "line": 48 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualLeftMetric_eq_fromConstPair", + "docstring": null, + "snippet": "lemma dualLeftMetric_eq_fromConstPair : εL' = fromConstPair Fermion.dualLeftMetric", + "line": 101 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualRightMetric_eq_fromConstPair", + "docstring": null, + "snippet": "lemma dualRightMetric_eq_fromConstPair : εR' = fromConstPair Fermion.dualRightMetric", + "line": 103 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualLeftMetric_eq_fromPairT", + "docstring": null, + "snippet": "lemma dualLeftMetric_eq_fromPairT : εL' = fromPairT (Fermion.dualLeftMetricVal)", + "line": 131 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualRightMetric_eq_fromPairT", + "docstring": null, + "snippet": "lemma dualRightMetric_eq_fromPairT : εR' = fromPairT (Fermion.dualRightMetricVal)", + "line": 136 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualLeftMetric_eq_dualLeftBasis", + "docstring": null, + "snippet": "lemma dualLeftMetric_eq_dualLeftBasis : εL' =\n fromPairT (dualLeftBasis 0 ⊗ₜ[ℂ] dualLeftBasis 1)\n - fromPairT (dualLeftBasis 1 ⊗ₜ[ℂ] dualLeftBasis 0)", + "line": 193 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualRightMetric_eq_dualRightBasis", + "docstring": null, + "snippet": "lemma dualRightMetric_eq_dualRightBasis : εR' =\n fromPairT (dualRightBasis 0 ⊗ₜ[ℂ] dualRightBasis 1)\n - fromPairT (dualRightBasis 1 ⊗ₜ[ℂ] dualRightBasis 0)", + "line": 207 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualLeftMetric_eq_basis", + "docstring": null, + "snippet": "lemma dualLeftMetric_eq_basis : εL' =\n (Tensor.basis (S := complexLorentzTensor) ![Color.downL, Color.downL]\n (fun | 0 => (0 : Fin 2) | 1 => (1 : Fin 2)))\n - (Tensor.basis (S := complexLorentzTensor)\n ![Color.downL, Color.downL] (fun | 0 => (1 : Fin 2) | 1 => (0 : Fin 2)))", + "line": 285 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualRightMetric_eq_basis", + "docstring": null, + "snippet": "lemma dualRightMetric_eq_basis : εR' =\n (Tensor.basis (S := complexLorentzTensor)\n ![Color.downR, Color.downR] (fun | 0 => (0 : Fin 2) | 1 => (1 : Fin 2)))\n - (Tensor.basis (S := complexLorentzTensor)\n ![Color.downR, Color.downR] (fun | 0 => (1 : Fin 2) | 1 => (0 : Fin 2)))", + "line": 315 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualLeftMetric_eq_ofRat", + "docstring": null, + "snippet": "lemma dualLeftMetric_eq_ofRat : εL' = ofRat fun f =>\n if f 0 = Fin.cast (by rfl) (0 : Fin 2) ∧ f 1 = Fin.cast (by rfl) (1 : Fin 2) then 1 else\n if f 1 = Fin.cast (by rfl) (0 : Fin 2) ∧ f 0 = Fin.cast (by rfl) (1 : Fin 2) then\n - 1 else 0", + "line": 366 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualRightMetric_eq_ofRat", + "docstring": null, + "snippet": "lemma dualRightMetric_eq_ofRat : εR' = ofRat fun f =>\n if f 0 = Fin.cast (by rfl) (0 : Fin 2) ∧ f 1 = Fin.cast (by rfl) (1 : Fin 2) then 1 else\n if f 1 = Fin.cast (by rfl) (0 : Fin 2) ∧ f 0 = Fin.cast (by rfl) (1 : Fin 2) then\n - 1 else 0", + "line": 387 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.actionT_dualLeftMetric", + "docstring": "The tensor `dualLeftMetric` is invariant under the action of `SL(2,ℂ)`.", + "snippet": "lemma actionT_dualLeftMetric (g : SL(2,ℂ)) : g • εL' = εL'", + "line": 428 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.actionT_dualRightMetric", + "docstring": "The tensor `dualRightMetric` is invariant under the action of `SL(2,ℂ)`.", + "snippet": "lemma actionT_dualRightMetric (g : SL(2,ℂ)) : g • εR' = εR'", + "line": 433 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Lemmas.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Lemmas.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualLeftMetric_antisymm", + "docstring": "The dual-left metric is antisymmetric `{εL' | α α' = - εL' | α' α}ᵀ`.", + "snippet": "lemma dualLeftMetric_antisymm : {εL' | α α' = - (εL' | α' α)}ᵀ", + "line": 78 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Lemmas.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualRightMetric_antisymm", + "docstring": "The dual-right metric is antisymmetric `{εR' | β β' = - εR' | β' β}ᵀ`.", + "snippet": "lemma dualRightMetric_antisymm : {εR' | α α' = - (εR' | α' α)}ᵀ", + "line": 89 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Lemmas.lean", + "kind": "lemma", + "name": "complexLorentzTensor.leftMetric_contr_dualLeftMetric", + "docstring": "The contraction of the left metric with the dual-left metric is the unit\n`{εL | α β ⊗ εL' | β γ = δL | α γ}ᵀ`.", + "snippet": "lemma leftMetric_contr_dualLeftMetric : {εL | α β ⊗ εL' | β γ = δL | α γ}ᵀ", + "line": 120 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Lemmas.lean", + "kind": "lemma", + "name": "complexLorentzTensor.rightMetric_contr_dualRightMetric", + "docstring": "The contraction of the right metric with the dual-right metric is the unit\n`{εR | α β ⊗ εR' | β γ = δR | α γ}ᵀ`.", + "snippet": "lemma rightMetric_contr_dualRightMetric : {εR | α β ⊗ εR' | β γ = δR | α γ}ᵀ", + "line": 126 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Lemmas.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualLeftMetric_contr_leftMetric", + "docstring": "The contraction of the dual-left metric with the left metric is the unit\n`{εL' | α β ⊗ εL | β γ = δL' | α γ}ᵀ`.", + "snippet": "lemma dualLeftMetric_contr_leftMetric : {εL' | α β ⊗ εL | β γ = δL' | α γ}ᵀ", + "line": 132 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Metrics/Lemmas.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualRightMetric_contr_rightMetric", + "docstring": "The contraction of the dual-right metric with the right metric is the unit\n`{εR' | α β ⊗ εR | β γ = δR' | α γ}ᵀ`.", + "snippet": "lemma dualRightMetric_contr_rightMetric : {εR' | α β ⊗ εR | β γ = δR' | α γ}ᵀ", + "line": 138 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "abbrev", + "name": "complexLorentzTensor.dualLeftLeftUnit", + "docstring": "The unit `δₐᵃ` as a complex Lorentz tensor.", + "snippet": "abbrev dualLeftLeftUnit : ℂT[.downL, .upL]", + "line": 40 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "abbrev", + "name": "complexLorentzTensor.leftDualLeftUnit", + "docstring": "The unit `δᵃₐ` as a complex Lorentz tensor.", + "snippet": "abbrev leftDualLeftUnit : ℂT[.upL, .downL]", + "line": 43 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "abbrev", + "name": "complexLorentzTensor.dualRightRightUnit", + "docstring": "The unit `δ_{dot a}^{dot a}` as a complex Lorentz tensor.", + "snippet": "abbrev dualRightRightUnit : ℂT[.downR, .upR]", + "line": 46 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "abbrev", + "name": "complexLorentzTensor.rightDualRightUnit", + "docstring": "The unit `δ^{dot a}_{dot a}` as a complex Lorentz tensor.", + "snippet": "abbrev rightDualRightUnit : ℂT[.upR, .downR]", + "line": 49 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualLeftLeftUnit_eq_fromConstPair", + "docstring": null, + "snippet": "lemma dualLeftLeftUnit_eq_fromConstPair : δL' = fromConstPair Fermion.dualLeftLeftUnit", + "line": 95 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.leftDualLeftUnit_eq_fromConstPair", + "docstring": null, + "snippet": "lemma leftDualLeftUnit_eq_fromConstPair : δL = fromConstPair Fermion.leftDualLeftUnit", + "line": 99 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualRightRightUnit_eq_fromConstPair", + "docstring": null, + "snippet": "lemma dualRightRightUnit_eq_fromConstPair : δR' = fromConstPair Fermion.dualRightRightUnit", + "line": 103 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.rightDualRightUnit_eq_fromConstPair", + "docstring": null, + "snippet": "lemma rightDualRightUnit_eq_fromConstPair : δR = fromConstPair Fermion.rightDualRightUnit", + "line": 107 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualLeftLeftUnit_eq_fromPairT", + "docstring": null, + "snippet": "lemma dualLeftLeftUnit_eq_fromPairT : δL' = fromPairT (Fermion.dualLeftLeftUnitVal)", + "line": 127 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.leftDualLeftUnit_eq_fromPairT", + "docstring": null, + "snippet": "lemma leftDualLeftUnit_eq_fromPairT : δL = fromPairT (Fermion.leftDualLeftUnitVal)", + "line": 132 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualRightRightUnit_eq_fromPairT", + "docstring": null, + "snippet": "lemma dualRightRightUnit_eq_fromPairT : δR' = fromPairT (Fermion.dualRightRightUnitVal)", + "line": 137 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.rightDualRightUnit_eq_fromPairT", + "docstring": null, + "snippet": "lemma rightDualRightUnit_eq_fromPairT : δR = fromPairT (Fermion.rightDualRightUnitVal)", + "line": 142 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualLeftLeftUnit_eq_dualLeftBasis_leftBasis", + "docstring": null, + "snippet": "lemma dualLeftLeftUnit_eq_dualLeftBasis_leftBasis : δL' =\n ∑ i, fromPairT (dualLeftBasis i ⊗ₜ[ℂ] leftBasis i)", + "line": 180 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.leftDualLeftUnit_eq_leftBasis_dualLeftBasis", + "docstring": null, + "snippet": "lemma leftDualLeftUnit_eq_leftBasis_dualLeftBasis : δL =\n ∑ i, fromPairT (leftBasis i ⊗ₜ[ℂ] dualLeftBasis i)", + "line": 186 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualRightRightUnit_eq_dualRightBasis_rightBasis", + "docstring": null, + "snippet": "lemma dualRightRightUnit_eq_dualRightBasis_rightBasis : δR' =\n ∑ i, fromPairT (dualRightBasis i ⊗ₜ[ℂ] rightBasis i)", + "line": 192 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.rightDualRightUnit_eq_rightBasis_dualRightBasis", + "docstring": null, + "snippet": "lemma rightDualRightUnit_eq_rightBasis_dualRightBasis : δR =\n ∑ i, fromPairT (rightBasis i ⊗ₜ[ℂ] dualRightBasis i)", + "line": 198 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualLeftLeftUnit_eq_basis", + "docstring": null, + "snippet": "lemma dualLeftLeftUnit_eq_basis : δL' =\n ∑ i, Tensor.basis (S := complexLorentzTensor)\n ![Color.downL, Color.upL] (fun | 0 => i | 1 => i)", + "line": 231 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.leftDualLeftUnit_eq_basis", + "docstring": null, + "snippet": "lemma leftDualLeftUnit_eq_basis : δL =\n ∑ i, Tensor.basis (S := complexLorentzTensor)\n ![Color.upL, Color.downL] (fun | 0 => i | 1 => i)", + "line": 242 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualRightRightUnit_eq_basis", + "docstring": null, + "snippet": "lemma dualRightRightUnit_eq_basis : δR' =\n ∑ i, Tensor.basis (S := complexLorentzTensor)\n ![Color.downR, Color.upR] (fun | 0 => i | 1 => i)", + "line": 253 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.rightDualRightUnit_eq_basis", + "docstring": null, + "snippet": "lemma rightDualRightUnit_eq_basis : δR =\n ∑ i, Tensor.basis (S := complexLorentzTensor)\n ![Color.upR, Color.downR] (fun | 0 => i | 1 => i)", + "line": 264 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualLeftLeftUnit_eq_ofRat", + "docstring": null, + "snippet": "lemma dualLeftLeftUnit_eq_ofRat : δL' = ofRat fun f =>\n if f 0 = f 1 then 1 else 0", + "line": 301 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.leftDualLeftUnit_eq_ofRat", + "docstring": null, + "snippet": "lemma leftDualLeftUnit_eq_ofRat : δL = ofRat fun f =>\n if f 0 = f 1 then 1 else 0", + "line": 311 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualRightRightUnit_eq_ofRat", + "docstring": null, + "snippet": "lemma dualRightRightUnit_eq_ofRat : δR' = ofRat fun f =>\n if f 0 = f 1 then 1 else 0", + "line": 321 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.rightDualRightUnit_eq_ofRat", + "docstring": null, + "snippet": "lemma rightDualRightUnit_eq_ofRat : δR = ofRat fun f =>\n if f 0 = f 1 then 1 else 0", + "line": 331 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.actionT_dualLeftLeftUnit", + "docstring": "The tensor `dualLeftLeftUnit` is invariant under the action of `SL(2,ℂ)`.", + "snippet": "lemma actionT_dualLeftLeftUnit (g : SL(2,ℂ)) : g • δL' = δL'", + "line": 359 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.actionT_leftDualLeftUnit", + "docstring": "The tensor `leftDualLeftUnit` is invariant under the action of `SL(2,ℂ)`.", + "snippet": "lemma actionT_leftDualLeftUnit (g : SL(2,ℂ)) : g • δL = δL", + "line": 364 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.actionT_dualRightRightUnit", + "docstring": "The tensor `dualRightRightUnit` is invariant under the action of `SL(2,ℂ)`.", + "snippet": "lemma actionT_dualRightRightUnit (g : SL(2,ℂ)) : g • δR' = δR'", + "line": 369 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Basic.lean", + "kind": "lemma", + "name": "complexLorentzTensor.actionT_rightDualRightUnit", + "docstring": "The tensor `rightDualRightUnit` is invariant under the action of `SL(2,ℂ)`.", + "snippet": "lemma actionT_rightDualRightUnit (g : SL(2,ℂ)) : g • δR = δR", + "line": 374 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComplexTensor/Units/Symm.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Symm.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualLeftLeftUnit_symm", + "docstring": "Swapping indices of `dualLeftLeftUnit` returns\n `leftDualLeftUnit`: `{δL' | α α' = δL | α' α}ᵀ`.", + "snippet": "lemma dualLeftLeftUnit_symm : {δL' | α α' = δL | α' α}ᵀ", + "line": 41 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Symm.lean", + "kind": "lemma", + "name": "complexLorentzTensor.leftDualLeftUnit_symm", + "docstring": "Swapping indices of `leftDualLeftUnit` returns\n `dualLeftLeftUnit`: `{δL | α α' = δL' | α' α}ᵀ`.", + "snippet": "lemma leftDualLeftUnit_symm : {δL | α α' = δL' | α' α}ᵀ", + "line": 47 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Symm.lean", + "kind": "lemma", + "name": "complexLorentzTensor.dualRightRightUnit_symm", + "docstring": "Swapping indices of `dualRightRightUnit` returns `rightDualRightUnit`:\n`{δR' | β β' = δR | β' β}ᵀ`.", + "snippet": "lemma dualRightRightUnit_symm : {δR' | β β' = δR | β' β}ᵀ", + "line": 54 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Units/Symm.lean", + "kind": "lemma", + "name": "complexLorentzTensor.rightDualRightUnit_symm", + "docstring": "Swapping indices of `rightDualRightUnit` returns `dualRightRightUnit`:\n`{δR | β β' = δR' | β' β}ᵀ`.", + "snippet": "lemma rightDualRightUnit_symm : {δR | β β' = δR' | β' β}ᵀ", + "line": 61 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Basic.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Basic.lean", + "kind": "lemma", + "name": "Lorentz.CoℂModule.SL2CRep_val", + "docstring": null, + "snippet": "lemma CoℂModule.SL2CRep_val (M : SL(2,ℂ)) (v : CoℂModule) :\n ((CoℂModule.SL2CRep M) v).val =\n (LorentzGroup.toComplex (SL2C.toLorentzGroup M))⁻¹ᵀ *ᵥ v.val", + "line": 113 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Basic.lean", + "kind": "def", + "name": "Lorentz.inclCoRealLorentz", + "docstring": "The semilinear map including real Lorentz co-vectors into complex covariant\n Lorentz vectors.", + "snippet": "def inclCoRealLorentz : CoMod 3 →ₛₗ[Complex.ofRealHom] CoℂModule where", + "line": 216 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Basic.lean", + "kind": "lemma", + "name": "Lorentz.inclCoRealLorentz_val", + "docstring": null, + "snippet": "lemma inclCoRealLorentz_val (v : CoMod 3) :\n (inclCoRealLorentz v).val = ofRealHom ∘ v.toFin1dℝ", + "line": 229 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Basic.lean", + "kind": "lemma", + "name": "Lorentz.complexCoBasis_of_real", + "docstring": null, + "snippet": "lemma complexCoBasis_of_real (i : Fin 1 ⊕ Fin 3) :\n (complexCoBasis i) = inclCoRealLorentz (CoMod.stdBasis i)", + "line": 232 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Basic.lean", + "kind": "lemma", + "name": "Lorentz.inclCoRealLorentz_ρ", + "docstring": null, + "snippet": "lemma inclCoRealLorentz_ρ (M : SL(2, ℂ)) (v : CoMod 3) :\n (CoℂModule.SL2CRep M) (inclCoRealLorentz v) =\n inclCoRealLorentz ((Co 3).ρ (SL2C.toLorentzGroup M) v)", + "line": 247 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Modules.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Modules.lean", + "kind": "lemma", + "name": "Lorentz.CoℂModule.ext", + "docstring": null, + "snippet": "@[ext]\nlemma ext (ψ ψ' : CoℂModule) (h : ψ.val = ψ'.val) : ψ = ψ'", + "line": 127 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Modules.lean", + "kind": "lemma", + "name": "Lorentz.CoℂModule.val_add", + "docstring": null, + "snippet": "@[simp]\nlemma val_add (ψ ψ' : CoℂModule) : (ψ + ψ').val = ψ.val + ψ'.val", + "line": 134 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Vector/Pre/Modules.lean", + "kind": "lemma", + "name": "Lorentz.CoℂModule.val_smul", + "docstring": null, + "snippet": "@[simp]\nlemma val_smul (r : ℂ) (ψ : CoℂModule) : (r • ψ).val = r • ψ.val", + "line": 137 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "def", + "name": "Fermion.leftHandedRep", + "docstring": "The vector space ℂ^2 carrying the fundamental representation of SL(2,C).\n In index notation corresponds to a Weyl fermion with indices ψ^a.", + "snippet": "def leftHandedRep : Representation ℂ SL(2,ℂ) LeftHandedWeyl where", + "line": 39 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "def", + "name": "Fermion.dualLeftHandedRep", + "docstring": "The vector space ℂ^2 carrying the representation of SL(2,C) given by\n M → (M⁻¹)ᵀ. In index notation corresponds to a Weyl fermion with indices ψ_a.", + "snippet": "def dualLeftHandedRep : Representation ℂ SL(2,ℂ) DualLeftHandedWeyl where", + "line": 77 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "def", + "name": "Fermion.dualLeftBasis", + "docstring": "The standard basis on dual-left-handed Weyl fermions.", + "snippet": "def dualLeftBasis : Basis (Fin 2) ℂ DualLeftHandedWeyl", + "line": 99 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "lemma", + "name": "Fermion.dualLeftBasis_toFin2ℂ", + "docstring": null, + "snippet": "@[simp]\nlemma dualLeftBasis_toFin2ℂ (i : Fin 2) : (dualLeftBasis i).toFin2ℂ = Pi.single i 1", + "line": 103 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "lemma", + "name": "Fermion.dualLeftBasis_ρ_apply", + "docstring": null, + "snippet": "@[simp]\nlemma dualLeftBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 2) :\n (LinearMap.toMatrix dualLeftBasis dualLeftBasis) (dualLeftHandedRep M) i j = (M.1⁻¹)ᵀ i j", + "line": 108 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "def", + "name": "Fermion.rightHandedRep", + "docstring": "The vector space ℂ^2 carrying the conjugate representation of SL(2,C).\n In index notation corresponds to a Weyl fermion with indices ψ^{dot a}.", + "snippet": "def rightHandedRep : Representation ℂ SL(2,ℂ) RightHandedWeyl where", + "line": 117 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "def", + "name": "Fermion.dualRightHandedRep", + "docstring": "The vector space ℂ^2 carrying the representation of SL(2,C) given by\n M → (M⁻¹)^†.\n In index notation this corresponds to a Weyl fermion with index `ψ_{dot a}`.", + "snippet": "def dualRightHandedRep : Representation ℂ SL(2,ℂ) DualRightHandedWeyl where", + "line": 155 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "def", + "name": "Fermion.dualRightBasis", + "docstring": "The standard basis on dual-right-handed Weyl fermions.", + "snippet": "def dualRightBasis : Basis (Fin 2) ℂ DualRightHandedWeyl", + "line": 177 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "lemma", + "name": "Fermion.dualRightBasis_toFin2ℂ", + "docstring": null, + "snippet": "@[simp]\nlemma dualRightBasis_toFin2ℂ (i : Fin 2) : (dualRightBasis i).toFin2ℂ = Pi.single i 1", + "line": 181 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "lemma", + "name": "Fermion.dualRightBasis_ρ_apply", + "docstring": null, + "snippet": "@[simp]\nlemma dualRightBasis_ρ_apply (M : SL(2,ℂ)) (i j : Fin 2) :\n (LinearMap.toMatrix dualRightBasis dualRightBasis) (dualRightHandedRep M) i j =\n ((M.1⁻¹).conjTranspose) i j", + "line": 186 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "def", + "name": "Fermion.leftHandedToDual", + "docstring": "The morphism between the representation `leftHanded` and the representation\n `dualLeftHanded` defined by multiplying an element of\n `leftHanded` by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`.", + "snippet": "def leftHandedToDual : leftHandedRep.IntertwiningMap dualLeftHandedRep where", + "line": 203 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "lemma", + "name": "Fermion.leftHandedToDual_hom_apply", + "docstring": null, + "snippet": "lemma leftHandedToDual_hom_apply (ψ : LeftHandedWeyl) :\n leftHandedToDual ψ =\n DualLeftHandedWeyl.toFin2ℂEquiv.symm (!![0, 1; -1, 0] *ᵥ ψ.toFin2ℂ)", + "line": 224 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "def", + "name": "Fermion.leftHandedDualTo", + "docstring": "The morphism from `dualLeftHanded` to\n `leftHanded` defined by multiplying an element of\n DualLeftHandedWeyl by the matrix `εₐ₁ₐ₂ = !![0, -1; 1, 0]`.", + "snippet": "def leftHandedDualTo : dualLeftHandedRep.IntertwiningMap leftHandedRep where", + "line": 231 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "lemma", + "name": "Fermion.leftHandedDualTo_hom_apply", + "docstring": null, + "snippet": "lemma leftHandedDualTo_hom_apply (ψ : DualLeftHandedWeyl) :\n leftHandedDualTo ψ =\n LeftHandedWeyl.toFin2ℂEquiv.symm (!![0, -1; 1, 0] *ᵥ ψ.toFin2ℂ)", + "line": 255 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "def", + "name": "Fermion.leftHandedDualEquiv", + "docstring": "The equivalence between the representation `leftHanded` and the representation\n `dualLeftHanded` defined by multiplying an element of\n `leftHanded` by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`.", + "snippet": "def leftHandedDualEquiv : leftHandedRep.Equiv dualLeftHandedRep", + "line": 262 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "lemma", + "name": "Fermion.leftHandedDualEquiv_hom_hom_apply", + "docstring": "`leftHandedDualEquiv` acting on an element `ψ : leftHanded` corresponds\n to multiplying `ψ` by the matrix `!![0, 1; -1, 0]`.", + "snippet": "lemma leftHandedDualEquiv_hom_hom_apply (ψ : LeftHandedWeyl) :\n leftHandedDualEquiv ψ =\n DualLeftHandedWeyl.toFin2ℂEquiv.symm (!![0, 1; -1, 0] *ᵥ ψ.toFin2ℂ)", + "line": 283 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "lemma", + "name": "Fermion.leftHandedDualEquiv_inv_hom_apply", + "docstring": "The inverse of `leftHandedDualEquiv` acting on an element`ψ : dualLeftHanded` corresponds\n to multiplying `ψ` by the matrix `!![0, -1; 1, 0]`.", + "snippet": "lemma leftHandedDualEquiv_inv_hom_apply (ψ : DualLeftHandedWeyl) :\n leftHandedDualEquiv.symm ψ =\n LeftHandedWeyl.toFin2ℂEquiv.symm (!![0, -1; 1, 0] *ᵥ ψ.toFin2ℂ)", + "line": 289 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "informal_definition", + "name": "Fermion.rightHandedWeylDualEquiv", + "docstring": "The linear equivalence between `rightHandedWeyl` and `DualRightHandedWeyl` given by multiplying\nan element of `rightHandedWeyl` by the matrix `εᵃ⁰ᵃ¹ = !![0, 1; -1, 0]]`.", + "snippet": "informal_definition rightHandedWeylDualEquiv where\n deps := [``rightHandedRep, ``dualRightHandedRep]\n tag := \"6VZR4\"", + "line": 296 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Basic.lean", + "kind": "informal_lemma", + "name": "Fermion.rightHandedWeylDualEquiv_equivariant", + "docstring": "The linear equivalence `rightHandedWeylDualEquiv` is equivariant with respect to the action of\n`SL(2,C)` on `rightHandedWeyl` and `DualRightHandedWeyl`.", + "snippet": "informal_lemma rightHandedWeylDualEquiv_equivariant where\n deps := [``rightHandedWeylDualEquiv]\n tag := \"6VZSG\"", + "line": 303 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "def", + "name": "Fermion.leftDualBi", + "docstring": "The bi-linear map corresponding to contraction of a left-handed Weyl fermion with a\n dual-left-handed Weyl fermion.", + "snippet": "def leftDualBi : LeftHandedWeyl →ₗ[ℂ] DualLeftHandedWeyl →ₗ[ℂ] ℂ where", + "line": 36 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "def", + "name": "Fermion.dualLeftBi", + "docstring": "The bi-linear map corresponding to contraction of a dual-left-handed Weyl fermion with a\n left-handed Weyl fermion.", + "snippet": "def dualLeftBi : DualLeftHandedWeyl →ₗ[ℂ] LeftHandedWeyl →ₗ[ℂ] ℂ where", + "line": 60 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "def", + "name": "Fermion.rightDualBi", + "docstring": "The bi-linear map corresponding to contraction of a right-handed Weyl fermion with a\n dual-right-handed Weyl fermion.", + "snippet": "def rightDualBi : RightHandedWeyl →ₗ[ℂ] DualRightHandedWeyl →ₗ[ℂ] ℂ where", + "line": 83 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "def", + "name": "Fermion.dualRightBi", + "docstring": "The bi-linear map corresponding to contraction of a dual-right-handed Weyl fermion with a\n right-handed Weyl fermion.", + "snippet": "def dualRightBi : DualRightHandedWeyl →ₗ[ℂ] RightHandedWeyl →ₗ[ℂ] ℂ where", + "line": 107 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "def", + "name": "Fermion.leftDualContraction", + "docstring": "The linear map from leftHandedWeyl ⊗ DualLeftHandedWeyl to ℂ given by\n summing over components of leftHandedWeyl and DualLeftHandedWeyl in the\n standard basis (i.e. the dot product).\n Physically, the contraction of a left-handed Weyl fermion with a dual-left-handed Weyl fermion.\n In index notation this is ψ^a φ_a.", + "snippet": "def leftDualContraction : (leftHandedRep.tprod dualLeftHandedRep).IntertwiningMap\n (Representation.trivial ℂ SL(2,ℂ) ℂ) where", + "line": 133 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "lemma", + "name": "Fermion.leftDualContraction_hom_tmul", + "docstring": null, + "snippet": "lemma leftDualContraction_hom_tmul (ψ : LeftHandedWeyl)\n (φ : DualLeftHandedWeyl) :\n leftDualContraction (ψ ⊗ₜ φ) = ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ", + "line": 141 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "lemma", + "name": "Fermion.leftDualContraction_basis", + "docstring": null, + "snippet": "lemma leftDualContraction_basis (i j : Fin 2) :\n leftDualContraction (leftBasis i ⊗ₜ dualLeftBasis j) = if i.1 = j.1 then (1 : ℂ) else 0", + "line": 146 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "def", + "name": "Fermion.dualLeftContraction", + "docstring": "The linear map from DualLeftHandedWeyl ⊗ leftHandedWeyl to ℂ given by\n summing over components of DualLeftHandedWeyl and leftHandedWeyl in the\n standard basis (i.e. the dot product).\n Physically, the contraction of a dual-left-handed Weyl fermion with a left-handed Weyl fermion.\n In index notation this is φ_a ψ^a.", + "snippet": "def dualLeftContraction : (dualLeftHandedRep.tprod leftHandedRep).IntertwiningMap\n (Representation.trivial ℂ SL(2,ℂ) ℂ) where", + "line": 160 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "lemma", + "name": "Fermion.dualLeftContraction_hom_tmul", + "docstring": null, + "snippet": "lemma dualLeftContraction_hom_tmul (φ : DualLeftHandedWeyl) (ψ : LeftHandedWeyl) :\n dualLeftContraction (φ ⊗ₜ ψ) = φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ", + "line": 168 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "lemma", + "name": "Fermion.dualLeftContraction_basis", + "docstring": null, + "snippet": "lemma dualLeftContraction_basis (i j : Fin 2) :\n dualLeftContraction (dualLeftBasis i ⊗ₜ leftBasis j) = if i.1 = j.1 then (1 : ℂ) else 0", + "line": 172 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "def", + "name": "Fermion.rightDualContraction", + "docstring": "The linear map from `rightHandedWeyl ⊗ DualRightHandedWeyl` to `ℂ` given by\n summing over components of `rightHandedWeyl` and `DualRightHandedWeyl` in the\n standard basis (i.e. the dot product).\n The contraction of a right-handed Weyl fermion with a left-handed Weyl fermion.\n In index notation this is `ψ^{dot a} φ_{dot a}`.", + "snippet": "def rightDualContraction : (rightHandedRep.tprod dualRightHandedRep).IntertwiningMap\n (Representation.trivial ℂ SL(2,ℂ) ℂ) where", + "line": 188 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "lemma", + "name": "Fermion.rightDualContraction_hom_tmul", + "docstring": null, + "snippet": "lemma rightDualContraction_hom_tmul (ψ : RightHandedWeyl)\n (φ : DualRightHandedWeyl) :\n rightDualContraction (ψ ⊗ₜ φ) = ψ.toFin2ℂ ⬝ᵥ φ.toFin2ℂ", + "line": 207 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "lemma", + "name": "Fermion.rightDualContraction_basis", + "docstring": null, + "snippet": "lemma rightDualContraction_basis (i j : Fin 2) :\n rightDualContraction (rightBasis i ⊗ₜ dualRightBasis j) =\n if i.1 = j.1 then (1 : ℂ) else 0", + "line": 212 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "def", + "name": "Fermion.dualRightContraction", + "docstring": "The linear map from DualRightHandedWeyl ⊗ rightHandedWeyl to ℂ given by\n summing over components of DualRightHandedWeyl and rightHandedWeyl in the\n standard basis (i.e. the dot product).\n The contraction of a right-handed Weyl fermion with a left-handed Weyl fermion.\n In index notation this is φ_{dot a} ψ^{dot a}.", + "snippet": "def dualRightContraction : (dualRightHandedRep.tprod rightHandedRep).IntertwiningMap\n (Representation.trivial ℂ SL(2,ℂ) ℂ) where", + "line": 229 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "lemma", + "name": "Fermion.dualRightContraction_hom_tmul", + "docstring": null, + "snippet": "lemma dualRightContraction_hom_tmul (φ : DualRightHandedWeyl)\n (ψ : RightHandedWeyl) :\n dualRightContraction (φ ⊗ₜ ψ) = φ.toFin2ℂ ⬝ᵥ ψ.toFin2ℂ", + "line": 248 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "lemma", + "name": "Fermion.dualRightContraction_basis", + "docstring": null, + "snippet": "lemma dualRightContraction_basis (i j : Fin 2) :\n dualRightContraction (dualRightBasis i ⊗ₜ rightBasis j) =\n if i.1 = j.1 then (1 : ℂ) else 0", + "line": 253 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "lemma", + "name": "Fermion.leftDualContraction_tmul_symm", + "docstring": null, + "snippet": "lemma leftDualContraction_tmul_symm (ψ : LeftHandedWeyl) (φ : DualLeftHandedWeyl) :\n leftDualContraction (ψ ⊗ₜ[ℂ] φ) = dualLeftContraction (φ ⊗ₜ[ℂ] ψ)", + "line": 269 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "lemma", + "name": "Fermion.dualLeftContraction_tmul_symm", + "docstring": null, + "snippet": "lemma dualLeftContraction_tmul_symm (φ : DualLeftHandedWeyl) (ψ : LeftHandedWeyl) :\n dualLeftContraction (φ ⊗ₜ[ℂ] ψ) = leftDualContraction (ψ ⊗ₜ[ℂ] φ)", + "line": 273 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "lemma", + "name": "Fermion.rightDualContraction_tmul_symm", + "docstring": null, + "snippet": "lemma rightDualContraction_tmul_symm (ψ : RightHandedWeyl) (φ : DualRightHandedWeyl) :\n rightDualContraction (ψ ⊗ₜ[ℂ] φ) = dualRightContraction (φ ⊗ₜ[ℂ] ψ)", + "line": 277 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Contraction.lean", + "kind": "lemma", + "name": "Fermion.dualRightContraction_tmul_symm", + "docstring": null, + "snippet": "lemma dualRightContraction_tmul_symm (φ : DualRightHandedWeyl) (ψ : RightHandedWeyl) :\n dualRightContraction (φ ⊗ₜ[ℂ] ψ) = rightDualContraction (ψ ⊗ₜ[ℂ] φ)", + "line": 281 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean", + "kind": "lemma", + "name": "Fermion.leftMetricVal_expand_tmul'", + "docstring": null, + "snippet": "lemma leftMetricVal_expand_tmul' : leftMetricVal = leftBasis 1 ⊗ₜ[ℂ] leftBasis 0\n - leftBasis 0 ⊗ₜ[ℂ] leftBasis 1", + "line": 87 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean", + "kind": "def", + "name": "Fermion.dualLeftMetricVal", + "docstring": "The metric `εₐₐ` as an element of `(dualLeftHanded ⊗ dualLeftHanded).V`.", + "snippet": "def dualLeftMetricVal : (DualLeftHandedWeyl ⊗[ℂ] DualLeftHandedWeyl)", + "line": 120 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean", + "kind": "lemma", + "name": "Fermion.dualLeftMetricVal_expand_tmul", + "docstring": "Expansion of `dualLeftMetricVal` into the left basis.", + "snippet": "lemma dualLeftMetricVal_expand_tmul : dualLeftMetricVal =\n dualLeftBasis 0 ⊗ₜ[ℂ] dualLeftBasis 1 - dualLeftBasis 1 ⊗ₜ[ℂ] dualLeftBasis 0", + "line": 125 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean", + "kind": "def", + "name": "Fermion.dualLeftMetric", + "docstring": "The metric `εₐₐ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ dualLeftHanded ⊗ dualLeftHanded`,\n making manifest its invariance under the action of `SL(2,ℂ)`.", + "snippet": "def dualLeftMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap\n (dualLeftHandedRep.tprod dualLeftHandedRep) where", + "line": 135 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean", + "kind": "lemma", + "name": "Fermion.dualLeftMetric_apply_one", + "docstring": null, + "snippet": "lemma dualLeftMetric_apply_one : dualLeftMetric (1 : ℂ) = dualLeftMetricVal", + "line": 158 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean", + "kind": "lemma", + "name": "Fermion.rightMetricVal_expand_tmul'", + "docstring": null, + "snippet": "lemma rightMetricVal_expand_tmul' : rightMetricVal = rightBasis 1 ⊗ₜ[ℂ] rightBasis 0\n - rightBasis 0 ⊗ₜ[ℂ] rightBasis 1", + "line": 177 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean", + "kind": "def", + "name": "Fermion.dualRightMetricVal", + "docstring": "The metric `ε_{dot a}_{dot a}` as an element of `(dualRightHanded ⊗ dualRightHanded).V`.", + "snippet": "def dualRightMetricVal : DualRightHandedWeyl ⊗[ℂ] DualRightHandedWeyl", + "line": 217 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean", + "kind": "lemma", + "name": "Fermion.dualRightMetricVal_expand_tmul", + "docstring": "Expansion of `rightMetricVal` into the left basis.", + "snippet": "lemma dualRightMetricVal_expand_tmul : dualRightMetricVal =\n dualRightBasis 0 ⊗ₜ[ℂ] dualRightBasis 1 - dualRightBasis 1 ⊗ₜ[ℂ] dualRightBasis 0", + "line": 222 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean", + "kind": "def", + "name": "Fermion.dualRightMetric", + "docstring": "The metric `ε_{dot a}_{dot a}` as a morphism\n `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ dualRightHanded ⊗ dualRightHanded`,\n making manifest its invariance under the action of `SL(2,ℂ)`.", + "snippet": "def dualRightMetric : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap\n (dualRightHandedRep.tprod dualRightHandedRep) where", + "line": 233 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean", + "kind": "lemma", + "name": "Fermion.dualRightMetric_apply_one", + "docstring": null, + "snippet": "lemma dualRightMetric_apply_one : dualRightMetric (1 : ℂ) = dualRightMetricVal", + "line": 265 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean", + "kind": "lemma", + "name": "Fermion.leftDualContraction_apply_metric", + "docstring": null, + "snippet": "lemma leftDualContraction_apply_metric :\n (TensorProduct.comm ℂ _ _ <|\n (TensorProduct.lid ℂ _).lTensor _ <|\n (leftDualContraction.toLinearMap.rTensor _).lTensor _ <|\n (TensorProduct.assoc ℂ _ _ _).symm.toLinearMap.lTensor _<|\n TensorProduct.assoc ℂ _ _ (_ ⊗[ℂ] _) <|\n (leftMetric 1) ⊗ₜ[ℂ] (dualLeftMetric 1)) = dualLeftLeftUnit (1 : ℂ)", + "line": 276 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean", + "kind": "lemma", + "name": "Fermion.dualLeftContraction_apply_metric", + "docstring": null, + "snippet": "lemma dualLeftContraction_apply_metric :\n (TensorProduct.comm ℂ _ _ <|\n (TensorProduct.lid ℂ _).lTensor _ <|\n (dualLeftContraction.toLinearMap.rTensor _).lTensor _ <|\n (TensorProduct.assoc ℂ _ _ _).symm.toLinearMap.lTensor _<|\n TensorProduct.assoc ℂ _ _ (_ ⊗[ℂ] _) <|\n (dualLeftMetric 1) ⊗ₜ[ℂ] (leftMetric 1)) = leftDualLeftUnit (1 : ℂ)", + "line": 295 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean", + "kind": "lemma", + "name": "Fermion.rightDualContraction_apply_metric", + "docstring": null, + "snippet": "lemma rightDualContraction_apply_metric :\n (TensorProduct.comm ℂ _ _ <|\n (TensorProduct.lid ℂ _).lTensor _ <|\n (rightDualContraction.toLinearMap.rTensor _).lTensor _ <|\n (TensorProduct.assoc ℂ _ _ _).symm.toLinearMap.lTensor _<|\n TensorProduct.assoc ℂ _ _ (_ ⊗[ℂ] _) <|\n (rightMetric 1) ⊗ₜ[ℂ] (dualRightMetric 1)) = dualRightRightUnit (1 : ℂ)", + "line": 314 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Metric.lean", + "kind": "lemma", + "name": "Fermion.dualRightContraction_apply_metric", + "docstring": null, + "snippet": "lemma dualRightContraction_apply_metric :\n (TensorProduct.comm ℂ _ _ <|\n (TensorProduct.lid ℂ _).lTensor _ <|\n (dualRightContraction.toLinearMap.rTensor _).lTensor _ <|\n (TensorProduct.assoc ℂ _ _ _).symm.toLinearMap.lTensor _<|\n TensorProduct.assoc ℂ _ _ (_ ⊗[ℂ] _) <|\n (dualRightMetric 1) ⊗ₜ[ℂ] (rightMetric 1)) = rightDualRightUnit (1 : ℂ)", + "line": 334 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "structure", + "name": "Fermion.LeftHandedWeyl", + "docstring": "The module in which left handed fermions live. This is equivalent to `Fin 2 → ℂ`.", + "snippet": "structure LeftHandedWeyl where\n val : Fin 2 → ℂ", + "line": 36 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "def", + "name": "Fermion.LeftHandedWeyl.toFin2ℂFun", + "docstring": "The equivalence between `LeftHandedWeyl` and `Fin 2 → ℂ`.", + "snippet": "def toFin2ℂFun : LeftHandedWeyl ≃ (Fin 2 → ℂ) where", + "line": 43 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "instance", + "name": "Fermion.LeftHandedWeyl.«instance@c5c29288»", + "docstring": "The instance of `AddCommMonoid` on `LeftHandedWeyl` defined via its equivalence\n with `Fin 2 → ℂ`.", + "snippet": "instance : AddCommMonoid LeftHandedWeyl", + "line": 51 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "instance", + "name": "Fermion.LeftHandedWeyl.«instance@c3de7b53»", + "docstring": "The instance of `AddCommGroup` on `LeftHandedWeyl` defined via its equivalence\n with `Fin 2 → ℂ`.", + "snippet": "instance : AddCommGroup LeftHandedWeyl", + "line": 55 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "instance", + "name": "Fermion.LeftHandedWeyl.«instance@db8c5fc4»", + "docstring": "The instance of `Module` on `LeftHandedWeyl` defined via its equivalence\n with `Fin 2 → ℂ`.", + "snippet": "instance : Module ℂ LeftHandedWeyl", + "line": 59 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "def", + "name": "Fermion.LeftHandedWeyl.toFin2ℂEquiv", + "docstring": "The linear equivalence between `LeftHandedWeyl` and `(Fin 2 → ℂ)`.", + "snippet": "@[simps!]\ndef toFin2ℂEquiv : LeftHandedWeyl ≃ₗ[ℂ] (Fin 2 → ℂ) where", + "line": 63 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "abbrev", + "name": "Fermion.LeftHandedWeyl.toFin2ℂ", + "docstring": "The underlying element of `Fin 2 → ℂ` of a element in `LeftHandedWeyl` defined\n through the linear equivalence `toFin2ℂEquiv`.", + "snippet": "abbrev toFin2ℂ (ψ : LeftHandedWeyl)", + "line": 73 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "structure", + "name": "Fermion.DualLeftHandedWeyl", + "docstring": "The module in which dual-left handed fermions live. This is equivalent to `Fin 2 → ℂ`.", + "snippet": "structure DualLeftHandedWeyl where\n val : Fin 2 → ℂ", + "line": 80 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "def", + "name": "Fermion.DualLeftHandedWeyl.toFin2ℂFun", + "docstring": "The equivalence between `DualLeftHandedWeyl` and `Fin 2 → ℂ`.", + "snippet": "def toFin2ℂFun : DualLeftHandedWeyl ≃ (Fin 2 → ℂ) where", + "line": 87 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "instance", + "name": "Fermion.DualLeftHandedWeyl.«instance@ed87b73f»", + "docstring": "The instance of `AddCommMonoid` on `DualLeftHandedWeyl` defined via its equivalence\n with `Fin 2 → ℂ`.", + "snippet": "instance : AddCommMonoid DualLeftHandedWeyl", + "line": 95 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "instance", + "name": "Fermion.DualLeftHandedWeyl.«instance@61470bec»", + "docstring": "The instance of `AddCommGroup` on `DualLeftHandedWeyl` defined via its equivalence\n with `Fin 2 → ℂ`.", + "snippet": "instance : AddCommGroup DualLeftHandedWeyl", + "line": 99 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "instance", + "name": "Fermion.DualLeftHandedWeyl.«instance@c5aed84d»", + "docstring": "The instance of `Module` on `DualLeftHandedWeyl` defined via its equivalence\n with `Fin 2 → ℂ`.", + "snippet": "instance : Module ℂ DualLeftHandedWeyl", + "line": 103 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "def", + "name": "Fermion.DualLeftHandedWeyl.toFin2ℂEquiv", + "docstring": "The linear equivalence between `DualLeftHandedWeyl` and `(Fin 2 → ℂ)`.", + "snippet": "@[simps!]\ndef toFin2ℂEquiv : DualLeftHandedWeyl ≃ₗ[ℂ] (Fin 2 → ℂ) where", + "line": 107 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "abbrev", + "name": "Fermion.DualLeftHandedWeyl.toFin2ℂ", + "docstring": "The underlying element of `Fin 2 → ℂ` of a element in `DualLeftHandedWeyl` defined\n through the linear equivalence `toFin2ℂEquiv`.", + "snippet": "abbrev toFin2ℂ (ψ : DualLeftHandedWeyl)", + "line": 117 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "structure", + "name": "Fermion.RightHandedWeyl", + "docstring": "The module in which right handed fermions live. This is equivalent to `Fin 2 → ℂ`.", + "snippet": "structure RightHandedWeyl where\n val : Fin 2 → ℂ", + "line": 125 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "def", + "name": "Fermion.RightHandedWeyl.toFin2ℂFun", + "docstring": "The equivalence between `RightHandedWeyl` and `Fin 2 → ℂ`.", + "snippet": "def toFin2ℂFun : RightHandedWeyl ≃ (Fin 2 → ℂ) where", + "line": 132 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "instance", + "name": "Fermion.RightHandedWeyl.«instance@97ccc5aa»", + "docstring": "The instance of `AddCommMonoid` on `RightHandedWeyl` defined via its equivalence\n with `Fin 2 → ℂ`.", + "snippet": "instance : AddCommMonoid RightHandedWeyl", + "line": 140 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "instance", + "name": "Fermion.RightHandedWeyl.«instance@4a60c666»", + "docstring": "The instance of `AddCommGroup` on `RightHandedWeyl` defined via its equivalence\n with `Fin 2 → ℂ`.", + "snippet": "instance : AddCommGroup RightHandedWeyl", + "line": 144 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "instance", + "name": "Fermion.RightHandedWeyl.«instance@753cf914»", + "docstring": "The instance of `Module` on `RightHandedWeyl` defined via its equivalence\n with `Fin 2 → ℂ`.", + "snippet": "instance : Module ℂ RightHandedWeyl", + "line": 148 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "def", + "name": "Fermion.RightHandedWeyl.toFin2ℂEquiv", + "docstring": "The linear equivalence between `RightHandedWeyl` and `(Fin 2 → ℂ)`.", + "snippet": "@[simps!]\ndef toFin2ℂEquiv : RightHandedWeyl ≃ₗ[ℂ] (Fin 2 → ℂ) where", + "line": 152 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "abbrev", + "name": "Fermion.RightHandedWeyl.toFin2ℂ", + "docstring": "The underlying element of `Fin 2 → ℂ` of a element in `RightHandedWeyl` defined\n through the linear equivalence `toFin2ℂEquiv`.", + "snippet": "abbrev toFin2ℂ (ψ : RightHandedWeyl)", + "line": 162 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "structure", + "name": "Fermion.DualRightHandedWeyl", + "docstring": "The module in which dual-right handed fermions live. This is equivalent to `Fin 2 → ℂ`.", + "snippet": "structure DualRightHandedWeyl where\n val : Fin 2 → ℂ", + "line": 171 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "def", + "name": "Fermion.DualRightHandedWeyl.toFin2ℂFun", + "docstring": "The equivalence between `DualRightHandedWeyl` and `Fin 2 → ℂ`.", + "snippet": "def toFin2ℂFun : DualRightHandedWeyl ≃ (Fin 2 → ℂ) where", + "line": 178 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "instance", + "name": "Fermion.DualRightHandedWeyl.«instance@c7559440»", + "docstring": "The instance of `AddCommMonoid` on `DualRightHandedWeyl` defined via its equivalence\n with `Fin 2 → ℂ`.", + "snippet": "instance : AddCommMonoid DualRightHandedWeyl", + "line": 186 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "instance", + "name": "Fermion.DualRightHandedWeyl.«instance@a8cda771»", + "docstring": "The instance of `AddCommGroup` on `DualRightHandedWeyl` defined via its equivalence\n with `Fin 2 → ℂ`.", + "snippet": "instance : AddCommGroup DualRightHandedWeyl", + "line": 190 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "instance", + "name": "Fermion.DualRightHandedWeyl.«instance@b26a3288»", + "docstring": "The instance of `Module` on `DualRightHandedWeyl` defined via its equivalence\n with `Fin 2 → ℂ`.", + "snippet": "instance : Module ℂ DualRightHandedWeyl", + "line": 194 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "def", + "name": "Fermion.DualRightHandedWeyl.toFin2ℂEquiv", + "docstring": "The linear equivalence between `DualRightHandedWeyl` and `(Fin 2 → ℂ)`.", + "snippet": "@[simps!]\ndef toFin2ℂEquiv : DualRightHandedWeyl ≃ₗ[ℂ] (Fin 2 → ℂ) where", + "line": 198 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Modules.lean", + "kind": "abbrev", + "name": "Fermion.DualRightHandedWeyl.toFin2ℂ", + "docstring": "The underlying element of `Fin 2 → ℂ` of a element in `DualRightHandedWeyl` defined\n through the linear equivalence `toFin2ℂEquiv`.", + "snippet": "abbrev toFin2ℂ (ψ : DualRightHandedWeyl)", + "line": 208 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "def", + "name": "Fermion.dualLeftdualLeftToMatrix", + "docstring": "Equivalence of `dualLeftHanded ⊗ dualLeftHanded` to `2 x 2` complex matrices.", + "snippet": "def dualLeftdualLeftToMatrix : (DualLeftHandedWeyl ⊗[ℂ] DualLeftHandedWeyl) ≃ₗ[ℂ]\n Matrix (Fin 2) (Fin 2) ℂ", + "line": 50 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualLeftdualLeftToMatrix_symm_expand_tmul", + "docstring": "Expanding `dualLeftdualLeftToMatrix` in terms of the standard basis.", + "snippet": "lemma dualLeftdualLeftToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :\n dualLeftdualLeftToMatrix.symm M = ∑ i, ∑ j, M i j •\n (dualLeftBasis i ⊗ₜ[ℂ] dualLeftBasis j)", + "line": 57 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "def", + "name": "Fermion.leftDualLeftToMatrix", + "docstring": "Equivalence of `leftHanded ⊗ dualLeftHanded` to `2 x 2` complex matrices.", + "snippet": "def leftDualLeftToMatrix : (LeftHandedWeyl ⊗[ℂ] DualLeftHandedWeyl) ≃ₗ[ℂ]\n Matrix (Fin 2) (Fin 2) ℂ", + "line": 69 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.leftDualLeftToMatrix_symm_expand_tmul", + "docstring": "Expanding `leftDualLeftToMatrix` in terms of the standard basis.", + "snippet": "lemma leftDualLeftToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :\n leftDualLeftToMatrix.symm M = ∑ i, ∑ j, M i j • (leftBasis i ⊗ₜ[ℂ] dualLeftBasis j)", + "line": 76 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "def", + "name": "Fermion.dualLeftLeftToMatrix", + "docstring": "Equivalence of `dualLeftHanded ⊗ leftHanded` to `2 x 2` complex matrices.", + "snippet": "def dualLeftLeftToMatrix : (DualLeftHandedWeyl ⊗[ℂ] LeftHandedWeyl) ≃ₗ[ℂ]\n Matrix (Fin 2) (Fin 2) ℂ", + "line": 87 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualLeftLeftToMatrix_symm_expand_tmul", + "docstring": "Expanding `dualLeftLeftToMatrix` in terms of the standard basis.", + "snippet": "lemma dualLeftLeftToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :\n dualLeftLeftToMatrix.symm M = ∑ i, ∑ j, M i j • (dualLeftBasis i ⊗ₜ[ℂ] leftBasis j)", + "line": 94 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "def", + "name": "Fermion.dualRightDualRightToMatrix", + "docstring": "Equivalence of `dualRightHanded ⊗ dualRightHanded` to `2 x 2` complex matrices.", + "snippet": "def dualRightDualRightToMatrix : (DualRightHandedWeyl ⊗[ℂ] DualRightHandedWeyl) ≃ₗ[ℂ]\n Matrix (Fin 2) (Fin 2) ℂ", + "line": 123 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualRightDualRightToMatrix_symm_expand_tmul", + "docstring": "Expanding `dualRightDualRightToMatrix` in terms of the standard basis.", + "snippet": "lemma dualRightDualRightToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :\n dualRightDualRightToMatrix.symm M =\n ∑ i, ∑ j, M i j • (dualRightBasis i ⊗ₜ[ℂ] dualRightBasis j)", + "line": 130 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "def", + "name": "Fermion.rightDualRightToMatrix", + "docstring": "Equivalence of `rightHanded ⊗ dualRightHanded` to `2 x 2` complex matrices.", + "snippet": "def rightDualRightToMatrix : (RightHandedWeyl ⊗[ℂ] DualRightHandedWeyl) ≃ₗ[ℂ]\n Matrix (Fin 2) (Fin 2) ℂ", + "line": 142 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.rightDualRightToMatrix_symm_expand_tmul", + "docstring": "Expanding `rightDualRightToMatrix` in terms of the standard basis.", + "snippet": "lemma rightDualRightToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :\n rightDualRightToMatrix.symm M = ∑ i, ∑ j, M i j • (rightBasis i ⊗ₜ[ℂ] dualRightBasis j)", + "line": 149 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "def", + "name": "Fermion.dualRightRightToMatrix", + "docstring": "Equivalence of `dualRightHanded ⊗ rightHanded` to `2 x 2` complex matrices.", + "snippet": "def dualRightRightToMatrix : (DualRightHandedWeyl ⊗[ℂ] RightHandedWeyl) ≃ₗ[ℂ]\n Matrix (Fin 2) (Fin 2) ℂ", + "line": 160 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualRightRightToMatrix_symm_expand_tmul", + "docstring": "Expanding `dualRightRightToMatrix` in terms of the standard basis.", + "snippet": "lemma dualRightRightToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :\n dualRightRightToMatrix.symm M = ∑ i, ∑ j, M i j • (dualRightBasis i ⊗ₜ[ℂ] rightBasis j)", + "line": 167 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "def", + "name": "Fermion.dualLeftDualRightToMatrix", + "docstring": "Equivalence of `dualLeftHanded ⊗ dualRightHanded` to `2 x 2` complex matrices.", + "snippet": "def dualLeftDualRightToMatrix : (DualLeftHandedWeyl ⊗[ℂ] DualRightHandedWeyl) ≃ₗ[ℂ]\n Matrix (Fin 2) (Fin 2) ℂ", + "line": 178 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualLeftDualRightToMatrix_symm_expand_tmul", + "docstring": "Expanding `dualLeftDualRightToMatrix` in terms of the standard basis.", + "snippet": "lemma dualLeftDualRightToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :\n dualLeftDualRightToMatrix.symm M = ∑ i, ∑ j, M i j •\n (dualLeftBasis i ⊗ₜ[ℂ] dualRightBasis j)", + "line": 185 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.coe_linearEquivFunOnFinite", + "docstring": "The coercion of `Finsupp.linearEquivFunOnFinite` to a function is the underlying\nfinitely-supported function, used to bridge it with `Matrix.mulVec`.", + "snippet": "private lemma coe_linearEquivFunOnFinite (g : (Fin 2 × Fin 2) →₀ ℂ) :\n Finsupp.linearEquivFunOnFinite ℂ ℂ (Fin 2 × Fin 2) g = ⇑g", + "line": 215 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualLeftdualLeftToMatrix_ρ", + "docstring": "The group action of `SL(2,ℂ)` on `dualLeftHanded ⊗ dualLeftHanded` is equivalent to\n `(M.1⁻¹)ᵀ * leftLeftToMatrix v * (M.1⁻¹)`.", + "snippet": "lemma dualLeftdualLeftToMatrix_ρ (v : (DualLeftHandedWeyl ⊗[ℂ] DualLeftHandedWeyl)) (M : SL(2,ℂ)) :\n dualLeftdualLeftToMatrix (TensorProduct.map (dualLeftHandedRep M) (dualLeftHandedRep M) v) =\n (M.1⁻¹)ᵀ * dualLeftdualLeftToMatrix v * (M.1⁻¹)", + "line": 269 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.leftDualLeftToMatrix_ρ", + "docstring": "The group action of `SL(2,ℂ)` on `leftHanded ⊗ dualLeftHanded` is equivalent to\n `M.1 * leftDualLeftToMatrix v * (M.1⁻¹)`.", + "snippet": "lemma leftDualLeftToMatrix_ρ (v : (LeftHandedWeyl ⊗[ℂ] DualLeftHandedWeyl)) (M : SL(2,ℂ)) :\n leftDualLeftToMatrix (TensorProduct.map (leftHandedRep M) (dualLeftHandedRep M) v) =\n M.1 * leftDualLeftToMatrix v * (M.1⁻¹)", + "line": 312 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualLeftLeftToMatrix_ρ", + "docstring": "The group action of `SL(2,ℂ)` on `dualLeftHanded ⊗ leftHanded` is equivalent to\n `(M.1⁻¹)ᵀ * leftDualLeftToMatrix v * (M.1)ᵀ`.", + "snippet": "lemma dualLeftLeftToMatrix_ρ (v : (DualLeftHandedWeyl ⊗[ℂ] LeftHandedWeyl)) (M : SL(2,ℂ)) :\n dualLeftLeftToMatrix (TensorProduct.map (dualLeftHandedRep M) (leftHandedRep M) v) =\n (M.1⁻¹)ᵀ * dualLeftLeftToMatrix v * (M.1)ᵀ", + "line": 353 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualRightDualRightToMatrix_ρ", + "docstring": "The group action of `SL(2,ℂ)` on `dualRightHanded ⊗ dualRightHanded` is equivalent to\n `((M.1⁻¹).conjTranspose * rightRightToMatrix v * (((M.1⁻¹).conjTranspose)ᵀ`.", + "snippet": "lemma dualRightDualRightToMatrix_ρ (v : (DualRightHandedWeyl ⊗[ℂ] DualRightHandedWeyl))\n (M : SL(2,ℂ)) :\n dualRightDualRightToMatrix (TensorProduct.map (dualRightHandedRep M) (dualRightHandedRep M) v) =\n ((M.1⁻¹).conjTranspose) * dualRightDualRightToMatrix v * (((M.1⁻¹).conjTranspose)ᵀ)", + "line": 436 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.rightDualRightToMatrix_ρ", + "docstring": "The group action of `SL(2,ℂ)` on `rightHanded ⊗ dualRightHanded` is equivalent to\n `(M.1.map star) * rightDualRightToMatrix v * (((M.1⁻¹).conjTranspose)ᵀ`.", + "snippet": "lemma rightDualRightToMatrix_ρ (v : (RightHandedWeyl ⊗[ℂ] DualRightHandedWeyl)) (M : SL(2,ℂ)) :\n rightDualRightToMatrix (TensorProduct.map (rightHandedRep M) (dualRightHandedRep M) v) =\n (M.1.map star) * rightDualRightToMatrix v * (((M.1⁻¹).conjTranspose)ᵀ)", + "line": 479 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualRightRightToMatrix_ρ", + "docstring": "The group action of `SL(2,ℂ)` on `dualRightHanded ⊗ rightHanded` is equivalent to\n `((M.1⁻¹).conjTranspose * rightDualRightToMatrix v * ((M.1.map star)).ᵀ`.", + "snippet": "lemma dualRightRightToMatrix_ρ (v : (DualRightHandedWeyl ⊗[ℂ] RightHandedWeyl)) (M : SL(2,ℂ)) :\n dualRightRightToMatrix (TensorProduct.map (dualRightHandedRep M) (rightHandedRep M) v) =\n ((M.1⁻¹).conjTranspose) * dualRightRightToMatrix v * (M.1.map star)ᵀ", + "line": 521 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualLeftDualRightToMatrix_ρ", + "docstring": null, + "snippet": "lemma dualLeftDualRightToMatrix_ρ (v : (DualLeftHandedWeyl ⊗[ℂ] DualRightHandedWeyl))\n (M : SL(2,ℂ)) :\n dualLeftDualRightToMatrix (TensorProduct.map (dualLeftHandedRep M) (dualRightHandedRep M) v) =\n (M.1⁻¹)ᵀ * dualLeftDualRightToMatrix v * ((M.1⁻¹).conjTranspose)ᵀ", + "line": 562 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualLeftdualLeftToMatrix_ρ_symm", + "docstring": null, + "snippet": "lemma dualLeftdualLeftToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :\n TensorProduct.map (dualLeftHandedRep M) (dualLeftHandedRep M)\n (dualLeftdualLeftToMatrix.symm v) =\n dualLeftdualLeftToMatrix.symm ((M.1⁻¹)ᵀ * v * (M.1⁻¹))", + "line": 656 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.leftDualLeftToMatrix_ρ_symm", + "docstring": null, + "snippet": "lemma leftDualLeftToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :\n TensorProduct.map (leftHandedRep M) (dualLeftHandedRep M) (leftDualLeftToMatrix.symm v) =\n leftDualLeftToMatrix.symm (M.1 * v * (M.1⁻¹))", + "line": 664 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualLeftLeftToMatrix_ρ_symm", + "docstring": null, + "snippet": "lemma dualLeftLeftToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :\n TensorProduct.map (dualLeftHandedRep M) (leftHandedRep M) (dualLeftLeftToMatrix.symm v) =\n dualLeftLeftToMatrix.symm ((M.1⁻¹)ᵀ * v * (M.1)ᵀ)", + "line": 671 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualRightDualRightToMatrix_ρ_symm", + "docstring": null, + "snippet": "lemma dualRightDualRightToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :\n TensorProduct.map (dualRightHandedRep M) (dualRightHandedRep M)\n (dualRightDualRightToMatrix.symm v) =\n dualRightDualRightToMatrix.symm (((M.1⁻¹).conjTranspose) * v * ((M.1⁻¹).conjTranspose)ᵀ)", + "line": 685 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.rightDualRightToMatrix_ρ_symm", + "docstring": null, + "snippet": "lemma rightDualRightToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :\n TensorProduct.map (rightHandedRep M) (dualRightHandedRep M) (rightDualRightToMatrix.symm v) =\n rightDualRightToMatrix.symm ((M.1.map star) * v * (((M.1⁻¹).conjTranspose)ᵀ))", + "line": 693 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualRightRightToMatrix_ρ_symm", + "docstring": null, + "snippet": "lemma dualRightRightToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :\n TensorProduct.map (dualRightHandedRep M) (rightHandedRep M) (dualRightRightToMatrix.symm v) =\n dualRightRightToMatrix.symm (((M.1⁻¹).conjTranspose) * v * (M.1.map star)ᵀ)", + "line": 700 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualLeftDualRightToMatrix_ρ_symm", + "docstring": null, + "snippet": "lemma dualLeftDualRightToMatrix_ρ_symm (v : Matrix (Fin 2) (Fin 2) ℂ) (M : SL(2,ℂ)) :\n TensorProduct.map (dualLeftHandedRep M) (dualRightHandedRep M)\n (dualLeftDualRightToMatrix.symm v) =\n dualLeftDualRightToMatrix.symm ((M.1⁻¹)ᵀ * v * ((M.1⁻¹).conjTranspose)ᵀ)", + "line": 707 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Two.lean", + "kind": "lemma", + "name": "Fermion.dualLeftDualRightToMatrix_ρ_symm_selfAdjoint", + "docstring": null, + "snippet": "lemma dualLeftDualRightToMatrix_ρ_symm_selfAdjoint (v : Matrix (Fin 2) (Fin 2) ℂ)\n (hv : IsSelfAdjoint v) (M : SL(2,ℂ)) :\n TensorProduct.map (dualLeftHandedRep M) (dualRightHandedRep M)\n (dualLeftDualRightToMatrix.symm v) =\n dualLeftDualRightToMatrix.symm (SL2C.toSelfAdjointMap (M.transpose⁻¹) ⟨v, hv⟩)", + "line": 724 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "def", + "name": "Fermion.leftDualLeftUnitVal", + "docstring": "The left-dual-left unit `δᵃₐ` as an element of `(leftHanded ⊗ dualLeftHanded).V`.", + "snippet": "def leftDualLeftUnitVal : (LeftHandedWeyl ⊗[ℂ] DualLeftHandedWeyl)", + "line": 30 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.leftDualLeftUnitVal_expand_tmul", + "docstring": "Expansion of `leftDualLeftUnitVal` into the basis.", + "snippet": "lemma leftDualLeftUnitVal_expand_tmul : leftDualLeftUnitVal =\n leftBasis 0 ⊗ₜ[ℂ] dualLeftBasis 0 + leftBasis 1 ⊗ₜ[ℂ] dualLeftBasis 1", + "line": 34 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "def", + "name": "Fermion.leftDualLeftUnit", + "docstring": "The left-dual-left unit `δᵃₐ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ leftHanded ⊗ dualLeftHanded `,\n manifesting the invariance under the `SL(2,ℂ)` action.", + "snippet": "def leftDualLeftUnit : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap\n (leftHandedRep.tprod dualLeftHandedRep) where", + "line": 43 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.leftDualLeftUnit_apply_one", + "docstring": null, + "snippet": "lemma leftDualLeftUnit_apply_one : leftDualLeftUnit (1 : ℂ) = leftDualLeftUnitVal", + "line": 64 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "def", + "name": "Fermion.dualLeftLeftUnitVal", + "docstring": "The dual-left-left unit `δₐᵃ` as an element of `(dualLeftHanded ⊗ leftHanded).V`.", + "snippet": "def dualLeftLeftUnitVal : (DualLeftHandedWeyl ⊗[ℂ] LeftHandedWeyl)", + "line": 69 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.dualLeftLeftUnitVal_expand_tmul", + "docstring": "Expansion of `dualLeftLeftUnitVal` into the basis.", + "snippet": "lemma dualLeftLeftUnitVal_expand_tmul : dualLeftLeftUnitVal =\n dualLeftBasis 0 ⊗ₜ[ℂ] leftBasis 0 + dualLeftBasis 1 ⊗ₜ[ℂ] leftBasis 1", + "line": 73 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "def", + "name": "Fermion.dualLeftLeftUnit", + "docstring": "The dual-left-left unit `δₐᵃ` as a morphism `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ dualLeftHanded ⊗ leftHanded `,\n manifesting the invariance under the `SL(2,ℂ)` action.", + "snippet": "def dualLeftLeftUnit :\n (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap\n (dualLeftHandedRep.tprod leftHandedRep) where", + "line": 82 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.dualLeftLeftUnit_apply_one", + "docstring": "Applying the morphism `dualLeftLeftUnit` to `1` returns `dualLeftLeftUnitVal`.", + "snippet": "lemma dualLeftLeftUnit_apply_one : dualLeftLeftUnit (1 : ℂ) = dualLeftLeftUnitVal", + "line": 106 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "def", + "name": "Fermion.rightDualRightUnitVal", + "docstring": "The right-dual-right unit `δ^{dot a}_{dot a}` as an element of\n `(rightHanded ⊗ dualRightHanded).V`.", + "snippet": "def rightDualRightUnitVal : RightHandedWeyl ⊗[ℂ] DualRightHandedWeyl", + "line": 112 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.rightDualRightUnitVal_expand_tmul", + "docstring": "Expansion of `rightDualRightUnitVal` into the basis.", + "snippet": "lemma rightDualRightUnitVal_expand_tmul : rightDualRightUnitVal =\n rightBasis 0 ⊗ₜ[ℂ] dualRightBasis 0 + rightBasis 1 ⊗ₜ[ℂ] dualRightBasis 1", + "line": 116 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "def", + "name": "Fermion.rightDualRightUnit", + "docstring": "The right-dual-right unit `δ^{dot a}_{dot a}` as a morphism\n `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ rightHanded ⊗ dualRightHanded`, manifesting\n the invariance under the `SL(2,ℂ)` action.", + "snippet": "def rightDualRightUnit : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap\n (rightHandedRep.tprod dualRightHandedRep) where", + "line": 126 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.rightDualRightUnit_apply_one", + "docstring": null, + "snippet": "lemma rightDualRightUnit_apply_one : rightDualRightUnit (1 : ℂ) = rightDualRightUnitVal", + "line": 153 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "def", + "name": "Fermion.dualRightRightUnitVal", + "docstring": "The dual-right-right unit `δ_{dot a}^{dot a}` as an element of\n `(rightHanded ⊗ dualRightHanded).V`.", + "snippet": "def dualRightRightUnitVal : (DualRightHandedWeyl ⊗[ℂ] RightHandedWeyl)", + "line": 159 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.dualRightRightUnitVal_expand_tmul", + "docstring": "Expansion of `dualRightRightUnitVal` into the basis.", + "snippet": "lemma dualRightRightUnitVal_expand_tmul : dualRightRightUnitVal =\n dualRightBasis 0 ⊗ₜ[ℂ] rightBasis 0 + dualRightBasis 1 ⊗ₜ[ℂ] rightBasis 1", + "line": 163 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "def", + "name": "Fermion.dualRightRightUnit", + "docstring": "The dual-right-right unit `δ_{dot a}^{dot a}` as a morphism\n `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ dualRightHanded ⊗ rightHanded`, manifesting\n the invariance under the `SL(2,ℂ)` action.", + "snippet": "def dualRightRightUnit : (Representation.trivial ℂ SL(2,ℂ) ℂ).IntertwiningMap\n (dualRightHandedRep.tprod rightHandedRep) where", + "line": 173 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.dualRightRightUnit_apply_one", + "docstring": null, + "snippet": "lemma dualRightRightUnit_apply_one : dualRightRightUnit (1 : ℂ) = dualRightRightUnitVal", + "line": 198 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.contr_dualLeftLeftUnit", + "docstring": "Contraction on the right with `dualLeftLeftUnit` does nothing.", + "snippet": "lemma contr_dualLeftLeftUnit (x : LeftHandedWeyl) :\n (TensorProduct.lid ℂ _ <|\n leftDualContraction.toLinearMap.rTensor _ <|\n (TensorProduct.assoc ℂ _ _ _).symm <|\n x ⊗ₜ[ℂ] (dualLeftLeftUnit (1 : ℂ))) = x", + "line": 209 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.contr_leftDualLeftUnit", + "docstring": "Contraction on the right with `leftDualLeftUnit` does nothing.", + "snippet": "lemma contr_leftDualLeftUnit (x : DualLeftHandedWeyl) :\n (TensorProduct.lid ℂ _ <|\n dualLeftContraction.toLinearMap.rTensor _ <|\n (TensorProduct.assoc ℂ _ _ _).symm <|\n x ⊗ₜ[ℂ] (leftDualLeftUnit (1 : ℂ))) = x", + "line": 220 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.contr_dualRightRightUnit", + "docstring": "Contraction on the right with `dualRightRightUnit` does nothing.", + "snippet": "lemma contr_dualRightRightUnit (x : RightHandedWeyl) :\n (TensorProduct.lid ℂ _ <|\n rightDualContraction.toLinearMap.rTensor _ <|\n (TensorProduct.assoc ℂ _ _ _).symm <|\n x ⊗ₜ[ℂ] (dualRightRightUnit (1 : ℂ))) = x", + "line": 231 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.contr_rightDualRightUnit", + "docstring": "Contraction on the right with `rightDualRightUnit` does nothing.", + "snippet": "lemma contr_rightDualRightUnit (x : DualRightHandedWeyl) :\n (TensorProduct.lid ℂ _ <|\n dualRightContraction.toLinearMap.rTensor _ <|\n (TensorProduct.assoc ℂ _ _ _).symm <|\n x ⊗ₜ[ℂ] (rightDualRightUnit (1 : ℂ))) = x", + "line": 242 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.dualLeftLeftUnit_symm", + "docstring": null, + "snippet": "lemma dualLeftLeftUnit_symm :\n dualLeftLeftUnit (1 : ℂ) = LinearMap.lTensor _ (LinearEquiv.refl _ _).toLinearMap\n (TensorProduct.comm ℂ _ _ (leftDualLeftUnit (1 : ℂ)))", + "line": 259 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.leftDualLeftUnit_symm", + "docstring": null, + "snippet": "lemma leftDualLeftUnit_symm :\n leftDualLeftUnit (1 : ℂ) = LinearMap.lTensor _ (LinearEquiv.refl _ _).toLinearMap\n (TensorProduct.comm ℂ _ _ (dualLeftLeftUnit (1 : ℂ)))", + "line": 266 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.dualRightRightUnit_symm", + "docstring": null, + "snippet": "lemma dualRightRightUnit_symm :\n dualRightRightUnit (1 : ℂ) = LinearMap.lTensor _ (LinearEquiv.refl _ _).toLinearMap\n (TensorProduct.comm ℂ _ _ (rightDualRightUnit (1 : ℂ)))", + "line": 273 + }, + { + "file": "Physlib/Relativity/Tensors/ComplexTensor/Weyl/Unit.lean", + "kind": "lemma", + "name": "Fermion.rightDualRightUnit_symm", + "docstring": null, + "snippet": "lemma rightDualRightUnit_symm :\n rightDualRightUnit (1 : ℂ) = LinearMap.lTensor _ (LinearEquiv.refl _ _).toLinearMap\n (TensorProduct.comm ℂ _ _ (dualRightRightUnit (1 : ℂ)))", + "line": 280 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComponentIdx/Basic.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Basic.lean", + "kind": "abbrev", + "name": "TensorSpecies.Tensor.ComponentIdx", + "docstring": "Given a list of indices `c : Fin n → C`, the type `ComponentIdx c` is the type of\ncomponent indices of a tensor with those colors. For instance, if `c = ![.up, .down]`,\nthen an element of `ComponentIdx c` specifies one basis index for `.up` and one for\n`.down`.", + "snippet": "@[nolint unusedArguments]\nabbrev ComponentIdx {n : ℕ} {S : TensorSpecies k C G V basisIdx rep b}\n (c : Fin n → C) : Type", + "line": 65 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Basic.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.ComponentIdx.congr_right", + "docstring": null, + "snippet": "lemma ComponentIdx.congr_right {n : ℕ} {c : Fin n → C} (b : ComponentIdx (S := S) c)\n (i j : Fin n) (h : i = j) : b i = basisIdxCongr (by simp [h]) (b j)", + "line": 69 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Basic.lean", + "kind": "def", + "name": "TensorSpecies.Tensor.ComponentIdx.cast", + "docstring": "Casting of a `ComponentIdx` through equivalent color maps.", + "snippet": "def ComponentIdx.cast {n m : ℕ} {c : Fin n → C} {cm : Fin m → C}\n (h : n = m) (hc : c = cm ∘ Fin.cast h) (b : ComponentIdx (S := S) c) :\n ComponentIdx (S := S) cm", + "line": 75 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean", + "kind": "def", + "name": "TensorSpecies.Tensor.ComponentIdx.dropPair", + "docstring": "The `ComponentIdx` obtained by dropping two components.", + "snippet": "def dropPair {n : ℕ} {c : Fin (n + 1 + 1) → C}\n (i j : Fin (n + 1 + 1)) (b : ComponentIdx (S := S) c) :\n ComponentIdx (S := S) (c ∘ Fin.succSuccAbove i j)", + "line": 65 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean", + "kind": "def", + "name": "TensorSpecies.Tensor.ComponentIdx.DropPairSection", + "docstring": "Given a coordinate parameter\n `b : Π k, Fin (S.repDim ((c ∘ i.succAbove ∘ j.succAbove) k)))`, the\n coordinate parameter `Π k, Fin (S.repDim (c k))` which map down to `b`.", + "snippet": "def DropPairSection {n : ℕ} {c : Fin (n + 1 + 1) → C}\n {i : Fin (n + 1 + 1)} {j : Fin (n + 1 + 1)}\n (b : ComponentIdx (S := S) (c ∘ Fin.succSuccAbove i j)) :\n Finset (ComponentIdx (S := S) c)", + "line": 79 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.ComponentIdx.DropPairSection.mem_iff_apply_succSuccAbove_eq", + "docstring": null, + "snippet": "lemma mem_iff_apply_succSuccAbove_eq {n : ℕ} {c : Fin (n + 1 + 1) → C}\n {i j : Fin (n + 1 + 1)}\n (b : ComponentIdx (c ∘ Fin.succSuccAbove i j))\n (b' : ComponentIdx c) :\n b' ∈ DropPairSection (S := S) b ↔\n ∀ m, b' (Fin.succSuccAbove i j m) = b m", + "line": 87 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.ComponentIdx.DropPairSection.mem_self_of_dropPair", + "docstring": null, + "snippet": "@[simp]\nlemma mem_self_of_dropPair {n : ℕ} {c : Fin (n + 1 + 1) → C}\n {i j : Fin (n + 1 + 1)}\n (b : ComponentIdx (c)) :\n b ∈ DropPairSection (S := S) (b.dropPair i j)", + "line": 98 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean", + "kind": "def", + "name": "TensorSpecies.Tensor.ComponentIdx.DropPairSection.ofFin", + "docstring": "Given a `b` in `ComponentIdx (c ∘ Fin.succSuccAbove i j))` and\n an `x` in `Fin (S.repDim (c i)) × Fin (S.repDim (c j))`, the corresponding\n coordinate parameter in `ComponentIdx c`.", + "snippet": "def ofFin {n : ℕ} {c : Fin (n + 1 + 1) → C}\n {i j : Fin (n + 1 + 1)} (hij : i ≠ j) (b : ComponentIdx (S := S) (c ∘ Fin.succSuccAbove i j))\n (x : basisIdx (c i) × basisIdx (c j)) :\n ComponentIdx (S := S) c", + "line": 107 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.ComponentIdx.DropPairSection.ofFin_apply_fst", + "docstring": null, + "snippet": "@[simp]\nlemma ofFin_apply_fst {n : ℕ} {c : Fin (n + 1 + 1) → C}\n {i j : Fin (n + 1 + 1)} (hij : i ≠ j) (b : ComponentIdx (c ∘ Fin.succSuccAbove i j))\n (x : basisIdx (c i) × basisIdx (c j)) :\n ofFin (S := S) hij b x i = x.1", + "line": 118 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.ComponentIdx.DropPairSection.ofFin_apply_snd", + "docstring": null, + "snippet": "@[simp]\nlemma ofFin_apply_snd {n : ℕ} {c : Fin (n + 1 + 1) → C}\n {i j : Fin (n + 1 + 1)} (hij : i ≠ j) (b : ComponentIdx (c ∘ Fin.succSuccAbove i j))\n (x : basisIdx (c i) × basisIdx (c j)) :\n ofFin (S := S) hij b x j = x.2", + "line": 125 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.ComponentIdx.DropPairSection.ofFin_mem_succSuccAboveSection", + "docstring": null, + "snippet": "lemma ofFin_mem_succSuccAboveSection {n : ℕ} {c : Fin (n + 1 + 1) → C}\n {i j : Fin (n + 1 + 1)} (hij : i ≠ j) (b : ComponentIdx (c ∘ Fin.succSuccAbove i j))\n (x : basisIdx (c i) × basisIdx (c j)) :\n ofFin (S := S) hij b x ∈ DropPairSection b", + "line": 133 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean", + "kind": "def", + "name": "TensorSpecies.Tensor.ComponentIdx.DropPairSection.ofFinEquiv", + "docstring": "The equivalence between `ContrSection b` and\n `basisIdx (c i) × basisIdx (c j)`.", + "snippet": "def ofFinEquiv {n : ℕ} {c : Fin n.succ.succ → C}\n {i j : Fin (n + 1 + 1)} (hij : i ≠ j)\n (b : ComponentIdx (c ∘ Fin.succSuccAbove i j)) :\n basisIdx (c i) × basisIdx (c j) ≃ DropPairSection (S := S) b where", + "line": 147 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.ComponentIdx.DropPairSection.ofFinEquiv_apply_fst", + "docstring": null, + "snippet": "@[simp]\nlemma ofFinEquiv_apply_fst {n : ℕ} {c : Fin (n + 1 + 1) → C}\n {i j : Fin (n + 1 + 1)} (hij : i ≠ j) (b : ComponentIdx (c ∘ Fin.succSuccAbove i j))\n (x : basisIdx (c i) × basisIdx (c j)) :\n (ofFinEquiv (S := S) hij b x).1 i = x.1", + "line": 170 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Contraction.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.ComponentIdx.DropPairSection.ofFinEquiv_apply_snd", + "docstring": null, + "snippet": "@[simp]\nlemma ofFinEquiv_apply_snd {n : ℕ} {c : Fin (n + 1 + 1) → C}\n {i j : Fin (n + 1 + 1)} (hij : i ≠ j) (b : ComponentIdx (c ∘ Fin.succSuccAbove i j))\n (x : basisIdx (c i) × basisIdx (c j)) :\n (ofFinEquiv (S := S) hij b x).1 j = x.2", + "line": 177 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComponentIdx/Product.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Product.lean", + "kind": "def", + "name": "TensorSpecies.Tensor.ComponentIdx.prod", + "docstring": "The equivalence between `ComponentIdx (Fin.append c c1)` and\n `ComponentIdx c × ComponentIdx c1` formed by products.", + "snippet": "def ComponentIdx.prod {n1 n2 : ℕ} {c : Fin n1 → C} {c1 : Fin n2 → C} :\n ComponentIdx (S := S) (Fin.append c c1) ≃\n ComponentIdx (S := S) c × ComponentIdx (S := S) c1 where", + "line": 56 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Product.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.ComponentIdx.prod_symm_natAdd", + "docstring": null, + "snippet": "@[simp]\nlemma ComponentIdx.prod_symm_natAdd {n1 n2 : ℕ} {c : Fin n1 → C} {c1 : Fin n2 → C}\n (p : ComponentIdx (S := S) c) (q : ComponentIdx (S := S) c1) (i : Fin n2) :\n ComponentIdx.prod.symm (p, q) (Fin.natAdd n1 i) =\n basisIdxCongr (by simp) (q i)", + "line": 70 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Product.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.ComponentIdx.prod_symm_castAdd", + "docstring": null, + "snippet": "@[simp]\nlemma ComponentIdx.prod_symm_castAdd {n1 n2 : ℕ} {c : Fin n1 → C} {c1 : Fin n2 → C}\n (p : ComponentIdx (S := S) c) (q : ComponentIdx (S := S) c1) (i : Fin n1) :\n ComponentIdx.prod.symm (p, q) (Fin.castAdd n2 i) =\n basisIdxCongr (by simp) (p i)", + "line": 76 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/ComponentIdx/Single.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Single.lean", + "kind": "def", + "name": "TensorSpecies.Tensor.ComponentIdx.single", + "docstring": "The equivalence between component indices for a single color and the basis indices\nof that color.", + "snippet": "def ComponentIdx.single {c : C} :\n ComponentIdx (S := S) ![c] ≃ basisIdx c where", + "line": 57 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Single.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.ComponentIdx.single_apply", + "docstring": null, + "snippet": "@[simp]\nlemma ComponentIdx.single_apply {c : C} (b : ComponentIdx (S := S) ![c]) :\n ComponentIdx.single (S := S) b = basisIdxCongr (by simp) (b 0)", + "line": 71 + }, + { + "file": "Physlib/Relativity/Tensors/ComponentIdx/Single.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.ComponentIdx.single_symm_apply", + "docstring": null, + "snippet": "@[simp]\nlemma ComponentIdx.single_symm_apply {c : C} (b : basisIdx c) (i : Fin 1) :\n (ComponentIdx.single (S := S) (c := c)).symm b i = basisIdxCongr (by simp) b", + "line": 75 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/Conjugation/Basic.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/Conjugation/Basic.lean", + "kind": "structure", + "name": "ConjTensorSpecies", + "docstring": "A tensor species bundled with a conjugation. Beyond the `TensorSpecies` it `extends`, it carries\nthe conjugate-colour involution `bar`, the index-set identification `barIdx_eq`, and the\ncontraction-coherence `conj_contrComm`. Also carries `bar_involution` and `bar_tau` (that `bar` is\ninvolutive and commutes with the variance dual).", + "snippet": "structure ConjTensorSpecies (k : Type) [CommRing k] [StarRing k] (C : Type) (G : Type) [Group G]\n (V : C → Type) [∀ c, AddCommGroup (V c)] [∀ c, Module k (V c)]\n (basisIdx : C → Type) [∀ c, Fintype (basisIdx c)] [∀ c, DecidableEq (basisIdx c)]\n (rep : (c : C) → Representation k G (V c)) (b : (c : C) → Basis (basisIdx c) k (V c))\n extends TensorSpecies k C G V basisIdx rep b where\n bar : C → C\n bar_involution : Function.Involutive bar\n bar_tau : ∀ c, bar (toTensorSpecies.τ c) = toTensorSpecies.τ (bar c)\n barIdx_eq : ∀ c, basisIdx (bar c) = basisIdx c\n conj_contrComm : ∀ (d : C) (x₁ : basisIdx d) (x₂ : basisIdx (toTensorSpecies.τ d)),\n star (toTensorSpecies.contr d (b d x₁ ⊗ₜ[k] b (toTensorSpecies.τ d) x₂))\n = toTensorSpecies.contr (bar d) (b (bar d) ((Equiv.cast (barIdx_eq d)).symm x₁) ⊗ₜ[k]\n b (toTensorSpecies.τ (bar d)) (basisIdxCongr (bar_tau d)\n ((Equiv.cast (barIdx_eq (toTensorSpecies.τ d))).symm x₂)))", + "line": 92 + }, + { + "file": "Physlib/Relativity/Tensors/Conjugation/Basic.lean", + "kind": "def", + "name": "ConjTensorSpecies.componentReindex", + "docstring": "Reindex component labels of `bar ∘ c` back to `c`, slotwise via the cast `barIdx_eq`.", + "snippet": "def componentReindex {n : ℕ} (c : Fin n → C) :\n ComponentIdx (S := S.toTensorSpecies) (S.bar ∘ c) ≃ ComponentIdx (S := S.toTensorSpecies) c", + "line": 137 + }, + { + "file": "Physlib/Relativity/Tensors/Conjugation/Basic.lean", + "kind": "def", + "name": "ConjTensorSpecies.conjT", + "docstring": "Conjugation of a tensor: conjugate the components and reindex the basis to the conjugate\ncolours. `conj`-semilinear by construction (see `conjT_smul`).", + "snippet": "def conjT {n : ℕ} {c : Fin n → C} (t : S.Tensor c) : S.Tensor (S.bar ∘ c)", + "line": 143 + }, + { + "file": "Physlib/Relativity/Tensors/Conjugation/Basic.lean", + "kind": "lemma", + "name": "ConjTensorSpecies.conjT_eq_permT_iff", + "docstring": "Componentwise criterion for `conjT t = permT σ h t'`. The conjugate of `t` equals the\nrecolouring `permT σ h t'` exactly when, at every component, the `star`-conjugated reindexed\ncomponent of `t` matches the corresponding component of `permT σ h t'`. This packages the\n`componentMap_conjT` expansion and the `repr`/`componentMap` bridge that the reality and Hermiticity\nproofs downstream would otherwise repeat by hand; the caller is left only with the species-specific\npermutation bookkeeping on the right-hand side.", + "snippet": "lemma conjT_eq_permT_iff {n m : ℕ} {c : Fin n → C} {c' : Fin m → C}\n {σ : Fin n → Fin m} (h : IsReindexing c' (S.bar ∘ c) σ)\n (t : S.Tensor c) (t' : S.Tensor c') :\n S.conjT t = permT σ h t' ↔\n ∀ φ : ComponentIdx (S := S.toTensorSpecies) (S.bar ∘ c),\n star (componentMap c t (S.componentReindex c φ))\n = (Tensor.basis (S.bar ∘ c)).repr (permT σ h t') φ", + "line": 174 + }, + { + "file": "Physlib/Relativity/Tensors/Conjugation/Basic.lean", + "kind": "lemma", + "name": "ConjTensorSpecies.barIdx_involutive_symm", + "docstring": "Undoing the two cast reindexings (at `bar c` then at `c`) on a label of the doubly-conjugated\ncolour returns the `bar_involution` cast. Free from `barIdx_eq` by `cast_cast` + proof\nirrelevance, with no coherence field needed.", + "snippet": "private lemma barIdx_involutive_symm (c : C) (y : basisIdx (S.bar (S.bar c))) :\n Equiv.cast (S.barIdx_eq c) (Equiv.cast (S.barIdx_eq (S.bar c)) y)\n = basisIdxCongr (S.bar_involution c) y", + "line": 202 + }, + { + "file": "Physlib/Relativity/Tensors/Conjugation/Basic.lean", + "kind": "lemma", + "name": "ConjTensorSpecies.isReindexing_bar_bar", + "docstring": "The identity permutation satisfies `IsReindexing` from `c` to `bar ∘ bar ∘ c`, as `bar` is an\ninvolution.", + "snippet": "lemma isReindexing_bar_bar {n : ℕ} (c : Fin n → C) :\n IsReindexing c (S.bar ∘ S.bar ∘ c) (id : Fin n → Fin n)", + "line": 209 + }, + { + "file": "Physlib/Relativity/Tensors/Conjugation/Basic.lean", + "kind": "lemma", + "name": "ConjTensorSpecies.conjT_conjT", + "docstring": "Conjugation is an involution: conjugating twice returns the original tensor, up to the\n`bar_involution` recolouring (the identity permutation `permT`).", + "snippet": "lemma conjT_conjT {n : ℕ} {c : Fin n → C} (t : S.Tensor c) :\n S.conjT (S.conjT t)\n = permT id (S.isReindexing_bar_bar c) t", + "line": 215 + }, + { + "file": "Physlib/Relativity/Tensors/Conjugation/Basic.lean", + "kind": "lemma", + "name": "ConjTensorSpecies.contrCond_bar", + "docstring": "Slots with dual colour in `c` have dual colour in `bar ∘ c`, as `bar` commutes with `τ`.", + "snippet": "lemma contrCond_bar {n : ℕ} {c : Fin (n + 1 + 1) → C} {i j : Fin (n + 1 + 1)}\n (h : i ≠ j ∧ S.τ (c i) = c j) :\n i ≠ j ∧ S.τ ((S.bar ∘ c) i) = (S.bar ∘ c) j", + "line": 242 + }, + { + "file": "Physlib/Relativity/Tensors/Conjugation/Basic.lean", + "kind": "lemma", + "name": "ConjTensorSpecies.conjT_contrT", + "docstring": "Conjugation commutes with contraction: conjugating a contracted tensor equals contracting the\nconjugate on the `bar`-images of the same two slots.", + "snippet": "lemma conjT_contrT {n : ℕ} {c : Fin (n + 1 + 1) → C} (i j : Fin (n + 1 + 1))\n (h : i ≠ j ∧ S.τ (c i) = c j) (t : S.Tensor c) :\n S.conjT (contrT n i j h t)\n = contrT n i j (S.contrCond_bar h) (S.conjT t)", + "line": 249 + }, + { + "file": "Physlib/Relativity/Tensors/Conjugation/Basic.lean", + "kind": "def", + "name": "ConjTensorSpecies.conjEquiv", + "docstring": "The conjugate-linear slot isomorphism `V c ≃ₛₗ[starRingEnd k] V (bar c)`: conjugate the\ncoordinates in the species basis and relabel to the conjugate colour via `barIdx_eq`.", + "snippet": "def conjEquiv {c : C} : V c ≃ₛₗ[starRingEnd k] V (S.bar c)", + "line": 301 + }, + { + "file": "Physlib/Relativity/Tensors/Conjugation/Basic.lean", + "kind": "def", + "name": "ConjTensorSpecies.IsHermitian", + "docstring": "A pairing `g : V c ⊗ V (bar c) → k` is Hermitian when conjugating and swapping its two slots\nvia `conjEquiv` returns its `star`: `g (x ⊗ y) = star (g (conjEquiv.symm y ⊗ conjEquiv x))`.", + "snippet": "def IsHermitian {c : C} (g : V c ⊗[k] V (S.bar c) →ₗ[k] k) : Prop", + "line": 326 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/Constructors.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/Constructors.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.fromPairT_eq_pure", + "docstring": null, + "snippet": "lemma fromPairT_eq_pure {c1 c2 : C} (x : V c1) (y : V c2) :\n fromPairT (S := S) (x ⊗ₜ[k] y) = Pure.toTensor (fun | 0 => x | 1 => y)", + "line": 132 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/Contraction/SuccSuccAbove.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/Contraction/SuccSuccAbove.lean", + "kind": "lemma", + "name": "Fin.apply_succSuccAbove_symm", + "docstring": null, + "snippet": "@[simp, nolint simpVarHead]\nlemma apply_succSuccAbove_symm {c : Fin (n + 1 + 1) → C} (i j : Fin (n + 1 + 1))\n (k : Fin n) : c (succSuccAbove i j k) = c (succSuccAbove j i k)", + "line": 169 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/Evaluation.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/Evaluation.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.evalT_basis_single", + "docstring": "Evaluating the single-index basis tensor `basis ![c] (single.symm b)` at the index `x`\n yields the field element `1` if `b = x` (transported across `![c] 0 = c`) and `0` otherwise:\n evaluation of a one-index basis tensor is the Kronecker delta.", + "snippet": "lemma evalT_basis_single {c : C} (b : basisIdx c) (x : basisIdx (![c] 0)) :\n (evalT 0 x (basis (S := S) ![c] (ComponentIdx.single.symm b))).toField =\n if basisIdxCongr (by simp) b = x then 1 else 0", + "line": 153 + }, + { + "file": "Physlib/Relativity/Tensors/Evaluation.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.eq_sum_evalT_of_single_tensor_basis", + "docstring": "Basis expansion of a one-index tensor: every `t : Tensor S ![c]` is the sum over basis\n indices `i` of its evaluation coefficient `toField (evalT 0 i t)` times the corresponding\n basis tensor.", + "snippet": "lemma eq_sum_evalT_of_single_tensor_basis {c : C} (t : Tensor S ![c]) :\n t = ∑ i, toField (evalT 0 i t) • basis ![c] (ComponentIdx.single.symm\n (basisIdxCongr (by simp) i))", + "line": 165 + }, + { + "file": "Physlib/Relativity/Tensors/Evaluation.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.eq_sum_evalT", + "docstring": "Reconstruction of a tensor from the evaluations of its last index: every `t : Tensor S c`\n is the sum over basis indices `i` of the evaluation `evalT (Fin.last n) i t` tensored with\n the basis covector `basis ![c (Fin.last n)] (single.symm i)`, with the appended index\n permuted back into the last slot.", + "snippet": "lemma eq_sum_evalT {n : ℕ} {c : Fin (n + 1) → C} (t : Tensor S c) :\n t = ∑ i, permT id (IsReindexing.append_succ_last c) (prodT (evalT (Fin.last n) i t)\n (basis ![c (Fin.last n)] (ComponentIdx.single.symm i)))", + "line": 182 + }, + { + "file": "Physlib/Relativity/Tensors/Evaluation.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.eq_sum_evalT_zero", + "docstring": "Reconstruction of a tensor from the evaluations of its first index: every `t : Tensor S c`\n is the sum over basis indices `i` of the basis covector `basis ![c 0] (single.symm i)` tensored\n with the evaluation `evalT 0 i t`, with the prepended index permuted back into the first slot.\n This is the first-index analogue of `eq_sum_evalT`.", + "snippet": "lemma eq_sum_evalT_zero {n : ℕ} {c : Fin (n + 1) → C} (t : Tensor S c) :\n t = ∑ i, permT _ (IsReindexing.append_of_first c)\n (prodT (basis ![c 0] (ComponentIdx.single.symm i)) (evalT 0 i t))", + "line": 217 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/LeviCivita/Basic.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/LeviCivita/Basic.lean", + "kind": "def", + "name": "realLorentzTensor.leviCivita", + "docstring": "The Levi-Civita tensor `εᵘᵛᵖᵟ` as a real Lorentz tensor in `d = 3`, with `ε⁰¹²³ = 1`.\n\nThe component on a multi-index `f` is the generalized Kronecker delta of `f` against the\nidentity, i.e. the sign of `f` when `f` is a permutation and `0` otherwise.", + "snippet": "noncomputable def leviCivita : ℝT[3, .up, .up, .up, .up]", + "line": 41 + }, + { + "file": "Physlib/Relativity/Tensors/LeviCivita/Basic.lean", + "kind": "lemma", + "name": "realLorentzTensor.leviCivita_eq_ofInt", + "docstring": "The `TensorInt.toTensor` form of the Levi-Civita tensor.", + "snippet": "lemma leviCivita_eq_ofInt : ε4 =\n TensorInt.toTensor (S := realLorentzTensor 3)\n (c := ![Color.up, Color.up, Color.up, Color.up]) fun f =>\n generalizedKroneckerDelta (fun i => finSumFinEquiv (f i)) (id : Fin 4 → Fin 4)", + "line": 50 + }, + { + "file": "Physlib/Relativity/Tensors/LeviCivita/Basic.lean", + "kind": "lemma", + "name": "realLorentzTensor.leviCivita_basis_repr_apply", + "docstring": "The components of the Levi-Civita tensor in the standard basis are the generalized\nKronecker delta of the multi-index against the identity.", + "snippet": "lemma leviCivita_basis_repr_apply\n (b : ComponentIdx (S := realLorentzTensor 3) ![Color.up, Color.up, Color.up, Color.up]) :\n (Tensor.basis _).repr ε4 b\n = (generalizedKroneckerDelta (fun i => finSumFinEquiv (b i)) (id : Fin 4 → Fin 4) : ℝ)", + "line": 58 + }, + { + "file": "Physlib/Relativity/Tensors/LeviCivita/Basic.lean", + "kind": "lemma", + "name": "realLorentzTensor.leviCivita_basis_repr_eq_zero_of_eq", + "docstring": "The Levi-Civita tensor vanishes on any multi-index with a repeated value: if two distinct\nindex positions `i ≠ j` carry the same basis index, the component is zero.", + "snippet": "lemma leviCivita_basis_repr_eq_zero_of_eq\n {b : ComponentIdx (S := realLorentzTensor 3) ![Color.up, Color.up, Color.up, Color.up]}\n {i j : Fin 4} (hij : i ≠ j) (h : b i = b j) :\n (Tensor.basis _).repr ε4 b = 0", + "line": 66 + }, + { + "file": "Physlib/Relativity/Tensors/LeviCivita/Basic.lean", + "kind": "lemma", + "name": "realLorentzTensor.leviCivita_antisymm", + "docstring": "The Levi-Civita tensor is antisymmetric in its first two indices\n`{ε4 | μ ν ρ σ = - ε4 | ν μ ρ σ}ᵀ`.", + "snippet": "lemma leviCivita_antisymm : {ε4 | μ ν ρ σ = - (ε4 | ν μ ρ σ)}ᵀ", + "line": 82 + }, + { + "file": "Physlib/Relativity/Tensors/LeviCivita/Basic.lean", + "kind": "lemma", + "name": "realLorentzTensor.leviCivita_antisymm_mid", + "docstring": "The Levi-Civita tensor is antisymmetric in its middle two indices\n`{ε4 | μ ν ρ σ = - ε4 | μ ρ ν σ}ᵀ`.", + "snippet": "lemma leviCivita_antisymm_mid : {ε4 | μ ν ρ σ = - (ε4 | μ ρ ν σ)}ᵀ", + "line": 96 + }, + { + "file": "Physlib/Relativity/Tensors/LeviCivita/Basic.lean", + "kind": "lemma", + "name": "realLorentzTensor.leviCivita_antisymm_last", + "docstring": "The Levi-Civita tensor is antisymmetric in its last two indices\n`{ε4 | μ ν ρ σ = - ε4 | μ ν σ ρ}ᵀ`.", + "snippet": "lemma leviCivita_antisymm_last : {ε4 | μ ν ρ σ = - (ε4 | μ ν σ ρ)}ᵀ", + "line": 110 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/Product.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/Product.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.Pure.prodP_update_left", + "docstring": null, + "snippet": "@[simp]\nlemma Pure.prodP_update_left {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C}\n (p1 : Pure S c) (p2 : Pure S c1) (i : Fin n1) (x : V (c i)) :\n Pure.prodP (Pure.update p1 i x) p2\n = Pure.update (Pure.prodP p1 p2) (Fin.castAdd n2 i)\n (LinearEquiv.cast (R := k) (by simp) x)", + "line": 353 + }, + { + "file": "Physlib/Relativity/Tensors/Product.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.Pure.prodP_update_right", + "docstring": null, + "snippet": "@[simp]\nlemma Pure.prodP_update_right {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C}\n (p1 : Pure S c) (p2 : Pure S c1) (i : Fin n2) (x : V (c1 i)) :\n Pure.prodP p1 (Pure.update p2 i x)\n = Pure.update (Pure.prodP p1 p2) (Fin.natAdd n1 i)\n (LinearEquiv.cast (R := k) (by simp) x)", + "line": 378 + }, + { + "file": "Physlib/Relativity/Tensors/Product.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.Pure.prodP_update_add_toTensor_left", + "docstring": null, + "snippet": "lemma Pure.prodP_update_add_toTensor_left {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C}\n (p1 : Pure S c) (p2 : Pure S c1) (i : Fin n2) (x y : V (c1 i)) :\n (Pure.prodP p1 (Pure.update p2 i (x + y))).toTensor\n = (Pure.prodP p1 (Pure.update p2 i x)).toTensor\n + (Pure.prodP p1 (Pure.update p2 i y)).toTensor", + "line": 402 + }, + { + "file": "Physlib/Relativity/Tensors/Product.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.prodT_basis'", + "docstring": null, + "snippet": "lemma prodT_basis' {n1 n2} {c : Fin n1 → C} {c1 : Fin n2 → C}\n (b : ComponentIdx c) (b1 : ComponentIdx (S := S) c1) :\n (basis c b).prodT (basis c1 b1) =\n basis (Fin.append c c1) (ComponentIdx.prod.symm (b, b1))", + "line": 467 + }, + { + "file": "Physlib/Relativity/Tensors/Product.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.prodT_zero_right", + "docstring": null, + "snippet": "lemma prodT_zero_right {n} {c : Fin n → C}\n {c1 : Fin 0 → C} (t : S.Tensor c) (t1 : S.Tensor c1) :\n prodT t t1 = (toField t1) • permT id (IsReindexing.append_zero_right) t", + "line": 636 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/RealTensor/Basic.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/RealTensor/Basic.lean", + "kind": "abbrev", + "name": "realLorentzTensor.modules", + "docstring": "The modules associated with each of the different types of real Lorentz vector space.", + "snippet": "abbrev modules (d : ℕ) : Color → Type\n | Color.up => Lorentz.ContrMod d\n | Color.down => Lorentz.CoMod d", + "line": 47 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Basic.lean", + "kind": "instance", + "name": "realLorentzTensor.modulesAddCommGroup", + "docstring": null, + "snippet": "instance modulesAddCommGroup (d : ℕ) : ∀ c, AddCommGroup (modules d c)\n | Color.up => inferInstance\n | Color.down => inferInstance", + "line": 51 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Basic.lean", + "kind": "instance", + "name": "realLorentzTensor.modulesModule", + "docstring": null, + "snippet": "instance modulesModule (d : ℕ) : ∀ c, Module ℝ (modules d c)\n | Color.up => inferInstance\n | Color.down => inferInstance", + "line": 55 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/RealTensor/CoVector/Basic.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Basic.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.eq_of_apply_eq", + "docstring": null, + "snippet": "@[ext]\nlemma eq_of_apply_eq {d : ℕ} {v w : CoVector d} (h : ∀ i, v i = w i) : v = w", + "line": 54 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Basic.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.apply_sum", + "docstring": null, + "snippet": "@[simp]\nlemma apply_sum {d : ℕ} {ι : Type} [Fintype ι] (f : ι → CoVector d) (i : Fin 1 ⊕ Fin d) :\n (∑ j, f j) i = ∑ j, f j i", + "line": 126 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/RealTensor/CoVector/Representation.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Representation.lean", + "kind": "def", + "name": "Lorentz.CoVector.rep", + "docstring": "The representation of the Lorentz group on Lorentz vectors.", + "snippet": "def rep {d : ℕ} : Representation ℝ (LorentzGroup d) (CoVector d) where", + "line": 34 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.rep_apply_eq_mulVec", + "docstring": null, + "snippet": "lemma rep_apply_eq_mulVec (d : ℕ) (Λ : LorentzGroup d) (v : CoVector d) :\n rep Λ v = (LorentzGroup.transpose Λ⁻¹) *ᵥ v", + "line": 48 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.rep_apply_eq_sum", + "docstring": null, + "snippet": "lemma rep_apply_eq_sum (d : ℕ) (Λ : LorentzGroup d) (v : CoVector d) (k : Fin 1 ⊕ Fin d) :\n rep Λ v k = ∑ j, (Λ⁻¹).1 j k • v j", + "line": 51 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.rep_apply_eq_sum_coe", + "docstring": null, + "snippet": "lemma rep_apply_eq_sum_coe (d : ℕ) (Λ : LorentzGroup d) (v : CoVector d) (k : Fin 1 ⊕ Fin d) :\n rep Λ v k = ∑ j, Λ.1⁻¹ j k • v j", + "line": 54 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.rep_apply_basis", + "docstring": null, + "snippet": "lemma rep_apply_basis {d} (μ : Fin 1 ⊕ Fin d) (Λ : LorentzGroup d) :\n rep Λ (basis μ) = ∑ j, Λ.1⁻¹ μ j • basis j", + "line": 58 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.rep_toMatrix", + "docstring": null, + "snippet": "lemma rep_toMatrix (d : ℕ) (Λ : LorentzGroup d) :\n LinearMap.toMatrix basis basis (rep Λ) = Λ.1⁻¹ᵀ", + "line": 63 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.rep_injective", + "docstring": null, + "snippet": "lemma rep_injective (d : ℕ) (Λ : LorentzGroup d) : Function.Injective (rep Λ)", + "line": 69 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.rep_surjective", + "docstring": null, + "snippet": "lemma rep_surjective (d : ℕ) (Λ : LorentzGroup d) : Function.Surjective (rep Λ)", + "line": 74 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.rep_bijective", + "docstring": null, + "snippet": "lemma rep_bijective (d : ℕ) (Λ : LorentzGroup d) : Function.Bijective (rep Λ)", + "line": 80 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "def", + "name": "Lorentz.CoVector.indexEquiv", + "docstring": "The equivalence between the type of indices of a Lorentz vector and\n `Fin 1 ⊕ Fin d`.", + "snippet": "def indexEquiv {d : ℕ} :\n ComponentIdx (S := (realLorentzTensor d)) ![Color.down] ≃ Fin 1 ⊕ Fin d", + "line": 46 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "instance", + "name": "Lorentz.CoVector.tensorial", + "docstring": null, + "snippet": "instance tensorial {d : ℕ} : Tensorial (realLorentzTensor d) ![.down] (CoVector d) where", + "line": 50 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.toTensor_symm_apply", + "docstring": null, + "snippet": "lemma toTensor_symm_apply {d : ℕ} (p : ℝT[d, .down]) :\n (toTensor (self := tensorial)).symm p =\n (Equiv.piCongrLeft' _ indexEquiv <|\n Finsupp.equivFunOnFinite <|\n (Tensor.basis (S := (realLorentzTensor d)) _).repr p)", + "line": 65 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.toTensor_symm_pure", + "docstring": null, + "snippet": "lemma toTensor_symm_pure {d : ℕ} (p : Pure (realLorentzTensor d) ![.down]) (i : Fin 1 ⊕ Fin d) :\n (toTensor (self := tensorial)).symm p.toTensor i =\n ((Lorentz.coBasis d).repr (p 0)) (indexEquiv.symm i 0)", + "line": 71 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.toTensor_symm_basis", + "docstring": null, + "snippet": "lemma toTensor_symm_basis {d : ℕ} (μ : Fin 1 ⊕ Fin d) :\n (toTensor (self := tensorial)).symm (Tensor.basis ![Color.down] (indexEquiv.symm μ)) =\n basis μ", + "line": 87 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.toTensor_basis_eq_tensor_basis", + "docstring": null, + "snippet": "lemma toTensor_basis_eq_tensor_basis {d : ℕ} (μ : Fin 1 ⊕ Fin d) :\n toTensor (basis μ) = Tensor.basis ![Color.down] (indexEquiv.symm μ)", + "line": 94 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.basis_eq_map_tensor_basis", + "docstring": null, + "snippet": "lemma basis_eq_map_tensor_basis {d} : basis =\n ((Tensor.basis (S := realLorentzTensor d) ![Color.down]).map\n toTensor.symm).reindex indexEquiv", + "line": 99 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.tensor_basis_map_eq_basis_reindex", + "docstring": null, + "snippet": "lemma tensor_basis_map_eq_basis_reindex {d} :\n (Tensor.basis (S := realLorentzTensor d) ![Color.down]).map toTensor.symm =\n basis.reindex indexEquiv.symm", + "line": 106 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.tensor_basis_repr_toTensor_apply", + "docstring": null, + "snippet": "lemma tensor_basis_repr_toTensor_apply {d : ℕ} (p : CoVector d) (μ : ComponentIdx ![Color.down]) :\n (Tensor.basis ![Color.down]).repr (toTensor p) μ =\n p (indexEquiv μ)", + "line": 113 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.smul_eq_sum", + "docstring": null, + "snippet": "lemma smul_eq_sum {d : ℕ} (i : Fin 1 ⊕ Fin d) (Λ : LorentzGroup d) (p : CoVector d) :\n (Λ • p) i = ∑ j, Λ⁻¹.1 j i * p j", + "line": 136 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.smul_eq_mulVec", + "docstring": null, + "snippet": "lemma smul_eq_mulVec {d} (Λ : LorentzGroup d) (p : CoVector d) :\n Λ • p = (LorentzGroup.transpose Λ⁻¹).1 *ᵥ p", + "line": 157 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.smul_add", + "docstring": null, + "snippet": "lemma smul_add {d : ℕ} (Λ : LorentzGroup d) (p q : CoVector d) :\n Λ • (p + q) = Λ • p + Λ • q", + "line": 165 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.smul_sub", + "docstring": null, + "snippet": "@[simp]\nlemma smul_sub {d : ℕ} (Λ : LorentzGroup d) (p q : CoVector d) :\n Λ • (p - q) = Λ • p - Λ • q", + "line": 170 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.smul_zero", + "docstring": null, + "snippet": "lemma smul_zero {d : ℕ} (Λ : LorentzGroup d) :\n Λ • (0 : CoVector d) = 0", + "line": 175 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.smul_neg", + "docstring": null, + "snippet": "lemma smul_neg {d : ℕ} (Λ : LorentzGroup d) (p : CoVector d) :\n Λ • (-p) = - (Λ • p)", + "line": 180 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "def", + "name": "Lorentz.CoVector.actionCLM", + "docstring": "The Lorentz action on vectors as a continuous linear map.", + "snippet": "def actionCLM {d : ℕ} (Λ : LorentzGroup d) :\n CoVector d →L[ℝ] CoVector d", + "line": 185 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.actionCLM_apply", + "docstring": null, + "snippet": "lemma actionCLM_apply {d : ℕ} (Λ : LorentzGroup d) (p : CoVector d) :\n actionCLM Λ p = Λ • p", + "line": 194 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/CoVector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.CoVector.smul_basis", + "docstring": null, + "snippet": "lemma smul_basis {d : ℕ} (Λ : LorentzGroup d) (μ : Fin 1 ⊕ Fin d) :\n Λ • basis μ = ∑ ν, Λ⁻¹.1 μ ν • basis ν", + "line": 198 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/RealTensor/ToComplex.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/RealTensor/ToComplex.lean", + "kind": "def", + "name": "realLorentzTensor.toComplexVector", + "docstring": "For a given color, the map turning a real Lorentz vector into a complex one.", + "snippet": "noncomputable def toComplexVector (c : realLorentzTensor.Color) :\n realLorentzTensor.modules 3 c →ₛₗ[Complex.ofRealHom] complexLorentzTensor.modules\n (colorToComplex c) where", + "line": 302 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/ToComplex.lean", + "kind": "lemma", + "name": "realLorentzTensor.toComplexVector_up_eq_inclCongrRealLorentz", + "docstring": null, + "snippet": "lemma toComplexVector_up_eq_inclCongrRealLorentz (v : Lorentz.ContrMod 3) :\n toComplexVector Color.up v = Lorentz.inclCongrRealLorentz v", + "line": 343 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/ToComplex.lean", + "kind": "lemma", + "name": "realLorentzTensor.toComplexVector_down_eq_inclCoRealLorentz", + "docstring": null, + "snippet": "lemma toComplexVector_down_eq_inclCoRealLorentz (v : Lorentz.CoMod 3) :\n toComplexVector Color.down v = Lorentz.inclCoRealLorentz v", + "line": 361 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/ToComplex.lean", + "kind": "def", + "name": "realLorentzTensor.toComplexPure", + "docstring": "The function which turns a real pure tensor into a complex one.", + "snippet": "noncomputable def toComplexPure {c : Fin n → Color} (p : Pure realLorentzTensor c) :\n Pure complexLorentzTensor (colorToComplex ∘ c)", + "line": 379 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/ToComplex.lean", + "kind": "lemma", + "name": "realLorentzTensor.toComplexPure_component", + "docstring": null, + "snippet": "lemma toComplexPure_component {c : Fin n → Color} (p : Pure realLorentzTensor c)\n (φ : ComponentIdx c) : (toComplexPure p).component (ComponentIdx.complexify φ) =\n p.component φ", + "line": 383 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/ToComplex.lean", + "kind": "lemma", + "name": "realLorentzTensor.actionP_toComplexPure", + "docstring": null, + "snippet": "lemma actionP_toComplexPure {n : ℕ} (c : Fin n → Color) (p : Pure realLorentzTensor c)\n (Λ : SL(2, ℂ)) :\n Λ • toComplexPure p = toComplexPure (toLorentzGroup Λ • p)", + "line": 446 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/ToComplex.lean", + "kind": "lemma", + "name": "realLorentzTensor.toComplex_pure", + "docstring": null, + "snippet": "lemma toComplex_pure {n : ℕ} (c : Fin n → Color) (p : Pure realLorentzTensor c) :\n toComplex p.toTensor = (toComplexPure p).toTensor", + "line": 492 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/ToComplex.lean", + "kind": "lemma", + "name": "realLorentzTensor.isReindexing_prodTColorToComplex", + "docstring": null, + "snippet": "lemma isReindexing_prodTColorToComplex {n m : ℕ}\n {c : Fin n → realLorentzTensor.Color} {c1 : Fin m → realLorentzTensor.Color} :\n IsReindexing (Fin.append (colorToComplex ∘ c) (colorToComplex ∘ c1))\n (colorToComplex ∘ Fin.append c c1)\n (id : Fin (n + m) → Fin (n + m))", + "line": 628 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/RealTensor/Vector/Basic.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Basic.lean", + "kind": "lemma", + "name": "Lorentz.Vector.eq_of_apply_eq", + "docstring": null, + "snippet": "@[ext]\nlemma eq_of_apply_eq {d : ℕ} {v w : Vector d} (h : ∀ i, v i = w i) : v = w", + "line": 58 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Basic.lean", + "kind": "def", + "name": "Lorentz.Vector.ofSpatialComponent", + "docstring": "The continuous linear map corresponding to the creation of a\n Lorentz Vector with only non-zero spatial components.", + "snippet": "def ofSpatialComponent {d : ℕ} : EuclideanSpace ℝ (Fin d) →L[ℝ] Vector d where", + "line": 410 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/RealTensor/Vector/Representation.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Representation.lean", + "kind": "def", + "name": "Lorentz.Vector.rep", + "docstring": "The representation of the Lorentz group on Lorentz vectors.", + "snippet": "def rep {d : ℕ} : Representation ℝ (LorentzGroup d) (Vector d) where", + "line": 34 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.Vector.rep_apply_eq_mulVec", + "docstring": null, + "snippet": "lemma rep_apply_eq_mulVec (d : ℕ) (Λ : LorentzGroup d) (v : Vector d) :\n rep Λ v = Λ *ᵥ v", + "line": 45 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.Vector.rep_apply_eq_sum", + "docstring": null, + "snippet": "lemma rep_apply_eq_sum (d : ℕ) (Λ : LorentzGroup d) (v : Vector d) (k : Fin 1 ⊕ Fin d) :\n rep Λ v k = ∑ j, Λ.1 k j • v j", + "line": 48 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.Vector.rep_apply_basis", + "docstring": null, + "snippet": "lemma rep_apply_basis {d} (μ : Fin 1 ⊕ Fin d) (Λ : LorentzGroup d) :\n rep Λ (basis μ) = ∑ j, Λ.1 j μ • basis j", + "line": 51 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.Vector.rep_toMatrix", + "docstring": null, + "snippet": "lemma rep_toMatrix (d : ℕ) (Λ : LorentzGroup d) :\n LinearMap.toMatrix basis basis (rep Λ) = Λ.1", + "line": 56 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.Vector.rep_injective", + "docstring": null, + "snippet": "lemma rep_injective (d : ℕ) (Λ : LorentzGroup d) : Function.Injective (rep Λ)", + "line": 61 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.Vector.rep_surjective", + "docstring": null, + "snippet": "lemma rep_surjective (d : ℕ) (Λ : LorentzGroup d) : Function.Surjective (rep Λ)", + "line": 66 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.Vector.rep_bijective", + "docstring": null, + "snippet": "lemma rep_bijective (d : ℕ) (Λ : LorentzGroup d) : Function.Bijective (rep Λ)", + "line": 72 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.Vector.rep_contDiff", + "docstring": null, + "snippet": "@[fun_prop]\nlemma rep_contDiff (d : ℕ) {n} (Λ : LorentzGroup d) : ContDiff ℝ n (rep Λ)", + "line": 76 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Representation.lean", + "kind": "lemma", + "name": "Lorentz.Vector.rep_left_injective", + "docstring": null, + "snippet": "lemma rep_left_injective (d : ℕ) : Function.Injective (rep (d := d))", + "line": 82 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "def", + "name": "Lorentz.Vector.indexEquiv", + "docstring": "The equivalence between the type of indices of a Lorentz vector and\n `Fin 1 ⊕ Fin d`.", + "snippet": "def indexEquiv {d : ℕ} :\n ComponentIdx (S := (realLorentzTensor d)) ![Color.up] ≃ Fin 1 ⊕ Fin d", + "line": 46 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "instance", + "name": "Lorentz.Vector.tensorial", + "docstring": null, + "snippet": "instance tensorial {d : ℕ} : Tensorial (realLorentzTensor d) ![.up] (Vector d) where", + "line": 50 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.toTensor_symm_apply", + "docstring": null, + "snippet": "lemma toTensor_symm_apply {d : ℕ} (p : ℝT[d, .up]) :\n (toTensor (self := tensorial)).symm p =\n (Equiv.piCongrLeft' _ indexEquiv <|\n Finsupp.equivFunOnFinite <|\n (Tensor.basis (S := (realLorentzTensor d)) _).repr p)", + "line": 65 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.toTensor_symm_pure", + "docstring": null, + "snippet": "lemma toTensor_symm_pure {d : ℕ} (p : Pure (realLorentzTensor d) ![.up]) (i : Fin 1 ⊕ Fin d) :\n (toTensor (self := tensorial)).symm p.toTensor i =\n ((Lorentz.contrBasis d).repr (p 0)) (indexEquiv.symm i 0)", + "line": 71 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.toTensor_symm_basis", + "docstring": null, + "snippet": "lemma toTensor_symm_basis {d : ℕ} (μ : Fin 1 ⊕ Fin d) :\n (toTensor (self := tensorial)).symm (Tensor.basis ![Color.up] (indexEquiv.symm μ)) =\n basis μ", + "line": 86 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.toTensor_basis_eq_tensor_basis", + "docstring": null, + "snippet": "lemma toTensor_basis_eq_tensor_basis {d : ℕ} (μ : Fin 1 ⊕ Fin d) :\n toTensor (basis μ) = Tensor.basis ![Color.up] (indexEquiv.symm μ)", + "line": 93 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.basis_eq_map_tensor_basis", + "docstring": null, + "snippet": "lemma basis_eq_map_tensor_basis {d} : basis =\n ((Tensor.basis\n (S := realLorentzTensor d) ![Color.up]).map toTensor.symm).reindex indexEquiv", + "line": 98 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.tensor_basis_map_eq_basis_reindex", + "docstring": null, + "snippet": "lemma tensor_basis_map_eq_basis_reindex {d} :\n (Tensor.basis (S := realLorentzTensor d) ![Color.up]).map toTensor.symm =\n basis.reindex indexEquiv.symm", + "line": 105 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.tensor_basis_repr_toTensor_apply", + "docstring": null, + "snippet": "lemma tensor_basis_repr_toTensor_apply {d : ℕ} (p : Vector d) (μ : ComponentIdx ![Color.up]) :\n (Tensor.basis ![Color.up]).repr (toTensor p) μ =\n p (indexEquiv μ)", + "line": 112 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.smul_eq_sum", + "docstring": null, + "snippet": "lemma smul_eq_sum {d : ℕ} (i : Fin 1 ⊕ Fin d) (Λ : LorentzGroup d) (p : Vector d) :\n (Λ • p) i = ∑ j, Λ.1 i j * p j", + "line": 133 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.smul_eq_mulVec", + "docstring": null, + "snippet": "lemma smul_eq_mulVec {d} (Λ : LorentzGroup d) (p : Vector d) :\n Λ • p = Λ.1 *ᵥ p", + "line": 154 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.smul_add", + "docstring": null, + "snippet": "lemma smul_add {d : ℕ} (Λ : LorentzGroup d) (p q : Vector d) :\n Λ • (p + q) = Λ • p + Λ • q", + "line": 161 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.smul_sub", + "docstring": null, + "snippet": "@[simp]\nlemma smul_sub {d : ℕ} (Λ : LorentzGroup d) (p q : Vector d) :\n Λ • (p - q) = Λ • p - Λ • q", + "line": 166 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.smul_zero", + "docstring": null, + "snippet": "lemma smul_zero {d : ℕ} (Λ : LorentzGroup d) :\n Λ • (0 : Vector d) = 0", + "line": 171 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.smul_neg", + "docstring": null, + "snippet": "lemma smul_neg {d : ℕ} (Λ : LorentzGroup d) (p : Vector d) :\n Λ • (-p) = - (Λ • p)", + "line": 176 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.neg_smul", + "docstring": null, + "snippet": "lemma neg_smul {d} (Λ : LorentzGroup d) (p : Vector d) :\n (-Λ) • p = - (Λ • p)", + "line": 180 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "LorentzGroup.eq_of_action_vector_eq", + "docstring": null, + "snippet": "lemma _root_.LorentzGroup.eq_of_action_vector_eq {d : ℕ}\n {Λ Λ' : LorentzGroup d} (h : ∀ p : Vector d, Λ • p = Λ' • p) :\n Λ = Λ'", + "line": 185 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "def", + "name": "Lorentz.Vector.actionCLM", + "docstring": "The Lorentz action on vectors as a continuous linear map.", + "snippet": "def actionCLM {d : ℕ} (Λ : LorentzGroup d) :\n Vector d →L[ℝ] Vector d", + "line": 198 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.actionCLM_apply", + "docstring": null, + "snippet": "lemma actionCLM_apply {d : ℕ} (Λ : LorentzGroup d) (p : Vector d) :\n actionCLM Λ p = Λ • p", + "line": 208 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.actionCLM_injective", + "docstring": null, + "snippet": "lemma actionCLM_injective {d : ℕ} (Λ : LorentzGroup d) :\n Function.Injective (actionCLM Λ)", + "line": 211 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.actionCLM_surjective", + "docstring": null, + "snippet": "lemma actionCLM_surjective {d : ℕ} (Λ : LorentzGroup d) :\n Function.Surjective (actionCLM Λ)", + "line": 216 + }, + { + "file": "Physlib/Relativity/Tensors/RealTensor/Vector/Tensorial.lean", + "kind": "lemma", + "name": "Lorentz.Vector.smul_basis", + "docstring": null, + "snippet": "lemma smul_basis {d : ℕ} (Λ : LorentzGroup d) (μ : Fin 1 ⊕ Fin d) :\n Λ • basis μ = ∑ ν, Λ.1 ν μ • basis ν", + "line": 223 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/Reindexing.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "def", + "name": "TensorSpecies.Tensor.IsReindexing", + "docstring": "Given two lists of indices `c : Fin n → C` and `c1 : Fin m → C` a map\n `σ : Fin m → Fin n` satisfies the condition `IsReindexing c c1 σ` if it is:\n- A bijection\n- Forms a commutative triangle with `c` and `c1`.", + "snippet": "def IsReindexing {n m : ℕ} (c : Fin n → C) (c1 : Fin m → C)\n (σ : Fin m → Fin n) : Prop", + "line": 49 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.injective", + "docstring": null, + "snippet": "lemma injective {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}\n {σ : Fin m → Fin n} (h : IsReindexing c c1 σ) : Function.Injective σ", + "line": 61 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.surjective", + "docstring": null, + "snippet": "lemma surjective {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}\n {σ : Fin m → Fin n} (h : IsReindexing c c1 σ) : Function.Surjective σ", + "line": 64 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.auto", + "docstring": null, + "snippet": "lemma auto {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}\n {σ : Fin m → Fin n} (h : IsReindexing c c1 σ := by {simp [IsReindexing]; try decide}) :\n IsReindexing c c1 σ", + "line": 68 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.on_id", + "docstring": null, + "snippet": "@[simp]\nlemma on_id {n : ℕ} {c c1 : Fin n → C} :\n IsReindexing c c1 (id : Fin n → Fin n) ↔ ∀ i, c i = c1 i", + "line": 74 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.on_id_symm", + "docstring": null, + "snippet": "lemma on_id_symm {n : ℕ} {c c1 : Fin n → C} (h : IsReindexing c1 c id) :\n IsReindexing c c1 (id : Fin n → Fin n)", + "line": 78 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "def", + "name": "TensorSpecies.Tensor.IsReindexing.inv", + "docstring": "For a map `σ` satisfying `IsReindexing c c1 σ`, the inverse of that map.", + "snippet": "def inv {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}\n (σ : Fin m → Fin n) (h : IsReindexing c c1 σ) : Fin n → Fin m", + "line": 84 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "def", + "name": "TensorSpecies.Tensor.IsReindexing.toEquiv", + "docstring": "For a map `σ : Fin m → Fin n` satisfying `IsReindexing c c1 σ`,\n that map lifted to an equivalence between\n `Fin n` and `Fin m`.", + "snippet": "def toEquiv {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}\n {σ : Fin m → Fin n} (h : IsReindexing c c1 σ) :\n Fin n ≃ Fin m where", + "line": 91 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.apply_inv_apply", + "docstring": null, + "snippet": "lemma apply_inv_apply {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}\n (σ : Fin m → Fin n) (h : IsReindexing c c1 σ) (x : Fin m) :\n h.inv σ (σ x) = x", + "line": 99 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.inv_apply_apply", + "docstring": null, + "snippet": "lemma inv_apply_apply {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}\n (σ : Fin m → Fin n) (h : IsReindexing c c1 σ) (x : Fin n) :\n σ (h.inv σ x) = x", + "line": 105 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.preserve_color", + "docstring": null, + "snippet": "lemma preserve_color {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}\n {σ : Fin m → Fin n} (h : IsReindexing c c1 σ) :\n ∀ (x : Fin m), c1 x = (c ∘ σ) x", + "line": 111 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.inv_perserve_color", + "docstring": null, + "snippet": "@[simp, nolint simpVarHead]\nlemma inv_perserve_color {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}\n {σ : Fin m → Fin n} (h : IsReindexing c c1 σ) (x : Fin n) :\n c1 (h.inv σ x) = c x", + "line": 121 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.toEquiv_symm_perserve_color", + "docstring": null, + "snippet": "@[simp, nolint simpVarHead]\nlemma toEquiv_symm_perserve_color {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}\n {σ : Fin m → Fin n} (h : IsReindexing c c1 σ) (x : Fin m) :\n c (h.toEquiv.symm x) = c1 x", + "line": 132 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.symm", + "docstring": "The inverse of a map satisfying `IsReindexing c c1 σ` is a reindexing of `c1` by `c`.", + "snippet": "lemma symm {n m : ℕ} {c : Fin n → C} {c1 : Fin m → C}\n {σ : Fin m → Fin n} (h : IsReindexing c c1 σ) : IsReindexing c1 c (h.inv σ)", + "line": 147 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.symm_of_id", + "docstring": null, + "snippet": "lemma symm_of_id {n : ℕ} {c c1 : Fin n → C} (h : IsReindexing c c1 id) :\n IsReindexing c1 c id", + "line": 151 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.comp", + "docstring": "The composition of two maps satisfying `IsReindexing` also satisfies the `IsReindexing`.", + "snippet": "lemma comp {n n1 n2 : ℕ} {c : Fin n → C} {c1 : Fin n1 → C}\n {c2 : Fin n2 → C} {σ : Fin n1 → Fin n} {σ2 : Fin n2 → Fin n1}\n (h : IsReindexing c c1 σ) (h2 : IsReindexing c1 c2 σ2) : IsReindexing c c2 (σ ∘ σ2)", + "line": 157 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.append_congr_left", + "docstring": null, + "snippet": "lemma append_congr_left {n n' n2 : ℕ} {c : Fin n → C} {c' : Fin n' → C}\n {σ : Fin n' → Fin n} (c2 : Fin n2 → C) (h : IsReindexing c c' σ) :\n IsReindexing (Fin.append c c2) (Fin.append c' c2)\n (Fin.append (Fin.castAdd n2 ∘ σ) (Fin.natAdd n))", + "line": 162 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.append_congr_right", + "docstring": null, + "snippet": "lemma append_congr_right {n n' n2 : ℕ} {c : Fin n → C} {c' : Fin n' → C}\n {σ : Fin n' → Fin n} (c2 : Fin n2 → C) (h : IsReindexing c c' σ) :\n IsReindexing (Fin.append c2 c) (Fin.append c2 c')\n (Fin.append (Fin.castAdd n) (Fin.natAdd n2 ∘ σ))", + "line": 178 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.append_zero_right", + "docstring": null, + "snippet": "lemma append_zero_right {n} {c : Fin n → C}\n {c1 : Fin 0 → C} : IsReindexing c (Fin.append c c1) id", + "line": 194 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.append_swap", + "docstring": null, + "snippet": "lemma append_swap {n n2 : ℕ} {c : Fin n → C} {c2 : Fin n2 → C} :\n IsReindexing (Fin.append c c2) (Fin.append c2 c)\n (Fin.append (Fin.natAdd n) (Fin.castAdd n2))", + "line": 203 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.append_assoc_right", + "docstring": null, + "snippet": "lemma append_assoc_right {n1 n2 n3 : ℕ} {c : Fin n1 → C} {c2 : Fin n2 → C} {c3 : Fin n3 → C} :\n IsReindexing (Fin.append c (Fin.append c2 c3)) (Fin.append (Fin.append c c2) c3)\n (Fin.cast (by grind))", + "line": 218 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.append_assoc_left", + "docstring": null, + "snippet": "lemma append_assoc_left {n1 n2 n3 : ℕ} {c : Fin n1 → C} {c2 : Fin n2 → C} {c3 : Fin n3 → C} :\n IsReindexing (Fin.append (Fin.append c c2) c3) (Fin.append c (Fin.append c2 c3))\n (Fin.cast (by grind))", + "line": 223 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.append_succAbove_natAdd", + "docstring": null, + "snippet": "lemma append_succAbove_natAdd {n n1 : ℕ} {c : Fin n → C} {c1 : Fin (n1 + 1) → C}\n (i : Fin (n1 + 1)) :\n IsReindexing (Fin.append c c1 ∘ (Fin.natAdd n i).succAbove)\n (Fin.append c (c1 ∘ i.succAbove)) id", + "line": 228 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.append_succAbove_castAdd", + "docstring": null, + "snippet": "lemma append_succAbove_castAdd {n n1 : ℕ} {c : Fin (n + 1) → C} {c1 : Fin (n1 + 1) → C}\n (i : Fin (n + 1)) :\n IsReindexing (Fin.append c c1 ∘ (Fin.castAdd (n1 + 1) i).succAbove)\n (Fin.append (c ∘ i.succAbove) c1) (Fin.cast (by grind))", + "line": 250 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.append_succSuccAbove_natAdd", + "docstring": "Removing two entries from the right component of `Fin.append c1 c` commutes with the\n append: removing the `i`-th and `j`-th entries of `c` and then appending `c1` matches\n removing the corresponding entries of `Fin.append c1 c`, via the identity permutation.\n This is used for the commutation of taking a *product* of tensors\n with *contraction* of indices.", + "snippet": "lemma append_succSuccAbove_natAdd {n n1 : ℕ} {c : Fin (n + 1 + 1) → C}\n {c1 : Fin n1 → C} (i j : Fin (n + 1 + 1)) :\n IsReindexing (Fin.append c1 c ∘ (Fin.natAdd n1 i).succSuccAbove (Fin.natAdd n1 j))\n (Fin.append c1 (c ∘ i.succSuccAbove j)) id", + "line": 279 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.succAbove_of_eq_zero", + "docstring": "Given a reindexing of `c` by `c1` via `σ` for which the index `i` is sent to `0`,\n removing the `i`-th entry of `c1` and the first entry of `c` yields a reindexing of\n `c ∘ Fin.succ` by `c1 ∘ i.succAbove` via the map sending `j` to the predecessor of\n `σ (i.succAbove j)`.", + "snippet": "lemma succAbove_of_eq_zero {n n1 : ℕ} {c : Fin (n + 1) → C} {c1 : Fin (n1 + 1) → C}\n {σ : Fin (n1 + 1) → Fin (n + 1)} (i : Fin (n1 + 1))\n (h : IsReindexing c c1 σ) (hi : σ i = 0) :\n IsReindexing (c ∘ Fin.succ) (c1 ∘ i.succAbove)\n (fun j => (σ (i.succAbove j)).pred (by simp [← hi, h.injective.eq_iff]))", + "line": 290 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.succAbove_of_neq_zero", + "docstring": "Given a reindexing of `c` by `c1` via `σ` for which the index `i` is not sent to `0`,\n removing the `i`-th entry of `c1` and the `(σ i)`-th entry of `c` yields a reindexing of\n `c ∘ (σ i).succAbove` by `c1 ∘ i.succAbove` via the map\n `(σ i).pred.predAbove ∘ σ ∘ i.succAbove`.", + "snippet": "lemma succAbove_of_neq_zero {n n1 : ℕ} {c : Fin (n + 1) → C} {c1 : Fin (n1 + 1) → C}\n {σ : Fin (n1 + 1) → Fin (n + 1)} (i : Fin (n1 + 1))\n (h : IsReindexing c c1 σ) (hi : σ i ≠ 0) :\n IsReindexing (c ∘ (σ i).succAbove) (c1 ∘ i.succAbove)\n ((Fin.pred (σ i) hi).predAbove ∘ σ ∘ i.succAbove)", + "line": 313 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.succAbove", + "docstring": "Given a reindexing of `c` by `c1` via `σ`, removing the `i`-th entry of `c1` and the\n `(σ i)`-th entry of `c` yields a reindexing of `c ∘ (σ i).succAbove` by `c1 ∘ i.succAbove`.\n This unifies `succAbove_of_eq_zero` and `succAbove_of_neq_zero` via a case split on\n whether `σ i = 0`. This is used for the commutation of *permutation*\n of indices with *evaluation* of indices.", + "snippet": "lemma succAbove {n n1 : ℕ} {c : Fin (n + 1) → C} {c1 : Fin (n1 + 1) → C}\n {σ : Fin (n1 + 1) → Fin (n + 1)} (i : Fin (n1 + 1))\n (h : IsReindexing c c1 σ) :\n IsReindexing (c ∘ (σ i).succAbove) (c1 ∘ i.succAbove)\n (if hi : σ i = 0 then fun j => (σ (i.succAbove j)).pred (by simp [← hi, h.injective.eq_iff])\n else (Fin.pred (σ i) hi).predAbove ∘ σ ∘ i.succAbove)", + "line": 352 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.succSuccAbove", + "docstring": "Given a reindexing of `c` by `c1` via `σ` and two distinct indices `i ≠ j`, removing the\n `i`-th and `j`-th entries of `c1` and the `(σ i)`-th and `(σ j)`-th entries of `c` yields a\n reindexing of `c ∘ (σ i).succSuccAbove (σ j)` by `c1 ∘ i.succSuccAbove j`.\n This is used for the commutation of *permutation* of indices with\n *contraction* of indices.", + "snippet": "lemma succSuccAbove {n n1 : ℕ} {c : Fin (n + 1 + 1) → C}\n {c1 : Fin (n1 + 1 + 1) → C}\n (i j : Fin (n1 + 1 + 1)) (hij : i ≠ j)\n {σ : Fin (n1 + 1 + 1) → Fin (n + 1 + 1)} (hσ : IsReindexing c c1 σ) :\n IsReindexing (c ∘ (σ i).succSuccAbove (σ j))\n (c1 ∘ i.succSuccAbove j) (i.funPredPredAbove j hij σ hσ.1)", + "line": 367 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.succSuccAbove_comm", + "docstring": "Removing two pairs of entries from `c` in either order gives the same colour list:\n removing the `i1`-th and `j1`-th entries and then the (shifted) `i2`-th and `j2`-th entries\n matches removing the `i2`-th and `j2`-th entries first and then the (shifted) `i1`-th and\n `j1`-th entries, via the identity permutation.\n This is used for the commutation of two *contractions*.", + "snippet": "lemma succSuccAbove_comm {n : ℕ} {c : Fin (n + 1 + 1 + 1 + 1) → C}\n (i1 j1 : Fin (n + 1 + 1 + 1 + 1)) (i2 j2 : Fin (n + 1 + 1))\n (hij1 : i1 ≠ j1) (hij2 : i2 ≠ j2) :\n let i2'", + "line": 383 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.succAbove_succSuccAbove_comm", + "docstring": "Removing one entry and then a pair of entries from `c` in either order gives the same\n colour list: removing the `k`-th entry and then the (shifted) `i`-th and `j`-th entries\n matches removing the `i`-th and `j`-th entries first and then the (shifted) `k`-th entry,\n via the identity permutation.\n This is used for the commutation of a *contraction* with an *evaluation*.", + "snippet": "lemma succAbove_succSuccAbove_comm {n : ℕ} {c : Fin (n + 1 + 1 + 1) → C}\n (k : Fin (n + 1 + 1 + 1)) (i j : Fin (n + 1 + 1)) (hij : i ≠ j) :\n let i'", + "line": 406 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.succAbove_succAbove_comm", + "docstring": "Removing two single entries from `c` in either order gives the same colour list:\n removing the `k1`-th entry and then the (shifted) `k2`-th entry matches removing the\n `k2`-th entry first and then the (shifted) `k1`-th entry, via the identity permutation.\n This is used for the commutation of two *evaluations* of indices.", + "snippet": "lemma succAbove_succAbove_comm {n : ℕ} {c : Fin (n + 1 + 1) → C}\n (k1 : Fin (n + 1 + 1)) (k2 : Fin (n + 1)) :\n let k2'", + "line": 428 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.append_succ_last", + "docstring": "Splitting a list of colours `c : Fin (n + 1) → C` into its first `n` entries and its\n last entry recovers `c`: the identity permutation matches\n `Fin.append (c ∘ (Fin.last n).succAbove) ![c (Fin.last n)]` with `c`.", + "snippet": "lemma append_succ_last {n : ℕ} (c : Fin (n + 1) → C) :\n IsReindexing (Fin.append (c ∘ (Fin.last n).succAbove) ![c (Fin.last n)]) c id", + "line": 444 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.append_of_first", + "docstring": "Splitting a list of colours `c : Fin (n + 1) → C` into its first entry and its remaining\n `n` entries recovers `c`: the canonical reindexing `Fin (1 + n) ≃ Fin (n + 1)` matches\n `Fin.append ![c 0] (c ∘ Fin.succAbove 0)` with `c`.", + "snippet": "lemma append_of_first {n : ℕ} (c : Fin (n + 1) → C) :\n IsReindexing (Fin.append ![c 0] (c ∘ Fin.succAbove 0)) c (Fin.cast (by grind))", + "line": 454 + }, + { + "file": "Physlib/Relativity/Tensors/Reindexing.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensor.IsReindexing.fin_cast_isReindexing", + "docstring": "Casting the domain along an equality `n1 = n` of lengths is a reindexing of `c` by\n `c ∘ Fin.cast h`.", + "snippet": "lemma fin_cast_isReindexing (n n1 : ℕ) {c : Fin n → C} (h : n1 = n) :\n IsReindexing c (c ∘ Fin.cast h) (Fin.cast h)", + "line": 464 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/Tensorial.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/Tensorial.lean", + "kind": "lemma", + "name": "TensorSpecies.Tensorial.prod_eq_sum_eval", + "docstring": "Double basis expansion of an element of a tensor product `M ⊗[k] M₂` of two `Tensorial`\n one-index spaces. Given bases `b`, `b2` of `M`, `M₂` coming from the single-index tensor bases,\n every `x : M ⊗[k] M₂` is the double sum over `i, j` of the iterated evaluation coefficient\n `toField (evalT 0 j (evalT 0 i (toTensor x)))` times `b i ⊗ₜ b2 j`.", + "snippet": "lemma prod_eq_sum_eval {c c2 : C} {M₂ : Type}\n [Tensorial S ![c] M] [AddCommMonoid M₂] [Module k M₂]\n [Tensorial S ![c2] M₂] {b : Module.Basis (basisIdx c) k M}\n {b2 : Module.Basis (basisIdx c2) k M₂}\n (h : b = ((Tensor.basis (S := S) ![c]).map toTensor.symm).reindex ComponentIdx.single)\n (h2 : b2 = ((Tensor.basis (S := S) ![c2]).map toTensor.symm).reindex ComponentIdx.single)\n (x : M ⊗[k] M₂) : x = ∑ i : (basisIdx c), ∑ j : (basisIdx c2),\n toField (evalT 0 (basisIdxCongr (by rfl) j)\n (evalT 0 (basisIdxCongr (by rfl) i) (toTensor x))) • b i ⊗ₜ[k] b2 j", + "line": 330 + } + ] + }, + { + "filename": "Physlib/Relativity/Tensors/TensorSpecies/Basic.lean", + "entries": [ + { + "file": "Physlib/Relativity/Tensors/TensorSpecies/Basic.lean", + "kind": "lemma", + "name": "TensorSpecies.basisIdxCongr_heq_arg", + "docstring": "`basisIdxCongr` only depends on its endpoints up to `HEq` of the arguments: equal target\ncolours and heterogeneously equal labels give equal casts.", + "snippet": "lemma basisIdxCongr_heq_arg {c1 c2 d : C} (h1 : c1 = d) (h2 : c2 = d)\n {x : basisIdx c1} {y : basisIdx c2} (hxy : HEq x y) :\n basisIdxCongr h1 x = basisIdxCongr h2 y", + "line": 98 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "structure", + "name": "GalileanGroup", + "docstring": "A Galilean transformation in `d` spatial dimensions. The fields are, in order, the spatial\northogonal part, boost velocity, spatial translation, and time translation.", + "snippet": "@[ext]\nstructure GalileanGroup (d : ℕ := 3) where\n rotation : Matrix.orthogonalGroup (Fin d) ℝ\n velocity : EuclideanSpace ℝ (Fin d)\n spaceTranslation : EuclideanSpace ℝ (Fin d)\n timeTranslation : Time", + "line": 30 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.orthogonal_smul_smul", + "docstring": null, + "snippet": "lemma orthogonal_smul_smul (R : Matrix.orthogonalGroup (Fin d) ℝ) (c : ℝ)\n (v : EuclideanSpace ℝ (Fin d)) :\n R • (c • v) = c • (R • v)", + "line": 44 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "instance", + "name": "GalileanGroup.«instance@f34d5bc7»", + "docstring": "The identity Galilean transformation.", + "snippet": "instance : One (GalileanGroup d) where", + "line": 54 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.one_rotation", + "docstring": null, + "snippet": "@[simp]\nlemma one_rotation : (1 : GalileanGroup d).rotation = 1", + "line": 58 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.one_velocity", + "docstring": null, + "snippet": "@[simp]\nlemma one_velocity : (1 : GalileanGroup d).velocity = 0", + "line": 61 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.one_spaceTranslation", + "docstring": null, + "snippet": "@[simp]\nlemma one_spaceTranslation : (1 : GalileanGroup d).spaceTranslation = 0", + "line": 64 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.one_timeTranslation", + "docstring": null, + "snippet": "@[simp]\nlemma one_timeTranslation : (1 : GalileanGroup d).timeTranslation = 0", + "line": 67 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "instance", + "name": "GalileanGroup.«instance@2d436bb6»", + "docstring": "The product whose action is composition: `(g * h) • tx = g • h • tx`.", + "snippet": "instance : Mul (GalileanGroup d) where", + "line": 70 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.mul_rotation", + "docstring": null, + "snippet": "@[simp]\nlemma mul_rotation (g h : GalileanGroup d) :\n (g * h).rotation = g.rotation * h.rotation", + "line": 78 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.mul_velocity", + "docstring": null, + "snippet": "@[simp]\nlemma mul_velocity (g h : GalileanGroup d) :\n (g * h).velocity = g.rotation • h.velocity + g.velocity", + "line": 82 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.mul_spaceTranslation", + "docstring": null, + "snippet": "@[simp]\nlemma mul_spaceTranslation (g h : GalileanGroup d) :\n (g * h).spaceTranslation =\n g.spaceTranslation + g.rotation • h.spaceTranslation + h.timeTranslation.val • g.velocity", + "line": 86 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.mul_timeTranslation", + "docstring": null, + "snippet": "@[simp]\nlemma mul_timeTranslation (g h : GalileanGroup d) :\n (g * h).timeTranslation = g.timeTranslation + h.timeTranslation", + "line": 92 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "instance", + "name": "GalileanGroup.«instance@3107b840»", + "docstring": "The inverse Galilean transformation.", + "snippet": "instance : Inv (GalileanGroup d) where", + "line": 96 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.inv_rotation", + "docstring": null, + "snippet": "@[simp]\nlemma inv_rotation (g : GalileanGroup d) :\n g⁻¹.rotation = g.rotation⁻¹", + "line": 104 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.inv_velocity", + "docstring": "The inverse boost velocity formula.", + "snippet": "@[simp]\nlemma inv_velocity (g : GalileanGroup d) :\n g⁻¹.velocity = -(g.rotation⁻¹ • g.velocity)", + "line": 109 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.inv_spaceTranslation", + "docstring": null, + "snippet": "@[simp]\nlemma inv_spaceTranslation (g : GalileanGroup d) :\n g⁻¹.spaceTranslation =\n -(g.rotation⁻¹ • g.spaceTranslation) +\n g.timeTranslation.val • (g.rotation⁻¹ • g.velocity)", + "line": 113 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.inv_timeTranslation", + "docstring": null, + "snippet": "@[simp]\nlemma inv_timeTranslation (g : GalileanGroup d) :\n g⁻¹.timeTranslation = -g.timeTranslation", + "line": 119 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "instance", + "name": "GalileanGroup.«instance@33c7352c»", + "docstring": "The Galilean transformations form a group under composition.", + "snippet": "instance : Group (GalileanGroup d) where", + "line": 123 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "instance", + "name": "GalileanGroup.«instance@5e5b6b3a»", + "docstring": null, + "snippet": "instance : Inhabited (GalileanGroup d) where", + "line": 156 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "def", + "name": "GalileanGroup.actSpace", + "docstring": "The Galilean action on space at a fixed time.", + "snippet": "noncomputable def actSpace (g : GalileanGroup d) (t : Time) (x : Space d) : Space d", + "line": 162 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "def", + "name": "GalileanGroup.act", + "docstring": "The Galilean action on `Time × Space d`.", + "snippet": "noncomputable def act (g : GalileanGroup d) (tx : Time × Space d) : Time × Space d", + "line": 167 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "instance", + "name": "GalileanGroup.«instance@c761f5d5»", + "docstring": null, + "snippet": "noncomputable instance : MulAction (GalileanGroup d) (Time × Space d) where", + "line": 170 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.smul_fst", + "docstring": null, + "snippet": "@[simp]\nlemma smul_fst (g : GalileanGroup d) (tx : Time × Space d) :\n (g • tx).1 = tx.1 + g.timeTranslation", + "line": 188 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.smul_snd", + "docstring": null, + "snippet": "@[simp]\nlemma smul_snd (g : GalileanGroup d) (tx : Time × Space d) :\n (g • tx).2 = g.actSpace tx.1 tx.2", + "line": 192 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.smul_mk", + "docstring": null, + "snippet": "@[simp]\nlemma smul_mk (g : GalileanGroup d) (t : Time) (x : Space d) :\n g • ((t, x) : Time × Space d) = (t + g.timeTranslation, g.actSpace t x)", + "line": 196 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.actSpace_apply", + "docstring": null, + "snippet": "@[simp]\nlemma actSpace_apply (g : GalileanGroup d) (t : Time) (x : Space d) (i : Fin d) :\n g.actSpace t x i =\n (g.rotation • (x -ᵥ (0 : Space d))) i + t.val * g.velocity i + g.spaceTranslation i", + "line": 200 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "def", + "name": "GalileanGroup.ofEuclidean", + "docstring": "A Euclidean spatial transformation as a Galilean transformation with zero boost and no time\ntranslation.", + "snippet": "def ofEuclidean (g : EuclideanGroup d) : GalileanGroup d", + "line": 209 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofEuclidean_rotation", + "docstring": null, + "snippet": "@[simp]\nlemma ofEuclidean_rotation (g : EuclideanGroup d) :\n (ofEuclidean g).rotation = g.linear", + "line": 213 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofEuclidean_velocity", + "docstring": null, + "snippet": "@[simp]\nlemma ofEuclidean_velocity (g : EuclideanGroup d) :\n (ofEuclidean g).velocity = 0", + "line": 217 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofEuclidean_spaceTranslation", + "docstring": null, + "snippet": "@[simp]\nlemma ofEuclidean_spaceTranslation (g : EuclideanGroup d) :\n (ofEuclidean g).spaceTranslation = g.translation", + "line": 221 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofEuclidean_timeTranslation", + "docstring": null, + "snippet": "@[simp]\nlemma ofEuclidean_timeTranslation (g : EuclideanGroup d) :\n (ofEuclidean g).timeTranslation = 0", + "line": 225 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "def", + "name": "GalileanGroup.euclidean.incl", + "docstring": "Inclusion of the Euclidean group into the Galilean group.", + "snippet": "def euclidean.incl : EuclideanGroup d →* GalileanGroup d where", + "line": 229 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "def", + "name": "GalileanGroup.ofOrthogonal", + "docstring": "A pure orthogonal spatial transformation as a Galilean transformation.", + "snippet": "def ofOrthogonal (R : Matrix.orthogonalGroup (Fin d) ℝ) : GalileanGroup d", + "line": 236 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofOrthogonal_rotation", + "docstring": null, + "snippet": "@[simp]\nlemma ofOrthogonal_rotation (R : Matrix.orthogonalGroup (Fin d) ℝ) :\n (ofOrthogonal R).rotation = R", + "line": 240 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofOrthogonal_velocity", + "docstring": null, + "snippet": "@[simp]\nlemma ofOrthogonal_velocity (R : Matrix.orthogonalGroup (Fin d) ℝ) :\n (ofOrthogonal R).velocity = 0", + "line": 244 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofOrthogonal_spaceTranslation", + "docstring": null, + "snippet": "@[simp]\nlemma ofOrthogonal_spaceTranslation (R : Matrix.orthogonalGroup (Fin d) ℝ) :\n (ofOrthogonal R).spaceTranslation = 0", + "line": 248 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofOrthogonal_timeTranslation", + "docstring": null, + "snippet": "@[simp]\nlemma ofOrthogonal_timeTranslation (R : Matrix.orthogonalGroup (Fin d) ℝ) :\n (ofOrthogonal R).timeTranslation = 0", + "line": 252 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "def", + "name": "GalileanGroup.orthogonal.incl", + "docstring": "Inclusion of the orthogonal group into the Galilean group.", + "snippet": "def orthogonal.incl : Matrix.orthogonalGroup (Fin d) ℝ →* GalileanGroup d where", + "line": 256 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "def", + "name": "GalileanGroup.rotation.incl", + "docstring": "Inclusion of the existing spatial rotation subgroup into the Galilean group.", + "snippet": "noncomputable def rotation.incl : EuclideanGroup.RotationGroup d →* GalileanGroup d where", + "line": 263 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "def", + "name": "GalileanGroup.ofSpaceTranslation", + "docstring": "A pure spatial translation as a Galilean transformation.", + "snippet": "def ofSpaceTranslation (a : EuclideanSpace ℝ (Fin d)) : GalileanGroup d", + "line": 270 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofSpaceTranslation_rotation", + "docstring": null, + "snippet": "@[simp]\nlemma ofSpaceTranslation_rotation (a : EuclideanSpace ℝ (Fin d)) :\n (ofSpaceTranslation a).rotation = 1", + "line": 274 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofSpaceTranslation_velocity", + "docstring": null, + "snippet": "@[simp]\nlemma ofSpaceTranslation_velocity (a : EuclideanSpace ℝ (Fin d)) :\n (ofSpaceTranslation a).velocity = 0", + "line": 278 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofSpaceTranslation_spaceTranslation", + "docstring": null, + "snippet": "@[simp]\nlemma ofSpaceTranslation_spaceTranslation (a : EuclideanSpace ℝ (Fin d)) :\n (ofSpaceTranslation a).spaceTranslation = a", + "line": 282 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofSpaceTranslation_timeTranslation", + "docstring": null, + "snippet": "@[simp]\nlemma ofSpaceTranslation_timeTranslation (a : EuclideanSpace ℝ (Fin d)) :\n (ofSpaceTranslation a).timeTranslation = 0", + "line": 286 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "def", + "name": "GalileanGroup.spaceTranslation.incl", + "docstring": "Inclusion of spatial translations into the Galilean group.", + "snippet": "def spaceTranslation.incl :\n Multiplicative (EuclideanSpace ℝ (Fin d)) →* GalileanGroup d where", + "line": 290 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "def", + "name": "GalileanGroup.spaceTranslationGroup.incl", + "docstring": "Inclusion of the existing spatial translation subgroup into the Galilean group.", + "snippet": "noncomputable def spaceTranslationGroup.incl :\n EuclideanGroup.TranslationGroup d →* GalileanGroup d", + "line": 298 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "def", + "name": "GalileanGroup.ofTimeTranslation", + "docstring": "A pure time translation as a Galilean transformation.", + "snippet": "def ofTimeTranslation (b : Time) : GalileanGroup d", + "line": 303 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofTimeTranslation_rotation", + "docstring": null, + "snippet": "@[simp]\nlemma ofTimeTranslation_rotation (b : Time) :\n (ofTimeTranslation (d := d) b).rotation = 1", + "line": 307 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofTimeTranslation_velocity", + "docstring": null, + "snippet": "@[simp]\nlemma ofTimeTranslation_velocity (b : Time) :\n (ofTimeTranslation (d := d) b).velocity = 0", + "line": 311 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofTimeTranslation_spaceTranslation", + "docstring": null, + "snippet": "@[simp]\nlemma ofTimeTranslation_spaceTranslation (b : Time) :\n (ofTimeTranslation (d := d) b).spaceTranslation = 0", + "line": 315 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "lemma", + "name": "GalileanGroup.ofTimeTranslation_timeTranslation", + "docstring": null, + "snippet": "@[simp]\nlemma ofTimeTranslation_timeTranslation (b : Time) :\n (ofTimeTranslation (d := d) b).timeTranslation = b", + "line": 319 + }, + { + "file": "Physlib/SpaceAndTime/GalileanGroup/Basic.lean", + "kind": "def", + "name": "GalileanGroup.timeTranslation.incl", + "docstring": "Inclusion of time translations into the Galilean group.", + "snippet": "def timeTranslation.incl : Multiplicative Time →* GalileanGroup d where", + "line": 323 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/Basic.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/Basic.lean", + "kind": "instance", + "name": "Space.instNontrivial", + "docstring": null, + "snippet": "instance instNontrivial {d : ℕ} [NeZero d] : Nontrivial (Space d) where", + "line": 243 + }, + { + "file": "Physlib/SpaceAndTime/Space/Basic.lean", + "kind": "def", + "name": "Space.homEuclideanSpaceSpace", + "docstring": "`Space d` is homoemorphic to d-dimensional euclidean space. This is used to define the `IsManifold`\ninstance on `Space d`.", + "snippet": "noncomputable def homEuclideanSpaceSpace (d : ℕ) : EuclideanSpace ℝ (Fin d) ≃ₜ Space d where", + "line": 262 + }, + { + "file": "Physlib/SpaceAndTime/Space/Basic.lean", + "kind": "instance", + "name": "Space.«instance@b9d497fa»", + "docstring": null, + "snippet": "noncomputable instance (d : ℕ) : ChartedSpace (EuclideanSpace ℝ (Fin d)) (Space d)", + "line": 276 + }, + { + "file": "Physlib/SpaceAndTime/Space/Basic.lean", + "kind": "instance", + "name": "Space.«instance@d376417d»", + "docstring": null, + "snippet": "instance (d : ℕ) : IsManifold (𝓡 d) ⊤ (Space d)", + "line": 279 + }, + { + "file": "Physlib/SpaceAndTime/Space/Basic.lean", + "kind": "lemma", + "name": "Space.chartAt_eq", + "docstring": null, + "snippet": "lemma chartAt_eq (d : ℕ) (p : Space d) :\n chartAt (EuclideanSpace ℝ (Fin d)) p =\n (homEuclideanSpaceSpace d).symm.toOpenPartialHomeomorph", + "line": 282 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/Derivatives/Basic.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/Derivatives/Basic.lean", + "kind": "lemma", + "name": "Space.deriv_sub", + "docstring": "Derivatives on space distribute over subtraction.", + "snippet": "@[to_fun]\nlemma deriv_sub [NormedAddCommGroup M] [NormedSpace ℝ M]\n (f1 f2 : Space d → M) (hf1 : Differentiable ℝ f1) (hf2 : Differentiable ℝ f2) :\n ∂[u] (f1 - f2) = ∂[u] f1 - ∂[u] f2", + "line": 192 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/Derivatives/Curl.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/Derivatives/Curl.lean", + "kind": "lemma", + "name": "Space.distCurl_coord_apply", + "docstring": null, + "snippet": "lemma distCurl_coord_apply {i : Fin 3}\n (f : Space →d[ℝ] (EuclideanSpace ℝ (Fin 3))) (η : 𝓢(Space, ℝ)) :\n (∇ᵈ ⨯ f) η i = - f (SchwartzMap.evalCLM ℝ Space ℝ (basis (i+2)) (fderivCLM ℝ Space ℝ η)) (i+1)\n + f (SchwartzMap.evalCLM ℝ Space ℝ (basis (i+1)) (fderivCLM ℝ Space ℝ η)) (i+2)", + "line": 735 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/Derivatives/Grad.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/Derivatives/Grad.lean", + "kind": "lemma", + "name": "Space.grad_eq_gradient", + "docstring": null, + "snippet": "lemma grad_eq_gradient {d} (f : Space d → ℝ) :\n ∇ f = basis.repr ∘ gradient f", + "line": 237 + }, + { + "file": "Physlib/SpaceAndTime/Space/Derivatives/Grad.lean", + "kind": "lemma", + "name": "Space.distGrad_const", + "docstring": null, + "snippet": "@[simp]\nlemma distGrad_const {d} (c : ℝ) :\n ∇ᵈ (Distribution.const ℝ (Space d) c) = 0", + "line": 617 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/Derivatives/Laplacian.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/Derivatives/Laplacian.lean", + "kind": "lemma", + "name": "Space.laplacian_eq_sum_snd_deriv", + "docstring": null, + "snippet": "lemma laplacian_eq_sum_snd_deriv {d} (f : Space d → ℝ) :\n Δ f = fun x => ∑ i, ∂[i] (∂[i] f) x", + "line": 59 + }, + { + "file": "Physlib/SpaceAndTime/Space/Derivatives/Laplacian.lean", + "kind": "def", + "name": "Space.distLaplacian", + "docstring": "The distributional `distLaplacian` operator.", + "snippet": "noncomputable def distLaplacian {d} :\n ((Space d) →d[ℝ] ℝ) →ₗ[ℝ] (Space d) →d[ℝ] ℝ", + "line": 88 + }, + { + "file": "Physlib/SpaceAndTime/Space/Derivatives/Laplacian.lean", + "kind": "lemma", + "name": "Space.distLaplacian_const", + "docstring": null, + "snippet": "@[simp]\nlemma distLaplacian_const {d : ℕ} (c : ℝ) :\n Δᵈ (Distribution.const ℝ (Space d) c) = 0", + "line": 102 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/DistOfFunction.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/DistOfFunction.lean", + "kind": "lemma", + "name": "Space.distDeriv_distOfFunction", + "docstring": "The distributional spatial derivative of the distribution associated to a differentiable\nfunction is the distribution associated to the corresponding pointwise spatial derivative.", + "snippet": "lemma distDeriv_distOfFunction {d : ℕ} (μ : Fin d) (f : Space d → F)\n (hf : IsDistBounded f) (hf_deriv : IsDistBounded (∂[μ] f))\n (hf_diff : Differentiable ℝ f) :\n ∂ᵈ[μ] (distOfFunction f hf) = distOfFunction (∂[μ] f) hf_deriv", + "line": 158 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/EuclideanGroup/Action.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Action.lean", + "kind": "instance", + "name": "EuclideanGroup.«instance@6372eb27»", + "docstring": "The action of the Euclidean group on the affine space of points `Space d`: `g = ⟨t, Q⟩` rotates\n`p` about the coordinate origin by the orthogonal part `Q` and then translates by `t`.", + "snippet": "noncomputable instance : MulAction (EuclideanGroup d) (Space d) where", + "line": 51 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Action.lean", + "kind": "lemma", + "name": "EuclideanGroup.dist_smul", + "docstring": "The Euclidean group acts on `Space d` **by isometries**: every rigid motion preserves distance.", + "snippet": "lemma dist_smul (g : EuclideanGroup d) (p q : Space d) :\n dist (g • p) (g • q) = dist p q", + "line": 82 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Action.lean", + "kind": "lemma", + "name": "EuclideanGroup.rotation_smul_vsub_origin", + "docstring": "A rotation acts on the displacement from the origin by its orthogonal part, for every `p`:\n`(r • p) -ᵥ origin = Q • (p -ᵥ origin)`.", + "snippet": "lemma rotation_smul_vsub_origin (r : RotationGroup d) (p : Space d) :\n (r • p) -ᵥ (0 : Space d) = (r : EuclideanGroup d).linear • (p -ᵥ (0 : Space d))", + "line": 116 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Action.lean", + "kind": "lemma", + "name": "EuclideanGroup.rotation_dist_smul", + "docstring": "The rotation group acts on `Space d` **by isometries** (inherited from `dist_smul`).", + "snippet": "lemma rotation_dist_smul (r : RotationGroup d) (p q : Space d) :\n dist (r • p) (r • q) = dist p q", + "line": 123 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Action.lean", + "kind": "lemma", + "name": "EuclideanGroup.chartEuclidean_smul", + "docstring": "Under the standard chart, the Euclidean action on `Space d` is the\ntransport of `toAffineIsometryMulEquiv` acting on `EuclideanSpace`:\n`chart (g • p) = (toAffineIsometryMulEquiv g) (chart p)`.", + "snippet": "lemma chartEuclidean_smul (g : EuclideanGroup d) (p : Space d) :\n Space.chartEuclidean d (g • p) = toAffineIsometryMulEquiv g (Space.chartEuclidean d p)", + "line": 136 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/EuclideanGroup/AffineGroup.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/AffineGroup.lean", + "kind": "def", + "name": "EuclideanGroup.orthogonalToLinearIsometryEquiv", + "docstring": "An orthogonal matrix viewed as a linear isometry equivalence of `EuclideanSpace ℝ (Fin n)`;\nthe linear ingredient of the first leg `toAffineIsometryHom`.", + "snippet": "noncomputable def orthogonalToLinearIsometryEquiv\n (Q : Matrix.orthogonalGroup (Fin n) ℝ) :\n EuclideanSpace ℝ (Fin n) ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n)", + "line": 58 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/AffineGroup.lean", + "kind": "def", + "name": "EuclideanGroup.toAffineIsometryHom", + "docstring": "The first leg of the inclusion: `⟨t, Q⟩ ↦ (x ↦ Q x + t)`, bundled as a monoid homomorphism\nfrom the Euclidean group into the affine isometry group of `EuclideanSpace ℝ (Fin n)`.", + "snippet": "noncomputable def toAffineIsometryHom :\n EuclideanGroup n →*\n AffineIsometryEquiv ℝ (EuclideanSpace ℝ (Fin n)) (EuclideanSpace ℝ (Fin n)) where", + "line": 74 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/AffineGroup.lean", + "kind": "def", + "name": "AffineIsometryEquiv.toAffineEquivHom", + "docstring": "The second leg of the inclusion: `AffineIsometryEquiv.toAffineEquiv` bundled as a monoid\nhomomorphism into the affine automorphism group.", + "snippet": "noncomputable def _root_.AffineIsometryEquiv.toAffineEquivHom :\n AffineIsometryEquiv ℝ (EuclideanSpace ℝ (Fin n)) (EuclideanSpace ℝ (Fin n)) →*\n AffineEquiv ℝ (EuclideanSpace ℝ (Fin n)) (EuclideanSpace ℝ (Fin n)) where", + "line": 95 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/AffineGroup.lean", + "kind": "def", + "name": "EuclideanGroup.toAffineEquiv", + "docstring": "The inclusion of the Euclidean group into Mathlib's affine automorphism group: the composite\nof the two legs `toAffineIsometryHom` and `AffineIsometryEquiv.toAffineEquivHom`.", + "snippet": "noncomputable def toAffineEquiv :\n EuclideanGroup n →* AffineEquiv ℝ (EuclideanSpace ℝ (Fin n)) (EuclideanSpace ℝ (Fin n))", + "line": 106 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/AffineGroup.lean", + "kind": "def", + "name": "EuclideanGroup.linearIsometryEquivToOrthogonal", + "docstring": "The `invFun` ingredient of `toAffineIsometryMulEquiv`: a linear isometry equivalence read back\nas an orthogonal matrix, inverse to the linear bridge `orthogonalToLinearIsometryEquiv` (see the\nround-trip `@[simp]` lemmas below).", + "snippet": "noncomputable def linearIsometryEquivToOrthogonal\n (L : EuclideanSpace ℝ (Fin n) ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n)) :\n Matrix.orthogonalGroup (Fin n) ℝ", + "line": 121 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/AffineGroup.lean", + "kind": "def", + "name": "EuclideanGroup.toAffineIsometryMulEquiv", + "docstring": "`EuclideanGroup n ≃* AffineIsometryEquiv ℝ (EuclideanSpace ℝ (Fin n)) _`: the Euclidean group\nis the full group of affine isometries of Euclidean space. This upgrades `toAffineIsometryHom` to\na group isomorphism with the same underlying map.", + "snippet": "noncomputable def toAffineIsometryMulEquiv :\n EuclideanGroup n ≃*\n AffineIsometryEquiv ℝ (EuclideanSpace ℝ (Fin n)) (EuclideanSpace ℝ (Fin n)) where", + "line": 185 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "structure", + "name": "EuclideanGroup", + "docstring": "An n-dimensional `Euclidean group` is a group of\nrotations, reflections, and translations.", + "snippet": "@[ext]\nstructure EuclideanGroup (n : ℕ) where\n translation : EuclideanSpace ℝ (Fin n)\n linear : Matrix.orthogonalGroup (Fin n) ℝ", + "line": 26 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "instance", + "name": "EuclideanGroup.«instance@134bed7b»", + "docstring": "Group structure on `E(n) = ℝ^n ⋊ O(n)`. The orthogonal component acts on translations by\nthe inherited matrix-vector action on `EuclideanSpace ℝ (Fin n)`, transported through `WithLp`\nfrom the coordinate action `Q.val *ᵥ v.ofLp`.", + "snippet": "noncomputable instance : Group (EuclideanGroup n) where", + "line": 37 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "lemma", + "name": "EuclideanGroup.det_coe_inv", + "docstring": null, + "snippet": "private lemma det_coe_inv {n : ℕ} (Q : Matrix.orthogonalGroup (Fin n) ℝ) :\n (Q⁻¹).val.det = (Q.val.det)⁻¹", + "line": 78 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "lemma", + "name": "EuclideanGroup.coe_inv", + "docstring": null, + "snippet": "private lemma coe_inv {n : ℕ} (Q : Matrix.orthogonalGroup (Fin n) ℝ) :\n (Q⁻¹).val = (Q.val)⁻¹", + "line": 84 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.SpecialEuclideanGroup", + "docstring": "Special Euclidean Group is the subgroup with det(Q) = 1 where Q ∈ O(n)", + "snippet": "def SpecialEuclideanGroup (n : ℕ) : Subgroup (EuclideanGroup n) where", + "line": 91 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.SpecialEuclideanGroup.incl", + "docstring": "The inclusion of the special Euclidean group into the Euclidean group.", + "snippet": "noncomputable def SpecialEuclideanGroup.incl (n : ℕ) :\n SpecialEuclideanGroup n →* EuclideanGroup n", + "line": 104 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.TranslationGroup", + "docstring": "Translation is the subgroup with Q = 1.", + "snippet": "def TranslationGroup (n : ℕ) : Subgroup (EuclideanGroup n) where", + "line": 108 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.TranslationGroup.incl", + "docstring": "The inclusion of the translation subgroup into the Euclidean group.", + "snippet": "noncomputable def TranslationGroup.incl (n : ℕ) :\n TranslationGroup n →* EuclideanGroup n", + "line": 121 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.translationVector.incl", + "docstring": "MonoidHom including a translation vector into the Euclidean Group.", + "snippet": "def translationVector.incl (n : ℕ) :\n Multiplicative (EuclideanSpace ℝ (Fin n)) →* EuclideanGroup n where", + "line": 125 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "lemma", + "name": "EuclideanGroup.translationVector.incl_range", + "docstring": "An API feature: the translation vector inclusion image is the `TranslationGroup` carrier.", + "snippet": "lemma translationVector.incl_range :\n Set.range (@translationVector.incl n) = (TranslationGroup n : Set (EuclideanGroup n))", + "line": 139 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "lemma", + "name": "EuclideanGroup.translation_zero", + "docstring": "The translation by the zero vector is the identity of the Euclidean group.", + "snippet": "lemma translation_zero : translationVector.incl n\n (Multiplicative.ofAdd (0 : EuclideanSpace ℝ (Fin n))) = 1", + "line": 156 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.OriginStabilizer", + "docstring": "The subgroup of `EuclideanGroup n` whose elements fix the origin\n(translation = 0). This is the copy of `O(n)` sitting inside `E(n)`.", + "snippet": "def OriginStabilizer (n : ℕ) : Subgroup (EuclideanGroup n) where", + "line": 162 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.RotationGroup", + "docstring": "Rotation Group is the subgroup of E(n) consisting of rotations about the origin:\nelements with `det = 1` (orientation-preserving) and `translation = 0` (origin-fixing).", + "snippet": "noncomputable def RotationGroup (n : ℕ) : Subgroup (EuclideanGroup n)", + "line": 174 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.RotationGroup.incl", + "docstring": "The inclusion of the rotation subgroup into the Euclidean group.", + "snippet": "noncomputable def RotationGroup.incl (n : ℕ) :\n RotationGroup n →* EuclideanGroup n", + "line": 178 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.RotationsAbout", + "docstring": "The subgroup of rotation about a spatial point `p : EuclideanSpace ℝ (Fin n)` consists of\nelements of the form T(p) * r * T(-p) with T(·) : translationVector.incl n (Multiplicative.ofAdd ·)\nand r : RotationGroup where r is viewed as a rotation about the origin. Note T(-p) = T(p)⁻¹.", + "snippet": "noncomputable def RotationsAbout : Subgroup (EuclideanGroup n) where", + "line": 186 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.RotationsAbout.incl", + "docstring": "The inclusion of rotations about `p` into the Euclidean group.", + "snippet": "noncomputable def RotationsAbout.incl : RotationsAbout p →* EuclideanGroup n", + "line": 205 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.RotationsAbout.toOrigin", + "docstring": "Conjugate a rotation about `p` back to a rotation about the origin.", + "snippet": "noncomputable def RotationsAbout.toOrigin :\n RotationsAbout p →* RotationGroup n where", + "line": 209 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.RotationsAbout.fromOrigin", + "docstring": "Conjugate a rotation about the origin to a rotation about `p`.", + "snippet": "noncomputable def RotationsAbout.fromOrigin :\n RotationGroup n →* RotationsAbout p where", + "line": 229 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "lemma", + "name": "EuclideanGroup.RotationsAbout.fromOrigin_comp_toOrigin", + "docstring": "`RotationsAbout.toOrigin p` followed by `RotationsAbout.fromOrigin p` is the identity; the\nforward leg of the isomorphism `RotationsAboutEquiv`.", + "snippet": "lemma RotationsAbout.fromOrigin_comp_toOrigin :\n (RotationsAbout.fromOrigin p).comp (RotationsAbout.toOrigin p) =\n MonoidHom.id (RotationsAbout p)", + "line": 244 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "lemma", + "name": "EuclideanGroup.RotationsAbout.toOrigin_comp_fromOrigin", + "docstring": "`RotationsAbout.fromOrigin p` followed by `RotationsAbout.toOrigin p` is the identity; the\nbackward leg of the isomorphism `RotationsAboutEquiv`.", + "snippet": "lemma RotationsAbout.toOrigin_comp_fromOrigin :\n (RotationsAbout.toOrigin p).comp (RotationsAbout.fromOrigin p) =\n MonoidHom.id (RotationGroup n)", + "line": 257 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.RotationsAboutEquiv", + "docstring": "API feature: conjugation by the translation `T(p)` exhibits the rotations about `p` as\nisomorphic to the rotations about the origin `RotationGroup n`.", + "snippet": "noncomputable def RotationsAboutEquiv : RotationsAbout p ≃* RotationGroup n", + "line": 270 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "lemma", + "name": "EuclideanGroup.RotationsAbout_zero", + "docstring": "API feature: the degenerate identity that `RotationsAbout 0 = RotationGroup n`", + "snippet": "lemma RotationsAbout_zero : RotationsAbout (0 : EuclideanSpace ℝ (Fin n)) = RotationGroup n", + "line": 275 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.specialOrthogonal.incl", + "docstring": "Rotations are members of special orthogonal groups and can be viewed as members of\northogonal groups.", + "snippet": "def specialOrthogonal.incl (n : ℕ) :\n Matrix.specialOrthogonalGroup (Fin n) ℝ →* Matrix.orthogonalGroup (Fin n) ℝ", + "line": 289 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.ofRotation", + "docstring": "The Euclidean group element given by a rotation about the origin (zero translation).", + "snippet": "def ofRotation (Q : Matrix.specialOrthogonalGroup (Fin n) ℝ) :\n EuclideanGroup n", + "line": 294 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.ofRotationTranslation", + "docstring": "Specialization to a group element from a rotation and a translation.", + "snippet": "def ofRotationTranslation (Q : Matrix.specialOrthogonalGroup (Fin n) ℝ)\n (t : EuclideanSpace ℝ (Fin n)) : EuclideanGroup n", + "line": 298 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "lemma", + "name": "EuclideanGroup.ofRotationTranslation_translation", + "docstring": "The specialization projects back to the translation component.", + "snippet": "@[simp]\nlemma ofRotationTranslation_translation (Q : Matrix.specialOrthogonalGroup (Fin n) ℝ)\n (t : EuclideanSpace ℝ (Fin n)) :\n (ofRotationTranslation Q t).translation = t", + "line": 304 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "lemma", + "name": "EuclideanGroup.ofRotationTranslation_linear", + "docstring": "The specialization projects back to the linear (rotation) component.", + "snippet": "@[simp]\nlemma ofRotationTranslation_linear (Q : Matrix.specialOrthogonalGroup (Fin n) ℝ)\n (t : EuclideanSpace ℝ (Fin n)) :\n (ofRotationTranslation Q t).linear = specialOrthogonal.incl n Q", + "line": 310 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "lemma", + "name": "EuclideanGroup.ofRotationTranslation_decompose", + "docstring": "API feature: the inclusion image decomposes as group product.", + "snippet": "@[simp]\nlemma ofRotationTranslation_decompose (Q : Matrix.specialOrthogonalGroup (Fin n) ℝ)\n (t : EuclideanSpace ℝ (Fin n)) :\n (ofRotationTranslation Q t) =\n (translationVector.incl n (Multiplicative.ofAdd t)) * (ofRotation (Q))", + "line": 316 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.specialOrthogonal.toRotation", + "docstring": "The isomorphism's forward map: a special orthogonal matrix as a rotation about the origin.", + "snippet": "noncomputable def specialOrthogonal.toRotation (n : ℕ) :\n Matrix.specialOrthogonalGroup (Fin n) ℝ →* RotationGroup n where", + "line": 327 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.specialOrthogonal.fromRotation", + "docstring": "The isomorphism's inverse map: the linear part of a rotation about the origin, as a special\northogonal matrix.", + "snippet": "noncomputable def specialOrthogonal.fromRotation (n : ℕ) :\n RotationGroup n →* Matrix.specialOrthogonalGroup (Fin n) ℝ where", + "line": 348 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "lemma", + "name": "EuclideanGroup.specialOrthogonal.fromRotation_comp_toRotation", + "docstring": "`specialOrthogonal.toRotation n` followed by `specialOrthogonal.fromRotation n` is the identity;\nthe forward leg of the isomorphism `specialOrthogonalEquiv`.", + "snippet": "lemma specialOrthogonal.fromRotation_comp_toRotation :\n (specialOrthogonal.fromRotation n).comp (specialOrthogonal.toRotation n) =\n MonoidHom.id (Matrix.specialOrthogonalGroup (Fin n) ℝ)", + "line": 356 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "lemma", + "name": "EuclideanGroup.specialOrthogonal.toRotation_comp_fromRotation", + "docstring": "`specialOrthogonal.fromRotation n` followed by `specialOrthogonal.toRotation n` is the identity;\nthe backward leg of the isomorphism `specialOrthogonalEquiv`.", + "snippet": "lemma specialOrthogonal.toRotation_comp_fromRotation :\n (specialOrthogonal.toRotation n).comp (specialOrthogonal.fromRotation n) =\n MonoidHom.id (RotationGroup n)", + "line": 365 + }, + { + "file": "Physlib/SpaceAndTime/Space/EuclideanGroup/Basic.lean", + "kind": "def", + "name": "EuclideanGroup.specialOrthogonalEquiv", + "docstring": "API feature: SO(n) ≃* RotationGroup n", + "snippet": "noncomputable def specialOrthogonalEquiv :\n Matrix.specialOrthogonalGroup (Fin n) ℝ ≃* RotationGroup n", + "line": 378 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/Integrals/Basic.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/Integrals/Basic.lean", + "kind": "lemma", + "name": "Space.integrable_spherical_of_integrable", + "docstring": null, + "snippet": "lemma integrable_spherical_of_integrable {d : ℕ} {F : Type*}\n [NormedAddCommGroup F] [NormedSpace ℝ F] {f : Space d → F}\n (hf : Integrable f volume) :\n Integrable\n (fun x : ↑(Metric.sphere (0 : Space d) 1) × Set.Ioi (0 : ℝ) =>\n f (x.2.1 • x.1.1))\n (volume (α := Space d).toSphere.prod\n (Measure.volumeIoiPow (Module.finrank ℝ (Space d) - 1)))", + "line": 123 + }, + { + "file": "Physlib/SpaceAndTime/Space/Integrals/Basic.lean", + "kind": "lemma", + "name": "Space.integral_volume_eq_spherical_iterated", + "docstring": null, + "snippet": "lemma integral_volume_eq_spherical_iterated {d : ℕ} [NeZero d] {F : Type*}\n [NormedAddCommGroup F] [NormedSpace ℝ F] (f : Space d → F)\n (hf : Integrable f volume) :\n ∫ x : Space d, f x =\n ∫ n : ↑(Metric.sphere (0 : Space d) 1),\n ∫ r : Set.Ioi (0 : ℝ), f (r.1 • n.1)\n ∂(Measure.volumeIoiPow (Module.finrank ℝ (Space d) - 1))\n ∂(volume (α := Space d).toSphere)", + "line": 152 + }, + { + "file": "Physlib/SpaceAndTime/Space/Integrals/Basic.lean", + "kind": "lemma", + "name": "Space.integral_volume_eq_spherical_integral", + "docstring": null, + "snippet": "lemma integral_volume_eq_spherical_integral {d : ℕ} [NeZero d] {F : Type*}\n [NormedAddCommGroup F] [NormedSpace ℝ F] (f : Space d → F)\n (hf : Integrable f volume) :\n ∫ x : Space d, f x =\n ∫ n : ↑(Metric.sphere (0 : Space d) 1),\n ∫ r : Set.Ioi (0 : ℝ), (r.1 ^ (d - 1)) • f (r.1 • n.1)\n ∂(.comap Subtype.val volume)\n ∂(volume (α := Space d).toSphere)", + "line": 194 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/Integrals/NormPow.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/Integrals/NormPow.lean", + "kind": "lemma", + "name": "Space.radial_jacobian_zpow_mul_self", + "docstring": null, + "snippet": "lemma radial_jacobian_zpow_mul_self\n {d p : ℕ} [NeZero d] {q : ℤ} (hp_int : (p : ℤ) = q + (d : ℤ))\n {r : ℝ} (hr : 0 < r) :\n r ^ (d - 1) * (r ^ q * r) = r ^ p", + "line": 54 + }, + { + "file": "Physlib/SpaceAndTime/Space/Integrals/NormPow.lean", + "kind": "lemma", + "name": "Space.radial_jacobian_zpow", + "docstring": null, + "snippet": "private lemma radial_jacobian_zpow\n {d p : ℕ} [NeZero d] {q : ℤ} (hp_int : (p : ℤ) = q + (d : ℤ))\n (hp_pos : 0 < p) {r : ℝ} (hr : 0 < r) :\n r ^ (d - 1) * r ^ q = r ^ (p - 1)", + "line": 70 + }, + { + "file": "Physlib/SpaceAndTime/Space/Integrals/NormPow.lean", + "kind": "lemma", + "name": "Space.radial_norm_power_spherical_integral_eq_space_integral", + "docstring": null, + "snippet": "lemma radial_norm_power_spherical_integral_eq_space_integral\n {d p : ℕ} [NeZero d] {q : ℤ} (hp_int : (p : ℤ) = q + (d : ℤ))\n (hp_pos : 0 < p) (η : 𝓢(Space d, ℝ)) :\n ∫ x : Space d, η x * ‖x‖ ^ q =\n ∫ (n : ↑(Metric.sphere (0 : Space d) 1)),\n ∫ (r : Set.Ioi (0 : ℝ)),\n r.1 ^ (p - 1) * η (r.1 • n.1)\n ∂(.comap Subtype.val volume)\n ∂(volume (α := Space d).toSphere)", + "line": 84 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/Module.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/Module.lean", + "kind": "lemma", + "name": "Space.norm_smul_sphere", + "docstring": null, + "snippet": "lemma norm_smul_sphere {d : ℕ} (n : ↑(Metric.sphere (0 : Space d) 1))\n {r : ℝ} (hr : 0 ≤ r) :\n ‖(r • (n : Space d))‖ = r", + "line": 293 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/Norm/Basic.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/Norm/Basic.lean", + "kind": "lemma", + "name": "Space.integrable_real_pow_mul_schwartz", + "docstring": null, + "snippet": "private lemma integrable_real_pow_mul_schwartz\n (ψ : 𝓢(ℝ, ℝ)) (k : ℕ) :\n Integrable (fun x : ℝ => x ^ k * ψ x) volume", + "line": 957 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/Basic.lean", + "kind": "lemma", + "name": "Space.radial_power_deriv_integral_by_parts", + "docstring": null, + "snippet": "private lemma radial_power_deriv_integral_by_parts\n {d : ℕ} (η : 𝓢(Space d, ℝ))\n (n : ↑(Metric.sphere (0 : Space d) 1))\n (p : ℕ) (hp : 0 < p) :\n - ∫ (r : Set.Ioi (0 : ℝ)),\n r.1 ^ p * (_root_.deriv (fun a => η (a • n.1)) r.1)\n ∂(.comap Subtype.val volume)\n =\n (p : ℝ) * ∫ (r : Set.Ioi (0 : ℝ)),\n r.1 ^ (p - 1) * η (r.1 • n.1)\n ∂(.comap Subtype.val volume)", + "line": 964 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/Basic.lean", + "kind": "lemma", + "name": "Space.distDiv_norm_zpow_smul_repr_self_apply_eq_radial_deriv", + "docstring": null, + "snippet": "private lemma distDiv_norm_zpow_smul_repr_self_apply_eq_radial_deriv\n {d p : ℕ} [NeZero d] (q : ℤ) (hq : 0 < q + (d : ℤ))\n (hp_int : (p : ℤ) = q + (d : ℤ))\n (η : 𝓢(Space d, ℝ)) :\n (∇ᵈ ⬝ (distOfFunction (fun x : Space d => ‖x‖ ^ q • basis.repr x)\n (IsDistBounded.zpow_smul_repr_self q (by omega)))) η =\n - ∫ n, (∫ (r : Set.Ioi (0 : ℝ)),\n r.1 ^ p * (_root_.deriv (fun a => η (a • n.1)) r.1)\n ∂(.comap Subtype.val volume))\n ∂(volume (α := Space d).toSphere)", + "line": 1109 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/Basic.lean", + "kind": "lemma", + "name": "Space.distDiv_norm_zpow_smul_repr_self_eq_smul", + "docstring": null, + "snippet": "lemma distDiv_norm_zpow_smul_repr_self_eq_smul\n {d : ℕ} [NeZero d] (q : ℤ) (hq : 0 < q + (d : ℤ)) :\n ∇ᵈ ⬝ (distOfFunction (fun x : Space d => ‖x‖ ^ q • basis.repr x)\n (IsDistBounded.zpow_smul_repr_self q (by omega))) =\n (((q + d : ℤ) : ℝ) •\n distOfFunction (fun x : Space d => ‖x‖ ^ q)\n (IsDistBounded.pow q (by omega)))", + "line": 1159 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/Basic.lean", + "kind": "lemma", + "name": "Space.distLaplacian_distOfFunction_norm_zpow", + "docstring": null, + "snippet": "lemma distLaplacian_distOfFunction_norm_zpow {d : ℕ} [NeZero d] (m : ℤ)\n (hdiv : 0 < m - 2 + (d : ℤ)) :\n Δᵈ (distOfFunction (fun x : Space d => ‖x‖ ^ m)\n (IsDistBounded.pow m (by omega))) =\n (((m : ℝ) * (((m - 2 + d : ℤ) : ℝ))) •\n distOfFunction (fun x : Space d => ‖x‖ ^ (m - 2))\n (IsDistBounded.pow (m - 2) (by omega)))", + "line": 1216 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/Basic.lean", + "kind": "lemma", + "name": "Space.distLaplacian_fundamentalSolution_norm_zpow", + "docstring": "The distributional Laplacian of `‖x‖ ^ (2 - d)` is `(2 - d) * d * volume (Metric.ball 0 1)`\ntimes the Dirac delta at the origin. For `d ≥ 3` this `‖x‖ ^ (2 - d)` is the (singular)\nfundamental solution of the Laplacian, and for `d = 1` it is `‖x‖`. When `d = 2` the exponent\nvanishes, so the identity collapses to the trivial `Δᵈ 1 = 0`; the genuine two-dimensional\nfundamental solution is the logarithm, proved in `distLaplacian_fundamentalSolution_log_norm`.\nThe statement also holds vacuously for `d = 0`, where the space is trivial.", + "snippet": "lemma distLaplacian_fundamentalSolution_norm_zpow {d : ℕ} :\n Δᵈ (distOfFunction (fun x : Space d => ‖x‖ ^ (- ((d : ℤ) - 2)))\n (IsDistBounded.pow _ (by omega))) =\n ((- ((d : ℝ) - 2)) * d *\n (volume (α := Space d)).real (Metric.ball 0 1)) • diracDelta ℝ 0", + "line": 1395 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/Basic.lean", + "kind": "lemma", + "name": "Space.distLaplacian_fundamentalSolution_log_norm", + "docstring": "In dimension two the fundamental solution of the Laplacian is the logarithm: the\ndistributional Laplacian of `Real.log ‖x‖` is `2 * volume (Metric.ball 0 1)` times the Dirac\ndelta at the origin.", + "snippet": "lemma distLaplacian_fundamentalSolution_log_norm :\n Δᵈ (distOfFunction (fun x : Space 2 => Real.log ‖x‖) IsDistBounded.log_norm) =\n (2 * (volume (α := Space 2)).real (Metric.ball 0 1)) • diracDelta ℝ 0", + "line": 1458 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/Norm/IteratedLaplacian.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/Norm/IteratedLaplacian.lean", + "kind": "def", + "name": "Space.oddNormIteratedLaplacianCoeff", + "docstring": "The scalar factor in the odd-dimensional iterated Laplacian of the norm.", + "snippet": "noncomputable def oddNormIteratedLaplacianCoeff (m : ℕ) : ℝ", + "line": 49 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/IteratedLaplacian.lean", + "kind": "lemma", + "name": "Space.oddNormIteratedLaplacianCoeff_factor_ne_zero", + "docstring": null, + "snippet": "private lemma oddNormIteratedLaplacianCoeff_factor_ne_zero {m k : ℕ} (hk : k < m) :\n ((((1 : ℤ) - 2 * (k : ℤ) : ℤ) : ℝ) *\n ((((1 : ℤ) - 2 * (k : ℤ) - 2 + (2 * m + 1 : ℤ) : ℤ) : ℝ))) ≠ 0", + "line": 56 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/IteratedLaplacian.lean", + "kind": "lemma", + "name": "Space.oddNormIteratedLaplacianCoeff_ne_zero", + "docstring": "The scalar factor in the odd-dimensional iterated Laplacian of the norm is nonzero.", + "snippet": "lemma oddNormIteratedLaplacianCoeff_ne_zero (m : ℕ) :\n oddNormIteratedLaplacianCoeff m ≠ 0", + "line": 66 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/IteratedLaplacian.lean", + "kind": "lemma", + "name": "Space.distLaplacian_norm_zpow_odd_boundary", + "docstring": null, + "snippet": "private lemma distLaplacian_norm_zpow_odd_boundary (m : ℕ) :\n Δᵈ (distOfFunction (fun x : Space (2 * m + 1) =>\n ‖x‖ ^ ((1 : ℤ) - 2 * (m : ℤ)))\n (IsDistBounded.pow _ (by omega))) =\n (((1 : ℤ) - 2 * (m : ℤ) : ℝ) * (2 * m + 1 : ℝ) *\n (volume (α := Space (2 * m + 1))).real (Metric.ball 0 1)) •\n diracDelta ℝ 0", + "line": 81 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/IteratedLaplacian.lean", + "kind": "lemma", + "name": "Space.iterated_distLaplacian_norm_zpow_odd_until_boundary", + "docstring": null, + "snippet": "private lemma iterated_distLaplacian_norm_zpow_odd_until_boundary\n (m k : ℕ) (hk : k ≤ m) :\n ((distLaplacian (d := 2 * m + 1))^[k])\n (distOfFunction (fun x : Space (2 * m + 1) => ‖x‖ ^ (1 : ℤ))\n (IsDistBounded.pow 1 (by omega))) =\n (∏ j ∈ Finset.range k,\n ((((1 : ℤ) - 2 * (j : ℤ) : ℤ) : ℝ) *\n ((((1 : ℤ) - 2 * (j : ℤ) - 2 + (2 * m + 1 : ℤ) : ℤ) : ℝ)))) •\n distOfFunction (fun x : Space (2 * m + 1) => ‖x‖ ^ ((1 : ℤ) - 2 * (k : ℤ)))\n (IsDistBounded.pow _ (by omega))", + "line": 103 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/IteratedLaplacian.lean", + "kind": "lemma", + "name": "Space.iterated_distLaplacian_norm_zpow_odd_eq_smul_diracDelta", + "docstring": "In dimension `2 * m + 1`, the `(m + 1)`-fold distributional Laplacian of the\ndistribution induced by the norm is a nonzero multiple of the Dirac delta at the origin.", + "snippet": "lemma iterated_distLaplacian_norm_zpow_odd_eq_smul_diracDelta (m : ℕ) :\n ((distLaplacian (d := 2 * m + 1))^[m + 1])\n (distOfFunction (fun x : Space (2 * m + 1) => ‖x‖ ^ (1 : ℤ))\n (IsDistBounded.pow 1 (by omega))) =\n oddNormIteratedLaplacianCoeff m • diracDelta ℝ 0", + "line": 144 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/Norm/Regularized.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/Norm/Regularized.lean", + "kind": "def", + "name": "Space.normRegularizedPow", + "docstring": "Power of regularized norm, `(‖x‖² + ε²)^(s/2)`.", + "snippet": "def normRegularizedPow (d : ℕ) (ε s : ℝ) : Space d → ℝ", + "line": 49 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/Regularized.lean", + "kind": "lemma", + "name": "Space.normRegularizedPow_eq", + "docstring": null, + "snippet": "lemma normRegularizedPow_eq (d : ℕ) (ε s : ℝ) :\n normRegularizedPow d ε s = fun x ↦ (‖x‖ ^ 2 + ε ^ 2) ^ (s / 2)", + "line": 52 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/Regularized.lean", + "kind": "lemma", + "name": "Space.norm_sq_add_unit_sq_pos", + "docstring": "For a nonzero regularization parameter, `‖x‖² + ε²` is positive.", + "snippet": "lemma norm_sq_add_unit_sq_pos {d : ℕ} (ε : ℝˣ) (x : Space d) : 0 < ‖x‖ ^ 2 + ε ^ 2", + "line": 56 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/Regularized.lean", + "kind": "lemma", + "name": "Space.normRegularizedPow_pos", + "docstring": "The regularized norm power is positive for nonzero regularization parameter.", + "snippet": "lemma normRegularizedPow_pos (d : ℕ) (ε : ℝˣ) (s : ℝ) (x : Space d) :\n 0 < normRegularizedPow d ε s x", + "line": 60 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/Regularized.lean", + "kind": "lemma", + "name": "Space.normRegularizedPow_hasTemperateGrowth", + "docstring": "The regularized norm power has temperate growth.", + "snippet": "lemma normRegularizedPow_hasTemperateGrowth (d : ℕ) (ε : ℝˣ) (s : ℝ) :\n HasTemperateGrowth (normRegularizedPow d ε s)", + "line": 65 + }, + { + "file": "Physlib/SpaceAndTime/Space/Norm/Regularized.lean", + "kind": "lemma", + "name": "Space.normRegularizedPow_measurable", + "docstring": null, + "snippet": "@[fun_prop]\nlemma normRegularizedPow_measurable (d : ℕ) (ε s : ℝ) :\n Measurable (normRegularizedPow d ε s)", + "line": 79 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Space/Origin.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Space/Origin.lean", + "kind": "instance", + "name": "Space.«instance@570dcd05»", + "docstring": null, + "snippet": "instance {d} : Zero (Space d) where", + "line": 28 + }, + { + "file": "Physlib/SpaceAndTime/Space/Origin.lean", + "kind": "lemma", + "name": "Space.zero_val", + "docstring": null, + "snippet": "@[simp]\nlemma zero_val {d : ℕ} : (0 : Space d).val = fun _ => 0", + "line": 32 + }, + { + "file": "Physlib/SpaceAndTime/Space/Origin.lean", + "kind": "lemma", + "name": "Space.zero_apply", + "docstring": null, + "snippet": "@[simp]\nlemma zero_apply {d : ℕ} (i : Fin d) :\n (0 : Space d) i = 0", + "line": 35 + }, + { + "file": "Physlib/SpaceAndTime/Space/Origin.lean", + "kind": "def", + "name": "Space.vectorToSpace", + "docstring": "A Euclidean vector, based at the chosen origin, viewed as a point of `Space d`.", + "snippet": "noncomputable def vectorToSpace {d : ℕ} (v : EuclideanSpace ℝ (Fin d)) : Space d", + "line": 40 + }, + { + "file": "Physlib/SpaceAndTime/Space/Origin.lean", + "kind": "lemma", + "name": "Space.vectorToSpace_apply", + "docstring": null, + "snippet": "@[simp]\nlemma vectorToSpace_apply {d : ℕ} (v : EuclideanSpace ℝ (Fin d)) (i : Fin d) :\n vectorToSpace v i = v i", + "line": 44 + }, + { + "file": "Physlib/SpaceAndTime/Space/Origin.lean", + "kind": "lemma", + "name": "Space.vectorToSpace_vsub_zero", + "docstring": null, + "snippet": "@[simp]\nlemma vectorToSpace_vsub_zero {d : ℕ} (v : EuclideanSpace ℝ (Fin d)) :\n vectorToSpace v -ᵥ (0 : Space d) = v", + "line": 49 + }, + { + "file": "Physlib/SpaceAndTime/Space/Origin.lean", + "kind": "def", + "name": "Space.chartEuclidean", + "docstring": "The standard chart `Space d ≃ᵃⁱ[ℝ] EuclideanSpace ℝ (Fin d)`, `p ↦ p -ᵥ 0`, identifying a point\nwith its coordinate vector relative to the origin (the vector-space zero `(0 : Space d)`).", + "snippet": "noncomputable def chartEuclidean (d : ℕ) :\n Space d ≃ᵃⁱ[ℝ] EuclideanSpace ℝ (Fin d)", + "line": 56 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/SpaceTime/Derivatives.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/SpaceTime/Derivatives.lean", + "kind": "def", + "name": "SpaceTime.manifoldDeriv", + "docstring": "The derivative of a function from `SpaceTime d` to a manifold along the `μ`\ncoordinate, as a tangent vector at the value of the function.", + "snippet": "noncomputable def manifoldDeriv {E H N : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]\n [TopologicalSpace H] (I : ModelWithCorners ℝ E H) [TopologicalSpace N]\n [ChartedSpace H N] {d : ℕ} (μ : Fin 1 ⊕ Fin d) (f : SpaceTime d → N) :\n (y : SpaceTime d) → TangentSpace I (f y)", + "line": 96 + }, + { + "file": "Physlib/SpaceAndTime/SpaceTime/Derivatives.lean", + "kind": "lemma", + "name": "SpaceTime.manifoldDeriv_eq", + "docstring": null, + "snippet": "lemma manifoldDeriv_eq {E H N : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]\n [TopologicalSpace H] (I : ModelWithCorners ℝ E H) [TopologicalSpace N]\n [ChartedSpace H N] {d : ℕ} (μ : Fin 1 ⊕ Fin d) (f : SpaceTime d → N)\n (y : SpaceTime d) :\n manifoldDeriv I μ f y =\n mfderiv 𝓘(ℝ, SpaceTime d) I f y\n ((Lorentz.Vector.basis μ : SpaceTime d) : TangentSpace 𝓘(ℝ, SpaceTime d) y)", + "line": 103 + }, + { + "file": "Physlib/SpaceAndTime/SpaceTime/Derivatives.lean", + "kind": "lemma", + "name": "SpaceTime.deriv_eq_mfderiv", + "docstring": "The spacetime derivative is the manifold derivative for functions into normed spaces.", + "snippet": "lemma deriv_eq_mfderiv {M : Type} [NormedAddCommGroup M] [NormedSpace ℝ M]\n {d : ℕ} (μ : Fin 1 ⊕ Fin d) (f : SpaceTime d → M) (y : SpaceTime d) :\n deriv μ f y =\n mfderiv 𝓘(ℝ, SpaceTime d) 𝓘(ℝ, M) f y\n ((Lorentz.Vector.basis μ : SpaceTime d) : TangentSpace 𝓘(ℝ, SpaceTime d) y)", + "line": 112 + }, + { + "file": "Physlib/SpaceAndTime/SpaceTime/Derivatives.lean", + "kind": "lemma", + "name": "SpaceTime.deriv_eq_manifoldDeriv", + "docstring": null, + "snippet": "lemma deriv_eq_manifoldDeriv {M : Type} [NormedAddCommGroup M] [NormedSpace ℝ M]\n {d : ℕ} (μ : Fin 1 ⊕ Fin d) (f : SpaceTime d → M) (y : SpaceTime d) :\n deriv μ f y = manifoldDeriv 𝓘(ℝ, M) μ f y", + "line": 120 + }, + { + "file": "Physlib/SpaceAndTime/SpaceTime/Derivatives.lean", + "kind": "lemma", + "name": "SpaceTime.manifoldDeriv_const", + "docstring": null, + "snippet": "@[simp]\nlemma manifoldDeriv_const {E H N : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]\n [TopologicalSpace H] (I : ModelWithCorners ℝ E H) [TopologicalSpace N]\n [ChartedSpace H N] {d : ℕ} (μ : Fin 1 ⊕ Fin d) (n : N) (y : SpaceTime d) :\n manifoldDeriv I μ (fun _ : SpaceTime d => n) y = 0", + "line": 126 + }, + { + "file": "Physlib/SpaceAndTime/SpaceTime/Derivatives.lean", + "kind": "lemma", + "name": "SpaceTime.contDiff_deriv", + "docstring": "If `f` is `C^{n+1}` then `∂_ μ f` is `C^n`.", + "snippet": "@[fun_prop]\nlemma contDiff_deriv {M : Type} [NormedAddCommGroup M] [NormedSpace ℝ M] {d : ℕ}\n {n : WithTop ℕ∞} (μ : Fin 1 ⊕ Fin d) (f : SpaceTime d → M) (hf : ContDiff ℝ (n + 1) f) :\n ContDiff ℝ n (∂_ μ f)", + "line": 215 + }, + { + "file": "Physlib/SpaceAndTime/SpaceTime/Derivatives.lean", + "kind": "lemma", + "name": "SpaceTime.differentiable_deriv", + "docstring": "If `f` is `C^2` then `∂_ μ f` is differentiable.", + "snippet": "@[fun_prop]\nlemma differentiable_deriv {M : Type} [NormedAddCommGroup M] [NormedSpace ℝ M] {d : ℕ}\n (μ : Fin 1 ⊕ Fin d) (f : SpaceTime d → M) (hf : ContDiff ℝ 2 f) :\n Differentiable ℝ (∂_ μ f)", + "line": 225 + }, + { + "file": "Physlib/SpaceAndTime/SpaceTime/Derivatives.lean", + "kind": "lemma", + "name": "SpaceTime.deriv_commute", + "docstring": "Derivatives on spacetime commute with one another (Clairaut's theorem).", + "snippet": "lemma deriv_commute {M : Type} [NormedAddCommGroup M] [NormedSpace ℝ M] {d : ℕ}\n (μ ν : Fin 1 ⊕ Fin d) (f : SpaceTime d → M) (hf : ContDiff ℝ 2 f) :\n ∂_ μ (∂_ ν f) = ∂_ ν (∂_ μ f)", + "line": 237 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/Time/Derivatives.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/Time/Derivatives.lean", + "kind": "def", + "name": "Time.manifoldDeriv", + "docstring": "The time derivative of a function from `Time` to a manifold, as a tangent vector at\nthe value of the function.", + "snippet": "noncomputable def manifoldDeriv {E H N : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]\n [TopologicalSpace H] (I : ModelWithCorners ℝ E H) [TopologicalSpace N]\n [ChartedSpace H N] (f : Time → N) : (t : Time) → TangentSpace I (f t)", + "line": 78 + }, + { + "file": "Physlib/SpaceAndTime/Time/Derivatives.lean", + "kind": "lemma", + "name": "Time.manifoldDeriv_eq", + "docstring": null, + "snippet": "lemma manifoldDeriv_eq {E H N : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]\n [TopologicalSpace H] (I : ModelWithCorners ℝ E H) [TopologicalSpace N]\n [ChartedSpace H N] (f : Time → N) (t : Time) :\n manifoldDeriv I f t =\n mfderiv 𝓘(ℝ, Time) I f t ((1 : Time) : TangentSpace 𝓘(ℝ, Time) t)", + "line": 84 + }, + { + "file": "Physlib/SpaceAndTime/Time/Derivatives.lean", + "kind": "lemma", + "name": "Time.deriv_eq_mfderiv", + "docstring": "The time derivative is the manifold derivative for functions into normed spaces.", + "snippet": "lemma deriv_eq_mfderiv [NormedAddCommGroup M] [NormedSpace ℝ M]\n (f : Time → M) (t : Time) :\n deriv f t =\n mfderiv 𝓘(ℝ, Time) 𝓘(ℝ, M) f t\n ((1 : Time) : TangentSpace 𝓘(ℝ, Time) t)", + "line": 92 + }, + { + "file": "Physlib/SpaceAndTime/Time/Derivatives.lean", + "kind": "lemma", + "name": "Time.deriv_eq_manifoldDeriv", + "docstring": null, + "snippet": "lemma deriv_eq_manifoldDeriv [NormedAddCommGroup M] [NormedSpace ℝ M]\n (f : Time → M) (t : Time) :\n deriv f t = manifoldDeriv 𝓘(ℝ, M) f t", + "line": 101 + }, + { + "file": "Physlib/SpaceAndTime/Time/Derivatives.lean", + "kind": "lemma", + "name": "Time.manifoldDeriv_const", + "docstring": null, + "snippet": "@[simp]\nlemma manifoldDeriv_const {E H N : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]\n [TopologicalSpace H] (I : ModelWithCorners ℝ E H) [TopologicalSpace N]\n [ChartedSpace H N] (n : N) :\n manifoldDeriv I (fun _ : Time => n) t = 0", + "line": 108 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/TimeAndSpace/Basic.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/Basic.lean", + "kind": "abbrev", + "name": "TimeAndSpace", + "docstring": "Euclidean spacetime as the product of time and `d`-dimensional space.", + "snippet": "abbrev TimeAndSpace (d : ℕ := 3)", + "line": 75 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/Basic.lean", + "kind": "def", + "name": "TimeAndSpace.time", + "docstring": "The time-coordinate projection from `TimeAndSpace d`.", + "snippet": "noncomputable def time {d : ℕ} : TimeAndSpace d →L[ℝ] Time", + "line": 82 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/Basic.lean", + "kind": "def", + "name": "TimeAndSpace.space", + "docstring": "The spatial-coordinate projection from `TimeAndSpace d`.", + "snippet": "noncomputable def space {d : ℕ} : TimeAndSpace d →L[ℝ] Space d", + "line": 86 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/Basic.lean", + "kind": "lemma", + "name": "TimeAndSpace.time_apply", + "docstring": null, + "snippet": "@[simp]\nlemma time_apply (tx : TimeAndSpace d) :\n time tx = tx.1", + "line": 90 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/Basic.lean", + "kind": "lemma", + "name": "TimeAndSpace.space_apply", + "docstring": null, + "snippet": "@[simp]\nlemma space_apply (tx : TimeAndSpace d) :\n space tx = tx.2", + "line": 94 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/Basic.lean", + "kind": "lemma", + "name": "TimeAndSpace.dist_time_le", + "docstring": "The time projection is nonexpanding for the product metric.", + "snippet": "lemma dist_time_le (tx ty : TimeAndSpace d) :\n dist (time tx) (time ty) ≤ dist tx ty", + "line": 98 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/Basic.lean", + "kind": "lemma", + "name": "TimeAndSpace.dist_space_le", + "docstring": "The spatial projection is nonexpanding for the product metric.", + "snippet": "lemma dist_space_le (tx ty : TimeAndSpace d) :\n dist (space tx) (space ty) ≤ dist tx ty", + "line": 105 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean", + "kind": "instance", + "name": "TimeAndSpace.instMulActionEuclideanGroup", + "docstring": "The Euclidean group acts on `TimeAndSpace d` by fixing time and acting on space.", + "snippet": "noncomputable instance instMulActionEuclideanGroup :\n MulAction (EuclideanGroup d) (TimeAndSpace d) where", + "line": 50 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean", + "kind": "lemma", + "name": "TimeAndSpace.smul_mk", + "docstring": "Coordinate formula for the Euclidean-group action on `TimeAndSpace d`.", + "snippet": "@[simp]\nlemma smul_mk (g : EuclideanGroup d) (t : Time) (x : Space d) :\n g • ((t, x) : TimeAndSpace d) = (t, g • x)", + "line": 64 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean", + "kind": "lemma", + "name": "TimeAndSpace.fst_smul", + "docstring": "The Euclidean-group action fixes the first coordinate.", + "snippet": "@[simp]\nlemma fst_smul (g : EuclideanGroup d) (tx : TimeAndSpace d) :\n (g • tx).1 = tx.1", + "line": 71 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean", + "kind": "lemma", + "name": "TimeAndSpace.snd_smul", + "docstring": "The Euclidean-group action applies to the second coordinate.", + "snippet": "@[simp]\nlemma snd_smul (g : EuclideanGroup d) (tx : TimeAndSpace d) :\n (g • tx).2 = g • tx.2", + "line": 78 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean", + "kind": "lemma", + "name": "TimeAndSpace.time_smul", + "docstring": "The Euclidean-group action fixes the time projection.", + "snippet": "lemma time_smul (g : EuclideanGroup d) (tx : TimeAndSpace d) :\n time (g • tx) = time tx", + "line": 84 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean", + "kind": "lemma", + "name": "TimeAndSpace.space_smul", + "docstring": "The Euclidean-group action applies to the space projection.", + "snippet": "lemma space_smul (g : EuclideanGroup d) (tx : TimeAndSpace d) :\n space (g • tx) = g • space tx", + "line": 89 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean", + "kind": "lemma", + "name": "TimeAndSpace.dist_smul", + "docstring": "The Euclidean-group action on `TimeAndSpace d` preserves product distance.", + "snippet": "lemma dist_smul (g : EuclideanGroup d) (tx ty : TimeAndSpace d) :\n dist (g • tx) (g • ty) = dist tx ty", + "line": 94 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean", + "kind": "def", + "name": "TimeAndSpace.actionLinearMap", + "docstring": "The linear part of the Euclidean-group action on `TimeAndSpace d`.", + "snippet": "noncomputable def actionLinearMap (g : EuclideanGroup d) :\n TimeAndSpace d →L[ℝ] TimeAndSpace d", + "line": 108 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean", + "kind": "def", + "name": "TimeAndSpace.actionTranslation", + "docstring": "The translation part of the Euclidean-group action on `TimeAndSpace d`.", + "snippet": "noncomputable def actionTranslation (g : EuclideanGroup d) : TimeAndSpace d", + "line": 117 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean", + "kind": "lemma", + "name": "TimeAndSpace.smul_eq_actionLinearMap_add", + "docstring": null, + "snippet": "lemma smul_eq_actionLinearMap_add (g : EuclideanGroup d) (tx : TimeAndSpace d) :\n g • tx = actionLinearMap g tx + actionTranslation g", + "line": 120 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean", + "kind": "lemma", + "name": "TimeAndSpace.smul_hasTemperateGrowth", + "docstring": "The Euclidean-group action on `TimeAndSpace d` has temperate growth.", + "snippet": "lemma smul_hasTemperateGrowth (g : EuclideanGroup d) :\n Function.HasTemperateGrowth (fun tx : TimeAndSpace d => g • tx)", + "line": 132 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean", + "kind": "lemma", + "name": "TimeAndSpace.isometry_smul", + "docstring": "The Euclidean-group action on `TimeAndSpace d` is an isometry.", + "snippet": "lemma isometry_smul (g : EuclideanGroup d) :\n Isometry (fun tx : TimeAndSpace d => g • tx)", + "line": 143 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/Action.lean", + "kind": "lemma", + "name": "TimeAndSpace.antilipschitz_smul", + "docstring": "The Euclidean-group action on `TimeAndSpace d` is antilipschitz.", + "snippet": "lemma antilipschitz_smul (g : EuclideanGroup d) :\n AntilipschitzWith 1 (fun tx : TimeAndSpace d => g • tx)", + "line": 148 + } + ] + }, + { + "filename": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/SchwartzAction.lean", + "entries": [ + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/SchwartzAction.lean", + "kind": "def", + "name": "TimeAndSpace.schwartzEuclideanAction", + "docstring": "The Euclidean-group pullback action on Schwartz maps over `TimeAndSpace d`, as a continuous\nrepresentation `ContRepresentation ℝ (EuclideanGroup d) 𝓢(TimeAndSpace d, F)`.\n\n`ContRepresentation R G V` unfolds to the monoid homomorphism `G →* V →L[R] V` into the\n*continuous*-linear maps: the builder `SchwartzMap.compCLMOfAntilipschitz` already yields a\ncontinuous-linear map for free, and the `→L` codomain lets the action compose under `∘L`. The\nplain mathlib `Representation` (`G →* V →ₗ[ℝ] V`) is recovered via\n`ContRepresentation.toRepresentation`.", + "snippet": "noncomputable def schwartzEuclideanAction {d : ℕ} :\n ContRepresentation ℝ (EuclideanGroup d) 𝓢(TimeAndSpace d, F) where", + "line": 60 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/SchwartzAction.lean", + "kind": "lemma", + "name": "TimeAndSpace.schwartzEuclideanAction_apply", + "docstring": "Pointwise formula for the monoid-homomorphism form of the Schwartz-map pullback action.", + "snippet": "@[simp]\nlemma schwartzEuclideanAction_apply {d : ℕ} (g : EuclideanGroup d) (η : 𝓢(TimeAndSpace d, F))\n (tx : TimeAndSpace d) :\n (schwartzEuclideanAction g η) tx = η (g⁻¹ • tx)", + "line": 78 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/SchwartzAction.lean", + "kind": "instance", + "name": "TimeAndSpace.instMulActionSchwartzMap", + "docstring": "The Euclidean group acts on Schwartz maps over `TimeAndSpace d` by pullback.", + "snippet": "noncomputable instance instMulActionSchwartzMap {d : ℕ} :\n MulAction (EuclideanGroup d) 𝓢(TimeAndSpace d, F) where", + "line": 83 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/SchwartzAction.lean", + "kind": "lemma", + "name": "TimeAndSpace.smul_schwartzMap_apply", + "docstring": "Pointwise formula for the `MulAction` instance on Schwartz maps.", + "snippet": "@[simp]\nlemma smul_schwartzMap_apply {d : ℕ} (g : EuclideanGroup d) (η : 𝓢(TimeAndSpace d, F))\n (tx : TimeAndSpace d) :\n (g • η) tx = η (g⁻¹ • tx)", + "line": 100 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/SchwartzAction.lean", + "kind": "lemma", + "name": "TimeAndSpace.schwartzEuclideanAction_mul_apply", + "docstring": "Applying `g` and then `h` to a Schwartz map is the pullback action of `h * g`.", + "snippet": "lemma schwartzEuclideanAction_mul_apply {d : ℕ} (g h : EuclideanGroup d)\n (η : 𝓢(TimeAndSpace d, F)) :\n schwartzEuclideanAction h (schwartzEuclideanAction g η) =\n schwartzEuclideanAction (h * g) η", + "line": 105 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/SchwartzAction.lean", + "kind": "lemma", + "name": "TimeAndSpace.schwartzEuclideanAction_injective", + "docstring": "Each Euclidean-group pullback action on Schwartz maps is injective.", + "snippet": "lemma schwartzEuclideanAction_injective {d : ℕ} (g : EuclideanGroup d) :\n Function.Injective (schwartzEuclideanAction (F := F) g)", + "line": 114 + }, + { + "file": "Physlib/SpaceAndTime/TimeAndSpace/EuclideanGroup/SchwartzAction.lean", + "kind": "lemma", + "name": "TimeAndSpace.schwartzEuclideanAction_surjective", + "docstring": "Each Euclidean-group pullback action on Schwartz maps is surjective.", + "snippet": "lemma schwartzEuclideanAction_surjective {d : ℕ} (g : EuclideanGroup d) :\n Function.Surjective (schwartzEuclideanAction (F := F) g)", + "line": 122 + } + ] + }, + { + "filename": "Physlib/Thermodynamics/Temperature/Basic.lean", + "entries": [ + { + "file": "Physlib/Thermodynamics/Temperature/Basic.lean", + "kind": "lemma", + "name": "Temperature.β_toReal", + "docstring": null, + "snippet": "lemma β_toReal (T : Temperature) : (β T : ℝ) = 1 / (kB * (T : ℝ))", + "line": 61 + }, + { + "file": "Physlib/Thermodynamics/Temperature/Basic.lean", + "kind": "lemma", + "name": "Temperature.ofβ_toReal", + "docstring": null, + "snippet": "lemma ofβ_toReal (β : ℝ≥0) : (ofβ β).toReal = 1 / (kB * (β : ℝ))", + "line": 81 + } + ] + }, + { + "filename": "Physlib/Units/WithDim/Basic.lean", + "entries": [ + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "instance", + "name": "WithDim.«instance@ec98a162»", + "docstring": null, + "snippet": "instance (d : Dimension) (M : Type) [Inhabited M] : Inhabited (WithDim d M) where", + "line": 45 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "instance", + "name": "WithDim.«instance@c39a3b54»", + "docstring": null, + "snippet": "instance (d : Dimension) (M : Type) [Zero M] : Zero (WithDim d M) where", + "line": 48 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "lemma", + "name": "WithDim.val_zero", + "docstring": null, + "snippet": "@[simp]\nlemma val_zero {d : Dimension} {M : Type} [Zero M] :\n (0 : WithDim d M).val = 0", + "line": 52 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "instance", + "name": "WithDim.«instance@ccb84cd6»", + "docstring": null, + "snippet": "instance (d : Dimension) (M : Type) [Add M] : Add (WithDim d M) where", + "line": 55 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "lemma", + "name": "WithDim.val_add", + "docstring": null, + "snippet": "@[simp]\nlemma val_add {d : Dimension} {M : Type} [Add M] (m1 m2 : WithDim d M) :\n (m1 + m2).val = m1.val + m2.val", + "line": 59 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "instance", + "name": "WithDim.«instance@61de96eb»", + "docstring": null, + "snippet": "instance (d : Dimension) (M : Type) [Neg M] : Neg (WithDim d M) where", + "line": 62 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "lemma", + "name": "WithDim.val_neg", + "docstring": null, + "snippet": "@[simp]\nlemma val_neg {d : Dimension} {M : Type} [Neg M] (m : WithDim d M) :\n (-m).val = -m.val", + "line": 66 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "instance", + "name": "WithDim.«instance@9b66d0ce»", + "docstring": null, + "snippet": "instance (d : Dimension) (M : Type) [Sub M] : Sub (WithDim d M) where", + "line": 69 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "lemma", + "name": "WithDim.val_sub", + "docstring": null, + "snippet": "@[simp]\nlemma val_sub {d : Dimension} {M : Type} [Sub M] (m1 m2 : WithDim d M) :\n (m1 - m2).val = m1.val - m2.val", + "line": 73 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "instance", + "name": "WithDim.«instance@a15a8cbf»", + "docstring": null, + "snippet": "instance (d : Dimension) (M : Type) [AddSemigroup M] :\n AddSemigroup (WithDim d M) where", + "line": 76 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "instance", + "name": "WithDim.«instance@e53aef60»", + "docstring": null, + "snippet": "instance (d : Dimension) (M : Type) [AddCommSemigroup M] :\n AddCommSemigroup (WithDim d M) where", + "line": 82 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "instance", + "name": "WithDim.«instance@a79c96d0»", + "docstring": null, + "snippet": "instance (d : Dimension) (M : Type) [AddMonoid M] :\n AddMonoid (WithDim d M) where", + "line": 88 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "instance", + "name": "WithDim.«instance@2b3671f3»", + "docstring": null, + "snippet": "instance (d : Dimension) (M : Type) [AddCommMonoid M] :\n AddCommMonoid (WithDim d M) where", + "line": 98 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "instance", + "name": "WithDim.«instance@17f8c5c8»", + "docstring": null, + "snippet": "instance (d : Dimension) (M : Type) [AddGroup M] :\n AddGroup (WithDim d M) where", + "line": 104 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "instance", + "name": "WithDim.«instance@060f3ddb»", + "docstring": null, + "snippet": "instance (d : Dimension) (M : Type) [AddCommGroup M] :\n AddCommGroup (WithDim d M) where", + "line": 114 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "instance", + "name": "WithDim.«instance@84bf74c6»", + "docstring": null, + "snippet": "instance (d : Dimension) (M : Type) [LE M] : LE (WithDim d M) where", + "line": 120 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "lemma", + "name": "WithDim.le_def", + "docstring": null, + "snippet": "@[simp]\nlemma le_def {d : Dimension} {M : Type} [LE M] (m1 m2 : WithDim d M) :\n m1 ≤ m2 ↔ m1.val ≤ m2.val", + "line": 124 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "instance", + "name": "WithDim.«instance@fac005f7»", + "docstring": null, + "snippet": "instance (d : Dimension) (M : Type) [LT M] : LT (WithDim d M) where", + "line": 127 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "lemma", + "name": "WithDim.lt_def", + "docstring": null, + "snippet": "@[simp]\nlemma lt_def {d : Dimension} {M : Type} [LT M] (m1 m2 : WithDim d M) :\n m1 < m2 ↔ m1.val < m2.val", + "line": 131 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "instance", + "name": "WithDim.«instance@0a2efc58»", + "docstring": null, + "snippet": "instance (d : Dimension) (M : Type) [Preorder M] :\n Preorder (WithDim d M) where", + "line": 134 + }, + { + "file": "Physlib/Units/WithDim/Basic.lean", + "kind": "instance", + "name": "WithDim.«instance@86c24b45»", + "docstring": null, + "snippet": "instance (d : Dimension) (M : Type) [PartialOrder M] :\n PartialOrder (WithDim d M) where", + "line": 145 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.actionDensity", + "docstring": "The action density determined by a local Lagrangian and a field.", + "snippet": "noncomputable def actionDensity (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Space d → ℝ", + "line": 61 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.HasFiniteAction", + "docstring": "The integrability condition for the action density of a field.", + "snippet": "def HasFiniteAction (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) : Prop", + "line": 66 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.IsAdmissibleForAction", + "docstring": "Symmetric admissibility predicate for the action functional: the pair `(L, f)` is admissible\nwhen the field is smooth and the corresponding action density is integrable.", + "snippet": "def IsAdmissibleForAction (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Prop", + "line": 71 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.action", + "docstring": "The action associated with a local Lagrangian.", + "snippet": "noncomputable def action (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : ℝ", + "line": 76 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.actionDensity_apply", + "docstring": null, + "snippet": "@[simp]\nlemma actionDensity_apply (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) (x : Space d) :\n actionDensity L f x = L (jetAt k f x)", + "line": 81 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.action_eq_integral", + "docstring": null, + "snippet": "@[simp]\nlemma action_eq_integral (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) :\n action L f = ∫ x, actionDensity L f x", + "line": 86 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.variedField", + "docstring": "The field obtained by varying `f` in the direction `η` with parameter `s`.", + "snippet": "noncomputable def variedField (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)))\n (s : ℝ) :\n Space d → EuclideanSpace ℝ (Fin m)", + "line": 95 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.variedField_contDiff", + "docstring": "Smoothness of the varied field for smooth `f` and admissible `η`.", + "snippet": "lemma variedField_contDiff (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)))\n (s : ℝ)\n (hf : ContDiff ℝ ∞ f) :\n ContDiff ℝ ∞ (variedField f η s)", + "line": 102 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.actionVariation", + "docstring": "The action of a field under an admissible variation.", + "snippet": "noncomputable def actionVariation (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) : ℝ → ℝ", + "line": 110 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.HasFiniteActionVariation", + "docstring": "Explicit integrability condition for the family of varied action densities.", + "snippet": "def HasFiniteActionVariation (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) : Prop", + "line": 116 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.HasCompactlySupportedActionVariationDifference", + "docstring": "Locality package for the action density under compactly supported variations: for every\nvariation parameter `s`, the change in the action density is a continuous compactly supported\nfunction. Combined with finiteness of the base action, this implies finiteness of the varied\naction.", + "snippet": "def HasCompactlySupportedActionVariationDifference (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) : Prop", + "line": 124 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasFiniteActionVariation_of_hasFiniteAction_of_compactlySupportedDifference", + "docstring": "If the base action is finite and every varied action density differs from the base density by\na continuous compactly supported function, then the whole varied family has finite action.", + "snippet": "lemma hasFiniteActionVariation_of_hasFiniteAction_of_compactlySupportedDifference\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)))\n (hbase : HasFiniteAction L f)\n (hdiff : HasCompactlySupportedActionVariationDifference L f η) :\n HasFiniteActionVariation L f η", + "line": 133 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.variedField_apply", + "docstring": null, + "snippet": "@[simp]\nlemma variedField_apply (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) (s : ℝ)\n (x : Space d) :\n variedField f η s x = f x + s • η x", + "line": 154 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.actionVariation_apply", + "docstring": null, + "snippet": "@[simp]\nlemma actionVariation_apply (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) (s : ℝ) :\n actionVariation L f η s = action L (variedField f η s)", + "line": 160 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetAt_variedField_eq_of_notMem_tsupport", + "docstring": null, + "snippet": "lemma jetAt_variedField_eq_of_notMem_tsupport\n (k : ℕ) (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) (s : ℝ)\n {x : Space d}\n (hf : ContDiff ℝ ∞ f) (hx : x ∉ tsupport η.toFun) :\n jetAt k (variedField f η s) x = jetAt k f x", + "line": 164 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.actionDensity_variedField_eq_of_notMem_tsupport", + "docstring": null, + "snippet": "lemma actionDensity_variedField_eq_of_notMem_tsupport\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)))\n (s : ℝ)\n {x : Space d} (hf : ContDiff ℝ ∞ f) (hx : x ∉ tsupport η.toFun) :\n actionDensity L (variedField f η s) x = actionDensity L f x", + "line": 178 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasCompactlySupportedActionVariationDifference_of_continuousInCoordinates", + "docstring": null, + "snippet": "lemma hasCompactlySupportedActionVariationDifference_of_continuousInCoordinates\n (L : Lagrangian d m k) (hcontL : Lagrangian.ContinuousInCoordinates L)\n (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) :\n HasCompactlySupportedActionVariationDifference L f η", + "line": 186 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Action.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.IsCritical", + "docstring": "A field is critical for a local action if every admissible compactly supported variation has\nvanishing first derivative at `s = 0`, under the explicit finite-action hypothesis for the varied\nfamily.", + "snippet": "def IsCritical (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) : Prop", + "line": 218 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/EulerLagrange.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/EulerLagrange.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.eulerLagrangeTerm", + "docstring": "The summand `(-1)^|I| D_I (∂L/∂u^a_I)` in the local Euler-Lagrange formula.", + "snippet": "noncomputable def eulerLagrangeTerm (L : Lagrangian d m k) (I : DerivativeIndex d k) (a : Fin m)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Space d → ℝ", + "line": 54 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/EulerLagrange.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.eulerLagrangeTerm_apply", + "docstring": null, + "snippet": "@[simp]\nlemma eulerLagrangeTerm_apply (L : Lagrangian d m k) (I : DerivativeIndex d k) (a : Fin m)\n (f : Space d → EuclideanSpace ℝ (Fin m)) (x : Space d) :\n eulerLagrangeTerm L I a f x =\n (-1 : ℝ) ^ I.1.order * iteratedTotalDerivative k I.1 (L.coordDeriv I a) f x", + "line": 60 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/EulerLagrange.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.eulerLagrangeComponent", + "docstring": "The `a`-th component of the local Euler-Lagrange operator.", + "snippet": "noncomputable def eulerLagrangeComponent (L : Lagrangian d m k) (a : Fin m)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Space d → ℝ", + "line": 71 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/EulerLagrange.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.eulerLagrangeComponent_apply", + "docstring": null, + "snippet": "@[simp]\nlemma eulerLagrangeComponent_apply (L : Lagrangian d m k) (a : Fin m)\n (f : Space d → EuclideanSpace ℝ (Fin m)) (x : Space d) :\n eulerLagrangeComponent L a f x = ∑ I : DerivativeIndex d k, eulerLagrangeTerm L I a f x", + "line": 76 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/EulerLagrange.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.eulerLagrangeOp", + "docstring": "The local Euler-Lagrange operator attached to a local Lagrangian.", + "snippet": "noncomputable def eulerLagrangeOp (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) :\n Space d → EuclideanSpace ℝ (Fin m)", + "line": 86 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/EulerLagrange.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.eulerLagrangeOp_apply", + "docstring": null, + "snippet": "lemma eulerLagrangeOp_apply (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (x : Space d) (a : Fin m) :\n eulerLagrangeOp L f x a = eulerLagrangeComponent L a f x", + "line": 90 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/EulerLagrange.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.eulerLagrangeOp_eq", + "docstring": null, + "snippet": "@[simp]\nlemma eulerLagrangeOp_eq (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) :\n eulerLagrangeOp L f = fun x => WithLp.toLp 2 fun a => eulerLagrangeComponent L a f x", + "line": 96 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "abbrev", + "name": "ClassicalFieldTheory.Local.FirstOrderJetCoordinates", + "docstring": "Coordinate data for first-order local jet points.", + "snippet": "abbrev FirstOrderJetCoordinates (d m : ℕ)", + "line": 63 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "abbrev", + "name": "ClassicalFieldTheory.Local.FirstOrderJetPoint", + "docstring": "Coordinate-level first-order jet points.", + "snippet": "abbrev FirstOrderJetPoint (d m : ℕ)", + "line": 66 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "abbrev", + "name": "ClassicalFieldTheory.Local.FirstOrderJetFiberData", + "docstring": "Fiber-direction data for first-order jet points.", + "snippet": "abbrev FirstOrderJetFiberData (d m : ℕ)", + "line": 69 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "abbrev", + "name": "ClassicalFieldTheory.Local.FirstOrderLagrangian", + "docstring": "First-order local lagrangians.", + "snippet": "abbrev FirstOrderLagrangian (d m : ℕ)", + "line": 72 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.firstDerivativeIndex", + "docstring": "The first-derivative coordinate index in the source direction `i`.", + "snippet": "def firstDerivativeIndex (d : ℕ) (i : Fin d) : DerivativeIndex d 1", + "line": 80 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.firstDerivativeIndex_coe", + "docstring": null, + "snippet": "@[simp]\nlemma firstDerivativeIndex_coe (d : ℕ) (i : Fin d) :\n ((firstDerivativeIndex d i : DerivativeIndex d 1) : MultiIndex d) =\n Physlib.MultiIndex.increment 0 i", + "line": 84 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.firstDerivativeIndex_order", + "docstring": null, + "snippet": "lemma firstDerivativeIndex_order (d : ℕ) (i : Fin d) :\n ((firstDerivativeIndex d i : DerivativeIndex d 1) : MultiIndex d).order = 1", + "line": 88 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.JetPoint.firstDerivCoord", + "docstring": "The first derivative coordinate `u^a_i` of a first-order jet point.", + "snippet": "def firstDerivCoord (J : FirstOrderJetPoint d m) (i : Fin d) (a : Fin m) : ℝ", + "line": 102 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.JetPoint.firstDerivVector", + "docstring": "The bundled target vector of first derivative coordinates in the source direction `i`.", + "snippet": "def firstDerivVector (J : FirstOrderJetPoint d m) (i : Fin d) : EuclideanSpace ℝ (Fin m)", + "line": 106 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetPoint.firstDerivVector_apply", + "docstring": null, + "snippet": "@[simp]\nlemma firstDerivVector_apply (J : FirstOrderJetPoint d m) (i : Fin d) (a : Fin m) :\n J.firstDerivVector i a = J.firstDerivCoord i a", + "line": 110 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetPoint.firstDerivCoord_ofBaseCoordinates", + "docstring": null, + "snippet": "@[simp]\nlemma firstDerivCoord_ofBaseCoordinates (x : Space d) (u : FirstOrderJetCoordinates d m)\n (i : Fin d) (a : Fin m) :\n (JetPoint.ofBaseCoordinates x u).firstDerivCoord i a = u (firstDerivativeIndex d i) a", + "line": 115 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "abbrev", + "name": "ClassicalFieldTheory.Local.firstOrderJetAt", + "docstring": "The first-order coordinate-level jet point of a field at a point.", + "snippet": "noncomputable abbrev firstOrderJetAt (f : Space d → EuclideanSpace ℝ (Fin m)) (x : Space d) :\n FirstOrderJetPoint d m", + "line": 127 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.firstOrderJetAt_base", + "docstring": null, + "snippet": "@[simp]\nlemma firstOrderJetAt_base (f : Space d → EuclideanSpace ℝ (Fin m)) (x : Space d) :\n (firstOrderJetAt f x).base = x", + "line": 132 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.firstOrderJetAt_value", + "docstring": null, + "snippet": "lemma firstOrderJetAt_value (f : Space d → EuclideanSpace ℝ (Fin m)) (x : Space d) :\n (firstOrderJetAt f x).value = f x", + "line": 135 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.firstOrderJetAt_firstDerivCoord", + "docstring": null, + "snippet": "@[simp]\nlemma firstOrderJetAt_firstDerivCoord (f : Space d → EuclideanSpace ℝ (Fin m))\n (x : Space d) (i : Fin d) (a : Fin m) :\n (firstOrderJetAt f x).firstDerivCoord i a =\n ∂^[Physlib.MultiIndex.increment 0 i] (fun y => (f y) a) x", + "line": 140 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "abbrev", + "name": "ClassicalFieldTheory.Local.firstOrderActionDensity", + "docstring": "Evaluate a first-order local lagrangian along the first jets of a field.", + "snippet": "noncomputable abbrev firstOrderActionDensity (L : FirstOrderLagrangian d m)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Space d → ℝ", + "line": 146 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "abbrev", + "name": "ClassicalFieldTheory.Local.firstOrderAction", + "docstring": "The action of a first-order local lagrangian.", + "snippet": "noncomputable abbrev firstOrderAction (L : FirstOrderLagrangian d m)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : ℝ", + "line": 151 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "abbrev", + "name": "ClassicalFieldTheory.Local.firstOrderEulerLagrangeOp", + "docstring": "The Euler-Lagrange operator of a first-order local lagrangian.", + "snippet": "noncomputable abbrev firstOrderEulerLagrangeOp (L : FirstOrderLagrangian d m)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Space d → EuclideanSpace ℝ (Fin m)", + "line": 156 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstOrder.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.firstOrder_isCritical_iff_eulerLagrange_zero", + "docstring": null, + "snippet": "theorem firstOrder_isCritical_iff_eulerLagrange_zero\n (L : FirstOrderLagrangian d m) (f : Space d → EuclideanSpace ℝ (Fin m))\n (hLf : IsAdmissibleForAction L f)\n (hL : Lagrangian.SmoothInCoordinates L) :\n IsCritical L f ↔ firstOrderEulerLagrangeOp L f = 0", + "line": 160 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.hasSmoothEulerLagrangeRegularity_of_smoothInCoordinates", + "docstring": null, + "snippet": "theorem hasSmoothEulerLagrangeRegularity_of_smoothInCoordinates\n (L : Lagrangian d m k) (hL : Lagrangian.SmoothInCoordinates L) :\n HasSmoothEulerLagrangeRegularity L", + "line": 51 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.isCritical_iff_eulerLagrange_zero_of_contDiff_and_smoothInCoordinates", + "docstring": null, + "snippet": "theorem isCritical_iff_eulerLagrange_zero_of_contDiff_and_smoothInCoordinates\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hL : Lagrangian.SmoothInCoordinates L)\n (hbase : HasFiniteAction L f) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 61 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.isCritical_iff_eulerLagrange_zero_of_admissibleForAction_and_smoothInCoordinates", + "docstring": null, + "snippet": "theorem isCritical_iff_eulerLagrange_zero_of_admissibleForAction_and_smoothInCoordinates\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (hLf : IsAdmissibleForAction L f)\n (hL : Lagrangian.SmoothInCoordinates L) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 70 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Basic.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Basic.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.firstVariationDensityTerm", + "docstring": "A single term in the linearized first-variation density before integration by parts.", + "snippet": "noncomputable def firstVariationDensityTerm (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) (I : DerivativeIndex d k)\n (a : Fin m) :\n Space d → ℝ", + "line": 51 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Basic.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.firstVariationDensity", + "docstring": "The pointwise linearized first-variation density before integration by parts.", + "snippet": "noncomputable def firstVariationDensity (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) : Space d → ℝ", + "line": 59 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Basic.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.firstVariationValue", + "docstring": "The value predicted by the first-variation formula for an admissible variation.", + "snippet": "noncomputable def firstVariationValue (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) : ℝ", + "line": 65 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Basic.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.firstVariationDensityTerm_apply", + "docstring": null, + "snippet": "@[simp]\nlemma firstVariationDensityTerm_apply (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) (I : DerivativeIndex d k)\n (a : Fin m) (x : Space d) :\n firstVariationDensityTerm L f η I a x =\n L.coordDeriv I a (jetAt k f x) * ∂^[I.1] (fun y => (η y) a) x", + "line": 71 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Basic.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.firstVariationDensity_apply", + "docstring": null, + "snippet": "@[simp]\nlemma firstVariationDensity_apply (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) (x : Space d) :\n firstVariationDensity L f η x =\n ∑ I : DerivativeIndex d k, ∑ a : Fin m, firstVariationDensityTerm L f η I a x", + "line": 79 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Basic.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.firstVariationValue_eq_integral", + "docstring": null, + "snippet": "@[simp]\nlemma firstVariationValue_eq_integral (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) :\n firstVariationValue L f η = ∫ x, ⟪eulerLagrangeOp L f x, η x⟫_ℝ", + "line": 86 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Basic.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.firstVariationValue_eq_zero_of_eulerLagrange_zero", + "docstring": null, + "snippet": "lemma firstVariationValue_eq_zero_of_eulerLagrange_zero\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) (hEuler : eulerLagrangeOp L f = 0) :\n firstVariationValue L f η = 0", + "line": 91 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.AllVariationsHaveFiniteAction", + "docstring": "Explicit global finiteness hypothesis for all admissible variations.", + "snippet": "def AllVariationsHaveFiniteAction (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Prop", + "line": 53 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.HasFirstVariationFormula", + "docstring": "The first-variation formula for the local action, packaged as a reusable hypothesis.", + "snippet": "def HasFirstVariationFormula (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Prop", + "line": 58 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.isCritical_of_eulerLagrange_zero", + "docstring": null, + "snippet": "lemma isCritical_of_eulerLagrange_zero (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (hfirst : HasFirstVariationFormula L f) (hEuler : eulerLagrangeOp L f = 0) :\n IsCritical L f", + "line": 63 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.eulerLagrange_zero_of_isCritical", + "docstring": null, + "snippet": "lemma eulerLagrange_zero_of_isCritical (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (hfirst : HasFirstVariationFormula L f)\n (hfin : AllVariationsHaveFiniteAction L f)\n (hcont : Continuous (eulerLagrangeOp L f))\n (hcrit : IsCritical L f) :\n eulerLagrangeOp L f = 0", + "line": 74 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.isCritical_iff_eulerLagrange_zero", + "docstring": null, + "snippet": "theorem isCritical_iff_eulerLagrange_zero (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (hfirst : HasFirstVariationFormula L f)\n (hfin : AllVariationsHaveFiniteAction L f)\n (hcont : Continuous (eulerLagrangeOp L f)) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 92 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasFirstVariationFormula_of_underIntegral_linearized_and_parts", + "docstring": null, + "snippet": "private lemma hasFirstVariationFormula_of_underIntegral_linearized_and_parts\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (hint : HasActionVariationDerivativeUnderIntegral L f)\n (hpoint : HasPointwiseLinearizedDensityFormula L f)\n (hibp : HasIntegratedByPartsFormula L f) :\n HasFirstVariationFormula L f", + "line": 102 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasFirstVariationFormula_of_contDiff_underIntegral_and_parts", + "docstring": null, + "snippet": "private lemma hasFirstVariationFormula_of_contDiff_underIntegral_and_parts\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hint : HasActionVariationDerivativeUnderIntegral L f)\n (hibp : HasIntegratedByPartsFormula L f) :\n HasFirstVariationFormula L f", + "line": 123 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasFirstVariationFormula_of_underIntegral_linearized_and_termwise", + "docstring": null, + "snippet": "private lemma hasFirstVariationFormula_of_underIntegral_linearized_and_termwise\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (hint : HasActionVariationDerivativeUnderIntegral L f)\n (hpoint : HasPointwiseLinearizedDensityFormula L f)\n (hterm : HasTermwiseIntegratedByPartsFormula L f) :\n HasFirstVariationFormula L f", + "line": 131 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasFirstVariationFormula_of_contDiff_underIntegral_and_termwise", + "docstring": null, + "snippet": "private lemma hasFirstVariationFormula_of_contDiff_underIntegral_and_termwise\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hint : HasActionVariationDerivativeUnderIntegral L f)\n (hterm : HasTermwiseIntegratedByPartsFormula L f) :\n HasFirstVariationFormula L f", + "line": 140 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.isCritical_iff_eulerLagrange_zero_of_underIntegral_linearized_and_parts", + "docstring": "The local Euler-Lagrange criterion obtained from the packaged analytic ingredients of the\nfirst-variation formula. This is the current formalized form of Theorem 5.2.", + "snippet": "private theorem isCritical_iff_eulerLagrange_zero_of_underIntegral_linearized_and_parts\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (hint : HasActionVariationDerivativeUnderIntegral L f)\n (hpoint : HasPointwiseLinearizedDensityFormula L f)\n (hibp : HasIntegratedByPartsFormula L f)\n (hfin : AllVariationsHaveFiniteAction L f)\n (hcont : Continuous (eulerLagrangeOp L f)) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 155 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.isCritical_iff_eulerLagrange_zero_of_underIntegral_linearized_and_termwise", + "docstring": "Variant of the local Euler-Lagrange criterion where the integration-by-parts input is reduced\nto a termwise hypothesis.", + "snippet": "private theorem isCritical_iff_eulerLagrange_zero_of_underIntegral_linearized_and_termwise\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (hint : HasActionVariationDerivativeUnderIntegral L f)\n (hpoint : HasPointwiseLinearizedDensityFormula L f)\n (hterm : HasTermwiseIntegratedByPartsFormula L f)\n (hfin : AllVariationsHaveFiniteAction L f)\n (hcont : Continuous (eulerLagrangeOp L f)) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 169 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.isCritical_iff_eulerLagrange_zero_of_contDiff_underIntegral_and_termwise", + "docstring": null, + "snippet": "private theorem isCritical_iff_eulerLagrange_zero_of_contDiff_underIntegral_and_termwise\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hint : HasActionVariationDerivativeUnderIntegral L f)\n (hterm : HasTermwiseIntegratedByPartsFormula L f)\n (hfin : AllVariationsHaveFiniteAction L f)\n (hcont : Continuous (eulerLagrangeOp L f)) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 180 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.isCritical_iff_eulerLagrange_zero_of_contDiff_continuous_coordDeriv_and_termwise", + "docstring": null, + "snippet": "private theorem isCritical_iff_eulerLagrange_zero_of_contDiff_continuous_coordDeriv_and_termwise\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hcontVar : ∀ η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)),\n ∀ I : DerivativeIndex d k, ∀ a : Fin m,\n Continuous (fun p : ℝ × Space d =>\n L.coordDeriv I a (jetAt k (variedField f η p.1) p.2)))\n (hterm : HasTermwiseIntegratedByPartsFormula L f)\n (hfin : AllVariationsHaveFiniteAction L f)\n (hcont : Continuous (eulerLagrangeOp L f)) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 191 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.isCritical_iff_eulerLagrange_zero_of_contDiff_and_regular", + "docstring": null, + "snippet": "private theorem isCritical_iff_eulerLagrange_zero_of_contDiff_and_regular\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hcontVar : ∀ η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)),\n Lagrangian.ContinuousCoordDerivAlongFamily L (fun s : ℝ => variedField f η s))\n (hcoeff : Lagrangian.ContDiffCoordDerivAlongField L f)\n (hfin : AllVariationsHaveFiniteAction L f)\n (hcont : Continuous (eulerLagrangeOp L f)) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 206 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.isCritical_iff_eulerLagrange_zero_of_contDiff_and_regularityAt", + "docstring": null, + "snippet": "private theorem isCritical_iff_eulerLagrange_zero_of_contDiff_and_regularityAt\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hreg : HasEulerLagrangeRegularityAt L f)\n (hfin : AllVariationsHaveFiniteAction L f)\n (hcont : Continuous (eulerLagrangeOp L f)) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 219 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.isCritical_iff_eulerLagrange_zero_of_contDiff_and_smoothRegularity", + "docstring": null, + "snippet": "private theorem isCritical_iff_eulerLagrange_zero_of_contDiff_and_smoothRegularity\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hreg : HasSmoothEulerLagrangeRegularity L)\n (hfin : AllVariationsHaveFiniteAction L f)\n (hcont : Continuous (eulerLagrangeOp L f)) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 229 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.isCritical_iff_eulerLagrange_zero_of_contDiff_and_coordinateRegularity", + "docstring": null, + "snippet": "private theorem isCritical_iff_eulerLagrange_zero_of_contDiff_and_coordinateRegularity\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hcoord : Lagrangian.ContDiffCoordDerivInCoordinates L)\n (hfin : AllVariationsHaveFiniteAction L f)\n (hcont : Continuous (eulerLagrangeOp L f)) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 238 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.isCritical_iff_eulerLagrange_zero_of_contDiff_and_regularityAt'", + "docstring": null, + "snippet": "private theorem isCritical_iff_eulerLagrange_zero_of_contDiff_and_regularityAt'\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hreg : HasEulerLagrangeRegularityAt L f)\n (hfin : AllVariationsHaveFiniteAction L f) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 247 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.isCritical_iff_eulerLagrange_zero_of_contDiff_and_smoothRegularity'", + "docstring": null, + "snippet": "private theorem isCritical_iff_eulerLagrange_zero_of_contDiff_and_smoothRegularity'\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hreg : HasSmoothEulerLagrangeRegularity L)\n (hfin : AllVariationsHaveFiniteAction L f) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 256 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.isCritical_iff_eulerLagrange_zero_of_contDiff_and_coordinateRegularity'", + "docstring": null, + "snippet": "private theorem isCritical_iff_eulerLagrange_zero_of_contDiff_and_coordinateRegularity'\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hcoord : Lagrangian.ContDiffCoordDerivInCoordinates L)\n (hfin : AllVariationsHaveFiniteAction L f) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 263 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.«theorem@4a640fdf»", + "docstring": null, + "snippet": "private theorem\n isCritical_iff_eulerLagrange_zero_of_contDiff_and_coordinateRegularity_of_hasFiniteAction\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hcoord : Lagrangian.ContDiffCoordDerivInCoordinates L)\n (hbase : HasFiniteAction L f)\n (hlocal :\n ∀ η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)),\n HasCompactlySupportedActionVariationDifference L f η) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 271 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Criterion.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.isCritical_iff_eulerLagrange_zero_of_hasFiniteAction_and_continuousInCoordinates", + "docstring": null, + "snippet": "theorem isCritical_iff_eulerLagrange_zero_of_hasFiniteAction_and_continuousInCoordinates\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hcoord : Lagrangian.ContDiffCoordDerivInCoordinates L)\n (hcontL : Lagrangian.ContinuousInCoordinates L)\n (hbase : HasFiniteAction L f) :\n IsCritical L f ↔ eulerLagrangeOp L f = 0", + "line": 286 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Density.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Density.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.HasActionVariationDerivativeUnderIntegral", + "docstring": "Differentiation of the varied action under the integral sign, packaged as a hypothesis.", + "snippet": "def HasActionVariationDerivativeUnderIntegral (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) :\n Prop", + "line": 52 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Density.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.HasPointwiseLinearizedDensityFormula", + "docstring": "Pointwise linearization of the action density before integration by parts.", + "snippet": "def HasPointwiseLinearizedDensityFormula (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Prop", + "line": 60 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Density.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.actionVariation_hasDerivAt_of_dominated_loc_of_deriv_le", + "docstring": "A direct bridge from Mathlib's dominated differentiation-under-the-integral theorem to the\nlocal action variation. This isolates the measure-theoretic input needed to prove\n`HasActionVariationDerivativeUnderIntegral`.", + "snippet": "lemma actionVariation_hasDerivAt_of_dominated_loc_of_deriv_le\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)))\n {F' : ℝ → Space d → ℝ} {bound : Space d → ℝ} {ε : ℝ} (hε : 0 < ε)\n (hmeas :\n ∀ᶠ s in nhds (0 : ℝ),\n AEStronglyMeasurable (actionDensity L (variedField f η s)) volume)\n (hfinite0 : Integrable (actionDensity L (variedField f η 0)))\n (hF'_meas : AEStronglyMeasurable (F' 0) volume)\n (hbound :\n ∀ᵐ x ∂volume, ∀ s ∈ Metric.ball (0 : ℝ) ε, ‖F' s x‖ ≤ bound x)\n (hbound_int : Integrable bound volume)\n (hderiv :\n ∀ᵐ x ∂volume, ∀ s ∈ Metric.ball (0 : ℝ) ε,\n HasDerivAt (fun r : ℝ => actionDensity L (variedField f η r) x) (F' s x) s) :\n HasDerivAt (actionVariation L f η) (∫ x, F' 0 x) 0", + "line": 69 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Density.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.HasDominatedVariationDerivativeNear", + "docstring": "A dominated bound for the pointwise first variation along the varied-field family near\n`s = 0`. This packages the analytic domination needed to differentiate the action under the\nintegral sign.", + "snippet": "def HasDominatedVariationDerivativeNear (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) : Prop", + "line": 95 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Density.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.continuous_coordDeriv_at_zero", + "docstring": null, + "snippet": "private lemma continuous_coordDeriv_at_zero\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)))\n (hcont : ∀ I : DerivativeIndex d k, ∀ a : Fin m,\n Continuous (fun p : ℝ × Space d =>\n L.coordDeriv I a (jetAt k (variedField f η p.1) p.2)))\n (I : DerivativeIndex d k) (a : Fin m) :\n Continuous (fun x : Space d => L.coordDeriv I a (jetAt k f x))", + "line": 104 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Density.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.firstVariationDensityTerm_norm_le_of_coeff_bound", + "docstring": null, + "snippet": "private lemma firstVariationDensityTerm_norm_le_of_coeff_bound\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)))\n (ψ : DerivativeIndex d k → Fin m → Space d → ℝ)\n (hψ : ∀ I : DerivativeIndex d k, ∀ a : Fin m, ∀ x : Space d,\n ψ I a x = ∂^[I.1] (fun y => (η y) a) x)\n (C : DerivativeIndex d k → Fin m → ℝ)\n (K : DerivativeIndex d k → Fin m → Set (Space d))\n (hKzero : ∀ I : DerivativeIndex d k, ∀ a : Fin m, ∀ x, x ∉ K I a → ψ I a x = 0)\n (hC : ∀ I : DerivativeIndex d k, ∀ a : Fin m,\n ∀ p ∈ Metric.closedBall (0 : ℝ) 1 ×ˢ K I a,\n ‖L.coordDeriv I a (jetAt k (variedField f η p.1) p.2)‖ ≤ C I a)\n {x : Space d} {s : ℝ} (hs : s ∈ Metric.ball (0 : ℝ) 1)\n (I : DerivativeIndex d k) (a : Fin m) :\n ‖firstVariationDensityTerm L (variedField f η s) η I a x‖ ≤ C I a * ‖ψ I a x‖", + "line": 119 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Density.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.norm_firstVariationDensity_le_bound", + "docstring": null, + "snippet": "private lemma norm_firstVariationDensity_le_bound\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)))\n (ψ : DerivativeIndex d k → Fin m → Space d → ℝ)\n (hψ : ∀ I : DerivativeIndex d k, ∀ a : Fin m, ∀ x : Space d,\n ψ I a x = ∂^[I.1] (fun y => (η y) a) x)\n (C : DerivativeIndex d k → Fin m → ℝ)\n (K : DerivativeIndex d k → Fin m → Set (Space d))\n (hKzero : ∀ I : DerivativeIndex d k, ∀ a : Fin m, ∀ x, x ∉ K I a → ψ I a x = 0)\n (hC : ∀ I : DerivativeIndex d k, ∀ a : Fin m,\n ∀ p ∈ Metric.closedBall (0 : ℝ) 1 ×ˢ K I a,\n ‖L.coordDeriv I a (jetAt k (variedField f η p.1) p.2)‖ ≤ C I a)\n {x : Space d} {s : ℝ} (hs : s ∈ Metric.ball (0 : ℝ) 1) :\n ‖firstVariationDensity L (variedField f η s) η x‖\n ≤ ∑ I : DerivativeIndex d k, ∑ a : Fin m, C I a * ‖ψ I a x‖", + "line": 151 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Density.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasDominatedVariationDerivativeNear_of_continuous_coordDeriv", + "docstring": "A concrete dominated bound for the varied first-variation density, obtained from continuity of\nthe jet-coordinate coefficient family in `(s, x)`. The compact support of the iterated derivatives\nof the admissible variation supplies the integrable majorant.", + "snippet": "lemma hasDominatedVariationDerivativeNear_of_continuous_coordDeriv\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)))\n (hcont : ∀ I : DerivativeIndex d k, ∀ a : Fin m,\n Continuous (fun p : ℝ × Space d =>\n L.coordDeriv I a (jetAt k (variedField f η p.1) p.2))) :\n HasDominatedVariationDerivativeNear L f η", + "line": 195 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Density.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasDerivAt_actionDensity_variedField_zero_of_contDiff", + "docstring": null, + "snippet": "lemma hasDerivAt_actionDensity_variedField_zero_of_contDiff\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f) :\n ∀ η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)), ∀ x : Space d,\n HasDerivAt (fun s : ℝ => actionDensity L (variedField f η s) x)\n (firstVariationDensity L f η x) 0", + "line": 257 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Density.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasDerivAt_actionDensity_variedField_of_contDiff", + "docstring": null, + "snippet": "lemma hasDerivAt_actionDensity_variedField_of_contDiff\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) (x : Space d) (s : ℝ) :\n HasDerivAt (fun r : ℝ => actionDensity L (variedField f η r) x)\n (firstVariationDensity L (variedField f η s) η x) s", + "line": 279 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Density.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasPointwiseLinearizedDensityFormula_of_contDiff", + "docstring": null, + "snippet": "lemma hasPointwiseLinearizedDensityFormula_of_contDiff\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f) :\n HasPointwiseLinearizedDensityFormula L f", + "line": 299 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Density.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasActionVariationDerivativeUnderIntegral_of_contDiff_of_dominatedVariation", + "docstring": null, + "snippet": "lemma hasActionVariationDerivativeUnderIntegral_of_contDiff_of_dominatedVariation\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hdom : ∀ η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)),\n HasFiniteActionVariation L f η →\n HasDominatedVariationDerivativeNear L f η) :\n HasActionVariationDerivativeUnderIntegral L f", + "line": 305 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Density.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasActionVariationDerivativeUnderIntegral_of_contDiff_of_continuous_coordDeriv", + "docstring": null, + "snippet": "lemma hasActionVariationDerivativeUnderIntegral_of_contDiff_of_continuous_coordDeriv\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hcontVar : ∀ η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)),\n ∀ I : DerivativeIndex d k, ∀ a : Fin m,\n Continuous (fun p : ℝ × Space d =>\n L.coordDeriv I a (jetAt k (variedField f η p.1) p.2))) :\n HasActionVariationDerivativeUnderIntegral L f", + "line": 340 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Density.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasActionVariationDerivativeUnderIntegral_of_contDiff_of_regular", + "docstring": null, + "snippet": "lemma hasActionVariationDerivativeUnderIntegral_of_contDiff_of_regular\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m)) (hf : ContDiff ℝ ∞ f)\n (hcontVar : ∀ η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)),\n Lagrangian.ContinuousCoordDerivAlongFamily L (fun s : ℝ => variedField f η s)) :\n HasActionVariationDerivativeUnderIntegral L f", + "line": 351 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/IntegrationByParts.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/IntegrationByParts.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.HasIntegratedByPartsFormula", + "docstring": "The integration-by-parts step sending the linearized density to the Euler-Lagrange pairing.", + "snippet": "def HasIntegratedByPartsFormula (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Prop", + "line": 52 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/IntegrationByParts.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.HasTermwiseIntegratedByPartsFormula", + "docstring": "Termwise integration-by-parts data for the linearized first-variation density.", + "snippet": "def HasTermwiseIntegratedByPartsFormula (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Prop", + "line": 58 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/IntegrationByParts.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.integral_mul_space_deriv_eq_neg_deriv_mul", + "docstring": null, + "snippet": "private lemma integral_mul_space_deriv_eq_neg_deriv_mul (i : Fin d) {g h : Space d → ℝ}\n (hg : ContDiff ℝ ∞ g) (hh : IsTestFunction h) :\n ∫ x, g x * ∂[i] h x = ∫ x, (-∂[i] g x) * h x", + "line": 67 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/IntegrationByParts.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.integral_mul_iteratedDerivList_eq_sign", + "docstring": null, + "snippet": "private lemma integral_mul_iteratedDerivList_eq_sign (L : List (Fin d)) {g h : Space d → ℝ}\n (hg : ContDiff ℝ ∞ g) (hh : IsTestFunction h) :\n ∫ x, g x * L.foldr (fun i f => ∂[i] f) h x =\n ∫ x, (((-1 : ℝ) ^ L.length) * L.foldr (fun i f => ∂[i] f) g x) * h x", + "line": 91 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/IntegrationByParts.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.integral_mul_iteratedDeriv_eq_sign", + "docstring": "Repeated integration by parts for iterated coordinate derivatives against a test function.", + "snippet": "lemma integral_mul_iteratedDeriv_eq_sign {g h : Space d → ℝ} (I : MultiIndex d)\n (hg : ContDiff ℝ ∞ g) (hh : IsTestFunction h) :\n ∫ x, g x * ∂^[I] h x =\n ∫ x, (((-1 : ℝ) ^ I.order) * ∂^[I] g x) * h x", + "line": 122 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/IntegrationByParts.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasTermwiseIntegratedByPartsFormula_of_contDiff_coordDeriv", + "docstring": "Concrete termwise integration by parts for the first-variation density, assuming the jet\ncoordinate coefficient functions are smooth along the field.", + "snippet": "lemma hasTermwiseIntegratedByPartsFormula_of_contDiff_coordDeriv\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (hcoeff : ∀ I : DerivativeIndex d k, ∀ a : Fin m,\n ContDiff ℝ ∞ (fun x => L.coordDeriv I a (jetAt k f x))) :\n HasTermwiseIntegratedByPartsFormula L f", + "line": 136 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/IntegrationByParts.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasTermwiseIntegratedByPartsFormula_of_regular", + "docstring": null, + "snippet": "lemma hasTermwiseIntegratedByPartsFormula_of_regular\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (hcoeff : Lagrangian.ContDiffCoordDerivAlongField L f) :\n HasTermwiseIntegratedByPartsFormula L f", + "line": 164 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/IntegrationByParts.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.integral_firstVariationDensity_eq_termwise_sum", + "docstring": null, + "snippet": "private lemma integral_firstVariationDensity_eq_termwise_sum\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)))\n (hterm : HasTermwiseIntegratedByPartsFormula L f) :\n ∫ x, firstVariationDensity L f η x =\n ∑ I : DerivativeIndex d k, ∑ a : Fin m,\n ∫ x, firstVariationDensityTerm L f η I a x", + "line": 170 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/IntegrationByParts.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.integral_eulerLagrange_termwise_sum_eq_pairing", + "docstring": null, + "snippet": "private lemma integral_eulerLagrange_termwise_sum_eq_pairing\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)))\n (hterm : HasTermwiseIntegratedByPartsFormula L f) :\n ∑ I : DerivativeIndex d k, ∑ a : Fin m, ∫ x, eulerLagrangeTerm L I a f x * (η x) a\n = firstVariationValue L f η", + "line": 190 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/IntegrationByParts.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.hasIntegratedByPartsFormula_of_termwise", + "docstring": null, + "snippet": "lemma hasIntegratedByPartsFormula_of_termwise (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (hterm : HasTermwiseIntegratedByPartsFormula L f) :\n HasIntegratedByPartsFormula L f", + "line": 233 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Regularity.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Regularity.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.HasEulerLagrangeRegularityAt", + "docstring": "The combined regularity package currently needed to derive the local Euler-Lagrange\ncriterion for a field `f`: smooth coefficient functions along `f`, and continuity of the\ncorresponding coefficient families along every admissible varied-field family based at `f`.", + "snippet": "def HasEulerLagrangeRegularityAt (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Prop", + "line": 53 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Regularity.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.HasSmoothEulerLagrangeRegularity", + "docstring": "A local Lagrangian has smooth Euler-Lagrange regularity if every smooth field satisfies the\nregularity package currently needed by the local first-variation proof.", + "snippet": "def HasSmoothEulerLagrangeRegularity (L : Lagrangian d m k) : Prop", + "line": 61 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Regularity.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.continuous_eulerLagrangeTerm_of_regular", + "docstring": null, + "snippet": "lemma continuous_eulerLagrangeTerm_of_regular\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (hcoeff : Lagrangian.ContDiffCoordDerivAlongField L f)\n (I : DerivativeIndex d k) (a : Fin m) :\n Continuous (eulerLagrangeTerm L I a f)", + "line": 65 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Regularity.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.continuous_eulerLagrangeComponent_of_regular", + "docstring": null, + "snippet": "lemma continuous_eulerLagrangeComponent_of_regular\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (hcoeff : Lagrangian.ContDiffCoordDerivAlongField L f)\n (a : Fin m) :\n Continuous (eulerLagrangeComponent L a f)", + "line": 79 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Regularity.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.continuous_eulerLagrangeOp_of_regular", + "docstring": null, + "snippet": "lemma continuous_eulerLagrangeOp_of_regular\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (hcoeff : Lagrangian.ContDiffCoordDerivAlongField L f) :\n Continuous (eulerLagrangeOp L f)", + "line": 88 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Regularity.lean", + "kind": "theorem", + "name": "ClassicalFieldTheory.Local.hasSmoothEulerLagrangeRegularity_of_contDiffCoordDerivInCoordinates", + "docstring": null, + "snippet": "theorem hasSmoothEulerLagrangeRegularity_of_contDiffCoordDerivInCoordinates\n (L : Lagrangian d m k) (hcoord : Lagrangian.ContDiffCoordDerivInCoordinates L) :\n HasSmoothEulerLagrangeRegularity L", + "line": 103 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Support.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Support.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.variedField_zero", + "docstring": null, + "snippet": "@[simp]\nlemma variedField_zero (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) :\n variedField f η 0 = f", + "line": 52 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Support.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.variedField_variedField", + "docstring": null, + "snippet": "@[simp]\nlemma variedField_variedField (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)))\n (s t : ℝ) :\n variedField (variedField f η s) η t = variedField f η (s + t)", + "line": 59 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Support.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.contDiff_space_deriv", + "docstring": null, + "snippet": "lemma contDiff_space_deriv {g : Space d → ℝ} (hg : ContDiff ℝ ∞ g)\n (i : Fin d) : ContDiff ℝ ∞ (∂[i] g)", + "line": 71 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Support.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.isTestFunction_space_deriv", + "docstring": null, + "snippet": "lemma isTestFunction_space_deriv {g : Space d → ℝ} (hg : IsTestFunction g) (i : Fin d) :\n IsTestFunction (∂[i] g)", + "line": 77 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Support.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.iteratedDerivList_contDiff", + "docstring": null, + "snippet": "lemma iteratedDerivList_contDiff (L : List (Fin d)) {g : Space d → ℝ}\n (hg : ContDiff ℝ ∞ g) :\n ContDiff ℝ ∞ (L.foldr (fun i h => ∂[i] h) g)", + "line": 81 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Support.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.iteratedDerivList_isTestFunction", + "docstring": null, + "snippet": "lemma iteratedDerivList_isTestFunction (L : List (Fin d)) {g : Space d → ℝ}\n (hg : IsTestFunction g) :\n IsTestFunction (L.foldr (fun i h => ∂[i] h) g)", + "line": 91 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Support.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.iteratedDerivList_commute_deriv", + "docstring": null, + "snippet": "lemma iteratedDerivList_commute_deriv (L : List (Fin d)) (i : Fin d)\n {g : Space d → ℝ} (hg : ContDiff ℝ ∞ g) :\n L.foldr (fun j h => ∂[j] h) (∂[i] g) = ∂[i] (L.foldr (fun j h => ∂[j] h) g)", + "line": 101 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Support.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.iteratedDeriv_coord_isTestFunction", + "docstring": null, + "snippet": "lemma iteratedDeriv_coord_isTestFunction (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)))\n (I : DerivativeIndex d k) (a : Fin m) :\n IsTestFunction (fun x => ∂^[I.1] (fun y => (η y) a) x)", + "line": 116 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Support.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.firstVariationDensityTerm_integrable_of_continuous_coordDeriv", + "docstring": null, + "snippet": "lemma firstVariationDensityTerm_integrable_of_continuous_coordDeriv\n (L : Lagrangian d m k) (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)))\n (I : DerivativeIndex d k) (a : Fin m)\n (hcont : Continuous (fun x : Space d =>\n L.coordDeriv I a (jetAt k f x))) :\n Integrable (firstVariationDensityTerm L f η I a)", + "line": 122 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/FirstVariation/Support.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.continuous_jetBaseCoordinates_variedField", + "docstring": null, + "snippet": "lemma continuous_jetBaseCoordinates_variedField\n (k : ℕ) (f : Space d → EuclideanSpace ℝ (Fin m))\n (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m)))\n (hf : ContDiff ℝ ∞ f) :\n Continuous\n (fun p : ℝ × Space d => (p.2, jetCoordinatesAt k (variedField f η p.1) p.2))", + "line": 144 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "abbrev", + "name": "ClassicalFieldTheory.Local.JetCoordinates", + "docstring": "Coordinate data `u^a_I` for `k`-th order jet points.\n\nFor each derivative index `I`, these are the scalar components of the corresponding\n`EuclideanSpace ℝ (Fin m)` fiber coordinate. We keep this componentwise form because the\nlocal Euler-Lagrange formulas are indexed by both `I` and `a`.", + "snippet": "abbrev JetCoordinates (d m k : ℕ)", + "line": 66 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "structure", + "name": "ClassicalFieldTheory.Local.JetPoint", + "docstring": "The coordinate-level points of the locally trivialized `k`-jet bundle for fields\n`Space d → EuclideanSpace ℝ (Fin m)`.", + "snippet": "structure JetPoint (d m k : ℕ) where\n base : Space d\n fiber : JetCoordinates d m k", + "line": 75 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.JetPoint.coord", + "docstring": "The coordinate `u^a_I` of a jet point.", + "snippet": "def coord (J : JetPoint d m k) (I : DerivativeIndex d k) (a : Fin m) : ℝ", + "line": 86 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.JetPoint.value", + "docstring": "The zero-th order value encoded by a jet point.", + "snippet": "def value (J : JetPoint d m k) : EuclideanSpace ℝ (Fin m)", + "line": 89 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetPoint.value_apply", + "docstring": null, + "snippet": "@[simp]\nlemma value_apply (J : JetPoint d m k) (a : Fin m) :\n J.value a = J.coord 0 a", + "line": 93 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetPoint.ext", + "docstring": null, + "snippet": "@[ext]\nlemma ext (J K : JetPoint d m k) (hbase : J.base = K.base)\n (hcoord : ∀ I a, J.coord I a = K.coord I a) :\n J = K", + "line": 98 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.JetPoint.ofBaseCoordinates", + "docstring": "Build a jet point from a base point and its jet coordinates.", + "snippet": "def ofBaseCoordinates (x : Space d) (u : JetCoordinates d m k) : JetPoint d m k where", + "line": 112 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.JetPoint.toBaseCoordinates", + "docstring": "The base point and jet coordinates of a jet point.", + "snippet": "def toBaseCoordinates (J : JetPoint d m k) : Space d × JetCoordinates d m k", + "line": 117 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetPoint.ofBaseCoordinates_base", + "docstring": null, + "snippet": "@[simp]\nlemma ofBaseCoordinates_base (x : Space d) (u : JetCoordinates d m k) :\n (ofBaseCoordinates x u).base = x", + "line": 120 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetPoint.ofBaseCoordinates_fiber", + "docstring": null, + "snippet": "@[simp]\nlemma ofBaseCoordinates_fiber (x : Space d) (u : JetCoordinates d m k) :\n (ofBaseCoordinates x u).fiber = u", + "line": 124 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetPoint.ofBaseCoordinates_coord", + "docstring": null, + "snippet": "@[simp]\nlemma ofBaseCoordinates_coord (x : Space d) (u : JetCoordinates d m k)\n (I : DerivativeIndex d k) (a : Fin m) :\n (ofBaseCoordinates x u).coord I a = u I a", + "line": 128 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetPoint.ofBaseCoordinates_base_fiber", + "docstring": null, + "snippet": "@[simp]\nlemma ofBaseCoordinates_base_fiber (J : JetPoint d m k) :\n ofBaseCoordinates J.base J.fiber = J", + "line": 133 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetPoint.toBaseCoordinates_ofBaseCoordinates", + "docstring": null, + "snippet": "@[simp]\nlemma toBaseCoordinates_ofBaseCoordinates (x : Space d) (u : JetCoordinates d m k) :\n (ofBaseCoordinates x u).toBaseCoordinates = (x, u)", + "line": 141 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.jetCoordinatesAt", + "docstring": "The jet-coordinate function determined by a field `g`.", + "snippet": "noncomputable def jetCoordinatesAt (k : ℕ) (g : Space d → EuclideanSpace ℝ (Fin m))\n (x : Space d) :\n JetCoordinates d m k", + "line": 152 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.jetAt", + "docstring": "The coordinate-level `k`-jet point of a field `f` at the point `x`.", + "snippet": "noncomputable def jetAt (k : ℕ) (f : Space d → EuclideanSpace ℝ (Fin m)) (x : Space d) :\n JetPoint d m k where", + "line": 158 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetAt_base", + "docstring": null, + "snippet": "@[simp]\nlemma jetAt_base (k : ℕ) (f : Space d → EuclideanSpace ℝ (Fin m)) (x : Space d) :\n (jetAt k f x).base = x", + "line": 164 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetAt_value", + "docstring": null, + "snippet": "@[simp]\nlemma jetAt_value (k : ℕ) (f : Space d → EuclideanSpace ℝ (Fin m)) (x : Space d) :\n (jetAt k f x).value = f x", + "line": 168 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetAt_coord", + "docstring": null, + "snippet": "@[simp]\nlemma jetAt_coord (k : ℕ) (f : Space d → EuclideanSpace ℝ (Fin m)) (x : Space d)\n (I : DerivativeIndex d k) (a : Fin m) :\n (jetAt k f x).coord I a = ∂^[I.1] (fun y => (f y) a) x", + "line": 174 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetAt_coord_zero", + "docstring": null, + "snippet": "lemma jetAt_coord_zero (k : ℕ) (f : Space d → EuclideanSpace ℝ (Fin m)) (x : Space d)\n (a : Fin m) :\n (jetAt k f x).coord 0 a = (f x) a", + "line": 178 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetCoordinatesAt_eq", + "docstring": null, + "snippet": "@[simp]\nlemma jetCoordinatesAt_eq (k : ℕ) (g : Space d → EuclideanSpace ℝ (Fin m)) (x : Space d)\n (I : DerivativeIndex d k) (a : Fin m) :\n jetCoordinatesAt k g x I a = ∂^[I.1] (fun y => (g y) a) x", + "line": 184 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetCoordinatesAt_zero", + "docstring": null, + "snippet": "lemma jetCoordinatesAt_zero (k : ℕ) (g : Space d → EuclideanSpace ℝ (Fin m))\n (x : Space d) (a : Fin m) :\n jetCoordinatesAt k g x 0 a = (g x) a", + "line": 188 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPoint.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetAt_eq_ofBaseCoordinates", + "docstring": null, + "snippet": "lemma jetAt_eq_ofBaseCoordinates (k : ℕ) (f : Space d → EuclideanSpace ℝ (Fin m))\n (x : Space d) :\n jetAt k f x = JetPoint.ofBaseCoordinates x (jetCoordinatesAt k f x)", + "line": 193 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "structure", + "name": "ClassicalFieldTheory.Local.JetFiberData", + "docstring": "Fiber-coordinate data for jet points of order `k`. This records only the coordinates `u^a_I`,\nnot the base point in `Space d`.", + "snippet": "structure JetFiberData (d m k : ℕ) where\n coord : JetCoordinates d m k", + "line": 54 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "instance", + "name": "ClassicalFieldTheory.Local.JetFiberData.«instance@24e9455d»", + "docstring": null, + "snippet": "instance : CoeFun (JetFiberData d m k) (fun _ => DerivativeIndex d k → Fin m → ℝ) where", + "line": 62 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetFiberData.ext", + "docstring": null, + "snippet": "@[ext]\nlemma ext (V W : JetFiberData d m k) (hcoord : ∀ I a, V.coord I a = W.coord I a) : V = W", + "line": 66 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.JetFiberData.value", + "docstring": "The zero-th order component of a jet-fiber direction, corresponding to the field value.", + "snippet": "def value (V : JetFiberData d m k) : EuclideanSpace ℝ (Fin m)", + "line": 79 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetFiberData.value_apply", + "docstring": null, + "snippet": "@[simp]\nlemma value_apply (V : JetFiberData d m k) (a : Fin m) :\n V.value a = V.coord 0 a", + "line": 83 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "instance", + "name": "ClassicalFieldTheory.Local.JetFiberData.«instance@14f5f97d»", + "docstring": null, + "snippet": "instance : Zero (JetFiberData d m k) where", + "line": 87 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "instance", + "name": "ClassicalFieldTheory.Local.JetFiberData.«instance@47407776»", + "docstring": null, + "snippet": "instance : Add (JetFiberData d m k) where", + "line": 90 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "instance", + "name": "ClassicalFieldTheory.Local.JetFiberData.«instance@cddd31c6»", + "docstring": null, + "snippet": "instance : SMul ℝ (JetFiberData d m k) where", + "line": 93 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetFiberData.zero_coord", + "docstring": null, + "snippet": "@[simp]\nlemma zero_coord (I : DerivativeIndex d k) (a : Fin m) :\n (0 : JetFiberData d m k).coord I a = 0", + "line": 97 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetFiberData.add_coord", + "docstring": null, + "snippet": "@[simp]\nlemma add_coord (V W : JetFiberData d m k) (I : DerivativeIndex d k) (a : Fin m) :\n (V + W).coord I a = V.coord I a + W.coord I a", + "line": 101 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetFiberData.smul_coord", + "docstring": null, + "snippet": "@[simp]\nlemma smul_coord (c : ℝ) (V : JetFiberData d m k) (I : DerivativeIndex d k) (a : Fin m) :\n (c • V).coord I a = c * V.coord I a", + "line": 105 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.JetPoint.addFiber", + "docstring": "Translate a jet point by a fiber-direction increment.", + "snippet": "def addFiber (J : JetPoint d m k) (V : JetFiberData d m k) : JetPoint d m k where", + "line": 120 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.JetPoint.lineMap", + "docstring": "The affine line in jet space through `J` in the fiber direction `V`.", + "snippet": "def lineMap (J : JetPoint d m k) (V : JetFiberData d m k) (s : ℝ) : JetPoint d m k", + "line": 125 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetPoint.addFiber_base", + "docstring": null, + "snippet": "@[simp]\nlemma addFiber_base (J : JetPoint d m k) (V : JetFiberData d m k) :\n (J.addFiber V).base = J.base", + "line": 129 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetPoint.addFiber_value", + "docstring": null, + "snippet": "@[simp]\nlemma addFiber_value (J : JetPoint d m k) (V : JetFiberData d m k) :\n (J.addFiber V).value = J.value + V.value", + "line": 133 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetPoint.addFiber_coord", + "docstring": null, + "snippet": "@[simp]\nlemma addFiber_coord (J : JetPoint d m k) (V : JetFiberData d m k)\n (I : DerivativeIndex d k) (a : Fin m) :\n (J.addFiber V).coord I a = J.coord I a + V.coord I a", + "line": 139 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetPoint.lineMap_base", + "docstring": null, + "snippet": "@[simp]\nlemma lineMap_base (J : JetPoint d m k) (V : JetFiberData d m k) (s : ℝ) :\n (J.lineMap V s).base = J.base", + "line": 144 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.JetPoint.lineMap_coord", + "docstring": null, + "snippet": "@[simp]\nlemma lineMap_coord (J : JetPoint d m k) (V : JetFiberData d m k) (s : ℝ)\n (I : DerivativeIndex d k) (a : Fin m) :\n (J.lineMap V s).coord I a = J.coord I a + s * V.coord I a", + "line": 148 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.jetDirectionAt", + "docstring": "The jet-fiber direction determined by a field `g` at a point `x`.", + "snippet": "noncomputable def jetDirectionAt (k : ℕ) (g : Space d → EuclideanSpace ℝ (Fin m))\n (x : Space d) :\n JetFiberData d m k where", + "line": 160 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetDirectionAt_coord", + "docstring": null, + "snippet": "@[simp]\nlemma jetDirectionAt_coord (k : ℕ) (g : Space d → EuclideanSpace ℝ (Fin m)) (x : Space d)\n (I : DerivativeIndex d k) (a : Fin m) :\n (jetDirectionAt k g x).coord I a = ∂^[I.1] (fun y => (g y) a) x", + "line": 166 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetDirectionAt_coord_zero", + "docstring": null, + "snippet": "lemma jetDirectionAt_coord_zero (k : ℕ) (g : Space d → EuclideanSpace ℝ (Fin m))\n (x : Space d) (a : Fin m) :\n (jetDirectionAt k g x).coord 0 a = (g x) a", + "line": 170 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetCoordinatesAt_add_smul", + "docstring": null, + "snippet": "lemma jetCoordinatesAt_add_smul (k : ℕ) (f g : Space d → EuclideanSpace ℝ (Fin m))\n (x : Space d) (s : ℝ)\n (hf : ContDiff ℝ ∞ f) (hg : ContDiff ℝ ∞ g) :\n jetCoordinatesAt k (fun y => f y + s • g y) x =\n jetCoordinatesAt k f x + s • jetCoordinatesAt k g x", + "line": 175 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointFiber.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetAt_add_smul", + "docstring": null, + "snippet": "lemma jetAt_add_smul (k : ℕ) (f g : Space d → EuclideanSpace ℝ (Fin m))\n (x : Space d) (s : ℝ)\n (hf : ContDiff ℝ ∞ f) (hg : ContDiff ℝ ∞ g) :\n jetAt k (fun y => f y + s • g y) x = (jetAt k f x).lineMap (jetDirectionAt k g x) s", + "line": 204 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointRegularity.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointRegularity.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetCoordinatesAt_contDiff", + "docstring": "Smoothness of the jet-coordinate map of a smooth field.", + "snippet": "lemma jetCoordinatesAt_contDiff (k : ℕ) (g : Space d → EuclideanSpace ℝ (Fin m))\n (hg : ContDiff ℝ ∞ g) :\n ContDiff ℝ ∞ (jetCoordinatesAt k g)", + "line": 53 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointRegularity.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetBaseCoordinates_contDiff", + "docstring": "Smoothness of the base-plus-coordinate jet map of a smooth field.", + "snippet": "lemma jetBaseCoordinates_contDiff (k : ℕ) (g : Space d → EuclideanSpace ℝ (Fin m))\n (hg : ContDiff ℝ ∞ g) :\n ContDiff ℝ ∞ (fun x : Space d => (x, jetCoordinatesAt k g x))", + "line": 66 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointRegularity.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.notMem_tsupport_coord", + "docstring": null, + "snippet": "private lemma notMem_tsupport_coord {g : Space d → EuclideanSpace ℝ (Fin m)} {x : Space d}\n (hx : x ∉ tsupport g) (a : Fin m) :\n x ∉ tsupport (fun y => (g y) a)", + "line": 76 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointRegularity.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetCoordinatesAt_eq_zero_of_notMem_tsupport", + "docstring": "Outside the topological support of a field, all of its jet coordinates vanish.", + "snippet": "lemma jetCoordinatesAt_eq_zero_of_notMem_tsupport (k : ℕ)\n (g : Space d → EuclideanSpace ℝ (Fin m)) {x : Space d}\n (hx : x ∉ tsupport g) :\n jetCoordinatesAt k g x = 0", + "line": 84 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/JetPointRegularity.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.jetDirectionAt_eq_zero_of_notMem_tsupport", + "docstring": "Outside the topological support of a field, the corresponding jet-fiber direction vanishes.", + "snippet": "lemma jetDirectionAt_eq_zero_of_notMem_tsupport (k : ℕ)\n (g : Space d → EuclideanSpace ℝ (Fin m)) {x : Space d}\n (hx : x ∉ tsupport g) :\n jetDirectionAt k g x = 0", + "line": 93 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/Lagrangian.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Lagrangian.lean", + "kind": "structure", + "name": "ClassicalFieldTheory.Local.Lagrangian", + "docstring": "A local `k`-th order Lagrangian for fields `Space d → EuclideanSpace ℝ (Fin m)`.", + "snippet": "structure Lagrangian (d m k : ℕ) where\n toFun : JetPoint d m k → ℝ\n coordDeriv : DerivativeIndex d k → Fin m → JetPoint d m k → ℝ\n hasDerivAlongFiber :\n ∀ J : JetPoint d m k, ∀ V : JetFiberData d m k,\n HasDerivAt (fun s : ℝ => toFun (J.lineMap V s))\n (∑ I : DerivativeIndex d k, ∑ a : Fin m, coordDeriv I a J * V.coord I a) 0", + "line": 60 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Lagrangian.lean", + "kind": "instance", + "name": "ClassicalFieldTheory.Local.Lagrangian.«instance@43bf9c65»", + "docstring": null, + "snippet": "instance : CoeFun (Lagrangian d m k) (fun _ => JetPoint d m k → ℝ) where", + "line": 76 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Lagrangian.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.Lagrangian.fiberDerivative", + "docstring": "The derivative of `L` along the fiber-jet direction `V` based at `J`.", + "snippet": "noncomputable def fiberDerivative (L : Lagrangian d m k) (J : JetPoint d m k)\n (V : JetFiberData d m k) : ℝ", + "line": 80 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Lagrangian.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.Lagrangian.hasDerivAt_lineMap", + "docstring": null, + "snippet": "lemma hasDerivAt_lineMap (L : Lagrangian d m k) (J : JetPoint d m k) (V : JetFiberData d m k) :\n HasDerivAt (fun s : ℝ => L (J.lineMap V s)) (L.fiberDerivative J V) 0", + "line": 84 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Lagrangian.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.Lagrangian.ContinuousInCoordinates", + "docstring": "Continuity of a local lagrangian in the explicit local jet coordinates `(x, u^a_I)`.", + "snippet": "def ContinuousInCoordinates (L : Lagrangian d m k) : Prop", + "line": 94 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Lagrangian.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.Lagrangian.ContDiffCoordDerivInCoordinates", + "docstring": "Intrinsic smoothness of the jet-coordinate derivatives of a local Lagrangian in the explicit\nlocal jet coordinates `(x, u^a_I)`.", + "snippet": "def ContDiffCoordDerivInCoordinates (L : Lagrangian d m k) : Prop", + "line": 100 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Lagrangian.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.Lagrangian.SmoothInCoordinates", + "docstring": "Public regularity package for a smooth local lagrangian in explicit local jet coordinates:\ncontinuity of the lagrangian itself together with smoothness of all jet-coordinate derivatives.", + "snippet": "def SmoothInCoordinates (L : Lagrangian d m k) : Prop", + "line": 108 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Lagrangian.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.Lagrangian.ContDiffCoordDerivAlongField", + "docstring": "Smoothness of all coefficient functions `x ↦ ∂L/∂u_I^a (j^k f(x))` along a field `f`.", + "snippet": "def ContDiffCoordDerivAlongField (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Prop", + "line": 112 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Lagrangian.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.Lagrangian.ContinuousCoordDerivAlongFamily", + "docstring": "Continuity of the coefficient family `(s, x) ↦ ∂L/∂u_I^a (j^k(F s)(x))` along a\none-parameter family of fields `F : ℝ → Space d → EuclideanSpace ℝ (Fin m)`.", + "snippet": "def ContinuousCoordDerivAlongFamily (L : Lagrangian d m k)\n (F : ℝ → Space d → EuclideanSpace ℝ (Fin m)) : Prop", + "line": 119 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Lagrangian.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.Lagrangian.continuousAlongField_of_inCoordinates", + "docstring": null, + "snippet": "lemma continuousAlongField_of_inCoordinates (L : Lagrangian d m k)\n (hcont : ContinuousInCoordinates L) (f : Space d → EuclideanSpace ℝ (Fin m))\n (hf : ContDiff ℝ ∞ f) :\n Continuous (fun x : Space d => L (jetAt k f x))", + "line": 124 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Lagrangian.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.Lagrangian.contDiffCoordDerivAlongField_of_inCoordinates", + "docstring": null, + "snippet": "lemma contDiffCoordDerivAlongField_of_inCoordinates (L : Lagrangian d m k)\n (hcoord : ContDiffCoordDerivInCoordinates L) (f : Space d → EuclideanSpace ℝ (Fin m))\n (hf : ContDiff ℝ ∞ f) :\n ContDiffCoordDerivAlongField L f", + "line": 131 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Lagrangian.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.Lagrangian.continuousCoordDerivAlongFamily_of_inCoordinates", + "docstring": null, + "snippet": "lemma continuousCoordDerivAlongFamily_of_inCoordinates (L : Lagrangian d m k)\n (hcoord : ContDiffCoordDerivInCoordinates L)\n (F : ℝ → Space d → EuclideanSpace ℝ (Fin m))\n (hjet : Continuous (fun p : ℝ × Space d => (p.2, jetCoordinatesAt k (F p.1) p.2))) :\n ContinuousCoordDerivAlongFamily L F", + "line": 142 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Lagrangian.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.Lagrangian.alongField", + "docstring": "Evaluate a local Lagrangian along the `k`-jets of a field.", + "snippet": "noncomputable def alongField (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Space d → ℝ", + "line": 161 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Lagrangian.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.Lagrangian.alongField_apply", + "docstring": null, + "snippet": "@[simp]\nlemma alongField_apply (L : Lagrangian d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) (x : Space d) :\n L.alongField f x = L (jetAt k f x)", + "line": 166 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDerivative.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDerivative.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.evalOnJet", + "docstring": "Evaluate a local jet-dependent function along the `k`-jet of a field.", + "snippet": "noncomputable def evalOnJet (k : ℕ) (F : JetPoint d m k → ℝ)\n (f : Space d → EuclideanSpace ℝ (Fin m)) :\n Space d → ℝ", + "line": 49 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDerivative.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.totalDerivative", + "docstring": "The total derivative of a jet-dependent function in the coordinate direction `i`.", + "snippet": "noncomputable def totalDerivative (k : ℕ) (i : Fin d) (F : JetPoint d m k → ℝ)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Space d → ℝ", + "line": 60 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDerivative.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.iteratedTotalDerivative", + "docstring": "The iterated total derivative indexed by a multi-index.", + "snippet": "noncomputable def iteratedTotalDerivative (k : ℕ) (I : MultiIndex d) (F : JetPoint d m k → ℝ)\n (f : Space d → EuclideanSpace ℝ (Fin m)) : Space d → ℝ", + "line": 65 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDerivative.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.evalOnJet_apply", + "docstring": null, + "snippet": "@[simp]\nlemma evalOnJet_apply (k : ℕ) (F : JetPoint d m k → ℝ)\n (f : Space d → EuclideanSpace ℝ (Fin m)) (x : Space d) :\n evalOnJet k F f x = F (jetAt k f x)", + "line": 70 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDerivative.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.totalDerivative_eq", + "docstring": null, + "snippet": "@[simp]\nlemma totalDerivative_eq (k : ℕ) (i : Fin d) (F : JetPoint d m k → ℝ)\n (f : Space d → EuclideanSpace ℝ (Fin m)) :\n totalDerivative k i F f = ∂[i] (fun x => F (jetAt k f x))", + "line": 75 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDerivative.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.iteratedTotalDerivative_eq", + "docstring": null, + "snippet": "@[simp]\nlemma iteratedTotalDerivative_eq (k : ℕ) (I : MultiIndex d) (F : JetPoint d m k → ℝ)\n (f : Space d → EuclideanSpace ℝ (Fin m)) :\n iteratedTotalDerivative k I F f = ∂^[I] (fun x => F (jetAt k f x))", + "line": 80 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDerivative.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.iteratedTotalDerivative_zero", + "docstring": null, + "snippet": "lemma iteratedTotalDerivative_zero (k : ℕ) (F : JetPoint d m k → ℝ)\n (f : Space d → EuclideanSpace ℝ (Fin m)) :\n iteratedTotalDerivative k (0 : MultiIndex d) F f = evalOnJet k F f", + "line": 84 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDerivative.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.iteratedTotalDerivative_single", + "docstring": null, + "snippet": "lemma iteratedTotalDerivative_single (k : ℕ) (i : Fin d) (F : JetPoint d m k → ℝ)\n (f : Space d → EuclideanSpace ℝ (Fin m)) :\n iteratedTotalDerivative k (Physlib.MultiIndex.increment 0 i) F f =\n totalDerivative k i F f", + "line": 89 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergence.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergence.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.HasTotalDivergenceDensity", + "docstring": "A lagrangian density has total-divergence form if, along every field, it is the sum of the\ntotal derivatives of a jet-dependent current. A current of order `k` gives a density of order\n`k + 1`.", + "snippet": "def HasTotalDivergenceDensity (L : Lagrangian d m (k + 1))\n (current : Fin d → JetPoint d m k → ℝ) : Prop", + "line": 66 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergence.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.IsEulerLagrangeTrivial", + "docstring": "A local lagrangian is Euler-Lagrange-trivial if its Euler-Lagrange operator vanishes along\nevery field.", + "snippet": "def IsEulerLagrangeTrivial (L : Lagrangian d m k) : Prop", + "line": 73 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergence.lean", + "kind": "structure", + "name": "ClassicalFieldTheory.Local.TotalDivergence", + "docstring": "A packaged local total divergence.\n\nThe current has order `k`, while the associated lagrangian has order `k + 1`, matching the\ncoordinate expression `L = ∑ i, Dᵢ Bᵢ`. The Euler-Lagrange-triviality field records the\nvariationally trivial direction needed by later equivalence results.", + "snippet": "structure TotalDivergence (d m k : ℕ) where\n lagrangian : Lagrangian d m (k + 1)\n current : Fin d → JetPoint d m k → ℝ\n hasTotalDivergenceDensity : HasTotalDivergenceDensity lagrangian current\n isEulerLagrangeTrivial : IsEulerLagrangeTrivial lagrangian", + "line": 86 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergence.lean", + "kind": "instance", + "name": "ClassicalFieldTheory.Local.TotalDivergence.«instance@5c9b98ce»", + "docstring": null, + "snippet": "instance : CoeFun (TotalDivergence d m k) (fun _ => Lagrangian d m (k + 1)) where", + "line": 100 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.TotalDivergence.actionDensity_eq", + "docstring": null, + "snippet": "@[simp]\nlemma actionDensity_eq (T : TotalDivergence d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) :\n actionDensity T.lagrangian f =\n fun x => ∑ i : Fin d, totalDerivative k i (T.current i) f x", + "line": 104 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.TotalDivergence.eulerLagrangeOp_eq_zero", + "docstring": null, + "snippet": "lemma eulerLagrangeOp_eq_zero (T : TotalDivergence d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) :\n eulerLagrangeOp T.lagrangian f = 0", + "line": 110 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.TotalDivergence.isCritical", + "docstring": null, + "snippet": "lemma isCritical (T : TotalDivergence d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (hTf : IsAdmissibleForAction T.lagrangian f)\n (hT : Lagrangian.SmoothInCoordinates T.lagrangian) :\n IsCritical T.lagrangian f", + "line": 115 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.HasTotalDivergenceDifference", + "docstring": "A lagrangian `target` differs from `source` by the density of a packaged total divergence.\n\nThe total divergence has current order `k`, hence its associated lagrangian and both compared\nlagrangians have order `k + 1`.", + "snippet": "def HasTotalDivergenceDifference (source target : Lagrangian d m (k + 1))\n (T : TotalDivergence d m k) : Prop", + "line": 64 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "def", + "name": "ClassicalFieldTheory.Local.IsEulerLagrangeEquivalent", + "docstring": "Two local lagrangians are Euler-Lagrange equivalent when they determine the same local\nEuler-Lagrange operator along every field.", + "snippet": "def IsEulerLagrangeEquivalent (source target : Lagrangian d m k) : Prop", + "line": 72 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.IsEulerLagrangeEquivalent.refl", + "docstring": null, + "snippet": "lemma refl (L : Lagrangian d m k) : IsEulerLagrangeEquivalent L L", + "line": 85 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.IsEulerLagrangeEquivalent.symm", + "docstring": null, + "snippet": "lemma symm {L1 L2 : Lagrangian d m k} (h : IsEulerLagrangeEquivalent L1 L2) :\n IsEulerLagrangeEquivalent L2 L1", + "line": 89 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.IsEulerLagrangeEquivalent.trans", + "docstring": null, + "snippet": "lemma trans {L1 L2 L3 : Lagrangian d m k}\n (h12 : IsEulerLagrangeEquivalent L1 L2) (h23 : IsEulerLagrangeEquivalent L2 L3) :\n IsEulerLagrangeEquivalent L1 L3", + "line": 94 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.IsEulerLagrangeEquivalent.eulerLagrangeOp_eq", + "docstring": null, + "snippet": "lemma eulerLagrangeOp_eq {source target : Lagrangian d m k}\n (h : IsEulerLagrangeEquivalent source target)\n (f : Space d → EuclideanSpace ℝ (Fin m)) :\n eulerLagrangeOp target f = eulerLagrangeOp source f", + "line": 100 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.IsEulerLagrangeEquivalent.eulerLagrangeOp_eq_zero_iff", + "docstring": null, + "snippet": "lemma eulerLagrangeOp_eq_zero_iff {source target : Lagrangian d m k}\n (h : IsEulerLagrangeEquivalent source target)\n (f : Space d → EuclideanSpace ℝ (Fin m)) :\n eulerLagrangeOp target f = 0 ↔ eulerLagrangeOp source f = 0", + "line": 106 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.IsEulerLagrangeEquivalent.isCritical_iff", + "docstring": null, + "snippet": "lemma isCritical_iff {source target : Lagrangian d m k}\n (h : IsEulerLagrangeEquivalent source target)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (hsourcef : IsAdmissibleForAction source f)\n (htargetf : IsAdmissibleForAction target f)\n (hsource : Lagrangian.SmoothInCoordinates source)\n (htarget : Lagrangian.SmoothInCoordinates target) :\n IsCritical target f ↔ IsCritical source f", + "line": 112 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.IsEulerLagrangeEquivalent.isCritical_target_of_source", + "docstring": null, + "snippet": "lemma isCritical_target_of_source {source target : Lagrangian d m k}\n (h : IsEulerLagrangeEquivalent source target)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (hsourcef : IsAdmissibleForAction source f)\n (htargetf : IsAdmissibleForAction target f)\n (hsource : Lagrangian.SmoothInCoordinates source)\n (htarget : Lagrangian.SmoothInCoordinates target)\n (hcrit : IsCritical source f) :\n IsCritical target f", + "line": 126 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.IsEulerLagrangeEquivalent.isCritical_source_of_target", + "docstring": null, + "snippet": "lemma isCritical_source_of_target {source target : Lagrangian d m k}\n (h : IsEulerLagrangeEquivalent source target)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (hsourcef : IsAdmissibleForAction source f)\n (htargetf : IsAdmissibleForAction target f)\n (hsource : Lagrangian.SmoothInCoordinates source)\n (htarget : Lagrangian.SmoothInCoordinates target)\n (hcrit : IsCritical target f) :\n IsCritical source f", + "line": 137 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "structure", + "name": "ClassicalFieldTheory.Local.TotalDivergenceEquivalence", + "docstring": "A packaged equivalence between two local lagrangians whose densities differ by a total\ndivergence.\n\nThe field `sameEulerLagrangeOp` records the variational consequence in the current Alpha API. It\nwill become a theorem once the stack has enough symbolic calculus for coordinate derivatives of\ntotal divergences.", + "snippet": "structure TotalDivergenceEquivalence (d m k : ℕ) where\n source : Lagrangian d m (k + 1)\n target : Lagrangian d m (k + 1)\n divergence : TotalDivergence d m k\n hasTotalDivergenceDifference :\n HasTotalDivergenceDifference source target divergence\n sameEulerLagrangeOp : IsEulerLagrangeEquivalent source target", + "line": 161 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.TotalDivergenceEquivalence.actionDensity_eq", + "docstring": null, + "snippet": "@[simp]\nlemma actionDensity_eq (E : TotalDivergenceEquivalence d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) :\n actionDensity E.target f =\n fun x => actionDensity E.source f x + actionDensity E.divergence.lagrangian f x", + "line": 179 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.TotalDivergenceEquivalence.eulerLagrangeOp_eq", + "docstring": null, + "snippet": "lemma eulerLagrangeOp_eq (E : TotalDivergenceEquivalence d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) :\n eulerLagrangeOp E.target f = eulerLagrangeOp E.source f", + "line": 185 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.TotalDivergenceEquivalence.eulerLagrangeOp_eq_zero_iff", + "docstring": null, + "snippet": "lemma eulerLagrangeOp_eq_zero_iff (E : TotalDivergenceEquivalence d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m)) :\n eulerLagrangeOp E.target f = 0 ↔ eulerLagrangeOp E.source f = 0", + "line": 190 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.TotalDivergenceEquivalence.isCritical_iff", + "docstring": null, + "snippet": "lemma isCritical_iff (E : TotalDivergenceEquivalence d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (hsourcef : IsAdmissibleForAction E.source f)\n (htargetf : IsAdmissibleForAction E.target f)\n (hsource : Lagrangian.SmoothInCoordinates E.source)\n (htarget : Lagrangian.SmoothInCoordinates E.target) :\n IsCritical E.target f ↔ IsCritical E.source f", + "line": 195 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.TotalDivergenceEquivalence.isCritical_target_of_source", + "docstring": null, + "snippet": "lemma isCritical_target_of_source (E : TotalDivergenceEquivalence d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (hsourcef : IsAdmissibleForAction E.source f)\n (htargetf : IsAdmissibleForAction E.target f)\n (hsource : Lagrangian.SmoothInCoordinates E.source)\n (htarget : Lagrangian.SmoothInCoordinates E.target)\n (hcrit : IsCritical E.source f) :\n IsCritical E.target f", + "line": 204 + }, + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/TotalDivergenceEquivalence.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.TotalDivergenceEquivalence.isCritical_source_of_target", + "docstring": null, + "snippet": "lemma isCritical_source_of_target (E : TotalDivergenceEquivalence d m k)\n (f : Space d → EuclideanSpace ℝ (Fin m))\n (hsourcef : IsAdmissibleForAction E.source f)\n (htargetf : IsAdmissibleForAction E.target f)\n (hsource : Lagrangian.SmoothInCoordinates E.source)\n (htarget : Lagrangian.SmoothInCoordinates E.target)\n (hcrit : IsCritical E.target f) :\n IsCritical E.source f", + "line": 214 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalFieldTheory/Local/Variation.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalFieldTheory/Local/Variation.lean", + "kind": "lemma", + "name": "ClassicalFieldTheory.Local.AdmissibleVariation.coord_euclidean", + "docstring": "A Euclidean component of an admissible variation is again a test function.", + "snippet": "@[fun_prop]\nlemma coord_euclidean (η : AdmissibleVariation d (EuclideanSpace ℝ (Fin m))) (a : Fin m) :\n IsTestFunction (fun x => (η.toFun x) a)", + "line": 39 + } + ] + }, + { + "filename": "PhyslibAlpha/ClassicalMechanics/CoupledSpringPotential.lean", + "entries": [ + { + "file": "PhyslibAlpha/ClassicalMechanics/CoupledSpringPotential.lean", + "kind": "def", + "name": "couplingPotential", + "docstring": "The potential energy of a pair of coupled springs.", + "snippet": "noncomputable def couplingPotential (x : EuclideanSpace ℝ (Fin 2)) : ℝ", + "line": 30 + }, + { + "file": "PhyslibAlpha/ClassicalMechanics/CoupledSpringPotential.lean", + "kind": "lemma", + "name": "couplingPotential_analyticAt", + "docstring": null, + "snippet": "lemma couplingPotential_analyticAt (z : EuclideanSpace ℝ (Fin 2)) :\n AnalyticAt ℝ couplingPotential z", + "line": 36 + }, + { + "file": "PhyslibAlpha/ClassicalMechanics/CoupledSpringPotential.lean", + "kind": "lemma", + "name": "couplingPotential_hasFDerivAt", + "docstring": null, + "snippet": "lemma couplingPotential_hasFDerivAt (x : EuclideanSpace ℝ (Fin 2)) :\n HasFDerivAt couplingPotential\n ((2 * x 0 + x 1) • (EuclideanSpace.proj (𝕜 := ℝ) 0)\n + (x 0 + 2 * x 1) • (EuclideanSpace.proj (𝕜 := ℝ) 1)) x", + "line": 49 + }, + { + "file": "PhyslibAlpha/ClassicalMechanics/CoupledSpringPotential.lean", + "kind": "lemma", + "name": "couplingPotential_gradient_zero", + "docstring": null, + "snippet": "lemma couplingPotential_gradient_zero :\n gradient couplingPotential (!₂[0, 0] : EuclideanSpace ℝ (Fin 2)) = 0", + "line": 65 + }, + { + "file": "PhyslibAlpha/ClassicalMechanics/CoupledSpringPotential.lean", + "kind": "lemma", + "name": "couplingPotential_iteratedFDeriv_two", + "docstring": null, + "snippet": "lemma couplingPotential_iteratedFDeriv_two (z : EuclideanSpace ℝ (Fin 2))\n (a b : EuclideanSpace ℝ (Fin 2)) :\n iteratedFDeriv ℝ 2 couplingPotential z ![a, b]\n = 2 * a 0 * b 0 + a 0 * b 1 + a 1 * b 0 + 2 * a 1 * b 1", + "line": 74 + }, + { + "file": "PhyslibAlpha/ClassicalMechanics/CoupledSpringPotential.lean", + "kind": "lemma", + "name": "couplingPotential_posDef", + "docstring": null, + "snippet": "lemma couplingPotential_posDef :\n (iteratedFDerivQuadraticMap couplingPotential\n (!₂[0, 0] : EuclideanSpace ℝ (Fin 2))).PosDef", + "line": 104 + }, + { + "file": "PhyslibAlpha/ClassicalMechanics/CoupledSpringPotential.lean", + "kind": "lemma", + "name": "coupled_spring_potential", + "docstring": null, + "snippet": "lemma coupled_spring_potential :\n let U", + "line": 115 + } + ] + }, + { + "filename": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "entries": [ + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "lemma", + "name": "Function.update₀", + "docstring": "Update a vector of length 2 in coordinate 0.", + "snippet": "@[simp]\nlemma Function.update₀ {α : Type*} {a b c : α} : Function.update ![a,b] 0 c = ![c,b]", + "line": 39 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "lemma", + "name": "Function.update₁", + "docstring": "Update a vector of length 2 in coordinate 1.", + "snippet": "@[simp]\nlemma Function.update₁ {α : Type*} {a b c : α} : Function.update ![a,b] 1 c = ![a,c]", + "line": 44 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "def", + "name": "hessianBilinearCompanion", + "docstring": "The Hessian companion as a bilinear map.", + "snippet": "noncomputable def hessianBilinearCompanion {V : Type*} [NormedAddCommGroup V]\n [NormedSpace ℝ V] (f : V → ℝ) (x₀ : V) : V →ₗ[ℝ] V →ₗ[ℝ] ℝ", + "line": 51 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "def", + "name": "iteratedFDerivQuadraticMap", + "docstring": "The second iterated Frechét derivative as a quadratic map.", + "snippet": "noncomputable def iteratedFDerivQuadraticMap {V : Type*} [NormedAddCommGroup V]\n [NormedSpace ℝ V] (f : V → ℝ) (x₀ : V) : QuadraticMap ℝ V ℝ", + "line": 78 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "def", + "name": "continuousBilinearMapOfContinuousMultilinearMap", + "docstring": "A continuous multilinear map is bilinear.", + "snippet": "noncomputable def continuousBilinearMapOfContinuousMultilinearMap\n {𝕜 : Type*} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]\n {V : Type*}\n [NormedAddCommGroup V] [NormedSpace 𝕜 V] [FiniteDimensional 𝕜 V]\n (g : ContinuousMultilinearMap 𝕜 (fun _ : Fin 2 => V) 𝕜) : V →L[𝕜] V →L[𝕜] 𝕜", + "line": 101 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "def", + "name": "QuadraticMap.toMultilinearMap", + "docstring": ".", + "snippet": "def QuadraticMap.toMultilinearMap {V : Type*} [AddCommGroup V] [Module ℝ V]\n (Q : QuadraticMap ℝ V ℝ) : MultilinearMap ℝ (fun _ : Fin 2 => V) ℝ", + "line": 118 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "def", + "name": "QuadraticMap.toMultilinearMapHalfPolarBilin", + "docstring": ".", + "snippet": "noncomputable def QuadraticMap.toMultilinearMapHalfPolarBilin\n {V : Type*} [AddCommGroup V] [Module ℝ V]\n (Q : QuadraticMap ℝ V ℝ) :\n MultilinearMap ℝ (fun _ : Fin 2 => V) ℝ", + "line": 126 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "theorem", + "name": "QuadraticMap.toMultilinearMap_continuous", + "docstring": "The multilinear map associated to a quadratic map is continuous.", + "snippet": "theorem QuadraticMap.toMultilinearMap_continuous {V : Type*}\n [NormedAddCommGroup V] [NormedSpace ℝ V]\n [FiniteDimensional ℝ V] (Q : QuadraticMap ℝ V ℝ) : Continuous Q.toMultilinearMap", + "line": 156 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "theorem", + "name": "QuadraticMap.toMultilinearMapHalfPolarBilin_continuous", + "docstring": ".", + "snippet": "theorem QuadraticMap.toMultilinearMapHalfPolarBilin_continuous {V : Type*}\n [NormedAddCommGroup V] [NormedSpace ℝ V]\n [FiniteDimensional ℝ V] (Q : QuadraticMap ℝ V ℝ) :\n Continuous Q.toMultilinearMapHalfPolarBilin", + "line": 185 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "def", + "name": "QuadraticMap.toContinuousMultilinearMap", + "docstring": "Construct a continuous multilinear map from a quadratic map on a finite dimensional space.", + "snippet": "def QuadraticMap.toContinuousMultilinearMap {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]\n [FiniteDimensional ℝ V] (Q : QuadraticMap ℝ V ℝ) :\n ContinuousMultilinearMap ℝ (fun _ : Fin 2 ↦ V) ℝ", + "line": 262 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "def", + "name": "QuadraticMap.toContinuousMultilinearMapHalfPolarBilin", + "docstring": ".", + "snippet": "noncomputable def QuadraticMap.toContinuousMultilinearMapHalfPolarBilin {V : Type*}\n [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V] (Q : QuadraticMap ℝ V ℝ) :\n ContinuousMultilinearMap ℝ (fun _ : Fin 2 ↦ V) ℝ", + "line": 269 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "theorem", + "name": "QuadraticMap.toContinuousMultilinearMap_apply", + "docstring": "The constructed continuous multilinear map agrees with the polar bilinear form.", + "snippet": "theorem QuadraticMap.toContinuousMultilinearMap_apply {V : Type*} [NormedAddCommGroup V]\n [NormedSpace ℝ V] [FiniteDimensional ℝ V] (Q : QuadraticMap ℝ V ℝ) (x y : V) :\n Q.toContinuousMultilinearMap ![x, y] = Q.polarBilin x y", + "line": 278 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "theorem", + "name": "QuadraticMap.toContinuousMultilinearMap_applyHalf", + "docstring": ".", + "snippet": "theorem QuadraticMap.toContinuousMultilinearMap_applyHalf {V : Type*} [NormedAddCommGroup V]\n [NormedSpace ℝ V] [FiniteDimensional ℝ V] (Q : QuadraticMap ℝ V ℝ) (x y : V) :\n Q.toContinuousMultilinearMapHalfPolarBilin ![x, y] = (1/2) * Q.polarBilin x y", + "line": 284 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "lemma", + "name": "coercive_of_posdefHalf", + "docstring": ".", + "snippet": "lemma coercive_of_posdefHalf {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]\n [FiniteDimensional ℝ V] {F : QuadraticMap ℝ V ℝ}\n (hf' : F.PosDef) :\n IsCoercive (continuousBilinearMapOfContinuousMultilinearMap\n F.toContinuousMultilinearMapHalfPolarBilin)", + "line": 291 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "lemma", + "name": "polarBilin_iteratedFDerivQuadraticMap", + "docstring": "The polar bilinear form of the second-derivative quadratic map is the symmetrized\nHessian.", + "snippet": "lemma polarBilin_iteratedFDerivQuadraticMap {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]\n (f : V → ℝ) (x₀ : V) (x y : V) :\n (iteratedFDerivQuadraticMap f x₀).polarBilin x y\n = iteratedFDeriv ℝ 2 f x₀ ![x, y] + iteratedFDeriv ℝ 2 f x₀ ![y, x]", + "line": 367 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "lemma", + "name": "iteratedFDeriv_two_swap", + "docstring": "Symmetry of the second iterated Fréchet derivative (Schwarz's theorem).", + "snippet": "lemma iteratedFDeriv_two_swap {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]\n {f : V → ℝ} {x₀ : V} (hf : ContDiffAt ℝ ⊤ f x₀) (x y : V) :\n iteratedFDeriv ℝ 2 f x₀ ![y, x] = iteratedFDeriv ℝ 2 f x₀ ![x, y]", + "line": 380 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "lemma", + "name": "coercive_of_posdef_of_contdiff", + "docstring": "Positive definiteness implies coercivity.\n The proof uses the general fact `coercive_of_posdefHalf`\n but requires a differentiability assumption on `f`.", + "snippet": "lemma coercive_of_posdef_of_contdiff {V : Type*}\n [NormedAddCommGroup V] [NormedSpace ℝ V]\n [FiniteDimensional ℝ V] {f : V → ℝ} {x₀ : V}\n (hf : ContDiffAt ℝ ⊤ f x₀)\n (hf' : (iteratedFDerivQuadraticMap f x₀).PosDef) :\n IsCoercive (continuousBilinearMapOfContinuousMultilinearMap\n (iteratedFDeriv ℝ 2 f x₀))", + "line": 397 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "lemma", + "name": "coercive_of_posdef", + "docstring": "Positive definiteness implies coercivity.", + "snippet": "lemma coercive_of_posdef {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]\n [FiniteDimensional ℝ V] {f : V → ℝ} {x₀ : V}\n (hf' : (iteratedFDerivQuadraticMap f x₀).PosDef) :\n IsCoercive (continuousBilinearMapOfContinuousMultilinearMap\n (iteratedFDeriv ℝ 2 f x₀))", + "line": 420 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "theorem", + "name": "le_of_littleO", + "docstring": ".", + "snippet": "theorem le_of_littleO {V : Type*}\n [NormedAddCommGroup V] [InnerProductSpace ℝ V]\n {f : V → ℝ} {x₀ x : V} {C : ℝ}\n (hx : C * (‖x - x₀‖ ^ 2) ≤ iteratedFDeriv ℝ 2 f x₀ fun _ ↦ x - x₀)\n (hx₀ : fderiv ℝ f x₀ x = fderiv ℝ f x₀ x₀)\n (h₁ : ‖f x - ∑ i ∈ range 3, 1 / ↑i ! * iteratedFDeriv ℝ i f x₀ fun _ ↦ x - x₀‖\n ≤ C / 2 * ‖x - x₀‖ ^ 2) :\n f x₀ ≤ f x", + "line": 473 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "theorem", + "name": "isLocalMin_of_posDef_of_littleo", + "docstring": "Second partial derivative test, \"little oh\" form.", + "snippet": "theorem isLocalMin_of_posDef_of_littleo {V : Type*} [NormedAddCommGroup V]\n [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] {f : V → ℝ} {x₀ : V}\n (h : (fun x => f x - ∑ i ∈ range 3, 1 / (i)! * iteratedFDeriv ℝ i f x₀ fun _ => x - x₀)\n =o[nhds x₀] fun x => ‖x - x₀‖ ^ 2) (h₀ : gradient f x₀ = 0) (hcd : ContDiffAt ℝ ⊤ f x₀)\n (hf : (iteratedFDerivQuadraticMap f x₀).PosDef) : IsLocalMin f x₀", + "line": 493 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "lemma", + "name": "littleO_of_powerseries", + "docstring": "Having a power series implies quadratic approximation.", + "snippet": "lemma littleO_of_powerseries {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]\n {f : V → ℝ} {x₀ : V} {p : FormalMultilinearSeries ℝ V ℝ}\n {r : NNReal} (h : HasFPowerSeriesOnBall f p x₀ r) :\n (fun x => f x - p.partialSum 3 (x - x₀)) =o[nhds x₀] fun x => ‖x - x₀‖ ^ 2", + "line": 517 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "lemma", + "name": "isLocalMin.of_subsingleton", + "docstring": null, + "snippet": "@[nontriviality]\nlemma isLocalMin.of_subsingleton {V : Type*} [TopologicalSpace V]\n [Subsingleton V] {f : V → ℝ} {x₀ : V} : IsLocalMin f x₀", + "line": 530 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "theorem", + "name": "second_derivative_test", + "docstring": "The second partial derivative test.", + "snippet": "theorem second_derivative_test {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]\n [FiniteDimensional ℝ V] {f : V → ℝ} {x₀ : V} {p : FormalMultilinearSeries ℝ V ℝ}\n (h₀ : gradient f x₀ = 0) {r : NNReal}\n (h₁ : HasFPowerSeriesOnBall f p x₀ r)\n (hf : (iteratedFDerivQuadraticMap f x₀).PosDef) : IsLocalMin f x₀", + "line": 535 + }, + { + "file": "PhyslibAlpha/Mathematics/PartialDerivativeTest.lean", + "kind": "theorem", + "name": "second_derivative_test_analyticAt", + "docstring": "A more convenient form of the second partial derivative test: it suffices that `f`\nis analytic at `x₀` (so that an explicit power series need not be produced by hand).", + "snippet": "theorem second_derivative_test_analyticAt {V : Type*} [NormedAddCommGroup V]\n [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] {f : V → ℝ} {x₀ : V}\n (h₀ : gradient f x₀ = 0) (han : AnalyticAt ℝ f x₀)\n (hf : (iteratedFDerivQuadraticMap f x₀).PosDef) : IsLocalMin f x₀", + "line": 559 + } + ] + }, + { + "filename": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "entries": [ + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "def", + "name": "a", + "docstring": "Annihilation operator.", + "snippet": "def a (x : ℕ → ℂ) : ℕ → ℂ", + "line": 17 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "def", + "name": "aLin", + "docstring": null, + "snippet": "def aLin : (ℕ → ℂ) →ₗ[ℂ] (ℕ → ℂ)", + "line": 19 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "def", + "name": "a_dag", + "docstring": "Creation operator.", + "snippet": "def a_dag (x : ℕ → ℂ) : ℕ → ℂ", + "line": 36 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "def", + "name": "a_dagLin", + "docstring": null, + "snippet": "def a_dagLin : (ℕ → ℂ) →ₗ[ℂ] (ℕ → ℂ)", + "line": 38 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "def", + "name": "ε", + "docstring": null, + "snippet": "def ε (n : ℕ) (c : ℂ) : ℕ → ℂ", + "line": 55 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "lemma", + "name": "a_dag_eq", + "docstring": "Verify that a_dag really is the transpose of a.", + "snippet": "lemma a_dag_eq (i j : ℕ) : a_dag (ε i 1) j = star (a (ε j 1) i)", + "line": 58 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "lemma", + "name": "verify_a", + "docstring": "Verify that a |n + 1⟩ = √(n + 1) |n ⟩", + "snippet": "lemma verify_a (n : ℕ) :\n a (ε (n + 1) 1) =\n ε n (√(n + 1))", + "line": 67 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "lemma", + "name": "verify_a_dag", + "docstring": "Verify that a† ∣n⟩ = √(n+1) ∣n+1⟩.", + "snippet": "lemma verify_a_dag (n : ℕ) :\n a_dag (ε n 1) = ε (n + 1) (√(n + 1))", + "line": 78 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "lemma", + "name": "verify_a_dag_a", + "docstring": null, + "snippet": "lemma verify_a_dag_a (n : ℕ) (x : ℕ → ℂ) :\n a_dag (a x) n = n * x n", + "line": 88 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "lemma", + "name": "verify_a_a_dag", + "docstring": null, + "snippet": "lemma verify_a_a_dag (n : ℕ) (x : ℕ → ℂ) :\n a (a_dag x) n = (n + 1) * x n", + "line": 103 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "lemma", + "name": "commutation_relation", + "docstring": null, + "snippet": "lemma commutation_relation :\n a ∘ a_dag - a_dag ∘ a = id", + "line": 115 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "def", + "name": "coherentState", + "docstring": null, + "snippet": "def coherentState (α : ℂ) : ℕ → ℂ", + "line": 138 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "def", + "name": "probabilityOf", + "docstring": null, + "snippet": "def probabilityOf (n : ℕ) (α : ℂ) : NNReal", + "line": 141 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "lemma", + "name": "probabilityOf_eq_poisson_C", + "docstring": "Coherent state has a Poisson distribution.", + "snippet": "lemma probabilityOf_eq_poisson_C (n : ℕ) (α : ℂ) :\n let Λ", + "line": 145 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "lemma", + "name": "coherentState_only_eigenvector", + "docstring": "The only eigenvectors of `a` are the coherent states.", + "snippet": "lemma coherentState_only_eigenvector (α : ℂ) (v : ℕ → ℂ) :\n a v = α • v ↔\n v = (Complex.exp (↑‖α‖ ^ 2 / 2) * v 0) • coherentState α", + "line": 201 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "lemma", + "name": "eigenvector_coherentState", + "docstring": null, + "snippet": "lemma eigenvector_coherentState (α : ℂ) :\n a (coherentState α) = α • coherentState α", + "line": 264 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "lemma", + "name": "distinct_eigenvectors_a", + "docstring": "None of the `coherentState` eigenvectors are proportional.", + "snippet": "lemma distinct_eigenvectors_a (α β c : ℂ)\n (hc : coherentState α = c • coherentState β) : α = β", + "line": 280 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "lemma", + "name": "a_dagalmost_eigenvector", + "docstring": "Formal eigenvectors for `a_dag` (not in `ℓ²(ℂ)`)\n(fails at n=0) .", + "snippet": "lemma a_dagalmost_eigenvector {α : ℂ} (hα : α ≠ 0) {n : ℕ} (hn : n ≠ 0) :\n a_dag (fun n => √(n.factorial) / α^n) n =\n α • (fun n => √(n.factorial) / α^n) n", + "line": 295 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "lemma", + "name": "a_dag_nullspace", + "docstring": null, + "snippet": "lemma a_dag_nullspace\n {v : ℕ → ℂ} (hv : a_dag v = 0) : v = 0", + "line": 314 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "lemma", + "name": "no_a_dag_eigenvector", + "docstring": null, + "snippet": "lemma no_a_dag_eigenvector (α : ℂ) (v : ℕ → ℂ) :\n a_dag v = α • v ↔ v = 0", + "line": 337 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "lemma", + "name": "commutationRelation", + "docstring": null, + "snippet": "lemma commutationRelation : ⁅aLin, a_dagLin⁆ = 1", + "line": 358 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/QuantumHarmonicOscillator.lean", + "kind": "def", + "name": "coherentState_ℓ2", + "docstring": "The coherent state belongs to `ℓ²(ℂ)`.", + "snippet": "def coherentState_ℓ2 (α : ℂ) : lp (fun _ : ℕ => ℂ) 2", + "line": 379 + } + ] + }, + { + "filename": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "entries": [ + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "krausApply", + "docstring": "Completely positive map given by a (not necessarily minimal) Kraus family.", + "snippet": "def krausApply {R : Type*} [Mul R] [Star R] [AddCommMonoid R]\n {q r : Type*} [Fintype q] [Fintype r]\n (K : r → Matrix q q R) (ρ : Matrix q q R) : Matrix q q R", + "line": 27 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "krausApply.posSemidef", + "docstring": "Kraus operator preserves PSD property.", + "snippet": "lemma krausApply.posSemidef {R : Type*} [Ring R] [PartialOrder R] [StarRing R]\n [AddLeftMono R]\n {q r : Type*} [Fintype q] [Fintype r]\n (K : r → Matrix q q R)\n {ρ : Matrix q q R} (hρ : ρ.PosSemidef) :\n (krausApply K ρ).PosSemidef", + "line": 33 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "QuantumChannel", + "docstring": "Quantum channel.", + "snippet": "def QuantumChannel {R : Type*} [Mul R] [One R] [Star R] [AddCommMonoid R]\n {q r : Type*} [Fintype q] [Fintype r] [DecidableEq q]\n (K : r → Matrix q q R)", + "line": 42 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "QuantumOperation", + "docstring": "Quantum operation.", + "snippet": "def QuantumOperation {R : Type*} [RCLike R]\n {q r : Type*} [Fintype q] [Fintype r] [DecidableEq q]\n (K : r → Matrix q q R)", + "line": 48 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "densityMatrix", + "docstring": "Density matrix.", + "snippet": "def densityMatrix {R : Type*} [Ring R] [PartialOrder R] [StarRing R] (d : Type*) [Fintype d]", + "line": 53 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "densityMatrix.convex_comb", + "docstring": "Density matrices are closed under real convex combinations.", + "snippet": "def densityMatrix.convex_comb {R : Type*} [RCLike R]\n {d : ℕ} (ρ₀ ρ₁ : densityMatrix (Fin d) (R := R)) {t : R}\n (hp₀ : 0 ≤ t) (hp₁ : 0 ≤ 1 - t) : densityMatrix (Fin d) (R := R)", + "line": 57 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "tr₂", + "docstring": "Also known as `partialTraceRight`.", + "snippet": "def tr₂ {R : Type*} [Ring R] {m n m' : Type*} [Fintype n]\n (ρ : Matrix (m × n) (m' × n) R) : Matrix m m' R", + "line": 67 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "stinespringOp", + "docstring": "`stinespringOp` is often written as `V`.", + "snippet": "def stinespringOp {R : Type*} [Ring R]\n {m r : Type*} [Fintype r] [DecidableEq r]\n (K : r → Matrix m m R) : Matrix (m × r) m R", + "line": 72 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "stinespringDilation", + "docstring": "The Stinespring dilation.", + "snippet": "def stinespringDilation {R : Type*} [Ring R] [StarRing R]\n {m r : Type*} [Fintype r] [DecidableEq r] [Fintype m]\n (K : r → Matrix m m R)\n (ρ : Matrix m m R)", + "line": 80 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "stinespringForm", + "docstring": "The partial trace of the Stinespring dilation.", + "snippet": "def stinespringForm {R : Type*} [Ring R] [StarRing R]\n {m r : Type*} [Fintype r] [DecidableEq r] [Fintype m]\n (K : r → Matrix m m R)", + "line": 88 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "stinespringOp_adjoint_mul_self", + "docstring": "A useful identity for Stinespring dilations.", + "snippet": "lemma stinespringOp_adjoint_mul_self {R : Type*} [Ring R] [StarRing R]\n {m r : Type*} [Fintype r] [DecidableEq r] [Fintype m]\n (K : r → Matrix m m R) :\n ∑ i, star K i * K i = (stinespringOp K)ᴴ * stinespringOp K", + "line": 94 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "stinespringForm_CPTNI", + "docstring": "A useful identity for completely positive, trace non-increasing maps.", + "snippet": "lemma stinespringForm_CPTNI {R : Type*} [RCLike R]\n {m r : Type*} [Fintype r] [DecidableEq r] [Fintype m] [DecidableEq m]\n (K : r → Matrix m m R)\n (hK : ∑ i, (K i)ᴴ * K i ≤ 1) :\n (stinespringOp K)ᴴ * (stinespringOp K) ≤ 1", + "line": 108 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "stinespringForm_CPTP_isometry", + "docstring": "The Stinespring operator of a completely positive, trace-preserving maps\nis an isometry. (Note that `stinespringOp K` is not a square matrix in general.)", + "snippet": "lemma stinespringForm_CPTP_isometry {R : Type*} [Ring R] [StarRing R]\n {m r : Type*} [Fintype r] [DecidableEq r] [Fintype m] [DecidableEq m]\n {K : r → Matrix m m R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) :\n (stinespringOp K)ᴴ * (stinespringOp K) = 1", + "line": 119 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "stinespringOrtho", + "docstring": "Proving the columns of `V = stinespringOp K` are independent is a step\non the way to constructing the unitary dilation.", + "snippet": "lemma stinespringOrtho {R : Type*} [RCLike R]\n {m r : Type*} [Fintype r] [DecidableEq r] [Fintype m] [DecidableEq m]\n {K : r → Matrix m m R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) :\n Orthonormal (𝕜 := R)\n fun j : m => WithLp.toLp 2 fun i : m × r => stinespringOp K i j", + "line": 131 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "basisCard", + "docstring": "`m` will of course be finite and bounded by `n` here,\nbut no need to assume or prove that.", + "snippet": "lemma basisCard {R : Type*} [RCLike R] {n m : Type*} [Fintype n] {s : Matrix n m R}\n (ho : Orthonormal R fun j ↦ WithLp.toLp 2 fun i ↦ s i j) :\n Fintype.card n =\n ho.toSubtypeRange.exists_orthonormalBasis_extension.choose.card", + "line": 166 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "stinespringCard", + "docstring": "Calculating the cardinality of the orthonormal basis obtained by\nextending the Stinespring orthonormal set, the columns of `stinespringOp K`.", + "snippet": "lemma stinespringCard {R : Type*} [RCLike R]\n {m r : Type*} [Fintype r] [DecidableEq r] [Fintype m] [DecidableEq m]\n {K : r → Matrix m m R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) :\n Fintype.card (m × r) = (stinespringOrtho\n hK).toSubtypeRange.exists_orthonormalBasis_extension.choose.card", + "line": 177 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "theorem", + "name": "complCard", + "docstring": "We need the 1 matrix, which we don't seem to have for an arbitrary\n`[Fintype m]`.\nSince we are comparing `Fin r` and `Fin (r-1)` we also cannot too\neasily use an arbitrary `[Fintype r] [Zero r]`.", + "snippet": "theorem complCard {R : Type*} [RCLike R] {m r : ℕ}\n {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) (z : Fin r) :\n let 𝓞", + "line": 193 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "Fin.predAbove_of_ne", + "docstring": "See discussion at https://leanprover.zulipchat.com/#narrow/channel/217875-Is-there-code-for-X.3F/topic/succAbove.20and.20predAbove.20lemmas/with/584270574", + "snippet": "def Fin.predAbove_of_ne {n : ℕ} {k i : Fin n}\n (h : i ≠ k) : Fin (n - 1)", + "line": 244 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "Fin.predAbove_of_ne_injective", + "docstring": "A \"missing lemma\" for `Fin` types.", + "snippet": "lemma Fin.predAbove_of_ne_injective (n : ℕ) (k x y : Fin n)\n (hx : x ≠ k) (hy : y ≠ k)\n (heq : Fin.predAbove_of_ne hx = Fin.predAbove_of_ne hy) : x = y", + "line": 251 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "onbPart", + "docstring": "The way this is written, `Fin r` and `Fin (r-1)` both occur\nso it is tricky to go to a general `Fintype`.", + "snippet": "def onbPart {R : Type*} [RCLike R]\n {m r : ℕ} {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) (x : Fin m × Fin r) {z : Fin r} (hx : ¬x.2 = z) :", + "line": 264 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "onbPart_inner", + "docstring": null, + "snippet": "lemma onbPart_inner {R : Type*} [RCLike R] {m r : ℕ} {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) {z : Fin r}\n {y : Fin m × Fin r} (hy : ¬y.2 = z)\n {x : Fin m × Fin r} (hx : ¬x.2 = z)\n (h : y ≠ x) :\n inner R (WithLp.toLp 2 <| onbPart hK y hy)\n (WithLp.toLp 2 <| onbPart hK x hx) = 0", + "line": 280 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "onbPart_norm", + "docstring": "The vectors in the Stinespring orthonormal basis have norm 1.", + "snippet": "lemma onbPart_norm {R : Type*} [RCLike R] {m r : ℕ} {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) (x : Fin m × Fin r)\n {z : Fin r} (hx : ¬x.2 = z) :\n ‖WithLp.toLp 2 <| onbPart hK x hx‖ = 1", + "line": 321 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "Ud", + "docstring": "Also known as `unitaryDilation`. Respects x,y order.", + "snippet": "def Ud {R : Type*} [RCLike R] {m r : ℕ}\n {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) (z : Fin r) :\n Matrix (Fin m × Fin r) (Fin m × Fin r) R", + "line": 332 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "general_dilation", + "docstring": "This generalization of Stinespring dilation has the right\n\"shape\" but otherwise nothing specific to it.", + "snippet": "def general_dilation {R : Type*}\n {m r : Type*} [DecidableEq r]\n (z : r)\n (S : Matrix (m × r) m R)\n (M : Matrix (m × r) (m × r) R) :\n Matrix (m × r) (m × r) R", + "line": 344 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "dilation", + "docstring": "A general, not necessarily unitary, dilation.", + "snippet": "def dilation {R : Type*} [Ring R]\n {m r : Type*} [Fintype r] [DecidableEq r]\n (K : r → Matrix m m R) (z : r) (M : Matrix (m × r) (m × r) R) :\n Matrix (m × r) (m × r) R", + "line": 354 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "theorem", + "name": "Ud_orthonormal₁", + "docstring": "One version of orthonormality of `stinespringOp`.", + "snippet": "theorem Ud_orthonormal₁ {R : Type*} [RCLike R] {m r : ℕ} {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) (z : Fin r) :\n Orthonormal R fun y ↦ if hy : y.2 = z then WithLp.toLp 2 fun i ↦ stinespringOp K i y.1\n else WithLp.toLp 2 fun i ↦ onbPart hK y hy i", + "line": 362 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "theorem", + "name": "Ud_orthonormal₂", + "docstring": "The Stinespring dilation columns form an orthonormal basis.", + "snippet": "theorem Ud_orthonormal₂ {R : Type*} [RCLike R]\n {m r : ℕ} {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) (z : Fin r) :\n Orthonormal R fun y ↦\n WithLp.toLp 2 fun i ↦ if hy : y.2 = z then stinespringOp K i y.1 else onbPart hK y hy i", + "line": 409 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "smul_self_one_of_norm_one", + "docstring": "If `β` has length 1 then the dot product of `β` with itself is 1.", + "snippet": "lemma smul_self_one_of_norm_one {R : Type*} [RCLike R]\n {t : Type*} [Fintype t] {β : t → R} (hj : ‖WithLp.toLp 2 β‖ = 1) :\n ∑ x, (starRingEnd R) (β x) * β x = 1", + "line": 430 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "theorem", + "name": "unitary_of_orthonormal", + "docstring": "A matrix whose columns are orthonormal is unitary.", + "snippet": "theorem unitary_of_orthonormal {R : Type*} [RCLike R]\n {m r : Type*} [Fintype r] [DecidableEq r] [Fintype m] [DecidableEq m]\n (α : Matrix (m × r) (m × r) R)\n (h₀ : Orthonormal R fun i ↦ WithLp.toLp 2 (α i)) : α * star α = 1", + "line": 447 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "Ud_unitaryT", + "docstring": "The transpose of the unitary dilation is unitary.", + "snippet": "lemma Ud_unitaryT {R : Type*} [RCLike R]\n {m r : ℕ} {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) (z : Fin r) :\n (Ud hK z)ᵀ ∈ unitary _", + "line": 469 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "Ud_unitary", + "docstring": "The unitary dilation `Ud` is in fact unitary.", + "snippet": "lemma Ud_unitary {R : Type*} [RCLike R]\n {m r : ℕ} {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) (z : Fin r) :\n (Ud hK z) ∈ unitary _", + "line": 481 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "tr₂_e₀Xe₀", + "docstring": "Taking the partial trace of a tensor product with a matrix of trace 1 is the\nidentity map.", + "snippet": "lemma tr₂_e₀Xe₀ {R : Type*} [RCLike R]\n {m w : Type*} [Fintype w] [Zero w]\n (e : Matrix w w R) (htr : e.trace = 1)\n (ρ : Matrix m m R) :\n tr₂ (ρ ⊗ₖ e) = ρ", + "line": 503 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "stinespringUnitaryForm", + "docstring": "The Stinespring unitary form.", + "snippet": "def stinespringUnitaryForm {R : Type*} [RCLike R] {m r : ℕ}\n {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) (z : Fin r)\n (ρ : Matrix (Fin m) (Fin m) R) :\n (Matrix (Fin m) (Fin m) R)", + "line": 520 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "stinespringUnitaryForm_e", + "docstring": "The Stinespring unitary form, general version.", + "snippet": "def stinespringUnitaryForm_e {R : Type*} [RCLike R] {m r : ℕ}\n {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) (z : Fin r) (e : Matrix (Fin r) (Fin r) R)\n (ρ : Matrix (Fin m) (Fin m) R) :\n (Matrix (Fin m) (Fin m) R)", + "line": 529 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "theorem", + "name": "tracefree_version", + "docstring": "Trace-free version of the Stinespring Dilation Theorem.", + "snippet": "theorem tracefree_version {R : Type*} [RCLike R]\n {m r : Type*} [Fintype r] [DecidableEq r] [Zero r] [Fintype m] [DecidableEq m]\n {K : r → Matrix m m R}\n (ρ : Matrix m m R) :\n let K'", + "line": 538 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "theorem", + "name": "heisenberg_schrõdinger", + "docstring": "A Heisberg picture / Schrödinger picture view of the Stinespring dilation.", + "snippet": "theorem heisenberg_schrõdinger {R : Type*} [RCLike R]\n {m r : Type*} [Fintype r] [DecidableEq r] [Zero r] [Fintype m] [DecidableEq m]\n {K : r → Matrix m m R}\n (ρ : Matrix m m R) :\n let K'", + "line": 558 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "generalForm", + "docstring": "A further generalization of `stinespringGeneralForm`.", + "snippet": "def generalForm {R : Type*} [RCLike R]\n {m r : Type*} [Fintype r] [DecidableEq r] [Fintype m]\n (z : r)\n (S : Matrix (m × r) m R)\n (M : Matrix (m × r) (m × r) R)", + "line": 573 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "stinespringGeneralForm", + "docstring": "General form of the Stinespring dilation.", + "snippet": "def stinespringGeneralForm {R : Type*} [RCLike R]\n {m r : Type*} [Fintype r] [DecidableEq r] [Fintype m]\n (K : r → Matrix m m R) (z : r)\n (M : Matrix (m × r) (m × r) R)", + "line": 582 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "stinespringGeneralForm_e", + "docstring": "Even more general form of the Stinespring dilation.", + "snippet": "def stinespringGeneralForm_e {R : Type*} [RCLike R]\n {m r : Type*} [Fintype r] [DecidableEq r] [Fintype m]\n (K : r → Matrix m m R) (z : r) (e : Matrix r r R)\n (M : Matrix (m × r) (m × r) R)", + "line": 590 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "theorem", + "name": "unitaryForm_of_general", + "docstring": "When we plug in `M = Ud hK`\ninto the general `stinespringGeneralForm`,\nthen we do get\n`stinespringUnitaryForm hK`.", + "snippet": "theorem unitaryForm_of_general {R : Type*} [RCLike R] {m r : ℕ}\n {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) (z : Fin r) :\n stinespringGeneralForm K z (Ud hK z) =\n stinespringUnitaryForm hK z", + "line": 603 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "theorem", + "name": "unitaryForm_of_general_e", + "docstring": "The Stinespring unitary form as a general form applied to the unitary dilation.", + "snippet": "theorem unitaryForm_of_general_e {R : Type*} [RCLike R] {m r : ℕ}\n {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) (z : Fin r) (e : Matrix (Fin r) (Fin r) R) :\n stinespringGeneralForm_e K z e (Ud hK z) =\n stinespringUnitaryForm_e hK z e", + "line": 636 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "stinespringGeneralForm_works", + "docstring": "Note we don't need any special properties of M,\nand we don't need K to be CPTP.\n\nUses `Fin` types because of the use of\n`Fin.sum_univ_succAbove` in the proof.", + "snippet": "lemma stinespringGeneralForm_works {R : Type*} [RCLike R] {m r : ℕ}\n (K : Fin r → Matrix (Fin m) (Fin m) R) (z : Fin r)\n (M : Matrix (Fin m × Fin r) (Fin m × Fin r) R) :\n stinespringGeneralForm K z M = krausApply K", + "line": 676 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "stinespringUnitaryForm_works", + "docstring": "Notice that unitarity is a side property, it is not why\nthe Stinespring form works.\n\nHere `z` is the coordinate used for the ancilla.", + "snippet": "lemma stinespringUnitaryForm_works {R : Type*} [RCLike R] {m r : ℕ}\n {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) (z : Fin r) :\n stinespringUnitaryForm hK z = krausApply K", + "line": 707 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "krausCompletion", + "docstring": "The \"orthogonal\" CPTP completion of a CPTNI map.\n`Vtilde` is an alternative name for `krausCompletion`.", + "snippet": "def krausCompletion {R : Type*} [RCLike R] {m r : ℕ}\n (K : Fin r → Matrix (Fin m) (Fin m) R) :\n Matrix (Fin m × Fin (r+1)) (Fin m) R", + "line": 717 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "theorem", + "name": "stinespringOp_apply", + "docstring": "Entrywise formula for the Stinespring isometry: its `((x₁, x₂), y)` entry is `K x₂ x₁ y`.", + "snippet": "theorem stinespringOp_apply {R : Type*} [Ring R] {m r : Type*} [Fintype r] [DecidableEq r]\n [Fintype m] [DecidableEq m] (K : r → Matrix m m R) (x : m × r) (y : m) :\n stinespringOp K x y = K x.2 x.1 y", + "line": 724 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "theorem", + "name": "stinespringOp_gram", + "docstring": "The Gram matrix of the Stinespring isometry is `∑ i, (K i)ᴴ * K i`.", + "snippet": "theorem stinespringOp_gram {R : Type*} [RCLike R] {m r : ℕ}\n (K : Fin r → Matrix (Fin m) (Fin m) R) :\n (stinespringOp K)ᴴ * stinespringOp K = ∑ i, star (K i) * K i", + "line": 731 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "krausCompletion_isometry_of_TNI", + "docstring": "Mar 14, 2026 by Bjørn for 4.27\nJune 13, 2026 by Aristotle for 4.31 including\n`stinespringOp_gram` and `stinespringOp_apply`.", + "snippet": "lemma krausCompletion_isometry_of_TNI {R : Type*} [RCLike R] {m r : ℕ}\n {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, star K i * K i ≤ 1) :\n (krausCompletion K)ᴴ * krausCompletion K = 1", + "line": 748 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "unital", + "docstring": "A unital operator.", + "snippet": "def unital {R : Type*} [RCLike R] {m r : ℕ}\n (K : Fin r → Matrix (Fin m) (Fin m) R)", + "line": 799 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "subunital", + "docstring": "A subunital operator.", + "snippet": "def subunital {R : Type*} [RCLike R] {m r : ℕ}\n (K : Fin r → Matrix (Fin m) (Fin m) R)", + "line": 803 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "partialTrace_tensor", + "docstring": "The identity `Tr_B (A ⨂ B) = Tr(B) · A`", + "snippet": "lemma partialTrace_tensor {R : Type*} [RCLike R] {m n : ℕ}\n (A : Matrix (Fin m) (Fin m) R) (B : Matrix (Fin n) (Fin n) R) :\n tr₂ (A ⊗ₖ B) = (trace B) • A", + "line": 808 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "krausApply_of_tensor", + "docstring": "A unitary dilation view of the application of a Kraus operator.", + "snippet": "lemma krausApply_of_tensor {R : Type*} [RCLike R] {m r : ℕ}\n {K : Fin r → Matrix (Fin m) (Fin m) R}\n (hK : ∑ i, (K i)ᴴ * K i = 1) (z : Fin r)\n (ρ α : Matrix (Fin m) (Fin m) R) (β : Matrix (Fin r) (Fin r) R)\n (hβ : β.trace = 1) (h : (Ud hK z) * (ρ ⊗ₖ (single z z 1)) * (Ud hK z)ᴴ = α ⊗ₖ β) :\n krausApply K ρ = α", + "line": 821 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "trace_tr₂", + "docstring": "Trace of partial trace equals trace.", + "snippet": "lemma trace_tr₂ {R : Type*} [RCLike R] {m n : ℕ}\n (ρ : Matrix (Fin m × Fin n) (Fin m × Fin n) R) :\n trace ρ = trace (tr₂ ρ)", + "line": 836 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "krausCompletionChannelMap", + "docstring": "The Kraus completion as a map from\noperations to channels.", + "snippet": "def krausCompletionChannelMap {R : Type*} [RCLike R] {q r : ℕ}\n {K : Fin r → Matrix (Fin q) (Fin q) R} (hK : QuantumOperation K) :\n {K : Fin (r+1) → Matrix (Fin q) (Fin q) R | QuantumChannel K}", + "line": 843 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "lemma", + "name": "CPTP_of_CPTNI", + "docstring": "The \"not orthogonal\" CPTP completion of a CPTNI map.", + "snippet": "lemma CPTP_of_CPTNI {R : Type*} [RCLike R]\n {q r : ℕ}\n {K : Fin r → Matrix (Fin q) (Fin q) R}\n (hq : QuantumOperation K) :\n ∃ K' : Fin (r+1) → Matrix (Fin q) (Fin q) R,\n QuantumChannel K' ∧\n ∀ i, ∀ H : i ≠ Fin.last r, K' i = K ⟨i.1, Fin.val_lt_last H⟩", + "line": 857 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "def", + "name": "partialTraceLeft", + "docstring": "Partial trace on the left of a tensor product.", + "snippet": "def partialTraceLeft {R : Type*} [RCLike R]\n {m n : Type*} [Fintype m]\n (ρ : Matrix (m × n)\n (m × n) R) : Matrix (n) (n) R", + "line": 885 + }, + { + "file": "PhyslibAlpha/QuantumMechanics/StinespringDilation.lean", + "kind": "theorem", + "name": "stinespringForm_eq", + "docstring": "A version of the Stinespring Dilation Theorem.", + "snippet": "theorem stinespringForm_eq {R : Type*} [RCLike R] {m r : ℕ}\n (K : Fin r → Matrix (Fin m) (Fin m) R)\n (ρ : Matrix (Fin m) (Fin m) R) :\n tr₂ (stinespringDilation K ρ) = krausApply K ρ", + "line": 892 + } + ] + }, + { + "filename": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "entries": [ + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "def", + "name": "Space.halfPlaneDomain", + "docstring": "The domain of the half-plane inside `Space 2`.", + "snippet": "def halfPlaneDomain : Set (Space 2)", + "line": 36 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "def", + "name": "Space.halfPlane", + "docstring": "The coordinate plane embedding used for the half-plane surface in `Space 3`.", + "snippet": "def halfPlane : Space 2 → Space 3", + "line": 39 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "lemma", + "name": "Space.halfPlane_eq", + "docstring": null, + "snippet": "lemma halfPlane_eq : halfPlane = (slice (2 : Fin 3)).symm ∘ (fun x : Space 2 => (0, x))", + "line": 41 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "lemma", + "name": "Space.halfPlane_injective", + "docstring": null, + "snippet": "lemma halfPlane_injective : Function.Injective halfPlane", + "line": 43 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "lemma", + "name": "Space.halfPlane_continuous", + "docstring": null, + "snippet": "@[fun_prop]\nlemma halfPlane_continuous : Continuous halfPlane", + "line": 50 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "lemma", + "name": "Space.halfPlane_measurableEmbedding", + "docstring": null, + "snippet": "lemma halfPlane_measurableEmbedding : MeasurableEmbedding halfPlane", + "line": 54 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "lemma", + "name": "Space.norm_halfPlane", + "docstring": null, + "snippet": "@[simp]\nlemma norm_halfPlane (x : Space 2) : ‖halfPlane x‖ = ‖x‖", + "line": 58 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "def", + "name": "Space.halfPlaneMeasure", + "docstring": "The measure on `Space 3` corresponding to integration over a half-plane.", + "snippet": "def halfPlaneMeasure : Measure (Space 3)", + "line": 69 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "instance", + "name": "Space.halfPlaneMeasure_hasTemperateGrowth", + "docstring": null, + "snippet": "instance halfPlaneMeasure_hasTemperateGrowth :\n halfPlaneMeasure.HasTemperateGrowth", + "line": 72 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "instance", + "name": "Space.halfPlaneMeasure_sFinite", + "docstring": null, + "snippet": "instance halfPlaneMeasure_sFinite : SFinite halfPlaneMeasure", + "line": 84 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "def", + "name": "Space.halfPlaneDist", + "docstring": "The distribution on `Space 3` corresponding to integration over a half-plane.", + "snippet": "def halfPlaneDist : (Space 3) →d[ℝ] ℝ", + "line": 95 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "lemma", + "name": "Space.halfPlaneDist_apply_eq_integral_halfPlaneMeasure", + "docstring": null, + "snippet": "lemma halfPlaneDist_apply_eq_integral_halfPlaneMeasure (f : 𝓢(Space 3, ℝ)) :\n halfPlaneDist f = ∫ x, f x ∂halfPlaneMeasure", + "line": 98 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "lemma", + "name": "Space.halfPlaneDist_apply_eq_integral_volume", + "docstring": null, + "snippet": "lemma halfPlaneDist_apply_eq_integral_volume (f : 𝓢(Space 3, ℝ)) :\n halfPlaneDist f = ∫ x, f (halfPlane x) ∂(volume.restrict halfPlaneDomain)", + "line": 102 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "def", + "name": "Space.halfPlaneSubmodule", + "docstring": "The coordinate plane containing the half-plane in `Space 3`.", + "snippet": "def halfPlaneSubmodule : Submodule ℝ (Space 3) where", + "line": 114 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "lemma", + "name": "Space.halfPlane_mem_halfPlaneSubmodule", + "docstring": null, + "snippet": "lemma halfPlane_mem_halfPlaneSubmodule (x : Space 2) : halfPlane x ∈ halfPlaneSubmodule", + "line": 124 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "lemma", + "name": "Space.halfPlane_image_domain_subset_halfPlaneSubmodule", + "docstring": null, + "snippet": "lemma halfPlane_image_domain_subset_halfPlaneSubmodule :\n halfPlane '' halfPlaneDomain ⊆ (halfPlaneSubmodule : Set (Space 3))", + "line": 127 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "lemma", + "name": "Space.halfPlaneSubmodule_ne_top", + "docstring": null, + "snippet": "lemma halfPlaneSubmodule_ne_top : halfPlaneSubmodule ≠ ⊤", + "line": 132 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/HalfPlane.lean", + "kind": "lemma", + "name": "Space.volume_halfPlane_image_domain", + "docstring": null, + "snippet": "lemma volume_halfPlane_image_domain :\n volume (halfPlane '' halfPlaneDomain : Set (Space 3)) = 0", + "line": 139 + } + ] + }, + { + "filename": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "entries": [ + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "def", + "name": "Space.line", + "docstring": "The coordinate line embedded in `Space d`.", + "snippet": "def line (d : ℕ) [NeZero d] : ℝ → Space d", + "line": 33 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "lemma", + "name": "Space.line_eq_smul_basis", + "docstring": null, + "snippet": "lemma line_eq_smul_basis (d : ℕ) [NeZero d] :\n line d = fun r => r • basis (0 : Fin d)", + "line": 36 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "lemma", + "name": "Space.line_injective", + "docstring": null, + "snippet": "lemma line_injective (d : ℕ) [NeZero d] : Function.Injective (line d)", + "line": 39 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "lemma", + "name": "Space.line_continuous", + "docstring": null, + "snippet": "@[fun_prop]\nlemma line_continuous (d : ℕ) [NeZero d] : Continuous (line d)", + "line": 45 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "lemma", + "name": "Space.line_measurableEmbedding", + "docstring": null, + "snippet": "lemma line_measurableEmbedding (d : ℕ) [NeZero d] : MeasurableEmbedding (line d)", + "line": 49 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "lemma", + "name": "Space.norm_line", + "docstring": null, + "snippet": "@[simp]\nlemma norm_line (d : ℕ) [NeZero d] (r : ℝ) : ‖line d r‖ = ‖r‖", + "line": 53 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "def", + "name": "Space.lineMeasure", + "docstring": "The measure on `Space d` corresponding to integration along a coordinate line.", + "snippet": "def lineMeasure (d : ℕ) [NeZero d] : Measure (Space d)", + "line": 64 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "instance", + "name": "Space.lineMeasure_hasTemperateGrowth", + "docstring": null, + "snippet": "instance lineMeasure_hasTemperateGrowth (d : ℕ) [NeZero d] :\n (lineMeasure d).HasTemperateGrowth", + "line": 67 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "def", + "name": "Space.lineDist", + "docstring": "The distribution on `Space d` corresponding to integration along a coordinate line.\n One can roughly think of this distribution as taking a test function `f` to its integral against\n a mass, charge or current density concentrated on a line.", + "snippet": "def lineDist (d : ℕ) [NeZero d] : (Space d) →d[ℝ] ℝ", + "line": 88 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "lemma", + "name": "Space.lineDist_apply_eq_integral_lineMeasure", + "docstring": null, + "snippet": "lemma lineDist_apply_eq_integral_lineMeasure (d : ℕ) [NeZero d] (f : 𝓢(Space d, ℝ)) :\n lineDist d f = ∫ x, f x ∂lineMeasure d", + "line": 91 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "lemma", + "name": "Space.lineDist_apply_eq_integral_volume", + "docstring": null, + "snippet": "lemma lineDist_apply_eq_integral_volume (d : ℕ) [NeZero d] (f : 𝓢(Space d, ℝ)) :\n lineDist d f = ∫ r : ℝ, f (line d r)", + "line": 95 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "def", + "name": "Space.lineSubmodule", + "docstring": "The linear subspace spanned by the coordinate line in `Space d`.", + "snippet": "def lineSubmodule (d : ℕ) [NeZero d] : Submodule ℝ (Space d)", + "line": 107 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "lemma", + "name": "Space.line_mem_lineSubmodule", + "docstring": null, + "snippet": "lemma line_mem_lineSubmodule (d : ℕ) [NeZero d] (r : ℝ) : line d r ∈ lineSubmodule d", + "line": 110 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "lemma", + "name": "Space.range_line_subset_lineSubmodule", + "docstring": null, + "snippet": "lemma range_line_subset_lineSubmodule (d : ℕ) [NeZero d] :\n Set.range (line d) ⊆ (lineSubmodule d : Set (Space d))", + "line": 114 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "lemma", + "name": "Space.lineSubmodule_ne_top", + "docstring": null, + "snippet": "lemma lineSubmodule_ne_top (d : ℕ) [NeZero d] (hd : 2 ≤ d) : lineSubmodule d ≠ ⊤", + "line": 119 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Line.lean", + "kind": "lemma", + "name": "Space.volume_line_range", + "docstring": null, + "snippet": "lemma volume_line_range (d : ℕ) [NeZero d] (hd : 2 ≤ d) :\n volume (Set.range (line d) : Set (Space d)) = 0", + "line": 129 + } + ] + }, + { + "filename": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "entries": [ + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "def", + "name": "Space.ring", + "docstring": "The embedding of the unit ring (a circle `S¹` in `Space 2`) into `Space 3`.", + "snippet": "def ring : Metric.sphere (0 : Space 2) 1 → Space 3", + "line": 33 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "lemma", + "name": "Space.ring_eq", + "docstring": null, + "snippet": "lemma ring_eq : ring = (slice 2).symm ∘ (fun x => (0, sphericalShell 2 x))", + "line": 36 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "lemma", + "name": "Space.ring_injective", + "docstring": null, + "snippet": "lemma ring_injective : Function.Injective ring", + "line": 38 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "lemma", + "name": "Space.ring_continuous", + "docstring": null, + "snippet": "@[fun_prop]\nlemma ring_continuous : Continuous ring", + "line": 44 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "lemma", + "name": "Space.ring_measurableEmbedding", + "docstring": null, + "snippet": "lemma ring_measurableEmbedding : MeasurableEmbedding ring", + "line": 49 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "def", + "name": "Space.ringMeasure", + "docstring": "The measure on `Space 3` corresponding to integration around a ring.", + "snippet": "def ringMeasure : Measure (Space 3)", + "line": 60 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "instance", + "name": "Space.ringMeasure_hasTemperateGrowth", + "docstring": null, + "snippet": "instance ringMeasure_hasTemperateGrowth :\n ringMeasure.HasTemperateGrowth", + "line": 63 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "instance", + "name": "Space.ringMeasure_prod_volume_hasTemperateGrowth", + "docstring": null, + "snippet": "instance ringMeasure_prod_volume_hasTemperateGrowth :\n (ringMeasure.prod (volume (α := Space))).HasTemperateGrowth", + "line": 70 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "instance", + "name": "Space.ringMeasure_sFinite", + "docstring": null, + "snippet": "instance ringMeasure_sFinite: SFinite ringMeasure", + "line": 74 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "instance", + "name": "Space.ringMeasure_finite", + "docstring": null, + "snippet": "instance ringMeasure_finite: IsFiniteMeasure ringMeasure", + "line": 78 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "lemma", + "name": "Space.integrable_ringMeasure_of_continuous", + "docstring": null, + "snippet": "lemma integrable_ringMeasure_of_continuous (f : Space → ℝ) (hf : Continuous (f ∘ ring)) :\n Integrable f ringMeasure", + "line": 82 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "lemma", + "name": "Space.integrable_ringMeasure_of_continuous_euclid", + "docstring": null, + "snippet": "lemma integrable_ringMeasure_of_continuous_euclid (f : Space → EuclideanSpace ℝ (Fin n))\n (hf : Continuous (f ∘ ring)) :\n Integrable f ringMeasure", + "line": 91 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "lemma", + "name": "Space.ringMeasure_prod_volume_map", + "docstring": null, + "snippet": "lemma ringMeasure_prod_volume_map :\n (ringMeasure.prod (volume (α := Space))).map (fun x : Space × Space => (x.1, x.2 + x.1))\n = (ringMeasure.prod (volume (α := Space)))", + "line": 100 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "lemma", + "name": "Space.ringMeasure_univ", + "docstring": null, + "snippet": "@[simp]\nlemma ringMeasure_univ : ringMeasure Set.univ = ENNReal.ofReal ((2 : ℝ) * π)", + "line": 111 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "def", + "name": "Space.ringDist", + "docstring": "The distribution on `Space 3` corresponding to integration around a ring.", + "snippet": "def ringDist : (Space 3) →d[ℝ] ℝ", + "line": 126 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "lemma", + "name": "Space.ringDist_apply_eq_integral_ringMeasure", + "docstring": null, + "snippet": "lemma ringDist_apply_eq_integral_ringMeasure (f : 𝓢(Space 3, ℝ)) :\n ringDist f = ∫ x, f x ∂ringMeasure", + "line": 129 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/Ring.lean", + "kind": "lemma", + "name": "Space.ringDist_eq_integral_delta", + "docstring": null, + "snippet": "lemma ringDist_eq_integral_delta (f : 𝓢(Space 3, ℝ)) :\n ringDist f = ∫ z, diracDelta ℝ z f ∂ringMeasure", + "line": 133 + } + ] + }, + { + "filename": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidCylinder.lean", + "entries": [ + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidCylinder.lean", + "kind": "def", + "name": "Space.solidCylinder", + "docstring": "The map embedding a cross-sectional disk extruded along the axis into `Space 3`. The disk is\n cut out by the solid-cylinder measure, which is supported on the closed unit disk.", + "snippet": "def solidCylinder : Space 2 × ℝ → Space 3", + "line": 44 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidCylinder.lean", + "kind": "lemma", + "name": "Space.solidCylinder_eq", + "docstring": null, + "snippet": "lemma solidCylinder_eq :\n solidCylinder = (slice 2).symm ∘ (fun x : Space 2 × ℝ => (x.2, x.1))", + "line": 47 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidCylinder.lean", + "kind": "lemma", + "name": "Space.solidCylinder_injective", + "docstring": null, + "snippet": "lemma solidCylinder_injective : Function.Injective solidCylinder", + "line": 50 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidCylinder.lean", + "kind": "lemma", + "name": "Space.solidCylinder_continuous", + "docstring": null, + "snippet": "@[fun_prop]\nlemma solidCylinder_continuous : Continuous solidCylinder", + "line": 57 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidCylinder.lean", + "kind": "lemma", + "name": "Space.solidCylinder_measurableEmbedding", + "docstring": null, + "snippet": "lemma solidCylinder_measurableEmbedding : MeasurableEmbedding solidCylinder", + "line": 61 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidCylinder.lean", + "kind": "lemma", + "name": "Space.norm_solidCylinder", + "docstring": null, + "snippet": "@[simp]\nlemma norm_solidCylinder (x : Space 2 × ℝ) :\n ‖solidCylinder x‖ = √(‖x.2‖ ^ 2 + ‖x.1‖ ^ 2)", + "line": 65 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidCylinder.lean", + "kind": "def", + "name": "Space.solidCylinderMeasure", + "docstring": "The measure on `Space 3` corresponding to integration over a solid cylinder, i.e. the\n pushforward of the product of the solid-sphere (closed unit disk) measure on `Space 2` with the\n line measure along the axis.", + "snippet": "def solidCylinderMeasure : Measure (Space 3)", + "line": 78 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidCylinder.lean", + "kind": "instance", + "name": "Space.solidCylinderMeasure_hasTemperateGrowth", + "docstring": null, + "snippet": "instance solidCylinderMeasure_hasTemperateGrowth :\n solidCylinderMeasure.HasTemperateGrowth", + "line": 81 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidCylinder.lean", + "kind": "instance", + "name": "Space.solidCylinderMeasure_sFinite", + "docstring": null, + "snippet": "instance solidCylinderMeasure_sFinite : SFinite solidCylinderMeasure", + "line": 106 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidCylinder.lean", + "kind": "def", + "name": "Space.solidCylinderDist", + "docstring": "The distribution on `Space 3` corresponding to integration over a solid cylinder.\n One can roughly think of this distribution as taking a test function `f` to its integral against\n a mass, charge or current density spread over a solid cylinder.", + "snippet": "def solidCylinderDist : (Space 3) →d[ℝ] ℝ", + "line": 119 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidCylinder.lean", + "kind": "lemma", + "name": "Space.solidCylinderDist_apply_eq_integral_solidCylinderMeasure", + "docstring": null, + "snippet": "lemma solidCylinderDist_apply_eq_integral_solidCylinderMeasure (f : 𝓢(Space 3, ℝ)) :\n solidCylinderDist f = ∫ x, f x ∂solidCylinderMeasure", + "line": 122 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidCylinder.lean", + "kind": "lemma", + "name": "Space.solidCylinderDist_apply_eq_integral_disk_volume", + "docstring": null, + "snippet": "lemma solidCylinderDist_apply_eq_integral_disk_volume (f : 𝓢(Space 3, ℝ)) :\n solidCylinderDist f =\n ∫ x, f (solidCylinder x) ∂((solidSphereMeasure 2).prod (volume (α := ℝ)))", + "line": 126 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidCylinder.lean", + "kind": "lemma", + "name": "Space.solidCylinderMeasure_univ_pos", + "docstring": null, + "snippet": "lemma solidCylinderMeasure_univ_pos : 0 < solidCylinderMeasure Set.univ", + "line": 138 + } + ] + }, + { + "filename": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidSphere.lean", + "entries": [ + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidSphere.lean", + "kind": "def", + "name": "Space.solidSphere", + "docstring": "The inclusion into `Space d` of its closed unit ball `B^d` (the subtype coercion `x ↦ x.1`).", + "snippet": "def solidSphere (d : ℕ) : Metric.closedBall (0 : Space d) 1 → Space d", + "line": 39 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidSphere.lean", + "kind": "lemma", + "name": "Space.solidSphere_injective", + "docstring": null, + "snippet": "lemma solidSphere_injective (d : ℕ) : Function.Injective (solidSphere d)", + "line": 41 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidSphere.lean", + "kind": "lemma", + "name": "Space.solidSphere_continuous", + "docstring": null, + "snippet": "lemma solidSphere_continuous (d : ℕ) : Continuous (solidSphere d)", + "line": 46 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidSphere.lean", + "kind": "lemma", + "name": "Space.solidSphere_measurableEmbedding", + "docstring": null, + "snippet": "lemma solidSphere_measurableEmbedding (d : ℕ) : MeasurableEmbedding (solidSphere d)", + "line": 48 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidSphere.lean", + "kind": "lemma", + "name": "Space.norm_solidSphere_le", + "docstring": null, + "snippet": "@[simp]\nlemma norm_solidSphere_le (d : ℕ) (x : Metric.closedBall (0 : Space d) 1) :\n ‖solidSphere d x‖ ≤ 1", + "line": 54 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidSphere.lean", + "kind": "def", + "name": "Space.solidSphereMeasure", + "docstring": "The measure on `Space d` corresponding to integration over a solid sphere, i.e. the\n ambient volume measure restricted to the closed unit ball.", + "snippet": "def solidSphereMeasure (d : ℕ) : Measure (Space d)", + "line": 68 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidSphere.lean", + "kind": "instance", + "name": "Space.solidSphereMeasure_isFiniteMeasure", + "docstring": null, + "snippet": "instance solidSphereMeasure_isFiniteMeasure (d : ℕ) :\n IsFiniteMeasure (solidSphereMeasure d)", + "line": 71 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidSphere.lean", + "kind": "instance", + "name": "Space.solidSphereMeasure_hasTemperateGrowth", + "docstring": null, + "snippet": "instance solidSphereMeasure_hasTemperateGrowth (d : ℕ) :\n (solidSphereMeasure d).HasTemperateGrowth", + "line": 76 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidSphere.lean", + "kind": "def", + "name": "Space.solidSphereDist", + "docstring": "The distribution on `Space d` corresponding to integration over a solid sphere.\n One can roughly think of this distribution as taking a test function `f` to its integral against\n a mass, charge or current density spread over a solid ball.", + "snippet": "def solidSphereDist (d : ℕ) : (Space d) →d[ℝ] ℝ", + "line": 89 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidSphere.lean", + "kind": "lemma", + "name": "Space.solidSphereDist_apply_eq_integral_solidSphereMeasure", + "docstring": null, + "snippet": "lemma solidSphereDist_apply_eq_integral_solidSphereMeasure (d : ℕ) (f : 𝓢(Space d, ℝ)) :\n solidSphereDist d f = ∫ x, f x ∂solidSphereMeasure d", + "line": 92 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidSphere.lean", + "kind": "lemma", + "name": "Space.solidSphereDist_apply_eq_integral_closedBall", + "docstring": null, + "snippet": "lemma solidSphereDist_apply_eq_integral_closedBall (d : ℕ) (f : 𝓢(Space d, ℝ)) :\n solidSphereDist d f = ∫ x in Metric.closedBall (0 : Space d) 1, f x", + "line": 96 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidSphere.lean", + "kind": "lemma", + "name": "Space.solidSphere_volume_pos", + "docstring": null, + "snippet": "lemma solidSphere_volume_pos (d : ℕ) :\n 0 < volume (Metric.closedBall (0 : Space d) 1)", + "line": 106 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SolidSphere.lean", + "kind": "lemma", + "name": "Space.solidSphereMeasure_univ_pos", + "docstring": null, + "snippet": "lemma solidSphereMeasure_univ_pos (d : ℕ) : 0 < solidSphereMeasure d Set.univ", + "line": 112 + } + ] + }, + { + "filename": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalCylinder.lean", + "entries": [ + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalCylinder.lean", + "kind": "def", + "name": "Space.sphericalCylinder", + "docstring": "The map embedding the unit circular shell extruded along the axis into `Space 3`.", + "snippet": "def sphericalCylinder : Metric.sphere (0 : Space 2) 1 × ℝ → Space 3", + "line": 38 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalCylinder.lean", + "kind": "lemma", + "name": "Space.sphericalCylinder_eq", + "docstring": null, + "snippet": "lemma sphericalCylinder_eq :\n sphericalCylinder = (slice 2).symm ∘ (fun x => (x.2, sphericalShell 2 x.1))", + "line": 41 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalCylinder.lean", + "kind": "lemma", + "name": "Space.sphericalCylinder_injective", + "docstring": null, + "snippet": "lemma sphericalCylinder_injective : Function.Injective sphericalCylinder", + "line": 44 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalCylinder.lean", + "kind": "lemma", + "name": "Space.sphericalCylinder_continuous", + "docstring": null, + "snippet": "@[fun_prop]\nlemma sphericalCylinder_continuous : Continuous sphericalCylinder", + "line": 51 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalCylinder.lean", + "kind": "lemma", + "name": "Space.sphericalCylinder_measurableEmbedding", + "docstring": null, + "snippet": "lemma sphericalCylinder_measurableEmbedding : MeasurableEmbedding sphericalCylinder", + "line": 56 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalCylinder.lean", + "kind": "lemma", + "name": "Space.norm_sphericalCylinder", + "docstring": null, + "snippet": "@[simp]\nlemma norm_sphericalCylinder (x : Metric.sphere (0 : Space 2) 1 × ℝ) :\n ‖sphericalCylinder x‖ = √(‖x.2‖ ^ 2 + 1)", + "line": 60 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalCylinder.lean", + "kind": "def", + "name": "Space.sphericalCylinderMeasure", + "docstring": "The measure on `Space 3` corresponding to integration over a spherical cylinder.", + "snippet": "def sphericalCylinderMeasure : Measure (Space 3)", + "line": 72 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalCylinder.lean", + "kind": "instance", + "name": "Space.sphericalCylinderMeasure_hasTemperateGrowth", + "docstring": null, + "snippet": "instance sphericalCylinderMeasure_hasTemperateGrowth :\n sphericalCylinderMeasure.HasTemperateGrowth", + "line": 76 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalCylinder.lean", + "kind": "instance", + "name": "Space.sphericalCylinderMeasure_sFinite", + "docstring": null, + "snippet": "instance sphericalCylinderMeasure_sFinite : SFinite sphericalCylinderMeasure", + "line": 100 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalCylinder.lean", + "kind": "def", + "name": "Space.sphericalCylinderDist", + "docstring": "The distribution on `Space 3` corresponding to integration over a spherical cylinder.", + "snippet": "def sphericalCylinderDist : (Space 3) →d[ℝ] ℝ", + "line": 112 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalCylinder.lean", + "kind": "lemma", + "name": "Space.sphericalCylinderDist_apply_eq_integral_sphericalCylinderMeasure", + "docstring": null, + "snippet": "lemma sphericalCylinderDist_apply_eq_integral_sphericalCylinderMeasure (f : 𝓢(Space 3, ℝ)) :\n sphericalCylinderDist f = ∫ x, f x ∂sphericalCylinderMeasure", + "line": 115 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalCylinder.lean", + "kind": "lemma", + "name": "Space.sphericalCylinderDist_apply_eq_integral_sphere_volume", + "docstring": null, + "snippet": "lemma sphericalCylinderDist_apply_eq_integral_sphere_volume (f : 𝓢(Space 3, ℝ)) :\n sphericalCylinderDist f =\n ∫ x, f (sphericalCylinder x)\n ∂((MeasureTheory.Measure.toSphere volume).prod (volume (α := ℝ)))", + "line": 119 + } + ] + }, + { + "filename": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalShell.lean", + "entries": [ + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalShell.lean", + "kind": "def", + "name": "Space.sphericalShell", + "docstring": "The inclusion into `Space d` of its unit sphere `S^{d-1}` (the subtype coercion `x ↦ x.1`).", + "snippet": "def sphericalShell (d : ℕ) : Metric.sphere (0 : Space d) 1 → Space d", + "line": 35 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalShell.lean", + "kind": "lemma", + "name": "Space.sphericalShell_injective", + "docstring": null, + "snippet": "lemma sphericalShell_injective (d : ℕ) : Function.Injective (sphericalShell d)", + "line": 37 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalShell.lean", + "kind": "lemma", + "name": "Space.sphericalShell_continuous", + "docstring": null, + "snippet": "lemma sphericalShell_continuous (d : ℕ) : Continuous (sphericalShell d)", + "line": 42 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalShell.lean", + "kind": "lemma", + "name": "Space.sphericalShell_measurableEmbedding", + "docstring": null, + "snippet": "lemma sphericalShell_measurableEmbedding (d : ℕ) : MeasurableEmbedding (sphericalShell d)", + "line": 44 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalShell.lean", + "kind": "lemma", + "name": "Space.norm_sphericalShell", + "docstring": null, + "snippet": "@[simp]\nlemma norm_sphericalShell (d : ℕ) (x : Metric.sphere (0 : Space d) 1) :\n ‖sphericalShell d x‖ = 1", + "line": 50 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalShell.lean", + "kind": "def", + "name": "Space.sphericalShellMeasure", + "docstring": "The measure on `Space d` corresponding to integration around a spherical shell.", + "snippet": "def sphericalShellMeasure (d : ℕ) : Measure (Space d)", + "line": 61 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalShell.lean", + "kind": "instance", + "name": "Space.sphericalShellMeasure_hasTemperateGrowth", + "docstring": null, + "snippet": "instance sphericalShellMeasure_hasTemperateGrowth (d : ℕ) :\n (sphericalShellMeasure d).HasTemperateGrowth", + "line": 64 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalShell.lean", + "kind": "def", + "name": "Space.sphericalShellDist", + "docstring": "The distribution associated with a spherical shell.\n One can roughly think of this distribution as the distribution which\n takes test functions `f (r)` to `∫ d³r f(r) ρ(r)` where `ρ(r)` is the\n mass, charge or current etc. distribution.", + "snippet": "def sphericalShellDist (d : ℕ) : (Space d) →d[ℝ] ℝ", + "line": 81 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalShell.lean", + "kind": "lemma", + "name": "Space.sphericalShellDist_apply_eq_integral_sphericalShellMeasure", + "docstring": null, + "snippet": "lemma sphericalShellDist_apply_eq_integral_sphericalShellMeasure (d : ℕ) (f : 𝓢(Space d, ℝ)) :\n sphericalShellDist d f = ∫ x, f x ∂sphericalShellMeasure d", + "line": 85 + }, + { + "file": "PhyslibAlpha/SpaceAndTime/Space/Surfaces/SphericalShell.lean", + "kind": "lemma", + "name": "Space.sphericalShellDist_apply_eq_integral_sphere_volume", + "docstring": null, + "snippet": "lemma sphericalShellDist_apply_eq_integral_sphere_volume (d : ℕ) (f : 𝓢(Space d, ℝ)) :\n sphericalShellDist d f =\n ∫ x, f (sphericalShell d x) ∂(MeasureTheory.Measure.toSphere volume)", + "line": 89 + } + ] + }, + { + "filename": "QuantumInfo/States/Mixed/MState.lean", + "entries": [ + { + "file": "QuantumInfo/States/Mixed/MState.lean", + "kind": "theorem", + "name": "MState.traceNorm_eq_one", + "docstring": null, + "snippet": "@[simp]\ntheorem traceNorm_eq_one (ρ : MState d) : ρ.m.traceNorm = 1", + "line": 1231 + } + ] + }, + { + "filename": "scripts/MetaPrograms/module_doc_lint.lean", + "entries": [ + { + "file": "scripts/MetaPrograms/module_doc_lint.lean", + "kind": "def", + "name": "linterExemptions", + "docstring": "The array of modules exempt from all linters, read from `scripts/LinterExemption.txt`.\n This is used to lint `QuantumInfo` file-by-file.", + "snippet": "def linterExemptions : IO (Array FilePath)", + "line": 218 + }, + { + "file": "scripts/MetaPrograms/module_doc_lint.lean", + "kind": "def", + "name": "importedFilePaths", + "docstring": "The file paths of the modules imported into the module `mods` (e.g. `Physlib`).", + "snippet": "def importedFilePaths (mods : Name) : IO (Array FilePath)", + "line": 225 + } + ] + }, + { + "filename": "scripts/MetaPrograms/runPhyslibLinters.lean", + "entries": [ + { + "file": "scripts/MetaPrograms/runPhyslibLinters.lean", + "kind": "def", + "name": "linterExemptions", + "docstring": "The file paths of modules exempt from all linters, read from `scripts/LinterExemption.txt`.\n This is used to lint `QuantumInfo` file-by-file.", + "snippet": "def linterExemptions : IO (Array String)", + "line": 19 + }, + { + "file": "scripts/MetaPrograms/runPhyslibLinters.lean", + "kind": "def", + "name": "moduleToFilePathString", + "docstring": "The file path (as a string e.g. `QuantumInfo/Foo/Bar.lean`) of the module `mod`.", + "snippet": "def moduleToFilePathString (mod : Name) : String", + "line": 26 + } + ] + }, + { + "filename": "scripts/MetaPrograms/style_lint.lean", + "entries": [ + { + "file": "scripts/MetaPrograms/style_lint.lean", + "kind": "def", + "name": "importedFilePaths", + "docstring": "The file paths of the modules imported into the module `mods` (e.g. `Physlib`),\n found by reading the module's `.olean` file.", + "snippet": "def importedFilePaths (mods : Name) : IO (Array FilePath)", + "line": 123 + }, + { + "file": "scripts/MetaPrograms/style_lint.lean", + "kind": "def", + "name": "linterExemptions", + "docstring": "The file paths of modules which should be skipped by the linters, read from\n `scripts/LinterExemption.txt`. This is used to lint `QuantumInfo` file-by-file.", + "snippet": "def linterExemptions : IO (Array String)", + "line": 137 + } + ] + }, + { + "filename": "scripts/PhyslibAlpha/runPhyslibAlphaLinters.lean", + "entries": [ + { + "file": "scripts/PhyslibAlpha/runPhyslibAlphaLinters.lean", + "kind": "def", + "name": "runLinterOnModule", + "docstring": "Run the Batteries linter on a given module and update the linter if `update` is `true`.", + "snippet": "unsafe def runLinterOnModule (module : Name): IO Unit", + "line": 18 + }, + { + "file": "scripts/PhyslibAlpha/runPhyslibAlphaLinters.lean", + "kind": "def", + "name": "main", + "docstring": null, + "snippet": "unsafe def main (_ : List String) : IO Unit", + "line": 64 + } + ] + } + ], + "pdfUrl": "/monthly-updates/2026-06.pdf", + "generatedAt": "2026-07-30T10:16:03.939Z" +} diff --git a/web2/data/monthly-updates/README.md b/web2/data/monthly-updates/README.md new file mode 100644 index 0000000..8e0c3f6 --- /dev/null +++ b/web2/data/monthly-updates/README.md @@ -0,0 +1,30 @@ +# Monthly updates + +Auto-generated per-month diff summaries for +[`leanprover-community/physlib`](https://github.com/leanprover-community/physlib). + +Each `YYYY-MM.json` file in this directory is produced by +[`../../scripts/generate-monthly-updates.js`](../../scripts/generate-monthly-updates.js), +which is run: + +- Automatically on the 1st of every month by the + [`monthly-updates.yml`](../../../.github/workflows/monthly-updates.yml) + GitHub Actions workflow (targets the previous month). +- Manually via `workflow_dispatch` (with an optional `month` or `backfill` + input) on the same workflow. +- Locally with: + + ```sh + cd web2 + # last completed month + GITHUB_TOKEN=... node scripts/generate-monthly-updates.js + # a specific month + GITHUB_TOKEN=... node scripts/generate-monthly-updates.js --month 2026-06 + # backfill the last N months + GITHUB_TOKEN=... node scripts/generate-monthly-updates.js --backfill 6 + # overwrite existing files + GITHUB_TOKEN=... node scripts/generate-monthly-updates.js --month 2026-06 --force + ``` + +Each JSON file drives the `/monthly-updates` listing table and its +per-month `/monthly-updates/` docs-style detail page. diff --git a/web2/public/monthly-updates/2026-06.pdf b/web2/public/monthly-updates/2026-06.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fa5e8e060ae3b720f2e7bd4e1d2b58dddd9f493f GIT binary patch literal 934876 zcma(2Ly#`OwuOt9ZQHhObCvC?ziivKZEF=)*|u%lHt#;~-Nro;r?Gq4oG}$&j*&^O zC?-MA%)kyqesp*J2E)cl#6)CoYz4!|$0%WG<6`Q>C}CsdVk%~8Vh=E7lryz6cd;O1 z;oxEy5P)%ZaWXZsh4I+9)|s>;>xADvrTu{XD@F1Q0>{XZpi9&V)(G2h7VfK|b@cN~ zQX(c(b&==T*m#pe@2bNkck%q#0}m;S{tZ0xj-|0~PsS3XOq;s5($ z`MTQMnJYWe8PKV~$a~uTum6?+DCV*L)8*r}xEc6wXgX!ROKwr~yiRncz|-~qIWWos z>eix^~V&j%zl7;iWPlNSaRc(U>qzXwwb zSt6u#o<~&rvD#9Hk7;VY zQ}X9|zoZtBWubnfTLR{w7Z)2mJO7+vJs)9wQCo0D^wMFRTrRZ7{y0v*3hsIjFYt=t zA@}3;K|4oaq$&p^YCmYBl~?h)t1;GKhGa6y)g`0)VQ{1njFS+I^7mlRyh(=x7KM#6 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b/web2/scripts/generate-monthly-updates.js @@ -0,0 +1,1166 @@ +#!/usr/bin/env node +/* + * Generate a monthly update report for the leanprover-community/physlib repo. + * + * For a given month (default: the previous UTC month, i.e. the standard + * "run on the 1st for the last month" case) this script: + * 1. Finds the first-parent commit of the default branch at the start of the + * month and at the start of the following month. + * 2. Diffs those two commits with local git to get exact per-file line + * counts and the list of changed .lean files (with rename detection). + * 3. Determines which declarations are *new* by parsing each changed .lean + * file in full at both commits and set-differencing by fully-qualified + * name (see scripts/lean-decls.js). + * 4. Writes the result to data/monthly-updates/<YYYY-MM>.json. + * + * Why local git and not GitHub's compare API: /compare caps its `files` array + * at 300 entries and omits `patch` entirely on large files. On a busy month + * both limits hit at once, and there is no truncation flag to detect it - the + * report just silently under-reports. A local clone has neither limit. + * + * Usage: + * node scripts/generate-monthly-updates.js # last month + * node scripts/generate-monthly-updates.js --month 2026-06 # specific month + * node scripts/generate-monthly-updates.js --backfill 6 # last 6 months + * node scripts/generate-monthly-updates.js --force # overwrite existing + * node scripts/generate-monthly-updates.js --no-pdf # skip PDF compile + * node scripts/generate-monthly-updates.js --repo <path> # reuse a clone + * + * Repo: needs a clone of leanprover-community/physlib. Pass --repo, or set + * PHYSLIB_REPO, or let the script clone into .cache/physlib.git and reuse it. + * + * Auth: GITHUB_TOKEN is optional and used only to attach GitHub logins and + * avatars to contributors. Commit counts and everything else come from git, so + * the script works fully offline against an existing clone. + * + * PDF: after writing the JSON report the script also writes a scientific-paper + * style LaTeX source to .monthly-updates-build/<slug>.tex and, if either + * `tectonic` or `xelatex` is on PATH, compiles it to + * public/monthly-updates/<slug>.pdf. Only the `.pdf` is published and + * committed; the `.tex` is a gitignored intermediate. + */ + +const https = require("node:https"); +const fs = require("node:fs"); +const os = require("node:os"); +const path = require("node:path"); +const { spawnSync, execFileSync } = require("node:child_process"); +const { findAddedDeclarations } = require("./lean-decls"); + +const OWNER = "leanprover-community"; +const REPO = "physlib"; +const CLONE_URL = `https://github.com/${OWNER}/${REPO}.git`; +const OUT_DIR = path.join(__dirname, "..", "data", "monthly-updates"); +// The .pdf is the published artifact and belongs in public/. The .tex is an +// intermediate - keep it out of public/ so it isn't served or committed. +const PDF_DIR = path.join(__dirname, "..", "public", "monthly-updates"); +const TEX_DIR = path.join(__dirname, "..", ".monthly-updates-build"); +const CACHE_REPO = path.join(__dirname, "..", ".cache", "physlib.git"); +const TOKEN = process.env.GITHUB_TOKEN || process.env.GH_TOKEN || ""; + +// ─── CLI ────────────────────────────────────────────────────────────────── +const args = process.argv.slice(2); +function argValue(name) { + const i = args.indexOf(name); + return i >= 0 && i + 1 < args.length ? args[i + 1] : null; +} +const flagForce = args.includes("--force"); +const flagNoPdf = args.includes("--no-pdf"); +const monthArg = argValue("--month"); +const backfillArg = argValue("--backfill"); +const repoArg = argValue("--repo") || process.env.PHYSLIB_REPO || ""; + +// ─── HTTP helpers ───────────────────────────────────────────────────────── +function ghRequest(pathAndQuery) { + const options = { + hostname: "api.github.com", + path: pathAndQuery, + method: "GET", + headers: { + "User-Agent": "physlib-monthly-updates", + Accept: "application/vnd.github+json", + "X-GitHub-Api-Version": "2022-11-28", + }, + }; + if (TOKEN) options.headers.Authorization = `Bearer ${TOKEN}`; + + return new Promise((resolve, reject) => { + const req = https.request(options, (res) => { + const chunks = []; + res.on("data", (c) => chunks.push(c)); + res.on("end", () => { + const body = Buffer.concat(chunks).toString("utf8"); + if (res.statusCode && res.statusCode >= 400) { + reject( + new Error( + `GitHub API ${res.statusCode} on ${pathAndQuery}: ${body.slice(0, 300)}`, + ), + ); + return; + } + try { + resolve(JSON.parse(body)); + } catch (err) { + reject(new Error(`Invalid JSON from ${pathAndQuery}: ${err.message}`)); + } + }); + }); + req.on("error", reject); + req.end(); + }); +} + +// ─── Date helpers ───────────────────────────────────────────────────────── +function monthBoundsUTC(year, monthIdx /* 0-based */) { + const start = new Date(Date.UTC(year, monthIdx, 1, 0, 0, 0)); + const end = new Date(Date.UTC(year, monthIdx + 1, 1, 0, 0, 0)); + return { start, end }; +} +function prevMonthUTC(now = new Date()) { + const y = now.getUTCFullYear(); + const m = now.getUTCMonth() - 1; + const d = new Date(Date.UTC(y, m, 1)); + return { year: d.getUTCFullYear(), monthIdx: d.getUTCMonth() }; +} +function formatMonthSlug(year, monthIdx) { + return `${year}-${String(monthIdx + 1).padStart(2, "0")}`; +} +function formatMonthLabel(year, monthIdx) { + return new Date(Date.UTC(year, monthIdx, 1)).toLocaleString("en-US", { + month: "short", + year: "numeric", + timeZone: "UTC", + }); +} + +// ─── Local git ──────────────────────────────────────────────────────────── +let REPO_DIR = ""; + +function git(gitArgs, opts = {}) { + return execFileSync("git", ["-C", REPO_DIR, ...gitArgs], { + encoding: "utf8", + // Whole Lean files, and diffs over thousands of them, blow the 1MB default. + maxBuffer: 1 << 28, + ...opts, + }); +} + +/** Ensure a usable clone exists and is up to date; sets REPO_DIR. */ +function ensureRepo() { + if (repoArg) { + REPO_DIR = path.resolve(repoArg); + if (!fs.existsSync(path.join(REPO_DIR, ".git")) && !fs.existsSync(path.join(REPO_DIR, "HEAD"))) { + throw new Error(`--repo ${REPO_DIR} is not a git repository`); + } + console.log(`• using existing clone at ${REPO_DIR}`); + } else { + REPO_DIR = CACHE_REPO; + if (!fs.existsSync(REPO_DIR)) { + console.log(`• cloning ${CLONE_URL} → ${path.relative(process.cwd(), REPO_DIR)} (once)`); + fs.mkdirSync(path.dirname(REPO_DIR), { recursive: true }); + // A bare mirror is all we need - we only ever read blobs via `git show`, + // never check anything out. + execFileSync("git", ["clone", "--bare", CLONE_URL, REPO_DIR], { + stdio: "inherit", + }); + } + } + // Refresh so a month that just ended is actually present. + try { + git(["fetch", "--quiet", "origin", "+refs/heads/*:refs/remotes/origin/*", "--tags"]); + } catch { + try { + git(["fetch", "--quiet", "--all"]); + } catch { + console.warn(" ⚠ could not fetch; working from whatever the clone already has"); + } + } +} + +/** + * The upstream default branch, as a rev this clone can resolve. + * + * Picks the remote whose URL actually points at OWNER/REPO rather than + * assuming `origin`: when --repo is a *fork* of physlib, `origin` is the fork + * and upstream is some other remote, and diffing the fork's branch would + * report the fork's commits instead of the library's. + */ +function resolveBranchRef() { + const candidates = []; + try { + const remotes = git(["remote", "-v"]) + .split("\n") + .filter(Boolean) + .map((l) => l.split(/\s+/)) + .filter(([, url]) => url && new RegExp(`[/:]${OWNER}/${REPO}(\\.git)?$`, "i").test(url)) + .map(([name]) => name); + for (const r of [...new Set(remotes)]) { + candidates.push(`${r}/master`, `${r}/main`); + } + } catch { + /* fall through to the generic candidates */ + } + candidates.push("origin/master", "origin/main", "master", "main"); + + for (const candidate of candidates) { + const res = spawnSync("git", ["-C", REPO_DIR, "rev-parse", "--verify", "--quiet", candidate]); + if (res.status === 0) return candidate; + } + throw new Error( + `could not resolve ${OWNER}/${REPO}'s default branch in ${REPO_DIR}`, + ); +} + +/** + * Latest commit on the branch's first-parent chain strictly before `iso`, or + * null if none. + * + * --first-parent matters: the branch's history is not linear, so two commits + * picked purely by date are not guaranteed to be ancestors of one another, and + * diffing an unrelated pair pulls in parallel-branch content and wildly + * overstates the month. The first-parent chain is a single ancestor line, so + * any two commits on it are always comparable. Callers still assert ancestry. + */ +function commitBefore(branchRef, iso) { + const out = git([ + "rev-list", + "-1", + "--first-parent", + `--before=${iso}`, + branchRef, + ]).trim(); + return out || null; +} + +function isAncestor(base, head) { + const res = spawnSync("git", ["-C", REPO_DIR, "merge-base", "--is-ancestor", base, head]); + return res.status === 0; +} + +/** Per-file additions/deletions for the whole diff. No 300-file ceiling. */ +function numstat(base, head) { + const out = git(["diff", "--numstat", "-z", "--find-renames", base, head]); + const fields = out.split("\0"); + const rows = []; + let i = 0; + while (i < fields.length) { + const rec = fields[i]; + if (!rec) { + i += 1; + continue; + } + const m = rec.match(/^(\d+|-)\t(\d+|-)\t(.*)$/); + if (!m) { + i += 1; + continue; + } + // With -z, a rename emits an empty path in the record and then two extra + // NUL-separated fields: old path, new path. + let filename = m[3]; + if (filename === "") { + filename = fields[i + 2] ?? ""; + i += 3; + } else { + i += 1; + } + rows.push({ + filename, + // "-" means binary; count it as zero lines rather than NaN. + additions: m[1] === "-" ? 0 : parseInt(m[1], 10), + deletions: m[2] === "-" ? 0 : parseInt(m[2], 10), + }); + } + return rows; +} + +/** + * Changed .lean files as { oldPath, newPath }. `oldPath` is null only for + * genuinely new files. + * + * Rename detection is essential, not a nicety: physlib renamed its entire + * source tree (PhysLean/... -> Physlib/...) in a single April 2026 commit, and + * without --find-renames every declaration in all 400+ moved files reads as + * brand new, because the new path simply does not exist at the base commit. + */ +function changedLeanFiles(base, head) { + const out = git([ + "diff", + "--name-status", + "--find-renames", + "-z", + "--diff-filter=ACMR", + base, + head, + "--", + "*.lean", + ]); + const fields = out.split("\0"); + const pairs = []; + let i = 0; + while (i < fields.length) { + const status = fields[i]; + if (!status) { + i += 1; + continue; + } + if ((status[0] === "R" || status[0] === "C") && i + 2 < fields.length) { + pairs.push({ oldPath: fields[i + 1], newPath: fields[i + 2] }); + i += 3; + } else if (i + 1 < fields.length) { + const p = fields[i + 1]; + pairs.push({ oldPath: status[0] === "A" ? null : p, newPath: p }); + i += 2; + } else { + break; + } + } + return pairs.sort((a, b) => a.newPath.localeCompare(b.newPath)); +} + +/** Contents of `filePath` at `sha`, or null if it didn't exist there. */ +function fileAt(sha, filePath) { + const res = spawnSync("git", ["-C", REPO_DIR, "show", `${sha}:${filePath}`], { + encoding: "utf8", + maxBuffer: 1 << 28, + }); + return res.status === 0 ? res.stdout : null; +} + +/** Commits in (base, head], from git - exact, and with no pagination cap. */ +function commitsInRange(base, head) { + const SEP = "\x1f"; + const out = git([ + "log", + `${base}..${head}`, + `--format=%H${SEP}%an${SEP}%ae`, + ]); + return out + .split("\n") + .filter(Boolean) + .map((line) => { + const [sha, name, email] = line.split(SEP); + return { sha, name, email }; + }); +} + +/** + * Best-effort email → { login, avatar_url, html_url } map from the GitHub API, + * used purely to decorate contributors. Returns an empty map without a token + * or on any failure: commit counts come from git either way. + */ +async function githubIdentities(branch, sinceIso, untilIso) { + const identities = new Map(); + if (!TOKEN) return identities; + try { + for (let page = 1; page <= 100; page++) { + const q = new URLSearchParams({ + sha: branch.replace(/^origin\//, ""), + since: sinceIso, + until: untilIso, + per_page: "100", + page: String(page), + }).toString(); + const batch = await ghRequest(`/repos/${OWNER}/${REPO}/commits?${q}`); + if (!Array.isArray(batch) || batch.length === 0) break; + for (const c of batch) { + const email = c.commit?.author?.email; + if (email && c.author?.login && !identities.has(email)) { + identities.set(email, { + login: c.author.login, + avatar_url: c.author.avatar_url ?? null, + html_url: c.author.html_url ?? null, + }); + } + } + if (batch.length < 100) break; + } + } catch (err) { + console.warn(` ⚠ could not fetch GitHub identities (${err.message}); using git names`); + } + return identities; +} + +// ─── Path helpers ───────────────────────────────────────────────────────── +/** Return the first two path components of a repo path, e.g. + * "Physlib/QuantumMechanics/Foo/Bar.lean" → "Physlib/QuantumMechanics". + * Single-component paths ("README.md") return the whole filename. + */ +function topSubfolder(filename) { + const parts = filename.split("/"); + if (parts.length <= 1) return parts[0]; + if (parts.length === 2) return parts[0]; + return `${parts[0]}/${parts[1]}`; +} + +// ─── LaTeX rendering ────────────────────────────────────────────────────── +// Characters that must be escaped when appearing in LaTeX body text (not in +// verbatim/listing blocks). Also substitutes common Lean 4 Unicode glyphs for +// their LaTeX-math equivalents so they render under XeLaTeX even when the +// current font lacks the glyph. +function texEscape(str) { + if (str == null) return ""; + let out = String(str) + .replace(/\\/g, "\\textbackslash{}") + .replace(/([&%$#_{}])/g, "\\$1") + .replace(/~/g, "\\textasciitilde{}") + .replace(/\^/g, "\\textasciicircum{}") + .replace(/</g, "\\textless{}") + .replace(/>/g, "\\textgreater{}"); + for (const [ch, rep] of TEXT_UNICODE_SUBST) { + if (out.includes(ch)) out = out.split(ch).join(rep); + } + return out; +} + +/** Escape a URL for use inside \href{...}{}. Only { } % \ need escaping. */ +function texEscapeUrl(url) { + return String(url) + .replace(/\\/g, "\\\\") + .replace(/([{}%])/g, "\\$1"); +} + +/** Escape a string for inclusion inside a fancyvrb `Verbatim` block via `\Verb`. + * We use `|...|` as the delimiter, so bar characters are the only concern. + * For block Verbatim we don't need this — content is truly verbatim. */ +function texEscapeInlineVerb(str) { + return String(str).replace(/\|/g, " "); +} + +const LEAN_KEYWORDS = [ + "theorem", "lemma", "def", "abbrev", "instance", "structure", "class", + "inductive", "axiom", "example", "opaque", "noncomputable", "private", + "protected", "unsafe", "partial", "scoped", "local", "by", "fun", "let", + "in", "if", "then", "else", "do", "match", "with", "where", "open", + "import", "namespace", "section", "end", "variable", "variables", "universe", + "universes", "attribute", "notation", "infix", "infixl", "infixr", "prefix", + "postfix", "informal_definition", "informal_lemma", "semiformal_result", + "TODO", "return", "have", "show", "from", "calc", "at", +]; + +// Unicode → LaTeX literate replacements for the `listings` package. +// +// Strategy: we set the monospace font to DejaVu Sans Mono (loaded by filename +// in the preamble), which has native glyphs for Greek letters (α–ω, Α–Ω), +// subscripts (₀–₉, ₐ, ₑ, ᵢ, ᵣ, ᵤ, ᵥ, ₒ, ₓ), superscripts (⁰–⁹, ⁺, ⁻), most +// math operators (∈, ∀, ∃, →, ↔, ≤, ≥, etc.), blackboard bold (ℝ, ℕ, ℤ, ℚ, +// ℂ), angle brackets, and common relations. XeLaTeX renders these directly, so +// we DO NOT map them through math mode — doing so injects zero-width math +// boxes whose subscript baseline shifts the glyph ordering in the PDF text +// stream (which produced things like `x₀` copying as `₀x` and `S.ω` as `Sω.`). +// +// Only characters DejaVu Sans Mono doesn't ship or that `listings` treats +// specially need to appear here. Each entry is [char, latex, columns]. +const LEAN_LITERATE = [ + // Script / calligraphic letters — DejaVu doesn't cover most of these. + ["𝓩", "{$\\mathcal{Z}$}", 1], ["𝒵", "{$\\mathcal{Z}$}", 1], + ["𝒜", "{$\\mathcal{A}$}", 1], ["ℬ", "{$\\mathcal{B}$}", 1], + ["𝒞", "{$\\mathcal{C}$}", 1], ["𝒟", "{$\\mathcal{D}$}", 1], + ["𝓔", "{$\\mathcal{E}$}", 1], ["ℱ", "{$\\mathcal{F}$}", 1], + ["𝒢", "{$\\mathcal{G}$}", 1], ["ℋ", "{$\\mathcal{H}$}", 1], + ["ℐ", "{$\\mathcal{I}$}", 1], ["𝒦", "{$\\mathcal{K}$}", 1], + ["ℒ", "{$\\mathcal{L}$}", 1], ["ℳ", "{$\\mathcal{M}$}", 1], + ["𝒩", "{$\\mathcal{N}$}", 1], ["𝒪", "{$\\mathcal{O}$}", 1], + ["𝒫", "{$\\mathcal{P}$}", 1], ["𝒬", "{$\\mathcal{Q}$}", 1], + ["ℛ", "{$\\mathcal{R}$}", 1], ["𝒮", "{$\\mathcal{S}$}", 1], + ["𝒯", "{$\\mathcal{T}$}", 1], ["𝒰", "{$\\mathcal{U}$}", 1], + ["𝒱", "{$\\mathcal{V}$}", 1], ["𝒲", "{$\\mathcal{W}$}", 1], + ["𝒳", "{$\\mathcal{X}$}", 1], ["𝒴", "{$\\mathcal{Y}$}", 1], + // Blackboard-bold letters DejaVu doesn't include. + ["𝔽", "{$\\mathbb{F}$}", 1], + // Rare glyphs sometimes seen in mathlib. + ["⦃", "{$\\{\\!|$}", 1], ["⦄", "{$|\\!\\}$}", 1], + ["◃", "{$\\triangleleft$}", 1], ["▹", "{$\\triangleright$}", 1], + ["★", "{$\\star$}", 1], ["♯", "{$\\sharp$}", 1], ["♭", "{$\\flat$}", 1], +]; + +// Text-mode Unicode → LaTeX substitution table. Applied by `texEscape` on any +// content that is rendered in normal LaTeX text mode (docstrings, section +// headers, table cells, etc.). Not applied inside listings, which use their +// own `literate` mapping. +const TEXT_UNICODE_SUBST = [ + ["α", "\\ensuremath{\\alpha}"], ["β", "\\ensuremath{\\beta}"], + ["γ", "\\ensuremath{\\gamma}"], ["δ", "\\ensuremath{\\delta}"], + ["ε", "\\ensuremath{\\varepsilon}"], ["ζ", "\\ensuremath{\\zeta}"], + ["η", "\\ensuremath{\\eta}"], ["θ", "\\ensuremath{\\theta}"], + ["ι", "\\ensuremath{\\iota}"], ["κ", "\\ensuremath{\\kappa}"], + ["λ", "\\ensuremath{\\lambda}"], ["μ", "\\ensuremath{\\mu}"], + ["ν", "\\ensuremath{\\nu}"], ["ξ", "\\ensuremath{\\xi}"], + ["π", "\\ensuremath{\\pi}"], ["ρ", "\\ensuremath{\\rho}"], + ["σ", "\\ensuremath{\\sigma}"], ["τ", "\\ensuremath{\\tau}"], + ["υ", "\\ensuremath{\\upsilon}"], ["φ", "\\ensuremath{\\varphi}"], + ["χ", "\\ensuremath{\\chi}"], ["ψ", "\\ensuremath{\\psi}"], + ["ω", "\\ensuremath{\\omega}"], + ["Γ", "\\ensuremath{\\Gamma}"], ["Δ", "\\ensuremath{\\Delta}"], + ["Θ", "\\ensuremath{\\Theta}"], ["Λ", "\\ensuremath{\\Lambda}"], + ["Ξ", "\\ensuremath{\\Xi}"], ["Π", "\\ensuremath{\\Pi}"], + ["Σ", "\\ensuremath{\\Sigma}"], ["Φ", "\\ensuremath{\\Phi}"], + ["Ψ", "\\ensuremath{\\Psi}"], ["Ω", "\\ensuremath{\\Omega}"], + ["→", "\\ensuremath{\\to}"], ["←", "\\ensuremath{\\gets}"], + ["↔", "\\ensuremath{\\leftrightarrow}"], ["↦", "\\ensuremath{\\mapsto}"], + ["⇒", "\\ensuremath{\\Rightarrow}"], ["⇐", "\\ensuremath{\\Leftarrow}"], + ["⇔", "\\ensuremath{\\Leftrightarrow}"], + ["∀", "\\ensuremath{\\forall}"], ["∃", "\\ensuremath{\\exists}"], + ["∈", "\\ensuremath{\\in}"], ["∉", "\\ensuremath{\\notin}"], + ["⊆", "\\ensuremath{\\subseteq}"], ["⊂", "\\ensuremath{\\subset}"], + ["⊇", "\\ensuremath{\\supseteq}"], ["⊃", "\\ensuremath{\\supset}"], + ["∪", "\\ensuremath{\\cup}"], ["∩", "\\ensuremath{\\cap}"], + ["∅", "\\ensuremath{\\emptyset}"], + ["≤", "\\ensuremath{\\leq}"], ["≥", "\\ensuremath{\\geq}"], + ["≠", "\\ensuremath{\\neq}"], ["≡", "\\ensuremath{\\equiv}"], + ["≃", "\\ensuremath{\\simeq}"], ["≅", "\\ensuremath{\\cong}"], + ["≈", "\\ensuremath{\\approx}"], + ["±", "\\ensuremath{\\pm}"], ["×", "\\ensuremath{\\times}"], + ["·", "\\ensuremath{\\cdot}"], ["∘", "\\ensuremath{\\circ}"], + ["⊕", "\\ensuremath{\\oplus}"], ["⊗", "\\ensuremath{\\otimes}"], + ["⊤", "\\ensuremath{\\top}"], ["⊥", "\\ensuremath{\\bot}"], + ["∇", "\\ensuremath{\\nabla}"], ["∂", "\\ensuremath{\\partial}"], + ["∫", "\\ensuremath{\\int}"], ["∑", "\\ensuremath{\\sum}"], + ["∏", "\\ensuremath{\\prod}"], ["∞", "\\ensuremath{\\infty}"], + ["ℝ", "\\ensuremath{\\mathbb{R}}"], ["ℕ", "\\ensuremath{\\mathbb{N}}"], + ["ℤ", "\\ensuremath{\\mathbb{Z}}"], ["ℚ", "\\ensuremath{\\mathbb{Q}}"], + ["ℂ", "\\ensuremath{\\mathbb{C}}"], ["𝔽", "\\ensuremath{\\mathbb{F}}"], + ["ℓ", "\\ensuremath{\\ell}"], ["ℏ", "\\ensuremath{\\hbar}"], + ["⟨", "\\ensuremath{\\langle}"], ["⟩", "\\ensuremath{\\rangle}"], + ["…", "\\ensuremath{\\ldots}"], + ["₀", "\\ensuremath{_{0}}"], ["₁", "\\ensuremath{_{1}}"], ["₂", "\\ensuremath{_{2}}"], + ["₃", "\\ensuremath{_{3}}"], ["₄", "\\ensuremath{_{4}}"], ["₅", "\\ensuremath{_{5}}"], + ["₆", "\\ensuremath{_{6}}"], ["₇", "\\ensuremath{_{7}}"], ["₈", "\\ensuremath{_{8}}"], + ["₉", "\\ensuremath{_{9}}"], ["ₙ", "\\ensuremath{_{n}}"], ["ₘ", "\\ensuremath{_{m}}"], + ["ₐ", "\\ensuremath{_{a}}"], ["ᵢ", "\\ensuremath{_{i}}"], ["ⱼ", "\\ensuremath{_{j}}"], + ["ₖ", "\\ensuremath{_{k}}"], ["ₜ", "\\ensuremath{_{t}}"], ["ₓ", "\\ensuremath{_{x}}"], + ["⁰", "\\ensuremath{^{0}}"], ["¹", "\\ensuremath{^{1}}"], ["²", "\\ensuremath{^{2}}"], + ["³", "\\ensuremath{^{3}}"], ["⁴", "\\ensuremath{^{4}}"], ["⁵", "\\ensuremath{^{5}}"], + ["⁶", "\\ensuremath{^{6}}"], ["⁷", "\\ensuremath{^{7}}"], ["⁸", "\\ensuremath{^{8}}"], + ["⁹", "\\ensuremath{^{9}}"], ["⁻", "\\ensuremath{^{-}}"], ["⁺", "\\ensuremath{^{+}}"], + ["¬", "\\ensuremath{\\neg}"], ["∧", "\\ensuremath{\\wedge}"], ["∨", "\\ensuremath{\\vee}"], +]; + +// Glyphs that XeLaTeX reported as missing from DejaVu Sans Mono when compiling +// real months. XeLaTeX has no font-fallback mechanism (that is a LuaTeX +// feature), so anything not mapped here silently typesets as nothing at all. +// +// Each entry is [char, math-mode body]. Both tables below are extended from +// this one list: `ℓ` was previously present in the text table but absent from +// the listings table, so it rendered in docstrings and vanished in code. One +// list, applied to both, is what stops that drift recurring. +const MISSING_GLYPH_MATH = [ + // Modifier letters used as super/subscripts. + ["ʲ", "^{j}"], ["ʳ", "^{r}"], ["ᴴ", "^{H}"], ["ᵀ", "^{T}"], + ["ᵃ", "^{a}"], ["ᵈ", "^{d}"], ["ᵖ", "^{p}"], ["ᵘ", "^{u}"], + ["ᵛ", "^{v}"], ["ᵟ", "^{\\delta}"], ["ᶜ", "^{c}"], ["ᶠ", "^{f}"], + ["ⁱ", "^{i}"], ["ᵣ", "_{r}"], ["ᵥ", "_{v}"], ["ₗ", "_{l}"], + ["ₛ", "_{s}"], + // Orthogonal-complement postfix, as used in mathlib. + ["ᗮ", "^{\\perp}"], + // Letterlike and blackboard forms. + ["ℓ", "\\ell"], ["𝑅", "R"], ["𝕜", "\\Bbbk"], ["𝟙", "\\mathbb{1}"], + // Bold-script capitals (U+1D4D0 block) - distinct codepoints from the + // plain script capitals already in LEAN_LITERATE. + ["𝓘", "\\mathcal{I}"], ["𝓞", "\\mathcal{O}"], ["𝓡", "\\mathcal{R}"], + ["𝓢", "\\mathcal{S}"], + // Operators and relations. + ["∣", "\\mid"], ["⊓", "\\sqcap"], ["⋊", "\\rtimes"], + ["⟪", "\\langle\\!\\langle"], ["⟫", "\\rangle\\!\\rangle"], + ["⟶", "\\longrightarrow"], ["⟹", "\\Longrightarrow"], + ["⨂", "\\bigotimes"], ["⨅", "\\bigsqcap"], ["⨯", "\\times"], + ["⬝", "\\cdot"], +]; + +for (const [ch, body] of MISSING_GLYPH_MATH) { + if (!LEAN_LITERATE.some(([c]) => c === ch)) { + LEAN_LITERATE.push([ch, `{$${body}$}`, 1]); + } + if (!TEXT_UNICODE_SUBST.some(([c]) => c === ch)) { + TEXT_UNICODE_SUBST.push([ch, `\\ensuremath{${body}}`]); + } +} + +function literateDirective() { + return LEAN_LITERATE + .map(([ch, rep, cols]) => `{${ch}}{${rep}}${cols}`) + .join(" "); +} + +function humanKind(kind) { + return kind.replace(/_/g, " "); +} + +function humanList(arr) { + if (arr.length === 0) return ""; + if (arr.length === 1) return arr[0]; + if (arr.length === 2) return `${arr[0]} and ${arr[1]}`; + return `${arr.slice(0, -1).join(", ")}, and ${arr[arr.length - 1]}`; +} + +function fmtN(n) { + return new Intl.NumberFormat("en-US").format(n); +} + +function renderLatex(report) { + const authors = report.contributors + .map((c) => texEscape(c.login || c.name || "unknown")) + .join(" \\and "); + + const totalDecls = Object.values(report.kindTotals).reduce( + (a, b) => a + b, + 0, + ); + + // Introduction paragraph + const kindPhrase = humanList( + Object.entries(report.kindTotals) + .sort((a, b) => b[1] - a[1]) + .map(([k, n]) => `${fmtN(n)} ${humanKind(k)}${n === 1 ? "" : "s"}`), + ); + const subfolderPhrase = report.subfolders.length > 0 + ? ` spanning ${report.subfolders.length} top-level ${report.subfolders.length === 1 ? "area" : "areas"} of the repository` + : ""; + const intro = + `During ${texEscape(report.label)}, ${fmtN(report.contributors.length)} ` + + `${report.contributors.length === 1 ? "contributor" : "contributors"} ` + + `submitted ${fmtN(report.totalCommits)} ` + + `${report.totalCommits === 1 ? "commit" : "commits"} that changed ` + + `${fmtN(report.linesChanged)} lines ` + + `(\\textcolor{addcol}{+${fmtN(report.totalAdditions)}} / ` + + `\\textcolor{delcol}{-${fmtN(report.totalDeletions)}}) ` + + `across ${fmtN(report.totalFilesChanged)} ` + + `${report.totalFilesChanged === 1 ? "file" : "files"}${subfolderPhrase}. ` + + (totalDecls > 0 + ? `This report catalogues ${fmtN(totalDecls)} newly added Lean declarations: ${kindPhrase}.` + : "No new Lean declarations were detected in this month's diff."); + + // Sections + const contributorRows = report.contributors + .map((c) => { + const label = c.login + ? `\\href{${texEscapeUrl(c.html_url)}}{\\texttt{${texEscape(c.login)}}}` + : `\\texttt{${texEscape(c.name || "unknown")}}`; + return `${label} & ${c.commits} \\\\`; + }) + .join("\n"); + + const subfolderRows = report.subfolders + .map( + (s) => + `\\texttt{${texEscape(s.path)}} & ${s.files} & \\textcolor{addcol}{+${fmtN(s.additions)}} & \\textcolor{delcol}{-${fmtN(s.deletions)}} \\\\`, + ) + .join("\n"); + + const kindRows = Object.entries(report.kindTotals) + .sort((a, b) => b[1] - a[1]) + .map(([k, n]) => `\\texttt{${texEscape(k)}} & ${fmtN(n)} \\\\`) + .join("\n"); + + // Per-file body sections + const fileSections = report.files + .map((f) => renderFileSection(f, report)) + .join("\n\n"); + + const noDeclsNote = report.files.length === 0 + ? "\\noindent\\emph{No new definitions, theorems, or lemmas were detected in the diff for this month. See the full compare link above for other changes.}\n" + : ""; + + return String.raw`% !TEX program = xelatex +% Auto-generated by web2/scripts/generate-monthly-updates.js +\documentclass[11pt,a4paper]{article} + +\usepackage[margin=1in]{geometry} +\usepackage{fontspec} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{xcolor} +\usepackage{hyperref} +\usepackage{listings} +\usepackage{parskip} +\usepackage{booktabs} +\usepackage{longtable} +\usepackage{titling} +\usepackage{fancyhdr} +\usepackage{tocloft} +\usepackage{microtype} + +% Use DejaVu Sans Mono for the monospace font -- it ships with TeX Live's +% dejavu package (which tectonic bundles) and has broad Unicode coverage for +% the Greek, subscripts, and math glyphs common in Lean 4 source. Loading by +% filename works both under system xelatex and inside tectonic's font bundle. +\setmonofont{DejaVuSansMono.ttf}[ + BoldFont = DejaVuSansMono-Bold.ttf, + ItalicFont = DejaVuSansMono-Oblique.ttf, + BoldItalicFont = DejaVuSansMono-BoldOblique.ttf, + Scale = MatchLowercase, +] + +\definecolor{addcol}{HTML}{047857} +\definecolor{delcol}{HTML}{be123c} +\definecolor{leankw}{HTML}{1e3a8a} +\definecolor{leancomment}{HTML}{047857} +\definecolor{leanstring}{HTML}{9a3412} +\definecolor{docbg}{HTML}{f8f9fc} +\definecolor{docrule}{HTML}{dbe1ec} + +\hypersetup{ + colorlinks=true, + linkcolor=blue!50!black, + urlcolor=blue!50!black, + citecolor=blue!50!black, + pdftitle={Physlib Monthly Report -- ${texEscape(report.label)}}, + pdfauthor={${report.contributors.map((c) => texEscape(c.login || c.name || "unknown")).join(", ")}}, + pdfsubject={Monthly diff summary of leanprover-community/physlib}, +} + +\lstdefinelanguage{Lean}{ + keywords={${LEAN_KEYWORDS.join(",")}}, + keywordstyle=\color{leankw}\bfseries, + sensitive=true, + morecomment=[l]{--}, + morecomment=[s]{/-}{-/}, + commentstyle=\color{leancomment}\itshape, + morestring=[b]", + stringstyle=\color{leanstring}, +} +\lstset{ + basicstyle=\ttfamily\small, + language=Lean, + breaklines=true, + breakatwhitespace=false, + showstringspaces=false, + frame=single, + rulecolor=\color{docrule}, + framerule=0.4pt, + numbers=none, + aboveskip=6pt, + belowskip=6pt, + extendedchars=true, + columns=fullflexible, + keepspaces=true, + inputencoding=utf8, + literate=${literateDirective()} +} + +\pagestyle{fancy} +\fancyhf{} +\fancyhead[L]{\small Physlib Monthly Report} +\fancyhead[R]{\small ${texEscape(report.label)}} +\fancyfoot[C]{\thepage} +\renewcommand{\headrulewidth}{0.4pt} + +\title{\vspace{-1cm}\textbf{Physlib Monthly Report} \\[0.3em] \Large ${texEscape(report.label)}} +\author{${authors || "\\emph{No authors recorded}"}} +\date{Generated ${texEscape(new Date(report.generatedAt).toUTCString())}} + +\begin{document} +\maketitle + +\begin{abstract} +\noindent Auto-generated summary of changes to +\href{https://github.com/${texEscapeUrl(report.repo)}}{\texttt{${texEscape(report.repo)}}} +on branch \texttt{${texEscape(report.branch)}} between +\texttt{${texEscape(report.baseSha.slice(0, 7))}} and +\texttt{${texEscape(report.headSha.slice(0, 7))}}. +The full diff is available at +\href{${texEscapeUrl(report.compareUrl)}}{GitHub compare}. +\end{abstract} + +\tableofcontents +\newpage + +\section{Introduction} +${intro} + +\vspace{0.5em} +\noindent See the +\href{${texEscapeUrl(report.compareUrl)}}{full compare diff} +and the +\href{${texEscapeUrl(report.commitsUrl)}}{list of commits} +on GitHub for the raw source. + +\subsection{Contributors} +${ + report.contributors.length === 0 + ? "\\emph{No contributors identified for this month.}" + : `\\begin{longtable}{l r}\n\\toprule\nContributor & Commits \\\\\n\\midrule\n${contributorRows}\n\\bottomrule\n\\end{longtable}` +} + +\subsection{Modified subfolders} +${ + report.subfolders.length === 0 + ? "\\emph{No modified files were detected.}" + : `\\begin{longtable}{l r r r}\n\\toprule\nSubfolder & Files & Additions & Deletions \\\\\n\\midrule\n${subfolderRows}\n\\bottomrule\n\\end{longtable}` +} + +\subsection{New declarations by kind} +${ + Object.keys(report.kindTotals).length === 0 + ? "\\emph{No new declarations were detected.}" + : `\\begin{longtable}{l r}\n\\toprule\nKind & Count \\\\\n\\midrule\n${kindRows}\n\\bottomrule\n\\end{longtable}` +} + +\section{Details by file} +${noDeclsNote}${fileSections} + +\end{document} +`; +} + +function renderFileSection(fileEntry, report) { + const module = fileEntry.filename.replace(/\.lean$/, "").replace(/\//g, "."); + const fileUrl = `https://github.com/${report.repo}/blob/${report.branch}/${fileEntry.filename}`; + const declBlocks = fileEntry.entries + .map((d) => renderDeclaration(d, report)) + .join("\n\n"); + // \subsection is used so the TOC shows every file. Names in monospace, escaped. + return `\\subsection{\\texorpdfstring{\\texttt{${texEscape(module)}}}{${texEscape(module)}}}\n` + + `\\noindent\\small \\href{${texEscapeUrl(fileUrl)}}{\\texttt{${texEscape(fileEntry.filename)}}} on GitHub.\\normalsize\n\n${declBlocks}`; +} + +function renderDeclaration(decl, report) { + const url = + `https://github.com/${report.repo}/blob/${report.branch}/${decl.file}` + + (decl.line ? `#L${decl.line}` : ""); + const header = `\\paragraph{\\texttt{${texEscape(decl.kind)}} \\texttt{${texEscape(decl.name)}}}` + + ` \\hfill \\href{${texEscapeUrl(url)}}{\\footnotesize [source]}`; + const doc = decl.docstring + ? `\n\\begin{quote}\\itshape\n${texEscape(decl.docstring)}\n\\end{quote}\n` + : "\n"; + const snippet = decl.snippet || ""; + // Emit code snippet as a Lean listing. Content is verbatim; no escaping. + const listing = `\\begin{lstlisting}[language=Lean]\n${snippet}\n\\end{lstlisting}`; + return `${header}\n${doc}${listing}`; +} + +// ─── PDF compilation ────────────────────────────────────────────────────── +function which(cmd) { + const res = spawnSync(process.platform === "win32" ? "where" : "which", [cmd], { + encoding: "utf8", + }); + if (res.status === 0) return res.stdout.split("\n")[0].trim(); + return null; +} + +/** + * Compile `texPath` → `pdfPath`. + * + * Returns { ok, missingGlyphs, overfull } on success, or { ok: false, + * noEngine: true } when neither engine is installed. Throws on genuine + * compilation errors so the caller can warn without aborting the whole run. + * + * The diagnostics matter: a glyph the font lacks does not fail the build, it + * just typesets as nothing, so the only way to notice is to read the log. + */ +function compileLatex(texPath, pdfPath) { + const tectonic = which("tectonic"); + const xelatex = which("xelatex"); + if (!tectonic && !xelatex) return { ok: false, noEngine: true }; + + const tmpDir = fs.mkdtempSync(path.join(os.tmpdir(), "physlib-monthly-")); + const base = path.basename(texPath, ".tex"); + try { + // Capture rather than inherit: a 200-page document produces thousands of + // lines of noise. On failure we print the tail, which is where the error is. + const run = (cmd, cmdArgs) => { + const res = spawnSync(cmd, cmdArgs, { encoding: "utf8", maxBuffer: 1 << 28 }); + if (res.status !== 0) { + const tail = String(res.stdout ?? "").split("\n").slice(-25).join("\n"); + throw new Error( + `${path.basename(cmd)} exited with status ${res.status}\n${tail}`, + ); + } + }; + + if (tectonic) { + // tectonic reruns internally as needed to settle the TOC. + run(tectonic, ["--outdir", tmpDir, "--keep-logs", "--chatter", "minimal", texPath]); + } else { + // xelatex needs two runs for the TOC. + for (let i = 0; i < 2; i++) { + run(xelatex, [ + "-interaction=nonstopmode", + "-halt-on-error", + "-output-directory", + tmpDir, + texPath, + ]); + } + } + + const producedPdf = path.join(tmpDir, `${base}.pdf`); + if (!fs.existsSync(producedPdf)) { + throw new Error(`compiler produced no PDF at ${producedPdf}`); + } + + // Scrape the log before the temp dir goes away. + let missingGlyphs = []; + let overfull = 0; + try { + const log = fs.readFileSync(path.join(tmpDir, `${base}.log`), "utf8"); + missingGlyphs = [ + ...new Set( + (log.match(/There is no \S+ \(U\+[0-9A-F]+\)/g) ?? []).map((m) => + m.replace(/^There is no /, ""), + ), + ), + ]; + overfull = (log.match(/Overfull \\hbox/g) ?? []).length; + } catch { + /* log is advisory only */ + } + + fs.copyFileSync(producedPdf, pdfPath); + return { ok: true, missingGlyphs, overfull }; + } finally { + try { + fs.rmSync(tmpDir, { recursive: true, force: true }); + } catch { /* best effort */ } + } +} + +// ─── Main per-month logic ───────────────────────────────────────────────── +async function generateMonth(year, monthIdx, defaultBranch, opts = {}) { + const slug = formatMonthSlug(year, monthIdx); + const label = formatMonthLabel(year, monthIdx); + const outPath = path.join(OUT_DIR, `${slug}.json`); + if (fs.existsSync(outPath) && !opts.force) { + console.log(`• ${slug}: exists, skipping (use --force to regenerate)`); + return { skipped: true, slug }; + } + + const { start, end } = monthBoundsUTC(year, monthIdx); + const now = new Date(); + if (end > now) { + console.log(`• ${slug}: month is not yet complete, skipping`); + return { skipped: true, slug }; + } + const startIso = start.toISOString(); + const endIso = end.toISOString(); + + console.log(`• ${slug}: locating boundary commits…`); + const baseSha = commitBefore(defaultBranch, startIso); + const headSha = commitBefore(defaultBranch, endIso); + if (!baseSha || !headSha) { + console.log(`• ${slug}: no commits found on ${defaultBranch}, skipping`); + return { skipped: true, slug }; + } + if (baseSha === headSha) { + console.log(`• ${slug}: no changes on ${defaultBranch}, writing empty report`); + } + // Both boundaries come off the first-parent chain, so this should always + // hold. Assert rather than assume: if it ever fails, the diff below would + // silently include unrelated parallel-branch work. + if (baseSha !== headSha && !isAncestor(baseSha, headSha)) { + throw new Error( + `boundary commits are not comparable: ${baseSha.slice(0, 7)} is not an ancestor of ${headSha.slice(0, 7)}`, + ); + } + + console.log(`• ${slug}: diffing ${baseSha.slice(0, 7)}…${headSha.slice(0, 7)}`); + const stats = numstat(baseSha, headSha); + let totalAdditions = 0; + let totalDeletions = 0; + const subfolderStats = new Map(); // subfolder -> { files: Set, additions, deletions } + for (const f of stats) { + totalAdditions += f.additions; + totalDeletions += f.deletions; + const sub = topSubfolder(f.filename); + if (!subfolderStats.has(sub)) { + subfolderStats.set(sub, { files: new Set(), additions: 0, deletions: 0 }); + } + const s = subfolderStats.get(sub); + s.files.add(f.filename); + s.additions += f.additions; + s.deletions += f.deletions; + } + + const subfolders = Array.from(subfolderStats.entries()) + .map(([p, s]) => ({ + path: p, + files: s.files.size, + additions: s.additions, + deletions: s.deletions, + })) + .sort((a, b) => b.additions + b.deletions - (a.additions + a.deletions)); + + // New declarations: parse each changed .lean file in full at both commits + // and set-difference by fully-qualified name. Never from diff hunks - a hunk + // is an arbitrary window, so a *modified* declaration is indistinguishable + // from a new one. See scripts/lean-decls.js. + const leanFiles = changedLeanFiles(baseSha, headSha); + console.log(`• ${slug}: parsing ${leanFiles.length} changed .lean files…`); + const filesReport = []; + for (const { oldPath, newPath } of leanFiles) { + const newContents = fileAt(headSha, newPath); + if (newContents === null) continue; // deleted again before the month ended + const oldContents = oldPath ? fileAt(baseSha, oldPath) : null; + const added = findAddedDeclarations(oldContents, newContents); + if (added.length === 0) continue; + filesReport.push({ + filename: newPath, + entries: added.map((d) => ({ + file: newPath, + kind: d.kind, + name: d.name, + docstring: d.doc, + snippet: d.signature, + line: d.line, + })), + }); + } + filesReport.sort((a, b) => a.filename.localeCompare(b.filename)); + + // Contributors: commit counts from git (exact, no pagination cap, and no + // double-counting of a commit that lands exactly on a month boundary the way + // the API's inclusive since/until does). GitHub is consulted only to attach + // logins and avatars, and is entirely optional. + const commits = commitsInRange(baseSha, headSha); + const identities = await githubIdentities(defaultBranch, startIso, endIso); + const contributors = new Map(); + for (const c of commits) { + const id = identities.get(c.email); + const key = id ? `login:${id.login}` : `name:${c.name}`; + if (!contributors.has(key)) { + contributors.set(key, { + login: id ? id.login : null, + name: id ? undefined : c.name, + html_url: id ? id.html_url : null, + avatar_url: id ? id.avatar_url : null, + commits: 0, + }); + } + contributors.get(key).commits += 1; + } + const contributorList = Array.from(contributors.values()).sort( + (a, b) => + b.commits - a.commits || + (a.login ?? a.name ?? "").localeCompare(b.login ?? b.name ?? ""), + ); + + const kindTotals = {}; + for (const { entries } of filesReport) { + for (const e of entries) kindTotals[e.kind] = (kindTotals[e.kind] || 0) + 1; + } + + const report = { + slug, + label, + year, + month: monthIdx + 1, + startDate: startIso, + endDate: endIso, + repo: `${OWNER}/${REPO}`, + branch: defaultBranch, + baseSha, + headSha, + compareUrl: `https://github.com/${OWNER}/${REPO}/compare/${baseSha}...${headSha}`, + commitsUrl: `https://github.com/${OWNER}/${REPO}/commits/${defaultBranch}?since=${startIso}&until=${endIso}`, + totalCommits: commits.length, + totalAdditions, + totalDeletions, + linesChanged: totalAdditions + totalDeletions, + totalFilesChanged: stats.length, + subfolders, + contributors: contributorList, + kindTotals, + files: filesReport, + pdfUrl: `/monthly-updates/${slug}.pdf`, + generatedAt: new Date().toISOString(), + }; + + fs.mkdirSync(OUT_DIR, { recursive: true }); + fs.writeFileSync(outPath, JSON.stringify(report, null, 2) + "\n", "utf8"); + const totalDecls = Object.values(kindTotals).reduce((a, b) => a + b, 0); + console.log( + `✓ ${slug}: ${filesReport.length} files w/ decls (${stats.length} touched), ${totalDecls} declarations, ${contributorList.length} contributors → ${path.relative(process.cwd(), outPath)}`, + ); + + // ─── LaTeX + PDF ──────────────────────────────────────────────────── + if (!opts.noPdf) { + try { + fs.mkdirSync(PDF_DIR, { recursive: true }); + fs.mkdirSync(TEX_DIR, { recursive: true }); + const texPath = path.join(TEX_DIR, `${slug}.tex`); + const tex = renderLatex(report); + fs.writeFileSync(texPath, tex, "utf8"); + console.log( + ` … wrote ${path.relative(process.cwd(), texPath)}`, + ); + const pdfPath = path.join(PDF_DIR, `${slug}.pdf`); + const compiled = compileLatex(texPath, pdfPath); + if (compiled.ok) { + console.log(` … compiled ${path.relative(process.cwd(), pdfPath)}`); + if (compiled.overfull) { + console.log(` … ${compiled.overfull} overfull line(s)`); + } + // Not fatal - the PDF is fine apart from these - but it must be + // visible, because the failure mode is a glyph silently rendering as + // nothing. Add any listed codepoint to MISSING_GLYPH_MATH. + if (compiled.missingGlyphs.length) { + console.warn( + ` ⚠ ${compiled.missingGlyphs.length} glyph(s) missing from the font and rendered as blank: ${compiled.missingGlyphs.join(", ")}`, + ); + } + } else if (compiled.noEngine) { + console.warn( + ` ⚠ neither tectonic nor xelatex was found on PATH — skipping PDF compile for ${slug}. Install tectonic (https://tectonic-typesetting.github.io) to enable local PDF builds.`, + ); + } + } catch (err) { + console.warn(` ⚠ PDF build failed for ${slug}: ${err.message}`); + } + } + + return { skipped: false, slug }; +} + +// ─── Entry point ────────────────────────────────────────────────────────── +(async function main() { + if (!TOKEN) { + console.warn( + "⚠️ GITHUB_TOKEN is not set. Contributors will be listed by git author name without GitHub logins or avatars; everything else is unaffected.", + ); + } + + try { + ensureRepo(); + const defaultBranch = resolveBranchRef(); + console.log(`• default branch: ${defaultBranch}`); + + const targets = []; + if (monthArg) { + const m = monthArg.match(/^(\d{4})-(\d{2})$/); + if (!m) { + console.error("--month must be YYYY-MM"); + process.exit(1); + } + targets.push({ year: parseInt(m[1], 10), monthIdx: parseInt(m[2], 10) - 1 }); + } else if (backfillArg) { + const n = parseInt(backfillArg, 10); + if (!Number.isFinite(n) || n <= 0) { + console.error("--backfill must be a positive integer"); + process.exit(1); + } + const { year, monthIdx } = prevMonthUTC(); + for (let i = 0; i < n; i++) { + const d = new Date(Date.UTC(year, monthIdx - i, 1)); + targets.push({ year: d.getUTCFullYear(), monthIdx: d.getUTCMonth() }); + } + } else { + targets.push(prevMonthUTC()); + } + + for (const t of targets) { + try { + await generateMonth(t.year, t.monthIdx, defaultBranch, { + force: flagForce, + noPdf: flagNoPdf, + }); + } catch (err) { + console.error( + `✗ ${formatMonthSlug(t.year, t.monthIdx)}: ${err.message}`, + ); + // Continue with next target on failure so backfills are resilient. + } + } + } catch (err) { + console.error(`✗ fatal: ${err.message}`); + process.exit(1); + } +})(); diff --git a/web2/scripts/lean-decls.js b/web2/scripts/lean-decls.js new file mode 100644 index 0000000..99e3bce --- /dev/null +++ b/web2/scripts/lean-decls.js @@ -0,0 +1,403 @@ +/* + * Extracts declarations (theorem/lemma/def/abbrev/instance/structure/ + * inductive/class/opaque/axiom) from Lean 4 source files, along with each + * one's docstring and type signature. + * + * This parses *complete files*, never diff fragments. That distinction is the + * whole design: to find what a month added, callers parse the full file at the + * month's first commit and at its last commit and set-difference the two + * results by fully-qualified name (see findAddedDeclarations). Parsing diff + * hunks instead is what earlier versions of this tooling did, and it is + * unreliable by construction - a hunk is an arbitrary window into the file, so + * docstrings and signatures are routinely truncated mid-token, and an edit to + * an existing proof looks identical to a brand-new declaration. It also can't + * see past GitHub's compare API, which caps at 300 files and omits patches on + * large diffs. + * + * The same whole-file-then-diff approach is what leanprover-community's own + * mathlib-changelog crawler uses, and for the same stated reason: actually + * elaborating Lean to get this information is far too slow to run over history. + * + * This is still a lexer, not a Lean parser - it does not resolve `open`, + * macros, or elaboration. It is deliberately conservative. + */ + +const crypto = require("node:crypto"); + +const DECL_KEYWORDS = [ + "theorem", + "lemma", + "def", + "abbrev", + "instance", + "structure", + "inductive", + "class", + "opaque", + "axiom", + // Physlib-specific declaration commands. These are genuine named results and + // belong in the changelog. Note `TODO` is deliberately NOT here: its syntax + // is `TODO "some free text"`, with no name at all, so treating it as a + // declaration yields garbage names like `"Split` or `"Define`. + "informal_definition", + "informal_lemma", + "semiformal_result", +]; + +// Bodies of these are field lists rather than a proof/term, so they have no +// `:=` to stop at - they run until the block dedents. The two `informal_*` +// commands take the same `name where` + indented-fields shape. +const BLOCK_BODY_KEYWORDS = new Set([ + "structure", + "inductive", + "class", + "informal_definition", + "informal_lemma", +]); + +// `semiformal_result` puts a quoted tag before the name: +// semiformal_result "FXNL5" restricted_isConnected {d : ℕ} : ... +// so the generic "first token after the keyword" rule would capture the tag. +const SEMIFORMAL_NAME_RE = + /^semiformal_result\s+"(?:[^"\\]|\\.)*"\s*([^\s:({[⦃⟨]*)/; + +const MOD = + "(?:(?:private|protected|noncomputable|scoped|local|unsafe|partial|nonrec)\\s+)*"; +const DECL_RE = new RegExp( + `^${MOD}(${DECL_KEYWORDS.join("|")})\\b[ \\t]*([^\\s:({\\[⦃⟨]*)`, +); +const NAMESPACE_RE = /^(namespace|section)\b[ \t]*(\S*)/; +const END_RE = /^end\b[ \t]*(\S*)/; +const ATTR_RE = /^\s*@\[/; + +const OPENERS = "([{⟨⦃"; +const CLOSERS = ")]}⟩⦄"; + +// How far a single declaration's signature may reasonably run before we give +// up. Real signatures are a few lines; a large structure body can be longer. +const MAX_SIG_LINES = 40; +const MAX_BLOCK_LINES = 60; + +/** + * Blanks out `--` line comments and `/- -/` block comments while keeping line + * numbering intact, and captures `/-- -/` docstrings separately. + * + * Returns { lines, docs } where `docs` maps the line index on which a + * docstring *ended* to that docstring's text. Line numbering is preserved so + * the caller can correlate the two. + */ +function stripComments(text) { + const n = text.length; + let i = 0; + let line = 0; + const out = [[]]; + const docs = new Map(); + + function newline() { + line += 1; + out.push([]); + } + + while (i < n) { + const c = text[i]; + if (c === "\n") { + newline(); + i += 1; + } else if (c === '"') { + // Copy string literals verbatim - a `--` or `/-` inside one is data, + // not a comment. + out[line].push(c); + i += 1; + while (i < n) { + if (text[i] === "\\" && i + 1 < n) { + out[line].push(text[i], text[i + 1]); + i += 2; + } else if (text[i] === "\n") { + newline(); + i += 1; + } else if (text[i] === '"') { + out[line].push('"'); + i += 1; + break; + } else { + out[line].push(text[i]); + i += 1; + } + } + } else if (text.startsWith("--", i)) { + while (i < n && text[i] !== "\n") i += 1; + } else if (text.startsWith("/--", i)) { + i += 3; + let depth = 1; + const buf = []; + while (i < n && depth > 0) { + if (text.startsWith("/-", i)) { + depth += 1; + buf.push(text[i], text[i + 1]); + i += 2; + } else if (text.startsWith("-/", i)) { + depth -= 1; + if (depth === 0) { + i += 2; + break; + } + buf.push(text[i], text[i + 1]); + i += 2; + } else if (text[i] === "\n") { + buf.push("\n"); + newline(); + i += 1; + } else { + buf.push(text[i]); + i += 1; + } + } + docs.set(line, buf.join("").trim()); + } else if (text.startsWith("/-", i)) { + i += 2; + let depth = 1; + while (i < n && depth > 0) { + if (text.startsWith("/-", i)) { + depth += 1; + i += 2; + } else if (text.startsWith("-/", i)) { + depth -= 1; + i += 2; + } else { + if (text[i] === "\n") newline(); + i += 1; + } + } + } else { + out[line].push(c); + i += 1; + } + } + + return { lines: out.map((chars) => chars.join("")), docs }; +} + +/** + * Scans `line` for `needle` at bracket depth 0, tracking nesting and skipping + * string literals. Returns { index, depth } with index -1 when not found. + */ +function findTopLevel(line, needle, depth) { + let i = 0; + const n = line.length; + let found = -1; + while (i < n) { + const c = line[i]; + if (c === '"') { + i += 1; + while (i < n) { + if (line[i] === "\\") { + i += 2; + continue; + } + if (line[i] === '"') { + i += 1; + break; + } + i += 1; + } + continue; + } + if (OPENERS.includes(c)) { + depth += 1; + } else if (CLOSERS.includes(c)) { + depth = Math.max(0, depth - 1); + } else if (found < 0 && depth === 0 && line.startsWith(needle, i)) { + found = i; + } + i += 1; + } + return { index: found, depth }; +} + +/** + * Collects the declaration's signature starting at `lines[start]`. + * + * For value declarations that is everything up to a top-level `:=` (or a + * trailing `where`). For structure/inductive/class it is the header plus the + * indented field block. Returns { text, next }. + */ +function signatureAt(lines, start, kind) { + if (BLOCK_BODY_KEYWORDS.has(kind)) { + const collected = [lines[start]]; + let j = start + 1; + while (j < lines.length && j - start < MAX_BLOCK_LINES) { + const nxt = lines[j]; + // Dedented back to top level: the block is over. + if (nxt.trim() && !/\s/.test(nxt[0])) break; + // Field docstrings were blanked out by stripComments, which would + // otherwise leave the field list full of gaps. + if (nxt.trim()) collected.push(nxt); + j += 1; + } + while (collected.length && !collected[collected.length - 1].trim()) { + collected.pop(); + } + return { text: collected.join("\n"), next: j }; + } + + const collected = []; + let depth = 0; + let j = start; + while (j < lines.length && j - start < MAX_SIG_LINES) { + const line = lines[j]; + const res = findTopLevel(line, ":=", depth); + depth = res.depth; + if (res.index >= 0) { + collected.push(line.slice(0, res.index).replace(/\s+$/, "")); + j += 1; + break; + } + collected.push(line); + j += 1; + if (/\bwhere\s*$/.test(line)) break; + // A blank line at depth 0 means the signature ended without a `:=` (e.g. + // `axiom`/`opaque`, or a declaration whose body follows a blank line) - + // don't run on into whatever comes next. + if (!line.trim() && depth === 0) break; + } + while (collected.length && !collected[collected.length - 1].trim()) { + collected.pop(); + } + return { text: collected.join("\n"), next: j }; +} + +/** + * Parses a complete Lean source file into a Map of fully-qualified name → + * { kind, name, doc, signature, line }. `line` is 1-based, for source links. + */ +function parseLean(contents) { + const { lines, docs } = stripComments(contents); + const scopes = []; // { kind, name } for namespace/section + const result = new Map(); + + let i = 0; + while (i < lines.length) { + const line = lines[i]; + if (!line.trim() || /\s/.test(line[0])) { + i += 1; + continue; + } + + const scopeMatch = NAMESPACE_RE.exec(line); + if (scopeMatch) { + scopes.push({ kind: scopeMatch[1], name: scopeMatch[2] }); + i += 1; + continue; + } + + const endMatch = END_RE.exec(line); + if (endMatch) { + const name = endMatch[1]; + let matched = false; + for (let k = scopes.length - 1; k >= 0; k--) { + if (scopes[k].name === name) { + scopes.length = k; + matched = true; + break; + } + } + if (!matched && scopes.length && !name) scopes.pop(); + i += 1; + continue; + } + + const declMatch = DECL_RE.exec(line); + if (!declMatch) { + i += 1; + continue; + } + + const kind = declMatch[1]; + let rawName = declMatch[2]; + if (kind === "semiformal_result") { + const m = SEMIFORMAL_NAME_RE.exec(line); + rawName = m ? m[1] : ""; + } + const sig = signatureAt(lines, i, kind); + let signature = sig.text; + + // Walk back over attribute lines to find this declaration's docstring. + // `docs` must be consulted at every step, not just at the end: extracting + // a docstring blanks the lines it occupied, so a walk-back that skips + // blank lines would step straight over it. + let doc = null; + let k = i - 1; + while (k >= 0) { + if (docs.has(k)) { + doc = docs.get(k); + break; + } + if (!lines[k].trim() || ATTR_RE.test(lines[k])) { + k -= 1; + continue; + } + break; + } + + const namespaces = scopes + .filter((s) => s.kind === "namespace" && s.name) + .map((s) => s.name); + let full; + if (rawName.startsWith("_root_.")) { + full = rawName.slice("_root_.".length); + } else if (rawName) { + full = [...namespaces, rawName].join("."); + } else { + // Anonymous instance - synthesise a key from its signature so two + // different anonymous instances don't collide. Must be a stable digest, + // not a per-process hash: a key that varies between the "old commit" + // and "new commit" parses would make every anonymous instance look + // newly added every month. + const digest = crypto + .createHash("md5") + .update(signature) + .digest("hex") + .slice(0, 8); + full = [...namespaces, `«${kind}@${digest}»`].join("."); + } + + // Re-attach any attribute lines directly above; they're part of how the + // declaration reads. + const attrs = []; + let a = i - 1; + while (a >= 0 && ATTR_RE.test(lines[a])) { + attrs.unshift(lines[a].replace(/\s+$/, "")); + a -= 1; + } + if (attrs.length) signature = [...attrs, signature].join("\n"); + + result.set(full, { + kind, + name: full, + doc, + signature: signature.trim(), + line: i + 1, + }); + + i = Math.max(sig.next, i + 1); + } + + return result; +} + +/** + * Declarations present in `newContents` but absent from `oldContents`, + * compared by fully-qualified name. Pass null/undefined for `oldContents` + * when the file is new. + */ +function findAddedDeclarations(oldContents, newContents) { + const oldDecls = oldContents ? parseLean(oldContents) : new Map(); + const newDecls = parseLean(newContents); + const added = []; + for (const [name, decl] of newDecls) { + if (!oldDecls.has(name)) added.push(decl); + } + return added; +} + +module.exports = { parseLean, findAddedDeclarations, DECL_KEYWORDS }; diff --git a/web2/src/app/monthly-updates/page.tsx b/web2/src/app/monthly-updates/page.tsx new file mode 100644 index 0000000..e280065 --- /dev/null +++ b/web2/src/app/monthly-updates/page.tsx @@ -0,0 +1,150 @@ +import type { Metadata } from "next"; +import { getMonthlyUpdates, type MonthlyContributor } from "@/lib/yaml"; +import { site } from "@/lib/site"; + +export const metadata: Metadata = { + title: "Monthly Updates | Physlib", + description: + "Auto-generated monthly reports of new definitions, theorems, and lemmas added to Physlib.", +}; + +function formatNumber(n: number): string { + return new Intl.NumberFormat("en-US").format(n); +} + +function ContributorPill({ c }: { c: MonthlyContributor }) { + const label = c.login ?? c.name ?? "unknown"; + const inner = ( + <span className="inline-flex items-center gap-1.5 rounded-full border border-border bg-surface-secondary/60 px-2 py-0.5 text-xs text-foreground/80 hover:bg-surface-secondary transition-colors"> + {c.avatar_url ? ( + // eslint-disable-next-line @next/next/no-img-element + <img + src={c.avatar_url} + alt="" + width={16} + height={16} + className="size-4 rounded-full" + /> + ) : null} + {label} + </span> + ); + if (c.html_url) { + return ( + <a + href={c.html_url} + target="_blank" + rel="noopener noreferrer" + className="no-underline" + > + {inner} + </a> + ); + } + return inner; +} + +export default async function MonthlyUpdatesPage() { + const updates = await getMonthlyUpdates(); + + return ( + <div className="mx-auto w-full max-w-5xl px-4 py-10 md:py-14"> + <h1 className="text-4xl font-bold tracking-tight mb-2">Monthly Updates</h1> + <p className="text-muted mb-8 leading-relaxed"> + A monthly changelog automatically generated on the 1st of each month + from{" "} + <a + href={site.github} + target="_blank" + rel="noopener noreferrer" + className="text-accent hover:underline underline-offset-2" + > + leanprover-community/physlib + </a> + . Each row links to a PDF report, formatted as a short scientific + paper, listing every new definition, theorem, and lemma added during + that month along with its docstring and code snippet. + </p> + + {updates.length === 0 ? ( + <div className="rounded-xl border border-border bg-surface p-8 text-center text-muted"> + No monthly updates have been generated yet. The first report will + appear on the 1st of the coming month. + </div> + ) : ( + <div className="rounded-xl border border-border overflow-x-auto"> + <table className="w-full text-sm"> + <thead> + <tr className="bg-surface-secondary"> + <th className="px-4 py-3 text-left font-semibold border-b border-border w-32"> + Date + </th> + <th className="px-4 py-3 text-left font-semibold border-b border-border"> + Contributors + </th> + <th className="px-4 py-3 text-right font-semibold border-b border-border w-36"> + Lines Changed + </th> + <th className="px-4 py-3 text-left font-semibold border-b border-border w-24"> + Link + </th> + </tr> + </thead> + <tbody> + {updates.map((u) => ( + <tr + key={u.slug} + className="border-b border-border last:border-0 hover:bg-surface-secondary/40 transition-colors" + > + <td className="px-4 py-3 align-top font-medium whitespace-nowrap"> + {u.label} + </td> + <td className="px-4 py-3 align-top"> + {u.contributors.length === 0 ? ( + <span className="text-muted text-xs">—</span> + ) : ( + <div className="flex flex-wrap gap-1.5"> + {u.contributors.map((c, i) => ( + <ContributorPill + // Index, not Math.random(): a fresh random key + // every render forces React to discard and rebuild + // the element each time. + key={c.login ?? c.name ?? `anon-${i}`} + c={c} + /> + ))} + </div> + )} + </td> + <td className="px-4 py-3 align-top text-right font-mono text-xs"> + <div> + <span className="text-emerald-600 dark:text-emerald-400"> + +{formatNumber(u.totalAdditions)} + </span>{" "} + <span className="text-rose-600 dark:text-rose-400"> + −{formatNumber(u.totalDeletions)} + </span> + </div> + <div className="text-muted text-[10px] mt-0.5"> + {formatNumber(u.linesChanged)} total + </div> + </td> + <td className="px-4 py-3 align-top"> + <a + href={u.pdfUrl ?? `/monthly-updates/${u.slug}.pdf`} + target="_blank" + rel="noopener noreferrer" + className="inline-flex items-center gap-1 text-accent hover:underline underline-offset-2 text-sm" + > + PDF ↓ + </a> + </td> + </tr> + ))} + </tbody> + </table> + </div> + )} + </div> + ); +} diff --git a/web2/src/lib/site.ts b/web2/src/lib/site.ts index 1b86ce1..c05ebb4 100644 --- a/web2/src/lib/site.ts +++ b/web2/src/lib/site.ts @@ -38,6 +38,7 @@ export const navSections: NavSection[] = [ { label: "PR triage", href: "/pr-tracker" }, { label: "Documentation", href: "/documentation-tracker" }, { label: "TODO list", href: "/todo" }, + { label: "Monthly updates", href: "/monthly-updates" }, ], }, { diff --git a/web2/src/lib/yaml.ts b/web2/src/lib/yaml.ts index 124185b..79c47dd 100644 --- a/web2/src/lib/yaml.ts +++ b/web2/src/lib/yaml.ts @@ -1,4 +1,4 @@ -import { readFile } from "node:fs/promises"; +import { readdir, readFile } from "node:fs/promises"; import path from "node:path"; import yaml from "js-yaml"; @@ -9,6 +9,11 @@ async function loadYaml<T>(file: string): Promise<T> { return yaml.load(raw) as T; } +async function loadJson<T>(file: string): Promise<T> { + const raw = await readFile(file, "utf8"); + return JSON.parse(raw) as T; +} + export type Maintainer = { name: string; summary: string; @@ -70,3 +75,80 @@ export type TodoData = { export async function getTodo(): Promise<TodoData> { return loadYaml<TodoData>("TODO.yml"); } + +// ─── Monthly updates ────────────────────────────────────────────────────── +export type MonthlyContributor = { + login: string | null; + name?: string; + html_url: string | null; + avatar_url: string | null; + commits: number; +}; + +export type MonthlyDeclaration = { + file: string; + kind: string; + name: string; + docstring: string | null; + snippet: string; + line: number; +}; + +export type MonthlyFileEntry = { + filename: string; + entries: MonthlyDeclaration[]; +}; + +export type MonthlyUpdate = { + slug: string; + label: string; + year: number; + month: number; + startDate: string; + endDate: string; + repo: string; + branch: string; + baseSha: string; + headSha: string; + compareUrl: string; + commitsUrl: string; + totalCommits: number; + totalAdditions: number; + totalDeletions: number; + linesChanged: number; + contributors: MonthlyContributor[]; + kindTotals: Record<string, number>; + files: MonthlyFileEntry[]; + /** Number of files that changed in this month's diff (superset of `files`, + * which only lists Lean files where new declarations were detected). */ + totalFilesChanged?: number; + subfolders?: Array<{ + path: string; + files: number; + additions: number; + deletions: number; + }>; + /** Public URL of the generated scientific-paper style PDF. */ + pdfUrl?: string; + generatedAt: string; +}; + +const monthlyDir = path.join(dataDir, "monthly-updates"); + +async function safeReaddir(dir: string): Promise<string[]> { + try { + return await readdir(dir); + } catch (err) { + if ((err as NodeJS.ErrnoException).code === "ENOENT") return []; + throw err; + } +} + +/** Returns all monthly updates, newest first. */ +export async function getMonthlyUpdates(): Promise<MonthlyUpdate[]> { + const files = (await safeReaddir(monthlyDir)).filter((f) => f.endsWith(".json")); + const items = await Promise.all( + files.map((f) => loadJson<MonthlyUpdate>(path.join(monthlyDir, f))), + ); + return items.sort((a, b) => b.slug.localeCompare(a.slug)); +} From 37f9364f9319a2e4bc9e362ddc5dcce4b7437e2c Mon Sep 17 00:00:00 2001 From: Alex-Zughaid <117576511+Alex-Zughaid@users.noreply.github.com> Date: Mon, 10 Aug 2026 10:40:07 +0100 Subject: [PATCH 2/3] generated new report --- web2/data/monthly-updates/2026-06.json | 191 ++++++++++------------- web2/scripts/generate-monthly-updates.js | 50 +++--- 2 files changed, 114 insertions(+), 127 deletions(-) diff --git a/web2/data/monthly-updates/2026-06.json b/web2/data/monthly-updates/2026-06.json index 2614fa6..ffd2e3a 100644 --- a/web2/data/monthly-updates/2026-06.json +++ b/web2/data/monthly-updates/2026-06.json @@ -302,192 +302,165 @@ ], "contributors": [ { - "login": null, - "name": "Joseph Tooby-Smith", - "html_url": null, - "avatar_url": null, + "login": "jstoobysmith", + "html_url": "https://github.com/jstoobysmith", + "avatar_url": "https://avatars.githubusercontent.com/u/72603918?v=4", "commits": 21 }, { - "login": null, - "name": "Nicola Bernini", - "html_url": null, - "avatar_url": null, + "login": "NicolaBernini", + "html_url": "https://github.com/NicolaBernini", + "avatar_url": "https://avatars.githubusercontent.com/u/6381645?v=4", "commits": 20 }, { - "login": null, - "name": "Robby Sneiderman", - "html_url": null, - "avatar_url": null, + "login": "Robby955", + "html_url": "https://github.com/Robby955", + "avatar_url": "https://avatars.githubusercontent.com/u/48200016?v=4", "commits": 15 }, { - "login": null, - "name": "Yidi Qi", - "html_url": null, - "avatar_url": null, + "login": "yidiq7", + "html_url": "https://github.com/yidiq7", + "avatar_url": "https://avatars.githubusercontent.com/u/29921686?v=4", "commits": 13 }, { - "login": null, - "name": "Gregory Loges", - "html_url": null, - "avatar_url": null, + "login": "gloges", + "html_url": "https://github.com/gloges", + "avatar_url": "https://avatars.githubusercontent.com/u/11429491?v=4", "commits": 12 }, { - "login": null, - "name": "Zhi Kai Pong", - "html_url": null, - "avatar_url": null, + "login": "zhikaip", + "html_url": "https://github.com/zhikaip", + "avatar_url": "https://avatars.githubusercontent.com/u/46996788?v=4", "commits": 9 }, { - "login": null, - "name": "giuseppe.sorge", - "html_url": null, - "avatar_url": null, + "login": "giuseppesorge", + "html_url": "https://github.com/giuseppesorge", + "avatar_url": "https://avatars.githubusercontent.com/u/10847456?v=4", "commits": 7 }, { - "login": null, - "name": "kastch", - "html_url": null, - "avatar_url": null, + "login": "kastch", + "html_url": "https://github.com/kastch", + "avatar_url": "https://avatars.githubusercontent.com/u/5752579?v=4", "commits": 7 }, { - "login": null, - "name": "nateabr", - "html_url": null, - "avatar_url": null, + "login": "nateabr", + "html_url": "https://github.com/nateabr", + "avatar_url": "https://avatars.githubusercontent.com/u/135662056?v=4", "commits": 7 }, { - "login": null, - "name": "juanjfndz", - "html_url": null, - "avatar_url": null, + "login": "juanjfndz", + "html_url": "https://github.com/juanjfndz", + "avatar_url": "https://avatars.githubusercontent.com/u/76520097?v=4", "commits": 6 }, { - "login": null, - "name": "Lazar Milikic", - "html_url": null, - "avatar_url": null, + "login": "Lemmy00", + "html_url": "https://github.com/Lemmy00", + "avatar_url": "https://avatars.githubusercontent.com/u/64690721?v=4", "commits": 5 }, { - "login": null, - "name": "Andrea Pari", - "html_url": null, - "avatar_url": null, + "login": "doxtor6", + "html_url": "https://github.com/doxtor6", + "avatar_url": "https://avatars.githubusercontent.com/u/91156808?v=4", "commits": 4 }, { - "login": null, - "name": "doxtor6", - "html_url": null, - "avatar_url": null, + "login": "pariandrea", + "html_url": "https://github.com/pariandrea", + "avatar_url": "https://avatars.githubusercontent.com/u/56761212?v=4", "commits": 4 }, { - "login": null, - "name": "Bjørn Kjos-Hanssen", - "html_url": null, - "avatar_url": null, + "login": "bjoernkjoshanssen", + "html_url": "https://github.com/bjoernkjoshanssen", + "avatar_url": "https://avatars.githubusercontent.com/u/1740286?v=4", "commits": 3 }, { - "login": null, - "name": "Justin Findlay", - "html_url": null, - "avatar_url": null, + "login": "jfindlay", + "html_url": "https://github.com/jfindlay", + "avatar_url": "https://avatars.githubusercontent.com/u/702905?v=4", "commits": 3 }, { - "login": null, - "name": "Shibashish", - "html_url": null, - "avatar_url": null, + "login": "mshibashish", + "html_url": "https://github.com/mshibashish", + "avatar_url": "https://avatars.githubusercontent.com/u/84513513?v=4", "commits": 3 }, { - "login": null, - "name": "abudjum", - "html_url": null, - "avatar_url": null, + "login": "abudjum", + "html_url": "https://github.com/abudjum", + "avatar_url": "https://avatars.githubusercontent.com/u/293911086?v=4", "commits": 2 }, { - "login": null, - "name": "Christian Krause", - "html_url": null, - "avatar_url": null, + "login": "Bergschaf", + "html_url": "https://github.com/Bergschaf", + "avatar_url": "https://avatars.githubusercontent.com/u/86738237?v=4", "commits": 2 }, { - "login": null, - "name": "Florian Wiesner", - "html_url": null, - "avatar_url": null, + "login": "FloWsnr", + "html_url": "https://github.com/FloWsnr", + "avatar_url": "https://avatars.githubusercontent.com/u/65916600?v=4", "commits": 2 }, { - "login": null, - "name": "LibertasSpZ", - "html_url": null, - "avatar_url": null, + "login": "LibertasSpZ", + "html_url": "https://github.com/LibertasSpZ", + "avatar_url": "https://avatars.githubusercontent.com/u/38987995?v=4", "commits": 2 }, { - "login": null, - "name": "Paul Lezeau", - "html_url": null, - "avatar_url": null, + "login": "Konseptt", + "html_url": "https://github.com/Konseptt", + "avatar_url": "https://avatars.githubusercontent.com/u/49203341?v=4", "commits": 1 }, { - "login": null, - "name": "Ranjan Sharma", - "html_url": null, - "avatar_url": null, + "login": "Paul-Lez", + "html_url": "https://github.com/Paul-Lez", + "avatar_url": "https://avatars.githubusercontent.com/u/72892199?v=4", "commits": 1 }, { - "login": null, - "name": "Shlok Vaibhav Singh", - "html_url": null, - "avatar_url": null, + "login": "ShlokVaibhav", + "html_url": "https://github.com/ShlokVaibhav", + "avatar_url": "https://avatars.githubusercontent.com/u/52267609?v=4", "commits": 1 }, { - "login": null, - "name": "Tracy McSheery", - "html_url": null, - "avatar_url": null, + "login": "teng10", + "html_url": "https://github.com/teng10", + "avatar_url": "https://avatars.githubusercontent.com/u/40032657?v=4", "commits": 1 }, { - "login": null, - "name": "Vasily Ilin", - "html_url": null, - "avatar_url": null, + "login": "tracyphasespace", + "html_url": "https://github.com/tracyphasespace", + "avatar_url": "https://avatars.githubusercontent.com/u/44329042?v=4", "commits": 1 }, { - "login": null, - "name": "Wouter Deconinck", - "html_url": null, - "avatar_url": null, + "login": "Vilin97", + "html_url": "https://github.com/Vilin97", + "avatar_url": "https://avatars.githubusercontent.com/u/17322558?v=4", "commits": 1 }, { - "login": null, - "name": "Yanting Teng", - "html_url": null, - "avatar_url": null, + "login": "wdconinc", + "html_url": "https://github.com/wdconinc", + "avatar_url": "https://avatars.githubusercontent.com/u/4656391?v=4", "commits": 1 } ], @@ -13098,5 +13071,5 @@ } ], "pdfUrl": "/monthly-updates/2026-06.pdf", - "generatedAt": "2026-07-30T10:16:03.939Z" + "generatedAt": "2026-07-30T11:23:48.425Z" } diff --git a/web2/scripts/generate-monthly-updates.js b/web2/scripts/generate-monthly-updates.js index 6a361d5..653e076 100644 --- a/web2/scripts/generate-monthly-updates.js +++ b/web2/scripts/generate-monthly-updates.js @@ -344,17 +344,28 @@ function commitsInRange(base, head) { } /** - * Best-effort email → { login, avatar_url, html_url } map from the GitHub API, - * used purely to decorate contributors. Returns an empty map without a token - * or on any failure: commit counts come from git either way. + * Best-effort GitHub identity lookup, used to turn git authors into linked, + * avatared accounts. Returns { bySha, byEmail } maps of + * { login, avatar_url, html_url }. + * + * Keyed primarily by commit SHA, which is an exact join against git's own + * commit list. Email is kept only as a secondary key for commits the API's + * date-windowed listing happens not to return: matching on email alone is + * unreliable because squash merges and web-UI commits can record a different + * address than the one git has locally. + * + * Runs without a token too - unauthenticated GitHub allows 60 requests/hour + * and a single month needs about two - so contributor links work locally. On + * rate-limit or any other failure we degrade to plain git names rather than + * failing the report. */ async function githubIdentities(branch, sinceIso, untilIso) { - const identities = new Map(); - if (!TOKEN) return identities; + const bySha = new Map(); + const byEmail = new Map(); try { for (let page = 1; page <= 100; page++) { const q = new URLSearchParams({ - sha: branch.replace(/^origin\//, ""), + sha: branch.replace(/^[^/]+\//, ""), since: sinceIso, until: untilIso, per_page: "100", @@ -363,21 +374,24 @@ async function githubIdentities(branch, sinceIso, untilIso) { const batch = await ghRequest(`/repos/${OWNER}/${REPO}/commits?${q}`); if (!Array.isArray(batch) || batch.length === 0) break; for (const c of batch) { + if (!c.author?.login) continue; + const identity = { + login: c.author.login, + avatar_url: c.author.avatar_url ?? null, + html_url: c.author.html_url ?? null, + }; + if (c.sha) bySha.set(c.sha, identity); const email = c.commit?.author?.email; - if (email && c.author?.login && !identities.has(email)) { - identities.set(email, { - login: c.author.login, - avatar_url: c.author.avatar_url ?? null, - html_url: c.author.html_url ?? null, - }); - } + if (email && !byEmail.has(email)) byEmail.set(email, identity); } if (batch.length < 100) break; } } catch (err) { - console.warn(` ⚠ could not fetch GitHub identities (${err.message}); using git names`); + console.warn( + ` ⚠ could not fetch GitHub identities (${err.message}); contributors will be listed by git name without links`, + ); } - return identities; + return { bySha, byEmail }; } // ─── Path helpers ───────────────────────────────────────────────────────── @@ -1011,10 +1025,10 @@ async function generateMonth(year, monthIdx, defaultBranch, opts = {}) { // the API's inclusive since/until does). GitHub is consulted only to attach // logins and avatars, and is entirely optional. const commits = commitsInRange(baseSha, headSha); - const identities = await githubIdentities(defaultBranch, startIso, endIso); + const { bySha, byEmail } = await githubIdentities(defaultBranch, startIso, endIso); const contributors = new Map(); for (const c of commits) { - const id = identities.get(c.email); + const id = bySha.get(c.sha) ?? byEmail.get(c.email); const key = id ? `login:${id.login}` : `name:${c.name}`; if (!contributors.has(key)) { contributors.set(key, { @@ -1114,7 +1128,7 @@ async function generateMonth(year, monthIdx, defaultBranch, opts = {}) { (async function main() { if (!TOKEN) { console.warn( - "⚠️ GITHUB_TOKEN is not set. Contributors will be listed by git author name without GitHub logins or avatars; everything else is unaffected.", + "⚠️ GITHUB_TOKEN is not set. Contributor identity lookups will be unauthenticated (60 req/h), which is enough for a month or two but will rate-limit on a long backfill. Everything else works offline.", ); } From c1e6920911baa4e694a650a3f9f3ea62402771ec Mon Sep 17 00:00:00 2001 From: Alex-Zughaid <117576511+Alex-Zughaid@users.noreply.github.com> Date: Mon, 10 Aug 2026 13:04:34 +0100 Subject: [PATCH 3/3] ... --- web2/data/monthly-updates/2026-06.json | 246 +++++++++++++-------- web2/public/monthly-updates/2026-06.pdf | Bin 934876 -> 935103 bytes web2/scripts/generate-monthly-updates.js | 270 ++++++++++++++++++++--- web2/src/lib/yaml.ts | 12 + 4 files changed, 410 insertions(+), 118 deletions(-) diff --git a/web2/data/monthly-updates/2026-06.json b/web2/data/monthly-updates/2026-06.json index ffd2e3a..5688bb1 100644 --- a/web2/data/monthly-updates/2026-06.json +++ b/web2/data/monthly-updates/2026-06.json @@ -6,11 +6,11 @@ "startDate": "2026-06-01T00:00:00.000Z", "endDate": "2026-07-01T00:00:00.000Z", "repo": "leanprover-community/physlib", - "branch": "upstream/master", + "branch": "origin/master", "baseSha": "d44dd713876a1cf7817747d9846cfd362353057f", "headSha": "05930fe0159096a15956f518d5917bc72a69175d", "compareUrl": "https://github.com/leanprover-community/physlib/compare/d44dd713876a1cf7817747d9846cfd362353057f...05930fe0159096a15956f518d5917bc72a69175d", - "commitsUrl": "https://github.com/leanprover-community/physlib/commits/upstream/master?since=2026-06-01T00:00:00.000Z&until=2026-07-01T00:00:00.000Z", + "commitsUrl": "https://github.com/leanprover-community/physlib/commits/origin/master?since=2026-06-01T00:00:00.000Z&until=2026-07-01T00:00:00.000Z", "totalCommits": 154, "totalAdditions": 28676, "totalDeletions": 18161, @@ -303,165 +303,231 @@ "contributors": [ { "login": "jstoobysmith", + "name": "Joseph Tooby-Smith", "html_url": "https://github.com/jstoobysmith", "avatar_url": "https://avatars.githubusercontent.com/u/72603918?v=4", - "commits": 21 + "commits": 21, + "linesChanged": 15350 }, { - "login": "NicolaBernini", - "html_url": "https://github.com/NicolaBernini", - "avatar_url": "https://avatars.githubusercontent.com/u/6381645?v=4", - "commits": 20 + "login": "Vilin97", + "name": "Vasily Ilin", + "html_url": "https://github.com/Vilin97", + "avatar_url": "https://avatars.githubusercontent.com/u/17322558?v=4", + "commits": 1, + "linesChanged": 8962 }, { - "login": "Robby955", - "html_url": "https://github.com/Robby955", - "avatar_url": "https://avatars.githubusercontent.com/u/48200016?v=4", - "commits": 15 + "login": "zhikaip", + "name": "Zhi Kai Pong", + "html_url": "https://github.com/zhikaip", + "avatar_url": "https://avatars.githubusercontent.com/u/46996788?v=4", + "commits": 9, + "linesChanged": 5540 }, { - "login": "yidiq7", - "html_url": "https://github.com/yidiq7", - "avatar_url": "https://avatars.githubusercontent.com/u/29921686?v=4", - "commits": 13 + "login": "juanjfndz", + "html_url": "https://github.com/juanjfndz", + "avatar_url": "https://avatars.githubusercontent.com/u/76520097?v=4", + "commits": 6, + "linesChanged": 3103 }, { "login": "gloges", + 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"https://github.com/kastch", - "avatar_url": "https://avatars.githubusercontent.com/u/5752579?v=4", - "commits": 7 + "login": "Robby955", + "name": "Robby Sneiderman", + "html_url": "https://github.com/Robby955", + "avatar_url": "https://avatars.githubusercontent.com/u/48200016?v=4", + "commits": 15, + "linesChanged": 1736 }, { "login": "nateabr", "html_url": "https://github.com/nateabr", "avatar_url": "https://avatars.githubusercontent.com/u/135662056?v=4", - "commits": 7 + "commits": 7, + "linesChanged": 1433 }, { - "login": "juanjfndz", - "html_url": "https://github.com/juanjfndz", - "avatar_url": "https://avatars.githubusercontent.com/u/76520097?v=4", - "commits": 6 + "login": "kastch", + "name": "Nikolai Kashcheev", + "html_url": "https://github.com/kastch", + "avatar_url": "https://avatars.githubusercontent.com/u/5752579?v=4", + "commits": 7, + "linesChanged": 1308 + }, + { + "login": "yidiq7", + "name": "Yidi Qi", + "html_url": "https://github.com/yidiq7", + "avatar_url": 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a/web2/scripts/generate-monthly-updates.js b/web2/scripts/generate-monthly-updates.js index 653e076..a840744 100644 --- a/web2/scripts/generate-monthly-updates.js +++ b/web2/scripts/generate-monthly-updates.js @@ -29,9 +29,14 @@ * Repo: needs a clone of leanprover-community/physlib. Pass --repo, or set * PHYSLIB_REPO, or let the script clone into .cache/physlib.git and reuse it. * - * Auth: GITHUB_TOKEN is optional and used only to attach GitHub logins and - * avatars to contributors. Commit counts and everything else come from git, so - * the script works fully offline against an existing clone. + * Auth: GITHUB_TOKEN is optional. Commit counts, lines changed, and everything + * else about contributors come from git, so the script works fully offline + * against an existing clone; without a token it just falls back to plain git + * names with no avatars. The reviewers list, though, comes entirely from the + * GitHub API (merged PRs + their reviews, one request each) since git has no + * concept of a review - without a token that's capped at 60 requests/hour, so + * on a busy month the reviewers section may come back empty rather than fail + * the report. * * PDF: after writing the JSON report the script also writes a scientific-paper * style LaTeX source to .monthly-updates-build/<slug>.tex and, if either @@ -326,21 +331,39 @@ function fileAt(sha, filePath) { return res.status === 0 ? res.stdout : null; } -/** Commits in (base, head], from git - exact, and with no pagination cap. */ +/** + * Commits in (base, head], from git - exact, and with no pagination cap. + * Each commit also carries its own additions/deletions (via --numstat) so + * callers can attribute lines changed to individual authors. Merge commits + * get no numstat lines from git by default (no -m/-c passed), which is what + * we want here: it avoids double-counting a merged branch's lines against + * whoever clicked "merge". + */ function commitsInRange(base, head) { + const MARK = "\x01"; const SEP = "\x1f"; const out = git([ "log", `${base}..${head}`, - `--format=%H${SEP}%an${SEP}%ae`, + `--format=${MARK}%H${SEP}%an${SEP}%ae`, + "--numstat", ]); - return out - .split("\n") - .filter(Boolean) - .map((line) => { - const [sha, name, email] = line.split(SEP); - return { sha, name, email }; - }); + const commits = []; + let current = null; + for (const line of out.split("\n")) { + if (line.startsWith(MARK)) { + const [sha, name, email] = line.slice(MARK.length).split(SEP); + current = { sha, name, email, additions: 0, deletions: 0 }; + commits.push(current); + continue; + } + if (!current) continue; + const m = line.match(/^(\d+|-)\t(\d+|-)\t/); + if (!m) continue; + current.additions += m[1] === "-" ? 0 : parseInt(m[1], 10); + current.deletions += m[2] === "-" ? 0 : parseInt(m[2], 10); + } + return commits; } /** @@ -394,6 +417,105 @@ async function githubIdentities(branch, sinceIso, untilIso) { return { bySha, byEmail }; } +/** + * Reviewers for the month: GitHub logins that submitted at least one review + * on a pull request merged in [sinceIso, untilIso), excluding the PR's own + * author (self-review), bot accounts, and anyone in `contributorLogins` - + * those authored commits this month too, so they belong in the Contributors + * list rather than here. + * + * Needs one Search API call (merged PRs in range) plus one call per merged + * PR (its reviews). Like `githubIdentities`, this is best-effort: without a + * token, or on any failure, we warn and return an empty list rather than + * failing the whole report - a busy month can exceed the 60/hr unauthenticated + * budget on the reviews calls alone. + */ +async function fetchReviewers(sinceIso, untilIso, contributorLogins) { + const reviewers = new Map(); // login -> { html_url, avatar_url, prs: Set<number> } + try { + // Search's merged: range is inclusive on both ends; back the upper bound + // off by one second so a PR merged exactly at the next month's boundary + // isn't double-counted, mirroring the git side's first-parent boundary + // handling above. + const untilInclusive = new Date( + new Date(untilIso).getTime() - 1000, + ).toISOString(); + const q = `repo:${OWNER}/${REPO} is:pr is:merged merged:${sinceIso}..${untilInclusive}`; + const prs = []; + for (let page = 1; page <= 10; page++) { + const params = new URLSearchParams({ + q, + per_page: "100", + page: String(page), + }).toString(); + const res = await ghRequest(`/search/issues?${params}`); + const items = Array.isArray(res.items) ? res.items : []; + for (const it of items) { + prs.push({ number: it.number, authorLogin: it.user?.login ?? null }); + } + if (items.length < 100) break; + } + for (const pr of prs) { + const reviews = await ghRequest( + `/repos/${OWNER}/${REPO}/pulls/${pr.number}/reviews?per_page=100`, + ); + if (!Array.isArray(reviews)) continue; + for (const r of reviews) { + const login = r.user?.login; + if (!login) continue; + if (login === pr.authorLogin) continue; + if (/\[bot\]$/.test(login)) continue; + if (contributorLogins.has(login)) continue; + if (!reviewers.has(login)) { + reviewers.set(login, { + html_url: r.user.html_url ?? null, + avatar_url: r.user.avatar_url ?? null, + prs: new Set(), + }); + } + reviewers.get(login).prs.add(pr.number); + } + } + } catch (err) { + console.warn( + ` ⚠ could not fetch PR reviewers (${err.message}); reviewers section will be empty`, + ); + } + return Array.from(reviewers.entries()) + .map(([login, r]) => ({ + login, + html_url: r.html_url, + avatar_url: r.avatar_url, + reviews: r.prs.size, + })) + .sort((a, b) => b.reviews - a.reviews || a.login.localeCompare(b.login)); +} + +/** + * GitHub profile display names for a set of logins - "Jane Smith" rather + * than "jsmith", for a byline that reads like a paper's author list instead + * of a commit log. One request per login (GitHub has no batch endpoint for + * this), so it's best-effort like the rest of the API integration: a login + * with no name set, a fetch failure, or exhausted rate limit just falls back + * to showing that person's username instead of failing the report. + */ +async function fetchProfileNames(logins) { + const names = new Map(); + for (const login of logins) { + try { + const user = await ghRequest(`/users/${encodeURIComponent(login)}`); + if (user && typeof user.name === "string" && user.name.trim()) { + names.set(login, user.name.trim()); + } + } catch (err) { + console.warn( + ` ⚠ could not fetch GitHub profile for ${login} (${err.message}); showing username instead`, + ); + } + } + return names; +} + // ─── Path helpers ───────────────────────────────────────────────────────── /** Return the first two path components of a repo path, e.g. * "Physlib/QuantumMechanics/Foo/Bar.lean" → "Physlib/QuantumMechanics". @@ -610,10 +732,33 @@ function fmtN(n) { return new Intl.NumberFormat("en-US").format(n); } +// Prefer the person's real GitHub profile name over their username - reads +// like a paper's author list rather than a commit log. Falls back to the +// username (or, for contributors git couldn't resolve to a GitHub account at +// all, their git commit name) when no profile name is on record. +function displayName(person) { + return person.name || person.login || "unknown"; +} + function renderLatex(report) { - const authors = report.contributors - .map((c) => texEscape(c.login || c.name || "unknown")) - .join(" \\and "); + const reviewers = report.reviewers ?? []; + + // Title-page byline: contributors (people who authored a commit/PR this + // month) and reviewers (people who only reviewed) as two labeled groups, + // rather than one flat \and-separated author list that doesn't distinguish + // the two roles. + const contributorNames = report.contributors + .map((c) => texEscape(displayName(c))) + .join(", "); + const reviewerNames = reviewers + .map((r) => texEscape(displayName(r))) + .join(", "); + const authors = [ + `\\textbf{Contributors:} ${contributorNames || "\\emph{none recorded}"}`, + reviewerNames ? `\\textbf{Reviewers:} ${reviewerNames}` : null, + ] + .filter(Boolean) + .join(" \\\\[0.3em]\n "); const totalDecls = Object.values(report.kindTotals).reduce( (a, b) => a + b, @@ -644,12 +789,25 @@ function renderLatex(report) { : "No new Lean declarations were detected in this month's diff."); // Sections + // Ordered by lines changed (see contributorList's sort in the generator), + // so the table order matches how much of the diff each person is behind. const contributorRows = report.contributors .map((c) => { - const label = c.login - ? `\\href{${texEscapeUrl(c.html_url)}}{\\texttt{${texEscape(c.login)}}}` - : `\\texttt{${texEscape(c.name || "unknown")}}`; - return `${label} & ${c.commits} \\\\`; + const name = texEscape(displayName(c)); + const label = c.html_url + ? `\\href{${texEscapeUrl(c.html_url)}}{${name}}` + : name; + return `${label} & ${fmtN(c.linesChanged)} & ${c.commits} \\\\`; + }) + .join("\n"); + + const reviewerRows = reviewers + .map((r) => { + const name = texEscape(displayName(r)); + const label = r.html_url + ? `\\href{${texEscapeUrl(r.html_url)}}{${name}}` + : name; + return `${label} & ${r.reviews} \\\\`; }) .join("\n"); @@ -718,7 +876,7 @@ function renderLatex(report) { urlcolor=blue!50!black, citecolor=blue!50!black, pdftitle={Physlib Monthly Report -- ${texEscape(report.label)}}, - pdfauthor={${report.contributors.map((c) => texEscape(c.login || c.name || "unknown")).join(", ")}}, + pdfauthor={${[...report.contributors.map(displayName), ...reviewers.map(displayName)].map(texEscape).join(", ")}}, pdfsubject={Monthly diff summary of leanprover-community/physlib}, } @@ -759,12 +917,26 @@ function renderLatex(report) { \renewcommand{\headrulewidth}{0.4pt} \title{\vspace{-1cm}\textbf{Physlib Monthly Report} \\[0.3em] \Large ${texEscape(report.label)}} -\author{${authors || "\\emph{No authors recorded}"}} +\author{} \date{Generated ${texEscape(new Date(report.generatedAt).toUTCString())}} \begin{document} \maketitle +% article.cls sets \author in a fixed-width single-column tabular, which +% doesn't wrap and centers every row on its widest line -- fine for one or +% two names, but a full contributor roster overflows the page, and a second +% \\ row (Reviewers) ends up centered on that off-page width and vanishes. +% A minipage wraps normally, so the byline is built here instead of via +% \author{}. +\begin{center} +\begin{minipage}{0.85\linewidth} +\centering +${authors} +\end{minipage} +\end{center} +\vspace{0.5em} + \begin{abstract} \noindent Auto-generated summary of changes to \href{https://github.com/${texEscapeUrl(report.repo)}}{\texttt{${texEscape(report.repo)}}} @@ -789,10 +961,21 @@ and the on GitHub for the raw source. \subsection{Contributors} +\noindent\small Ordered by lines changed. People who authored a commit or pull request merged this month.\normalsize + ${ report.contributors.length === 0 ? "\\emph{No contributors identified for this month.}" - : `\\begin{longtable}{l r}\n\\toprule\nContributor & Commits \\\\\n\\midrule\n${contributorRows}\n\\bottomrule\n\\end{longtable}` + : `\\begin{longtable}{l r r}\n\\toprule\nContributor & Lines Changed & Commits \\\\\n\\midrule\n${contributorRows}\n\\bottomrule\n\\end{longtable}` +} + +\subsection{Reviewers} +\noindent\small People who reviewed a pull request merged this month but did not author one themselves.\normalsize + +${ + reviewers.length === 0 + ? "\\emph{No reviewers identified for this month.}" + : `\\begin{longtable}{l r}\n\\toprule\nReviewer & PRs Reviewed \\\\\n\\midrule\n${reviewerRows}\n\\bottomrule\n\\end{longtable}` } \subsection{Modified subfolders} @@ -1020,10 +1203,10 @@ async function generateMonth(year, monthIdx, defaultBranch, opts = {}) { } filesReport.sort((a, b) => a.filename.localeCompare(b.filename)); - // Contributors: commit counts from git (exact, no pagination cap, and no - // double-counting of a commit that lands exactly on a month boundary the way - // the API's inclusive since/until does). GitHub is consulted only to attach - // logins and avatars, and is entirely optional. + // Contributors: commit counts and lines changed from git (exact, no + // pagination cap, and no double-counting of a commit that lands exactly on + // a month boundary the way the API's inclusive since/until does). GitHub is + // consulted only to attach logins and avatars, and is entirely optional. const commits = commitsInRange(baseSha, headSha); const { bySha, byEmail } = await githubIdentities(defaultBranch, startIso, endIso); const contributors = new Map(); @@ -1037,16 +1220,46 @@ async function generateMonth(year, monthIdx, defaultBranch, opts = {}) { html_url: id ? id.html_url : null, avatar_url: id ? id.avatar_url : null, commits: 0, + linesChanged: 0, }); } - contributors.get(key).commits += 1; + const entry = contributors.get(key); + entry.commits += 1; + entry.linesChanged += c.additions + c.deletions; } + // Ordered by lines changed rather than commit count: a handful of large + // commits should outrank a long tail of one-line tweaks. const contributorList = Array.from(contributors.values()).sort( (a, b) => + b.linesChanged - a.linesChanged || b.commits - a.commits || (a.login ?? a.name ?? "").localeCompare(b.login ?? b.name ?? ""), ); + // Reviewers: GitHub logins who reviewed a PR merged this month but didn't + // author any commits this month (those people are already contributors). + const contributorLogins = new Set( + contributorList.map((c) => c.login).filter(Boolean), + ); + const reviewerList = await fetchReviewers( + startIso, + endIso, + contributorLogins, + ); + + // Real names for the byline - "Jane Smith", not "jsmith" - for everyone we + // have a GitHub login for. Contributors we couldn't resolve to a login keep + // their git commit name (already in `.name`) unchanged. + const profileNames = await fetchProfileNames( + new Set([...contributorLogins, ...reviewerList.map((r) => r.login)]), + ); + for (const c of contributorList) { + if (c.login && profileNames.has(c.login)) c.name = profileNames.get(c.login); + } + for (const r of reviewerList) { + if (profileNames.has(r.login)) r.name = profileNames.get(r.login); + } + const kindTotals = {}; for (const { entries } of filesReport) { for (const e of entries) kindTotals[e.kind] = (kindTotals[e.kind] || 0) + 1; @@ -1072,6 +1285,7 @@ async function generateMonth(year, monthIdx, defaultBranch, opts = {}) { totalFilesChanged: stats.length, subfolders, contributors: contributorList, + reviewers: reviewerList, kindTotals, files: filesReport, pdfUrl: `/monthly-updates/${slug}.pdf`, @@ -1082,7 +1296,7 @@ async function generateMonth(year, monthIdx, defaultBranch, opts = {}) { fs.writeFileSync(outPath, JSON.stringify(report, null, 2) + "\n", "utf8"); const totalDecls = Object.values(kindTotals).reduce((a, b) => a + b, 0); console.log( - `✓ ${slug}: ${filesReport.length} files w/ decls (${stats.length} touched), ${totalDecls} declarations, ${contributorList.length} contributors → ${path.relative(process.cwd(), outPath)}`, + `✓ ${slug}: ${filesReport.length} files w/ decls (${stats.length} touched), ${totalDecls} declarations, ${contributorList.length} contributors, ${reviewerList.length} reviewers → ${path.relative(process.cwd(), outPath)}`, ); // ─── LaTeX + PDF ──────────────────────────────────────────────────── diff --git a/web2/src/lib/yaml.ts b/web2/src/lib/yaml.ts index 79c47dd..49da0d7 100644 --- a/web2/src/lib/yaml.ts +++ b/web2/src/lib/yaml.ts @@ -83,6 +83,17 @@ export type MonthlyContributor = { html_url: string | null; avatar_url: string | null; commits: number; + linesChanged: number; +}; + +/** Someone who reviewed a PR merged this month but didn't author a commit. */ +export type MonthlyReviewer = { + login: string; + /** GitHub profile display name, when set. */ + name?: string; + html_url: string | null; + avatar_url: string | null; + reviews: number; }; export type MonthlyDeclaration = { @@ -117,6 +128,7 @@ export type MonthlyUpdate = { totalDeletions: number; linesChanged: number; contributors: MonthlyContributor[]; + reviewers?: MonthlyReviewer[]; kindTotals: Record<string, number>; files: MonthlyFileEntry[]; /** Number of files that changed in this month's diff (superset of `files`,