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affeldt-aist merged 3 commits into from
Sep 30, 2026
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unif cont -> cont #2113

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86154d7
unif cont -> cont
affeldt-aist Sep 25, 2026
b926f2f
Co-authored-by: Takafumi Saikawa <tscompor@gmail.com>
affeldt-aist Sep 29, 2026
d68873b
fix
affeldt-aist Sep 29, 2026
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2 changes: 2 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -11,6 +11,8 @@
+ lemma `limn_einf_shift_new`
+ lemma `limn_einf_shiftS`
+ lemma `limn_einf_cst`
- in `uniform_structure.v`:
+ lemma `unif_continuous_continuous`

### Changed

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2 changes: 1 addition & 1 deletion classical/unstable.v
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Expand Up @@ -424,7 +424,7 @@ End reassociate_products.
Lemma swapK {T1 T2 : Type} : cancel (@swap T1 T2) (@swap T2 T1).
Proof. by case=> ? ?. Qed.

Definition map_pair {S U : Type} (f : S -> U) (x : (S * S)) : (U * U) :=
Definition map_pair {S U : Type} (f : S -> U) (x : S * S) : (U * U) :=
(f x.1, f x.2).

Section order_min.
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13 changes: 7 additions & 6 deletions theories/normedtype_theory/tvs.v
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Expand Up @@ -299,7 +299,8 @@ have unif : unif_continuous (fun x => (0, x) : M * M).
by rewrite inE/= => -[[[a1 a2] [b1 b2]]]/= /[swap]-[] -> -> <-.
move=> /= U /sub_unif_continuous /unif /=.
rewrite -comp_preimage/= /comp/= /nbhs/=.
by congr entourage => /=; rewrite eqEsubset; split=> x /=; rewrite !sub0r.
congr entourage => /=; rewrite eqEsubset.
by split=> x; rewrite /map_pair/= !sub0r.
Qed.

Lemma add_unif_continuous : unif_continuous (fun x : M * M => x.1 + x.2).
Expand All @@ -311,12 +312,12 @@ have unif: unif_continuous (fun x => (x.1, -x.2) : M * M).
((fun xy => (xy.1.1, xy.2.1, (-xy.1.2, -xy.2.2))) @^-1` (U1 `*` U2))).
move=> /= [] [] a1 a2 [] b1 b2/= [] ab1 ab2.
have /U12 : (a1, b1, (-a2, -b2)) \in U1 `*` U2 by rewrite !inE.
by rewrite inE/= => [] [] [] [] c1 c2 [] d1 d2/= cd [] <- <- <- <-.
by rewrite /map_pair inE/= => [] [] [] [] c1 c2 [] d1 d2/= cd [] <- <- <- <-.
exists (U1, ((fun xy : M * M => (- xy.1, - xy.2)) @^-1` U2)); first by split.
by move=> /= [] [] a1 a2 [] b1 b2/= [] aU bU; exists (a1, b1, (a2, b2)).
move=> /= U /sub_unif_continuous/unif; rewrite /nbhs/=.
rewrite -comp_preimage/=/comp/=.
by congr entourage; rewrite eqEsubset; split=> x /=; rewrite !opprK.
by congr entourage; rewrite eqEsubset; split=> x /=; rewrite /map_pair !opprK.
Qed.

HB.instance Definition _ :=
Expand All @@ -339,7 +340,7 @@ apply: (@filterS _ _ entourage_filter
((fun xy => (xy.1.1, xy.2.1, (- xy.1.2, - xy.2.2))) @^-1` (U1 `*` U2))).
move=> /= [] [] a1 a2 [] b1 b2/= [] ab1 ab2.
have /U12 : (a1, b1, (-a2, -b2)) \in U1 `*` U2 by rewrite !inE.
by rewrite inE/= => [] [] [] [] c1 c2 [] d1 d2/= cd [] <- <- <- <-.
by rewrite /map_pair inE/= => [] [] [] [] c1 c2 [] d1 d2/= cd [] <- <- <- <-.
exists (U1, ((fun xy : M * M => (- xy.1, - xy.2)) @^-1` U2)); first by split.
by move=> /= [] [] a1 a2 [] b1 b2/= [] aU bU; exists (a1, b1, (a2, b2)).
Qed.
Expand Down Expand Up @@ -374,11 +375,11 @@ have unif: unif_continuous (fun x => (-1, x) : R^o * M).
have /U12 : ((-1, -1), x) \in U1 `*` U2.
rewrite in_setX/= (mem_set xU2) andbT.
by apply/mem_set; exact: entourage_refl.
by rewrite inE/= => [[[]]] [] a1 a2 [] b1 b2/= abU [] {2}<- <- <-/=.
by rewrite /map_pair inE/= => [[[]]] [] a1 a2 [] b1 b2/= abU [] {2}<- <- <-/=.
move=> /= U /scale_unif_continuous/unif/=.
rewrite /nbhs/=.
rewrite -comp_preimage/=/comp/=.
by congr entourage; rewrite eqEsubset; split=> x /=; rewrite !scaleN1r.
by congr entourage; rewrite eqEsubset; split=> x /=; rewrite /map_pair !scaleN1r.
Qed.

#[warning="-HB.no-new-instance"]
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16 changes: 15 additions & 1 deletion theories/topology_theory/uniform_structure.v
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@@ -1,6 +1,8 @@
(* mathcomp analysis (c) 2017 Inria and AIST. License: CeCILL-C. *)
From HB Require Import structures.
From mathcomp Require Import boot order algebra all_classical.
#[warning="-warn-library-file-internal-analysis"]
From mathcomp Require Import unstable.
From mathcomp Require Import topology_structure.

(**md**************************************************************************)
Expand Down Expand Up @@ -326,7 +328,19 @@ by apply: nbhs_singleton; apply: nbhs_interior; exact: nbhs_entourage.
Qed.

Definition unif_continuous (U V : uniformType) (f : U -> V) :=
(fun xy => (f xy.1, f xy.2)) @ entourage --> entourage.
(map_pair f) @ entourage --> entourage.

Lemma unif_continuous_continuous (U V : uniformType) (f : U -> V) :
unif_continuous f -> continuous f.
Proof.
rewrite /unif_continuous /cvg_to !nbhs_simpl => ucf.
rewrite /cvg_to /= => u X; rewrite !nbhs_simpl /= -!nbhs_entourageE.
case => Y /ucf /=; set Y' := (Y' in entourage Y') => eY' YX.
exists Y' => //.
rewrite /Y' /=.
rewrite -image_sub => v [] u' /= Yfuu' <-.
exact: YX.
Qed.

Definition entourage_set (U : uniformType) (A : set ((set U) * (set U))) :=
exists2 B, entourage B & forall PQ, A PQ -> forall p q,
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