feat(PFunctor): redefine PFunctor.FreeM in terms of PFunctor.W

Cslib.lean

@@ -72,6 +72,7 @@ public import Cslib.Foundations.Data.OmegaSequence.Flatten
 public import Cslib.Foundations.Data.OmegaSequence.InfOcc
 public import Cslib.Foundations.Data.OmegaSequence.Init
 public import Cslib.Foundations.Data.OmegaSequence.Temporal
+public import Cslib.Foundations.Data.PFunctor.Basic
 public import Cslib.Foundations.Data.PFunctor.Free
 public import Cslib.Foundations.Data.RelatesInSteps
 public import Cslib.Foundations.Data.Set.Saturation

Cslib/Foundations/Data/PFunctor/Basic.lean

@@ -0,0 +1,101 @@
+/-
+Copyright (c) 2026 PolyFun Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Devon Tuma, Quang Dao
+-/
+module
+
+public import Cslib.Init
+public import Mathlib.Data.PFunctor.Univariate.Basic
+
+/-!
+# Polynomial Functors
+
+This file defines additional constructions on `PFunctor` that don't belong in core mathlib.
+The main definitions is `monomial A B` for the `PFunctor` with constant family `B` over `A`,
+as well as special cases of this such as a canonical choice of `0` and `1`.
+
+We also define the sum `P + Q` whose shapes are a sum of the shapes of `P` and `Q`,
+with a type family defined by sum elimination into the individual child types of `P` and `Q`.
+-/
+
+@[expose] public section
+
+universe uA uB uA₁ uA₂
+
+namespace PFunctor
+
+section monomial
+
+/-- The monomial `PFunctor` with head type `A` and constant `B` for any `a : A`. -/
+@[reducible] def monomial (A : Type uA) (B : Type uB) : PFunctor := ⟨A, fun _ => B⟩
+
+lemma monomial_A (A : Type uA) (B : Type uB) : (monomial A B).A = A := rfl
+
+lemma monomial_B (A : Type uA) (B : Type uB) (a : (monomial A B).A) :
+    (monomial A B).B a = B := rfl
+
+end monomial
+
+section zero
+
+/-- The zero polynomial functor, defined as `A = PEmpty` and `B _ = PEmpty`, is the identity with
+  respect to sum (up to equivalence) -/
+@[reducible] protected def zero : PFunctor := monomial PEmpty PEmpty
+
+instance instZeroPFunctor : Zero PFunctor where zero := PFunctor.zero
+
+@[simp] lemma zero_A : (0 : PFunctor).A = PEmpty := rfl
+
+@[simp] lemma zero_B (a : (0 : PFunctor).A) : (0 : PFunctor).B a = PEmpty := rfl
+
+end zero
+
+section one
+
+/-- The unit polynomial functor, defined as `A = PUnit` and `B _ = PEmpty`, is the identity with
+  respect to product (up to equivalence) -/
+@[reducible] protected def one : PFunctor := monomial PUnit PEmpty
+
+instance instOnePFunctor : One PFunctor where one := PFunctor.one
+
+@[simp] lemma one_A : (1 : PFunctor).A = PUnit := rfl
+
+@[simp] lemma one_B (a : (1 : PFunctor).A) : (1 : PFunctor).B a = PEmpty := rfl
+
+end one
+
+/-- The constant polynomial functor `P(X) = A X^ PEmpty = A` -/
+abbrev const (A : Type uA) : PFunctor := monomial A PEmpty
+
+/-- The linear polynomial functor `P(X) = A X` -/
+abbrev linear (A : Type uA) : PFunctor := monomial A PUnit
+
+/-- The self monomial polynomial functor `P(X) = S X^ S` -/
+abbrev selfMonomial (S : Type uA) : PFunctor.{uA, uA} := monomial S S
+
+/-- The pure power polynomial functor `P(X) = X^ B` -/
+abbrev purePower (B : Type uB) : PFunctor := monomial PUnit B
+
+section add
+
+/-- The sum of two polynomial functors `P` and `Q`, written as `P + Q`,
+defined as the sum of the head types and the sum case analysis for the child types. -/
+def add (P : PFunctor.{uA₁, uB}) (Q : PFunctor.{uA₂, uB}) :
+    PFunctor.{max uA₁ uA₂, uB} := ⟨P.A ⊕ Q.A, Sum.elim P.B Q.B⟩
+
+instance instHAddPFunctor :
+  HAdd PFunctor.{uA₁, uB} PFunctor.{uA₂, uB} PFunctor.{max uA₁ uA₂, uB} where
+  hAdd := add
+
+@[simp] lemma add_A (P Q : PFunctor) : (add P Q).A = (P.A ⊕ Q.A) := rfl
+
+@[simp] lemma add_B_inl (P : PFunctor.{uA₁, uB}) (Q : PFunctor.{uA₂, uB}) (a : P.A) :
+    (add P Q).B (.inl a) = P.B a := rfl
+
+@[simp] lemma add_B_inr (P : PFunctor.{uA₁, uB}) (Q : PFunctor.{uA₂, uB}) (a : Q.A) :
+    (add P Q).B (.inr a) = Q.B a := rfl
+
+end add
+
+end PFunctor