feat port: Algebra.FreeMonoid.Count (#1483)

It necessary to use `Bool` instead of `Prop` to match the changes made in `List.countp` so I add to change some `if` to `bif` at  a couple of places. 



Co-authored-by: Johan Commelin <johan@commelin.net>

Mathlib.lean

@@ -14,6 +14,7 @@ import Mathlib.Algebra.Field.Defs
 import Mathlib.Algebra.Field.Opposite
 import Mathlib.Algebra.Field.Power
 import Mathlib.Algebra.FreeMonoid.Basic
+import Mathlib.Algebra.FreeMonoid.Count
 import Mathlib.Algebra.GCDMonoid.Basic
 import Mathlib.Algebra.Group.Basic
 import Mathlib.Algebra.Group.Commutator

Mathlib/Algebra/FreeMonoid/Count.lean

@@ -0,0 +1,100 @@
+/-
+Copyright (c) 2022 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+
+! This file was ported from Lean 3 source module algebra.free_monoid.count
+! leanprover-community/mathlib commit a2d2e18906e2b62627646b5d5be856e6a642062f
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Algebra.FreeMonoid.Basic
+import Mathlib.Data.List.Count
+
+/-!
+# `List.count` as a bundled homomorphism
+
+In this file we define `FreeMonoid.countp`, `FreeMonoid.count`, `FreeAddMonoid.countp`, and
+`FreeAddMonoid.count`. These are `List.countp` and `List.count` bundled as multiplicative and
+additive homomorphisms from `FreeMonoid` and `FreeAddMonoid`.
+
+We do not use `to_additive` because it can't map `Multiplicative ℕ` to `ℕ`.
+-/
+
+variable {α : Type _} (p : α → Prop) [DecidablePred p]
+
+namespace FreeAddMonoid
+
+/-- `List.countp` as a bundled additive monoid homomorphism. -/
+-- Porting note: changed the type of `p : α → Prop` to `p : α → Bool` to match the
+-- change in `List.countp`
+def countp (p : α → Bool): FreeAddMonoid α →+ ℕ where
+  toFun := List.countp p
+  map_zero' := List.countp_nil p
+  map_add' := List.countp_append p
+#align free_add_monoid.countp FreeAddMonoid.countp
+
+theorem countp_of (x : α): countp p (of x) = if p x = true then 1 else 0 := by
+  simp [countp, List.countp, List.countp.go]
+#align free_add_monoid.countp_of FreeAddMonoid.countp_of
+
+theorem countp_apply (l : FreeAddMonoid α) : countp p l = List.countp p l := rfl
+#align free_add_monoid.countp_apply FreeAddMonoid.countp_apply
+
+/-- `List.count` as a bundled additive monoid homomorphism. -/
+-- Porting note: changed from `countp (Eq x)` to match the definition of `List.count` and thus
+-- we can prove `count_apply` by `rfl`
+def count [DecidableEq α] (x : α) : FreeAddMonoid α →+ ℕ := countp (· == x)
+#align free_add_monoid.count FreeAddMonoid.count
+
+theorem count_of [DecidableEq α] (x y : α) : count x (of y) = (Pi.single x 1 : α → ℕ) y := by
+  simp [Pi.single, Function.update, count, countp, List.countp, List.countp.go,
+    Bool.beq_eq_decide_eq]
+#align free_add_monoid.count_of FreeAddMonoid.count_of
+
+theorem count_apply [DecidableEq α] (x : α) (l : FreeAddMonoid α) : count x l = List.count x l :=
+  rfl
+#align free_add_monoid.count_apply FreeAddMonoid.count_apply
+
+end FreeAddMonoid
+
+namespace FreeMonoid
+
+/-- `list.countp` as a bundled multiplicative monoid homomorphism. -/
+-- Porting note: changed the type of `p : α → Prop` to `p : α → Bool` to match the
+-- definition of `FreeAddMonoid.countp`
+def countp (p : α → Bool): FreeMonoid α →* Multiplicative ℕ :=
+    AddMonoidHom.toMultiplicative (FreeAddMonoid.countp p)
+#align free_monoid.countp FreeMonoid.countp
+
+-- Porting note: changed the type of `p : α → Prop` to `p : α → Bool` and `if` to `bif`
+theorem countp_of' (x : α) (p : α → Bool):
+    countp p (of x) = bif p x then Multiplicative.ofAdd 1 else Multiplicative.ofAdd 0 := by
+    simp [countp]
+    exact AddMonoidHom.toMultiplicative_apply_apply (FreeAddMonoid.countp p) (of x)
+#align free_monoid.countp_of' FreeMonoid.countp_of'
+
+-- Porting note: changed `if` to `bif`
+theorem countp_of (x : α) : countp p (of x) = bif p x then Multiplicative.ofAdd 1 else 1 := by
+  rw [countp_of', ofAdd_zero]
+#align free_monoid.countp_of FreeMonoid.countp_of
+
+-- `rfl` is not transitive
+theorem countp_apply (l : FreeAddMonoid α) : countp p l = Multiplicative.ofAdd (List.countp p l) :=
+  rfl
+#align free_monoid.countp_apply FreeMonoid.countp_apply
+
+/-- `List.count` as a bundled additive monoid homomorphism. -/
+def count [DecidableEq α] (x : α) : FreeMonoid α →* Multiplicative ℕ := countp (· == x)
+#align free_monoid.count FreeMonoid.count
+
+theorem count_apply [DecidableEq α] (x : α) (l : FreeAddMonoid α) :
+    count x l = Multiplicative.ofAdd (List.count x l) := rfl
+#align free_monoid.count_apply FreeMonoid.count_apply
+
+theorem count_of [DecidableEq α] (x y : α) :
+    count x (of y) = @Pi.mulSingle α (fun _ => Multiplicative ℕ) _ _ x (Multiplicative.ofAdd 1) y :=
+  by simp [count, countp_of, Pi.mulSingle_apply, eq_comm, Bool.beq_eq_decide_eq]
+#align free_monoid.count_of FreeMonoid.count_of
+
+end FreeMonoid