feat: port Data.Finsupp.Multiset (#1988)

Some proofs needed rewriting because or different unfolding behaviour of `Equiv`s and `toFun`s, but in the end it was not too bad. Revisiting Hom/Equiv/Basic might be valuable at some point

Author: LukasMias

Mathlib.lean

@@ -329,6 +329,7 @@ import Mathlib.Data.Finsupp.Defs
 import Mathlib.Data.Finsupp.Fin
 import Mathlib.Data.Finsupp.Fintype
 import Mathlib.Data.Finsupp.Indicator
+import Mathlib.Data.Finsupp.Multiset
 import Mathlib.Data.Finsupp.NeLocus
 import Mathlib.Data.Finsupp.Order
 import Mathlib.Data.Finsupp.Pointwise

Mathlib/Data/Finsupp/Multiset.lean

@@ -0,0 +1,244 @@
+/-
+Copyright (c) 2018 Johannes Hölzl. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Johannes Hölzl
+
+! This file was ported from Lean 3 source module data.finsupp.multiset
+! leanprover-community/mathlib commit 59694bd07f0a39c5beccba34bd9f413a160782bf
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Data.Finsupp.Basic
+import Mathlib.Data.Finsupp.Order
+
+/-!
+# Equivalence between `multiset` and `ℕ`-valued finitely supported functions
+
+This defines `Finsupp.toMultiset` the equivalence between `α →₀ ℕ` and `Multiset α`, along
+with `Multiset.toFinsupp` the reverse equivalence and `Finsupp.orderIsoMultiset` the equivalence
+promoted to an order isomorphism.
+-/
+
+
+open Finset
+
+open BigOperators Classical
+
+noncomputable section
+
+variable {α β ι : Type _}
+
+namespace Finsupp
+/-- Given `f : α →₀ ℕ`, `f.toMultiset` is the multiset with multiplicities given by the values of
+`f` on the elements of `α`. We define this function as an `AddEquiv`. -/
+def toMultiset : (α →₀ ℕ) ≃+ Multiset α
+    where
+  toFun f := Finsupp.sum f fun a n => n • {a}
+  invFun s := ⟨s.toFinset, fun a => s.count a, fun a => by simp⟩
+  left_inv f :=
+    ext fun a =>
+      by
+      simp only [sum, Multiset.count_sum', Multiset.count_singleton, mul_boole, coe_mk,
+        mem_support_iff, Multiset.count_nsmul, Finset.sum_ite_eq, ite_not, ite_eq_right_iff]
+      exact Eq.symm
+  right_inv s := by simp only [sum, coe_mk, Multiset.toFinset_sum_count_nsmul_eq]
+  -- Porting note: times out if h is not specified
+  map_add' f g := sum_add_index' (h := fun a n => n • ({a} : Multiset α))
+    (fun _ ↦ zero_nsmul _) (fun _ ↦ add_nsmul _)
+#align finsupp.to_multiset Finsupp.toMultiset
+
+theorem toMultiset_zero : toMultiset (0 : α →₀ ℕ) = 0 :=
+  rfl
+#align finsupp.to_multiset_zero Finsupp.toMultiset_zero
+
+theorem toMultiset_add (m n : α →₀ ℕ) : toMultiset (m + n) = toMultiset m + toMultiset n :=
+  toMultiset.map_add m n
+#align finsupp.to_multiset_add Finsupp.toMultiset_add
+
+theorem toMultiset_apply (f : α →₀ ℕ) : toMultiset f = f.sum fun a n => n • {a} :=
+  rfl
+#align finsupp.to_multiset_apply Finsupp.toMultiset_apply
+
+@[simp]
+theorem toMultiset_symm_apply [DecidableEq α] (s : Multiset α) (x : α) :
+    Finsupp.toMultiset.symm s x = s.count x := by
+    -- Porting note: proof used to be `convert rfl`
+    have : Finsupp.toMultiset.symm s x = Finsupp.toMultiset.invFun s x := rfl
+    simp_rw [this, toMultiset, coe_mk]
+    congr
+#align finsupp.to_multiset_symm_apply Finsupp.toMultiset_symm_apply
+
+@[simp]
+theorem toMultiset_single (a : α) (n : ℕ) : toMultiset (single a n) = n • {a} := by
+  rw [toMultiset_apply, sum_single_index]; apply zero_nsmul
+#align finsupp.to_multiset_single Finsupp.toMultiset_single
+
+theorem toMultiset_sum {f : ι → α →₀ ℕ} (s : Finset ι) :
+    Finsupp.toMultiset (∑ i in s, f i) = ∑ i in s, Finsupp.toMultiset (f i) :=
+  map_sum _ _ _
+#align finsupp.to_multiset_sum Finsupp.toMultiset_sum
+
+theorem toMultiset_sum_single (s : Finset ι) (n : ℕ) :
+    Finsupp.toMultiset (∑ i in s, single i n) = n • s.val := by
+  simp_rw [toMultiset_sum, Finsupp.toMultiset_single, sum_nsmul, sum_multiset_singleton]
+#align finsupp.to_multiset_sum_single Finsupp.toMultiset_sum_single
+
+theorem card_toMultiset (f : α →₀ ℕ) : Multiset.card (toMultiset f) = f.sum fun _ => id := by
+  simp [toMultiset_apply, map_finsupp_sum, Function.id_def]
+#align finsupp.card_to_multiset Finsupp.card_toMultiset
+
+theorem toMultiset_map (f : α →₀ ℕ) (g : α → β) :
+    f.toMultiset.map g = toMultiset (f.mapDomain g) := by
+  refine' f.induction _ _
+  · rw [toMultiset_zero, Multiset.map_zero, mapDomain_zero, toMultiset_zero]
+  · intro a n f _ _ ih
+    rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
+      toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
+      (Multiset.mapAddMonoidHom g).map_nsmul]
+    rfl
+#align finsupp.to_multiset_map Finsupp.toMultiset_map
+
+@[simp]
+theorem prod_toMultiset [CommMonoid α] (f : α →₀ ℕ) :
+    f.toMultiset.prod = f.prod fun a n => a ^ n := by
+  refine' f.induction _ _
+  · rw [toMultiset_zero, Multiset.prod_zero, Finsupp.prod_zero_index]
+  · intro a n f _ _ ih
+    rw [toMultiset_add, Multiset.prod_add, ih, toMultiset_single, Multiset.prod_nsmul,
+      Finsupp.prod_add_index' pow_zero pow_add, Finsupp.prod_single_index, Multiset.prod_singleton]
+    · exact pow_zero a
+#align finsupp.prod_to_multiset Finsupp.prod_toMultiset
+
+@[simp]
+theorem toFinset_toMultiset [DecidableEq α] (f : α →₀ ℕ) : f.toMultiset.toFinset = f.support := by
+  refine' f.induction _ _
+  · rw [toMultiset_zero, Multiset.toFinset_zero, support_zero]
+  · intro a n f ha hn ih
+    rw [toMultiset_add, Multiset.toFinset_add, ih, toMultiset_single, support_add_eq,
+      support_single_ne_zero _ hn, Multiset.toFinset_nsmul _ _ hn, Multiset.toFinset_singleton]
+    refine' Disjoint.mono_left support_single_subset _
+    rwa [Finset.disjoint_singleton_left]
+#align finsupp.to_finset_to_multiset Finsupp.toFinset_toMultiset
+
+@[simp]
+theorem count_toMultiset [DecidableEq α] (f : α →₀ ℕ) (a : α) : (toMultiset f).count a = f a :=
+  calc
+    (toMultiset f).count a = Finsupp.sum f (fun x n => (n • {x} : Multiset α).count a) :=
+      by rw [toMultiset_apply]; exact map_sum (Multiset.countAddMonoidHom a) _ f.support
+    _ = f.sum fun x n => n * ({x} : Multiset α).count a := by simp only [Multiset.count_nsmul]
+    _ = f a * ({a} : Multiset α).count a :=
+      sum_eq_single _
+        (fun a' _ H => by simp only [Multiset.count_singleton, if_false, H.symm, mul_zero]) fun H =>
+        by simp only [not_mem_support_iff.1 H, zero_mul]
+    _ = f a := by rw [Multiset.count_singleton_self, mul_one]
+
+#align finsupp.count_to_multiset Finsupp.count_toMultiset
+
+@[simp]
+theorem mem_toMultiset (f : α →₀ ℕ) (i : α) : i ∈ toMultiset f ↔ i ∈ f.support := by
+  rw [← Multiset.count_ne_zero, Finsupp.count_toMultiset, Finsupp.mem_support_iff]
+#align finsupp.mem_to_multiset Finsupp.mem_toMultiset
+
+end Finsupp
+
+namespace Multiset
+
+/-- Given a multiset `s`, `s.toFinsupp` returns the finitely supported function on `ℕ` given by
+the multiplicities of the elements of `s`. -/
+def toFinsupp : Multiset α ≃+ (α →₀ ℕ) :=
+  Finsupp.toMultiset.symm
+#align multiset.to_finsupp Multiset.toFinsupp
+
+@[simp]
+theorem toFinsupp_support [DecidableEq α] (s : Multiset α) : s.toFinsupp.support = s.toFinset := by
+  -- Porting note: used to be `convert rfl`
+  ext
+  simp [toFinsupp]
+#align multiset.to_finsupp_support Multiset.toFinsupp_support
+
+@[simp]
+theorem toFinsupp_apply [DecidableEq α] (s : Multiset α) (a : α) : toFinsupp s a = s.count a := by
+  -- Porting note: used to be `convert rfl`
+  exact Finsupp.toMultiset_symm_apply s a
+#align multiset.to_finsupp_apply Multiset.toFinsupp_apply
+
+theorem toFinsupp_zero : toFinsupp (0 : Multiset α) = 0 :=
+  AddEquiv.map_zero _
+#align multiset.to_finsupp_zero Multiset.toFinsupp_zero
+
+theorem toFinsupp_add (s t : Multiset α) : toFinsupp (s + t) = toFinsupp s + toFinsupp t :=
+  toFinsupp.map_add s t
+#align multiset.to_finsupp_add Multiset.toFinsupp_add
+
+@[simp]
+theorem toFinsupp_singleton (a : α) : toFinsupp ({a} : Multiset α) = Finsupp.single a 1 :=
+  Finsupp.toMultiset.symm_apply_eq.2 <| by simp
+#align multiset.to_finsupp_singleton Multiset.toFinsupp_singleton
+
+@[simp]
+theorem toFinsupp_toMultiset (s : Multiset α) : Finsupp.toMultiset (toFinsupp s) = s :=
+  Finsupp.toMultiset.apply_symm_apply s
+#align multiset.to_finsupp_to_multiset Multiset.toFinsupp_toMultiset
+
+theorem toFinsupp_eq_iff {s : Multiset α} {f : α →₀ ℕ} :
+    toFinsupp s = f ↔ s = Finsupp.toMultiset f :=
+  Finsupp.toMultiset.symm_apply_eq
+#align multiset.to_finsupp_eq_iff Multiset.toFinsupp_eq_iff
+
+end Multiset
+
+@[simp]
+theorem Finsupp.toMultiset_toFinsupp (f : α →₀ ℕ) :
+    Multiset.toFinsupp (Finsupp.toMultiset f) = f :=
+  Finsupp.toMultiset.symm_apply_apply f
+#align finsupp.to_multiset_to_finsupp Finsupp.toMultiset_toFinsupp
+
+/-! ### As an order isomorphism -/
+
+
+namespace Finsupp
+/-- `Finsupp.toMultiset` as an order isomorphism. -/
+def orderIsoMultiset : (ι →₀ ℕ) ≃o Multiset ι
+    where
+  toEquiv := toMultiset.toEquiv
+  map_rel_iff' := by
+    -- Porting note: This proof used to be simp [Multiset.le_iff_count, le_def]
+    intro f g;
+    -- Porting note: the following should probably be a simp lemma somewhere;
+    -- maybe coe_toEquiv in Hom/Equiv/Basic?
+    have : ⇑ (toMultiset (α := ι)).toEquiv = toMultiset := rfl
+    simp [Multiset.le_iff_count, le_def, ← toMultiset_symm_apply, this]
+#align finsupp.order_iso_multiset Finsupp.orderIsoMultiset
+
+@[simp]
+theorem coe_orderIsoMultiset : ⇑(@orderIsoMultiset ι) = toMultiset :=
+  rfl
+#align finsupp.coe_order_iso_multiset Finsupp.coe_orderIsoMultiset
+
+@[simp]
+theorem coe_orderIsoMultiset_symm : ⇑(@orderIsoMultiset ι).symm = Multiset.toFinsupp :=
+  rfl
+#align finsupp.coe_order_iso_multiset_symm Finsupp.coe_orderIsoMultiset_symm
+
+theorem toMultiset_strictMono : StrictMono (@toMultiset ι) :=
+  (@orderIsoMultiset ι).strictMono
+#align finsupp.to_multiset_strict_mono Finsupp.toMultiset_strictMono
+
+theorem sum_id_lt_of_lt (m n : ι →₀ ℕ) (h : m < n) : (m.sum fun _ => id) < n.sum fun _ => id := by
+  rw [← card_toMultiset, ← card_toMultiset]
+  apply Multiset.card_lt_of_lt
+  exact toMultiset_strictMono h
+#align finsupp.sum_id_lt_of_lt Finsupp.sum_id_lt_of_lt
+
+variable (ι)
+
+/-- The order on `ι →₀ ℕ` is well-founded. -/
+theorem lt_wf : WellFounded (@LT.lt (ι →₀ ℕ) _) :=
+  Subrelation.wf (sum_id_lt_of_lt _ _) <| InvImage.wf _ Nat.lt_wfRel.2
+#align finsupp.lt_wf Finsupp.lt_wf
+
+end Finsupp
+
+theorem Multiset.toFinsupp_strictMono : StrictMono (@Multiset.toFinsupp ι) :=
+  (@Finsupp.orderIsoMultiset ι).symm.strictMono
+#align multiset.to_finsupp_strict_mono Multiset.toFinsupp_strictMono