[Merged by Bors] - feat port: Data.Set.Pointwise.BigOperators

Mathlib.lean

@@ -429,6 +429,7 @@ import Mathlib.Data.Set.NAry
 import Mathlib.Data.Set.Opposite
 import Mathlib.Data.Set.Pairwise
 import Mathlib.Data.Set.Pointwise.Basic
+import Mathlib.Data.Set.Pointwise.BigOperators
 import Mathlib.Data.Set.Pointwise.Iterate
 import Mathlib.Data.Set.Pointwise.SMul
 import Mathlib.Data.Set.Prod

Mathlib/Data/Set/Pointwise/BigOperators.lean

@@ -0,0 +1,152 @@
+/-
+Copyright (c) 2021 Eric Wieser. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eric Wieser
+
+! This file was ported from Lean 3 source module data.set.pointwise.big_operators
+! leanprover-community/mathlib commit 008205aa645b3f194c1da47025c5f110c8406eab
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Algebra.BigOperators.Basic
+import Mathlib.Data.Set.Pointwise.Basic
+
+/-!
+# Results about pointwise operations on sets and big operators.
+-/
+
+
+namespace Set
+
+section BigOperators
+
+-- Porting note: commented out the next line
+-- open BigOperators
+
+open Pointwise Function
+
+variable {α : Type _} {ι : Type _} [CommMonoid α]
+
+/-- The n-ary version of `Set.mem_mul`. -/
+@[to_additive " The n-ary version of `Set.mem_add`. "]
+theorem mem_finset_prod (t : Finset ι) (f : ι → Set α) (a : α) :
+    (a ∈ ∏ i in t, f i) ↔ ∃ (g : ι → α)(_ : ∀ {i}, i ∈ t → g i ∈ f i), (∏ i in t, g i) = a := by
+  classical
+    induction' t using Finset.induction_on with i is hi ih generalizing a
+    · simp_rw [Finset.prod_empty, Set.mem_one]
+      exact ⟨fun h ↦ ⟨fun _ ↦ a, fun hi ↦ False.elim (Finset.not_mem_empty _ hi), h.symm⟩,
+        fun ⟨_, _, hf⟩ ↦ hf.symm⟩
+    rw [Finset.prod_insert hi, Set.mem_mul]
+    simp_rw [Finset.prod_insert hi]
+    simp_rw [ih]
+    constructor
+    · rintro ⟨x, y, hx, ⟨g, hg, rfl⟩, rfl⟩
+      refine ⟨Function.update g i x, ?_, ?_⟩
+      · intro j hj
+        obtain rfl | hj := Finset.mem_insert.mp hj
+        · rwa [Function.update_same]
+        · rw [update_noteq (ne_of_mem_of_not_mem hj hi)]
+          exact hg hj
+      · rw [Finset.prod_update_of_not_mem hi, Function.update_same]
+    · rintro ⟨g, hg, rfl⟩
+      exact ⟨g i, is.prod g, hg (is.mem_insert_self _),
+        ⟨⟨g, fun hi ↦ hg (Finset.mem_insert_of_mem hi), rfl⟩, rfl⟩⟩
+#align set.mem_finset_prod Set.mem_finset_prod
+#align set.mem_finset_sum Set.mem_finset_sum
+
+/-- A version of `Set.mem_finset_prod` with a simpler RHS for products over a Fintype. -/
+@[to_additive " A version of `Set.mem_finset_sum` with a simpler RHS for sums over a Fintype. "]
+theorem mem_fintype_prod [Fintype ι] (f : ι → Set α) (a : α) :
+    (a ∈ ∏ i, f i) ↔ ∃ (g : ι → α)(_ : ∀ i, g i ∈ f i), (∏ i, g i) = a := by
+  rw [mem_finset_prod]
+  simp
+#align set.mem_fintype_prod Set.mem_fintype_prod
+#align set.mem_fintype_sum Set.mem_fintype_sum
+
+/-- An n-ary version of `Set.mul_mem_mul`. -/
+@[to_additive " An n-ary version of `Set.add_mem_add`. "]
+theorem list_prod_mem_list_prod (t : List ι) (f : ι → Set α) (g : ι → α) (hg : ∀ i ∈ t, g i ∈ f i) :
+    (t.map g).prod ∈ (t.map f).prod := by
+  induction' t with h tl ih
+  · simp_rw [List.map_nil, List.prod_nil, Set.mem_one]
+  · simp_rw [List.map_cons, List.prod_cons]
+    exact mul_mem_mul (hg h <| List.mem_cons_self _ _)
+      (ih fun i hi ↦ hg i <| List.mem_cons_of_mem _ hi)
+#align set.list_prod_mem_list_prod Set.list_prod_mem_list_prod
+#align set.list_sum_mem_list_sum Set.list_sum_mem_list_sum
+
+/-- An n-ary version of `Set.mul_subset_mul`. -/
+@[to_additive " An n-ary version of `Set.add_subset_add`. "]
+theorem list_prod_subset_list_prod (t : List ι) (f₁ f₂ : ι → Set α) (hf : ∀ i ∈ t, f₁ i ⊆ f₂ i) :
+    (t.map f₁).prod ⊆ (t.map f₂).prod := by
+  induction' t with h tl ih
+  · rfl
+  · simp_rw [List.map_cons, List.prod_cons]
+    exact mul_subset_mul (hf h <| List.mem_cons_self _ _)
+      (ih fun i hi ↦ hf i <| List.mem_cons_of_mem _ hi)
+#align set.list_prod_subset_list_prod Set.list_prod_subset_list_prod
+#align set.list_sum_subset_list_sum Set.list_sum_subset_list_sum
+
+@[to_additive]
+theorem list_prod_singleton {M : Type _} [CommMonoid M] (s : List M) :
+    (s.map fun i ↦ ({i} : Set M)).prod = {s.prod} :=
+  (map_list_prod (singletonMonoidHom : M →* Set M) _).symm
+#align set.list_prod_singleton Set.list_prod_singleton
+#align set.list_sum_singleton Set.list_sum_singleton
+
+/-- An n-ary version of `Set.mul_mem_mul`. -/
+@[to_additive " An n-ary version of `Set.add_mem_add`. "]
+theorem multiset_prod_mem_multiset_prod (t : Multiset ι) (f : ι → Set α) (g : ι → α)
+    (hg : ∀ i ∈ t, g i ∈ f i) : (t.map g).prod ∈ (t.map f).prod := by
+  induction t using Quotient.inductionOn
+  simp_rw [Multiset.quot_mk_to_coe, Multiset.coe_map, Multiset.coe_prod]
+  exact list_prod_mem_list_prod _ _ _ hg
+#align set.multiset_prod_mem_multiset_prod Set.multiset_prod_mem_multiset_prod
+#align set.multiset_sum_mem_multiset_sum Set.multiset_sum_mem_multiset_sum
+
+/-- An n-ary version of `Set.mul_subset_mul`. -/
+@[to_additive " An n-ary version of `Set.add_subset_add`. "]
+theorem multiset_prod_subset_multiset_prod (t : Multiset ι) (f₁ f₂ : ι → Set α)
+    (hf : ∀ i ∈ t, f₁ i ⊆ f₂ i) : (t.map f₁).prod ⊆ (t.map f₂).prod := by
+  induction t using Quotient.inductionOn
+  simp_rw [Multiset.quot_mk_to_coe, Multiset.coe_map, Multiset.coe_prod]
+  exact list_prod_subset_list_prod _ _ _ hf
+#align set.multiset_prod_subset_multiset_prod Set.multiset_prod_subset_multiset_prod
+#align set.multiset_sum_subset_multiset_sum Set.multiset_sum_subset_multiset_sum
+
+@[to_additive]
+theorem multiset_prod_singleton {M : Type _} [CommMonoid M] (s : Multiset M) :
+    (s.map fun i ↦ ({i} : Set M)).prod = {s.prod} :=
+  (map_multiset_prod (singletonMonoidHom : M →* Set M) _).symm
+#align set.multiset_prod_singleton Set.multiset_prod_singleton
+#align set.multiset_sum_singleton Set.multiset_sum_singleton
+
+/-- An n-ary version of `Set.mul_mem_mul`. -/
+@[to_additive " An n-ary version of `Set.add_mem_add`. "]
+theorem finset_prod_mem_finset_prod (t : Finset ι) (f : ι → Set α) (g : ι → α)
+    (hg : ∀ i ∈ t, g i ∈ f i) : (∏ i in t, g i) ∈ ∏ i in t, f i :=
+  multiset_prod_mem_multiset_prod _ _ _ hg
+#align set.finset_prod_mem_finset_prod Set.finset_prod_mem_finset_prod
+#align set.finset_sum_mem_finset_sum Set.finset_sum_mem_finset_sum
+
+/-- An n-ary version of `Set.mul_subset_mul`. -/
+@[to_additive " An n-ary version of `Set.add_subset_add`. "]
+theorem finset_prod_subset_finset_prod (t : Finset ι) (f₁ f₂ : ι → Set α)
+    (hf : ∀ i ∈ t, f₁ i ⊆ f₂ i) : (∏ i in t, f₁ i) ⊆ ∏ i in t, f₂ i :=
+  multiset_prod_subset_multiset_prod _ _ _ hf
+#align set.finset_prod_subset_finset_prod Set.finset_prod_subset_finset_prod
+#align set.finset_sum_subset_finset_sum Set.finset_sum_subset_finset_sum
+
+@[to_additive]
+theorem finset_prod_singleton {M ι : Type _} [CommMonoid M] (s : Finset ι) (I : ι → M) :
+    (∏ i : ι in s, ({I i} : Set M)) = {∏ i : ι in s, I i} :=
+  (map_prod (singletonMonoidHom : M →* Set M) _ _).symm
+#align set.finset_prod_singleton Set.finset_prod_singleton
+#align set.finset_sum_singleton Set.finset_sum_singleton
+
+/-! TODO: define `decidable_mem_finset_prod` and `decidable_mem_finset_sum`. -/
+
+
+end BigOperators
+
+end Set