chore(data/finset): add a few lemmas (#4901)

src/data/finset/basic.lean

@@ -815,6 +815,18 @@ begin
   exact set.insert_diff_of_mem ↑s h
 end
 
+@[simp] lemma insert_sdiff_insert (s t : finset α) (x : α) :
+  (insert x s) \ (insert x t) = s \ insert x t :=
+insert_sdiff_of_mem _ (mem_insert_self _ _)
+
+lemma sdiff_insert_of_not_mem {s : finset α} {x : α} (h : x ∉ s) (t : finset α) :
+  s \ (insert x t) = s \ t :=
+begin
+  refine subset.antisymm (sdiff_subset_sdiff (subset.refl _) (subset_insert _ _)) (λ y hy, _),
+  simp only [mem_sdiff, mem_insert, not_or_distrib] at hy ⊢,
+  exact ⟨hy.1, λ hxy, h $ hxy ▸ hy.1, hy.2⟩
+end
+
 @[simp] lemma sdiff_subset (s t : finset α) : s \ t ⊆ s :=
 by simp [subset_iff, mem_sdiff] {contextual := tt}
 
@@ -915,6 +927,24 @@ lemma piecewise_singleton [decidable_eq α] (i : α) :
   piecewise {i} f g = function.update g i (f i) :=
 by rw [← insert_emptyc_eq, piecewise_insert, piecewise_empty]
 
+lemma piecewise_piecewise_of_subset_left {s t : finset α} [Π i, decidable (i ∈ s)]
+  [Π i, decidable (i ∈ t)] (h : s ⊆ t) (f₁ f₂ g : Π a, δ a) :
+  s.piecewise (t.piecewise f₁ f₂) g = s.piecewise f₁ g :=
+s.piecewise_congr (λ i hi, piecewise_eq_of_mem _ _ _ (h hi)) (λ _ _, rfl)
+
+@[simp] lemma piecewise_idem_left (f₁ f₂ g : Π a, δ a) :
+  s.piecewise (s.piecewise f₁ f₂) g = s.piecewise f₁ g :=
+piecewise_piecewise_of_subset_left (subset.refl _) _ _ _
+
+lemma piecewise_piecewise_of_subset_right {s t : finset α} [Π i, decidable (i ∈ s)]
+  [Π i, decidable (i ∈ t)] (h : t ⊆ s) (f g₁ g₂ : Π a, δ a) :
+  s.piecewise f (t.piecewise g₁ g₂) = s.piecewise f g₂ :=
+s.piecewise_congr (λ _ _, rfl) (λ i hi, t.piecewise_eq_of_not_mem _ _ (mt (@h _) hi))
+
+@[simp] lemma piecewise_idem_right (f g₁ g₂ : Π a, δ a) :
+  s.piecewise f (s.piecewise g₁ g₂) = s.piecewise f g₂ :=
+piecewise_piecewise_of_subset_right (subset.refl _) f g₁ g₂
+
 lemma update_eq_piecewise {β : Type*} [decidable_eq α] (f : α → β) (i : α) (v : β) :
   function.update f i v = piecewise (singleton i) (λj, v) f :=
 (piecewise_singleton _ _ _).symm
@@ -951,6 +981,19 @@ lemma le_piecewise_of_le_of_le {δ : α → Type*} [Π i, preorder (δ i)] {f g
   (Hf : h ≤ f) (Hg : h ≤ g) : h ≤ s.piecewise f g :=
 λ x, piecewise_cases s f g (λ y, h x ≤ y) (Hf x) (Hg x)
 
+lemma piecewise_mem_Icc_of_mem_of_mem {δ : α → Type*} [Π i, preorder (δ i)] {f f₁ g g₁ : Π i, δ i}
+  (hf : f ∈ set.Icc f₁ g₁) (hg : g ∈ set.Icc f₁ g₁) :
+  s.piecewise f g ∈ set.Icc f₁ g₁ :=
+⟨le_piecewise_of_le_of_le _ hf.1 hg.1, piecewise_le_of_le_of_le _ hf.2 hg.2⟩
+
+lemma piecewise_mem_Icc {δ : α → Type*} [Π i, preorder (δ i)] {f g : Π i, δ i} (h : f ≤ g) :
+  s.piecewise f g ∈ set.Icc f g :=
+piecewise_mem_Icc_of_mem_of_mem _ (set.left_mem_Icc.2 h) (set.right_mem_Icc.2 h)
+
+lemma piecewise_mem_Icc' {δ : α → Type*} [Π i, preorder (δ i)] {f g : Π i, δ i} (h : g ≤ f) :
+  s.piecewise f g ∈ set.Icc g f :=
+piecewise_mem_Icc_of_mem_of_mem _ (set.right_mem_Icc.2 h) (set.left_mem_Icc.2 h)
+
 end piecewise
 
 section decidable_pi_exists