feat(library/data/finset/{bigops,comb}): add two theorems for Haitao

library/data/finset/bigops.lean

@@ -117,6 +117,19 @@ section deceqA
       finset.induction_on s
         (by rewrite Union_empty)
         (take s1 a Pa IH, by rewrite [image_insert, *Union_insert, IH])
+
+  lemma Union_const [deceqB : decidable_eq B] {f : A → finset B} {s : finset A} {t : finset B}
+    (a : A) : s ≠ ∅ → (∀ x, x ∈ s → f x = t) → Union s f = t :=
+  begin
+    induction s with a' s' H IH,
+      {intros [H1, H2], exfalso, apply H1 !rfl},
+    intros [H1, H2],
+    rewrite [Union_insert, H2 _ !mem_insert],
+    cases (decidable.em (s' = ∅)) with [seq, sne],
+      {rewrite [seq, Union_empty, union_empty] },
+    have H3 : ∀ x, x ∈ s' → f x = t, from (λ x H', H2 x (mem_insert_of_mem _ H')),
+    rewrite [IH sne H3, union_self]
+  end
 end deceqA
 
 end union

library/data/finset/comb.lean

@@ -74,6 +74,18 @@ ext (take y, iff.intro
         show y ∈ image f (insert a s), from eq.subst (eq.symm H2) H3)
       (assume H2 : y ∈ image f s,
         show y ∈ image f (insert a s), from mem_image_of_mem_image_of_subset H2 !subset_insert)))
+
+lemma image_compose {C : Type} [deceqC : decidable_eq C] {f : B → C} {g : A → B} {s : finset A} :
+  image (f∘g) s = image f (image g s) :=
+ext (take z, iff.intro
+  (assume Hz : z ∈ image (f∘g) s,
+    obtain x (Hx : x ∈ s) (Hgfx : f (g x) = z), from exists_of_mem_image Hz,
+    by rewrite -Hgfx; apply mem_image_of_mem _ (mem_image_of_mem _ Hx))
+  (assume Hz : z ∈ image f (image g s),
+    obtain y (Hy : y ∈ image g s) (Hfy : f y = z), from exists_of_mem_image Hz,
+    obtain x (Hx : x ∈ s) (Hgx : g x = y), from exists_of_mem_image Hy,
+    mem_image_of_mem_of_eq Hx (by esimp; rewrite [Hgx, Hfy])))
+
 end image
 
 /- filter and set-builder notation -/