refactor(library/data/finset/basic.lean): change order of arguments to induction tactic

library/algebra/group_bigops.lean

@@ -130,7 +130,7 @@ section comm_monoid
       (∀{x}, x ∈ s → f x = g x) → Prod s f = Prod s g :=
     finset.induction_on s
       (assume H, rfl)
-      (take s' x, assume H1 : x ∉ s',
+      (take x s', assume H1 : x ∉ s',
         assume IH : (∀ {x : A}, x ∈ s' → f x = g x) → Prod s' f = Prod s' g,
         assume H2 : ∀{y}, y ∈ insert x s' → f y = g y,
         assert H3 : ∀y, y ∈ s' → f y = g y, from

library/data/finset/basic.lean

@@ -21,22 +21,22 @@ definition to_nodup_list [h : decidable_eq A] (l : list A) : nodup_list A :=
 
 namespace finset
 
-private definition eqv (l₁ l₂ : nodup_list A) :=
+private definition equiv (l₁ l₂ : nodup_list A) :=
 perm (elt_of l₁) (elt_of l₂)
 
-local infix ~ := eqv
+local infix ~ := equiv
 
-private definition eqv.refl (l : nodup_list A) : l ~ l :=
+private definition equiv.refl (l : nodup_list A) : l ~ l :=
 !perm.refl
 
-private definition eqv.symm {l₁ l₂ : nodup_list A} : l₁ ~ l₂ → l₂ ~ l₁ :=
+private definition equiv.symm {l₁ l₂ : nodup_list A} : l₁ ~ l₂ → l₂ ~ l₁ :=
 assume p, perm.symm p
 
-private definition eqv.trans {l₁ l₂ l₃ : nodup_list A} : l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ :=
+private definition equiv.trans {l₁ l₂ l₃ : nodup_list A} : l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ :=
 assume p₁ p₂, perm.trans p₁ p₂
 
 definition nodup_list_setoid [instance] (A : Type) : setoid (nodup_list A) :=
-setoid.mk (@eqv A) (mk_equivalence (@eqv A) (@eqv.refl A) (@eqv.symm A) (@eqv.trans A))
+setoid.mk (@equiv A) (mk_equivalence (@equiv A) (@equiv.refl A) (@equiv.symm A) (@equiv.trans A))
 
 definition finset (A : Type) : Type :=
 quot (nodup_list_setoid A)
@@ -196,7 +196,7 @@ quot.induction_on s
 
 protected theorem induction [recursor 6] {P : finset A → Prop}
     (H1 : P empty)
-    (H2 : ∀⦃s : finset A⦄, ∀{a : A}, a ∉ s → P s → P (insert a s)) :
+    (H2 : ∀ ⦃a : A⦄, ∀{s : finset A}, a ∉ s → P s → P (insert a s)) :
   ∀s, P s :=
 take s,
 quot.induction_on s
@@ -221,7 +221,7 @@ quot.induction_on s
 
 protected theorem induction_on {P : finset A → Prop} (s : finset A)
     (H1 : P empty)
-    (H2 : ∀⦃s : finset A⦄, ∀{a : A}, a ∉ s → P s → P (insert a s)) :
+    (H2 : ∀ ⦃a : A⦄, ∀ {s : finset A}, a ∉ s → P s → P (insert a s)) :
   P s :=
 finset.induction H1 H2 s
 end insert

library/data/finset/card.lean

@@ -16,28 +16,28 @@ include deceqA
 
 theorem card_add_card (s₁ s₂ : finset A) : card s₁ + card s₂ = card (s₁ ∪ s₂) + card (s₁ ∩ s₂) :=
 begin
-  induction s₂ with s₂ a ans2 IH,
-   show card s₁ + card (∅:finset A) = card (s₁ ∪ ∅) + card (s₁ ∩ ∅),
-    by rewrite [union_empty, card_empty, inter_empty],
-   show card s₁ + card (insert a s₂) = card (s₁ ∪ (insert a s₂)) + card (s₁ ∩ (insert a s₂)),
-    from decidable.by_cases
-      (assume as1 : a ∈ s₁,
-        assert H : a ∉ s₁ ∩ s₂, from assume H', ans2 (mem_of_mem_inter_right H'),
-        begin
-          rewrite [card_insert_of_not_mem ans2, union.comm, -insert_union, union.comm],
-          rewrite [insert_union, insert_eq_of_mem as1, insert_eq, inter.distrib_left, inter.comm],
-          rewrite [singleton_inter_of_mem as1, -insert_eq, card_insert_of_not_mem H, -*add.assoc],
-          rewrite IH
-        end)
-      (assume ans1 : a ∉ s₁,
-        assert H : a ∉ s₁ ∪ s₂, from assume H',
-          or.elim (mem_or_mem_of_mem_union H') (assume as1, ans1 as1) (assume as2, ans2 as2),
-        begin
-          rewrite [card_insert_of_not_mem ans2, union.comm, -insert_union, union.comm],
-          rewrite [card_insert_of_not_mem H, insert_eq, inter.distrib_left, inter.comm],
-          rewrite [singleton_inter_of_not_mem ans1, empty_union, add.right_comm],
-          rewrite [-add.assoc, IH]
-        end)
+  induction s₂ with a s₂ ans2 IH,
+    show card s₁ + card (∅:finset A) = card (s₁ ∪ ∅) + card (s₁ ∩ ∅),
+      by rewrite [union_empty, card_empty, inter_empty],
+    show card s₁ + card (insert a s₂) = card (s₁ ∪ (insert a s₂)) + card (s₁ ∩ (insert a s₂)),
+      from decidable.by_cases
+        (assume as1 : a ∈ s₁,
+          assert H : a ∉ s₁ ∩ s₂, from assume H', ans2 (mem_of_mem_inter_right H'),
+          begin
+            rewrite [card_insert_of_not_mem ans2, union.comm, -insert_union, union.comm],
+            rewrite [insert_union, insert_eq_of_mem as1, insert_eq, inter.distrib_left, inter.comm],
+            rewrite [singleton_inter_of_mem as1, -insert_eq, card_insert_of_not_mem H, -*add.assoc],
+            rewrite IH
+          end)
+        (assume ans1 : a ∉ s₁,
+          assert H : a ∉ s₁ ∪ s₂, from assume H',
+            or.elim (mem_or_mem_of_mem_union H') (assume as1, ans1 as1) (assume as2, ans2 as2),
+          begin
+            rewrite [card_insert_of_not_mem ans2, union.comm, -insert_union, union.comm],
+            rewrite [card_insert_of_not_mem H, insert_eq, inter.distrib_left, inter.comm],
+            rewrite [singleton_inter_of_not_mem ans1, empty_union, add.right_comm],
+            rewrite [-add.assoc, IH]
+          end)
 end
 
 theorem card_union (s₁ s₂ : finset A) : card (s₁ ∪ s₂) = card s₁ + card s₂ - card (s₁ ∩ s₂) :=
@@ -63,7 +63,7 @@ include deceqB
 
 theorem card_image_eq_of_inj_on {f : A → B} {s : finset A} (H1 : inj_on f (ts s)) : card (image f s) = card s :=
 begin
-  induction s with t a H IH,
+  induction s with a t H IH,
     { rewrite [card_empty] },
     { have H2 : ts t ⊆ ts (insert a t), by rewrite [-subset_eq_to_set_subset]; apply subset_insert,
       have H3 : card (image f t) = card t, from IH (inj_on_of_inj_on_of_subset H1 H2),
@@ -85,7 +85,7 @@ end card_image
 
 theorem Sum_const_eq_card_mul (s : finset A) (n : nat) : (∑ x ∈ s, n) = card s * n :=
 begin
-  induction s with s' a H IH,
+  induction s with a s' H IH,
     rewrite [Sum_empty, card_empty, zero_mul],
     rewrite [Sum_insert_of_not_mem _ H, IH, card_insert_of_not_mem H, add.comm,
              mul.right_distrib, one_mul]
@@ -102,7 +102,7 @@ theorem card_Union_of_disjoint (s : finset A) (f : A → finset B) :
     card (⋃ x ∈ s, f x) = ∑ x ∈ s, card (f x) :=
 finset.induction_on s
   (assume H, by rewrite [Union_empty, Sum_empty, card_empty])
-  (take s' a, assume H : a ∉ s',
+  (take a s', assume H : a ∉ s',
     assume IH,
     assume H1 : ∀ {a₁ a₂ : A}, a₁ ∈ insert a s' → a₂ ∈ insert a s' → a₁ ≠ a₂ → f a₁ ∩ f a₂ = ∅,
     have H2  : ∀ a₁ a₂ : A, a₁ ∈ s' → a₂ ∈ s' → a₁ ≠ a₂ → f a₁ ∩ f a₂ = ∅, from