From 7c92161e49c81fed64600b6b4ede3d784986a516 Mon Sep 17 00:00:00 2001 From: Jeremy Avigad Date: Mon, 25 May 2015 18:37:07 +1000 Subject: [PATCH] refactor(library/data/finset/basic.lean): change order of arguments to induction tactic --- library/algebra/group_bigops.lean | 2 +- library/data/finset/basic.lean | 16 +++++----- library/data/finset/card.lean | 50 +++++++++++++++---------------- 3 files changed, 34 insertions(+), 34 deletions(-) diff --git a/library/algebra/group_bigops.lean b/library/algebra/group_bigops.lean index 456cf02598..9c6f3b53db 100644 --- a/library/algebra/group_bigops.lean +++ b/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 diff --git a/library/data/finset/basic.lean b/library/data/finset/basic.lean index e98ca6848f..69654f0d2a 100644 --- a/library/data/finset/basic.lean +++ b/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 diff --git a/library/data/finset/card.lean b/library/data/finset/card.lean index 4bd04da64b..7481251f78 100644 --- a/library/data/finset/card.lean +++ b/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