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fix/feat(library/logic/connectives,library/data/{finset,set}): fix names or.left_distrib etc., fix implicit arguments, and make small additions

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Jeremy Avigad 2015-07-27 06:22:05 +03:00 committed by Leonardo de Moura
parent a124bc246a
commit 471c9058eb
4 changed files with 31 additions and 17 deletions
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10
library/data/finset/basic.lean
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@ -297,7 +297,7 @@ theorem mem_erase_eq (a b : A) (s : finset A) : a ∈ erase b s = (a ∈ s ∧ a
propext !mem_erase_iff
open decidable
theorem erase_insert (a : A) (s : finset A) : a ∉ s → erase a (insert a s) = s :=
theorem erase_insert {a : A} {s : finset A} : a ∉ s → erase a (insert a s) = s :=
λ anins, finset.ext (λ b, by_cases
(λ beqa : b = a, iff.intro
(λ bin, by subst b; exact absurd bin !mem_erase)
@ -481,16 +481,16 @@ variable [h : decidable_eq A]
include h
theorem inter.distrib_left (s t u : finset A) : s ∩ (t ∪ u) = (s ∩ t) ∪ (s ∩ u) :=
ext (take x, by rewrite [mem_inter_eq, *mem_union_eq, *mem_inter_eq]; apply and.distrib_left)
ext (take x, by rewrite [mem_inter_eq, *mem_union_eq, *mem_inter_eq]; apply and.left_distrib)
theorem inter.distrib_right (s t u : finset A) : (s ∪ t) ∩ u = (s ∩ u) ∪ (t ∩ u) :=
ext (take x, by rewrite [mem_inter_eq, *mem_union_eq, *mem_inter_eq]; apply and.distrib_right)
ext (take x, by rewrite [mem_inter_eq, *mem_union_eq, *mem_inter_eq]; apply and.right_distrib)
theorem union.distrib_left (s t u : finset A) : s ∪ (t ∩ u) = (s ∪ t) ∩ (s ∪ u) :=
ext (take x, by rewrite [mem_union_eq, *mem_inter_eq, *mem_union_eq]; apply or.distrib_left)
ext (take x, by rewrite [mem_union_eq, *mem_inter_eq, *mem_union_eq]; apply or.left_distrib)
theorem union.distrib_right (s t u : finset A) : (s ∩ t) ∪ u = (s ∪ u) ∩ (t ∪ u) :=
ext (take x, by rewrite [mem_union_eq, *mem_inter_eq, *mem_union_eq]; apply or.distrib_right)
ext (take x, by rewrite [mem_union_eq, *mem_inter_eq, *mem_union_eq]; apply or.right_distrib)
end inter
/- disjoint -/

14
library/data/finset/comb.lean
View file

@ -150,6 +150,11 @@ iff.intro
theorem mem_filter_eq : x ∈ filter p s = (x ∈ s ∧ p x) :=
propext !mem_filter_iff
variable t : finset A
theorem mem_filter_union_iff : x ∈ filter p (s ∪ t) ↔ x ∈ filter p s ∨ x ∈ filter p t :=
by rewrite [*mem_filter_iff, mem_union_iff, and.right_distrib]
end filter
theorem mem_singleton_eq' {A : Type} [deceq : decidable_eq A] (x a : A) : x ∈ '{a} = (x = a) :=
@ -423,6 +428,15 @@ begin
(show x ⊆ s, by rewrite [(erase_eq_of_not_mem this) at H']; apply H'))))
end
theorem subset_of_mem_powerset {s t : finset A} (H : s ∈ powerset t) : s ⊆ t :=
by rewrite mem_powerset_iff_subset at H; exact H
theorem mem_powerset_of_subset {s t : finset A} (H : s ⊆ t) : s ∈ powerset t :=
by rewrite -mem_powerset_iff_subset at H; exact H
theorem empty_mem_powerset (s : finset A) : ∅ ∈ powerset s :=
by rewrite mem_powerset_iff_subset; apply empty_subset
end powerset
end finset

8
library/data/set/basic.lean
View file

@ -157,16 +157,16 @@ setext (take x, !true_and)
/- distributivity laws -/
theorem inter.distrib_left (s t u : set X) : s ∩ (t ∪ u) = (s ∩ t) ∪ (s ∩ u) :=
setext (take x, !and.distrib_left)
setext (take x, !and.left_distrib)
theorem inter.distrib_right (s t u : set X) : (s ∪ t) ∩ u = (s ∩ u) ∪ (t ∩ u) :=
setext (take x, !and.distrib_right)
setext (take x, !and.right_distrib)
theorem union.distrib_left (s t u : set X) : s ∪ (t ∩ u) = (s ∪ t) ∩ (s ∪ u) :=
setext (take x, !or.distrib_left)
setext (take x, !or.left_distrib)
theorem union.distrib_right (s t u : set X) : (s ∩ t) ∪ u = (s ∪ u) ∩ (t ∪ u) :=
setext (take x, !or.distrib_right)
setext (take x, !or.right_distrib)
/- set-builder notation -/

16
library/logic/connectives.lean
View file

@ -152,7 +152,7 @@ theorem and_imp_eq (a b c : Prop) : (a ∧ b → c) = (a → b → c) :=
propext !and_imp_iff
theorem and_iff_and (H1 : a ↔ c) (H2 : b ↔ d) : (a ∧ b) ↔ (c ∧ d) :=
iff.intro (and.imp (iff.mp H1) (iff.mp H2)) (and.imp (iff.mpr H1) (iff.mpr H2))
iff.intro (and.imp (iff.mp H1) (iff.mp H2)) (and.imp (iff.mpr H1) (iff.mpr H2))
/- or -/
@ -221,25 +221,25 @@ theorem or_self (a : Prop) : a ∨ a ↔ a :=
iff.intro (or.rec imp.id imp.id) or.inl
theorem or_iff_or (H1 : a ↔ c) (H2 : b ↔ d) : (a ∨ b) ↔ (c ∨ d) :=
iff.intro (or.imp (iff.mp H1) (iff.mp H2)) (or.imp (iff.mpr H1) (iff.mpr H2))
iff.intro (or.imp (iff.mp H1) (iff.mp H2)) (or.imp (iff.mpr H1) (iff.mpr H2))
/- distributivity -/
theorem and.distrib_left (a b c : Prop) : a ∧ (b ∨ c) ↔ (a ∧ b) ∨ (a ∧ c) :=
theorem and.left_distrib (a b c : Prop) : a ∧ (b ∨ c) ↔ (a ∧ b) ∨ (a ∧ c) :=
iff.intro
(and.rec (λH, or.imp (and.intro H) (and.intro H)))
(or.rec (and.imp_right or.inl) (and.imp_right or.inr))
theorem and.distrib_right (a b c : Prop) : (a ∨ b) ∧ c ↔ (a ∧ c) ∨ (b ∧ c) :=
iff.trans (iff.trans !and.comm !and.distrib_left) (or_iff_or !and.comm !and.comm)
theorem and.right_distrib (a b c : Prop) : (a ∨ b) ∧ c ↔ (a ∧ c) ∨ (b ∧ c) :=
iff.trans (iff.trans !and.comm !and.left_distrib) (or_iff_or !and.comm !and.comm)
theorem or.distrib_left (a b c : Prop) : a ∨ (b ∧ c) ↔ (a ∨ b) ∧ (a ∨ c) :=
theorem or.left_distrib (a b c : Prop) : a ∨ (b ∧ c) ↔ (a ∨ b) ∧ (a ∨ c) :=
iff.intro
(or.rec (λH, and.intro (or.inl H) (or.inl H)) (and.imp or.inr or.inr))
(and.rec (or.rec (imp.syl imp.intro or.inl) (imp.syl or.imp_right and.intro)))
theorem or.distrib_right (a b c : Prop) : (a ∧ b) ∨ c ↔ (a ∨ c) ∧ (b ∨ c) :=
iff.trans (iff.trans !or.comm !or.distrib_left) (and_iff_and !or.comm !or.comm)
theorem or.right_distrib (a b c : Prop) : (a ∧ b) ∨ c ↔ (a ∨ c) ∧ (b ∨ c) :=
iff.trans (iff.trans !or.comm !or.left_distrib) (and_iff_and !or.comm !or.comm)
/- iff -/