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refactor(library/data): use new 'obtain' expression

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Leonardo de Moura 2015-05-11 09:14:48 -07:00
parent a009cf24e3
commit f59a81d744
3 changed files with 20 additions and 21 deletions
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6
library/data/finset/card.lean
View file

@ -77,9 +77,9 @@ finset.induction_on s
have H4 : f a ∉ image f t, have H4 : f a ∉ image f t,
proof proof
assume H5 : f a ∈ image f t, assume H5 : f a ∈ image f t,
obtain x (H6 : x ∈ t ∧ f x = f a), from exists_of_mem_image H5, obtain x (H6l : x ∈ t) (H6r : f x = f a), from exists_of_mem_image H5,
have H7 : x = a, from H1 (mem_insert_of_mem _ (and.left H6)) !mem_insert (and.right H6), have H7 : x = a, from H1 (mem_insert_of_mem _ H6l) !mem_insert H6r,
show false, from H (H7 ▸ and.left H6) show false, from H (H7 ▸ H6l)
qed, qed,
calc calc
card (image f (insert a t)) = card (insert (f a) (image f t)) : image_insert card (image f (insert a t)) = card (insert (f a) (image f t)) : image_insert

27
library/data/finset/comb.lean
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@ -41,32 +41,31 @@ quot.induction_on s
theorem mem_image_iff (f : A → B) {s : finset A} {y : B} : y ∈ image f s ↔ ∃x, x ∈ s ∧ f x = y := theorem mem_image_iff (f : A → B) {s : finset A} {y : B} : y ∈ image f s ↔ ∃x, x ∈ s ∧ f x = y :=
iff.intro exists_of_mem_image iff.intro exists_of_mem_image
(assume H, (assume H,
obtain x (H1 : x ∈ s ∧ f x = y), from H, obtain x (H₁ : x ∈ s) (H₂ : f x = y), from H,
mem_image_of_mem_of_eq (and.left H1) (and.right H1)) mem_image_of_mem_of_eq H₁ H₂)
theorem mem_image_eq (f : A → B) {s : finset A} {y : B} : y ∈ image f s = ∃x, x ∈ s ∧ f x = y := theorem mem_image_eq (f : A → B) {s : finset A} {y : B} : y ∈ image f s = ∃x, x ∈ s ∧ f x = y :=
propext (mem_image_iff f) propext (mem_image_iff f)
theorem mem_image_of_mem_image_of_subset {f : A → B} {s t : finset A} {y : B} theorem mem_image_of_mem_image_of_subset {f : A → B} {s t : finset A} {y : B}
(H1 : y ∈ image f s) (H2 : s ⊆ t) : y ∈ image f t := (H1 : y ∈ image f s) (H2 : s ⊆ t) : y ∈ image f t :=
obtain x (H3: x ∈ s ∧ f x = y), from exists_of_mem_image H1, obtain x (H3: x ∈ s) (H4 : f x = y), from exists_of_mem_image H1,
have H4 : x ∈ t, from mem_of_subset_of_mem H2 (and.left H3), have H5 : x ∈ t, from mem_of_subset_of_mem H2 H3,
show y ∈ image f t, from mem_image_of_mem_of_eq H4 (and.right H3) show y ∈ image f t, from mem_image_of_mem_of_eq H5 H4
theorem image_insert [h' : decidable_eq A] (f : A → B) (s : finset A) (a : A) : theorem image_insert [h' : decidable_eq A] (f : A → B) (s : finset A) (a : A) :
image f (insert a s) = insert (f a) (image f s) := image f (insert a s) = insert (f a) (image f s) :=
ext (take y, iff.intro ext (take y, iff.intro
(assume H : y ∈ image f (insert a s), (assume H : y ∈ image f (insert a s),
obtain x (H1 : x ∈ insert a s ∧ f x = y), from exists_of_mem_image H, obtain x (H1l : x ∈ insert a s) (H1r :f x = y), from exists_of_mem_image H,
have H2 : x = a ∨ x ∈ s, from eq_or_mem_of_mem_insert (and.left H1), have H2 : x = a ∨ x ∈ s, from eq_or_mem_of_mem_insert H1l,
or.elim H2 or.elim H2
(assume H3 : x = a, (assume H3 : x = a,
have H4 : f a = y, from eq.subst H3 (and.right H1), have H4 : f a = y, from eq.subst H3 H1r,
show y ∈ insert (f a) (image f s), from eq.subst H4 !mem_insert) show y ∈ insert (f a) (image f s), from eq.subst H4 !mem_insert)
(assume H3 : x ∈ s, (assume H3 : x ∈ s,
have H4 : f x = y, from and.right H1,
have H5 : f x ∈ image f s, from mem_image_of_mem f H3, have H5 : f x ∈ image f s, from mem_image_of_mem f H3,
show y ∈ insert (f a) (image f s), from eq.subst H4 (mem_insert_of_mem _ H5))) show y ∈ insert (f a) (image f s), from eq.subst H1r (mem_insert_of_mem _ H5)))
(assume H : y ∈ insert (f a) (image f s), (assume H : y ∈ insert (f a) (image f s),
have H1 : y = f a ∨ y ∈ image f s, from eq_or_mem_of_mem_insert H, have H1 : y = f a ∨ y ∈ image f s, from eq_or_mem_of_mem_insert H,
or.elim H1 or.elim H1
@ -237,8 +236,8 @@ theorem any_of_mem {p : A → Prop} {s : finset A} {a : A} : a ∈ s → p a →
quot.induction_on s (λ l H1 H2, list.any_of_mem H1 H2) quot.induction_on s (λ l H1 H2, list.any_of_mem H1 H2)
theorem any_of_exists {p : A → Prop} {s : finset A} (H : ∃a, a ∈ s ∧ p a) : any s p := theorem any_of_exists {p : A → Prop} {s : finset A} (H : ∃a, a ∈ s ∧ p a) : any s p :=
obtain a H', from H, obtain a H₁ H₂, from H,
any_of_mem (and.left H') (and.right H') any_of_mem H₁ H₂
theorem any_iff_exists (p : A → Prop) (s : finset A) : any s p ↔ (∃a, a ∈ s ∧ p a) := theorem any_iff_exists (p : A → Prop) (s : finset A) : any s p ↔ (∃a, a ∈ s ∧ p a) :=
iff.intro exists_of_any any_of_exists iff.intro exists_of_any any_of_exists
@ -249,8 +248,8 @@ any_of_mem (mem_insert a s) H
theorem any_of_insert_right [h : decidable_eq A] {p : A → Prop} {s : finset A} (a : A) (H : any s p) : theorem any_of_insert_right [h : decidable_eq A] {p : A → Prop} {s : finset A} (a : A) (H : any s p) :
any (insert a s) p := any (insert a s) p :=
obtain b (H' : b ∈ s ∧ p b), from exists_of_any H, obtain b (H₁ : b ∈ s) (H₂ : p b), from exists_of_any H,
any_of_mem (mem_insert_of_mem a (and.left H')) (and.right H') any_of_mem (mem_insert_of_mem a H₁) H₂
definition decidable_any [instance] (p : A → Prop) [h : decidable_pred p] (s : finset A) : definition decidable_any [instance] (p : A → Prop) [h : decidable_pred p] (s : finset A) :
decidable (any s p) := decidable (any s p) :=

8
library/data/list/comb.lean
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@ -48,8 +48,8 @@ theorem exists_of_mem_map {A B : Type} {f : A → B} {b : B} :
(assume H1 : b = f c, (assume H1 : b = f c,
exists.intro c (and.intro !mem_cons (eq.symm H1))) exists.intro c (and.intro !mem_cons (eq.symm H1)))
(assume H1 : b ∈ map f l, (assume H1 : b ∈ map f l,
obtain a (H : a ∈ l ∧ f a = b), from exists_of_mem_map H1, obtain a (Hl : a ∈ l) (Hr : f a = b), from exists_of_mem_map H1,
exists.intro a (and.intro (mem_cons_of_mem _ (and.left H)) (and.right H))) exists.intro a (and.intro (mem_cons_of_mem _ Hl) Hr))
theorem eq_of_map_const {A B : Type} {b₁ b₂ : B} : ∀ {l : list A}, b₁ ∈ map (const A b₂) l → b₁ = b₂ theorem eq_of_map_const {A B : Type} {b₁ b₂ : B} : ∀ {l : list A}, b₁ ∈ map (const A b₂) l → b₁ = b₂
| [] h := absurd h !not_mem_nil | [] h := absurd h !not_mem_nil
@ -234,8 +234,8 @@ theorem exists_of_any {p : A → Prop} : ∀{l : list A}, any l p → ∃a, a
| (b::l) H := or.elim H | (b::l) H := or.elim H
(assume H1 : p b, exists.intro b (and.intro !mem_cons H1)) (assume H1 : p b, exists.intro b (and.intro !mem_cons H1))
(assume H1 : any l p, (assume H1 : any l p,
obtain a (H2 : a ∈ l ∧ p a), from exists_of_any H1, obtain a (H2l : a ∈ l) (H2r : p a), from exists_of_any H1,
exists.intro a (and.intro (mem_cons_of_mem b (and.left H2)) (and.right H2))) exists.intro a (and.intro (mem_cons_of_mem b H2l) H2r))
definition decidable_all (p : A → Prop) [H : decidable_pred p] : ∀ l, decidable (all l p) definition decidable_all (p : A → Prop) [H : decidable_pred p] : ∀ l, decidable (all l p)
| [] := decidable_true | [] := decidable_true