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refactor(library/data/quotient): make proofs more robust

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Leonardo de Moura 2015-06-24 18:41:40 -07:00
parent c9f3b766f8
commit d2e64d30e8
2 changed files with 8 additions and 8 deletions
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6
library/data/quotient/basic.lean
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@ -208,7 +208,7 @@ theorem fun_image_def {A B : Type} (f : A → B) (a : A) :
fun_image f a = tag (f a) (exists.intro a rfl) := rfl fun_image f a = tag (f a) (exists.intro a rfl) := rfl
theorem elt_of_fun_image {A B : Type} (f : A → B) (a : A) : elt_of (fun_image f a) = f a := theorem elt_of_fun_image {A B : Type} (f : A → B) (a : A) : elt_of (fun_image f a) = f a :=
elt_of.tag _ _ by esimp
theorem image_elt_of {A B : Type} {f : A → B} (u : image f) : ∃a, f a = elt_of u := theorem image_elt_of {A B : Type} {f : A → B} (u : image f) : ∃a, f a = elt_of u :=
has_property u has_property u
@ -230,13 +230,13 @@ theorem fun_image_eq {A B : Type} (f : A → B) (a a' : A)
iff.intro iff.intro
(assume H : f a = f a', tag_eq H) (assume H : f a = f a', tag_eq H)
(assume H : fun_image f a = fun_image f a', (assume H : fun_image f a = fun_image f a',
(congr_arg elt_of H ▸ elt_of_fun_image f a) ▸ elt_of_fun_image f a') by injection H; assumption)
theorem idempotent_image_elt_of {A : Type} {f : A → A} (H : ∀a, f (f a) = f a) (u : image f) theorem idempotent_image_elt_of {A : Type} {f : A → A} (H : ∀a, f (f a) = f a) (u : image f)
: fun_image f (elt_of u) = u := : fun_image f (elt_of u) = u :=
obtain (a : A) (Ha : fun_image f a = u), from fun_image_surj u, obtain (a : A) (Ha : fun_image f a = u), from fun_image_surj u,
calc calc
fun_image f (elt_of u) = fun_image f (elt_of (fun_image f a)) : {Ha⁻¹} fun_image f (elt_of u) = fun_image f (elt_of (fun_image f a)) : by rewrite Ha
... = fun_image f (f a) : {elt_of_fun_image f a} ... = fun_image f (f a) : {elt_of_fun_image f a}
... = fun_image f a : {iff.elim_left (fun_image_eq f (f a) a) (H a)} ... = fun_image f a : {iff.elim_left (fun_image_eq f (f a) a) (H a)}
... = u : Ha ... = u : Ha

10
library/data/quotient/util.lean
View file

@ -48,10 +48,10 @@ namespace quotient
(pr1.mk a a') ▸ (pr2.mk a a') ▸ rfl (pr1.mk a a') ▸ (pr2.mk a a') ▸ rfl
theorem map_pair_pr1 (f : A → B) (a : A × A) : pr1 (map_pair f a) = f (pr1 a) := theorem map_pair_pr1 (f : A → B) (a : A × A) : pr1 (map_pair f a) = f (pr1 a) :=
!pr1.mk by esimp
theorem map_pair_pr2 (f : A → B) (a : A × A) : pr2 (map_pair f a) = f (pr2 a) := theorem map_pair_pr2 (f : A → B) (a : A × A) : pr2 (map_pair f a) = f (pr2 a) :=
!pr2.mk by esimp
/- coordinatewise binary maps -/ /- coordinatewise binary maps -/
@ -72,10 +72,10 @@ namespace quotient
... = pair (f a b) (f a' b') : {pr1.mk a a'} ... = pair (f a b) (f a' b') : {pr1.mk a a'}
theorem map_pair2_pr1 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) : theorem map_pair2_pr1 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
pr1 (map_pair2 f a b) = f (pr1 a) (pr1 b) := !pr1.mk pr1 (map_pair2 f a b) = f (pr1 a) (pr1 b) := by esimp
theorem map_pair2_pr2 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) : theorem map_pair2_pr2 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
pr2 (map_pair2 f a b) = f (pr2 a) (pr2 b) := !pr2.mk pr2 (map_pair2 f a b) = f (pr2 a) (pr2 b) := by esimp
theorem map_pair2_flip {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) : theorem map_pair2_flip {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
flip (map_pair2 f a b) = map_pair2 f (flip a) (flip b) := flip (map_pair2 f a b) = map_pair2 f (flip a) (flip b) :=
@ -123,7 +123,7 @@ have Hx : pr1 (map_pair2 f (map_pair2 f u v) w) =
= f (pr1 (map_pair2 f u v)) (pr1 w) : map_pair2_pr1 f _ _ = f (pr1 (map_pair2 f u v)) (pr1 w) : map_pair2_pr1 f _ _
... = f (f (pr1 u) (pr1 v)) (pr1 w) : {map_pair2_pr1 f _ _} ... = f (f (pr1 u) (pr1 v)) (pr1 w) : {map_pair2_pr1 f _ _}
... = f (pr1 u) (f (pr1 v) (pr1 w)) : Hassoc (pr1 u) (pr1 v) (pr1 w) ... = f (pr1 u) (f (pr1 v) (pr1 w)) : Hassoc (pr1 u) (pr1 v) (pr1 w)
... = f (pr1 u) (pr1 (map_pair2 f v w)) : {(map_pair2_pr1 f _ _)⁻¹} ... = f (pr1 u) (pr1 (map_pair2 f v w)) : by rewrite [map_pair2_pr1 f]
... = pr1 (map_pair2 f u (map_pair2 f v w)) : (map_pair2_pr1 f _ _)⁻¹, ... = pr1 (map_pair2 f u (map_pair2 f v w)) : (map_pair2_pr1 f _ _)⁻¹,
have Hy : pr2 (map_pair2 f (map_pair2 f u v) w) = have Hy : pr2 (map_pair2 f (map_pair2 f u v) w) =
pr2 (map_pair2 f u (map_pair2 f v w)), from pr2 (map_pair2 f u (map_pair2 f v w)), from