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fix(library/data/category): minor problem that was being masked by bug #182, fixes #183

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Leonardo de Moura 2014-09-11 16:58:32 -07:00
parent 7f1977694f
commit c8e20ff3c0
1 changed files with 11 additions and 11 deletions
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22
library/data/category.lean
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@ -43,7 +43,7 @@ namespace category
theorem id_left : Π {A B : ob} {f : mor A B}, id ∘ f = f :=
rec (λ mor comp id assoc idr idl, idl) Cat
theorem id_compose : (ID A) ∘ id = id :=
theorem id_compose {A : ob} : (ID A) ∘ id = id :=
id_left
theorem left_id_unique (i : mor A A) (H : Π{B} {f : mor B A}, i ∘ f = f) : i = id :=
@ -125,18 +125,18 @@ namespace category
theorem iso_of_id {A : ob} : ID A⁻¹ = id :=
left_inverse_eq_right_inverse inverse_compose id_compose
theorem composition_is_section [instance] {f : mor A B} {g : mor B C}
theorem composition_is_section [instance] {f : mor A B} {g : mor B C}
(Hf : is_section f) (Hg : is_section g) : is_section (g ∘ f) :=
is_section.mk
(calc
(retraction_of f ∘ retraction_of g) ∘ g ∘ f
(retraction_of f ∘ retraction_of g) ∘ g ∘ f
= retraction_of f ∘ retraction_of g ∘ g ∘ f : symm assoc
... = retraction_of f ∘ (retraction_of g ∘ g) ∘ f : {assoc}
... = retraction_of f ∘ id ∘ f : {retraction_compose}
... = retraction_of f ∘ f : {id_left}
... = id : retraction_compose)
theorem composition_is_retraction [instance] {f : mor A B} {g : mor B C}
theorem composition_is_retraction [instance] {f : mor A B} {g : mor B C}
(Hf : is_retraction f) (Hg : is_retraction g) : is_retraction (g ∘ f) :=
is_retraction.mk
(calc
@ -146,7 +146,7 @@ namespace category
... = g ∘ section_of g : {id_left}
... = id : compose_section)
theorem composition_is_inverse [instance] {f : mor A B} {g : mor B C}
theorem composition_is_inverse [instance] {f : mor A B} {g : mor B C}
(Hf : is_iso f) (Hg : is_iso g) : is_iso (g ∘ f) :=
section_retraction_imp_iso _ _
@ -156,7 +156,7 @@ namespace category
∀⦃C⦄ {g h : mor B C}, g ∘ f = h ∘ f → g = h
theorem section_is_mono {f : mor A B} (H : is_section f) : mono f :=
λ C g h H,
λ C g h H,
calc
g = id ∘ g : symm id_left
... = (retraction_of f ∘ f) ∘ g : {symm retraction_compose}
@ -167,7 +167,7 @@ namespace category
... = h : id_left
theorem retraction_is_epi {f : mor A B} (H : is_retraction f) : epi f :=
λ C g h H,
λ C g h H,
calc
g = g ∘ id : symm id_right
... = g ∘ f ∘ section_of f : {symm compose_section}
@ -200,14 +200,14 @@ namespace category
-- check g ∘ f
-- check f ∘op g
-- check f ∘op g = g ∘ f
-- theorem compose_op : f ∘op g = g ∘ f :=
-- theorem compose_op : f ∘op g = g ∘ f :=
-- rfl
-- theorem compose_op {C : category ob} {a b c : ob} {f : @mor ob C a b} {g : @mor ob C b c} : f ∘op g = g ∘ f :=
-- theorem compose_op {C : category ob} {a b c : ob} {f : @mor ob C a b} {g : @mor ob C b c} : f ∘op g = g ∘ f :=
-- rfl
-- theorem op_op {C : category ob} : opposite (opposite C) = C :=
-- rec (λ mor comp id assoc idl idr, sorry) C
end
end
section
--need extensionality