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feat(hott): standardize the naming of definitions proving equality of elements of a structure

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examples:
foo_eq : Pi {A B : foo}, _ -> A = B
foo_mk_eq : Pi _, foo.mk _ = foo.mk _ (if constructor is called "bar", then this becomes "bar_eq")
foo_eq_equiv : Pi {A B : foo}, (A = B) ≃ _

also changed priority of some instances of is_trunc
This commit is contained in:
Floris van Doorn 2015-04-27 17:29:56 -04:00 committed by Leonardo de Moura
parent b70841171a
commit 40086d0084
14 changed files with 61 additions and 54 deletions
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6
hott/algebra/category/constructions/functor.hlean
View file

@ -96,7 +96,7 @@ namespace category
(@componentwise_is_iso _ _ _ _ (to_hom η) (struct η) c)
definition componentwise_iso_id (c : C) : componentwise_iso (iso.refl F) c = iso.refl (F c) :=
iso.eq_mk (idpath (ID (F c)))
iso_eq (idpath (ID (F c)))
definition componentwise_iso_iso_of_eq (p : F = G) (c : C)
: componentwise_iso (iso_of_eq p) c = iso_of_eq (ap010 to_fun_ob p c) :=
@ -133,8 +133,8 @@ namespace category
definition iso_of_eq_eq_of_iso (η : F ≅ G) : iso_of_eq (eq_of_iso η) = η :=
begin
apply iso.eq_mk,
apply nat_trans_eq_mk,
apply iso_eq,
apply nat_trans_eq,
intro c,
rewrite natural_map_hom_of_eq, esimp [eq_of_iso],
rewrite ap010_functor_eq, esimp [hom_of_eq,eq_of_iso_ob],

8
hott/algebra/category/constructions/hset.hlean
View file

@ -42,8 +42,8 @@ namespace category
definition is_equiv_iso_of_equiv (A B : Precategory_hset) : is_equiv (@iso_of_equiv A B) :=
adjointify _ (λf, equiv_of_iso f)
(λf, iso.eq_mk idp)
(λf, equiv.eq_mk idp)
(λf, iso_eq idp)
(λf, equiv_eq idp)
local attribute is_equiv_iso_of_equiv [instance]
open sigma.ops
@ -66,8 +66,8 @@ namespace category
(iso.to_inv f)
(ap10 (right_inverse (to_hom f)))
(ap10 (left_inverse (to_hom f))))
(λf, iso.eq_mk idp)
(λf, equiv.eq_mk idp)
(λf, iso_eq idp)
(λf, equiv_eq idp)
definition equiv_eq_iso (A B : Precategory_hset) : (A ≃ B) = (A ≅ B) :=
ua !equiv_equiv_iso

4
hott/algebra/category/functor.hlean
View file

@ -191,9 +191,9 @@ namespace functor
exact r,
end
-- definition ap010_functor_eq_mk' {F₁ F₂ : C ⇒ D} (p : to_fun_ob F₁ = to_fun_ob F₂)
-- definition ap010_functor_eq' {F₁ F₂ : C ⇒ D} (p : to_fun_ob F₁ = to_fun_ob F₂)
-- (q : p ▹ F₁ = F₂) (c : C) :
-- ap to_fun_ob (functor_eq_mk (apd10 p) (λa b f, _)) = p := sorry
-- ap to_fun_ob (functor_eq (apd10 p) (λa b f, _)) = p := sorry
-- begin
-- cases F₂, revert q, apply (homotopy.rec_on p), clear p, esimp, intros (p, q),
-- cases p, clears (e_1, e_2),

8
hott/algebra/category/iso.hlean
View file

@ -154,12 +154,12 @@ namespace iso
protected definition trans ⦃a b c : ob⦄ (H1 : a ≅ b) (H2 : b ≅ c) : a ≅ c :=
mk (to_hom H2 ∘ to_hom H1)
protected definition eq_mk' {f f' : a ⟶ b} [H : is_iso f] [H' : is_iso f'] (p : f = f')
definition iso_mk_eq {f f' : a ⟶ b} [H : is_iso f] [H' : is_iso f'] (p : f = f')
: iso.mk f = iso.mk f' :=
apd011 iso.mk p !is_hprop.elim
protected definition eq_mk {f f' : a ≅ b} (p : to_hom f = to_hom f') : f = f' :=
by (cases f; cases f'; apply (iso.eq_mk' p))
definition iso_eq {f f' : a ≅ b} (p : to_hom f = to_hom f') : f = f' :=
by (cases f; cases f'; apply (iso_mk_eq p))
-- The structure for isomorphism can be characterized up to equivalence by a sigma type.
protected definition sigma_char ⦃a b : ob⦄ : (Σ (f : hom a b), is_iso f) ≃ (a ≅ b) :=
@ -193,7 +193,7 @@ namespace iso
definition iso_of_eq_con (p : a = b) (q : b = c)
: iso_of_eq (p ⬝ q) = iso.trans (iso_of_eq p) (iso_of_eq q) :=
eq.rec_on q (eq.rec_on p (iso.eq_mk !id_comp⁻¹))
eq.rec_on q (eq.rec_on p (iso_eq !id_comp⁻¹))
section
open funext

24
hott/algebra/category/nat_trans.hlean
View file

@ -38,25 +38,25 @@ namespace nat_trans
protected definition ID [reducible] {C D : Precategory} (F : functor C D) : nat_trans F F :=
id
definition nat_trans_eq_mk' {η₁ η₂ : Π (a : C), hom (F a) (G a)}
definition nat_trans_mk_eq {η₁ η₂ : Π (a : C), hom (F a) (G a)}
(nat₁ : Π (a b : C) (f : hom a b), G f ∘ η₁ a = η₁ b ∘ F f)
(nat₂ : Π (a b : C) (f : hom a b), G f ∘ η₂ a = η₂ b ∘ F f)
(p : η₁ ∼ η₂)
: nat_trans.mk η₁ nat₁ = nat_trans.mk η₂ nat₂ :=
apd011 nat_trans.mk (eq_of_homotopy p) !is_hprop.elim
definition nat_trans_eq_mk {η₁ η₂ : F ⟹ G} : natural_map η₁ ∼ natural_map η₂ → η₁ = η₂ :=
nat_trans.rec_on η₁ (λf₁ nat₁, nat_trans.rec_on η₂ (λf₂ nat₂ p, !nat_trans_eq_mk' p))
definition nat_trans_eq {η₁ η₂ : F ⟹ G} : natural_map η₁ ∼ natural_map η₂ → η₁ = η₂ :=
nat_trans.rec_on η₁ (λf₁ nat₁, nat_trans.rec_on η₂ (λf₂ nat₂ p, !nat_trans_mk_eq p))
protected definition assoc (η₃ : H ⟹ I) (η₂ : G ⟹ H) (η₁ : F ⟹ G) :
η₃ ∘n (η₂ ∘n η₁) = (η₃ ∘n η₂) ∘n η₁ :=
nat_trans_eq_mk (λa, !assoc)
nat_trans_eq (λa, !assoc)
protected definition id_left (η : F ⟹ G) : id ∘n η = η :=
nat_trans_eq_mk (λa, !id_left)
nat_trans_eq (λa, !id_left)
protected definition id_right (η : F ⟹ G) : η ∘n id = η :=
nat_trans_eq_mk (λa, !id_right)
nat_trans_eq (λa, !id_right)
protected definition sigma_char (F G : C ⇒ D) :
(Σ (η : Π (a : C), hom (F a) (G a)), Π (a b : C) (f : hom a b), G f ∘ η a = η b ∘ F f) ≃ (F ⟹ G) :=
@ -69,7 +69,7 @@ namespace nat_trans
fapply sigma.mk,
intro a, exact (H a),
intros [a, b, f], exact (naturality H f),
intro η, apply nat_trans_eq_mk, intro a, apply idp,
intro η, apply nat_trans_eq, intro a, apply idp,
intro S,
fapply sigma_eq,
apply eq_of_homotopy, intro a,
@ -98,21 +98,21 @@ namespace nat_trans
definition functor_nat_trans_compose_commute (η : F ⟹ G) (θ : F' ⟹ G')
: (θ ∘nf G) ∘n (F' ∘fn η) = (G' ∘fn η) ∘n (θ ∘nf F) :=
nat_trans_eq_mk (λc, (naturality θ (η c))⁻¹)
nat_trans_eq (λc, (naturality θ (η c))⁻¹)
definition fn_n_distrib (F' : D ⇒ E) (η : G ⟹ H) (θ : F ⟹ G)
: F' ∘fn (η ∘n θ) = (F' ∘fn η) ∘n (F' ∘fn θ) :=
nat_trans_eq_mk (λc, !respect_comp)
nat_trans_eq (λc, !respect_comp)
definition n_nf_distrib (η : G ⟹ H) (θ : F ⟹ G) (F' : B ⇒ C)
: (η ∘n θ) ∘nf F' = (η ∘nf F') ∘n (θ ∘nf F') :=
nat_trans_eq_mk (λc, idp)
nat_trans_eq (λc, idp)
definition fn_id (F' : D ⇒ E) : F' ∘fn nat_trans.ID F = nat_trans.id :=
nat_trans_eq_mk (λc, !respect_id)
nat_trans_eq (λc, !respect_id)
definition id_nf (F' : B ⇒ C) : nat_trans.ID F ∘nf F' = nat_trans.id :=
nat_trans_eq_mk (λc, idp)
nat_trans_eq (λc, idp)
definition id_fn (η : G ⟹ H) (c : C) : (functor.id ∘fn η) c = η c :=
idp

19
hott/algebra/category/precategory.hlean
View file

@ -85,9 +85,6 @@ namespace category
definition homset [reducible] (x y : ob) : hset :=
hset.mk (hom x y) _
-- definition is_hprop_eq_hom [instance] : is_hprop (f = f') :=
-- !is_trunc_eq
end basic_lemmas
section squares
parameters {ob : Type} [C : precategory ob]
@ -165,11 +162,7 @@ namespace category
/-Characterization of paths between precategories-/
-- auxiliary definition for speeding up precategory_eq_mk
private definition is_hprop_pi (A : Type) (B : A → Type) (H : Π (a : A), is_hprop (B a)) : is_hprop (Π (x : A), B x) :=
is_trunc_pi B (-2 .+1)
definition precategory_eq_mk (ob : Type)
definition precategory_mk'_eq (ob : Type)
(hom1 : ob → ob → Type)
(hom2 : ob → ob → Type)
(homH1 : Π(a b : ob), is_hset (hom1 a b))
@ -203,7 +196,7 @@ namespace category
repeat (apply is_hprop.elim),
end
definition precategory_eq_mk' (ob : Type)
definition precategory_eq (ob : Type)
(C D : precategory ob)
(p : @hom ob C = @hom ob D)
(q : transport (λ x, Πa b c, x b c → x a b → x a c) p
@ -211,10 +204,10 @@ namespace category
(r : transport (λ x, Πa, x a a) p (@ID ob C) = @ID ob D) : C = D :=
begin
cases C, cases D,
apply precategory_eq_mk, apply q, apply r,
apply precategory_mk'_eq, apply q, apply r,
end
definition precategory_eq_mk'' (ob : Type)
definition precategory_mk_eq (ob : Type)
(hom1 : ob → ob → Type)
(hom2 : ob → ob → Type)
(homH1 : Π(a b : ob), is_hset (hom1 a b))
@ -240,7 +233,7 @@ namespace category
precategory.mk hom1 comp1 ID1 assoc1 id_left1 id_right1
= precategory.mk hom2 comp2 ID2 assoc2 id_left2 id_right2 :=
begin
fapply precategory_eq_mk,
fapply precategory_eq,
apply eq_of_homotopy, intros,
apply eq_of_homotopy, intros,
exact (p _ _),
@ -248,7 +241,7 @@ namespace category
exact r,
end
definition Precategory_eq_mk (C D : Precategory)
definition Precategory_eq (C D : Precategory)
(p : carrier C = carrier D)
(q : p ▹ (Precategory.struct C) = Precategory.struct D) : C = D :=
begin

6
hott/algebra/category/yoneda.hlean
View file

@ -68,11 +68,11 @@ namespace functor
(Fhom F f) d = to_fun_hom F (f, id) := idp
theorem functor_curry_id (c : C) : Fhom F (ID c) = nat_trans.id :=
nat_trans_eq_mk (λd, respect_id F _)
nat_trans_eq (λd, respect_id F _)
theorem functor_curry_comp ⦃c c' c'' : C⦄ (f' : c' ⟶ c'') (f : c ⟶ c')
: Fhom F (f' ∘ f) = Fhom F f' ∘n Fhom F f :=
nat_trans_eq_mk (λd, calc
nat_trans_eq (λd, calc
natural_map (Fhom F (f' ∘ f)) d = F (f' ∘ f, id) : functor_curry_hom_def
... = F (f' ∘ f, id ∘ id) : by rewrite id_comp
... = F ((f',id) ∘ (f, id)) : by esimp
@ -158,7 +158,7 @@ namespace functor
begin
fapply functor_eq, exact (functor_curry_functor_uncurry_ob G),
intros [c, c', f],
fapply nat_trans_eq_mk,
fapply nat_trans_eq,
intro d,
apply concat,
{apply (ap (λx, x ∘ _)),

3
hott/init/funext.hlean
View file

@ -54,7 +54,7 @@ section
universe variables l k
parameters [wf : weak_funext.{l k}] {A : Type.{l}} {B : A → Type.{k}} (f : Π x, B x)
definition is_contr_sigma_homotopy [instance] : is_contr (Σ (g : Π x, B x), f ∼ g) :=
definition is_contr_sigma_homotopy : is_contr (Σ (g : Π x, B x), f ∼ g) :=
is_contr.mk (sigma.mk f (homotopy.refl f))
(λ dp, sigma.rec_on dp
(λ (g : Π x, B x) (h : f ∼ g),
@ -75,6 +75,7 @@ section
endt
)
)
local attribute is_contr_sigma_homotopy [instance]
parameters (Q : Π g (h : f ∼ g), Type) (d : Q f (homotopy.refl f))

2
hott/init/hedberg.hlean
View file

@ -44,4 +44,4 @@ section
... = q : aux)
end
attribute is_hset_of_decidable_eq [instance]
attribute is_hset_of_decidable_eq [instance] [priority 600]

5
hott/init/trunc.hlean
View file

@ -118,7 +118,7 @@ namespace is_trunc
/- truncation is upward close -/
-- n-types are also (n+1)-types
definition is_trunc_succ [instance] [priority 100] (A : Type) (n : trunc_index)
definition is_trunc_succ [instance] [priority 900] (A : Type) (n : trunc_index)
[H : is_trunc n A] : is_trunc (n.+1) A :=
trunc_index.rec_on n
(λ A (H : is_contr A), !is_trunc_succ_intro)
@ -182,7 +182,8 @@ namespace is_trunc
/- instances -/
definition is_contr_sigma_eq [instance] {A : Type} (a : A) : is_contr (Σ(x : A), a = x) :=
definition is_contr_sigma_eq [instance] [priority 800] {A : Type} (a : A)
: is_contr (Σ(x : A), a = x) :=
is_contr.mk (sigma.mk a idp) (λp, sigma.rec_on p (λ b q, eq.rec_on q idp))
definition is_contr_unit [instance] : is_contr unit :=

21
hott/types/equiv.hlean
View file

@ -48,7 +48,7 @@ namespace is_equiv
{apply equiv.symm, apply sigma_pi_equiv_pi_sigma},
fapply is_trunc_equiv_closed,
{apply pi_equiv_pi_id, intro a,
apply (equiv_fiber_eq (fiber.mk (u.1 (f a)) (u.2 (f a))) (fiber.mk a idp))},
apply (fiber_eq_equiv (fiber.mk (u.1 (f a)) (u.2 (f a))) (fiber.mk a idp))},
fapply is_trunc_pi,
intro a, fapply @is_contr_eq,
apply is_contr_fiber_of_is_equiv
@ -83,16 +83,29 @@ namespace is_equiv
theorem is_hprop_is_equiv [instance] : is_hprop (is_equiv f) :=
is_hprop_of_imp_is_contr (λ(H : is_equiv f), is_trunc_equiv_closed -2 (equiv.symm !sigma_char'))
/- contractible fibers -/ -- TODO: uncomment this (needs to import instance is_hprop_is_trunc)
-- definition is_contr_fun [reducible] {A B : Type} (f : A → B) := Π(b : B), is_contr (fiber f b)
-- definition is_hprop_is_contr_fun (f : A → B) : is_hprop (is_contr_fun f) := _
-- definition is_equiv_of_is_contr_fun [instance] [H : is_contr_fun f] : is_equiv f :=
-- adjointify _ (λb, point (center (fiber f b)))
-- (λb, point_eq (center (fiber f b)))
-- (λa, ap point (contr (fiber.mk a idp)))
-- definition is_equiv_of_imp_is_equiv (H : B → is_equiv f) : is_equiv f :=
-- @is_equiv_of_is_contr_fun _ _ f (is_contr_fun.mk (λb, @is_contr_fiber_of_is_equiv _ _ _ (H b) _))
end is_equiv
namespace equiv
open is_equiv
variables {A B : Type}
protected definition eq_mk' {f f' : A → B} [H : is_equiv f] [H' : is_equiv f'] (p : f = f')
definition equiv_mk_eq {f f' : A → B} [H : is_equiv f] [H' : is_equiv f'] (p : f = f')
: equiv.mk f H = equiv.mk f' H' :=
apd011 equiv.mk p !is_hprop.elim
protected definition eq_mk {f f' : A ≃ B} (p : to_fun f = to_fun f') : f = f' :=
by (cases f; cases f'; apply (equiv.eq_mk' p))
definition equiv_eq {f f' : A ≃ B} (p : to_fun f = to_fun f') : f = f' :=
by (cases f; cases f'; apply (equiv_mk_eq p))
end equiv

4
hott/types/fiber.hlean
View file

@ -29,7 +29,7 @@ namespace fiber
{intro x, cases x, apply idp},
end
definition equiv_fiber_eq (x y : fiber f b)
definition fiber_eq_equiv (x y : fiber f b)
: (x = y) ≃ (Σ(p : point x = point y), point_eq x = ap f p ⬝ point_eq y) :=
begin
apply equiv.trans,
@ -46,6 +46,6 @@ namespace fiber
definition fiber_eq {x y : fiber f b} (p : point x = point y)
(q : point_eq x = ap f p ⬝ point_eq y) : x = y :=
to_inv !equiv_fiber_eq ⟨p, q⟩
to_inv !fiber_eq_equiv ⟨p, q⟩
end fiber

2
hott/types/sigma.hlean
View file

@ -387,6 +387,6 @@ attribute sigma.is_trunc_sigma [instance] [priority 1505]
open is_trunc sigma prod
/- truncatedness -/
definition prod.is_trunc_prod [instance] [priority 1510] (A B : Type) (n : trunc_index)
definition prod.is_trunc_prod [instance] [priority 1490] (A B : Type) (n : trunc_index)
[HA : is_trunc n A] [HB : is_trunc n B] : is_trunc n (A × B) :=
is_trunc.is_trunc_equiv_closed n !equiv_prod

3
hott/types/trunc.hlean
View file

@ -128,8 +128,7 @@ namespace is_trunc
intro A, apply (trunctype.rec_on A), intros [A1, A2], apply idp,
end
-- set_option pp.notation false
protected definition trunctype.eq (n : trunc_index) (A B : n-Type) :
protected definition trunctype_eq_equiv (n : trunc_index) (A B : n-Type) :
(A = B) ≃ (carrier A = carrier B) :=
calc
(A = B) ≃ (trunctype.sigma_char n A = trunctype.sigma_char n B) : eq_equiv_fn_eq_of_equiv