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style(*): rename is_hprop/is_hset to is_prop/is_set

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Floris van Doorn 2016-02-15 15:18:07 -05:00 committed by Leonardo de Moura
parent 816237315c
commit 4e2cc66061
70 changed files with 384 additions and 384 deletions
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2
hott/algebra/category/category.hlean
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@ -116,7 +116,7 @@ namespace category
apply eq_of_fn_eq_fn !category.sigma_char,
fapply sigma_eq,
{ induction C, induction D, esimp, exact precategory_eq @p q},
{ unfold is_univalent, apply is_hprop.elimo},
{ unfold is_univalent, apply is_prop.elimo},
end
definition category_eq_of_equiv {ob : Type}

6
hott/algebra/category/constructions/comma.hlean
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@ -98,7 +98,7 @@ namespace category
{ intro u, exact (comma_morphism.mk u.1 u.2.1 u.2.2)},
{ intro f, cases f with g h p p', exact ⟨g, h, p⟩},
{ intro f, cases f with g h p p', esimp,
apply ap (comma_morphism.mk' g h p), apply is_hprop.elim},
apply ap (comma_morphism.mk' g h p), apply is_prop.elim},
{ intro u, cases u with u1 u2, cases u2 with u2 u3, reflexivity},
end
@ -113,8 +113,8 @@ namespace category
begin
cases f with g h p₁ p₁', cases f' with g' h' p₂ p₂', cases p, cases q,
apply ap011 (comma_morphism.mk' g' h'),
apply is_hprop.elim,
apply is_hprop.elim
apply is_prop.elim,
apply is_prop.elim
end
definition comma_compose (g : comma_morphism y z) (f : comma_morphism x y) : comma_morphism x z :=

8
hott/algebra/category/constructions/cone.hlean
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@ -60,7 +60,7 @@ namespace category
theorem cone_hom_eq {f f' : cone_hom x y} (q : cone_to_hom f = cone_to_hom f') : f = f' :=
begin
induction f, induction f', esimp at *, induction q, apply ap (cone_hom.mk f),
apply @is_hprop.elim, apply pi.is_trunc_pi, intro x, apply is_trunc_eq, -- type class fails
apply @is_prop.elim, apply pi.is_trunc_pi, intro x, apply is_trunc_eq, -- type class fails
end
variable (F)
@ -120,7 +120,7 @@ namespace category
{ intro v, exact cone_iso.mk v.1 v.2},
{ intro v, induction v with f p, fapply sigma_eq: esimp,
{ apply iso_eq, reflexivity},
{ apply is_hprop.elimo, apply is_trunc_pi, intro i, apply is_hprop_hom_eq}},
{ apply is_prop.elimo, apply is_trunc_pi, intro i, apply is_prop_hom_eq}},
{ intro h, esimp, apply iso_eq, apply cone_hom_eq, reflexivity},
end
@ -134,8 +134,8 @@ namespace category
{ intro v, induction v with p q, induction x with c η, induction y with c' η', esimp at *,
induction p, esimp, fapply sigma_eq: esimp,
{ apply c_cone_obj_eq},
{ apply is_hprop.elimo, apply is_trunc_pi, intro i, apply is_hprop_hom_eq}},
{ intro r, induction r, esimp, induction x, esimp, apply ap02, apply is_hprop.elim},
{ apply is_prop.elimo, apply is_trunc_pi, intro i, apply is_prop_hom_eq}},
{ intro r, induction r, esimp, induction x, esimp, apply ap02, apply is_prop.elim},
end
section is_univalent

8
hott/algebra/category/constructions/discrete.hlean
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@ -31,16 +31,16 @@ namespace category
definition Groupoid_of_1_type [constructor] (A : Type) [H : is_trunc 1 A] : Groupoid :=
groupoid.Mk _ (groupoid_of_1_type A)
definition discrete_precategory [constructor] (A : Type) [H : is_hset A] : precategory A :=
definition discrete_precategory [constructor] (A : Type) [H : is_set A] : precategory A :=
precategory_of_1_type A
definition discrete_groupoid [constructor] (A : Type) [H : is_hset A] : groupoid A :=
definition discrete_groupoid [constructor] (A : Type) [H : is_set A] : groupoid A :=
groupoid_of_1_type A
definition Discrete_precategory [constructor] (A : Type) [H : is_hset A] : Precategory :=
definition Discrete_precategory [constructor] (A : Type) [H : is_set A] : Precategory :=
precategory.Mk (discrete_precategory A)
definition Discrete_groupoid [constructor] (A : Type) [H : is_hset A] : Groupoid :=
definition Discrete_groupoid [constructor] (A : Type) [H : is_set A] : Groupoid :=
groupoid.Mk _ (discrete_groupoid A)
definition c2 [constructor] : Precategory := Discrete_precategory bool

12
hott/algebra/category/constructions/finite_cats.hlean
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@ -14,7 +14,7 @@ open bool unit is_trunc sum eq functor equiv
namespace category
variables {A : Type} (R : A → A → Type) (H : Π⦃a b c⦄, R a b → R b c → empty)
[HR : Πa b, is_hset (R a b)] [HA : is_trunc 1 A]
[HR : Πa b, is_set (R a b)] [HA : is_trunc 1 A]
include H HR HA
@ -63,7 +63,7 @@ namespace category
| f2 : equalizer_category_hom ff tt
open equalizer_category_hom
theorem is_hset_equalizer_category_hom (b₁ b₂ : bool) : is_hset (equalizer_category_hom b₁ b₂) :=
theorem is_set_equalizer_category_hom (b₁ b₂ : bool) : is_set (equalizer_category_hom b₁ b₂) :=
begin
assert H : Πb b', equalizer_category_hom b b' ≃ bool.rec (bool.rec empty bool) (λb, empty) b b',
{ intro b b', fapply equiv.MK,
@ -75,7 +75,7 @@ namespace category
induction b₁: induction b₂: exact _
end
local attribute is_hset_equalizer_category_hom [instance]
local attribute is_set_equalizer_category_hom [instance]
definition equalizer_category [constructor] : Precategory :=
sparse_category
equalizer_category_hom
@ -107,8 +107,8 @@ namespace category
| f2 : pullback_category_hom BL BR
open pullback_category_hom
theorem is_hset_pullback_category_hom (b₁ b₂ : pullback_category_ob)
: is_hset (pullback_category_hom b₁ b₂) :=
theorem is_set_pullback_category_hom (b₁ b₂ : pullback_category_ob)
: is_set (pullback_category_hom b₁ b₂) :=
begin
assert H : Πb b', pullback_category_hom b b' ≃
pullback_category_ob.rec (λb, empty) (λb, empty)
@ -122,7 +122,7 @@ namespace category
induction b₁: induction b₂: exact _
end
local attribute is_hset_pullback_category_hom pullback_category_ob_decidable_equality [instance]
local attribute is_set_pullback_category_hom pullback_category_ob_decidable_equality [instance]
definition pullback_category [constructor] : Precategory :=
sparse_category
pullback_category_hom

4
hott/algebra/category/constructions/functor.hlean
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@ -47,7 +47,7 @@ namespace functor
fapply (apd011 nat_trans.mk),
apply eq_of_homotopy, intro c, apply left_inverse,
apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros,
apply is_hset.elim
apply is_set.elim
end
definition nat_trans_right_inverse : η ∘n nat_trans_inverse η = 1 :=
@ -55,7 +55,7 @@ namespace functor
fapply (apd011 nat_trans.mk),
apply eq_of_homotopy, intro c, apply right_inverse,
apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros,
apply is_hset.elim
apply is_set.elim
end
definition is_natural_iso [constructor] : is_iso η :=

4
hott/algebra/category/constructions/initial.hlean
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@ -38,10 +38,10 @@ namespace category
definition initial_functor_op (C : Precategory)
: (initial_functor C)ᵒᵖᶠ = initial_functor Cᵒᵖ :=
by apply @is_hprop.elim (0 ⇒ Cᵒᵖ)
by apply @is_prop.elim (0 ⇒ Cᵒᵖ)
definition initial_functor_comp {C D : Precategory} (F : C ⇒ D)
: F ∘f initial_functor C = initial_functor D :=
by apply @is_hprop.elim (0 ⇒ D)
by apply @is_prop.elim (0 ⇒ D)
end category

2
hott/algebra/category/constructions/opposite.hlean
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@ -21,7 +21,7 @@ namespace category
(λ a b f, !id_right)
(λ a b f, !id_left)
(λ a, !id_id)
(λ a b, !is_hset_hom)
(λ a b, !is_set_hom)
definition Opposite [reducible] [constructor] (C : Precategory) : Precategory :=
precategory.Mk (opposite C)

6
hott/algebra/category/constructions/sum.hlean
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@ -19,12 +19,12 @@ namespace category
sum.rec (λc, sum.rec (λc', lift (c ⟶ c')) (λd, lift empty))
(λd, sum.rec (λc, lift empty) (λd', lift (d ⟶ d')))
theorem is_hset_sum_hom {obC : Type} {obD : Type}
theorem is_set_sum_hom {obC : Type} {obD : Type}
(C : precategory obC) (D : precategory obD) (x y : obC + obD)
: is_hset (sum_hom C D x y) :=
: is_set (sum_hom C D x y) :=
by induction x: induction y: esimp at *: exact _
local attribute is_hset_sum_hom [instance]
local attribute is_set_sum_hom [instance]
definition precategory_sum [constructor] [instance] (obC obD : Type)
[C : precategory obC] [D : precategory obD] : precategory (obC + obD) :=

4
hott/algebra/category/constructions/terminal.hlean
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@ -32,8 +32,8 @@ namespace category
is_contr.mk (terminal_functor C)
begin
intro F, fapply functor_eq,
{ intro x, apply @is_hprop.elim unit},
{ intro x y f, apply @is_hprop.elim unit}
{ intro x, apply @is_prop.elim unit},
{ intro x y f, apply @is_prop.elim unit}
end
definition terminal_functor_op (C : Precategory)

8
hott/algebra/category/functor/adjoint.hlean
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@ -43,15 +43,15 @@ namespace category
abbreviation counit_unit_eq [unfold 4] := @is_left_adjoint.H
abbreviation unit_counit_eq [unfold 4] := @is_left_adjoint.K
theorem is_hprop_is_left_adjoint [instance] {C : Category} {D : Precategory} (F : C ⇒ D)
: is_hprop (is_left_adjoint F) :=
theorem is_prop_is_left_adjoint [instance] {C : Category} {D : Precategory} (F : C ⇒ D)
: is_prop (is_left_adjoint F) :=
begin
apply is_hprop.mk,
apply is_prop.mk,
intro G G', cases G with G η ε H K, cases G' with G' η' ε' H' K',
assert lem₁ : Π(p : G = G'), p ▸ η = η' → p ▸ ε = ε'
→ is_left_adjoint.mk G η ε H K = is_left_adjoint.mk G' η' ε' H' K',
{ intros p q r, induction p, induction q, induction r, esimp,
apply apd011 (is_left_adjoint.mk G η ε) !is_hprop.elim !is_hprop.elim},
apply apd011 (is_left_adjoint.mk G η ε) !is_prop.elim !is_prop.elim},
assert lem₂ : Π (d : carrier D),
(to_fun_hom G (natural_map ε' d) ∘
natural_map η (to_fun_ob G' d)) ∘

14
hott/algebra/category/functor/attributes.hlean
View file

@ -117,24 +117,24 @@ namespace category
apply is_equiv_of_is_surjective_of_is_embedding,
end
theorem is_hprop_fully_faithful [instance] (F : C ⇒ D) : is_hprop (fully_faithful F) :=
theorem is_prop_fully_faithful [instance] (F : C ⇒ D) : is_prop (fully_faithful F) :=
by unfold fully_faithful; exact _
theorem is_hprop_full [instance] (F : C ⇒ D) : is_hprop (full F) :=
theorem is_prop_full [instance] (F : C ⇒ D) : is_prop (full F) :=
by unfold full; exact _
theorem is_hprop_faithful [instance] (F : C ⇒ D) : is_hprop (faithful F) :=
theorem is_prop_faithful [instance] (F : C ⇒ D) : is_prop (faithful F) :=
by unfold faithful; exact _
theorem is_hprop_essentially_surjective [instance] (F : C ⇒ D)
: is_hprop (essentially_surjective F) :=
theorem is_prop_essentially_surjective [instance] (F : C ⇒ D)
: is_prop (essentially_surjective F) :=
by unfold essentially_surjective; exact _
theorem is_hprop_is_weak_equivalence [instance] (F : C ⇒ D) : is_hprop (is_weak_equivalence F) :=
theorem is_prop_is_weak_equivalence [instance] (F : C ⇒ D) : is_prop (is_weak_equivalence F) :=
by unfold is_weak_equivalence; exact _
definition fully_faithful_equiv (F : C ⇒ D) : fully_faithful F ≃ (faithful F × full F) :=
equiv_of_is_hprop (λH, (faithful_of_fully_faithful F, full_of_fully_faithful F))
equiv_of_is_prop (λH, (faithful_of_fully_faithful F, full_of_fully_faithful F))
(λH, fully_faithful_of_full_of_faithful (pr1 H) (pr2 H))
/- alternative proof using direct calculation with equivalences

14
hott/algebra/category/functor/basic.hlean
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@ -56,7 +56,7 @@ namespace functor
{H₂ : Π(a b : C), hom a b → hom (F₂ a) (F₂ b)} (id₁ id₂ comp₁ comp₂)
(pF : F₁ = F₂) (pH : pF ▸ H₁ = H₂)
: functor.mk F₁ H₁ id₁ comp₁ = functor.mk F₂ H₂ id₂ comp₂ :=
apd01111 functor.mk pF pH !is_hprop.elim !is_hprop.elim
apd01111 functor.mk pF pH !is_prop.elim !is_prop.elim
definition functor_eq' {F₁ F₂ : C ⇒ D} : Π(p : to_fun_ob F₁ = to_fun_ob F₂),
(transport (λx, Πa b f, hom (x a) (x b)) p @(to_fun_hom F₁) = @(to_fun_hom F₂)) → F₁ = F₂ :=
@ -181,10 +181,10 @@ namespace functor
q (respect_comp F g f) end qed
section
local attribute precategory.is_hset_hom [instance] [priority 1001]
local attribute precategory.is_set_hom [instance] [priority 1001]
local attribute trunctype.struct [instance] [priority 1] -- remove after #842 is closed
protected theorem is_hset_functor [instance]
[HD : is_hset D] : is_hset (functor C D) :=
protected theorem is_set_functor [instance]
[HD : is_set D] : is_set (functor C D) :=
by apply is_trunc_equiv_closed; apply functor.sigma_char
end
@ -193,8 +193,8 @@ namespace functor
(id comp) : functor_mk_eq' id id comp comp (idpath F) (idpath H) = idp :=
begin
fapply apd011 (apd01111 functor.mk idp idp),
apply is_hset.elim,
apply is_hset.elim
apply is_set.elim,
apply is_set.elim
end
definition functor_eq'_idp (F : C ⇒ D) : functor_eq' idp idp = (idpath F) :=
@ -210,7 +210,7 @@ namespace functor
theorem functor_eq2' {F₁ F₂ : C ⇒ D} {p₁ p₂ : to_fun_ob F₁ = to_fun_ob F₂} (q₁ q₂)
(r : p₁ = p₂) : functor_eq' p₁ q₁ = functor_eq' p₂ q₂ :=
by cases r; apply (ap (functor_eq' p₂)); apply is_hprop.elim
by cases r; apply (ap (functor_eq' p₂)); apply is_prop.elim
theorem functor_eq2 {F₁ F₂ : C ⇒ D} (p q : F₁ = F₂) (r : ap010 to_fun_ob p ~ ap010 to_fun_ob q)
: p = q :=

10
hott/algebra/category/functor/equivalence.hlean
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@ -228,8 +228,8 @@ namespace category
definition equivalence_of_isomorphism [constructor] (F : C ≅c D) : C ≃c D :=
equivalence.mk F _
theorem is_hprop_is_equivalence [instance] {C : Category} {D : Precategory} (F : C ⇒ D)
: is_hprop (is_equivalence F) :=
theorem is_prop_is_equivalence [instance] {C : Category} {D : Precategory} (F : C ⇒ D)
: is_prop (is_equivalence F) :=
begin
assert f : is_equivalence F ≃ Σ(H : is_left_adjoint F), is_iso (unit F) × is_iso (counit F),
{ fapply equiv.MK,
@ -241,7 +241,7 @@ namespace category
apply is_trunc_equiv_closed_rev, exact f,
end
theorem is_hprop_is_isomorphism [instance] (F : C ⇒ D) : is_hprop (is_isomorphism F) :=
theorem is_prop_is_isomorphism [instance] (F : C ⇒ D) : is_prop (is_isomorphism F) :=
by unfold is_isomorphism; exact _
/- closure properties -/
@ -332,13 +332,13 @@ namespace category
definition equivalence_eq {C : Category} {D : Precategory} {F F' : C ≃c D}
(p : equivalence.to_functor F = equivalence.to_functor F') : F = F' :=
begin
induction F, induction F', exact apd011 equivalence.mk p !is_hprop.elim
induction F, induction F', exact apd011 equivalence.mk p !is_prop.elim
end
definition isomorphism_eq {F F' : C ≅c D}
(p : isomorphism.to_functor F = isomorphism.to_functor F') : F = F' :=
begin
induction F, induction F', exact apd011 isomorphism.mk p !is_hprop.elim
induction F, induction F', exact apd011 isomorphism.mk p !is_prop.elim
end
definition is_equiv_isomorphism_of_equivalence [constructor] (C D : Category)

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

@ -138,12 +138,12 @@ namespace yoneda
section
set_option apply.class_instance false
definition is_hprop_representable {C : Category} (F : Cᵒᵖ ⇒ cset)
: is_hprop (is_representable F) :=
definition is_prop_representable {C : Category} (F : Cᵒᵖ ⇒ cset)
: is_prop (is_representable F) :=
begin
fapply is_trunc_equiv_closed,
{ exact proof fiber.sigma_char ɏ F qed ⬝e sigma.sigma_equiv_sigma_id (λc, !eq_equiv_iso)},
{ apply function.is_hprop_fiber_of_is_embedding, apply is_embedding_yoneda_embedding}
{ apply function.is_prop_fiber_of_is_embedding, apply is_embedding_yoneda_embedding}
end
end

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

@ -27,7 +27,7 @@ namespace category
begin
fapply groupoid.mk, fapply precategory.mk,
intros, exact A,
intros, apply (@group.is_hset_carrier A G),
intros, apply (@group.is_set_carrier A G),
intros [a, b, c, g, h], exact (@group.mul A G g h),
intro a, exact (@group.one A G),
intros, exact (@group.mul_assoc A G h g f)⁻¹,
@ -43,7 +43,7 @@ namespace category
begin
fapply group.mk,
intro f g, apply (comp f g),
apply is_hset_hom,
apply is_set_hom,
intros f g h, apply (assoc f g h)⁻¹,
apply (ID a),
intro f, apply id_left,

12
hott/algebra/category/iso.hlean
View file

@ -114,16 +114,16 @@ namespace iso
[Hf : is_iso f] [Hg : is_iso g] : is_iso (g ∘ f) :=
!is_iso_of_split_epi_of_split_mono
theorem is_hprop_is_iso [instance] (f : hom a b) : is_hprop (is_iso f) :=
theorem is_prop_is_iso [instance] (f : hom a b) : is_prop (is_iso f) :=
begin
apply is_hprop.mk, intro H H',
apply is_prop.mk, intro H H',
cases H with g li ri, cases H' with g' li' ri',
fapply (apd0111 (@is_iso.mk ob C a b f)),
apply left_inverse_eq_right_inverse,
apply li,
apply ri',
apply is_hprop.elim,
apply is_hprop.elim,
apply is_prop.elim,
apply is_prop.elim,
end
end iso open iso
@ -175,7 +175,7 @@ namespace iso
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
apd011 iso.mk p !is_prop.elim
variable {C}
definition iso_eq {f f' : a ≅ b} (p : to_hom f = to_hom f') : f = f' :=
@ -194,7 +194,7 @@ namespace iso
end
-- The type of isomorphisms between two objects is a set
definition is_hset_iso [instance] : is_hset (a ≅ b) :=
definition is_set_iso [instance] : is_set (a ≅ b) :=
begin
apply is_trunc_is_equiv_closed,
apply equiv.to_is_equiv (!iso.sigma_char),

22
hott/algebra/category/limits/colimits.hlean
View file

@ -29,15 +29,15 @@ namespace category
!center
definition hom_initial_eq [H : is_initial c'] (f f' : c' ⟶ c) : f = f' :=
!is_hprop.elim
!is_prop.elim
definition eq_initial_morphism [H : is_initial c'] (f : c' ⟶ c) : f = initial_morphism c c' :=
!is_hprop.elim
!is_prop.elim
definition initial_iso_initial {c c' : ob} (H : is_initial c) (K : is_initial c') : c ≅ c' :=
iso_of_opposite_iso (@terminal_iso_terminal _ (opposite C) _ _ H K)
theorem is_hprop_is_initial [instance] : is_hprop (is_initial c) := _
theorem is_prop_is_initial [instance] : is_prop (is_initial c) := _
omit C
@ -56,9 +56,9 @@ namespace category
initial_iso_initial (@has_initial_object.is_initial D H₁) (@has_initial_object.is_initial D H₂)
set_option pp.coercions true
theorem is_hprop_has_initial_object [instance] (D : Category)
: is_hprop (has_initial_object D) :=
is_hprop_has_terminal_object (Category_opposite D)
theorem is_prop_has_initial_object [instance] (D : Category)
: is_prop (has_initial_object D) :=
is_prop_has_terminal_object (Category_opposite D)
variable (D)
abbreviation has_colimits_of_shape := has_limits_of_shape Dᵒᵖ Iᵒᵖ
@ -76,12 +76,12 @@ namespace category
section
open pi
theorem is_hprop_has_colimits_of_shape [instance] (D : Category) (I : Precategory)
: is_hprop (has_colimits_of_shape D I) :=
is_hprop_has_limits_of_shape (Category_opposite D) _
theorem is_prop_has_colimits_of_shape [instance] (D : Category) (I : Precategory)
: is_prop (has_colimits_of_shape D I) :=
is_prop_has_limits_of_shape (Category_opposite D) _
theorem is_hprop_is_cocomplete [instance] (D : Category) : is_hprop (is_cocomplete D) :=
is_hprop_is_complete (Category_opposite D)
theorem is_prop_is_cocomplete [instance] (D : Category) : is_prop (is_cocomplete D) :=
is_prop_is_complete (Category_opposite D)
end
variables {D I} (F : I ⇒ D) [H : has_colimits_of_shape D I] {i j : I}

22
hott/algebra/category/limits/limits.hlean
View file

@ -26,17 +26,17 @@ namespace category
!center
definition hom_terminal_eq [H : is_terminal c'] (f f' : c ⟶ c') : f = f' :=
!is_hprop.elim
!is_prop.elim
definition eq_terminal_morphism [H : is_terminal c'] (f : c ⟶ c') : f = terminal_morphism c c' :=
!is_hprop.elim
!is_prop.elim
definition terminal_iso_terminal (c c' : ob) [H : is_terminal c] [K : is_terminal c']
: c ≅ c' :=
iso.MK !terminal_morphism !terminal_morphism !hom_terminal_eq !hom_terminal_eq
local attribute is_terminal [reducible]
theorem is_hprop_is_terminal [instance] : is_hprop (is_terminal c) :=
theorem is_prop_is_terminal [instance] : is_prop (is_terminal c) :=
_
omit C
@ -53,13 +53,13 @@ namespace category
: @terminal_object D H₁ ≅ @terminal_object D H₂ :=
!terminal_iso_terminal
theorem is_hprop_has_terminal_object [instance] (D : Category)
: is_hprop (has_terminal_object D) :=
theorem is_prop_has_terminal_object [instance] (D : Category)
: is_prop (has_terminal_object D) :=
begin
apply is_hprop.mk, intro t₁ t₂, induction t₁ with d₁ H₁, induction t₂ with d₂ H₂,
apply is_prop.mk, intro t₁ t₂, induction t₁ with d₁ H₁, induction t₂ with d₂ H₂,
assert p : d₁ = d₂,
{ apply eq_of_iso, apply terminal_iso_terminal},
induction p, exact ap _ !is_hprop.elim
induction p, exact ap _ !is_prop.elim
end
variable (D)
@ -79,12 +79,12 @@ namespace category
section
open pi
theorem is_hprop_has_limits_of_shape [instance] (D : Category) (I : Precategory)
: is_hprop (has_limits_of_shape D I) :=
by apply is_trunc_pi; intro F; exact is_hprop_has_terminal_object (Category_cone F)
theorem is_prop_has_limits_of_shape [instance] (D : Category) (I : Precategory)
: is_prop (has_limits_of_shape D I) :=
by apply is_trunc_pi; intro F; exact is_prop_has_terminal_object (Category_cone F)
local attribute is_complete [reducible]
theorem is_hprop_is_complete [instance] (D : Category) : is_hprop (is_complete D) := _
theorem is_prop_is_complete [instance] (D : Category) : is_prop (is_complete D) := _
end
variables {D I}

6
hott/algebra/category/limits/set.hlean
View file

@ -49,7 +49,7 @@ namespace category
apply eq_of_homotopy, intro x, fapply sigma_eq: esimp,
{ apply eq_of_homotopy, intro i, exact (ap10 (q i) x)⁻¹},
{ with_options [elaborator.ignore_instances true] -- TODO: fix
( refine is_hprop.elimo _ _ _;
( refine is_prop.elimo _ _ _;
refine is_trunc_pi _ _; intro i;
refine is_trunc_pi _ _; intro j;
refine is_trunc_pi _ _; intro f;
@ -73,7 +73,7 @@ namespace category
fapply cone_obj.mk,
{ fapply trunctype.mk,
{ apply set_quotient (is_cocomplete_set_cone_rel.{u v w} I F)},
{ apply is_hset_set_quotient}},
{ apply is_set_set_quotient}},
{ fapply nat_trans.mk,
{ intro i x, esimp, apply class_of, exact ⟨i, x⟩},
{ intro i j f, esimp, apply eq_of_homotopy, intro y, apply eq_of_rel, esimp,
@ -99,7 +99,7 @@ namespace category
{ esimp, intro h, induction h with f q, apply cone_hom_eq, esimp at *,
apply eq_of_homotopy, refine set_quotient.rec _ _,
{ intro v, induction v with i x, esimp, exact (ap10 (q i) x)⁻¹},
{ intro v w r, apply is_hprop.elimo}}},
{ intro v w r, apply is_prop.elimo}}},
end
end category

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

@ -52,7 +52,7 @@ namespace nat_trans
(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
apd011 nat_trans.mk (eq_of_homotopy p) !is_prop.elim
definition nat_trans_eq {η₁ η₂ : F ⟹ G} : natural_map η₁ ~ natural_map η₂ → η₁ = η₂ :=
by induction η₁; induction η₂; apply nat_trans_mk_eq
@ -82,10 +82,10 @@ namespace nat_trans
intro S,
fapply sigma_eq,
{ apply eq_of_homotopy, intro a, apply idp},
{ apply is_hprop.elimo}
{ apply is_prop.elimo}
end
definition is_hset_nat_trans [instance] : is_hset (F ⟹ G) :=
definition is_set_nat_trans [instance] : is_set (F ⟹ G) :=
by apply is_trunc_is_equiv_closed; apply (equiv.to_is_equiv !nat_trans.sigma_char)
definition change_natural_map [constructor] (η : F ⟹ G) (f : Π (a : C), F a ⟶ G a)

16
hott/algebra/category/precategory.hlean
View file

@ -28,10 +28,10 @@ namespace category
(id_left : Π ⦃a b : ob⦄ (f : hom a b), comp !ID f = f)
(id_right : Π ⦃a b : ob⦄ (f : hom a b), comp f !ID = f)
(id_id : Π (a : ob), comp !ID !ID = ID a)
(is_hset_hom : Π(a b : ob), is_hset (hom a b))
(is_set_hom : Π(a b : ob), is_set (hom a b))
-- attribute precategory [multiple-instances] --this is not used anywhere
attribute precategory.is_hset_hom [instance]
attribute precategory.is_set_hom [instance]
infixr ∘ := precategory.comp
-- input ⟶ using \--> (this is a different arrow than \-> (→))
@ -48,15 +48,15 @@ namespace category
abbreviation id_left [unfold 2] := @precategory.id_left
abbreviation id_right [unfold 2] := @precategory.id_right
abbreviation id_id [unfold 2] := @precategory.id_id
abbreviation is_hset_hom [unfold 2] := @precategory.is_hset_hom
abbreviation is_set_hom [unfold 2] := @precategory.is_set_hom
definition is_hprop_hom_eq {ob : Type} [C : precategory ob] {x y : ob} (f g : x ⟶ y)
: is_hprop (f = g) :=
definition is_prop_hom_eq {ob : Type} [C : precategory ob] {x y : ob} (f g : x ⟶ y)
: is_prop (f = g) :=
_
-- the constructor you want to use in practice
protected definition precategory.mk [constructor] {ob : Type} (hom : ob → ob → Type)
[hset : Π (a b : ob), is_hset (hom a b)]
[hset : Π (a b : ob), is_set (hom a b)]
(comp : Π ⦃a b c : ob⦄, hom b c → hom a b → hom a c) (ID : Π (a : ob), hom a a)
(ass : Π ⦃a b c d : ob⦄ (h : hom c d) (g : hom b c) (f : hom a b),
comp h (comp g f) = comp (comp h g) f)
@ -179,7 +179,7 @@ namespace category
assert K : ID1 = ID2,
{ apply eq_of_homotopy, intro a, exact !ir'⁻¹ ⬝ !il},
induction K,
apply ap0111111 (precategory.mk' hom1 comp1 ID1): apply is_hprop.elim
apply ap0111111 (precategory.mk' hom1 comp1 ID1): apply is_prop.elim
end
@ -211,7 +211,7 @@ namespace category
assert K : ID1 = ID2,
{ apply eq_of_homotopy, intros, apply r},
induction H, induction K,
apply ap0111111 (precategory.mk' hom1 comp1 ID1): apply is_hprop.elim
apply ap0111111 (precategory.mk' hom1 comp1 ID1): apply is_prop.elim
end
-/

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

@ -10,12 +10,12 @@ open is_trunc eq
namespace category
structure strict_precategory [class] (ob : Type) extends precategory ob :=
mk' :: (is_hset_ob : is_hset ob)
mk' :: (is_set_ob : is_set ob)
attribute strict_precategory.is_hset_ob [instance]
attribute strict_precategory.is_set_ob [instance]
definition strict_precategory.mk [reducible] {ob : Type} (C : precategory ob)
(H : is_hset ob) : strict_precategory ob :=
(H : is_set ob) : strict_precategory ob :=
precategory.rec_on C strict_precategory.mk' H
structure Strict_precategory : Type :=

12
hott/algebra/group.hlean
View file

@ -19,10 +19,10 @@ variable {A : Type}
namespace algebra
structure semigroup [class] (A : Type) extends has_mul A :=
(is_hset_carrier : is_hset A)
(is_set_carrier : is_set A)
(mul_assoc : Πa b c, mul (mul a b) c = mul a (mul b c))
attribute semigroup.is_hset_carrier [instance]
attribute semigroup.is_set_carrier [instance]
definition mul.assoc [s : semigroup A] (a b c : A) : a * b * c = a * (b * c) :=
!semigroup.mul_assoc
@ -60,10 +60,10 @@ abbreviation eq_of_mul_eq_mul_right' := @mul.right_cancel
/- additive semigroup -/
structure add_semigroup [class] (A : Type) extends has_add A :=
(is_hset_carrier : is_hset A)
(is_set_carrier : is_set A)
(add_assoc : Πa b c, add (add a b) c = add a (add b c))
attribute add_semigroup.is_hset_carrier [instance]
attribute add_semigroup.is_set_carrier [instance]
definition add.assoc [s : add_semigroup A] (a b c : A) : a + b + c = a + (b + c) :=
!add_semigroup.add_assoc
@ -128,7 +128,7 @@ definition add_monoid.to_monoid {A : Type} [s : add_monoid A] : monoid A :=
one := add_monoid.zero A,
mul_one := add_monoid.add_zero,
one_mul := add_monoid.zero_add,
is_hset_carrier := _
is_set_carrier := _
⦄
definition add_comm_monoid.to_comm_monoid {A : Type} [s : add_comm_monoid A] : comm_monoid A :=
@ -585,7 +585,7 @@ definition group_of_add_group (A : Type) [G : add_group A] : group A :=
mul_one := add_zero,
inv := has_neg.neg,
mul_left_inv := add.left_inv,
is_hset_carrier := _⦄
is_set_carrier := _⦄
namespace norm_num
reveal add.assoc

8
hott/algebra/homotopy_group.hlean
View file

@ -50,12 +50,12 @@ namespace eq
prefix `π₁`:95 := fundamental_group
open equiv unit
theorem trivial_homotopy_of_is_hset (A : Type*) [H : is_hset A] (n : ℕ) : πG[n+1] A = G0 :=
theorem trivial_homotopy_of_is_set (A : Type*) [H : is_set A] (n : ℕ) : πG[n+1] A = G0 :=
begin
apply trivial_group_of_is_contr,
apply is_trunc_trunc_of_is_trunc,
apply is_contr_loop_of_is_trunc,
apply is_trunc_succ_succ_of_is_hset
apply is_trunc_succ_succ_of_is_set
end
definition homotopy_group_succ_out (A : Type*) (n : ℕ) : πG[ n +1] A = π₁ Ω[n] A := idp
@ -77,8 +77,8 @@ namespace eq
exact ap (Group_homotopy_group n) !loop_space_succ_eq_in⁻¹}
end
theorem trivial_homotopy_of_is_hset_loop_space {A : Type*} {n : ℕ} (m : ℕ) (H : is_hset (Ω[n] A))
theorem trivial_homotopy_of_is_set_loop_space {A : Type*} {n : ℕ} (m : ℕ) (H : is_set (Ω[n] A))
: πG[m+n+1] A = G0 :=
!homotopy_group_add ⬝ !trivial_homotopy_of_is_hset
!homotopy_group_add ⬝ !trivial_homotopy_of_is_set
end eq

10
hott/algebra/hott.hlean
View file

@ -50,11 +50,11 @@ namespace algebra
... = Hm G1 (Hi g) : by rewrite Gh4
... = Hi g : Gh2),
cases same_one, cases same_inv,
have ps : Gs = Hs, from !is_hprop.elim,
have ph1 : Gh1 = Hh1, from !is_hprop.elim,
have ph2 : Gh2 = Hh2, from !is_hprop.elim,
have ph3 : Gh3 = Hh3, from !is_hprop.elim,
have ph4 : Gh4 = Hh4, from !is_hprop.elim,
have ps : Gs = Hs, from !is_prop.elim,
have ph1 : Gh1 = Hh1, from !is_prop.elim,
have ph2 : Gh2 = Hh2, from !is_prop.elim,
have ph3 : Gh3 = Hh3, from !is_prop.elim,
have ph4 : Gh4 = Hh4, from !is_prop.elim,
cases ps, cases ph1, cases ph2, cases ph3, cases ph4, reflexivity
end

2
hott/algebra/trunc_group.hlean
View file

@ -86,7 +86,7 @@ namespace algebra
mul_one := trunc_mul_one,
inv := trunc_inv,
mul_left_inv := trunc_mul_left_inv,
is_hset_carrier := _⦄
is_set_carrier := _⦄
definition trunc_comm_group [constructor] (mul_comm : ∀a b, mul a b = mul b a) : comm_group G :=

26
hott/choice.hlean
View file

@ -7,22 +7,22 @@ namespace choice
-- 3.8.1. The axiom of choice.
definition AC [reducible] := Π (X : Type.{u}) (A : X -> Type.{u}) (P : Π x, A x -> Type.{u}),
is_hset X -> (Π x, is_hset (A x)) -> (Π x a, is_hprop (P x a)) ->
is_set X -> (Π x, is_set (A x)) -> (Π x a, is_prop (P x a)) ->
(Π x, ∥ Σ a, P x a ∥) -> ∥ Σ f, Π x, P x (f x) ∥
-- 3.8.3. Corresponds to the assertion that
-- "the cartesian product of a family of nonempty sets is nonempty".
definition AC_cart [reducible] := Π (X : Type.{u}) (A : X -> Type.{u}),
is_hset X -> (Π x, is_hset (A x)) -> (Π x, ∥ A x ∥) -> ∥ Π x, A x ∥
is_set X -> (Π x, is_set (A x)) -> (Π x, ∥ A x ∥) -> ∥ Π x, A x ∥
-- A slight variant of AC with a modified (equivalent) codomain.
definition AC' [reducible] := Π (X : Type.{u}) (A : X -> Type.{u}) (P : Π x, A x -> Type.{u}),
is_hset X -> (Π x, is_hset (A x)) -> (Π x a, is_hprop (P x a))
is_set X -> (Π x, is_set (A x)) -> (Π x a, is_prop (P x a))
-> (Π x, ∥ Σ a, P x a ∥) -> ∥ Π x, Σ a : A x, P x a ∥
-- The equivalence of AC and AC' follows from the equivalence of their codomains.
definition AC_equiv_AC' : AC.{u} ≃ AC'.{u} :=
equiv_of_is_hprop
equiv_of_is_prop
(λ H X A P HX HA HP HI, trunc_functor _ (to_fun !sigma_pi_equiv_pi_sigma) (H X A P HX HA HP HI))
(λ H X A P HX HA HP HI, trunc_functor _ (to_inv !sigma_pi_equiv_pi_sigma) (H X A P HX HA HP HI))
@ -39,7 +39,7 @@ namespace choice
-- Which is enough to show AC' ≃ AC_cart, since both are hprops.
definition AC'_equiv_AC_cart : AC'.{u} ≃ AC_cart.{u} :=
equiv_of_is_hprop AC_cart_of_AC'.{u} AC'_of_AC_cart.{u}
equiv_of_is_prop AC_cart_of_AC'.{u} AC'_of_AC_cart.{u}
-- 3.8.2. AC ≃ AC_cart follows by transitivity.
definition AC_equiv_AC_cart : AC.{u} ≃ AC_cart.{u} :=
@ -52,21 +52,21 @@ namespace choice
definition Y : X -> Type.{1} := λ x, x0 = x
definition not_is_hset_X : ¬ is_hset X :=
definition not_is_set_X : ¬ is_set X :=
begin
intro H, apply not_is_hprop_bool_eq_bool,
intro H, apply not_is_prop_bool_eq_bool,
apply @is_trunc_equiv_closed (x0 = x0),
apply equiv.symm !equiv_subtype
end
definition is_hset_x1 (x : X) : is_hset x.1 :=
definition is_set_x1 (x : X) : is_set x.1 :=
by cases x; induction a_1; cases a_1; exact _
definition is_hset_Yx (x : X) : is_hset (Y x) :=
definition is_set_Yx (x : X) : is_set (Y x) :=
begin
apply @is_trunc_equiv_closed _ _ _ !equiv_subtype,
apply @is_trunc_equiv_closed _ _ _ (equiv.symm !eq_equiv_equiv),
apply is_trunc_equiv; repeat (apply is_hset_x1)
apply is_trunc_equiv; repeat (apply is_set_x1)
end
definition trunc_Yx (x : X) : ∥ Y x ∥ :=
@ -81,8 +81,8 @@ namespace choice
-- 3.8.5. There exists a type X and a family Y : X → U such that each Y(x) is a set,
-- but such that (3.8.3) is false.
definition X_must_be_hset : Σ (X : Type.{1}) (Y : X -> Type.{1})
(HA : Π x : X, is_hset (Y x)), ¬ ((Π x : X, ∥ Y x ∥) -> ∥ Π x : X, Y x ∥) :=
⟨X, Y, is_hset_Yx, λ H, trunc.rec_on (H trunc_Yx)
(λ H0, not_is_hset_X (@is_trunc_of_is_contr _ _ (is_contr.mk x0 H0)))⟩
(HA : Π x : X, is_set (Y x)), ¬ ((Π x : X, ∥ Y x ∥) -> ∥ Π x : X, Y x ∥) :=
⟨X, Y, is_set_Yx, λ H, trunc.rec_on (H trunc_Yx)
(λ H0, not_is_set_X (@is_trunc_of_is_contr _ _ (is_contr.mk x0 H0)))⟩
end choice

4
hott/cubical/square.hlean
View file

@ -483,8 +483,8 @@ namespace eq
by induction s₂;induction s₁;constructor
open is_trunc
definition is_hset.elims [H : is_hset A] : square p₁₀ p₁₂ p₀₁ p₂₁ :=
square_of_eq !is_hset.elim
definition is_set.elims [H : is_set A] : square p₁₀ p₁₂ p₀₁ p₂₁ :=
square_of_eq !is_set.elim
-- definition square_of_con_inv_hsquare {p₁ p₂ p₃ p₄ : a₁ = a₂}
-- {t : p₁ = p₂} {b : p₃ = p₄} {l : p₁ = p₃} {r : p₂ = p₄}

46
hott/function.hlean
View file

@ -57,54 +57,54 @@ namespace function
: f a = f a' → a = a' :=
(ap f)⁻¹
definition is_embedding_of_is_injective [HA : is_hset A] [HB : is_hset B]
definition is_embedding_of_is_injective [HA : is_set A] [HB : is_set B]
(H : Π(a a' : A), f a = f a' → a = a') : is_embedding f :=
begin
intro a a',
fapply adjointify,
{exact (H a a')},
{intro p, apply is_hset.elim},
{intro p, apply is_hset.elim}
{intro p, apply is_set.elim},
{intro p, apply is_set.elim}
end
variable (f)
definition is_hprop_is_embedding [instance] : is_hprop (is_embedding f) :=
definition is_prop_is_embedding [instance] : is_prop (is_embedding f) :=
by unfold is_embedding; exact _
definition is_embedding_equiv_is_injective [HA : is_hset A] [HB : is_hset B]
definition is_embedding_equiv_is_injective [HA : is_set A] [HB : is_set B]
: is_embedding f ≃ (Π(a a' : A), f a = f a' → a = a') :=
begin
fapply equiv.MK,
{ apply @is_injective_of_is_embedding},
{ apply is_embedding_of_is_injective},
{ intro H, apply is_hprop.elim},
{ intro H, apply is_hprop.elim, }
{ intro H, apply is_prop.elim},
{ intro H, apply is_prop.elim, }
end
definition is_hprop_fiber_of_is_embedding [H : is_embedding f] (b : B) :
is_hprop (fiber f b) :=
definition is_prop_fiber_of_is_embedding [H : is_embedding f] (b : B) :
is_prop (fiber f b) :=
begin
apply is_hprop.mk, intro v w,
apply is_prop.mk, intro v w,
induction v with a p, induction w with a' q, induction q,
fapply fiber_eq,
{ esimp, apply is_injective_of_is_embedding p},
{ esimp [is_injective_of_is_embedding], symmetry, apply right_inv}
end
definition is_hprop_fun_of_is_embedding [H : is_embedding f] : is_trunc_fun -1 f :=
is_hprop_fiber_of_is_embedding f
definition is_prop_fun_of_is_embedding [H : is_embedding f] : is_trunc_fun -1 f :=
is_prop_fiber_of_is_embedding f
definition is_embedding_of_is_hprop_fun [constructor] [H : is_trunc_fun -1 f] : is_embedding f :=
definition is_embedding_of_is_prop_fun [constructor] [H : is_trunc_fun -1 f] : is_embedding f :=
begin
intro a a', fapply adjointify,
{ intro p, exact ap point (@is_hprop.elim (fiber f (f a')) _ (fiber.mk a p) (fiber.mk a' idp))},
{ intro p, exact ap point (@is_prop.elim (fiber f (f a')) _ (fiber.mk a p) (fiber.mk a' idp))},
{ intro p, rewrite [-ap_compose], esimp, apply ap_con_eq (@point_eq _ _ f (f a'))},
{ intro p, induction p, apply ap (ap point), apply is_hprop_elim_self}
{ intro p, induction p, apply ap (ap point), apply is_prop_elim_self}
end
variable {f}
definition is_surjective_rec_on {P : Type} (H : is_surjective f) (b : B) [Pt : is_hprop P]
definition is_surjective_rec_on {P : Type} (H : is_surjective f) (b : B) [Pt : is_prop P]
(IH : fiber f b → P) : P :=
trunc.rec_on (H b) IH
variable (f)
@ -113,7 +113,7 @@ namespace function
: is_surjective f :=
λb, tr (H b)
definition is_hprop_is_surjective [instance] : is_hprop (is_surjective f) :=
definition is_prop_is_surjective [instance] : is_prop (is_surjective f) :=
by unfold is_surjective; exact _
definition is_weakly_constant_ap [instance] [H : is_weakly_constant f] (a a' : A) :
@ -184,8 +184,8 @@ namespace function
: is_constant (f ∘ point : fiber f b → B) :=
is_constant.mk b (λv, by induction v with a p;exact p)
definition is_embedding_of_is_hprop_fiber [H : Π(b : B), is_hprop (fiber f b)] : is_embedding f :=
is_embedding_of_is_hprop_fun _
definition is_embedding_of_is_prop_fiber [H : Π(b : B), is_prop (fiber f b)] : is_embedding f :=
is_embedding_of_is_prop_fun _
definition is_retraction_of_is_equiv [instance] [H : is_equiv f] : is_retraction f :=
is_retraction.mk f⁻¹ (right_inv f)
@ -208,9 +208,9 @@ namespace function
local attribute is_equiv_of_is_section_of_is_retraction [instance] [priority 10000]
local attribute trunctype.struct [instance] [priority 1] -- remove after #842 is closed
variable (f)
definition is_hprop_is_retraction_prod_is_section : is_hprop (is_retraction f × is_section f) :=
definition is_prop_is_retraction_prod_is_section : is_prop (is_retraction f × is_section f) :=
begin
apply is_hprop_of_imp_is_contr, intro H, induction H with H1 H2,
apply is_prop_of_imp_is_contr, intro H, induction H with H1 H2,
exact _,
end
end
@ -233,11 +233,11 @@ namespace function
exact ap r (center_eq (is_retraction.sect r b)) ⬝ (is_retraction.right_inverse r b)
end
local attribute is_hprop_is_retraction_prod_is_section [instance]
local attribute is_prop_is_retraction_prod_is_section [instance]
definition is_retraction_prod_is_section_equiv_is_equiv [constructor]
: (is_retraction f × is_section f) ≃ is_equiv f :=
begin
apply equiv_of_is_hprop,
apply equiv_of_is_prop,
intro H, induction H, apply is_equiv_of_is_section_of_is_retraction,
intro H, split, repeat exact _
end

8
hott/hit/coeq.hlean
View file

@ -74,12 +74,12 @@ parameters {A B : Type.{u}} (f g : A → B)
(x : A) : transport (elim_type P_i Pcp) (cp x) = Pcp x :=
by rewrite [tr_eq_cast_ap_fn,↑elim_type,elim_cp];apply cast_ua_fn
protected definition rec_hprop {P : coeq → Type} [H : Πx, is_hprop (P x)]
protected definition rec_hprop {P : coeq → Type} [H : Πx, is_prop (P x)]
(P_i : Π(x : B), P (coeq_i x)) (y : coeq) : P y :=
rec P_i (λa, !is_hprop.elimo) y
rec P_i (λa, !is_prop.elimo) y
protected definition elim_hprop {P : Type} [H : is_hprop P] (P_i : B → P) (y : coeq) : P :=
elim P_i (λa, !is_hprop.elim) y
protected definition elim_hprop {P : Type} [H : is_prop P] (P_i : B → P) (y : coeq) : P :=
elim P_i (λa, !is_prop.elim) y
end

16
hott/hit/colimit.hlean
View file

@ -85,13 +85,13 @@ section
{j : J} (x : A (dom j)) : transport (elim_type Pincl Pglue) (cglue j x) = Pglue j x :=
by rewrite [tr_eq_cast_ap_fn,↑elim_type,elim_cglue];apply cast_ua_fn
protected definition rec_hprop {P : colimit → Type} [H : Πx, is_hprop (P x)]
protected definition rec_hprop {P : colimit → Type} [H : Πx, is_prop (P x)]
(Pincl : Π⦃i : I⦄ (x : A i), P (ι x)) (y : colimit) : P y :=
rec Pincl (λa b, !is_hprop.elimo) y
rec Pincl (λa b, !is_prop.elimo) y
protected definition elim_hprop {P : Type} [H : is_hprop P] (Pincl : Π⦃i : I⦄ (x : A i), P)
protected definition elim_hprop {P : Type} [H : is_prop P] (Pincl : Π⦃i : I⦄ (x : A i), P)
(y : colimit) : P :=
elim Pincl (λa b, !is_hprop.elim) y
elim Pincl (λa b, !is_prop.elim) y
end
end colimit
@ -175,13 +175,13 @@ section
: transport (elim_type Pincl Pglue) (glue a) = Pglue a :=
by rewrite [tr_eq_cast_ap_fn,↑elim_type,elim_glue];apply cast_ua_fn
protected definition rec_hprop {P : seq_colim → Type} [H : Πx, is_hprop (P x)]
protected definition rec_hprop {P : seq_colim → Type} [H : Πx, is_prop (P x)]
(Pincl : Π⦃n : ℕ⦄ (a : A n), P (sι a)) (aa : seq_colim) : P aa :=
rec Pincl (λa b, !is_hprop.elimo) aa
rec Pincl (λa b, !is_prop.elimo) aa
protected definition elim_hprop {P : Type} [H : is_hprop P] (Pincl : Π⦃n : ℕ⦄ (a : A n), P)
protected definition elim_hprop {P : Type} [H : is_prop P] (Pincl : Π⦃n : ℕ⦄ (a : A n), P)
: seq_colim → P :=
elim Pincl (λa b, !is_hprop.elim)
elim Pincl (λa b, !is_prop.elim)
end

8
hott/hit/pushout.hlean
View file

@ -83,13 +83,13 @@ parameters {TL BL TR : Type} (f : TL → BL) (g : TL → TR)
: transport (elim_type Pinl Pinr Pglue) (glue x) = Pglue x :=
by rewrite [tr_eq_cast_ap_fn,↑elim_type,elim_glue];apply cast_ua_fn
protected definition rec_hprop {P : pushout → Type} [H : Πx, is_hprop (P x)]
protected definition rec_hprop {P : pushout → Type} [H : Πx, is_prop (P x)]
(Pinl : Π(x : BL), P (inl x)) (Pinr : Π(x : TR), P (inr x)) (y : pushout) :=
rec Pinl Pinr (λx, !is_hprop.elimo) y
rec Pinl Pinr (λx, !is_prop.elimo) y
protected definition elim_hprop {P : Type} [H : is_hprop P] (Pinl : BL → P) (Pinr : TR → P)
protected definition elim_hprop {P : Type} [H : is_prop P] (Pinl : BL → P) (Pinr : TR → P)
(y : pushout) : P :=
elim Pinl Pinr (λa, !is_hprop.elim) y
elim Pinl Pinr (λa, !is_prop.elim) y
end
end pushout

8
hott/hit/quotient.hlean
View file

@ -39,11 +39,11 @@ namespace quotient
end
protected definition rec_hprop {A : Type} {R : A → A → Type} {P : quotient R → Type}
[H : Πx, is_hprop (P x)] (Pc : Π(a : A), P (class_of R a)) (x : quotient R) : P x :=
quotient.rec Pc (λa a' H, !is_hprop.elimo) x
[H : Πx, is_prop (P x)] (Pc : Π(a : A), P (class_of R a)) (x : quotient R) : P x :=
quotient.rec Pc (λa a' H, !is_prop.elimo) x
protected definition elim_hprop {P : Type} [H : is_hprop P] (Pc : A → P) (x : quotient R) : P :=
quotient.elim Pc (λa a' H, !is_hprop.elim) x
protected definition elim_hprop {P : Type} [H : is_prop P] (Pc : A → P) (x : quotient R) : P :=
quotient.elim Pc (λa a' H, !is_prop.elim) x
protected definition elim_type (Pc : A → Type)
(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a ≃ Pc a') : quotient R → Type :=

32
hott/hit/set_quotient.hlean
View file

@ -23,10 +23,10 @@ section
definition eq_of_rel {a a' : A} (H : R a a') : class_of a = class_of a' :=
ap tr (eq_of_rel _ H)
theorem is_hset_set_quotient [instance] : is_hset set_quotient :=
theorem is_set_set_quotient [instance] : is_set set_quotient :=
begin unfold set_quotient, exact _ end
protected definition rec {P : set_quotient → Type} [Pt : Πaa, is_hset (P aa)]
protected definition rec {P : set_quotient → Type} [Pt : Πaa, is_set (P aa)]
(Pc : Π(a : A), P (class_of a)) (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a =[eq_of_rel H] Pc a')
(x : set_quotient) : P x :=
begin
@ -38,24 +38,24 @@ section
end
protected definition rec_on [reducible] {P : set_quotient → Type} (x : set_quotient)
[Pt : Πaa, is_hset (P aa)] (Pc : Π(a : A), P (class_of a))
[Pt : Πaa, is_set (P aa)] (Pc : Π(a : A), P (class_of a))
(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a =[eq_of_rel H] Pc a') : P x :=
rec Pc Pp x
theorem rec_eq_of_rel {P : set_quotient → Type} [Pt : Πaa, is_hset (P aa)]
theorem rec_eq_of_rel {P : set_quotient → Type} [Pt : Πaa, is_set (P aa)]
(Pc : Π(a : A), P (class_of a)) (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a =[eq_of_rel H] Pc a')
{a a' : A} (H : R a a') : apdo (rec Pc Pp) (eq_of_rel H) = Pp H :=
!is_hset.elimo
!is_set.elimo
protected definition elim {P : Type} [Pt : is_hset P] (Pc : A → P)
protected definition elim {P : Type} [Pt : is_set P] (Pc : A → P)
(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a') (x : set_quotient) : P :=
rec Pc (λa a' H, pathover_of_eq (Pp H)) x
protected definition elim_on [reducible] {P : Type} (x : set_quotient) [Pt : is_hset P]
protected definition elim_on [reducible] {P : Type} (x : set_quotient) [Pt : is_set P]
(Pc : A → P) (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a') : P :=
elim Pc Pp x
theorem elim_eq_of_rel {P : Type} [Pt : is_hset P] (Pc : A → P)
theorem elim_eq_of_rel {P : Type} [Pt : is_set P] (Pc : A → P)
(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a') {a a' : A} (H : R a a')
: ap (elim Pc Pp) (eq_of_rel H) = Pp H :=
begin
@ -63,13 +63,13 @@ section
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑elim,rec_eq_of_rel],
end
protected definition rec_hprop {P : set_quotient → Type} [Pt : Πaa, is_hprop (P aa)]
protected definition rec_hprop {P : set_quotient → Type} [Pt : Πaa, is_prop (P aa)]
(Pc : Π(a : A), P (class_of a)) (x : set_quotient) : P x :=
rec Pc (λa a' H, !is_hprop.elimo) x
rec Pc (λa a' H, !is_prop.elimo) x
protected definition elim_hprop {P : Type} [Pt : is_hprop P] (Pc : A → P) (x : set_quotient)
protected definition elim_hprop {P : Type} [Pt : is_prop P] (Pc : A → P) (x : set_quotient)
: P :=
elim Pc (λa a' H, !is_hprop.elim) x
elim Pc (λa a' H, !is_prop.elim) x
end
end set_quotient
@ -84,11 +84,11 @@ namespace set_quotient
variables {A B C : Type} (R : A → A → hprop) (S : B → B → hprop) (T : C → C → hprop)
definition is_surjective_class_of : is_surjective (class_of : A → set_quotient R) :=
λx, set_quotient.rec_on x (λa, tr (fiber.mk a idp)) (λa a' r, !is_hprop.elimo)
λx, set_quotient.rec_on x (λa, tr (fiber.mk a idp)) (λa a' r, !is_prop.elimo)
/- non-dependent universal property -/
definition set_quotient_arrow_equiv (B : Type) [H : is_hset B] :
definition set_quotient_arrow_equiv (B : Type) [H : is_set B] :
(set_quotient R → B) ≃ (Σ(f : A → B), Π(a a' : A), R a a' → f a = f a') :=
begin
fapply equiv.MK,
@ -108,7 +108,7 @@ namespace set_quotient
intros a' a'' H1,
refine to_inv !trunctype_eq_equiv _, esimp,
apply ua,
apply equiv_of_is_hprop,
apply equiv_of_is_prop,
{ intro H2, exact is_transitive.trans R H2 H1},
{ intro H2, apply is_transitive.trans R H2, exact is_symmetric.symm R H1}
end
@ -139,7 +139,7 @@ namespace set_quotient
{ intro b b' s, apply eq_of_rel, apply H, apply is_reflexive.refl R, exact s}},
{ intro a a' r, apply eq_of_homotopy, refine set_quotient.rec _ _,
{ intro b, esimp, apply eq_of_rel, apply H, exact r, apply is_reflexive.refl S},
{ intro b b' s, apply eq_pathover, esimp, apply is_hset.elims}}
{ intro b b' s, apply eq_pathover, esimp, apply is_set.elims}}
end
end set_quotient

2
hott/hit/trunc.hlean
View file

@ -117,7 +117,7 @@ namespace trunc
definition or.intro_left [reducible] [constructor] (x : X) : X ∨ Y := tr (inl x)
definition or.intro_right [reducible] [constructor] (y : Y) : X ∨ Y := tr (inr y)
definition is_contr_of_merely_hprop [H : is_hprop A] (aa : merely A) : is_contr A :=
definition is_contr_of_merely_hprop [H : is_prop A] (aa : merely A) : is_contr A :=
is_contr_of_inhabited_hprop (trunc.rec_on aa id)
section

4
hott/homotopy/circle.hlean
View file

@ -219,7 +219,7 @@ namespace circle
exact ap succ p},
{ intros n p, rewrite [↑circle.encode, nat_succ_eq_int_succ, neg_succ, -power_con_inv,
@con_tr _ circle.code, transport_code_loop_inv, ↑[circle.encode] at p, p, -neg_succ] }},
{ apply pathover_of_tr_eq, apply eq_of_homotopy, intro a, apply @is_hset.elim,
{ apply pathover_of_tr_eq, apply eq_of_homotopy, intro a, apply @is_set.elim,
esimp, exact _} end end},
{ intro p, cases p, exact idp},
end
@ -254,7 +254,7 @@ namespace circle
open nat
definition homotopy_group_of_circle (n : ℕ) : πG[n+1 +1] S¹. = G0 :=
begin
refine @trivial_homotopy_of_is_hset_loop_space S¹. 1 n _,
refine @trivial_homotopy_of_is_set_loop_space S¹. 1 n _,
apply is_trunc_equiv_closed_rev, apply base_eq_base_equiv
end

2
hott/homotopy/connectedness.hlean
View file

@ -255,7 +255,7 @@ namespace homotopy
apply pathover_of_tr_eq,
rewrite [transport_eq_Fr,idp_con],
revert H, induction n with [n, IH],
{ intro H, apply is_hprop.elim },
{ intro H, apply is_prop.elim },
{ intros H,
change ap (@tr n .+2 (susp A)) (merid a) = ap tr (merid a'),
generalize a',

2
hott/homotopy/sphere.hlean
View file

@ -190,7 +190,7 @@ namespace is_trunc
let H' := iff.elim_left (is_trunc_iff_is_contr_loop n A) H a,
note H'' := @is_trunc_equiv_closed_rev _ _ _ !pmap_sphere H',
assert p : (f = pmap.mk (λx, f base) (respect_pt f)),
apply is_hprop.elim,
apply is_prop.elim,
exact ap10 (ap pmap.to_fun p) x
end

8
hott/hprop_trunc.hlean
View file

@ -35,16 +35,16 @@ namespace is_trunc
induction (P a b), apply idp},
end
definition is_hprop_is_trunc [instance] (n : trunc_index) :
Π (A : Type), is_hprop (is_trunc n A) :=
definition is_prop_is_trunc [instance] (n : trunc_index) :
Π (A : Type), is_prop (is_trunc n A) :=
begin
induction n,
{ intro A,
apply is_trunc_is_equiv_closed,
{ apply equiv.to_is_equiv, apply is_contr.sigma_char},
apply is_hprop.mk, intros,
apply is_prop.mk, intros,
fapply sigma_eq, apply x.2,
apply is_hprop.elimo},
apply is_prop.elimo},
{ intro A,
apply is_trunc_is_equiv_closed,
apply equiv.to_is_equiv,

6
hott/init/hedberg.hlean
View file

@ -37,11 +37,11 @@ section
eq.rec_on (f (refl x)) rfl,
eq.rec_on p aux
definition is_hset_of_decidable_eq : is_hset A :=
is_hset.mk A (λ x y p q, calc
definition is_set_of_decidable_eq : is_set A :=
is_set.mk A (λ x y p q, calc
p = (f (refl x))⁻¹ ⬝ (f p) : aux
... = (f (refl x))⁻¹ ⬝ (f q) : f_const
... = q : aux)
end
attribute is_hset_of_decidable_eq [instance] [priority 600]
attribute is_set_of_decidable_eq [instance] [priority 600]

60
hott/init/trunc.hlean
View file

@ -108,8 +108,8 @@ open nat num is_trunc.trunc_index
namespace is_trunc
abbreviation is_contr := is_trunc -2
abbreviation is_hprop := is_trunc -1
abbreviation is_hset := is_trunc 0
abbreviation is_prop := is_trunc -1
abbreviation is_set := is_trunc 0
variables {A B : Type}
@ -184,41 +184,41 @@ namespace is_trunc
definition is_trunc_of_is_contr (A : Type) (n : trunc_index) [H : is_contr A] : is_trunc n A :=
trunc_index.rec_on n H (λn H, _)
definition is_trunc_succ_of_is_hprop (A : Type) (n : trunc_index) [H : is_hprop A]
definition is_trunc_succ_of_is_prop (A : Type) (n : trunc_index) [H : is_prop A]
: is_trunc (n.+1) A :=
is_trunc_of_leq A (show -1 ≤ n.+1, from star)
definition is_trunc_succ_succ_of_is_hset (A : Type) (n : trunc_index) [H : is_hset A]
definition is_trunc_succ_succ_of_is_set (A : Type) (n : trunc_index) [H : is_set A]
: is_trunc (n.+2) A :=
@(is_trunc_of_leq A (show 0 ≤ n.+2, from proof star qed)) H
/- hprops -/
definition is_hprop.elim [H : is_hprop A] (x y : A) : x = y :=
definition is_prop.elim [H : is_prop A] (x y : A) : x = y :=
!center
definition is_contr_of_inhabited_hprop {A : Type} [H : is_hprop A] (x : A) : is_contr A :=
is_contr.mk x (λy, !is_hprop.elim)
definition is_contr_of_inhabited_hprop {A : Type} [H : is_prop A] (x : A) : is_contr A :=
is_contr.mk x (λy, !is_prop.elim)
theorem is_hprop_of_imp_is_contr {A : Type} (H : A → is_contr A) : is_hprop A :=
theorem is_prop_of_imp_is_contr {A : Type} (H : A → is_contr A) : is_prop A :=
@is_trunc_succ_intro A -2
(λx y,
have H2 [visible] : is_contr A, from H x,
!is_contr_eq)
theorem is_hprop.mk {A : Type} (H : ∀x y : A, x = y) : is_hprop A :=
is_hprop_of_imp_is_contr (λ x, is_contr.mk x (H x))
theorem is_prop.mk {A : Type} (H : ∀x y : A, x = y) : is_prop A :=
is_prop_of_imp_is_contr (λ x, is_contr.mk x (H x))
theorem is_hprop_elim_self {A : Type} {H : is_hprop A} (x : A) : is_hprop.elim x x = idp :=
!is_hprop.elim
theorem is_prop_elim_self {A : Type} {H : is_prop A} (x : A) : is_prop.elim x x = idp :=
!is_prop.elim
/- hsets -/
theorem is_hset.mk (A : Type) (H : ∀(x y : A) (p q : x = y), p = q) : is_hset A :=
@is_trunc_succ_intro _ _ (λ x y, is_hprop.mk (H x y))
theorem is_set.mk (A : Type) (H : ∀(x y : A) (p q : x = y), p = q) : is_set A :=
@is_trunc_succ_intro _ _ (λ x y, is_prop.mk (H x y))
definition is_hset.elim [H : is_hset A] ⦃x y : A⦄ (p q : x = y) : p = q :=
!is_hprop.elim
definition is_set.elim [H : is_set A] ⦃x y : A⦄ (p q : x = y) : p = q :=
!is_prop.elim
/- instances -/
@ -241,16 +241,16 @@ namespace is_trunc
definition is_contr_unit : is_contr unit :=
is_contr.mk star (λp, unit.rec_on p idp)
definition is_hprop_empty : is_hprop empty :=
is_hprop.mk (λx, !empty.elim x)
definition is_prop_empty : is_prop empty :=
is_prop.mk (λx, !empty.elim x)
local attribute is_contr_unit is_hprop_empty [instance]
local attribute is_contr_unit is_prop_empty [instance]
definition is_trunc_unit [instance] (n : trunc_index) : is_trunc n unit :=
!is_trunc_of_is_contr
definition is_trunc_empty [instance] (n : trunc_index) : is_trunc (n.+1) empty :=
!is_trunc_succ_of_is_hprop
!is_trunc_succ_of_is_prop
/- interaction with equivalences -/
@ -289,16 +289,16 @@ namespace is_trunc
: is_trunc n A :=
is_trunc_is_equiv_closed n (to_inv f)
definition is_equiv_of_is_hprop [constructor] [HA : is_hprop A] [HB : is_hprop B]
definition is_equiv_of_is_prop [constructor] [HA : is_prop A] [HB : is_prop B]
(f : A → B) (g : B → A) : is_equiv f :=
is_equiv.mk f g (λb, !is_hprop.elim) (λa, !is_hprop.elim) (λa, !is_hset.elim)
is_equiv.mk f g (λb, !is_prop.elim) (λa, !is_prop.elim) (λa, !is_set.elim)
definition equiv_of_is_hprop [constructor] [HA : is_hprop A] [HB : is_hprop B]
definition equiv_of_is_prop [constructor] [HA : is_prop A] [HB : is_prop B]
(f : A → B) (g : B → A) : A ≃ B :=
equiv.mk f (is_equiv_of_is_hprop f g)
equiv.mk f (is_equiv_of_is_prop f g)
definition equiv_of_iff_of_is_hprop [unfold 5] [HA : is_hprop A] [HB : is_hprop B] (H : A ↔ B) : A ≃ B :=
equiv_of_is_hprop (iff.elim_left H) (iff.elim_right H)
definition equiv_of_iff_of_is_prop [unfold 5] [HA : is_prop A] [HB : is_prop B] (H : A ↔ B) : A ≃ B :=
equiv_of_is_prop (iff.elim_left H) (iff.elim_right H)
/- truncatedness of lift -/
definition is_trunc_lift [instance] [priority 1450] (A : Type) (n : trunc_index)
@ -324,16 +324,16 @@ namespace is_trunc
{a a₂ : A} (p : a = a₂)
(c : C a) (c₂ : C a₂)
definition is_hprop.elimo [H : is_hprop (C a)] : c =[p] c₂ :=
pathover_of_eq_tr !is_hprop.elim
definition is_prop.elimo [H : is_prop (C a)] : c =[p] c₂ :=
pathover_of_eq_tr !is_prop.elim
definition is_trunc_pathover [instance]
(n : trunc_index) [H : is_trunc (n.+1) (C a)] : is_trunc n (c =[p] c₂) :=
is_trunc_equiv_closed_rev n !pathover_equiv_eq_tr
variables {p c c₂}
theorem is_hset.elimo (q q' : c =[p] c₂) [H : is_hset (C a)] : q = q' :=
!is_hprop.elim
theorem is_set.elimo (q q' : c =[p] c₂) [H : is_set (C a)] : q = q' :=
!is_prop.elim
-- TODO: port "Truncated morphisms"

4
hott/init/util.hlean
View file

@ -11,6 +11,6 @@ import init.trunc
open is_trunc
definition eq_rec_eq.{l₁ l₂} {A : Type.{l₁}} {B : A → Type.{l₂}} [h : is_hset A] {a : A} (b : B a) (p : a = a) :
definition eq_rec_eq.{l₁ l₂} {A : Type.{l₁}} {B : A → Type.{l₂}} [h : is_set A] {a : A} (b : B a) (p : a = a) :
b = @eq.rec.{l₂ l₁} A a (λ (a' : A) (h : a = a'), B a') b a p :=
eq.rec_on (is_hset.elim (eq.refl a) p) (eq.refl (eq.rec_on (eq.refl a) b))
eq.rec_on (is_set.elim (eq.refl a) p) (eq.refl (eq.rec_on (eq.refl a) b))

6
hott/logic.hlean
View file

@ -9,10 +9,10 @@ import types.pi
open eq is_trunc decidable
theorem dif_pos {c : Type} [H : decidable c] [P : is_hprop c] (Hc : c)
theorem dif_pos {c : Type} [H : decidable c] [P : is_prop c] (Hc : c)
{A : Type} {t : c → A} {e : ¬ c → A} : dite c t e = t Hc :=
by induction H with Hc Hnc; apply ap t; apply is_hprop.elim; apply absurd Hc Hnc
by induction H with Hc Hnc; apply ap t; apply is_prop.elim; apply absurd Hc Hnc
theorem dif_neg {c : Type} [H : decidable c] (Hnc : ¬c)
{A : Type} {t : c → A} {e : ¬ c → A} : dite c t e = e Hnc :=
by induction H with Hc Hnc; apply absurd Hc Hnc; apply ap e; apply is_hprop.elim
by induction H with Hc Hnc; apply absurd Hc Hnc; apply ap e; apply is_prop.elim

6
hott/types/bool.hlean
View file

@ -155,9 +155,9 @@ namespace bool
assert H2 : bnot = id, from !cast_ua_fn⁻¹ ⬝ ap cast H,
absurd (ap10 H2 tt) ff_ne_tt
theorem is_hset_bool : is_hset bool := _
theorem not_is_hprop_bool_eq_bool : ¬ is_hprop (bool = bool) :=
λ H, eq_bnot_ne_idp !is_hprop.elim
theorem is_set_bool : is_set bool := _
theorem not_is_prop_bool_eq_bool : ¬ is_prop (bool = bool) :=
λ H, eq_bnot_ne_idp !is_prop.elim
definition bool_equiv_option_unit [constructor] : bool ≃ option unit :=
begin

6
hott/types/eq.hlean
View file

@ -460,7 +460,7 @@ namespace eq
(p ⬝ q) ▸ (center (Σa, code a)).2
definition decode' {a : A} (c : code a) : a₀ = a :=
(is_hprop.elim ⟨a₀, encode idp⟩ ⟨a, c⟩)..1
(is_prop.elim ⟨a₀, encode idp⟩ ⟨a, c⟩)..1
definition decode {a : A} (c : code a) : a₀ = a :=
(decode' (encode idp))⁻¹ ⬝ decode' c
@ -472,8 +472,8 @@ namespace eq
{ exact decode},
{ intro c,
unfold [encode, decode, decode'],
induction p, esimp, rewrite [is_hprop_elim_self,▸*,+idp_con], apply tr_eq_of_pathover,
eapply @sigma.rec_on _ _ (λx, x.2 =[(is_hprop.elim ⟨x.1, x.2⟩ ⟨a, c⟩)..1] c)
induction p, esimp, rewrite [is_prop_elim_self,▸*,+idp_con], apply tr_eq_of_pathover,
eapply @sigma.rec_on _ _ (λx, x.2 =[(is_prop.elim ⟨x.1, x.2⟩ ⟨a, c⟩)..1] c)
(center (sigma code)), -- BUG(?): induction fails
intro a c, apply eq_pr2},
{ intro q, induction q, esimp, apply con.left_inv, },

28
hott/types/equiv.hlean
View file

@ -75,19 +75,19 @@ namespace is_equiv
local attribute is_contr_right_inverse [instance] [priority 1600]
local attribute is_contr_right_coherence [instance] [priority 1600]
theorem is_hprop_is_equiv [instance] : is_hprop (is_equiv f) :=
is_hprop_of_imp_is_contr
theorem is_prop_is_equiv [instance] : is_prop (is_equiv f) :=
is_prop_of_imp_is_contr
(λ(H : is_equiv f), is_trunc_equiv_closed -2 (equiv.symm !is_equiv.sigma_char'))
definition inv_eq_inv {A B : Type} {f f' : A → B} {Hf : is_equiv f} {Hf' : is_equiv f'}
(p : f = f') : f⁻¹ = f'⁻¹ :=
apd011 inv p !is_hprop.elim
apd011 inv p !is_prop.elim
/- contractible fibers -/
definition is_contr_fun_of_is_equiv [H : is_equiv f] : is_contr_fun f :=
is_contr_fiber_of_is_equiv f
definition is_hprop_is_contr_fun (f : A → B) : is_hprop (is_contr_fun f) := _
definition is_prop_is_contr_fun (f : A → B) : is_prop (is_contr_fun f) := _
definition is_equiv_of_is_contr_fun [H : is_contr_fun f] : is_equiv f :=
adjointify _ (λb, point (center (fiber f b)))
@ -98,7 +98,7 @@ namespace is_equiv
@is_equiv_of_is_contr_fun _ _ f (λb, @is_contr_fiber_of_is_equiv _ _ _ (H b) _)
definition is_equiv_equiv_is_contr_fun : is_equiv f ≃ is_contr_fun f :=
equiv_of_is_hprop _ (λH, !is_equiv_of_is_contr_fun)
equiv_of_is_prop _ (λH, !is_equiv_of_is_contr_fun)
end is_equiv
@ -129,7 +129,7 @@ namespace equiv
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
apd011 equiv.mk p !is_prop.elim
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))
@ -164,11 +164,11 @@ namespace equiv
: is_equiv (ap to_fun : f = f' → to_fun f = to_fun f') :=
begin
fapply adjointify,
{intro p, cases f with f H, cases f' with f' H', cases p, apply ap (mk f'), apply is_hprop.elim},
{intro p, cases f with f H, cases f' with f' H', cases p, apply ap (mk f'), apply is_prop.elim},
{intro p, cases f with f H, cases f' with f' H', cases p,
apply @concat _ _ (ap to_fun (ap (equiv.mk f') (is_hprop.elim H H'))), {apply idp},
generalize is_hprop.elim H H', intro q, cases q, apply idp},
{intro p, cases p, cases f with f H, apply ap (ap (equiv.mk f)), apply is_hset.elim}
apply @concat _ _ (ap to_fun (ap (equiv.mk f') (is_prop.elim H H'))), {apply idp},
generalize is_prop.elim H H', intro q, cases q, apply idp},
{intro p, cases p, cases f with f H, apply ap (ap (equiv.mk f)), apply is_set.elim}
end
definition equiv_pathover {A : Type} {a a' : A} (p : a = a')
@ -179,21 +179,21 @@ namespace equiv
{ intro a, apply equiv.sigma_char},
{ fapply sigma_pathover,
esimp, apply arrow_pathover, exact r,
apply is_hprop.elimo}
apply is_prop.elimo}
end
definition is_contr_equiv (A B : Type) [HA : is_contr A] [HB : is_contr B] : is_contr (A ≃ B) :=
begin
apply @is_contr_of_inhabited_hprop, apply is_hprop.mk,
apply @is_contr_of_inhabited_hprop, apply is_prop.mk,
intro x y, cases x with fx Hx, cases y with fy Hy, generalize Hy,
apply (eq_of_homotopy (λ a, !eq_of_is_contr)) ▸ (λ Hy, !is_hprop.elim ▸ rfl),
apply (eq_of_homotopy (λ a, !eq_of_is_contr)) ▸ (λ Hy, !is_prop.elim ▸ rfl),
apply equiv_of_is_contr_of_is_contr
end
definition is_trunc_succ_equiv (n : trunc_index) (A B : Type)
[HA : is_trunc n.+1 A] [HB : is_trunc n.+1 B] : is_trunc n.+1 (A ≃ B) :=
@is_trunc_equiv_closed _ _ n.+1 (equiv.symm !equiv.sigma_char)
(@is_trunc_sigma _ _ _ _ (λ f, !is_trunc_succ_of_is_hprop))
(@is_trunc_sigma _ _ _ _ (λ f, !is_trunc_succ_of_is_prop))
definition is_trunc_equiv (n : trunc_index) (A B : Type)
[HA : is_trunc n A] [HB : is_trunc n B] : is_trunc n (A ≃ B) :=

2
hott/types/fiber.hlean
View file

@ -105,7 +105,7 @@ namespace fiber
{ intro a, apply fiber.mk a, reflexivity },
{ intro a, reflexivity },
{ intro f, cases f with a H, change fiber.mk a (refl star) = fiber.mk a H,
rewrite [is_hset.elim H (refl star)] }
rewrite [is_set.elim H (refl star)] }
end
definition fiber_const_equiv (A : Type) (a₀ : A) (a : A)

14
hott/types/fin.hlean
View file

@ -28,7 +28,7 @@ begin
intro i, cases i with i ilt, reflexivity
end
definition is_hset_fin [instance] : is_hset (fin n) :=
definition is_set_fin [instance] : is_set (fin n) :=
begin
apply is_trunc_equiv_closed, apply equiv.symm, apply sigma_char,
end
@ -36,11 +36,11 @@ end
definition eq_of_veq : Π {i j : fin n}, (val i) = j → i = j :=
begin
intro i j, cases i with i ilt, cases j with j jlt, esimp,
intro p, induction p, apply ap (mk i), apply !is_hprop.elim
intro p, induction p, apply ap (mk i), apply !is_prop.elim
end
definition eq_of_veq_refl (i : fin n) : eq_of_veq (refl (val i)) = idp :=
!is_hprop.elim
!is_prop.elim
definition veq_of_eq : Π {i j : fin n}, i = j → (val i) = j :=
by intro i j P; apply ap val; exact P
@ -379,9 +379,9 @@ begin
apply decidable.rec,
{ intro ilt', esimp[val_inj], apply concat,
apply ap (λ x, eq.rec_on x _), esimp[eq_of_veq, rfl], reflexivity,
assert H : ilt = ilt', apply is_hprop.elim, cases H,
assert H' : is_hprop.elim (lt_add_of_lt_right ilt 1) (lt_add_of_lt_right ilt 1) = idp,
apply is_hprop.elim,
assert H : ilt = ilt', apply is_prop.elim, cases H,
assert H' : is_prop.elim (lt_add_of_lt_right ilt 1) (lt_add_of_lt_right ilt 1) = idp,
apply is_prop.elim,
krewrite H' },
{ intro a, contradiction },
end
@ -395,7 +395,7 @@ begin
{ intro a, apply absurd a !nat.lt_irrefl },
{ intro a, esimp[val_inj], apply concat,
assert H : (le.antisymm (le_of_lt_succ (lt_succ_self n)) (le_of_not_gt a))⁻¹ = idp,
apply is_hprop.elim,
apply is_prop.elim,
apply ap _ H, krewrite eq_of_veq_refl },
end

2
hott/types/int/basic.hlean
View file

@ -573,7 +573,7 @@ protected definition integral_domain [reducible] [trans_instance] : integral_dom
mul_comm := int.mul_comm,
zero_ne_one := int.zero_ne_one,
eq_zero_sum_eq_zero_of_mul_eq_zero := @int.eq_zero_sum_eq_zero_of_mul_eq_zero,
is_hset_carrier := is_hset_of_decidable_eq⦄
is_set_carrier := is_set_of_decidable_eq⦄
definition int_has_sub [reducible] [instance] [priority int.prio] : has_sub int :=
has_sub.mk has_sub.sub

2
hott/types/list.hlean
View file

@ -137,7 +137,7 @@ rfl
theorem last_congr {l₁ l₂ : list T} (h₁ : l₁ ≠ []) (h₂ : l₂ ≠ []) (h₃ : l₁ = l₂)
: last l₁ h₁ = last l₂ h₂ :=
apd011 last h₃ !is_hprop.elim
apd011 last h₃ !is_prop.elim
theorem last_concat {x : T} : ∀ {l : list T} (h : concat x l ≠ []), last (concat x l) h = x
| [] h := rfl

2
hott/types/nat/basic.hlean
View file

@ -305,7 +305,7 @@ protected definition comm_semiring [reducible] [trans_instance] : comm_semiring
zero_mul := nat.zero_mul,
mul_zero := nat.mul_zero,
mul_comm := nat.mul_comm,
is_hset_carrier:= _⦄
is_set_carrier:= _⦄
end nat
section

24
hott/types/nat/hott.hlean
View file

@ -11,32 +11,32 @@ import .order
open is_trunc unit empty eq equiv algebra
namespace nat
definition is_hprop_le [instance] (n m : ℕ) : is_hprop (n ≤ m) :=
definition is_prop_le [instance] (n m : ℕ) : is_prop (n ≤ m) :=
begin
assert lem : Π{n m : ℕ} (p : n ≤ m) (q : n = m), p = q ▸ le.refl n,
{ intros, cases p,
{ assert H' : q = idp, apply is_hset.elim,
{ assert H' : q = idp, apply is_set.elim,
cases H', reflexivity},
{ cases q, exfalso, apply not_succ_le_self a}},
apply is_hprop.mk, intro H1 H2, induction H2,
apply is_prop.mk, intro H1 H2, induction H2,
{ apply lem},
{ cases H1,
{ exfalso, apply not_succ_le_self a},
{ exact ap le.step !v_0}},
end
definition is_hprop_lt [instance] (n m : ℕ) : is_hprop (n < m) := !is_hprop_le
definition is_prop_lt [instance] (n m : ℕ) : is_prop (n < m) := !is_prop_le
definition le_equiv_succ_le_succ (n m : ℕ) : (n ≤ m) ≃ (succ n ≤ succ m) :=
equiv_of_is_hprop succ_le_succ le_of_succ_le_succ
equiv_of_is_prop succ_le_succ le_of_succ_le_succ
definition le_succ_equiv_pred_le (n m : ℕ) : (n ≤ succ m) ≃ (pred n ≤ m) :=
equiv_of_is_hprop pred_le_pred le_succ_of_pred_le
equiv_of_is_prop pred_le_pred le_succ_of_pred_le
theorem lt_by_cases_lt {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a = b → P)
(H3 : a > b → P) (H : a < b) : lt.by_cases H1 H2 H3 = H1 H :=
begin
unfold lt.by_cases, induction (lt.trichotomy a b) with H' H',
{ esimp, exact ap H1 !is_hprop.elim},
{ esimp, exact ap H1 !is_prop.elim},
{ exfalso, cases H' with H' H', apply lt.irrefl, exact H' ▸ H, exact lt.asymm H H'}
end
@ -45,7 +45,7 @@ namespace nat
begin
unfold lt.by_cases, induction (lt.trichotomy a b) with H' H',
{ exfalso, apply lt.irrefl, exact H ▸ H'},
{ cases H' with H' H', esimp, exact ap H2 !is_hprop.elim, exfalso, apply lt.irrefl, exact H ▸ H'}
{ cases H' with H' H', esimp, exact ap H2 !is_prop.elim, exfalso, apply lt.irrefl, exact H ▸ H'}
end
theorem lt_by_cases_ge {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a = b → P)
@ -53,7 +53,7 @@ namespace nat
begin
unfold lt.by_cases, induction (lt.trichotomy a b) with H' H',
{ exfalso, exact lt.asymm H H'},
{ cases H' with H' H', exfalso, apply lt.irrefl, exact H' ▸ H, esimp, exact ap H3 !is_hprop.elim}
{ cases H' with H' H', exfalso, apply lt.irrefl, exact H' ▸ H, esimp, exact ap H3 !is_prop.elim}
end
theorem lt_ge_by_cases_lt {n m : ℕ} {P : Type} (H1 : n < m → P) (H2 : n ≥ m → P)
@ -65,7 +65,7 @@ namespace nat
begin
unfold [lt_ge_by_cases,lt.by_cases], induction (lt.trichotomy n m) with H' H',
{ exfalso, apply lt.irrefl, exact lt_of_le_of_lt H H'},
{ cases H' with H' H'; all_goals (esimp; apply ap H2 !is_hprop.elim)}
{ cases H' with H' H'; all_goals (esimp; apply ap H2 !is_prop.elim)}
end
theorem lt_ge_by_cases_le {n m : ℕ} {P : Type} {H1 : n ≤ m → P} {H2 : n ≥ m → P}
@ -73,10 +73,10 @@ namespace nat
: lt_ge_by_cases (λH', H1 (le_of_lt H')) H2 = H1 H :=
begin
unfold [lt_ge_by_cases,lt.by_cases], induction (lt.trichotomy n m) with H' H',
{ esimp, apply ap H1 !is_hprop.elim},
{ esimp, apply ap H1 !is_prop.elim},
{ cases H' with H' H',
{ esimp, induction H', esimp, symmetry,
exact ap H1 !is_hprop.elim ⬝ Heq idp ⬝ ap H2 !is_hprop.elim},
exact ap H1 !is_prop.elim ⬝ Heq idp ⬝ ap H2 !is_prop.elim},
{ exfalso, apply lt.irrefl, apply lt_of_le_of_lt H H'}}
end

10
hott/types/pi.hlean
View file

@ -238,7 +238,7 @@ namespace pi
{ intro f, exact star},
{ intro u a, exact !center},
{ intro u, induction u, reflexivity},
{ intro f, apply eq_of_homotopy, intro a, apply is_hprop.elim}
{ intro f, apply eq_of_homotopy, intro a, apply is_prop.elim}
end
/- Interaction with other type constructors -/
@ -298,15 +298,15 @@ namespace pi
theorem is_trunc_not [instance] (n : trunc_index) (A : Type) : is_trunc (n.+1) ¬A :=
by unfold not;exact _
theorem is_hprop_pi_eq [instance] [priority 490] (a : A) : is_hprop (Π(a' : A), a = a') :=
is_hprop_of_imp_is_contr
theorem is_prop_pi_eq [instance] [priority 490] (a : A) : is_prop (Π(a' : A), a = a') :=
is_prop_of_imp_is_contr
( assume (f : Πa', a = a'),
assert is_contr A, from is_contr.mk a f,
by exact _) /- force type clas resolution -/
theorem is_hprop_neg (A : Type) : is_hprop (¬A) := _
theorem is_prop_neg (A : Type) : is_prop (¬A) := _
local attribute ne [reducible]
theorem is_hprop_ne [instance] {A : Type} (a b : A) : is_hprop (a ≠ b) := _
theorem is_prop_ne [instance] {A : Type} (a b : A) : is_prop (a ≠ b) := _
/- Symmetry of Π -/
definition is_equiv_flip [instance] {P : A → A' → Type}

2
hott/types/pointed2.hlean
View file

@ -87,7 +87,7 @@ namespace pointed
definition pequiv_eq {p q : A ≃* B} (H : p = q :> (A →* B)) : p = q :=
begin
cases p with f Hf, cases q with g Hg, esimp at *,
exact apd011 pequiv_of_pmap H !is_hprop.elim
exact apd011 pequiv_of_pmap H !is_prop.elim
end
definition loop_space_pequiv_rfl

2
hott/types/pullback.hlean
View file

@ -58,7 +58,7 @@ namespace pullback
{ intro v, induction v with x y p, exact (x, y)},
{ intro v, induction v with x y, exact pullback.mk x y idp},
{ intro v, induction v, reflexivity},
{ intro v, induction v, esimp, apply ap _ !is_hprop.elim},
{ intro v, induction v, esimp, apply ap _ !is_prop.elim},
end
definition pullback_along {f : A₂₀ → A₂₂} (g : A₀₂ → A₂₂) : pullback f g → A₂₀ :=

14
hott/types/sigma.hlean
View file

@ -423,28 +423,28 @@ namespace sigma
/- Subtypes (sigma types whose second components are hprops) -/
definition subtype [reducible] {A : Type} (P : A → Type) [H : Πa, is_hprop (P a)] :=
definition subtype [reducible] {A : Type} (P : A → Type) [H : Πa, is_prop (P a)] :=
Σ(a : A), P a
notation [parsing_only] `{` binder `|` r:(scoped:1 P, subtype P) `}` := r
/- To prove equality in a subtype, we only need equality of the first component. -/
definition subtype_eq [H : Πa, is_hprop (B a)] {u v : {a | B a}} : u.1 = v.1 → u = v :=
definition subtype_eq [H : Πa, is_prop (B a)] {u v : {a | B a}} : u.1 = v.1 → u = v :=
sigma_eq_unc ∘ inv pr1
definition is_equiv_subtype_eq [H : Πa, is_hprop (B a)] (u v : {a | B a})
definition is_equiv_subtype_eq [H : Πa, is_prop (B a)] (u v : {a | B a})
: is_equiv (subtype_eq : u.1 = v.1 → u = v) :=
!is_equiv_compose
local attribute is_equiv_subtype_eq [instance]
definition equiv_subtype [H : Πa, is_hprop (B a)] (u v : {a | B a}) : (u.1 = v.1) ≃ (u = v) :=
definition equiv_subtype [H : Πa, is_prop (B a)] (u v : {a | B a}) : (u.1 = v.1) ≃ (u = v) :=
equiv.mk !subtype_eq _
definition subtype_eq_inv {A : Type} {B : A → Type} [H : Πa, is_hprop (B a)] (u v : Σa, B a)
definition subtype_eq_inv {A : Type} {B : A → Type} [H : Πa, is_prop (B a)] (u v : Σa, B a)
: u = v → u.1 = v.1 :=
subtype_eq⁻¹ᶠ
local attribute subtype_eq_inv [reducible]
definition is_equiv_subtype_eq_inv {A : Type} {B : A → Type} [H : Πa, is_hprop (B a)]
definition is_equiv_subtype_eq_inv {A : Type} {B : A → Type} [H : Πa, is_prop (B a)]
(u v : Σa, B a) : is_equiv (subtype_eq_inv u v) :=
_
@ -463,7 +463,7 @@ namespace sigma
theorem is_trunc_subtype (B : A → hprop) (n : trunc_index)
[HA : is_trunc (n.+1) A] : is_trunc (n.+1) (Σa, B a) :=
@(is_trunc_sigma B (n.+1)) _ (λa, !is_trunc_succ_of_is_hprop)
@(is_trunc_sigma B (n.+1)) _ (λa, !is_trunc_succ_of_is_prop)
end sigma

10
hott/types/sum.hlean
View file

@ -215,7 +215,7 @@ namespace sum
cases u' with u' Hu', exfalso, apply empty_of_inl_eq_inr,
calc inl ⋆ = H⁻¹ (H (inl ⋆)) : (to_left_inv H (inl ⋆))⁻¹
... = H⁻¹ (inl u') : {Hu'}
... = H⁻¹ (inl u) : is_hprop.elim
... = H⁻¹ (inl u) : is_prop.elim
... = H⁻¹ (H (inr a)) : {Hu⁻¹}
... = inr a : to_left_inv H (inr a)
end
@ -229,7 +229,7 @@ namespace sum
{ intro x, exfalso, cases x with u' Hu', apply empty_of_inl_eq_inr,
calc inl ⋆ = H⁻¹ (H (inl ⋆)) : (to_left_inv H (inl ⋆))⁻¹
... = H⁻¹ (inl u') : ap H⁻¹ Hu'
... = H⁻¹ (inl u) : {!is_hprop.elim}
... = H⁻¹ (inl u) : {!is_prop.elim}
... = H⁻¹ (H (inr _)) : {Hu⁻¹}
... = inr _ : to_left_inv H },
{ intro x, cases x with b' Hb', esimp, cases sum.mem_cases (H⁻¹ (inr b)) with x x,
@ -250,7 +250,7 @@ namespace sum
{ cases x with u' Hu', exfalso, apply empty_of_inl_eq_inr,
calc inl ⋆ = H (H⁻¹ (inl ⋆)) : (to_right_inv H (inl ⋆))⁻¹
... = H (inl u') : {ap H Hu'}
... = H (inl u) : {!is_hprop.elim}
... = H (inl u) : {!is_prop.elim}
... = H (H⁻¹ (inr b)) : {ap H Hu⁻¹}
... = inr b : to_right_inv H (inr b) },
{ cases x with a Ha, exfalso, apply empty_of_inl_eq_inr,
@ -318,9 +318,9 @@ namespace sum
induction n with n IH,
{ exfalso, exact H !center !center},
{ clear IH, induction n with n IH,
{ apply is_hprop.mk, intros x y,
{ apply is_prop.mk, intros x y,
induction x, all_goals induction y, all_goals esimp,
all_goals try (exfalso;apply H;assumption;assumption), all_goals apply ap _ !is_hprop.elim},
all_goals try (exfalso;apply H;assumption;assumption), all_goals apply ap _ !is_prop.elim},
{ apply is_trunc_sum}}
end

48
hott/types/trunc.hlean
View file

@ -87,13 +87,13 @@ namespace is_trunc
/- theorems about decidable equality and axiom K -/
theorem is_hset_of_axiom_K {A : Type} (K : Π{a : A} (p : a = a), p = idp) : is_hset A :=
is_hset.mk _ (λa b p q, eq.rec_on q K p)
theorem is_set_of_axiom_K {A : Type} (K : Π{a : A} (p : a = a), p = idp) : is_set A :=
is_set.mk _ (λa b p q, eq.rec_on q K p)
theorem is_hset_of_relation.{u} {A : Type.{u}} (R : A → A → Type.{u})
(mere : Π(a b : A), is_hprop (R a b)) (refl : Π(a : A), R a a)
(imp : Π{a b : A}, R a b → a = b) : is_hset A :=
is_hset_of_axiom_K
theorem is_set_of_relation.{u} {A : Type.{u}} (R : A → A → Type.{u})
(mere : Π(a b : A), is_prop (R a b)) (refl : Π(a : A), R a a)
(imp : Π{a b : A}, R a b → a = b) : is_set A :=
is_set_of_axiom_K
(λa p,
have H2 : transport (λx, R a x → a = x) p (@imp a a) = @imp a a, from !apd,
have H3 : Π(r : R a a), transport (λx, a = x) p (imp r)
@ -103,27 +103,27 @@ namespace is_trunc
calc
imp (refl a) ⬝ p = transport (λx, a = x) p (imp (refl a)) : transport_eq_r
... = imp (transport (λx, R a x) p (refl a)) : H3
... = imp (refl a) : is_hprop.elim,
... = imp (refl a) : is_prop.elim,
cancel_left (imp (refl a)) H4)
definition relation_equiv_eq {A : Type} (R : A → A → Type)
(mere : Π(a b : A), is_hprop (R a b)) (refl : Π(a : A), R a a)
(mere : Π(a b : A), is_prop (R a b)) (refl : Π(a : A), R a a)
(imp : Π{a b : A}, R a b → a = b) (a b : A) : R a b ≃ a = b :=
@equiv_of_is_hprop _ _ _
(@is_trunc_eq _ _ (is_hset_of_relation R mere refl @imp) a b)
@equiv_of_is_prop _ _ _
(@is_trunc_eq _ _ (is_set_of_relation R mere refl @imp) a b)
imp
(λp, p ▸ refl a)
local attribute not [reducible]
theorem is_hset_of_double_neg_elim {A : Type} (H : Π(a b : A), ¬¬a = b → a = b)
: is_hset A :=
is_hset_of_relation (λa b, ¬¬a = b) _ (λa n, n idp) H
theorem is_set_of_double_neg_elim {A : Type} (H : Π(a b : A), ¬¬a = b → a = b)
: is_set A :=
is_set_of_relation (λa b, ¬¬a = b) _ (λa n, n idp) H
section
open decidable
--this is proven differently in init.hedberg
theorem is_hset_of_decidable_eq (A : Type) [H : decidable_eq A] : is_hset A :=
is_hset_of_double_neg_elim (λa b, by_contradiction)
theorem is_set_of_decidable_eq (A : Type) [H : decidable_eq A] : is_set A :=
is_set_of_double_neg_elim (λa b, by_contradiction)
end
theorem is_trunc_of_axiom_K_of_leq {A : Type} (n : trunc_index) (H : -1 ≤ n)
@ -138,9 +138,9 @@ namespace is_trunc
induction p, apply Hp
end
theorem is_hprop_iff_is_contr {A : Type} (a : A) :
is_hprop A ↔ is_contr A :=
iff.intro (λH, is_contr.mk a (is_hprop.elim a)) _
theorem is_prop_iff_is_contr {A : Type} (a : A) :
is_prop A ↔ is_contr A :=
iff.intro (λH, is_contr.mk a (is_prop.elim a)) _
theorem is_trunc_succ_iff_is_trunc_loop (A : Type) (Hn : -1 ≤ n) :
is_trunc (n.+1) A ↔ Π(a : A), is_trunc n (a = a) :=
@ -152,7 +152,7 @@ namespace is_trunc
revert A, induction n with n IH,
{ intro A, esimp [Iterated_loop_space], transitivity _,
{ apply is_trunc_succ_iff_is_trunc_loop, apply le.refl},
{ apply pi_iff_pi, intro a, esimp, apply is_hprop_iff_is_contr, reflexivity}},
{ apply pi_iff_pi, intro a, esimp, apply is_prop_iff_is_contr, reflexivity}},
{ intro A, esimp [Iterated_loop_space],
transitivity _, apply @is_trunc_succ_iff_is_trunc_loop @n, esimp, constructor,
apply pi_iff_pi, intro a, transitivity _, apply IH,
@ -167,7 +167,7 @@ namespace is_trunc
induction n with n,
{ esimp [sub_two,Iterated_loop_space], apply iff.intro,
intro H a, exact is_contr_of_inhabited_hprop a,
intro H, apply is_hprop_of_imp_is_contr, exact H},
intro H, apply is_prop_of_imp_is_contr, exact H},
{ apply is_trunc_iff_is_contr_loop_succ},
end
@ -225,15 +225,15 @@ namespace trunc
{ clear n, intro n IH A m H, induction m with m,
{ apply (@is_trunc_of_leq _ -2), exact unit.star},
{ apply is_trunc_succ_intro, intro aa aa',
apply (@trunc.rec_on _ _ _ aa (λy, !is_trunc_succ_of_is_hprop)),
eapply (@trunc.rec_on _ _ _ aa' (λy, !is_trunc_succ_of_is_hprop)),
apply (@trunc.rec_on _ _ _ aa (λy, !is_trunc_succ_of_is_prop)),
eapply (@trunc.rec_on _ _ _ aa' (λy, !is_trunc_succ_of_is_prop)),
intro a a', apply (is_trunc_equiv_closed_rev),
{ apply tr_eq_tr_equiv},
{ exact (IH _ _ _)}}}
end
open equiv.ops
definition unique_choice {P : A → Type} [H : Πa, is_hprop (P a)] (f : Πa, ∥ P a ∥) (a : A)
definition unique_choice {P : A → Type} [H : Πa, is_prop (P a)] (f : Πa, ∥ P a ∥) (a : A)
: P a :=
!trunc_equiv (f a)
@ -262,7 +262,7 @@ namespace function
definition is_equiv_equiv_is_embedding_times_is_surjective [constructor] (f : A → B)
: is_equiv f ≃ (is_embedding f × is_surjective f) :=
equiv_of_is_hprop (λH, (_, _))
equiv_of_is_prop (λH, (_, _))
(λP, prod.rec_on P (λH₁ H₂, !is_equiv_of_is_surjective_of_is_embedding))
end function

14
hott/types/univ.hlean
View file

@ -39,13 +39,13 @@ namespace univ
/- Properties which can be disproven for the universe -/
definition not_is_hset_type0 : ¬is_hset Type₀ :=
assume H : is_hset Type₀,
absurd !is_hset.elim eq_bnot_ne_idp
definition not_is_set_type0 : ¬is_set Type₀ :=
assume H : is_set Type₀,
absurd !is_set.elim eq_bnot_ne_idp
definition not_is_hset_type : ¬is_hset Type.{u} :=
assume H : is_hset Type,
absurd (is_trunc_is_embedding_closed lift star) not_is_hset_type0
definition not_is_set_type : ¬is_set Type.{u} :=
assume H : is_set Type,
absurd (is_trunc_is_embedding_closed lift star) not_is_set_type0
definition not_double_negation_elimination0 : ¬Π(A : Type₀), ¬¬A → A :=
begin
@ -53,7 +53,7 @@ namespace univ
have u : ¬¬bool, by exact (λg, g tt),
let H1 := apdo f eq_bnot,
note H2 := apo10 H1 u,
have p : eq_bnot ▸ u = u, from !is_hprop.elim,
have p : eq_bnot ▸ u = u, from !is_prop.elim,
rewrite p at H2,
note H3 := eq_of_pathover_ua H2, esimp at H3, --TODO: use apply ... at after #700
exact absurd H3 (bnot_ne (f bool u)),

6
src/library/class_instance_resolution.cpp
View file

@ -189,10 +189,10 @@ optional<expr> mk_class_instance(environment const & env, old_local_context cons
return mk_class_instance(env, ctx.get_data(), type, nullptr);
}
optional<expr> mk_hset_instance(type_checker & tc, options const & o, list<expr> const & ctx, expr const & type) {
optional<expr> mk_set_instance(type_checker & tc, options const & o, list<expr> const & ctx, expr const & type) {
level lvl = sort_level(tc.ensure_type(type).first);
expr is_hset = tc.whnf(mk_app(mk_constant(get_is_trunc_is_hset_name(), {lvl}), type)).first;
return mk_class_instance(tc.env(), o, ctx, is_hset);
expr is_set = tc.whnf(mk_app(mk_constant(get_is_trunc_is_set_name(), {lvl}), type)).first;
return mk_class_instance(tc.env(), o, ctx, is_set);
}
optional<expr> mk_subsingleton_instance(environment const & env, options const & o, list<expr> const & ctx, expr const & type) {

2
src/library/class_instance_resolution.h
View file

@ -39,7 +39,7 @@ pair<expr, constraint> mk_class_instance_elaborator(
optional<expr> mk_class_instance(environment const & env, io_state const & ios, old_local_context const & ctx, expr const & type, bool use_local_instances);
optional<expr> mk_class_instance(environment const & env, old_local_context const & ctx, expr const & type);
optional<expr> mk_hset_instance(type_checker & tc, options const & o, list<expr> const & ctx, expr const & type);
optional<expr> mk_set_instance(type_checker & tc, options const & o, list<expr> const & ctx, expr const & type);
optional<expr> mk_subsingleton_instance(environment const & env, options const & o,
list<expr> const & ctx, expr const & type);

24
src/library/constants.cpp
View file

@ -84,9 +84,9 @@ name const * g_implies = nullptr;
name const * g_implies_of_if_neg = nullptr;
name const * g_implies_of_if_pos = nullptr;
name const * g_implies_resolve = nullptr;
name const * g_is_trunc_is_hprop = nullptr;
name const * g_is_trunc_is_hprop_elim = nullptr;
name const * g_is_trunc_is_hset = nullptr;
name const * g_is_trunc_is_prop = nullptr;
name const * g_is_trunc_is_prop_elim = nullptr;
name const * g_is_trunc_is_set = nullptr;
name const * g_ite = nullptr;
name const * g_left_distrib = nullptr;
name const * g_le_refl = nullptr;
@ -359,9 +359,9 @@ void initialize_constants() {
g_implies_of_if_neg = new name{"implies_of_if_neg"};
g_implies_of_if_pos = new name{"implies_of_if_pos"};
g_implies_resolve = new name{"implies", "resolve"};
g_is_trunc_is_hprop = new name{"is_trunc", "is_hprop"};
g_is_trunc_is_hprop_elim = new name{"is_trunc", "is_hprop", "elim"};
g_is_trunc_is_hset = new name{"is_trunc", "is_hset"};
g_is_trunc_is_prop = new name{"is_trunc", "is_prop"};
g_is_trunc_is_prop_elim = new name{"is_trunc", "is_prop", "elim"};
g_is_trunc_is_set = new name{"is_trunc", "is_set"};
g_ite = new name{"ite"};
g_left_distrib = new name{"left_distrib"};
g_le_refl = new name{"le", "refl"};
@ -635,9 +635,9 @@ void finalize_constants() {
delete g_implies_of_if_neg;
delete g_implies_of_if_pos;
delete g_implies_resolve;
delete g_is_trunc_is_hprop;
delete g_is_trunc_is_hprop_elim;
delete g_is_trunc_is_hset;
delete g_is_trunc_is_prop;
delete g_is_trunc_is_prop_elim;
delete g_is_trunc_is_set;
delete g_ite;
delete g_left_distrib;
delete g_le_refl;
@ -910,9 +910,9 @@ name const & get_implies_name() { return *g_implies; }
name const & get_implies_of_if_neg_name() { return *g_implies_of_if_neg; }
name const & get_implies_of_if_pos_name() { return *g_implies_of_if_pos; }
name const & get_implies_resolve_name() { return *g_implies_resolve; }
name const & get_is_trunc_is_hprop_name() { return *g_is_trunc_is_hprop; }
name const & get_is_trunc_is_hprop_elim_name() { return *g_is_trunc_is_hprop_elim; }
name const & get_is_trunc_is_hset_name() { return *g_is_trunc_is_hset; }
name const & get_is_trunc_is_prop_name() { return *g_is_trunc_is_prop; }
name const & get_is_trunc_is_prop_elim_name() { return *g_is_trunc_is_prop_elim; }
name const & get_is_trunc_is_set_name() { return *g_is_trunc_is_set; }
name const & get_ite_name() { return *g_ite; }
name const & get_left_distrib_name() { return *g_left_distrib; }
name const & get_le_refl_name() { return *g_le_refl; }

6
src/library/constants.h
View file

@ -86,9 +86,9 @@ name const & get_implies_name();
name const & get_implies_of_if_neg_name();
name const & get_implies_of_if_pos_name();
name const & get_implies_resolve_name();
name const & get_is_trunc_is_hprop_name();
name const & get_is_trunc_is_hprop_elim_name();
name const & get_is_trunc_is_hset_name();
name const & get_is_trunc_is_prop_name();
name const & get_is_trunc_is_prop_elim_name();
name const & get_is_trunc_is_set_name();
name const & get_ite_name();
name const & get_left_distrib_name();
name const & get_le_refl_name();

6
src/library/constants.txt
View file

@ -79,9 +79,9 @@ implies
implies_of_if_neg
implies_of_if_pos
implies.resolve
is_trunc.is_hprop
is_trunc.is_hprop.elim
is_trunc.is_hset
is_trunc.is_prop
is_trunc.is_prop.elim
is_trunc.is_set
ite
left_distrib
le.refl

10
src/library/tactic/inversion_tactic.cpp
View file

@ -43,7 +43,7 @@ result::result(list<goal> const & gs, list<list<expr>> const & args, list<implem
The eq_rec_eq definition is of the form
definition eq_rec_eq.{l₁ l₂} {A : Type.{l₁}} {B : A → Type.{l₂}} [h : is_hset A] {a : A} (b : B a) (p : a = a) :
definition eq_rec_eq.{l₁ l₂} {A : Type.{l₁}} {B : A → Type.{l₂}} [h : is_set A] {a : A} (b : B a) (p : a = a) :
b = @eq.rec.{l₂ l₁} A a (λ (a' : A) (h : a = a'), B a') b a p :=
...
*/
@ -56,8 +56,8 @@ optional<expr> apply_eq_rec_eq(type_checker & tc, io_state const & ios, list<exp
if (!is_local(p) || !is_eq_a_a(tc, mlocal_type(p)))
return none_expr();
expr const & A = args[0];
auto is_hset_A = mk_hset_instance(tc, ios.get_options(), ctx, A);
if (!is_hset_A)
auto is_set_A = mk_set_instance(tc, ios.get_options(), ctx, A);
if (!is_set_A)
return none_expr();
levels ls = const_levels(eq_rec_fn);
level l2 = head(ls);
@ -76,7 +76,7 @@ optional<expr> apply_eq_rec_eq(type_checker & tc, io_state const & ios, list<exp
expr a = args[1];
expr b = args[3];
expr r = mk_constant(get_eq_rec_eq_name(), {l1, l2});
return some_expr(mk_app({r, A, B, *is_hset_A, a, b, p}));
return some_expr(mk_app({r, A, B, *is_set_A, a, b, p}));
}
typedef inversion::implementation_ptr implementation_ptr;
@ -540,7 +540,7 @@ class inversion_tac {
/** \brief Process goal of the form: Pi (H : eq.rec A s C a s p = b), R
The idea is to reduce it to
Pi (H : a = b), R
when A is a hset
when A is a set
and then invoke intro_next_eq recursively.
\remark \c type is the type of \c g after some normalization

2
src/library/type_context.cpp
View file

@ -1874,7 +1874,7 @@ optional<expr> type_context::mk_subsingleton_instance(expr const & type) {
if (is_standard(m_env))
subsingleton = mk_app(mk_constant(get_subsingleton_name(), {lvl}), type);
else
subsingleton = whnf(mk_app(mk_constant(get_is_trunc_is_hprop_name(), {lvl}), type));
subsingleton = whnf(mk_app(mk_constant(get_is_trunc_is_prop_name(), {lvl}), type));
auto r = mk_class_instance(subsingleton);
m_ss_cache.insert(mk_pair(type, r));
return r;

2
src/library/util.cpp
View file

@ -632,7 +632,7 @@ expr mk_subsingleton_elim(type_checker & tc, expr const & h, expr const & x, exp
if (is_standard(tc.env())) {
r = mk_constant(get_subsingleton_elim_name(), {l});
} else {
r = mk_constant(get_is_trunc_is_hprop_elim_name(), {l});
r = mk_constant(get_is_trunc_is_prop_elim_name(), {l});
}
return mk_app({r, A, h, x, y});
}

2
src/library/util.h
View file

@ -170,7 +170,7 @@ expr mk_symm(type_checker & tc, expr const & H);
expr mk_trans(type_checker & tc, expr const & H1, expr const & H2);
expr mk_subst(type_checker & tc, expr const & motive, expr const & x, expr const & y, expr const & xeqy, expr const & h);
expr mk_subst(type_checker & tc, expr const & motive, expr const & xeqy, expr const & h);
/** \brief Create an proof for x = y using subsingleton.elim (in standard mode) and is_trunc.is_hprop.elim (in HoTT mode) */
/** \brief Create an proof for x = y using subsingleton.elim (in standard mode) and is_trunc.is_prop.elim (in HoTT mode) */
expr mk_subsingleton_elim(type_checker & tc, expr const & h, expr const & x, expr const & y);
/** \brief Return true iff \c e is a term of the form (eq.rec ....) */

2
tests/lean/simplifier1.hlean
View file

@ -21,7 +21,7 @@ constants H₁ : a ≠ b
constants H₂ : a = c
namespace tst attribute H₂ [simp] end tst
definition ne_is_hprop [instance] (a b : nat) : is_hprop (a ≠ b) :=
definition ne_is_prop [instance] (a b : nat) : is_prop (a ≠ b) :=
sorry
-- TODO(Daniel): simplifier seems to have applied unnecessary step (see: (eq.nrec ... (eq.refl ..)))
#simplify eq tst 0 (g a b H₁)