michael/lean2
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

style(hott): replace all other occurrences of hprop/hset

Browse source 
They are replaced by either Prop/Set or prop/set
This commit is contained in:
Floris van Doorn 2016-02-15 15:55:29 -05:00 committed by Leonardo de Moura
parent 4e2cc66061
commit 43cf2ad23d
27 changed files with 79 additions and 80 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

24
hott/algebra/category/constructions/set.hlean
View file

@ -12,18 +12,18 @@ open eq category equiv iso is_equiv is_trunc function sigma
namespace category
definition precategory_hset.{u} [reducible] [constructor] : precategory hset.{u} :=
precategory.mk (λx y : hset, x → y)
definition precategory_Set.{u} [reducible] [constructor] : precategory Set.{u} :=
precategory.mk (λx y : Set, x → y)
(λx y z g f a, g (f a))
(λx a, a)
(λx y z w h g f, eq_of_homotopy (λa, idp))
(λx y f, eq_of_homotopy (λa, idp))
(λx y f, eq_of_homotopy (λa, idp))
definition Precategory_hset [reducible] [constructor] : Precategory :=
Precategory.mk hset precategory_hset
definition Precategory_Set [reducible] [constructor] : Precategory :=
Precategory.mk Set precategory_Set
abbreviation set [constructor] := Precategory_hset
abbreviation set [constructor] := Precategory_Set
namespace set
local attribute is_equiv_subtype_eq [instance]
@ -48,7 +48,7 @@ namespace category
local attribute is_equiv_iso_of_equiv [instance]
definition iso_of_eq_eq_compose (A B : hset) : @iso_of_eq _ _ A B =
definition iso_of_eq_eq_compose (A B : Set) : @iso_of_eq _ _ A B =
@iso_of_equiv A B ∘ @equiv_of_eq A B ∘ subtype_eq_inv _ _ ∘
@ap _ _ (to_fun (trunctype.sigma_char 0)) A B :=
eq_of_homotopy (λp, eq.rec_on p idp)
@ -65,7 +65,7 @@ namespace category
definition equiv_eq_iso (A B : set) : (A ≃ B) = (A ≅ B) :=
ua !equiv_equiv_iso
definition is_univalent_hset (A B : set) : is_equiv (iso_of_eq : A = B → A ≅ B) :=
definition is_univalent_Set (A B : set) : is_equiv (iso_of_eq : A = B → A ≅ B) :=
assert H₁ : is_equiv (@iso_of_equiv A B ∘ @equiv_of_eq A B ∘ subtype_eq_inv _ _ ∘
@ap _ _ (to_fun (trunctype.sigma_char 0)) A B), from
@is_equiv_compose _ _ _ _ _
@ -82,13 +82,13 @@ namespace category
end
end set
definition category_hset [instance] [constructor] [reducible] : category hset :=
category.mk precategory_hset set.is_univalent_hset
definition category_Set [instance] [constructor] [reducible] : category Set :=
category.mk precategory_Set set.is_univalent_Set
definition Category_hset [reducible] [constructor] : Category :=
Category.mk hset category_hset
definition Category_Set [reducible] [constructor] : Category :=
Category.mk Set category_Set
abbreviation cset [constructor] := Category_hset
abbreviation cset [constructor] := Category_Set
open functor lift
definition functor_lift.{u v} [constructor] : set.{u} ⇒ set.{max u v} :=

2
hott/algebra/category/functor/yoneda.hlean
View file

@ -86,7 +86,7 @@ namespace yoneda
-- TODO: Investigate what is the bottleneck to type check the next theorem
-- attribute yoneda_lemma functor_lift Precategory_hset precategory_hset homset
-- attribute yoneda_lemma functor_lift Precategory_Set precategory_Set homset
-- yoneda_embedding nat_trans.compose functor_nat_trans_compose [reducible]
-- attribute tlift functor.compose [reducible]
theorem yoneda_lemma_natural_functor.{u v} {C : Precategory.{u v}} (c : C) (F F' : Cᵒᵖ ⇒ cset)

2
hott/algebra/category/limits/set.hlean
View file

@ -58,7 +58,7 @@ namespace category
definition is_cocomplete_set_cone_rel.{u v w} [unfold 3 4]
(I : Precategory.{v w}) (F : I ⇒ set.{max u v w}ᵒᵖ) : (Σ(i : I), trunctype.carrier (F i)) →
(Σ(i : I), trunctype.carrier (F i)) → hprop.{max u v w} :=
(Σ(i : I), trunctype.carrier (F i)) → Prop.{max u v w} :=
begin
intro v w, induction v with i x, induction w with j y,
fapply trunctype.mk,

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

@ -56,13 +56,13 @@ namespace category
-- 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_set (hom a b)]
[set : 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)
(idl : Π ⦃a b : ob⦄ (f : hom a b), comp (ID b) f = f)
(idr : Π ⦃a b : ob⦄ (f : hom a b), comp f (ID a) = f) : precategory ob :=
precategory.mk' hom comp ID ass (λa b c d h g f, !ass⁻¹) idl idr (λa, !idl) hset
precategory.mk' hom comp ID ass (λa b c d h g f, !ass⁻¹) idl idr (λa, !idl) set
section basic_lemmas
variables {ob : Type} [C : precategory ob]
@ -83,8 +83,8 @@ namespace category
calc i = id ∘ i : by rewrite id_left
... = id : by rewrite H
definition homset [reducible] [constructor] (x y : ob) : hset :=
hset.mk (hom x y) _
definition homset [reducible] [constructor] (x y : ob) : Set :=
Set.mk (hom x y) _
end basic_lemmas
section squares

2
hott/algebra/hott.hlean
View file

@ -6,7 +6,7 @@ Author: Floris van Doorn
Theorems about algebra specific to HoTT
-/
import .group arity types.pi hprop_trunc types.unit .bundled
import .group arity types.pi prop_trunc types.unit .bundled
open equiv eq equiv.ops is_trunc unit

4
hott/book.md
View file

@ -73,7 +73,7 @@ Chapter 3: Sets and logic
- 3.1 (Sets and n-types): [init.trunc](init/trunc.hlean). Example 3.1.9 in [types.univ](types/univ.hlean)
- 3.2 (Propositions as types?): [types.univ](types/univ.hlean)
- 3.3 (Mere propositions): [init.trunc](init/trunc.hlean) and [hprop_trunc](hprop_trunc.hlean) (Lemma 3.3.5).
- 3.3 (Mere propositions): [init.trunc](init/trunc.hlean) and [prop_trunc](prop_trunc.hlean) (Lemma 3.3.5).
- 3.4 (Classical vs. intuitionistic logic): decidable is defined in [init.logic](init/logic.hlean)
- 3.5 (Subsets and propositional resizing): Lemma 3.5.1 is subtype_eq in [types.sigma](types/sigma.hlean), we don't have propositional resizing as axiom yet.
- 3.6 (The logic of mere propositions): in the corresponding file in the [types](types/types.md) folder. (is_trunc_prod is defined in [types.sigma](types/sigma.hlean))
@ -128,7 +128,7 @@ Chapter 6: Higher inductive types
Chapter 7: Homotopy n-types
---------
- 7.1 (Definition of n-types): [init.trunc](init/trunc.hlean), [types.trunc](types/trunc.hlean), [types.sigma](types/sigma.hlean) (Theorem 7.1.8), [types.pi](types/pi.hlean) (Theorem 7.1.9), [hprop_trunc](hprop_trunc.hlean) (Theorem 7.1.10)
- 7.1 (Definition of n-types): [init.trunc](init/trunc.hlean), [types.trunc](types/trunc.hlean), [types.sigma](types/sigma.hlean) (Theorem 7.1.8), [types.pi](types/pi.hlean) (Theorem 7.1.9), [prop_trunc](prop_trunc.hlean) (Theorem 7.1.10)
- 7.2 (Uniqueness of identity proofs and Hedberg’s theorem): [init.hedberg](init/hedberg.hlean) and [types.trunc](types/trunc.hlean)
- 7.3 (Truncations): [init.hit](init/hit.hlean), [hit.trunc](hit/trunc.hlean) and [types.trunc](types/trunc.hlean)
- 7.4 (Colimits of n-types): not formalized

4
hott/choice.hlean
View file

@ -37,7 +37,7 @@ namespace choice
by intro H X A P HX HA HP HI;
apply H X (λ x, Σ a, P x a) HX (λ x, !is_trunc_sigma) HI
-- Which is enough to show AC' ≃ AC_cart, since both are hprops.
-- Which is enough to show AC' ≃ AC_cart, since both are props.
definition AC'_equiv_AC_cart : AC'.{u} ≃ AC_cart.{u} :=
equiv_of_is_prop AC_cart_of_AC'.{u} AC'_of_AC_cart.{u}
@ -80,7 +80,7 @@ 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})
definition X_must_be_set : Σ (X : Type.{1}) (Y : X -> Type.{1})
(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)))⟩

4
hott/hit/coeq.hlean
View file

@ -74,11 +74,11 @@ 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_prop (P x)]
protected definition rec_prop {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_prop.elimo) y
protected definition elim_hprop {P : Type} [H : is_prop P] (P_i : B → P) (y : coeq) : P :=
protected definition elim_prop {P : Type} [H : is_prop P] (P_i : B → P) (y : coeq) : P :=
elim P_i (λa, !is_prop.elim) y
end

8
hott/hit/colimit.hlean
View file

@ -85,11 +85,11 @@ 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_prop (P x)]
protected definition rec_prop {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_prop.elimo) y
protected definition elim_hprop {P : Type} [H : is_prop P] (Pincl : Π⦃i : I⦄ (x : A i), P)
protected definition elim_prop {P : Type} [H : is_prop P] (Pincl : Π⦃i : I⦄ (x : A i), P)
(y : colimit) : P :=
elim Pincl (λa b, !is_prop.elim) y
@ -175,11 +175,11 @@ 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_prop (P x)]
protected definition rec_prop {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_prop.elimo) aa
protected definition elim_hprop {P : Type} [H : is_prop P] (Pincl : Π⦃n : ℕ⦄ (a : A n), P)
protected definition elim_prop {P : Type} [H : is_prop P] (Pincl : Π⦃n : ℕ⦄ (a : A n), P)
: seq_colim → P :=
elim Pincl (λa b, !is_prop.elim)

4
hott/hit/pushout.hlean
View file

@ -83,11 +83,11 @@ 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_prop (P x)]
protected definition rec_prop {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_prop.elimo) y
protected definition elim_hprop {P : Type} [H : is_prop P] (Pinl : BL → P) (Pinr : TR → P)
protected definition elim_prop {P : Type} [H : is_prop P] (Pinl : BL → P) (Pinr : TR → P)
(y : pushout) : P :=
elim Pinl Pinr (λa, !is_prop.elim) y

4
hott/hit/quotient.hlean
View file

@ -38,11 +38,11 @@ namespace quotient
rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑quotient.elim,rec_eq_of_rel],
end
protected definition rec_hprop {A : Type} {R : A → A → Type} {P : quotient R → Type}
protected definition rec_prop {A : Type} {R : A → A → Type} {P : quotient R → Type}
[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_prop P] (Pc : A → P) (x : quotient R) : P :=
protected definition elim_prop {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)

10
hott/hit/set_quotient.hlean
View file

@ -12,7 +12,7 @@ open eq is_trunc trunc quotient equiv
namespace set_quotient
section
parameters {A : Type} (R : A → A → hprop)
parameters {A : Type} (R : A → A → Prop)
-- set-quotients are just set-truncations of (type) quotients
definition set_quotient : Type := trunc 0 (quotient R)
@ -63,11 +63,11 @@ 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_prop (P aa)]
protected definition rec_prop {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_prop.elimo) x
protected definition elim_hprop {P : Type} [Pt : is_prop P] (Pc : A → P) (x : set_quotient)
protected definition elim_prop {P : Type} [Pt : is_prop P] (Pc : A → P) (x : set_quotient)
: P :=
elim Pc (λa a' H, !is_prop.elim) x
@ -81,7 +81,7 @@ attribute set_quotient.rec_on set_quotient.elim_on [unfold 4]
open sigma relation function
namespace set_quotient
variables {A B C : Type} (R : A → A → hprop) (S : B → B → hprop) (T : C → C → hprop)
variables {A B C : Type} (R : A → A → Prop) (S : B → B → Prop) (T : C → C → Prop)
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_prop.elimo)
@ -102,7 +102,7 @@ namespace set_quotient
end
protected definition code [unfold 4] (a : A) (x : set_quotient R) [H : is_equivalence R]
: hprop :=
: Prop :=
set_quotient.elim_on x (R a)
begin
intros a' a'' H1,

12
hott/hit/trunc.hlean
View file

@ -98,14 +98,14 @@ namespace trunc
/- Propositional truncation -/
-- should this live in hprop?
definition merely [reducible] [constructor] (A : Type) : hprop := trunctype.mk (trunc -1 A) _
-- should this live in Prop?
definition merely [reducible] [constructor] (A : Type) : Prop := trunctype.mk (trunc -1 A) _
notation `||`:max A `||`:0 := merely A
notation `∥`:max A `∥`:0 := merely A
definition Exists [reducible] [constructor] (P : X → Type) : hprop := ∥ sigma P ∥
definition or [reducible] [constructor] (A B : Type) : hprop := ∥ A ⊎ B ∥
definition Exists [reducible] [constructor] (P : X → Type) : Prop := ∥ sigma P ∥
definition or [reducible] [constructor] (A B : Type) : Prop := ∥ A ⊎ B ∥
notation `exists` binders `,` r:(scoped P, Exists P) := r
notation `∃` binders `,` r:(scoped P, Exists P) := r
@ -117,8 +117,8 @@ 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_prop A] (aa : merely A) : is_contr A :=
is_contr_of_inhabited_hprop (trunc.rec_on aa id)
definition is_contr_of_merely_prop [H : is_prop A] (aa : merely A) : is_contr A :=
is_contr_of_inhabited_prop (trunc.rec_on aa id)
section
open sigma.ops

10
hott/homotopy/cellcomplex.hlean
View file

@ -36,14 +36,14 @@ namespace cellcomplex
definition fdcc_family [reducible] : ℕ → family :=
nat.rec
-- a zero-dimensional cell complex is just an hset
-- a zero-dimensional cell complex is just an set
-- with realization the identity map
⟨hset , λA, trunctype.carrier A⟩
⟨Set , λA, trunctype.carrier A⟩
(λn fdcc_family_n, -- sigma.rec (λ fdcc_n realize_n,
/- a (succ n)-dimensional cell complex is a triple of
an n-dimensional cell complex X, an hset of (succ n)-cells A,
an n-dimensional cell complex X, an set of (succ n)-cells A,
and an attaching map f : A × sphere n → |X| -/
⟨Σ X : pr1 fdcc_family_n , Σ A : hset, A × sphere n → pr2 fdcc_family_n X ,
⟨Σ X : pr1 fdcc_family_n , Σ A : Set, A × sphere n → pr2 fdcc_family_n X ,
/- the realization of such is the pushout of f with
canonical map A × sphere n → unit -/
sigma.rec (λX , sigma.rec (λA f, pushout (λx , star) f))
@ -51,7 +51,7 @@ namespace cellcomplex
definition fdcc (n : ℕ) : Type := pr1 (fdcc_family n)
definition cell : Πn, fdcc n → hset :=
definition cell : Πn, fdcc n → Set :=
nat.cases (λA, A) (λn T, pr1 (pr2 T))
end cellcomplex

4
hott/homotopy/connectedness.hlean
View file

@ -196,7 +196,7 @@ namespace homotopy
: is_surjective f → is_conn_map -1 f :=
begin
intro H, intro b,
exact @is_contr_of_inhabited_hprop (∥fiber f b∥) (is_trunc_trunc -1 (fiber f b)) (H b),
exact @is_contr_of_inhabited_prop (∥fiber f b∥) (is_trunc_trunc -1 (fiber f b)) (H b),
end
definition is_surjection_of_minus_one_conn {A B : Type} (f : A → B)
@ -210,7 +210,7 @@ namespace homotopy
λH, @center (∥A∥) H
definition minus_one_conn_of_merely {A : Type} : ∥A∥ → is_conn -1 A :=
@is_contr_of_inhabited_hprop (∥A∥) (is_trunc_trunc -1 A)
@is_contr_of_inhabited_prop (∥A∥) (is_trunc_trunc -1 A)
section
open arrow

2
hott/hott.md
View file

@ -10,7 +10,7 @@ The Lean Homotopy Type Theory library consists of the following directories:
* [cubical](cubical/cubical.md): cubical types
The following files don't fit in any of the subfolders:
* [hprop_trunc](hprop_trunc.hlean): in this file we prove that `is_trunc n A` is a mere proposition. We separate this from [types.trunc](types/trunc.hlean) to avoid circularity in imports.
* [prop_trunc](prop_trunc.hlean): in this file we prove that `is_trunc n A` is a mere proposition. We separate this from [types.trunc](types/trunc.hlean) to avoid circularity in imports.
* [eq2](eq2.hlean): coherence rules for the higher dimensional structure of equality
* [function](function.hlean): embeddings, (split) surjections, retractions
* [arity](arity.hlean) : equality theorems about functions with arity 2 or higher

2
hott/init/default.hlean
View file

@ -23,5 +23,5 @@ namespace core
export [declaration] function
export equiv (to_inv to_right_inv to_left_inv)
export is_equiv (inv right_inv left_inv adjointify)
export [abbreviation] [declaration] is_trunc (hprop.mk hset.mk)
export [abbreviation] [declaration] is_trunc (Prop.mk Set.mk)
end core

2
hott/init/funext.hlean
View file

@ -84,7 +84,7 @@ section
local attribute weak_funext [reducible]
local attribute homotopy_ind [reducible]
definition homotopy_ind_comp : homotopy_ind f (homotopy.refl f) = d :=
(@hprop_eq_of_is_contr _ _ _ _ !eq_of_is_contr idp)⁻¹ ▸ idp
(@prop_eq_of_is_contr _ _ _ _ !eq_of_is_contr idp)⁻¹ ▸ idp
end
/- Now the proof is fairly easy; we can just use the same induction principle on both sides. -/

2
hott/init/hedberg.hlean
View file

@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
Hedberg's Theorem: every type with decidable equality is a hset
Hedberg's Theorem: every type with decidable equality is a set
-/
prelude

20
hott/init/trunc.hlean
View file

@ -8,8 +8,6 @@ Definition of is_trunc (n-truncatedness)
Ported from Coq HoTT.
-/
--TODO: can we replace some definitions with a hprop as codomain by theorems?
prelude
import .nat .logic .equiv .pathover
open eq nat sigma unit sigma.ops
@ -135,12 +133,12 @@ namespace is_trunc
definition eq_of_is_contr [H : is_contr A] (x y : A) : x = y :=
(center_eq x)⁻¹ ⬝ (center_eq y)
definition hprop_eq_of_is_contr {A : Type} [H : is_contr A] {x y : A} (p q : x = y) : p = q :=
definition prop_eq_of_is_contr {A : Type} [H : is_contr A] {x y : A} (p q : x = y) : p = q :=
have K : ∀ (r : x = y), eq_of_is_contr x y = r, from (λ r, eq.rec_on r !con.left_inv),
(K p)⁻¹ ⬝ K q
theorem is_contr_eq {A : Type} [H : is_contr A] (x y : A) : is_contr (x = y) :=
is_contr.mk !eq_of_is_contr (λ p, !hprop_eq_of_is_contr)
is_contr.mk !eq_of_is_contr (λ p, !prop_eq_of_is_contr)
local attribute is_contr_eq [instance]
/- truncation is upward close -/
@ -192,12 +190,12 @@ namespace is_trunc
: is_trunc (n.+2) A :=
@(is_trunc_of_leq A (show 0 ≤ n.+2, from proof star qed)) H
/- hprops -/
/- props -/
definition is_prop.elim [H : is_prop A] (x y : A) : x = y :=
!center
definition is_contr_of_inhabited_hprop {A : Type} [H : is_prop A] (x : A) : is_contr A :=
definition is_contr_of_inhabited_prop {A : Type} [H : is_prop A] (x : A) : is_contr A :=
is_contr.mk x (λy, !is_prop.elim)
theorem is_prop_of_imp_is_contr {A : Type} (H : A → is_contr A) : is_prop A :=
@ -212,7 +210,7 @@ namespace is_trunc
theorem is_prop_elim_self {A : Type} {H : is_prop A} (x : A) : is_prop.elim x x = idp :=
!is_prop.elim
/- hsets -/
/- sets -/
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))
@ -351,11 +349,11 @@ namespace is_trunc
attribute trunctype.struct [instance] [priority 1400]
notation n `-Type` := trunctype n
abbreviation hprop := -1-Type
abbreviation hset := 0-Type
abbreviation Prop := -1-Type
abbreviation Set := 0-Type
protected abbreviation hprop.mk := @trunctype.mk -1
protected abbreviation hset.mk := @trunctype.mk (-1.+1)
protected abbreviation Prop.mk := @trunctype.mk -1
protected abbreviation Set.mk := @trunctype.mk (-1.+1)
protected abbreviation trunctype.mk' [parsing_only] (n : trunc_index) (A : Type)
[H : is_trunc n A] : n-Type :=

0
hott/hprop_trunc.hlean → hott/prop_trunc.hlean
View file

2
hott/types/bool.hlean
View file

@ -168,6 +168,6 @@ namespace bool
{ intro b, cases b, reflexivity, reflexivity},
end
definition tbool [constructor] : hset := trunctype.mk bool _
definition tbool [constructor] : Set := trunctype.mk bool _
end bool

4
hott/types/equiv.hlean
View file

@ -7,7 +7,7 @@ Ported from Coq HoTT
Theorems about the types equiv and is_equiv
-/
import .fiber .arrow arity ..hprop_trunc
import .fiber .arrow arity ..prop_trunc
open eq is_trunc sigma sigma.ops pi fiber function equiv equiv.ops
@ -184,7 +184,7 @@ namespace equiv
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_prop.mk,
apply @is_contr_of_inhabited_prop, 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_prop.elim ▸ rfl),
apply equiv_of_is_contr_of_is_contr

4
hott/types/pi.hlean
View file

@ -226,9 +226,9 @@ namespace pi
fapply equiv.MK,
{ intro f, exact f (center A)},
{ intro b a, exact (center_eq a) ▸ b},
{ intro b, rewrite [hprop_eq_of_is_contr (center_eq (center A)) idp]},
{ intro b, rewrite [prop_eq_of_is_contr (center_eq (center A)) idp]},
{ intro f, apply eq_of_homotopy, intro a, induction (center_eq a),
rewrite [hprop_eq_of_is_contr (center_eq (center A)) idp]}
rewrite [prop_eq_of_is_contr (center_eq (center A)) idp]}
end
definition pi_equiv_of_is_contr_right [constructor] [H : Πa, is_contr (B a)]

6
hott/types/sigma.hlean
View file

@ -280,7 +280,7 @@ namespace sigma
equiv.MK
(λu, (center_eq u.1)⁻¹ ▸ u.2)
(λb, ⟨!center, b⟩)
abstract (λb, ap (λx, x ▸ b) !hprop_eq_of_is_contr) end
abstract (λb, ap (λx, x ▸ b) !prop_eq_of_is_contr) end
abstract (λu, sigma_eq !center_eq !tr_pathover) end
/- Associativity -/
@ -421,7 +421,7 @@ namespace sigma
equiv.mk sigma.coind_unc _
end
/- Subtypes (sigma types whose second components are hprops) -/
/- Subtypes (sigma types whose second components are props) -/
definition subtype [reducible] {A : Type} (P : A → Type) [H : Πa, is_prop (P a)] :=
Σ(a : A), P a
@ -461,7 +461,7 @@ namespace sigma
exact IH _ _ _ _}
end
theorem is_trunc_subtype (B : A → hprop) (n : trunc_index)
theorem is_trunc_subtype (B : A → Prop) (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_prop)

8
hott/types/trunc.hlean
View file

@ -6,7 +6,7 @@ Authors: Floris van Doorn
Properties of is_trunc and trunctype
-/
-- NOTE: the fact that (is_trunc n A) is a mere proposition is proved in .hprop_trunc
-- NOTE: the fact that (is_trunc n A) is a mere proposition is proved in .prop_trunc
import types.pi types.eq types.equiv ..function
@ -75,7 +75,7 @@ namespace is_trunc
fapply is_trunc_equiv_closed,
{apply equiv.symm, apply eq_equiv_equiv},
induction n,
{apply @is_contr_of_inhabited_hprop,
{apply @is_contr_of_inhabited_prop,
{apply is_trunc_is_embedding_closed,
{apply is_embedding_to_fun} ,
{exact unit.star}},
@ -166,7 +166,7 @@ namespace is_trunc
begin
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 a, exact is_contr_of_inhabited_prop a,
intro H, apply is_prop_of_imp_is_contr, exact H},
{ apply is_trunc_iff_is_contr_loop_succ},
end
@ -242,7 +242,7 @@ namespace trunc
: transport (λa, trunc n (P a)) p (tr x) = tr (p ▸ x) :=
by induction p; reflexivity
definition image {A B : Type} (f : A → B) (b : B) : hprop := ∥ fiber f b ∥
definition image {A B : Type} (f : A → B) (b : B) : Prop := ∥ fiber f b ∥
-- truncation of pointed types
definition ptrunc [constructor] (n : trunc_index) (X : Type*) : Type* :=

5
src/emacs/lean-syntax.el
View file

@ -27,7 +27,7 @@
(defconst lean-keywords1-regexp
(eval `(rx word-start (or ,@lean-keywords1) word-end)))
(defconst lean-keywords2
'("by+" "begin+" "example.")
'("by+" "begin+")
"lean keywords ending with 'symbol'")
(defconst lean-keywords2-regexp
(eval `(rx word-start (or ,@lean-keywords2) symbol-end)))
@ -172,10 +172,11 @@
("\\(set_option\\)[ \t]*\\([^ \t\n]*\\)" (2 'font-lock-constant-face))
(,lean-keywords2-regexp . 'font-lock-keyword-face)
(,lean-keywords1-regexp . 'font-lock-keyword-face)
(,(rx word-start (group "example") ".") (1 'font-lock-keyword-face))
(,(rx (or "∎")) . 'font-lock-keyword-face)
;; Types
(,(rx word-start (or "Prop" "Type" "Type'" "Type₊" "Type₀" "Type₁" "Type₂" "Type₃" "Type*" "Set") symbol-end) . 'font-lock-type-face)
(,(rx word-start (group "Type") ".") (1 'font-lock-type-face))
(,(rx word-start (group (or "Prop" "Type" "Set")) ".") (1 'font-lock-type-face))
;; String
("\"[^\"]*\"" . 'font-lock-string-face)
;; ;; Constants