lean2/hott/init/hit.hlean

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/-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: init.hit
Authors: Floris van Doorn
Declaration of the primitive hits in Lean
-/
prelude
import .trunc
open is_trunc eq
/-
We take two higher inductive types (hits) as primitive notions in Lean. We define all other hits
in terms of these two hits. The hits which are primitive are
- n-truncation
- type quotients (non-truncated quotients)
For each of the hits we add the following constants:
- the type formation
- the term and path constructors
- the dependent recursor
We add the computation rule for point constructors judgmentally to the kernel of Lean.
The computation rules for the path constructors are added (propositionally) as axioms
In this file we only define the dependent recursor. For the nondependent recursor and all other
uses of these hits, see the folder ../hit/
-/
constant trunc.{u} (n : trunc_index) (A : Type.{u}) : Type.{u}
namespace trunc
constant tr {n : trunc_index} {A : Type} (a : A) : trunc n A
constant is_trunc_trunc (n : trunc_index) (A : Type) : is_trunc n (trunc n A)
attribute is_trunc_trunc [instance]
protected constant rec {n : trunc_index} {A : Type} {P : trunc n A → Type}
[Pt : Πaa, is_trunc n (P aa)] (H : Πa, P (tr a)) : Πaa, P aa
protected definition rec_on [reducible] {n : trunc_index} {A : Type} {P : trunc n A → Type}
(aa : trunc n A) [Pt : Πaa, is_trunc n (P aa)] (H : Πa, P (tr a)) : P aa :=
trunc.rec H aa
end trunc
constant type_quotient.{u v} {A : Type.{u}} (R : A → A → Type.{v}) : Type.{max u v}
namespace type_quotient
constant class_of {A : Type} (R : A → A → Type) (a : A) : type_quotient R
constant eq_of_rel {A : Type} (R : A → A → Type) {a a' : A} (H : R a a')
: class_of R a = class_of R a'
protected constant rec {A : Type} {R : A → A → Type} {P : type_quotient R → Type}
(Pc : Π(a : A), P (class_of R a)) (Pp : Π⦃a a' : A⦄ (H : R a a'), eq_of_rel R H ▸ Pc a = Pc a')
(x : type_quotient R) : P x
protected definition rec_on [reducible] {A : Type} {R : A → A → Type} {P : type_quotient R → Type}
(x : type_quotient R) (Pc : Π(a : A), P (class_of R a))
(Pp : Π⦃a a' : A⦄ (H : R a a'), eq_of_rel R H ▸ Pc a = Pc a') : P x :=
rec Pc Pp x
end type_quotient
init_hits -- Initialize builtin computational rules for trunc and type_quotient
namespace trunc
definition rec_tr [reducible] {n : trunc_index} {A : Type} {P : trunc n A → Type}
[Pt : Πaa, is_trunc n (P aa)] (H : Πa, P (tr a)) (a : A) : trunc.rec H (tr a) = H a :=
idp
end trunc
namespace type_quotient
definition rec_class_of {A : Type} {R : A → A → Type} {P : type_quotient R → Type}
(Pc : Π(a : A), P (class_of R a)) (Pp : Π⦃a a' : A⦄ (H : R a a'), eq_of_rel R H ▸ Pc a = Pc a')
(a : A) : type_quotient.rec Pc Pp (class_of R a) = Pc a :=
idp
constant rec_eq_of_rel {A : Type} {R : A → A → Type} {P : type_quotient R → Type}
(Pc : Π(a : A), P (class_of R a)) (Pp : Π⦃a a' : A⦄ (H : R a a'), eq_of_rel R H ▸ Pc a = Pc a')
{a a' : A} (H : R a a') : apd (type_quotient.rec Pc Pp) (eq_of_rel R H) = Pp H
end type_quotient
attribute type_quotient.class_of trunc.tr [constructor]
attribute type_quotient.rec trunc.rec [unfold-c 6]
attribute type_quotient.rec_on trunc.rec_on [unfold-c 4]