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