lean2/hott/hit/suspension.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: hit.suspension
Authors: Floris van Doorn
Declaration of suspension
-/
import .pushout
open pushout unit eq equiv equiv.ops
definition suspension (A : Type) : Type := pushout (λ(a : A), star.{0}) (λ(a : A), star.{0})
namespace suspension
variable {A : Type}
definition north (A : Type) : suspension A :=
inl _ _ star
definition south (A : Type) : suspension A :=
inr _ _ star
definition merid (a : A) : north A = south A :=
glue _ _ a
protected definition rec {P : suspension A → Type} (PN : P !north) (PS : P !south)
(Pm : Π(a : A), merid a ▹ PN = PS) (x : suspension A) : P x :=
begin
fapply (pushout.rec_on _ _ x),
{ intro u, cases u, exact PN},
{ intro u, cases u, exact PS},
{ exact Pm},
end
protected definition rec_on [reducible] {P : suspension A → Type} (y : suspension A)
(PN : P !north) (PS : P !south) (Pm : Π(a : A), merid a ▹ PN = PS) : P y :=
rec PN PS Pm y
theorem rec_merid {P : suspension A → Type} (PN : P !north) (PS : P !south)
(Pm : Π(a : A), merid a ▹ PN = PS) (a : A)
: apd (rec PN PS Pm) (merid a) = Pm a :=
!rec_glue
protected definition elim {P : Type} (PN : P) (PS : P) (Pm : A → PN = PS)
(x : suspension A) : P :=
rec PN PS (λa, !tr_constant ⬝ Pm a) x
protected definition elim_on [reducible] {P : Type} (x : suspension A)
(PN : P) (PS : P) (Pm : A → PN = PS) : P :=
elim PN PS Pm x
theorem elim_merid {P : Type} (PN : P) (PS : P) (Pm : A → PN = PS) (a : A)
: ap (elim PN PS Pm) (merid a) = Pm a :=
begin
apply (@cancel_left _ _ _ _ (tr_constant (merid a) (elim PN PS Pm !north))),
rewrite [-apd_eq_tr_constant_con_ap,↑elim,rec_merid],
end
protected definition elim_type (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
(x : suspension A) : Type :=
elim PN PS (λa, ua (Pm a)) x
protected definition elim_type_on [reducible] (x : suspension A)
(PN : Type) (PS : Type) (Pm : A → PN ≃ PS) : Type :=
elim_type PN PS Pm x
theorem elim_type_merid (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
(x : suspension A) (a : A) : transport (elim_type PN PS Pm) (merid a) = Pm a :=
by rewrite [tr_eq_cast_ap_fn,↑elim_type,elim_merid];apply cast_ua_fn
end suspension