lean2/hott/hit/circle.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.circle
Authors: Floris van Doorn
Declaration of the circle
-/
import .sphere
open eq suspension bool sphere_index
definition circle [reducible] := suspension bool --redefine this as sphere 1
namespace circle
definition base1 : circle := !north
definition base2 : circle := !south
definition seg1 : base1 = base2 := merid tt
definition seg2 : base2 = base1 := (merid ff)⁻¹
definition base : circle := base1
definition loop : base = base := seg1 ⬝ seg2
definition rec2 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2)
(Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1) (x : circle) : P x :=
begin
fapply (suspension.rec_on x),
{ exact Pb1},
{ exact Pb2},
{ intro b, cases b,
apply tr_eq_of_eq_inv_tr, exact Ps2⁻¹,
exact Ps1},
end
definition rec2_on [reducible] {P : circle → Type} (x : circle) (Pb1 : P base1) (Pb2 : P base2)
(Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1) : P x :=
circle.rec2 Pb1 Pb2 Ps1 Ps2 x
definition rec2_seg1 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2)
(Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1)
: apD (rec2 Pb1 Pb2 Ps1 Ps2) seg1 = sorry ⬝ Ps1 ⬝ sorry :=
sorry
definition rec2_seg2 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2)
(Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1)
: apD (rec2 Pb1 Pb2 Ps1 Ps2) seg2 = sorry ⬝ Ps2 ⬝ sorry :=
sorry
definition elim2 {P : Type} (Pb1 Pb2 : P) (Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1) (x : circle) : P :=
rec2 Pb1 Pb2 (!tr_constant ⬝ Ps1) (!tr_constant ⬝ Ps2) x
definition elim2_on [reducible] {P : Type} (x : circle) (Pb1 Pb2 : P)
(Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1) : P :=
elim2 Pb1 Pb2 Ps1 Ps2 x
definition elim2_seg1 {P : Type} (Pb1 Pb2 : P) (Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1)
: ap (elim2 Pb1 Pb2 Ps1 Ps2) seg1 = sorry ⬝ Ps1 ⬝ sorry :=
sorry
definition elim2_seg2 {P : Type} (Pb1 Pb2 : P) (Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1)
: ap (elim2 Pb1 Pb2 Ps1 Ps2) seg2 = sorry ⬝ Ps2 ⬝ sorry :=
sorry
protected definition rec {P : circle → Type} (Pbase : P base) (Ploop : loop ▹ Pbase = Pbase)
(x : circle) : P x :=
begin
fapply (rec2_on x),
{ exact Pbase},
{ exact (transport P seg1 Pbase)},
{ apply idp},
{ apply eq_tr_of_inv_tr_eq, rewrite -tr_con, apply Ploop},
end
protected definition rec_on [reducible] {P : circle → Type} (x : circle) (Pbase : P base)
(Ploop : loop ▹ Pbase = Pbase) : P x :=
circle.rec Pbase Ploop x
protected definition elim {P : Type} (Pbase : P) (Ploop : Pbase = Pbase)
(x : circle) : P :=
circle.rec Pbase (tr_constant loop Pbase ⬝ Ploop) x
protected definition elim_on [reducible] {P : Type} (x : circle) (Pbase : P)
(Ploop : Pbase = Pbase) : P :=
elim Pbase Ploop x
definition rec_loop {P : circle → Type} (Pbase : P base) (Ploop : loop ▹ Pbase = Pbase) :
ap (circle.rec Pbase Ploop) loop = sorry ⬝ Ploop ⬝ sorry :=
sorry
definition elim_loop {P : Type} (Pbase : P) (Ploop : Pbase = Pbase) :
ap (circle.elim Pbase Ploop) loop = sorry ⬝ Ploop ⬝ sorry :=
sorry
end circle