120 lines
4.1 KiB
Text
120 lines
4.1 KiB
Text
/-
|
|
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 equiv equiv.ops
|
|
|
|
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
|
|
|
|
theorem 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 = Ps1 :=
|
|
!rec_merid
|
|
|
|
theorem 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 = Ps2 :=
|
|
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
|
|
|
|
theorem elim2_seg1 {P : Type} (Pb1 Pb2 : P) (Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1)
|
|
: ap (elim2 Pb1 Pb2 Ps1 Ps2) seg1 = Ps1 :=
|
|
begin
|
|
apply (@cancel_left _ _ _ _ (tr_constant seg1 (elim2 Pb1 Pb2 Ps1 Ps2 base1))),
|
|
rewrite [-apd_eq_tr_constant_con_ap,↑elim2,rec2_seg1],
|
|
end
|
|
|
|
theorem elim2_seg2 {P : Type} (Pb1 Pb2 : P) (Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1)
|
|
: ap (elim2 Pb1 Pb2 Ps1 Ps2) seg2 = Ps2 :=
|
|
begin
|
|
apply (@cancel_left _ _ _ _ (tr_constant seg2 (elim2 Pb1 Pb2 Ps1 Ps2 base2))),
|
|
rewrite [-apd_eq_tr_constant_con_ap,↑elim2,rec2_seg2],
|
|
end
|
|
|
|
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
|
|
|
|
example {P : circle → Type} (Pbase : P base) (Ploop : loop ▹ Pbase = Pbase) : rec Pbase Ploop base = Pbase := idp
|
|
|
|
protected definition rec_on [reducible] {P : circle → Type} (x : circle) (Pbase : P base)
|
|
(Ploop : loop ▹ Pbase = Pbase) : P x :=
|
|
rec Pbase Ploop x
|
|
|
|
theorem rec_loop {P : circle → Type} (Pbase : P base) (Ploop : loop ▹ Pbase = Pbase) :
|
|
apd (rec Pbase Ploop) loop = Ploop :=
|
|
sorry
|
|
|
|
protected definition elim {P : Type} (Pbase : P) (Ploop : Pbase = Pbase)
|
|
(x : circle) : P :=
|
|
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
|
|
|
|
theorem elim_loop {P : Type} (Pbase : P) (Ploop : Pbase = Pbase) :
|
|
ap (elim Pbase Ploop) loop = Ploop :=
|
|
begin
|
|
apply (@cancel_left _ _ _ _ (tr_constant loop (elim Pbase Ploop base))),
|
|
rewrite [-apd_eq_tr_constant_con_ap,↑elim,rec_loop],
|
|
end
|
|
|
|
protected definition elim_type (Pbase : Type) (Ploop : Pbase ≃ Pbase)
|
|
(x : circle) : Type :=
|
|
elim Pbase (ua Ploop) x
|
|
|
|
protected definition elim_type_on [reducible] (x : circle) (Pbase : Type)
|
|
(Ploop : Pbase ≃ Pbase) : Type :=
|
|
elim_type Pbase Ploop x
|
|
|
|
theorem elim_type_loop (Pbase : Type) (Ploop : Pbase ≃ Pbase) :
|
|
transport (elim_type Pbase Ploop) loop = Ploop :=
|
|
by rewrite [tr_eq_cast_ap_fn,↑elim_type,elim_loop];apply cast_ua_fn
|
|
|
|
end circle
|