lean2/hott/hit/torus.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.
Authors: Floris van Doorn
Declaration of the torus
-/
import .two_quotient
open two_quotient eq bool unit relation
namespace torus
definition torus_R (x y : unit) := bool
local infix `⬝r`:75 := @e_closure.trans unit torus_R star star star
local postfix `⁻¹ʳ`:(max+10) := @e_closure.symm unit torus_R star star
local notation `[`:max a `]`:0 := @e_closure.of_rel unit torus_R star star a
inductive torus_Q : Π⦃x y : unit⦄, e_closure torus_R x y → e_closure torus_R x y → Type :=
| Qmk : torus_Q ([ff] ⬝r [tt]) ([tt] ⬝r [ff])
definition torus := two_quotient torus_R torus_Q
definition base : torus := incl0 _ _ star
definition loop1 : base = base := incl1 _ _ ff
definition loop2 : base = base := incl1 _ _ tt
definition surf : loop1 ⬝ loop2 = loop2 ⬝ loop1 :=
incl2 _ _ torus_Q.Qmk
-- protected definition rec {P : torus → Type} (Pb : P base) (Pl1 : Pb =[loop1] Pb)
-- (Pl2 : Pb =[loop2] Pb) (Pf : squareover P fill Pl1 Pl1 Pl2 Pl2)
-- (x : torus) : P x :=
-- sorry
-- example {P : torus → Type} (Pb : P base) (Pl1 : Pb =[loop1] Pb) (Pl2 : Pb =[loop2] Pb)
-- (Pf : squareover P fill Pl1 Pl1 Pl2 Pl2) : torus.rec Pb Pl1 Pl2 Pf base = Pb := idp
-- definition rec_loop1 {P : torus → Type} (Pb : P base) (Pl1 : Pb =[loop1] Pb)
-- (Pl2 : Pb =[loop2] Pb) (Pf : squareover P fill Pl1 Pl1 Pl2 Pl2)
-- : apdo (torus.rec Pb Pl1 Pl2 Pf) loop1 = Pl1 :=
-- sorry
-- definition rec_loop2 {P : torus → Type} (Pb : P base) (Pl1 : Pb =[loop1] Pb)
-- (Pl2 : Pb =[loop2] Pb) (Pf : squareover P fill Pl1 Pl1 Pl2 Pl2)
-- : apdo (torus.rec Pb Pl1 Pl2 Pf) loop2 = Pl2 :=
-- sorry
-- definition rec_surf {P : torus → Type} (Pb : P base) (Pl1 : Pb =[loop1] Pb)
-- (Pl2 : Pb =[loop2] Pb) (Pf : squareover P fill Pl1 Pl1 Pl2 Pl2)
-- : cubeover P rfl1 (apds (torus.rec Pb Pl1 Pl2 Pf) fill) Pf
-- (vdeg_squareover !rec_loop2) (vdeg_squareover !rec_loop2)
-- (vdeg_squareover !rec_loop1) (vdeg_squareover !rec_loop1) :=
-- sorry
protected definition elim {P : Type} (Pb : P) (Pl1 : Pb = Pb) (Pl2 : Pb = Pb)
(Ps : Pl1 ⬝ Pl2 = Pl2 ⬝ Pl1) (x : torus) : P :=
begin
induction x,
{ exact Pb},
{ induction s,
{ exact Pl1},
{ exact Pl2}},
{ induction q, exact Ps},
end
protected definition elim_on [reducible] {P : Type} (x : torus) (Pb : P)
(Pl1 : Pb = Pb) (Pl2 : Pb = Pb) (Ps : Pl1 ⬝ Pl2 = Pl2 ⬝ Pl1) : P :=
torus.elim Pb Pl1 Pl2 Ps x
definition elim_loop1 {P : Type} (Pb : P) (Pl1 : Pb = Pb) (Pl2 : Pb = Pb)
(Ps : Pl1 ⬝ Pl2 = Pl2 ⬝ Pl1) : ap (torus.elim Pb Pl1 Pl2 Ps) loop1 = Pl1 :=
!elim_incl1
definition elim_loop2 {P : Type} (Pb : P) (Pl1 : Pb = Pb) (Pl2 : Pb = Pb)
(Ps : Pl1 ⬝ Pl2 = Pl2 ⬝ Pl1) : ap (torus.elim Pb Pl1 Pl2 Ps) loop2 = Pl2 :=
!elim_incl1
theorem elim_surf {P : Type} (Pb : P) (Pl1 : Pb = Pb) (Pl2 : Pb = Pb)
(Ps : Pl1 ⬝ Pl2 = Pl2 ⬝ Pl1)
: square (ap02 (torus.elim Pb Pl1 Pl2 Ps) surf)
Ps
(!ap_con ⬝ (!elim_loop1 ◾ !elim_loop2))
(!ap_con ⬝ (!elim_loop2 ◾ !elim_loop1)) :=
!elim_incl2
end torus
attribute torus.base [constructor]
attribute /-torus.rec-/ torus.elim [unfold 6] [recursor 6]
--attribute torus.elim_type [unfold 9]
attribute /-torus.rec_on-/ torus.elim_on [unfold 2]
--attribute torus.elim_type_on [unfold 6]