100 lines
3.5 KiB
Text
100 lines
3.5 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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Authors: Floris van Doorn
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Declaration of the torus
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-/
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import hit.two_quotient
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open two_quotient eq bool unit equiv
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namespace torus
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open e_closure relation
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definition torus_R (x y : unit) := bool
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local infix `⬝r`:75 := @e_closure.trans unit torus_R star star star
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local postfix `⁻¹ʳ`:(max+10) := @e_closure.symm unit torus_R star star
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local notation `[`:max a `]`:0 := @e_closure.of_rel unit torus_R star star a
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inductive torus_Q : Π⦃x y : unit⦄, e_closure torus_R x y → e_closure torus_R x y → Type :=
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| Qmk : torus_Q ([ff] ⬝r [tt]) ([tt] ⬝r [ff])
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open torus_Q
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definition torus := two_quotient torus_R torus_Q
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notation `T²` := torus
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definition base : torus := incl0 _ _ star
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definition loop1 : base = base := incl1 _ _ ff
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definition loop2 : base = base := incl1 _ _ tt
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definition surf' : loop1 ⬝ loop2 = loop2 ⬝ loop1 :=
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incl2 _ _ Qmk
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definition surf : square loop1 loop1 loop2 loop2 :=
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square_of_eq (incl2 _ _ Qmk)
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protected definition rec {P : torus → Type} (Pb : P base) (Pl1 : Pb =[loop1] Pb)
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(Pl2 : Pb =[loop2] Pb) (Ps : squareover P surf Pl1 Pl1 Pl2 Pl2) (x : torus) : P x :=
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begin
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induction x,
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{ induction a, exact Pb},
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{ induction s: induction a; induction a',
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{ exact Pl1},
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{ exact Pl2}},
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{ induction q, esimp, apply change_path_of_pathover, apply pathover_of_squareover, exact Ps},
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end
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protected definition rec_on [reducible] {P : torus → Type} (x : torus) (Pb : P base)
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(Pl1 : Pb =[loop1] Pb) (Pl2 : Pb =[loop2] Pb) (Ps : squareover P surf Pl1 Pl1 Pl2 Pl2) : P x :=
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torus.rec Pb Pl1 Pl2 Ps x
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theorem rec_loop1 {P : torus → Type} (Pb : P base) (Pl1 : Pb =[loop1] Pb)
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(Pl2 : Pb =[loop2] Pb) (Ps : squareover P surf Pl1 Pl1 Pl2 Pl2)
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: apdo (torus.rec Pb Pl1 Pl2 Ps) loop1 = Pl1 :=
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!rec_incl1
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theorem rec_loop2 {P : torus → Type} (Pb : P base) (Pl1 : Pb =[loop1] Pb)
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(Pl2 : Pb =[loop2] Pb) (Ps : squareover P surf Pl1 Pl1 Pl2 Pl2)
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: apdo (torus.rec Pb Pl1 Pl2 Ps) loop2 = Pl2 :=
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!rec_incl1
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protected definition elim {P : Type} (Pb : P) (Pl1 : Pb = Pb) (Pl2 : Pb = Pb)
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(Ps : square Pl1 Pl1 Pl2 Pl2) (x : torus) : P :=
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begin
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induction x,
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{ exact Pb},
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{ induction s,
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{ exact Pl1},
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{ exact Pl2}},
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{ induction q, apply eq_of_square, exact Ps},
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end
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protected definition elim_on [reducible] {P : Type} (x : torus) (Pb : P)
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(Pl1 : Pb = Pb) (Pl2 : Pb = Pb) (Ps : square Pl1 Pl1 Pl2 Pl2) : P :=
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torus.elim Pb Pl1 Pl2 Ps x
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definition elim_loop1 {P : Type} {Pb : P} {Pl1 : Pb = Pb} {Pl2 : Pb = Pb}
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(Ps : square Pl1 Pl1 Pl2 Pl2) : ap (torus.elim Pb Pl1 Pl2 Ps) loop1 = Pl1 :=
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!elim_incl1
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definition elim_loop2 {P : Type} {Pb : P} {Pl1 : Pb = Pb} {Pl2 : Pb = Pb}
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(Ps : square Pl1 Pl1 Pl2 Pl2) : ap (torus.elim Pb Pl1 Pl2 Ps) loop2 = Pl2 :=
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!elim_incl1
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theorem elim_surf {P : Type} {Pb : P} {Pl1 : Pb = Pb} {Pl2 : Pb = Pb}
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(Ps : square Pl1 Pl1 Pl2 Pl2)
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: whisker_square (elim_loop1 Ps) (elim_loop1 Ps) (elim_loop2 Ps) (elim_loop2 Ps)
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(aps (torus.elim Pb Pl1 Pl2 Ps) surf) = Ps :=
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begin
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apply whisker_square_aps_eq,
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apply elim_incl2
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end
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end torus
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attribute torus.base [constructor]
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attribute torus.rec torus.elim [unfold 6] [recursor 6]
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--attribute torus.elim_type [unfold 5]
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attribute torus.rec_on torus.elim_on [unfold 2]
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--attribute torus.elim_type_on [unfold 1]
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