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feat(hit): add hits with 2-path constructors In hit.two_quotient we define a general construction to define hits with 2-dimensional path constructors, similar to quotients. We can add 2-paths between any two 'words', where a word consists of 1-path constructors, concatenation and inverses. We use this to define the torus, reflexive quotients and the reduced suspension. There is still one 'sorry' in the construction
2015-06-26 02:27:03 +00:00
/-
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
-/
refactor(hott): move homotopy hits to new homotopy folder
2015-09-13 18:58:11 +00:00
import hit.two_quotient
feat(hit): add hits with 2-path constructors In hit.two_quotient we define a general construction to define hits with 2-dimensional path constructors, similar to quotients. We can add 2-paths between any two 'words', where a word consists of 1-path constructors, concatenation and inverses. We use this to define the torus, reflexive quotients and the reduced suspension. There is still one 'sorry' in the construction
2015-06-26 02:27:03 +00:00
feat(homotopy/torus): give recursion and induction principle for the torus also change the surface of the torus to a square instead of an equality between paths
2015-11-22 22:46:54 +00:00
open two_quotient eq bool unit equiv
feat(hit): add hits with 2-path constructors In hit.two_quotient we define a general construction to define hits with 2-dimensional path constructors, similar to quotients. We can add 2-paths between any two 'words', where a word consists of 1-path constructors, concatenation and inverses. We use this to define the torus, reflexive quotients and the reduced suspension. There is still one 'sorry' in the construction
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namespace torus
feat(homotopy/torus): give recursion and induction principle for the torus also change the surface of the torus to a square instead of an equality between paths
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open e_closure relation
feat(hit): add hits with 2-path constructors In hit.two_quotient we define a general construction to define hits with 2-dimensional path constructors, similar to quotients. We can add 2-paths between any two 'words', where a word consists of 1-path constructors, concatenation and inverses. We use this to define the torus, reflexive quotients and the reduced suspension. There is still one 'sorry' in the construction
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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])
feat(homotopy/torus): give recursion and induction principle for the torus also change the surface of the torus to a square instead of an equality between paths
2015-11-22 22:46:54 +00:00
open torus_Q
feat(hit): add hits with 2-path constructors In hit.two_quotient we define a general construction to define hits with 2-dimensional path constructors, similar to quotients. We can add 2-paths between any two 'words', where a word consists of 1-path constructors, concatenation and inverses. We use this to define the torus, reflexive quotients and the reduced suspension. There is still one 'sorry' in the construction
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definition torus := two_quotient torus_R torus_Q
feat(homotopy/torus): give recursion and induction principle for the torus also change the surface of the torus to a square instead of an equality between paths
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notation `T²` := torus
feat(hit): add hits with 2-path constructors In hit.two_quotient we define a general construction to define hits with 2-dimensional path constructors, similar to quotients. We can add 2-paths between any two 'words', where a word consists of 1-path constructors, concatenation and inverses. We use this to define the torus, reflexive quotients and the reduced suspension. There is still one 'sorry' in the construction
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definition base : torus := incl0 _ _ star
definition loop1 : base = base := incl1 _ _ ff
definition loop2 : base = base := incl1 _ _ tt
feat(homotopy/torus): give recursion and induction principle for the torus also change the surface of the torus to a square instead of an equality between paths
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definition surf' : loop1 ⬝ loop2 = loop2 ⬝ loop1 :=
incl2 _ _ Qmk
definition surf : square loop1 loop1 loop2 loop2 :=
square_of_eq (incl2 _ _ Qmk)
protected definition rec {P : torus → Type} (Pb : P base) (Pl1 : Pb =[loop1] Pb)
(Pl2 : Pb =[loop2] Pb) (Ps : squareover P surf Pl1 Pl1 Pl2 Pl2) (x : torus) : P x :=
begin
induction x,
{ induction a, exact Pb},
{ induction s: induction a; induction a',
{ exact Pl1},
{ exact Pl2}},
{ induction q, esimp, apply change_path_of_pathover, apply pathover_of_squareover, exact Ps},
end
protected definition rec_on [reducible] {P : torus → Type} (x : torus) (Pb : P base)
(Pl1 : Pb =[loop1] Pb) (Pl2 : Pb =[loop2] Pb) (Ps : squareover P surf Pl1 Pl1 Pl2 Pl2) : P x :=
torus.rec Pb Pl1 Pl2 Ps x
theorem rec_loop1 {P : torus → Type} (Pb : P base) (Pl1 : Pb =[loop1] Pb)
(Pl2 : Pb =[loop2] Pb) (Ps : squareover P surf Pl1 Pl1 Pl2 Pl2)
: apdo (torus.rec Pb Pl1 Pl2 Ps) loop1 = Pl1 :=
!rec_incl1
theorem rec_loop2 {P : torus → Type} (Pb : P base) (Pl1 : Pb =[loop1] Pb)
(Pl2 : Pb =[loop2] Pb) (Ps : squareover P surf Pl1 Pl1 Pl2 Pl2)
: apdo (torus.rec Pb Pl1 Pl2 Ps) loop2 = Pl2 :=
!rec_incl1
feat(hit): add hits with 2-path constructors In hit.two_quotient we define a general construction to define hits with 2-dimensional path constructors, similar to quotients. We can add 2-paths between any two 'words', where a word consists of 1-path constructors, concatenation and inverses. We use this to define the torus, reflexive quotients and the reduced suspension. There is still one 'sorry' in the construction
2015-06-26 02:27:03 +00:00
protected definition elim {P : Type} (Pb : P) (Pl1 : Pb = Pb) (Pl2 : Pb = Pb)
feat(homotopy/torus): give recursion and induction principle for the torus also change the surface of the torus to a square instead of an equality between paths
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(Ps : square Pl1 Pl1 Pl2 Pl2) (x : torus) : P :=
feat(hit): add hits with 2-path constructors In hit.two_quotient we define a general construction to define hits with 2-dimensional path constructors, similar to quotients. We can add 2-paths between any two 'words', where a word consists of 1-path constructors, concatenation and inverses. We use this to define the torus, reflexive quotients and the reduced suspension. There is still one 'sorry' in the construction
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begin
induction x,
{ exact Pb},
{ induction s,
{ exact Pl1},
{ exact Pl2}},
feat(homotopy/torus): give recursion and induction principle for the torus also change the surface of the torus to a square instead of an equality between paths
2015-11-22 22:46:54 +00:00
{ induction q, apply eq_of_square, exact Ps},
feat(hit): add hits with 2-path constructors In hit.two_quotient we define a general construction to define hits with 2-dimensional path constructors, similar to quotients. We can add 2-paths between any two 'words', where a word consists of 1-path constructors, concatenation and inverses. We use this to define the torus, reflexive quotients and the reduced suspension. There is still one 'sorry' in the construction
2015-06-26 02:27:03 +00:00
end
protected definition elim_on [reducible] {P : Type} (x : torus) (Pb : P)
feat(homotopy/torus): give recursion and induction principle for the torus also change the surface of the torus to a square instead of an equality between paths
2015-11-22 22:46:54 +00:00
(Pl1 : Pb = Pb) (Pl2 : Pb = Pb) (Ps : square Pl1 Pl1 Pl2 Pl2) : P :=
feat(hit): add hits with 2-path constructors In hit.two_quotient we define a general construction to define hits with 2-dimensional path constructors, similar to quotients. We can add 2-paths between any two 'words', where a word consists of 1-path constructors, concatenation and inverses. We use this to define the torus, reflexive quotients and the reduced suspension. There is still one 'sorry' in the construction
2015-06-26 02:27:03 +00:00
torus.elim Pb Pl1 Pl2 Ps x
feat(homotopy/torus): give recursion and induction principle for the torus also change the surface of the torus to a square instead of an equality between paths
2015-11-22 22:46:54 +00:00
definition elim_loop1 {P : Type} {Pb : P} {Pl1 : Pb = Pb} {Pl2 : Pb = Pb}
(Ps : square Pl1 Pl1 Pl2 Pl2) : ap (torus.elim Pb Pl1 Pl2 Ps) loop1 = Pl1 :=
feat(hit): add hits with 2-path constructors In hit.two_quotient we define a general construction to define hits with 2-dimensional path constructors, similar to quotients. We can add 2-paths between any two 'words', where a word consists of 1-path constructors, concatenation and inverses. We use this to define the torus, reflexive quotients and the reduced suspension. There is still one 'sorry' in the construction
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!elim_incl1
feat(homotopy/torus): give recursion and induction principle for the torus also change the surface of the torus to a square instead of an equality between paths
2015-11-22 22:46:54 +00:00
definition elim_loop2 {P : Type} {Pb : P} {Pl1 : Pb = Pb} {Pl2 : Pb = Pb}
(Ps : square Pl1 Pl1 Pl2 Pl2) : ap (torus.elim Pb Pl1 Pl2 Ps) loop2 = Pl2 :=
feat(hit): add hits with 2-path constructors In hit.two_quotient we define a general construction to define hits with 2-dimensional path constructors, similar to quotients. We can add 2-paths between any two 'words', where a word consists of 1-path constructors, concatenation and inverses. We use this to define the torus, reflexive quotients and the reduced suspension. There is still one 'sorry' in the construction
2015-06-26 02:27:03 +00:00
!elim_incl1
feat(homotopy/torus): give recursion and induction principle for the torus also change the surface of the torus to a square instead of an equality between paths
2015-11-22 22:46:54 +00:00
theorem elim_surf {P : Type} {Pb : P} {Pl1 : Pb = Pb} {Pl2 : Pb = Pb}
(Ps : square Pl1 Pl1 Pl2 Pl2)
: whisker_square (elim_loop1 Ps) (elim_loop1 Ps) (elim_loop2 Ps) (elim_loop2 Ps)
(aps (torus.elim Pb Pl1 Pl2 Ps) surf) = Ps :=
begin
apply whisker_square_aps_eq,
apply elim_incl2
end
feat(hit): add hits with 2-path constructors In hit.two_quotient we define a general construction to define hits with 2-dimensional path constructors, similar to quotients. We can add 2-paths between any two 'words', where a word consists of 1-path constructors, concatenation and inverses. We use this to define the torus, reflexive quotients and the reduced suspension. There is still one 'sorry' in the construction
2015-06-26 02:27:03 +00:00
end torus
attribute torus.base [constructor]
feat(homotopy/torus): give recursion and induction principle for the torus also change the surface of the torus to a square instead of an equality between paths
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attribute torus.rec torus.elim [unfold 6] [recursor 6]
--attribute torus.elim_type [unfold 5]
attribute torus.rec_on torus.elim_on [unfold 2]
--attribute torus.elim_type_on [unfold 1]
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