lean2/hott/hit/red_susp.hlean
Floris van Doorn 24762fe843 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
2015-06-25 22:31:41 -04:00

87 lines
2.7 KiB
Text

/-
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 reduced suspension
-/
import .two_quotient types.pointed algebra.e_closure
open simple_two_quotient eq unit pointed e_closure
namespace red_susp
section
parameter {A : Pointed}
inductive red_susp_R : unit → unit → Type :=
| Rmk : Π(a : A), red_susp_R star star
open red_susp_R
inductive red_susp_Q : Π⦃x : unit⦄, e_closure red_susp_R x x → Type :=
| Qmk : red_susp_Q [Rmk pt]
open red_susp_Q
local abbreviation R := red_susp_R
local abbreviation Q := red_susp_Q
definition red_susp : Type := simple_two_quotient R Q -- TODO: define this in root namespace
definition base : red_susp :=
incl0 R Q star
definition merid (a : A) : base = base :=
incl1 R Q (Rmk a)
definition merid_pt : merid pt = idp :=
incl2 R Q Qmk
-- protected definition rec {P : red_susp → Type} (Pb : P base) (Pm : Π(a : A), Pb =[merid a] Pb)
-- (Pe : Pm pt =[merid_pt] idpo) (x : red_susp) : P x :=
-- begin
-- induction x,
-- end
-- protected definition rec_on [reducible] {P : red_susp → Type} (x : red_susp) (Pb : P base)
-- (Pm : Π(a : A), Pb =[merid a] Pb) (Pe : Pm pt =[merid_pt] idpo) : P x :=
-- rec Pb Pm Pe x
-- definition rec_merid {P : red_susp → Type} (Pb : P base) (Pm : Π(a : A), Pb =[merid a] Pb)
-- (Pe : Pm pt =[merid_pt] idpo) (a : A)
-- : apdo (rec Pb Pm Pe) (merid a) = Pm a :=
-- !rec_incl1
-- theorem elim_merid_pt {P : red_susp → Type} (Pb : P base) (Pm : Π(a : A), Pb =[merid a] Pb)
-- (Pe : Pm pt =[merid_pt] idpo)
-- : square (ap02 (rec Pb Pm Pe) merid_pt) Pe (rec_merid Pe pt) idp :=
-- !rec_incl2
protected definition elim {P : Type} (Pb : P) (Pm : Π(a : A), Pb = Pb)
(Pe : Pm pt = idp) (x : red_susp) : P :=
begin
induction x,
exact Pb,
induction s, exact Pm a,
induction q, exact Pe
end
protected definition elim_on [reducible] {P : Type} (x : red_susp) (Pb : P)
(Pm : Π(a : A), Pb = Pb) (Pe : Pm pt = idp) : P :=
elim Pb Pm Pe x
definition elim_merid {P : Type} {Pb : P} {Pm : Π(a : A), Pb = Pb}
(Pe : Pm pt = idp) (a : A)
: ap (elim Pb Pm Pe) (merid a) = Pm a :=
!elim_incl1
theorem elim_merid_pt {P : Type} (Pb : P) (Pm : Π(a : A), Pb = Pb)
(Pe : Pm pt = idp) : square (ap02 (elim Pb Pm Pe) merid_pt) Pe (elim_merid Pe pt) idp :=
!elim_incl2
end
end red_susp
attribute red_susp.base [constructor]
attribute /-red_susp.rec-/ red_susp.elim [unfold-c 6] [recursor 6]
--attribute red_susp.elim_type [unfold-c 9]
attribute /-red_susp.rec_on-/ red_susp.elim_on [unfold-c 3]
--attribute red_susp.elim_type_on [unfold-c 6]