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