84 lines
3 KiB
Text
84 lines
3 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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Quotient of a reflexive relation
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-/
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import homotopy.circle cubical.squareover .two_quotient
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open eq simple_two_quotient e_closure
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namespace refl_quotient
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section
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parameters {A : Type} (R : A → A → Type) (ρ : Πa, R a a)
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inductive refl_quotient_Q : Π⦃a : A⦄, e_closure R a a → Type :=
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| Qmk {} : Π(a : A), refl_quotient_Q [ρ a]
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open refl_quotient_Q
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local abbreviation Q := refl_quotient_Q
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definition refl_quotient : Type := simple_two_quotient R Q
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definition rclass_of (a : A) : refl_quotient := incl0 R Q a
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definition req_of_rel ⦃a a' : A⦄ (r : R a a') : rclass_of a = rclass_of a' :=
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incl1 R Q r
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definition pρ (a : A) : req_of_rel (ρ a) = idp :=
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incl2 R Q (Qmk a)
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protected definition rec {P : refl_quotient → Type} (Pc : Π(a : A), P (rclass_of a))
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(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a =[req_of_rel H] Pc a')
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(Pr : Π(a : A), change_path (pρ a) (Pp (ρ a)) = idpo) (x : refl_quotient) : P x :=
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begin
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induction x,
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exact Pc a,
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exact Pp s,
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induction q, apply Pr
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end
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protected definition rec_on [reducible] {P : refl_quotient → Type} (x : refl_quotient)
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(Pc : Π(a : A), P (rclass_of a)) (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a =[req_of_rel H] Pc a')
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(Pr : Π(a : A), change_path (pρ a) (Pp (ρ a)) = idpo) : P x :=
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rec Pc Pp Pr x
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definition rec_req_of_rel {P : Type} {P : refl_quotient → Type} (Pc : Π(a : A), P (rclass_of a))
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(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a =[req_of_rel H] Pc a')
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(Pr : Π(a : A), change_path (pρ a) (Pp (ρ a)) = idpo) ⦃a a' : A⦄ (r : R a a')
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: apd (rec Pc Pp Pr) (req_of_rel r) = Pp r :=
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!rec_incl1
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protected definition elim {P : Type} (Pc : Π(a : A), P)
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(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a') (Pr : Π(a : A), Pp (ρ a) = idp)
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(x : refl_quotient) : P :=
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begin
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induction x,
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exact Pc a,
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exact Pp s,
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induction q, apply Pr
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end
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protected definition elim_on [reducible] {P : Type} (x : refl_quotient) (Pc : Π(a : A), P)
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(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a') (Pr : Π(a : A), Pp (ρ a) = idp) : P :=
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elim Pc Pp Pr x
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definition elim_req_of_rel {P : Type} {Pc : Π(a : A), P}
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{Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a'} (Pr : Π(a : A), Pp (ρ a) = idp)
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⦃a a' : A⦄ (r : R a a') : ap (elim Pc Pp Pr) (req_of_rel r) = Pp r :=
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!elim_incl1
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theorem elim_pρ {P : Type} (Pc : Π(a : A), P)
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(Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a') (Pr : Π(a : A), Pp (ρ a) = idp) (a : A)
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: square (ap02 (elim Pc Pp Pr) (pρ a)) (Pr a) (elim_req_of_rel Pr (ρ a)) idp :=
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!elim_incl2
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end
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end refl_quotient
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attribute refl_quotient.rclass_of [constructor]
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attribute refl_quotient.rec refl_quotient.elim [unfold 8] [recursor 8]
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--attribute refl_quotient.elim_type [unfold 9]
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attribute refl_quotient.rec_on refl_quotient.elim_on [unfold 5]
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--attribute refl_quotient.elim_type_on [unfold 6]
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