74 lines
2.3 KiB
Text
74 lines
2.3 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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Module: hit.suspension
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Authors: Floris van Doorn
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Declaration of suspension
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-/
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import .pushout
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open pushout unit eq equiv equiv.ops
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definition suspension (A : Type) : Type := pushout (λ(a : A), star.{0}) (λ(a : A), star.{0})
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namespace suspension
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variable {A : Type}
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definition north (A : Type) : suspension A :=
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inl _ _ star
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definition south (A : Type) : suspension A :=
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inr _ _ star
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definition merid (a : A) : north A = south A :=
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glue _ _ a
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protected definition rec {P : suspension A → Type} (PN : P !north) (PS : P !south)
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(Pm : Π(a : A), merid a ▸ PN = PS) (x : suspension A) : P x :=
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begin
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fapply (pushout.rec_on _ _ x),
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{ intro u, cases u, exact PN},
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{ intro u, cases u, exact PS},
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{ exact Pm},
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end
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protected definition rec_on [reducible] {P : suspension A → Type} (y : suspension A)
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(PN : P !north) (PS : P !south) (Pm : Π(a : A), merid a ▸ PN = PS) : P y :=
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rec PN PS Pm y
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theorem rec_merid {P : suspension A → Type} (PN : P !north) (PS : P !south)
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(Pm : Π(a : A), merid a ▸ PN = PS) (a : A)
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: apd (rec PN PS Pm) (merid a) = Pm a :=
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!rec_glue
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protected definition elim {P : Type} (PN : P) (PS : P) (Pm : A → PN = PS)
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(x : suspension A) : P :=
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rec PN PS (λa, !tr_constant ⬝ Pm a) x
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protected definition elim_on [reducible] {P : Type} (x : suspension A)
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(PN : P) (PS : P) (Pm : A → PN = PS) : P :=
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elim PN PS Pm x
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theorem elim_merid {P : Type} (PN : P) (PS : P) (Pm : A → PN = PS) (a : A)
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: ap (elim PN PS Pm) (merid a) = Pm a :=
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begin
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apply (@cancel_left _ _ _ _ (tr_constant (merid a) (elim PN PS Pm !north))),
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rewrite [-apd_eq_tr_constant_con_ap,↑elim,rec_merid],
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end
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protected definition elim_type (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
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(x : suspension A) : Type :=
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elim PN PS (λa, ua (Pm a)) x
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protected definition elim_type_on [reducible] (x : suspension A)
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(PN : Type) (PS : Type) (Pm : A → PN ≃ PS) : Type :=
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elim_type PN PS Pm x
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theorem elim_type_merid (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
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(x : suspension A) (a : A) : transport (elim_type PN PS Pm) (merid a) = Pm a :=
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by rewrite [tr_eq_cast_ap_fn,↑elim_type,elim_merid];apply cast_ua_fn
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end suspension
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