91 lines
3 KiB
Text
91 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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Module: hit.cylinder
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Authors: Floris van Doorn
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Declaration of mapping cylinders
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-/
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import .type_quotient
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open type_quotient eq sum equiv equiv.ops
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namespace cylinder
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section
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universe u
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parameters {A B : Type.{u}} (f : A → B)
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local abbreviation C := B + A
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inductive cylinder_rel : C → C → Type :=
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| Rmk : Π(a : A), cylinder_rel (inl (f a)) (inr a)
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open cylinder_rel
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local abbreviation R := cylinder_rel
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definition cylinder := type_quotient cylinder_rel -- TODO: define this in root namespace
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definition base (b : B) : cylinder :=
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class_of R (inl b)
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definition top (a : A) : cylinder :=
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class_of R (inr a)
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definition seg (a : A) : base (f a) = top a :=
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eq_of_rel cylinder_rel (Rmk f a)
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protected definition rec {P : cylinder → Type}
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(Pbase : Π(b : B), P (base b)) (Ptop : Π(a : A), P (top a))
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(Pseg : Π(a : A), seg a ▹ Pbase (f a) = Ptop a) (x : cylinder) : P x :=
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begin
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fapply (type_quotient.rec_on x),
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{ intro a, cases a,
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apply Pbase,
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apply Ptop},
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{ intros [a, a', H], cases H, apply Pseg}
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end
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protected definition rec_on [reducible] {P : cylinder → Type} (x : cylinder)
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(Pbase : Π(b : B), P (base b)) (Ptop : Π(a : A), P (top a))
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(Pseg : Π(a : A), seg a ▹ Pbase (f a) = Ptop a) : P x :=
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rec Pbase Ptop Pseg x
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theorem rec_seg {P : cylinder → Type}
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(Pbase : Π(b : B), P (base b)) (Ptop : Π(a : A), P (top a))
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(Pseg : Π(a : A), seg a ▹ Pbase (f a) = Ptop a)
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(a : A) : apd (rec Pbase Ptop Pseg) (seg a) = Pseg a :=
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!rec_eq_of_rel
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protected definition elim {P : Type} (Pbase : B → P) (Ptop : A → P)
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(Pseg : Π(a : A), Pbase (f a) = Ptop a) (x : cylinder) : P :=
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rec Pbase Ptop (λa, !tr_constant ⬝ Pseg a) x
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protected definition elim_on [reducible] {P : Type} (x : cylinder) (Pbase : B → P) (Ptop : A → P)
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(Pseg : Π(a : A), Pbase (f a) = Ptop a) : P :=
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elim Pbase Ptop Pseg x
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theorem elim_seg {P : Type} (Pbase : B → P) (Ptop : A → P)
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(Pseg : Π(a : A), Pbase (f a) = Ptop a)
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(a : A) : ap (elim Pbase Ptop Pseg) (seg a) = Pseg a :=
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begin
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apply (@cancel_left _ _ _ _ (tr_constant (seg a) (elim Pbase Ptop Pseg (base (f a))))),
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rewrite [-apd_eq_tr_constant_con_ap,↑elim,rec_seg],
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end
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protected definition elim_type (Pbase : B → Type) (Ptop : A → Type)
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(Pseg : Π(a : A), Pbase (f a) ≃ Ptop a) (x : cylinder) : Type :=
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elim Pbase Ptop (λa, ua (Pseg a)) x
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protected definition elim_type_on [reducible] (x : cylinder) (Pbase : B → Type) (Ptop : A → Type)
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(Pseg : Π(a : A), Pbase (f a) ≃ Ptop a) : Type :=
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elim_type Pbase Ptop Pseg x
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theorem elim_type_seg (Pbase : B → Type) (Ptop : A → Type)
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(Pseg : Π(a : A), Pbase (f a) ≃ Ptop a)
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(a : A) : transport (elim_type Pbase Ptop Pseg) (seg a) = Pseg a :=
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by rewrite [tr_eq_cast_ap_fn,↑elim_type,elim_seg];apply cast_ua_fn
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end
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end cylinder
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