8db4676c46
Add more theorems about mapping cylinders, fibers, truncated 2-quotient, truncated univalence, pre/postcomposition with an iso in a precategory. renamings: equiv.refl -> equiv.rfl and equiv_eq <-> equiv_eq'
143 lines
4.6 KiB
Text
143 lines
4.6 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 mapping cylinders
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-/
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import hit.quotient
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open quotient eq sum equiv fiber
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namespace cylinder
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section
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parameters {A B : Type} (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 := quotient cylinder_rel -- TODO: define this in root namespace
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parameter {f}
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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), Pbase (f a) =[seg a] Ptop a) (x : cylinder) : P x :=
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begin
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induction x,
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{ cases a,
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apply Pbase,
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apply Ptop},
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{ 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), Pbase (f a) =[seg 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), Pbase (f a) =[seg 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, pathover_of_eq (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 eq_of_fn_eq_fn_inv !(pathover_constant (seg a)),
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rewrite [▸*,-apd_eq_pathover_of_eq_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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attribute cylinder.base cylinder.top [constructor]
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attribute cylinder.rec cylinder.elim [unfold 8] [recursor 8]
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attribute cylinder.elim_type [unfold 7]
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attribute cylinder.rec_on cylinder.elim_on [unfold 5]
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attribute cylinder.elim_type_on [unfold 4]
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namespace cylinder
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open sigma sigma.ops
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variables {A B : Type} (f : A → B)
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/- cylinder as a dependent family -/
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definition pr1 [unfold 4] : cylinder f → B :=
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cylinder.elim id f (λa, idp)
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definition fcylinder : B → Type := fiber (pr1 f)
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definition cylinder_equiv_sigma_fcylinder [constructor] : cylinder f ≃ Σb, fcylinder f b :=
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!sigma_fiber_equiv⁻¹ᵉ
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variable {f}
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definition fbase (b : B) : fcylinder f b :=
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fiber.mk (base b) idp
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definition ftop (a : A) : fcylinder f (f a) :=
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fiber.mk (top a) idp
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definition fseg (a : A) : fbase (f a) = ftop a :=
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fiber_eq (seg a) !elim_seg⁻¹
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-- set_option pp.notation false
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-- -- The induction principle for the dependent mapping cylinder (TODO)
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-- protected definition frec {P : Π(b), fcylinder f b → Type}
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-- (Pbase : Π(b : B), P _ (fbase b)) (Ptop : Π(a : A), P _ (ftop a))
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-- (Pseg : Π(a : A), Pbase (f a) =[fseg a] Ptop a) {b : B} (x : fcylinder f b) : P _ x :=
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-- begin
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-- cases x with x p, induction p,
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-- induction x: esimp,
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-- { apply Pbase},
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-- { apply Ptop},
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-- { esimp, --fapply fiber_pathover,
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-- --refine pathover_of_pathover_ap P (λx, fiber.mk x idp),
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-- exact sorry}
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-- end
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-- theorem frec_fseg {P : Π(b), fcylinder f b → Type}
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-- (Pbase : Π(b : B), P _ (fbase b)) (Ptop : Π(a : A), P _ (ftop a))
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-- (Pseg : Π(a : A), Pbase (f a) =[fseg a] Ptop a) (a : A)
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-- : apd (cylinder.frec Pbase Ptop Pseg) (fseg a) = Pseg a :=
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-- sorry
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end cylinder
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