449 lines
18 KiB
Text
449 lines
18 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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Author: Floris van Doorn
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Basic theorems about pathovers
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-/
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prelude
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import .path .equiv
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open equiv is_equiv function
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variables {A A' : Type} {B B' : A → Type} {B'' : A' → Type} {C : Π⦃a⦄, B a → Type}
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{a a₂ a₃ a₄ : A} {p p' : a = a₂} {p₂ : a₂ = a₃} {p₃ : a₃ = a₄} {p₁₃ : a = a₃}
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{b b' : B a} {b₂ b₂' : B a₂} {b₃ : B a₃} {b₄ : B a₄}
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{c : C b} {c₂ : C b₂}
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namespace eq
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inductive pathover.{l} (B : A → Type.{l}) (b : B a) : Π{a₂ : A}, a = a₂ → B a₂ → Type.{l} :=
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idpatho : pathover B b (refl a) b
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notation b ` =[`:50 p:0 `] `:0 b₂:50 := pathover _ b p b₂
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definition idpo [reducible] [constructor] : b =[refl a] b :=
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pathover.idpatho b
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definition idpatho [reducible] [constructor] (b : B a) : b =[refl a] b :=
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pathover.idpatho b
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/- equivalences with equality using transport -/
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definition pathover_of_tr_eq [unfold 5 8] (r : p ▸ b = b₂) : b =[p] b₂ :=
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by cases p; cases r; constructor
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definition pathover_of_eq_tr [unfold 5 8] (r : b = p⁻¹ ▸ b₂) : b =[p] b₂ :=
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by cases p; cases r; constructor
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definition tr_eq_of_pathover [unfold 8] (r : b =[p] b₂) : p ▸ b = b₂ :=
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by cases r; reflexivity
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definition eq_tr_of_pathover [unfold 8] (r : b =[p] b₂) : b = p⁻¹ ▸ b₂ :=
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by cases r; reflexivity
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definition pathover_equiv_tr_eq [constructor] (p : a = a₂) (b : B a) (b₂ : B a₂)
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: (b =[p] b₂) ≃ (p ▸ b = b₂) :=
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begin
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fapply equiv.MK,
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{ exact tr_eq_of_pathover},
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{ exact pathover_of_tr_eq},
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{ intro r, cases p, cases r, apply idp},
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{ intro r, cases r, apply idp},
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end
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definition pathover_equiv_eq_tr [constructor] (p : a = a₂) (b : B a) (b₂ : B a₂)
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: (b =[p] b₂) ≃ (b = p⁻¹ ▸ b₂) :=
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begin
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fapply equiv.MK,
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{ exact eq_tr_of_pathover},
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{ exact pathover_of_eq_tr},
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{ intro r, cases p, cases r, apply idp},
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{ intro r, cases r, apply idp},
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end
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definition pathover_tr [unfold 5] (p : a = a₂) (b : B a) : b =[p] p ▸ b :=
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by cases p; constructor
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definition tr_pathover [unfold 5] (p : a = a₂) (b : B a₂) : p⁻¹ ▸ b =[p] b :=
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by cases p; constructor
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definition concato [unfold 12] (r : b =[p] b₂) (r₂ : b₂ =[p₂] b₃) : b =[p ⬝ p₂] b₃ :=
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by induction r₂; exact r
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definition inverseo [unfold 8] (r : b =[p] b₂) : b₂ =[p⁻¹] b :=
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by induction r; constructor
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definition concato_eq [unfold 10] (r : b =[p] b₂) (q : b₂ = b₂') : b =[p] b₂' :=
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by induction q; exact r
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definition eq_concato [unfold 9] (q : b = b') (r : b' =[p] b₂) : b =[p] b₂ :=
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by induction q; exact r
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definition change_path [unfold 9] (q : p = p') (r : b =[p] b₂) : b =[p'] b₂ :=
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q ▸ r
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-- infix ` ⬝ ` := concato
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infix ` ⬝o `:72 := concato
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infix ` ⬝op `:73 := concato_eq
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infix ` ⬝po `:74 := eq_concato
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-- postfix `⁻¹` := inverseo
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postfix `⁻¹ᵒ`:(max+10) := inverseo
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definition pathover_cancel_right (q : b =[p ⬝ p₂] b₃) (r : b₃ =[p₂⁻¹] b₂) : b =[p] b₂ :=
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change_path !con_inv_cancel_right (q ⬝o r)
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definition pathover_cancel_right' (q : b =[p₁₃ ⬝ p₂⁻¹] b₂) (r : b₂ =[p₂] b₃) : b =[p₁₃] b₃ :=
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change_path !inv_con_cancel_right (q ⬝o r)
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definition pathover_cancel_left (q : b₂ =[p⁻¹] b) (r : b =[p ⬝ p₂] b₃) : b₂ =[p₂] b₃ :=
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change_path !inv_con_cancel_left (q ⬝o r)
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definition pathover_cancel_left' (q : b =[p] b₂) (r : b₂ =[p⁻¹ ⬝ p₁₃] b₃) : b =[p₁₃] b₃ :=
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change_path !con_inv_cancel_left (q ⬝o r)
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/- Some of the theorems analogous to theorems for = in init.path -/
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definition cono_idpo (r : b =[p] b₂) : r ⬝o idpo =[con_idp p] r :=
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pathover.rec_on r idpo
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definition idpo_cono (r : b =[p] b₂) : idpo ⬝o r =[idp_con p] r :=
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pathover.rec_on r idpo
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definition cono.assoc' (r : b =[p] b₂) (r₂ : b₂ =[p₂] b₃) (r₃ : b₃ =[p₃] b₄) :
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r ⬝o (r₂ ⬝o r₃) =[!con.assoc'] (r ⬝o r₂) ⬝o r₃ :=
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pathover.rec_on r₃ (pathover.rec_on r₂ (pathover.rec_on r idpo))
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definition cono.assoc (r : b =[p] b₂) (r₂ : b₂ =[p₂] b₃) (r₃ : b₃ =[p₃] b₄) :
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(r ⬝o r₂) ⬝o r₃ =[!con.assoc] r ⬝o (r₂ ⬝o r₃) :=
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pathover.rec_on r₃ (pathover.rec_on r₂ (pathover.rec_on r idpo))
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definition cono.right_inv (r : b =[p] b₂) : r ⬝o r⁻¹ᵒ =[!con.right_inv] idpo :=
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pathover.rec_on r idpo
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definition cono.left_inv (r : b =[p] b₂) : r⁻¹ᵒ ⬝o r =[!con.left_inv] idpo :=
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pathover.rec_on r idpo
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definition eq_of_pathover {a' a₂' : A'} (q : a' =[p] a₂') : a' = a₂' :=
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by cases q;reflexivity
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definition pathover_of_eq [unfold 5 8] (p : a = a₂) {a' a₂' : A'} (q : a' = a₂') : a' =[p] a₂' :=
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by cases p;cases q;constructor
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definition pathover_constant [constructor] (p : a = a₂) (a' a₂' : A') : a' =[p] a₂' ≃ a' = a₂' :=
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begin
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fapply equiv.MK,
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{ exact eq_of_pathover},
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{ exact pathover_of_eq p},
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{ intro r, cases p, cases r, reflexivity},
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{ intro r, cases r, reflexivity},
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end
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definition pathover_of_eq_tr_constant_inv (p : a = a₂) (a' : A')
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: pathover_of_eq p (tr_constant p a')⁻¹ = pathover_tr p a' :=
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by cases p; constructor
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definition eq_of_pathover_idp [unfold 6] {b' : B a} (q : b =[idpath a] b') : b = b' :=
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tr_eq_of_pathover q
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--should B be explicit in the next two definitions?
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definition pathover_idp_of_eq [unfold 6] {b' : B a} (q : b = b') : b =[idpath a] b' :=
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pathover_of_tr_eq q
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definition pathover_idp [constructor] (b : B a) (b' : B a) : b =[idpath a] b' ≃ b = b' :=
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equiv.MK eq_of_pathover_idp
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(pathover_idp_of_eq)
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(to_right_inv !pathover_equiv_tr_eq)
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(to_left_inv !pathover_equiv_tr_eq)
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definition eq_of_pathover_idp_pathover_of_eq {A X : Type} (x : X) {a a' : A} (p : a = a') :
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eq_of_pathover_idp (pathover_of_eq (idpath x) p) = p :=
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by induction p; reflexivity
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-- definition pathover_idp (b : B a) (b' : B a) : b =[idpath a] b' ≃ b = b' :=
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-- pathover_equiv_tr_eq idp b b'
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-- definition eq_of_pathover_idp [reducible] {b' : B a} (q : b =[idpath a] b') : b = b' :=
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-- to_fun !pathover_idp q
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-- definition pathover_idp_of_eq [reducible] {b' : B a} (q : b = b') : b =[idpath a] b' :=
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-- to_inv !pathover_idp q
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definition idp_rec_on [recursor] [unfold 7] {P : Π⦃b₂ : B a⦄, b =[idpath a] b₂ → Type}
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{b₂ : B a} (r : b =[idpath a] b₂) (H : P idpo) : P r :=
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have H2 : P (pathover_idp_of_eq (eq_of_pathover_idp r)), from
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eq.rec_on (eq_of_pathover_idp r) H,
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proof left_inv !pathover_idp r ▸ H2 qed
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definition rec_on_right [recursor] {P : Π⦃b₂ : B a₂⦄, b =[p] b₂ → Type}
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{b₂ : B a₂} (r : b =[p] b₂) (H : P !pathover_tr) : P r :=
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by cases r; exact H
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definition rec_on_left [recursor] {P : Π⦃b : B a⦄, b =[p] b₂ → Type}
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{b : B a} (r : b =[p] b₂) (H : P !tr_pathover) : P r :=
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by cases r; exact H
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--pathover with fibration B' ∘ f
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definition pathover_ap [unfold 10] (B' : A' → Type) (f : A → A') {p : a = a₂}
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{b : B' (f a)} {b₂ : B' (f a₂)} (q : b =[p] b₂) : b =[ap f p] b₂ :=
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by cases q; constructor
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definition pathover_of_pathover_ap (B' : A' → Type) (f : A → A') {p : a = a₂}
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{b : B' (f a)} {b₂ : B' (f a₂)} (q : b =[ap f p] b₂) : b =[p] b₂ :=
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by cases p; apply (idp_rec_on q); apply idpo
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definition pathover_compose [constructor] (B' : A' → Type) (f : A → A') (p : a = a₂)
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(b : B' (f a)) (b₂ : B' (f a₂)) : b =[p] b₂ ≃ b =[ap f p] b₂ :=
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begin
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fapply equiv.MK,
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{ exact pathover_ap B' f},
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{ exact pathover_of_pathover_ap B' f},
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{ intro q, cases p, esimp, apply (idp_rec_on q), apply idp},
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{ intro q, cases q, reflexivity},
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end
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definition pathover_of_pathover_tr (q : b =[p ⬝ p₂] p₂ ▸ b₂) : b =[p] b₂ :=
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pathover_cancel_right q !pathover_tr⁻¹ᵒ
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definition pathover_tr_of_pathover (q : b =[p₁₃ ⬝ p₂⁻¹] b₂) : b =[p₁₃] p₂ ▸ b₂ :=
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pathover_cancel_right' q !pathover_tr
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definition pathover_of_tr_pathover (q : p ▸ b =[p⁻¹ ⬝ p₁₃] b₃) : b =[p₁₃] b₃ :=
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pathover_cancel_left' !pathover_tr q
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definition tr_pathover_of_pathover (q : b =[p ⬝ p₂] b₃) : p ▸ b =[p₂] b₃ :=
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pathover_cancel_left !pathover_tr⁻¹ᵒ q
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definition pathover_tr_of_eq (q : b = b') : b =[p] p ▸ b' :=
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by cases q;apply pathover_tr
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definition tr_pathover_of_eq (q : b₂ = b₂') : p⁻¹ ▸ b₂ =[p] b₂' :=
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by cases q;apply tr_pathover
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definition eq_of_parallel_po_right (q : b =[p] b₂) (q' : b =[p] b₂') : b₂ = b₂' :=
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begin
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apply @eq_of_pathover_idp A B, apply change_path (con.left_inv p),
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exact q⁻¹ᵒ ⬝o q'
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end
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definition eq_of_parallel_po_left (q : b =[p] b₂) (q' : b' =[p] b₂) : b = b' :=
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begin
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apply @eq_of_pathover_idp A B, apply change_path (con.right_inv p),
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exact q ⬝o q'⁻¹ᵒ
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end
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variable (C)
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definition transporto (r : b =[p] b₂) (c : C b) : C b₂ :=
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by induction r;exact c
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infix ` ▸o `:75 := transporto _
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definition fn_tro_eq_tro_fn {C' : Π ⦃a : A⦄, B a → Type} (q : b =[p] b₂)
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(f : Π⦃a : A⦄ (b : B a), C b → C' b) (c : C b) : f b₂ (q ▸o c) = q ▸o (f b c) :=
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by induction q; reflexivity
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variable {C}
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/- various variants of ap for pathovers -/
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definition apd [unfold 6] (f : Πa, B a) (p : a = a₂) : f a =[p] f a₂ :=
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by induction p; constructor
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definition apo [unfold 12] {f : A → A'} (g : Πa, B a → B'' (f a)) (q : b =[p] b₂) :
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g a b =[p] g a₂ b₂ :=
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by induction q; constructor
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definition apd011 [unfold 10] (f : Πa, B a → A') (Ha : a = a₂) (Hb : b =[Ha] b₂)
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: f a b = f a₂ b₂ :=
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by cases Hb; reflexivity
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definition apd0111 [unfold 13 14] (f : Πa b, C b → A') (Ha : a = a₂) (Hb : b =[Ha] b₂)
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(Hc : c =[apd011 C Ha Hb] c₂) : f a b c = f a₂ b₂ c₂ :=
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by cases Hb; apply (idp_rec_on Hc); apply idp
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definition apod11 {f : Πb, C b} {g : Πb₂, C b₂} (r : f =[p] g)
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{b : B a} {b₂ : B a₂} (q : b =[p] b₂) : f b =[apd011 C p q] g b₂ :=
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by cases r; apply (idp_rec_on q); constructor
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definition apdo10 {f : Πb, C b} {g : Πb₂, C b₂} (r : f =[p] g)
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(b : B a) : f b =[apd011 C p !pathover_tr] g (p ▸ b) :=
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by cases r; constructor
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definition apo10 [unfold 9] {f : B a → B' a} {g : B a₂ → B' a₂} (r : f =[p] g)
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(b : B a) : f b =[p] g (p ▸ b) :=
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by cases r; constructor
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definition apo10_constant_right [unfold 9] {f : B a → A'} {g : B a₂ → A'} (r : f =[p] g)
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(b : B a) : f b = g (p ▸ b) :=
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by cases r; constructor
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definition apo10_constant_left [unfold 9] {f : A' → B a} {g : A' → B a₂} (r : f =[p] g)
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(a' : A') : f a' =[p] g a' :=
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by cases r; constructor
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definition apo11 {f : B a → B' a} {g : B a₂ → B' a₂} (r : f =[p] g)
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(q : b =[p] b₂) : f b =[p] g b₂ :=
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by induction q; exact apo10 r b
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definition apdo011 {A : Type} {B : A → Type} {C : Π⦃a⦄, B a → Type}
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(f : Π⦃a⦄ (b : B a), C b) {a a' : A} (p : a = a') {b : B a} {b' : B a'} (q : b =[p] b')
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: f b =[apd011 C p q] f b' :=
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by cases q; constructor
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definition apdo0111 {A : Type} {B : A → Type} {C C' : Π⦃a⦄, B a → Type}
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(f : Π⦃a⦄ {b : B a}, C b → C' b) {a a' : A} (p : a = a') {b : B a} {b' : B a'} (q : b =[p] b')
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{c : C b} {c' : C b'} (r : c =[apd011 C p q] c')
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: f c =[apd011 C' p q] f c' :=
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begin
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induction q, esimp at r, induction r using idp_rec_on, constructor
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end
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definition apo11_constant_right [unfold 12] {f : B a → A'} {g : B a₂ → A'}
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(q : f =[p] g) (r : b =[p] b₂) : f b = g b₂ :=
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eq_of_pathover (apo11 q r)
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/- properties about these "ap"s, transporto and pathover_ap -/
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definition apd_con (f : Πa, B a) (p : a = a₂) (q : a₂ = a₃)
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: apd f (p ⬝ q) = apd f p ⬝o apd f q :=
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by cases p; cases q; reflexivity
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definition apd_inv (f : Πa, B a) (p : a = a₂) : apd f p⁻¹ = (apd f p)⁻¹ᵒ :=
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by cases p; reflexivity
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definition apd_eq_pathover_of_eq_ap (f : A → A') (p : a = a₂) :
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apd f p = pathover_of_eq p (ap f p) :=
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eq.rec_on p idp
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definition apo_invo (f : Πa, B a → B' a) {Ha : a = a₂} (Hb : b =[Ha] b₂)
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: (apo f Hb)⁻¹ᵒ = apo f Hb⁻¹ᵒ :=
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by induction Hb; reflexivity
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definition apd011_inv (f : Πa, B a → A') (Ha : a = a₂) (Hb : b =[Ha] b₂)
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: (apd011 f Ha Hb)⁻¹ = (apd011 f Ha⁻¹ Hb⁻¹ᵒ) :=
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by induction Hb; reflexivity
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definition cast_apd011 (q : b =[p] b₂) (c : C b) : cast (apd011 C p q) c = q ▸o c :=
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by induction q; reflexivity
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definition apd_compose1 (g : Πa, B a → B' a) (f : Πa, B a) (p : a = a₂)
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: apd (g ∘' f) p = apo g (apd f p) :=
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by induction p; reflexivity
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definition apd_compose2 (g : Πa', B'' a') (f : A → A') (p : a = a₂)
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: apd (λa, g (f a)) p = pathover_of_pathover_ap B'' f (apd g (ap f p)) :=
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by induction p; reflexivity
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definition apo_tro (C : Π⦃a⦄, B' a → Type) (f : Π⦃a⦄, B a → B' a) (q : b =[p] b₂)
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(c : C (f b)) : apo f q ▸o c = q ▸o c :=
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by induction q; reflexivity
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definition pathover_ap_tro {B' : A' → Type} (C : Π⦃a'⦄, B' a' → Type) (f : A → A')
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{b : B' (f a)} {b₂ : B' (f a₂)} (q : b =[p] b₂) (c : C b) : pathover_ap B' f q ▸o c = q ▸o c :=
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by induction q; reflexivity
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definition pathover_ap_invo_tro {B' : A' → Type} (C : Π⦃a'⦄, B' a' → Type) (f : A → A')
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{b : B' (f a)} {b₂ : B' (f a₂)} (q : b =[p] b₂) (c : C b₂)
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: (pathover_ap B' f q)⁻¹ᵒ ▸o c = q⁻¹ᵒ ▸o c :=
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by induction q; reflexivity
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definition pathover_tro (q : b =[p] b₂) (c : C b) : c =[apd011 C p q] q ▸o c :=
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by induction q; constructor
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definition pathover_ap_invo {B' : A' → Type} (f : A → A') {p : a = a₂}
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{b : B' (f a)} {b₂ : B' (f a₂)} (q : b =[p] b₂)
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: pathover_ap B' f q⁻¹ᵒ =[ap_inv f p] (pathover_ap B' f q)⁻¹ᵒ :=
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by induction q; exact idpo
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definition tro_invo_tro {A : Type} {B : A → Type} (C : Π⦃a⦄, B a → Type)
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{a a' : A} {p : a = a'} {b : B a} {b' : B a'} (q : b =[p] b') (c : C b') :
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q ▸o (q⁻¹ᵒ ▸o c) = c :=
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by induction q; reflexivity
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definition invo_tro_tro {A : Type} {B : A → Type} (C : Π⦃a⦄, B a → Type)
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{a a' : A} {p : a = a'} {b : B a} {b' : B a'} (q : b =[p] b') (c : C b) :
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q⁻¹ᵒ ▸o (q ▸o c) = c :=
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by induction q; reflexivity
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definition cono_tro {A : Type} {B : A → Type} (C : Π⦃a⦄, B a → Type)
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{a₁ a₂ a₃ : A} {p₁ : a₁ = a₂} {p₂ : a₂ = a₃} {b₁ : B a₁} {b₂ : B a₂} {b₃ : B a₃}
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(q₁ : b₁ =[p₁] b₂) (q₂ : b₂ =[p₂] b₃) (c : C b₁) :
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transporto C (q₁ ⬝o q₂) c = transporto C q₂ (transporto C q₁ c) :=
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by induction q₂; reflexivity
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definition is_equiv_transporto [constructor] {A : Type} {B : A → Type} (C : Π⦃a⦄, B a → Type)
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{a a' : A} {p : a = a'} {b : B a} {b' : B a'} (q : b =[p] b') : is_equiv (transporto C q) :=
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begin
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fapply adjointify,
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{ exact transporto C q⁻¹ᵒ},
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{ exact tro_invo_tro C q},
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{ exact invo_tro_tro C q}
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end
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definition equiv_apd011 [constructor] {A : Type} {B : A → Type} (C : Π⦃a⦄, B a → Type)
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{a a' : A} {p : a = a'} {b : B a} {b' : B a'} (q : b =[p] b') : C b ≃ C b' :=
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equiv.mk (transporto C q) !is_equiv_transporto
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/- some cancellation laws for concato_eq an variants -/
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definition cono.right_inv_eq (q : b = b') :
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pathover_idp_of_eq q ⬝op q⁻¹ = (idpo : b =[refl a] b) :=
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by induction q;constructor
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definition cono.right_inv_eq' (q : b = b') :
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q ⬝po (pathover_idp_of_eq q⁻¹) = (idpo : b =[refl a] b) :=
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by induction q;constructor
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definition cono.left_inv_eq (q : b = b') :
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pathover_idp_of_eq q⁻¹ ⬝op q = (idpo : b' =[refl a] b') :=
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by induction q;constructor
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definition cono.left_inv_eq' (q : b = b') :
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q⁻¹ ⬝po pathover_idp_of_eq q = (idpo : b' =[refl a] b') :=
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by induction q;constructor
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definition pathover_of_fn_pathover_fn (f : Π{a}, B a ≃ B' a) (r : f b =[p] f b₂) : b =[p] b₂ :=
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(left_inv f b)⁻¹ ⬝po apo (λa, f⁻¹ᵉ) r ⬝op left_inv f b₂
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/- a pathover in a pathover type where the only thing which varies is the path is the same as
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an equality with a change_path -/
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definition change_path_of_pathover (s : p = p') (r : b =[p] b₂) (r' : b =[p'] b₂)
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(q : r =[s] r') : change_path s r = r' :=
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by induction s; eapply idp_rec_on q; reflexivity
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definition pathover_of_change_path (s : p = p') (r : b =[p] b₂) (r' : b =[p'] b₂)
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(q : change_path s r = r') : r =[s] r' :=
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by induction s; induction q; constructor
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definition pathover_pathover_path [constructor] (s : p = p') (r : b =[p] b₂) (r' : b =[p'] b₂) :
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(r =[s] r') ≃ change_path s r = r' :=
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begin
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fapply equiv.MK,
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{ apply change_path_of_pathover},
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{ apply pathover_of_change_path},
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{ intro q, induction s, induction q, reflexivity},
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{ intro q, induction s, eapply idp_rec_on q, reflexivity},
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end
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/- variants of inverse2 and concat2 -/
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definition inverseo2 [unfold 10] {r r' : b =[p] b₂} (s : r = r') : r⁻¹ᵒ = r'⁻¹ᵒ :=
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by induction s; reflexivity
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definition concato2 [unfold 15 16] {r r' : b =[p] b₂} {r₂ r₂' : b₂ =[p₂] b₃}
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(s : r = r') (s₂ : r₂ = r₂') : r ⬝o r₂ = r' ⬝o r₂' :=
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by induction s; induction s₂; reflexivity
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infixl ` ◾o `:75 := concato2
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postfix [parsing_only] `⁻²ᵒ`:(max+10) := inverseo2 --this notation is abusive, should we use it?
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-- find a better name for this
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definition fn_tro_eq_tro_fn2 (q : b =[p] b₂)
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{k : A → A} {l : Π⦃a⦄, B a → B (k a)} (m : Π⦃a⦄ {b : B a}, C b → C (l b))
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(c : C b) :
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m (q ▸o c) = (pathover_ap B k (apo l q)) ▸o (m c) :=
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by induction q; reflexivity
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definition apd0111_precompose (f : Π⦃a⦄ {b : B a}, C b → A')
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{k : A → A} {l : Π⦃a⦄, B a → B (k a)} (m : Π⦃a⦄ {b : B a}, C b → C (l b))
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{q : b =[p] b₂} (c : C b)
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: apd0111 (λa b c, f (m c)) p q (pathover_tro q c) ⬝ ap (@f _ _) (fn_tro_eq_tro_fn2 q m c) =
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apd0111 f (ap k p) (pathover_ap B k (apo l q)) (pathover_tro _ (m c)) :=
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by induction q; reflexivity
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end eq
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