196 lines
7 KiB
Text
196 lines
7 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 equiv.ops
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variables {A A' : Type} {B : A → Type} {C : Πa, B a → Type}
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{a a₂ a₃ a₄ : A} {p : a = a₂} {p₂ : a₂ = a₃} {p₃ : a₃ = a₄}
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{b b' : B a} {b₂ b₂' : B a₂} {b₃ : B a₃} {b₄ : B a₄}
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{c : C a b} {c₂ : C a₂ 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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/- equivalences with equality using transport -/
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definition pathover_of_tr_eq (r : p ▸ b = b₂) : b =[p] b₂ :=
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by cases p; cases r; exact idpo
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definition pathover_of_eq_tr (r : b = p⁻¹ ▸ b₂) : b =[p] b₂ :=
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by cases p; cases r; exact idpo
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definition tr_eq_of_pathover (r : b =[p] b₂) : p ▸ b = b₂ :=
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by cases r; exact idp
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definition eq_tr_of_pathover (r : b =[p] b₂) : b = p⁻¹ ▸ b₂ :=
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by cases r; exact idp
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definition pathover_equiv_tr_eq (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 (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 (p : a = a₂) (b : B a) : b =[p] p ▸ b :=
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pathover_of_tr_eq idp
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definition tr_pathover (p : a = a₂) (b : B a₂) : p⁻¹ ▸ b =[p] b :=
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pathover_of_eq_tr idp
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definition concato (r : b =[p] b₂) (r₂ : b₂ =[p₂] b₃) : b =[p ⬝ p₂] b₃ :=
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pathover.rec_on r₂ (pathover.rec_on r idpo)
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definition concato_eq (r : b =[p] b₂) (q : b₂ = b₂') : b =[p] b₂' :=
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eq.rec_on q r
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definition eq_concato (q : b = b') (r : b' =[p] b₂) : b =[p] b₂ :=
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eq.rec_on q (λr, r) r
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definition inverseo (r : b =[p] b₂) : b₂ =[p⁻¹] b :=
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pathover.rec_on r idpo
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definition apdo (f : Πa, B a) (p : a = a₂) : f a =[p] f a₂ :=
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eq.rec_on p idpo
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-- infix `⬝` := concato
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infix `⬝o`:75 := concato
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-- postfix `⁻¹` := inverseo
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postfix `⁻¹ᵒ`:(max+10) := inverseo
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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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-- the left inverse law.
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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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-- the right inverse law.
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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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/- Some of the theorems analogous to theorems for transport in init.path -/
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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 {a' a₂' : A'} (q : a' = a₂') : a' =[p] a₂' :=
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by cases p;cases q;exact idpo
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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},
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{ intro r, cases p, cases r, exact idp},
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{ intro r, cases r, exact idp},
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end
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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 {b' : B a} (q : b =[idpath a] b') : b = b' :=
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tr_eq_of_pathover q
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definition pathover_idp_of_eq {b' : B a} (q : b = b') : b =[idpath a] b' :=
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pathover_of_tr_eq q
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definition idp_rec_on [recursor] {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_compose (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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{ intro r, cases r, exact idpo},
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{ intro r, cases p, apply (idp_rec_on r), apply idpo},
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{ intro r, cases p, esimp [function.compose,function.id], apply (idp_rec_on r), apply idp},
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{ intro r, cases r, exact idp},
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end
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definition apdo_con (f : Πa, B a) (p : a = a₂) (q : a₂ = a₃)
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: apdo f (p ⬝ q) = apdo f p ⬝o apdo f q :=
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by cases p; cases q; exact idp
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definition apdo_inv (f : Πa, B a) (p : a = a₂) : apdo f p⁻¹ = (apdo f p)⁻¹ᵒ :=
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by cases p; exact idp
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definition apdo_eq_pathover_of_eq_ap (f : A → A') (p : a = a₂) :
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apdo f p = pathover_of_eq (ap f p) :=
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eq.rec_on p idp
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definition pathover_of_pathover_tr (q : b =[p ⬝ p₂] p₂ ▸ b₂) : b =[p] b₂ :=
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by cases p₂;exact q
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definition pathover_tr_of_pathover {p : a = a₃} (q : b =[p ⬝ p₂⁻¹] b₂) : b =[p] p₂ ▸ b₂ :=
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by cases p₂;exact q
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definition apo011 (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; exact idp
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definition apo0111 (f : Πa b, C a b → A') (Ha : a = a₂) (Hb : b =[Ha] b₂)
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(Hc : c =[apo011 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 apo11 {f : Πb, C a b} {g : Πb₂, C a₂ b₂} (r : f =[p] g)
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{b : B a} {b₂ : B a₂} (q : b =[p] b₂) : f b =[apo011 C p q] g b₂ :=
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by cases r; apply (idp_rec_on q); exact idpo
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definition apo10 {f : Πb, C a b} {g : Πb₂, C a₂ b₂} (r : f =[p] g)
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{b : B a} : f b =[apo011 C p !pathover_tr] g (p ▸ b) :=
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by cases r; exact idpo
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end eq
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