270 lines
9.9 KiB
Text
270 lines
9.9 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: types.pathover
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Author: Floris van Doorn
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Theorems about pathovers
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-/
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import types.sigma arity
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open eq equiv is_equiv equiv.ops
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namespace cubical
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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 a} {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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{u v w : Σa, B a}
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inductive pathover (B : A → Type) (b : B a) : Π{a₂ : A} (p : a = a₂) (b₂ : B a₂), Type :=
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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] {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_transport_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_transport (r : b = p⁻¹ ▹ b₂) : b =[p] b₂ :=
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by cases p; cases r; exact idpo
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definition transport_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_transport_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_transport_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 transport_eq_of_pathover},
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{ exact pathover_of_transport_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_transport (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_transport_of_pathover},
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{ exact pathover_of_eq_transport},
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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_transport (p : a = a₂) (b : B a) : b =[p] p ▹ b :=
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pathover_of_transport_eq idp
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definition transport_pathover (p : a = a₂) (b : B a) : p⁻¹ ▹ b =[p] b :=
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pathover_of_eq_transport 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 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 `⬝o`:75 := concato
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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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--set_option pp.notation false
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definition pathover_constant (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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{ intro r, cases r, exact idp},
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{ intro r, cases p, cases r, exact idpo},
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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_transport_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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transport_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_transport_eq q
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definition idp_rec_on {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)),
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from eq.rec_on (eq_of_pathover_idp r) H,
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left_inv !pathover_idp r ▹ H2
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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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open sigma sigma.ops
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namespace sigma
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/- pathovers used for sigma types -/
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definition dpair_eq_dpair (p : a = a₂) (q : b =[p] b₂) : ⟨a, b⟩ = ⟨a₂, b₂⟩ :=
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by cases q; apply idp
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definition sigma_eq (p : u.1 = v.1) (q : u.2 =[p] v.2) : u = v :=
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by cases u; cases v; apply (dpair_eq_dpair p q)
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/- Projections of paths from a total space -/
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definition pathover_pr2 (p : u = v) : u.2 =[p..1] v.2 :=
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by cases p; apply idpo
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postfix `..2o`:(max+1) := pathover_pr2
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--superfluous notation, but useful if you want an 'o' on both projections
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postfix [parsing-only] `..1o`:(max+1) := eq_pr1
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private definition dpair_sigma_eq (p : u.1 = v.1) (q : u.2 =[p] v.2)
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: ⟨(sigma_eq p q)..1, (sigma_eq p q)..2o⟩ = ⟨p, q⟩ :=
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by cases u; cases v; cases q; apply idp
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definition sigma_eq_pr1 (p : u.1 = v.1) (q : u.2 =[p] v.2) : (sigma_eq p q)..1 = p :=
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(dpair_sigma_eq p q)..1
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definition sigma_eq_pr2 (p : u.1 = v.1) (q : u.2 =[p] v.2)
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: (sigma_eq p q)..2o =[sigma_eq_pr1 p q] q :=
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(dpair_sigma_eq p q)..2o
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definition sigma_eq_eta (p : u = v) : sigma_eq (p..1) (p..2o) = p :=
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by cases p; cases u; apply idp
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/- the uncurried version of sigma_eq. We will prove that this is an equivalence -/
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definition sigma_eq_uncurried : Π (pq : Σ(p : u.1 = v.1), u.2 =[p] v.2), u = v
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| sigma_eq_uncurried ⟨pq₁, pq₂⟩ := sigma_eq pq₁ pq₂
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definition dpair_sigma_eq_uncurried : Π (pq : Σ(p : u.1 = v.1), u.2 =[p] v.2),
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⟨(sigma_eq_uncurried pq)..1, (sigma_eq_uncurried pq)..2o⟩ = pq
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| dpair_sigma_eq_uncurried ⟨pq₁, pq₂⟩ := dpair_sigma_eq pq₁ pq₂
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definition sigma_eq_pr1_uncurried (pq : Σ(p : u.1 = v.1), u.2 =[p] v.2)
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: (sigma_eq_uncurried pq)..1 = pq.1 :=
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(dpair_sigma_eq_uncurried pq)..1
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definition sigma_eq_pr2_uncurried (pq : Σ(p : u.1 = v.1), u.2 =[p] v.2)
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: (sigma_eq_uncurried pq)..2o =[sigma_eq_pr1_uncurried pq] pq.2 :=
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(dpair_sigma_eq_uncurried pq)..2o
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definition sigma_eq_eta_uncurried (p : u = v) : sigma_eq_uncurried (sigma.mk p..1 p..2o) = p :=
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sigma_eq_eta p
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definition is_equiv_sigma_eq [instance] (u v : Σa, B a)
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: is_equiv (@sigma_eq_uncurried A B u v) :=
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adjointify sigma_eq_uncurried
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(λp, ⟨p..1, p..2o⟩)
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sigma_eq_eta_uncurried
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dpair_sigma_eq_uncurried
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definition equiv_sigma_eq (u v : Σa, B a) : (Σ(p : u.1 = v.1), u.2 =[p] v.2) ≃ (u = v) :=
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equiv.mk sigma_eq_uncurried !is_equiv_sigma_eq
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end sigma
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definition apD011o (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 apD0111o (f : Πa b, C a b → A') (Ha : a = a₂) (Hb : b =[Ha] b₂)
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(Hc : c =[apD011o 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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namespace pi
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--the most 'natural' version here needs a notion of "path over a pathover"
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definition pi_pathover {f : Πb, C a b} {g : Πb₂, C a₂ b₂}
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(r : Π(b : B a) (b₂ : B a₂) (q : b =[p] b₂), f b =[apD011o C p q] g b₂) : f =[p] g :=
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begin
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cases p, apply pathover_idp_of_eq,
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apply eq_of_homotopy, intro b,
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apply eq_of_pathover_idp, apply r
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end
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definition pi_pathover' {C : (Σa, B a) → Type} {f : Πb, C ⟨a, b⟩} {g : Πb₂, C ⟨a₂, b₂⟩}
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(r : Π(b : B a) (b₂ : B a₂) (q : b =[p] b₂), f b =[sigma.dpair_eq_dpair p q] g b₂)
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: f =[p] g :=
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begin
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cases p, apply pathover_idp_of_eq,
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apply eq_of_homotopy, intro b,
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apply (@eq_of_pathover_idp _ C), exact (r b b (pathover.idpatho b)),
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end
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definition ap11o {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 =[apD011o C p q] g b₂ :=
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by cases r; apply (idp_rec_on q); exact idpo
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definition ap10o {f : Πb, C a b} {g : Πb₂, C a₂ b₂} (r : f =[p] g)
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{b : B a} : f b =[apD011o C p !pathover_transport] g (p ▹ b) :=
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by cases r; exact idpo
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-- definition equiv_pi_pathover' (f : Πb, C a b) (g : Πb₂, C a₂ b₂) :
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-- (f =[p] g) ≃ (Π(b : B a), f b =[apD011o C p !pathover_transport] g (p ▹ b)) :=
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-- begin
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-- fapply equiv.MK,
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-- { exact ap10o},
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-- { exact pi_pathover'},
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-- { cases p, exact sorry},
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-- { intro r, cases r, },
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-- end
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-- definition equiv_pi_pathover (f : Πb, C a b) (g : Πb₂, C a₂ b₂) :
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-- (f =[p] g) ≃ (Π(b : B a) (b₂ : B a₂) (q : b =[p] b₂), f b =[apD011o C p q] g b₂) :=
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-- begin
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-- fapply equiv.MK,
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-- { exact ap11o},
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-- { exact pi_pathover},
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-- { cases p, exact sorry},
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-- { intro r, cases r, },
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-- end
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end pi
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end cubical
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