409 lines
18 KiB
Text
409 lines
18 KiB
Text
/-
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Copyright (c) 2014 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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Partially ported from Coq HoTT
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Theorems about path types (identity types)
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-/
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open eq sigma sigma.ops equiv is_equiv equiv.ops
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-- TODO: Rename transport_eq_... and pathover_eq_... to eq_transport_... and eq_pathover_...
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namespace eq
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/- Path spaces -/
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variables {A B : Type} {a a₁ a₂ a₃ a₄ a' : A} {b b1 b2 : B} {f g : A → B} {h : B → A}
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{p p' p'' : a₁ = a₂}
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/- The path spaces of a path space are not, of course, determined; they are just the
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higher-dimensional structure of the original space. -/
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/- some lemmas about whiskering or other higher paths -/
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theorem whisker_left_con_right (p : a₁ = a₂) {q q' q'' : a₂ = a₃} (r : q = q') (s : q' = q'')
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: whisker_left p (r ⬝ s) = whisker_left p r ⬝ whisker_left p s :=
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begin
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induction p, induction r, induction s, reflexivity
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end
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theorem whisker_right_con_right (q : a₂ = a₃) (r : p = p') (s : p' = p'')
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: whisker_right (r ⬝ s) q = whisker_right r q ⬝ whisker_right s q :=
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begin
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induction q, induction r, induction s, reflexivity
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end
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theorem whisker_left_con_left (p : a₁ = a₂) (p' : a₂ = a₃) {q q' : a₃ = a₄} (r : q = q')
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: whisker_left (p ⬝ p') r = !con.assoc ⬝ whisker_left p (whisker_left p' r) ⬝ !con.assoc' :=
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begin
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induction p', induction p, induction r, induction q, reflexivity
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end
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theorem whisker_right_con_left {p p' : a₁ = a₂} (q : a₂ = a₃) (q' : a₃ = a₄) (r : p = p')
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: whisker_right r (q ⬝ q') = !con.assoc' ⬝ whisker_right (whisker_right r q) q' ⬝ !con.assoc :=
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begin
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induction q', induction q, induction r, induction p, reflexivity
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end
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theorem whisker_left_inv_left (p : a₂ = a₁) {q q' : a₂ = a₃} (r : q = q')
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: !con_inv_cancel_left⁻¹ ⬝ whisker_left p (whisker_left p⁻¹ r) ⬝ !con_inv_cancel_left = r :=
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begin
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induction p, induction r, induction q, reflexivity
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end
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theorem whisker_left_inv (p : a₁ = a₂) {q q' : a₂ = a₃} (r : q = q')
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: whisker_left p r⁻¹ = (whisker_left p r)⁻¹ :=
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by induction r; reflexivity
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theorem whisker_right_inv {p p' : a₁ = a₂} (q : a₂ = a₃) (r : p = p')
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: whisker_right r⁻¹ q = (whisker_right r q)⁻¹ :=
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by induction r; reflexivity
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theorem ap_eq_ap10 {f g : A → B} (p : f = g) (a : A) : ap (λh, h a) p = ap10 p a :=
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by induction p;reflexivity
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theorem inverse2_right_inv (r : p = p') : r ◾ inverse2 r ⬝ con.right_inv p' = con.right_inv p :=
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by induction r;induction p;reflexivity
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theorem inverse2_left_inv (r : p = p') : inverse2 r ◾ r ⬝ con.left_inv p' = con.left_inv p :=
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by induction r;induction p;reflexivity
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theorem ap_con_right_inv (f : A → B) (p : a₁ = a₂)
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: ap_con f p p⁻¹ ⬝ whisker_left _ (ap_inv f p) ⬝ con.right_inv (ap f p)
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= ap (ap f) (con.right_inv p) :=
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by induction p;reflexivity
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theorem ap_con_left_inv (f : A → B) (p : a₁ = a₂)
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: ap_con f p⁻¹ p ⬝ whisker_right (ap_inv f p) _ ⬝ con.left_inv (ap f p)
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= ap (ap f) (con.left_inv p) :=
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by induction p;reflexivity
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theorem idp_con_whisker_left {q q' : a₂ = a₃} (r : q = q') :
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!idp_con⁻¹ ⬝ whisker_left idp r = r ⬝ !idp_con⁻¹ :=
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by induction r;induction q;reflexivity
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theorem whisker_left_idp_con {q q' : a₂ = a₃} (r : q = q') :
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whisker_left idp r ⬝ !idp_con = !idp_con ⬝ r :=
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by induction r;induction q;reflexivity
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theorem idp_con_idp {p : a = a} (q : p = idp) : idp_con p ⬝ q = ap (λp, idp ⬝ p) q :=
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by cases q;reflexivity
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definition ap_weakly_constant [unfold-c 8] {A B : Type} {f : A → B} {b : B} (p : Πx, f x = b)
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{x y : A} (q : x = y) : ap f q = p x ⬝ (p y)⁻¹ :=
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by induction q;exact !con.right_inv⁻¹
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definition inv2_inv {p q : a = a'} (r : p = q) : inverse2 r⁻¹ = (inverse2 r)⁻¹ :=
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by induction r;reflexivity
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definition inv2_con {p p' p'' : a = a'} (r : p = p') (r' : p' = p'')
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: inverse2 (r ⬝ r') = inverse2 r ⬝ inverse2 r' :=
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by induction r';induction r;reflexivity
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definition con2_inv {p₁ q₁ : a₁ = a₂} {p₂ q₂ : a₂ = a₃} (r₁ : p₁ = q₁) (r₂ : p₂ = q₂)
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: (r₁ ◾ r₂)⁻¹ = r₁⁻¹ ◾ r₂⁻¹ :=
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by induction r₁;induction r₂;reflexivity
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theorem eq_con_inv_of_con_eq_whisker_left {A : Type} {a a₂ a₃ : A}
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{p : a = a₂} {q q' : a₂ = a₃} {r : a = a₃} (s' : q = q') (s : p ⬝ q' = r) :
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eq_con_inv_of_con_eq (whisker_left p s' ⬝ s)
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= eq_con_inv_of_con_eq s ⬝ whisker_left r (inverse2 s')⁻¹ :=
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by induction s';induction q;induction s;reflexivity
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theorem right_inv_eq_idp {A : Type} {a : A} {p : a = a} (r : p = idpath a) :
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con.right_inv p = r ◾ inverse2 r :=
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by cases r;reflexivity
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/- Transporting in path spaces.
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There are potentially a lot of these lemmas, so we adopt a uniform naming scheme:
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- `l` means the left endpoint varies
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- `r` means the right endpoint varies
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- `F` means application of a function to that (varying) endpoint.
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-/
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definition transport_eq_l (p : a₁ = a₂) (q : a₁ = a₃)
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: transport (λx, x = a₃) p q = p⁻¹ ⬝ q :=
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by induction p; induction q; reflexivity
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definition transport_eq_r (p : a₂ = a₃) (q : a₁ = a₂)
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: transport (λx, a₁ = x) p q = q ⬝ p :=
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by induction p; induction q; reflexivity
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definition transport_eq_lr (p : a₁ = a₂) (q : a₁ = a₁)
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: transport (λx, x = x) p q = p⁻¹ ⬝ q ⬝ p :=
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by induction p; rewrite [▸*,idp_con]
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definition transport_eq_Fl (p : a₁ = a₂) (q : f a₁ = b)
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: transport (λx, f x = b) p q = (ap f p)⁻¹ ⬝ q :=
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by induction p; induction q; reflexivity
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definition transport_eq_Fr (p : a₁ = a₂) (q : b = f a₁)
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: transport (λx, b = f x) p q = q ⬝ (ap f p) :=
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by induction p; reflexivity
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definition transport_eq_FlFr (p : a₁ = a₂) (q : f a₁ = g a₁)
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: transport (λx, f x = g x) p q = (ap f p)⁻¹ ⬝ q ⬝ (ap g p) :=
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by induction p; rewrite [▸*,idp_con]
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definition transport_eq_FlFr_D {B : A → Type} {f g : Πa, B a}
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(p : a₁ = a₂) (q : f a₁ = g a₁)
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: transport (λx, f x = g x) p q = (apd f p)⁻¹ ⬝ ap (transport B p) q ⬝ (apd g p) :=
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by induction p; rewrite [▸*,idp_con,ap_id]
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definition transport_eq_FFlr (p : a₁ = a₂) (q : h (f a₁) = a₁)
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: transport (λx, h (f x) = x) p q = (ap h (ap f p))⁻¹ ⬝ q ⬝ p :=
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by induction p; rewrite [▸*,idp_con]
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definition transport_eq_lFFr (p : a₁ = a₂) (q : a₁ = h (f a₁))
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: transport (λx, x = h (f x)) p q = p⁻¹ ⬝ q ⬝ (ap h (ap f p)) :=
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by induction p; rewrite [▸*,idp_con]
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/- Pathovers -/
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-- In the comment we give the fibration of the pathover
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-- we should probably try to do everything just with pathover_eq (defined in cubical.square),
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-- the following definitions may be removed in future.
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definition pathover_eq_l (p : a₁ = a₂) (q : a₁ = a₃) : q =[p] p⁻¹ ⬝ q := /-(λx, x = a₃)-/
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by induction p; induction q; exact idpo
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definition pathover_eq_r (p : a₂ = a₃) (q : a₁ = a₂) : q =[p] q ⬝ p := /-(λx, a₁ = x)-/
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by induction p; induction q; exact idpo
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definition pathover_eq_lr (p : a₁ = a₂) (q : a₁ = a₁) : q =[p] p⁻¹ ⬝ q ⬝ p := /-(λx, x = x)-/
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by induction p; rewrite [▸*,idp_con]; exact idpo
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definition pathover_eq_Fl (p : a₁ = a₂) (q : f a₁ = b) : q =[p] (ap f p)⁻¹ ⬝ q := /-(λx, f x = b)-/
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by induction p; induction q; exact idpo
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definition pathover_eq_Fr (p : a₁ = a₂) (q : b = f a₁) : q =[p] q ⬝ (ap f p) := /-(λx, b = f x)-/
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by induction p; exact idpo
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definition pathover_eq_FlFr (p : a₁ = a₂) (q : f a₁ = g a₁) : q =[p] (ap f p)⁻¹ ⬝ q ⬝ (ap g p) :=
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/-(λx, f x = g x)-/
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by induction p; rewrite [▸*,idp_con]; exact idpo
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definition pathover_eq_FlFr_D {B : A → Type} {f g : Πa, B a} (p : a₁ = a₂) (q : f a₁ = g a₁)
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: q =[p] (apd f p)⁻¹ ⬝ ap (transport B p) q ⬝ (apd g p) := /-(λx, f x = g x)-/
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by induction p; rewrite [▸*,idp_con,ap_id];exact idpo
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definition pathover_eq_FFlr (p : a₁ = a₂) (q : h (f a₁) = a₁) : q =[p] (ap h (ap f p))⁻¹ ⬝ q ⬝ p :=
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/-(λx, h (f x) = x)-/
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by induction p; rewrite [▸*,idp_con];exact idpo
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definition pathover_eq_lFFr (p : a₁ = a₂) (q : a₁ = h (f a₁)) : q =[p] p⁻¹ ⬝ q ⬝ (ap h (ap f p)) :=
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/-(λx, x = h (f x))-/
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by induction p; rewrite [▸*,idp_con];exact idpo
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definition pathover_eq_r_idp (p : a₁ = a₂) : idp =[p] p := /-(λx, a₁ = x)-/
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by induction p; exact idpo
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definition pathover_eq_l_idp (p : a₁ = a₂) : idp =[p] p⁻¹ := /-(λx, x = a₁)-/
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by induction p; exact idpo
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definition pathover_eq_l_idp' (p : a₁ = a₂) : idp =[p⁻¹] p := /-(λx, x = a₂)-/
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by induction p; exact idpo
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-- The Functorial action of paths is [ap].
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/- Equivalences between path spaces -/
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/- [ap_closed] is in init.equiv -/
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definition equiv_ap (f : A → B) [H : is_equiv f] (a₁ a₂ : A)
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: (a₁ = a₂) ≃ (f a₁ = f a₂) :=
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equiv.mk (ap f) _
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/- Path operations are equivalences -/
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definition is_equiv_eq_inverse (a₁ a₂ : A) : is_equiv (@inverse A a₁ a₂) :=
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is_equiv.mk inverse inverse inv_inv inv_inv (λp, eq.rec_on p idp)
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local attribute is_equiv_eq_inverse [instance]
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definition eq_equiv_eq_symm (a₁ a₂ : A) : (a₁ = a₂) ≃ (a₂ = a₁) :=
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equiv.mk inverse _
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definition is_equiv_concat_left [constructor] [instance] (p : a₁ = a₂) (a₃ : A)
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: is_equiv (concat p : a₂ = a₃ → a₁ = a₃) :=
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is_equiv.mk (concat p) (concat p⁻¹)
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(con_inv_cancel_left p)
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(inv_con_cancel_left p)
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(λq, by induction p;induction q;reflexivity)
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local attribute is_equiv_concat_left [instance]
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definition equiv_eq_closed_left [constructor] (a₃ : A) (p : a₁ = a₂) : (a₁ = a₃) ≃ (a₂ = a₃) :=
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equiv.mk (concat p⁻¹) _
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definition is_equiv_concat_right [constructor] [instance] (p : a₂ = a₃) (a₁ : A)
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: is_equiv (λq : a₁ = a₂, q ⬝ p) :=
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is_equiv.mk (λq, q ⬝ p) (λq, q ⬝ p⁻¹)
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(λq, inv_con_cancel_right q p)
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(λq, con_inv_cancel_right q p)
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(λq, by induction p;induction q;reflexivity)
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local attribute is_equiv_concat_right [instance]
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definition equiv_eq_closed_right [constructor] (a₁ : A) (p : a₂ = a₃) : (a₁ = a₂) ≃ (a₁ = a₃) :=
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equiv.mk (λq, q ⬝ p) _
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definition eq_equiv_eq_closed [constructor] (p : a₁ = a₂) (q : a₃ = a₄) : (a₁ = a₃) ≃ (a₂ = a₄) :=
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equiv.trans (equiv_eq_closed_left a₃ p) (equiv_eq_closed_right a₂ q)
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definition is_equiv_whisker_left (p : a₁ = a₂) (q r : a₂ = a₃)
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: is_equiv (@whisker_left A a₁ a₂ a₃ p q r) :=
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begin
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fapply adjointify,
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{intro s, apply (!cancel_left s)},
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{intro s,
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apply concat, {apply whisker_left_con_right},
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apply concat, rotate_left 1, apply (whisker_left_inv_left p s),
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apply concat2,
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{apply concat, {apply whisker_left_con_right},
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apply concat2,
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{induction p, induction q, reflexivity},
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{reflexivity}},
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{induction p, induction r, reflexivity}},
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{intro s, induction s, induction q, induction p, reflexivity}
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end
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definition eq_equiv_con_eq_con_left (p : a₁ = a₂) (q r : a₂ = a₃) : (q = r) ≃ (p ⬝ q = p ⬝ r) :=
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equiv.mk _ !is_equiv_whisker_left
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definition is_equiv_whisker_right {p q : a₁ = a₂} (r : a₂ = a₃)
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: is_equiv (λs, @whisker_right A a₁ a₂ a₃ p q s r) :=
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begin
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fapply adjointify,
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{intro s, apply (!cancel_right s)},
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{intro s, induction r, cases s, induction q, reflexivity},
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{intro s, induction s, induction r, induction p, reflexivity}
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end
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definition eq_equiv_con_eq_con_right (p q : a₁ = a₂) (r : a₂ = a₃) : (p = q) ≃ (p ⬝ r = q ⬝ r) :=
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equiv.mk _ !is_equiv_whisker_right
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/-
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The following proofs can be simplified a bit by concatenating previous equivalences.
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However, these proofs have the advantage that the inverse is definitionally equal to
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what we would expect
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-/
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definition is_equiv_con_eq_of_eq_inv_con (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
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: is_equiv (con_eq_of_eq_inv_con : p = r⁻¹ ⬝ q → r ⬝ p = q) :=
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begin
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fapply adjointify,
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{ apply eq_inv_con_of_con_eq},
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{ intro s, induction r, rewrite [↑[con_eq_of_eq_inv_con,eq_inv_con_of_con_eq],
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con.assoc,con.assoc,con.left_inv,▸*,-con.assoc,con.right_inv,▸* at *,idp_con s]},
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{ intro s, induction r, rewrite [↑[con_eq_of_eq_inv_con,eq_inv_con_of_con_eq],
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con.assoc,con.assoc,con.right_inv,▸*,-con.assoc,con.left_inv,▸* at *,idp_con s] },
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end
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definition eq_inv_con_equiv_con_eq (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
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: (p = r⁻¹ ⬝ q) ≃ (r ⬝ p = q) :=
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equiv.mk _ !is_equiv_con_eq_of_eq_inv_con
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definition is_equiv_con_eq_of_eq_con_inv (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
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: is_equiv (con_eq_of_eq_con_inv : r = q ⬝ p⁻¹ → r ⬝ p = q) :=
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begin
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fapply adjointify,
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{ apply eq_con_inv_of_con_eq},
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{ intro s, induction p, rewrite [↑[con_eq_of_eq_con_inv,eq_con_inv_of_con_eq]]},
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{ intro s, induction p, rewrite [↑[con_eq_of_eq_con_inv,eq_con_inv_of_con_eq]] },
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end
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definition eq_con_inv_equiv_con_eq (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
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: (r = q ⬝ p⁻¹) ≃ (r ⬝ p = q) :=
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equiv.mk _ !is_equiv_con_eq_of_eq_con_inv
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definition is_equiv_inv_con_eq_of_eq_con (p : a₁ = a₃) (q : a₂ = a₃) (r : a₁ = a₂)
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: is_equiv (inv_con_eq_of_eq_con : p = r ⬝ q → r⁻¹ ⬝ p = q) :=
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begin
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fapply adjointify,
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{ apply eq_con_of_inv_con_eq},
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{ intro s, induction r, rewrite [↑[inv_con_eq_of_eq_con,eq_con_of_inv_con_eq],
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con.assoc,con.assoc,con.left_inv,▸*,-con.assoc,con.right_inv,▸* at *,idp_con s]},
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{ intro s, induction r, rewrite [↑[inv_con_eq_of_eq_con,eq_con_of_inv_con_eq],
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con.assoc,con.assoc,con.right_inv,▸*,-con.assoc,con.left_inv,▸* at *,idp_con s] },
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end
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definition eq_con_equiv_inv_con_eq (p : a₁ = a₃) (q : a₂ = a₃) (r : a₁ = a₂)
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: (p = r ⬝ q) ≃ (r⁻¹ ⬝ p = q) :=
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equiv.mk _ !is_equiv_inv_con_eq_of_eq_con
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definition is_equiv_con_inv_eq_of_eq_con (p : a₃ = a₁) (q : a₂ = a₃) (r : a₂ = a₁)
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: is_equiv (con_inv_eq_of_eq_con : r = q ⬝ p → r ⬝ p⁻¹ = q) :=
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begin
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fapply adjointify,
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{ apply eq_con_of_con_inv_eq},
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{ intro s, induction p, rewrite [↑[con_inv_eq_of_eq_con,eq_con_of_con_inv_eq]]},
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{ intro s, induction p, rewrite [↑[con_inv_eq_of_eq_con,eq_con_of_con_inv_eq]] },
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end
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definition eq_con_equiv_con_inv_eq (p : a₃ = a₁) (q : a₂ = a₃) (r : a₂ = a₁)
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: (r = q ⬝ p) ≃ (r ⬝ p⁻¹ = q) :=
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equiv.mk _ !is_equiv_con_inv_eq_of_eq_con
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local attribute is_equiv_inv_con_eq_of_eq_con
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is_equiv_con_inv_eq_of_eq_con
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is_equiv_con_eq_of_eq_con_inv
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is_equiv_con_eq_of_eq_inv_con [instance]
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definition is_equiv_eq_con_of_inv_con_eq (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
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: is_equiv (eq_con_of_inv_con_eq : r⁻¹ ⬝ q = p → q = r ⬝ p) :=
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is_equiv_inv inv_con_eq_of_eq_con
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definition is_equiv_eq_con_of_con_inv_eq (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
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: is_equiv (eq_con_of_con_inv_eq : q ⬝ p⁻¹ = r → q = r ⬝ p) :=
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is_equiv_inv con_inv_eq_of_eq_con
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definition is_equiv_eq_con_inv_of_con_eq (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
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: is_equiv (eq_con_inv_of_con_eq : r ⬝ p = q → r = q ⬝ p⁻¹) :=
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is_equiv_inv con_eq_of_eq_con_inv
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definition is_equiv_eq_inv_con_of_con_eq (p : a₁ = a₃) (q : a₂ = a₃) (r : a₂ = a₁)
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: is_equiv (eq_inv_con_of_con_eq : r ⬝ p = q → p = r⁻¹ ⬝ q) :=
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is_equiv_inv con_eq_of_eq_inv_con
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/- Pathover Equivalences -/
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definition pathover_eq_equiv_l (p : a₁ = a₂) (q : a₁ = a₃) (r : a₂ = a₃) : q =[p] r ≃ q = p ⬝ r :=
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/-(λx, x = a₃)-/
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by induction p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
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definition pathover_eq_equiv_r (p : a₂ = a₃) (q : a₁ = a₂) (r : a₁ = a₃) : q =[p] r ≃ q ⬝ p = r :=
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/-(λx, a₁ = x)-/
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by induction p; apply pathover_idp
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definition pathover_eq_equiv_lr (p : a₁ = a₂) (q : a₁ = a₁) (r : a₂ = a₂)
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: q =[p] r ≃ q ⬝ p = p ⬝ r := /-(λx, x = x)-/
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by induction p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
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definition pathover_eq_equiv_Fl (p : a₁ = a₂) (q : f a₁ = b) (r : f a₂ = b)
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: q =[p] r ≃ q = ap f p ⬝ r := /-(λx, f x = b)-/
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by induction p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
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definition pathover_eq_equiv_Fr (p : a₁ = a₂) (q : b = f a₁) (r : b = f a₂)
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: q =[p] r ≃ q ⬝ ap f p = r := /-(λx, b = f x)-/
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by induction p; apply pathover_idp
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definition pathover_eq_equiv_FlFr (p : a₁ = a₂) (q : f a₁ = g a₁) (r : f a₂ = g a₂)
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: q =[p] r ≃ q ⬝ ap g p = ap f p ⬝ r := /-(λx, f x = g x)-/
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by induction p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
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definition pathover_eq_equiv_FFlr (p : a₁ = a₂) (q : h (f a₁) = a₁) (r : h (f a₂) = a₂)
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: q =[p] r ≃ q ⬝ p = ap h (ap f p) ⬝ r :=
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/-(λx, h (f x) = x)-/
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by induction p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
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definition pathover_eq_equiv_lFFr (p : a₁ = a₂) (q : a₁ = h (f a₁)) (r : a₂ = h (f a₂))
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: q =[p] r ≃ q ⬝ ap h (ap f p) = p ⬝ r :=
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/-(λx, x = h (f x))-/
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by induction p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹
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-- a lot of this library still needs to be ported from Coq HoTT
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end eq
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