341 lines
13 KiB
Text
341 lines
13 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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import .cubical.square
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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} {b b1 b2 : B} {f g : A → B} {h : B → A}
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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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definition 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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cases p, cases r, cases s, exact idp
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end
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definition whisker_right_con_right {p p' p'' : a1 = a2} (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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cases q, cases r, cases s, exact idp
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end
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definition 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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cases p', cases p, cases r, cases q, exact idp
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end
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definition 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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cases q', cases q, cases r, cases p, exact idp
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end
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definition 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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cases p, cases r, cases q, exact idp
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end
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definition con_right_inv2 (p : a1 = a2) : (con.right_inv p)⁻¹ ⬝ con.right_inv p = idp :=
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by cases p;exact idp
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definition con_left_inv2 (p : a1 = a2) : (con.left_inv p)⁻¹ ⬝ con.left_inv p = idp :=
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by cases p;exact idp
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definition 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 cases p;exact idp
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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 cases p; cases 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 cases p; cases 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 cases 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 cases p; cases 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 cases 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 cases 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 cases 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 cases 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 cases 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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definition pathover_eq {f g : A → B} {p : a1 = a2} {q : f a1 = g a1} {r : f a2 = g a2}
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(s : square q r (ap f p) (ap g p)) : q =[p] r :=
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by cases p;apply pathover_idp_of_eq;exact eq_of_vdeg_square s
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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 cases p; cases 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 cases p; cases 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 cases 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 cases p; cases 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 cases 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 cases 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 cases 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 cases 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 cases 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 cases 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 cases 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 cases 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 cases p;cases q;exact idp)
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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 cases p;cases q;exact idp)
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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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{cases p, cases q, exact idp},
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{exact idp}},
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{cases p, cases r, exact idp}},
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{intro s, cases s, cases q, cases p, exact idp}
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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, cases r, cases s, cases q, reflexivity},
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{intro s, cases s, cases r, cases 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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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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cases r,
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apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right),
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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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cases p,
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apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right)
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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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cases r,
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apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right)
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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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cases p,
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apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right)
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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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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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begin
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cases r,
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apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right)
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end
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definition inv_con_eq_equiv_eq_con (p : a1 = a3) (q : a2 = a3) (r : a2 = a1)
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: (r⁻¹ ⬝ q = p) ≃ (q = r ⬝ p) :=
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equiv.mk _ !is_equiv_eq_con_of_inv_con_eq
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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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begin
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cases p,
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apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right)
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end
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definition con_inv_eq_equiv_eq_con (p : a1 = a3) (q : a2 = a3) (r : a2 = a1)
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: (q ⬝ p⁻¹ = r) ≃ (q = r ⬝ p) :=
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equiv.mk _ !is_equiv_eq_con_of_con_inv_eq
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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 cases 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 cases 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 cases 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 cases 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 cases 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 cases 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 cases 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 cases 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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