388 lines
19 KiB
Text
388 lines
19 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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Squares in a type
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-/
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import types.eq
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open eq equiv is_equiv
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namespace eq
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variables {A B : Type} {a a' a'' a₀₀ a₂₀ a₄₀ a₀₂ a₂₂ a₂₄ a₀₄ a₄₂ a₄₄ a₁ a₂ a₃ a₄ : A}
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/-a₀₀-/ {p₁₀ p₁₀' : a₀₀ = a₂₀} /-a₂₀-/ {p₃₀ : a₂₀ = a₄₀} /-a₄₀-/
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{p₀₁ p₀₁' : a₀₀ = a₀₂} /-s₁₁-/ {p₂₁ p₂₁' : a₂₀ = a₂₂} /-s₃₁-/ {p₄₁ : a₄₀ = a₄₂}
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/-a₀₂-/ {p₁₂ p₁₂' : a₀₂ = a₂₂} /-a₂₂-/ {p₃₂ : a₂₂ = a₄₂} /-a₄₂-/
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{p₀₃ : a₀₂ = a₀₄} /-s₁₃-/ {p₂₃ : a₂₂ = a₂₄} /-s₃₃-/ {p₄₃ : a₄₂ = a₄₄}
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/-a₀₄-/ {p₁₄ : a₀₄ = a₂₄} /-a₂₄-/ {p₃₄ : a₂₄ = a₄₄} /-a₄₄-/
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inductive square {A : Type} {a₀₀ : A}
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: Π{a₂₀ a₀₂ a₂₂ : A}, a₀₀ = a₂₀ → a₀₂ = a₂₂ → a₀₀ = a₀₂ → a₂₀ = a₂₂ → Type :=
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ids : square idp idp idp idp
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/- square top bottom left right -/
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variables {s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁} {s₃₁ : square p₃₀ p₃₂ p₂₁ p₄₁}
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{s₁₃ : square p₁₂ p₁₄ p₀₃ p₂₃} {s₃₃ : square p₃₂ p₃₄ p₂₃ p₄₃}
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definition ids [reducible] [constructor] := @square.ids
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definition idsquare [reducible] [constructor] (a : A) := @square.ids A a
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definition hrefl [unfold 4] (p : a = a') : square idp idp p p :=
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by induction p; exact ids
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definition vrefl [unfold 4] (p : a = a') : square p p idp idp :=
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by induction p; exact ids
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definition hrfl [reducible] [unfold 4] {p : a = a'} : square idp idp p p :=
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!hrefl
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definition vrfl [reducible] [unfold 4] {p : a = a'} : square p p idp idp :=
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!vrefl
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definition hdeg_square [unfold 6] {p q : a = a'} (r : p = q) : square idp idp p q :=
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by induction r;apply hrefl
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definition vdeg_square [unfold 6] {p q : a = a'} (r : p = q) : square p q idp idp :=
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by induction r;apply vrefl
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definition hconcat [unfold 16] (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (s₃₁ : square p₃₀ p₃₂ p₂₁ p₄₁)
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: square (p₁₀ ⬝ p₃₀) (p₁₂ ⬝ p₃₂) p₀₁ p₄₁ :=
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by induction s₃₁; exact s₁₁
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definition vconcat [unfold 16] (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (s₁₃ : square p₁₂ p₁₄ p₀₃ p₂₃)
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: square p₁₀ p₁₄ (p₀₁ ⬝ p₀₃) (p₂₁ ⬝ p₂₃) :=
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by induction s₁₃; exact s₁₁
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definition hinverse [unfold 10] (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) : square p₁₀⁻¹ p₁₂⁻¹ p₂₁ p₀₁ :=
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by induction s₁₁;exact ids
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definition vinverse [unfold 10] (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) : square p₁₂ p₁₀ p₀₁⁻¹ p₂₁⁻¹ :=
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by induction s₁₁;exact ids
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definition eq_vconcat [unfold 11] {p : a₀₀ = a₂₀} (r : p = p₁₀) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) :
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square p p₁₂ p₀₁ p₂₁ :=
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by induction r; exact s₁₁
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definition vconcat_eq [unfold 11] {p : a₀₂ = a₂₂} (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (r : p₁₂ = p) :
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square p₁₀ p p₀₁ p₂₁ :=
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by induction r; exact s₁₁
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definition eq_hconcat [unfold 11] {p : a₀₀ = a₀₂} (r : p = p₀₁) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) :
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square p₁₀ p₁₂ p p₂₁ :=
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by induction r; exact s₁₁
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definition hconcat_eq [unfold 11] {p : a₂₀ = a₂₂} (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (r : p₂₁ = p) :
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square p₁₀ p₁₂ p₀₁ p :=
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by induction r; exact s₁₁
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infix `⬝h`:75 := hconcat
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infix `⬝v`:75 := vconcat
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infix `⬝hp`:75 := hconcat_eq
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infix `⬝vp`:75 := vconcat_eq
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infix `⬝ph`:75 := eq_hconcat
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infix `⬝pv`:75 := eq_vconcat
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postfix `⁻¹ʰ`:(max+1) := hinverse
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postfix `⁻¹ᵛ`:(max+1) := vinverse
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definition transpose [unfold 10] (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) : square p₀₁ p₂₁ p₁₀ p₁₂ :=
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by induction s₁₁;exact ids
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definition aps {B : Type} (f : A → B) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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: square (ap f p₁₀) (ap f p₁₂) (ap f p₀₁) (ap f p₂₁) :=
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by induction s₁₁;exact ids
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definition natural_square [unfold 8] {f g : A → B} (p : f ~ g) (q : a = a') :
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square (ap f q) (ap g q) (p a) (p a') :=
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eq.rec_on q hrfl
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definition natural_square_tr [unfold 8] {f g : A → B} (p : f ~ g) (q : a = a') :
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square (p a) (p a') (ap f q) (ap g q) :=
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eq.rec_on q vrfl
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definition whisker_tl (p : a = a₀₀) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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: square (p ⬝ p₁₀) p₁₂ (p ⬝ p₀₁) p₂₁ :=
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by induction s₁₁;induction p;exact ids
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definition whisker_br (p : a₂₂ = a) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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: square p₁₀ (p₁₂ ⬝ p) p₀₁ (p₂₁ ⬝ p) :=
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by induction p;exact s₁₁
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/- some higher ∞-groupoid operations -/
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definition vconcat_vrfl (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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: s₁₁ ⬝v vrefl p₁₂ = s₁₁ :=
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by induction s₁₁; reflexivity
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definition hconcat_hrfl (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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: s₁₁ ⬝h hrefl p₂₁ = s₁₁ :=
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by induction s₁₁; reflexivity
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/- equivalences -/
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definition eq_of_square [unfold 10] (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) : p₁₀ ⬝ p₂₁ = p₀₁ ⬝ p₁₂ :=
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by induction s₁₁; apply idp
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definition square_of_eq (r : p₁₀ ⬝ p₂₁ = p₀₁ ⬝ p₁₂) : square p₁₀ p₁₂ p₀₁ p₂₁ :=
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by induction p₁₂; esimp [concat] at r; induction r; induction p₂₁; induction p₁₀; exact ids
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definition eq_top_of_square [unfold 10] (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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: p₁₀ = p₀₁ ⬝ p₁₂ ⬝ p₂₁⁻¹ :=
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by induction s₁₁; apply idp
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definition square_of_eq_top (r : p₁₀ = p₀₁ ⬝ p₁₂ ⬝ p₂₁⁻¹) : square p₁₀ p₁₂ p₀₁ p₂₁ :=
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by induction p₂₁; induction p₁₂; esimp at r;induction r;induction p₁₀;exact ids
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definition eq_bot_of_square [unfold 10] (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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: p₁₂ = p₀₁⁻¹ ⬝ p₁₀ ⬝ p₂₁ :=
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by induction s₁₁; apply idp
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definition square_equiv_eq [constructor] (t : a₀₀ = a₀₂) (b : a₂₀ = a₂₂)
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(l : a₀₀ = a₂₀) (r : a₀₂ = a₂₂) : square t b l r ≃ t ⬝ r = l ⬝ b :=
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begin
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fapply equiv.MK,
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{ exact eq_of_square},
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{ exact square_of_eq},
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{ intro s, induction b, esimp [concat] at s, induction s, induction r, induction t, apply idp},
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{ intro s, induction s, apply idp},
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end
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definition hdeg_square_equiv' [constructor] (p q : a = a') : square idp idp p q ≃ p = q :=
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by transitivity _;apply square_equiv_eq;transitivity _;apply eq_equiv_eq_symm;
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apply equiv_eq_closed_right;apply idp_con
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definition vdeg_square_equiv' [constructor] (p q : a = a') : square p q idp idp ≃ p = q :=
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by transitivity _;apply square_equiv_eq;apply equiv_eq_closed_right; apply idp_con
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definition eq_of_hdeg_square [reducible] {p q : a = a'} (s : square idp idp p q) : p = q :=
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to_fun !hdeg_square_equiv' s
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definition eq_of_vdeg_square [reducible] {p q : a = a'} (s : square p q idp idp) : p = q :=
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to_fun !vdeg_square_equiv' s
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definition top_deg_square (l : a₁ = a₂) (b : a₂ = a₃) (r : a₄ = a₃)
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: square (l ⬝ b ⬝ r⁻¹) b l r :=
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by induction r;induction b;induction l;constructor
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definition bot_deg_square (l : a₁ = a₂) (t : a₁ = a₃) (r : a₃ = a₄)
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: square t (l⁻¹ ⬝ t ⬝ r) l r :=
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by induction r;induction t;induction l;constructor
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/-
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the following two equivalences have as underlying inverse function the functions
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hdeg_square and vdeg_square, respectively.
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See examples below the definition
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-/
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definition hdeg_square_equiv [constructor] (p q : a = a') : square idp idp p q ≃ p = q :=
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begin
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fapply equiv_change_fun,
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{ fapply equiv_change_inv, apply hdeg_square_equiv', exact hdeg_square,
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intro s, induction s, induction p, reflexivity},
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{ exact eq_of_hdeg_square},
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{ reflexivity}
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end
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definition vdeg_square_equiv [constructor] (p q : a = a') : square p q idp idp ≃ p = q :=
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begin
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fapply equiv_change_fun,
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{ fapply equiv_change_inv, apply vdeg_square_equiv',exact vdeg_square,
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intro s, induction s, induction p, reflexivity},
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{ exact eq_of_vdeg_square},
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{ reflexivity}
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end
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-- example (p q : a = a') : to_inv (hdeg_square_equiv' p q) = hdeg_square := idp -- this fails
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example (p q : a = a') : to_inv (hdeg_square_equiv p q) = hdeg_square := idp
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definition eq_pathover [unfold 7] {f g : A → B} {p : a = a'} {q : f a = g a} {r : f a' = g a'}
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(s : square q r (ap f p) (ap g p)) : q =[p] r :=
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by induction p;apply pathover_idp_of_eq;exact eq_of_vdeg_square s
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definition square_of_pathover [unfold 7]
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{f g : A → B} {p : a = a'} {q : f a = g a} {r : f a' = g a'}
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(s : q =[p] r) : square q r (ap f p) (ap g p) :=
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by induction p;apply vdeg_square;exact eq_of_pathover_idp s
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/- interaction of equivalences with operations on squares -/
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definition eq_pathover_equiv_square [constructor] {f g : A → B}
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(p : a = a') (q : f a = g a) (r : f a' = g a') : q =[p] r ≃ square q r (ap f p) (ap g p) :=
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equiv.MK square_of_pathover
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eq_pathover
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begin
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intro s, induction p, esimp [square_of_pathover,eq_pathover],
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exact ap vdeg_square (to_right_inv !pathover_idp (eq_of_vdeg_square s))
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⬝ to_left_inv !vdeg_square_equiv s
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end
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begin
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intro s, induction p, esimp [square_of_pathover,eq_pathover],
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exact ap pathover_idp_of_eq (to_right_inv !vdeg_square_equiv (eq_of_pathover_idp s))
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⬝ to_left_inv !pathover_idp s
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end
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definition square_of_pathover_eq_concato {f g : A → B} {p : a = a'} {q q' : f a = g a}
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{r : f a' = g a'} (s' : q = q') (s : q' =[p] r)
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: square_of_pathover (s' ⬝po s) = s' ⬝pv square_of_pathover s :=
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by induction s;induction s';reflexivity
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definition square_of_pathover_concato_eq {f g : A → B} {p : a = a'} {q : f a = g a}
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{r r' : f a' = g a'} (s' : r = r') (s : q =[p] r)
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: square_of_pathover (s ⬝op s') = square_of_pathover s ⬝vp s' :=
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by induction s;induction s';reflexivity
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definition square_of_pathover_concato {f g : A → B} {p : a = a'} {p' : a' = a''} {q : f a = g a}
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{q' : f a' = g a'} {q'' : f a'' = g a''} (s : q =[p] q') (s' : q' =[p'] q'')
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: square_of_pathover (s ⬝o s')
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= ap_con f p p' ⬝ph (square_of_pathover s ⬝v square_of_pathover s') ⬝hp (ap_con g p p')⁻¹ :=
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by induction s';induction s;esimp [ap_con,hconcat_eq];exact !vconcat_vrfl⁻¹
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definition eq_of_square_hrfl [unfold 4] (p : a = a') : eq_of_square hrfl = idp_con p :=
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by induction p;reflexivity
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definition eq_of_square_vrfl [unfold 4] (p : a = a') : eq_of_square vrfl = (idp_con p)⁻¹ :=
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by induction p;reflexivity
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definition eq_of_square_hdeg_square {p q : a = a'} (r : p = q)
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: eq_of_square (hdeg_square r) = !idp_con ⬝ r⁻¹ :=
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by induction r;induction p;reflexivity
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definition eq_of_square_vdeg_square {p q : a = a'} (r : p = q)
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: eq_of_square (vdeg_square r) = r ⬝ !idp_con⁻¹ :=
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by induction r;induction p;reflexivity
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definition eq_of_square_eq_vconcat {p : a₀₀ = a₂₀} (r : p = p₁₀) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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: eq_of_square (r ⬝pv s₁₁) = whisker_right r p₂₁ ⬝ eq_of_square s₁₁ :=
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by induction s₁₁;cases r;reflexivity
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definition eq_of_square_eq_hconcat {p : a₀₀ = a₀₂} (r : p = p₀₁) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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: eq_of_square (r ⬝ph s₁₁) = eq_of_square s₁₁ ⬝ (whisker_right r p₁₂)⁻¹ :=
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by induction r;reflexivity
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definition eq_of_square_vconcat_eq {p : a₀₂ = a₂₂} (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (r : p₁₂ = p)
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: eq_of_square (s₁₁ ⬝vp r) = eq_of_square s₁₁ ⬝ whisker_left p₀₁ r :=
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by induction r;reflexivity
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definition eq_of_square_hconcat_eq {p : a₂₀ = a₂₂} (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (r : p₂₁ = p)
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: eq_of_square (s₁₁ ⬝hp r) = (whisker_left p₁₀ r)⁻¹ ⬝ eq_of_square s₁₁ :=
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by induction s₁₁; induction r;reflexivity
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-- definition vconcat_eq [unfold 11] {p : a₀₂ = a₂₂} (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (r : p₁₂ = p) :
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-- square p₁₀ p p₀₁ p₂₁ :=
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-- by induction r; exact s₁₁
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-- definition eq_hconcat [unfold 11] {p : a₀₀ = a₀₂} (r : p = p₀₁)
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-- (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) : square p₁₀ p₁₂ p p₂₁ :=
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-- by induction r; exact s₁₁
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-- definition hconcat_eq [unfold 11] {p : a₂₀ = a₂₂}
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-- (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (r : p₂₁ = p) : square p₁₀ p₁₂ p₀₁ p :=
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-- by induction r; exact s₁₁
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-- the following definition is very slow, maybe it's interesting to see why?
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-- definition eq_pathover_equiv_square' {f g : A → B}(p : a = a') (q : f a = g a) (r : f a' = g a')
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-- : square q r (ap f p) (ap g p) ≃ q =[p] r :=
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-- equiv.MK eq_pathover
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-- square_of_pathover
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-- (λs, begin
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-- induction p, rewrite [↑[square_of_pathover,eq_pathover],
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-- to_right_inv !vdeg_square_equiv (eq_of_pathover_idp s),
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-- to_left_inv !pathover_idp s]
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-- end)
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-- (λs, begin
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-- induction p, rewrite [↑[square_of_pathover,eq_pathover],▸*,
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-- to_right_inv !(@pathover_idp A) (eq_of_vdeg_square s),
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-- to_left_inv !vdeg_square_equiv s]
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-- end)
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/- recursors for squares where some sides are reflexivity -/
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definition rec_on_b [recursor] {a₀₀ : A}
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{P : Π{a₂₀ a₁₂ : A} {t : a₀₀ = a₂₀} {l : a₀₀ = a₁₂} {r : a₂₀ = a₁₂}, square t idp l r → Type}
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{a₂₀ a₁₂ : A} {t : a₀₀ = a₂₀} {l : a₀₀ = a₁₂} {r : a₂₀ = a₁₂}
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(s : square t idp l r) (H : P ids) : P s :=
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have H2 : P (square_of_eq (eq_of_square s)),
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from eq.rec_on (eq_of_square s : t ⬝ r = l) (by induction r; induction t; exact H),
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left_inv (to_fun !square_equiv_eq) s ▸ H2
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definition rec_on_r [recursor] {a₀₀ : A}
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{P : Π{a₀₂ a₂₁ : A} {t : a₀₀ = a₂₁} {b : a₀₂ = a₂₁} {l : a₀₀ = a₀₂}, square t b l idp → Type}
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{a₀₂ a₂₁ : A} {t : a₀₀ = a₂₁} {b : a₀₂ = a₂₁} {l : a₀₀ = a₀₂}
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(s : square t b l idp) (H : P ids) : P s :=
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let p : l ⬝ b = t := (eq_of_square s)⁻¹ in
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have H2 : P (square_of_eq (eq_of_square s)⁻¹⁻¹),
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from @eq.rec_on _ _ (λx p, P (square_of_eq p⁻¹)) _ p (by induction b; induction l; exact H),
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left_inv (to_fun !square_equiv_eq) s ▸ !inv_inv ▸ H2
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definition rec_on_l [recursor] {a₀₁ : A}
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{P : Π {a₂₀ a₂₂ : A} {t : a₀₁ = a₂₀} {b : a₀₁ = a₂₂} {r : a₂₀ = a₂₂},
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square t b idp r → Type}
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{a₂₀ a₂₂ : A} {t : a₀₁ = a₂₀} {b : a₀₁ = a₂₂} {r : a₂₀ = a₂₂}
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(s : square t b idp r) (H : P ids) : P s :=
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let p : t ⬝ r = b := eq_of_square s ⬝ !idp_con in
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have H2 : P (square_of_eq (p ⬝ !idp_con⁻¹)),
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from eq.rec_on p (by induction r; induction t; exact H),
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left_inv (to_fun !square_equiv_eq) s ▸ !con_inv_cancel_right ▸ H2
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definition rec_on_t [recursor] {a₁₀ : A}
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{P : Π {a₀₂ a₂₂ : A} {b : a₀₂ = a₂₂} {l : a₁₀ = a₀₂} {r : a₁₀ = a₂₂}, square idp b l r → Type}
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{a₀₂ a₂₂ : A} {b : a₀₂ = a₂₂} {l : a₁₀ = a₀₂} {r : a₁₀ = a₂₂}
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(s : square idp b l r) (H : P ids) : P s :=
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let p : l ⬝ b = r := (eq_of_square s)⁻¹ ⬝ !idp_con in
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assert H2 : P (square_of_eq ((p ⬝ !idp_con⁻¹)⁻¹)),
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from eq.rec_on p (by induction b; induction l; exact H),
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assert H3 : P (square_of_eq ((eq_of_square s)⁻¹⁻¹)),
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from eq.rec_on !con_inv_cancel_right H2,
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assert H4 : P (square_of_eq (eq_of_square s)),
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from eq.rec_on !inv_inv H3,
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proof
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left_inv (to_fun !square_equiv_eq) s ▸ H4
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qed
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definition rec_on_tb [recursor] {a : A}
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{P : Π{b : A} {l : a = b} {r : a = b}, square idp idp l r → Type}
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{b : A} {l : a = b} {r : a = b}
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(s : square idp idp l r) (H : P ids) : P s :=
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have H2 : P (square_of_eq (eq_of_square s)),
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from eq.rec_on (eq_of_square s : idp ⬝ r = l) (by induction r; exact H),
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left_inv (to_fun !square_equiv_eq) s ▸ H2
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definition rec_on_lr [recursor] {a : A}
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{P : Π{a' : A} {t : a = a'} {b : a = a'}, square t b idp idp → Type}
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{a' : A} {t : a = a'} {b : a = a'}
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(s : square t b idp idp) (H : P ids) : P s :=
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let p : idp ⬝ b = t := (eq_of_square s)⁻¹ in
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assert H2 : P (square_of_eq (eq_of_square s)⁻¹⁻¹),
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from @eq.rec_on _ _ (λx q, P (square_of_eq q⁻¹)) _ p (by induction b; exact H),
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to_left_inv (!square_equiv_eq) s ▸ !inv_inv ▸ H2
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--we can also do the other recursors (tl, tr, bl, br, tbl, tbr, tlr, blr), but let's postpone this until they are needed
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definition whisker_square [unfold 14 15 16 17] (r₁₀ : p₁₀ = p₁₀') (r₁₂ : p₁₂ = p₁₂')
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(r₀₁ : p₀₁ = p₀₁') (r₂₁ : p₂₁ = p₂₁') (s : square p₁₀ p₁₂ p₀₁ p₂₁)
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: square p₁₀' p₁₂' p₀₁' p₂₁' :=
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by induction r₁₀; induction r₁₂; induction r₀₁; induction r₂₁; exact s
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/- squares commute with some operations on 2-paths -/
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definition square_inv2 {p₁ p₂ p₃ p₄ : a = a'}
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{t : p₁ = p₂} {b : p₃ = p₄} {l : p₁ = p₃} {r : p₂ = p₄} (s : square t b l r)
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: square (inverse2 t) (inverse2 b) (inverse2 l) (inverse2 r) :=
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by induction s;constructor
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definition square_con2 {p₁ p₂ p₃ p₄ : a₁ = a₂} {q₁ q₂ q₃ q₄ : a₂ = a₃}
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{t₁ : p₁ = p₂} {b₁ : p₃ = p₄} {l₁ : p₁ = p₃} {r₁ : p₂ = p₄}
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{t₂ : q₁ = q₂} {b₂ : q₃ = q₄} {l₂ : q₁ = q₃} {r₂ : q₂ = q₄}
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(s₁ : square t₁ b₁ l₁ r₁) (s₂ : square t₂ b₂ l₂ r₂)
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: square (t₁ ◾ t₂) (b₁ ◾ b₂) (l₁ ◾ l₂) (r₁ ◾ r₂) :=
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by induction s₂;induction s₁;constructor
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-- definition square_of_con_inv_hsquare {p₁ p₂ p₃ p₄ : a₁ = a₂}
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-- {t : p₁ = p₂} {b : p₃ = p₄} {l : p₁ = p₃} {r : p₂ = p₄}
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-- (s : square (con_inv_eq_idp t) (con_inv_eq_idp b) (l ◾ r⁻²) idp)
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-- : square t b l r :=
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-- sorry --by induction s
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end eq
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