636 lines
31 KiB
Text
636 lines
31 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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Authors: Floris van Doorn, Jakob von Raumer
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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 sigma
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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 hdeg_square_idp (p : a = a') : hdeg_square (refl p) = hrfl :=
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by reflexivity
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definition vdeg_square_idp (p : a = a') : vdeg_square (refl p) = vrfl :=
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by reflexivity
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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 dconcat [unfold 14] {p₀₀ : a₀₀ = a} {p₂₂ : a = a₂₂}
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(s₂₁ : square p₀₀ p₁₂ p₀₁ p₂₂) (s₁₂ : square p₁₀ p₂₂ p₀₀ p₂₁) : square p₁₀ p₁₂ p₀₁ p₂₁ :=
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by induction s₁₂; exact s₂₁
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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 12] {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 12] {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 `:69 := hconcat --type using \tr
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infix ` ⬝v `:70 := vconcat --type using \tr
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infix ` ⬝hp `:72 := hconcat_eq --type using \tr
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infix ` ⬝vp `:74 := vconcat_eq --type using \tr
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infix ` ⬝ph `:71 := eq_hconcat --type using \tr
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infix ` ⬝pv `:73 := eq_vconcat --type using \tr
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postfix `⁻¹ʰ`:(max+1) := hinverse --type using \-1h
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postfix `⁻¹ᵛ`:(max+1) := vinverse --type using \-1v
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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 [unfold 12] (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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/- canceling, whiskering and moving thinks along the sides of the square -/
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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;constructor
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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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definition whisker_rt (p : a = a₂₀) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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: square (p₁₀ ⬝ p⁻¹) p₁₂ p₀₁ (p ⬝ p₂₁) :=
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by induction s₁₁;induction p;constructor
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definition whisker_tr (p : a₂₀ = a) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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: square (p₁₀ ⬝ p) p₁₂ p₀₁ (p⁻¹ ⬝ p₂₁) :=
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by induction s₁₁;induction p;constructor
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definition whisker_bl (p : a = a₀₂) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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: square p₁₀ (p ⬝ p₁₂) (p₀₁ ⬝ p⁻¹) p₂₁ :=
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by induction s₁₁;induction p;constructor
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definition whisker_lb (p : a₀₂ = a) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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: square p₁₀ (p⁻¹ ⬝ p₁₂) (p₀₁ ⬝ p) p₂₁ :=
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by induction s₁₁;induction p;constructor
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definition cancel_tl (p : a = a₀₀) (s₁₁ : square (p ⬝ p₁₀) p₁₂ (p ⬝ p₀₁) p₂₁)
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: square p₁₀ p₁₂ p₀₁ p₂₁ :=
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by induction p; rewrite +idp_con at s₁₁; exact s₁₁
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definition cancel_br (p : a₂₂ = a) (s₁₁ : square p₁₀ (p₁₂ ⬝ p) p₀₁ (p₂₁ ⬝ p))
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: square p₁₀ p₁₂ p₀₁ p₂₁ :=
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by induction p;exact s₁₁
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definition cancel_rt (p : a = a₂₀) (s₁₁ : square (p₁₀ ⬝ p⁻¹) p₁₂ p₀₁ (p ⬝ p₂₁))
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: square p₁₀ p₁₂ p₀₁ p₂₁ :=
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by induction p; rewrite idp_con at s₁₁; exact s₁₁
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definition cancel_tr (p : a₂₀ = a) (s₁₁ : square (p₁₀ ⬝ p) p₁₂ p₀₁ (p⁻¹ ⬝ p₂₁))
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: square p₁₀ p₁₂ p₀₁ p₂₁ :=
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by induction p; rewrite [▸* at s₁₁,idp_con at s₁₁]; exact s₁₁
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definition cancel_bl (p : a = a₀₂) (s₁₁ : square p₁₀ (p ⬝ p₁₂) (p₀₁ ⬝ p⁻¹) p₂₁)
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: square p₁₀ p₁₂ p₀₁ p₂₁ :=
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by induction p; rewrite idp_con at s₁₁; exact s₁₁
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definition cancel_lb (p : a₀₂ = a) (s₁₁ : square p₁₀ (p⁻¹ ⬝ p₁₂) (p₀₁ ⬝ p) p₂₁)
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: square p₁₀ p₁₂ p₀₁ p₂₁ :=
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by induction p; rewrite [▸* at s₁₁,idp_con at s₁₁]; exact s₁₁
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definition move_top_of_left {p : a₀₀ = a} {q : a = a₀₂} (s : square p₁₀ p₁₂ (p ⬝ q) p₂₁)
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: square (p⁻¹ ⬝ p₁₀) p₁₂ q p₂₁ :=
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by apply cancel_tl p; rewrite con_inv_cancel_left; exact s
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definition move_top_of_left' {p : a = a₀₀} {q : a = a₀₂} (s : square p₁₀ p₁₂ (p⁻¹ ⬝ q) p₂₁)
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: square (p ⬝ p₁₀) p₁₂ q p₂₁ :=
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by apply cancel_tl p⁻¹; rewrite inv_con_cancel_left; exact s
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definition move_left_of_top {p : a₀₀ = a} {q : a = a₂₀} (s : square (p ⬝ q) p₁₂ p₀₁ p₂₁)
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: square q p₁₂ (p⁻¹ ⬝ p₀₁) p₂₁ :=
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by apply cancel_tl p; rewrite con_inv_cancel_left; exact s
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definition move_left_of_top' {p : a = a₀₀} {q : a = a₂₀} (s : square (p⁻¹ ⬝ q) p₁₂ p₀₁ p₂₁)
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: square q p₁₂ (p ⬝ p₀₁) p₂₁ :=
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by apply cancel_tl p⁻¹; rewrite inv_con_cancel_left; exact s
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definition move_bot_of_right {p : a₂₀ = a} {q : a = a₂₂} (s : square p₁₀ p₁₂ p₀₁ (p ⬝ q))
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: square p₁₀ (p₁₂ ⬝ q⁻¹) p₀₁ p :=
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by apply cancel_br q; rewrite inv_con_cancel_right; exact s
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definition move_bot_of_right' {p : a₂₀ = a} {q : a₂₂ = a} (s : square p₁₀ p₁₂ p₀₁ (p ⬝ q⁻¹))
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: square p₁₀ (p₁₂ ⬝ q) p₀₁ p :=
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by apply cancel_br q⁻¹; rewrite con_inv_cancel_right; exact s
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definition move_right_of_bot {p : a₀₂ = a} {q : a = a₂₂} (s : square p₁₀ (p ⬝ q) p₀₁ p₂₁)
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: square p₁₀ p p₀₁ (p₂₁ ⬝ q⁻¹) :=
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by apply cancel_br q; rewrite inv_con_cancel_right; exact s
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definition move_right_of_bot' {p : a₀₂ = a} {q : a₂₂ = a} (s : square p₁₀ (p ⬝ q⁻¹) p₀₁ p₂₁)
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: square p₁₀ p p₀₁ (p₂₁ ⬝ q) :=
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by apply cancel_br q⁻¹; rewrite con_inv_cancel_right; exact s
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definition move_top_of_right {p : a₂₀ = a} {q : a = a₂₂} (s : square p₁₀ p₁₂ p₀₁ (p ⬝ q))
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: square (p₁₀ ⬝ p) p₁₂ p₀₁ q :=
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by apply cancel_rt p; rewrite con_inv_cancel_right; exact s
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definition move_right_of_top {p : a₀₀ = a} {q : a = a₂₀} (s : square (p ⬝ q) p₁₂ p₀₁ p₂₁)
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: square p p₁₂ p₀₁ (q ⬝ p₂₁) :=
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by apply cancel_tr q; rewrite inv_con_cancel_left; exact s
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definition move_bot_of_left {p : a₀₀ = a} {q : a = a₀₂} (s : square p₁₀ p₁₂ (p ⬝ q) p₂₁)
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: square p₁₀ (q ⬝ p₁₂) p p₂₁ :=
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by apply cancel_lb q; rewrite inv_con_cancel_left; exact s
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definition move_left_of_bot {p : a₀₂ = a} {q : a = a₂₂} (s : square p₁₀ (p ⬝ q) p₀₁ p₂₁)
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: square p₁₀ q (p₀₁ ⬝ p) p₂₁ :=
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by apply cancel_bl p; rewrite con_inv_cancel_right; exact s
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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 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_of_eq_bot (r : p₀₁⁻¹ ⬝ p₁₀ ⬝ p₂₁ = p₁₂) : square p₁₀ p₁₂ p₀₁ p₂₁ :=
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by induction p₂₁; induction p₁₀; esimp at r; induction r; induction p₀₁; exact ids
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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 example below the definition
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-/
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definition hdeg_square_equiv [constructor] (p q : a = a') :
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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') :
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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
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/-
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characterization of pathovers in a equality type. The type B of the equality is fixed here.
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A version where B may also varies over the path p is given in the file squareover
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-/
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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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begin induction p, apply pathover_idp_of_eq, exact eq_of_vdeg_square s end
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definition eq_pathover_constant_left {g : A → B} {p : a = a'} {b : B} {q : b = g a} {r : b = g a'}
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(s : square q r idp (ap g p)) : q =[p] r :=
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eq_pathover (ap_constant p b ⬝ph s)
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definition eq_pathover_id_left {g : A → A} {p : a = a'} {q : a = g a} {r : a' = g a'}
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(s : square q r p (ap g p)) : q =[p] r :=
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eq_pathover (ap_id p ⬝ph s)
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definition eq_pathover_constant_right {f : A → B} {p : a = a'} {b : B} {q : f a = b} {r : f a' = b}
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(s : square q r (ap f p) idp) : q =[p] r :=
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eq_pathover (s ⬝hp (ap_constant p b)⁻¹)
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definition eq_pathover_id_right {f : A → A} {p : a = a'} {q : f a = a} {r : f a' = a'}
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(s : square q r (ap f p) p) : q =[p] r :=
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eq_pathover (s ⬝hp (ap_id p)⁻¹)
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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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definition eq_pathover_constant_left_id_right {p : a = a'} {a₀ : A} {q : a₀ = a} {r : a₀ = a'}
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(s : square q r idp p) : q =[p] r :=
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eq_pathover (ap_constant p a₀ ⬝ph s ⬝hp (ap_id p)⁻¹)
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definition eq_pathover_id_left_constant_right {p : a = a'} {a₀ : A} {q : a = a₀} {r : a' = a₀}
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(s : square q r p idp) : q =[p] r :=
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eq_pathover (ap_id p ⬝ph s ⬝hp (ap_constant p a₀)⁻¹)
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definition loop_pathover {p : a = a'} {q : a = a} {r : a' = a'} (s : square q r p p) : q =[p] r :=
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eq_pathover (ap_id p ⬝ph s ⬝hp (ap_id p)⁻¹)
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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 p₂₁ r ⬝ 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 p₁₂ r)⁻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 :=
|
|
by induction r;reflexivity
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|
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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 :=
|
|
by induction s₁₁; induction r;reflexivity
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definition change_path_eq_pathover {A B : Type} {a a' : A} {f g : A → B}
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|
{p p' : a = a'} (r : p = p')
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{q : f a = g a} {q' : f a' = g a'}
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(s : square q q' (ap f p) (ap g p)) :
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|
change_path r (eq_pathover s) = eq_pathover ((ap02 f r)⁻¹ ⬝ph s ⬝hp (ap02 g r)) :=
|
|
by induction r; reflexivity
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definition eq_hconcat_hdeg_square {A : Type} {a a' : A} {p₁ p₂ p₃ : a = a'} (q₁ : p₁ = p₂)
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(q₂ : p₂ = p₃) : q₁ ⬝ph hdeg_square q₂ = hdeg_square (q₁ ⬝ q₂) :=
|
|
by induction q₁; induction q₂; reflexivity
|
|
|
|
definition hdeg_square_hconcat_eq {A : Type} {a a' : A} {p₁ p₂ p₃ : a = a'} (q₁ : p₁ = p₂)
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(q₂ : p₂ = p₃) : hdeg_square q₁ ⬝hp q₂ = hdeg_square (q₁ ⬝ q₂) :=
|
|
by induction q₁; induction q₂; reflexivity
|
|
|
|
definition eq_hconcat_eq_hdeg_square {A : Type} {a a' : A} {p₁ p₂ p₃ p₄ : a = a'} (q₁ : p₁ = p₂)
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|
(q₂ : p₂ = p₃) (q₃ : p₃ = p₄) : q₁ ⬝ph hdeg_square q₂ ⬝hp q₃ = hdeg_square (q₁ ⬝ q₂ ⬝ q₃) :=
|
|
by induction q₃; apply eq_hconcat_hdeg_square
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|
|
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-- definition vconcat_eq [unfold 11] {p : a₀₂ = a₂₂} (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (r : p₁₂ = p) :
|
|
-- square p₁₀ p p₀₁ p₂₁ :=
|
|
-- 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 :=
|
|
-- by induction r; exact s₁₁
|
|
|
|
-- definition hconcat_eq [unfold 11] {p : a₂₀ = a₂₂}
|
|
-- (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (r : p₂₁ = p) : square p₁₀ p₁₂ p₀₁ p :=
|
|
-- by induction r; exact s₁₁
|
|
|
|
|
|
-- the following definition is very slow, maybe it's interesting to see why?
|
|
-- definition eq_pathover_equiv_square' {f g : A → B}(p : a = a') (q : f a = g a) (r : f a' = g a')
|
|
-- : square q r (ap f p) (ap g p) ≃ q =[p] r :=
|
|
-- equiv.MK eq_pathover
|
|
-- square_of_pathover
|
|
-- (λs, begin
|
|
-- induction p, rewrite [↑[square_of_pathover,eq_pathover],
|
|
-- to_right_inv !vdeg_square_equiv (eq_of_pathover_idp s),
|
|
-- to_left_inv !pathover_idp s]
|
|
-- end)
|
|
-- (λs, begin
|
|
-- induction p, rewrite [↑[square_of_pathover,eq_pathover],▸*,
|
|
-- to_right_inv !(@pathover_idp A) (eq_of_vdeg_square s),
|
|
-- to_left_inv !vdeg_square_equiv s]
|
|
-- end)
|
|
|
|
/- recursors for squares where some sides are reflexivity -/
|
|
|
|
definition rec_on_b [recursor] {a₀₀ : A}
|
|
{P : Π{a₂₀ a₁₂ : A} {t : a₀₀ = a₂₀} {l : a₀₀ = a₁₂} {r : a₂₀ = a₁₂}, square t idp l r → Type}
|
|
{a₂₀ a₁₂ : A} {t : a₀₀ = a₂₀} {l : a₀₀ = a₁₂} {r : a₂₀ = a₁₂}
|
|
(s : square t idp l r) (H : P ids) : P s :=
|
|
have H2 : P (square_of_eq (eq_of_square s)),
|
|
from eq.rec_on (eq_of_square s : t ⬝ r = l) (by induction r; induction t; exact H),
|
|
left_inv (to_fun !square_equiv_eq) s ▸ H2
|
|
|
|
definition rec_on_r [recursor] {a₀₀ : A}
|
|
{P : Π{a₀₂ a₂₁ : A} {t : a₀₀ = a₂₁} {b : a₀₂ = a₂₁} {l : a₀₀ = a₀₂}, square t b l idp → Type}
|
|
{a₀₂ a₂₁ : A} {t : a₀₀ = a₂₁} {b : a₀₂ = a₂₁} {l : a₀₀ = a₀₂}
|
|
(s : square t b l idp) (H : P ids) : P s :=
|
|
let p : l ⬝ b = t := (eq_of_square s)⁻¹ in
|
|
have H2 : P (square_of_eq (eq_of_square s)⁻¹⁻¹),
|
|
from @eq.rec_on _ _ (λx p, P (square_of_eq p⁻¹)) _ p (by induction b; induction l; exact H),
|
|
left_inv (to_fun !square_equiv_eq) s ▸ !inv_inv ▸ H2
|
|
|
|
definition rec_on_l [recursor] {a₀₁ : A}
|
|
{P : Π {a₂₀ a₂₂ : A} {t : a₀₁ = a₂₀} {b : a₀₁ = a₂₂} {r : a₂₀ = a₂₂},
|
|
square t b idp r → Type}
|
|
{a₂₀ a₂₂ : A} {t : a₀₁ = a₂₀} {b : a₀₁ = a₂₂} {r : a₂₀ = a₂₂}
|
|
(s : square t b idp r) (H : P ids) : P s :=
|
|
let p : t ⬝ r = b := eq_of_square s ⬝ !idp_con in
|
|
have H2 : P (square_of_eq (p ⬝ !idp_con⁻¹)),
|
|
from eq.rec_on p (by induction r; induction t; exact H),
|
|
left_inv (to_fun !square_equiv_eq) s ▸ !con_inv_cancel_right ▸ H2
|
|
|
|
definition rec_on_t [recursor] {a₁₀ : A}
|
|
{P : Π {a₀₂ a₂₂ : A} {b : a₀₂ = a₂₂} {l : a₁₀ = a₀₂} {r : a₁₀ = a₂₂}, square idp b l r → Type}
|
|
{a₀₂ a₂₂ : A} {b : a₀₂ = a₂₂} {l : a₁₀ = a₀₂} {r : a₁₀ = a₂₂}
|
|
(s : square idp b l r) (H : P ids) : P s :=
|
|
let p : l ⬝ b = r := (eq_of_square s)⁻¹ ⬝ !idp_con in
|
|
have H2 : P (square_of_eq ((p ⬝ !idp_con⁻¹)⁻¹)),
|
|
from eq.rec_on p (by induction b; induction l; exact H),
|
|
have H3 : P (square_of_eq ((eq_of_square s)⁻¹⁻¹)),
|
|
from eq.rec_on !con_inv_cancel_right H2,
|
|
have H4 : P (square_of_eq (eq_of_square s)),
|
|
from eq.rec_on !inv_inv H3,
|
|
proof
|
|
left_inv (to_fun !square_equiv_eq) s ▸ H4
|
|
qed
|
|
|
|
definition rec_on_tb [recursor] {a : A}
|
|
{P : Π{b : A} {l : a = b} {r : a = b}, square idp idp l r → Type}
|
|
{b : A} {l : a = b} {r : a = b}
|
|
(s : square idp idp l r) (H : P ids) : P s :=
|
|
have H2 : P (square_of_eq (eq_of_square s)),
|
|
from eq.rec_on (eq_of_square s : idp ⬝ r = l) (by induction r; exact H),
|
|
left_inv (to_fun !square_equiv_eq) s ▸ H2
|
|
|
|
definition rec_on_lr [recursor] {a : A}
|
|
{P : Π{a' : A} {t : a = a'} {b : a = a'}, square t b idp idp → Type}
|
|
{a' : A} {t : a = a'} {b : a = a'}
|
|
(s : square t b idp idp) (H : P ids) : P s :=
|
|
let p : idp ⬝ b = t := (eq_of_square s)⁻¹ in
|
|
have H2 : P (square_of_eq (eq_of_square s)⁻¹⁻¹),
|
|
from @eq.rec_on _ _ (λx q, P (square_of_eq q⁻¹)) _ p (by induction b; exact H),
|
|
to_left_inv (!square_equiv_eq) s ▸ !inv_inv ▸ H2
|
|
|
|
--we can also do the other recursors (tl, tr, bl, br, tbl, tbr, tlr, blr), but let's postpone this until they are needed
|
|
|
|
definition whisker_square [unfold 14 15 16 17] (r₁₀ : p₁₀ = p₁₀') (r₁₂ : p₁₂ = p₁₂')
|
|
(r₀₁ : p₀₁ = p₀₁') (r₂₁ : p₂₁ = p₂₁') (s : square p₁₀ p₁₂ p₀₁ p₂₁)
|
|
: square p₁₀' p₁₂' p₀₁' p₂₁' :=
|
|
by induction r₁₀; induction r₁₂; induction r₀₁; induction r₂₁; exact s
|
|
|
|
/- squares commute with some operations on 2-paths -/
|
|
|
|
definition square_inv2 {p₁ p₂ p₃ p₄ : a = a'}
|
|
{t : p₁ = p₂} {b : p₃ = p₄} {l : p₁ = p₃} {r : p₂ = p₄} (s : square t b l r)
|
|
: square (inverse2 t) (inverse2 b) (inverse2 l) (inverse2 r) :=
|
|
by induction s;constructor
|
|
|
|
definition square_con2 {p₁ p₂ p₃ p₄ : a₁ = a₂} {q₁ q₂ q₃ q₄ : a₂ = a₃}
|
|
{t₁ : p₁ = p₂} {b₁ : p₃ = p₄} {l₁ : p₁ = p₃} {r₁ : p₂ = p₄}
|
|
{t₂ : q₁ = q₂} {b₂ : q₃ = q₄} {l₂ : q₁ = q₃} {r₂ : q₂ = q₄}
|
|
(s₁ : square t₁ b₁ l₁ r₁) (s₂ : square t₂ b₂ l₂ r₂)
|
|
: square (t₁ ◾ t₂) (b₁ ◾ b₂) (l₁ ◾ l₂) (r₁ ◾ r₂) :=
|
|
by induction s₂;induction s₁;constructor
|
|
|
|
open is_trunc
|
|
definition is_set.elims [H : is_set A] : square p₁₀ p₁₂ p₀₁ p₂₁ :=
|
|
square_of_eq !is_set.elim
|
|
|
|
definition is_trunc_square [instance] (n : trunc_index) [H : is_trunc n .+2 A]
|
|
: is_trunc n (square p₁₀ p₁₂ p₀₁ p₂₁) :=
|
|
is_trunc_equiv_closed_rev n !square_equiv_eq
|
|
|
|
-- definition square_of_con_inv_hsquare {p₁ p₂ p₃ p₄ : a₁ = a₂}
|
|
-- {t : p₁ = p₂} {b : p₃ = p₄} {l : p₁ = p₃} {r : p₂ = p₄}
|
|
-- (s : square (con_inv_eq_idp t) (con_inv_eq_idp b) (l ◾ r⁻²) idp)
|
|
-- : square t b l r :=
|
|
-- sorry --by induction s
|
|
|
|
/- Square fillers -/
|
|
-- TODO replace by "more algebraic" fillers?
|
|
|
|
variables (p₁₀ p₁₂ p₀₁ p₂₁)
|
|
definition square_fill_t : Σ (p : a₀₀ = a₂₀), square p p₁₂ p₀₁ p₂₁ :=
|
|
by induction p₀₁; induction p₂₁; exact ⟨_, !vrefl⟩
|
|
|
|
definition square_fill_b : Σ (p : a₀₂ = a₂₂), square p₁₀ p p₀₁ p₂₁ :=
|
|
by induction p₀₁; induction p₂₁; exact ⟨_, !vrefl⟩
|
|
|
|
definition square_fill_l : Σ (p : a₀₀ = a₀₂), square p₁₀ p₁₂ p p₂₁ :=
|
|
by induction p₁₀; induction p₁₂; exact ⟨_, !hrefl⟩
|
|
|
|
definition square_fill_r : Σ (p : a₂₀ = a₂₂) , square p₁₀ p₁₂ p₀₁ p :=
|
|
by induction p₁₀; induction p₁₂; exact ⟨_, !hrefl⟩
|
|
|
|
/- Squares having an 'ap' term on one face -/
|
|
--TODO find better names
|
|
definition square_Flr_ap_idp {c : B} {f : A → B} (p : Π a, f a = c)
|
|
{a b : A} (q : a = b) : square (p a) (p b) (ap f q) idp :=
|
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by induction q; apply vrfl
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definition square_Flr_idp_ap {c : B} {f : A → B} (p : Π a, c = f a)
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{a b : A} (q : a = b) : square (p a) (p b) idp (ap f q) :=
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by induction q; apply vrfl
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definition square_ap_idp_Flr {b : B} {f : A → B} (p : Π a, f a = b)
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{a b : A} (q : a = b) : square (ap f q) idp (p a) (p b) :=
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by induction q; apply hrfl
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/- Matching eq_hconcat with hconcat etc. -/
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-- TODO maybe rename hconcat_eq and the like?
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variable (s₁₁)
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definition ph_eq_pv_h_vp {p : a₀₀ = a₀₂} (r : p = p₀₁) :
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r ⬝ph s₁₁ = !idp_con⁻¹ ⬝pv ((hdeg_square r) ⬝h s₁₁) ⬝vp !idp_con :=
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by cases r; cases s₁₁; esimp
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definition hdeg_h_eq_pv_ph_vp {p : a₀₀ = a₀₂} (r : p = p₀₁) :
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hdeg_square r ⬝h s₁₁ = !idp_con ⬝pv (r ⬝ph s₁₁) ⬝vp !idp_con⁻¹ :=
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by cases r; cases s₁₁; esimp
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definition hp_eq_h {p : a₂₀ = a₂₂} (r : p₂₁ = p) :
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s₁₁ ⬝hp r = s₁₁ ⬝h hdeg_square r :=
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by cases r; cases s₁₁; esimp
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definition pv_eq_ph_vdeg_v_vh {p : a₀₀ = a₂₀} (r : p = p₁₀) :
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r ⬝pv s₁₁ = !idp_con⁻¹ ⬝ph ((vdeg_square r) ⬝v s₁₁) ⬝hp !idp_con :=
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by cases r; cases s₁₁; esimp
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definition vdeg_v_eq_ph_pv_hp {p : a₀₀ = a₂₀} (r : p = p₁₀) :
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vdeg_square r ⬝v s₁₁ = !idp_con ⬝ph (r ⬝pv s₁₁) ⬝hp !idp_con⁻¹ :=
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by cases r; cases s₁₁; esimp
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definition vp_eq_v {p : a₀₂ = a₂₂} (r : p₁₂ = p) :
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s₁₁ ⬝vp r = s₁₁ ⬝v vdeg_square r :=
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by cases r; cases s₁₁; esimp
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definition natural_square [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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square_of_pathover (apd p q)
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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 (ap f q) (ap g q) (p a) (p a') :=
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transpose (natural_square p q)
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definition natural_square011 {A A' : Type} {B : A → Type}
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{a a' : A} {p : a = a'} {b : B a} {b' : B a'} (q : b =[p] b')
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{l r : Π⦃a⦄, B a → A'} (g : Π⦃a⦄ (b : B a), l b = r b)
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: square (apd011 l p q) (apd011 r p q) (g b) (g b') :=
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begin
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induction q, exact hrfl
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end
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definition natural_square0111' {A A' : Type} {B : A → Type} (C : Π⦃a⦄, B a → Type)
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{a a' : A} {p : a = a'} {b : B a} {b' : B a'} {q : b =[p] b'}
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{c : C b} {c' : C b'} (s : c =[apd011 C p q] c')
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{l r : Π⦃a⦄ {b : B a}, C b → A'}
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(g : Π⦃a⦄ {b : B a} (c : C b), l c = r c)
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: square (apd0111 l p q s) (apd0111 r p q s) (g c) (g c') :=
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begin
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induction q, esimp at s, induction s using idp_rec_on, exact hrfl
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end
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-- this can be generalized a bit, by making the domain and codomain of k different, and also have
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-- a function at the RHS of s (similar to m)
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definition natural_square0111 {A A' : Type} {B : A → Type} (C : Π⦃a⦄, B a → Type)
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{a a' : A} {p : a = a'} {b : B a} {b' : B a'} {q : b =[p] b'}
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{c : C b} {c' : C b'} (r : c =[apd011 C p q] c')
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{k : A → A} {l : Π⦃a⦄, B a → B (k a)} (m : Π⦃a⦄ {b : B a}, C b → C (l b))
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{f : Π⦃a⦄ {b : B a}, C b → A'}
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(s : Π⦃a⦄ {b : B a} (c : C b), f (m c) = f c)
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: square (apd0111 (λa b c, f (m c)) p q r) (apd0111 f p q r) (s c) (s c') :=
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begin
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induction q, esimp at r, induction r using idp_rec_on, exact hrfl
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end
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end eq
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