307 lines
17 KiB
Text
307 lines
17 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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Squareovers
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-/
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import .square
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open eq equiv is_equiv sigma
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namespace eq
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-- we give the argument B explicitly, because Lean would find (λa, B a) by itself, which
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-- makes the type uglier (of course the two terms are definitionally equal)
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inductive squareover {A : Type} (B : A → Type) {a₀₀ : A} {b₀₀ : B a₀₀} :
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Π{a₂₀ a₀₂ a₂₂ : A}
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{p₁₀ : a₀₀ = a₂₀} {p₁₂ : a₀₂ = a₂₂} {p₀₁ : a₀₀ = a₀₂} {p₂₁ : a₂₀ = a₂₂}
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(s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
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{b₂₀ : B a₂₀} {b₀₂ : B a₀₂} {b₂₂ : B a₂₂}
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(q₁₀ : pathover B b₀₀ p₁₀ b₂₀) (q₁₂ : pathover B b₀₂ p₁₂ b₂₂)
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(q₀₁ : pathover B b₀₀ p₀₁ b₀₂) (q₂₁ : pathover B b₂₀ p₂₁ b₂₂),
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Type :=
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idsquareo : squareover B ids idpo idpo idpo idpo
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variables {A A' : Type} {B : A → Type}
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{a a' a'' a₀₀ a₂₀ a₄₀ a₀₂ a₂₂ a₂₄ a₀₄ a₄₂ a₄₄ : A}
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/-a₀₀-/ {p₁₀ : a₀₀ = a₂₀} /-a₂₀-/ {p₃₀ : a₂₀ = a₄₀} /-a₄₀-/
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{p₀₁ : a₀₀ = a₀₂} /-s₁₁-/ {p₂₁ : a₂₀ = a₂₂} /-s₃₁-/ {p₄₁ : a₄₀ = a₄₂}
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/-a₀₂-/ {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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{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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{b₀₀ : B a₀₀} {b₂₀ : B a₂₀} {b₄₀ : B a₄₀}
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{b₀₂ : B a₀₂} {b₂₂ : B a₂₂} {b₄₂ : B a₄₂}
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{b₀₄ : B a₀₄} {b₂₄ : B a₂₄} {b₄₄ : B a₄₄}
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/-b₀₀-/ {q₁₀ : b₀₀ =[p₁₀] b₂₀} /-b₂₀-/ {q₃₀ : b₂₀ =[p₃₀] b₄₀} /-b₄₀-/
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{q₀₁ : b₀₀ =[p₀₁] b₀₂} /-t₁₁-/ {q₂₁ : b₂₀ =[p₂₁] b₂₂} /-t₃₁-/ {q₄₁ : b₄₀ =[p₄₁] b₄₂}
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/-b₀₂-/ {q₁₂ : b₀₂ =[p₁₂] b₂₂} /-b₂₂-/ {q₃₂ : b₂₂ =[p₃₂] b₄₂} /-b₄₂-/
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{q₀₃ : b₀₂ =[p₀₃] b₀₄} /-t₁₃-/ {q₂₃ : b₂₂ =[p₂₃] b₂₄} /-t₃₃-/ {q₄₃ : b₄₂ =[p₄₃] b₄₄}
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/-b₀₄-/ {q₁₄ : b₀₄ =[p₁₄] b₂₄} /-b₂₄-/ {q₃₄ : b₂₄ =[p₃₄] b₄₄} /-b₄₄-/
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definition squareo := @squareover A B a₀₀
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definition idsquareo [reducible] [constructor] (b₀₀ : B a₀₀) := @squareover.idsquareo A B a₀₀ b₀₀
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definition idso [reducible] [constructor] := @squareover.idsquareo A B a₀₀ b₀₀
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definition apds (f : Πa, B a) (s : square p₁₀ p₁₂ p₀₁ p₂₁)
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: squareover B s (apd f p₁₀) (apd f p₁₂) (apd f p₀₁) (apd f p₂₁) :=
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square.rec_on s idso
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definition vrflo : squareover B vrfl q₁₀ q₁₀ idpo idpo :=
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by induction q₁₀; exact idso
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definition hrflo : squareover B hrfl idpo idpo q₁₀ q₁₀ :=
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by induction q₁₀; exact idso
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definition vdeg_squareover {p₁₀'} {s : p₁₀ = p₁₀'} {q₁₀' : b₀₀ =[p₁₀'] b₂₀}
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(r : change_path s q₁₀ = q₁₀')
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: squareover B (vdeg_square s) q₁₀ q₁₀' idpo idpo :=
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by induction s; esimp at *; induction r; exact vrflo
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definition hdeg_squareover {p₀₁'} {s : p₀₁ = p₀₁'} {q₀₁' : b₀₀ =[p₀₁'] b₀₂}
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(r : change_path s q₀₁ = q₀₁')
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: squareover B (hdeg_square s) idpo idpo q₀₁ q₀₁' :=
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by induction s; esimp at *; induction r; exact hrflo
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definition hconcato
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(t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁) (t₃₁ : squareover B s₃₁ q₃₀ q₃₂ q₂₁ q₄₁)
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: squareover B (hconcat s₁₁ s₃₁) (q₁₀ ⬝o q₃₀) (q₁₂ ⬝o q₃₂) q₀₁ q₄₁ :=
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by induction t₃₁; exact t₁₁
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definition vconcato
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(t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁) (t₁₃ : squareover B s₁₃ q₁₂ q₁₄ q₀₃ q₂₃)
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: squareover B (vconcat s₁₁ s₁₃) q₁₀ q₁₄ (q₀₁ ⬝o q₀₃) (q₂₁ ⬝o q₂₃) :=
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by induction t₁₃; exact t₁₁
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definition hinverseo (t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁)
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: squareover B (hinverse s₁₁) q₁₀⁻¹ᵒ q₁₂⁻¹ᵒ q₂₁ q₀₁ :=
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by induction t₁₁; constructor
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definition vinverseo (t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁)
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: squareover B (vinverse s₁₁) q₁₂ q₁₀ q₀₁⁻¹ᵒ q₂₁⁻¹ᵒ :=
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by induction t₁₁; constructor
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definition eq_vconcato {q : b₀₀ =[p₁₀] b₂₀}
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(r : q = q₁₀) (t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁) : squareover B s₁₁ q q₁₂ q₀₁ q₂₁ :=
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by induction r; exact t₁₁
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definition vconcato_eq {q : b₀₂ =[p₁₂] b₂₂}
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(t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁) (r : q₁₂ = q) : squareover B s₁₁ q₁₀ q q₀₁ q₂₁ :=
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by induction r; exact t₁₁
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definition eq_hconcato {q : b₀₀ =[p₀₁] b₀₂}
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(r : q = q₀₁) (t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁) : squareover B s₁₁ q₁₀ q₁₂ q q₂₁ :=
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by induction r; exact t₁₁
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definition hconcato_eq {q : b₂₀ =[p₂₁] b₂₂}
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(t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁) (r : q₂₁ = q) : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q :=
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by induction r; exact t₁₁
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definition pathover_vconcato {p : a₀₀ = a₂₀} {sp : p = p₁₀} {q : b₀₀ =[p] b₂₀}
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(r : change_path sp q = q₁₀) (t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁)
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: squareover B (sp ⬝pv s₁₁) q q₁₂ q₀₁ q₂₁ :=
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by induction sp; induction r; exact t₁₁
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definition vconcato_pathover {p : a₀₂ = a₂₂} {sp : p₁₂ = p} {q : b₀₂ =[p] b₂₂}
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(t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁) (r : change_path sp q₁₂ = q)
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: squareover B (s₁₁ ⬝vp sp) q₁₀ q q₀₁ q₂₁ :=
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by induction sp; induction r; exact t₁₁
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definition pathover_hconcato {p : a₀₀ = a₀₂} {sp : p = p₀₁} {q : b₀₀ =[p] b₀₂}
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(r : change_path sp q = q₀₁) (t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁) :
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squareover B (sp ⬝ph s₁₁) q₁₀ q₁₂ q q₂₁ :=
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by induction sp; induction r; exact t₁₁
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definition hconcato_pathover {p : a₂₀ = a₂₂} {sp : p₂₁ = p} {q : b₂₀ =[p] b₂₂}
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(t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁) (r : change_path sp q₂₁ = q) :
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squareover B (s₁₁ ⬝hp sp) q₁₀ q₁₂ q₀₁ q :=
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by induction sp; induction r; exact t₁₁
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infix ` ⬝ho `:69 := hconcato --type using \tr
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infix ` ⬝vo `:70 := vconcato --type using \tr
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infix ` ⬝hop `:72 := hconcato_eq --type using \tr
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infix ` ⬝vop `:74 := vconcato_eq --type using \tr
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infix ` ⬝pho `:71 := eq_hconcato --type using \tr
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infix ` ⬝pvo `:73 := eq_vconcato --type using \tr
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-- relating squareovers to squares
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definition square_of_squareover (t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁) :
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square (!con_tr ⬝ ap (λa, p₂₁ ▸ a) (tr_eq_of_pathover q₁₀))
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(tr_eq_of_pathover q₁₂)
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(ap (λq, q ▸ b₀₀) (eq_of_square s₁₁) ⬝ !con_tr ⬝ ap (λa, p₁₂ ▸ a) (tr_eq_of_pathover q₀₁))
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(tr_eq_of_pathover q₂₁) :=
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by induction t₁₁; esimp; constructor
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/-
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definition squareover_of_square
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(q : square (!con_tr ⬝ ap (λa, p₂₁ ▸ a) (tr_eq_of_pathover q₁₀))
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(tr_eq_of_pathover q₁₂)
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(ap (λq, q ▸ b₀₀) (eq_of_square s₁₁) ⬝ !con_tr ⬝ ap (λa, p₁₂ ▸ a) (tr_eq_of_pathover q₀₁))
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(tr_eq_of_pathover q₂₁))
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: squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁ :=
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sorry
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-/
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definition square_of_squareover_ids {b₀₀ b₀₂ b₂₀ b₂₂ : B a}
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{t : b₀₀ = b₂₀} {b : b₀₂ = b₂₂} {l : b₀₀ = b₀₂} {r : b₂₀ = b₂₂}
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(so : squareover B ids (pathover_idp_of_eq t)
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(pathover_idp_of_eq b)
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(pathover_idp_of_eq l)
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(pathover_idp_of_eq r)) : square t b l r :=
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begin
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note H := square_of_squareover so, -- use apply ... in
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rewrite [▸* at H,+idp_con at H,+ap_id at H,↑pathover_idp_of_eq at H],
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rewrite [+to_right_inv !(pathover_equiv_tr_eq (refl a)) at H],
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exact H
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end
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definition squareover_ids_of_square {b₀₀ b₀₂ b₂₀ b₂₂ : B a}
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{t : b₀₀ = b₂₀} {b : b₀₂ = b₂₂} {l : b₀₀ = b₀₂} {r : b₂₀ = b₂₂} (q : square t b l r)
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: squareover B ids (pathover_idp_of_eq t)
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(pathover_idp_of_eq b)
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(pathover_idp_of_eq l)
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(pathover_idp_of_eq r) :=
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square.rec_on q idso
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-- relating pathovers to squareovers
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definition pathover_of_squareover' (t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁)
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: q₁₀ ⬝o q₂₁ =[eq_of_square s₁₁] q₀₁ ⬝o q₁₂ :=
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by induction t₁₁; constructor
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definition pathover_of_squareover {s : p₁₀ ⬝ p₂₁ = p₀₁ ⬝ p₁₂}
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(t₁₁ : squareover B (square_of_eq s) q₁₀ q₁₂ q₀₁ q₂₁)
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: q₁₀ ⬝o q₂₁ =[s] q₀₁ ⬝o q₁₂ :=
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begin
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revert s t₁₁, refine equiv_rect' !square_equiv_eq⁻¹ᵉ (λa b, squareover B b _ _ _ _ → _) _,
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intro s, exact pathover_of_squareover'
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end
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definition squareover_of_pathover {s : p₁₀ ⬝ p₂₁ = p₀₁ ⬝ p₁₂}
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(r : q₁₀ ⬝o q₂₁ =[s] q₀₁ ⬝o q₁₂) : squareover B (square_of_eq s) q₁₀ q₁₂ q₀₁ q₂₁ :=
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by induction q₁₂; esimp [concato] at r;induction r;induction q₂₁;induction q₁₀;constructor
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definition pathover_top_of_squareover (t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁)
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: q₁₀ =[eq_top_of_square s₁₁] q₀₁ ⬝o q₁₂ ⬝o q₂₁⁻¹ᵒ :=
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by induction t₁₁; constructor
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definition squareover_of_pathover_top {s : p₁₀ = p₀₁ ⬝ p₁₂ ⬝ p₂₁⁻¹}
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(r : q₁₀ =[s] q₀₁ ⬝o q₁₂ ⬝o q₂₁⁻¹ᵒ)
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: squareover B (square_of_eq_top s) q₁₀ q₁₂ q₀₁ q₂₁ :=
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by induction q₂₁; induction q₁₂; esimp at r;induction r;induction q₁₀;constructor
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definition pathover_of_hdeg_squareover {p₀₁' : a₀₀ = a₀₂} {r : p₀₁ = p₀₁'} {q₀₁' : b₀₀ =[p₀₁'] b₀₂}
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(t : squareover B (hdeg_square r) idpo idpo q₀₁ q₀₁') : q₀₁ =[r] q₀₁' :=
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by induction r; induction q₀₁'; exact (pathover_of_squareover' t)⁻¹ᵒ
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definition pathover_of_vdeg_squareover {p₁₀' : a₀₀ = a₂₀} {r : p₁₀ = p₁₀'} {q₁₀' : b₀₀ =[p₁₀'] b₂₀}
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(t : squareover B (vdeg_square r) q₁₀ q₁₀' idpo idpo) : q₁₀ =[r] q₁₀' :=
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by induction r; induction q₁₀'; exact pathover_of_squareover' t
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definition squareover_of_eq_top (r : change_path (eq_top_of_square s₁₁) q₁₀ = q₀₁ ⬝o q₁₂ ⬝o q₂₁⁻¹ᵒ)
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: squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁ :=
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begin
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induction s₁₁, revert q₁₂ q₁₀ r,
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eapply idp_rec_on q₂₁, clear q₂₁,
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intro q₁₂,
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eapply idp_rec_on q₁₂, clear q₁₂,
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esimp, intros,
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induction r, eapply idp_rec_on q₁₀,
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constructor
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end
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definition eq_top_of_squareover (r : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁)
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: change_path (eq_top_of_square s₁₁) q₁₀ = q₀₁ ⬝o q₁₂ ⬝o q₂₁⁻¹ᵒ :=
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by induction r; reflexivity
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definition change_square {s₁₁'} (p : s₁₁ = s₁₁') (r : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁)
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: squareover B s₁₁' q₁₀ q₁₂ q₀₁ q₂₁ :=
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p ▸ r
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/-
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definition squareover_equiv_pathover (q₁₀ : b₀₀ =[p₁₀] b₂₀) (q₁₂ : b₀₂ =[p₁₂] b₂₂)
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(q₀₁ : b₀₀ =[p₀₁] b₀₂) (q₂₁ : b₂₀ =[p₂₁] b₂₂)
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: squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁ ≃ q₁₀ ⬝o q₂₁ =[eq_of_square s₁₁] q₀₁ ⬝o q₁₂ :=
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begin
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fapply equiv.MK,
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{ exact pathover_of_squareover},
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{ intro r, rewrite [-to_left_inv !square_equiv_eq s₁₁], apply squareover_of_pathover, exact r},
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{ intro r, }, --need characterization of squareover lying over ids.
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{ intro s, induction s, apply idp},
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end
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-/
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definition eq_of_vdeg_squareover {q₁₀' : b₀₀ =[p₁₀] b₂₀}
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(p : squareover B vrfl q₁₀ q₁₀' idpo idpo) : q₁₀ = q₁₀' :=
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begin
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note H := square_of_squareover p, -- use apply ... in
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induction p₁₀, -- if needed we can remove this induction and use con_tr_idp in types/eq2
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rewrite [▸* at H,idp_con at H,+ap_id at H],
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let H' := eq_of_vdeg_square H,
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exact eq_of_fn_eq_fn !pathover_equiv_tr_eq H'
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end
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-- definition vdeg_tr_squareover {q₁₂ : p₀₁ ▸ b₀₀ =[p₁₂] p₂₁ ▸ b₂₀} (r : q₁₀ =[_] q₁₂)
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-- : squareover B s₁₁ q₁₀ q₁₂ !pathover_tr !pathover_tr :=
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-- by induction p;exact vrflo
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/- A version of eq_pathover where the type of the equality also varies -/
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definition eq_pathover_dep {f g : Πa, B a} {p : a = a'} {q : f a = g a}
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{r : f a' = g a'} (s : squareover B hrfl (pathover_idp_of_eq q) (pathover_idp_of_eq r)
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(apd f p) (apd g p)) : q =[p] r :=
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begin
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induction p, apply pathover_idp_of_eq, apply eq_of_vdeg_square, exact square_of_squareover_ids s
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end
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/- charcaterization of pathovers in pathovers -/
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-- in this version the fibration (B) of the pathover does not depend on the variable (a)
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definition pathover_pathover {a' a₂' : A'} {p : a' = a₂'} {f g : A' → A}
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{b : Πa, B (f a)} {b₂ : Πa, B (g a)} {q : Π(a' : A'), f a' = g a'}
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(r : pathover B (b a') (q a') (b₂ a'))
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(r₂ : pathover B (b a₂') (q a₂') (b₂ a₂'))
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(s : squareover B (natural_square q p) r r₂
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(pathover_ap B f (apd b p)) (pathover_ap B g (apd b₂ p)))
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: pathover (λa, pathover B (b a) (q a) (b₂ a)) r p r₂ :=
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begin
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induction p, esimp at s, apply pathover_idp_of_eq, apply eq_of_vdeg_squareover, exact s
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end
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definition squareover_change_path_left {p₀₁' : a₀₀ = a₀₂} (r : p₀₁' = p₀₁)
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{q₀₁ : b₀₀ =[p₀₁'] b₀₂} (t : squareover B (r ⬝ph s₁₁) q₁₀ q₁₂ q₀₁ q₂₁)
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: squareover B s₁₁ q₁₀ q₁₂ (change_path r q₀₁) q₂₁ :=
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by induction r; exact t
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definition squareover_change_path_right {p₂₁' : a₂₀ = a₂₂} (r : p₂₁' = p₂₁)
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{q₂₁ : b₂₀ =[p₂₁'] b₂₂} (t : squareover B (s₁₁ ⬝hp r⁻¹) q₁₀ q₁₂ q₀₁ q₂₁)
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: squareover B s₁₁ q₁₀ q₁₂ q₀₁ (change_path r q₂₁) :=
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by induction r; exact t
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definition squareover_change_path_right' {p₂₁' : a₂₀ = a₂₂} (r : p₂₁ = p₂₁')
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{q₂₁ : b₂₀ =[p₂₁'] b₂₂} (t : squareover B (s₁₁ ⬝hp r) q₁₀ q₁₂ q₀₁ q₂₁)
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: squareover B s₁₁ q₁₀ q₁₂ q₀₁ (change_path r⁻¹ q₂₁) :=
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by induction r; exact t
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/- You can construct a square in a sigma-type by giving a squareover -/
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definition square_dpair_eq_dpair {a₀₀ a₂₀ a₀₂ a₂₂ : A}
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{p₁₀ : a₀₀ = a₂₀} {p₀₁ : a₀₀ = a₀₂} {p₂₁ : a₂₀ = a₂₂} {p₁₂ : a₀₂ = a₂₂}
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(s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) {b₀₀ : B a₀₀} {b₂₀ : B a₂₀} {b₀₂ : B a₀₂} {b₂₂ : B a₂₂}
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{q₁₀ : b₀₀ =[p₁₀] b₂₀} {q₀₁ : b₀₀ =[p₀₁] b₀₂} {q₂₁ : b₂₀ =[p₂₁] b₂₂} {q₁₂ : b₀₂ =[p₁₂] b₂₂}
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(t₁₁ : squareover B s₁₁ q₁₀ q₁₂ q₀₁ q₂₁) :
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square (dpair_eq_dpair p₁₀ q₁₀) (dpair_eq_dpair p₁₂ q₁₂)
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(dpair_eq_dpair p₀₁ q₀₁) (dpair_eq_dpair p₂₁ q₂₁) :=
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by induction t₁₁; constructor
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definition sigma_square {v₀₀ v₂₀ v₀₂ v₂₂ : Σa, B a}
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{p₁₀ : v₀₀ = v₂₀} {p₀₁ : v₀₀ = v₀₂} {p₂₁ : v₂₀ = v₂₂} {p₁₂ : v₀₂ = v₂₂}
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(s₁₁ : square p₁₀..1 p₁₂..1 p₀₁..1 p₂₁..1)
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(t₁₁ : squareover B s₁₁ p₁₀..2 p₁₂..2 p₀₁..2 p₂₁..2) : square p₁₀ p₁₂ p₀₁ p₂₁ :=
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begin
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induction v₀₀, induction v₂₀, induction v₀₂, induction v₂₂,
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rewrite [▸* at *, -sigma_eq_eta p₁₀, -sigma_eq_eta p₁₂, -sigma_eq_eta p₀₁, -sigma_eq_eta p₂₁],
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exact square_dpair_eq_dpair s₁₁ t₁₁
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end
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end eq
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