lean2/hott/cubical/cube.hlean

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/-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Floris van Doorn
Cubes
-/
import .square
open equiv is_equiv
namespace eq
inductive cube {A : Type} {a₀₀₀ : A}
: Π{a₂₀₀ a₀₂₀ a₂₂₀ a₀₀₂ a₂₀₂ a₀₂₂ a₂₂₂ : A}
{p₁₀₀ : a₀₀₀ = a₂₀₀} {p₀₁₀ : a₀₀₀ = a₀₂₀} {p₀₀₁ : a₀₀₀ = a₀₀₂}
{p₁₂₀ : a₀₂₀ = a₂₂₀} {p₂₁₀ : a₂₀₀ = a₂₂₀} {p₂₀₁ : a₂₀₀ = a₂₀₂}
{p₁₀₂ : a₀₀₂ = a₂₀₂} {p₀₁₂ : a₀₀₂ = a₀₂₂} {p₀₂₁ : a₀₂₀ = a₀₂₂}
{p₁₂₂ : a₀₂₂ = a₂₂₂} {p₂₁₂ : a₂₀₂ = a₂₂₂} {p₂₂₁ : a₂₂₀ = a₂₂₂}
(s₁₁₀ : square p₀₁₀ p₂₁₀ p₁₀₀ p₁₂₀)
(s₁₁₂ : square p₀₁₂ p₂₁₂ p₁₀₂ p₁₂₂)
(s₀₁₁ : square p₀₁₀ p₀₁₂ p₀₀₁ p₀₂₁)
(s₂₁₁ : square p₂₁₀ p₂₁₂ p₂₀₁ p₂₂₁)
(s₁₀₁ : square p₁₀₀ p₁₀₂ p₀₀₁ p₂₀₁)
(s₁₂₁ : square p₁₂₀ p₁₂₂ p₀₂₁ p₂₂₁), Type :=
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idc : cube ids ids ids ids ids ids
variables {A B : Type} {a₀₀₀ a₂₀₀ a₀₂₀ a₂₂₀ a₀₀₂ a₂₀₂ a₀₂₂ a₂₂₂ a a' : A}
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{p₁₀₀ : a₀₀₀ = a₂₀₀} {p₀₁₀ : a₀₀₀ = a₀₂₀} {p₀₀₁ : a₀₀₀ = a₀₀₂}
{p₁₂₀ : a₀₂₀ = a₂₂₀} {p₂₁₀ : a₂₀₀ = a₂₂₀} {p₂₀₁ : a₂₀₀ = a₂₀₂}
{p₁₀₂ : a₀₀₂ = a₂₀₂} {p₀₁₂ : a₀₀₂ = a₀₂₂} {p₀₂₁ : a₀₂₀ = a₀₂₂}
{p₁₂₂ : a₀₂₂ = a₂₂₂} {p₂₁₂ : a₂₀₂ = a₂₂₂} {p₂₂₁ : a₂₂₀ = a₂₂₂}
{s₁₁₀ : square p₀₁₀ p₂₁₀ p₁₀₀ p₁₂₀}
{s₁₁₂ : square p₀₁₂ p₂₁₂ p₁₀₂ p₁₂₂}
{s₀₁₁ : square p₀₁₀ p₀₁₂ p₀₀₁ p₀₂₁}
{s₂₁₁ : square p₂₁₀ p₂₁₂ p₂₀₁ p₂₂₁}
{s₁₀₁ : square p₁₀₀ p₁₀₂ p₀₀₁ p₂₀₁}
{s₁₂₁ : square p₁₂₀ p₁₂₂ p₀₂₁ p₂₂₁}
{b₁ b₂ b₃ b₄ : B}
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definition idc [reducible] [constructor] := @cube.idc
definition idcube [reducible] [constructor] (a : A) := @cube.idc A a
definition rfl1 : cube s₁₁₀ s₁₁₀ vrfl vrfl vrfl vrfl := by induction s₁₁₀; exact idc
definition rfl2 : cube vrfl vrfl s₁₁₀ s₁₁₀ hrfl hrfl := by induction s₁₁₀; exact idc
definition rfl3 : cube hrfl hrfl hrfl hrfl s₁₀₁ s₁₀₁ := by induction s₁₀₁; exact idc
definition eq_of_cube (c : cube s₁₁₀ s₁₁₂ s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁) :
transpose s₁₀₁⁻¹ᵛ ⬝h s₁₁₀ ⬝h transpose s₁₂₁ =
whisker_square (eq_bot_of_square s₀₁₁) (eq_bot_of_square s₂₁₁) idp idp s₁₁₂ :=
by induction c; reflexivity
--set_option pp.implicit true
definition eq_of_vdeg_cube {s s' : square p₀₁₀ p₂₁₀ p₁₀₀ p₁₂₀}
(c : cube s s' vrfl vrfl vrfl vrfl) : s = s' :=
begin
induction s, exact eq_of_cube c
end
definition square_pathover [unfold 7]
{f₁ : A → b₁ = b₂} {f₂ : A → b₃ = b₄} {f₃ : A → b₁ = b₃} {f₄ : A → b₂ = b₄}
{p : a = a'}
{q : square (f₁ a) (f₂ a) (f₃ a) (f₄ a)} {r : square (f₁ a') (f₂ a') (f₃ a') (f₄ a')}
(s : cube q r (vdeg_square (ap f₁ p)) (vdeg_square (ap f₂ p))
(vdeg_square (ap f₃ p)) (vdeg_square (ap f₄ p))) : q =[p] r :=
by induction p;apply pathover_idp_of_eq;exact eq_of_vdeg_cube s
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end eq