47 lines
2.7 KiB
Text
47 lines
2.7 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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Theorems about squareovers
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-/
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import cubical.pathover cubical.square
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open eq equiv is_equiv equiv.ops
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namespace cubical
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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}
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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} /-s₁₁-/ {q₂₁ : b₂₀ =[p₂₁] b₂₂} /-s₃₁-/ {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} /-s₁₃-/ {q₂₃ : b₂₂ =[p₂₃] b₂₄} /-s₃₃-/ {q₄₃ : b₄₂ =[p₄₃] b₄₄}
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/-b₀₄-/ {q₁₄ : b₀₄ =[p₁₄] b₂₄} /-b₂₄-/ {q₃₄ : b₂₄ =[p₃₄] b₄₄} /-b₄₄-/
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inductive squareover (B : A → Type) {b₀₀ : B a₀₀} :
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Π{a₂₀ a₀₂ a₂₂ : A} {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 : b₀₀ =[p₁₀] b₂₀) (q₁₂ : b₀₂ =[p₁₂] b₂₂) (q₀₁ : b₀₀ =[p₀₁] b₀₂) (q₂₁ : b₂₀ =[p₂₁] b₂₂),
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Type :=
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idsquareo : squareover B ids idpo idpo idpo idpo
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definition squareo := @squareover A a₀₀ B
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definition idsquareo [reducible] [constructor] (b₀₀ : B a₀₀) := @squareover.idsquareo A a₀₀ B b₀₀
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definition idso [reducible] [constructor] := @squareover.idsquareo A a₀₀ B b₀₀
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end cubical
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