44 lines
2.7 KiB
Text
44 lines
2.7 KiB
Text
/-
|
|
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
|
|
|
|
Theorems about squareovers
|
|
-/
|
|
|
|
import types.square
|
|
|
|
open eq equiv is_equiv equiv.ops
|
|
|
|
namespace cubical
|
|
|
|
variables {A A' : Type} {B : A → Type}
|
|
{a a' a'' a₀₀ a₂₀ a₄₀ a₀₂ a₂₂ a₂₄ a₀₄ a₄₂ a₄₄ : A}
|
|
/-a₀₀-/ {p₁₀ : a₀₀ = a₂₀} /-a₂₀-/ {p₃₀ : a₂₀ = a₄₀} /-a₄₀-/
|
|
{p₀₁ : a₀₀ = a₀₂} /-s₁₁-/ {p₂₁ : a₂₀ = a₂₂} /-s₃₁-/ {p₄₁ : a₄₀ = a₄₂}
|
|
/-a₀₂-/ {p₁₂ : a₀₂ = a₂₂} /-a₂₂-/ {p₃₂ : a₂₂ = a₄₂} /-a₄₂-/
|
|
{p₀₃ : a₀₂ = a₀₄} /-s₁₃-/ {p₂₃ : a₂₂ = a₂₄} /-s₃₃-/ {p₄₃ : a₄₂ = a₄₄}
|
|
/-a₀₄-/ {p₁₄ : a₀₄ = a₂₄} /-a₂₄-/ {p₃₄ : a₂₄ = a₄₄} /-a₄₄-/
|
|
{s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁}
|
|
{b₀₀ : B a₀₀} {b₂₀ : B a₂₀} {b₄₀ : B a₄₀}
|
|
{b₀₂ : B a₀₂} {b₂₂ : B a₂₂} {b₄₂ : B a₄₂}
|
|
{b₀₄ : B a₀₄} {b₂₄ : B a₂₄} {b₄₄ : B a₄₄}
|
|
/-b₀₀-/ {q₁₀ : b₀₀ =[p₁₀] b₂₀} /-b₂₀-/ {q₃₀ : b₂₀ =[p₃₀] b₄₀} /-b₄₀-/
|
|
{q₀₁ : b₀₀ =[p₀₁] b₀₂} /-s₁₁-/ {q₂₁ : b₂₀ =[p₂₁] b₂₂} /-s₃₁-/ {q₄₁ : b₄₀ =[p₄₁] b₄₂}
|
|
/-b₀₂-/ {q₁₂ : b₀₂ =[p₁₂] b₂₂} /-b₂₂-/ {q₃₂ : b₂₂ =[p₃₂] b₄₂} /-b₄₂-/
|
|
{q₀₃ : b₀₂ =[p₀₃] b₀₄} /-s₁₃-/ {q₂₃ : b₂₂ =[p₂₃] b₂₄} /-s₃₃-/ {q₄₃ : b₄₂ =[p₄₃] b₄₄}
|
|
/-b₀₄-/ {q₁₄ : b₀₄ =[p₁₄] b₂₄} /-b₂₄-/ {q₃₄ : b₂₄ =[p₃₄] b₄₄} /-b₄₄-/
|
|
|
|
inductive squareover (B : A → Type) {b₀₀ : B a₀₀} :
|
|
Π{a₂₀ a₀₂ a₂₂ : A} {p₁₀ : a₀₀ = a₂₀} {p₁₂ : a₀₂ = a₂₂} {p₀₁ : a₀₀ = a₀₂} {p₂₁ : a₂₀ = a₂₂}
|
|
(s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
|
|
{b₂₀ : B a₂₀} {b₀₂ : B a₀₂} {b₂₂ : B a₂₂}
|
|
(q₁₀ : b₀₀ =[p₁₀] b₂₀) (q₁₂ : b₀₂ =[p₁₂] b₂₂) (q₀₁ : b₀₀ =[p₀₁] b₀₂) (q₂₁ : b₂₀ =[p₂₁] b₂₂),
|
|
Type :=
|
|
idsquareo : squareover B ids idpo idpo idpo idpo
|
|
|
|
definition squareo := @squareover A a₀₀ B
|
|
definition idsquareo [reducible] [constructor] (b₀₀ : B a₀₀) := @squareover.idsquareo A a₀₀ B b₀₀
|
|
definition idso [reducible] [constructor] := @squareover.idsquareo A a₀₀ B b₀₀
|
|
|
|
end cubical
|