statement stringlengths 5 10.7k | proof stringlengths 0 9.25k | type stringclasses 4
values | symbolic_name stringlengths 1 80 | library stringclasses 103
values | filename stringclasses 746
values | imports listlengths 0 49 | deps listlengths 0 64 | docstring stringclasses 1
value | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
eqM' : l' R' ＝ r' L' | eqM' = refl | function | eqM' | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"l'",
"r'",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eqR' : R' ＝ r' R' | eqR' = refl | function | eqR' | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"r'",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eql' : (i : I) → L' ＝ eqL' i | eql' i = refl | function | eql' | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"eqL'",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eqc' : (i : I) → l' R' ＝ eqM' i | eqc' i = refl | function | eqc' | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"eqM'",
"l'",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eqr' : (i : I) → R' ＝ eqR' i | eqr' i = refl | function | eqr' | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"eqR'",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
φ : 𝔹 → 𝔹' | φ L = L'
φ R = R'
φ (l x) = l' (φ x)
φ (r x) = r' (φ x)
φ (eqL i) = eqL' i
φ (eqM i) = eqM' i
φ (eqR i) = eqR' i | function | φ | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"eqL'",
"eqM'",
"eqR'",
"l'",
"r'",
"𝔹'"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
γ : 𝔹' → 𝔹 | γ L' = L
γ R' = R
γ (η middle) = M
γ (η (left y)) = l (γ (η y))
γ (η (right y)) = r (γ (η y)) | function | γ | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"left",
"middle",
"right",
"𝔹'"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
φγ : (y : 𝔹') → φ (γ y) ＝ y | φγ L' = refl
φγ R' = refl
φγ (η y) = δ y
where
δ : (y : 𝔻) → φ (γ (η y)) ＝ η y
δ middle = refl
δ (left y) = cong l' (δ y)
δ (right y) = cong r' (δ y) | function | φγ | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"l'",
"left",
"middle",
"r'",
"refl",
"right",
"𝔹'"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
square-l : (y : 𝔹') → γ (l' y) ＝ l (γ y) | square-l L' = eqL
square-l R' = refl
square-l (η x) = refl | function | square-l | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"l'",
"refl",
"𝔹'"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
square-r : (y : 𝔹') → γ (r' y) ＝ r (γ y) | square-r L' = eqM
square-r R' = eqR
square-r (η x) = refl | function | square-r | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"r'",
"refl",
"𝔹'"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
path-construction : {X : Type ℓ}
(x y : X)
(p : x ＝ y)
→ PathP (λ i → x ＝ p i) (𝑟𝒆𝑓𝑙 x ∙ 𝑟𝒆𝑓𝑙 x) (p ∙ 𝑟𝒆𝑓𝑙 y) | path-construction x y p i j = hcomp (λ k → λ { (j = i0) → x
; (j = i1) → p i})
(p (i ∧ j)) | function | path-construction | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"Type",
"𝑟𝒆𝑓𝑙"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
fixed-point-construction : {X : Type ℓ}
(x : X)
(f : X → X)
(p : x ＝ f x)
→ PathP (λ i → x ＝ p i) (𝑟𝒆𝑓𝑙 x) (p ∙ 𝑟𝒆𝑓𝑙 (f x)) | fixed-point-construction x f p i j = hcomp (λ k → λ { (i = i0) → x
; (j = i0) → x
; (j = i1) → p i})
(p (i ∧ j)) | function | fixed-point-construction | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"Type",
"𝑟𝒆𝑓𝑙"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eql : PathP (λ i → L ＝ eqL i) refl (eqL ∙ refl) | eql = fixed-point-construction L l eqL | function | eql | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"fixed-point-construction",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eqc : PathP (λ i → M ＝ eqM i) (refl ∙ refl) (eqM ∙ refl) | eqc = path-construction M (r L) eqM | function | eqc | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"path-construction",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eqr : PathP (λ i → R ＝ eqR i) refl (eqR ∙ refl) | eqr = fixed-point-construction R r eqR | function | eqr | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"fixed-point-construction",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
γφ : (x : 𝔹) → γ (φ x) ＝ x | γφ L = refl
γφ R = refl
γφ (l x) = square-l (φ x) ∙ cong l (γφ x)
γφ (r x) = square-r (φ x) ∙ cong r (γφ x)
γφ (eqL i) = eql i
γφ (eqM i) = eqc i
γφ (eqR i) = eqr i | function | γφ | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"eqc",
"eql",
"eqr",
"refl",
"square-l",
"square-r"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
cancellr : 𝔻 → 𝔻 | cancellr middle = middle -- arbitrary choice
cancellr (left x) = x
cancellr (right x) = x | function | cancellr | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"left",
"middle",
"right"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
cancelη : 𝔹' → 𝔻 | cancelη L' = middle -- arbitrary choice
cancelη R' = middle -- arbitrary choice
cancelη (η x) = x | function | cancelη | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"middle",
"𝔹'"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
left-lc : {x y : 𝔻} → left x ＝ left y → x ＝ y | left-lc = cong cancellr | function | left-lc | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"cancellr",
"left"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
right-lc : {x y : 𝔻} → right x ＝ right y → x ＝ y | right-lc = cong cancellr | function | right-lc | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"cancellr",
"right"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
isLeft : 𝔻 → Type₀ | isLeft middle = 𝟘
isLeft (left x) = 𝟙
isLeft (right x) = 𝟘 | function | isLeft | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"left",
"middle",
"right"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
isMiddle : 𝔻 → Type₀ | isMiddle middle = 𝟙
isMiddle (left x) = 𝟘
isMiddle (right x) = 𝟘 | function | isMiddle | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"left",
"middle",
"right"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
left-is-not-right : {x y : 𝔻} → ¬ left x ＝ right y | left-is-not-right p = transport (cong isLeft p) * | function | left-is-not-right | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"isLeft",
"left",
"right",
"transport"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
middle-is-not-left : {x : 𝔻} → ¬ middle ＝ left x | middle-is-not-left p = transport (cong isMiddle p) * | function | middle-is-not-left | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"isMiddle",
"left",
"middle",
"transport"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
middle-is-not-right : {x : 𝔻} → ¬ middle ＝ right x | middle-is-not-right p = transport (cong isMiddle p) * | function | middle-is-not-right | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"isMiddle",
"middle",
"right",
"transport"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
𝔻-is-discrete : Discrete 𝔻 | 𝔻-is-discrete middle middle = yes refl
𝔻-is-discrete middle (left y) = no middle-is-not-left
𝔻-is-discrete middle (right y) = no middle-is-not-right
𝔻-is-discrete (left x) middle = no (middle-is-not-left ∘ sym)
𝔻-is-discrete (left x) (left y) = mapDec (cong left) (λ ν p → ν (left-lc p)) (𝔻-is-d... | function | 𝔻-is-discrete | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"left",
"left-is-not-right",
"left-lc",
"middle",
"middle-is-not-left",
"middle-is-not-right",
"refl",
"right",
"right-lc",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
η-lc : {x y : 𝔻} → η x ＝ η y → x ＝ y | η-lc = cong cancelη | function | η-lc | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"cancelη"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
is-L' : 𝔹' → Type₀ | is-L' L' = 𝟙
is-L' R' = 𝟘
is-L' (η x) = 𝟘 | function | is-L' | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"𝔹'"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
is-η : 𝔹' → Type₀ | is-η L' = 𝟘
is-η R' = 𝟘
is-η (η x) = 𝟙 | function | is-η | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"𝔹'"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
L'-is-not-R' : ¬ L' ＝ R' | L'-is-not-R' p = transport (cong is-L' p) * | function | L'-is-not-R' | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"is-L'",
"transport"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
L'-is-not-η : {x : 𝔻} → ¬ L' ＝ η x | L'-is-not-η p = transport (cong is-L' p) * | function | L'-is-not-η | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"is-L'",
"transport"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
η-is-not-R' : {x : 𝔻} → ¬ η x ＝ R' | η-is-not-R' p = transport (cong is-η p) * | function | η-is-not-R' | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"is-η",
"transport"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
𝔹'-is-discrete : Discrete 𝔹' | 𝔹'-is-discrete L' L' = yes refl
𝔹'-is-discrete L' R' = no L'-is-not-R'
𝔹'-is-discrete L' (η x) = no L'-is-not-η
𝔹'-is-discrete R' L' = no (L'-is-not-R' ∘ sym)
𝔹'-is-discrete R' R' = yes refl
𝔹'-is-discrete R' (η x) = no (η-is-not-R' ∘ sym)
𝔹'-is-discrete (η x) L' = no (L'-is-not-... | function | 𝔹'-is-discrete | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"L'-is-not-R'",
"L'-is-not-η",
"refl",
"sym",
"η-is-not-R'",
"η-lc",
"𝔹'",
"𝔻-is-discrete"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
𝔹'-is-set : isSet 𝔹' | 𝔹'-is-set = Discrete→isSet 𝔹'-is-discrete | function | 𝔹'-is-set | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"𝔹'",
"𝔹'-is-discrete"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
𝔹'-is-equiv-to-𝔹 : 𝔹' ≃ 𝔹 | 𝔹'-is-equiv-to-𝔹 = isoToEquiv (iso γ φ γφ φγ) | function | 𝔹'-is-equiv-to-𝔹 | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"γφ",
"φγ",
"𝔹'"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
𝔹-is-set : isSet 𝔹 | 𝔹-is-set = isOfHLevelRespectEquiv 2 𝔹'-is-equiv-to-𝔹 𝔹'-is-set | function | 𝔹-is-set | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"𝔹'-is-equiv-to-𝔹",
"𝔹'-is-set"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
𝔹-is-discrete : Discrete 𝔹 | 𝔹-is-discrete x y = mapDec
(isoFunInjective' (iso φ γ φγ γφ) x y)
(λ f p → f (cong φ p))
(𝔹'-is-discrete (φ x) (φ y)) | function | 𝔹-is-discrete | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"isoFunInjective'",
"γφ",
"φγ",
"𝔹'-is-discrete"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
compatible : {X : Type ℓ} (f g : 𝔹 → X) → Type ℓ | compatible f g = f R ＝ g L | function | compatible | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"Type"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
cases : {X : Type ℓ} (f g : 𝔹 → X) → compatible f g → (𝔹 → X) | cases f g p L = f L
cases f g p R = g R
cases f g p (l b) = f b
cases f g p (r b) = g b
cases f g p (eqL i) = f L
cases f g p (eqM i) = p i
cases f g p (eqR i) = g R | function | cases | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"Type",
"compatible"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
NB-cases : {X : Type ℓ} (f g : 𝔹 → X) (p : compatible f g)
→ cases f g p ∼ 𝔹-ind (λ _ → X) (f L) (g R) (λ b _ → f b) (λ b _ → g b) (λ _ → f L) p (λ _ → g R) | NB-cases f g p L = refl
NB-cases f g p R = refl
NB-cases f g p (l b) = refl
NB-cases f g p (r b) = refl
NB-cases f g p (eqL i) = λ _ → f L
NB-cases f g p (eqM i) = λ _ → p i
NB-cases f g p (eqR i) = λ _ → g R | function | NB-cases | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"Type",
"cases",
"compatible",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
path-lemma : {X : Type ℓ}
(h : 𝔹 → X)
{x y : 𝔹}
{a : X}
(p : x ＝ y)
(q : h y ＝ a)
→ PathP (λ i → h (p i) ＝ a) (cong h p ∙ q) q | path-lemma h p q i j = hcomp (λ k → λ { (i = i1) → q (j ∧ k)
; (j = i0) → h (p i)
; (j = i1) → q k})
(h (p (i ∨ j))) | function | path-lemma | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"Type"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
compatible-higher : {X : Type ℓ}
(f g : 𝔹 → X)
(p : compatible f g)
(h : 𝔹 → X)
(u : h ∘ l ∼ f)
(v : h ∘ r ∼ g)
→ Type ℓ | compatible-higher f g p h u v = Square (u R) (v L) (cong h eqM) p | function | compatible-higher | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"Type",
"compatible"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
cases-uniqueness : {X : Type ℓ}
(f g : 𝔹 → X)
(p : compatible f g)
(h : 𝔹 → X)
(u : h ∘ l ∼ f)
(v : h ∘ r ∼ g)
→ compatible-higher f g p h u v
→ h ∼ cases f g p | cases-uniqueness f g p h u v c L = cong h eqL ∙ u L
cases-uniqueness f g p h u v c R = cong h eqR ∙ v R
cases-uniqueness f g p h u v c (l x) = u x
cases-uniqueness f g p h u v c (r x) = v x
cases-uniqueness f g p h u v c (eqL i) = path-lemma h eqL (u L) i
cases-uniqueness f g p h u v c (eqM i) = c i
cas... | function | cases-uniqueness | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"Type",
"cases",
"compatible",
"compatible-higher",
"path-lemma"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
cases-uniqueness-set : {X : Type ℓ}
(f g : 𝔹 → X)
(p : compatible f g)
(h : 𝔹 → X)
(u : h ∘ l ∼ f)
(v : h ∘ r ∼ g)
→ isSet X
→ h ∼ cases f g p | cases-uniqueness-set f g p h u v isSetX = cases-uniqueness f g p h u v c
where
c : Square (u R) (v L) (λ i → h (eqM i)) p
c = isSet→isSet' isSetX (u R) (v L) (cong h eqM) p | function | cases-uniqueness-set | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"Type",
"cases",
"cases-uniqueness",
"compatible"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
mirror : 𝔹 → 𝔹 | mirror = 𝔹-rec R L r l eqR (sym eqM) eqL | function | mirror | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
mirror-defining-equations :
(mirror L ＝ R)
× (mirror R ＝ L)
× (mirror ∘ l ＝ r ∘ mirror)
× (mirror ∘ r ＝ l ∘ mirror) | mirror-defining-equations = refl , refl , refl , refl | function | mirror-defining-equations | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"mirror",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
mirror-involutive : (x : 𝔹) → mirror (mirror x) ＝ x | mirror-involutive = 𝔹-ind-eq _ _ refl refl (λ x → cong l) (λ y → cong r) | function | mirror-involutive | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"mirror",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
linv : 𝔹 → 𝔹 | linv = cases id (λ _ → R) refl | function | linv | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"cases",
"id",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
linv-defining-equations :
(linv L ＝ L)
× (linv R ＝ R)
× (linv ∘ l ＝ id )
× (linv ∘ r ＝ λ _ → R) | linv-defining-equations = refl , refl , refl , refl | function | linv-defining-equations | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"id",
"linv",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
rinv : 𝔹 → 𝔹 | rinv = cases (λ _ → L) id refl | function | rinv | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"cases",
"id",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
rinv-defining-equations :
(rinv L ＝ L)
× (rinv R ＝ R)
× (rinv ∘ l ＝ λ _ → L)
× (rinv ∘ r ＝ id) | rinv-defining-equations = refl , refl , refl , refl | function | rinv-defining-equations | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"id",
"refl",
"rinv"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
l-lc : {x y : 𝔹} → l x ＝ l y → x ＝ y | l-lc = cong linv | function | l-lc | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"linv"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
r-lc : {x y : 𝔹} → r x ＝ r y → x ＝ y | r-lc = cong rinv | function | r-lc | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"rinv"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
the-only-point-mapped-to-L-by-l-is-L : {x : 𝔹} → l x ＝ L → x ＝ L | the-only-point-mapped-to-L-by-l-is-L = cong linv | function | the-only-point-mapped-to-L-by-l-is-L | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"linv"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
the-only-point-mapped-to-R-by-r-is-R : {x : 𝔹} → r x ＝ R → x ＝ R | the-only-point-mapped-to-R-by-r-is-R = cong rinv | function | the-only-point-mapped-to-R-by-r-is-R | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"rinv"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
the-only-point-mapped-to-M-by-l-is-R : {x : 𝔹} → l x ＝ M → x ＝ R | the-only-point-mapped-to-M-by-l-is-R = l-lc | function | the-only-point-mapped-to-M-by-l-is-R | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"l-lc"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
the-only-point-mapped-to-M-by-r-is-L : {x : 𝔹} → r x ＝ M → x ＝ L | the-only-point-mapped-to-M-by-r-is-L p = r-lc (p ∙ eqM) | function | the-only-point-mapped-to-M-by-r-is-L | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"r-lc"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
lr-common-image : {x y : 𝔹} → l x ＝ r y → (x ＝ R) × (y ＝ L) | lr-common-image p = cong linv p , cong rinv (sym p) | function | lr-common-image | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"linv",
"rinv",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
the-only-fixed-point-of-l-is-L : (x : 𝔹) → l x ＝ x → x ＝ L | the-only-fixed-point-of-l-is-L = 𝔹-ind-prop _ (λ x → isPropΠ (λ _ → 𝔹-is-set _ _)) a b f g
where
a : l L ＝ L → L ＝ L
a _ = refl
b : l R ＝ R → R ＝ L
b p = snd s
where
q : l R ＝ r R
q = p ∙ eqR
s : (R ＝ R) × (R ＝ L)
s = lr-common-image q
f : (x : 𝔹) → (l x ＝ x → x ＝ L) → l (l x) ＝ l x →... | function | the-only-fixed-point-of-l-is-L | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"fst",
"l-lc",
"lr-common-image",
"r-lc",
"refl",
"sym",
"𝔹-is-set"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
the-only-fixed-point-of-r-is-R : (x : 𝔹) → r x ＝ x → x ＝ R | the-only-fixed-point-of-r-is-R x p = sym (mirror-involutive x) ∙ t
where
q : l (mirror x) ＝ mirror x
q = cong mirror p
s : mirror x ＝ L
s = the-only-fixed-point-of-l-is-L (mirror x) q
t : mirror (mirror x) ＝ R
t = cong mirror s | function | the-only-fixed-point-of-r-is-R | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"mirror",
"mirror-involutive",
"sym",
"the-only-fixed-point-of-l-is-L"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
is-L : 𝔹 → Type₀ | is-L = 𝔹-rec 𝟙 𝟘 id (λ X → 𝟘) refl refl refl | function | is-L | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"id",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
is-L-defining-equations :
(is-L L ＝ 𝟙)
× (is-L R ＝ 𝟘)
× (is-L ∘ l ＝ is-L)
× (is-L ∘ r ＝ λ _ → 𝟘) | is-L-defining-equations = refl , refl , refl , refl | function | is-L-defining-equations | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"is-L",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
L-is-not-R : ¬ L ＝ R | L-is-not-R p = transport (cong is-L p) * | function | L-is-not-R | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"is-L",
"transport"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
is-R : 𝔹 → Type₀ | is-R = 𝔹-rec 𝟘 𝟙 (λ X → 𝟘) id refl refl refl | function | is-R | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"id",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
is-R-defining-equations :
(is-R L ＝ 𝟘)
× (is-R R ＝ 𝟙)
× (is-R ∘ l ＝ λ _ → 𝟘)
× (is-R ∘ r ＝ is-R) | is-R-defining-equations = refl , refl , refl , refl | function | is-R-defining-equations | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"is-R",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
m-compatibility : l (r R) ＝ r (l L) | m-compatibility = cong l (sym eqR) ∙∙ eqM ∙∙ cong r eqL | function | m-compatibility | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
m : 𝔹 → 𝔹 | m = cases (l ∘ r) (r ∘ l) m-compatibility | function | m | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"cases",
"m-compatibility"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
cases-lc-l : (f g f' g' : 𝔹 → 𝔹)
(c : compatible f g)
(c' : compatible f' g')
→ cases f g c ＝ cases f' g' c'
→ f ∼ f' | cases-lc-l f g f' g' c c' p x = cong (λ - → - (l x)) p | function | cases-lc-l | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"cases",
"compatible",
"f'"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
cases-lc-r : (f g f' g' : 𝔹 → 𝔹)
(c : compatible f g)
(c' : compatible f' g')
→ cases f g c ＝ cases f' g' c'
→ g ∼ g' | cases-lc-r f g f' g' c c' p x = cong (λ - → - (r x)) p | function | cases-lc-r | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"cases",
"compatible",
"f'"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
m-defining-equations : (m L ＝ l (r L))
× (m R ＝ r (l R))
× (m ∘ l ＝ l ∘ r)
× (m ∘ r ＝ r ∘ l) | m-defining-equations = refl , refl , refl , refl | function | m-defining-equations | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
l-by-cases : l ∼ cases (l ∘ l) (m ∘ l) (cong l eqM) | l-by-cases = cases-uniqueness (l ∘ l) (m ∘ l)
(cong l eqM) l (λ x → refl) (λ x → refl) (λ i → refl) | function | l-by-cases | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"cases",
"cases-uniqueness",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
r-by-cases : r ∼ cases (m ∘ r) (r ∘ r) (cong r eqM) | r-by-cases = cases-uniqueness (r ∘ l) (r ∘ r)
(cong r eqM) r (λ x → refl) (λ x → refl) (λ i → refl) | function | r-by-cases | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"cases",
"cases-uniqueness",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
is-𝓛-function : (𝔹 → 𝔹) → Type₀ | is-𝓛-function f = compatible (l ∘ f) (m ∘ f) | function | is-𝓛-function | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"compatible"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
is-𝓡-function : (𝔹 → 𝔹) → Type₀ | is-𝓡-function f = compatible (m ∘ f) (r ∘ f) | function | is-𝓡-function | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"compatible"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
𝓛 : (f : 𝔹 → 𝔹) → is-𝓛-function f → (𝔹 → 𝔹) | 𝓛 f = cases (l ∘ f) (m ∘ f) | function | 𝓛 | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"cases",
"is-𝓛-function"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
𝓡 : (f : 𝔹 → 𝔹) → is-𝓡-function f → (𝔹 → 𝔹) | 𝓡 f = cases (m ∘ f) (r ∘ f) | function | 𝓡 | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"cases",
"is-𝓡-function"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
is-𝓛𝓡-function : (𝔹 → 𝔹) → Type₀ | is-𝓛𝓡-function f = is-𝓛-function f × is-𝓡-function f | function | is-𝓛𝓡-function | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"is-𝓛-function",
"is-𝓡-function"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
being-𝓛𝓡-function-is-prop : (f : 𝔹 → 𝔹) → isProp (is-𝓛𝓡-function f) | being-𝓛𝓡-function-is-prop f = isProp× (𝔹-is-set _ _) (𝔹-is-set _ _) | function | being-𝓛𝓡-function-is-prop | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"is-𝓛𝓡-function",
"𝔹-is-set"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
F : Type₀ | F = Σ f ꞉ (𝔹 → 𝔹) , is-𝓛𝓡-function f | function | F | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"is-𝓛𝓡-function"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
𝐿 𝑅 : F | 𝐿 = l , cong l eqM , m-compatibility | function | 𝐿 | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"m-compatibility"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
𝑙 𝑟 : F → F | 𝑙 (f , a , b) = 𝓛 f a , preservation-𝓛𝓛 f a b , preservation-𝓡𝓛 f a b | function | 𝑙 | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"preservation-𝓛𝓛",
"preservation-𝓡𝓛"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq𝐿 : 𝐿 ＝ 𝑙 𝐿 | eq𝐿 = Σ＝Prop being-𝓛𝓡-function-is-prop (funExt l-by-cases) | function | eq𝐿 | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"being-𝓛𝓡-function-is-prop",
"l-by-cases"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq𝑀 : 𝑙 𝑅 ＝ 𝑟 𝐿 | eq𝑀 = refl | function | eq𝑀 | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq𝑅 : 𝑅 ＝ 𝑟 𝑅 | eq𝑅 = Σ＝Prop being-𝓛𝓡-function-is-prop (funExt r-by-cases) | function | eq𝑅 | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"being-𝓛𝓡-function-is-prop",
"r-by-cases"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
mid : 𝔹 → F | mid = 𝔹-rec 𝐿 𝑅 𝑙 𝑟 eq𝐿 eq𝑀 eq𝑅 | function | mid | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"eq𝐿",
"eq𝑀",
"eq𝑅"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
mid-definition-equations :
(mid L ＝ 𝐿 )
× (mid R ＝ 𝑅)
× (mid ∘ l ＝ 𝑙 ∘ mid)
× (mid ∘ r ＝ 𝑟 ∘ mid) | mid-definition-equations = refl , refl , refl , refl | function | mid-definition-equations | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"mid",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
_⊕_ : 𝔹 → 𝔹 → 𝔹 | function | _⊕_ | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
⊕-defining-equations : (x y : 𝔹)
→ ( l x ⊕ y ＝ cases (l ∘ (x ⊕_)) (m ∘ (x ⊕_)) (fst (snd (mid x))) y )
× ( r x ⊕ y ＝ cases (m ∘ (x ⊕_)) (r ∘ (x ⊕_)) (snd (snd (mid x))) y ) | ⊕-defining-equations x y = refl , refl | function | ⊕-defining-equations | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"cases",
"fst",
"mid",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
⊕-property : (x : 𝔹)
→ (l (x ⊕ R) ＝ m (x ⊕ L))
× (m (x ⊕ R) ＝ r (x ⊕ L)) | ⊕-property x = snd (mid x) | function | ⊕-property | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"mid"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
⊕-defining-equations' : (x y : 𝔹)
→ ( L ⊕ y ＝ l y )
× ( R ⊕ y ＝ r y )
× ( l x ⊕ L ＝ l (x ⊕ L) )
× ( l x ⊕ R ＝ m (x ⊕ R) )
× ( l x ⊕ l y ＝ l (x ⊕ y) )
× ( l x ⊕ r y ＝ m (x ⊕ y) )
× ( r x ⊕ R ＝ r (x ⊕ R) )
× ( r x ⊕ L ＝ m (x ⊕ L) )
× ( r x ⊕ l y ＝ m (x ⊕ y) )
× ( ... | ⊕-defining-equations' x y = refl , refl , refl , refl , refl , refl , refl , refl , refl , refl | function | ⊕-defining-equations' | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
minv : 𝔹 → 𝔹 | minv = cases
(cases (λ _ → L) l eqL)
(cases r (λ _ → R) (sym eqR))
eqM | function | minv | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"cases",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
minv-defining-equations :
(minv L ＝ L)
× (minv R ＝ R)
× (minv ∘ l ∘ l ＝ λ _ → L)
× (minv ∘ l ∘ r ＝ l)
× (minv ∘ r ∘ l ＝ r)
× (minv ∘ r ∘ r ＝ λ _ → R) | minv-defining-equations = refl , refl , refl , refl , refl , refl | function | minv-defining-equations | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"minv",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
minv-is-left-inv : {x : 𝔹} → minv (m x) ＝ x | minv-is-left-inv {x} = 𝔹-cases-eq (minv ∘ m) id (λ b → refl) (λ b → refl) x | function | minv-is-left-inv | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"id",
"minv",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
m-lc : {x y : 𝔹} → m x ＝ m y → x ＝ y | m-lc p = sym (minv-is-left-inv) ∙ cong minv p ∙ minv-is-left-inv | function | m-lc | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"minv",
"minv-is-left-inv",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
m-ll-common-image : (x y : 𝔹) → m x ＝ l (l y) → (x ＝ L) × (y ＝ R) | m-ll-common-image x y p = u , v
where
u : x ＝ L
u = sym minv-is-left-inv ∙ cong minv p
q : l (l y) ＝ m L
q = sym p ∙ cong m u
v : y ＝ R
v = cong (linv ∘ linv) q | function | m-ll-common-image | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"linv",
"minv",
"minv-is-left-inv",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
m-rr-common-image : (x y : 𝔹) → m x ＝ r (r y) → (x ＝ R) × (y ＝ L) | m-rr-common-image x y p = u , v
where
u : x ＝ R
u = sym minv-is-left-inv ∙ cong minv p
q : r (r y) ＝ m R
q = sym p ∙ cong m u
v : y ＝ L
v = cong (rinv ∘ rinv) q | function | m-rr-common-image | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"minv",
"minv-is-left-inv",
"rinv",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
minv-M : (x : 𝔹) → minv ((L ⊕ R) ⊕ x) ＝ x | minv-M = 𝔹-cases-eq _ _ (λ x → refl) (λ x → refl) | function | minv-M | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"minv",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
minv-L : (x : 𝔹) → minv (L ⊕ (L ⊕ x)) ＝ L | minv-L x = refl | function | minv-L | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"minv",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
minv-R : (x : 𝔹) → minv (R ⊕ (R ⊕ x)) ＝ R | minv-R x = refl | function | minv-R | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"minv",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
_+ᵗ_ : 𝔹 → 𝔹 → 𝔹 | function | _+ᵗ_ | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e |
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