statement stringlengths 5 10.9k | proof stringlengths 0 30.2k | type stringclasses 3
values | symbolic_name stringlengths 1 45 | library stringclasses 254
values | filename stringclasses 731
values | imports listlengths 1 62 | deps listlengths 0 64 | docstring stringclasses 1
value | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
0⌣ : ∀ {ℓ} {A : Type ℓ} (n m : ℕ) (x : coHom n A)
→ x ⌣ (0ₕ m) ≡ 0ₕ _ | 0⌣ n m = ST.elim (λ _ → isOfHLevelPath 2 squash₂ _ _)
λ f → cong ∣_∣₂ (funExt λ x → ⌣ₖ-0ₖ n m (f x)) | function | 0⌣ | ZCohomology.RingStructure | Cubical/ZCohomology/RingStructure/RingLaws.agda | [
"Cubical.Foundations.HLevels",
"Cubical.Foundations.Function",
"Cubical.Foundations.Path",
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Pointed",
"Cubical.Foundations.Pointed.Homogeneous",
"Cubical.Foundations.GroupoidLaws",
"Cubical.Data.Nat",
"Cubical.Data.Int",
"Cubical.Data.Sigma",
"C... | [
"0ₕ",
"coHom",
"cong",
"funExt",
"isOfHLevelPath",
"⌣ₖ-0ₖ"
] | https://github.com/agda/cubical | d4a2af62de40a6ca9a0b51981e41f804d879a1b9 | |
⌣0 : ∀ {ℓ} {A : Type ℓ} (n m : ℕ) (x : coHom n A)
→ (0ₕ m) ⌣ x ≡ 0ₕ _ | ⌣0 n m = ST.elim (λ _ → isOfHLevelPath 2 squash₂ _ _)
λ f → cong ∣_∣₂ (funExt λ x → 0ₖ-⌣ₖ m n (f x))
-- Multiplicative unit | function | ⌣0 | ZCohomology.RingStructure | Cubical/ZCohomology/RingStructure/RingLaws.agda | [
"Cubical.Foundations.HLevels",
"Cubical.Foundations.Function",
"Cubical.Foundations.Path",
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Pointed",
"Cubical.Foundations.Pointed.Homogeneous",
"Cubical.Foundations.GroupoidLaws",
"Cubical.Data.Nat",
"Cubical.Data.Int",
"Cubical.Data.Sigma",
"C... | [
"0ₕ",
"0ₖ-⌣ₖ",
"coHom",
"cong",
"funExt",
"isOfHLevelPath"
] | https://github.com/agda/cubical | d4a2af62de40a6ca9a0b51981e41f804d879a1b9 | |
1⌣ : ∀ {ℓ} {A : Type ℓ} → coHom 0 A | 1⌣ = ∣ (λ _ → 1) ∣₂ | function | 1⌣ | ZCohomology.RingStructure | Cubical/ZCohomology/RingStructure/RingLaws.agda | [
"Cubical.Foundations.HLevels",
"Cubical.Foundations.Function",
"Cubical.Foundations.Path",
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Pointed",
"Cubical.Foundations.Pointed.Homogeneous",
"Cubical.Foundations.GroupoidLaws",
"Cubical.Data.Nat",
"Cubical.Data.Int",
"Cubical.Data.Sigma",
"C... | [
"coHom"
] | https://github.com/agda/cubical | d4a2af62de40a6ca9a0b51981e41f804d879a1b9 | |
lUnit⌣ₖ : (n : ℕ) (x : coHomK n) → _⌣ₖ_ {n = 0} (pos 1) x ≡ x | lUnit⌣ₖ zero = λ _ → refl
lUnit⌣ₖ (suc n) x = rUnitₖ _ x | function | lUnit⌣ₖ | ZCohomology.RingStructure | Cubical/ZCohomology/RingStructure/RingLaws.agda | [
"Cubical.Foundations.HLevels",
"Cubical.Foundations.Function",
"Cubical.Foundations.Path",
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Pointed",
"Cubical.Foundations.Pointed.Homogeneous",
"Cubical.Foundations.GroupoidLaws",
"Cubical.Data.Nat",
"Cubical.Data.Int",
"Cubical.Data.Sigma",
"C... | [
"_⌣ₖ_",
"coHomK",
"pos",
"rUnitₖ",
"refl",
"suc",
"zero"
] | https://github.com/agda/cubical | d4a2af62de40a6ca9a0b51981e41f804d879a1b9 | |
lUnit⌣ : ∀ {ℓ} {A : Type ℓ} (n : ℕ) (x : coHom n A)
→ x ⌣ 1⌣ ≡ subst (λ n → coHom n A) (sym (+'-rid n)) x | lUnit⌣ zero = ST.elim (λ _ → isOfHLevelPath 2 squash₂ _ _)
λ f → cong ∣_∣₂ (funExt (λ x → comm-·₀ (f x) (pos 1))) ∙ sym (transportRefl ∣ f ∣₂)
lUnit⌣ (suc n) =
ST.elim (λ _ → isOfHLevelPath 2 squash₂ _ _)
λ f → cong ∣_∣₂ (funExt (λ x → cong (f x +ₖ_) (0ₖ-⌣ₖ zero _ (f x)) ∙ rUnitₖ _ (f x)))
∙ sym (transport... | function | lUnit⌣ | ZCohomology.RingStructure | Cubical/ZCohomology/RingStructure/RingLaws.agda | [
"Cubical.Foundations.HLevels",
"Cubical.Foundations.Function",
"Cubical.Foundations.Path",
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Pointed",
"Cubical.Foundations.Pointed.Homogeneous",
"Cubical.Foundations.GroupoidLaws",
"Cubical.Data.Nat",
"Cubical.Data.Int",
"Cubical.Data.Sigma",
"C... | [
"0ₖ-⌣ₖ",
"1⌣",
"coHom",
"comm-·₀",
"cong",
"funExt",
"isOfHLevelPath",
"pos",
"rUnitₖ",
"subst",
"suc",
"sym",
"transportRefl",
"zero"
] | https://github.com/agda/cubical | d4a2af62de40a6ca9a0b51981e41f804d879a1b9 | |
rUnit⌣ : ∀ {ℓ} {A : Type ℓ} (n : ℕ) (x : coHom n A)
→ 1⌣ ⌣ x ≡ x | rUnit⌣ zero = ST.elim (λ _ → isOfHLevelPath 2 squash₂ _ _)
λ f → refl
rUnit⌣ (suc n) =
ST.elim (λ _ → isOfHLevelPath 2 squash₂ _ _)
λ f → cong ∣_∣₂ (funExt λ x → rUnitₖ _ (f x)) | function | rUnit⌣ | ZCohomology.RingStructure | Cubical/ZCohomology/RingStructure/RingLaws.agda | [
"Cubical.Foundations.HLevels",
"Cubical.Foundations.Function",
"Cubical.Foundations.Path",
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Pointed",
"Cubical.Foundations.Pointed.Homogeneous",
"Cubical.Foundations.GroupoidLaws",
"Cubical.Data.Nat",
"Cubical.Data.Int",
"Cubical.Data.Sigma",
"C... | [
"1⌣",
"coHom",
"cong",
"funExt",
"isOfHLevelPath",
"rUnitₖ",
"refl",
"suc",
"zero"
] | https://github.com/agda/cubical | d4a2af62de40a6ca9a0b51981e41f804d879a1b9 | |
-ₕDistᵣ : ∀ {ℓ} {A : Type ℓ} (n m : ℕ)
(x : coHom n A) (y : coHom m A) → (-ₕ (x ⌣ y)) ≡ x ⌣ (-ₕ y) | -ₕDistᵣ n m =
ST.elim2 (λ _ _ → isOfHLevelPath 2 squash₂ _ _)
λ f g → cong ∣_∣₂ (funExt λ x → -Distᵣ n m (f x) (g x)) | function | -ₕDistᵣ | ZCohomology.RingStructure | Cubical/ZCohomology/RingStructure/RingLaws.agda | [
"Cubical.Foundations.HLevels",
"Cubical.Foundations.Function",
"Cubical.Foundations.Path",
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Pointed",
"Cubical.Foundations.Pointed.Homogeneous",
"Cubical.Foundations.GroupoidLaws",
"Cubical.Data.Nat",
"Cubical.Data.Int",
"Cubical.Data.Sigma",
"C... | [
"-Distᵣ",
"coHom",
"cong",
"funExt",
"isOfHLevelPath"
] | https://github.com/agda/cubical | d4a2af62de40a6ca9a0b51981e41f804d879a1b9 | |
-ₕDistₗ : ∀ {ℓ} {A : Type ℓ} (n m : ℕ)
(x : coHom n A) (y : coHom m A) → (-ₕ (x ⌣ y)) ≡ (-ₕ x) ⌣ y | -ₕDistₗ n m =
ST.elim2 (λ _ _ → isOfHLevelPath 2 squash₂ _ _)
λ f g → cong ∣_∣₂ (funExt λ x → -Distₗ n m (f x) (g x)) | function | -ₕDistₗ | ZCohomology.RingStructure | Cubical/ZCohomology/RingStructure/RingLaws.agda | [
"Cubical.Foundations.HLevels",
"Cubical.Foundations.Function",
"Cubical.Foundations.Path",
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Pointed",
"Cubical.Foundations.Pointed.Homogeneous",
"Cubical.Foundations.GroupoidLaws",
"Cubical.Data.Nat",
"Cubical.Data.Int",
"Cubical.Data.Sigma",
"C... | [
"-Distₗ",
"coHom",
"cong",
"funExt",
"isOfHLevelPath"
] | https://github.com/agda/cubical | d4a2af62de40a6ca9a0b51981e41f804d879a1b9 | |
-ₕDistₗᵣ : ∀ {ℓ} {A : Type ℓ} (n m : ℕ)
(x : coHom n A) (y : coHom m A) → (-ₕ x) ⌣ (-ₕ y) ≡ x ⌣ y | -ₕDistₗᵣ n m x y =
sym (-ₕDistₗ n m x (-ₕ y))
∙∙ cong -ₕ_ (sym (-ₕDistᵣ n m x y))
∙∙ ST.elim2 {C = λ x y → (-ₕ (-ₕ (x ⌣ y))) ≡ x ⌣ y}
(λ _ _ → isSetPathImplicit)
(λ f g → cong ∣_∣₂ (funExt λ _ → -ₖ^2 _)) x y | function | -ₕDistₗᵣ | ZCohomology.RingStructure | Cubical/ZCohomology/RingStructure/RingLaws.agda | [
"Cubical.Foundations.HLevels",
"Cubical.Foundations.Function",
"Cubical.Foundations.Path",
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Pointed",
"Cubical.Foundations.Pointed.Homogeneous",
"Cubical.Foundations.GroupoidLaws",
"Cubical.Data.Nat",
"Cubical.Data.Int",
"Cubical.Data.Sigma",
"C... | [
"-ₕDistᵣ",
"-ₕDistₗ",
"-ₕ_",
"-ₖ^2",
"coHom",
"cong",
"funExt",
"isSetPathImplicit",
"sym"
] | https://github.com/agda/cubical | d4a2af62de40a6ca9a0b51981e41f804d879a1b9 | |
fst (⌣ₖ-distrFun n m x y) z =
z ⌣ₖ (x +ₖ y) | fst (⌣ₖ-distrFun2 n m x y) z =
z ⌣ₖ x +ₖ z ⌣ₖ y
fst (⌣ₖ-distrFun-r n m x y) z =
(x +ₖ y) ⌣ₖ z
fst (⌣ₖ-distrFun2-r n m x y) z =
x ⌣ₖ z +ₖ y ⌣ₖ z | function | fst | ZCohomology.RingStructure | Cubical/ZCohomology/RingStructure/RingLaws.agda | [
"Cubical.Foundations.HLevels",
"Cubical.Foundations.Function",
"Cubical.Foundations.Path",
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Pointed",
"Cubical.Foundations.Pointed.Homogeneous",
"Cubical.Foundations.GroupoidLaws",
"Cubical.Data.Nat",
"Cubical.Data.Int",
"Cubical.Data.Sigma",
"C... | [
"⌣ₖ-distrFun",
"⌣ₖ-distrFun-r",
"⌣ₖ-distrFun2",
"⌣ₖ-distrFun2-r"
] | https://github.com/agda/cubical | d4a2af62de40a6ca9a0b51981e41f804d879a1b9 | |
snd (⌣ₖ-distrFun n m x y) =
0ₖ-⌣ₖ m n (x +ₖ y) | snd (⌣ₖ-distrFun2 n m x y) =
cong₂ _+ₖ_ (0ₖ-⌣ₖ m n x) (0ₖ-⌣ₖ m n y) ∙ rUnitₖ _ _
snd (⌣ₖ-distrFun-r n m x y) =
⌣ₖ-0ₖ n m (x +ₖ y) -- ⌣ₖ-0ₖ m n (x +ₖ y)
snd (⌣ₖ-distrFun2-r n m x y) =
cong₂ _+ₖ_ (⌣ₖ-0ₖ n m x) (⌣ₖ-0ₖ n m y) ∙ rUnitₖ _ _ | function | snd | ZCohomology.RingStructure | Cubical/ZCohomology/RingStructure/RingLaws.agda | [
"Cubical.Foundations.HLevels",
"Cubical.Foundations.Function",
"Cubical.Foundations.Path",
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Pointed",
"Cubical.Foundations.Pointed.Homogeneous",
"Cubical.Foundations.GroupoidLaws",
"Cubical.Data.Nat",
"Cubical.Data.Int",
"Cubical.Data.Sigma",
"C... | [
"0ₖ-⌣ₖ",
"_+ₖ_",
"cong₂",
"rUnitₖ",
"⌣ₖ-0ₖ",
"⌣ₖ-distrFun",
"⌣ₖ-distrFun-r",
"⌣ₖ-distrFun2",
"⌣ₖ-distrFun2-r"
] | https://github.com/agda/cubical | d4a2af62de40a6ca9a0b51981e41f804d879a1b9 |
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