Datasets:

phanerozoic
/

Agda-Cubical

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	source_url stringclasses 1 value	commit stringclasses 1 value
Formula : Type where const : ℕ → Formula _∧_ : Formula → Formula → Formula _∨_ : Formula → Formula → Formula ¬_ : Formula → Formula ⊥ : Formula ⊤ : Formula		data	Formula	Algebra	Cubical/Algebra/LindenbaumTarski.agda	[ "Cubical.HITs.SetQuotients.Base", "Cubical.HITs.SetQuotients.Properties", "Cubical.Foundations.Prelude", "Cubical.Relation.Binary.Base", "Cubical.Data.Nat.Base", "Cubical.Data.Prod.Base", "Cubical.Algebra.DistLattice.Base" ]	[ "const", "¬_" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
ctxt : Type where ∅ : ctxt _∷_ : ctxt → Formula → ctxt		data	ctxt	Algebra	Cubical/Algebra/LindenbaumTarski.agda	[ "Cubical.HITs.SetQuotients.Base", "Cubical.HITs.SetQuotients.Properties", "Cubical.Foundations.Prelude", "Cubical.Relation.Binary.Base", "Cubical.Data.Nat.Base", "Cubical.Data.Prod.Base", "Cubical.Algebra.DistLattice.Base" ]	[ "Formula" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
_∈_ : Formula → ctxt → Type where Z : ∀ {Γ ϕ} → ϕ ∈ Γ ∷ ϕ S : ∀ {Γ ϕ ψ} → ϕ ∈ Γ → ϕ ∈ Γ ∷ ψ -- Provability		data	_∈_	Algebra	Cubical/Algebra/LindenbaumTarski.agda	[ "Cubical.HITs.SetQuotients.Base", "Cubical.HITs.SetQuotients.Properties", "Cubical.Foundations.Prelude", "Cubical.Relation.Binary.Base", "Cubical.Data.Nat.Base", "Cubical.Data.Prod.Base", "Cubical.Algebra.DistLattice.Base" ]	[ "Formula", "ctxt" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
_⊢_ : ctxt → Formula → Type where ∧-I : {Γ : ctxt} {ϕ ψ : Formula} → Γ ⊢ ϕ → Γ ⊢ ψ → Γ ⊢ ϕ ∧ ψ ∧-E₁ : {Γ : ctxt} {ϕ ψ : Formula} → Γ ⊢ ϕ ∧ ψ → Γ ⊢ ϕ ∧-E₂ : {Γ : ctxt} {ϕ ψ : Formula} → Γ ⊢ ϕ ∧ ψ → Γ ⊢ ψ ∨-I₁ : {Γ : ctxt} {ϕ ψ : Formula} → Γ ⊢ ψ → Γ ...		data	_⊢_	Algebra	Cubical/Algebra/LindenbaumTarski.agda	[ "Cubical.HITs.SetQuotients.Base", "Cubical.HITs.SetQuotients.Properties", "Cubical.Foundations.Prelude", "Cubical.Relation.Binary.Base", "Cubical.Data.Nat.Base", "Cubical.Data.Prod.Base", "Cubical.Algebra.DistLattice.Base" ]	[ "Formula", "LEM", "ctxt" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
FinMatrix : (A : Type ℓ) (m n : ℕ) → Type ℓ	FinMatrix A m n = FinVec (FinVec A n) m	function	FinMatrix	Algebra	Cubical/Algebra/Matrix.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Equiv", "Cubical.Foundations.Structure", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.HLevels", "Cubical.Foundations.Transport", "Cubical.Functions.FunExtEquiv", "Cubica...	[ "FinVec" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
VecMatrix : (A : Type ℓ) (m n : ℕ) → Type ℓ	VecMatrix A m n = Vec (Vec A n) m	function	VecMatrix	Algebra	Cubical/Algebra/Matrix.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Equiv", "Cubical.Foundations.Structure", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.HLevels", "Cubical.Foundations.Transport", "Cubical.Functions.FunExtEquiv", "Cubica...	[ "Vec" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
FinMatrix→VecMatrix : {m n : ℕ} → FinMatrix A m n → VecMatrix A m n	FinMatrix→VecMatrix M = FinVec→Vec (λ fm → FinVec→Vec (M fm))	function	FinMatrix→VecMatrix	Algebra	Cubical/Algebra/Matrix.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Equiv", "Cubical.Foundations.Structure", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.HLevels", "Cubical.Foundations.Transport", "Cubical.Functions.FunExtEquiv", "Cubica...	[ "FinMatrix", "FinVec→Vec", "VecMatrix" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
VecMatrix→FinMatrix : {m n : ℕ} → VecMatrix A m n → FinMatrix A m n	VecMatrix→FinMatrix M fn fm = lookup fm (lookup fn M)	function	VecMatrix→FinMatrix	Algebra	Cubical/Algebra/Matrix.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Equiv", "Cubical.Foundations.Structure", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.HLevels", "Cubical.Foundations.Transport", "Cubical.Functions.FunExtEquiv", "Cubica...	[ "FinMatrix", "VecMatrix", "lookup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
FinMatrix→VecMatrix→FinMatrix : {m n : ℕ} (M : FinMatrix A m n) → VecMatrix→FinMatrix (FinMatrix→VecMatrix M) ≡ M	FinMatrix→VecMatrix→FinMatrix {m = zero} M = funExt (⊥.rec ∘ ¬Fin0) FinMatrix→VecMatrix→FinMatrix {n = zero} M = funExt₂ (λ _ → ⊥.rec ∘ ¬Fin0) FinMatrix→VecMatrix→FinMatrix {m = suc m} {n = suc n} M = funExt₂ goal where goal : (fm : Fin (suc m)) (fn : Fin (suc n)) → VecMatrix→FinMatrix (_ ∷ FinMatrix→VecMatr...	function	FinMatrix→VecMatrix→FinMatrix	Algebra	Cubical/Algebra/Matrix.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Equiv", "Cubical.Foundations.Structure", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.HLevels", "Cubical.Foundations.Transport", "Cubical.Functions.FunExtEquiv", "Cubica...	[ "Fin", "FinMatrix", "FinMatrix→VecMatrix", "FinVec→Vec→FinVec", "VecMatrix→FinMatrix", "funExt", "funExt₂", "refl", "suc", "zero", "¬Fin0" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
VecMatrix→FinMatrix→VecMatrix : {m n : ℕ} (M : VecMatrix A m n) → FinMatrix→VecMatrix (VecMatrix→FinMatrix M) ≡ M	VecMatrix→FinMatrix→VecMatrix {m = zero} [] = refl VecMatrix→FinMatrix→VecMatrix {m = suc m} (M ∷ MS) i = Vec→FinVec→Vec M i ∷ VecMatrix→FinMatrix→VecMatrix MS i	function	VecMatrix→FinMatrix→VecMatrix	Algebra	Cubical/Algebra/Matrix.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Equiv", "Cubical.Foundations.Structure", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.HLevels", "Cubical.Foundations.Transport", "Cubical.Functions.FunExtEquiv", "Cubica...	[ "FinMatrix→VecMatrix", "VecMatrix", "VecMatrix→FinMatrix", "Vec→FinVec→Vec", "[]", "refl", "suc", "zero" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
FinMatrixIsoVecMatrix : (A : Type ℓ) (m n : ℕ) → Iso (FinMatrix A m n) (VecMatrix A m n)		function	FinMatrixIsoVecMatrix	Algebra	Cubical/Algebra/Matrix.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Equiv", "Cubical.Foundations.Structure", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.HLevels", "Cubical.Foundations.Transport", "Cubical.Functions.FunExtEquiv", "Cubica...	[ "FinMatrix", "Iso", "VecMatrix" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
FinMatrix≃VecMatrix : {m n : ℕ} → FinMatrix A m n ≃ VecMatrix A m n	FinMatrix≃VecMatrix {_} {A} {m} {n} = isoToEquiv (FinMatrixIsoVecMatrix A m n)	function	FinMatrix≃VecMatrix	Algebra	Cubical/Algebra/Matrix.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Equiv", "Cubical.Foundations.Structure", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.HLevels", "Cubical.Foundations.Transport", "Cubical.Functions.FunExtEquiv", "Cubica...	[ "FinMatrix", "FinMatrixIsoVecMatrix", "VecMatrix", "isoToEquiv" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
FinMatrix≡VecMatrix : (A : Type ℓ) (m n : ℕ) → FinMatrix A m n ≡ VecMatrix A m n	FinMatrix≡VecMatrix _ _ _ = ua FinMatrix≃VecMatrix -- Define abelian group structure on matrices	function	FinMatrix≡VecMatrix	Algebra	Cubical/Algebra/Matrix.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Equiv", "Cubical.Foundations.Structure", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.HLevels", "Cubical.Foundations.Transport", "Cubical.Functions.FunExtEquiv", "Cubica...	[ "FinMatrix", "FinMatrix≃VecMatrix", "VecMatrix", "ua" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
flatten : {n m : ℕ} → FinMatrix A n m → FinVec A (n ·ℕ m)	flatten {n = suc n} M = M zero ++Fin flatten (M ∘ suc)	function	flatten	Algebra	Cubical/Algebra/Matrix.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Equiv", "Cubical.Foundations.Structure", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.HLevels", "Cubical.Foundations.Transport", "Cubical.Functions.FunExtEquiv", "Cubica...	[ "FinMatrix", "FinVec", "suc", "zero" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
flattenElim : {P : A → Type ℓ'} {n m : ℕ} (M : FinMatrix A n m) → (∀ i j → P (M i j)) → (∀ i → P (flatten M i))	flattenElim {n = suc n} {m = zero} M PMHyp ind = ⊥.rec (¬Fin0 (transport (λ i → Fin (0≡m·0 n (~ i))) ind)) flattenElim {n = suc n} {m = suc m} M PMHyp zero = PMHyp zero zero flattenElim {P = P} {n = suc n} {m = suc m} M PMHyp (suc i) = ++FinElim {P = P} (M zero ∘ suc) (flatten (M ∘ suc)) (PMHyp zero ∘ suc) (fla...	function	flattenElim	Algebra	Cubical/Algebra/Matrix.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Equiv", "Cubical.Foundations.Structure", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.HLevels", "Cubical.Foundations.Transport", "Cubical.Functions.FunExtEquiv", "Cubica...	[ "++FinElim", "0≡m·0", "Fin", "FinMatrix", "flatten", "suc", "transport", "zero", "¬Fin0" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
fun (FinMatrixIsoVecMatrix A m n) = FinMatrix→VecMatrix		function	fun	Algebra	Cubical/Algebra/Matrix.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Equiv", "Cubical.Foundations.Structure", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.HLevels", "Cubical.Foundations.Transport", "Cubical.Functions.FunExtEquiv", "Cubica...	[ "FinMatrixIsoVecMatrix", "FinMatrix→VecMatrix" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
inv (FinMatrixIsoVecMatrix A m n) = VecMatrix→FinMatrix		function	inv	Algebra	Cubical/Algebra/Matrix.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Equiv", "Cubical.Foundations.Structure", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.HLevels", "Cubical.Foundations.Transport", "Cubical.Functions.FunExtEquiv", "Cubica...	[ "FinMatrixIsoVecMatrix", "VecMatrix→FinMatrix" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
sec (FinMatrixIsoVecMatrix A m n) = VecMatrix→FinMatrix→VecMatrix		function	sec	Algebra	Cubical/Algebra/Matrix.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Equiv", "Cubical.Foundations.Structure", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.HLevels", "Cubical.Foundations.Transport", "Cubical.Functions.FunExtEquiv", "Cubica...	[ "FinMatrixIsoVecMatrix", "VecMatrix→FinMatrix→VecMatrix" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
ret (FinMatrixIsoVecMatrix A m n) = FinMatrix→VecMatrix→FinMatrix		function	ret	Algebra	Cubical/Algebra/Matrix.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Equiv", "Cubical.Foundations.Structure", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.HLevels", "Cubical.Foundations.Transport", "Cubical.Functions.FunExtEquiv", "Cubica...	[ "FinMatrixIsoVecMatrix", "FinMatrix→VecMatrix→FinMatrix" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
SymGroup : (X : Type ℓ) → isSet X → Group ℓ	SymGroup X isSetX = makeGroup {G = X ≃ X} (idEquiv X) compEquiv invEquiv (isOfHLevel≃ 2 isSetX isSetX) compEquiv-assoc compEquivEquivId compEquivIdEquiv invEquiv-is-rinv invEquiv-is-linv	function	SymGroup	Algebra	Cubical/Algebra/SymmetricGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Structure", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.SumFin", "Cubic...	[ "Group", "compEquiv", "compEquiv-assoc", "compEquivEquivId", "compEquivIdEquiv", "idEquiv", "invEquiv", "invEquiv-is-linv", "invEquiv-is-rinv", "isOfHLevel≃", "isSet", "makeGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
FinSymGroup : ℕ → Group₀	FinSymGroup n = SymGroup (Fin n) isSetFin	function	FinSymGroup	Algebra	Cubical/Algebra/SymmetricGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Structure", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.SumFin", "Cubic...	[ "Fin", "Group₀", "SymGroup", "isSetFin" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
Sym0≃1 : GroupEquiv (FinSymGroup 0) UnitGroup₀	Sym0≃1 = contrGroupEquivUnit $ inhProp→isContr (idEquiv _) $ isOfHLevel≃ 1 isProp⊥ isProp⊥	function	Sym0≃1	Algebra	Cubical/Algebra/SymmetricGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Structure", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.SumFin", "Cubic...	[ "FinSymGroup", "GroupEquiv", "UnitGroup₀", "contrGroupEquivUnit", "idEquiv", "inhProp→isContr", "isOfHLevel≃", "isProp⊥" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
Sym0≡1 : FinSymGroup 0 ≡ UnitGroup₀	Sym0≡1 = uaGroup Sym0≃1	function	Sym0≡1	Algebra	Cubical/Algebra/SymmetricGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Structure", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.SumFin", "Cubic...	[ "FinSymGroup", "Sym0≃1", "UnitGroup₀", "uaGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
Sym1≃1 : GroupEquiv (FinSymGroup 1) UnitGroup₀	Sym1≃1 = contrGroupEquivUnit $ isOfHLevel≃ 0 isContrSumFin1 isContrSumFin1	function	Sym1≃1	Algebra	Cubical/Algebra/SymmetricGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Structure", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.SumFin", "Cubic...	[ "FinSymGroup", "GroupEquiv", "UnitGroup₀", "contrGroupEquivUnit", "isContrSumFin1", "isOfHLevel≃" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
Sym1≡1 : FinSymGroup 1 ≡ UnitGroup₀	Sym1≡1 = uaGroup Sym1≃1	function	Sym1≡1	Algebra	Cubical/Algebra/SymmetricGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Structure", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.SumFin", "Cubic...	[ "FinSymGroup", "Sym1≃1", "UnitGroup₀", "uaGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
Sym2≃Bool : GroupEquiv (FinSymGroup 2) BoolGroup	Sym2≃Bool = GroupIso→GroupEquiv $ invGroupIso $ ≅Bool $ Fin 2 ≃ Fin 2 Iso⟨ equivCompIso SumFin2≃Bool SumFin2≃Bool ⟩ Bool ≃ Bool Iso⟨ invIso univalenceIso ⟩ Bool ≡ Bool Iso⟨ invIso reflectIso ⟩ Bool ∎Iso where open BoolReflection	function	Sym2≃Bool	Algebra	Cubical/Algebra/SymmetricGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Structure", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.SumFin", "Cubic...	[ "Bool", "BoolGroup", "Fin", "FinSymGroup", "GroupEquiv", "GroupIso→GroupEquiv", "SumFin2≃Bool", "equivCompIso", "invGroupIso", "invIso", "univalenceIso" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
Sym2≡Bool : FinSymGroup 2 ≡ BoolGroup	Sym2≡Bool = uaGroup Sym2≃Bool	function	Sym2≡Bool	Algebra	Cubical/Algebra/SymmetricGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Structure", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.SumFin", "Cubic...	[ "BoolGroup", "FinSymGroup", "Sym2≃Bool", "uaGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
SymUnit≃Unit : GroupEquiv (SymGroup Unit isSetUnit) UnitGroup₀	SymUnit≃Unit = contrGroupEquivUnit $ isOfHLevel≃ 0 isContrUnit isContrUnit	function	SymUnit≃Unit	Algebra	Cubical/Algebra/SymmetricGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Structure", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.SumFin", "Cubic...	[ "GroupEquiv", "SymGroup", "UnitGroup₀", "contrGroupEquivUnit", "isContrUnit", "isOfHLevel≃", "isSetUnit" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
SymUnit≡Unit : SymGroup Unit isSetUnit ≡ UnitGroup₀	SymUnit≡Unit = uaGroup SymUnit≃Unit	function	SymUnit≡Unit	Algebra	Cubical/Algebra/SymmetricGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Structure", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.SumFin", "Cubic...	[ "SymGroup", "SymUnit≃Unit", "UnitGroup₀", "isSetUnit", "uaGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
SymBool≃Bool : GroupEquiv (SymGroup Bool isSetBool) BoolGroup	SymBool≃Bool = GroupIso→GroupEquiv $ invGroupIso $ ≅Bool $ Bool ≃ Bool Iso⟨ invIso univalenceIso ⟩ Bool ≡ Bool Iso⟨ invIso reflectIso ⟩ Bool ∎Iso where open BoolReflection	function	SymBool≃Bool	Algebra	Cubical/Algebra/SymmetricGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Structure", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.SumFin", "Cubic...	[ "Bool", "BoolGroup", "GroupEquiv", "GroupIso→GroupEquiv", "SymGroup", "invGroupIso", "invIso", "isSetBool", "univalenceIso" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
SymBool≡Bool : SymGroup Bool isSetBool ≡ BoolGroup	SymBool≡Bool = uaGroup SymBool≃Bool	function	SymBool≡Bool	Algebra	Cubical/Algebra/SymmetricGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Structure", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.SumFin", "Cubic...	[ "Bool", "BoolGroup", "SymBool≃Bool", "SymGroup", "isSetBool", "uaGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
Sym-cong-≃ : ∀ isSetX isSetY → X ≃ Y → GroupEquiv (SymGroup X isSetX) (SymGroup Y isSetY)	Sym-cong-≃ isSetX isSetY e .fst = equivComp e e Sym-cong-≃ isSetX isSetY e .snd = makeIsGroupHom λ g h → sym $ equivEq $ funExt λ x → cong (e .fst ∘ h .fst) (retEq e _)	function	Sym-cong-≃	Algebra	Cubical/Algebra/SymmetricGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Structure", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.SumFin", "Cubic...	[ "GroupEquiv", "SymGroup", "cong", "equivComp", "equivEq", "funExt", "sym" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
IsAbGroup {A : Type ℓ} (0g : A) (_+_ : A → A → A) (-_ : A → A) : Type ℓ where constructor isabgroup field isGroup : IsGroup 0g _+_ -_ +Comm : (x y : A) → x + y ≡ y + x open IsGroup isGroup public renaming ( ·Assoc to +Assoc ; ·IdL to +IdL ; ·IdR ...		record	IsAbGroup	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "+Assoc", "+Comm", "+IdL", "+IdR", "+InvL", "+InvR", "-_", "0g", "IsGroup", "_+_", "_-_", "isGroup", "·Assoc", "·IdL", "·IdR", "·InvL", "·InvR" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
AbGroupStr (A : Type ℓ) : Type (ℓ-suc ℓ) where constructor abgroupstr field 0g : A _+_ : A → A → A -_ : A → A isAbGroup : IsAbGroup 0g _+_ -_ infixr 7 _+_ infix 8 -_ open IsAbGroup isAbGroup public		record	AbGroupStr	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "-_", "0g", "IsAbGroup", "_+_", "isAbGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
AbGroup : ∀ ℓ → Type (ℓ-suc ℓ)	AbGroup ℓ = TypeWithStr ℓ AbGroupStr	function	AbGroup	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "AbGroupStr", "TypeWithStr" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
AbGroupStr→GroupStr : {G : Type ℓ} → AbGroupStr G → GroupStr G	AbGroupStr→GroupStr A .1g = A .0g AbGroupStr→GroupStr A ._·_ = A ._+_ AbGroupStr→GroupStr A .inv = A .-_ AbGroupStr→GroupStr A .isGroup = A .isAbGroup .isGroup	function	AbGroupStr→GroupStr	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "AbGroupStr", "GroupStr" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
AbGroup→Group : AbGroup ℓ → Group ℓ		function	AbGroup→Group	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "AbGroup", "Group" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
Group→AbGroup : (G : Group ℓ) → ((x y : fst G) → _·_ (snd G) x y ≡ _·_ (snd G) y x) → AbGroup ℓ		function	Group→AbGroup	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "AbGroup", "Group", "_·_", "fst", "snd" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
AbGroupHom : (G : AbGroup ℓ) (H : AbGroup ℓ') → Type (ℓ-max ℓ ℓ')	AbGroupHom G H = GroupHom (AbGroup→Group G) (AbGroup→Group H)	function	AbGroupHom	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "AbGroup", "AbGroup→Group", "GroupHom" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
AbGroupIso : (G : AbGroup ℓ) (H : AbGroup ℓ') → Type (ℓ-max ℓ ℓ')	AbGroupIso G H = GroupIso (AbGroup→Group G) (AbGroup→Group H)	function	AbGroupIso	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "AbGroup", "AbGroup→Group", "GroupIso" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
IsAbGroupEquiv : {A : Type ℓ} {B : Type ℓ'} (G : AbGroupStr A) (e : A ≃ B) (H : AbGroupStr B) → Type (ℓ-max ℓ ℓ')	IsAbGroupEquiv G e H = IsGroupHom (AbGroupStr→GroupStr G) (e .fst) (AbGroupStr→GroupStr H)	function	IsAbGroupEquiv	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "AbGroupStr", "AbGroupStr→GroupStr", "IsGroupHom" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
AbGroupEquiv : (G : AbGroup ℓ) (H : AbGroup ℓ') → Type (ℓ-max ℓ ℓ')	AbGroupEquiv G H = Σ[ e ∈ (G .fst ≃ H .fst) ] IsAbGroupEquiv (G .snd) e (H .snd)	function	AbGroupEquiv	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "AbGroup", "IsAbGroupEquiv" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
isPropIsAbGroup : {G : Type ℓ} (0g : G) (_+_ : G → G → G) (-_ : G → G) → isProp (IsAbGroup 0g _+_ (-_))	isPropIsAbGroup 0g _+_ -_ = isOfHLevelRetractFromIso 1 IsAbGroupIsoΣ (isPropΣ (isPropIsGroup 0g _+_ (-_)) (λ grp → isPropΠ2 (λ _ _ → grp .is-set _ _))) where open IsGroup	function	isPropIsAbGroup	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "-_", "0g", "IsAbGroup", "IsGroup", "_+_", "isOfHLevelRetractFromIso", "isProp", "isPropIsGroup", "isPropΠ2", "isPropΣ" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
𝒮ᴰ-AbGroup : DUARel (𝒮-Univ ℓ) AbGroupStr ℓ	𝒮ᴰ-AbGroup = 𝒮ᴰ-Record (𝒮-Univ _) IsAbGroupEquiv (fields: data[ _+_ ∣ autoDUARel _ _ ∣ pres· ] data[ 0g ∣ autoDUARel _ _ ∣ pres1 ] data[ -_ ∣ autoDUARel _ _ ∣ presinv ] prop[ isAbGroup ∣ (λ _ _ → isPropIsAbGroup _ _ _) ]) where open AbGroupStr open IsGroupHom -- Extract the chara...	function	𝒮ᴰ-AbGroup	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "-_", "0g", "AbGroupStr", "DUARel", "IsAbGroupEquiv", "IsGroupHom", "_+_", "autoDUARel", "isAbGroup", "isPropIsAbGroup", "𝒮-Univ" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
AbGroupPath : (G H : AbGroup ℓ) → (AbGroupEquiv G H) ≃ (G ≡ H)	AbGroupPath = ∫ 𝒮ᴰ-AbGroup .UARel.ua -- The module below defines an abelian group induced from an -- equivalence between an abelian group G and a type A which preserves -- the full raw group structure from G to A. This version is useful -- when proving that some type equivalent to an abelian group is an -- abelian g...	function	AbGroupPath	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "AbGroup", "AbGroupEquiv", "𝒮ᴰ-AbGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
dirProdAb : AbGroup ℓ → AbGroup ℓ' → AbGroup (ℓ-max ℓ ℓ')	dirProdAb A B = Group→AbGroup (DirProd (AbGroup→Group A) (AbGroup→Group B)) λ p q → ΣPathP (+Comm (isAbGroup (snd A)) _ _ , +Comm (isAbGroup (snd B)) _ _)	function	dirProdAb	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "+Comm", "AbGroup", "AbGroup→Group", "DirProd", "Group→AbGroup", "isAbGroup", "snd" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
trivialAbGroup : ∀ {ℓ} → AbGroup ℓ		function	trivialAbGroup	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "AbGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
move4 : ∀ {ℓ} {A : Type ℓ} (x y z w : A) (_+_ : A → A → A) → ((x y z : A) → x + (y + z) ≡ (x + y) + z) → ((x y : A) → x + y ≡ y + x) → (x + y) + (z + w) ≡ ((x + z) + (y + w))	move4 x y z w _+_ assoc +Comm = sym (assoc x y (z + w)) ∙∙ cong (x +_) (assoc y z w ∙∙ cong (_+ w) (+Comm y z) ∙∙ sym (assoc z y w)) ∙∙ assoc x z (y + w) ---- The type of homomorphisms A → B is an AbGroup if B is -----	function	move4	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "+Comm", "_+_", "assoc", "cong", "sym" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
fst (AbGroup→Group A) = fst A	fst (Group→AbGroup G +Comm) = fst G fst trivialAbGroup = Unit*	function	fst	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "+Comm", "AbGroup→Group", "Group→AbGroup", "Unit*", "trivialAbGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
snd (AbGroup→Group A) = AbGroupStr→GroupStr (snd A)		function	snd	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "AbGroupStr→GroupStr", "AbGroup→Group" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
AbGroupStr.0g (snd (Group→AbGroup G +Comm)) = 1g (snd G)		function	AbGroupStr.0g	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "+Comm", "1g", "Group→AbGroup", "snd" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
AbGroupStr._+_ (snd (Group→AbGroup G +Comm)) = _·_ (snd G)		function	AbGroupStr._+_	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "+Comm", "Group→AbGroup", "_·_", "snd" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
AbGroupStr.- snd (Group→AbGroup G +Comm) = inv (snd G)		function	AbGroupStr.-	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "+Comm", "Group→AbGroup", "inv", "snd" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
IsAbGroup.isGroup (AbGroupStr.isAbGroup (snd (Group→AbGroup G +Comm))) = isGroup (snd G)		function	IsAbGroup.isGroup	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "+Comm", "Group→AbGroup", "isGroup", "snd" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
IsAbGroup.+Comm (AbGroupStr.isAbGroup (snd (Group→AbGroup G +Comm))) = +Comm		function	IsAbGroup.+Comm	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "+Comm", "Group→AbGroup", "snd" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
0g (snd trivialAbGroup) = tt*		function	0g	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "snd", "trivialAbGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
_+_ (snd trivialAbGroup) _ _ = tt*		function	_+_	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "snd", "trivialAbGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
(- snd trivialAbGroup) _ = tt*		function	-	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "snd", "trivialAbGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
isAbGroup (snd trivialAbGroup) = makeIsAbGroup (isProp→isSet isPropUnit*) (λ _ _ _ → refl) (λ _ → refl) (λ _ → refl) (λ _ _ → refl) -- useful lemma -- dup...		function	isAbGroup	Algebra.AbGroup	Cubical/Algebra/AbGroup/Base.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Equiv", "Cubical.Foundations.Equiv.HalfAdjoint", "Cubical.Foundations.HLevels", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Univalence", "Cubical.Foundations.Transport", "Cubical.Foundations.SIP", "Cubical.Data.Sigma", "Cubical.Data.U...	[ "isPropUnit*", "isProp→isSet", "refl", "snd", "trivialAbGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
FinitePresentation (A : AbGroup ℓ) : Type ℓ where field nGens : ℕ nRels : ℕ rels : AbGroupHom ℤ[Fin nRels ] ℤ[Fin nGens ] fpiso : AbGroupIso A (ℤ[Fin nGens ] /Im rels)		record	FinitePresentation	Algebra.AbGroup	Cubical/Algebra/AbGroup/FinitePresentation.agda	[ "Cubical.Foundations.Prelude", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.HITs.PropositionalTruncation", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.AbGroup", "Cubical.Algebra.AbGroup.Instances.FreeAbGroup", "Cubica...	[ "AbGroup", "AbGroupHom", "AbGroupIso", "fpiso", "nGens", "nRels", "rels", "ℤ[Fin" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
isFinitelyPresented : AbGroup ℓ → Type ℓ	isFinitelyPresented G = ∥ FinitePresentation G ∥₁	function	isFinitelyPresented	Algebra.AbGroup	Cubical/Algebra/AbGroup/FinitePresentation.agda	[ "Cubical.Foundations.Prelude", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.HITs.PropositionalTruncation", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.AbGroup", "Cubical.Algebra.AbGroup.Instances.FreeAbGroup", "Cubica...	[ "AbGroup", "FinitePresentation" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
GrIsoPresFinitePresentation : ∀ {ℓ ℓ'} {A : AbGroup ℓ} {B : AbGroup ℓ'} → AbGroupIso A B → FinitePresentation A → FinitePresentation B		function	GrIsoPresFinitePresentation	Algebra.AbGroup	Cubical/Algebra/AbGroup/FinitePresentation.agda	[ "Cubical.Foundations.Prelude", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.HITs.PropositionalTruncation", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.AbGroup", "Cubical.Algebra.AbGroup.Instances.FreeAbGroup", "Cubica...	[ "AbGroup", "AbGroupIso", "FinitePresentation" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
nGens (GrIsoPresFinitePresentation abG fpA) = nGens fpA		function	nGens	Algebra.AbGroup	Cubical/Algebra/AbGroup/FinitePresentation.agda	[ "Cubical.Foundations.Prelude", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.HITs.PropositionalTruncation", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.AbGroup", "Cubical.Algebra.AbGroup.Instances.FreeAbGroup", "Cubica...	[ "GrIsoPresFinitePresentation" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
nRels (GrIsoPresFinitePresentation abG fpA) = nRels fpA		function	nRels	Algebra.AbGroup	Cubical/Algebra/AbGroup/FinitePresentation.agda	[ "Cubical.Foundations.Prelude", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.HITs.PropositionalTruncation", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.AbGroup", "Cubical.Algebra.AbGroup.Instances.FreeAbGroup", "Cubica...	[ "GrIsoPresFinitePresentation" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
rels (GrIsoPresFinitePresentation abG fpA) = rels fpA		function	rels	Algebra.AbGroup	Cubical/Algebra/AbGroup/FinitePresentation.agda	[ "Cubical.Foundations.Prelude", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.HITs.PropositionalTruncation", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.AbGroup", "Cubical.Algebra.AbGroup.Instances.FreeAbGroup", "Cubica...	[ "GrIsoPresFinitePresentation" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
fpiso (GrIsoPresFinitePresentation abG fpA) = compGroupIso (invGroupIso abG) (fpiso fpA)		function	fpiso	Algebra.AbGroup	Cubical/Algebra/AbGroup/FinitePresentation.agda	[ "Cubical.Foundations.Prelude", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.HITs.PropositionalTruncation", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.AbGroup", "Cubical.Algebra.AbGroup.Instances.FreeAbGroup", "Cubica...	[ "GrIsoPresFinitePresentation", "compGroupIso", "invGroupIso" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
addGroupHom : (A : AbGroup ℓ) (B : AbGroup ℓ') (ϕ ψ : AbGroupHom A B) → AbGroupHom A B		function	addGroupHom	Algebra.AbGroup	Cubical/Algebra/AbGroup/Properties.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Algebra.AbGroup.Base", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.Group.QuotientGroup", "Cubical.Algebra.Group.Subgroup", "Cubical.Algebra.Group.ZActi...	[ "AbGroup", "AbGroupHom" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
negGroupHom : (A : AbGroup ℓ) (B : AbGroup ℓ') (ϕ : AbGroupHom A B) → AbGroupHom A B		function	negGroupHom	Algebra.AbGroup	Cubical/Algebra/AbGroup/Properties.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Algebra.AbGroup.Base", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.Group.QuotientGroup", "Cubical.Algebra.Group.Subgroup", "Cubical.Algebra.Group.ZActi...	[ "AbGroup", "AbGroupHom" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
subtrGroupHom : (A : AbGroup ℓ) (B : AbGroup ℓ') (ϕ ψ : AbGroupHom A B) → AbGroupHom A B	subtrGroupHom A B ϕ ψ = addGroupHom A B ϕ (negGroupHom A B ψ) -- Abelian groups quotiented by image of a map	function	subtrGroupHom	Algebra.AbGroup	Cubical/Algebra/AbGroup/Properties.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Algebra.AbGroup.Base", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.Group.QuotientGroup", "Cubical.Algebra.Group.Subgroup", "Cubical.Algebra.Group.ZActi...	[ "AbGroup", "AbGroupHom", "addGroupHom", "negGroupHom" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
_/Im_ : {H : Group ℓ} (G : AbGroup ℓ) (ϕ : GroupHom H (AbGroup→Group G)) → AbGroup ℓ		function	_/Im_	Algebra.AbGroup	Cubical/Algebra/AbGroup/Properties.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Algebra.AbGroup.Base", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.Group.QuotientGroup", "Cubical.Algebra.Group.Subgroup", "Cubical.Algebra.Group.ZActi...	[ "AbGroup", "AbGroup→Group", "Group", "GroupHom" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
ℤ·isHom : (n : ℤ) (G : AbGroup ℓ) (x y : fst G) → (n ℤ[ (AbGroup→Group G) ]· (AbGroupStr._+_ (snd G) x y)) ≡ AbGroupStr._+_ (snd G) (n ℤ[ (AbGroup→Group G) ]· x) (n ℤ[ (AbGroup→Group G) ]· y)	ℤ·isHom (pos n) G = ℤ·isHom-pos G n ℤ·isHom (negsuc n) G = ℤ·isHom-neg G n -- left multiplication as a group homomorphism	function	ℤ·isHom	Algebra.AbGroup	Cubical/Algebra/AbGroup/Properties.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Algebra.AbGroup.Base", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.Group.QuotientGroup", "Cubical.Algebra.Group.Subgroup", "Cubical.Algebra.Group.ZActi...	[ "AbGroup", "AbGroupStr._+_", "AbGroup→Group", "fst", "negsuc", "pos", "snd" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
multₗHom : (G : AbGroup ℓ) (n : ℤ) → AbGroupHom G G		function	multₗHom	Algebra.AbGroup	Cubical/Algebra/AbGroup/Properties.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Algebra.AbGroup.Base", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.Group.QuotientGroup", "Cubical.Algebra.Group.Subgroup", "Cubical.Algebra.Group.ZActi...	[ "AbGroup", "AbGroupHom" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
_/^_ : (G : AbGroup ℓ) (n : ℕ) → AbGroup ℓ		function	_/^_	Algebra.AbGroup	Cubical/Algebra/AbGroup/Properties.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Algebra.AbGroup.Base", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.Group.QuotientGroup", "Cubical.Algebra.Group.Subgroup", "Cubical.Algebra.Group.ZActi...	[ "AbGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
_[_]ₜ : (G : AbGroup ℓ) (n : ℕ) → AbGroup ℓ		function	_[_]ₜ	Algebra.AbGroup	Cubical/Algebra/AbGroup/Properties.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Algebra.AbGroup.Base", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.Group.QuotientGroup", "Cubical.Algebra.Group.Subgroup", "Cubical.Algebra.Group.ZActi...	[ "AbGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
sumFinG-neg : (n : ℕ) {A : AbGroup ℓ} → (f : Fin n → fst A) → sumFinGroup (AbGroup→Group A) {n} (λ x → AbGroupStr.-_ (snd A) (f x)) ≡ AbGroupStr.-_ (snd A) (sumFinGroup (AbGroup→Group A) {n} f)	sumFinG-neg zero {A} f = sym (GroupTheory.inv1g (AbGroup→Group A)) sumFinG-neg (suc n) {A} f = cong₂ compA refl (sumFinG-neg n {A = A} (f ∘ injectSuc)) ∙∙ AbGroupStr.+Comm (snd A) _ _ ∙∙ sym (GroupTheory.invDistr (AbGroup→Group A) _ _) where -A = AbGroupStr.-_ (snd A) 0A = AbGroupStr.0g (snd A) compA = Ab...	function	sumFinG-neg	Algebra.AbGroup	Cubical/Algebra/AbGroup/Properties.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Algebra.AbGroup.Base", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.Group.QuotientGroup", "Cubical.Algebra.Group.Subgroup", "Cubical.Algebra.Group.ZActi...	[ "AbGroup", "AbGroupStr.0g", "AbGroupStr._+_", "AbGroup→Group", "Fin", "cong₂", "fst", "injectSuc", "refl", "snd", "suc", "sumFinGroup", "sym", "zero" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
fst (addGroupHom A B ϕ ψ) x = AbGroupStr._+_ (snd B) (ϕ .fst x) (ψ .fst x)	fst (negGroupHom A B ϕ) x = AbGroupStr.-_ (snd B) (ϕ .fst x) fst (multₗHom G n) g = n ℤ[ (AbGroup→Group G) ]· g	function	fst	Algebra.AbGroup	Cubical/Algebra/AbGroup/Properties.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Algebra.AbGroup.Base", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.Group.QuotientGroup", "Cubical.Algebra.Group.Subgroup", "Cubical.Algebra.Group.ZActi...	[ "AbGroupStr._+_", "AbGroup→Group", "addGroupHom", "multₗHom", "negGroupHom", "snd" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
snd (addGroupHom A B ϕ ψ) = makeIsGroupHom λ x y → cong₂ (AbGroupStr._+_ (snd B)) (IsGroupHom.pres· (snd ϕ) x y) (IsGroupHom.pres· (snd ψ) x y) ∙ AbGroupTheory.comm-4 B (fst ϕ x) (fst ϕ y) (fst ψ x) (fst ψ y)	snd (negGroupHom A B ϕ) = makeIsGroupHom λ x y → sym (IsGroupHom.presinv (snd ϕ) (AbGroupStr._+_ (snd A) x y)) ∙ cong (fst ϕ) (GroupTheory.invDistr (AbGroup→Group A) x y ∙ AbGroupStr.+Comm (snd A) _ _) ∙ IsGroupHom.pres· (snd ϕ) _ _ ∙ cong₂ (AbGroupStr._+_ (snd B)) (I...	function	snd	Algebra.AbGroup	Cubical/Algebra/AbGroup/Properties.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Algebra.AbGroup.Base", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.Group.QuotientGroup", "Cubical.Algebra.Group.Subgroup", "Cubical.Algebra.Group.ZActi...	[ "AbGroupStr._+_", "AbGroup→Group", "addGroupHom", "cong", "cong₂", "fst", "multₗHom", "negGroupHom", "sym", "ℤ·isHom" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
G /Im ϕ = Group→AbGroup (G /' ϕ) (elimProp2 (λ _ _ → squash/ _ _) λ a b → cong [_] (AbGroupStr.+Comm (snd G) _ _)) where _/'_ : {H : Group ℓ} (G : AbGroup ℓ) (ϕ : GroupHom H (AbGroup→Group G)) → Group ℓ G /' ϕ = AbGroup→Group G / (imSubgroup ϕ , isNormalIm ϕ λ _ _ → AbGroupStr.+Comm (snd G) _ _) ...	G /^ n = G /Im multₗHom G (pos n) -- Torsion subgrous G [ n ]ₜ = Group→AbGroup (Subgroup→Group (AbGroup→Group G) (kerSubgroup (multₗHom G (pos n)))) λ {(x , p) (y , q) → Σ≡Prop (λ _ → isPropIsInKer (multₗHom G (pos n)) _) (AbGroupStr.+Comm (snd G) _ _)} -- finite sums commute with negations	function	G	Algebra.AbGroup	Cubical/Algebra/AbGroup/Properties.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Algebra.AbGroup.Base", "Cubical.Algebra.Group", "Cubical.Algebra.Group.Morphisms", "Cubical.Algebra.Group.MorphismProperties", "Cubical.Algebra.Group.QuotientGroup", "Cubical.Algebra.Group.Subgroup", "Cubical.Algebra.Group.ZActi...	[ "AbGroup", "AbGroup→Group", "Group", "GroupHom", "Group→AbGroup", "[_]", "cong", "elimProp2", "isPropIsInKer", "multₗHom", "pos", "snd", "Σ≡Prop" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
tensorFun : (A : Group ℓ) (B : Group ℓ') (T C : AbGroup (ℓ-max ℓ ℓ')) (f : GroupHom A (HomGroup B T )) → GroupHom (T ) (C ) → GroupHom A (HomGroup B C )		function	tensorFun	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "AbGroup", "Group", "GroupHom" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
isTensorProductOf_and_ : AbGroup ℓ → AbGroup ℓ' → AbGroup (ℓ-max ℓ ℓ')→ Type _	isTensorProductOf_and_ {ℓ} {ℓ'} A B T = Σ[ f ∈ GroupHom (A ) ((HomGroup (B ) T) ) ] ((C : AbGroup (ℓ-max ℓ ℓ')) → isEquiv {A = GroupHom (T ) (C )} {B = GroupHom (A ) ((HomGroup (B ) C) )} (tensorFun (A ) (B ) T C f)) ...	function	isTensorProductOf_and_	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "AbGroup", "GroupHom", "isEquiv", "tensorFun" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
commFun : ∀ {ℓ ℓ'} {A : AbGroup ℓ} {B : AbGroup ℓ'} → A ⨂₁ B → B ⨂₁ A	commFun (a ⊗ b) = b ⊗ a commFun (x +⊗ x₁) = commFun x +⊗ commFun x₁ commFun (+⊗Comm x x₁ i) = +⊗Comm (commFun x) (commFun x₁) i commFun (+⊗Assoc x x₁ x₂ i) = +⊗Assoc (commFun x) (commFun x₁) (commFun x₂) i commFun (+⊗IdL x i) = +⊗IdL (commFun x) i commFun (⊗DistR+⊗ x y z i) = ⊗DistL+⊗ y z x i commFun (⊗DistL+⊗ x y z i)...	function	commFun	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "AbGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
commFun²≡id : ∀ {ℓ ℓ'} {A : AbGroup ℓ} {B : AbGroup ℓ'} (x : A ⨂₁ B) → commFun (commFun x) ≡ x	commFun²≡id = ⊗elimProp (λ _ → ⊗squash _ _) (λ _ _ → refl) (λ _ _ p q → cong₂ _+⊗_ p q)	function	commFun²≡id	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "AbGroup", "commFun", "cong₂", "refl" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
⨂-commIso : ∀ {ℓ ℓ'} {A : AbGroup ℓ} {B : AbGroup ℓ'} → GroupIso ((A ⨂ B) ) ((B ⨂ A) )		function	⨂-commIso	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "AbGroup", "GroupIso" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
⨂-comm : ∀ {ℓ ℓ'} {A : AbGroup ℓ} {B : AbGroup ℓ'} → AbGroupEquiv (A ⨂ B) (B ⨂ A)		function	⨂-comm	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "AbGroup", "AbGroupEquiv" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
lIncl⨂ : {G : AbGroup ℓ} {H : AbGroup ℓ'} → (h : fst H) → AbGroupHom G (G ⨂ H)		function	lIncl⨂	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "AbGroup", "AbGroupHom", "fst" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
rIncl⨂ : {G : AbGroup ℓ} {H : AbGroup ℓ'} → (g : fst G) → AbGroupHom H (G ⨂ H)		function	rIncl⨂	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "AbGroup", "AbGroupHom", "fst" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
G→G⨂G→Gₗ : {G : Ring ℓ} → Path (AbGroupHom (Ring→AbGroup G) (Ring→AbGroup G)) ((compGroupHom (lIncl⨂ (RingStr.1r (snd G))) TensorMultHom)) idGroupHom	G→G⨂G→Gₗ {G = G} = Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt (RingStr.·IdR (snd G)))	function	G→G⨂G→Gₗ	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "AbGroupHom", "Path", "Ring", "compGroupHom", "funExt", "idGroupHom", "isPropIsGroupHom", "lIncl⨂", "snd", "Σ≡Prop" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
G→G⨂G→Gᵣ : {G : Ring ℓ} → Path (AbGroupHom (Ring→AbGroup G) (Ring→AbGroup G)) ((compGroupHom (rIncl⨂ (RingStr.1r (snd G))) TensorMultHom)) idGroupHom	G→G⨂G→Gᵣ {G = G} = Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt (RingStr.·IdL (snd G)))	function	G→G⨂G→Gᵣ	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "AbGroupHom", "Path", "Ring", "compGroupHom", "funExt", "idGroupHom", "isPropIsGroupHom", "rIncl⨂", "snd", "Σ≡Prop" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
fst (fst (tensorFun A B T C (f , p) (g , q)) a) b = g (fst (f a) b)	fst ⨂-comm = isoToEquiv (fst (⨂-commIso)) fst (lIncl⨂ h) g = g ⊗ h fst (rIncl⨂ g) h = g ⊗ h	function	fst	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "isoToEquiv", "lIncl⨂", "rIncl⨂", "tensorFun", "⨂-comm", "⨂-commIso" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
snd (fst (tensorFun A B T C (f , p) (g , q)) a) = makeIsGroupHom λ x y → cong g (IsGroupHom.pres· (snd (f a)) x y) ∙ IsGroupHom.pres· q _ _	snd (tensorFun A B T C (f , p) (g , q)) = makeIsGroupHom λ x y → Σ≡Prop (λ _ → isPropIsGroupHom _ _) (funExt λ b → cong g (funExt⁻ (cong fst (IsGroupHom.pres· p x y)) b) ∙ IsGroupHom.pres· q _ _) snd ⨂-commIso = makeIsGroupHom λ x y → refl snd ⨂-comm = ...	function	snd	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "cong", "fst", "funExt", "funExt⁻", "isPropIsGroupHom", "lIncl⨂", "rIncl⨂", "refl", "tensorFun", "Σ≡Prop", "⨂-comm", "⨂-commIso" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
Iso.fun (fst ⨂-commIso) = commFun		function	Iso.fun	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "commFun", "fst", "⨂-commIso" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
Iso.inv (fst ⨂-commIso) = commFun		function	Iso.inv	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "commFun", "fst", "⨂-commIso" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
Iso.sec (fst ⨂-commIso) = commFun²≡id		function	Iso.sec	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "commFun²≡id", "fst", "⨂-commIso" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
Iso.ret (fst ⨂-commIso) = commFun²≡id		function	Iso.ret	Algebra.AbGroup	Cubical/Algebra/AbGroup/TensorProduct.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.Equiv", "Cubical.Foundations.HLevels", "Cubical.Relation.Binary", "Cubical.Data.Nat", "Cubical.Data.Int", "Cubical.Data.List", "Cubical.Data.Sigma", "Cubical.Data.Sum", "Cubic...	[ "commFun²≡id", "fst", "⨂-commIso" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
DiffℤasAbGroup : AbGroup ℓ-zero	DiffℤasAbGroup = makeAbGroup {G = ℤ} [ (0 , 0) ] _+ℤ_ -ℤ_ ℤ-isSet +ℤ-assoc (λ x → zero-identityʳ 0 x) (λ x → -ℤ-invʳ ...	function	DiffℤasAbGroup	Algebra.AbGroup.Instances	Cubical/Algebra/AbGroup/Instances/DiffInt.agda	[ "Cubical.Foundations.Prelude", "Cubical.HITs.SetQuotients", "Cubical.Algebra.AbGroup.Base", "Cubical.Data.Int.MoreInts.DiffInt" ]	[ "+ℤ-assoc", "+ℤ-comm", "-ℤ-invʳ", "-ℤ_", "AbGroup", "zero-identityʳ", "ℤ-isSet" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
AbDirProd : ∀ {ℓ ℓ'} → AbGroup ℓ → AbGroup ℓ' → AbGroup (ℓ-max ℓ ℓ')	AbDirProd G H = Group→AbGroup (DirProd (AbGroup→Group G) (AbGroup→Group H)) comm where comm : (x y : fst G × fst H) → _ ≡ _ comm (g1 , h1) (g2 , h2) = ΣPathP (AbGroupStr.+Comm (snd G) g1 g2 , AbGroupStr.+Comm (snd H) h1 h2)	function	AbDirProd	Algebra.AbGroup.Instances	Cubical/Algebra/AbGroup/Instances/DirectProduct.agda	[ "Cubical.Data.Sigma", "Cubical.Algebra.AbGroup.Base", "Cubical.Algebra.Group.DirProd" ]	[ "AbGroup", "AbGroup→Group", "DirProd", "Group→AbGroup", "comm", "fst", "g1", "h1", "h2", "snd" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
FAGAbGroup→AbGroupHom : ∀ {ℓ ℓ'} {A : Type ℓ} {G : AbGroup ℓ'} → (A → fst G) → AbGroupHom (FAGAbGroup {A = A}) G		function	FAGAbGroup→AbGroupHom	Algebra.AbGroup.Instances	Cubical/Algebra/AbGroup/Instances/FreeAbGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.HLevels", "Cubical.Foundations.Equiv", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.Nat.Order", "Cubical.Data.Nat.Order.Inductive", "Cubical.Data.Int", "Cubical.Data....	[ "AbGroup", "AbGroupHom", "fst" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
FAGAbGroupGroupHom≡ : ∀ {ℓ ℓ'} {A : Type ℓ} {G : AbGroup ℓ'} (f g : AbGroupHom (FAGAbGroup {A = A}) G) → (∀ a → (fst f) (⟦ a ⟧) ≡ (fst g) (⟦ a ⟧)) → f ≡ g	FAGAbGroupGroupHom≡ {G = G} f g p = GroupHom≡ (funExt (ElimProp.f (AbGroupStr.is-set (snd G) _ _) p (IsGroupHom.pres1 (snd f) ∙ sym (IsGroupHom.pres1 (snd g))) (λ p q → IsGroupHom.pres· (snd f) _ _ ∙ cong₂ (AbGroupStr._+_ (snd G)) p q ∙ sym (IsGroupHom.pres· (snd g) _ _)) λ p → IsG...	function	FAGAbGroupGroupHom≡	Algebra.AbGroup.Instances	Cubical/Algebra/AbGroup/Instances/FreeAbGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.HLevels", "Cubical.Foundations.Equiv", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.Nat.Order", "Cubical.Data.Nat.Order.Inductive", "Cubical.Data.Int", "Cubical.Data....	[ "AbGroup", "AbGroupHom", "AbGroupStr._+_", "GroupHom≡", "cong", "cong₂", "fst", "funExt", "snd", "sym" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
ℤ[Fin_] : (n : ℕ) → AbGroup ℓ-zero		function	ℤ[Fin_]	Algebra.AbGroup.Instances	Cubical/Algebra/AbGroup/Instances/FreeAbGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.HLevels", "Cubical.Foundations.Equiv", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.Nat.Order", "Cubical.Data.Nat.Order.Inductive", "Cubical.Data.Int", "Cubical.Data....	[ "AbGroup" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9
ℤFinGenerator : {n : ℕ} (k : Fin n) → ℤ[Fin n ] .fst		function	ℤFinGenerator	Algebra.AbGroup.Instances	Cubical/Algebra/AbGroup/Instances/FreeAbGroup.agda	[ "Cubical.Foundations.Prelude", "Cubical.Foundations.Function", "Cubical.Foundations.Isomorphism", "Cubical.Foundations.HLevels", "Cubical.Foundations.Equiv", "Cubical.Data.Sigma", "Cubical.Data.Nat", "Cubical.Data.Nat.Order", "Cubical.Data.Nat.Order.Inductive", "Cubical.Data.Int", "Cubical.Data....	[ "Fin", "ℤ[Fin" ]	https://github.com/agda/cubical	d4a2af62de40a6ca9a0b51981e41f804d879a1b9

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Agda-Cubical

Structured dataset from agda/cubical — Cubical Agda library for HoTT and univalent mathematics.

Source

Repository: https://github.com/agda/cubical
Commit: d4a2af62de40a6ca9a0b51981e41f804d879a1b9
Files: 1192
License: mit

Schema

Column	Type	Description
statement	string	Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof
proof	string	Verbatim proof/body, empty if the declaration has none
type	string	Declaration keyword
symbolic_name	string	Declaration identifier
library	string	Sub-library
filename	string	Repository-relative source path
imports	list[string]	File-level `Require`/`Import` modules
deps	list[string]	Intra-corpus identifiers referenced
docstring	string	Preceding documentation comment, empty if absent
source_url	string	Upstream repository
commit	string	Upstream commit extracted

Statistics

Entries: 9,811
With proof: 7,432 (75.8%)
With docstring: 0 (0.0%)
Libraries: 254

By type

Type	Count
function	9,466
record	206
data	139

Example

FinMatrix→VecMatrix : {m n : ℕ} → FinMatrix A m n → VecMatrix A m n
FinMatrix→VecMatrix M = FinVec→Vec (λ fm → FinVec→Vec (M fm))

type: function | symbolic_name: FinMatrix→VecMatrix | Cubical/Algebra/Matrix.agda

Use

Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint and together form the complete declaration, for proof modeling, autoformalization, retrieval, and dependency analysis via deps.

Citation

@misc{agda_cubical_dataset,
  title  = {Agda-Cubical},
  author = {Norton, Charles},
  year   = {2026},
  note   = {Extracted from https://github.com/agda/cubical, commit d4a2af62de40},
  url    = {https://huggingface.co/datasets/phanerozoic/Agda-Cubical}
}

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