Datasets:

phanerozoic
/

Agda-Cubical

fact stringlengths 5 16.9k	type stringclasses 3 values	library stringclasses 1 value	imports listlengths 1 62	filename stringclasses 545 values	symbolic_name stringlengths 1 34
Formula : Type where const : ℕ → Formula _∧_ : Formula → Formula → Formula _∨_ : Formula → Formula → Formula ¬_ : Formula → Formula ⊥ : Formula ⊤ : Formula	data	Cubical	[ "open import Cubical.HITs.SetQuotients.Base", "open import Cubical.HITs.SetQuotients.Properties", "open import Cubical.Foundations.Prelude hiding (_∧_; _∨_)", "open import Cubical.Relation.Binary.Base", "open import Cubical.Data.Nat.Base", "open import Cubical.Data.Prod.Base", "open import Cubical.Algeb...	Cubical/Algebra/LindenbaumTarski.agda	Formula
ctxt : Type where ∅ : ctxt _∷_ : ctxt → Formula → ctxt	data	Cubical	[ "open import Cubical.HITs.SetQuotients.Base", "open import Cubical.HITs.SetQuotients.Properties", "open import Cubical.Foundations.Prelude hiding (_∧_; _∨_)", "open import Cubical.Relation.Binary.Base", "open import Cubical.Data.Nat.Base", "open import Cubical.Data.Prod.Base", "open import Cubical.Algeb...	Cubical/Algebra/LindenbaumTarski.agda	ctxt
_ ∈_ : Formula → ctxt → Type where Z : ∀ {Γ ϕ} → ϕ ∈ Γ ∷ ϕ S : ∀ {Γ ϕ ψ} → ϕ ∈ Γ → ϕ ∈ Γ ∷ ψ -- Provability	data	Cubical	[ "open import Cubical.HITs.SetQuotients.Base", "open import Cubical.HITs.SetQuotients.Properties", "open import Cubical.Foundations.Prelude hiding (_∧_; _∨_)", "open import Cubical.Relation.Binary.Base", "open import Cubical.Data.Nat.Base", "open import Cubical.Data.Prod.Base", "open import Cubical.Algeb...	Cubical/Algebra/LindenbaumTarski.agda	_
_ ⊢_ : 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 ⊢ ϕ ∨ ψ ∨-I₂ : {Γ : ctxt} {ϕ ψ : Formula} → Γ ⊢ ϕ → Γ ⊢ ϕ ∨ ψ ∨-E : {Γ : ctxt} {ϕ ψ γ : Formula} → Γ ⊢ ϕ ∨ ψ → Γ ∷ ϕ ⊢ γ → Γ ∷ ψ ⊢ γ → Γ ⊢ γ ¬-I : {Γ : ctxt} {ϕ : Formula} → Γ ∷ ϕ ⊢ ⊥ → Γ ⊢ ¬ ϕ ¬-E : {Γ : ctxt} {ϕ : Formula} → Γ ⊢ ϕ → Γ ⊢ ¬ ϕ → Γ ⊢ ⊥ ⊥-E : {Γ : ctxt} {ϕ : Formula} → Γ ⊢ ⊥ → Γ ⊢ ϕ ⊤-I : {Γ : ctxt} → Γ ⊢ ⊤ axiom : {Γ : ctxt} {ϕ : Formula} → ϕ ∈ Γ → Γ ⊢ ϕ LEM : {Γ : ctxt} {ϕ : Formula} → Γ ⊢ ¬ ϕ ∨ ϕ weakening : {Γ : ctxt} {ϕ ψ : Formula} → Γ ⊢ ψ → Γ ∷ ϕ ⊢ ψ	data	Cubical	[ "open import Cubical.HITs.SetQuotients.Base", "open import Cubical.HITs.SetQuotients.Properties", "open import Cubical.Foundations.Prelude hiding (_∧_; _∨_)", "open import Cubical.Relation.Binary.Base", "open import Cubical.Data.Nat.Base", "open import Cubical.Data.Prod.Base", "open import Cubical.Algeb...	Cubical/Algebra/LindenbaumTarski.agda	_
FinMatrix : (A : Type ℓ) (m n : ℕ) → Type ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Structure", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open import Cubical.Foundations.HLeve...	Cubical/Algebra/Matrix.agda	FinMatrix
VecMatrix : (A : Type ℓ) (m n : ℕ) → Type ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Structure", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open import Cubical.Foundations.HLeve...	Cubical/Algebra/Matrix.agda	VecMatrix
FinMatrixIsoVecMatrix : (A : Type ℓ) (m n : ℕ) → Iso (FinMatrix A m n) (VecMatrix A m n)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Structure", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open import Cubical.Foundations.HLeve...	Cubical/Algebra/Matrix.agda	FinMatrixIsoVecMatrix
flatten : {n m : ℕ} → FinMatrix A n m → FinVec A (n ·ℕ m)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Structure", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open import Cubical.Foundations.HLeve...	Cubical/Algebra/Matrix.agda	flatten
SymGroup : (X : Type ℓ) → isSet X → Group ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open import Cubical.Foundations.Structu...	Cubical/Algebra/SymmetricGroup.agda	SymGroup
FinSymGroup : ℕ → Group₀	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open import Cubical.Foundations.Structu...	Cubical/Algebra/SymmetricGroup.agda	FinSymGroup
satAC : (ℓ' : Level) (n : ℕ) (A : Type ℓ) → Type (ℓ-max ℓ (ℓ-suc ℓ'))	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Data.Nat", "open import Cubical.Data.Sigma", "open import Cubic...	Cubical/Axiom/Choice.agda	satAC
satAC₀ : {A : Type ℓ} → satAC ℓ' 0 A	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Data.Nat", "open import Cubical.Data.Sigma", "open import Cubic...	Cubical/Axiom/Choice.agda	satAC₀
FinSatAC : (n m : ℕ) → ∀ {ℓ} → satAC ℓ n (Fin m)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Data.Nat", "open import Cubical.Data.Sigma", "open import Cubic...	Cubical/Axiom/Choice.agda	FinSatAC
InductiveFinSatAC : (n m : ℕ) → ∀ {ℓ} → satAC ℓ n (Fin m)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Data.Nat", "open import Cubical.Data.Sigma", "open import Cubic...	Cubical/Axiom/Choice.agda	InductiveFinSatAC
LLPO : Type ℓ → Type ℓ	function	Cubical	[ "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Transport", "open import Cubical.Data.Bool", "open import Cubical.Data.Empty as Empty", "ope...	Cubical/Axiom/Omniscience.agda	LLPO
isPropLLPO : isProp (LLPO A)	function	Cubical	[ "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Transport", "open import Cubical.Data.Bool", "open import Cubical.Data.Empty as Empty", "ope...	Cubical/Axiom/Omniscience.agda	isPropLLPO
WLPO : Type ℓ → Type ℓ	function	Cubical	[ "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Transport", "open import Cubical.Data.Bool", "open import Cubical.Data.Empty as Empty", "ope...	Cubical/Axiom/Omniscience.agda	WLPO
WLPO' : Type ℓ → Type ℓ	function	Cubical	[ "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Transport", "open import Cubical.Data.Bool", "open import Cubical.Data.Empty as Empty", "ope...	Cubical/Axiom/Omniscience.agda	WLPO'
isPropWLPO : isProp (WLPO A)	function	Cubical	[ "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Transport", "open import Cubical.Data.Bool", "open import Cubical.Data.Empty as Empty", "ope...	Cubical/Axiom/Omniscience.agda	isPropWLPO
isPropWLPO' : isProp (WLPO' A)	function	Cubical	[ "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Transport", "open import Cubical.Data.Bool", "open import Cubical.Data.Empty as Empty", "ope...	Cubical/Axiom/Omniscience.agda	isPropWLPO'
LPO : Type ℓ → Type ℓ	function	Cubical	[ "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Transport", "open import Cubical.Data.Bool", "open import Cubical.Data.Empty as Empty", "ope...	Cubical/Axiom/Omniscience.agda	LPO
Path : ∀ {ℓ} (A : Type ℓ) → A → A → Type ℓ	function	Cubical	[ "open import Agda.Builtin.Cubical.Path public", "open import Agda.Builtin.Cubical.Sub public", "open import Agda.Primitive.Cubical public", "import Agda.Builtin.Cubical.Glue", "open import Agda.Primitive public", "open import Agda.Builtin.Sigma public" ]	Cubical/Core/Primitives.agda	Path
Σ-syntax : ∀ {ℓ ℓ'} (A : Type ℓ) (B : A → Type ℓ') → Type (ℓ-max ℓ ℓ')	function	Cubical	[ "open import Agda.Builtin.Cubical.Path public", "open import Agda.Builtin.Cubical.Sub public", "open import Agda.Primitive.Cubical public", "import Agda.Builtin.Cubical.Glue", "open import Agda.Primitive public", "open import Agda.Builtin.Sigma public" ]	Cubical/Core/Primitives.agda	Σ-syntax
yieldsCWskel : (ℕ → Type ℓ) → Type ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	yieldsCWskel
CWskel : (ℓ : Level) → Type (ℓ-suc ℓ)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	CWskel
yieldsFinCWskel : (n : ℕ) (X : ℕ → Type ℓ) → Type ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	yieldsFinCWskel
finCWskel : (ℓ : Level) → (n : ℕ) → Type (ℓ-suc ℓ)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	finCWskel
isFinCWskel : ∀ {ℓ} (C : CWskel ℓ) → Type ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	isFinCWskel
realiseSeq : CWskel ℓ → Sequence ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	realiseSeq
realiseFinSeq : (m : ℕ) → CWskel ℓ → FinSequence m ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	realiseFinSeq
realise : CWskel ℓ → Type ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	realise
hasCWskel : (X : Type ℓ) → Type (ℓ-suc ℓ)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	hasCWskel
isCW : (X : Type ℓ) → Type (ℓ-suc ℓ)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	isCW
CW : (ℓ : Level) → Type (ℓ-suc ℓ)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	CW
CWexplicit : (ℓ : Level) → Type (ℓ-suc ℓ)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	CWexplicit
isFinIsCW : {X : Type ℓ} → hasCWskel X → Type ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	isFinIsCW
hasFinCWskel : (X : Type ℓ) → Type (ℓ-suc ℓ)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	hasFinCWskel
isFinCW : (X : Type ℓ) → Type (ℓ-suc ℓ)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	isFinCW
finCW : (ℓ : Level) → Type (ℓ-suc ℓ)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	finCW
finCWexplicit : (ℓ : Level) → Type (ℓ-suc ℓ)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	finCWexplicit
cofibCW : ∀ {ℓ} (n : ℕ) (C : CWskel ℓ) → Type ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	cofibCW
to_cofibCW : (n : ℕ) (C : CWskel ℓ) → fst C (suc n) → cofibCW n C	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	to_cofibCW
δ : (n : ℕ) (C : CWskel ℓ) → cofibCW n C → Susp (fst C n)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Pointed", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Fin.Base", "...	Cubical/CW/Base.agda	δ
yieldsConnectedCWskel : (A : ℕ → Type ℓ) (n : ℕ) → Type _	function	Cubical	[ "open import Cubical.CW.Base", "open import Cubical.CW.Properties", "open import Cubical.CW.Instances.SphereBouquetMap", "open import Cubical.CW.Subcomplex", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cub...	Cubical/CW/Connected.agda	yieldsConnectedCWskel
yieldsCombinatorialConnectedCWskel : (A : ℕ → Type ℓ) (n : ℕ) → Type _	function	Cubical	[ "open import Cubical.CW.Base", "open import Cubical.CW.Properties", "open import Cubical.CW.Instances.SphereBouquetMap", "open import Cubical.CW.Subcomplex", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cub...	Cubical/CW/Connected.agda	yieldsCombinatorialConnectedCWskel
connectedCWskel : (ℓ : Level) (n : ℕ) → Type (ℓ-suc ℓ)	function	Cubical	[ "open import Cubical.CW.Base", "open import Cubical.CW.Properties", "open import Cubical.CW.Instances.SphereBouquetMap", "open import Cubical.CW.Subcomplex", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cub...	Cubical/CW/Connected.agda	connectedCWskel
combinatorialConnectedCWskel : (ℓ : Level) (n : ℕ) → Type (ℓ-suc ℓ)	function	Cubical	[ "open import Cubical.CW.Base", "open import Cubical.CW.Properties", "open import Cubical.CW.Instances.SphereBouquetMap", "open import Cubical.CW.Subcomplex", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cub...	Cubical/CW/Connected.agda	combinatorialConnectedCWskel
isConnectedCW : ∀ {ℓ} (n : ℕ) → Type ℓ → Type (ℓ-suc ℓ)	function	Cubical	[ "open import Cubical.CW.Base", "open import Cubical.CW.Properties", "open import Cubical.CW.Instances.SphereBouquetMap", "open import Cubical.CW.Subcomplex", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cub...	Cubical/CW/Connected.agda	isConnectedCW
isConnectedCW' : ∀ {ℓ} (n : ℕ) → Type ℓ → Type (ℓ-suc ℓ)	function	Cubical	[ "open import Cubical.CW.Base", "open import Cubical.CW.Properties", "open import Cubical.CW.Instances.SphereBouquetMap", "open import Cubical.CW.Subcomplex", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cub...	Cubical/CW/Connected.agda	isConnectedCW'
ConnectedCW : (ℓ : Level) (n : ℕ) → Type (ℓ-suc ℓ)	function	Cubical	[ "open import Cubical.CW.Base", "open import Cubical.CW.Properties", "open import Cubical.CW.Instances.SphereBouquetMap", "open import Cubical.CW.Subcomplex", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cub...	Cubical/CW/Connected.agda	ConnectedCW
cellHom {C : CWskel ℓ} {D : CWskel ℓ'} (f g : cellMap C D) : Type (ℓ-max ℓ ℓ') where constructor cellhom open SequenceMap field hom : (n : ℕ) → (x : C .fst n) → CW↪ D n (f .map n x) ≡ CW↪ D n (g .map n x) coh : (n : ℕ) → (c : C .fst n) → Square (cong (CW↪ D (suc n)) (hom n c)) (hom (suc n) (CW↪ C n c)) (cong (CW↪ D (suc n)) (f .comm n c)) (cong (CW↪ D (suc n)) (g .comm n c)) -- Finite cellular homotopies	record	Cubical	[ "open import Cubical.CW.Base", "open import Cubical.CW.Properties", "open import Cubical.CW.Map", "open import Cubical.CW.ChainComplex", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Path", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Fu...	Cubical/CW/Homotopy.agda	cellHom
finCellHom (m : ℕ) {C : CWskel ℓ} {D : CWskel ℓ'} (f g : finCellMap m C D) : Type (ℓ-max ℓ ℓ') where constructor fincellhom open FinSequenceMap field fhom : (n : Fin (suc m)) (x : C .fst (fst n)) → CW↪ D (fst n) (f .fmap n x) ≡ CW↪ D (fst n) (g .fmap n x) fcoh : (n : Fin m) (c : C .fst (fst n)) → Square (cong (CW↪ D (suc (fst n))) (fhom (injectSuc n) c)) (fhom (fsuc n) (CW↪ C (fst n) c)) (cong (CW↪ D (suc (fst n))) (f .fcomm n c)) (cong (CW↪ D (suc (fst n))) (g .fcomm n c))	record	Cubical	[ "open import Cubical.CW.Base", "open import Cubical.CW.Properties", "open import Cubical.CW.Map", "open import Cubical.CW.ChainComplex", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Path", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Fu...	Cubical/CW/Homotopy.agda	finCellHom
cellMap : (C : CWskel ℓ) (D : CWskel ℓ') → Type (ℓ-max ℓ ℓ')	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.GroupoidLaws", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data....	Cubical/CW/Map.agda	cellMap
realiseCellMap : cellMap C D → realise C → realise D	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.GroupoidLaws", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data....	Cubical/CW/Map.agda	realiseCellMap
idCellMap : (C : CWskel ℓ) → cellMap C C	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.GroupoidLaws", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data....	Cubical/CW/Map.agda	idCellMap
composeCellMap : (g : cellMap D E) (f : cellMap C D) → cellMap C E	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.GroupoidLaws", "open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)", "open import Cubical.Data....	Cubical/CW/Map.agda	composeCellMap
CW₀-empty : (C : CWskel ℓ) → ¬ fst C 0	function	Cubical	[ "open import Cubical.CW.Base", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Pointed", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open im...	Cubical/CW/Properties.agda	CW₀-empty
CW₁-discrete : (C : CWskel ℓ) → fst C 1 ≃ Fin (snd C .fst 0)	function	Cubical	[ "open import Cubical.CW.Base", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Pointed", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open im...	Cubical/CW/Properties.agda	CW₁-discrete
isSet-CW₀ : (C : CWskel ℓ) → isSet (fst C (suc zero))	function	Cubical	[ "open import Cubical.CW.Base", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Pointed", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open im...	Cubical/CW/Properties.agda	isSet-CW₀
realiseFin : (n : ℕ) (C : finCWskel ℓ n) → Iso (fst C n) (realise (finCWskel→CWskel n C))	function	Cubical	[ "open import Cubical.CW.Base", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Pointed", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open im...	Cubical/CW/Properties.agda	realiseFin
satAC-CW₁ : (n : ℕ) (C : CWskel ℓ) → satAC ℓ' n (fst C (suc zero))	function	Cubical	[ "open import Cubical.CW.Base", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Pointed", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open im...	Cubical/CW/Properties.agda	satAC-CW₁
subCWExplicit : ∀ {ℓ} (n : ℕ) → CWexplicit ℓ → CWexplicit ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Transpo...	Cubical/CW/Subcomplex.agda	subCWExplicit
subCW : ∀ {ℓ} (n : ℕ) → CWexplicit ℓ → CW ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Transpo...	Cubical/CW/Subcomplex.agda	subCW
niceFinCWskel : ∀ {ℓ} (n : ℕ) → finCWskel ℓ n → finCWskel ℓ n	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Transpo...	Cubical/CW/Subcomplex.agda	niceFinCWskel
makeNiceFinCWskel : ∀ {ℓ} {A : Type ℓ} → hasFinCWskel A → hasFinCWskel A	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Transpo...	Cubical/CW/Subcomplex.agda	makeNiceFinCWskel
makeNiceFinCW : ∀ {ℓ} → finCW ℓ → finCW ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Foundations.Transpo...	Cubical/CW/Subcomplex.agda	makeNiceFinCW
isFinInd : Type ℓ → Type ℓ	function	Cubical	[ "open import Cubical.Data.Nat", "open import Cubical.Data.SumFin", "open import Cubical.Data.Sigma", "open import Cubical.Data.FinSet", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Functions.Surjection", "open import Cubical.HITs.Propositiona...	Cubical/Data/FinInd.agda	isFinInd
isFinInd-S¹ : isFinInd S¹	function	Cubical	[ "open import Cubical.Data.Nat", "open import Cubical.Data.SumFin", "open import Cubical.Data.Sigma", "open import Cubical.Data.FinSet", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Functions.Surjection", "open import Cubical.HITs.Propositiona...	Cubical/Data/FinInd.agda	isFinInd-S¹
isSubFinSet : Type ℓ → Type ℓ	function	Cubical	[ "open import Cubical.Data.Nat", "open import Cubical.Data.SumFin", "open import Cubical.Data.Sigma", "open import Cubical.Data.FinSet", "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Functions.Embed...	Cubical/Data/SubFinSet.agda	isSubFinSet
UARel (A : Type ℓA) (ℓ≅A : Level) : Type (ℓ-max ℓA (ℓ-suc ℓ≅A)) where no-eta-equality constructor uarel field _≅_ : A → A → Type ℓ≅A ua : (a a' : A) → (a ≅ a') ≃ (a ≡ a') uaIso : (a a' : A) → Iso (a ≅ a') (a ≡ a') uaIso a a' = equivToIso (ua a a') ≅→≡ : {a a' : A} (p : a ≅ a') → a ≡ a' ≅→≡ {a} {a'} = Iso.fun (uaIso a a') ≡→≅ : {a a' : A} (p : a ≡ a') → a ≅ a' ≡→≅ {a} {a'} = Iso.inv (uaIso a a') ρ : (a : A) → a ≅ a ρ a = ≡→≅ refl	record	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Transport", "open import Cubical.Data.Sigma", "open import Cubical.Relation.Binary" ]	Cubical/Displayed/Base.agda	UARel
DUARel {A : Type ℓA} (𝒮-A : UARel A ℓ≅A) (B : A → Type ℓB) (ℓ≅B : Level) : Type (ℓ-max (ℓ-max ℓA ℓB) (ℓ-max ℓ≅A (ℓ-suc ℓ≅B))) where no-eta-equality constructor duarel open UARel 𝒮-A field _≅ᴰ⟨_⟩_ : {a a' : A} → B a → a ≅ a' → B a' → Type ℓ≅B uaᴰ : {a a' : A} (b : B a) (p : a ≅ a') (b' : B a') → (b ≅ᴰ⟨ p ⟩ b') ≃ PathP (λ i → B (≅→≡ p i)) b b' fiberRel : (a : A) → Rel (B a) (B a) ℓ≅B fiberRel a = _≅ᴰ⟨ ρ a ⟩_ uaᴰρ : {a : A} (b b' : B a) → b ≅ᴰ⟨ ρ a ⟩ b' ≃ (b ≡ b') uaᴰρ {a} b b' = compEquiv (uaᴰ b (ρ _) b') (substEquiv (λ q → PathP (λ i → B (q i)) b b') (secEq (ua a a) refl)) ρᴰ : {a : A} → (b : B a) → b ≅ᴰ⟨ ρ a ⟩ b ρᴰ {a} b = invEq (uaᴰρ b b) refl -- total UARel induced by a DUARel	record	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Transport", "open import Cubical.Data.Sigma", "open import Cubical.Relation.Binary" ]	Cubical/Displayed/Base.agda	DUARel
𝒮-generic : (A : Type ℓA) → UARel A ℓA	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Transport", "open import Cubical.Displayed.Base", "open import Cubical.Displayed.Subst" ]	Cubical/Displayed/Generic.agda	𝒮-generic
𝒮ᴰ-generic : {A : Type ℓA} (𝒮-A : UARel A ℓ≅A) (B : A → Type ℓB) → DUARel 𝒮-A B ℓB	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Transport", "open import Cubical.Displayed.Base", "open import Cubical.Displayed.Subst" ]	Cubical/Displayed/Generic.agda	𝒮ᴰ-generic
𝒮ˢ-generic : {A : Type ℓA} (𝒮-A : UARel A ℓ≅A) (B : A → Type ℓB) → SubstRel 𝒮-A B	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Transport", "open import Cubical.Displayed.Base", "open import Cubical.Displayed.Subst" ]	Cubical/Displayed/Generic.agda	𝒮ˢ-generic
UARelHom {A : Type ℓA} {B : Type ℓB} (𝒮-A : UARel A ℓ≅A) (𝒮-B : UARel B ℓ≅B) : Type (ℓ-max (ℓ-max ℓA ℓ≅A) (ℓ-max ℓB ℓ≅B)) where no-eta-equality constructor uarelhom field fun : A → B rel : ∀ {a a'} → UARel._≅_ 𝒮-A a a' → UARel._≅_ 𝒮-B (fun a) (fun a') ua : ∀ {a a'} (p : UARel._≅_ 𝒮-A a a') → cong fun (UARel.≅→≡ 𝒮-A p) ≡ UARel.≅→≡ 𝒮-B (rel p)	record	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Transport", "open import Cubical.Displayed.Base", "open import Cubical.Displayed.Subst" ]	Cubical/Displayed/Morphism.agda	UARelHom
𝒮-id : {A : Type ℓA} (𝒮-A : UARel A ℓ≅A) → UARelHom 𝒮-A 𝒮-A	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Transport", "open import Cubical.Displayed.Base", "open import Cubical.Displayed.Subst" ]	Cubical/Displayed/Morphism.agda	𝒮-id
𝒮-prop : (P : hProp ℓP) → UARel (P .fst) ℓ-zero	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.HLevels", "open import Cubical.Data.Sigma", "open import Cubical.Data.Unit", "open import Cubical.Displayed.Base" ]	Cubical/Displayed/Prop.agda	𝒮-prop
DUAFields {ℓA ℓ≅A ℓR ℓ≅R} {A : Type ℓA} (𝒮-A : UARel A ℓ≅A) (R : A → Type ℓR) (_≅R⟨_⟩_ : {a a' : A} → R a → UARel._≅_ 𝒮-A a a' → R a' → Type ℓ≅R) : ∀ {ℓS ℓ≅S} {S : A → Type ℓS} (πS : ∀ {a} → R a → S a) (𝒮ᴰ-S : DUARel 𝒮-A S ℓ≅S) (πS≅ : ∀ {a} {r : R a} {e} {r' : R a} → r ≅R⟨ e ⟩ r' → DUARel._≅ᴰ⟨_⟩_ 𝒮ᴰ-S (πS r) e (πS r')) → Typeω where -- `fields:` -- Base case, no fields yet recorded in `𝒮ᴰ-S`. fields: : DUAFields 𝒮-A R _≅R⟨_⟩_ (λ _ → tt) (𝒮ᴰ-Unit 𝒮-A) (λ _ → tt) -- `… data[ πF ∣ 𝒮ᴰ-F ∣ πF≅ ]` -- Add a new field with a DURG. `πF` should be the name of the field in the structure record `R` and `πF≅` -- the name of the corresponding field in the equivalence record `_≅R⟨_⟩_`, while `𝒮ᴰ-F` is a DURG for the -- field's type over `𝒮-A`. Data fields that depend on previous fields of the record are not currently -- supported. _data[_∣_∣_] : ∀ {ℓS ℓ≅S} {S : A → Type ℓS} {πS : ∀ {a} → R a → S a} {𝒮ᴰ-S : DUARel 𝒮-A S ℓ≅S} {πS≅ : ∀ {a} {r : R a} {e} {r' : R a} → r ≅R⟨ e ⟩ r' → DUARel._≅ᴰ⟨_⟩_ 𝒮ᴰ-S (πS r) e (πS r')} → DUAFields 𝒮-A R _≅R⟨_⟩_ πS 𝒮ᴰ-S πS≅ → ∀ {ℓF ℓ≅F} {F : A → Type ℓF} (πF : ∀ {a} → (r : R a) → F a) (𝒮ᴰ-F : DUARel 𝒮-A F ℓ≅F) (πF≅ : ∀ {a} {r : R a} {e} {r' : R a} (p : r ≅R⟨ e ⟩ r') → DUARel._≅ᴰ⟨_⟩_ 𝒮ᴰ-F (πF r) e (πF r')) → DUAFields 𝒮-A R _≅R⟨_⟩_ (λ r → πS r , πF r) (𝒮ᴰ-S ×𝒮ᴰ 𝒮ᴰ-F) (λ p → πS≅ p , πF≅ p) -- `… prop[ πF ∣ propF ]` -- Add a new propositional field. `πF` should be the name of the field in the structure record `R`, while -- propF is a proof that this field is a proposition. _prop[_∣_] : ∀ {ℓS ℓ≅S} {S : A → Type ℓS} {πS : ∀ {a} → R a → S a} {𝒮ᴰ-S : DUARel 𝒮-A S ℓ≅S} {πS≅ : ∀ {a} {r : R a} {e} {r' : R a} → r ≅R⟨ e ⟩ r' → DUARel._≅ᴰ⟨_⟩_ 𝒮ᴰ-S (πS r) e (πS r')} → DUAFields 𝒮-A R _≅R⟨_⟩_ πS 𝒮ᴰ-S πS≅ → ∀ {ℓF} {F : (a : A) → S a → Type ℓF} (πF : ∀ {a} → (r : R a) → F a (πS r)) (propF : ∀ a s → isProp (F a s)) → DUAFields 𝒮-A R _≅R⟨_⟩_ (λ r → πS r , πF r) (𝒮ᴰ-subtype 𝒮ᴰ-S propF) (λ p → πS≅ p)	data	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Function", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Path", "open import Cubical.Data.Sigma", "open import Cubical.Data.List as List", "open import...	Cubical/Displayed/Record.agda	DUAFields
SubstRel {A : Type ℓA} {ℓ≅A : Level} (𝒮-A : UARel A ℓ≅A) (B : A → Type ℓB) : Type (ℓ-max (ℓ-max ℓA ℓB) ℓ≅A) where no-eta-equality constructor substrel open UARel 𝒮-A field act : {a a' : A} → a ≅ a' → B a ≃ B a' uaˢ : {a a' : A} (p : a ≅ a') (b : B a) → subst B (≅→≡ p) b ≡ equivFun (act p) b uaˢ⁻ : {a a' : A} (p : a ≅ a') (b : B a') → subst B (sym (≅→≡ p)) b ≡ invEq (act p) b uaˢ⁻ p b = subst B (sym (≅→≡ p)) b ≡⟨ cong (subst B (sym (≅→≡ p))) (sym (secEq (act p) b)) ⟩ subst B (sym (≅→≡ p)) (equivFun (act p) (invEq (act p) b)) ≡⟨ cong (subst B (sym (≅→≡ p))) (sym (uaˢ p (invEq (act p) b))) ⟩ subst B (sym (≅→≡ p)) (subst B (≅→≡ p) (invEq (act p) b)) ≡⟨ pathToIso (cong B (≅→≡ p)) .Iso.ret (invEq (act p) b) ⟩ invEq (act p) b ∎ Subst→DUA : {A : Type ℓA} {ℓ≅A : Level} {𝒮-A : UARel A ℓ≅A} {B : A → Type ℓB} → SubstRel 𝒮-A B → DUARel 𝒮-A B ℓB DUARel._≅ᴰ⟨_⟩_ (Subst→DUA 𝒮ˢ-B) b p b' = equivFun (SubstRel.act 𝒮ˢ-B p) b ≡ b' DUARel.uaᴰ (Subst→DUA {𝒮-A = 𝒮-A} {B = B} 𝒮ˢ-B) b p b' = equivFun (SubstRel.act 𝒮ˢ-B p) b ≡ b' ≃⟨ invEquiv (compPathlEquiv (sym (SubstRel.uaˢ 𝒮ˢ-B p b))) ⟩ subst B (≅→≡ p) b ≡ b' ≃⟨ invEquiv (PathP≃Path (λ i → B (≅→≡ p i)) b b') ⟩ PathP (λ i → B (≅→≡ p i)) b b' ■ where open UARel 𝒮-A	record	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Path", "open import Cubical.Foundations.Transport", "open import Cubical.Foundations.Univalence", "open import Cubical.Displayed.Base" ]	Cubical/Displayed/Subst.agda	SubstRel
𝒮-Unit : UARel Unit ℓ-zero	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.HLevels", "open import Cubical.Data.Unit", "open import Cubical.Displayed.Base", "open import Cubical.Displayed.Constant" ]	Cubical/Displayed/Unit.agda	𝒮-Unit
𝒮ᴰ-Unit : {A : Type ℓA} (𝒮-A : UARel A ℓ≅A) → DUARel 𝒮-A (λ _ → Unit) ℓ-zero	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.HLevels", "open import Cubical.Data.Unit", "open import Cubical.Displayed.Base", "open import Cubical.Displayed.Constant" ]	Cubical/Displayed/Unit.agda	𝒮ᴰ-Unit
𝒮-Univ : ∀ ℓ → UARel (Type ℓ) ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open import Cubical.Displayed.Base", "open import Cubical.Displayed.Subst" ]	Cubical/Displayed/Universe.agda	𝒮-Univ
𝒮ˢ-El : ∀ ℓ → SubstRel (𝒮-Univ ℓ) (λ X → X)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open import Cubical.Displayed.Base", "open import Cubical.Displayed.Subst" ]	Cubical/Displayed/Universe.agda	𝒮ˢ-El
𝒮ᴰ-El : ∀ ℓ → DUARel (𝒮-Univ ℓ) (λ X → X) ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Foundations.Isomorphism", "open import Cubical.Foundations.Univalence", "open import Cubical.Displayed.Base", "open import Cubical.Displayed.Subst" ]	Cubical/Displayed/Universe.agda	𝒮ᴰ-El
Fin : ℕ → FinSet ℓ-zero	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Bool", "open import Cubical.Data.Sum", "open import Cubical.Data.Sigma", "open import Cubical.Data.Vec", "open import Cu...	Cubical/Experiments/Combinatorics.agda	Fin
s2 : card (_ , isFinSet≃ (Fin 2) (Fin 2)) ≡ 2	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Bool", "open import Cubical.Data.Sum", "open import Cubical.Data.Sigma", "open import Cubical.Data.Vec", "open import Cu...	Cubical/Experiments/Combinatorics.agda	s2
s3 : card (_ , isFinSet≃ (Fin 3) (Fin 3)) ≡ 6	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Bool", "open import Cubical.Data.Sum", "open import Cubical.Data.Sigma", "open import Cubical.Data.Vec", "open import Cu...	Cubical/Experiments/Combinatorics.agda	s3
getFun : {n : ℕ} → Vec ℕ n → Fin n .fst → ℕ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Bool", "open import Cubical.Data.Sum", "open import Cubical.Data.Sigma", "open import Cubical.Data.Vec", "open import Cu...	Cubical/Experiments/Combinatorics.agda	getFun
s : sum (Fin _) f ≡ 31	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Bool", "open import Cubical.Data.Sum", "open import Cubical.Data.Sigma", "open import Cubical.Data.Vec", "open import Cu...	Cubical/Experiments/Combinatorics.agda	s
p : prod (Fin _) f ≡ 6480	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Bool", "open import Cubical.Data.Sum", "open import Cubical.Data.Sigma", "open import Cubical.Data.Vec", "open import Cu...	Cubical/Experiments/Combinatorics.agda	p
m : maxValue (Fin _) f ∣ fzero ∣ ≡ 9	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Bool", "open import Cubical.Data.Sum", "open import Cubical.Data.Sigma", "open import Cubical.Data.Vec", "open import Cu...	Cubical/Experiments/Combinatorics.agda	m
n1 : card (_ , isFinSetFiberDisc (Fin _) ℕ discreteℕ f 1) ≡ 2	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Bool", "open import Cubical.Data.Sum", "open import Cubical.Data.Sigma", "open import Cubical.Data.Vec", "open import Cu...	Cubical/Experiments/Combinatorics.agda	n1
R : {n : ℕ} → Fin n .fst → Fin n .fst → Type	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Bool", "open import Cubical.Data.Sum", "open import Cubical.Data.Sigma", "open import Cubical.Data.Vec", "open import Cu...	Cubical/Experiments/Combinatorics.agda	R
isDecR : {n : ℕ} → (x y : Fin n .fst) → isDecProp (R {n = n} x y)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Bool", "open import Cubical.Data.Sum", "open import Cubical.Data.Sigma", "open import Cubical.Data.Vec", "open import Cu...	Cubical/Experiments/Combinatorics.agda	isDecR
isEquivRelR : {n : ℕ} → isEquivRel (R {n = n})	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Bool", "open import Cubical.Data.Sum", "open import Cubical.Data.Sigma", "open import Cubical.Data.Vec", "open import Cu...	Cubical/Experiments/Combinatorics.agda	isEquivRelR
collapsed : (n : ℕ) → FinSet ℓ-zero	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Nat.Order", "open import Cubical.Data.Bool", "open import Cubical.Data.Sum", "open import Cubical.Data.Sigma", "open import Cubical.Data.Vec", "open import Cu...	Cubical/Experiments/Combinatorics.agda	collapsed
https : //github.com/EgbertRijke/OEIS-A000001	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Bool", "open import Cubical.Data.Sigma", "open import Cubical.Data.FinSet", "open import Cubical.Data.FinSet.Induction", "open import Cubical.Data.FinSet.Constr...	Cubical/Experiments/CountingFiniteStructure.agda	https
isFinStrCard : (S : FinSet ℓ-zero → FinSet ℓ) (n : ℕ) → isFinType 0 (FinSetWithStrOfCard S n)	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Bool", "open import Cubical.Data.Sigma", "open import Cubical.Data.FinSet", "open import Cubical.Data.FinSet.Induction", "open import Cubical.Data.FinSet.Constr...	Cubical/Experiments/CountingFiniteStructure.agda	isFinStrCard
TrivialStr : FinSet ℓ → FinSet ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Bool", "open import Cubical.Data.Sigma", "open import Cubical.Data.FinSet", "open import Cubical.Data.FinSet.Induction", "open import Cubical.Data.FinSet.Constr...	Cubical/Experiments/CountingFiniteStructure.agda	TrivialStr
IdentityStr : FinSet ℓ → FinSet ℓ	function	Cubical	[ "open import Cubical.Foundations.Prelude", "open import Cubical.Foundations.Equiv", "open import Cubical.Data.Nat", "open import Cubical.Data.Bool", "open import Cubical.Data.Sigma", "open import Cubical.Data.FinSet", "open import Cubical.Data.FinSet.Induction", "open import Cubical.Data.FinSet.Constr...	Cubical/Experiments/CountingFiniteStructure.agda	IdentityStr

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Agda-Cubical

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

3,028 declarations extracted from Agda source files.

Applications

Training language models on formal proofs
Fine-tuning theorem provers
Retrieval-augmented generation for proof assistants
Learning proof embeddings and representations

Source

Repository: https://github.com/agda/cubical

Schema

Column	Type	Description
fact	string	Declaration body
type	string	data, record, function
library	string	Source module
imports	list	Required imports
filename	string	Source file path
symbolic_name	string	Identifier

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Size of downloaded dataset files:

300 kB

Size of the auto-converted Parquet files:

300 kB

Number of rows:

3,028

Collection including phanerozoic/Agda-Cubical

Proof Assistant Projects

Collection

Digesting proof assistant libraries for AI ingestion. • 84 items • Updated Jan 15 • 3