fact stringlengths 5 670 | type stringclasses 3 values | library stringclasses 18 values | imports listlengths 0 48 | filename stringclasses 782 values | symbolic_name stringlengths 1 35 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
RecStruct : Set a → (ℓ₁ ℓ₂ : Level) → Set _ | function | Root | [
"Level",
"Relation.Unary",
"Relation.Unary.PredicateTransformer"
] | Induction.agda | RecStruct | |
RecursorBuilder : RecStruct A ℓ₁ ℓ₂ → Set _ | function | Root | [
"Level",
"Relation.Unary",
"Relation.Unary.PredicateTransformer"
] | Induction.agda | RecursorBuilder | |
Recursor : RecStruct A ℓ₁ ℓ₂ → Set _ | function | Root | [
"Level",
"Relation.Unary",
"Relation.Unary.PredicateTransformer"
] | Induction.agda | Recursor | |
build : RecursorBuilder Rec → Recursor Rec | function | Root | [
"Level",
"Relation.Unary",
"Relation.Unary.PredicateTransformer"
] | Induction.agda | build | |
SubsetRecursorBuilder : Pred A ℓ → RecStruct A ℓ₁ ℓ₂ → Set _ | function | Root | [
"Level",
"Relation.Unary",
"Relation.Unary.PredicateTransformer"
] | Induction.agda | SubsetRecursorBuilder | |
SubsetRecursor : Pred A ℓ → RecStruct A ℓ₁ ℓ₂ → Set _ | function | Root | [
"Level",
"Relation.Unary",
"Relation.Unary.PredicateTransformer"
] | Induction.agda | SubsetRecursor | |
subsetBuild : SubsetRecursorBuilder Q Rec → SubsetRecursor Q Rec | function | Root | [
"Level",
"Relation.Unary",
"Relation.Unary.PredicateTransformer"
] | Induction.agda | subsetBuild | |
Lift {a} ℓ (A: Set a) : Set (a ⊔ ℓ)
where
constructor lift
field lower : A
open Lift public
-- Synonyms
0ℓ : Level | record | Root | [
"Agda.Primitive as Prim public"
] | Level.agda | Lift | |
0ℓ : Level | function | Root | [
"Agda.Primitive as Prim public"
] | Level.agda | 0ℓ | |
levelOfType : ∀ {a} → Set a → Level | function | Root | [
"Agda.Primitive as Prim public"
] | Level.agda | levelOfType | |
levelOfTerm : ∀ {a} {A : Set a} → A → Level | function | Root | [
"Agda.Primitive as Prim public"
] | Level.agda | levelOfTerm | |
SizedSet : (ℓ : Level) → Set (suc ℓ) | function | Root | [
"Level",
"Agda.Builtin.Size public"
] | Size.agda | SizedSet | |
SuccessorSet c ℓ: Set (suc (c ⊔ ℓ))
where
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
suc# : Op₁ Carrier
zero# : Carrier
isSuccessorSet : IsSuccessorSet _≈_ suc# zero#
open IsSuccessorSet isSuccessorSet public
| record | Algebra | [
"Algebra.Bundles.Raw",
"Algebra.Core",
"Algebra.Structures",
"Relation.Binary.Core",
"Level"
] | Algebra/Bundles.agda | SuccessorSet | |
Magma c ℓ: Set (suc (c ⊔ ℓ))
where
infixl 7 _∙_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
isMagma : IsMagma _≈_ _∙_
open IsMagma isMagma public
| record | Algebra | [
"Algebra.Bundles.Raw",
"Algebra.Core",
"Algebra.Structures",
"Relation.Binary.Core",
"Level"
] | Algebra/Bundles.agda | Magma | |
UnitalMagma c ℓ: Set (suc (c ⊔ ℓ))
where
infixl 7 _∙_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
ε : Carrier
isUnitalMagma : IsUnitalMagma _≈_ _∙_ ε
open IsUnitalMagma isUnitalMagma public | record | Algebra | [
"Algebra.Bundles.Raw",
"Algebra.Core",
"Algebra.Structures",
"Relation.Binary.Core",
"Level"
] | Algebra/Bundles.agda | UnitalMagma | |
InvertibleMagma c ℓ: Set (suc (c ⊔ ℓ))
where
infix 8 _⁻¹
infixl 7 _∙_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
ε : Carrier
_⁻¹ : Op₁ Carrier
isInvertibleMagma : IsInvertibleMagma _≈_ _∙_ ε _⁻¹ | record | Algebra | [
"Algebra.Bundles.Raw",
"Algebra.Core",
"Algebra.Structures",
"Relation.Binary.Core",
"Level"
] | Algebra/Bundles.agda | InvertibleMagma | |
NearSemiring c ℓ: Set (suc (c ⊔ ℓ))
where
infixl 7 _*_
infixl 6 _+_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_+_ : Op₂ Carrier
_*_ : Op₂ Carrier
0# : Carrier
isNearSemiring : IsNearSemiring _≈_ _+_ _*_ 0# | record | Algebra | [
"Algebra.Bundles.Raw",
"Algebra.Core",
"Algebra.Structures",
"Relation.Binary.Core",
"Level"
] | Algebra/Bundles.agda | NearSemiring | |
SemiringWithoutAnnihilatingZero c ℓ: Set (suc (c ⊔ ℓ))
where
infixl 7 _*_
infixl 6 _+_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_+_ : Op₂ Carrier
_*_ : Op₂ Carrier
0# : Carrier
1# : Carrier | record | Algebra | [
"Algebra.Bundles.Raw",
"Algebra.Core",
"Algebra.Structures",
"Relation.Binary.Core",
"Level"
] | Algebra/Bundles.agda | SemiringWithoutAnnihilatingZero | |
RingWithoutOne c ℓ: Set (suc (c ⊔ ℓ))
where
infix 8 -_
infixl 7 _*_
infixl 6 _+_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_+_ : Op₂ Carrier
_*_ : Op₂ Carrier
-_ : Op₁ Carrier | record | Algebra | [
"Algebra.Bundles.Raw",
"Algebra.Core",
"Algebra.Structures",
"Relation.Binary.Core",
"Level"
] | Algebra/Bundles.agda | RingWithoutOne | |
NonAssociativeRing c ℓ: Set (suc (c ⊔ ℓ))
where
infix 8 -_
infixl 7 _*_
infixl 6 _+_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_+_ : Op₂ Carrier
_*_ : Op₂ Carrier
-_ : Op₁ Carrier | record | Algebra | [
"Algebra.Bundles.Raw",
"Algebra.Core",
"Algebra.Structures",
"Relation.Binary.Core",
"Level"
] | Algebra/Bundles.agda | NonAssociativeRing | |
Quasigroup c ℓ: Set (suc (c ⊔ ℓ))
where
infixl 7 _∙_
infixl 7 _\\_
infixl 7 _//_
infix 4 _≈_
field
Carrier : Set c
_≈_ : Rel Carrier ℓ
_∙_ : Op₂ Carrier
_\\_ : Op₂ Carrier
_//_ : Op₂ Carrier | record | Algebra | [
"Algebra.Bundles.Raw",
"Algebra.Core",
"Algebra.Structures",
"Relation.Binary.Core",
"Level"
] | Algebra/Bundles.agda | Quasigroup | |
Op₁ : ∀ {ℓ} → Set ℓ → Set ℓ | function | Algebra | [
"Level"
] | Algebra/Core.agda | Op₁ | |
Op₂ : ∀ {ℓ} → Set ℓ → Set ℓ | function | Algebra | [
"Level"
] | Algebra/Core.agda | Op₂ | |
Congruent₁ : Op₁ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Congruent₁ | |
Congruent₂ : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Congruent₂ | |
LeftCongruent : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | LeftCongruent | |
RightCongruent : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | RightCongruent | |
Associative : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Associative | |
Commutative : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Commutative | |
LeftIdentity : A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | LeftIdentity | |
RightIdentity : A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | RightIdentity | |
Identity : A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Identity | |
LeftZero : A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | LeftZero | |
RightZero : A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | RightZero | |
Zero : A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Zero | |
LeftInverse : A → Op₁ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | LeftInverse | |
RightInverse : A → Op₁ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | RightInverse | |
Inverse : A → Op₁ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Inverse | |
LeftInvertible : A → Op₂ A → A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | LeftInvertible | |
RightInvertible : A → Op₂ A → A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | RightInvertible | |
Invertible : A → Op₂ A → A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Invertible | |
LeftConical : A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | LeftConical | |
RightConical : A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | RightConical | |
Conical : A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Conical | |
_DistributesOverˡ_ : Op₂ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | _DistributesOverˡ_ | |
_DistributesOverʳ_ : Op₂ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | _DistributesOverʳ_ | |
_DistributesOver_ : Op₂ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | _DistributesOver_ | |
_MiddleFourExchange_ : Op₂ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | _MiddleFourExchange_ | |
_IdempotentOn_ : Op₂ A → A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | _IdempotentOn_ | |
Idempotent : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Idempotent | |
IdempotentFun : Op₁ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | IdempotentFun | |
Selective : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Selective | |
_Absorbs_ : Op₂ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | _Absorbs_ | |
Absorptive : Op₂ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Absorptive | |
SelfInverse : Op₁ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | SelfInverse | |
Involutive : Op₁ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Involutive | |
LeftCancellative : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | LeftCancellative | |
RightCancellative : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | RightCancellative | |
Cancellative : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Cancellative | |
AlmostLeftCancellative : A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | AlmostLeftCancellative | |
AlmostRightCancellative : A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | AlmostRightCancellative | |
AlmostCancellative : A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | AlmostCancellative | |
Interchangable : Op₂ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Interchangable | |
LeftDividesˡ : Op₂ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | LeftDividesˡ | |
LeftDividesʳ : Op₂ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | LeftDividesʳ | |
RightDividesˡ : Op₂ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | RightDividesˡ | |
RightDividesʳ : Op₂ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | RightDividesʳ | |
LeftDivides : Op₂ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | LeftDivides | |
RightDivides : Op₂ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | RightDivides | |
StarRightExpansive : A → Op₂ A → Op₂ A → Op₁ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | StarRightExpansive | |
StarLeftExpansive : A → Op₂ A → Op₂ A → Op₁ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | StarLeftExpansive | |
StarExpansive : A → Op₂ A → Op₂ A → Op₁ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | StarExpansive | |
StarLeftDestructive : Op₂ A → Op₂ A → Op₁ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | StarLeftDestructive | |
StarRightDestructive : Op₂ A → Op₂ A → Op₁ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | StarRightDestructive | |
StarDestructive : Op₂ A → Op₂ A → Op₁ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | StarDestructive | |
LeftAlternative : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | LeftAlternative | |
RightAlternative : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | RightAlternative | |
Alternative : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Alternative | |
Flexible : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Flexible | |
Medial : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Medial | |
LeftSemimedial : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | LeftSemimedial | |
RightSemimedial : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | RightSemimedial | |
Semimedial : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Semimedial | |
LeftBol : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | LeftBol | |
RightBol : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | RightBol | |
MiddleBol : Op₂ A → Op₂ A → Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | MiddleBol | |
Identical : Op₂ A → Set _ | function | Algebra | [
"Relation.Binary.Core",
"Algebra.Core",
"Data.Product.Base",
"Data.Sum.Base",
"Relation.Binary.Definitions",
"Relation.Nullary.Negation.Core"
] | Algebra/Definitions.agda | Identical | |
IsSuccessorSet (suc#: Op₁ A) (zero# : A) : Set (a ⊔ ℓ)
where
field
isEquivalence : IsEquivalence _≈_
suc#-cong : Congruent₁ suc#
open IsEquivalence isEquivalence public
setoid : Setoid a ℓ
setoid = record { isEquivalence = isEquivalence }
------------------------------------------------------------------------ | record | Algebra | [
"Relation.Binary.Core",
"Relation.Binary.Bundles",
"Relation.Binary.Structures",
"Algebra.Core",
"Algebra.Definitions _≈_",
"Algebra.Consequences.Setoid",
"Data.Product.Base",
"Level"
] | Algebra/Structures.agda | IsSuccessorSet | |
IsMagma (∙: Op₂ A) : Set (a ⊔ ℓ)
where
field
isEquivalence : IsEquivalence _≈_
∙-cong : Congruent₂ ∙
open IsEquivalence isEquivalence public
setoid : Setoid a ℓ
setoid = record { isEquivalence = isEquivalence }
open Consequences.Congruence setoid ∙-cong public | record | Algebra | [
"Relation.Binary.Core",
"Relation.Binary.Bundles",
"Relation.Binary.Structures",
"Algebra.Core",
"Algebra.Definitions _≈_",
"Algebra.Consequences.Setoid",
"Data.Product.Base",
"Level"
] | Algebra/Structures.agda | IsMagma | |
IsUnitalMagma (∙: Op₂ A) (ε : A) : Set (a ⊔ ℓ)
where
field
isMagma : IsMagma ∙
identity : Identity ε ∙
open IsMagma isMagma public
identityˡ : LeftIdentity ε ∙
identityˡ = proj₁ identity
identityʳ : RightIdentity ε ∙ | record | Algebra | [
"Relation.Binary.Core",
"Relation.Binary.Bundles",
"Relation.Binary.Structures",
"Algebra.Core",
"Algebra.Definitions _≈_",
"Algebra.Consequences.Setoid",
"Data.Product.Base",
"Level"
] | Algebra/Structures.agda | IsUnitalMagma | |
IsInvertibleMagma (_∙_: Op₂ A) (ε : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ)
where
field
isMagma : IsMagma _∙_
inverse : Inverse ε _⁻¹ _∙_
⁻¹-cong : Congruent₁ _⁻¹
open IsMagma isMagma public
inverseˡ : LeftInverse ε _⁻¹ _∙_
inverseˡ = proj₁ inverse
| record | Algebra | [
"Relation.Binary.Core",
"Relation.Binary.Bundles",
"Relation.Binary.Structures",
"Algebra.Core",
"Algebra.Definitions _≈_",
"Algebra.Consequences.Setoid",
"Data.Product.Base",
"Level"
] | Algebra/Structures.agda | IsInvertibleMagma | |
IsNearSemiring (+ *: Op₂ A) (0# : A) : Set (a ⊔ ℓ)
where
field
+-isMonoid : IsMonoid + 0#
*-cong : Congruent₂ *
*-assoc : Associative *
distribʳ : * DistributesOverʳ +
zeroˡ : LeftZero 0# *
open IsMonoid +-isMonoid public
renaming
( assoc to +-assoc | record | Algebra | [
"Relation.Binary.Core",
"Relation.Binary.Bundles",
"Relation.Binary.Structures",
"Algebra.Core",
"Algebra.Definitions _≈_",
"Algebra.Consequences.Setoid",
"Data.Product.Base",
"Level"
] | Algebra/Structures.agda | IsNearSemiring | |
IsSemiring (+ *: Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ)
where
field
isSemiringWithoutAnnihilatingZero :
IsSemiringWithoutAnnihilatingZero + * 0# 1#
zero : Zero 0# *
open IsSemiringWithoutAnnihilatingZero
isSemiringWithoutAnnihilatingZero public
isSemiringWithoutOne : IsSemiringWithoutOne + * 0#
isSemiringWithoutOne = record | record | Algebra | [
"Relation.Binary.Core",
"Relation.Binary.Bundles",
"Relation.Binary.Structures",
"Algebra.Core",
"Algebra.Definitions _≈_",
"Algebra.Consequences.Setoid",
"Data.Product.Base",
"Level"
] | Algebra/Structures.agda | IsSemiring | |
IsRingWithoutOne (+ *: Op₂ A) (-_ : Op₁ A) (0# : A) : Set (a ⊔ ℓ)
where
field
+-isAbelianGroup : IsAbelianGroup + 0# -_
*-cong : Congruent₂ *
*-assoc : Associative *
distrib : * DistributesOver +
open IsAbelianGroup +-isAbelianGroup public
renaming
( assoc to +-assoc
; ∙-cong to +-cong | record | Algebra | [
"Relation.Binary.Core",
"Relation.Binary.Bundles",
"Relation.Binary.Structures",
"Algebra.Core",
"Algebra.Definitions _≈_",
"Algebra.Consequences.Setoid",
"Data.Product.Base",
"Level"
] | Algebra/Structures.agda | IsRingWithoutOne | |
IsNonAssociativeRing (+ *: Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ)
where
field
+-isAbelianGroup : IsAbelianGroup + 0# -_
*-cong : Congruent₂ *
*-identity : Identity 1# *
distrib : * DistributesOver +
zero : Zero 0# *
open IsAbelianGroup +-isAbelianGroup public
renaming
( assoc to +-assoc | record | Algebra | [
"Relation.Binary.Core",
"Relation.Binary.Bundles",
"Relation.Binary.Structures",
"Algebra.Core",
"Algebra.Definitions _≈_",
"Algebra.Consequences.Setoid",
"Data.Product.Base",
"Level"
] | Algebra/Structures.agda | IsNonAssociativeRing | |
IsQuasigroup (∙ \\ //: Op₂ A) : Set (a ⊔ ℓ)
where
field
isMagma : IsMagma ∙
\\-cong : Congruent₂ \\
//-cong : Congruent₂ //
leftDivides : LeftDivides ∙ \\
rightDivides : RightDivides ∙ //
open IsMagma isMagma public
\\-congˡ : LeftCongruent \\ | record | Algebra | [
"Relation.Binary.Core",
"Relation.Binary.Bundles",
"Relation.Binary.Structures",
"Algebra.Core",
"Algebra.Definitions _≈_",
"Algebra.Consequences.Setoid",
"Data.Product.Base",
"Level"
] | Algebra/Structures.agda | IsQuasigroup | |
DoubleNegationElimination : ∀ ℓ → Set (suc ℓ) | function | Axiom | [
"Axiom.ExcludedMiddle",
"Level",
"Relation.Nullary.Decidable.Core",
"Relation.Nullary.Negation.Core"
] | Axiom/DoubleNegationElimination.agda | DoubleNegationElimination | |
em⇒dne : ExcludedMiddle ℓ → DoubleNegationElimination ℓ | function | Axiom | [
"Axiom.ExcludedMiddle",
"Level",
"Relation.Nullary.Decidable.Core",
"Relation.Nullary.Negation.Core"
] | Axiom/DoubleNegationElimination.agda | em⇒dne | |
dne⇒em : DoubleNegationElimination ℓ → ExcludedMiddle ℓ | function | Axiom | [
"Axiom.ExcludedMiddle",
"Level",
"Relation.Nullary.Decidable.Core",
"Relation.Nullary.Negation.Core"
] | Axiom/DoubleNegationElimination.agda | dne⇒em | |
ExcludedMiddle : ∀ ℓ → Set (suc ℓ) | function | Axiom | [
"Level",
"Relation.Nullary.Decidable.Core"
] | Axiom/ExcludedMiddle.agda | ExcludedMiddle |
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