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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