80 lines
2.4 KiB
Idris
80 lines
2.4 KiB
Idris
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module My.Structures
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%hide Semigroup
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%hide Monoid
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%hide empty
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infix 5 <->
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public export
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interface Setoid t where
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constructor MkSetoid
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(<->) : t -> t -> Type
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reflexivity : {0 a: t} -> a <-> a
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transitivity : {a,b,c: t} -> (a <-> b) -> (b <-> c) -> (a <-> c)
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symmetry : {a, b: t} -> (a <-> b) -> (b <-> a)
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public export
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trivialSetoid : (t: Type) -> Setoid t
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trivialSetoid _ = MkSetoid Equal Refl (\a,b => trans a b) (\a => sym a)
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public export
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AssociativityProof : {x: Type} -> Setoid x => (x -> x -> x) -> Type
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AssociativityProof {x} t = (a, b, c: x) -> (t (t a b) c) <-> (t a (t b c))
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public export
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RightIdentityProof : {x: Type} -> Setoid x => (x -> x -> x) -> x -> Type
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RightIdentityProof t e = (a: x) -> t a e <-> a
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public export
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LeftIdentityProof : {x: Type} -> Setoid x => (x -> x -> x) -> x -> Type
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LeftIdentityProof t e = (a: x) -> a <-> t e a
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public export
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RightInverseProof : {x: Type} -> Setoid x => (x -> x -> x) -> (x -> x) -> x -> Type
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RightInverseProof t inverse e = (a: x) -> t a (inverse a) <-> e
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public export
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LeftInverseProof : {x: Type} -> Setoid x => (x -> x -> x) -> (x -> x) -> x -> Type
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LeftInverseProof t inverse e = (a: x) -> t (inverse a) a <-> e
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public export
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CommutativityProof : {x: Type} -> Setoid x => (x -> x -> x) -> Type
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CommutativityProof t = (a, b: x) -> t a b <-> t b a
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public export
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rightToLeftIdentity :
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{x: Type} -> Setoid x =>
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(f: x -> x -> x) -> (e: x) ->
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CommutativityProof f ->
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RightIdentityProof f e ->
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LeftIdentityProof f e
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rightToLeftIdentity f e commutativity rightIdentity x = symmetry $ transitivity (commutativity e x) (rightIdentity x)
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public export
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leftToRightIdentity :
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{x: Type} -> Setoid x =>
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(f: x -> x -> x) -> (e: x) ->
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CommutativityProof f ->
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LeftIdentityProof f e ->
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RightIdentityProof f e
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leftToRightIdentity f e commutativity leftIdentity x = symmetry $ transitivity (leftIdentity x) (commutativity e x)
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public export
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interface Setoid t => Semigroup t where
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∘ : t -> t -> t
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associativityProof : AssociativityProof ∘
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public export
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interface Semigroup t => Monoid t where
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empty : t
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rightIdentityProof : RightIdentityProof ∘ empty
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leftIdentityProof : LeftIdentityProof ∘ empty
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public export
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interface Monoid t => Group t where
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⁻¹ : t -> t
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rightInverseProof : RightInverseProof ∘ (⁻¹) My.Structures.empty
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leftInverseProof : LeftInverseProof ∘ (⁻¹) My.Structures.empty
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