Data.Linear.Inverse(source)

This module is based on the content of the functional pearl
How to Take the Inverse of a Type
by Daniel Marshall and Dominic Orchard

Definitions

Inverse : Type -> Type

  Inverse is a shorthand for 'multiplicative skew inverse'

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mkInverse : (a -@ ()) -@ Inverse a

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divide : a -@ (Inverse a -@ ())

  And this is the morphism witnessing the multiplicative skew inverse

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doubleNegation : a -@ Inverse (Inverse a)

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maybeNeg : LMaybe a -@ (Inverse a -@ LMaybe (Inverse a))

  We only use the inverse of a if the LMaybe is actually Just.m

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lid : a -@ a

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lcompose : (b -@ c) -@ ((a -@ b) -@ (a -@ c))

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comap : (b -@ a) -@ (Inverse a -@ Inverse b)

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comapId : (u : Inverse a) -> comap lid u = lid u

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comapComp : (u : Inverse a) -> comap (lcompose g f) u = lcompose (comap f) (comap g) u

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munit : () -@ Inverse ()

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mmult : Inverse a -@ (Inverse b -@ Inverse (LPair a b))

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Pow : Type -> INTEGER -> Type

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zip : Pow a n -@ (Pow b n -@ Pow (LPair a b) n)

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Fixity Declaration: infixr operator, level 6

powPositiveL : (m : Nat) -> (n : Nat) -> Pow a (cast (m + n)) -@ LPair (Pow a (cast m)) (Pow a (cast n))

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powPositiveR : (m : Nat) -> (n : Nat) -> Pow a (cast m) -@ (Pow a (cast n) -@ Pow a (cast (m + n)))

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powNegateL : (m : Nat) -> Pow a (negate (cast m)) -> Pow (Inverse a) (cast m)

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powNegateR : (m : Nat) -> Pow (Inverse a) (cast m) -> Pow a (negate (cast m))

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powNegativeL : (m : Nat) -> (n : Nat) -> Pow a (negate (cast (m + n))) -@ LPair (Pow a (negate (cast m))) (Pow a (negate (cast n)))

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powNegativeR : (m : Nat) -> (n : Nat) -> Pow a (negate (cast m)) -@ (Pow a (negate (cast n)) -@ Pow a (negate (cast (m + n))))

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powAnnihilate : (m : Nat) -> Pow a (cast m) -@ (Pow a (negate (cast m)) -@ ())

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powMinus : (m : Nat) -> (n : Nat) -> Pow a (cast m) -@ (Pow a (negate (cast n)) -@ Pow a (cast m - cast n))

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