Data.Container.Base.Morphism.Instances

Definitions

pushDown : Cont -> Cont

  If we model the idea of a container (S !> P) as a box
   ┌──────┐
   │ s:S  │
   ├──────┤
   │  Ps  │
   └──────┘
  then `pushDown` is interpreted as keeping the box frame in place while
  pushing down the container: pruning any part of the container that goes 
  out of the box, and using `Unit` for anything new that appears:
   ┌──────┐
   │ Unit │
   ├──────┤
   │ s:S  │
   └──────┘
      Ps

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pushIntoContinuation : (d >< p) =%> l -> p =%> (pushDown d >@ l)

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terminal : c =%> UnitCont

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leftUnit : (Scalar >< c) =%> c

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rightUnit : (c >< Scalar) =%> c

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leftUnitInv : c =%> (Scalar >< c)

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rightUnitInv : c =%> (c >< Scalar)

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assocL : ((a >< b) >< c) =%> (a >< (b >< c))

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assocR : (a >< (b >< c)) =%> ((a >< b) >< c)

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swap : (a >< b) =%> (b >< a)

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swapMiddle : ((c1 >< c2) >< (c3 >< c4)) =%> ((c1 >< c3) >< (c2 >< c4))

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leftUnit : (Scalar >@ c) =%> c

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rightUnit : (c >@ Scalar) =%> c

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leftUnitInv : c =%> (Scalar >@ c)

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rightUnitInv : c =%> (c >@ Scalar)

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assocL : ((a >@ b) >@ c) =%> (a >@ (b >@ c))

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assocR : (a >@ (b >@ c)) =%> ((a >@ b) >@ c)

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elim : (c >+< c) =%> c

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initial : Empty =%> c

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cojoin : c =%> z -> d =%> z -> (c >+< d) =%> z

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direct : (a .Shp -> Bool) -> a =%> (a >+< a)

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duoidal : ((c >@ d) >< (e >@ f)) =%> ((c >< e) >@ (d >< f))

  Interaction between composition and tensor product
  Swaps the operations, and middle two containers
  Not an isomorphism!

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tensorToComp : (c >< f) =%> (c >@ f)

  Tensor product embeds into composition
  A special case of `duoidal`

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distribute : (c >< e) =%> s -> (c >< (e >@ g)) =%> (s >@ g)

  Specific distributive law we need

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State : Cont -> Type

  "State" as defined in https://arxiv.org/abs/2403.13001 and open games 
  
        ┌─────────────┐
        │             ├──► (x : c.Shp)
        │    State    │
        │             ├◄── c.Pos x
        └─────────────┘

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toState : c .Shp -> State c

  Given a shape of any container, state can be defined

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fromState : State c -> c .Shp

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mapState : State c -> c =%> d -> State d

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Costate : Cont -> Type

  "Costate" as defined in https://arxiv.org/abs/2403.13001 and open games 
  
                   ┌─────────────┐
   (x : c.Shp)  ──►┤             │
                   │   Costate   │
      c.Pos x   ◄──┤             │
                   └─────────────┘

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toCostate : ((x : c .Shp) -> c .Pos x) -> Costate c

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fromCostate : Costate c -> (x : c .Shp) -> c .Pos x

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mapCostate : Costate d -> c =%> d -> Costate c

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pairCostate : Costate c -> Costate d -> Costate (c >< d)

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fromNapCostateToState : Costate (Nap (c .Shp)) -> State c

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fromStateToNapCostate : State c -> Costate (Nap (c .Shp))

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wrapIntoVector : c =%> d -> Tensor [c] =%> Tensor [d]

  Wraps a dependent lens `c =%> d`
  into one of type `c >@ Scalar =%> d >@ Scalar`
  Needed because `c >@ Scalar` isn't automatically reduced to `c`

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wrapIntoMatrix : (c >@ c') =%> (d >@ d') -> Tensor [c, c'] =%> Tensor [d, d']

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wrapIntoVectorHancock : c =%> d -> HancockTensor [c] =%> HancockTensor [d]

  Wraps a dependent lens `c =%> d`
  into one of type `c >< Scalar =%> d >< Scalar`
  Needed because `c >< Scalar` isn't automatically reduced to `c`

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pick : Pick c a =%> c

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atShape : (s : c .Shp) -> At c s =%> c

  A lens from a container specialised to a shape to the container itself

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vectToList1 : Vect (S k) =%> List1

  A vector of positive length is a non-empty list

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handleEffect : Monad m => (m <!> effect) =%> Scalar -> a =%> effect -> (m <!> a) =%> Scalar

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inorderBackward : (b : BinTreeShape) -> Fin (numNodesAndLeaves b) -> BinTreePos b

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inorder : BinTree =%> List

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inorderBackward : (b : BinTreeShape) -> Fin (numNodes b) -> BinTreePosNode b

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inorder : BinTreeNode =%> List

  Traverses a binary tree container in order, producing a list container

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preorderBinTreeNode : (b : BinTreeShape) -> Fin (numNodes b) -> BinTreePosNode b

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inorderBackward : (b : BinTreeShape) -> Fin (numLeaves b) -> BinTreePosLeaf b

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inorder : BinTreeLeaf =%> List

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vectToList : Vect n =%> List

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maybeToList : Maybe =%> List

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