### view agda/deltaM.agda @ 118:53cb21845dea

Prove association-law for DeltaM
author Yasutaka Higa Mon, 02 Feb 2015 11:54:23 +0900 f02c5ad4a327 0f9ecd118a03
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open import Level

open import basic
open import delta
open import delta.functor
open import nat
open import laws

module deltaM where

-- DeltaM definitions

data DeltaM {l : Level}
(M : Set l -> Set l)
{functorM : Functor M}
(A : Set l)
: (Nat -> Set l) where
deltaM : {n : Nat} -> Delta (M A) (S n) -> DeltaM M {functorM} {monadM} A (S n)

-- DeltaM utils

unDeltaM : {l : Level} {A : Set l} {n : Nat}
{M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} ->
(DeltaM M {functorM} {monadM} A (S n)) -> Delta (M A) (S n)
unDeltaM (deltaM d) = d

headDeltaM : {l : Level} {A : Set l} {n : Nat}
{M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM}
-> DeltaM M {functorM} {monadM} A (S n) -> M A

tailDeltaM :  {l : Level} {A : Set l} {n : Nat}
{M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM}
-> DeltaM {l} M {functorM} {monadM} A (S (S n)) -> DeltaM M {functorM} {monadM} A (S n)
tailDeltaM {_} {n} (deltaM d) = deltaM (tailDelta d)

appendDeltaM : {l : Level} {A : Set l} {n m : Nat}
{M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} ->
DeltaM M {functorM} {monadM} A (S n) ->
DeltaM M {functorM} {monadM} A (S m) ->
DeltaM M {functorM} {monadM} A ((S n) + (S m))
appendDeltaM (deltaM d) (deltaM dd) = deltaM (deltaAppend d dd)

dmap : {l : Level} {A B : Set l} {n : Nat}
{M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM} ->
(M A -> B) -> DeltaM M {functorM} {monadM} A (S n) -> Delta B (S n)
dmap f (deltaM d) = delta-fmap f d

-- functor definitions
open Functor
deltaM-fmap : {l : Level} {A B : Set l} {n : Nat}
{M : Set l -> Set l} {functorM : Functor M} {monadM : Monad M functorM}
-> (A -> B) -> DeltaM M {functorM} {monadM} A (S n) -> DeltaM M {functorM} {monadM} B (S n)
deltaM-fmap {l} {A} {B} {n} {M} {functorM} f (deltaM d) = deltaM (fmap delta-is-functor (fmap functorM f) d)

deltaM-eta : {l : Level} {A : Set l} {n : Nat}
{M : Set l -> Set l} {functorM : Functor M} {monadM   : Monad M functorM} ->
A -> (DeltaM M {functorM} {monadM} A (S n))
deltaM-eta {n = n} {monadM = mm} x = deltaM (delta-eta {n = n} (eta mm x))

deltaM-mu : {l : Level} {A : Set l} {n : Nat}
{M : Set l -> Set l} {functorM : Functor  M} {monadM   : Monad  M functorM} ->
DeltaM M {functorM} {monadM} (DeltaM M {functorM} {monadM} A (S n)) (S n)  ->
DeltaM M {functorM} {monadM} A (S n)
deltaM-mu {n = O}   {functorM = fm} {monadM = mm} d = deltaM (mono  (mu mm (fmap fm headDeltaM (headDeltaM d))))
deltaM-mu {n = S n} {functorM = fm} {monadM = mm} d = deltaM (delta (mu mm (fmap fm headDeltaM (headDeltaM d)))
(unDeltaM (deltaM-mu (deltaM-fmap tailDeltaM (tailDeltaM d)))))

deltaM-bind : {l : Level} {A B : Set l}
{n : Nat}
{M : Set l -> Set l}
{functorM : Functor M}