open import Level
open import Relation.Binary.PropositionalEquality

module subtype {l : Level} (Context : Set l) where


record DataSegment {ll : Level} (A : Set ll) : Set (l !$\sqcup$! ll) where
  field
    get : Context !$\rightarrow$! A
    set : Context !$\rightarrow$! A !$\rightarrow$! Context
open DataSegment

data CodeSegment {l1 l2 : Level} (A : Set l1) (B : Set l2) : Set (l !$\sqcup$! l1 !$\sqcup$! l2) where
  cs : {{_ : DataSegment A}} {{_ : DataSegment B}} !$\rightarrow$! (A !$\rightarrow$! B) !$\rightarrow$! CodeSegment A B

goto : {l1 l2 : Level} {I : Set l1} {O : Set l2} !$\rightarrow$! CodeSegment I O !$\rightarrow$! I !$\rightarrow$! O
goto (cs b) i = b i

exec : {l1 l2 : Level} {I : Set l1} {O : Set l2} {{_ : DataSegment I}} {{_ : DataSegment O}}
     !$\rightarrow$! CodeSegment I O !$\rightarrow$! Context !$\rightarrow$! Context
exec {l} {{i}} {{o}}  (cs b) c = set o c (b (get i c))


comp : {con : Context} !$\rightarrow$! {l1 l2 l3 l4 : Level}
       {A : Set l1} {B : Set l2} {C : Set l3} {D : Set l4}
       {{_ : DataSegment A}} {{_ : DataSegment B}} {{_ : DataSegment C}} {{_ : DataSegment D}}
       !$\rightarrow$! (C !$\rightarrow$! D) !$\rightarrow$! (A !$\rightarrow$! B) !$\rightarrow$! A !$\rightarrow$! D
comp {con} {{i}} {{io}} {{oi}} {{o}} g f x = g (get oi (set io con (f x)))

csComp : {con : Context} !$\rightarrow$! {l1 l2 l3 l4 : Level}
        {A : Set l1} {B : Set l2} {C : Set l3} {D : Set l4}
         {{_ : DataSegment A}} {{_ : DataSegment B}} {{_ : DataSegment C}} {{_ : DataSegment D}}
       !$\rightarrow$! CodeSegment C D !$\rightarrow$! CodeSegment A B !$\rightarrow$! CodeSegment A D
csComp {con} {A} {B} {C} {D} {{da}} {{db}} {{dc}} {{dd}} (cs g) (cs f)
      = cs {{da}} {{dd}} (comp {con} {{da}} {{db}} {{dc}} {{dd}} g f)



comp-associative : {A B C D E F : Set l} {con : Context}
                   {{da : DataSegment A}} {{db : DataSegment B}} {{dc : DataSegment C}}
                   {{dd : DataSegment D}} {{de : DataSegment E}} {{df : DataSegment F}}
                   !$\rightarrow$! (a : CodeSegment A B) (b : CodeSegment C D) (c : CodeSegment E F)
                   !$\rightarrow$! csComp {con} c (csComp {con} b a)  !$\equiv$! csComp {con} (csComp {con} c b) a
comp-associative (cs _) (cs _) (cs _) = refl