module stack-subtype-sample where

open import Level renaming (suc to S ; zero to O)
open import Function
open import Data.Nat
open import Data.Maybe
open import Relation.Binary.PropositionalEquality

open import stack-subtype @$\mathbb{N}$@
open import subtype Context  as N
open import subtype Meta     as M


record Num : Set where
  field
    num : @$\mathbb{N}$@

instance
  NumIsNormalDataSegment : N.DataSegment Num
  NumIsNormalDataSegment = record { get = (\c @$\rightarrow$@ record { num = Context.n c})
                                  ; set = (\c n @$\rightarrow$@ record c {n = Num.num n})}
  NumIsMetaDataSegment : M.DataSegment Num
  NumIsMetaDataSegment = record { get = (\m @$\rightarrow$@ record {num = Context.n (Meta.context m)})
                                ; set = (\m n @$\rightarrow$@ record m {context = record (Meta.context m) {n = Num.num n}})}


plus3 : Num @$\rightarrow$@ Num
plus3 record { num = n } = record {num = n + 3}

plus3CS : N.CodeSegment Num Num
plus3CS = N.cs plus3



plus5AndPushWithPlus3 : {mc : Meta} {{_ : N.DataSegment Num}}
               @$\rightarrow$@ M.CodeSegment Num (Meta)
plus5AndPushWithPlus3 {mc} {{nn}} = M.cs (\n @$\rightarrow$@ record {context = con n ; nextCS = (liftContext {{nn}} {{nn}} plus3CS) ; stack = st} )
  where
    co    = Meta.context mc
    con : Num @$\rightarrow$@ Context
    con record { num = num } = N.DataSegment.set nn co record {num = num + 5}
    st    = Meta.stack mc




push-sample : {{_ : N.DataSegment Num}} {{_ : M.DataSegment Num}} @$\rightarrow$@  Meta
push-sample {{nd}} {{md}} = M.exec {{md}} (plus5AndPushWithPlus3 {mc} {{nd}}) mc
  where
    con  = record { n = 4 ; element = just 0}
    code = N.cs (\c @$\rightarrow$@ c)
    mc   = record {context = con ; stack = emptySingleLinkedStack ; nextCS = code}


push-sample-equiv : push-sample @$\equiv$@ record { nextCS  = liftContext plus3CS
                                          ; stack   = record { top = nothing}
                                          ; context = record { n = 9} }
push-sample-equiv = refl


pushed-sample : {m : Meta} {{_ : N.DataSegment Num}} {{_ : M.DataSegment Num}} @$\rightarrow$@  Meta
pushed-sample {m} {{nd}} {{md}} = M.exec {{md}} (M.csComp {m} {{md}} pushSingleLinkedStackCS (plus5AndPushWithPlus3 {mc} {{nd}})) mc
  where
    con  = record { n = 4 ; element = just 0}
    code = N.cs (\c @$\rightarrow$@ c)
    mc   = record {context = con ; stack = emptySingleLinkedStack ; nextCS = code}



pushed-sample-equiv : {m : Meta} @$\rightarrow$@
                      pushed-sample {m} @$\equiv$@ record { nextCS  = liftContext plus3CS
                                                  ; stack   = record { top = just (cons 0 nothing) }
                                                  ; context = record { n   = 12} }
pushed-sample-equiv = refl



pushNum : N.CodeSegment Context Context
pushNum = N.cs pn
  where
    pn : Context @$\rightarrow$@ Context
    pn record { n = n } = record { n = pred n  ; element = just n}


pushOnce : Meta @$\rightarrow$@ Meta
pushOnce m = M.exec pushSingleLinkedStackCS m

n-push : {m : Meta} {{_ : M.DataSegment Meta}} (n : @$\mathbb{N}$@) @$\rightarrow$@ M.CodeSegment Meta Meta
n-push {{mm}} (zero)      = M.cs {{mm}} {{mm}} id
n-push {m} {{mm}} (suc n) = M.cs {{mm}} {{mm}} (\m @$\rightarrow$@ M.exec {{mm}} {{mm}} (n-push {m} {{mm}} n) (pushOnce m))

popOnce : Meta @$\rightarrow$@ Meta
popOnce m = M.exec popSingleLinkedStackCS m

n-pop : {m : Meta} {{_ : M.DataSegment Meta}} (n : @$\mathbb{N}$@) @$\rightarrow$@ M.CodeSegment Meta Meta
n-pop {{mm}} (zero)      = M.cs {{mm}} {{mm}} id
n-pop {m} {{mm}} (suc n) = M.cs {{mm}} {{mm}} (\m @$\rightarrow$@ M.exec {{mm}} {{mm}} (n-pop {m} {{mm}} n) (popOnce m))



initMeta : @$\mathbb{N}$@  @$\rightarrow$@ Maybe @$\mathbb{N}$@ @$\rightarrow$@ N.CodeSegment Context Context @$\rightarrow$@ Meta
initMeta n mn code = record { context = record { n = n ; element = mn}
                         ; stack   = emptySingleLinkedStack
                         ; nextCS  = code
                         }

n-push-cs-exec = M.exec (n-push {meta} 3) meta
  where
    meta = (initMeta 5 (just 9) pushNum)


n-push-cs-exec-equiv : n-push-cs-exec @$\equiv$@ record { nextCS  = pushNum
                                                ; context = record {n = 2 ; element = just 3}
                                                ; stack   = record {top = just (cons 4 (just (cons 5 (just (cons 9 nothing)))))}}
n-push-cs-exec-equiv = refl


n-pop-cs-exec = M.exec (n-pop {meta} 4) meta
  where
    meta = record { nextCS  = N.cs id
                  ; context = record { n = 0 ; element = nothing}
                  ; stack   = record {top = just (cons 9 (just (cons 8 (just (cons 7 (just (cons 6 (just (cons 5 nothing)))))))))}
                  }

n-pop-cs-exec-equiv : n-pop-cs-exec @$\equiv$@ record { nextCS  = N.cs id
                                              ; context = record { n = 0 ; element = just 6}
                                              ; stack   = record { top = just (cons 5 nothing)}
                                              }

n-pop-cs-exec-equiv = refl


open @$\equiv$@-Reasoning

id-meta : @$\mathbb{N}$@ @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ SingleLinkedStack @$\mathbb{N}$@ @$\rightarrow$@ Meta
id-meta n e s = record { context = record {n = n ; element = just e}
                       ; nextCS = (N.cs id) ; stack = s}

exec-comp : (f g : M.CodeSegment Meta Meta) (m : Meta) @$\rightarrow$@ M.exec (M.csComp {m} f g) m @$\equiv$@ M.exec f (M.exec g m)
exec-comp (M.cs x) (M.cs _) m = refl


push-pop-type : @$\mathbb{N}$@ @$\rightarrow$@ @$\mathbb{N}$@  @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ Element @$\mathbb{N}$@ @$\rightarrow$@ Set@$\text{1}$@
push-pop-type n e x s = M.exec (M.csComp {meta} (M.cs popOnce) (M.cs pushOnce)) meta @$\equiv$@ meta
  where
    meta = id-meta n e record {top = just (cons x (just s))}

push-pop : (n e x : @$\mathbb{N}$@) @$\rightarrow$@ (s : Element @$\mathbb{N}$@) @$\rightarrow$@  push-pop-type n e x s
push-pop n e x s = refl



pop-n-push-type : @$\mathbb{N}$@ @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ SingleLinkedStack @$\mathbb{N}$@ @$\rightarrow$@ Set@$\text{1}$@
pop-n-push-type n cn ce s = M.exec (M.csComp {meta} (M.cs popOnce) (n-push {meta} (suc n))) meta
                         @$\equiv$@ M.exec (n-push {meta} n) meta
  where
    meta = id-meta cn ce s

pop-n-push : (n cn ce : @$\mathbb{N}$@) @$\rightarrow$@ (s : SingleLinkedStack @$\mathbb{N}$@) @$\rightarrow$@ pop-n-push-type n cn ce s

pop-n-push zero cn ce s    = refl
pop-n-push (suc n) cn ce s = begin
  M.exec (M.csComp {id-meta cn ce s} (M.cs popOnce) (n-push {id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))})} (suc (suc n)))) (id-meta cn ce s)
  @$\equiv$@@$\langle$@ refl @$\rangle$@
  M.exec (M.csComp {id-meta cn ce s} (M.cs popOnce) (M.csComp {id-meta cn ce s} (n-push {id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))})} (suc n)) (M.cs pushOnce))) (id-meta cn ce s)
  @$\equiv$@@$\langle$@ exec-comp (M.cs popOnce) (M.csComp {id-meta cn ce s} (n-push {id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))})} (suc n)) (M.cs pushOnce)) (id-meta cn ce s) @$\rangle$@                        
  M.exec (M.cs popOnce) (M.exec (M.csComp {id-meta cn ce s} (n-push {id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))})} (suc n)) (M.cs pushOnce)) (id-meta cn ce s))
  @$\equiv$@@$\langle$@ cong (\x @$\rightarrow$@ M.exec (M.cs popOnce) x) (exec-comp (n-push {id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))})} (suc n)) (M.cs pushOnce) (id-meta cn ce s)) @$\rangle$@
  M.exec (M.cs popOnce) (M.exec (n-push {id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))})} (suc n))(M.exec (M.cs pushOnce) (id-meta cn ce s)))
  @$\equiv$@@$\langle$@ refl @$\rangle$@                                                                                                          
  M.exec (M.cs popOnce) (M.exec (n-push {id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))})} (suc n)) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))})))
  @$\equiv$@@$\langle$@ sym (exec-comp (M.cs popOnce) (n-push {id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))})} (suc n)) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}))) @$\rangle$@
  M.exec (M.csComp {id-meta cn ce s} (M.cs popOnce) (n-push {id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))})} (suc n))) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}))
  @$\equiv$@@$\langle$@ pop-n-push n cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}) @$\rangle$@
  M.exec (n-push n) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}))
  @$\equiv$@@$\langle$@ refl @$\rangle$@
  M.exec (n-push n) (pushOnce (id-meta cn ce s))
  @$\equiv$@@$\langle$@ refl @$\rangle$@
  M.exec (n-push n) (M.exec (M.cs pushOnce) (id-meta cn ce s))
  @$\equiv$@@$\langle$@ refl @$\rangle$@
  M.exec (n-push {id-meta cn ce s} (suc n)) (id-meta cn ce s)
  @$\blacksquare$@



n-push-pop-type : @$\mathbb{N}$@ @$\rightarrow$@  @$\mathbb{N}$@  @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ SingleLinkedStack @$\mathbb{N}$@ @$\rightarrow$@ Set@$\text{1}$@
n-push-pop-type n cn ce st = M.exec (M.csComp {meta} (n-pop {meta} n) (n-push {meta} n)) meta @$\equiv$@ meta
  where
    meta = id-meta cn ce st

n-push-pop : (n cn ce : @$\mathbb{N}$@) @$\rightarrow$@ (s : SingleLinkedStack @$\mathbb{N}$@) @$\rightarrow$@ n-push-pop-type n cn ce s
n-push-pop zero    cn ce s = refl
n-push-pop (suc n) cn ce s = begin
  M.exec (M.csComp {id-meta cn ce s} (n-pop {id-meta cn ce s}  (suc n)) (n-push {id-meta cn ce s} (suc n))) (id-meta cn ce s)
  @$\equiv$@@$\langle$@ refl @$\rangle$@
  M.exec (M.csComp  {id-meta cn ce s}  (M.cs (\m @$\rightarrow$@ M.exec (n-pop {id-meta cn ce s} n) (popOnce m))) (n-push {id-meta cn ce s} (suc n))) (id-meta cn ce s)
  @$\equiv$@@$\langle$@ exec-comp (M.cs (\m @$\rightarrow$@ M.exec (n-pop n) (popOnce m))) (n-push {id-meta cn ce s}  (suc n))  (id-meta cn ce s) @$\rangle$@
  M.exec (M.cs (\m @$\rightarrow$@ M.exec (n-pop  {id-meta cn ce s}  n) (popOnce m))) (M.exec (n-push {id-meta cn ce s} (suc n)) (id-meta cn ce s))
  @$\equiv$@@$\langle$@ refl @$\rangle$@
  M.exec (n-pop n) (popOnce (M.exec (n-push {id-meta cn ce s} (suc n)) (id-meta cn ce s)))
  @$\equiv$@@$\langle$@ refl @$\rangle$@
  M.exec (n-pop n) (M.exec (M.cs popOnce) (M.exec (n-push {id-meta cn ce s} (suc n)) (id-meta cn ce s)))
  @$\equiv$@@$\langle$@ cong (\x @$\rightarrow$@ M.exec (n-pop  {id-meta cn ce s}  n) x) (sym (exec-comp (M.cs popOnce) (n-push {id-meta cn ce s}  (suc n)) (id-meta cn ce s))) @$\rangle$@
  M.exec (n-pop n) (M.exec (M.csComp {id-meta cn ce s} (M.cs popOnce) (n-push {id-meta cn ce s} (suc n))) (id-meta cn ce s))
  @$\equiv$@@$\langle$@ cong (\x @$\rightarrow$@ M.exec (n-pop {id-meta cn ce s}  n) x) (pop-n-push n cn ce s) @$\rangle$@
  M.exec (n-pop n) (M.exec (n-push n) (id-meta cn ce s))
  @$\equiv$@@$\langle$@ sym (exec-comp (n-pop n) (n-push n) (id-meta cn ce s)) @$\rangle$@
  M.exec (M.csComp (n-pop n) (n-push n)) (id-meta cn ce s)
  @$\equiv$@@$\langle$@ n-push-pop n cn ce s @$\rangle$@
  id-meta cn ce s
  @$\blacksquare$@