module stack where

open import Relation.Binary.PropositionalEquality

data Nat : Set where
  zero : Nat
  suc  : Nat -> Nat

data Bool : Set where
  True  : Bool
  False : Bool

data Maybe (a : Set) : Set  where
  Nothing : Maybe a
  Just    : a -> Maybe a

record Stack {a t : Set} (stackImpl : Set) : Set  where
  field
    stack : stackImpl
    push : stackImpl -> a -> (stackImpl -> t) -> t
    pop  : stackImpl -> (stackImpl -> Maybe a -> t) -> t

data Element (a : Set) : Set where
  cons : a -> Maybe (Element a) -> Element a

datum : {a : Set} -> Element a -> a
datum (cons a _) = a

next : {a : Set} -> Element a -> Maybe (Element a)
next (cons _ n) = n


{-
-- cannot define recrusive record definition. so use linked list with maybe.
record Element {l : Level} (a : Set l) : Set (suc l) where
  field
    datum : a  -- `data` is reserved by Agda.
    next : Maybe (Element a)
-}



record SingleLinkedStack (a : Set) : Set where
  field
    top : Maybe (Element a)
open SingleLinkedStack

pushSingleLinkedStack : {Data t : Set} -> SingleLinkedStack Data -> Data -> (Code : SingleLinkedStack Data -> t) -> t
pushSingleLinkedStack stack datum next = next stack1
  where
    element = cons datum (top stack)
    stack1  = record {top = Just element}


popSingleLinkedStack : {a t : Set} -> SingleLinkedStack a -> (Code : SingleLinkedStack a -> (Maybe a) -> t) -> t
popSingleLinkedStack stack cs with (top stack)
...                                | Nothing = cs stack  Nothing
...                                | Just d  = cs stack1 (Just data1)
  where
    data1  = datum d
    stack1 = record { top = (next d) }


emptySingleLinkedStack : {a : Set} -> SingleLinkedStack a
emptySingleLinkedStack = record {top = Nothing}

createSingleLinkedStack : {a b : Set} -> Stack {a} {b} (SingleLinkedStack a)
createSingleLinkedStack = record { stack = emptySingleLinkedStack
                                 ; push = pushSingleLinkedStack
                                 ; pop  = popSingleLinkedStack
                                 }



test01 : {a : Set} -> SingleLinkedStack a -> Maybe a -> Bool
test01 stack _ with (top stack)
...                  | (Just _) = True
...                  | Nothing  = False


test02 : {a : Set} -> SingleLinkedStack a -> Bool
test02 stack = (popSingleLinkedStack stack) test01

test03 : {a : Set} -> a ->  Bool
test03 v = pushSingleLinkedStack emptySingleLinkedStack v test02


lemma : {A : Set} {a : A} -> test03 a ≡ False
lemma = refl

id : {A : Set} -> A -> A
id a = a


n-push : {A : Set} {a : A} -> Nat -> SingleLinkedStack A -> SingleLinkedStack A
n-push zero s            = s
n-push {A} {a} (suc n) s = pushSingleLinkedStack (n-push {A} {a} n s) a (\s -> s)

n-pop : {A : Set} {a : A} -> Nat -> SingleLinkedStack A -> SingleLinkedStack A
n-pop zero    s         = s
n-pop {A} {a} (suc n) s = popSingleLinkedStack (n-pop {A} {a} n s) (\s _ -> s)

open ≡-Reasoning

push-pop-equiv : {A : Set} {a : A} (s : SingleLinkedStack A) -> popSingleLinkedStack (pushSingleLinkedStack s a (\s -> s)) (\s _ -> s) ≡ s
push-pop-equiv s = refl

push-and-n-pop : {A : Set} {a : A} (n : Nat) (s : SingleLinkedStack A) -> n-pop {A} {a} (suc n) (pushSingleLinkedStack s a id) ≡ n-pop {A} {a} n s
push-and-n-pop zero s            = refl
push-and-n-pop {A} {a} (suc n) s = begin
  n-pop (suc (suc n)) (pushSingleLinkedStack s a id)
  ≡⟨ refl ⟩
  popSingleLinkedStack (n-pop (suc n) (pushSingleLinkedStack s a id)) (\s _ -> s)
  ≡⟨ cong (\s -> popSingleLinkedStack s (\s _ -> s)) (push-and-n-pop n s) ⟩ 
  popSingleLinkedStack (n-pop n s) (\s _ -> s)
  ≡⟨ refl ⟩
  n-pop (suc n) s
  ∎
  

n-push-pop-equiv : {A : Set} {a : A} (n : Nat) (s : SingleLinkedStack A) -> (n-pop {A} {a} n (n-push {A} {a} n s)) ≡ s
n-push-pop-equiv zero s            = refl
n-push-pop-equiv {A} {a} (suc n) s = begin
  n-pop (suc n) (n-push (suc n) s)
  ≡⟨ refl ⟩
  n-pop (suc n) (pushSingleLinkedStack (n-push n s) a (\s -> s))
  ≡⟨ push-and-n-pop n (n-push n s)  ⟩
  n-pop n (n-push n s)
  ≡⟨ n-push-pop-equiv n s\ ⟩
  s
  ∎