open import Level renaming (suc to succ ; zero to Zero )
module stackTest where

open import stack

open import Relation.Binary.PropositionalEquality
open import Relation.Binary.Core
open import Data.Nat
open import Function


open SingleLinkedStack
open Stack

----
--
-- proof of properties ( concrete cases )
--

test01 : {n : Level } {a : Set n} !$\rightarrow$! SingleLinkedStack a !$\rightarrow$! Maybe a !$\rightarrow$! Bool {n}
test01 stack _ with (top stack)
...                  | (Just _) = True
...                  | Nothing  = False


test02 : {n : Level } {a : Set n} !$\rightarrow$! SingleLinkedStack a !$\rightarrow$! Bool
test02 stack = popSingleLinkedStack stack test01

test03 : {n : Level } {a : Set n} !$\rightarrow$! a !$\rightarrow$!  Bool
test03 v = pushSingleLinkedStack emptySingleLinkedStack v test02

-- after a push and a pop, the stack is empty
lemma : {n : Level} {A : Set n} {a : A} !$\rightarrow$! test03 a !$\equiv$! False
lemma = refl

testStack01 : {n m : Level } {a : Set n} !$\rightarrow$! a !$\rightarrow$! Bool {m}
testStack01 v = pushStack createSingleLinkedStack v (
   \s !$\rightarrow$! popStack s (\s1 d1 !$\rightarrow$! True))

-- after push 1 and 2, pop2 get 1 and 2

testStack02 : {m : Level } !$\rightarrow$!  ( Stack  !$\mathbb{N}$! (SingleLinkedStack !$\mathbb{N}$!) !$\rightarrow$! Bool {m} ) !$\rightarrow$! Bool {m}
testStack02 cs = pushStack createSingleLinkedStack 1 (
   \s !$\rightarrow$! pushStack s 2 cs)


testStack031 : (d1 d2 : !$\mathbb{N}$! ) !$\rightarrow$! Bool {Zero}
testStack031 2 1 = True
testStack031 _ _ = False

testStack032 : (d1 d2 : Maybe !$\mathbb{N}$!) !$\rightarrow$! Bool {Zero}
testStack032  (Just d1) (Just d2) = testStack031 d1 d2
testStack032  _ _ = False

testStack03 : {m : Level } !$\rightarrow$! Stack  !$\mathbb{N}$! (SingleLinkedStack !$\mathbb{N}$!) !$\rightarrow$! ((Maybe !$\mathbb{N}$!) !$\rightarrow$! (Maybe !$\mathbb{N}$!) !$\rightarrow$! Bool {m} ) !$\rightarrow$! Bool {m}
testStack03 s cs = pop2Stack s (
   \s d1 d2 !$\rightarrow$! cs d1 d2 )

testStack04 : Bool
testStack04 = testStack02 (\s !$\rightarrow$! testStack03 s testStack032)

testStack05 : testStack04 !$\equiv$! True
testStack05 = refl

testStack06 : {m : Level } !$\rightarrow$! Maybe (Element !$\mathbb{N}$!)
testStack06 = pushStack createSingleLinkedStack 1 (
   \s !$\rightarrow$! pushStack s 2 (\s !$\rightarrow$! top (stack s)))

testStack07 : {m : Level } !$\rightarrow$! Maybe (Element !$\mathbb{N}$!)
testStack07 = pushSingleLinkedStack emptySingleLinkedStack 1 (
   \s !$\rightarrow$! pushSingleLinkedStack s 2 (\s !$\rightarrow$! top s))

testStack08 = pushSingleLinkedStack emptySingleLinkedStack 1 
   $ \s !$\rightarrow$! pushSingleLinkedStack s 2 
   $ \s !$\rightarrow$! pushSingleLinkedStack s 3 
   $ \s !$\rightarrow$! pushSingleLinkedStack s 4 
   $ \s !$\rightarrow$! pushSingleLinkedStack s 5 
   $ \s !$\rightarrow$! top s

------
--
-- proof of properties with indefinite state of stack
--
-- this should be proved by properties of the stack inteface, not only by the implementation,
--    and the implementation have to provides the properties.
--
--    we cannot write "s !$\equiv$! s3", since level of the Set does not fit , but use stack s !$\equiv$! stack s3 is ok.
--    anyway some implementations may result s != s3
--  

stackInSomeState : {l m : Level } {D : Set l} {t : Set m } (s : SingleLinkedStack D ) !$\rightarrow$! Stack {l} {m} D {t}  ( SingleLinkedStack  D )
stackInSomeState s =  record { stack = s ; stackMethods = singleLinkedStackSpec }

push!$\rightarrow$!push!$\rightarrow$!pop2 : {l : Level } {D : Set l} (x y : D ) (s : SingleLinkedStack D ) !$\rightarrow$!
    pushStack ( stackInSomeState s )  x ( \s1 !$\rightarrow$! pushStack s1 y ( \s2 !$\rightarrow$! pop2Stack s2 ( \s3 y1 x1 !$\rightarrow$! (Just x !$\equiv$! x1 ) !$\wedge$! (Just y !$\equiv$! y1 ) ) ))
push!$\rightarrow$!push!$\rightarrow$!pop2 {l} {D} x y s = record { pi1 = refl ; pi2 = refl }


-- id : {n : Level} {A : Set n} !$\rightarrow$! A !$\rightarrow$! A
-- id a = a

-- push a, n times

n-push : {n : Level} {A : Set n} {a : A} !$\rightarrow$! !$\mathbb{N}$! !$\rightarrow$! SingleLinkedStack A !$\rightarrow$! SingleLinkedStack A
n-push zero s            = s
n-push {l} {A} {a} (suc n) s = pushSingleLinkedStack (n-push {l} {A} {a} n s) a (\s !$\rightarrow$! s ) 

n-pop :  {n : Level}{A : Set n} {a : A} !$\rightarrow$! !$\mathbb{N}$! !$\rightarrow$! SingleLinkedStack A !$\rightarrow$! SingleLinkedStack A
n-pop zero    s         = s
n-pop  {_} {A} {a} (suc n) s = popSingleLinkedStack (n-pop {_} {A} {a} n s) (\s _ !$\rightarrow$! s )

open !$\equiv$!-Reasoning

push-pop-equiv : {n : Level} {A : Set n} {a : A} (s : SingleLinkedStack A) !$\rightarrow$! (popSingleLinkedStack (pushSingleLinkedStack s a (\s !$\rightarrow$! s)) (\s _ !$\rightarrow$! s) ) !$\equiv$! s
push-pop-equiv s = refl

push-and-n-pop : {n : Level} {A : Set n} {a : A} (n : !$\mathbb{N}$!) (s : SingleLinkedStack A) !$\rightarrow$! n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack s a id) !$\equiv$! 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 {_} {A} {a} (suc (suc n)) (pushSingleLinkedStack s a id)
  !$\equiv$!!$\langle$! refl !$\rangle$!
   popSingleLinkedStack (n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack s a id)) (\s _ !$\rightarrow$! s)
  !$\equiv$!!$\langle$! cong (\s !$\rightarrow$! popSingleLinkedStack s (\s _ !$\rightarrow$! s )) (push-and-n-pop n s) !$\rangle$! 
   popSingleLinkedStack (n-pop {_} {A} {a} n s) (\s _ !$\rightarrow$! s)
  !$\equiv$!!$\langle$! refl !$\rangle$!
    n-pop {_} {A} {a} (suc n) s
  !$\blacksquare$!
  

n-push-pop-equiv : {n : Level} {A : Set n} {a : A} (n : !$\mathbb{N}$!) (s : SingleLinkedStack A) !$\rightarrow$! (n-pop {_} {A} {a} n (n-push {_} {A} {a} n s)) !$\equiv$! s
n-push-pop-equiv zero s            = refl
n-push-pop-equiv {_} {A} {a} (suc n) s = begin
    n-pop {_} {A} {a} (suc n) (n-push (suc n) s)
  !$\equiv$!!$\langle$! refl !$\rangle$!
    n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack (n-push n s) a (\s !$\rightarrow$! s))
  !$\equiv$!!$\langle$! push-and-n-pop n (n-push n s)  !$\rangle$!
    n-pop {_} {A} {a} n (n-push n s)
  !$\equiv$!!$\langle$! n-push-pop-equiv n s !$\rangle$!
    s
  !$\blacksquare$!


n-push-pop-equiv-empty : {n : Level} {A : Set n} {a : A} !$\rightarrow$! (n : !$\mathbb{N}$!) !$\rightarrow$! n-pop {_} {A} {a} n (n-push {_} {A} {a} n emptySingleLinkedStack)  !$\equiv$! emptySingleLinkedStack
n-push-pop-equiv-empty n = n-push-pop-equiv n emptySingleLinkedStack