module whileTestGears where

open import Function
open import Data.Nat
open import Data.Bool hiding ( _≟_ ; _≤?_ ; _≤_ ; _<_)
open import Level renaming ( suc to succ ; zero to Zero )
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Relation.Binary.PropositionalEquality

open import utilities
open  _/\_

record Env : Set (succ Zero) where
  field
    varn : ℕ
    vari : ℕ
open Env 

whileTest : {l : Level} {t : Set l}  → (c10 : ℕ) → (Code : Env → t) → t
whileTest c10 next = next (record {varn = c10 ; vari = 0 } )

{-# TERMINATING #-}
whileLoop : {l : Level} {t : Set l} → Env → (Code : Env → t) → t
whileLoop env next with lt 0 (varn env)
whileLoop env next | false = next env
whileLoop env next | true =
    whileLoop (record env {varn = (varn env) - 1 ; vari = (vari env) + 1}) next

test1 : Env 
test1 = whileTest 10 (λ env → whileLoop env (λ env1 → env1 ))


proof1 : whileTest 10 (λ env → whileLoop env (λ e → (vari e) ≡ 10 ))
proof1 = refl

--                                                                              ↓PostCondition
whileTest' : {l : Level} {t : Set l}  →  {c10 :  ℕ } → (Code : (env : Env )  → ((vari env) ≡ 0) /\ ((varn env) ≡ c10) → t) → t
whileTest' {_} {_}  {c10} next = next env proof2
  where
    env : Env 
    env = record {vari = 0 ; varn = c10 }
    proof2 : ((vari env) ≡ 0) /\ ((varn env) ≡ c10) -- PostCondition
    proof2 = record {pi1 = refl ; pi2 = refl}

open import Data.Empty
open import Data.Nat.Properties


{-# TERMINATING #-} --                                                  ↓PreCondition(Invaliant)
whileLoop' : {l : Level} {t : Set l} → (env : Env ) → {c10 :  ℕ } → ((varn env) + (vari env) ≡ c10) → (Code : Env  → t) → t
whileLoop' env proof next with  ( suc zero  ≤? (varn  env) )
whileLoop' env proof next | no p = next env 
whileLoop' env {c10} proof next | yes p = whileLoop' env1 (proof3 p ) next
    where
      env1 = record env {varn = (varn  env) - 1 ; vari = (vari env) + 1}
      1<0 : 1 ≤ zero → ⊥
      1<0 ()
      proof3 : (suc zero  ≤ (varn  env))  → varn env1 + vari env1 ≡ c10
      proof3 (s≤s lt) with varn  env
      proof3 (s≤s z≤n) | zero = ⊥-elim (1<0 p)
      proof3 (s≤s (z≤n {n'}) ) | suc n =  let open ≡-Reasoning  in
          begin
             n' + (vari env + 1) 
          ≡⟨ cong ( λ z → n' + z ) ( +-sym  {vari env} {1} )  ⟩
             n' + (1 + vari env ) 
          ≡⟨ sym ( +-assoc (n')  1 (vari env) ) ⟩
             (n' + 1) + vari env 
          ≡⟨ cong ( λ z → z + vari env )  +1≡suc  ⟩
             (suc n' ) + vari env 
          ≡⟨⟩
             varn env + vari env
          ≡⟨ proof  ⟩
             c10
          ∎

-- Condition to Invariant
conversion1 : {l : Level} {t : Set l } → (env : Env ) → {c10 :  ℕ } → ((vari env) ≡ 0) /\ ((varn env) ≡ c10)
               → (Code : (env1 : Env ) → (varn env1 + vari env1 ≡ c10) → t) → t
conversion1 env {c10} p1 next = next env proof4
   where
      proof4 : varn env + vari env ≡ c10
      proof4 = let open ≡-Reasoning  in
          begin
            varn env + vari env
          ≡⟨ cong ( λ n → n + vari env ) (pi2 p1 ) ⟩
            c10 + vari env
          ≡⟨ cong ( λ n → c10 + n ) (pi1 p1 ) ⟩
            c10 + 0
          ≡⟨ +-sym {c10} {0} ⟩
            c10
          ∎


proofGears : {c10 :  ℕ } → Set
proofGears {c10} = whileTest' {_} {_} {c10} (λ n p1 →  conversion1 n p1 (λ n1 p2 → whileLoop' n1 p2 (λ n2 →  ( vari n2 ≡ c10 )))) 


-- proofGearsMeta : {c10 :  ℕ } →  proofGears {c10}
-- proofGearsMeta {c10} = {!!} -- net yet done

--
--      openended Env c  <=>  Context
--

open import Relation.Nullary
open import Relation.Binary

record Envc : Set (succ Zero) where
  field
    c10 : ℕ
    varn : ℕ
    vari : ℕ
open Envc 

whileTestP : {l : Level} {t : Set l} → (c10 : ℕ) → (Code : Envc → t) → t
whileTestP c10 next = next (record {varn = c10 ; vari = 0 ; c10 = c10 } )

whileLoopP : {l : Level} {t : Set l} → Envc → (next : Envc → t) → (exit : Envc → t) → t
whileLoopP env next exit with <-cmp 0 (varn env)
whileLoopP env next exit | tri≈ ¬a b ¬c = exit env
whileLoopP env next exit | tri< a ¬b ¬c = 
     next (record env {varn = (varn env) - 1 ; vari = (vari env) + 1 }) 

{-# TERMINATING #-}
loopP : {l : Level} {t : Set l} → Envc → (exit : Envc → t) → t
loopP env exit = whileLoopP env (λ env → loopP env exit ) exit

whileTestPCall : (c10 :  ℕ ) → Envc
whileTestPCall c10 = whileTestP {_} {_} c10 (λ env → loopP env (λ env →  env))

data whileTestState  : Set where
  s1 : whileTestState
  s2 : whileTestState
  sf : whileTestState

whileTestStateP : whileTestState → Envc →  Set 
whileTestStateP s1 env = (vari env ≡ 0) /\ (varn env ≡ c10 env) 
whileTestStateP s2 env = (varn env + vari env ≡ c10 env) 
whileTestStateP sf env = (vari env ≡ c10 env) 

whileTestPwP : {l : Level} {t : Set l} → (c10 : ℕ) → ((env : Envc ) → whileTestStateP s1 env → t) → t
whileTestPwP c10 next = next env record { pi1 = refl ; pi2 = refl } where
   env : Envc
   env = whileTestP c10 ( λ env → env )

whileLoopPwP : {l : Level} {t : Set l}   → (env : Envc ) → whileTestStateP s2 env
    → (next : (env : Envc ) → whileTestStateP s2 env  → t)
    → (exit : (env : Envc ) → whileTestStateP sf env  → t) → t
whileLoopPwP env s next exit with <-cmp 0 (varn env)
whileLoopPwP env s next exit | tri≈ ¬a b ¬c = exit env (lem (sym b) s)
  where
    lem : (varn env ≡ 0) → (varn env + vari env ≡ c10 env) → vari env ≡ c10 env
    lem p1 p2 rewrite p1 = p2

whileLoopPwP env s next exit | tri< a ¬b ¬c  = next (record env {varn = (varn env) - 1 ; vari = (vari env) + 1 }) (proof5 a)
  where
    1<0 : 1 ≤ zero → ⊥
    1<0 ()
    proof5 : (suc zero  ≤ (varn  env))  → (varn env - 1) + (vari env + 1) ≡ c10 env
    proof5 (s≤s lt) with varn  env
    proof5 (s≤s z≤n) | zero = ⊥-elim (1<0 a)
    proof5 (s≤s (z≤n {n'}) ) | suc n = let open ≡-Reasoning in
      begin
        n' + (vari env + 1)
      ≡⟨ cong ( λ z → n' + z ) ( +-sym  {vari env} {1} )  ⟩
        n' + (1 + vari env )
      ≡⟨ sym ( +-assoc (n')  1 (vari env) ) ⟩
        (n' + 1) + vari env
      ≡⟨ cong ( λ z → z + vari env )  +1≡suc  ⟩
        (suc n' ) + vari env
      ≡⟨⟩
        varn env + vari env
      ≡⟨ s  ⟩
         c10 env
      ∎

{-# TERMINATING #-}
loopPwP : {l : Level} {t : Set l} → (env : Envc ) → whileTestStateP s2 env → (exit : (env : Envc ) → whileTestStateP sf env → t) → t
loopPwP env s exit = whileLoopPwP env s (λ env s → loopPwP env s exit ) exit

whileTestPCallwP : (c :  ℕ ) → Set
whileTestPCallwP c = whileTestPwP {_} {_} c ( λ env s → loopPwP env (conv env s) ( λ env s → vari env ≡ c )  ) where
    conv : (env : Envc ) → (vari env ≡ 0) /\ (varn env ≡ c10 env) → varn env + vari env ≡ c10 env
    conv e record { pi1 = refl ; pi2 = refl } = +zero


conv1 : (env : Envc ) → (vari env ≡ 0) /\ (varn env ≡ c10 env) → varn env + vari env ≡ c10 env
conv1 e record { pi1 = refl ; pi2 = refl } = +zero

-- = whileTestPwP (suc c) (λ env s → loopPwP env (conv1 env s) (λ env₁ s₁ → {!!}))

{-# TERMINATING #-}
Proof : (c :  ℕ ) → whileTestPCallwP c
Proof zero = whileTestPwP {_} {_} zero ( λ env s → loopPwP env (conv env s) ( λ env s → refl)  )
  where
    conv : (env : Envc ) → (vari env ≡ 0) /\ (varn env ≡ c10 env) → varn env + vari env ≡ c10 env
    conv e record { pi1 = refl ; pi2 = refl } = +zero
Proof (suc c) = whileTestPwP {_} {_} c ( λ env s → loopPwP env (conv env s) ( λ env s → lem ))
  where
    lem : whileLoopPwP (record { c10 = suc c ; varn = c ; vari = 0 + 1 }) ({!!})
      (λ env s → loopPwP env s (λ env₁ s₁ → vari env₁ ≡ suc c)) (λ env s3 → {!!})
    lem = {!!}
    conv : (env : Envc ) → (vari env ≡ 0) /\ (varn env ≡ c10 env) → varn env + vari env ≡ c10 env
    conv e record { pi1 = refl ; pi2 = refl } = +zero


{--
-- whileLoopPwP : {l : Level} {t : Set l}   → (env : Envc ) → whileTestStateP s2 env
--     → (next : (env : Envc ) → whileTestStateP s2 env  → t)
--     → (exit : (env : Envc ) → whileTestStateP sf env  → t) → t

next : (whileTestGears.proof5
     (record { c10 = suc c ; varn = suc c ; vari = 0 })
     (<ᵇ⇒< 0 (suc c) Agda.Builtin.Unit.⊤.tt)
     (λ x →
        Relation.Nullary.Reflects.invert (ofⁿ (λ ())) (≡⇒≡ᵇ 0 (suc c) x))
     (<⇒≯ (<ᵇ⇒< 0 (suc c) Agda.Builtin.Unit.⊤.tt))
     (whileTestGears.conv (suc c) (whileTestP (suc c) (λ env₁ → env₁))
      (record { pi1 = refl ; pi2 = refl }))
     (λ env₁ s₁ → loopPwP env₁ s₁ (λ env₂ s₂ → vari env₂ ≡ suc c))
     (λ env₁ s₁ → vari env₁ ≡ suc c)
     (<ᵇ⇒< 0 (suc c) Agda.Builtin.Unit.⊤.tt))

exit : (whileTestGears.conv (suc c) (whileTestP (suc c) (λ env₁ → env₁))
      (record { pi1 = refl ; pi2 = refl }))
     (λ env₁ s₁ → loopPwP env₁ s₁ (λ env₂ s₂ → vari env₂ ≡ suc c))
     (λ env₁ s₁ → vari env₁ ≡ suc c)
     (<ᵇ⇒< 0 (suc c) Agda.Builtin.Unit.⊤.tt))
    (λ env₁ s₁ → loopPwP env₁ s₁ (λ env₂ s₂ → vari env₂ ≡ suc c))
    (λ env₁ s₁ → vari env₁ ≡ suc c)

    | (<-cmp 0 c
       | Relation.Nullary.Decidable.Core.map′ (≡ᵇ⇒≡ 0 c) (≡⇒≡ᵇ 0 c)
         (Data.Bool.Properties.T? (0 ≡ᵇ c))
       | Data.Bool.Properties.T? (0 <ᵇ c))

--}