module whileTestGears1 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 where
  field
    varn : ℕ
    vari : ℕ
open Env

whileTestS : { m : Level}  → (c10 : ℕ) → (Code : Env → Set m) → Set m
whileTestS c10 next = next (record {varn = c10 ; vari = 0} )

whileTestS1 :  (c10 : ℕ) →  whileTestS c10 (λ e → ((varn e ≡ c10) /\ (vari e ≡ 0 )) )
whileTestS1 c10 = record { pi1 = refl ; pi2 = refl }


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 {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

lemma1 : {i : ℕ} →  ¬ 1 ≤ i → i ≡ 0
lemma1 {zero} not = refl
lemma1 {suc i} not = ⊥-elim ( not (s≤s z≤n) )

{-# TERMINATING #-} --                                                  ↓PreCondition(Invaliant)
whileLoop' : {l : Level} {t : Set l} → (env : Env) → {c10 :  ℕ } → ((varn env) + (vari env) ≡ c10)
   → (Code : (e1 : Env )→ vari e1 ≡ c10 → t) → t
whileLoop' env proof next with  ( suc zero  ≤? (varn  env) )
whileLoop' env {c10} proof next | no p = next env ( begin
       vari env            ≡⟨ refl ⟩
       0 + vari env        ≡⟨ cong (λ k → k + vari env) (sym (lemma1 p )) ⟩
       varn env + vari env ≡⟨ proof ⟩
       c10 ∎ ) where open ≡-Reasoning  
whileLoop' env {c10} proof next | yes p = whileLoop' env1 (proof3 p ) next where
      env1 = record {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 Invaliant
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
          ∎

open import Data.Unit hiding ( _≟_ ;  _≤?_ ; _≤_)

whileTestSpec1 : (c10 : ℕ) →  (e1 : Env ) → vari e1 ≡ c10 → ⊤
whileTestSpec1 _ _ x = tt

proofGears : {c10 :  ℕ } → ⊤
proofGears {c10} = whileTest' {_} {_}  {c10} (λ n p1 →  conversion1 n p1 (λ n1 p2 → whileLoop' n1 p2 (λ n2 p3 → whileTestSpec1 c10 n2 p3 ))) 

--                                                                      ↓PreCondition(Invaliant)
whileLoopSeg : {l : Level} {t : Set l} → {c10 :  ℕ } → (env : Env) → ((varn env) + (vari env) ≡ c10)
   → (next : (e1 : Env )→ varn e1 + vari e1 ≡ c10 → varn e1 < varn env → t) 
   → (exit : (e1 : Env )→ vari e1 ≡ c10 → t) → t
whileLoopSeg env proof next exit with  ( suc zero  ≤? (varn  env) )
whileLoopSeg {_} {_} {c10} env proof next exit | no p = exit env ( begin
       vari env            ≡⟨ refl ⟩
       0 + vari env        ≡⟨ cong (λ k → k + vari env) (sym (lemma1 p )) ⟩
       varn env + vari env ≡⟨ proof ⟩
       c10 ∎ ) where open ≡-Reasoning  
whileLoopSeg {_} {_} {c10} env proof next exit | yes p = next env1 (proof3 p ) proof4 where
      env1 = record {varn = (varn  env) - 1 ; vari = (vari env) + 1}
      1<0 : 1 ≤ zero → ⊥
      1<0 ()
      proof4 : varn env1 < varn env
      proof4 = {!!}
      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
          ∎

open import Relation.Binary.Definitions

nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥
nat-≤>  (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x

TerminatingLoop : {l : Level} {t : Set l} {c10 :  ℕ } → (i : ℕ) → (env : Env) → i ≡ varn env
   →  varn env + vari env ≡ c10 
   →  (exit : (e1 : Env )→ vari e1 ≡ c10 → t) → t
TerminatingLoop {_} {t} {c10} i env refl p exit with <-cmp 0 i
... | tri≈ ¬a b ¬c = whileLoopSeg {_} {t} {c10} env p (λ e1 eq lt → ⊥-elim (lemma3 e1 b lt) ) exit where
    lemma3 : (e1 : Env) → 0 ≡ varn env → varn e1 < varn env → ⊥
    lemma3 e refl ()
... | tri< a ¬b ¬c = whileLoopSeg {_} {t} {c10} env p (TerminatingLoop1 i) exit where
    TerminatingLoop1 : (j : ℕ) → (e1 : Env) → varn e1 + vari e1 ≡ c10 → varn e1 < varn env → t
    TerminatingLoop1 zero e1 eq lt = whileLoopSeg {_} {t} {c10} env p {!!} exit 
    TerminatingLoop1 (suc j) e1 eq lt with <-cmp j (varn e1)
    ... | tri< (s≤s a) ¬b ¬c = TerminatingLoop1 j e1 {!!} {!!}
    ... | tri≈ ¬a b ¬c = whileLoopSeg {_} {t} {c10} e1 {!!} lemma4 exit where
       lemma4 : (e2 : Env) → varn e2 + vari e2 ≡ c10 → varn e2 < varn e1 → t
       lemma4 e2 eq lt = TerminatingLoop1 j {!!} {!!} {!!}
    ... | tri> ¬a ¬b c =  ⊥-elim ( nat-≤> lt {!!} )