module whileTestPrim 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.Core 



record Env : Set where
  field
    varn : ℕ
    vari : ℕ
open Env

PrimComm : Set
PrimComm = Env → Env

Cond : Set
Cond = (Env → Bool) 

data Comm : Set where
  Skip  : Comm
  Abort : Comm
  PComm : PrimComm -> Comm
  Seq   : Comm -> Comm -> Comm
  If    : Cond -> Comm -> Comm -> Comm
  While : Cond -> Comm -> Comm

_-_ : ℕ -> ℕ -> ℕ
x - zero  = x
zero - _  = zero
(suc x) - (suc y)  = x - y

lt : ℕ -> ℕ -> Bool
lt x y with (suc x ) ≤? y
lt x y | yes p = true
lt x y | no ¬p = false

Equal : ℕ -> ℕ -> Bool
Equal x y with x ≟ y
Equal x y | yes p = true
Equal x y | no ¬p = false

program : Comm
program = 
    Seq ( PComm (λ env → record env {varn = 10}))
    $ Seq ( PComm (λ env → record env {vari = 0}))
    $ While (λ env → lt (varn env ) zero )
      (Seq (PComm (λ env → record env {vari = ((vari env) + 1)} ))
        $ PComm (λ env → record env {varn = ((varn env) - 1)} ))

simple : Comm
simple = 
    Seq ( PComm (λ env → record env {varn = 10}))
    $  PComm (λ env → record env {vari = 0})

{-# TERMINATING #-}
interpret : Env → Comm → Env
interpret env Skip = env
interpret env Abort = env
interpret env (PComm x) = x env
interpret env (Seq comm comm1) = interpret (interpret env comm) comm1
interpret env (If x then else) with x env
... | true = interpret env then
... | false = interpret env else
interpret env (While x comm) with x env
... | true = interpret (interpret env comm) (While x comm)
... | false = env

test1 : Env
test1 =  interpret ( record { vari = 0  ; varn = 0 } ) program


empty-case : (env : Env) → (( λ e → true ) env ) ≡ true 
empty-case _ = refl

implies : Bool → Bool → Bool
implies false _ = true
implies true true = true
implies true false = false

Axiom : Cond -> PrimComm -> Cond -> Set
Axiom pre comm post = ∀ (env : Env) →  implies (pre env) ( post (comm env)) ≡ true

Tautology : Cond -> Cond -> Set
Tautology pre post = ∀ (env : Env) →  implies (pre env) (post env) ≡ true

_/\_ :  Cond -> Cond -> Cond
x /\ y =  λ env → x env ∧ y env 

neg :  Cond -> Cond 
neg x  =  λ env → not ( x env )

data HTProof : Cond -> Comm -> Cond -> Set where
  PrimRule : {bPre : Cond} -> {pcm : PrimComm} -> {bPost : Cond} ->
             (pr : Axiom bPre pcm bPost) ->
             HTProof bPre (PComm pcm) bPost
  SkipRule : (b : Cond) -> HTProof b Skip b
  AbortRule : (bPre : Cond) -> (bPost : Cond) ->
              HTProof bPre Abort bPost
  WeakeningRule : {bPre : Cond} -> {bPre' : Cond} -> {cm : Comm} ->
                {bPost' : Cond} -> {bPost : Cond} ->
                Tautology bPre bPre' ->
                HTProof bPre' cm bPost' ->
                Tautology bPost' bPost ->
                HTProof bPre cm bPost
  SeqRule : {bPre : Cond} -> {cm1 : Comm} -> {bMid : Cond} ->
            {cm2 : Comm} -> {bPost : Cond} ->
            HTProof bPre cm1 bMid ->
            HTProof bMid cm2 bPost ->
            HTProof bPre (Seq cm1 cm2) bPost
  IfRule : {cmThen : Comm} -> {cmElse : Comm} ->
           {bPre : Cond} -> {bPost : Cond} ->
           {b : Cond} ->
           HTProof (bPre /\ b) cmThen bPost ->
           HTProof (bPre /\ neg b) cmElse bPost ->
           HTProof bPre (If b cmThen cmElse) bPost
  WhileRule : {cm : Comm} -> {bInv : Cond} -> {b : Cond} ->
              HTProof (bInv /\ b) cm bInv ->
              HTProof bInv (While b cm) (bInv /\ neg b)

initCond : Cond
initCond env = true

stmt1Cond : Cond
stmt1Cond env = Equal (varn env) 10

stmt2Cond : Cond
stmt2Cond env = (Equal (varn env) 10) ∧ (Equal (vari env) 0)

whileInv : Cond
whileInv env = Equal ((varn env) + (vari env)) 10

whileInv' : Cond
whileInv' env = Equal ((varn env) + (vari env)) 11

termCond : Cond
termCond env = Equal (vari env) 10

eqlemma : { x y : ℕ } →  Equal x y ≡ true → x ≡ y
eqlemma {x} {y} eq with x ≟ y
eqlemma {x} {y} refl | yes refl = refl
eqlemma {x} {y} () | no ¬p 

proofs : HTProof initCond simple stmt2Cond
proofs =
      SeqRule {initCond} ( PrimRule empty-case )
    $ PrimRule {stmt1Cond} {_} {stmt2Cond} lemma
  where
     lemma : Axiom stmt1Cond (λ env → record { varn = varn env ; vari = 0 }) stmt2Cond
     lemma env with stmt1Cond env
     lemma env | false = refl
     lemma env | true = refl


proof1 : HTProof initCond program termCond
proof1 =
      SeqRule {λ e → true} ( PrimRule empty-case )
    $ SeqRule {λ e →  Equal (varn e) 10} ( PrimRule lemma1   )
    $ WeakeningRule {λ e → (Equal (varn e) 10) ∧ (Equal (vari e) 0)}  lemma2 (
            WhileRule {_} {λ e → Equal ((varn e) + (vari e)) 10}
            $ SeqRule (PrimRule {λ e →  whileInv e  ∧ lt (varn e) zero } lemma3)
                     $ PrimRule {whileInv'} {_} {whileInv}  lemma4  ) lemma5
  where
     lemma1 : Axiom stmt1Cond (λ env → record { varn = varn env ; vari = 0 }) stmt2Cond
     lemma1 env with stmt1Cond env
     lemma1 env | false = refl
     lemma1 env | true = refl
     lemma2 :  Tautology stmt2Cond whileInv
     lemma2 env with stmt2Cond env | Equal (varn env + vari env) 10
     lemma2 env | false | false = refl
     lemma2 env | false | true = refl
     lemma2 env | true | true = refl
     lemma2 env | true | false = {!!}
     lemma3 : Axiom (whileInv /\ (λ env → lt (varn env) zero)) (λ env → record { varn = varn env ; vari = vari env + 1 }) whileInv'
     lemma3 = {!!}
     lemma4 :  Axiom whileInv' (λ env → record { varn = varn env - 1 ; vari = vari env }) whileInv
     lemma4 = {!!}
     lemma5 :  Tautology (whileInv /\ neg (λ z → lt (varn z) zero)) termCond
     lemma5 = {!!}