```{-# LANGUAGE ExistentialQuantification #-}
{-# LANGUAGE StandaloneDeriving        #-}

-----------------------------------------------------------------------------
-- |
-- Module      :  Test.StateMachine.Logic
-- Copyright   :  (C) 2017, ATS Advanced Telematic Systems GmbH
-- License     :  BSD-style (see the file LICENSE)
--
-- Maintainer  :  Stevan Andjelkovic <stevan@advancedtelematic.com>
-- Stability   :  provisional
-- Portability :  non-portable (GHC extensions)
--
-- This module provides a propositional logic which gives counterexamples when
-- the proposition is false.
--
-----------------------------------------------------------------------------

module Test.StateMachine.Logic where

infixr 1 :=>
infixr 2 :||
infixr 3 :&&

data Logic
= Bot
| Top
| Logic :&& Logic
| Logic :|| Logic
| Logic :=> Logic
| Not Logic
| Predicate Predicate
| Annotate String Logic
deriving Show

data Predicate
= forall a. (Eq  a, Show a) => a :== a
| forall a. (Eq  a, Show a) => a :/= a
| forall a. (Ord a, Show a) => a :<  a
| forall a. (Ord a, Show a) => a :<= a
| forall a. (Ord a, Show a) => a :>  a
| forall a. (Ord a, Show a) => a :>= a
| forall a. (Eq  a, Show a) => Elem    a [a]
| forall a. (Eq  a, Show a) => NotElem a [a]

deriving instance Show Predicate

dual :: Predicate -> Predicate
dual p = case p of
x :== y        -> x :/= y
x :/= y        -> x :== y
x :<  y        -> x :>= y
x :<= y        -> x :>  y
x :>  y        -> x :<= y
x :>= y        -> x :<  y
x `Elem`    xs -> x `NotElem` xs
x `NotElem` xs -> x `Elem`    xs

-- See Yuri Gurevich's "Intuitionistic logic with strong negation" (1977).
strongNeg :: Logic -> Logic
strongNeg l = case l of
Bot          -> Top
Top          -> Bot
l :&& r      -> strongNeg l :|| strongNeg r
l :|| r      -> strongNeg l :&& strongNeg r
l :=> r      ->           l :&& strongNeg r
Not l        -> l
Predicate p  -> Predicate (dual p)
Annotate s l -> Annotate s (strongNeg l)

data Counterexample
= BotC
| Fst Counterexample
| Snd Counterexample
| EitherC Counterexample Counterexample
| ImpliesC Counterexample
| NotC Counterexample
| PredicateC Predicate
| AnnotateC String Counterexample
deriving Show

data Value
= VFalse Counterexample
| VTrue
deriving Show

logic :: Logic -> Value
logic Bot            = VFalse BotC
logic Top            = VTrue
logic (l :&& r)      = case logic l of
VFalse ce -> VFalse (Fst ce)
VTrue     -> case logic r of
VFalse ce' -> VFalse (Snd ce')
VTrue      -> VTrue
logic (l :|| r)      = case logic l of
VTrue     -> VTrue
VFalse ce -> case logic r of
VTrue      -> VTrue
VFalse ce' -> VFalse (EitherC ce ce')
logic (l :=> r)      = case logic l of
VFalse _ -> VTrue
VTrue    -> case logic r of
VTrue     -> VTrue
VFalse ce -> VFalse (ImpliesC ce)
logic (Not l)        = case logic (strongNeg l) of
VTrue     -> VTrue
VFalse ce -> VFalse (NotC ce)
logic (Predicate p)  = predicate p
logic (Annotate s l) = case logic l of
VTrue     -> VTrue
VFalse ce -> VFalse (AnnotateC s ce)

predicate :: Predicate -> Value
predicate p = let b = boolean p in case p of
x :== y        -> b (x == y)
x :/= y        -> b (x /= y)
x :<  y        -> b (x <  y)
x :<= y        -> b (x <= y)
x :>  y        -> b (x >  y)
x :>= y        -> b (x >= y)
x `Elem`    xs -> b (x `elem`    xs)
x `NotElem` xs -> b (x `notElem` xs)
where
boolean :: Predicate -> Bool -> Value
boolean p True  = VTrue
boolean p False = VFalse (PredicateC (dual p))
```