sbv-13.4: Data/SBV/TP/Kernel.hs
-----------------------------------------------------------------------------
-- |
-- Module : Data.SBV.TP.Kernel
-- Copyright : (c) Levent Erkok
-- License : BSD3
-- Maintainer: erkokl@gmail.com
-- Stability : experimental
--
-- Kernel of the TP prover API.
-----------------------------------------------------------------------------
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE NamedFieldPuns #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeApplications #-}
{-# OPTIONS_GHC -Wall -Werror #-}
module Data.SBV.TP.Kernel (
Proposition, Proof(..)
, axiom
, lemma, lemmaWith
, inductiveLemma, inductiveLemmaWith
, internalAxiom
, TPProofContext (..), smtProofStep, HasInductionSchema(..)
, tpMergeCfg
) where
import Control.Monad.Trans (liftIO, MonadIO)
import Data.List (intercalate)
import Data.Maybe (catMaybes)
import Data.SBV.Core.Data hiding (None)
import Data.SBV.Trans.Control hiding (getProof)
import Data.SBV.Core.Symbolic (MonadSymbolic)
import Data.SBV.SMT.SMT
import Data.SBV.Core.Model
import Data.SBV.Provers.Prover
import Data.SBV.TP.Utils
import Data.Time (NominalDiffTime)
import Data.SBV.Utils.TDiff
import Data.Dynamic
import Type.Reflection (typeRep)
import qualified Data.SBV.List as SL (nil, (.:))
-- | A proposition is something SBV is capable of proving/disproving in TP.
type Proposition a = ( QNot a
, QuantifiedBool a
, QSaturate Symbolic a
, Skolemize (NegatesTo a)
, Satisfiable (Symbolic (SkolemsTo (NegatesTo a)))
, Constraint Symbolic (SkolemsTo (NegatesTo a))
, Typeable a
)
-- | An inductive proposition is a proposition that has an induction scheme associated with it.
type Inductive a = (HasInductionSchema a, Proposition a)
-- | A class of values that has an associated induction schema. SBV manages this internally.
class HasInductionSchema a where
inductionSchema :: a -> ProofObj
-- | Induction schema for integers. Note that this is good for proving properties over naturals really.
-- There are other instances that would apply to all integers, but this one is really the most useful
-- in practice.
instance HasInductionSchema (Forall nm Integer -> SBool) where
inductionSchema p = proofOf $ internalAxiom "inductInteger" ax
where pf = p . Forall
ax = sAnd [pf 0, quantifiedBool (\(Forall i) -> (i .>= 0 .=> pf i) .=> pf (i + 1))]
.=> quantifiedBool (\(Forall i) -> pf i)
-- | Induction schema for integers with one extra argument
instance SymVal at => HasInductionSchema (Forall nm Integer -> Forall an at -> SBool) where
inductionSchema p = proofOf $ internalAxiom "inductInteger1" ax
where pf i a = p (Forall i) (Forall a)
ax = sAnd [ quantifiedBool (\ (Forall a) -> pf 0 a)
, quantifiedBool (\(Forall i) (Forall a) -> (i .>= 0 .=> pf i a) .=> pf (i + 1) a)]
.=> quantifiedBool (\(Forall i) (Forall a) -> pf i a)
-- | Induction schema for integers with two extra arguments
instance (SymVal at, SymVal bt) => HasInductionSchema (Forall nm Integer -> Forall an at -> Forall bn bt -> SBool) where
inductionSchema p = proofOf $ internalAxiom "inductInteger2" ax
where pf i a b = p (Forall i) (Forall a) (Forall b)
ax = sAnd [ quantifiedBool (\ (Forall a) (Forall b) -> pf 0 a b)
, quantifiedBool (\(Forall i) (Forall a) (Forall b) -> (i .>= 0 .=> pf i a b) .=> pf (i + 1) a b)]
.=> quantifiedBool (\(Forall i) (Forall a) (Forall b) -> pf i a b)
-- | Induction schema for integers with three extra arguments
instance (SymVal at, SymVal bt, SymVal ct) => HasInductionSchema (Forall nm Integer -> Forall an at -> Forall bn bt -> Forall cn ct -> SBool) where
inductionSchema p = proofOf $ internalAxiom "inductInteger3" ax
where pf i a b c = p (Forall i) (Forall a) (Forall b) (Forall c)
ax = sAnd [ quantifiedBool (\ (Forall a) (Forall b) (Forall c) -> pf 0 a b c)
, quantifiedBool (\(Forall i) (Forall a) (Forall b) (Forall c) -> (i .>= 0 .=> pf i a b c) .=> pf (i + 1) a b c)]
.=> quantifiedBool (\(Forall i) (Forall a) (Forall b) (Forall c) -> pf i a b c)
-- | Induction schema for integers with four extra arguments
instance (SymVal at, SymVal bt, SymVal ct, SymVal dt) => HasInductionSchema (Forall nm Integer -> Forall an at -> Forall bn bt -> Forall cn ct -> Forall dn dt -> SBool) where
inductionSchema p = proofOf $ internalAxiom "inductInteger4" ax
where pf i a b c d = p (Forall i) (Forall a) (Forall b) (Forall c) (Forall d)
ax = sAnd [ quantifiedBool (\ (Forall a) (Forall b) (Forall c) (Forall d) -> pf 0 a b c d)
, quantifiedBool (\(Forall i) (Forall a) (Forall b) (Forall c) (Forall d) -> (i .>= 0 .=> pf i a b c d) .=> pf (i + 1) a b c d)]
.=> quantifiedBool (\(Forall i) (Forall a) (Forall b) (Forall c) (Forall d) -> pf i a b c d)
-- | Induction schema for integers with five extra arguments
instance (SymVal at, SymVal bt, SymVal ct, SymVal dt, SymVal et) => HasInductionSchema (Forall nm Integer -> Forall an at -> Forall bn bt -> Forall cn ct -> Forall dn dt -> Forall en et -> SBool) where
inductionSchema p = proofOf $ internalAxiom "inductInteger5" ax
where pf i a b c d e = p (Forall i) (Forall a) (Forall b) (Forall c) (Forall d) (Forall e)
ax = sAnd [ quantifiedBool (\ (Forall a) (Forall b) (Forall c) (Forall d) (Forall e) -> pf 0 a b c d e)
, quantifiedBool (\(Forall i) (Forall a) (Forall b) (Forall c) (Forall d) (Forall e) -> (i .>= 0 .=> pf i a b c d e) .=> pf (i + 1) a b c d e)]
.=> quantifiedBool (\(Forall i) (Forall a) (Forall b) (Forall c) (Forall d) (Forall e) -> pf i a b c d e)
-- | Induction schema for lists.
instance SymVal x => HasInductionSchema (Forall nm [x] -> SBool) where
inductionSchema p = proofOf $ internalAxiom ("induct" ++ show (typeRep @[x])) ax
where pf = p . Forall
ax = sAnd [pf SL.nil, quantifiedBool (\(Forall x) (Forall xs) -> pf xs .=> pf (x SL..: xs))]
.=> quantifiedBool (\(Forall xs) -> pf xs)
-- | Induction schema for lists with one extra argument
instance (SymVal x, SymVal at) => HasInductionSchema (Forall nm [x] -> Forall an at -> SBool) where
inductionSchema p = proofOf $ internalAxiom ("induct" ++ show (typeRep @[x]) ++ "1") ax
where pf xs a = p (Forall xs) (Forall a)
ax = sAnd [ quantifiedBool (\ (Forall a) -> pf SL.nil a)
, quantifiedBool (\(Forall x) (Forall xs) (Forall a) -> pf xs a .=> pf (x SL..: xs) a)]
.=> quantifiedBool (\(Forall xs) (Forall a) -> pf xs a)
-- | Induction schema for lists with two extra arguments
instance (SymVal x, SymVal at, SymVal bt) => HasInductionSchema (Forall nm [x] -> Forall an at -> Forall bn bt -> SBool) where
inductionSchema p = proofOf $ internalAxiom ("induct" ++ show (typeRep @[x]) ++ "2") ax
where pf xs a b = p (Forall xs) (Forall a) (Forall b)
ax = sAnd [ quantifiedBool (\ (Forall a) (Forall b) -> pf SL.nil a b)
, quantifiedBool (\(Forall x) (Forall xs) (Forall a) (Forall b) -> pf xs a b .=> pf (x SL..: xs) a b)]
.=> quantifiedBool (\(Forall xs) (Forall a) (Forall b) -> pf xs a b)
-- | Induction schema for lists with three extra arguments
instance (SymVal x, SymVal at, SymVal bt, SymVal ct) => HasInductionSchema (Forall nm [x] -> Forall an at -> Forall bn bt -> Forall cn ct -> SBool) where
inductionSchema p = proofOf $ internalAxiom ("induct" ++ show (typeRep @[x]) ++ "3") ax
where pf xs a b c = p (Forall xs) (Forall a) (Forall b) (Forall c)
ax = sAnd [ quantifiedBool (\ (Forall a) (Forall b) (Forall c) -> pf SL.nil a b c)
, quantifiedBool (\(Forall x) (Forall xs) (Forall a) (Forall b) (Forall c) -> pf xs a b c .=> pf (x SL..: xs) a b c)]
.=> quantifiedBool (\(Forall xs) (Forall a) (Forall b) (Forall c) -> pf xs a b c)
-- | Induction schema for lists with four extra arguments
instance (SymVal x, SymVal at, SymVal bt, SymVal ct, SymVal dt) => HasInductionSchema (Forall nm [x] -> Forall an at -> Forall bn bt -> Forall cn ct -> Forall dn dt -> SBool) where
inductionSchema p = proofOf $ internalAxiom ("induct" ++ show (typeRep @[x]) ++ "4") ax
where pf xs a b c d = p (Forall xs) (Forall a) (Forall b) (Forall c) (Forall d)
ax = sAnd [ quantifiedBool (\ (Forall a) (Forall b) (Forall c) (Forall d) -> pf SL.nil a b c d)
, quantifiedBool (\(Forall x) (Forall xs) (Forall a) (Forall b) (Forall c) (Forall d) -> pf xs a b c d .=> pf (x SL..: xs) a b c d)]
.=> quantifiedBool (\(Forall xs) (Forall a) (Forall b) (Forall c) (Forall d) -> pf xs a b c d)
-- | Induction schema for lists with five extra arguments
instance (SymVal x, SymVal at, SymVal bt, SymVal ct, SymVal dt, SymVal et) => HasInductionSchema (Forall nm [x] -> Forall an at -> Forall bn bt -> Forall cn ct -> Forall dn dt -> Forall en et -> SBool) where
inductionSchema p = proofOf $ internalAxiom ("induct" ++ show (typeRep @[x]) ++ "5") ax
where pf xs a b c d e = p (Forall xs) (Forall a) (Forall b) (Forall c) (Forall d) (Forall e)
ax = sAnd [ quantifiedBool (\ (Forall a) (Forall b) (Forall c) (Forall d) (Forall e) -> pf SL.nil a b c d e)
, quantifiedBool (\(Forall x) (Forall xs) (Forall a) (Forall b) (Forall c) (Forall d) (Forall e) -> pf xs a b c d e .=> pf (x SL..: xs) a b c d e)]
.=> quantifiedBool (\(Forall xs) (Forall a) (Forall b) (Forall c) (Forall d) (Forall e) -> pf xs a b c d e)
-- | Accept the given definition as a fact. Usually used to introduce definitial axioms,
-- giving meaning to uninterpreted symbols. Note that we perform no checks on these propositions,
-- if you assert nonsense, then you get nonsense back. So, calls to 'axiom' should be limited to
-- definitions, or basic axioms (like commutativity, associativity) of uninterpreted function symbols.
axiom :: Proposition a => String -> a -> TP (Proof a)
axiom nm p = do cfg <- getTPConfig
u <- tpGetNextUnique
_ <- liftIO $ startTP cfg True "Axiom" 0 (TPProofOneShot nm [])
let Proof iax = internalAxiom nm p
pure $ Proof (iax { isUserAxiom = True, uniqId = u })
-- | Internal axiom generator; so we can keep truck of TP's trusted axioms, vs. user given axioms.
internalAxiom :: Proposition a => String -> a -> Proof a
internalAxiom nm p = Proof $ ProofObj { dependencies = []
, isUserAxiom = False
, getObjProof = label nm (quantifiedBool p)
, getProp = toDyn p
, proofName = nm
, uniqId = TPInternal
, isCached = False
}
-- | Propagate the settings for ribbon/timing from top to current. Because in any subsequent configuration
-- in a lemmaWith, inductWith etc., we just want to change the solver, not the actual settings for TP.
tpMergeCfg :: SMTConfig -> SMTConfig -> SMTConfig
tpMergeCfg cur top = cur{tpOptions = tpOptions top}
-- | Prove a given statement, using auxiliaries as helpers. Using the default solver.
lemma :: Proposition a => String -> a -> [ProofObj] -> TP (Proof a)
lemma nm f by = do cfg <- getTPConfig
lemmaWith cfg nm f by
-- | Prove a lemma, using the given configuration.
lemmaWith :: Proposition a => SMTConfig -> String -> a -> [ProofObj] -> TP (Proof a)
lemmaWith cfgIn nm inputProp by = withProofCache nm $ do
topCfg <- getTPConfig
let cfg@SMTConfig{tpOptions = TPOptions{printStats}} = cfgIn `tpMergeCfg` topCfg
tpSt <- getTPState
u <- tpGetNextUnique
liftIO $ getTimeStampIf printStats >>= runSMTWith cfg . go tpSt cfg u
where go tpSt cfg u mbStartTime = do qSaturateSavingObservables inputProp
mapM_ (constrain . getObjProof) by
query $ smtProofStep cfg tpSt "Lemma" 0 (TPProofOneShot nm by) Nothing inputProp [] (good cfg mbStartTime u)
-- What to do if all goes well
good cfg mbStart u d = do mbElapsed <- getElapsedTime mbStart
liftIO $ finishTP cfg ("Q.E.D." ++ concludeModulo by) d $ catMaybes [mbElapsed]
pure $ Proof $ ProofObj { dependencies = by
, isUserAxiom = False
, getObjProof = label nm (quantifiedBool inputProp)
, getProp = toDyn inputProp
, proofName = nm
, uniqId = u
, isCached = False
}
-- | Prove a given statement, using the induction schema for the proposition. Using the default solver.
inductiveLemma :: Inductive a => String -> a -> [ProofObj] -> TP (Proof a)
inductiveLemma nm f by = do cfg <- getTPConfig
inductiveLemmaWith cfg nm f by
-- | Prove a given statement, using the induction schema for the proposition. Using the default solver.
inductiveLemmaWith :: Inductive a => SMTConfig -> String -> a -> [ProofObj] -> TP (Proof a)
inductiveLemmaWith cfg nm f by = lemmaWith cfg nm f (inductionSchema f : by)
-- | Capture the general flow of a proof-step. Note that this is the only point where we call the backend solver
-- in a TP proof.
smtProofStep :: (SolverContext m, MonadIO m, MonadQuery m, MonadSymbolic m, Proposition a)
=> SMTConfig -- ^ config
-> TPState -- ^ TPState
-> String -- ^ tag
-> Int -- ^ level
-> TPProofContext -- ^ the context in which we're doing the proof
-> Maybe SBool -- ^ Assumptions under which we do the check-sat. If there's one we'll push/pop
-> a -- ^ what we want to prove
-> [(String, SVal)] -- ^ Values to display in case of failure
-> ((Int, Maybe NominalDiffTime) -> IO r) -- ^ what to do when unsat, with the tab amount and time elapsed (if asked)
-> m r
smtProofStep cfg@SMTConfig{verbose, tpOptions = TPOptions{printStats}} tpState tag level ctx mbAssumptions prop disps unsat = do
case mbAssumptions of
Nothing -> do queryDebug ["; smtProofStep: No context value to push."]
check
Just asm -> do queryDebug ["; smtProofStep: Pushing in the context: " ++ show asm]
inNewAssertionStack $ do constrain asm
check
where check = do
tab <- liftIO $ startTP cfg verbose tag level ctx
-- It's tempting to skolemize here.. But skolemization creates fresh constants
-- based on the name given, and they mess with all else. So, don't skolemize!
constrain $ sNot (quantifiedBool prop)
(mbT, r) <- timeIf printStats checkSat
updStats tpState (\s -> s{noOfCheckSats = noOfCheckSats s + 1})
case mbT of
Nothing -> pure ()
Just t -> updStats tpState (\s -> s{solverElapsed = solverElapsed s + t})
case r of
Unk -> unknown
Sat -> cex
DSat{} -> cex
Unsat -> liftIO $ unsat (tab, mbT)
die = error "Failed"
fullNm = case ctx of
TPProofOneShot s _ -> s
TPProofStep True s _ ss -> "assumptions for " ++ intercalate "." (s : ss)
TPProofStep False s _ ss -> intercalate "." (s : ss)
unknown = do r <- getUnknownReason
liftIO $ do putStrLn $ "\n*** Failed to prove " ++ fullNm ++ "."
putStrLn $ "\n*** Solver reported: " ++ show r
die
-- What to do if the proof fails
cex = do
liftIO $ putStrLn $ "\n*** Failed to prove " ++ fullNm ++ "."
res <- case ctx of
TPProofStep{} -> do mapM_ (uncurry sObserve) disps
Satisfiable cfg <$> getModel
TPProofOneShot _ by ->
-- When trying to get a counter-example not in query mode, we
-- do a skolemized sat call, which gets better counter-examples.
-- We only include the those facts that are user-given axioms. This
-- way our counter-example will be more likely to be relevant
-- to the proposition we're currently proving. (Hopefully.)
-- Remember that we first have to negate, and then skolemize!
do SatResult res <- liftIO $ satWith cfg $ do
qSaturateSavingObservables prop
mapM_ constrain [getObjProof | ProofObj{isUserAxiom, getObjProof} <- by, isUserAxiom] :: Symbolic ()
mapM_ (uncurry sObserve) disps
pure $ skolemize (qNot prop)
pure res
liftIO $ print $ ThmResult res
die