sbv-7.7: SMT Based Verification: Symbolic Haskell theorem prover using SMT solving.

Copyright(c) Levent Erkok
LicenseBSD3
Maintainererkokl@gmail.com
Stabilityexperimental
Safe HaskellNone
LanguageHaskell2010

Documentation.SBV.Examples.Uninterpreted.Deduce

Contents

Description

Demonstrates uninterpreted sorts and how they can be used for deduction. This example is inspired by the discussion at http://stackoverflow.com/questions/10635783/using-axioms-for-deductions-in-z3, essentially showing how to show the required deduction using SBV.

Synopsis

Representing uninterpreted booleans

newtype B Source #

The uninterpreted sort B, corresponding to the carrier. To prevent SBV from translating it to an enumerated type, we simply attach an unused field

Constructors

B () 

Instances

Eq B Source # 

Methods

(==) :: B -> B -> Bool #

(/=) :: B -> B -> Bool #

Data B Source # 

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> B -> c B #

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c B #

toConstr :: B -> Constr #

dataTypeOf :: B -> DataType #

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c B) #

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c B) #

gmapT :: (forall b. Data b => b -> b) -> B -> B #

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> B -> r #

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> B -> r #

gmapQ :: (forall d. Data d => d -> u) -> B -> [u] #

gmapQi :: Int -> (forall d. Data d => d -> u) -> B -> u #

gmapM :: Monad m => (forall d. Data d => d -> m d) -> B -> m B #

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> B -> m B #

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> B -> m B #

Ord B Source # 

Methods

compare :: B -> B -> Ordering #

(<) :: B -> B -> Bool #

(<=) :: B -> B -> Bool #

(>) :: B -> B -> Bool #

(>=) :: B -> B -> Bool #

max :: B -> B -> B #

min :: B -> B -> B #

Read B Source # 
Show B Source # 

Methods

showsPrec :: Int -> B -> ShowS #

show :: B -> String #

showList :: [B] -> ShowS #

HasKind B Source # 
SymWord B Source # 

type SB = SBV B Source #

Handy shortcut for the type of symbolic values over B

Uninterpreted connectives over B

and :: SB -> SB -> SB Source #

Uninterpreted logical connective and

or :: SB -> SB -> SB Source #

Uninterpreted logical connective or

not :: SB -> SB Source #

Uninterpreted logical connective not

Axioms of the logical system

ax1 :: [String] Source #

Distributivity of OR over AND, as an axiom in terms of the uninterpreted functions we have introduced. Note how variables range over the uninterpreted sort B.

ax2 :: [String] Source #

One of De Morgan's laws, again as an axiom in terms of our uninterpeted logical connectives.

ax3 :: [String] Source #

Double negation axiom, similar to the above.

Demonstrated deduction

test :: IO ThmResult Source #

Proves the equivalence NOT (p OR (q AND r)) == (NOT p AND NOT q) OR (NOT p AND NOT r), following from the axioms we have specified above. We have:

>>> test
Q.E.D.