sbv-5.2: SMT Based Verification: Symbolic Haskell theorem prover using SMT solving.

Data.SBV.Examples.Uninterpreted.Deduce

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

data 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

 Source Source Source Source Source Source 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

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.
```