sbv-8.2: SMT Based Verification: Symbolic Haskell theorem prover using SMT solving.

Documentation.SBV.Examples.Queries.Interpolants

Description

Demonstrates extraction of interpolants via queries.

N.B. As of Z3 version 4.8.0; Z3 no longer supports interpolants. You need to use the MathSAT backend for this example to work.

Synopsis

# Documentation

Compute the interpolant for the following sets of formulas:

{x - 3y >= -1, x + y >= 0}

AND

{z - 2x >= 3, 2z <= 1}

where the variables are integers. Note that these sets of formulas are themselves satisfiable, but not taken all together. The pair (x, y) = (0, 0) satisfies the first set. The pair (x, z) = (-2, 0) satisfies the second. However, there's no triple (x, y, z) that satisfies all these four formulas together. We can use SBV to check this fact:

>>> sat $\x y z -> sAnd [x - 3*y .>= -1, x + y .>= 0, z - 2*x .>= 3, 2 * z .<= (1::SInteger)] Unsatisfiable  An interpolant for these sets would only talk about the variable x that is common to both. We have: >>> runSMTWith mathSAT example "(<= 0 s0)"  Notice that we get a string back, not a term; so there's some back-translation we need to do. We know that s0 is x through our translation mechanism, so the interpolant is saying that x >= 0 is entailed by the first set of formulas, and is inconsistent with the second. Let's use SBV to indeed show that this is the case: >>> prove$ \x y -> (x - 3*y .>= -1 .&& x + y .>= 0) .=> (x .>= (0::SInteger))
Q.E.D.


And:

>>> prove \$ \x z -> (z - 2*x .>= 3 .&& 2 * z .<= 1) .=> sNot (x .>= (0::SInteger))
Q.E.D.


This establishes that we indeed have an interpolant!