Is there anyone, who can help me with my smt homework? - smt

Question

Homework:
Lab 4
In the second SMT lab exercise (Lab 4) we are going to solve the Chicken McNugget problem with the help of an SMT solver (an instance of the Frobenious Coin Problem).
Assume that a fast food chain sells boxes with fried chicken pieces in the sizes A=7, B= 9, C=16. Your friends and you are hungry and want to eat X chicken pieces (a natural number). The first question we are going to address is whether it is possible to buy U boxes of size A, V boxes of size B and W boxes of size C such that you get exactly X chicken pieces (without left over). In the first part (1 point) formulate this as an SMT problem and try to solve it for the given A, B, C values and X=40. First determine the logic you need and then give a manual incoding into SMT lib. In the second part (1.5 point) check that X0 = X+0, ..., X6 = X + 6 also have similar representations in terms of U0,V0,W0, ..., U6,V6,W6. With the help of this encoding determine the minimal number X which gives a satisfiable solution. Then Y=X-1 is the solution of the Chicken McNuggets problem for the fixed A, B, C values above, which means that it is the largest number Y which can NOT be represented this way, or in terms of chicken pieces, no matter how many (U,V,W) boxes of the given sizes (A,B,C) you buy and your friends do eat exactly Y pieces, then there has to be some left over pieces. Generating these two SMT LIB encodings for these two parts for the given concrete A,B,C and being able to present the solution (and modify and check it manually for other values) will give you thus half the points (2.5) for the Lab. You will be required to bring your own laptop on which you have installed and can use the SMT solver for the presentation (the web interface is not enough).
My SMT Code:
(set-logic QF_AUFNIA)
(declare-const u Int)
(declare-const v Int)
(declare-const w Int)
(declare-const x Int)
(assert (>= u 0))
(assert (>= v 0))
(assert (>= w 0))
(assert (= 40(+ (* u 7)(* v 9)(* w 16))))
(assert (= 40(+ (* u 7)(* v 9))))
(assert (= 40(+ (* v 9)(* w 16))))
(assert (= 40(+ (* u 7)(* w 16))))
(assert (= 40(* v 9)))
(assert (= 40(* u 7)))
(assert (= 40(* w 16)))
(declare-const x0 Int)
(declare-const x1 Int)
(declare-const x2 Int)
(declare-const x3 Int)
(declare-const x4 Int)
(declare-const x5 Int)
(declare-const x6 Int)
(declare-const u0 Int)
(declare-const u1 Int)
(declare-const u2 Int)
(declare-const u3 Int)
(declare-const u4 Int)
(declare-const u5 Int)
(declare-const u6 Int)
(declare-const v0 Int)
(declare-const v1 Int)
(declare-const v2 Int)
(declare-const v3 Int)
(declare-const v4 Int)
(declare-const v5 Int)
(declare-const v6 Int)
(declare-const w0 Int)
(declare-const w1 Int)
(declare-const w2 Int)
(declare-const w3 Int)
(declare-const w4 Int)
(declare-const w5 Int)
(declare-const w6 Int)
(declare-const y Int)
(assert (= x0 (+ x 0)))
(assert (= x1 (+ x 1)))
(assert (= x2 (+ x 2)))
(assert (= x3 (+ x 3)))
(assert (= x4 (+ x 4)))
(assert (= x5 (+ x 5)))
(assert (= x6 (+ x 6)))
(assert (= u0 (+ u 0)))
(assert (= u1 (+ u 1)))
(assert (= u2 (+ u 2)))
(assert (= u3 (+ u 3)))
(assert (= u4 (+ u 4)))
(assert (= u5 (+ u 5)))
(assert (= u6 (+ u 6)))
(assert (= v0 (+ v 0)))
(assert (= v1 (+ v 1)))
(assert (= v2 (+ v 2)))
(assert (= v3 (+ v 3)))
(assert (= v4 (+ v 4)))
(assert (= v5 (+ v 5)))
(assert (= v6 (+ v 6)))
(assert (= w0 (+ w 0)))
(assert (= w1 (+ w 1)))
(assert (= w2 (+ w 2)))
(assert (= w3 (+ w 3)))
(assert (= w4 (+ w 4)))
(assert (= w5 (+ w 5)))
(assert (= w6 (+ w 6)))
(assert (= x 40))
(assert (= x (+ (* u 7)(* v 9)(* w 16))))
(check-sat)
(exit)
I wrote my code, but it is always unsatisfiable. It is said, that X should be satisfiable. I tried X almost with every figure to get it satisfiable, but it doesn't work.
Is there anyone who could help me?

Answer 1

You're getting unsat for a very simple reason. Consider this line:
(assert (= 40(* u 7)))
where u is an Int value. Z3 can easily determine that there is no such integer value, as 7 does not divide 40. Similarly, you have many other constraints that are just not satisfiable for similar reasons.
Here's a simple idea to push forward. If someone gave you u, v, and w values and asked you to just check that they are indeed a correct solution, how would you program that? Write down what properties these values would have to satisfy, and simply assert them. Through the magic of constraint solving, what you'd consider "checking" a solution will become "finding" the solution. Hope that helps!
Also, stack-overflow works the best when you ask specific and pointed questions, instead of dumping an entire problem set. You'll get much better answers if you ask specific questions. Best of luck!