Unexpected unknown with ill-defined function definition axiom

1
votes

The SMT program further down encodes the (ill-defined) function definition ∀ s · wild(s) = 1 + wild(s) in a slightly roundabout way (applying Dafny's "limited functions" encoding of recursive functions) and then attempts to prove that wild(emp) = 1 + wild(emp). However, Z3 4.6.0 (and also a recent 4.7.0 nightly) unexpectedly yield unknown instead of unsat.

(set-option :auto_config false) ;; true -> no change in behaviour
(set-option :smt.mbqi false) ;; true -> no change in behaviour

(declare-sort Snap)      ;; In the full example, this is ...
(declare-const emp Snap) ;; ... declared using declare-datatypes

(declare-fun wild (Snap) Int)
(declare-fun wild%limited (Snap) Int)

(assert (forall ((s Snap)) (! ;; AX-1
  (= (wild%limited s) (wild s))
  :pattern ((wild s))
)))

(assert (forall ((s Snap)) (! ;; AX-2
  (=
    (wild s)
    (+ 1 (wild%limited emp)))
  :pattern ((wild s))
)))

(push) ;; Full examples uses incremental mode
(assert
  (not 
    (= 
      (wild emp)
      (+ 1 (wild emp)))))
(check-sat) ;; UNKNOWN --- but why?
(pop)

Given my understanding of Z3 and triggers, I would expect the following proof steps to happen:

  ¬(wild(emp) = 1 + wild(emp))  // Source assertion
≡ ¬(1 + wild%limited(emp) = 1 + wild(emp))  // By AX-2
≡ ¬(1 + wild%limited(emp) = 1 + wild%limited(emp))  // By AX-1
≡ ¬(true)  // Done: UNSAT

But that doesn't appear to happen. My guess is that the axioms aren't instantiated — and indeed, get-info :all-statistics reports no quantified instantiation.

Can anybody shed some light on this?

1
z3

1 Answers

1
votes

the last assert simplifies to "true", so there is no ground occurrence of (wild emp) that would trigger quantifier instantiation.