Question

我试图使用以下Z3-SMT-LIB代码证明Z3组满足关联定律

(set-option :mbqi true)
(declare-sort S)
(declare-fun f (S S) S)
(declare-const a S)
(declare-const b S)
(declare-const c S)
(assert (forall ((x S) (y S))
            (= (f x y)  (f y x))))
(assert (forall ((x S))
            (= (f x a) x)))
(assert (= (f b b) c))
(assert (= (f b c) a))
(assert (= (f c c) b))
(check-sat)
(get-model)
(declare-fun x () S)
(declare-fun y () S)
(declare-fun z () S)
(assert (not (=> (and (or (= x a) (= x b) (= x c)) (or (= y a) (= y b) (= y c)) 
                  (or (= z a) (= z b) (= z c))) 
            (=  (f x (f y z)) (f (f x y) z)))))
(check-sat)

相应的输出是

sat 
(model 
;; universe for S: 
;; S!val!1 S!val!0 S!val!2 
;; ----------- 
;; definitions for universe elements: 
(declare-fun S!val!1 () S) 
(declare-fun S!val!0 () S) 
(declare-fun S!val!2 () S) 
;; cardinality constraint: 
(forall ((x S)) (or (= x S!val!1) (= x S!val!0) (= x S!val!2))) 
;; ----------- 
(define-fun b () S S!val!0) 
(define-fun c () S S!val!1) 
(define-fun a () S S!val!2) 
(define-fun f ((x!1 S) (x!2 S)) S 
  (ite (and (= x!1 S!val!0) (= x!2 S!val!0)) S!val!1 
  (ite (and (= x!1 S!val!0) (= x!2 S!val!1)) S!val!2 
  (ite (and (= x!1 S!val!1) (= x!2 S!val!1)) S!val!0 
  (ite (and (= x!1 S!val!1) (= x!2 S!val!0)) S!val!2 
  (ite (and (= x!1 S!val!0) (= x!2 S!val!2)) S!val!0 
  (ite (and (= x!1 S!val!2) (= x!2 S!val!0)) S!val!0 
  (ite (and (= x!1 S!val!2) (= x!2 S!val!1)) S!val!1 
  (ite (and (= x!1 S!val!1) (= x!2 S!val!2)) S!val!1 x!1))))))))) 
 ) 

 unsat

在线here

运行此示例

问题是：

此证明是否正确？
有更优雅的证据吗？

Answer 1

每个群体都必须满足相关性公理。我猜你试图证明第一个断言块的每个模型都满足关联公理。为了完成这个例子，我们还应该证明逆元素公理。

为了证明这两个属性，我们可以使用以下命令：

(push)
;; prove the inverse axiom
(assert (not (forall ((x S)) (exists ((y S)) (= (f x y) a)))))
(check-sat)                
(pop)

(push)
;; prove the associativity axiom
(assert (not (forall ((x S) (y S) (z S)) (= (f x (f y z)) (f (f x y) z)))))
(check-sat)                
(pop)

Z3无法证明这一点。但是，如果我们断言我们只对大小最多为3的模型感兴趣，它就会成功。

(assert (forall ((x S)) (or (= x a) (= x b) (= x c))))

Here就是完整的例子。

如何用Z3-SMT-LIB证明Z3组满足关联律

1 个答案: