Some proofs of validity using Z3 SMT 2.0 online -

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lemma: forall x : r, x <> 0 -> (x / x) = 1.

proof:

(declare-const x real) (push) (assert (or (> x 0) (< x 0))) (assert (not (= (/ x x) 1))) (check-sat)

and output : 

unsat

qed.

lemma: forall x y : r, x <> 0, y <> 0 -> (x / x + y / y) = 2.

proof:

(declare-const x real) (declare-const y real) (push) (assert (or (> x 0) (< x 0))) (assert (or (> y 0) (< y 0))) (assert (not (= (+ (/ x x) (/ y y)) 2))) (check-sat)

and output is:

unsat

qed.

lemma: forall x y : r, x <> 0, y <> 0 -> (x / x + x / y) = ((x + y) / y).

proof:

(declare-const x real) (declare-const y real) (push) (assert (or (> x 0) (< x 0))) (assert (or (> y 0) (< y 0))) (assert (not (= (+ (/ x x) (/ x y)) (/ (+ x y) y)))) (check-sat)

and output is:

unsat

qed.

these lemmas proved using coq + maple at

http://coq.inria.fr/v8.2pl1/contribs/maplemode.examples.html

and using z3py online at

some proofs of validity using z3py online , strategy proposed nikolaj bjorner

please let me know if proofs z3 smt 2.0 correct , if know more direct form prove them using z3 smt 2.0. many thanks.

there nothing wrong encodings. may consider following encoding closer lemmas trying prove (also available online here):

(assert (not (forall ((x real)) (=> (not (= x 0)) (= (/ x x) 1))))) (check-sat)  (reset) (assert (not (forall ((x real) (y real)) (=> (and (not (= x 0))                                                   (not (= y 0)))                                              (= (+ (/ x x) (/ y y)) 2))))) (check-sat)  (reset) (assert (not (forall ((x real) (y real)) (=> (and (not (= x 0))                                                   (not (= y 0)))                                              (= (+ (/ x x) (/ x y)) (/ (+ x y) y)))))) (check-sat)

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