intros.auto paramodulation.
qed.
-theorem bool4:
+theorem bool266:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall x,y:A. (mult x (add (inv x) y)) = (mult x y).
+intros.auto paramodulation.
+qed.
+
+theorem bool507:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall x,y:A. zero = (mult x (mult (inv x) y)).
+intros.auto paramodulation.
+qed.
+
+theorem bool515:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall x,y:A. zero = mult (inv x) (mult x y).
+intros.auto paramodulation.
+qed.
+
+theorem bool304:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall x,y:A. x = (mult (add y x) x).
+intros.auto paramodulation.
+qed.
+
+theorem bool531:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall x,y:A. zero = (mult (inv (add x y)) y).
+intros.auto paramodulation.
+qed.
+
+theorem bool253:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall x,y:A. (add (inv x) (mult y x)) = (add (inv x) y).
+intros.auto paramodulation.
+qed.
+
+theorem bool557:
\forall A:Set.
\forall one:A.
\forall zero:A.
\forall i2: (\forall x:A. (mult x one) = x).
\forall inv1: (\forall x:A. (add x (inv x)) = one).
\forall inv2: (\forall x:A. (mult x (inv x)) = zero).
- \forall x,y,z:A.
- (add x y) = (add (add x (mult y z)) y).
-intros.
-exact (
- ((eq_ind A (add y (add x (mult y z)))
- (\lambda x_DemodGoal_193:A.(eq A (add x y) x_DemodGoal_193))
- (eq_ind A x
- (\lambda x_DemodGoal_893:A.(eq A x_DemodGoal_893 (add y (add x (mult y z)))))
- (eq_ind A y (\lambda x_DemodGoal_894:A.(eq A x x_DemodGoal_894))
- (eq_ind A (add y zero) (\lambda x_Demod_554:A.(eq A x x_Demod_554))
- (eq_ind_r A (add zero x) (\lambda x_SupR_809:A.(eq A x_SupR_809 (add y zero)))
- (eq_ind_r A (add y (mult (add zero y) zero))
- (\lambda x_Demod_553:A.(eq A (add zero x) x_Demod_553))
- (eq_ind A (add (add zero x) zero)
- (\lambda x_Demod_540:A.(eq A x_Demod_540 (add x (mult (add zero x) zero))))
- (eq_ind_r A (mult zero x))
- (\lambda x_Demod_526:A.(eq A (add (add zero x) x_Demod_526) (add x (mult (add zero x) zero))))
- (eq_ind_r A (mult (add zero x) (add (add zero x) x))
- (\lambda x_SupR_606:A.(eq A (add (add zero x) (mult zero x)) x_SupR_606))
- (eq_ind A (add (add zero x) zero)
- (\lambda x_SupR_296:A.(eq A (add (add zero x) (mult zero x)) (mult x_SupR_296 (add (add zero x) x))))
- (H2 (add zero x) zero x) (add zero x) (H4 (add zero x))) (add x (mult (add zero x) zero))
- (eq_ind A (add x zero)
- (\lambda x_SupR_357:A.(eq A (add x (mult (add zero x) zero)) (mult x_SupR_357 (add (add zero x) x))))
- (eq_ind A (mult (add (add zero x) x) (add x zero))
- (\lambda x_SupR_310:A.(eq A (add x (mult (add zero x) zero)) x_SupR_310))
- (eq_ind A (add x (add zero x))
- (\lambda x_SupR_302:A.(eq A (add x (mult (add zero x) zero)) (mult x_SupR_302 (add x zero))))
- (H2 x (add zero x) zero) (add (add zero x) x) (H x (add zero x))) (mult (add x zero) (add (add zero x) x)) (H1 (add (add zero x) x) (add x zero))) (add zero x) (H x zero))) zero
- (eq_ind A (mult zero one)
- (\lambda x_Demod_507:A.(eq A x_Demod_507 (mult zero x)))
- (eq_ind_r A (add x one)
- (\lambda x_Demod_506:A.(eq A (mult zero x_Demod_506) (mult zero x)))
- (eq_ind_r A (inv zero)
- (\lambda x_SupR_785:A.(eq A (mult zero (add x x_SupR_785)) (mult zero x)))
- (eq_ind A (add (mult zero x) zero)
- (\lambda x_Demod_223:A.(eq A (mult zero (add x (inv zero))) x_Demod_223))
- (eq_ind A (mult zero (inv zero))
- (\lambda x_SupR_337:A.(eq A (mult zero (add x (inv zero))) (add (mult zero x) x_SupR_337)))
- (H3 zero x (inv zero)) zero (H7 zero)) (mult zero x) (H4 (mult zero x))) one
- (eq_ind A (add (inv zero) zero)
- (\lambda x_SupR_463:A.(eq A one x_SupR_463))
- (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_298:A.(eq A x_SupR_298 (add (inv zero) zero))) (H zero (inv zero)) one (H6 zero)) (inv zero) (H4 (inv zero)))) one
- (eq_ind A (add x (inv x))
- (\lambda x_Demod_268:A.(eq A x_Demod_268 (add x one)))
- (eq_ind A (mult (inv x) one)
- (\lambda x_SupR_396:A.(eq A (add x x_SupR_396) (add x one)))
- (eq_ind_r A (mult one (add x one))
- (\lambda x_Demod_199:A.(eq A (add x (mult (inv x) one)) x_Demod_199))
- (eq_ind A (add x (inv x))
- (\lambda x_SupR_268:A.(eq A (add x (mult (inv x) one)) (mult x_SupR_268 (add x one)))) (H2 x (inv x) one) one (H6 x)) (add x one)
- (eq_ind A (mult (add x one) one)
- (\lambda x_SupR_271:A.(eq A x_SupR_271 (mult one (add x one))))
- (H1 (add x one) one) (add x one) (H5 (add x one)))) (inv x) (H5 (inv x))) one (H6 x))) zero (H5 zero))) (add zero x) (H4 (add zero x))) (add y zero)
- (eq_ind A (add zero zero)
- (\lambda x_Demod_543:A.(eq A (add y x_Demod_543) (add y (mult (add zero y) zero))))
- (eq_ind_r A (mult zero y)
- (\lambda x_Demod_524:A.(eq A (add y (add zero x_Demod_524)) (add y (mult (add zero y) zero))))
- (eq_ind_r A (mult zero (add zero y))
- (\lambda x_SupR_628:A.(eq A (add y x_SupR_628) (add y (mult (add zero y) zero))))
- (eq_ind_r A (mult (add y (add zero y)) (add y zero))
- (\lambda x_Demod_195:A.(eq A (add y (mult zero (add zero y))) x_Demod_195))
- (eq_ind_r A (mult (add y zero) (add y (add zero y)))
- (\lambda x_SupR_272:A.(eq A x_SupR_272 (mult (add y (add zero y)) (add y zero)))) (H1 (add y zero) (add y (add zero y))) (add y (mult zero (add zero y))) (H2 y zero (add zero y))) (add y (mult (add zero y) zero)) (H2 y (add zero y) zero)) (add zero (mult zero y))
- (eq_ind A (add zero zero)
- (\lambda x_SupR_296:A.(eq A (add zero (mult zero y)) (mult x_SupR_296 (add zero y)))) (H2 zero zero y) zero (H4 zero))) zero
- (eq_ind A (mult zero one)
- (\lambda x_Demod_507:A.(eq A x_Demod_507 (mult zero y)))
- (eq_ind_r A (add y one)
- (\lambda x_Demod_506:A.(eq A (mult zero x_Demod_506) (mult zero y)))
- (eq_ind_r A (inv zero)
- (\lambda x_SupR_785:A.(eq A (mult zero (add y x_SupR_785)) (mult zero y)))
- (eq_ind A (add (mult zero y) zero)
- (\lambda x_Demod_223:A.(eq A (mult zero (add y (inv zero))) x_Demod_223))
- (eq_ind A (mult zero (inv zero))
- (\lambda x_SupR_337:A.(eq A (mult zero (add y (inv zero))) (add (mult zero y) x_SupR_337))) (H3 zero y (inv zero)) zero (H7 zero)) (mult zero y) (H4 (mult zero y))) one
- (eq_ind A (add (inv zero) zero)
- (\lambda x_SupR_463:A.(eq A one x_SupR_463)) (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_298:A.(eq A x_SupR_298 (add (inv zero) zero))) (H zero (inv zero)) one (H6 zero)) (inv zero) (H4 (inv zero)))) one
- (eq_ind A (add y (inv y))
- (\lambda x_Demod_268:A.(eq A x_Demod_268 (add y one)))
- (eq_ind A (mult (inv y) one)
- (\lambda x_SupR_396:A.(eq A (add y x_SupR_396) (add y one)))
- (eq_ind_r A (mult one (add y one))
- (\lambda x_Demod_199:A.(eq A (add y (mult (inv y) one)) x_Demod_199))
- (eq_ind A (add y (inv y))
- (\lambda x_SupR_268:A.(eq A (add y (mult (inv y) one)) (mult x_SupR_268 (add y one)))) (H2 y (inv y) one) one (H6 y)) (add y one)
- (eq_ind A (mult (add y one) one)
- (\lambda x_SupR_271:A.(eq A x_SupR_271 (mult one (add y one)))) (H1 (add y one) one) (add y one) (H5 (add y one)))) (inv y) (H5 (inv y))) one (H6 y))) zero (H5 zero))) zero (H4 zero))) x
- (eq_ind A (add x zero)
- (\lambda x_SupR_299:A.(eq A x_SupR_299 (add zero x))) (H x zero) x (H4 x))) y (H4 y)) (add y (add x (mult y z)))
- (sym_eq\subst[A \Assign A ; x \Assign (add y (add x (mult y z))) ; y \Assign y]
- (eq_ind_r A (add zero y)
- (\lambda x_SupR_826:A.(eq A (add y (add x (mult y z))) x_SupR_826))
- (eq_ind_r A (add zero (mult (add y zero) y))
- (\lambda x_Demod_553:A.(eq A (add y (add x (mult y z))) x_Demod_553))
- (eq_ind A (add (add y (add x (mult y z))) zero)
- (\lambda x_Demod_540:A.(eq A x_Demod_540 (add (add x (mult y z)) (mult (add y (add x (mult y z))) y))))
- (eq_ind_r A (mult zero (add x (mult y z)))
- (\lambda x_Demod_526:A.(eq A (add (add y (add x (mult y z))) x_Demod_526) (add (add x (mult y z)) (mult (add y (add x (mult y z))) y))))
- (eq_ind_r A (mult (add y (add x (mult y z))) (add (add y (add x (mult y z))) (add x (mult y z))))
- (\lambda x_SupR_606:A.(eq A (add (add y (add x (mult y z))) (mult zero (add x (mult y z)))) x_SupR_606))
- (eq_ind A (add (add y (add x (mult y z))) zero)
- (\lambda x_SupR_296:A.(eq A (add (add y (add x (mult y z))) (mult zero (add x (mult y z)))) (mult x_SupR_296 (add (add y (add x (mult y z))) (add x (mult y z))))))
- (H2 (add y (add x (mult y z))) zero (add x (mult y z))) (add y (add x (mult y z))) (H4 (add y (add x (mult y z))))) (add (add x (mult y z)) (mult (add y (add x (mult y z))) y))
- (eq_ind A (add (add x (mult y z)) y)
- (\lambda x_SupR_357:A.(eq A (add (add x (mult y z)) (mult (add y (add x (mult y z))) y)) (mult x_SupR_357 (add (add y (add x (mult y z))) (add x (mult y z)))))) (eq_ind A (mult (add (add y (add x (mult y z))) (add x (mult y z))) (add (add x (mult y z)) y))
- (\lambda x_SupR_310:A.(eq A (add (add x (mult y z)) (mult (add y (add x (mult y z))) y)) x_SupR_310)) (eq_ind A (add (add x (mult y z)) (add y (add x (mult y z))))
- (\lambda x_SupR_302:A.(eq A (add (add x (mult y z)) (mult (add y (add x (mult y z))) y)) (mult x_SupR_302 (add (add x (mult y z)) y)))) (H2 (add x (mult y z)) (add y (add x (mult y z))) y) (add (add y (add x (mult y z))) (add x (mult y z))) (H (add x (mult y z)) (add y (add x (mult y z))))) (mult (add (add x (mult y z)) y) (add (add y (add x (mult y z))) (add x (mult y z)))) (H1 (add (add y (add x (mult y z))) (add x (mult y z))) (add (add x (mult y z)) y))) (add y (add x (mult y z))) (H (add x (mult y z)) y))) zero (eq_ind A (mult zero one)
- (\lambda x_Demod_507:A.(eq A x_Demod_507 (mult zero (add x (mult y z)))))
- (eq_ind_r A (add (add x (mult y z)) one)
- (\lambda x_Demod_506:A.(eq A (mult zero x_Demod_506) (mult zero (add x (mult y z)))))
- (eq_ind_r A (inv zero)
- (\lambda x_SupR_785:A.(eq A (mult zero (add (add x (mult y z)) x_SupR_785)) (mult zero (add x (mult y z)))))
- (eq_ind A (add (mult zero (add x (mult y z))) zero)
- (\lambda x_Demod_223:A.(eq A (mult zero (add (add x (mult y z)) (inv zero))) x_Demod_223))
- (eq_ind A (mult zero (inv zero))
- (\lambda x_SupR_337:A.(eq A (mult zero (add (add x (mult y z)) (inv zero))) (add (mult zero (add x (mult y z))) x_SupR_337))) (H3 zero (add x (mult y z)) (inv zero)) zero (H7 zero)) (mult zero (add x (mult y z))) (H4 (mult zero (add x (mult y z))))) one
- (eq_ind A (add (inv zero) zero)
- (\lambda x_SupR_463:A.(eq A one x_SupR_463))
- (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_298:A.(eq A x_SupR_298 (add (inv zero) zero))) (H zero (inv zero)) one (H6 zero)) (inv zero) (H4 (inv zero)))) one
- (eq_ind A (add (add x (mult y z)) (inv (add x (mult y z))))
- (\lambda x_Demod_268:A.(eq A x_Demod_268 (add (add x (mult y z)) one)))
- (eq_ind A (mult (inv (add x (mult y z))) one)
- (\lambda x_SupR_396:A.(eq A (add (add x (mult y z)) x_SupR_396) (add (add x (mult y z)) one)))
- (eq_ind_r A (mult one (add (add x (mult y z)) one))
- (\lambda x_Demod_199:A.(eq A (add (add x (mult y z)) (mult (inv (add x (mult y z))) one)) x_Demod_199))
- (eq_ind A (add (add x (mult y z)) (inv (add x (mult y z))))
- (\lambda x_SupR_268:A.(eq A (add (add x (mult y z)) (mult (inv (add x (mult y z))) one)) (mult x_SupR_268 (add (add x (mult y z)) one)))) (H2 (add x (mult y z)) (inv (add x (mult y z))) one) one (H6 (add x (mult y z)))) (add (add x (mult y z)) one) (eq_ind A (mult (add (add x (mult y z)) one) one)
- (\lambda x_SupR_271:A.(eq A x_SupR_271 (mult one (add (add x (mult y z)) one)))) (H1 (add (add x (mult y z)) one) one) (add (add x (mult y z)) one) (H5 (add (add x (mult y z)) one)))) (inv (add x (mult y z))) (H5 (inv (add x (mult y z))))) one (H6 (add x (mult y z))))) zero (H5 zero))) (add y (add x (mult y z))) (H4 (add y (add x (mult y z))))) (add zero y)
- (eq_ind A (add y zero)
- (\lambda x_Demod_543:A.(eq A (add zero x_Demod_543) (add zero (mult (add y zero) y))))
- (eq_ind_r A (mult zero zero)
- (\lambda x_Demod_524:A.(eq A (add zero (add y x_Demod_524)) (add zero (mult (add y zero) y))))
- (eq_ind_r A (mult y (add y zero))
- (\lambda x_SupR_628:A.(eq A (add zero x_SupR_628) (add zero (mult (add y zero) y))))
- (eq_ind_r A (mult (add zero (add y zero)) (add zero y))
- (\lambda x_Demod_195:A.(eq A (add zero (mult y (add y zero))) x_Demod_195))
- (eq_ind_r A (mult (add zero y) (add zero (add y zero)))
- (\lambda x_SupR_272:A.(eq A x_SupR_272 (mult (add zero (add y zero)) (add zero y)))) (H1 (add zero y) (add zero (add y zero))) (add zero (mult y (add y zero))) (H2 zero y (add y zero))) (add zero (mult (add y zero) y)) (H2 zero (add y zero) y)) (add y (mult zero zero))
- (eq_ind A (add y zero)
- (\lambda x_SupR_296:A.(eq A (add y (mult zero zero)) (mult x_SupR_296 (add y zero)))) (H2 y zero zero) y (H4 y))) zero
- (eq_ind A (mult zero one)
- (\lambda x_Demod_507:A.(eq A x_Demod_507 (mult zero zero)))
- (eq_ind_r A (add zero one)
- (\lambda x_Demod_506:A.(eq A (mult zero x_Demod_506) (mult zero zero)))
- (eq_ind_r A (inv zero)
- (\lambda x_SupR_785:A.(eq A (mult zero (add zero x_SupR_785)) (mult zero zero)))
- (eq_ind A (add (mult zero zero) zero)
- (\lambda x_Demod_223:A.(eq A (mult zero (add zero (inv zero))) x_Demod_223))
- (eq_ind A (mult zero (inv zero))
- (\lambda x_SupR_337:A.(eq A (mult zero (add zero (inv zero))) (add (mult zero zero) x_SupR_337))) (H3 zero zero (inv zero)) zero (H7 zero)) (mult zero zero) (H4 (mult zero zero))) one
- (eq_ind A (add (inv zero) zero)
- (\lambda x_SupR_463:A.(eq A one x_SupR_463))
- (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_298:A.(eq A x_SupR_298 (add (inv zero) zero))) (H zero (inv zero)) one (H6 zero)) (inv zero) (H4 (inv zero)))) one
- (eq_ind A (add zero (inv zero))
- (\lambda x_Demod_268:A.(eq A x_Demod_268 (add zero one)))
- (eq_ind A (mult (inv zero) one) (\lambda x_SupR_396:A.(eq A (add zero x_SupR_396) (add zero one)))
- (eq_ind_r A (mult one (add zero one))
- (\lambda x_Demod_199:A.(eq A (add zero (mult (inv zero) one)) x_Demod_199))
- (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_268:A.(eq A (add zero (mult (inv zero) one)) (mult x_SupR_268 (add zero one)))) (H2 zero (inv zero) one) one (H6 zero)) (add zero one)
- (eq_ind A (mult (add zero one) one)
- (\lambda x_SupR_271:A.(eq A x_SupR_271 (mult one (add zero one)))) (H1 (add zero one) one) (add zero one) (H5 (add zero one)))) (inv zero) (H5 (inv zero))) one (H6 zero))) zero (H5 zero))) y (H4 y))) y
- (eq_ind A (add y zero)
- (\lambda x_SupR_299:A.(eq A x_SupR_299 (add zero y))) (H y zero) y (H4 y))))) (add x y)
- (sym_eq\subst[A \Assign A ; x \Assign (add x y) ; y \Assign x]
- (eq_ind_r A (add zero x)
- (\lambda x_SupR_826:A.(eq A (add x y) x_SupR_826))
- (eq_ind_r A (add zero (mult (add x zero) x))
- (\lambda x_Demod_553:A.(eq A (add x y) x_Demod_553))
- (eq_ind A (add (add x y) zero)
- (\lambda x_Demod_540:A.(eq A x_Demod_540 (add y (mult (add x y) x))))
- (eq_ind_r A (mult zero y)
- (\lambda x_Demod_526:A.(eq A (add (add x y) x_Demod_526) (add y (mult (add x y) x))))
- (eq_ind_r A (mult (add x y) (add (add x y) y))
- (\lambda x_SupR_606:A.(eq A (add (add x y) (mult zero y)) x_SupR_606))
- (eq_ind A (add (add x y) zero)
- (\lambda x_SupR_296:A.(eq A (add (add x y) (mult zero y)) (mult x_SupR_296 (add (add x y) y)))) (H2 (add x y) zero y) (add x y) (H4 (add x y))) (add y (mult (add x y) x))
- (eq_ind A (add y x)
- (\lambda x_SupR_357:A.(eq A (add y (mult (add x y) x)) (mult x_SupR_357 (add (add x y) y))))
- (eq_ind A (mult (add (add x y) y) (add y x))
- (\lambda x_SupR_310:A.(eq A (add y (mult (add x y) x)) x_SupR_310))
- (eq_ind A (add y (add x y))
- (\lambda x_SupR_302:A.(eq A (add y (mult (add x y) x)) (mult x_SupR_302 (add y x)))) (H2 y (add x y) x) (add (add x y) y) (H y (add x y))) (mult (add y x) (add (add x y) y)) (H1 (add (add x y) y) (add y x))) (add x y) (H y x))) zero
- (eq_ind A (mult zero one)
- (\lambda x_Demod_507:A.(eq A x_Demod_507 (mult zero y)))
- (eq_ind_r A (add y one)
- (\lambda x_Demod_506:A.(eq A (mult zero x_Demod_506) (mult zero y)))
- (eq_ind_r A (inv zero)
- (\lambda x_SupR_785:A.(eq A (mult zero (add y x_SupR_785)) (mult zero y)))
- (eq_ind A (add (mult zero y) zero)
- (\lambda x_Demod_223:A.(eq A (mult zero (add y (inv zero))) x_Demod_223))
- (eq_ind A (mult zero (inv zero))
- (\lambda x_SupR_337:A.(eq A (mult zero (add y (inv zero))) (add (mult zero y) x_SupR_337))) (H3 zero y (inv zero)) zero (H7 zero)) (mult zero y) (H4 (mult zero y))) one
- (eq_ind A (add (inv zero) zero)
- (\lambda x_SupR_463:A.(eq A one x_SupR_463))
- (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_298:A.(eq A x_SupR_298 (add (inv zero) zero))) (H zero (inv zero)) one (H6 zero)) (inv zero) (H4 (inv zero)))) one
- (eq_ind A (add y (inv y))
- (\lambda x_Demod_268:A.(eq A x_Demod_268 (add y one)))
- (eq_ind A (mult (inv y) one)
- (\lambda x_SupR_396:A.(eq A (add y x_SupR_396) (add y one)))
- (eq_ind_r A (mult one (add y one))
- (\lambda x_Demod_199:A.(eq A (add y (mult (inv y) one)) x_Demod_199))
- (eq_ind A (add y (inv y))
- (\lambda x_SupR_268:A.(eq A (add y (mult (inv y) one)) (mult x_SupR_268 (add y one)))) (H2 y (inv y) one) one (H6 y)) (add y one)
- (eq_ind A (mult (add y one) one)
- (\lambda x_SupR_271:A.(eq A x_SupR_271 (mult one (add y one)))) (H1 (add y one) one) (add y one) (H5 (add y one)))) (inv y) (H5 (inv y))) one (H6 y))) zero (H5 zero))) (add x y) (H4 (add x y))) (add zero x)
- (eq_ind A (add x zero)
- (\lambda x_Demod_543:A.(eq A (add zero x_Demod_543) (add zero (mult (add x zero) x))))
- (eq_ind_r A (mult zero zero)
- (\lambda x_Demod_524:A.(eq A (add zero (add x x_Demod_524)) (add zero (mult (add x zero) x))))
- (eq_ind_r A (mult x (add x zero))
- (\lambda x_SupR_628:A.(eq A (add zero x_SupR_628) (add zero (mult (add x zero) x))))
- (eq_ind_r A (mult (add zero (add x zero)) (add zero x))
- (\lambda x_Demod_195:A.(eq A (add zero (mult x (add x zero))) x_Demod_195))
- (eq_ind_r A (mult (add zero x) (add zero (add x zero)))
- (\lambda x_SupR_272:A.(eq A x_SupR_272 (mult (add zero (add x zero)) (add zero x)))) (H1 (add zero x) (add zero (add x zero))) (add zero (mult x (add x zero))) (H2 zero x (add x zero))) (add zero (mult (add x zero) x)) (H2 zero (add x zero) x)) (add x (mult zero zero))
- (eq_ind A (add x zero)
- (\lambda x_SupR_296:A.(eq A (add x (mult zero zero)) (mult x_SupR_296 (add x zero)))) (H2 x zero zero) x (H4 x))) zero
- (eq_ind A (mult zero one)
- (\lambda x_Demod_507:A.(eq A x_Demod_507 (mult zero zero)))
- (eq_ind_r A (add zero one)
- (\lambda x_Demod_506:A.(eq A (mult zero x_Demod_506) (mult zero zero)))
- (eq_ind_r A (inv zero)
- (\lambda x_SupR_785:A.(eq A (mult zero (add zero x_SupR_785)) (mult zero zero)))
- (eq_ind A (add (mult zero zero) zero)
- (\lambda x_Demod_223:A.(eq A (mult zero (add zero (inv zero))) x_Demod_223))
- (eq_ind A (mult zero (inv zero))
- (\lambda x_SupR_337:A.(eq A (mult zero (add zero (inv zero))) (add (mult zero zero) x_SupR_337))) (H3 zero zero (inv zero)) zero (H7 zero)) (mult zero zero) (H4 (mult zero zero))) one
- (eq_ind A (add (inv zero) zero)
- (\lambda x_SupR_463:A.(eq A one x_SupR_463))
- (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_298:A.(eq A x_SupR_298 (add (inv zero) zero))) (H zero (inv zero)) one (H6 zero)) (inv zero) (H4 (inv zero)))) one
- (eq_ind A (add zero (inv zero))
- (\lambda x_Demod_268:A.(eq A x_Demod_268 (add zero one)))
- (eq_ind A (mult (inv zero) one)
- (\lambda x_SupR_396:A.(eq A (add zero x_SupR_396) (add zero one)))
- (eq_ind_r A (mult one (add zero one))
- (\lambda x_Demod_199:A.(eq A (add zero (mult (inv zero) one)) x_Demod_199))
- (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_268:A.(eq A (add zero (mult (inv zero) one)) (mult x_SupR_268 (add zero one)))) (H2 zero (inv zero) one) one (H6 zero)) (add zero one)
- (eq_ind A (mult (add zero one) one)
- (\lambda x_SupR_271:A.(eq A x_SupR_271 (mult one (add zero one)))) (H1 (add zero one) one) (add zero one) (H5 (add zero one)))) (inv zero) (H5 (inv zero))) one (H6 zero))) zero (H5 zero))) x (H4 x))) x
- (eq_ind A (add x zero)
- (\lambda x_SupR_299:A.(eq A x_SupR_299 (add zero x))) (H x zero) x (H4 x))))) (add (add x (mult y z)) y)
- (sym_eq\subst[A \Assign A ; x \Assign (add (add x (mult y z)) y) ; y \Assign (add y (add x (mult y z)))] (H (add x (mult y z)) y)))
-). qed.
+ (*
+ \forall hint1: (\forall x,y:A. (add (inv x) (mult y x)) = (add (inv x) y)).
+ \forall hint2: \forall x,y:A.zero = (mult (inv x) (mult x y)).
+ \forall hint2: (\forall x,y:A. zero = (mult (inv (add x y)) y)).
+ *)
+ \forall x,y:A.
+ inv x = (add (inv x) (inv (add y x))).
+intros.auto paramodulation.
+qed.
-STOP
+theorem bool609:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ (*
+ \forall hint1: (\forall x,y:A. (add (inv x) (mult y x)) = (add (inv x) y)).
+ \forall hint2: \forall x,y:A.zero = (mult (inv x) (mult x y)).
+ \forall hint2: (\forall x,y:A. zero = (mult (inv (add x y)) y)).
+ *)
+ \forall x,y:A.
+ inv x = (add (inv (add y x)) (inv x)).
+intros.auto paramodulation.
+qed.
+(*
+theorem bool260:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall x,y,z:A.
+ add x (mult x y) = mult x (add x y).
+intros.auto paramodulation.
+qed.
+
+theorem bool276:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall x,y,z,u:A.
+ (mult (add x y) (add z (add x u))) = (add (mult (add x y) z) (add x (mult y u))).
+intros.auto paramodulation.
+qed.
+
+theorem bool250:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall x,y,z:A.
+ add x (mult y z) = mult (add y x) (add x z).
+intros.auto paramodulation.
+qed.
+theorem bool756minimal:
+ \forall A:Set.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall hint1: (\forall x,y,z,u:A.
+ add y (add x (mult x u)) = (add (mult (add x y) z) (add x (mult y u)))).
+ \forall hint2: (\forall x,y:A. x = (mult (add y x) x)).
+ \forall x,y,z:A.
+ add x (add y (mult y z)) = add x (add y (mult x z)).
+intros;
auto paramodulation.
+qed.
+theorem bool756simplified:
+ \forall A:Set.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall hint1: (\forall x,y,z,u:A.
+ (mult (add x y) (add z (add x u))) = (add (mult (add x y) z) (add x (mult y u)))).
+ \forall hint2: (\forall x,y:A. x = (mult (add y x) x)).
+ \forall hint3: (\forall x,y,z:A.
+ add x (mult y z) = mult (add y x) (add x z)).
+ \forall hint4: (\forall x,y:A.
+ add x (mult x y) = mult x (add x y)).
+ \forall x,y,z:A.
+ add x (add y (mult y z)) = add x (add y (mult x z)).
+intros;
+auto paramodulation.
qed.
+(* 46 sec. *)
-theorem bool4:
+theorem bool756:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall hint1: (\forall x,y,z,u:A.
+ (mult (add x y) (add z (add x u))) = (add (mult (add x y) z) (add x (mult y u)))).
+ \forall hint2: (\forall x,y:A. x = (mult (add y x) x)).
+ \forall hint3: (\forall x,y,z:A.
+ add x (mult y z) = mult (add y x) (add x z)).
+ \forall hint4: (\forall x,y:A.
+ add x (mult x y) = mult x (add x y)).
+ \forall x,y,z:A.
+ add x y = add x (add y (mult x z)).
+intros;
+cut (mult (add y x) (add x (add y z)) = add x (add y (mult x z)));
+[auto paramodulation
+|auto paramodulation]
+qed.
+(* 186 sec *)
+*)
+theorem bool756full:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall x,y,z:A.
+ add x y = add x (add y (mult x z)).
+intros;auto paramodulation.
+qed.
+(* war=5; active 225, maxmeta 172568 *)
+(* war=4; active 249, maxmeta 223220 *)
+(*
+theorem bool1164:
\forall A:Set.
\forall one:A.
\forall zero:A.
\forall inv2: (\forall x:A. (mult x (inv x)) = zero).
\forall x,y,z:A.
(add x y) = (add (add x (mult y z)) y).
-intros.
-exact
-((eq_ind A (add y (add x (mult y z)))
- (\lambda x_DemodGoal_193:A.(eq A (add x y) x_DemodGoal_193))
- (eq_ind A x
- (\lambda x_DemodGoal_893:A.(eq A x_DemodGoal_893 (add y (add x (mult y z)))))
- (eq_ind A y (\lambda x_DemodGoal_894:A.(eq A x x_DemodGoal_894))
- (eq_ind A (add y zero) (\lambda x_Demod_554:A.(eq A x x_Demod_554))
- (eq_ind_r A (add zero x) (\lambda x_SupR_809:A.(eq A x_SupR_809 (add y zero)))
- (eq_ind_r A (add y (mult (add zero y) zero))
- (\lambda x_Demod_553:A.(eq A (add zero x) x_Demod_553))
- (eq_ind A (add (add zero x) zero)
- (\lambda x_Demod_540:A.(eq A x_Demod_540 (add x (mult (add zero x) zero))))
- (eq_ind_r A (mult zero x)
- (\lambda x_Demod_526:A.(eq A (add (add zero x) x_Demod_526) (add x (mult (add zero x) zero))))
- (eq_ind_r A (mult (add zero x) (add (add zero x) x))
- (\lambda x_SupR_606:A.(eq A (add (add zero x) (mult zero x)) x_SupR_606))
- (eq_ind A (add (add zero x) zero)
- (\lambda x_SupR_296:A.(eq A (add (add zero x) (mult zero x)) (mult x_SupR_296 (add (add zero x) x))))
- (H2 (add zero x) zero x) (add zero x) (H4 (add zero x))) (add x (mult (add zero x) zero))
- (eq_ind A (add x zero)
- (\lambda x_SupR_357:A.(eq A (add x (mult (add zero x) zero)) (mult x_SupR_357 (add (add zero x) x))))
- (eq_ind A (mult (add (add zero x) x) (add x zero))
- (\lambda x_SupR_310:A.(eq A (add x (mult (add zero x) zero)) x_SupR_310))
- (eq_ind A (add x (add zero x))
- (\lambda x_SupR_302:A.(eq A (add x (mult (add zero x) zero)) (mult x_SupR_302 (add x zero))))
- (H2 x (add zero x) zero) (add (add zero x) x) (H x (add zero x))) (mult (add x zero) (add (add zero x) x)) (H1 (add (add zero x) x) (add x zero))) (add zero x) (H x zero))) zero
- (eq_ind A (mult zero one)
- (\lambda x_Demod_507:A.(eq A x_Demod_507 (mult zero x)))
- (eq_ind_r A (add x one)
- (\lambda x_Demod_506:A.(eq A (mult zero x_Demod_506) (mult zero x)))
- (eq_ind_r A (inv zero)
- (\lambda x_SupR_785:A.(eq A (mult zero (add x x_SupR_785)) (mult zero x)))
- (eq_ind A (add (mult zero x) zero)
- (\lambda x_Demod_223:A.(eq A (mult zero (add x (inv zero))) x_Demod_223))
- (eq_ind A (mult zero (inv zero))
- (\lambda x_SupR_337:A.(eq A (mult zero (add x (inv zero))) (add (mult zero x) x_SupR_337)))
- (H3 zero x (inv zero)) zero (H7 zero)) (mult zero x) (H4 (mult zero x))) one
- (eq_ind A (add (inv zero) zero)
- (\lambda x_SupR_463:A.(eq A one x_SupR_463))
- (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_298:A.(eq A x_SupR_298 (add (inv zero) zero))) (H zero (inv zero)) one (H6 zero)) (inv zero) (H4 (inv zero)))) one
- (eq_ind A (add x (inv x))
- (\lambda x_Demod_268:A.(eq A x_Demod_268 (add x one)))
- (eq_ind A (mult (inv x) one)
- (\lambda x_SupR_396:A.(eq A (add x x_SupR_396) (add x one)))
- (eq_ind_r A (mult one (add x one))
- (\lambda x_Demod_199:A.(eq A (add x (mult (inv x) one)) x_Demod_199))
- (eq_ind A (add x (inv x))
- (\lambda x_SupR_268:A.(eq A (add x (mult (inv x) one)) (mult x_SupR_268 (add x one)))) (H2 x (inv x) one) one (H6 x)) (add x one)
- (eq_ind A (mult (add x one) one)
- (\lambda x_SupR_271:A.(eq A x_SupR_271 (mult one (add x one))))
- (H1 (add x one) one) (add x one) (H5 (add x one)))) (inv x) (H5 (inv x))) one (H6 x))) zero (H5 zero))) (add zero x) (H4 (add zero x))) (add y zero)
- (eq_ind A (add zero zero)
- (\lambda x_Demod_543:A.(eq A (add y x_Demod_543) (add y (mult (add zero y) zero))))
- (eq_ind_r A (mult zero y)
- (\lambda x_Demod_524:A.(eq A (add y (add zero x_Demod_524)) (add y (mult (add zero y) zero))))
- (eq_ind_r A (mult zero (add zero y))
- (\lambda x_SupR_628:A.(eq A (add y x_SupR_628) (add y (mult (add zero y) zero))))
- (eq_ind_r A (mult (add y (add zero y)) (add y zero))
- (\lambda x_Demod_195:A.(eq A (add y (mult zero (add zero y))) x_Demod_195))
- (eq_ind_r A (mult (add y zero) (add y (add zero y)))
- (\lambda x_SupR_272:A.(eq A x_SupR_272 (mult (add y (add zero y)) (add y zero)))) (H1 (add y zero) (add y (add zero y))) (add y (mult zero (add zero y))) (H2 y zero (add zero y))) (add y (mult (add zero y) zero)) (H2 y (add zero y) zero)) (add zero (mult zero y))
- (eq_ind A (add zero zero)
- (\lambda x_SupR_296:A.(eq A (add zero (mult zero y)) (mult x_SupR_296 (add zero y)))) (H2 zero zero y) zero (H4 zero))) zero
- (eq_ind A (mult zero one)
- (\lambda x_Demod_507:A.(eq A x_Demod_507 (mult zero y)))
- (eq_ind_r A (add y one)
- (\lambda x_Demod_506:A.(eq A (mult zero x_Demod_506) (mult zero y)))
- (eq_ind_r A (inv zero)
- (\lambda x_SupR_785:A.(eq A (mult zero (add y x_SupR_785)) (mult zero y)))
- (eq_ind A (add (mult zero y) zero)
- (\lambda x_Demod_223:A.(eq A (mult zero (add y (inv zero))) x_Demod_223))
- (eq_ind A (mult zero (inv zero))
- (\lambda x_SupR_337:A.(eq A (mult zero (add y (inv zero))) (add (mult zero y) x_SupR_337))) (H3 zero y (inv zero)) zero (H7 zero)) (mult zero y) (H4 (mult zero y))) one
- (eq_ind A (add (inv zero) zero)
- (\lambda x_SupR_463:A.(eq A one x_SupR_463)) (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_298:A.(eq A x_SupR_298 (add (inv zero) zero))) (H zero (inv zero)) one (H6 zero)) (inv zero) (H4 (inv zero)))) one
- (eq_ind A (add y (inv y))
- (\lambda x_Demod_268:A.(eq A x_Demod_268 (add y one)))
- (eq_ind A (mult (inv y) one)
- (\lambda x_SupR_396:A.(eq A (add y x_SupR_396) (add y one)))
- (eq_ind_r A (mult one (add y one))
- (\lambda x_Demod_199:A.(eq A (add y (mult (inv y) one)) x_Demod_199))
- (eq_ind A (add y (inv y))
- (\lambda x_SupR_268:A.(eq A (add y (mult (inv y) one)) (mult x_SupR_268 (add y one)))) (H2 y (inv y) one) one (H6 y)) (add y one)
- (eq_ind A (mult (add y one) one)
- (\lambda x_SupR_271:A.(eq A x_SupR_271 (mult one (add y one)))) (H1 (add y one) one) (add y one) (H5 (add y one)))) (inv y) (H5 (inv y))) one (H6 y))) zero (H5 zero))) zero (H4 zero))) x
- (eq_ind A (add x zero)
- (\lambda x_SupR_299:A.(eq A x_SupR_299 (add zero x))) (H x zero) x (H4 x))) y (H4 y)) (add y (add x (mult y z)))
- (sym_eq\subst[A \Assign A ; x \Assign (add y (add x (mult y z))) ; y \Assign y]
- (eq_ind_r A (add zero y)
- (\lambda x_SupR_826:A.(eq A (add y (add x (mult y z))) x_SupR_826))
- (eq_ind_r A (add zero (mult (add y zero) y))
- (\lambda x_Demod_553:A.(eq A (add y (add x (mult y z))) x_Demod_553))
- (eq_ind A (add (add y (add x (mult y z))) zero)
- (\lambda x_Demod_540:A.(eq A x_Demod_540 (add (add x (mult y z)) (mult (add y (add x (mult y z))) y))))
- (eq_ind_r A (mult zero (add x (mult y z)))
- (\lambda x_Demod_526:A.(eq A (add (add y (add x (mult y z))) x_Demod_526) (add (add x (mult y z)) (mult (add y (add x (mult y z))) y))))
- (eq_ind_r A (mult (add y (add x (mult y z))) (add (add y (add x (mult y z))) (add x (mult y z))))
- (\lambda x_SupR_606:A.(eq A (add (add y (add x (mult y z))) (mult zero (add x (mult y z)))) x_SupR_606))
- (eq_ind A (add (add y (add x (mult y z))) zero)
- (\lambda x_SupR_296:A.(eq A (add (add y (add x (mult y z))) (mult zero (add x (mult y z)))) (mult x_SupR_296 (add (add y (add x (mult y z))) (add x (mult y z))))))
- (H2 (add y (add x (mult y z))) zero (add x (mult y z))) (add y (add x (mult y z))) (H4 (add y (add x (mult y z))))) (add (add x (mult y z)) (mult (add y (add x (mult y z))) y))
- (eq_ind A (add (add x (mult y z)) y)
- (\lambda x_SupR_357:A.(eq A (add (add x (mult y z)) (mult (add y (add x (mult y z))) y)) (mult x_SupR_357 (add (add y (add x (mult y z))) (add x (mult y z)))))) (eq_ind A (mult (add (add y (add x (mult y z))) (add x (mult y z))) (add (add x (mult y z)) y))
- (\lambda x_SupR_310:A.(eq A (add (add x (mult y z)) (mult (add y (add x (mult y z))) y)) x_SupR_310)) (eq_ind A (add (add x (mult y z)) (add y (add x (mult y z))))
- (\lambda x_SupR_302:A.(eq A (add (add x (mult y z)) (mult (add y (add x (mult y z))) y)) (mult x_SupR_302 (add (add x (mult y z)) y)))) (H2 (add x (mult y z)) (add y (add x (mult y z))) y) (add (add y (add x (mult y z))) (add x (mult y z))) (H (add x (mult y z)) (add y (add x (mult y z))))) (mult (add (add x (mult y z)) y) (add (add y (add x (mult y z))) (add x (mult y z)))) (H1 (add (add y (add x (mult y z))) (add x (mult y z))) (add (add x (mult y z)) y))) (add y (add x (mult y z))) (H (add x (mult y z)) y))) zero (eq_ind A (mult zero one)
- (\lambda x_Demod_507:A.(eq A x_Demod_507 (mult zero (add x (mult y z)))))
- (eq_ind_r A (add (add x (mult y z)) one)
- (\lambda x_Demod_506:A.(eq A (mult zero x_Demod_506) (mult zero (add x (mult y z)))))
- (eq_ind_r A (inv zero)
- (\lambda x_SupR_785:A.(eq A (mult zero (add (add x (mult y z)) x_SupR_785)) (mult zero (add x (mult y z)))))
- (eq_ind A (add (mult zero (add x (mult y z))) zero)
- (\lambda x_Demod_223:A.(eq A (mult zero (add (add x (mult y z)) (inv zero))) x_Demod_223))
- (eq_ind A (mult zero (inv zero))
- (\lambda x_SupR_337:A.(eq A (mult zero (add (add x (mult y z)) (inv zero))) (add (mult zero (add x (mult y z))) x_SupR_337))) (H3 zero (add x (mult y z)) (inv zero)) zero (H7 zero)) (mult zero (add x (mult y z))) (H4 (mult zero (add x (mult y z))))) one
- (eq_ind A (add (inv zero) zero)
- (\lambda x_SupR_463:A.(eq A one x_SupR_463))
- (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_298:A.(eq A x_SupR_298 (add (inv zero) zero))) (H zero (inv zero)) one (H6 zero)) (inv zero) (H4 (inv zero)))) one
- (eq_ind A (add (add x (mult y z)) (inv (add x (mult y z))))
- (\lambda x_Demod_268:A.(eq A x_Demod_268 (add (add x (mult y z)) one)))
- (eq_ind A (mult (inv (add x (mult y z))) one)
- (\lambda x_SupR_396:A.(eq A (add (add x (mult y z)) x_SupR_396) (add (add x (mult y z)) one)))
- (eq_ind_r A (mult one (add (add x (mult y z)) one))
- (\lambda x_Demod_199:A.(eq A (add (add x (mult y z)) (mult (inv (add x (mult y z))) one)) x_Demod_199))
- (eq_ind A (add (add x (mult y z)) (inv (add x (mult y z))))
- (\lambda x_SupR_268:A.(eq A (add (add x (mult y z)) (mult (inv (add x (mult y z))) one)) (mult x_SupR_268 (add (add x (mult y z)) one)))) (H2 (add x (mult y z)) (inv (add x (mult y z))) one) one (H6 (add x (mult y z)))) (add (add x (mult y z)) one) (eq_ind A (mult (add (add x (mult y z)) one) one)
- (\lambda x_SupR_271:A.(eq A x_SupR_271 (mult one (add (add x (mult y z)) one)))) (H1 (add (add x (mult y z)) one) one) (add (add x (mult y z)) one) (H5 (add (add x (mult y z)) one)))) (inv (add x (mult y z))) (H5 (inv (add x (mult y z))))) one (H6 (add x (mult y z))))) zero (H5 zero))) (add y (add x (mult y z))) (H4 (add y (add x (mult y z))))) (add zero y)
- (eq_ind A (add y zero)
- (\lambda x_Demod_543:A.(eq A (add zero x_Demod_543) (add zero (mult (add y zero) y))))
- (eq_ind_r A (mult zero zero)
- (\lambda x_Demod_524:A.(eq A (add zero (add y x_Demod_524)) (add zero (mult (add y zero) y))))
- (eq_ind_r A (mult y (add y zero))
- (\lambda x_SupR_628:A.(eq A (add zero x_SupR_628) (add zero (mult (add y zero) y))))
- (eq_ind_r A (mult (add zero (add y zero)) (add zero y))
- (\lambda x_Demod_195:A.(eq A (add zero (mult y (add y zero))) x_Demod_195))
- (eq_ind_r A (mult (add zero y) (add zero (add y zero)))
- (\lambda x_SupR_272:A.(eq A x_SupR_272 (mult (add zero (add y zero)) (add zero y)))) (H1 (add zero y) (add zero (add y zero))) (add zero (mult y (add y zero))) (H2 zero y (add y zero))) (add zero (mult (add y zero) y)) (H2 zero (add y zero) y)) (add y (mult zero zero))
- (eq_ind A (add y zero)
- (\lambda x_SupR_296:A.(eq A (add y (mult zero zero)) (mult x_SupR_296 (add y zero)))) (H2 y zero zero) y (H4 y))) zero
- (eq_ind A (mult zero one)
- (\lambda x_Demod_507:A.(eq A x_Demod_507 (mult zero zero)))
- (eq_ind_r A (add zero one)
- (\lambda x_Demod_506:A.(eq A (mult zero x_Demod_506) (mult zero zero)))
- (eq_ind_r A (inv zero)
- (\lambda x_SupR_785:A.(eq A (mult zero (add zero x_SupR_785)) (mult zero zero)))
- (eq_ind A (add (mult zero zero) zero)
- (\lambda x_Demod_223:A.(eq A (mult zero (add zero (inv zero))) x_Demod_223))
- (eq_ind A (mult zero (inv zero))
- (\lambda x_SupR_337:A.(eq A (mult zero (add zero (inv zero))) (add (mult zero zero) x_SupR_337))) (H3 zero zero (inv zero)) zero (H7 zero)) (mult zero zero) (H4 (mult zero zero))) one
- (eq_ind A (add (inv zero) zero)
- (\lambda x_SupR_463:A.(eq A one x_SupR_463))
- (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_298:A.(eq A x_SupR_298 (add (inv zero) zero))) (H zero (inv zero)) one (H6 zero)) (inv zero) (H4 (inv zero)))) one
- (eq_ind A (add zero (inv zero))
- (\lambda x_Demod_268:A.(eq A x_Demod_268 (add zero one)))
- (eq_ind A (mult (inv zero) one) (\lambda x_SupR_396:A.(eq A (add zero x_SupR_396) (add zero one)))
- (eq_ind_r A (mult one (add zero one))
- (\lambda x_Demod_199:A.(eq A (add zero (mult (inv zero) one)) x_Demod_199))
- (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_268:A.(eq A (add zero (mult (inv zero) one)) (mult x_SupR_268 (add zero one)))) (H2 zero (inv zero) one) one (H6 zero)) (add zero one)
- (eq_ind A (mult (add zero one) one)
- (\lambda x_SupR_271:A.(eq A x_SupR_271 (mult one (add zero one)))) (H1 (add zero one) one) (add zero one) (H5 (add zero one)))) (inv zero) (H5 (inv zero))) one (H6 zero))) zero (H5 zero))) y (H4 y))) y
- (eq_ind A (add y zero)
- (\lambda x_SupR_299:A.(eq A x_SupR_299 (add zero y))) (H y zero) y (H4 y))))) (add x y)
- (sym_eq\subst[A \Assign A ; x \Assign (add x y) ; y \Assign x]
- (eq_ind_r A (add zero x)
- (\lambda x_SupR_826:A.(eq A (add x y) x_SupR_826))
- (eq_ind_r A (add zero (mult (add x zero) x))
- (\lambda x_Demod_553:A.(eq A (add x y) x_Demod_553))
- (eq_ind A (add (add x y) zero)
- (\lambda x_Demod_540:A.(eq A x_Demod_540 (add y (mult (add x y) x))))
- (eq_ind_r A (mult zero y)
- (\lambda x_Demod_526:A.(eq A (add (add x y) x_Demod_526) (add y (mult (add x y) x))))
- (eq_ind_r A (mult (add x y) (add (add x y) y))
- (\lambda x_SupR_606:A.(eq A (add (add x y) (mult zero y)) x_SupR_606))
- (eq_ind A (add (add x y) zero)
- (\lambda x_SupR_296:A.(eq A (add (add x y) (mult zero y)) (mult x_SupR_296 (add (add x y) y)))) (H2 (add x y) zero y) (add x y) (H4 (add x y))) (add y (mult (add x y) x))
- (eq_ind A (add y x)
- (\lambda x_SupR_357:A.(eq A (add y (mult (add x y) x)) (mult x_SupR_357 (add (add x y) y))))
- (eq_ind A (mult (add (add x y) y) (add y x))
- (\lambda x_SupR_310:A.(eq A (add y (mult (add x y) x)) x_SupR_310))
- (eq_ind A (add y (add x y))
- (\lambda x_SupR_302:A.(eq A (add y (mult (add x y) x)) (mult x_SupR_302 (add y x)))) (H2 y (add x y) x) (add (add x y) y) (H y (add x y))) (mult (add y x) (add (add x y) y)) (H1 (add (add x y) y) (add y x))) (add x y) (H y x))) zero
- (eq_ind A (mult zero one)
- (\lambda x_Demod_507:A.(eq A x_Demod_507 (mult zero y)))
- (eq_ind_r A (add y one)
- (\lambda x_Demod_506:A.(eq A (mult zero x_Demod_506) (mult zero y)))
- (eq_ind_r A (inv zero)
- (\lambda x_SupR_785:A.(eq A (mult zero (add y x_SupR_785)) (mult zero y)))
- (eq_ind A (add (mult zero y) zero)
- (\lambda x_Demod_223:A.(eq A (mult zero (add y (inv zero))) x_Demod_223))
- (eq_ind A (mult zero (inv zero))
- (\lambda x_SupR_337:A.(eq A (mult zero (add y (inv zero))) (add (mult zero y) x_SupR_337))) (H3 zero y (inv zero)) zero (H7 zero)) (mult zero y) (H4 (mult zero y))) one
- (eq_ind A (add (inv zero) zero)
- (\lambda x_SupR_463:A.(eq A one x_SupR_463))
- (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_298:A.(eq A x_SupR_298 (add (inv zero) zero))) (H zero (inv zero)) one (H6 zero)) (inv zero) (H4 (inv zero)))) one
- (eq_ind A (add y (inv y))
- (\lambda x_Demod_268:A.(eq A x_Demod_268 (add y one)))
- (eq_ind A (mult (inv y) one)
- (\lambda x_SupR_396:A.(eq A (add y x_SupR_396) (add y one)))
- (eq_ind_r A (mult one (add y one))
- (\lambda x_Demod_199:A.(eq A (add y (mult (inv y) one)) x_Demod_199))
- (eq_ind A (add y (inv y))
- (\lambda x_SupR_268:A.(eq A (add y (mult (inv y) one)) (mult x_SupR_268 (add y one)))) (H2 y (inv y) one) one (H6 y)) (add y one)
- (eq_ind A (mult (add y one) one)
- (\lambda x_SupR_271:A.(eq A x_SupR_271 (mult one (add y one)))) (H1 (add y one) one) (add y one) (H5 (add y one)))) (inv y) (H5 (inv y))) one (H6 y))) zero (H5 zero))) (add x y) (H4 (add x y))) (add zero x)
- (eq_ind A (add x zero)
- (\lambda x_Demod_543:A.(eq A (add zero x_Demod_543) (add zero (mult (add x zero) x))))
- (eq_ind_r A (mult zero zero)
- (\lambda x_Demod_524:A.(eq A (add zero (add x x_Demod_524)) (add zero (mult (add x zero) x))))
- (eq_ind_r A (mult x (add x zero))
- (\lambda x_SupR_628:A.(eq A (add zero x_SupR_628) (add zero (mult (add x zero) x))))
- (eq_ind_r A (mult (add zero (add x zero)) (add zero x))
- (\lambda x_Demod_195:A.(eq A (add zero (mult x (add x zero))) x_Demod_195))
- (eq_ind_r A (mult (add zero x) (add zero (add x zero)))
- (\lambda x_SupR_272:A.(eq A x_SupR_272 (mult (add zero (add x zero)) (add zero x)))) (H1 (add zero x) (add zero (add x zero))) (add zero (mult x (add x zero))) (H2 zero x (add x zero))) (add zero (mult (add x zero) x)) (H2 zero (add x zero) x)) (add x (mult zero zero))
- (eq_ind A (add x zero)
- (\lambda x_SupR_296:A.(eq A (add x (mult zero zero)) (mult x_SupR_296 (add x zero)))) (H2 x zero zero) x (H4 x))) zero
- (eq_ind A (mult zero one)
- (\lambda x_Demod_507:A.(eq A x_Demod_507 (mult zero zero)))
- (eq_ind_r A (add zero one)
- (\lambda x_Demod_506:A.(eq A (mult zero x_Demod_506) (mult zero zero)))
- (eq_ind_r A (inv zero)
- (\lambda x_SupR_785:A.(eq A (mult zero (add zero x_SupR_785)) (mult zero zero)))
- (eq_ind A (add (mult zero zero) zero)
- (\lambda x_Demod_223:A.(eq A (mult zero (add zero (inv zero))) x_Demod_223))
- (eq_ind A (mult zero (inv zero))
- (\lambda x_SupR_337:A.(eq A (mult zero (add zero (inv zero))) (add (mult zero zero) x_SupR_337))) (H3 zero zero (inv zero)) zero (H7 zero)) (mult zero zero) (H4 (mult zero zero))) one
- (eq_ind A (add (inv zero) zero)
- (\lambda x_SupR_463:A.(eq A one x_SupR_463))
- (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_298:A.(eq A x_SupR_298 (add (inv zero) zero))) (H zero (inv zero)) one (H6 zero)) (inv zero) (H4 (inv zero)))) one
- (eq_ind A (add zero (inv zero))
- (\lambda x_Demod_268:A.(eq A x_Demod_268 (add zero one)))
- (eq_ind A (mult (inv zero) one)
- (\lambda x_SupR_396:A.(eq A (add zero x_SupR_396) (add zero one)))
- (eq_ind_r A (mult one (add zero one))
- (\lambda x_Demod_199:A.(eq A (add zero (mult (inv zero) one)) x_Demod_199))
- (eq_ind A (add zero (inv zero))
- (\lambda x_SupR_268:A.(eq A (add zero (mult (inv zero) one)) (mult x_SupR_268 (add zero one)))) (H2 zero (inv zero) one) one (H6 zero)) (add zero one)
- (eq_ind A (mult (add zero one) one)
- (\lambda x_SupR_271:A.(eq A x_SupR_271 (mult one (add zero one)))) (H1 (add zero one) one) (add zero one) (H5 (add zero one)))) (inv zero) (H5 (inv zero))) one (H6 zero))) zero (H5 zero))) x (H4 x))) x
- (eq_ind A (add x zero)
- (\lambda x_SupR_299:A.(eq A x_SupR_299 (add zero x))) (H x zero) x (H4 x))))) (add (add x (mult y z)) y)
- (sym_eq\subst[A \Assign A ; x \Assign (add (add x (mult y z)) y) ; y \Assign (add y (add x (mult y z)))] (H (add x (mult y z)) y)))
-).
-auto paramodulation.
+intros.auto paramodulation.
+qed.
+theorem bool1230:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall x,y,z:A.
+ \forall c1z: (\forall x:A.(add x z) = (add z x)).
+ add (add x y) z = add (add x y) (add z y).
+intros.auto paramodulation.
qed.
+
+theorem bool1230:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall x,y,z:A.
+ add (add x y) z = add (add x y) (add z y).
+intros.auto paramodulation.
+qed.
+
+theorem bool1372:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall x,y,z:A.
+ add x (add y z) = add (add x z) y.
+intros.auto paramodulation.
+qed.*)
+
+theorem bool381:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall x,y:A.
+ add (inv x) y = add (mult x y) (inv x).
+intros.auto paramodulation.
+qed.
+
+theorem bool5hint1:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall hint1731:(\forall x,y:A. add (inv (add x y)) y = add y (inv x)).
+ \forall hint1735:(\forall x,y:A. add (inv (add x y)) x = add x (inv y)).
+ \forall hint623:(\forall x,y:A. inv (mult x y) = add (inv x) (inv (mult x y))).
+ \forall x,y:A.
+ (inv (mult x y)) = (add (inv x) (inv y)).
+intros.auto paramodulation.
+qed.
+(* 90 *)
+
+theorem bool5hint2:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall hint1731:(\forall x,y:A. add (inv (add x y)) y = add y (inv x)).
+ \forall hint623:(\forall x,y:A. inv (mult x y) = add (inv x) (inv (mult x y))).
+ \forall x,y:A.
+ (inv (mult x y)) = (add (inv x) (inv y)).
+intros.auto paramodulation.
+qed.
+(* 41 *)
+
+theorem bool5hint3:
+ \forall A:Set.
+ \forall one:A.
+ \forall zero:A.
+ \forall add: A \to A \to A.
+ \forall mult: A \to A \to A.
+ \forall inv: A \to A.
+ \forall c1:(\forall x,y:A.(add x y) = (add y x)).
+ \forall c2:(\forall x,y:A.(mult x y) = (mult y x)).
+ \forall d1: (\forall x,y,z:A.
+ (add x (mult y z)) = (mult (add x y) (add x z))).
+ \forall d2: (\forall x,y,z:A.
+ (mult x (add y z)) = (add (mult x y) (mult x z))).
+ \forall i1: (\forall x:A. (add x zero) = x).
+ \forall i2: (\forall x:A. (mult x one) = x).
+ \forall inv1: (\forall x:A. (add x (inv x)) = one).
+ \forall inv2: (\forall x:A. (mult x (inv x)) = zero).
+ \forall hint1731:(\forall x,y:A. add (inv (add x y)) y = add y (inv x)).
+ \forall hint609:(\forall x,y:A. inv x = add (inv (add y x)) (inv x)).
+ \forall x,y:A.
+ (inv (mult x y)) = (add (inv x) (inv y)).
+intros.auto paramodulation.
+qed.
+(* 41 *)
+
theorem bool5:
\forall A:Set.
\forall one:A.
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