From: Andrea Asperti Date: Mon, 6 Feb 2006 08:16:41 +0000 (+0000) Subject: Bug? X-Git-Tag: 0.4.95@7852~1728 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=b86caba4f52e441e0456f7afa404750ff0ba1109;p=helm.git Bug? --- diff --git a/matita/tests/SK.ma b/matita/tests/SK.ma index 708f92f30..2aa5f51e9 100644 --- a/matita/tests/SK.ma +++ b/matita/tests/SK.ma @@ -92,8 +92,281 @@ theorem bool3: \forall x:A. (inv (inv x)) = x. intros.auto paramodulation. qed. - -theorem bool2: + +theorem bool4: + \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 (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. + +qed. + +theorem bool5: \forall A:Set. \forall one:A. \forall zero:A.