%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SWC185+1 : TPTP v9.0.0. Released v2.4.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n010.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:30:27 AM UTC 2025
% Result : Theorem 1.23s 1.39s
% Output : Proof 1.23s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SWC185+1 : TPTP v9.0.0. Released v2.4.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n010.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Fri Jun 20 08:34:26 EDT 2025
% 0.11/0.33 % CPUTime :
% 1.23/1.39 % SZS status Theorem
% 1.23/1.39 % SZS output start Proof
% 1.23/1.39 tff(memberP_type, type, (
% 1.23/1.39 memberP: ( $i * $i ) > $o)).
% 1.23/1.39 tff(tptp_fun_X2_47_type, type, (
% 1.23/1.39 tptp_fun_X2_47: $i)).
% 1.23/1.39 tff(tptp_fun_X3_48_type, type, (
% 1.23/1.39 tptp_fun_X3_48: $i)).
% 1.23/1.39 tff(tptp_fun_X4_49_type, type, (
% 1.23/1.39 tptp_fun_X4_49: $i)).
% 1.23/1.39 tff(tptp_fun_U_45_type, type, (
% 1.23/1.39 tptp_fun_U_45: $i)).
% 1.23/1.39 tff(app_type, type, (
% 1.23/1.39 app: ( $i * $i ) > $i)).
% 1.23/1.39 tff(cons_type, type, (
% 1.23/1.39 cons: ( $i * $i ) > $i)).
% 1.23/1.39 tff(nil_type, type, (
% 1.23/1.39 nil: $i)).
% 1.23/1.39 tff(ssList_type, type, (
% 1.23/1.39 ssList: $i > $o)).
% 1.23/1.39 tff(ssItem_type, type, (
% 1.23/1.39 ssItem: $i > $o)).
% 1.23/1.39 tff(tptp_fun_V_46_type, type, (
% 1.23/1.39 tptp_fun_V_46: $i)).
% 1.23/1.39 tff(1,plain,
% 1.23/1.39 ((ssList(U!45) & (ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y)))))) & ssList(U!45) & ssItem(X2!47) & ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))))) <=> (ssList(U!45) & ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y)))))) & ssItem(X2!47) & ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))))),
% 1.23/1.39 inference(rewrite,[status(thm)],[])).
% 1.23/1.39 tff(2,plain,
% 1.23/1.39 ((ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y)))))) & ssList(U!45) & (ssItem(X2!47) & ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))))) <=> (ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y)))))) & ssList(U!45) & ssItem(X2!47) & ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))))),
% 1.23/1.39 inference(rewrite,[status(thm)],[])).
% 1.23/1.39 tff(3,plain,
% 1.23/1.39 ((ssItem(X2!47) & (ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))))) <=> (ssItem(X2!47) & ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))))),
% 1.23/1.39 inference(rewrite,[status(thm)],[])).
% 1.23/1.39 tff(4,plain,
% 1.23/1.39 ((~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))) <=> (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45))))),
% 1.23/1.39 inference(rewrite,[status(thm)],[])).
% 1.23/1.39 tff(5,plain,
% 1.23/1.39 ((ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45))))) <=> (ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))))),
% 1.23/1.39 inference(monotonicity,[status(thm)],[4])).
% 1.23/1.39 tff(6,plain,
% 1.23/1.39 ((ssItem(X2!47) & (ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))))) <=> (ssItem(X2!47) & (ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45))))))),
% 1.23/1.39 inference(monotonicity,[status(thm)],[5])).
% 1.23/1.39 tff(7,plain,
% 1.23/1.39 ((ssItem(X2!47) & (ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))))) <=> (ssItem(X2!47) & ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))))),
% 1.23/1.39 inference(transitivity,[status(thm)],[6, 3])).
% 1.23/1.40 tff(8,plain,
% 1.23/1.40 ((ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y)))))) & ssList(U!45) & (ssItem(X2!47) & (ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45))))))) <=> (ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y)))))) & ssList(U!45) & (ssItem(X2!47) & ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45))))))),
% 1.23/1.40 inference(monotonicity,[status(thm)],[7])).
% 1.23/1.40 tff(9,plain,
% 1.23/1.40 ((ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y)))))) & ssList(U!45) & (ssItem(X2!47) & (ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45))))))) <=> (ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y)))))) & ssList(U!45) & ssItem(X2!47) & ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))))),
% 1.23/1.40 inference(transitivity,[status(thm)],[8, 2])).
% 1.23/1.40 tff(10,plain,
% 1.23/1.40 ((ssList(U!45) & (ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y)))))) & ssList(U!45) & (ssItem(X2!47) & (ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))))))) <=> (ssList(U!45) & (ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y)))))) & ssList(U!45) & ssItem(X2!47) & ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45))))))),
% 1.23/1.40 inference(monotonicity,[status(thm)],[9])).
% 1.23/1.40 tff(11,plain,
% 1.23/1.40 ((ssList(U!45) & (ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y)))))) & ssList(U!45) & (ssItem(X2!47) & (ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))))))) <=> (ssList(U!45) & ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y)))))) & ssItem(X2!47) & ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))))),
% 1.23/1.40 inference(transitivity,[status(thm)],[10, 1])).
% 1.23/1.40 tff(12,plain,
% 1.23/1.40 ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | ?[Y: $i] : (ssItem(Y) & ?[Z: $i] : (ssList(Z) & ?[X1: $i] : (ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U) & (memberP(X1, Y) | memberP(Z, Y))))) | (~ssList(U)) | ![X2: $i] : ((~ssItem(X2)) | ![X3: $i] : ((~ssList(X3)) | ![X4: $i] : (((~memberP(X3, X2)) & (~memberP(X4, X2))) | (~ssList(X4)) | (~(app(app(X3, cons(X2, nil)), X4) = U)))))))) <=> (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | ?[Y: $i] : (ssItem(Y) & ?[Z: $i] : (ssList(Z) & ?[X1: $i] : (ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U) & (memberP(X1, Y) | memberP(Z, Y))))) | (~ssList(U)) | ![X2: $i] : ((~ssItem(X2)) | ![X3: $i] : ((~ssList(X3)) | ![X4: $i] : (((~memberP(X3, X2)) & (~memberP(X4, X2))) | (~ssList(X4)) | (~(app(app(X3, cons(X2, nil)), X4) = U))))))))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(13,plain,
% 1.23/1.40 ((~![U: $i] : (ssList(U) => ![V: $i] : (ssList(V) => ![W: $i] : (ssList(W) => ![X: $i] : (ssList(X) => ((((~(V = X)) | (~(U = W))) | ?[Y: $i] : (ssItem(Y) & ?[Z: $i] : (ssList(Z) & ?[X1: $i] : ((ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = W)) & (memberP(Z, Y) | memberP(X1, Y)))))) | ![X2: $i] : (ssItem(X2) => ![X3: $i] : (ssList(X3) => ![X4: $i] : (ssList(X4) => ((~(app(app(X3, cons(X2, nil)), X4) = U)) | ((~memberP(X3, X2)) & (~memberP(X4, X2))))))))))))) <=> (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | ?[Y: $i] : (ssItem(Y) & ?[Z: $i] : (ssList(Z) & ?[X1: $i] : (ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U) & (memberP(X1, Y) | memberP(Z, Y))))) | (~ssList(U)) | ![X2: $i] : ((~ssItem(X2)) | ![X3: $i] : ((~ssList(X3)) | ![X4: $i] : (((~memberP(X3, X2)) & (~memberP(X4, X2))) | (~ssList(X4)) | (~(app(app(X3, cons(X2, nil)), X4) = U))))))))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(14,axiom,(~![U: $i] : (ssList(U) => ![V: $i] : (ssList(V) => ![W: $i] : (ssList(W) => ![X: $i] : (ssList(X) => ((((~(V = X)) | (~(U = W))) | ?[Y: $i] : (ssItem(Y) & ?[Z: $i] : (ssList(Z) & ?[X1: $i] : ((ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = W)) & (memberP(Z, Y) | memberP(X1, Y)))))) | ![X2: $i] : (ssItem(X2) => ![X3: $i] : (ssList(X3) => ![X4: $i] : (ssList(X4) => ((~(app(app(X3, cons(X2, nil)), X4) = U)) | ((~memberP(X3, X2)) & (~memberP(X4, X2))))))))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','co1')).
% 1.23/1.40 tff(15,plain,
% 1.23/1.40 (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | ?[Y: $i] : (ssItem(Y) & ?[Z: $i] : (ssList(Z) & ?[X1: $i] : (ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U) & (memberP(X1, Y) | memberP(Z, Y))))) | (~ssList(U)) | ![X2: $i] : ((~ssItem(X2)) | ![X3: $i] : ((~ssList(X3)) | ![X4: $i] : (((~memberP(X3, X2)) & (~memberP(X4, X2))) | (~ssList(X4)) | (~(app(app(X3, cons(X2, nil)), X4) = U)))))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[14, 13])).
% 1.23/1.40 tff(16,plain,
% 1.23/1.40 (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | ?[Y: $i] : (ssItem(Y) & ?[Z: $i] : (ssList(Z) & ?[X1: $i] : (ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U) & (memberP(X1, Y) | memberP(Z, Y))))) | (~ssList(U)) | ![X2: $i] : ((~ssItem(X2)) | ![X3: $i] : ((~ssList(X3)) | ![X4: $i] : (((~memberP(X3, X2)) & (~memberP(X4, X2))) | (~ssList(X4)) | (~(app(app(X3, cons(X2, nil)), X4) = U)))))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[15, 12])).
% 1.23/1.40 tff(17,plain,
% 1.23/1.40 (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | ?[Y: $i] : (ssItem(Y) & ?[Z: $i] : (ssList(Z) & ?[X1: $i] : (ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U) & (memberP(X1, Y) | memberP(Z, Y))))) | (~ssList(U)) | ![X2: $i] : ((~ssItem(X2)) | ![X3: $i] : ((~ssList(X3)) | ![X4: $i] : (((~memberP(X3, X2)) & (~memberP(X4, X2))) | (~ssList(X4)) | (~(app(app(X3, cons(X2, nil)), X4) = U)))))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[16, 12])).
% 1.23/1.40 tff(18,plain,
% 1.23/1.40 (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | ?[Y: $i] : (ssItem(Y) & ?[Z: $i] : (ssList(Z) & ?[X1: $i] : (ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U) & (memberP(X1, Y) | memberP(Z, Y))))) | (~ssList(U)) | ![X2: $i] : ((~ssItem(X2)) | ![X3: $i] : ((~ssList(X3)) | ![X4: $i] : (((~memberP(X3, X2)) & (~memberP(X4, X2))) | (~ssList(X4)) | (~(app(app(X3, cons(X2, nil)), X4) = U)))))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[17, 12])).
% 1.23/1.40 tff(19,plain,
% 1.23/1.40 (ssList(U!45) & ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y)))))) & ssItem(X2!47) & ssList(X3!48) & (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[18, 11])).
% 1.23/1.40 tff(20,plain,
% 1.23/1.40 (ssItem(X2!47)),
% 1.23/1.40 inference(and_elim,[status(thm)],[19])).
% 1.23/1.40 tff(21,plain,
% 1.23/1.40 (^[Y: $i] : refl(((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y)))))) <=> ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y)))))))),
% 1.23/1.40 inference(bind,[status(th)],[])).
% 1.23/1.40 tff(22,plain,
% 1.23/1.40 (![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y)))))) <=> ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y))))))),
% 1.23/1.40 inference(quant_intro,[status(thm)],[21])).
% 1.23/1.40 tff(23,plain,
% 1.23/1.40 (^[Y: $i] : rewrite(((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y)))))) <=> ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y)))))))),
% 1.23/1.40 inference(bind,[status(th)],[])).
% 1.23/1.40 tff(24,plain,
% 1.23/1.40 (![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y)))))) <=> ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y))))))),
% 1.23/1.40 inference(quant_intro,[status(thm)],[23])).
% 1.23/1.40 tff(25,plain,
% 1.23/1.40 (![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y)))))) <=> ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y))))))),
% 1.23/1.40 inference(transitivity,[status(thm)],[24, 22])).
% 1.23/1.40 tff(26,plain,
% 1.23/1.40 (^[Y: $i] : rewrite(((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y)))))) <=> ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y)))))))),
% 1.23/1.40 inference(bind,[status(th)],[])).
% 1.23/1.40 tff(27,plain,
% 1.23/1.40 (![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y)))))) <=> ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y))))))),
% 1.23/1.40 inference(quant_intro,[status(thm)],[26])).
% 1.23/1.40 tff(28,plain,
% 1.23/1.40 (![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : (~(ssList(X1) & (app(app(Z, cons(Y, nil)), X1) = U!45) & (memberP(X1, Y) | memberP(Z, Y))))))),
% 1.23/1.40 inference(and_elim,[status(thm)],[19])).
% 1.23/1.40 tff(29,plain,
% 1.23/1.40 (![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y))))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[28, 27])).
% 1.23/1.40 tff(30,plain,
% 1.23/1.40 (![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y))))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[29, 25])).
% 1.23/1.40 tff(31,plain,
% 1.23/1.40 (((~![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y))))))) | ((~ssItem(X2!47)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(Z, X2!47))))))) <=> ((~![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y))))))) | (~ssItem(X2!47)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(Z, X2!47))))))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(32,plain,
% 1.23/1.40 ((~![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y))))))) | ((~ssItem(X2!47)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(Z, X2!47))))))),
% 1.23/1.40 inference(quant_inst,[status(thm)],[])).
% 1.23/1.40 tff(33,plain,
% 1.23/1.40 ((~![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | (~(memberP(X1, Y) | memberP(Z, Y))))))) | (~ssItem(X2!47)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(Z, X2!47)))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[32, 31])).
% 1.23/1.40 tff(34,plain,
% 1.23/1.40 (![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(Z, X2!47)))))),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[33, 30, 20])).
% 1.23/1.40 tff(35,plain,
% 1.23/1.40 (ssList(X3!48)),
% 1.23/1.40 inference(and_elim,[status(thm)],[19])).
% 1.23/1.40 tff(36,plain,
% 1.23/1.40 (((~![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(Z, X2!47)))))) | ((~ssList(X3!48)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(X3!48, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(X3!48, X2!47)))))) <=> ((~![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(Z, X2!47)))))) | (~ssList(X3!48)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(X3!48, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(X3!48, X2!47)))))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(37,plain,
% 1.23/1.40 ((~![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(Z, X2!47)))))) | ((~ssList(X3!48)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(X3!48, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(X3!48, X2!47)))))),
% 1.23/1.40 inference(quant_inst,[status(thm)],[])).
% 1.23/1.40 tff(38,plain,
% 1.23/1.40 ((~![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(Z, X2!47)))))) | (~ssList(X3!48)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(X3!48, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(X3!48, X2!47))))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[37, 36])).
% 1.23/1.40 tff(39,plain,
% 1.23/1.40 (![X1: $i] : ((~ssList(X1)) | (~(app(app(X3!48, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(X3!48, X2!47))))),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[38, 35, 34])).
% 1.23/1.40 tff(40,plain,
% 1.23/1.40 (~(((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)))),
% 1.23/1.40 inference(and_elim,[status(thm)],[19])).
% 1.23/1.40 tff(41,plain,
% 1.23/1.40 (app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45),
% 1.23/1.40 inference(or_elim,[status(thm)],[40])).
% 1.23/1.40 tff(42,plain,
% 1.23/1.40 (ssList(X4!49)),
% 1.23/1.40 inference(or_elim,[status(thm)],[40])).
% 1.23/1.40 tff(43,plain,
% 1.23/1.40 (((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X3!48, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(X3!48, X2!47))))) | ((~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)) | (~(memberP(X3!48, X2!47) | memberP(X4!49, X2!47))))) <=> ((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X3!48, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(X3!48, X2!47))))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)) | (~(memberP(X3!48, X2!47) | memberP(X4!49, X2!47))))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(44,plain,
% 1.23/1.40 (((~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)) | (~(memberP(X3!48, X2!47) | memberP(X4!49, X2!47)))) <=> ((~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)) | (~(memberP(X3!48, X2!47) | memberP(X4!49, X2!47))))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(45,plain,
% 1.23/1.40 ((~(memberP(X4!49, X2!47) | memberP(X3!48, X2!47))) <=> (~(memberP(X3!48, X2!47) | memberP(X4!49, X2!47)))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(46,plain,
% 1.23/1.40 (((~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)) | (~(memberP(X4!49, X2!47) | memberP(X3!48, X2!47)))) <=> ((~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)) | (~(memberP(X3!48, X2!47) | memberP(X4!49, X2!47))))),
% 1.23/1.40 inference(monotonicity,[status(thm)],[45])).
% 1.23/1.40 tff(47,plain,
% 1.23/1.40 (((~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)) | (~(memberP(X4!49, X2!47) | memberP(X3!48, X2!47)))) <=> ((~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)) | (~(memberP(X3!48, X2!47) | memberP(X4!49, X2!47))))),
% 1.23/1.40 inference(transitivity,[status(thm)],[46, 44])).
% 1.23/1.40 tff(48,plain,
% 1.23/1.40 (((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X3!48, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(X3!48, X2!47))))) | ((~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)) | (~(memberP(X4!49, X2!47) | memberP(X3!48, X2!47))))) <=> ((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X3!48, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(X3!48, X2!47))))) | ((~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)) | (~(memberP(X3!48, X2!47) | memberP(X4!49, X2!47)))))),
% 1.23/1.40 inference(monotonicity,[status(thm)],[47])).
% 1.23/1.40 tff(49,plain,
% 1.23/1.40 (((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X3!48, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(X3!48, X2!47))))) | ((~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)) | (~(memberP(X4!49, X2!47) | memberP(X3!48, X2!47))))) <=> ((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X3!48, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(X3!48, X2!47))))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)) | (~(memberP(X3!48, X2!47) | memberP(X4!49, X2!47))))),
% 1.23/1.40 inference(transitivity,[status(thm)],[48, 43])).
% 1.23/1.40 tff(50,plain,
% 1.23/1.40 ((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X3!48, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(X3!48, X2!47))))) | ((~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)) | (~(memberP(X4!49, X2!47) | memberP(X3!48, X2!47))))),
% 1.23/1.40 inference(quant_inst,[status(thm)],[])).
% 1.23/1.40 tff(51,plain,
% 1.23/1.40 ((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X3!48, cons(X2!47, nil)), X1) = U!45)) | (~(memberP(X1, X2!47) | memberP(X3!48, X2!47))))) | (~ssList(X4!49)) | (~(app(app(X3!48, cons(X2!47, nil)), X4!49) = U!45)) | (~(memberP(X3!48, X2!47) | memberP(X4!49, X2!47)))),
% 1.23/1.40 inference(modus_ponens,[status(thm)],[50, 49])).
% 1.23/1.40 tff(52,plain,
% 1.23/1.40 (~(memberP(X3!48, X2!47) | memberP(X4!49, X2!47))),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[51, 42, 41, 39])).
% 1.23/1.40 tff(53,plain,
% 1.23/1.40 ((memberP(X3!48, X2!47) | memberP(X4!49, X2!47)) | (~memberP(X3!48, X2!47))),
% 1.23/1.40 inference(tautology,[status(thm)],[])).
% 1.23/1.40 tff(54,plain,
% 1.23/1.40 (~memberP(X3!48, X2!47)),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[53, 52])).
% 1.23/1.40 tff(55,plain,
% 1.23/1.40 ((memberP(X3!48, X2!47) | memberP(X4!49, X2!47)) | (~memberP(X4!49, X2!47))),
% 1.23/1.40 inference(tautology,[status(thm)],[])).
% 1.23/1.40 tff(56,plain,
% 1.23/1.40 (~memberP(X4!49, X2!47)),
% 1.23/1.40 inference(unit_resolution,[status(thm)],[55, 52])).
% 1.23/1.40 tff(57,plain,
% 1.23/1.40 ((~(~(memberP(X3!48, X2!47) | memberP(X4!49, X2!47)))) <=> (memberP(X3!48, X2!47) | memberP(X4!49, X2!47))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(58,plain,
% 1.23/1.40 (((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47))) <=> (~(memberP(X3!48, X2!47) | memberP(X4!49, X2!47)))),
% 1.23/1.40 inference(rewrite,[status(thm)],[])).
% 1.23/1.40 tff(59,plain,
% 1.23/1.40 ((~((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47)))) <=> (~(~(memberP(X3!48, X2!47) | memberP(X4!49, X2!47))))),
% 1.23/1.40 inference(monotonicity,[status(thm)],[58])).
% 1.23/1.40 tff(60,plain,
% 1.23/1.40 ((~((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47)))) <=> (memberP(X3!48, X2!47) | memberP(X4!49, X2!47))),
% 1.23/1.40 inference(transitivity,[status(thm)],[59, 57])).
% 1.23/1.40 tff(61,plain,
% 1.23/1.40 (~((~memberP(X3!48, X2!47)) & (~memberP(X4!49, X2!47)))),
% 1.23/1.40 inference(or_elim,[status(thm)],[40])).
% 1.23/1.40 tff(62,plain,
% 1.23/1.41 (memberP(X3!48, X2!47) | memberP(X4!49, X2!47)),
% 1.23/1.41 inference(modus_ponens,[status(thm)],[61, 60])).
% 1.23/1.41 tff(63,plain,
% 1.23/1.41 ($false),
% 1.23/1.41 inference(unit_resolution,[status(thm)],[62, 56, 54])).
% 1.23/1.41 % SZS output end Proof
% 1.23/1.41 % E exiting
%------------------------------------------------------------------------------