%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SWC279+1 : TPTP v9.0.0. Released v2.4.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n011.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 05:30:46 AM UTC 2025
% Result : Theorem 22.55s 22.73s
% Output : Proof 22.55s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SWC279+1 : TPTP v9.0.0. Released v2.4.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n011.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Jun 20 08:49:11 EDT 2025
% 0.12/0.34 % CPUTime :
% 22.55/22.73 % SZS status Theorem
% 22.55/22.73 % SZS output start Proof
% 22.55/22.73 tff(leq_type, type, (
% 22.55/22.73 leq: ( $i * $i ) > $o)).
% 22.55/22.73 tff(tptp_fun_X3_47_type, type, (
% 22.55/22.73 tptp_fun_X3_47: $i)).
% 22.55/22.73 tff(tptp_fun_X6_50_type, type, (
% 22.55/22.73 tptp_fun_X6_50: $i)).
% 22.55/22.73 tff(memberP_type, type, (
% 22.55/22.73 memberP: ( $i * $i ) > $o)).
% 22.55/22.73 tff(tptp_fun_X4_48_type, type, (
% 22.55/22.73 tptp_fun_X4_48: $i)).
% 22.55/22.73 tff(tptp_fun_X5_49_type, type, (
% 22.55/22.73 tptp_fun_X5_49: $i)).
% 22.55/22.73 tff(ssItem_type, type, (
% 22.55/22.73 ssItem: $i > $o)).
% 22.55/22.73 tff(tptp_fun_U_45_type, type, (
% 22.55/22.73 tptp_fun_U_45: $i)).
% 22.55/22.73 tff(app_type, type, (
% 22.55/22.73 app: ( $i * $i ) > $i)).
% 22.55/22.73 tff(cons_type, type, (
% 22.55/22.73 cons: ( $i * $i ) > $i)).
% 22.55/22.73 tff(nil_type, type, (
% 22.55/22.73 nil: $i)).
% 22.55/22.73 tff(ssList_type, type, (
% 22.55/22.73 ssList: $i > $o)).
% 22.55/22.73 tff(tptp_fun_V_46_type, type, (
% 22.55/22.73 tptp_fun_V_46: $i)).
% 22.55/22.73 tff(1,plain,
% 22.55/22.73 ((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)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2)))))))) & ssList(U!45) & ssItem(X3!47) & ssList(X4!48) & ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!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)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2)))))))) & ssItem(X3!47) & ssList(X4!48) & ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45))),
% 22.55/22.73 inference(rewrite,[status(thm)],[])).
% 22.55/22.73 tff(2,plain,
% 22.55/22.73 ((ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2)))))))) & ssList(U!45) & (ssItem(X3!47) & ssList(X4!48) & ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!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)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2)))))))) & ssList(U!45) & ssItem(X3!47) & ssList(X4!48) & ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45))),
% 22.55/22.73 inference(rewrite,[status(thm)],[])).
% 22.55/22.73 tff(3,plain,
% 22.55/22.73 ((ssItem(X3!47) & (ssList(X4!48) & ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45))) <=> (ssItem(X3!47) & ssList(X4!48) & ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45))),
% 22.55/22.73 inference(rewrite,[status(thm)],[])).
% 22.55/22.73 tff(4,plain,
% 22.55/22.73 ((ssList(X4!48) & (ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45))) <=> (ssList(X4!48) & ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45))),
% 22.55/22.73 inference(rewrite,[status(thm)],[])).
% 22.55/22.73 tff(5,plain,
% 22.55/22.73 ((ssItem(X3!47) & (ssList(X4!48) & (ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)))) <=> (ssItem(X3!47) & (ssList(X4!48) & ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)))),
% 22.55/22.73 inference(monotonicity,[status(thm)],[4])).
% 22.55/22.73 tff(6,plain,
% 22.55/22.73 ((ssItem(X3!47) & (ssList(X4!48) & (ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)))) <=> (ssItem(X3!47) & ssList(X4!48) & ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45))),
% 22.55/22.73 inference(transitivity,[status(thm)],[5, 3])).
% 22.55/22.73 tff(7,plain,
% 22.55/22.73 ((ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2)))))))) & ssList(U!45) & (ssItem(X3!47) & (ssList(X4!48) & (ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!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)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2)))))))) & ssList(U!45) & (ssItem(X3!47) & ssList(X4!48) & ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)))),
% 22.55/22.73 inference(monotonicity,[status(thm)],[6])).
% 22.55/22.73 tff(8,plain,
% 22.55/22.73 ((ssList(V!46) & ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2)))))))) & ssList(U!45) & (ssItem(X3!47) & (ssList(X4!48) & (ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!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)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2)))))))) & ssList(U!45) & ssItem(X3!47) & ssList(X4!48) & ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45))),
% 22.55/22.73 inference(transitivity,[status(thm)],[7, 2])).
% 22.55/22.73 tff(9,plain,
% 22.55/22.73 ((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)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2)))))))) & ssList(U!45) & (ssItem(X3!47) & (ssList(X4!48) & (ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!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)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2)))))))) & ssList(U!45) & ssItem(X3!47) & ssList(X4!48) & ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)))),
% 22.55/22.73 inference(monotonicity,[status(thm)],[8])).
% 22.55/22.73 tff(10,plain,
% 22.55/22.73 ((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)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2)))))))) & ssList(U!45) & (ssItem(X3!47) & (ssList(X4!48) & (ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!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)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2)))))))) & ssItem(X3!47) & ssList(X4!48) & ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45))),
% 22.55/22.73 inference(transitivity,[status(thm)],[9, 1])).
% 22.55/22.73 tff(11,plain,
% 22.55/22.73 ((~![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) & ?[X2: $i] : (ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2))))))) | (~ssList(U)) | ![X3: $i] : ((~ssItem(X3)) | ![X4: $i] : ((~ssList(X4)) | ![X5: $i] : ((~ssList(X5)) | ![X6: $i] : ((~ssItem(X6)) | (((~memberP(X4, X6)) | leq(X6, X3)) & ((~memberP(X5, X6)) | leq(X3, X6)))) | (~(app(app(X4, cons(X3, nil)), X5) = 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) & ?[X2: $i] : (ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2))))))) | (~ssList(U)) | ![X3: $i] : ((~ssItem(X3)) | ![X4: $i] : ((~ssList(X4)) | ![X5: $i] : ((~ssList(X5)) | ![X6: $i] : ((~ssItem(X6)) | (((~memberP(X4, X6)) | leq(X6, X3)) & ((~memberP(X5, X6)) | leq(X3, X6)))) | (~(app(app(X4, cons(X3, nil)), X5) = U))))))))),
% 22.55/22.73 inference(rewrite,[status(thm)],[])).
% 22.55/22.73 tff(12,plain,
% 22.55/22.73 ((~![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)) & ?[X2: $i] : (ssItem(X2) & (((~leq(Y, X2)) & memberP(X1, X2)) | ((~leq(X2, Y)) & memberP(Z, X2)))))))) | ![X3: $i] : (ssItem(X3) => ![X4: $i] : (ssList(X4) => ![X5: $i] : (ssList(X5) => ((~(app(app(X4, cons(X3, nil)), X5) = U)) | ![X6: $i] : (ssItem(X6) => (((~memberP(X4, X6)) | leq(X6, X3)) & ((~memberP(X5, X6)) | leq(X3, X6)))))))))))))) <=> (~![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) & ?[X2: $i] : (ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2))))))) | (~ssList(U)) | ![X3: $i] : ((~ssItem(X3)) | ![X4: $i] : ((~ssList(X4)) | ![X5: $i] : ((~ssList(X5)) | ![X6: $i] : ((~ssItem(X6)) | (((~memberP(X4, X6)) | leq(X6, X3)) & ((~memberP(X5, X6)) | leq(X3, X6)))) | (~(app(app(X4, cons(X3, nil)), X5) = U))))))))),
% 22.55/22.73 inference(rewrite,[status(thm)],[])).
% 22.55/22.73 tff(13,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)) & ?[X2: $i] : (ssItem(X2) & (((~leq(Y, X2)) & memberP(X1, X2)) | ((~leq(X2, Y)) & memberP(Z, X2)))))))) | ![X3: $i] : (ssItem(X3) => ![X4: $i] : (ssList(X4) => ![X5: $i] : (ssList(X5) => ((~(app(app(X4, cons(X3, nil)), X5) = U)) | ![X6: $i] : (ssItem(X6) => (((~memberP(X4, X6)) | leq(X6, X3)) & ((~memberP(X5, X6)) | leq(X3, X6)))))))))))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','co1')).
% 22.55/22.73 tff(14,plain,
% 22.55/22.73 (~![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) & ?[X2: $i] : (ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2))))))) | (~ssList(U)) | ![X3: $i] : ((~ssItem(X3)) | ![X4: $i] : ((~ssList(X4)) | ![X5: $i] : ((~ssList(X5)) | ![X6: $i] : ((~ssItem(X6)) | (((~memberP(X4, X6)) | leq(X6, X3)) & ((~memberP(X5, X6)) | leq(X3, X6)))) | (~(app(app(X4, cons(X3, nil)), X5) = U)))))))),
% 22.55/22.73 inference(modus_ponens,[status(thm)],[13, 12])).
% 22.55/22.73 tff(15,plain,
% 22.55/22.73 (~![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) & ?[X2: $i] : (ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2))))))) | (~ssList(U)) | ![X3: $i] : ((~ssItem(X3)) | ![X4: $i] : ((~ssList(X4)) | ![X5: $i] : ((~ssList(X5)) | ![X6: $i] : ((~ssItem(X6)) | (((~memberP(X4, X6)) | leq(X6, X3)) & ((~memberP(X5, X6)) | leq(X3, X6)))) | (~(app(app(X4, cons(X3, nil)), X5) = U)))))))),
% 22.55/22.73 inference(modus_ponens,[status(thm)],[14, 11])).
% 22.55/22.73 tff(16,plain,
% 22.55/22.73 (~![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) & ?[X2: $i] : (ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2))))))) | (~ssList(U)) | ![X3: $i] : ((~ssItem(X3)) | ![X4: $i] : ((~ssList(X4)) | ![X5: $i] : ((~ssList(X5)) | ![X6: $i] : ((~ssItem(X6)) | (((~memberP(X4, X6)) | leq(X6, X3)) & ((~memberP(X5, X6)) | leq(X3, X6)))) | (~(app(app(X4, cons(X3, nil)), X5) = U)))))))),
% 22.55/22.73 inference(modus_ponens,[status(thm)],[15, 11])).
% 22.55/22.73 tff(17,plain,
% 22.55/22.73 (~![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) & ?[X2: $i] : (ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2))))))) | (~ssList(U)) | ![X3: $i] : ((~ssItem(X3)) | ![X4: $i] : ((~ssList(X4)) | ![X5: $i] : ((~ssList(X5)) | ![X6: $i] : ((~ssItem(X6)) | (((~memberP(X4, X6)) | leq(X6, X3)) & ((~memberP(X5, X6)) | leq(X3, X6)))) | (~(app(app(X4, cons(X3, nil)), X5) = U)))))))),
% 22.55/22.73 inference(modus_ponens,[status(thm)],[16, 11])).
% 22.55/22.73 tff(18,plain,
% 22.55/22.73 (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)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2)))))))) & ssItem(X3!47) & ssList(X4!48) & ssList(X5!49) & (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))) & (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)),
% 22.55/22.73 inference(modus_ponens,[status(thm)],[17, 10])).
% 22.55/22.73 tff(19,plain,
% 22.55/22.73 (ssItem(X3!47)),
% 22.55/22.73 inference(and_elim,[status(thm)],[18])).
% 22.55/22.73 tff(20,plain,
% 22.55/22.73 (^[Y: $i] : refl(((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2))))))))) <=> ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2))))))))))),
% 22.55/22.73 inference(bind,[status(th)],[])).
% 22.55/22.73 tff(21,plain,
% 22.55/22.73 (![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2))))))))) <=> ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2)))))))))),
% 22.55/22.73 inference(quant_intro,[status(thm)],[20])).
% 22.55/22.73 tff(22,plain,
% 22.55/22.73 (^[Y: $i] : rewrite(((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2))))))))) <=> ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2))))))))))),
% 22.55/22.73 inference(bind,[status(th)],[])).
% 22.55/22.73 tff(23,plain,
% 22.55/22.73 (![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2))))))))) <=> ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2)))))))))),
% 22.55/22.73 inference(quant_intro,[status(thm)],[22])).
% 22.55/22.73 tff(24,plain,
% 22.55/22.73 (![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2))))))))) <=> ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2)))))))))),
% 22.55/22.73 inference(transitivity,[status(thm)],[23, 21])).
% 22.55/22.73 tff(25,plain,
% 22.55/22.73 (^[Y: $i] : rewrite(((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2)))))))) <=> ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2))))))))))),
% 22.55/22.73 inference(bind,[status(th)],[])).
% 22.55/22.73 tff(26,plain,
% 22.55/22.73 (![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2)))))))) <=> ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2)))))))))),
% 22.55/22.73 inference(quant_intro,[status(thm)],[25])).
% 22.55/22.73 tff(27,plain,
% 22.55/22.73 (![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : (~(ssItem(X2) & (((~leq(X2, Y)) & memberP(Z, X2)) | ((~leq(Y, X2)) & memberP(X1, X2))))))))),
% 22.55/22.73 inference(and_elim,[status(thm)],[18])).
% 22.55/22.73 tff(28,plain,
% 22.55/22.73 (![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2)))))))))),
% 22.55/22.73 inference(modus_ponens,[status(thm)],[27, 26])).
% 22.55/22.73 tff(29,plain,
% 22.55/22.73 (![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2)))))))))),
% 22.55/22.73 inference(modus_ponens,[status(thm)],[28, 24])).
% 22.55/22.73 tff(30,plain,
% 22.55/22.73 (((~![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2)))))))))) | ((~ssItem(X3!47)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(Z, X2)) | leq(X2, X3!47)))))))))) <=> ((~![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2)))))))))) | (~ssItem(X3!47)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(Z, X2)) | leq(X2, X3!47)))))))))),
% 22.55/22.73 inference(rewrite,[status(thm)],[])).
% 22.55/22.73 tff(31,plain,
% 22.55/22.73 (((~ssItem(X3!47)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, X3!47) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(X3!47, X2))))))))) <=> ((~ssItem(X3!47)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(Z, X2)) | leq(X2, X3!47)))))))))),
% 22.55/22.73 inference(rewrite,[status(thm)],[])).
% 22.55/22.73 tff(32,plain,
% 22.55/22.73 (((~![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2)))))))))) | ((~ssItem(X3!47)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, X3!47) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(X3!47, X2)))))))))) <=> ((~![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2)))))))))) | ((~ssItem(X3!47)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(Z, X2)) | leq(X2, X3!47))))))))))),
% 22.55/22.73 inference(monotonicity,[status(thm)],[31])).
% 22.55/22.73 tff(33,plain,
% 22.55/22.73 (((~![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2)))))))))) | ((~ssItem(X3!47)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, X3!47) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(X3!47, X2)))))))))) <=> ((~![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2)))))))))) | (~ssItem(X3!47)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(Z, X2)) | leq(X2, X3!47)))))))))),
% 22.55/22.73 inference(transitivity,[status(thm)],[32, 30])).
% 22.55/22.73 tff(34,plain,
% 22.55/22.73 ((~![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2)))))))))) | ((~ssItem(X3!47)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, X3!47) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(X3!47, X2)))))))))),
% 22.55/22.73 inference(quant_inst,[status(thm)],[])).
% 22.55/22.73 tff(35,plain,
% 22.55/22.73 ((~![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(Y, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~(leq(X2, Y) | (~memberP(Z, X2)))) | (~((~memberP(X1, X2)) | leq(Y, X2)))))))))) | (~ssItem(X3!47)) | ![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(Z, X2)) | leq(X2, X3!47))))))))),
% 22.55/22.74 inference(modus_ponens,[status(thm)],[34, 33])).
% 22.55/22.74 tff(36,plain,
% 22.55/22.74 (![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(Z, X2)) | leq(X2, X3!47))))))))),
% 22.55/22.74 inference(unit_resolution,[status(thm)],[35, 29, 19])).
% 22.55/22.74 tff(37,plain,
% 22.55/22.74 (ssList(X4!48)),
% 22.55/22.74 inference(and_elim,[status(thm)],[18])).
% 22.55/22.74 tff(38,plain,
% 22.55/22.74 (((~![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(Z, X2)) | leq(X2, X3!47))))))))) | ((~ssList(X4!48)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(X4!48, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47))))))))) <=> ((~![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(Z, X2)) | leq(X2, X3!47))))))))) | (~ssList(X4!48)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(X4!48, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47))))))))),
% 22.55/22.74 inference(rewrite,[status(thm)],[])).
% 22.55/22.74 tff(39,plain,
% 22.55/22.74 ((~![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(Z, X2)) | leq(X2, X3!47))))))))) | ((~ssList(X4!48)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(X4!48, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47))))))))),
% 22.55/22.74 inference(quant_inst,[status(thm)],[])).
% 22.55/22.74 tff(40,plain,
% 22.55/22.74 ((~![Z: $i] : ((~ssList(Z)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(Z, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(Z, X2)) | leq(X2, X3!47))))))))) | (~ssList(X4!48)) | ![X1: $i] : ((~ssList(X1)) | (~(app(app(X4!48, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47)))))))),
% 22.55/22.74 inference(modus_ponens,[status(thm)],[39, 38])).
% 22.55/22.74 tff(41,plain,
% 22.55/22.74 (![X1: $i] : ((~ssList(X1)) | (~(app(app(X4!48, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47)))))))),
% 22.55/22.74 inference(unit_resolution,[status(thm)],[40, 37, 36])).
% 22.55/22.74 tff(42,plain,
% 22.55/22.74 (app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45),
% 22.55/22.74 inference(and_elim,[status(thm)],[18])).
% 22.55/22.74 tff(43,plain,
% 22.55/22.74 (ssList(X5!49)),
% 22.55/22.74 inference(and_elim,[status(thm)],[18])).
% 22.55/22.74 tff(44,plain,
% 22.55/22.74 (((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X4!48, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47)))))))) | ((~ssList(X5!49)) | (~(app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X4!48, X2)) | leq(X2, X3!47))) | (~((~memberP(X5!49, X2)) | leq(X3!47, X2)))))))) <=> ((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X4!48, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47)))))))) | (~ssList(X5!49)) | (~(app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X4!48, X2)) | leq(X2, X3!47))) | (~((~memberP(X5!49, X2)) | leq(X3!47, X2)))))))),
% 22.55/22.74 inference(rewrite,[status(thm)],[])).
% 22.55/22.74 tff(45,plain,
% 22.55/22.74 (((~ssList(X5!49)) | (~(app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X5!49, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47))))))) <=> ((~ssList(X5!49)) | (~(app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X4!48, X2)) | leq(X2, X3!47))) | (~((~memberP(X5!49, X2)) | leq(X3!47, X2)))))))),
% 22.55/22.74 inference(rewrite,[status(thm)],[])).
% 22.55/22.74 tff(46,plain,
% 22.55/22.74 (((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X4!48, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47)))))))) | ((~ssList(X5!49)) | (~(app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X5!49, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47)))))))) <=> ((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X4!48, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47)))))))) | ((~ssList(X5!49)) | (~(app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X4!48, X2)) | leq(X2, X3!47))) | (~((~memberP(X5!49, X2)) | leq(X3!47, X2))))))))),
% 22.55/22.74 inference(monotonicity,[status(thm)],[45])).
% 22.55/22.74 tff(47,plain,
% 22.55/22.74 (((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X4!48, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47)))))))) | ((~ssList(X5!49)) | (~(app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X5!49, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47)))))))) <=> ((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X4!48, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47)))))))) | (~ssList(X5!49)) | (~(app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X4!48, X2)) | leq(X2, X3!47))) | (~((~memberP(X5!49, X2)) | leq(X3!47, X2)))))))),
% 22.55/22.74 inference(transitivity,[status(thm)],[46, 44])).
% 22.55/22.74 tff(48,plain,
% 22.55/22.74 ((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X4!48, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47)))))))) | ((~ssList(X5!49)) | (~(app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X5!49, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47)))))))),
% 22.55/22.74 inference(quant_inst,[status(thm)],[])).
% 22.55/22.74 tff(49,plain,
% 22.55/22.74 ((~![X1: $i] : ((~ssList(X1)) | (~(app(app(X4!48, cons(X3!47, nil)), X1) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X1, X2)) | leq(X3!47, X2))) | (~((~memberP(X4!48, X2)) | leq(X2, X3!47)))))))) | (~ssList(X5!49)) | (~(app(app(X4!48, cons(X3!47, nil)), X5!49) = U!45)) | ![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X4!48, X2)) | leq(X2, X3!47))) | (~((~memberP(X5!49, X2)) | leq(X3!47, X2))))))),
% 22.55/22.74 inference(modus_ponens,[status(thm)],[48, 47])).
% 22.55/22.74 tff(50,plain,
% 22.55/22.74 (![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X4!48, X2)) | leq(X2, X3!47))) | (~((~memberP(X5!49, X2)) | leq(X3!47, X2))))))),
% 22.55/22.74 inference(unit_resolution,[status(thm)],[49, 43, 42, 41])).
% 22.55/22.74 tff(51,plain,
% 22.55/22.74 (~((~ssItem(X6!50)) | (((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))),
% 22.55/22.74 inference(and_elim,[status(thm)],[18])).
% 22.55/22.74 tff(52,plain,
% 22.55/22.74 (ssItem(X6!50)),
% 22.55/22.74 inference(or_elim,[status(thm)],[51])).
% 22.55/22.74 tff(53,plain,
% 22.55/22.74 (((~![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X4!48, X2)) | leq(X2, X3!47))) | (~((~memberP(X5!49, X2)) | leq(X3!47, X2))))))) | ((~ssItem(X6!50)) | (~((~((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47))) | (~((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))))) <=> ((~![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X4!48, X2)) | leq(X2, X3!47))) | (~((~memberP(X5!49, X2)) | leq(X3!47, X2))))))) | (~ssItem(X6!50)) | (~((~((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47))) | (~((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))))),
% 22.55/22.74 inference(rewrite,[status(thm)],[])).
% 22.55/22.81 tff(54,plain,
% 22.55/22.81 ((~![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X4!48, X2)) | leq(X2, X3!47))) | (~((~memberP(X5!49, X2)) | leq(X3!47, X2))))))) | ((~ssItem(X6!50)) | (~((~((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47))) | (~((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))))),
% 22.55/22.81 inference(quant_inst,[status(thm)],[])).
% 22.55/22.81 tff(55,plain,
% 22.55/22.81 ((~![X2: $i] : ((~ssItem(X2)) | (~((~((~memberP(X4!48, X2)) | leq(X2, X3!47))) | (~((~memberP(X5!49, X2)) | leq(X3!47, X2))))))) | (~ssItem(X6!50)) | (~((~((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47))) | (~((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50)))))),
% 22.55/22.81 inference(modus_ponens,[status(thm)],[54, 53])).
% 22.55/22.81 tff(56,plain,
% 22.55/22.81 (~((~((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47))) | (~((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))),
% 22.55/22.81 inference(unit_resolution,[status(thm)],[55, 52, 50])).
% 22.55/22.81 tff(57,plain,
% 22.55/22.81 (((~((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47))) | (~((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50)))) | ((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47))),
% 22.55/22.81 inference(tautology,[status(thm)],[])).
% 22.55/22.81 tff(58,plain,
% 22.55/22.81 ((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)),
% 22.55/22.81 inference(unit_resolution,[status(thm)],[57, 56])).
% 22.55/22.81 tff(59,plain,
% 22.55/22.81 (((~((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47))) | (~((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50)))) | ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))),
% 22.55/22.81 inference(tautology,[status(thm)],[])).
% 22.55/22.81 tff(60,plain,
% 22.55/22.81 ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50)),
% 22.55/22.81 inference(unit_resolution,[status(thm)],[59, 56])).
% 22.55/22.81 tff(61,plain,
% 22.55/22.81 ((~(~((~((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47))) | (~((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50)))))) <=> ((~((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47))) | (~((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))),
% 22.55/22.81 inference(rewrite,[status(thm)],[])).
% 22.55/22.81 tff(62,plain,
% 22.55/22.81 ((((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))) <=> (~((~((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47))) | (~((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50)))))),
% 22.55/22.81 inference(rewrite,[status(thm)],[])).
% 22.55/22.81 tff(63,plain,
% 22.55/22.81 ((~(((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50)))) <=> (~(~((~((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47))) | (~((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))))),
% 22.55/22.81 inference(monotonicity,[status(thm)],[62])).
% 22.55/22.81 tff(64,plain,
% 22.55/22.81 ((~(((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50)))) <=> ((~((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47))) | (~((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50))))),
% 22.55/22.81 inference(transitivity,[status(thm)],[63, 61])).
% 22.55/22.81 tff(65,plain,
% 22.55/22.81 (~(((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47)) & ((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50)))),
% 22.55/22.81 inference(or_elim,[status(thm)],[51])).
% 22.55/22.81 tff(66,plain,
% 22.55/22.81 ((~((~memberP(X4!48, X6!50)) | leq(X6!50, X3!47))) | (~((~memberP(X5!49, X6!50)) | leq(X3!47, X6!50)))),
% 22.55/22.81 inference(modus_ponens,[status(thm)],[65, 64])).
% 22.55/22.81 tff(67,plain,
% 22.55/22.81 ($false),
% 22.55/22.81 inference(unit_resolution,[status(thm)],[66, 60, 58])).
% 22.55/22.81 % SZS output end Proof
% 22.55/22.81 % E exiting
%------------------------------------------------------------------------------