%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SWC254+1 : TPTP v9.0.0. Released v2.4.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n007.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 05:30:41 AM UTC 2025
% Result : Theorem 0.19s 0.49s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SWC254+1 : TPTP v9.0.0. Released v2.4.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.12/0.32 % Computer : n007.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 08:46:35 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.19/0.49 % SZS status Theorem
% 0.19/0.49 % SZS output start Proof
% 0.19/0.49 tff(tptp_fun_U_45_type, type, (
% 0.19/0.49 tptp_fun_U_45: $i)).
% 0.19/0.49 tff(cons_type, type, (
% 0.19/0.49 cons: ( $i * $i ) > $i)).
% 0.19/0.49 tff(nil_type, type, (
% 0.19/0.49 nil: $i)).
% 0.19/0.49 tff(ssItem_type, type, (
% 0.19/0.49 ssItem: $i > $o)).
% 0.19/0.49 tff(singletonP_type, type, (
% 0.19/0.49 singletonP: $i > $o)).
% 0.19/0.49 tff(tptp_fun_V_4_type, type, (
% 0.19/0.49 tptp_fun_V_4: $i > $i)).
% 0.19/0.49 tff(ssList_type, type, (
% 0.19/0.49 ssList: $i > $o)).
% 0.19/0.49 tff(tptp_fun_V_46_type, type, (
% 0.19/0.49 tptp_fun_V_46: $i)).
% 0.19/0.49 tff(app_type, type, (
% 0.19/0.49 app: ( $i * $i ) > $i)).
% 0.19/0.49 tff(tptp_fun_Y_47_type, type, (
% 0.19/0.49 tptp_fun_Y_47: $i)).
% 0.19/0.49 tff(tptp_fun_Z_48_type, type, (
% 0.19/0.49 tptp_fun_Z_48: $i)).
% 0.19/0.49 tff(neq_type, type, (
% 0.19/0.49 neq: ( $i * $i ) > $o)).
% 0.19/0.49 tff(1,plain,
% 0.19/0.49 ((ssList(U!45) & (ssList(V!46) & ssList(U!45) & (~singletonP(U!45)) & neq(V!46, nil) & ssItem(Y!47) & (~((~ssList(Z!48)) | (~(cons(Y!47, nil) = U!45)) | (~(app(Z!48, cons(Y!47, nil)) = V!46)))))) <=> (ssList(U!45) & ssList(V!46) & (~singletonP(U!45)) & neq(V!46, nil) & ssItem(Y!47) & (~((~ssList(Z!48)) | (~(cons(Y!47, nil) = U!45)) | (~(app(Z!48, cons(Y!47, nil)) = V!46)))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(2,plain,
% 0.19/0.49 ((ssList(V!46) & ssList(U!45) & (~singletonP(U!45)) & neq(V!46, nil) & (ssItem(Y!47) & (~((~ssList(Z!48)) | (~(cons(Y!47, nil) = U!45)) | (~(app(Z!48, cons(Y!47, nil)) = V!46)))))) <=> (ssList(V!46) & ssList(U!45) & (~singletonP(U!45)) & neq(V!46, nil) & ssItem(Y!47) & (~((~ssList(Z!48)) | (~(cons(Y!47, nil) = U!45)) | (~(app(Z!48, cons(Y!47, nil)) = V!46)))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(3,plain,
% 0.19/0.49 ((ssList(U!45) & (ssList(V!46) & ssList(U!45) & (~singletonP(U!45)) & neq(V!46, nil) & (ssItem(Y!47) & (~((~ssList(Z!48)) | (~(cons(Y!47, nil) = U!45)) | (~(app(Z!48, cons(Y!47, nil)) = V!46))))))) <=> (ssList(U!45) & (ssList(V!46) & ssList(U!45) & (~singletonP(U!45)) & neq(V!46, nil) & ssItem(Y!47) & (~((~ssList(Z!48)) | (~(cons(Y!47, nil) = U!45)) | (~(app(Z!48, cons(Y!47, nil)) = V!46))))))),
% 0.19/0.49 inference(monotonicity,[status(thm)],[2])).
% 0.19/0.49 tff(4,plain,
% 0.19/0.49 ((ssList(U!45) & (ssList(V!46) & ssList(U!45) & (~singletonP(U!45)) & neq(V!46, nil) & (ssItem(Y!47) & (~((~ssList(Z!48)) | (~(cons(Y!47, nil) = U!45)) | (~(app(Z!48, cons(Y!47, nil)) = V!46))))))) <=> (ssList(U!45) & ssList(V!46) & (~singletonP(U!45)) & neq(V!46, nil) & ssItem(Y!47) & (~((~ssList(Z!48)) | (~(cons(Y!47, nil) = U!45)) | (~(app(Z!48, cons(Y!47, nil)) = V!46)))))),
% 0.19/0.49 inference(transitivity,[status(thm)],[3, 1])).
% 0.19/0.49 tff(5,plain,
% 0.19/0.49 ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~ssList(U)) | singletonP(U) | (~neq(V, nil)) | ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | (~(cons(Y, nil) = U)) | (~(app(Z, cons(Y, nil)) = V))))))) <=> (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~ssList(U)) | singletonP(U) | (~neq(V, nil)) | ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | (~(cons(Y, nil) = U)) | (~(app(Z, cons(Y, nil)) = V)))))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(6,plain,
% 0.19/0.49 ((~![U: $i] : (ssList(U) => ![V: $i] : (ssList(V) => ![W: $i] : (ssList(W) => ![X: $i] : (ssList(X) => (((~(V = X)) | (~(U = W))) | ((((~neq(V, nil)) | ![Y: $i] : (ssItem(Y) => ![Z: $i] : (ssList(Z) => ((~(cons(Y, nil) = W)) | (~(app(Z, cons(Y, nil)) = X)))))) | singletonP(U)) & ((~neq(V, nil)) | neq(X, nil))))))))) <=> (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~ssList(U)) | singletonP(U) | (~neq(V, nil)) | ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | (~(cons(Y, nil) = U)) | (~(app(Z, cons(Y, nil)) = V)))))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(7,axiom,(~![U: $i] : (ssList(U) => ![V: $i] : (ssList(V) => ![W: $i] : (ssList(W) => ![X: $i] : (ssList(X) => (((~(V = X)) | (~(U = W))) | ((((~neq(V, nil)) | ![Y: $i] : (ssItem(Y) => ![Z: $i] : (ssList(Z) => ((~(cons(Y, nil) = W)) | (~(app(Z, cons(Y, nil)) = X)))))) | singletonP(U)) & ((~neq(V, nil)) | neq(X, nil))))))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','co1')).
% 0.19/0.49 tff(8,plain,
% 0.19/0.49 (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~ssList(U)) | singletonP(U) | (~neq(V, nil)) | ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | (~(cons(Y, nil) = U)) | (~(app(Z, cons(Y, nil)) = V))))))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[7, 6])).
% 0.19/0.49 tff(9,plain,
% 0.19/0.49 (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~ssList(U)) | singletonP(U) | (~neq(V, nil)) | ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | (~(cons(Y, nil) = U)) | (~(app(Z, cons(Y, nil)) = V))))))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[8, 5])).
% 0.19/0.49 tff(10,plain,
% 0.19/0.49 (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~ssList(U)) | singletonP(U) | (~neq(V, nil)) | ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | (~(cons(Y, nil) = U)) | (~(app(Z, cons(Y, nil)) = V))))))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[9, 5])).
% 0.19/0.49 tff(11,plain,
% 0.19/0.49 (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~ssList(U)) | singletonP(U) | (~neq(V, nil)) | ![Y: $i] : ((~ssItem(Y)) | ![Z: $i] : ((~ssList(Z)) | (~(cons(Y, nil) = U)) | (~(app(Z, cons(Y, nil)) = V))))))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[10, 5])).
% 0.19/0.49 tff(12,plain,
% 0.19/0.49 (ssList(U!45) & ssList(V!46) & (~singletonP(U!45)) & neq(V!46, nil) & ssItem(Y!47) & (~((~ssList(Z!48)) | (~(cons(Y!47, nil) = U!45)) | (~(app(Z!48, cons(Y!47, nil)) = V!46))))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[11, 4])).
% 0.19/0.49 tff(13,plain,
% 0.19/0.49 (ssList(U!45)),
% 0.19/0.49 inference(and_elim,[status(thm)],[12])).
% 0.19/0.49 tff(14,plain,
% 0.19/0.49 (^[U: $i] : refl(((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U)))))))) <=> ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U)))))))))),
% 0.19/0.49 inference(bind,[status(th)],[])).
% 0.19/0.49 tff(15,plain,
% 0.19/0.49 (![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U)))))))) <=> ![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U))))))))),
% 0.19/0.49 inference(quant_intro,[status(thm)],[14])).
% 0.19/0.49 tff(16,plain,
% 0.19/0.49 (^[U: $i] : rewrite(((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U)))))))) <=> ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U)))))))))),
% 0.19/0.49 inference(bind,[status(th)],[])).
% 0.19/0.49 tff(17,plain,
% 0.19/0.49 (![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U)))))))) <=> ![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U))))))))),
% 0.19/0.49 inference(quant_intro,[status(thm)],[16])).
% 0.19/0.49 tff(18,plain,
% 0.19/0.49 (![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U)))))))) <=> ![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U))))))))),
% 0.19/0.49 inference(transitivity,[status(thm)],[17, 15])).
% 0.19/0.49 tff(19,plain,
% 0.19/0.49 (^[U: $i] : rewrite(((~ssList(U)) | (((~singletonP(U)) | (ssItem(tptp_fun_V_4(U)) & (cons(tptp_fun_V_4(U), nil) = U))) & (singletonP(U) | ![V: $i] : (~(ssItem(V) & (cons(V, nil) = U)))))) <=> ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U)))))))))),
% 0.19/0.49 inference(bind,[status(th)],[])).
% 0.19/0.49 tff(20,plain,
% 0.19/0.49 (![U: $i] : ((~ssList(U)) | (((~singletonP(U)) | (ssItem(tptp_fun_V_4(U)) & (cons(tptp_fun_V_4(U), nil) = U))) & (singletonP(U) | ![V: $i] : (~(ssItem(V) & (cons(V, nil) = U)))))) <=> ![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U))))))))),
% 0.19/0.49 inference(quant_intro,[status(thm)],[19])).
% 0.19/0.49 tff(21,plain,
% 0.19/0.49 (![U: $i] : ((~ssList(U)) | (singletonP(U) <=> ?[V: $i] : (ssItem(V) & (cons(V, nil) = U)))) <=> ![U: $i] : ((~ssList(U)) | (singletonP(U) <=> ?[V: $i] : (ssItem(V) & (cons(V, nil) = U))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(22,plain,
% 0.19/0.49 (^[U: $i] : trans(monotonicity(rewrite((singletonP(U) <=> ?[V: $i] : (ssItem(V) & (cons(V, nil) = U))) <=> (singletonP(U) <=> ?[V: $i] : (ssItem(V) & (cons(V, nil) = U)))), ((ssList(U) => (singletonP(U) <=> ?[V: $i] : (ssItem(V) & (cons(V, nil) = U)))) <=> (ssList(U) => (singletonP(U) <=> ?[V: $i] : (ssItem(V) & (cons(V, nil) = U)))))), rewrite((ssList(U) => (singletonP(U) <=> ?[V: $i] : (ssItem(V) & (cons(V, nil) = U)))) <=> ((~ssList(U)) | (singletonP(U) <=> ?[V: $i] : (ssItem(V) & (cons(V, nil) = U))))), ((ssList(U) => (singletonP(U) <=> ?[V: $i] : (ssItem(V) & (cons(V, nil) = U)))) <=> ((~ssList(U)) | (singletonP(U) <=> ?[V: $i] : (ssItem(V) & (cons(V, nil) = U))))))),
% 0.19/0.49 inference(bind,[status(th)],[])).
% 0.19/0.49 tff(23,plain,
% 0.19/0.49 (![U: $i] : (ssList(U) => (singletonP(U) <=> ?[V: $i] : (ssItem(V) & (cons(V, nil) = U)))) <=> ![U: $i] : ((~ssList(U)) | (singletonP(U) <=> ?[V: $i] : (ssItem(V) & (cons(V, nil) = U))))),
% 0.19/0.49 inference(quant_intro,[status(thm)],[22])).
% 0.19/0.49 tff(24,axiom,(![U: $i] : (ssList(U) => (singletonP(U) <=> ?[V: $i] : (ssItem(V) & (cons(V, nil) = U))))), file('/export/starexec/sandbox2/benchmark/Axioms/SWC001+0.ax','ax4')).
% 0.19/0.49 tff(25,plain,
% 0.19/0.49 (![U: $i] : ((~ssList(U)) | (singletonP(U) <=> ?[V: $i] : (ssItem(V) & (cons(V, nil) = U))))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[24, 23])).
% 0.19/0.49 tff(26,plain,
% 0.19/0.49 (![U: $i] : ((~ssList(U)) | (singletonP(U) <=> ?[V: $i] : (ssItem(V) & (cons(V, nil) = U))))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[25, 21])).
% 0.19/0.49 tff(27,plain,(
% 0.19/0.49 ![U: $i] : ((~ssList(U)) | (((~singletonP(U)) | (ssItem(tptp_fun_V_4(U)) & (cons(tptp_fun_V_4(U), nil) = U))) & (singletonP(U) | ![V: $i] : (~(ssItem(V) & (cons(V, nil) = U))))))),
% 0.19/0.49 inference(skolemize,[status(sab)],[26])).
% 0.19/0.49 tff(28,plain,
% 0.19/0.49 (![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U))))))))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[27, 20])).
% 0.19/0.49 tff(29,plain,
% 0.19/0.49 (![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U))))))))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[28, 18])).
% 0.19/0.49 tff(30,plain,
% 0.19/0.49 (((~![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U))))))))) | ((~ssList(U!45)) | (~((~(singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45))))) | (~((~singletonP(U!45)) | (~((~ssItem(tptp_fun_V_4(U!45))) | (~(cons(tptp_fun_V_4(U!45), nil) = U!45)))))))))) <=> ((~![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U))))))))) | (~ssList(U!45)) | (~((~(singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45))))) | (~((~singletonP(U!45)) | (~((~ssItem(tptp_fun_V_4(U!45))) | (~(cons(tptp_fun_V_4(U!45), nil) = U!45)))))))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(31,plain,
% 0.19/0.49 (((~ssList(U!45)) | (~((~((~singletonP(U!45)) | (~((~ssItem(tptp_fun_V_4(U!45))) | (~(cons(tptp_fun_V_4(U!45), nil) = U!45)))))) | (~(singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45)))))))) <=> ((~ssList(U!45)) | (~((~(singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45))))) | (~((~singletonP(U!45)) | (~((~ssItem(tptp_fun_V_4(U!45))) | (~(cons(tptp_fun_V_4(U!45), nil) = U!45)))))))))),
% 0.19/0.49 inference(rewrite,[status(thm)],[])).
% 0.19/0.49 tff(32,plain,
% 0.19/0.49 (((~![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U))))))))) | ((~ssList(U!45)) | (~((~((~singletonP(U!45)) | (~((~ssItem(tptp_fun_V_4(U!45))) | (~(cons(tptp_fun_V_4(U!45), nil) = U!45)))))) | (~(singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45))))))))) <=> ((~![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U))))))))) | ((~ssList(U!45)) | (~((~(singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45))))) | (~((~singletonP(U!45)) | (~((~ssItem(tptp_fun_V_4(U!45))) | (~(cons(tptp_fun_V_4(U!45), nil) = U!45))))))))))),
% 0.19/0.49 inference(monotonicity,[status(thm)],[31])).
% 0.19/0.49 tff(33,plain,
% 0.19/0.49 (((~![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U))))))))) | ((~ssList(U!45)) | (~((~((~singletonP(U!45)) | (~((~ssItem(tptp_fun_V_4(U!45))) | (~(cons(tptp_fun_V_4(U!45), nil) = U!45)))))) | (~(singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45))))))))) <=> ((~![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U))))))))) | (~ssList(U!45)) | (~((~(singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45))))) | (~((~singletonP(U!45)) | (~((~ssItem(tptp_fun_V_4(U!45))) | (~(cons(tptp_fun_V_4(U!45), nil) = U!45)))))))))),
% 0.19/0.49 inference(transitivity,[status(thm)],[32, 30])).
% 0.19/0.49 tff(34,plain,
% 0.19/0.49 ((~![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U))))))))) | ((~ssList(U!45)) | (~((~((~singletonP(U!45)) | (~((~ssItem(tptp_fun_V_4(U!45))) | (~(cons(tptp_fun_V_4(U!45), nil) = U!45)))))) | (~(singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45))))))))),
% 0.19/0.49 inference(quant_inst,[status(thm)],[])).
% 0.19/0.49 tff(35,plain,
% 0.19/0.49 ((~![U: $i] : ((~ssList(U)) | (~((~((~singletonP(U)) | (~((~ssItem(tptp_fun_V_4(U))) | (~(cons(tptp_fun_V_4(U), nil) = U)))))) | (~(singletonP(U) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U))))))))) | (~ssList(U!45)) | (~((~(singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45))))) | (~((~singletonP(U!45)) | (~((~ssItem(tptp_fun_V_4(U!45))) | (~(cons(tptp_fun_V_4(U!45), nil) = U!45))))))))),
% 0.19/0.49 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.19/0.49 tff(36,plain,
% 0.19/0.49 (~((~(singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45))))) | (~((~singletonP(U!45)) | (~((~ssItem(tptp_fun_V_4(U!45))) | (~(cons(tptp_fun_V_4(U!45), nil) = U!45)))))))),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[35, 29, 13])).
% 0.19/0.49 tff(37,plain,
% 0.19/0.49 (((~(singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45))))) | (~((~singletonP(U!45)) | (~((~ssItem(tptp_fun_V_4(U!45))) | (~(cons(tptp_fun_V_4(U!45), nil) = U!45))))))) | (singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45))))),
% 0.19/0.49 inference(tautology,[status(thm)],[])).
% 0.19/0.49 tff(38,plain,
% 0.19/0.49 (singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45)))),
% 0.19/0.49 inference(unit_resolution,[status(thm)],[37, 36])).
% 0.19/0.49 tff(39,plain,
% 0.19/0.49 (~singletonP(U!45)),
% 0.19/0.49 inference(and_elim,[status(thm)],[12])).
% 0.19/0.49 tff(40,plain,
% 0.19/0.49 ((~(singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45))))) | singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45)))),
% 0.19/0.49 inference(tautology,[status(thm)],[])).
% 0.19/0.50 tff(41,plain,
% 0.19/0.50 ((~(singletonP(U!45) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45))))) | ![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45)))),
% 0.19/0.50 inference(unit_resolution,[status(thm)],[40, 39])).
% 0.19/0.50 tff(42,plain,
% 0.19/0.50 (![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45)))),
% 0.19/0.50 inference(unit_resolution,[status(thm)],[41, 38])).
% 0.19/0.50 tff(43,plain,
% 0.19/0.50 (~((~ssList(Z!48)) | (~(cons(Y!47, nil) = U!45)) | (~(app(Z!48, cons(Y!47, nil)) = V!46)))),
% 0.19/0.50 inference(and_elim,[status(thm)],[12])).
% 0.19/0.50 tff(44,plain,
% 0.19/0.50 (cons(Y!47, nil) = U!45),
% 0.19/0.50 inference(or_elim,[status(thm)],[43])).
% 0.19/0.50 tff(45,plain,
% 0.19/0.50 (ssItem(Y!47)),
% 0.19/0.50 inference(and_elim,[status(thm)],[12])).
% 0.19/0.50 tff(46,plain,
% 0.19/0.50 (((~![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45)))) | ((~ssItem(Y!47)) | (~(cons(Y!47, nil) = U!45)))) <=> ((~![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45)))) | (~ssItem(Y!47)) | (~(cons(Y!47, nil) = U!45)))),
% 0.19/0.50 inference(rewrite,[status(thm)],[])).
% 0.19/0.50 tff(47,plain,
% 0.19/0.50 ((~![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45)))) | ((~ssItem(Y!47)) | (~(cons(Y!47, nil) = U!45)))),
% 0.19/0.50 inference(quant_inst,[status(thm)],[])).
% 0.19/0.50 tff(48,plain,
% 0.19/0.50 ((~![V: $i] : ((~ssItem(V)) | (~(cons(V, nil) = U!45)))) | (~ssItem(Y!47)) | (~(cons(Y!47, nil) = U!45))),
% 0.19/0.50 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.19/0.50 tff(49,plain,
% 0.19/0.50 ($false),
% 0.19/0.50 inference(unit_resolution,[status(thm)],[48, 45, 44, 42])).
% 0.19/0.50 % SZS output end Proof
% 0.19/0.50 % E exiting
%------------------------------------------------------------------------------