%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SWC416+1 : TPTP v9.0.0. Released v2.4.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n015.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 05:31:11 AM UTC 2025
% Result : Theorem 0.60s 0.82s
% Output : Proof 0.68s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : SWC416+1 : TPTP v9.0.0. Released v2.4.0.
% 0.04/0.12 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n015.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Jun 20 09:17:10 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.60/0.82 % SZS status Theorem
% 0.60/0.82 % SZS output start Proof
% 0.60/0.82 tff(neq_type, type, (
% 0.60/0.82 neq: ( $i * $i ) > $o)).
% 0.60/0.82 tff(tptp_fun_V_46_type, type, (
% 0.60/0.82 tptp_fun_V_46: $i)).
% 0.60/0.82 tff(nil_type, type, (
% 0.60/0.82 nil: $i)).
% 0.60/0.82 tff(ssList_type, type, (
% 0.60/0.82 ssList: $i > $o)).
% 0.60/0.82 tff(app_type, type, (
% 0.60/0.82 app: ( $i * $i ) > $i)).
% 0.60/0.82 tff(tptp_fun_U_45_type, type, (
% 0.60/0.82 tptp_fun_U_45: $i)).
% 0.60/0.82 tff(cons_type, type, (
% 0.60/0.82 cons: ( $i * $i ) > $i)).
% 0.60/0.82 tff(hd_type, type, (
% 0.60/0.82 hd: $i > $i)).
% 0.60/0.82 tff(ssItem_type, type, (
% 0.60/0.82 ssItem: $i > $o)).
% 0.60/0.82 tff(tptp_fun_X2_47_type, type, (
% 0.60/0.82 tptp_fun_X2_47: $i)).
% 0.60/0.82 tff(segmentP_type, type, (
% 0.60/0.82 segmentP: ( $i * $i ) > $o)).
% 0.60/0.82 tff(1,plain,
% 0.60/0.82 (ssList(nil) <=> ssList(nil)),
% 0.60/0.82 inference(rewrite,[status(thm)],[])).
% 0.60/0.82 tff(2,axiom,(ssList(nil)), file('/export/starexec/sandbox2/benchmark/Axioms/SWC001+0.ax','ax17')).
% 0.60/0.82 tff(3,plain,
% 0.60/0.82 (ssList(nil)),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[2, 1])).
% 0.60/0.82 tff(4,plain,
% 0.60/0.82 (^[U: $i] : refl(((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V))))) <=> ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V))))))),
% 0.60/0.82 inference(bind,[status(th)],[])).
% 0.60/0.82 tff(5,plain,
% 0.60/0.82 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V))))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))),
% 0.60/0.82 inference(quant_intro,[status(thm)],[4])).
% 0.60/0.82 tff(6,plain,
% 0.60/0.82 (^[U: $i] : rewrite(((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V))))) <=> ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V))))))),
% 0.60/0.82 inference(bind,[status(th)],[])).
% 0.60/0.82 tff(7,plain,
% 0.60/0.82 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V))))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))),
% 0.60/0.82 inference(quant_intro,[status(thm)],[6])).
% 0.60/0.82 tff(8,plain,
% 0.60/0.82 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V))))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))),
% 0.60/0.82 inference(transitivity,[status(thm)],[7, 5])).
% 0.60/0.82 tff(9,plain,
% 0.60/0.82 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V))))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))),
% 0.60/0.82 inference(rewrite,[status(thm)],[])).
% 0.60/0.82 tff(10,plain,
% 0.60/0.82 (^[U: $i] : trans(monotonicity(quant_intro(proof_bind(^[V: $i] : trans(monotonicity(trans(rewrite((neq(U, V) <=> (~(U = V))) <=> (~((U = V) <=> neq(U, V)))), rewrite((~((U = V) <=> neq(U, V))) <=> (~((U = V) <=> neq(U, V)))), ((neq(U, V) <=> (~(U = V))) <=> (~((U = V) <=> neq(U, V))))), ((ssList(V) => (neq(U, V) <=> (~(U = V)))) <=> (ssList(V) => (~((U = V) <=> neq(U, V)))))), rewrite((ssList(V) => (~((U = V) <=> neq(U, V)))) <=> ((~ssList(V)) | (~((U = V) <=> neq(U, V))))), ((ssList(V) => (neq(U, V) <=> (~(U = V)))) <=> ((~ssList(V)) | (~((U = V) <=> neq(U, V))))))), (![V: $i] : (ssList(V) => (neq(U, V) <=> (~(U = V)))) <=> ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))), ((ssList(U) => ![V: $i] : (ssList(V) => (neq(U, V) <=> (~(U = V))))) <=> (ssList(U) => ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V))))))), rewrite((ssList(U) => ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V))))) <=> ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))), ((ssList(U) => ![V: $i] : (ssList(V) => (neq(U, V) <=> (~(U = V))))) <=> ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))))),
% 0.60/0.82 inference(bind,[status(th)],[])).
% 0.60/0.82 tff(11,plain,
% 0.60/0.82 (![U: $i] : (ssList(U) => ![V: $i] : (ssList(V) => (neq(U, V) <=> (~(U = V))))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))),
% 0.60/0.82 inference(quant_intro,[status(thm)],[10])).
% 0.60/0.82 tff(12,axiom,(![U: $i] : (ssList(U) => ![V: $i] : (ssList(V) => (neq(U, V) <=> (~(U = V)))))), file('/export/starexec/sandbox2/benchmark/Axioms/SWC001+0.ax','ax15')).
% 0.60/0.82 tff(13,plain,
% 0.60/0.82 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[12, 11])).
% 0.60/0.82 tff(14,plain,
% 0.60/0.82 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[13, 9])).
% 0.60/0.82 tff(15,plain,(
% 0.60/0.82 ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))),
% 0.60/0.82 inference(skolemize,[status(sab)],[14])).
% 0.60/0.82 tff(16,plain,
% 0.60/0.82 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[15, 8])).
% 0.60/0.82 tff(17,plain,
% 0.60/0.82 (((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))) | ((~ssList(nil)) | ![V: $i] : ((~ssList(V)) | (~((nil = V) <=> neq(nil, V)))))) <=> ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))) | (~ssList(nil)) | ![V: $i] : ((~ssList(V)) | (~((nil = V) <=> neq(nil, V)))))),
% 0.60/0.82 inference(rewrite,[status(thm)],[])).
% 0.60/0.82 tff(18,plain,
% 0.60/0.82 ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))) | ((~ssList(nil)) | ![V: $i] : ((~ssList(V)) | (~((nil = V) <=> neq(nil, V)))))),
% 0.60/0.82 inference(quant_inst,[status(thm)],[])).
% 0.60/0.82 tff(19,plain,
% 0.60/0.82 ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))) | (~ssList(nil)) | ![V: $i] : ((~ssList(V)) | (~((nil = V) <=> neq(nil, V))))),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[18, 17])).
% 0.60/0.82 tff(20,plain,
% 0.60/0.82 (![V: $i] : ((~ssList(V)) | (~((nil = V) <=> neq(nil, V))))),
% 0.60/0.82 inference(unit_resolution,[status(thm)],[19, 16, 3])).
% 0.60/0.82 tff(21,plain,
% 0.60/0.82 ((ssList(U!45) & (ssList(V!46) & (~((~ssItem(X2!47)) | (~(app(cons(X2!47, nil), U!45) = V!46)))) & ssList(U!45) & (~(ssItem(hd(V!46)) & ssList(V!46) & ssList(cons(hd(V!46), nil)) & (app(cons(hd(V!46), nil), U!45) = V!46) & neq(nil, V!46))) & neq(V!46, nil))) <=> (ssList(U!45) & ssList(V!46) & (~((~ssItem(X2!47)) | (~(app(cons(X2!47, nil), U!45) = V!46)))) & (~(ssItem(hd(V!46)) & ssList(V!46) & ssList(cons(hd(V!46), nil)) & (app(cons(hd(V!46), nil), U!45) = V!46) & neq(nil, V!46))) & neq(V!46, nil))),
% 0.60/0.82 inference(rewrite,[status(thm)],[])).
% 0.60/0.82 tff(22,plain,
% 0.60/0.82 ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | ![X2: $i] : ((~ssItem(X2)) | (~(app(cons(X2, nil), U) = V))) | (~ssList(U)) | (ssItem(hd(V)) & ssList(V) & ssList(cons(hd(V), nil)) & (app(cons(hd(V), nil), U) = V) & neq(nil, V)) | (~neq(V, nil))))) <=> (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | ![X2: $i] : ((~ssItem(X2)) | (~(app(cons(X2, nil), U) = V))) | (~ssList(U)) | (ssItem(hd(V)) & ssList(V) & ssList(cons(hd(V), nil)) & (app(cons(hd(V), nil), U) = V) & neq(nil, V)) | (~neq(V, nil)))))),
% 0.60/0.82 inference(rewrite,[status(thm)],[])).
% 0.60/0.82 tff(23,plain,
% 0.60/0.82 ((~![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] : ((ssList(Y) & (V = Y)) & ?[Z: $i] : ((ssList(Z) & (app(Z, U) = Y)) & ?[X1: $i] : (((ssItem(X1) & (cons(X1, nil) = Z)) & (hd(V) = X1)) & neq(nil, V))))) | ![X2: $i] : (ssItem(X2) => (~(app(cons(X2, nil), W) = X)))) & ((~neq(V, nil)) | neq(X, nil))))))))) <=> (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | ![X2: $i] : ((~ssItem(X2)) | (~(app(cons(X2, nil), U) = V))) | (~ssList(U)) | (ssItem(hd(V)) & ssList(V) & ssList(cons(hd(V), nil)) & (app(cons(hd(V), nil), U) = V) & neq(nil, V)) | (~neq(V, nil)))))),
% 0.60/0.82 inference(rewrite,[status(thm)],[])).
% 0.60/0.82 tff(24,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] : ((ssList(Y) & (V = Y)) & ?[Z: $i] : ((ssList(Z) & (app(Z, U) = Y)) & ?[X1: $i] : (((ssItem(X1) & (cons(X1, nil) = Z)) & (hd(V) = X1)) & neq(nil, V))))) | ![X2: $i] : (ssItem(X2) => (~(app(cons(X2, nil), W) = X)))) & ((~neq(V, nil)) | neq(X, nil))))))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','co1')).
% 0.60/0.82 tff(25,plain,
% 0.60/0.82 (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | ![X2: $i] : ((~ssItem(X2)) | (~(app(cons(X2, nil), U) = V))) | (~ssList(U)) | (ssItem(hd(V)) & ssList(V) & ssList(cons(hd(V), nil)) & (app(cons(hd(V), nil), U) = V) & neq(nil, V)) | (~neq(V, nil))))),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[24, 23])).
% 0.60/0.82 tff(26,plain,
% 0.60/0.82 (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | ![X2: $i] : ((~ssItem(X2)) | (~(app(cons(X2, nil), U) = V))) | (~ssList(U)) | (ssItem(hd(V)) & ssList(V) & ssList(cons(hd(V), nil)) & (app(cons(hd(V), nil), U) = V) & neq(nil, V)) | (~neq(V, nil))))),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[25, 22])).
% 0.60/0.82 tff(27,plain,
% 0.60/0.82 (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | ![X2: $i] : ((~ssItem(X2)) | (~(app(cons(X2, nil), U) = V))) | (~ssList(U)) | (ssItem(hd(V)) & ssList(V) & ssList(cons(hd(V), nil)) & (app(cons(hd(V), nil), U) = V) & neq(nil, V)) | (~neq(V, nil))))),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[26, 22])).
% 0.60/0.82 tff(28,plain,
% 0.60/0.82 (~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | ![X2: $i] : ((~ssItem(X2)) | (~(app(cons(X2, nil), U) = V))) | (~ssList(U)) | (ssItem(hd(V)) & ssList(V) & ssList(cons(hd(V), nil)) & (app(cons(hd(V), nil), U) = V) & neq(nil, V)) | (~neq(V, nil))))),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[27, 22])).
% 0.60/0.82 tff(29,plain,
% 0.60/0.82 (ssList(U!45) & ssList(V!46) & (~((~ssItem(X2!47)) | (~(app(cons(X2!47, nil), U!45) = V!46)))) & (~(ssItem(hd(V!46)) & ssList(V!46) & ssList(cons(hd(V!46), nil)) & (app(cons(hd(V!46), nil), U!45) = V!46) & neq(nil, V!46))) & neq(V!46, nil)),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[28, 21])).
% 0.60/0.82 tff(30,plain,
% 0.60/0.82 (ssList(V!46)),
% 0.60/0.82 inference(and_elim,[status(thm)],[29])).
% 0.60/0.82 tff(31,plain,
% 0.60/0.82 (((~![V: $i] : ((~ssList(V)) | (~((nil = V) <=> neq(nil, V))))) | ((~ssList(V!46)) | (~((nil = V!46) <=> neq(nil, V!46))))) <=> ((~![V: $i] : ((~ssList(V)) | (~((nil = V) <=> neq(nil, V))))) | (~ssList(V!46)) | (~((nil = V!46) <=> neq(nil, V!46))))),
% 0.60/0.82 inference(rewrite,[status(thm)],[])).
% 0.60/0.82 tff(32,plain,
% 0.60/0.82 ((~![V: $i] : ((~ssList(V)) | (~((nil = V) <=> neq(nil, V))))) | ((~ssList(V!46)) | (~((nil = V!46) <=> neq(nil, V!46))))),
% 0.60/0.82 inference(quant_inst,[status(thm)],[])).
% 0.60/0.82 tff(33,plain,
% 0.60/0.82 ((~![V: $i] : ((~ssList(V)) | (~((nil = V) <=> neq(nil, V))))) | (~ssList(V!46)) | (~((nil = V!46) <=> neq(nil, V!46)))),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.60/0.82 tff(34,plain,
% 0.60/0.82 (~((nil = V!46) <=> neq(nil, V!46))),
% 0.60/0.82 inference(unit_resolution,[status(thm)],[33, 30, 20])).
% 0.60/0.82 tff(35,plain,
% 0.60/0.82 (^[U: $i] : refl(((~ssList(U)) | (segmentP(nil, U) <=> (nil = U))) <=> ((~ssList(U)) | (segmentP(nil, U) <=> (nil = U))))),
% 0.60/0.82 inference(bind,[status(th)],[])).
% 0.60/0.82 tff(36,plain,
% 0.60/0.82 (![U: $i] : ((~ssList(U)) | (segmentP(nil, U) <=> (nil = U))) <=> ![U: $i] : ((~ssList(U)) | (segmentP(nil, U) <=> (nil = U)))),
% 0.60/0.82 inference(quant_intro,[status(thm)],[35])).
% 0.60/0.82 tff(37,plain,
% 0.60/0.82 (![U: $i] : ((~ssList(U)) | (segmentP(nil, U) <=> (nil = U))) <=> ![U: $i] : ((~ssList(U)) | (segmentP(nil, U) <=> (nil = U)))),
% 0.60/0.82 inference(rewrite,[status(thm)],[])).
% 0.60/0.82 tff(38,plain,
% 0.60/0.82 (^[U: $i] : rewrite((ssList(U) => (segmentP(nil, U) <=> (nil = U))) <=> ((~ssList(U)) | (segmentP(nil, U) <=> (nil = U))))),
% 0.60/0.82 inference(bind,[status(th)],[])).
% 0.60/0.82 tff(39,plain,
% 0.60/0.82 (![U: $i] : (ssList(U) => (segmentP(nil, U) <=> (nil = U))) <=> ![U: $i] : ((~ssList(U)) | (segmentP(nil, U) <=> (nil = U)))),
% 0.60/0.82 inference(quant_intro,[status(thm)],[38])).
% 0.60/0.82 tff(40,axiom,(![U: $i] : (ssList(U) => (segmentP(nil, U) <=> (nil = U)))), file('/export/starexec/sandbox2/benchmark/Axioms/SWC001+0.ax','ax58')).
% 0.60/0.82 tff(41,plain,
% 0.60/0.82 (![U: $i] : ((~ssList(U)) | (segmentP(nil, U) <=> (nil = U)))),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.60/0.82 tff(42,plain,
% 0.60/0.82 (![U: $i] : ((~ssList(U)) | (segmentP(nil, U) <=> (nil = U)))),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[41, 37])).
% 0.60/0.82 tff(43,plain,(
% 0.60/0.82 ![U: $i] : ((~ssList(U)) | (segmentP(nil, U) <=> (nil = U)))),
% 0.60/0.82 inference(skolemize,[status(sab)],[42])).
% 0.60/0.82 tff(44,plain,
% 0.60/0.82 (![U: $i] : ((~ssList(U)) | (segmentP(nil, U) <=> (nil = U)))),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[43, 36])).
% 0.60/0.82 tff(45,plain,
% 0.60/0.82 (((~![U: $i] : ((~ssList(U)) | (segmentP(nil, U) <=> (nil = U)))) | ((~ssList(V!46)) | (segmentP(nil, V!46) <=> (nil = V!46)))) <=> ((~![U: $i] : ((~ssList(U)) | (segmentP(nil, U) <=> (nil = U)))) | (~ssList(V!46)) | (segmentP(nil, V!46) <=> (nil = V!46)))),
% 0.60/0.82 inference(rewrite,[status(thm)],[])).
% 0.60/0.82 tff(46,plain,
% 0.60/0.82 ((~![U: $i] : ((~ssList(U)) | (segmentP(nil, U) <=> (nil = U)))) | ((~ssList(V!46)) | (segmentP(nil, V!46) <=> (nil = V!46)))),
% 0.60/0.82 inference(quant_inst,[status(thm)],[])).
% 0.60/0.82 tff(47,plain,
% 0.60/0.82 ((~![U: $i] : ((~ssList(U)) | (segmentP(nil, U) <=> (nil = U)))) | (~ssList(V!46)) | (segmentP(nil, V!46) <=> (nil = V!46))),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.60/0.82 tff(48,plain,
% 0.60/0.82 (segmentP(nil, V!46) <=> (nil = V!46)),
% 0.60/0.82 inference(unit_resolution,[status(thm)],[47, 44, 30])).
% 0.60/0.82 tff(49,plain,
% 0.60/0.82 (((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))) | ((~ssList(V!46)) | ![V: $i] : ((~ssList(V)) | (~((V!46 = V) <=> neq(V!46, V)))))) <=> ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))) | (~ssList(V!46)) | ![V: $i] : ((~ssList(V)) | (~((V!46 = V) <=> neq(V!46, V)))))),
% 0.60/0.82 inference(rewrite,[status(thm)],[])).
% 0.60/0.82 tff(50,plain,
% 0.60/0.82 ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))) | ((~ssList(V!46)) | ![V: $i] : ((~ssList(V)) | (~((V!46 = V) <=> neq(V!46, V)))))),
% 0.60/0.82 inference(quant_inst,[status(thm)],[])).
% 0.60/0.82 tff(51,plain,
% 0.60/0.82 ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssList(V)) | (~((U = V) <=> neq(U, V)))))) | (~ssList(V!46)) | ![V: $i] : ((~ssList(V)) | (~((V!46 = V) <=> neq(V!46, V))))),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.60/0.82 tff(52,plain,
% 0.60/0.82 (![V: $i] : ((~ssList(V)) | (~((V!46 = V) <=> neq(V!46, V))))),
% 0.60/0.82 inference(unit_resolution,[status(thm)],[51, 16, 30])).
% 0.60/0.82 tff(53,plain,
% 0.60/0.82 (((~![V: $i] : ((~ssList(V)) | (~((V!46 = V) <=> neq(V!46, V))))) | ((~ssList(nil)) | (~((V!46 = nil) <=> neq(V!46, nil))))) <=> ((~![V: $i] : ((~ssList(V)) | (~((V!46 = V) <=> neq(V!46, V))))) | (~ssList(nil)) | (~((V!46 = nil) <=> neq(V!46, nil))))),
% 0.60/0.82 inference(rewrite,[status(thm)],[])).
% 0.60/0.82 tff(54,plain,
% 0.60/0.82 ((~![V: $i] : ((~ssList(V)) | (~((V!46 = V) <=> neq(V!46, V))))) | ((~ssList(nil)) | (~((V!46 = nil) <=> neq(V!46, nil))))),
% 0.60/0.82 inference(quant_inst,[status(thm)],[])).
% 0.60/0.82 tff(55,plain,
% 0.60/0.82 ((~![V: $i] : ((~ssList(V)) | (~((V!46 = V) <=> neq(V!46, V))))) | (~ssList(nil)) | (~((V!46 = nil) <=> neq(V!46, nil)))),
% 0.60/0.82 inference(modus_ponens,[status(thm)],[54, 53])).
% 0.60/0.82 tff(56,plain,
% 0.60/0.82 (~((V!46 = nil) <=> neq(V!46, nil))),
% 0.60/0.82 inference(unit_resolution,[status(thm)],[55, 3, 52])).
% 0.60/0.82 tff(57,plain,
% 0.60/0.82 (neq(V!46, nil)),
% 0.60/0.82 inference(and_elim,[status(thm)],[29])).
% 0.60/0.82 tff(58,plain,
% 0.60/0.82 (((V!46 = nil) <=> neq(V!46, nil)) | (~(V!46 = nil)) | (~neq(V!46, nil))),
% 0.60/0.82 inference(tautology,[status(thm)],[])).
% 0.60/0.82 tff(59,plain,
% 0.60/0.82 (((V!46 = nil) <=> neq(V!46, nil)) | (~(V!46 = nil))),
% 0.60/0.82 inference(unit_resolution,[status(thm)],[58, 57])).
% 0.60/0.82 tff(60,plain,
% 0.60/0.82 (~(V!46 = nil)),
% 0.60/0.82 inference(unit_resolution,[status(thm)],[59, 56])).
% 0.60/0.82 tff(61,plain,
% 0.60/0.82 (^[U: $i] : refl(((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U)))) <=> ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U)))))),
% 0.60/0.82 inference(bind,[status(th)],[])).
% 0.60/0.82 tff(62,plain,
% 0.60/0.82 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U)))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U))))),
% 0.60/0.82 inference(quant_intro,[status(thm)],[61])).
% 0.60/0.82 tff(63,plain,
% 0.60/0.82 (^[U: $i] : rewrite(((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U)))) <=> ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U)))))),
% 0.60/0.82 inference(bind,[status(th)],[])).
% 0.60/0.82 tff(64,plain,
% 0.60/0.82 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U)))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U))))),
% 0.60/0.82 inference(quant_intro,[status(thm)],[63])).
% 0.60/0.82 tff(65,plain,
% 0.60/0.82 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U)))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U))))),
% 0.60/0.83 inference(transitivity,[status(thm)],[64, 62])).
% 0.60/0.83 tff(66,plain,
% 0.60/0.83 (^[U: $i] : rewrite(((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~(segmentP(U, V) & segmentP(V, U))))) <=> ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U)))))),
% 0.60/0.83 inference(bind,[status(th)],[])).
% 0.60/0.83 tff(67,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~(segmentP(U, V) & segmentP(V, U))))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U))))),
% 0.60/0.83 inference(quant_intro,[status(thm)],[66])).
% 0.60/0.83 tff(68,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~(segmentP(U, V) & segmentP(V, U))))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~(segmentP(U, V) & segmentP(V, U)))))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(69,plain,
% 0.60/0.83 (^[U: $i] : trans(monotonicity(quant_intro(proof_bind(^[V: $i] : trans(monotonicity(rewrite(((segmentP(U, V) & segmentP(V, U)) => (U = V)) <=> ((~(segmentP(U, V) & segmentP(V, U))) | (U = V))), ((ssList(V) => ((segmentP(U, V) & segmentP(V, U)) => (U = V))) <=> (ssList(V) => ((~(segmentP(U, V) & segmentP(V, U))) | (U = V))))), rewrite((ssList(V) => ((~(segmentP(U, V) & segmentP(V, U))) | (U = V))) <=> ((U = V) | (~ssList(V)) | (~(segmentP(U, V) & segmentP(V, U))))), ((ssList(V) => ((segmentP(U, V) & segmentP(V, U)) => (U = V))) <=> ((U = V) | (~ssList(V)) | (~(segmentP(U, V) & segmentP(V, U))))))), (![V: $i] : (ssList(V) => ((segmentP(U, V) & segmentP(V, U)) => (U = V))) <=> ![V: $i] : ((U = V) | (~ssList(V)) | (~(segmentP(U, V) & segmentP(V, U)))))), ((ssList(U) => ![V: $i] : (ssList(V) => ((segmentP(U, V) & segmentP(V, U)) => (U = V)))) <=> (ssList(U) => ![V: $i] : ((U = V) | (~ssList(V)) | (~(segmentP(U, V) & segmentP(V, U))))))), rewrite((ssList(U) => ![V: $i] : ((U = V) | (~ssList(V)) | (~(segmentP(U, V) & segmentP(V, U))))) <=> ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~(segmentP(U, V) & segmentP(V, U)))))), ((ssList(U) => ![V: $i] : (ssList(V) => ((segmentP(U, V) & segmentP(V, U)) => (U = V)))) <=> ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~(segmentP(U, V) & segmentP(V, U)))))))),
% 0.60/0.83 inference(bind,[status(th)],[])).
% 0.60/0.83 tff(70,plain,
% 0.60/0.83 (![U: $i] : (ssList(U) => ![V: $i] : (ssList(V) => ((segmentP(U, V) & segmentP(V, U)) => (U = V)))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~(segmentP(U, V) & segmentP(V, U)))))),
% 0.60/0.83 inference(quant_intro,[status(thm)],[69])).
% 0.60/0.83 tff(71,axiom,(![U: $i] : (ssList(U) => ![V: $i] : (ssList(V) => ((segmentP(U, V) & segmentP(V, U)) => (U = V))))), file('/export/starexec/sandbox2/benchmark/Axioms/SWC001+0.ax','ax54')).
% 0.60/0.83 tff(72,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~(segmentP(U, V) & segmentP(V, U)))))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.60/0.83 tff(73,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~(segmentP(U, V) & segmentP(V, U)))))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[72, 68])).
% 0.60/0.83 tff(74,plain,(
% 0.60/0.83 ![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~(segmentP(U, V) & segmentP(V, U)))))),
% 0.60/0.83 inference(skolemize,[status(sab)],[73])).
% 0.60/0.83 tff(75,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U))))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[74, 67])).
% 0.60/0.83 tff(76,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U))))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[75, 65])).
% 0.60/0.83 tff(77,plain,
% 0.60/0.83 (((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U))))) | ((~ssList(V!46)) | ![V: $i] : ((~ssList(V)) | (V!46 = V) | (~segmentP(V!46, V)) | (~segmentP(V, V!46))))) <=> ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U))))) | (~ssList(V!46)) | ![V: $i] : ((~ssList(V)) | (V!46 = V) | (~segmentP(V!46, V)) | (~segmentP(V, V!46))))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(78,plain,
% 0.60/0.83 (((~ssList(V!46)) | ![V: $i] : ((V!46 = V) | (~ssList(V)) | (~segmentP(V!46, V)) | (~segmentP(V, V!46)))) <=> ((~ssList(V!46)) | ![V: $i] : ((~ssList(V)) | (V!46 = V) | (~segmentP(V!46, V)) | (~segmentP(V, V!46))))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(79,plain,
% 0.60/0.83 (((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U))))) | ((~ssList(V!46)) | ![V: $i] : ((V!46 = V) | (~ssList(V)) | (~segmentP(V!46, V)) | (~segmentP(V, V!46))))) <=> ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U))))) | ((~ssList(V!46)) | ![V: $i] : ((~ssList(V)) | (V!46 = V) | (~segmentP(V!46, V)) | (~segmentP(V, V!46)))))),
% 0.60/0.83 inference(monotonicity,[status(thm)],[78])).
% 0.60/0.83 tff(80,plain,
% 0.60/0.83 (((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U))))) | ((~ssList(V!46)) | ![V: $i] : ((V!46 = V) | (~ssList(V)) | (~segmentP(V!46, V)) | (~segmentP(V, V!46))))) <=> ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U))))) | (~ssList(V!46)) | ![V: $i] : ((~ssList(V)) | (V!46 = V) | (~segmentP(V!46, V)) | (~segmentP(V, V!46))))),
% 0.60/0.83 inference(transitivity,[status(thm)],[79, 77])).
% 0.60/0.83 tff(81,plain,
% 0.60/0.83 ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U))))) | ((~ssList(V!46)) | ![V: $i] : ((V!46 = V) | (~ssList(V)) | (~segmentP(V!46, V)) | (~segmentP(V, V!46))))),
% 0.60/0.83 inference(quant_inst,[status(thm)],[])).
% 0.60/0.83 tff(82,plain,
% 0.60/0.83 ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((U = V) | (~ssList(V)) | (~segmentP(U, V)) | (~segmentP(V, U))))) | (~ssList(V!46)) | ![V: $i] : ((~ssList(V)) | (V!46 = V) | (~segmentP(V!46, V)) | (~segmentP(V, V!46)))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[81, 80])).
% 0.60/0.83 tff(83,plain,
% 0.60/0.83 (![V: $i] : ((~ssList(V)) | (V!46 = V) | (~segmentP(V!46, V)) | (~segmentP(V, V!46)))),
% 0.60/0.83 inference(unit_resolution,[status(thm)],[82, 76, 30])).
% 0.60/0.83 tff(84,plain,
% 0.60/0.83 (^[U: $i] : refl(((~ssList(U)) | segmentP(U, nil)) <=> ((~ssList(U)) | segmentP(U, nil)))),
% 0.60/0.83 inference(bind,[status(th)],[])).
% 0.60/0.83 tff(85,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | segmentP(U, nil)) <=> ![U: $i] : ((~ssList(U)) | segmentP(U, nil))),
% 0.60/0.83 inference(quant_intro,[status(thm)],[84])).
% 0.60/0.83 tff(86,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | segmentP(U, nil)) <=> ![U: $i] : ((~ssList(U)) | segmentP(U, nil))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(87,plain,
% 0.60/0.83 (^[U: $i] : rewrite((ssList(U) => segmentP(U, nil)) <=> ((~ssList(U)) | segmentP(U, nil)))),
% 0.60/0.83 inference(bind,[status(th)],[])).
% 0.60/0.83 tff(88,plain,
% 0.60/0.83 (![U: $i] : (ssList(U) => segmentP(U, nil)) <=> ![U: $i] : ((~ssList(U)) | segmentP(U, nil))),
% 0.60/0.83 inference(quant_intro,[status(thm)],[87])).
% 0.60/0.83 tff(89,axiom,(![U: $i] : (ssList(U) => segmentP(U, nil))), file('/export/starexec/sandbox2/benchmark/Axioms/SWC001+0.ax','ax57')).
% 0.60/0.83 tff(90,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | segmentP(U, nil))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[89, 88])).
% 0.60/0.83 tff(91,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | segmentP(U, nil))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[90, 86])).
% 0.60/0.83 tff(92,plain,(
% 0.60/0.83 ![U: $i] : ((~ssList(U)) | segmentP(U, nil))),
% 0.60/0.83 inference(skolemize,[status(sab)],[91])).
% 0.60/0.83 tff(93,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | segmentP(U, nil))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[92, 85])).
% 0.60/0.83 tff(94,plain,
% 0.60/0.83 (((~![U: $i] : ((~ssList(U)) | segmentP(U, nil))) | ((~ssList(V!46)) | segmentP(V!46, nil))) <=> ((~![U: $i] : ((~ssList(U)) | segmentP(U, nil))) | (~ssList(V!46)) | segmentP(V!46, nil))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(95,plain,
% 0.60/0.83 ((~![U: $i] : ((~ssList(U)) | segmentP(U, nil))) | ((~ssList(V!46)) | segmentP(V!46, nil))),
% 0.60/0.83 inference(quant_inst,[status(thm)],[])).
% 0.60/0.83 tff(96,plain,
% 0.60/0.83 ((~![U: $i] : ((~ssList(U)) | segmentP(U, nil))) | (~ssList(V!46)) | segmentP(V!46, nil)),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[95, 94])).
% 0.60/0.83 tff(97,plain,
% 0.60/0.83 (segmentP(V!46, nil)),
% 0.60/0.83 inference(unit_resolution,[status(thm)],[96, 93, 30])).
% 0.60/0.83 tff(98,plain,
% 0.60/0.83 (((~![V: $i] : ((~ssList(V)) | (V!46 = V) | (~segmentP(V!46, V)) | (~segmentP(V, V!46)))) | ((~ssList(nil)) | (~segmentP(nil, V!46)) | (V!46 = nil) | (~segmentP(V!46, nil)))) <=> ((~![V: $i] : ((~ssList(V)) | (V!46 = V) | (~segmentP(V!46, V)) | (~segmentP(V, V!46)))) | (~ssList(nil)) | (~segmentP(nil, V!46)) | (V!46 = nil) | (~segmentP(V!46, nil)))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(99,plain,
% 0.60/0.83 (((~ssList(nil)) | (V!46 = nil) | (~segmentP(V!46, nil)) | (~segmentP(nil, V!46))) <=> ((~ssList(nil)) | (~segmentP(nil, V!46)) | (V!46 = nil) | (~segmentP(V!46, nil)))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(100,plain,
% 0.60/0.83 (((~![V: $i] : ((~ssList(V)) | (V!46 = V) | (~segmentP(V!46, V)) | (~segmentP(V, V!46)))) | ((~ssList(nil)) | (V!46 = nil) | (~segmentP(V!46, nil)) | (~segmentP(nil, V!46)))) <=> ((~![V: $i] : ((~ssList(V)) | (V!46 = V) | (~segmentP(V!46, V)) | (~segmentP(V, V!46)))) | ((~ssList(nil)) | (~segmentP(nil, V!46)) | (V!46 = nil) | (~segmentP(V!46, nil))))),
% 0.60/0.83 inference(monotonicity,[status(thm)],[99])).
% 0.60/0.83 tff(101,plain,
% 0.60/0.83 (((~![V: $i] : ((~ssList(V)) | (V!46 = V) | (~segmentP(V!46, V)) | (~segmentP(V, V!46)))) | ((~ssList(nil)) | (V!46 = nil) | (~segmentP(V!46, nil)) | (~segmentP(nil, V!46)))) <=> ((~![V: $i] : ((~ssList(V)) | (V!46 = V) | (~segmentP(V!46, V)) | (~segmentP(V, V!46)))) | (~ssList(nil)) | (~segmentP(nil, V!46)) | (V!46 = nil) | (~segmentP(V!46, nil)))),
% 0.60/0.83 inference(transitivity,[status(thm)],[100, 98])).
% 0.60/0.83 tff(102,plain,
% 0.60/0.83 ((~![V: $i] : ((~ssList(V)) | (V!46 = V) | (~segmentP(V!46, V)) | (~segmentP(V, V!46)))) | ((~ssList(nil)) | (V!46 = nil) | (~segmentP(V!46, nil)) | (~segmentP(nil, V!46)))),
% 0.60/0.83 inference(quant_inst,[status(thm)],[])).
% 0.60/0.83 tff(103,plain,
% 0.60/0.83 ((~![V: $i] : ((~ssList(V)) | (V!46 = V) | (~segmentP(V!46, V)) | (~segmentP(V, V!46)))) | (~ssList(nil)) | (~segmentP(nil, V!46)) | (V!46 = nil) | (~segmentP(V!46, nil))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[102, 101])).
% 0.60/0.83 tff(104,plain,
% 0.60/0.83 ((~segmentP(nil, V!46)) | (V!46 = nil)),
% 0.60/0.83 inference(unit_resolution,[status(thm)],[103, 3, 97, 83])).
% 0.60/0.83 tff(105,plain,
% 0.60/0.83 (~segmentP(nil, V!46)),
% 0.60/0.83 inference(unit_resolution,[status(thm)],[104, 60])).
% 0.60/0.83 tff(106,plain,
% 0.60/0.83 ((~(segmentP(nil, V!46) <=> (nil = V!46))) | segmentP(nil, V!46) | (~(nil = V!46))),
% 0.60/0.83 inference(tautology,[status(thm)],[])).
% 0.60/0.83 tff(107,plain,
% 0.60/0.83 (~(nil = V!46)),
% 0.60/0.83 inference(unit_resolution,[status(thm)],[106, 105, 48])).
% 0.60/0.83 tff(108,plain,
% 0.60/0.83 (((nil = V!46) <=> neq(nil, V!46)) | (nil = V!46) | neq(nil, V!46)),
% 0.60/0.83 inference(tautology,[status(thm)],[])).
% 0.60/0.83 tff(109,plain,
% 0.60/0.83 (neq(nil, V!46)),
% 0.60/0.83 inference(unit_resolution,[status(thm)],[108, 107, 34])).
% 0.60/0.83 tff(110,plain,
% 0.60/0.83 (^[U: $i] : refl(((nil = U) | ssItem(hd(U)) | (~ssList(U))) <=> ((nil = U) | ssItem(hd(U)) | (~ssList(U))))),
% 0.60/0.83 inference(bind,[status(th)],[])).
% 0.60/0.83 tff(111,plain,
% 0.60/0.83 (![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U))) <=> ![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U)))),
% 0.60/0.83 inference(quant_intro,[status(thm)],[110])).
% 0.60/0.83 tff(112,plain,
% 0.60/0.83 (![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U))) <=> ![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U)))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(113,plain,
% 0.60/0.83 (^[U: $i] : trans(monotonicity(rewrite(((~(nil = U)) => ssItem(hd(U))) <=> ((nil = U) | ssItem(hd(U)))), ((ssList(U) => ((~(nil = U)) => ssItem(hd(U)))) <=> (ssList(U) => ((nil = U) | ssItem(hd(U)))))), rewrite((ssList(U) => ((nil = U) | ssItem(hd(U)))) <=> ((nil = U) | ssItem(hd(U)) | (~ssList(U)))), ((ssList(U) => ((~(nil = U)) => ssItem(hd(U)))) <=> ((nil = U) | ssItem(hd(U)) | (~ssList(U)))))),
% 0.60/0.83 inference(bind,[status(th)],[])).
% 0.60/0.83 tff(114,plain,
% 0.60/0.83 (![U: $i] : (ssList(U) => ((~(nil = U)) => ssItem(hd(U)))) <=> ![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U)))),
% 0.60/0.83 inference(quant_intro,[status(thm)],[113])).
% 0.60/0.83 tff(115,axiom,(![U: $i] : (ssList(U) => ((~(nil = U)) => ssItem(hd(U))))), file('/export/starexec/sandbox2/benchmark/Axioms/SWC001+0.ax','ax22')).
% 0.60/0.83 tff(116,plain,
% 0.60/0.83 (![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U)))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[115, 114])).
% 0.60/0.83 tff(117,plain,
% 0.60/0.83 (![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U)))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[116, 112])).
% 0.60/0.83 tff(118,plain,(
% 0.60/0.83 ![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U)))),
% 0.60/0.83 inference(skolemize,[status(sab)],[117])).
% 0.60/0.83 tff(119,plain,
% 0.60/0.83 (![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U)))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[118, 111])).
% 0.60/0.83 tff(120,plain,
% 0.60/0.83 (((~![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U)))) | (ssItem(hd(V!46)) | (~ssList(V!46)) | (nil = V!46))) <=> ((~![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U)))) | ssItem(hd(V!46)) | (~ssList(V!46)) | (nil = V!46))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(121,plain,
% 0.60/0.83 (((nil = V!46) | ssItem(hd(V!46)) | (~ssList(V!46))) <=> (ssItem(hd(V!46)) | (~ssList(V!46)) | (nil = V!46))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(122,plain,
% 0.60/0.83 (((~![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U)))) | ((nil = V!46) | ssItem(hd(V!46)) | (~ssList(V!46)))) <=> ((~![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U)))) | (ssItem(hd(V!46)) | (~ssList(V!46)) | (nil = V!46)))),
% 0.60/0.83 inference(monotonicity,[status(thm)],[121])).
% 0.60/0.83 tff(123,plain,
% 0.60/0.83 (((~![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U)))) | ((nil = V!46) | ssItem(hd(V!46)) | (~ssList(V!46)))) <=> ((~![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U)))) | ssItem(hd(V!46)) | (~ssList(V!46)) | (nil = V!46))),
% 0.60/0.83 inference(transitivity,[status(thm)],[122, 120])).
% 0.60/0.83 tff(124,plain,
% 0.60/0.83 ((~![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U)))) | ((nil = V!46) | ssItem(hd(V!46)) | (~ssList(V!46)))),
% 0.60/0.83 inference(quant_inst,[status(thm)],[])).
% 0.60/0.83 tff(125,plain,
% 0.60/0.83 ((~![U: $i] : ((nil = U) | ssItem(hd(U)) | (~ssList(U)))) | ssItem(hd(V!46)) | (~ssList(V!46)) | (nil = V!46)),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[124, 123])).
% 0.60/0.83 tff(126,plain,
% 0.60/0.83 (ssItem(hd(V!46)) | (nil = V!46)),
% 0.60/0.83 inference(unit_resolution,[status(thm)],[125, 119, 30])).
% 0.60/0.83 tff(127,plain,
% 0.60/0.83 (ssItem(hd(V!46))),
% 0.60/0.83 inference(unit_resolution,[status(thm)],[126, 107])).
% 0.60/0.83 tff(128,plain,
% 0.60/0.83 (ssList(U!45)),
% 0.60/0.83 inference(and_elim,[status(thm)],[29])).
% 0.60/0.83 tff(129,plain,
% 0.60/0.83 (^[U: $i] : refl(((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V))) <=> ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V))))),
% 0.60/0.83 inference(bind,[status(th)],[])).
% 0.60/0.83 tff(130,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))),
% 0.60/0.83 inference(quant_intro,[status(thm)],[129])).
% 0.60/0.83 tff(131,plain,
% 0.60/0.83 (^[U: $i] : rewrite(((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V))) <=> ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V))))),
% 0.60/0.83 inference(bind,[status(th)],[])).
% 0.60/0.83 tff(132,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))),
% 0.60/0.83 inference(quant_intro,[status(thm)],[131])).
% 0.60/0.83 tff(133,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))),
% 0.60/0.83 inference(transitivity,[status(thm)],[132, 130])).
% 0.60/0.83 tff(134,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(135,plain,
% 0.60/0.83 (^[U: $i] : trans(monotonicity(quant_intro(proof_bind(^[V: $i] : rewrite((ssItem(V) => (hd(cons(V, U)) = V)) <=> ((~ssItem(V)) | (hd(cons(V, U)) = V)))), (![V: $i] : (ssItem(V) => (hd(cons(V, U)) = V)) <=> ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))), ((ssList(U) => ![V: $i] : (ssItem(V) => (hd(cons(V, U)) = V))) <=> (ssList(U) => ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V))))), rewrite((ssList(U) => ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V))) <=> ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))), ((ssList(U) => ![V: $i] : (ssItem(V) => (hd(cons(V, U)) = V))) <=> ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))))),
% 0.60/0.83 inference(bind,[status(th)],[])).
% 0.60/0.83 tff(136,plain,
% 0.60/0.83 (![U: $i] : (ssList(U) => ![V: $i] : (ssItem(V) => (hd(cons(V, U)) = V))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))),
% 0.60/0.83 inference(quant_intro,[status(thm)],[135])).
% 0.60/0.83 tff(137,axiom,(![U: $i] : (ssList(U) => ![V: $i] : (ssItem(V) => (hd(cons(V, U)) = V)))), file('/export/starexec/sandbox2/benchmark/Axioms/SWC001+0.ax','ax23')).
% 0.60/0.83 tff(138,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[137, 136])).
% 0.60/0.83 tff(139,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[138, 134])).
% 0.60/0.83 tff(140,plain,(
% 0.60/0.83 ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))),
% 0.60/0.83 inference(skolemize,[status(sab)],[139])).
% 0.60/0.83 tff(141,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[140, 133])).
% 0.60/0.83 tff(142,plain,
% 0.60/0.83 (((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))) | ((~ssList(U!45)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U!45)) = V)))) <=> ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))) | (~ssList(U!45)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U!45)) = V)))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(143,plain,
% 0.60/0.83 ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))) | ((~ssList(U!45)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U!45)) = V)))),
% 0.60/0.83 inference(quant_inst,[status(thm)],[])).
% 0.60/0.83 tff(144,plain,
% 0.60/0.83 ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U)) = V)))) | (~ssList(U!45)) | ![V: $i] : ((~ssItem(V)) | (hd(cons(V, U!45)) = V))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[143, 142])).
% 0.60/0.83 tff(145,plain,
% 0.60/0.83 (![V: $i] : ((~ssItem(V)) | (hd(cons(V, U!45)) = V))),
% 0.60/0.83 inference(unit_resolution,[status(thm)],[144, 141, 128])).
% 0.60/0.83 tff(146,plain,
% 0.60/0.83 (~((~ssItem(X2!47)) | (~(app(cons(X2!47, nil), U!45) = V!46)))),
% 0.60/0.83 inference(and_elim,[status(thm)],[29])).
% 0.60/0.83 tff(147,plain,
% 0.60/0.83 (ssItem(X2!47)),
% 0.60/0.83 inference(or_elim,[status(thm)],[146])).
% 0.60/0.83 tff(148,plain,
% 0.60/0.83 (((~![V: $i] : ((~ssItem(V)) | (hd(cons(V, U!45)) = V))) | ((~ssItem(X2!47)) | (hd(cons(X2!47, U!45)) = X2!47))) <=> ((~![V: $i] : ((~ssItem(V)) | (hd(cons(V, U!45)) = V))) | (~ssItem(X2!47)) | (hd(cons(X2!47, U!45)) = X2!47))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(149,plain,
% 0.60/0.83 ((~![V: $i] : ((~ssItem(V)) | (hd(cons(V, U!45)) = V))) | ((~ssItem(X2!47)) | (hd(cons(X2!47, U!45)) = X2!47))),
% 0.60/0.83 inference(quant_inst,[status(thm)],[])).
% 0.60/0.83 tff(150,plain,
% 0.60/0.83 ((~![V: $i] : ((~ssItem(V)) | (hd(cons(V, U!45)) = V))) | (~ssItem(X2!47)) | (hd(cons(X2!47, U!45)) = X2!47)),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[149, 148])).
% 0.60/0.83 tff(151,plain,
% 0.60/0.83 (hd(cons(X2!47, U!45)) = X2!47),
% 0.60/0.83 inference(unit_resolution,[status(thm)],[150, 147, 145])).
% 0.60/0.83 tff(152,plain,
% 0.60/0.83 (app(cons(X2!47, nil), U!45) = V!46),
% 0.60/0.83 inference(or_elim,[status(thm)],[146])).
% 0.60/0.83 tff(153,plain,
% 0.60/0.83 (^[U: $i] : refl(((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U)))) <=> ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U)))))),
% 0.60/0.83 inference(bind,[status(th)],[])).
% 0.60/0.83 tff(154,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U)))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))),
% 0.60/0.83 inference(quant_intro,[status(thm)],[153])).
% 0.60/0.83 tff(155,plain,
% 0.60/0.83 (^[U: $i] : rewrite(((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U)))) <=> ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U)))))),
% 0.60/0.83 inference(bind,[status(th)],[])).
% 0.60/0.83 tff(156,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U)))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))),
% 0.60/0.83 inference(quant_intro,[status(thm)],[155])).
% 0.60/0.83 tff(157,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U)))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))),
% 0.60/0.83 inference(transitivity,[status(thm)],[156, 154])).
% 0.60/0.83 tff(158,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U)))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(159,plain,
% 0.60/0.83 (^[U: $i] : trans(monotonicity(quant_intro(proof_bind(^[V: $i] : rewrite((ssItem(V) => (cons(V, U) = app(cons(V, nil), U))) <=> ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))), (![V: $i] : (ssItem(V) => (cons(V, U) = app(cons(V, nil), U))) <=> ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))), ((ssList(U) => ![V: $i] : (ssItem(V) => (cons(V, U) = app(cons(V, nil), U)))) <=> (ssList(U) => ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U)))))), rewrite((ssList(U) => ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U)))) <=> ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))), ((ssList(U) => ![V: $i] : (ssItem(V) => (cons(V, U) = app(cons(V, nil), U)))) <=> ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))))),
% 0.60/0.83 inference(bind,[status(th)],[])).
% 0.60/0.83 tff(160,plain,
% 0.60/0.83 (![U: $i] : (ssList(U) => ![V: $i] : (ssItem(V) => (cons(V, U) = app(cons(V, nil), U)))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))),
% 0.60/0.83 inference(quant_intro,[status(thm)],[159])).
% 0.60/0.83 tff(161,axiom,(![U: $i] : (ssList(U) => ![V: $i] : (ssItem(V) => (cons(V, U) = app(cons(V, nil), U))))), file('/export/starexec/sandbox2/benchmark/Axioms/SWC001+0.ax','ax81')).
% 0.60/0.83 tff(162,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[161, 160])).
% 0.60/0.83 tff(163,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[162, 158])).
% 0.60/0.83 tff(164,plain,(
% 0.60/0.83 ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))),
% 0.60/0.83 inference(skolemize,[status(sab)],[163])).
% 0.60/0.83 tff(165,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[164, 157])).
% 0.60/0.83 tff(166,plain,
% 0.60/0.83 (((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))) | ((~ssList(U!45)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U!45) = app(cons(V, nil), U!45))))) <=> ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))) | (~ssList(U!45)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U!45) = app(cons(V, nil), U!45))))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(167,plain,
% 0.60/0.83 ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))) | ((~ssList(U!45)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U!45) = app(cons(V, nil), U!45))))),
% 0.60/0.83 inference(quant_inst,[status(thm)],[])).
% 0.60/0.83 tff(168,plain,
% 0.60/0.83 ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U) = app(cons(V, nil), U))))) | (~ssList(U!45)) | ![V: $i] : ((~ssItem(V)) | (cons(V, U!45) = app(cons(V, nil), U!45)))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[167, 166])).
% 0.60/0.83 tff(169,plain,
% 0.60/0.83 (![V: $i] : ((~ssItem(V)) | (cons(V, U!45) = app(cons(V, nil), U!45)))),
% 0.60/0.83 inference(unit_resolution,[status(thm)],[168, 165, 128])).
% 0.60/0.83 tff(170,plain,
% 0.60/0.83 (((~![V: $i] : ((~ssItem(V)) | (cons(V, U!45) = app(cons(V, nil), U!45)))) | ((~ssItem(X2!47)) | (cons(X2!47, U!45) = app(cons(X2!47, nil), U!45)))) <=> ((~![V: $i] : ((~ssItem(V)) | (cons(V, U!45) = app(cons(V, nil), U!45)))) | (~ssItem(X2!47)) | (cons(X2!47, U!45) = app(cons(X2!47, nil), U!45)))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(171,plain,
% 0.60/0.83 ((~![V: $i] : ((~ssItem(V)) | (cons(V, U!45) = app(cons(V, nil), U!45)))) | ((~ssItem(X2!47)) | (cons(X2!47, U!45) = app(cons(X2!47, nil), U!45)))),
% 0.60/0.83 inference(quant_inst,[status(thm)],[])).
% 0.60/0.83 tff(172,plain,
% 0.60/0.83 ((~![V: $i] : ((~ssItem(V)) | (cons(V, U!45) = app(cons(V, nil), U!45)))) | (~ssItem(X2!47)) | (cons(X2!47, U!45) = app(cons(X2!47, nil), U!45))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[171, 170])).
% 0.60/0.83 tff(173,plain,
% 0.60/0.83 (cons(X2!47, U!45) = app(cons(X2!47, nil), U!45)),
% 0.60/0.83 inference(unit_resolution,[status(thm)],[172, 147, 169])).
% 0.60/0.83 tff(174,plain,
% 0.60/0.83 (cons(X2!47, U!45) = V!46),
% 0.60/0.83 inference(transitivity,[status(thm)],[173, 152])).
% 0.60/0.83 tff(175,plain,
% 0.60/0.83 (hd(cons(X2!47, U!45)) = hd(V!46)),
% 0.60/0.83 inference(monotonicity,[status(thm)],[174])).
% 0.60/0.83 tff(176,plain,
% 0.60/0.83 (hd(V!46) = hd(cons(X2!47, U!45))),
% 0.60/0.83 inference(symmetry,[status(thm)],[175])).
% 0.60/0.83 tff(177,plain,
% 0.60/0.83 (hd(V!46) = X2!47),
% 0.60/0.83 inference(transitivity,[status(thm)],[176, 151])).
% 0.60/0.83 tff(178,plain,
% 0.60/0.83 (cons(hd(V!46), nil) = cons(X2!47, nil)),
% 0.60/0.83 inference(monotonicity,[status(thm)],[177])).
% 0.60/0.83 tff(179,plain,
% 0.60/0.83 (ssList(cons(hd(V!46), nil)) <=> ssList(cons(X2!47, nil))),
% 0.60/0.83 inference(monotonicity,[status(thm)],[178])).
% 0.60/0.83 tff(180,plain,
% 0.60/0.83 (ssList(cons(X2!47, nil)) <=> ssList(cons(hd(V!46), nil))),
% 0.60/0.83 inference(symmetry,[status(thm)],[179])).
% 0.60/0.83 tff(181,plain,
% 0.60/0.83 (^[U: $i] : refl(((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U)))) <=> ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U)))))),
% 0.60/0.83 inference(bind,[status(th)],[])).
% 0.60/0.83 tff(182,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U)))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))),
% 0.60/0.83 inference(quant_intro,[status(thm)],[181])).
% 0.60/0.83 tff(183,plain,
% 0.60/0.83 (^[U: $i] : rewrite(((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U)))) <=> ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U)))))),
% 0.60/0.83 inference(bind,[status(th)],[])).
% 0.60/0.83 tff(184,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U)))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))),
% 0.60/0.83 inference(quant_intro,[status(thm)],[183])).
% 0.60/0.83 tff(185,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U)))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))),
% 0.60/0.83 inference(transitivity,[status(thm)],[184, 182])).
% 0.60/0.83 tff(186,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U)))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))),
% 0.60/0.83 inference(rewrite,[status(thm)],[])).
% 0.60/0.83 tff(187,plain,
% 0.60/0.83 (^[U: $i] : trans(monotonicity(quant_intro(proof_bind(^[V: $i] : rewrite((ssItem(V) => ssList(cons(V, U))) <=> ((~ssItem(V)) | ssList(cons(V, U))))), (![V: $i] : (ssItem(V) => ssList(cons(V, U))) <=> ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))), ((ssList(U) => ![V: $i] : (ssItem(V) => ssList(cons(V, U)))) <=> (ssList(U) => ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U)))))), rewrite((ssList(U) => ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U)))) <=> ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))), ((ssList(U) => ![V: $i] : (ssItem(V) => ssList(cons(V, U)))) <=> ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))))),
% 0.60/0.83 inference(bind,[status(th)],[])).
% 0.60/0.83 tff(188,plain,
% 0.60/0.83 (![U: $i] : (ssList(U) => ![V: $i] : (ssItem(V) => ssList(cons(V, U)))) <=> ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))),
% 0.60/0.83 inference(quant_intro,[status(thm)],[187])).
% 0.60/0.83 tff(189,axiom,(![U: $i] : (ssList(U) => ![V: $i] : (ssItem(V) => ssList(cons(V, U))))), file('/export/starexec/sandbox2/benchmark/Axioms/SWC001+0.ax','ax16')).
% 0.60/0.83 tff(190,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[189, 188])).
% 0.60/0.83 tff(191,plain,
% 0.60/0.83 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))),
% 0.60/0.83 inference(modus_ponens,[status(thm)],[190, 186])).
% 0.60/0.83 tff(192,plain,(
% 0.60/0.83 ![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))),
% 0.60/0.84 inference(skolemize,[status(sab)],[191])).
% 0.60/0.84 tff(193,plain,
% 0.60/0.84 (![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))),
% 0.60/0.84 inference(modus_ponens,[status(thm)],[192, 185])).
% 0.60/0.84 tff(194,plain,
% 0.60/0.84 (((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))) | ((~ssList(nil)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, nil))))) <=> ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))) | (~ssList(nil)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, nil))))),
% 0.60/0.84 inference(rewrite,[status(thm)],[])).
% 0.60/0.84 tff(195,plain,
% 0.60/0.84 ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))) | ((~ssList(nil)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, nil))))),
% 0.60/0.84 inference(quant_inst,[status(thm)],[])).
% 0.60/0.84 tff(196,plain,
% 0.60/0.84 ((~![U: $i] : ((~ssList(U)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, U))))) | (~ssList(nil)) | ![V: $i] : ((~ssItem(V)) | ssList(cons(V, nil)))),
% 0.60/0.84 inference(modus_ponens,[status(thm)],[195, 194])).
% 0.60/0.84 tff(197,plain,
% 0.60/0.84 (![V: $i] : ((~ssItem(V)) | ssList(cons(V, nil)))),
% 0.60/0.84 inference(unit_resolution,[status(thm)],[196, 193, 3])).
% 0.60/0.84 tff(198,plain,
% 0.60/0.84 (((~![V: $i] : ((~ssItem(V)) | ssList(cons(V, nil)))) | ((~ssItem(X2!47)) | ssList(cons(X2!47, nil)))) <=> ((~![V: $i] : ((~ssItem(V)) | ssList(cons(V, nil)))) | (~ssItem(X2!47)) | ssList(cons(X2!47, nil)))),
% 0.60/0.84 inference(rewrite,[status(thm)],[])).
% 0.60/0.84 tff(199,plain,
% 0.60/0.84 ((~![V: $i] : ((~ssItem(V)) | ssList(cons(V, nil)))) | ((~ssItem(X2!47)) | ssList(cons(X2!47, nil)))),
% 0.60/0.84 inference(quant_inst,[status(thm)],[])).
% 0.60/0.84 tff(200,plain,
% 0.60/0.84 ((~![V: $i] : ((~ssItem(V)) | ssList(cons(V, nil)))) | (~ssItem(X2!47)) | ssList(cons(X2!47, nil))),
% 0.60/0.84 inference(modus_ponens,[status(thm)],[199, 198])).
% 0.60/0.84 tff(201,plain,
% 0.60/0.84 (ssList(cons(X2!47, nil))),
% 0.60/0.84 inference(unit_resolution,[status(thm)],[200, 147, 197])).
% 0.60/0.84 tff(202,plain,
% 0.60/0.84 (ssList(cons(hd(V!46), nil))),
% 0.60/0.84 inference(modus_ponens,[status(thm)],[201, 180])).
% 0.60/0.84 tff(203,plain,
% 0.60/0.84 (app(cons(hd(V!46), nil), U!45) = app(cons(X2!47, nil), U!45)),
% 0.60/0.84 inference(monotonicity,[status(thm)],[178])).
% 0.60/0.84 tff(204,plain,
% 0.60/0.84 (app(cons(hd(V!46), nil), U!45) = V!46),
% 0.60/0.84 inference(transitivity,[status(thm)],[203, 152])).
% 0.60/0.84 tff(205,plain,
% 0.60/0.84 ((~(~((~ssList(V!46)) | (~ssItem(hd(V!46))) | (~ssList(cons(hd(V!46), nil))) | (~(app(cons(hd(V!46), nil), U!45) = V!46)) | (~neq(nil, V!46))))) <=> ((~ssList(V!46)) | (~ssItem(hd(V!46))) | (~ssList(cons(hd(V!46), nil))) | (~(app(cons(hd(V!46), nil), U!45) = V!46)) | (~neq(nil, V!46)))),
% 0.60/0.84 inference(rewrite,[status(thm)],[])).
% 0.60/0.84 tff(206,plain,
% 0.60/0.84 ((ssItem(hd(V!46)) & ssList(V!46) & ssList(cons(hd(V!46), nil)) & (app(cons(hd(V!46), nil), U!45) = V!46) & neq(nil, V!46)) <=> (~((~ssList(V!46)) | (~ssItem(hd(V!46))) | (~ssList(cons(hd(V!46), nil))) | (~(app(cons(hd(V!46), nil), U!45) = V!46)) | (~neq(nil, V!46))))),
% 0.60/0.84 inference(rewrite,[status(thm)],[])).
% 0.60/0.84 tff(207,plain,
% 0.60/0.84 ((~(ssItem(hd(V!46)) & ssList(V!46) & ssList(cons(hd(V!46), nil)) & (app(cons(hd(V!46), nil), U!45) = V!46) & neq(nil, V!46))) <=> (~(~((~ssList(V!46)) | (~ssItem(hd(V!46))) | (~ssList(cons(hd(V!46), nil))) | (~(app(cons(hd(V!46), nil), U!45) = V!46)) | (~neq(nil, V!46)))))),
% 0.60/0.84 inference(monotonicity,[status(thm)],[206])).
% 0.60/0.84 tff(208,plain,
% 0.60/0.84 ((~(ssItem(hd(V!46)) & ssList(V!46) & ssList(cons(hd(V!46), nil)) & (app(cons(hd(V!46), nil), U!45) = V!46) & neq(nil, V!46))) <=> ((~ssList(V!46)) | (~ssItem(hd(V!46))) | (~ssList(cons(hd(V!46), nil))) | (~(app(cons(hd(V!46), nil), U!45) = V!46)) | (~neq(nil, V!46)))),
% 0.60/0.84 inference(transitivity,[status(thm)],[207, 205])).
% 0.60/0.84 tff(209,plain,
% 0.60/0.84 (~(ssItem(hd(V!46)) & ssList(V!46) & ssList(cons(hd(V!46), nil)) & (app(cons(hd(V!46), nil), U!45) = V!46) & neq(nil, V!46))),
% 0.60/0.84 inference(and_elim,[status(thm)],[29])).
% 0.60/0.84 tff(210,plain,
% 0.60/0.84 ((~ssList(V!46)) | (~ssItem(hd(V!46))) | (~ssList(cons(hd(V!46), nil))) | (~(app(cons(hd(V!46), nil), U!45) = V!46)) | (~neq(nil, V!46))),
% 0.60/0.84 inference(modus_ponens,[status(thm)],[209, 208])).
% 0.60/0.84 tff(211,plain,
% 0.60/0.84 ((~ssItem(hd(V!46))) | (~ssList(cons(hd(V!46), nil))) | (~(app(cons(hd(V!46), nil), U!45) = V!46)) | (~neq(nil, V!46))),
% 0.68/0.84 inference(unit_resolution,[status(thm)],[210, 30])).
% 0.68/0.84 tff(212,plain,
% 0.68/0.84 ($false),
% 0.68/0.84 inference(unit_resolution,[status(thm)],[211, 204, 202, 127, 109])).
% 0.68/0.84 % SZS output end Proof
% 0.68/0.85 % E exiting
%------------------------------------------------------------------------------