%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : ALG176+1 : TPTP v9.0.0. Released v2.7.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.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 04:58:33 AM UTC 2025
% Result : Theorem 0.12s 0.38s
% Output : Proof 0.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : ALG176+1 : TPTP v9.0.0. Released v2.7.0.
% 0.06/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n001.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % 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 : Thu Jun 19 10:52:18 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.12/0.38 % SZS status Theorem
% 0.12/0.38 % SZS output start Proof
% 0.12/0.38 tff(j_type, type, (
% 0.12/0.38 j: $i > $i)).
% 0.12/0.38 tff(tptp_fun_V_1_type, type, (
% 0.12/0.38 tptp_fun_V_1: $i > $i)).
% 0.12/0.38 tff(h_type, type, (
% 0.12/0.38 h: $i > $i)).
% 0.12/0.38 tff(tptp_fun_U_0_type, type, (
% 0.12/0.38 tptp_fun_U_0: $i)).
% 0.12/0.38 tff(op1_type, type, (
% 0.12/0.38 op1: ( $i * $i ) > $i)).
% 0.12/0.38 tff(op2_type, type, (
% 0.12/0.38 op2: ( $i * $i ) > $i)).
% 0.12/0.38 tff(sorti2_type, type, (
% 0.12/0.38 sorti2: $i > $o)).
% 0.12/0.38 tff(sorti1_type, type, (
% 0.12/0.38 sorti1: $i > $o)).
% 0.12/0.38 tff(1,plain,
% 0.12/0.38 (^[U: $i] : refl(((~sorti1(U)) | sorti2(h(U))) <=> ((~sorti1(U)) | sorti2(h(U))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(2,plain,
% 0.12/0.38 (![U: $i] : ((~sorti1(U)) | sorti2(h(U))) <=> ![U: $i] : ((~sorti1(U)) | sorti2(h(U)))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.38 tff(3,plain,
% 0.12/0.38 (![U: $i] : ((~sorti1(U)) | sorti2(h(U))) <=> ![U: $i] : ((~sorti1(U)) | sorti2(h(U)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(4,plain,
% 0.12/0.38 ((~((![U: $i] : (sorti1(U) => sorti2(h(U))) & ![V: $i] : (sorti2(V) => sorti1(j(V)))) => (~(((![W: $i] : (sorti1(W) => ![X: $i] : (sorti1(X) => (h(op1(W, X)) = op2(h(W), h(X))))) & ![Y: $i] : (sorti2(Y) => ![Z: $i] : (sorti2(Z) => (j(op2(Y, Z)) = op1(j(Y), j(Z)))))) & ![X1: $i] : (sorti2(X1) => (h(j(X1)) = X1))) & ![X2: $i] : (sorti1(X2) => (j(h(X2)) = X2)))))) <=> (~((~(![U: $i] : ((~sorti1(U)) | sorti2(h(U))) & ![V: $i] : ((~sorti2(V)) | sorti1(j(V))))) | (~(![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X))))) & ![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z))))) & ![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1)) & ![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(5,axiom,(~((![U: $i] : (sorti1(U) => sorti2(h(U))) & ![V: $i] : (sorti2(V) => sorti1(j(V)))) => (~(((![W: $i] : (sorti1(W) => ![X: $i] : (sorti1(X) => (h(op1(W, X)) = op2(h(W), h(X))))) & ![Y: $i] : (sorti2(Y) => ![Z: $i] : (sorti2(Z) => (j(op2(Y, Z)) = op1(j(Y), j(Z)))))) & ![X1: $i] : (sorti2(X1) => (h(j(X1)) = X1))) & ![X2: $i] : (sorti1(X2) => (j(h(X2)) = X2)))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','co1')).
% 0.12/0.38 tff(6,plain,
% 0.12/0.38 (~((~(![U: $i] : ((~sorti1(U)) | sorti2(h(U))) & ![V: $i] : ((~sorti2(V)) | sorti1(j(V))))) | (~(![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X))))) & ![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z))))) & ![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1)) & ![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2)))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.12/0.38 tff(7,plain,
% 0.12/0.38 (![U: $i] : ((~sorti1(U)) | sorti2(h(U))) & ![V: $i] : ((~sorti2(V)) | sorti1(j(V)))),
% 0.12/0.38 inference(or_elim,[status(thm)],[6])).
% 0.12/0.38 tff(8,plain,
% 0.12/0.38 (![U: $i] : ((~sorti1(U)) | sorti2(h(U)))),
% 0.12/0.38 inference(and_elim,[status(thm)],[7])).
% 0.12/0.38 tff(9,plain,
% 0.12/0.38 (![U: $i] : ((~sorti1(U)) | sorti2(h(U)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[8, 3])).
% 0.12/0.38 tff(10,plain,(
% 0.12/0.38 ![U: $i] : ((~sorti1(U)) | sorti2(h(U)))),
% 0.12/0.38 inference(skolemize,[status(sab)],[9])).
% 0.12/0.38 tff(11,plain,
% 0.12/0.38 (![U: $i] : ((~sorti1(U)) | sorti2(h(U)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[10, 2])).
% 0.12/0.38 tff(12,plain,
% 0.12/0.38 ((sorti1(U!0) & ![V: $i] : ((~sorti1(V)) | (~(op1(V, U!0) = V)))) <=> (sorti1(U!0) & ![V: $i] : ((~sorti1(V)) | (~(op1(V, U!0) = V))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(13,plain,
% 0.12/0.38 (?[U: $i] : (sorti1(U) & ![V: $i] : ((~sorti1(V)) | (~(op1(V, U) = V)))) <=> ?[U: $i] : (sorti1(U) & ![V: $i] : ((~sorti1(V)) | (~(op1(V, U) = V))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(14,plain,
% 0.12/0.38 (^[U: $i] : rewrite((sorti1(U) & ![V: $i] : (sorti1(V) => (~(op1(V, U) = V)))) <=> (sorti1(U) & ![V: $i] : ((~sorti1(V)) | (~(op1(V, U) = V)))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(15,plain,
% 0.12/0.38 (?[U: $i] : (sorti1(U) & ![V: $i] : (sorti1(V) => (~(op1(V, U) = V)))) <=> ?[U: $i] : (sorti1(U) & ![V: $i] : ((~sorti1(V)) | (~(op1(V, U) = V))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[14])).
% 0.12/0.38 tff(16,axiom,(?[U: $i] : (sorti1(U) & ![V: $i] : (sorti1(V) => (~(op1(V, U) = V))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','ax3')).
% 0.12/0.38 tff(17,plain,
% 0.12/0.38 (?[U: $i] : (sorti1(U) & ![V: $i] : ((~sorti1(V)) | (~(op1(V, U) = V))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.12/0.38 tff(18,plain,
% 0.12/0.38 (?[U: $i] : (sorti1(U) & ![V: $i] : ((~sorti1(V)) | (~(op1(V, U) = V))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.12/0.38 tff(19,plain,
% 0.12/0.38 (sorti1(U!0) & ![V: $i] : ((~sorti1(V)) | (~(op1(V, U!0) = V)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[18, 12])).
% 0.12/0.38 tff(20,plain,
% 0.12/0.38 (sorti1(U!0)),
% 0.12/0.38 inference(and_elim,[status(thm)],[19])).
% 0.12/0.38 tff(21,plain,
% 0.12/0.38 (((~![U: $i] : ((~sorti1(U)) | sorti2(h(U)))) | ((~sorti1(U!0)) | sorti2(h(U!0)))) <=> ((~![U: $i] : ((~sorti1(U)) | sorti2(h(U)))) | (~sorti1(U!0)) | sorti2(h(U!0)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(22,plain,
% 0.12/0.38 ((~![U: $i] : ((~sorti1(U)) | sorti2(h(U)))) | ((~sorti1(U!0)) | sorti2(h(U!0)))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(23,plain,
% 0.12/0.38 ((~![U: $i] : ((~sorti1(U)) | sorti2(h(U)))) | (~sorti1(U!0)) | sorti2(h(U!0))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[22, 21])).
% 0.12/0.38 tff(24,plain,
% 0.12/0.38 (sorti2(h(U!0))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[23, 20, 11])).
% 0.12/0.38 tff(25,plain,
% 0.12/0.38 (^[U: $i] : refl(((~sorti2(U)) | (~((~sorti2(tptp_fun_V_1(U))) | (~(op2(tptp_fun_V_1(U), U) = tptp_fun_V_1(U)))))) <=> ((~sorti2(U)) | (~((~sorti2(tptp_fun_V_1(U))) | (~(op2(tptp_fun_V_1(U), U) = tptp_fun_V_1(U)))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(26,plain,
% 0.12/0.38 (![U: $i] : ((~sorti2(U)) | (~((~sorti2(tptp_fun_V_1(U))) | (~(op2(tptp_fun_V_1(U), U) = tptp_fun_V_1(U)))))) <=> ![U: $i] : ((~sorti2(U)) | (~((~sorti2(tptp_fun_V_1(U))) | (~(op2(tptp_fun_V_1(U), U) = tptp_fun_V_1(U))))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[25])).
% 0.12/0.38 tff(27,plain,
% 0.12/0.38 ((~?[U: $i] : (sorti2(U) & ![V: $i] : ((~sorti2(V)) | (~(op2(V, U) = V))))) <=> (~?[U: $i] : (sorti2(U) & ![V: $i] : ((~sorti2(V)) | (~(op2(V, U) = V)))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(28,plain,
% 0.12/0.38 ((~?[U: $i] : (sorti2(U) & ![V: $i] : (sorti2(V) => (~(op2(V, U) = V))))) <=> (~?[U: $i] : (sorti2(U) & ![V: $i] : ((~sorti2(V)) | (~(op2(V, U) = V)))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(29,axiom,(~?[U: $i] : (sorti2(U) & ![V: $i] : (sorti2(V) => (~(op2(V, U) = V))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','ax4')).
% 0.12/0.38 tff(30,plain,
% 0.12/0.38 (~?[U: $i] : (sorti2(U) & ![V: $i] : ((~sorti2(V)) | (~(op2(V, U) = V))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[29, 28])).
% 0.12/0.38 tff(31,plain,
% 0.12/0.38 (~?[U: $i] : (sorti2(U) & ![V: $i] : ((~sorti2(V)) | (~(op2(V, U) = V))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[30, 27])).
% 0.12/0.38 tff(32,plain,
% 0.12/0.38 (~?[U: $i] : (sorti2(U) & ![V: $i] : ((~sorti2(V)) | (~(op2(V, U) = V))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[31, 27])).
% 0.12/0.38 tff(33,plain,
% 0.12/0.38 (~?[U: $i] : (sorti2(U) & ![V: $i] : ((~sorti2(V)) | (~(op2(V, U) = V))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[32, 27])).
% 0.12/0.38 tff(34,plain,
% 0.12/0.38 (^[U: $i] : nnf_neg(refl($oeq((~sorti2(U)), (~sorti2(U)))), sk($oeq((~![V: $i] : ((~sorti2(V)) | (~(op2(V, U) = V)))), (~((~sorti2(tptp_fun_V_1(U))) | (~(op2(tptp_fun_V_1(U), U) = tptp_fun_V_1(U))))))), $oeq((~(sorti2(U) & ![V: $i] : ((~sorti2(V)) | (~(op2(V, U) = V))))), ((~sorti2(U)) | (~((~sorti2(tptp_fun_V_1(U))) | (~(op2(tptp_fun_V_1(U), U) = tptp_fun_V_1(U))))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(35,plain,(
% 0.12/0.38 ![U: $i] : ((~sorti2(U)) | (~((~sorti2(tptp_fun_V_1(U))) | (~(op2(tptp_fun_V_1(U), U) = tptp_fun_V_1(U))))))),
% 0.12/0.38 inference(nnf-neg,[status(sab)],[33, 34])).
% 0.12/0.38 tff(36,plain,
% 0.12/0.38 (![U: $i] : ((~sorti2(U)) | (~((~sorti2(tptp_fun_V_1(U))) | (~(op2(tptp_fun_V_1(U), U) = tptp_fun_V_1(U))))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[35, 26])).
% 0.12/0.38 tff(37,plain,
% 0.12/0.38 (((~![U: $i] : ((~sorti2(U)) | (~((~sorti2(tptp_fun_V_1(U))) | (~(op2(tptp_fun_V_1(U), U) = tptp_fun_V_1(U))))))) | ((~sorti2(h(U!0))) | (~((~sorti2(tptp_fun_V_1(h(U!0)))) | (~(op2(tptp_fun_V_1(h(U!0)), h(U!0)) = tptp_fun_V_1(h(U!0)))))))) <=> ((~![U: $i] : ((~sorti2(U)) | (~((~sorti2(tptp_fun_V_1(U))) | (~(op2(tptp_fun_V_1(U), U) = tptp_fun_V_1(U))))))) | (~sorti2(h(U!0))) | (~((~sorti2(tptp_fun_V_1(h(U!0)))) | (~(op2(tptp_fun_V_1(h(U!0)), h(U!0)) = tptp_fun_V_1(h(U!0)))))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(38,plain,
% 0.12/0.38 ((~![U: $i] : ((~sorti2(U)) | (~((~sorti2(tptp_fun_V_1(U))) | (~(op2(tptp_fun_V_1(U), U) = tptp_fun_V_1(U))))))) | ((~sorti2(h(U!0))) | (~((~sorti2(tptp_fun_V_1(h(U!0)))) | (~(op2(tptp_fun_V_1(h(U!0)), h(U!0)) = tptp_fun_V_1(h(U!0)))))))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(39,plain,
% 0.12/0.38 ((~![U: $i] : ((~sorti2(U)) | (~((~sorti2(tptp_fun_V_1(U))) | (~(op2(tptp_fun_V_1(U), U) = tptp_fun_V_1(U))))))) | (~sorti2(h(U!0))) | (~((~sorti2(tptp_fun_V_1(h(U!0)))) | (~(op2(tptp_fun_V_1(h(U!0)), h(U!0)) = tptp_fun_V_1(h(U!0))))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.12/0.38 tff(40,plain,
% 0.12/0.38 (~((~sorti2(tptp_fun_V_1(h(U!0)))) | (~(op2(tptp_fun_V_1(h(U!0)), h(U!0)) = tptp_fun_V_1(h(U!0)))))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[39, 36, 24])).
% 0.12/0.38 tff(41,plain,
% 0.12/0.38 (((~sorti2(tptp_fun_V_1(h(U!0)))) | (~(op2(tptp_fun_V_1(h(U!0)), h(U!0)) = tptp_fun_V_1(h(U!0))))) | (op2(tptp_fun_V_1(h(U!0)), h(U!0)) = tptp_fun_V_1(h(U!0)))),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(42,plain,
% 0.12/0.38 (op2(tptp_fun_V_1(h(U!0)), h(U!0)) = tptp_fun_V_1(h(U!0))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[41, 40])).
% 0.12/0.38 tff(43,plain,
% 0.12/0.38 (tptp_fun_V_1(h(U!0)) = op2(tptp_fun_V_1(h(U!0)), h(U!0))),
% 0.12/0.38 inference(symmetry,[status(thm)],[42])).
% 0.12/0.38 tff(44,plain,
% 0.12/0.38 (j(tptp_fun_V_1(h(U!0))) = j(op2(tptp_fun_V_1(h(U!0)), h(U!0)))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[43])).
% 0.12/0.38 tff(45,plain,
% 0.12/0.38 (j(op2(tptp_fun_V_1(h(U!0)), h(U!0))) = j(tptp_fun_V_1(h(U!0)))),
% 0.12/0.38 inference(symmetry,[status(thm)],[44])).
% 0.12/0.38 tff(46,plain,
% 0.12/0.38 (((~sorti2(tptp_fun_V_1(h(U!0)))) | (~(op2(tptp_fun_V_1(h(U!0)), h(U!0)) = tptp_fun_V_1(h(U!0))))) | sorti2(tptp_fun_V_1(h(U!0)))),
% 0.12/0.38 inference(tautology,[status(thm)],[])).
% 0.12/0.38 tff(47,plain,
% 0.12/0.38 (sorti2(tptp_fun_V_1(h(U!0)))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[46, 40])).
% 0.12/0.38 tff(48,plain,
% 0.12/0.38 (^[Y: $i] : refl(((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z))))) <=> ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(49,plain,
% 0.12/0.38 (![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z))))) <=> ![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[48])).
% 0.12/0.38 tff(50,plain,
% 0.12/0.38 (^[Y: $i] : rewrite(((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z))))) <=> ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(51,plain,
% 0.12/0.38 (![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z))))) <=> ![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[50])).
% 0.12/0.38 tff(52,plain,
% 0.12/0.38 (![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z))))) <=> ![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[51, 49])).
% 0.12/0.38 tff(53,plain,
% 0.12/0.38 (![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z))))) <=> ![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(54,plain,
% 0.12/0.38 (![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X))))) & ![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z))))) & ![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1)) & ![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))),
% 0.12/0.38 inference(or_elim,[status(thm)],[6])).
% 0.12/0.38 tff(55,plain,
% 0.12/0.38 (![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))),
% 0.12/0.38 inference(and_elim,[status(thm)],[54])).
% 0.12/0.38 tff(56,plain,
% 0.12/0.38 (![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[55, 53])).
% 0.12/0.38 tff(57,plain,(
% 0.12/0.38 ![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))),
% 0.12/0.38 inference(skolemize,[status(sab)],[56])).
% 0.12/0.38 tff(58,plain,
% 0.12/0.38 (![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[57, 52])).
% 0.12/0.38 tff(59,plain,
% 0.12/0.38 (((~![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))) | ((~sorti2(tptp_fun_V_1(h(U!0)))) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(tptp_fun_V_1(h(U!0)), Z)) = op1(j(tptp_fun_V_1(h(U!0))), j(Z)))))) <=> ((~![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))) | (~sorti2(tptp_fun_V_1(h(U!0)))) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(tptp_fun_V_1(h(U!0)), Z)) = op1(j(tptp_fun_V_1(h(U!0))), j(Z)))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(60,plain,
% 0.12/0.38 ((~![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))) | ((~sorti2(tptp_fun_V_1(h(U!0)))) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(tptp_fun_V_1(h(U!0)), Z)) = op1(j(tptp_fun_V_1(h(U!0))), j(Z)))))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(61,plain,
% 0.12/0.38 ((~![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))) | (~sorti2(tptp_fun_V_1(h(U!0)))) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(tptp_fun_V_1(h(U!0)), Z)) = op1(j(tptp_fun_V_1(h(U!0))), j(Z))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[60, 59])).
% 0.12/0.38 tff(62,plain,
% 0.12/0.38 (![Z: $i] : ((~sorti2(Z)) | (j(op2(tptp_fun_V_1(h(U!0)), Z)) = op1(j(tptp_fun_V_1(h(U!0))), j(Z))))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[61, 58, 47])).
% 0.12/0.38 tff(63,plain,
% 0.12/0.38 (((~![Z: $i] : ((~sorti2(Z)) | (j(op2(tptp_fun_V_1(h(U!0)), Z)) = op1(j(tptp_fun_V_1(h(U!0))), j(Z))))) | ((~sorti2(h(U!0))) | (j(op2(tptp_fun_V_1(h(U!0)), h(U!0))) = op1(j(tptp_fun_V_1(h(U!0))), j(h(U!0)))))) <=> ((~![Z: $i] : ((~sorti2(Z)) | (j(op2(tptp_fun_V_1(h(U!0)), Z)) = op1(j(tptp_fun_V_1(h(U!0))), j(Z))))) | (~sorti2(h(U!0))) | (j(op2(tptp_fun_V_1(h(U!0)), h(U!0))) = op1(j(tptp_fun_V_1(h(U!0))), j(h(U!0)))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(64,plain,
% 0.12/0.38 ((~![Z: $i] : ((~sorti2(Z)) | (j(op2(tptp_fun_V_1(h(U!0)), Z)) = op1(j(tptp_fun_V_1(h(U!0))), j(Z))))) | ((~sorti2(h(U!0))) | (j(op2(tptp_fun_V_1(h(U!0)), h(U!0))) = op1(j(tptp_fun_V_1(h(U!0))), j(h(U!0)))))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(65,plain,
% 0.12/0.38 ((~![Z: $i] : ((~sorti2(Z)) | (j(op2(tptp_fun_V_1(h(U!0)), Z)) = op1(j(tptp_fun_V_1(h(U!0))), j(Z))))) | (~sorti2(h(U!0))) | (j(op2(tptp_fun_V_1(h(U!0)), h(U!0))) = op1(j(tptp_fun_V_1(h(U!0))), j(h(U!0))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.12/0.38 tff(66,plain,
% 0.12/0.38 (j(op2(tptp_fun_V_1(h(U!0)), h(U!0))) = op1(j(tptp_fun_V_1(h(U!0))), j(h(U!0)))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[65, 24, 62])).
% 0.12/0.38 tff(67,plain,
% 0.12/0.38 (op1(j(tptp_fun_V_1(h(U!0))), j(h(U!0))) = j(op2(tptp_fun_V_1(h(U!0)), h(U!0)))),
% 0.12/0.38 inference(symmetry,[status(thm)],[66])).
% 0.12/0.38 tff(68,plain,
% 0.12/0.38 (^[X2: $i] : refl(((~sorti1(X2)) | (j(h(X2)) = X2)) <=> ((~sorti1(X2)) | (j(h(X2)) = X2)))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(69,plain,
% 0.12/0.38 (![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2)) <=> ![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[68])).
% 0.12/0.38 tff(70,plain,
% 0.12/0.38 (![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2)) <=> ![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(71,plain,
% 0.12/0.38 (![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))),
% 0.12/0.38 inference(and_elim,[status(thm)],[54])).
% 0.12/0.38 tff(72,plain,
% 0.12/0.38 (![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.12/0.38 tff(73,plain,(
% 0.12/0.38 ![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))),
% 0.12/0.38 inference(skolemize,[status(sab)],[72])).
% 0.12/0.38 tff(74,plain,
% 0.12/0.38 (![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[73, 69])).
% 0.12/0.38 tff(75,plain,
% 0.12/0.38 (((~![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))) | ((~sorti1(U!0)) | (j(h(U!0)) = U!0))) <=> ((~![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))) | (~sorti1(U!0)) | (j(h(U!0)) = U!0))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(76,plain,
% 0.12/0.38 ((~![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))) | ((~sorti1(U!0)) | (j(h(U!0)) = U!0))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(77,plain,
% 0.12/0.38 ((~![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))) | (~sorti1(U!0)) | (j(h(U!0)) = U!0)),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.12/0.38 tff(78,plain,
% 0.12/0.38 (j(h(U!0)) = U!0),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[77, 20, 74])).
% 0.12/0.38 tff(79,plain,
% 0.12/0.38 (U!0 = j(h(U!0))),
% 0.12/0.38 inference(symmetry,[status(thm)],[78])).
% 0.12/0.38 tff(80,plain,
% 0.12/0.38 (op1(j(tptp_fun_V_1(h(U!0))), U!0) = op1(j(tptp_fun_V_1(h(U!0))), j(h(U!0)))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[79])).
% 0.12/0.38 tff(81,plain,
% 0.12/0.38 (op1(j(tptp_fun_V_1(h(U!0))), U!0) = j(tptp_fun_V_1(h(U!0)))),
% 0.12/0.38 inference(transitivity,[status(thm)],[80, 67, 45])).
% 0.12/0.38 tff(82,plain,
% 0.12/0.38 (^[V: $i] : refl(((~sorti2(V)) | sorti1(j(V))) <=> ((~sorti2(V)) | sorti1(j(V))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(83,plain,
% 0.12/0.38 (![V: $i] : ((~sorti2(V)) | sorti1(j(V))) <=> ![V: $i] : ((~sorti2(V)) | sorti1(j(V)))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[82])).
% 0.12/0.38 tff(84,plain,
% 0.12/0.38 (![V: $i] : ((~sorti2(V)) | sorti1(j(V))) <=> ![V: $i] : ((~sorti2(V)) | sorti1(j(V)))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(85,plain,
% 0.12/0.38 (![V: $i] : ((~sorti2(V)) | sorti1(j(V)))),
% 0.12/0.38 inference(and_elim,[status(thm)],[7])).
% 0.12/0.38 tff(86,plain,
% 0.12/0.38 (![V: $i] : ((~sorti2(V)) | sorti1(j(V)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.12/0.38 tff(87,plain,(
% 0.12/0.38 ![V: $i] : ((~sorti2(V)) | sorti1(j(V)))),
% 0.12/0.38 inference(skolemize,[status(sab)],[86])).
% 0.12/0.38 tff(88,plain,
% 0.12/0.38 (![V: $i] : ((~sorti2(V)) | sorti1(j(V)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[87, 83])).
% 0.12/0.38 tff(89,plain,
% 0.12/0.38 (((~![V: $i] : ((~sorti2(V)) | sorti1(j(V)))) | ((~sorti2(tptp_fun_V_1(h(U!0)))) | sorti1(j(tptp_fun_V_1(h(U!0)))))) <=> ((~![V: $i] : ((~sorti2(V)) | sorti1(j(V)))) | (~sorti2(tptp_fun_V_1(h(U!0)))) | sorti1(j(tptp_fun_V_1(h(U!0)))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(90,plain,
% 0.12/0.38 ((~![V: $i] : ((~sorti2(V)) | sorti1(j(V)))) | ((~sorti2(tptp_fun_V_1(h(U!0)))) | sorti1(j(tptp_fun_V_1(h(U!0)))))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(91,plain,
% 0.12/0.39 ((~![V: $i] : ((~sorti2(V)) | sorti1(j(V)))) | (~sorti2(tptp_fun_V_1(h(U!0)))) | sorti1(j(tptp_fun_V_1(h(U!0))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[90, 89])).
% 0.12/0.39 tff(92,plain,
% 0.12/0.39 (sorti1(j(tptp_fun_V_1(h(U!0))))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[91, 88, 47])).
% 0.12/0.39 tff(93,plain,
% 0.12/0.39 (^[V: $i] : refl(((~sorti1(V)) | (~(op1(V, U!0) = V))) <=> ((~sorti1(V)) | (~(op1(V, U!0) = V))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(94,plain,
% 0.12/0.39 (![V: $i] : ((~sorti1(V)) | (~(op1(V, U!0) = V))) <=> ![V: $i] : ((~sorti1(V)) | (~(op1(V, U!0) = V)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[93])).
% 0.12/0.39 tff(95,plain,
% 0.12/0.39 (![V: $i] : ((~sorti1(V)) | (~(op1(V, U!0) = V)))),
% 0.12/0.39 inference(and_elim,[status(thm)],[19])).
% 0.12/0.39 tff(96,plain,
% 0.12/0.39 (![V: $i] : ((~sorti1(V)) | (~(op1(V, U!0) = V)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[95, 94])).
% 0.12/0.39 tff(97,plain,
% 0.12/0.39 (((~![V: $i] : ((~sorti1(V)) | (~(op1(V, U!0) = V)))) | ((~sorti1(j(tptp_fun_V_1(h(U!0))))) | (~(op1(j(tptp_fun_V_1(h(U!0))), U!0) = j(tptp_fun_V_1(h(U!0))))))) <=> ((~![V: $i] : ((~sorti1(V)) | (~(op1(V, U!0) = V)))) | (~sorti1(j(tptp_fun_V_1(h(U!0))))) | (~(op1(j(tptp_fun_V_1(h(U!0))), U!0) = j(tptp_fun_V_1(h(U!0))))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(98,plain,
% 0.12/0.39 ((~![V: $i] : ((~sorti1(V)) | (~(op1(V, U!0) = V)))) | ((~sorti1(j(tptp_fun_V_1(h(U!0))))) | (~(op1(j(tptp_fun_V_1(h(U!0))), U!0) = j(tptp_fun_V_1(h(U!0))))))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(99,plain,
% 0.12/0.39 ((~![V: $i] : ((~sorti1(V)) | (~(op1(V, U!0) = V)))) | (~sorti1(j(tptp_fun_V_1(h(U!0))))) | (~(op1(j(tptp_fun_V_1(h(U!0))), U!0) = j(tptp_fun_V_1(h(U!0)))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[98, 97])).
% 0.12/0.39 tff(100,plain,
% 0.12/0.39 (~(op1(j(tptp_fun_V_1(h(U!0))), U!0) = j(tptp_fun_V_1(h(U!0))))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[99, 96, 92])).
% 0.12/0.39 tff(101,plain,
% 0.12/0.39 ($false),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[100, 81])).
% 0.12/0.39 % SZS output end Proof
% 0.12/0.39 % E exiting
%------------------------------------------------------------------------------