%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : ALG070+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 04:58:05 AM UTC 2025
% Result : Theorem 0.19s 0.37s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : ALG070+1 : TPTP v9.0.0. Released v2.7.0.
% 0.07/0.12 % 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 10:51:33 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.19/0.37 % SZS status Theorem
% 0.19/0.37 % SZS output start Proof
% 0.19/0.37 tff(tptp_fun_U_0_type, type, (
% 0.19/0.37 tptp_fun_U_0: $i)).
% 0.19/0.37 tff(op2_type, type, (
% 0.19/0.37 op2: ( $i * $i ) > $i)).
% 0.19/0.37 tff(h_type, type, (
% 0.19/0.37 h: $i > $i)).
% 0.19/0.37 tff(j_type, type, (
% 0.19/0.37 j: $i > $i)).
% 0.19/0.37 tff(sorti2_type, type, (
% 0.19/0.37 sorti2: $i > $o)).
% 0.19/0.37 tff(sorti1_type, type, (
% 0.19/0.37 sorti1: $i > $o)).
% 0.19/0.37 tff(op1_type, type, (
% 0.19/0.37 op1: ( $i * $i ) > $i)).
% 0.19/0.37 tff(1,plain,
% 0.19/0.37 (^[X1: $i] : refl(((~sorti2(X1)) | (h(j(X1)) = X1)) <=> ((~sorti2(X1)) | (h(j(X1)) = X1)))),
% 0.19/0.37 inference(bind,[status(th)],[])).
% 0.19/0.37 tff(2,plain,
% 0.19/0.37 (![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1)) <=> ![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1))),
% 0.19/0.37 inference(quant_intro,[status(thm)],[1])).
% 0.19/0.37 tff(3,plain,
% 0.19/0.37 (![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1)) <=> ![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1))),
% 0.19/0.37 inference(rewrite,[status(thm)],[])).
% 0.19/0.37 tff(4,plain,
% 0.19/0.37 ((~((![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))))))),
% 0.19/0.37 inference(rewrite,[status(thm)],[])).
% 0.19/0.37 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/sandbox2/benchmark/theBenchmark.p','co1')).
% 0.19/0.37 tff(6,plain,
% 0.19/0.37 (~((~(![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)))))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.19/0.37 tff(7,plain,
% 0.19/0.37 (![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))),
% 0.19/0.37 inference(or_elim,[status(thm)],[6])).
% 0.19/0.37 tff(8,plain,
% 0.19/0.37 (![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1))),
% 0.19/0.37 inference(and_elim,[status(thm)],[7])).
% 0.19/0.37 tff(9,plain,
% 0.19/0.37 (![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[8, 3])).
% 0.19/0.37 tff(10,plain,(
% 0.19/0.37 ![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1))),
% 0.19/0.37 inference(skolemize,[status(sab)],[9])).
% 0.19/0.37 tff(11,plain,
% 0.19/0.37 (![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[10, 2])).
% 0.19/0.37 tff(12,plain,
% 0.19/0.37 ((~![U: $i] : ((~sorti2(U)) | (~sorti2(op2(U, U))) | (op2(U, op2(U, U)) = U))) <=> (~![U: $i] : ((~sorti2(U)) | (~sorti2(op2(U, U))) | (op2(U, op2(U, U)) = U)))),
% 0.19/0.37 inference(rewrite,[status(thm)],[])).
% 0.19/0.37 tff(13,plain,
% 0.19/0.37 ((~![U: $i] : (sorti2(U) => ![V: $i] : (sorti2(V) => ((~(op2(U, U) = V)) | (op2(U, V) = U))))) <=> (~![U: $i] : ((~sorti2(U)) | (~sorti2(op2(U, U))) | (op2(U, op2(U, U)) = U)))),
% 0.19/0.37 inference(rewrite,[status(thm)],[])).
% 0.19/0.37 tff(14,axiom,(~![U: $i] : (sorti2(U) => ![V: $i] : (sorti2(V) => ((~(op2(U, U) = V)) | (op2(U, V) = U))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','ax4')).
% 0.19/0.37 tff(15,plain,
% 0.19/0.37 (~![U: $i] : ((~sorti2(U)) | (~sorti2(op2(U, U))) | (op2(U, op2(U, U)) = U))),
% 0.19/0.37 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.19/0.38 tff(16,plain,
% 0.19/0.38 (~![U: $i] : ((~sorti2(U)) | (~sorti2(op2(U, U))) | (op2(U, op2(U, U)) = U))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[15, 12])).
% 0.19/0.38 tff(17,plain,
% 0.19/0.38 (~![U: $i] : ((~sorti2(U)) | (~sorti2(op2(U, U))) | (op2(U, op2(U, U)) = U))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.19/0.38 tff(18,plain,
% 0.19/0.38 (~![U: $i] : ((~sorti2(U)) | (~sorti2(op2(U, U))) | (op2(U, op2(U, U)) = U))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[17, 12])).
% 0.19/0.38 tff(19,plain,(
% 0.19/0.38 ~((~sorti2(U!0)) | (~sorti2(op2(U!0, U!0))) | (op2(U!0, op2(U!0, U!0)) = U!0))),
% 0.19/0.38 inference(skolemize,[status(sab)],[18])).
% 0.19/0.38 tff(20,plain,
% 0.19/0.38 (sorti2(U!0)),
% 0.19/0.38 inference(or_elim,[status(thm)],[19])).
% 0.19/0.38 tff(21,plain,
% 0.19/0.38 (((~![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1))) | ((~sorti2(U!0)) | (h(j(U!0)) = U!0))) <=> ((~![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1))) | (~sorti2(U!0)) | (h(j(U!0)) = U!0))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(22,plain,
% 0.19/0.38 ((~![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1))) | ((~sorti2(U!0)) | (h(j(U!0)) = U!0))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(23,plain,
% 0.19/0.38 ((~![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1))) | (~sorti2(U!0)) | (h(j(U!0)) = U!0)),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[22, 21])).
% 0.19/0.38 tff(24,plain,
% 0.19/0.38 (h(j(U!0)) = U!0),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[23, 20, 11])).
% 0.19/0.38 tff(25,plain,
% 0.19/0.38 (^[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))))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(26,plain,
% 0.19/0.38 (![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)))))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[25])).
% 0.19/0.38 tff(27,plain,
% 0.19/0.38 (^[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))))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(28,plain,
% 0.19/0.38 (![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)))))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[27])).
% 0.19/0.38 tff(29,plain,
% 0.19/0.38 (![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)))))),
% 0.19/0.38 inference(transitivity,[status(thm)],[28, 26])).
% 0.19/0.38 tff(30,plain,
% 0.19/0.38 (![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)))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(31,plain,
% 0.19/0.38 (![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))),
% 0.19/0.38 inference(and_elim,[status(thm)],[7])).
% 0.19/0.38 tff(32,plain,
% 0.19/0.38 (![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[31, 30])).
% 0.19/0.38 tff(33,plain,(
% 0.19/0.38 ![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))),
% 0.19/0.38 inference(skolemize,[status(sab)],[32])).
% 0.19/0.38 tff(34,plain,
% 0.19/0.38 (![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[33, 29])).
% 0.19/0.38 tff(35,plain,
% 0.19/0.38 (((~![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))) | ((~sorti2(U!0)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(U!0, Z)) = op1(j(U!0), j(Z)))))) <=> ((~![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))) | (~sorti2(U!0)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(U!0, Z)) = op1(j(U!0), j(Z)))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(36,plain,
% 0.19/0.38 ((~![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))) | ((~sorti2(U!0)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(U!0, Z)) = op1(j(U!0), j(Z)))))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(37,plain,
% 0.19/0.38 ((~![Y: $i] : ((~sorti2(Y)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(Y, Z)) = op1(j(Y), j(Z)))))) | (~sorti2(U!0)) | ![Z: $i] : ((~sorti2(Z)) | (j(op2(U!0, Z)) = op1(j(U!0), j(Z))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.19/0.38 tff(38,plain,
% 0.19/0.38 (![Z: $i] : ((~sorti2(Z)) | (j(op2(U!0, Z)) = op1(j(U!0), j(Z))))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[37, 20, 34])).
% 0.19/0.38 tff(39,plain,
% 0.19/0.38 (sorti2(op2(U!0, U!0))),
% 0.19/0.38 inference(or_elim,[status(thm)],[19])).
% 0.19/0.38 tff(40,plain,
% 0.19/0.38 (((~![Z: $i] : ((~sorti2(Z)) | (j(op2(U!0, Z)) = op1(j(U!0), j(Z))))) | ((~sorti2(op2(U!0, U!0))) | (j(op2(U!0, op2(U!0, U!0))) = op1(j(U!0), j(op2(U!0, U!0)))))) <=> ((~![Z: $i] : ((~sorti2(Z)) | (j(op2(U!0, Z)) = op1(j(U!0), j(Z))))) | (~sorti2(op2(U!0, U!0))) | (j(op2(U!0, op2(U!0, U!0))) = op1(j(U!0), j(op2(U!0, U!0)))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(41,plain,
% 0.19/0.38 ((~![Z: $i] : ((~sorti2(Z)) | (j(op2(U!0, Z)) = op1(j(U!0), j(Z))))) | ((~sorti2(op2(U!0, U!0))) | (j(op2(U!0, op2(U!0, U!0))) = op1(j(U!0), j(op2(U!0, U!0)))))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(42,plain,
% 0.19/0.38 ((~![Z: $i] : ((~sorti2(Z)) | (j(op2(U!0, Z)) = op1(j(U!0), j(Z))))) | (~sorti2(op2(U!0, U!0))) | (j(op2(U!0, op2(U!0, U!0))) = op1(j(U!0), j(op2(U!0, U!0))))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[41, 40])).
% 0.19/0.38 tff(43,plain,
% 0.19/0.38 (j(op2(U!0, op2(U!0, U!0))) = op1(j(U!0), j(op2(U!0, U!0)))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[42, 39, 38])).
% 0.19/0.38 tff(44,plain,
% 0.19/0.38 (op1(j(U!0), j(op2(U!0, U!0))) = j(op2(U!0, op2(U!0, U!0)))),
% 0.19/0.38 inference(symmetry,[status(thm)],[43])).
% 0.19/0.38 tff(45,plain,
% 0.19/0.38 (((~![Z: $i] : ((~sorti2(Z)) | (j(op2(U!0, Z)) = op1(j(U!0), j(Z))))) | ((~sorti2(U!0)) | (j(op2(U!0, U!0)) = op1(j(U!0), j(U!0))))) <=> ((~![Z: $i] : ((~sorti2(Z)) | (j(op2(U!0, Z)) = op1(j(U!0), j(Z))))) | (~sorti2(U!0)) | (j(op2(U!0, U!0)) = op1(j(U!0), j(U!0))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(46,plain,
% 0.19/0.38 ((~![Z: $i] : ((~sorti2(Z)) | (j(op2(U!0, Z)) = op1(j(U!0), j(Z))))) | ((~sorti2(U!0)) | (j(op2(U!0, U!0)) = op1(j(U!0), j(U!0))))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(47,plain,
% 0.19/0.38 ((~![Z: $i] : ((~sorti2(Z)) | (j(op2(U!0, Z)) = op1(j(U!0), j(Z))))) | (~sorti2(U!0)) | (j(op2(U!0, U!0)) = op1(j(U!0), j(U!0)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.19/0.38 tff(48,plain,
% 0.19/0.38 (j(op2(U!0, U!0)) = op1(j(U!0), j(U!0))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[47, 20, 38])).
% 0.19/0.38 tff(49,plain,
% 0.19/0.38 (op1(j(U!0), j(U!0)) = j(op2(U!0, U!0))),
% 0.19/0.38 inference(symmetry,[status(thm)],[48])).
% 0.19/0.38 tff(50,plain,
% 0.19/0.38 (op1(j(U!0), op1(j(U!0), j(U!0))) = op1(j(U!0), j(op2(U!0, U!0)))),
% 0.19/0.38 inference(monotonicity,[status(thm)],[49])).
% 0.19/0.38 tff(51,plain,
% 0.19/0.38 (sorti1(op1(j(U!0), j(U!0))) <=> sorti1(j(op2(U!0, U!0)))),
% 0.19/0.38 inference(monotonicity,[status(thm)],[49])).
% 0.19/0.38 tff(52,plain,
% 0.19/0.38 (sorti1(j(op2(U!0, U!0))) <=> sorti1(op1(j(U!0), j(U!0)))),
% 0.19/0.38 inference(symmetry,[status(thm)],[51])).
% 0.19/0.38 tff(53,plain,
% 0.19/0.38 (^[V: $i] : refl(((~sorti2(V)) | sorti1(j(V))) <=> ((~sorti2(V)) | sorti1(j(V))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(54,plain,
% 0.19/0.38 (![V: $i] : ((~sorti2(V)) | sorti1(j(V))) <=> ![V: $i] : ((~sorti2(V)) | sorti1(j(V)))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[53])).
% 0.19/0.38 tff(55,plain,
% 0.19/0.38 (![V: $i] : ((~sorti2(V)) | sorti1(j(V))) <=> ![V: $i] : ((~sorti2(V)) | sorti1(j(V)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(56,plain,
% 0.19/0.38 (![U: $i] : ((~sorti1(U)) | sorti2(h(U))) & ![V: $i] : ((~sorti2(V)) | sorti1(j(V)))),
% 0.19/0.38 inference(or_elim,[status(thm)],[6])).
% 0.19/0.38 tff(57,plain,
% 0.19/0.38 (![V: $i] : ((~sorti2(V)) | sorti1(j(V)))),
% 0.19/0.38 inference(and_elim,[status(thm)],[56])).
% 0.19/0.38 tff(58,plain,
% 0.19/0.38 (![V: $i] : ((~sorti2(V)) | sorti1(j(V)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[57, 55])).
% 0.19/0.38 tff(59,plain,(
% 0.19/0.38 ![V: $i] : ((~sorti2(V)) | sorti1(j(V)))),
% 0.19/0.38 inference(skolemize,[status(sab)],[58])).
% 0.19/0.38 tff(60,plain,
% 0.19/0.38 (![V: $i] : ((~sorti2(V)) | sorti1(j(V)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[59, 54])).
% 0.19/0.38 tff(61,plain,
% 0.19/0.38 (((~![V: $i] : ((~sorti2(V)) | sorti1(j(V)))) | ((~sorti2(op2(U!0, U!0))) | sorti1(j(op2(U!0, U!0))))) <=> ((~![V: $i] : ((~sorti2(V)) | sorti1(j(V)))) | (~sorti2(op2(U!0, U!0))) | sorti1(j(op2(U!0, U!0))))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(62,plain,
% 0.19/0.38 ((~![V: $i] : ((~sorti2(V)) | sorti1(j(V)))) | ((~sorti2(op2(U!0, U!0))) | sorti1(j(op2(U!0, U!0))))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(63,plain,
% 0.19/0.38 ((~![V: $i] : ((~sorti2(V)) | sorti1(j(V)))) | (~sorti2(op2(U!0, U!0))) | sorti1(j(op2(U!0, U!0)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[62, 61])).
% 0.19/0.38 tff(64,plain,
% 0.19/0.38 (sorti1(j(op2(U!0, U!0)))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[63, 39, 60])).
% 0.19/0.38 tff(65,plain,
% 0.19/0.38 (sorti1(op1(j(U!0), j(U!0)))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[64, 52])).
% 0.19/0.38 tff(66,plain,
% 0.19/0.38 (((~![V: $i] : ((~sorti2(V)) | sorti1(j(V)))) | ((~sorti2(U!0)) | sorti1(j(U!0)))) <=> ((~![V: $i] : ((~sorti2(V)) | sorti1(j(V)))) | (~sorti2(U!0)) | sorti1(j(U!0)))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(67,plain,
% 0.19/0.38 ((~![V: $i] : ((~sorti2(V)) | sorti1(j(V)))) | ((~sorti2(U!0)) | sorti1(j(U!0)))),
% 0.19/0.38 inference(quant_inst,[status(thm)],[])).
% 0.19/0.38 tff(68,plain,
% 0.19/0.38 ((~![V: $i] : ((~sorti2(V)) | sorti1(j(V)))) | (~sorti2(U!0)) | sorti1(j(U!0))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[67, 66])).
% 0.19/0.38 tff(69,plain,
% 0.19/0.38 (sorti1(j(U!0))),
% 0.19/0.38 inference(unit_resolution,[status(thm)],[68, 20, 60])).
% 0.19/0.38 tff(70,plain,
% 0.19/0.38 (^[U: $i] : refl(((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U)) <=> ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U)))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(71,plain,
% 0.19/0.38 (![U: $i] : ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U)) <=> ![U: $i] : ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[70])).
% 0.19/0.38 tff(72,plain,
% 0.19/0.38 (![U: $i] : ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U)) <=> ![U: $i] : ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))),
% 0.19/0.38 inference(rewrite,[status(thm)],[])).
% 0.19/0.38 tff(73,plain,
% 0.19/0.38 (^[U: $i] : trans(monotonicity(trans(quant_intro(proof_bind(^[V: $i] : rewrite((sorti1(V) => ((~(op1(U, U) = V)) | (op1(U, V) = U))) <=> ((op1(U, V) = U) | (~(op1(U, U) = V)) | (~sorti1(V))))), (![V: $i] : (sorti1(V) => ((~(op1(U, U) = V)) | (op1(U, V) = U))) <=> ![V: $i] : ((op1(U, V) = U) | (~(op1(U, U) = V)) | (~sorti1(V))))), trans(der(![V: $i] : ((op1(U, V) = U) | (~(op1(U, U) = V)) | (~sorti1(V))) <=> ![V: $i] : ((~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))), elim_unused(![V: $i] : ((~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U)) <=> ((~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))), (![V: $i] : ((op1(U, V) = U) | (~(op1(U, U) = V)) | (~sorti1(V))) <=> ((~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U)))), (![V: $i] : (sorti1(V) => ((~(op1(U, U) = V)) | (op1(U, V) = U))) <=> ((~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U)))), ((sorti1(U) => ![V: $i] : (sorti1(V) => ((~(op1(U, U) = V)) | (op1(U, V) = U)))) <=> (sorti1(U) => ((~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))))), rewrite((sorti1(U) => ((~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))) <=> ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))), ((sorti1(U) => ![V: $i] : (sorti1(V) => ((~(op1(U, U) = V)) | (op1(U, V) = U)))) <=> ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))))),
% 0.19/0.38 inference(bind,[status(th)],[])).
% 0.19/0.38 tff(74,plain,
% 0.19/0.38 (![U: $i] : (sorti1(U) => ![V: $i] : (sorti1(V) => ((~(op1(U, U) = V)) | (op1(U, V) = U)))) <=> ![U: $i] : ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))),
% 0.19/0.38 inference(quant_intro,[status(thm)],[73])).
% 0.19/0.38 tff(75,axiom,(![U: $i] : (sorti1(U) => ![V: $i] : (sorti1(V) => ((~(op1(U, U) = V)) | (op1(U, V) = U))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','ax3')).
% 0.19/0.38 tff(76,plain,
% 0.19/0.38 (![U: $i] : ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[75, 74])).
% 0.19/0.38 tff(77,plain,
% 0.19/0.38 (![U: $i] : ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))),
% 0.19/0.38 inference(modus_ponens,[status(thm)],[76, 72])).
% 0.19/0.38 tff(78,plain,(
% 0.19/0.38 ![U: $i] : ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))),
% 0.19/0.39 inference(skolemize,[status(sab)],[77])).
% 0.19/0.39 tff(79,plain,
% 0.19/0.39 (![U: $i] : ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[78, 71])).
% 0.19/0.39 tff(80,plain,
% 0.19/0.39 (((~![U: $i] : ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))) | ((~sorti1(j(U!0))) | (~sorti1(op1(j(U!0), j(U!0)))) | (op1(j(U!0), op1(j(U!0), j(U!0))) = j(U!0)))) <=> ((~![U: $i] : ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))) | (~sorti1(j(U!0))) | (~sorti1(op1(j(U!0), j(U!0)))) | (op1(j(U!0), op1(j(U!0), j(U!0))) = j(U!0)))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(81,plain,
% 0.19/0.39 ((~![U: $i] : ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))) | ((~sorti1(j(U!0))) | (~sorti1(op1(j(U!0), j(U!0)))) | (op1(j(U!0), op1(j(U!0), j(U!0))) = j(U!0)))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(82,plain,
% 0.19/0.39 ((~![U: $i] : ((~sorti1(U)) | (~sorti1(op1(U, U))) | (op1(U, op1(U, U)) = U))) | (~sorti1(j(U!0))) | (~sorti1(op1(j(U!0), j(U!0)))) | (op1(j(U!0), op1(j(U!0), j(U!0))) = j(U!0))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[81, 80])).
% 0.19/0.39 tff(83,plain,
% 0.19/0.39 ((~sorti1(op1(j(U!0), j(U!0)))) | (op1(j(U!0), op1(j(U!0), j(U!0))) = j(U!0))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[82, 79, 69])).
% 0.19/0.39 tff(84,plain,
% 0.19/0.39 (op1(j(U!0), op1(j(U!0), j(U!0))) = j(U!0)),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[83, 65])).
% 0.19/0.39 tff(85,plain,
% 0.19/0.39 (j(U!0) = op1(j(U!0), op1(j(U!0), j(U!0)))),
% 0.19/0.39 inference(symmetry,[status(thm)],[84])).
% 0.19/0.39 tff(86,plain,
% 0.19/0.39 (j(U!0) = j(op2(U!0, op2(U!0, U!0)))),
% 0.19/0.39 inference(transitivity,[status(thm)],[85, 50, 44])).
% 0.19/0.39 tff(87,plain,
% 0.19/0.39 (h(j(U!0)) = h(j(op2(U!0, op2(U!0, U!0))))),
% 0.19/0.39 inference(monotonicity,[status(thm)],[86])).
% 0.19/0.39 tff(88,plain,
% 0.19/0.39 (h(j(op2(U!0, op2(U!0, U!0)))) = h(j(U!0))),
% 0.19/0.39 inference(symmetry,[status(thm)],[87])).
% 0.19/0.39 tff(89,plain,
% 0.19/0.39 (^[U: $i] : refl(((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V)))) <=> ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(90,plain,
% 0.19/0.39 (![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V)))) <=> ![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[89])).
% 0.19/0.39 tff(91,plain,
% 0.19/0.39 (^[U: $i] : rewrite(((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V)))) <=> ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V)))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(92,plain,
% 0.19/0.39 (![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V)))) <=> ![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[91])).
% 0.19/0.39 tff(93,plain,
% 0.19/0.39 (![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V)))) <=> ![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))),
% 0.19/0.39 inference(transitivity,[status(thm)],[92, 90])).
% 0.19/0.39 tff(94,plain,
% 0.19/0.39 (![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V)))) <=> ![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(95,plain,
% 0.19/0.39 (^[U: $i] : trans(monotonicity(quant_intro(proof_bind(^[V: $i] : rewrite((sorti2(V) => sorti2(op2(U, V))) <=> ((~sorti2(V)) | sorti2(op2(U, V))))), (![V: $i] : (sorti2(V) => sorti2(op2(U, V))) <=> ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))), ((sorti2(U) => ![V: $i] : (sorti2(V) => sorti2(op2(U, V)))) <=> (sorti2(U) => ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V)))))), rewrite((sorti2(U) => ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V)))) <=> ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))), ((sorti2(U) => ![V: $i] : (sorti2(V) => sorti2(op2(U, V)))) <=> ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))))),
% 0.19/0.39 inference(bind,[status(th)],[])).
% 0.19/0.39 tff(96,plain,
% 0.19/0.39 (![U: $i] : (sorti2(U) => ![V: $i] : (sorti2(V) => sorti2(op2(U, V)))) <=> ![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))),
% 0.19/0.39 inference(quant_intro,[status(thm)],[95])).
% 0.19/0.39 tff(97,axiom,(![U: $i] : (sorti2(U) => ![V: $i] : (sorti2(V) => sorti2(op2(U, V))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','ax2')).
% 0.19/0.39 tff(98,plain,
% 0.19/0.39 (![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[97, 96])).
% 0.19/0.39 tff(99,plain,
% 0.19/0.39 (![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[98, 94])).
% 0.19/0.39 tff(100,plain,(
% 0.19/0.39 ![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))),
% 0.19/0.39 inference(skolemize,[status(sab)],[99])).
% 0.19/0.39 tff(101,plain,
% 0.19/0.39 (![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[100, 93])).
% 0.19/0.39 tff(102,plain,
% 0.19/0.39 (((~![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))) | ((~sorti2(U!0)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U!0, V))))) <=> ((~![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))) | (~sorti2(U!0)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U!0, V))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(103,plain,
% 0.19/0.39 ((~![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))) | ((~sorti2(U!0)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U!0, V))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(104,plain,
% 0.19/0.39 ((~![U: $i] : ((~sorti2(U)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U, V))))) | (~sorti2(U!0)) | ![V: $i] : ((~sorti2(V)) | sorti2(op2(U!0, V)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[103, 102])).
% 0.19/0.39 tff(105,plain,
% 0.19/0.39 (![V: $i] : ((~sorti2(V)) | sorti2(op2(U!0, V)))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[104, 101, 20])).
% 0.19/0.39 tff(106,plain,
% 0.19/0.39 (((~![V: $i] : ((~sorti2(V)) | sorti2(op2(U!0, V)))) | ((~sorti2(op2(U!0, U!0))) | sorti2(op2(U!0, op2(U!0, U!0))))) <=> ((~![V: $i] : ((~sorti2(V)) | sorti2(op2(U!0, V)))) | (~sorti2(op2(U!0, U!0))) | sorti2(op2(U!0, op2(U!0, U!0))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(107,plain,
% 0.19/0.39 ((~![V: $i] : ((~sorti2(V)) | sorti2(op2(U!0, V)))) | ((~sorti2(op2(U!0, U!0))) | sorti2(op2(U!0, op2(U!0, U!0))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(108,plain,
% 0.19/0.39 ((~![V: $i] : ((~sorti2(V)) | sorti2(op2(U!0, V)))) | (~sorti2(op2(U!0, U!0))) | sorti2(op2(U!0, op2(U!0, U!0)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[107, 106])).
% 0.19/0.39 tff(109,plain,
% 0.19/0.39 (sorti2(op2(U!0, op2(U!0, U!0)))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[108, 39, 105])).
% 0.19/0.39 tff(110,plain,
% 0.19/0.39 (((~![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1))) | ((~sorti2(op2(U!0, op2(U!0, U!0)))) | (h(j(op2(U!0, op2(U!0, U!0)))) = op2(U!0, op2(U!0, U!0))))) <=> ((~![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1))) | (~sorti2(op2(U!0, op2(U!0, U!0)))) | (h(j(op2(U!0, op2(U!0, U!0)))) = op2(U!0, op2(U!0, U!0))))),
% 0.19/0.39 inference(rewrite,[status(thm)],[])).
% 0.19/0.39 tff(111,plain,
% 0.19/0.39 ((~![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1))) | ((~sorti2(op2(U!0, op2(U!0, U!0)))) | (h(j(op2(U!0, op2(U!0, U!0)))) = op2(U!0, op2(U!0, U!0))))),
% 0.19/0.39 inference(quant_inst,[status(thm)],[])).
% 0.19/0.39 tff(112,plain,
% 0.19/0.39 ((~![X1: $i] : ((~sorti2(X1)) | (h(j(X1)) = X1))) | (~sorti2(op2(U!0, op2(U!0, U!0)))) | (h(j(op2(U!0, op2(U!0, U!0)))) = op2(U!0, op2(U!0, U!0)))),
% 0.19/0.39 inference(modus_ponens,[status(thm)],[111, 110])).
% 0.19/0.39 tff(113,plain,
% 0.19/0.39 (h(j(op2(U!0, op2(U!0, U!0)))) = op2(U!0, op2(U!0, U!0))),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[112, 11, 109])).
% 0.19/0.39 tff(114,plain,
% 0.19/0.39 (op2(U!0, op2(U!0, U!0)) = h(j(op2(U!0, op2(U!0, U!0))))),
% 0.19/0.39 inference(symmetry,[status(thm)],[113])).
% 0.19/0.39 tff(115,plain,
% 0.19/0.39 (op2(U!0, op2(U!0, U!0)) = U!0),
% 0.19/0.39 inference(transitivity,[status(thm)],[114, 88, 24])).
% 0.19/0.39 tff(116,plain,
% 0.19/0.39 (~(op2(U!0, op2(U!0, U!0)) = U!0)),
% 0.19/0.39 inference(or_elim,[status(thm)],[19])).
% 0.19/0.39 tff(117,plain,
% 0.19/0.39 ($false),
% 0.19/0.39 inference(unit_resolution,[status(thm)],[116, 115])).
% 0.19/0.39 % SZS output end Proof
% 0.19/0.39 % E exiting
%------------------------------------------------------------------------------