%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : ALG073+1 : TPTP v9.0.0. Released v2.7.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n008.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 0.12s 0.37s
% Output : Proof 0.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : ALG073+1 : TPTP v9.0.0. Released v2.7.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n008.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 10:47:33 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.12/0.37 % SZS status Theorem
% 0.12/0.37 % SZS output start Proof
% 0.12/0.37 tff(tptp_fun_U_0_type, type, (
% 0.12/0.37 tptp_fun_U_0: $i)).
% 0.12/0.37 tff(op1_type, type, (
% 0.12/0.37 op1: ( $i * $i ) > $i)).
% 0.12/0.37 tff(j_type, type, (
% 0.12/0.37 j: $i > $i)).
% 0.12/0.37 tff(h_type, type, (
% 0.12/0.37 h: $i > $i)).
% 0.12/0.37 tff(sorti1_type, type, (
% 0.12/0.37 sorti1: $i > $o)).
% 0.12/0.37 tff(sorti2_type, type, (
% 0.12/0.37 sorti2: $i > $o)).
% 0.12/0.37 tff(op2_type, type, (
% 0.12/0.37 op2: ( $i * $i ) > $i)).
% 0.12/0.37 tff(1,plain,
% 0.12/0.37 (^[X2: $i] : refl(((~sorti1(X2)) | (j(h(X2)) = X2)) <=> ((~sorti1(X2)) | (j(h(X2)) = X2)))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(2,plain,
% 0.12/0.37 (![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2)) <=> ![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.37 tff(3,plain,
% 0.12/0.37 (![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2)) <=> ![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(4,plain,
% 0.12/0.37 ((~((![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))))))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 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/sandbox/benchmark/theBenchmark.p','co1')).
% 0.12/0.37 tff(6,plain,
% 0.12/0.37 (~((~(![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)))))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.12/0.37 tff(7,plain,
% 0.12/0.37 (![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))),
% 0.12/0.37 inference(or_elim,[status(thm)],[6])).
% 0.12/0.37 tff(8,plain,
% 0.12/0.37 (![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))),
% 0.12/0.37 inference(and_elim,[status(thm)],[7])).
% 0.12/0.37 tff(9,plain,
% 0.12/0.37 (![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[8, 3])).
% 0.12/0.37 tff(10,plain,(
% 0.12/0.37 ![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))),
% 0.12/0.37 inference(skolemize,[status(sab)],[9])).
% 0.12/0.37 tff(11,plain,
% 0.12/0.37 (![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[10, 2])).
% 0.12/0.37 tff(12,plain,
% 0.12/0.37 (?[U: $i] : (sorti1(U) & (~(op1(U, op1(U, U)) = U)) & sorti1(op1(U, U)) & (op1(op1(U, U), op1(U, U)) = U)) <=> ?[U: $i] : (sorti1(U) & (~(op1(U, op1(U, U)) = U)) & sorti1(op1(U, U)) & (op1(op1(U, U), op1(U, U)) = U))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(13,plain,
% 0.12/0.37 (^[U: $i] : trans(monotonicity(trans(quant_intro(proof_bind(^[V: $i] : trans(monotonicity(rewrite(((sorti1(V) & (op1(U, U) = V)) & (op1(V, V) = U)) <=> (sorti1(V) & (op1(U, U) = V) & (op1(V, V) = U))), ((((sorti1(V) & (op1(U, U) = V)) & (op1(V, V) = U)) & (~(op1(U, V) = U))) <=> ((sorti1(V) & (op1(U, U) = V) & (op1(V, V) = U)) & (~(op1(U, V) = U))))), rewrite(((sorti1(V) & (op1(U, U) = V) & (op1(V, V) = U)) & (~(op1(U, V) = U))) <=> (sorti1(V) & (op1(U, U) = V) & (op1(V, V) = U) & (~(op1(U, V) = U)))), ((((sorti1(V) & (op1(U, U) = V)) & (op1(V, V) = U)) & (~(op1(U, V) = U))) <=> (sorti1(V) & (op1(U, U) = V) & (op1(V, V) = U) & (~(op1(U, V) = U)))))), (?[V: $i] : (((sorti1(V) & (op1(U, U) = V)) & (op1(V, V) = U)) & (~(op1(U, V) = U))) <=> ?[V: $i] : (sorti1(V) & (op1(U, U) = V) & (op1(V, V) = U) & (~(op1(U, V) = U))))), trans(der(?[V: $i] : (sorti1(V) & (op1(U, U) = V) & (op1(V, V) = U) & (~(op1(U, V) = U))) <=> ?[V: $i] : ((~(op1(U, op1(U, U)) = U)) & sorti1(op1(U, U)) & (op1(op1(U, U), op1(U, U)) = U))), elim_unused(?[V: $i] : ((~(op1(U, op1(U, U)) = U)) & sorti1(op1(U, U)) & (op1(op1(U, U), op1(U, U)) = U)) <=> ((~(op1(U, op1(U, U)) = U)) & sorti1(op1(U, U)) & (op1(op1(U, U), op1(U, U)) = U))), (?[V: $i] : (sorti1(V) & (op1(U, U) = V) & (op1(V, V) = U) & (~(op1(U, V) = U))) <=> ((~(op1(U, op1(U, U)) = U)) & sorti1(op1(U, U)) & (op1(op1(U, U), op1(U, U)) = U)))), (?[V: $i] : (((sorti1(V) & (op1(U, U) = V)) & (op1(V, V) = U)) & (~(op1(U, V) = U))) <=> ((~(op1(U, op1(U, U)) = U)) & sorti1(op1(U, U)) & (op1(op1(U, U), op1(U, U)) = U)))), ((sorti1(U) & ?[V: $i] : (((sorti1(V) & (op1(U, U) = V)) & (op1(V, V) = U)) & (~(op1(U, V) = U)))) <=> (sorti1(U) & ((~(op1(U, op1(U, U)) = U)) & sorti1(op1(U, U)) & (op1(op1(U, U), op1(U, U)) = U))))), rewrite((sorti1(U) & ((~(op1(U, op1(U, U)) = U)) & sorti1(op1(U, U)) & (op1(op1(U, U), op1(U, U)) = U))) <=> (sorti1(U) & (~(op1(U, op1(U, U)) = U)) & sorti1(op1(U, U)) & (op1(op1(U, U), op1(U, U)) = U))), ((sorti1(U) & ?[V: $i] : (((sorti1(V) & (op1(U, U) = V)) & (op1(V, V) = U)) & (~(op1(U, V) = U)))) <=> (sorti1(U) & (~(op1(U, op1(U, U)) = U)) & sorti1(op1(U, U)) & (op1(op1(U, U), op1(U, U)) = U))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(14,plain,
% 0.12/0.37 (?[U: $i] : (sorti1(U) & ?[V: $i] : (((sorti1(V) & (op1(U, U) = V)) & (op1(V, V) = U)) & (~(op1(U, V) = U)))) <=> ?[U: $i] : (sorti1(U) & (~(op1(U, op1(U, U)) = U)) & sorti1(op1(U, U)) & (op1(op1(U, U), op1(U, U)) = U))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[13])).
% 0.12/0.37 tff(15,axiom,(?[U: $i] : (sorti1(U) & ?[V: $i] : (((sorti1(V) & (op1(U, U) = V)) & (op1(V, V) = U)) & (~(op1(U, V) = U))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','ax3')).
% 0.12/0.37 tff(16,plain,
% 0.12/0.37 (?[U: $i] : (sorti1(U) & (~(op1(U, op1(U, U)) = U)) & sorti1(op1(U, U)) & (op1(op1(U, U), op1(U, U)) = U))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.12/0.37 tff(17,plain,
% 0.12/0.37 (?[U: $i] : (sorti1(U) & (~(op1(U, op1(U, U)) = U)) & sorti1(op1(U, U)) & (op1(op1(U, U), op1(U, U)) = U))),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.12/0.37 tff(18,plain,(
% 0.12/0.37 sorti1(U!0) & (~(op1(U!0, op1(U!0, U!0)) = U!0)) & sorti1(op1(U!0, U!0)) & (op1(op1(U!0, U!0), op1(U!0, U!0)) = U!0)),
% 0.12/0.37 inference(skolemize,[status(sab)],[17])).
% 0.12/0.37 tff(19,plain,
% 0.12/0.37 (sorti1(U!0)),
% 0.12/0.37 inference(and_elim,[status(thm)],[18])).
% 0.12/0.37 tff(20,plain,
% 0.12/0.37 (((~![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))),
% 0.12/0.37 inference(rewrite,[status(thm)],[])).
% 0.12/0.37 tff(21,plain,
% 0.12/0.37 ((~![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))) | ((~sorti1(U!0)) | (j(h(U!0)) = U!0))),
% 0.12/0.37 inference(quant_inst,[status(thm)],[])).
% 0.12/0.37 tff(22,plain,
% 0.12/0.37 ((~![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))) | (~sorti1(U!0)) | (j(h(U!0)) = U!0)),
% 0.12/0.37 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.12/0.37 tff(23,plain,
% 0.12/0.37 (j(h(U!0)) = U!0),
% 0.12/0.37 inference(unit_resolution,[status(thm)],[22, 19, 11])).
% 0.12/0.37 tff(24,plain,
% 0.12/0.37 (^[W: $i] : refl(((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X))))) <=> ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X))))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(25,plain,
% 0.12/0.37 (![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X))))) <=> ![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[24])).
% 0.12/0.37 tff(26,plain,
% 0.12/0.37 (^[W: $i] : rewrite(((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X))))) <=> ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(27,plain,
% 0.12/0.38 (![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X))))) <=> ![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[26])).
% 0.12/0.38 tff(28,plain,
% 0.12/0.38 (![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X))))) <=> ![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[27, 25])).
% 0.12/0.38 tff(29,plain,
% 0.12/0.38 (![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X))))) <=> ![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(30,plain,
% 0.12/0.38 (![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))),
% 0.12/0.38 inference(and_elim,[status(thm)],[7])).
% 0.12/0.38 tff(31,plain,
% 0.12/0.38 (![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[30, 29])).
% 0.12/0.38 tff(32,plain,(
% 0.12/0.38 ![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))),
% 0.12/0.38 inference(skolemize,[status(sab)],[31])).
% 0.12/0.38 tff(33,plain,
% 0.12/0.38 (![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[32, 28])).
% 0.12/0.38 tff(34,plain,
% 0.12/0.38 (((~![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))) | ((~sorti1(U!0)) | ![X: $i] : ((~sorti1(X)) | (h(op1(U!0, X)) = op2(h(U!0), h(X)))))) <=> ((~![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))) | (~sorti1(U!0)) | ![X: $i] : ((~sorti1(X)) | (h(op1(U!0, X)) = op2(h(U!0), h(X)))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(35,plain,
% 0.12/0.38 ((~![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))) | ((~sorti1(U!0)) | ![X: $i] : ((~sorti1(X)) | (h(op1(U!0, X)) = op2(h(U!0), h(X)))))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(36,plain,
% 0.12/0.38 ((~![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))) | (~sorti1(U!0)) | ![X: $i] : ((~sorti1(X)) | (h(op1(U!0, X)) = op2(h(U!0), h(X))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.12/0.38 tff(37,plain,
% 0.12/0.38 (![X: $i] : ((~sorti1(X)) | (h(op1(U!0, X)) = op2(h(U!0), h(X))))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[36, 19, 33])).
% 0.12/0.38 tff(38,plain,
% 0.12/0.38 (sorti1(op1(U!0, U!0))),
% 0.12/0.38 inference(and_elim,[status(thm)],[18])).
% 0.12/0.38 tff(39,plain,
% 0.12/0.38 (((~![X: $i] : ((~sorti1(X)) | (h(op1(U!0, X)) = op2(h(U!0), h(X))))) | ((~sorti1(op1(U!0, U!0))) | (h(op1(U!0, op1(U!0, U!0))) = op2(h(U!0), h(op1(U!0, U!0)))))) <=> ((~![X: $i] : ((~sorti1(X)) | (h(op1(U!0, X)) = op2(h(U!0), h(X))))) | (~sorti1(op1(U!0, U!0))) | (h(op1(U!0, op1(U!0, U!0))) = op2(h(U!0), h(op1(U!0, U!0)))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(40,plain,
% 0.12/0.38 ((~![X: $i] : ((~sorti1(X)) | (h(op1(U!0, X)) = op2(h(U!0), h(X))))) | ((~sorti1(op1(U!0, U!0))) | (h(op1(U!0, op1(U!0, U!0))) = op2(h(U!0), h(op1(U!0, U!0)))))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(41,plain,
% 0.12/0.38 ((~![X: $i] : ((~sorti1(X)) | (h(op1(U!0, X)) = op2(h(U!0), h(X))))) | (~sorti1(op1(U!0, U!0))) | (h(op1(U!0, op1(U!0, U!0))) = op2(h(U!0), h(op1(U!0, U!0))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.12/0.38 tff(42,plain,
% 0.12/0.38 (h(op1(U!0, op1(U!0, U!0))) = op2(h(U!0), h(op1(U!0, U!0)))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[41, 38, 37])).
% 0.12/0.38 tff(43,plain,
% 0.12/0.38 (op2(h(U!0), h(op1(U!0, U!0))) = h(op1(U!0, op1(U!0, U!0)))),
% 0.12/0.38 inference(symmetry,[status(thm)],[42])).
% 0.12/0.38 tff(44,plain,
% 0.12/0.38 (op1(op1(U!0, U!0), op1(U!0, U!0)) = U!0),
% 0.12/0.38 inference(and_elim,[status(thm)],[18])).
% 0.12/0.38 tff(45,plain,
% 0.12/0.38 (U!0 = op1(op1(U!0, U!0), op1(U!0, U!0))),
% 0.12/0.38 inference(symmetry,[status(thm)],[44])).
% 0.12/0.38 tff(46,plain,
% 0.12/0.38 (h(U!0) = h(op1(op1(U!0, U!0), op1(U!0, U!0)))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[45])).
% 0.12/0.38 tff(47,plain,
% 0.12/0.38 (h(op1(op1(U!0, U!0), op1(U!0, U!0))) = h(U!0)),
% 0.12/0.38 inference(symmetry,[status(thm)],[46])).
% 0.12/0.38 tff(48,plain,
% 0.12/0.38 (((~![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))) | ((~sorti1(op1(U!0, U!0))) | ![X: $i] : ((~sorti1(X)) | (h(op1(op1(U!0, U!0), X)) = op2(h(op1(U!0, U!0)), h(X)))))) <=> ((~![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))) | (~sorti1(op1(U!0, U!0))) | ![X: $i] : ((~sorti1(X)) | (h(op1(op1(U!0, U!0), X)) = op2(h(op1(U!0, U!0)), h(X)))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(49,plain,
% 0.12/0.38 ((~![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))) | ((~sorti1(op1(U!0, U!0))) | ![X: $i] : ((~sorti1(X)) | (h(op1(op1(U!0, U!0), X)) = op2(h(op1(U!0, U!0)), h(X)))))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(50,plain,
% 0.12/0.38 ((~![W: $i] : ((~sorti1(W)) | ![X: $i] : ((~sorti1(X)) | (h(op1(W, X)) = op2(h(W), h(X)))))) | (~sorti1(op1(U!0, U!0))) | ![X: $i] : ((~sorti1(X)) | (h(op1(op1(U!0, U!0), X)) = op2(h(op1(U!0, U!0)), h(X))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[49, 48])).
% 0.12/0.38 tff(51,plain,
% 0.12/0.38 (![X: $i] : ((~sorti1(X)) | (h(op1(op1(U!0, U!0), X)) = op2(h(op1(U!0, U!0)), h(X))))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[50, 38, 33])).
% 0.12/0.38 tff(52,plain,
% 0.12/0.38 (((~![X: $i] : ((~sorti1(X)) | (h(op1(op1(U!0, U!0), X)) = op2(h(op1(U!0, U!0)), h(X))))) | ((~sorti1(op1(U!0, U!0))) | (h(op1(op1(U!0, U!0), op1(U!0, U!0))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))))) <=> ((~![X: $i] : ((~sorti1(X)) | (h(op1(op1(U!0, U!0), X)) = op2(h(op1(U!0, U!0)), h(X))))) | (~sorti1(op1(U!0, U!0))) | (h(op1(op1(U!0, U!0), op1(U!0, U!0))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(53,plain,
% 0.12/0.38 ((~![X: $i] : ((~sorti1(X)) | (h(op1(op1(U!0, U!0), X)) = op2(h(op1(U!0, U!0)), h(X))))) | ((~sorti1(op1(U!0, U!0))) | (h(op1(op1(U!0, U!0), op1(U!0, U!0))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(54,plain,
% 0.12/0.38 ((~![X: $i] : ((~sorti1(X)) | (h(op1(op1(U!0, U!0), X)) = op2(h(op1(U!0, U!0)), h(X))))) | (~sorti1(op1(U!0, U!0))) | (h(op1(op1(U!0, U!0), op1(U!0, U!0))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[53, 52])).
% 0.12/0.38 tff(55,plain,
% 0.12/0.38 (h(op1(op1(U!0, U!0), op1(U!0, U!0))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[54, 38, 51])).
% 0.12/0.38 tff(56,plain,
% 0.12/0.38 (op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))) = h(op1(op1(U!0, U!0), op1(U!0, U!0)))),
% 0.12/0.38 inference(symmetry,[status(thm)],[55])).
% 0.12/0.38 tff(57,plain,
% 0.12/0.38 (op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))) = h(U!0)),
% 0.12/0.38 inference(transitivity,[status(thm)],[56, 47])).
% 0.12/0.38 tff(58,plain,
% 0.12/0.38 (op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))) = op2(h(U!0), h(U!0))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[57, 57])).
% 0.12/0.38 tff(59,plain,
% 0.12/0.38 (op2(h(U!0), h(U!0)) = op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))),
% 0.12/0.38 inference(symmetry,[status(thm)],[58])).
% 0.12/0.38 tff(60,plain,
% 0.12/0.38 (((~![X: $i] : ((~sorti1(X)) | (h(op1(U!0, X)) = op2(h(U!0), h(X))))) | ((~sorti1(U!0)) | (h(op1(U!0, U!0)) = op2(h(U!0), h(U!0))))) <=> ((~![X: $i] : ((~sorti1(X)) | (h(op1(U!0, X)) = op2(h(U!0), h(X))))) | (~sorti1(U!0)) | (h(op1(U!0, U!0)) = op2(h(U!0), h(U!0))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(61,plain,
% 0.12/0.38 ((~![X: $i] : ((~sorti1(X)) | (h(op1(U!0, X)) = op2(h(U!0), h(X))))) | ((~sorti1(U!0)) | (h(op1(U!0, U!0)) = op2(h(U!0), h(U!0))))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(62,plain,
% 0.12/0.38 ((~![X: $i] : ((~sorti1(X)) | (h(op1(U!0, X)) = op2(h(U!0), h(X))))) | (~sorti1(U!0)) | (h(op1(U!0, U!0)) = op2(h(U!0), h(U!0)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[61, 60])).
% 0.12/0.38 tff(63,plain,
% 0.12/0.38 (h(op1(U!0, U!0)) = op2(h(U!0), h(U!0))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[62, 19, 37])).
% 0.12/0.38 tff(64,plain,
% 0.12/0.38 (h(op1(U!0, U!0)) = op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[63, 59])).
% 0.12/0.38 tff(65,plain,
% 0.12/0.38 (h(U!0) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))),
% 0.12/0.38 inference(transitivity,[status(thm)],[46, 55])).
% 0.12/0.38 tff(66,plain,
% 0.12/0.38 (op2(h(U!0), h(op1(U!0, U!0))) = op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[65, 64])).
% 0.12/0.38 tff(67,plain,
% 0.12/0.38 (op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) = op2(h(U!0), h(op1(U!0, U!0)))),
% 0.12/0.38 inference(symmetry,[status(thm)],[66])).
% 0.12/0.38 tff(68,plain,
% 0.12/0.38 (op2(h(U!0), h(U!0)) = h(op1(U!0, U!0))),
% 0.12/0.38 inference(symmetry,[status(thm)],[63])).
% 0.12/0.38 tff(69,plain,
% 0.12/0.38 (op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))) = h(op1(U!0, U!0))),
% 0.12/0.38 inference(transitivity,[status(thm)],[58, 68])).
% 0.12/0.38 tff(70,plain,
% 0.12/0.38 (op2(op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))), op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[69, 69])).
% 0.12/0.38 tff(71,plain,
% 0.12/0.38 (sorti2(op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) <=> sorti2(h(op1(U!0, U!0)))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[69])).
% 0.12/0.38 tff(72,plain,
% 0.12/0.38 (sorti2(h(op1(U!0, U!0))) <=> sorti2(op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))))),
% 0.12/0.38 inference(symmetry,[status(thm)],[71])).
% 0.12/0.38 tff(73,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(74,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)],[73])).
% 0.12/0.38 tff(75,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(76,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(77,plain,
% 0.12/0.38 (![U: $i] : ((~sorti1(U)) | sorti2(h(U)))),
% 0.12/0.38 inference(and_elim,[status(thm)],[76])).
% 0.12/0.38 tff(78,plain,
% 0.12/0.38 (![U: $i] : ((~sorti1(U)) | sorti2(h(U)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[77, 75])).
% 0.12/0.38 tff(79,plain,(
% 0.12/0.38 ![U: $i] : ((~sorti1(U)) | sorti2(h(U)))),
% 0.12/0.38 inference(skolemize,[status(sab)],[78])).
% 0.12/0.38 tff(80,plain,
% 0.12/0.38 (![U: $i] : ((~sorti1(U)) | sorti2(h(U)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[79, 74])).
% 0.12/0.38 tff(81,plain,
% 0.12/0.38 (((~![U: $i] : ((~sorti1(U)) | sorti2(h(U)))) | ((~sorti1(op1(U!0, U!0))) | sorti2(h(op1(U!0, U!0))))) <=> ((~![U: $i] : ((~sorti1(U)) | sorti2(h(U)))) | (~sorti1(op1(U!0, U!0))) | sorti2(h(op1(U!0, U!0))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(82,plain,
% 0.12/0.38 ((~![U: $i] : ((~sorti1(U)) | sorti2(h(U)))) | ((~sorti1(op1(U!0, U!0))) | sorti2(h(op1(U!0, U!0))))),
% 0.12/0.38 inference(quant_inst,[status(thm)],[])).
% 0.12/0.38 tff(83,plain,
% 0.12/0.38 ((~![U: $i] : ((~sorti1(U)) | sorti2(h(U)))) | (~sorti1(op1(U!0, U!0))) | sorti2(h(op1(U!0, U!0)))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[82, 81])).
% 0.12/0.38 tff(84,plain,
% 0.12/0.38 (sorti2(h(op1(U!0, U!0)))),
% 0.12/0.38 inference(unit_resolution,[status(thm)],[83, 38, 80])).
% 0.12/0.38 tff(85,plain,
% 0.12/0.38 (sorti2(op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[84, 72])).
% 0.12/0.38 tff(86,plain,
% 0.12/0.38 (sorti2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))) <=> sorti2(h(U!0))),
% 0.12/0.38 inference(monotonicity,[status(thm)],[57])).
% 0.12/0.38 tff(87,plain,
% 0.12/0.38 (sorti2(h(U!0)) <=> sorti2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))),
% 0.12/0.38 inference(symmetry,[status(thm)],[86])).
% 0.12/0.38 tff(88,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(89,plain,
% 0.12/0.38 ((~![U: $i] : ((~sorti1(U)) | sorti2(h(U)))) | ((~sorti1(U!0)) | sorti2(h(U!0)))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(90,plain,
% 0.12/0.39 ((~![U: $i] : ((~sorti1(U)) | sorti2(h(U)))) | (~sorti1(U!0)) | sorti2(h(U!0))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[89, 88])).
% 0.12/0.39 tff(91,plain,
% 0.12/0.39 (sorti2(h(U!0))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[90, 19, 80])).
% 0.12/0.39 tff(92,plain,
% 0.12/0.39 (sorti2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[91, 87])).
% 0.12/0.39 tff(93,plain,
% 0.12/0.39 (^[U: $i] : refl(((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U))) <=> ((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(94,plain,
% 0.12/0.39 (![U: $i] : ((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U))) <=> ![U: $i] : ((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[93])).
% 0.12/0.39 tff(95,plain,
% 0.12/0.39 (^[U: $i] : trans(monotonicity(rewrite((sorti2(U) & (~(op2(U, op2(U, U)) = U)) & sorti2(op2(U, U)) & (op2(op2(U, U), op2(U, U)) = U)) <=> (~((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U))))), ((~(sorti2(U) & (~(op2(U, op2(U, U)) = U)) & sorti2(op2(U, U)) & (op2(op2(U, U), op2(U, U)) = U))) <=> (~(~((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U))))))), rewrite((~(~((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U))))) <=> ((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U)))), ((~(sorti2(U) & (~(op2(U, op2(U, U)) = U)) & sorti2(op2(U, U)) & (op2(op2(U, U), op2(U, U)) = U))) <=> ((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U)))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(96,plain,
% 0.12/0.39 (![U: $i] : (~(sorti2(U) & (~(op2(U, op2(U, U)) = U)) & sorti2(op2(U, U)) & (op2(op2(U, U), op2(U, U)) = U))) <=> ![U: $i] : ((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[95])).
% 0.12/0.39 tff(97,plain,
% 0.12/0.39 ((~?[U: $i] : (sorti2(U) & (~(op2(U, op2(U, U)) = U)) & sorti2(op2(U, U)) & (op2(op2(U, U), op2(U, U)) = U))) <=> (~?[U: $i] : (sorti2(U) & (~(op2(U, op2(U, U)) = U)) & sorti2(op2(U, U)) & (op2(op2(U, U), op2(U, U)) = U)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(98,plain,
% 0.12/0.39 ((~?[U: $i] : (sorti2(U) & ?[V: $i] : (((sorti2(V) & (op2(U, U) = V)) & (op2(V, V) = U)) & (~(op2(U, V) = U))))) <=> (~?[U: $i] : (sorti2(U) & (~(op2(U, op2(U, U)) = U)) & sorti2(op2(U, U)) & (op2(op2(U, U), op2(U, U)) = U)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(99,axiom,(~?[U: $i] : (sorti2(U) & ?[V: $i] : (((sorti2(V) & (op2(U, U) = V)) & (op2(V, V) = U)) & (~(op2(U, V) = U))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','ax4')).
% 0.12/0.39 tff(100,plain,
% 0.12/0.39 (~?[U: $i] : (sorti2(U) & (~(op2(U, op2(U, U)) = U)) & sorti2(op2(U, U)) & (op2(op2(U, U), op2(U, U)) = U))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[99, 98])).
% 0.12/0.39 tff(101,plain,
% 0.12/0.39 (~?[U: $i] : (sorti2(U) & (~(op2(U, op2(U, U)) = U)) & sorti2(op2(U, U)) & (op2(op2(U, U), op2(U, U)) = U))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[100, 97])).
% 0.12/0.39 tff(102,plain,
% 0.12/0.39 (~?[U: $i] : (sorti2(U) & (~(op2(U, op2(U, U)) = U)) & sorti2(op2(U, U)) & (op2(op2(U, U), op2(U, U)) = U))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[101, 97])).
% 0.12/0.39 tff(103,plain,
% 0.12/0.39 (~?[U: $i] : (sorti2(U) & (~(op2(U, op2(U, U)) = U)) & sorti2(op2(U, U)) & (op2(op2(U, U), op2(U, U)) = U))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[102, 97])).
% 0.12/0.39 tff(104,plain,
% 0.12/0.39 (^[U: $i] : refl($oeq((~(sorti2(U) & (~(op2(U, op2(U, U)) = U)) & sorti2(op2(U, U)) & (op2(op2(U, U), op2(U, U)) = U))), (~(sorti2(U) & (~(op2(U, op2(U, U)) = U)) & sorti2(op2(U, U)) & (op2(op2(U, U), op2(U, U)) = U)))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(105,plain,(
% 0.12/0.39 ![U: $i] : (~(sorti2(U) & (~(op2(U, op2(U, U)) = U)) & sorti2(op2(U, U)) & (op2(op2(U, U), op2(U, U)) = U)))),
% 0.12/0.39 inference(nnf-neg,[status(sab)],[103, 104])).
% 0.12/0.39 tff(106,plain,
% 0.12/0.39 (![U: $i] : ((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[105, 96])).
% 0.12/0.39 tff(107,plain,
% 0.12/0.39 (![U: $i] : ((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[106, 94])).
% 0.12/0.39 tff(108,plain,
% 0.12/0.39 (((~![U: $i] : ((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U)))) | ((~sorti2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) | (op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))) | (~sorti2(op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))))) | (~(op2(op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))), op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))))) <=> ((~![U: $i] : ((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U)))) | (~sorti2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) | (op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))) | (~sorti2(op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))))) | (~(op2(op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))), op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(109,plain,
% 0.12/0.39 ((~![U: $i] : ((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U)))) | ((~sorti2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) | (op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))) | (~sorti2(op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))))) | (~(op2(op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))), op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(110,plain,
% 0.12/0.39 ((~![U: $i] : ((~sorti2(U)) | (op2(U, op2(U, U)) = U) | (~sorti2(op2(U, U))) | (~(op2(op2(U, U), op2(U, U)) = U)))) | (~sorti2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) | (op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))) | (~sorti2(op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))))) | (~(op2(op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))), op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[109, 108])).
% 0.12/0.39 tff(111,plain,
% 0.12/0.39 ((~sorti2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) | (op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))) | (~sorti2(op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))))) | (~(op2(op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))), op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[110, 107])).
% 0.12/0.39 tff(112,plain,
% 0.12/0.39 (op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))))) = op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[111, 92, 85, 70])).
% 0.12/0.39 tff(113,plain,
% 0.12/0.39 (op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))) = op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(op2(h(op1(U!0, U!0)), h(op1(U!0, U!0))), op2(h(op1(U!0, U!0)), h(op1(U!0, U!0)))))),
% 0.12/0.39 inference(symmetry,[status(thm)],[112])).
% 0.12/0.39 tff(114,plain,
% 0.12/0.39 (h(U!0) = h(op1(U!0, op1(U!0, U!0)))),
% 0.12/0.39 inference(transitivity,[status(thm)],[46, 55, 113, 67, 43])).
% 0.12/0.39 tff(115,plain,
% 0.12/0.39 (j(h(U!0)) = j(h(op1(U!0, op1(U!0, U!0))))),
% 0.12/0.39 inference(monotonicity,[status(thm)],[114])).
% 0.12/0.39 tff(116,plain,
% 0.12/0.39 (j(h(op1(U!0, op1(U!0, U!0)))) = j(h(U!0))),
% 0.12/0.39 inference(symmetry,[status(thm)],[115])).
% 0.12/0.39 tff(117,plain,
% 0.12/0.39 (^[U: $i] : refl(((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V)))) <=> ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V)))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(118,plain,
% 0.12/0.39 (![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V)))) <=> ![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[117])).
% 0.12/0.39 tff(119,plain,
% 0.12/0.39 (^[U: $i] : rewrite(((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V)))) <=> ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V)))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(120,plain,
% 0.12/0.39 (![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V)))) <=> ![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[119])).
% 0.12/0.39 tff(121,plain,
% 0.12/0.39 (![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V)))) <=> ![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[120, 118])).
% 0.12/0.39 tff(122,plain,
% 0.12/0.39 (![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V)))) <=> ![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(123,plain,
% 0.12/0.39 (^[U: $i] : trans(monotonicity(quant_intro(proof_bind(^[V: $i] : rewrite((sorti1(V) => sorti1(op1(U, V))) <=> ((~sorti1(V)) | sorti1(op1(U, V))))), (![V: $i] : (sorti1(V) => sorti1(op1(U, V))) <=> ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))), ((sorti1(U) => ![V: $i] : (sorti1(V) => sorti1(op1(U, V)))) <=> (sorti1(U) => ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V)))))), rewrite((sorti1(U) => ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V)))) <=> ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))), ((sorti1(U) => ![V: $i] : (sorti1(V) => sorti1(op1(U, V)))) <=> ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(124,plain,
% 0.12/0.39 (![U: $i] : (sorti1(U) => ![V: $i] : (sorti1(V) => sorti1(op1(U, V)))) <=> ![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[123])).
% 0.12/0.39 tff(125,axiom,(![U: $i] : (sorti1(U) => ![V: $i] : (sorti1(V) => sorti1(op1(U, V))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','ax1')).
% 0.12/0.39 tff(126,plain,
% 0.12/0.39 (![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[125, 124])).
% 0.12/0.39 tff(127,plain,
% 0.12/0.39 (![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[126, 122])).
% 0.12/0.39 tff(128,plain,(
% 0.12/0.39 ![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))),
% 0.12/0.39 inference(skolemize,[status(sab)],[127])).
% 0.12/0.39 tff(129,plain,
% 0.12/0.39 (![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[128, 121])).
% 0.12/0.39 tff(130,plain,
% 0.12/0.39 (((~![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))) | ((~sorti1(U!0)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U!0, V))))) <=> ((~![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))) | (~sorti1(U!0)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U!0, V))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(131,plain,
% 0.12/0.40 ((~![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))) | ((~sorti1(U!0)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U!0, V))))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(132,plain,
% 0.12/0.40 ((~![U: $i] : ((~sorti1(U)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U, V))))) | (~sorti1(U!0)) | ![V: $i] : ((~sorti1(V)) | sorti1(op1(U!0, V)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[131, 130])).
% 0.12/0.40 tff(133,plain,
% 0.12/0.40 (![V: $i] : ((~sorti1(V)) | sorti1(op1(U!0, V)))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[132, 129, 19])).
% 0.12/0.40 tff(134,plain,
% 0.12/0.40 (((~![V: $i] : ((~sorti1(V)) | sorti1(op1(U!0, V)))) | ((~sorti1(op1(U!0, U!0))) | sorti1(op1(U!0, op1(U!0, U!0))))) <=> ((~![V: $i] : ((~sorti1(V)) | sorti1(op1(U!0, V)))) | (~sorti1(op1(U!0, U!0))) | sorti1(op1(U!0, op1(U!0, U!0))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(135,plain,
% 0.12/0.40 ((~![V: $i] : ((~sorti1(V)) | sorti1(op1(U!0, V)))) | ((~sorti1(op1(U!0, U!0))) | sorti1(op1(U!0, op1(U!0, U!0))))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(136,plain,
% 0.12/0.40 ((~![V: $i] : ((~sorti1(V)) | sorti1(op1(U!0, V)))) | (~sorti1(op1(U!0, U!0))) | sorti1(op1(U!0, op1(U!0, U!0)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[135, 134])).
% 0.12/0.40 tff(137,plain,
% 0.12/0.40 (sorti1(op1(U!0, op1(U!0, U!0)))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[136, 38, 133])).
% 0.12/0.40 tff(138,plain,
% 0.12/0.40 (((~![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))) | ((~sorti1(op1(U!0, op1(U!0, U!0)))) | (j(h(op1(U!0, op1(U!0, U!0)))) = op1(U!0, op1(U!0, U!0))))) <=> ((~![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))) | (~sorti1(op1(U!0, op1(U!0, U!0)))) | (j(h(op1(U!0, op1(U!0, U!0)))) = op1(U!0, op1(U!0, U!0))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(139,plain,
% 0.12/0.40 ((~![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))) | ((~sorti1(op1(U!0, op1(U!0, U!0)))) | (j(h(op1(U!0, op1(U!0, U!0)))) = op1(U!0, op1(U!0, U!0))))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(140,plain,
% 0.12/0.40 ((~![X2: $i] : ((~sorti1(X2)) | (j(h(X2)) = X2))) | (~sorti1(op1(U!0, op1(U!0, U!0)))) | (j(h(op1(U!0, op1(U!0, U!0)))) = op1(U!0, op1(U!0, U!0)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[139, 138])).
% 0.12/0.40 tff(141,plain,
% 0.12/0.40 (j(h(op1(U!0, op1(U!0, U!0)))) = op1(U!0, op1(U!0, U!0))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[140, 11, 137])).
% 0.12/0.40 tff(142,plain,
% 0.12/0.40 (op1(U!0, op1(U!0, U!0)) = j(h(op1(U!0, op1(U!0, U!0))))),
% 0.12/0.40 inference(symmetry,[status(thm)],[141])).
% 0.12/0.40 tff(143,plain,
% 0.12/0.40 (op1(U!0, op1(U!0, U!0)) = U!0),
% 0.12/0.40 inference(transitivity,[status(thm)],[142, 116, 23])).
% 0.12/0.40 tff(144,plain,
% 0.12/0.40 (~(op1(U!0, op1(U!0, U!0)) = U!0)),
% 0.12/0.40 inference(and_elim,[status(thm)],[18])).
% 0.12/0.40 tff(145,plain,
% 0.12/0.40 ($false),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[144, 143])).
% 0.12/0.40 % SZS output end Proof
% 0.12/0.40 % E exiting
%------------------------------------------------------------------------------