TSTP Solution File: LCL109-5 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : LCL109-5 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n006.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 : Sun Sep 18 04:54:58 EDT 2022
% Result : Unsatisfiable 35.68s 22.45s
% Output : Proof 35.87s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : LCL109-5 : TPTP v8.1.0. Released v1.0.0.
% 0.07/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n006.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Thu Sep 1 17:55:39 EDT 2022
% 0.13/0.35 % CPUTime :
% 0.13/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.35 Usage: tptp [options] [-file:]file
% 0.13/0.35 -h, -? prints this message.
% 0.13/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.35 -m, -model generate model.
% 0.13/0.35 -p, -proof generate proof.
% 0.13/0.35 -c, -core generate unsat core of named formulas.
% 0.13/0.35 -st, -statistics display statistics.
% 0.13/0.35 -t:timeout set timeout (in second).
% 0.13/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.35 -<param>:<value> configuration parameter and value.
% 0.13/0.35 -o:<output-file> file to place output in.
% 35.68/22.45 % SZS status Unsatisfiable
% 35.68/22.45 % SZS output start Proof
% 35.68/22.45 tff(truth_type, type, (
% 35.68/22.45 truth: $i)).
% 35.68/22.45 tff(implies_type, type, (
% 35.68/22.45 implies: ( $i * $i ) > $i)).
% 35.68/22.45 tff(not_type, type, (
% 35.68/22.45 not: $i > $i)).
% 35.68/22.45 tff(y_type, type, (
% 35.68/22.45 y: $i)).
% 35.68/22.45 tff(x_type, type, (
% 35.68/22.45 x: $i)).
% 35.68/22.45 tff(big_V_type, type, (
% 35.68/22.45 big_V: ( $i * $i ) > $i)).
% 35.68/22.45 tff(ordered_type, type, (
% 35.68/22.45 ordered: ( $i * $i ) > $o)).
% 35.68/22.45 tff(big_hat_type, type, (
% 35.68/22.45 big_hat: ( $i * $i ) > $i)).
% 35.68/22.45 tff(1,plain,
% 35.68/22.45 (^[Z: $i, Y: $i, X: $i] : refl((implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth) <=> (implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth))),
% 35.68/22.45 inference(bind,[status(th)],[])).
% 35.68/22.45 tff(2,plain,
% 35.68/22.45 (![Z: $i, Y: $i, X: $i] : (implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth) <=> ![Z: $i, Y: $i, X: $i] : (implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth)),
% 35.68/22.45 inference(quant_intro,[status(thm)],[1])).
% 35.68/22.45 tff(3,plain,
% 35.68/22.45 (![Z: $i, Y: $i, X: $i] : (implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth) <=> ![Z: $i, Y: $i, X: $i] : (implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth)),
% 35.68/22.45 inference(rewrite,[status(thm)],[])).
% 35.68/22.45 tff(4,axiom,(![Z: $i, Y: $i, X: $i] : (implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth)), file('/export/starexec/sandbox/benchmark/Axioms/LCL001-0.ax','wajsberg_2')).
% 35.68/22.45 tff(5,plain,
% 35.68/22.45 (![Z: $i, Y: $i, X: $i] : (implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth)),
% 35.68/22.45 inference(modus_ponens,[status(thm)],[4, 3])).
% 35.68/22.45 tff(6,plain,(
% 35.68/22.45 ![Z: $i, Y: $i, X: $i] : (implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth)),
% 35.68/22.45 inference(skolemize,[status(sab)],[5])).
% 35.68/22.45 tff(7,plain,
% 35.68/22.45 (![Z: $i, Y: $i, X: $i] : (implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth)),
% 35.68/22.45 inference(modus_ponens,[status(thm)],[6, 2])).
% 35.68/22.45 tff(8,plain,
% 35.68/22.45 ((~![Z: $i, Y: $i, X: $i] : (implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth)) | (implies(implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), not(not(implies(x, y)))), implies(implies(not(not(implies(x, y))), truth), implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), truth))) = truth)),
% 35.68/22.45 inference(quant_inst,[status(thm)],[])).
% 35.68/22.45 tff(9,plain,
% 35.68/22.45 (implies(implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), not(not(implies(x, y)))), implies(implies(not(not(implies(x, y))), truth), implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), truth))) = truth),
% 35.68/22.45 inference(unit_resolution,[status(thm)],[8, 7])).
% 35.68/22.45 tff(10,plain,
% 35.68/22.45 (^[Y: $i, X: $i] : refl((implies(implies(not(X), not(Y)), implies(Y, X)) = truth) <=> (implies(implies(not(X), not(Y)), implies(Y, X)) = truth))),
% 35.68/22.45 inference(bind,[status(th)],[])).
% 35.68/22.45 tff(11,plain,
% 35.68/22.45 (![Y: $i, X: $i] : (implies(implies(not(X), not(Y)), implies(Y, X)) = truth) <=> ![Y: $i, X: $i] : (implies(implies(not(X), not(Y)), implies(Y, X)) = truth)),
% 35.68/22.45 inference(quant_intro,[status(thm)],[10])).
% 35.68/22.45 tff(12,plain,
% 35.68/22.45 (![Y: $i, X: $i] : (implies(implies(not(X), not(Y)), implies(Y, X)) = truth) <=> ![Y: $i, X: $i] : (implies(implies(not(X), not(Y)), implies(Y, X)) = truth)),
% 35.68/22.45 inference(rewrite,[status(thm)],[])).
% 35.68/22.45 tff(13,axiom,(![Y: $i, X: $i] : (implies(implies(not(X), not(Y)), implies(Y, X)) = truth)), file('/export/starexec/sandbox/benchmark/Axioms/LCL001-0.ax','wajsberg_4')).
% 35.68/22.45 tff(14,plain,
% 35.68/22.45 (![Y: $i, X: $i] : (implies(implies(not(X), not(Y)), implies(Y, X)) = truth)),
% 35.68/22.45 inference(modus_ponens,[status(thm)],[13, 12])).
% 35.68/22.45 tff(15,plain,(
% 35.68/22.45 ![Y: $i, X: $i] : (implies(implies(not(X), not(Y)), implies(Y, X)) = truth)),
% 35.68/22.45 inference(skolemize,[status(sab)],[14])).
% 35.68/22.45 tff(16,plain,
% 35.68/22.45 (![Y: $i, X: $i] : (implies(implies(not(X), not(Y)), implies(Y, X)) = truth)),
% 35.68/22.45 inference(modus_ponens,[status(thm)],[15, 11])).
% 35.68/22.46 tff(17,plain,
% 35.68/22.46 ((~![Y: $i, X: $i] : (implies(implies(not(X), not(Y)), implies(Y, X)) = truth)) | (implies(implies(not(implies(y, x)), not(implies(x, y))), implies(implies(x, y), implies(y, x))) = truth)),
% 35.68/22.46 inference(quant_inst,[status(thm)],[])).
% 35.68/22.46 tff(18,plain,
% 35.68/22.46 (implies(implies(not(implies(y, x)), not(implies(x, y))), implies(implies(x, y), implies(y, x))) = truth),
% 35.68/22.46 inference(unit_resolution,[status(thm)],[17, 16])).
% 35.68/22.46 tff(19,plain,
% 35.68/22.46 (^[X: $i] : refl((implies(truth, X) = X) <=> (implies(truth, X) = X))),
% 35.68/22.46 inference(bind,[status(th)],[])).
% 35.68/22.46 tff(20,plain,
% 35.68/22.46 (![X: $i] : (implies(truth, X) = X) <=> ![X: $i] : (implies(truth, X) = X)),
% 35.68/22.46 inference(quant_intro,[status(thm)],[19])).
% 35.68/22.46 tff(21,plain,
% 35.68/22.46 (![X: $i] : (implies(truth, X) = X) <=> ![X: $i] : (implies(truth, X) = X)),
% 35.68/22.46 inference(rewrite,[status(thm)],[])).
% 35.68/22.46 tff(22,axiom,(![X: $i] : (implies(truth, X) = X)), file('/export/starexec/sandbox/benchmark/Axioms/LCL001-0.ax','wajsberg_1')).
% 35.68/22.46 tff(23,plain,
% 35.68/22.46 (![X: $i] : (implies(truth, X) = X)),
% 35.68/22.46 inference(modus_ponens,[status(thm)],[22, 21])).
% 35.68/22.46 tff(24,plain,(
% 35.68/22.46 ![X: $i] : (implies(truth, X) = X)),
% 35.68/22.46 inference(skolemize,[status(sab)],[23])).
% 35.68/22.46 tff(25,plain,
% 35.68/22.46 (![X: $i] : (implies(truth, X) = X)),
% 35.68/22.46 inference(modus_ponens,[status(thm)],[24, 20])).
% 35.68/22.46 tff(26,plain,
% 35.68/22.46 ((~![X: $i] : (implies(truth, X) = X)) | (implies(truth, implies(not(implies(y, x)), not(implies(x, y)))) = implies(not(implies(y, x)), not(implies(x, y))))),
% 35.68/22.46 inference(quant_inst,[status(thm)],[])).
% 35.68/22.46 tff(27,plain,
% 35.68/22.46 (implies(truth, implies(not(implies(y, x)), not(implies(x, y)))) = implies(not(implies(y, x)), not(implies(x, y)))),
% 35.68/22.46 inference(unit_resolution,[status(thm)],[26, 25])).
% 35.68/22.46 tff(28,assumption,(~(implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(not(not(implies(x, y))), not(not(implies(y, x)))))), introduced(assumption)).
% 35.68/22.46 tff(29,plain,
% 35.68/22.46 ((~![X: $i] : (implies(truth, X) = X)) | (implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.68/22.46 inference(quant_inst,[status(thm)],[])).
% 35.68/22.46 tff(30,plain,
% 35.68/22.46 ($false),
% 35.68/22.46 inference(unit_resolution,[status(thm)],[29, 25, 28])).
% 35.68/22.46 tff(31,plain,(implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(not(not(implies(x, y))), not(not(implies(y, x))))), inference(lemma,lemma(discharge,[]))).
% 35.68/22.46 tff(32,plain,
% 35.68/22.46 ((~![Z: $i, Y: $i, X: $i] : (implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth)) | (implies(implies(not(implies(y, x)), not(implies(x, y))), implies(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x))))) = truth)),
% 35.68/22.46 inference(quant_inst,[status(thm)],[])).
% 35.68/22.46 tff(33,plain,
% 35.68/22.46 (implies(implies(not(implies(y, x)), not(implies(x, y))), implies(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x))))) = truth),
% 35.68/22.46 inference(unit_resolution,[status(thm)],[32, 7])).
% 35.68/22.46 tff(34,assumption,(implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(not(not(implies(x, y))), not(not(implies(y, x))))), introduced(assumption)).
% 35.68/22.46 tff(35,assumption,(~ordered(implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))), implies(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))))), introduced(assumption)).
% 35.68/22.46 tff(36,plain,
% 35.68/22.46 ((~![Y: $i, X: $i] : (implies(implies(not(X), not(Y)), implies(Y, X)) = truth)) | (implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(implies(x, y)))))))), not(not(not(not(not(not(not(implies(y, x)))))))))) = truth)),
% 35.68/22.46 inference(quant_inst,[status(thm)],[])).
% 35.68/22.46 tff(37,plain,
% 35.68/22.46 (implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(implies(x, y)))))))), not(not(not(not(not(not(not(implies(y, x)))))))))) = truth),
% 35.68/22.46 inference(unit_resolution,[status(thm)],[36, 16])).
% 35.68/22.46 tff(38,plain,
% 35.68/22.46 (truth = implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(implies(x, y)))))))), not(not(not(not(not(not(not(implies(y, x))))))))))),
% 35.68/22.46 inference(symmetry,[status(thm)],[37])).
% 35.68/22.46 tff(39,plain,
% 35.68/22.46 (ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth) <=> ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(implies(x, y)))))))), not(not(not(not(not(not(not(implies(y, x)))))))))))),
% 35.68/22.46 inference(monotonicity,[status(thm)],[38])).
% 35.68/22.46 tff(40,plain,
% 35.68/22.46 (ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(implies(x, y)))))))), not(not(not(not(not(not(not(implies(y, x))))))))))) <=> ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth)),
% 35.68/22.46 inference(symmetry,[status(thm)],[39])).
% 35.68/22.46 tff(41,assumption,(~ordered(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(implies(x, y)))))))), not(not(not(not(not(not(not(implies(y, x))))))))))), introduced(assumption)).
% 35.68/22.46 tff(42,plain,
% 35.68/22.46 (^[Y: $i, X: $i] : refl(((~(implies(X, Y) = truth)) | ordered(X, Y)) <=> ((~(implies(X, Y) = truth)) | ordered(X, Y)))),
% 35.68/22.46 inference(bind,[status(th)],[])).
% 35.68/22.46 tff(43,plain,
% 35.68/22.46 (![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y)) <=> ![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))),
% 35.68/22.46 inference(quant_intro,[status(thm)],[42])).
% 35.68/22.46 tff(44,plain,
% 35.68/22.46 (![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y)) <=> ![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))),
% 35.68/22.46 inference(rewrite,[status(thm)],[])).
% 35.68/22.46 tff(45,axiom,(![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))), file('/export/starexec/sandbox/benchmark/Axioms/LCL001-1.ax','partial_order_definition2')).
% 35.68/22.46 tff(46,plain,
% 35.68/22.46 (![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))),
% 35.68/22.46 inference(modus_ponens,[status(thm)],[45, 44])).
% 35.68/22.46 tff(47,plain,(
% 35.68/22.46 ![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))),
% 35.68/22.46 inference(skolemize,[status(sab)],[46])).
% 35.68/22.46 tff(48,plain,
% 35.68/22.46 (![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))),
% 35.68/22.46 inference(modus_ponens,[status(thm)],[47, 43])).
% 35.68/22.46 tff(49,plain,
% 35.68/22.46 (((~![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))) | ((~(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(implies(x, y)))))))), not(not(not(not(not(not(not(implies(y, x)))))))))) = truth)) | ordered(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(implies(x, y)))))))), not(not(not(not(not(not(not(implies(y, x)))))))))))) <=> ((~![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))) | (~(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(implies(x, y)))))))), not(not(not(not(not(not(not(implies(y, x)))))))))) = truth)) | ordered(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(implies(x, y)))))))), not(not(not(not(not(not(not(implies(y, x)))))))))))),
% 35.68/22.46 inference(rewrite,[status(thm)],[])).
% 35.68/22.46 tff(50,plain,
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% 35.68/22.46 tff(85,assumption,(~(implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth) = truth)), introduced(assumption)).
% 35.68/22.46 tff(86,plain,
% 35.68/22.46 (^[Y: $i, X: $i] : refl(((~ordered(X, Y)) | (implies(X, Y) = truth)) <=> ((~ordered(X, Y)) | (implies(X, Y) = truth)))),
% 35.68/22.46 inference(bind,[status(th)],[])).
% 35.68/22.46 tff(87,plain,
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% 35.68/22.46 inference(quant_intro,[status(thm)],[86])).
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% 35.68/22.46 inference(rewrite,[status(thm)],[])).
% 35.68/22.46 tff(89,axiom,(![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))), file('/export/starexec/sandbox/benchmark/Axioms/LCL001-1.ax','partial_order_definition1')).
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% 35.68/22.46 inference(skolemize,[status(sab)],[90])).
% 35.68/22.46 tff(92,plain,
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% 35.68/22.46 inference(modus_ponens,[status(thm)],[91, 87])).
% 35.68/22.46 tff(93,plain,
% 35.68/22.46 (((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | ((~ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth)) | (implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth) = truth))) <=> ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | (~ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth)) | (implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth) = truth))),
% 35.68/22.47 inference(rewrite,[status(thm)],[])).
% 35.68/22.47 tff(94,plain,
% 35.68/22.47 ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | ((~ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth)) | (implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth) = truth))),
% 35.68/22.47 inference(quant_inst,[status(thm)],[])).
% 35.68/22.47 tff(95,plain,
% 35.68/22.47 ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | (~ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth)) | (implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth) = truth)),
% 35.68/22.47 inference(modus_ponens,[status(thm)],[94, 93])).
% 35.68/22.47 tff(96,plain,
% 35.68/22.47 ($false),
% 35.68/22.47 inference(unit_resolution,[status(thm)],[95, 92, 85, 68])).
% 35.68/22.47 tff(97,plain,(implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth) = truth), inference(lemma,lemma(discharge,[]))).
% 35.68/22.47 tff(98,plain,
% 35.68/22.47 ((~![Z: $i, Y: $i, X: $i] : (implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth)) | (implies(implies(implies(not(not(not(not(not(not(not(not(not(implies(x, y)))))))))), not(not(not(not(not(not(not(not(not(implies(y, x))))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(not(not(not(not(not(not(not(implies(x, y)))))))))), not(not(not(not(not(not(not(not(not(implies(y, x))))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))))) = truth)),
% 35.68/22.47 inference(quant_inst,[status(thm)],[])).
% 35.68/22.47 tff(99,plain,
% 35.68/22.47 (implies(implies(implies(not(not(not(not(not(not(not(not(not(implies(x, y)))))))))), not(not(not(not(not(not(not(not(not(implies(y, x))))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(not(not(not(not(not(not(not(implies(x, y)))))))))), not(not(not(not(not(not(not(not(not(implies(y, x))))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))))) = truth),
% 35.68/22.47 inference(unit_resolution,[status(thm)],[98, 7])).
% 35.68/22.47 tff(100,plain,
% 35.68/22.47 ((~![Y: $i, X: $i] : (implies(implies(not(X), not(Y)), implies(Y, X)) = truth)) | (implies(implies(not(not(not(not(not(not(not(not(not(implies(x, y)))))))))), not(not(not(not(not(not(not(not(not(implies(y, x))))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))) = truth)),
% 35.68/22.47 inference(quant_inst,[status(thm)],[])).
% 35.68/22.47 tff(101,plain,
% 35.68/22.47 (implies(implies(not(not(not(not(not(not(not(not(not(implies(x, y)))))))))), not(not(not(not(not(not(not(not(not(implies(y, x))))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))) = truth),
% 35.68/22.47 inference(unit_resolution,[status(thm)],[100, 16])).
% 35.68/22.47 tff(102,plain,
% 35.68/22.47 (implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(not(not(not(not(not(not(not(implies(x, y)))))))))), not(not(not(not(not(not(not(not(not(implies(y, x))))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))) = implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth)),
% 35.68/22.47 inference(monotonicity,[status(thm)],[101])).
% 35.68/22.47 tff(103,plain,
% 35.68/22.47 (implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth) = implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(not(not(not(not(not(not(not(implies(x, y)))))))))), not(not(not(not(not(not(not(not(not(implies(y, x))))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))))),
% 35.68/22.47 inference(symmetry,[status(thm)],[102])).
% 35.68/22.47 tff(104,assumption,(implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth) = truth), introduced(assumption)).
% 35.68/22.47 tff(105,plain,
% 35.68/22.47 (truth = implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth)),
% 35.68/22.47 inference(symmetry,[status(thm)],[104])).
% 35.68/22.47 tff(106,plain,
% 35.68/22.47 (truth = implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(not(not(not(not(not(not(not(implies(x, y)))))))))), not(not(not(not(not(not(not(not(not(implies(y, x))))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))))),
% 35.68/22.47 inference(transitivity,[status(thm)],[105, 103])).
% 35.68/22.47 tff(107,plain,
% 35.68/22.47 (truth = implies(implies(not(not(not(not(not(not(not(not(not(implies(x, y)))))))))), not(not(not(not(not(not(not(not(not(implies(y, x))))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))),
% 35.68/22.47 inference(symmetry,[status(thm)],[101])).
% 35.68/22.47 tff(108,plain,
% 35.68/22.47 (implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth) = implies(implies(not(not(not(not(not(not(not(not(not(implies(x, y)))))))))), not(not(not(not(not(not(not(not(not(implies(y, x))))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))),
% 35.68/22.47 inference(transitivity,[status(thm)],[104, 107])).
% 35.68/22.47 tff(109,plain,
% 35.68/22.47 (implies(implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth), truth) = implies(implies(implies(not(not(not(not(not(not(not(not(not(implies(x, y)))))))))), not(not(not(not(not(not(not(not(not(implies(y, x))))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(not(not(not(not(not(not(not(implies(x, y)))))))))), not(not(not(not(not(not(not(not(not(implies(y, x))))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))))),
% 35.68/22.47 inference(monotonicity,[status(thm)],[108, 106])).
% 35.68/22.47 tff(110,plain,
% 35.68/22.47 (^[Y: $i, X: $i] : refl((implies(implies(X, Y), Y) = implies(implies(Y, X), X)) <=> (implies(implies(X, Y), Y) = implies(implies(Y, X), X)))),
% 35.68/22.47 inference(bind,[status(th)],[])).
% 35.68/22.47 tff(111,plain,
% 35.68/22.47 (![Y: $i, X: $i] : (implies(implies(X, Y), Y) = implies(implies(Y, X), X)) <=> ![Y: $i, X: $i] : (implies(implies(X, Y), Y) = implies(implies(Y, X), X))),
% 35.68/22.47 inference(quant_intro,[status(thm)],[110])).
% 35.68/22.47 tff(112,plain,
% 35.68/22.47 (![Y: $i, X: $i] : (implies(implies(X, Y), Y) = implies(implies(Y, X), X)) <=> ![Y: $i, X: $i] : (implies(implies(X, Y), Y) = implies(implies(Y, X), X))),
% 35.68/22.47 inference(rewrite,[status(thm)],[])).
% 35.68/22.47 tff(113,axiom,(![Y: $i, X: $i] : (implies(implies(X, Y), Y) = implies(implies(Y, X), X))), file('/export/starexec/sandbox/benchmark/Axioms/LCL001-0.ax','wajsberg_3')).
% 35.68/22.47 tff(114,plain,
% 35.68/22.47 (![Y: $i, X: $i] : (implies(implies(X, Y), Y) = implies(implies(Y, X), X))),
% 35.68/22.47 inference(modus_ponens,[status(thm)],[113, 112])).
% 35.68/22.47 tff(115,plain,(
% 35.68/22.47 ![Y: $i, X: $i] : (implies(implies(X, Y), Y) = implies(implies(Y, X), X))),
% 35.68/22.47 inference(skolemize,[status(sab)],[114])).
% 35.68/22.47 tff(116,plain,
% 35.68/22.47 (![Y: $i, X: $i] : (implies(implies(X, Y), Y) = implies(implies(Y, X), X))),
% 35.68/22.47 inference(modus_ponens,[status(thm)],[115, 111])).
% 35.68/22.47 tff(117,plain,
% 35.68/22.47 ((~![Y: $i, X: $i] : (implies(implies(X, Y), Y) = implies(implies(Y, X), X))) | (implies(implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth), truth) = implies(implies(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))), implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))))),
% 35.68/22.47 inference(quant_inst,[status(thm)],[])).
% 35.68/22.47 tff(118,plain,
% 35.68/22.47 (implies(implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth), truth) = implies(implies(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))), implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))))),
% 35.68/22.47 inference(unit_resolution,[status(thm)],[117, 116])).
% 35.68/22.47 tff(119,plain,
% 35.68/22.47 (implies(implies(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))), implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))) = implies(implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth), truth)),
% 35.68/22.47 inference(symmetry,[status(thm)],[118])).
% 35.68/22.47 tff(120,assumption,(implies(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))) = implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))), introduced(assumption)).
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% 35.68/22.47 inference(symmetry,[status(thm)],[120])).
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% 35.68/22.47 tff(123,plain,
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% 35.68/22.47 inference(transitivity,[status(thm)],[122, 119, 109, 99])).
% 35.68/22.47 tff(124,assumption,(~(implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))) = truth)), introduced(assumption)).
% 35.68/22.47 tff(125,plain,
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% 35.68/22.47 inference(unit_resolution,[status(thm)],[124, 123])).
% 35.68/22.47 tff(126,plain,((~(implies(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))) = implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))))) | (implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))) = truth) | (~(implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth) = truth))), inference(lemma,lemma(discharge,[]))).
% 35.68/22.47 tff(127,plain,
% 35.68/22.47 ((implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))) = truth) | (~(implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth) = truth))),
% 35.68/22.47 inference(unit_resolution,[status(thm)],[126, 84])).
% 35.68/22.47 tff(128,plain,
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% 35.68/22.47 inference(unit_resolution,[status(thm)],[127, 97])).
% 35.68/22.47 tff(129,plain,
% 35.68/22.47 (ordered(implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))), implies(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))))) <=> ordered(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))))),
% 35.68/22.47 inference(monotonicity,[status(thm)],[128, 84])).
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% 35.68/22.47 inference(symmetry,[status(thm)],[129])).
% 35.68/22.47 tff(131,plain,
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% 35.68/22.47 inference(monotonicity,[status(thm)],[130])).
% 35.68/22.47 tff(132,assumption,(~ordered(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))))), introduced(assumption)).
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% 35.68/22.47 inference(modus_ponens,[status(thm)],[132, 131])).
% 35.68/22.47 tff(134,assumption,(ordered(implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))), implies(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))))), introduced(assumption)).
% 35.68/22.48 tff(135,plain,
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% 35.68/22.48 inference(unit_resolution,[status(thm)],[134, 133])).
% 35.68/22.48 tff(136,plain,((~ordered(implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))), implies(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))))) | ordered(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))))), inference(lemma,lemma(discharge,[]))).
% 35.68/22.48 tff(137,plain,
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% 35.68/22.48 inference(unit_resolution,[status(thm)],[136, 80])).
% 35.68/22.48 tff(138,plain,
% 35.68/22.48 (((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | ((~ordered(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))))) | (implies(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))) = truth))) <=> ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | (~ordered(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))))) | (implies(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))) = truth))),
% 35.68/22.48 inference(rewrite,[status(thm)],[])).
% 35.68/22.48 tff(139,plain,
% 35.68/22.48 ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | ((~ordered(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))))) | (implies(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))) = truth))),
% 35.68/22.48 inference(quant_inst,[status(thm)],[])).
% 35.68/22.48 tff(140,plain,
% 35.68/22.48 ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | (~ordered(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))))) | (implies(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))) = truth)),
% 35.68/22.48 inference(modus_ponens,[status(thm)],[139, 138])).
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% 35.68/22.48 inference(unit_resolution,[status(thm)],[140, 92, 137])).
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% 35.68/22.48 inference(symmetry,[status(thm)],[84])).
% 35.68/22.48 tff(143,plain,
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% 35.68/22.48 inference(transitivity,[status(thm)],[142, 141])).
% 35.68/22.48 tff(144,plain,
% 35.68/22.48 (implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.68/22.48 inference(monotonicity,[status(thm)],[143])).
% 35.68/22.48 tff(145,plain,
% 35.68/22.48 (truth = implies(truth, implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))))),
% 35.68/22.48 inference(symmetry,[status(thm)],[141])).
% 35.68/22.48 tff(146,plain,
% 35.68/22.48 ((~![X: $i] : (implies(truth, X) = X)) | (implies(truth, implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))) = implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y))))))),
% 35.68/22.48 inference(quant_inst,[status(thm)],[])).
% 35.68/22.48 tff(147,plain,
% 35.68/22.48 (implies(truth, implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))) = implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))),
% 35.68/22.48 inference(unit_resolution,[status(thm)],[146, 25])).
% 35.68/22.48 tff(148,plain,
% 35.68/22.48 (implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))), implies(truth, implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y))))))) = implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y))))))),
% 35.68/22.48 inference(monotonicity,[status(thm)],[147])).
% 35.68/22.48 tff(149,plain,
% 35.68/22.48 (implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))) = implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))), implies(truth, implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))))),
% 35.68/22.48 inference(symmetry,[status(thm)],[148])).
% 35.68/22.48 tff(150,plain,
% 35.68/22.48 ((~![Y: $i, X: $i] : (implies(implies(X, Y), Y) = implies(implies(Y, X), X))) | (implies(implies(implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))))),
% 35.68/22.48 inference(quant_inst,[status(thm)],[])).
% 35.68/22.48 tff(151,plain,
% 35.68/22.48 (implies(implies(implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y))))))),
% 35.68/22.48 inference(unit_resolution,[status(thm)],[150, 116])).
% 35.68/22.48 tff(152,plain,
% 35.68/22.48 ((~![Y: $i, X: $i] : (implies(implies(not(X), not(Y)), implies(Y, X)) = truth)) | (implies(implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth)),
% 35.68/22.48 inference(quant_inst,[status(thm)],[])).
% 35.68/22.48 tff(153,plain,
% 35.68/22.48 (implies(implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth),
% 35.68/22.48 inference(unit_resolution,[status(thm)],[152, 16])).
% 35.68/22.48 tff(154,plain,
% 35.68/22.48 (implies(implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))),
% 35.68/22.48 inference(transitivity,[status(thm)],[153, 145, 84])).
% 35.68/22.48 tff(155,plain,
% 35.68/22.48 (implies(implies(implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.68/22.48 inference(monotonicity,[status(thm)],[154])).
% 35.68/22.48 tff(156,plain,
% 35.68/22.48 (implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.68/22.48 inference(symmetry,[status(thm)],[155])).
% 35.68/22.48 tff(157,plain,
% 35.68/22.48 (implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.68/22.48 inference(symmetry,[status(thm)],[144])).
% 35.68/22.48 tff(158,plain,
% 35.68/22.48 (implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(not(not(implies(x, y))), not(not(implies(y, x))))),
% 35.68/22.48 inference(unit_resolution,[status(thm)],[29, 25])).
% 35.68/22.48 tff(159,plain,
% 35.68/22.48 (implies(not(not(implies(x, y))), not(not(implies(y, x)))) = implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.81/22.48 inference(symmetry,[status(thm)],[158])).
% 35.81/22.48 tff(160,plain,
% 35.81/22.48 (implies(not(not(implies(x, y))), not(not(implies(y, x)))) = implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))), implies(truth, implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))))),
% 35.81/22.48 inference(transitivity,[status(thm)],[159, 157, 156, 151, 149])).
% 35.81/22.48 tff(161,plain,
% 35.81/22.48 (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))), implies(truth, implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y))))))))),
% 35.81/22.48 inference(monotonicity,[status(thm)],[159, 160])).
% 35.81/22.48 tff(162,plain,
% 35.81/22.48 (implies(implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))), implies(truth, implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))))) = implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.81/22.48 inference(symmetry,[status(thm)],[161])).
% 35.81/22.48 tff(163,plain,
% 35.81/22.48 ((~![Z: $i, Y: $i, X: $i] : (implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth)) | (implies(implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))), implies(truth, implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))))) = truth)),
% 35.81/22.48 inference(quant_inst,[status(thm)],[])).
% 35.81/22.48 tff(164,plain,
% 35.81/22.48 (implies(implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))), implies(truth, implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))))) = truth),
% 35.81/22.48 inference(unit_resolution,[status(thm)],[163, 7])).
% 35.81/22.48 tff(165,plain,
% 35.81/22.48 (truth = implies(implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y)))))), implies(truth, implies(not(not(not(implies(y, x)))), not(not(not(implies(x, y))))))))),
% 35.81/22.48 inference(symmetry,[status(thm)],[164])).
% 35.81/22.48 tff(166,plain,
% 35.81/22.48 (truth = implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.81/22.48 inference(transitivity,[status(thm)],[165, 162])).
% 35.81/22.48 tff(167,plain,
% 35.81/22.48 (^[Y: $i, X: $i] : refl((big_V(X, Y) = implies(implies(X, Y), Y)) <=> (big_V(X, Y) = implies(implies(X, Y), Y)))),
% 35.81/22.48 inference(bind,[status(th)],[])).
% 35.81/22.48 tff(168,plain,
% 35.81/22.48 (![Y: $i, X: $i] : (big_V(X, Y) = implies(implies(X, Y), Y)) <=> ![Y: $i, X: $i] : (big_V(X, Y) = implies(implies(X, Y), Y))),
% 35.81/22.48 inference(quant_intro,[status(thm)],[167])).
% 35.81/22.48 tff(169,plain,
% 35.81/22.48 (![Y: $i, X: $i] : (big_V(X, Y) = implies(implies(X, Y), Y)) <=> ![Y: $i, X: $i] : (big_V(X, Y) = implies(implies(X, Y), Y))),
% 35.81/22.48 inference(rewrite,[status(thm)],[])).
% 35.81/22.48 tff(170,axiom,(![Y: $i, X: $i] : (big_V(X, Y) = implies(implies(X, Y), Y))), file('/export/starexec/sandbox/benchmark/Axioms/LCL001-1.ax','big_V_definition')).
% 35.81/22.48 tff(171,plain,
% 35.81/22.48 (![Y: $i, X: $i] : (big_V(X, Y) = implies(implies(X, Y), Y))),
% 35.81/22.48 inference(modus_ponens,[status(thm)],[170, 169])).
% 35.81/22.48 tff(172,plain,(
% 35.81/22.48 ![Y: $i, X: $i] : (big_V(X, Y) = implies(implies(X, Y), Y))),
% 35.81/22.48 inference(skolemize,[status(sab)],[171])).
% 35.81/22.48 tff(173,plain,
% 35.81/22.48 (![Y: $i, X: $i] : (big_V(X, Y) = implies(implies(X, Y), Y))),
% 35.81/22.48 inference(modus_ponens,[status(thm)],[172, 168])).
% 35.81/22.48 tff(174,plain,
% 35.81/22.48 ((~![Y: $i, X: $i] : (big_V(X, Y) = implies(implies(X, Y), Y))) | (big_V(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y)))))),
% 35.81/22.48 inference(quant_inst,[status(thm)],[])).
% 35.81/22.48 tff(175,plain,
% 35.81/22.48 (big_V(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.81/22.48 inference(unit_resolution,[status(thm)],[174, 173])).
% 35.81/22.48 tff(176,plain,
% 35.81/22.48 (implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y)))) = big_V(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.81/22.48 inference(symmetry,[status(thm)],[175])).
% 35.81/22.48 tff(177,plain,
% 35.81/22.48 ((~![Y: $i, X: $i] : (implies(implies(not(X), not(Y)), implies(Y, X)) = truth)) | (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))) = truth)),
% 35.81/22.48 inference(quant_inst,[status(thm)],[])).
% 35.81/22.48 tff(178,plain,
% 35.81/22.48 (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))) = truth),
% 35.81/22.48 inference(unit_resolution,[status(thm)],[177, 16])).
% 35.81/22.48 tff(179,plain,
% 35.81/22.48 (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))),
% 35.81/22.48 inference(transitivity,[status(thm)],[178, 145, 84])).
% 35.81/22.48 tff(180,plain,
% 35.81/22.48 (implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.81/22.48 inference(monotonicity,[status(thm)],[179])).
% 35.81/22.48 tff(181,plain,
% 35.81/22.48 (implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.81/22.48 inference(symmetry,[status(thm)],[180])).
% 35.81/22.48 tff(182,plain,
% 35.81/22.48 (implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y)))) = implies(truth, implies(not(implies(y, x)), not(implies(x, y))))),
% 35.81/22.48 inference(monotonicity,[status(thm)],[143])).
% 35.81/22.48 tff(183,plain,
% 35.81/22.48 (implies(truth, implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.81/22.48 inference(symmetry,[status(thm)],[182])).
% 35.81/22.48 tff(184,plain,
% 35.81/22.48 (implies(not(implies(y, x)), not(implies(x, y))) = implies(truth, implies(not(implies(y, x)), not(implies(x, y))))),
% 35.81/22.48 inference(symmetry,[status(thm)],[27])).
% 35.81/22.48 tff(185,plain,
% 35.81/22.48 (implies(not(implies(y, x)), not(implies(x, y))) = big_V(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.81/22.48 inference(transitivity,[status(thm)],[184, 183, 181, 176])).
% 35.81/22.48 tff(186,plain,
% 35.81/22.48 (implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(big_V(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.81/22.48 inference(monotonicity,[status(thm)],[185])).
% 35.81/22.48 tff(187,plain,
% 35.81/22.48 (implies(big_V(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.81/22.48 inference(symmetry,[status(thm)],[186])).
% 35.81/22.48 tff(188,plain,
% 35.81/22.48 (^[Z: $i, Y: $i, X: $i] : refl((implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z))) <=> (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z))))),
% 35.81/22.48 inference(bind,[status(th)],[])).
% 35.81/22.48 tff(189,plain,
% 35.81/22.48 (![Z: $i, Y: $i, X: $i] : (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z))) <=> ![Z: $i, Y: $i, X: $i] : (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z)))),
% 35.81/22.48 inference(quant_intro,[status(thm)],[188])).
% 35.81/22.48 tff(190,plain,
% 35.81/22.48 (![Z: $i, Y: $i, X: $i] : (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z))) <=> ![Z: $i, Y: $i, X: $i] : (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z)))),
% 35.81/22.48 inference(rewrite,[status(thm)],[])).
% 35.81/22.48 tff(191,axiom,(![Z: $i, Y: $i, X: $i] : (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','lemma_6')).
% 35.81/22.48 tff(192,plain,
% 35.81/22.48 (![Z: $i, Y: $i, X: $i] : (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z)))),
% 35.81/22.48 inference(modus_ponens,[status(thm)],[191, 190])).
% 35.81/22.48 tff(193,plain,(
% 35.81/22.48 ![Z: $i, Y: $i, X: $i] : (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z)))),
% 35.81/22.48 inference(skolemize,[status(sab)],[192])).
% 35.81/22.48 tff(194,plain,
% 35.81/22.48 (![Z: $i, Y: $i, X: $i] : (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z)))),
% 35.81/22.48 inference(modus_ponens,[status(thm)],[193, 189])).
% 35.81/22.48 tff(195,plain,
% 35.81/22.48 ((~![Z: $i, Y: $i, X: $i] : (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z)))) | (implies(big_V(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))))),
% 35.81/22.48 inference(quant_inst,[status(thm)],[])).
% 35.81/22.48 tff(196,plain,
% 35.81/22.48 (implies(big_V(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.81/22.48 inference(unit_resolution,[status(thm)],[195, 194])).
% 35.81/22.48 tff(197,plain,
% 35.81/22.48 (big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))) = implies(big_V(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.81/22.49 inference(symmetry,[status(thm)],[196])).
% 35.81/22.49 tff(198,plain,
% 35.81/22.49 (big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))) = implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.81/22.49 inference(transitivity,[status(thm)],[197, 187])).
% 35.81/22.49 tff(199,plain,
% 35.81/22.49 (ordered(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), truth) <=> ordered(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.81/22.49 inference(monotonicity,[status(thm)],[198, 166])).
% 35.81/22.49 tff(200,plain,
% 35.81/22.49 (ordered(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))) <=> ordered(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), truth)),
% 35.81/22.49 inference(symmetry,[status(thm)],[199])).
% 35.81/22.49 tff(201,plain,
% 35.81/22.49 ((~![Y: $i, X: $i] : (implies(implies(X, Y), Y) = implies(implies(Y, X), X))) | (implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.81/22.49 inference(quant_inst,[status(thm)],[])).
% 35.81/22.49 tff(202,plain,
% 35.81/22.49 (implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.81/22.49 inference(unit_resolution,[status(thm)],[201, 116])).
% 35.81/22.49 tff(203,plain,
% 35.81/22.49 (implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.81/22.49 inference(symmetry,[status(thm)],[202])).
% 35.81/22.49 tff(204,plain,
% 35.81/22.49 (implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(not(implies(y, x)), not(implies(x, y)))),
% 35.81/22.49 inference(transitivity,[status(thm)],[203, 180, 182, 27])).
% 35.81/22.49 tff(205,plain,
% 35.81/22.49 (ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))) <=> ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.81/22.49 inference(monotonicity,[status(thm)],[204])).
% 35.81/22.49 tff(206,plain,
% 35.81/22.49 (ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))) <=> ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.81/22.49 inference(symmetry,[status(thm)],[205])).
% 35.81/22.49 tff(207,plain,
% 35.81/22.49 (((~![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))) | ((~(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))) = truth)) | ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))))) <=> ((~![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))) | (~(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))) = truth)) | ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))))),
% 35.81/22.49 inference(rewrite,[status(thm)],[])).
% 35.81/22.49 tff(208,plain,
% 35.81/22.49 ((~![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))) | ((~(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))) = truth)) | ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))))),
% 35.81/22.49 inference(quant_inst,[status(thm)],[])).
% 35.81/22.49 tff(209,plain,
% 35.81/22.49 ((~![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))) | (~(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))) = truth)) | ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.81/22.49 inference(modus_ponens,[status(thm)],[208, 207])).
% 35.81/22.49 tff(210,plain,
% 35.81/22.49 (ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.81/22.49 inference(unit_resolution,[status(thm)],[209, 48, 178])).
% 35.81/22.49 tff(211,plain,
% 35.81/22.49 (ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.81/22.49 inference(modus_ponens,[status(thm)],[210, 206])).
% 35.81/22.49 tff(212,plain,
% 35.81/22.49 (^[Z: $i, Y: $i, X: $i] : refl(((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z))) <=> ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z))))),
% 35.81/22.49 inference(bind,[status(th)],[])).
% 35.81/22.49 tff(213,plain,
% 35.81/22.49 (![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z))) <=> ![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))),
% 35.81/22.49 inference(quant_intro,[status(thm)],[212])).
% 35.81/22.49 tff(214,plain,
% 35.81/22.49 (![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z))) <=> ![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))),
% 35.81/22.49 inference(rewrite,[status(thm)],[])).
% 35.81/22.49 tff(215,axiom,(![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','lemma_3_2')).
% 35.81/22.49 tff(216,plain,
% 35.81/22.49 (![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))),
% 35.81/22.49 inference(modus_ponens,[status(thm)],[215, 214])).
% 35.81/22.49 tff(217,plain,(
% 35.81/22.49 ![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))),
% 35.81/22.49 inference(skolemize,[status(sab)],[216])).
% 35.81/22.49 tff(218,plain,
% 35.81/22.49 (![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))),
% 35.81/22.49 inference(modus_ponens,[status(thm)],[217, 213])).
% 35.81/22.49 tff(219,plain,
% 35.81/22.49 (((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | ((~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))))) | ordered(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | (~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))))) | ordered(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))))),
% 35.81/22.49 inference(rewrite,[status(thm)],[])).
% 35.81/22.49 tff(220,plain,
% 35.81/22.49 ((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | ((~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))))) | ordered(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))))),
% 35.81/22.49 inference(quant_inst,[status(thm)],[])).
% 35.81/22.49 tff(221,plain,
% 35.81/22.49 ((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | (~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))))) | ordered(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.81/22.49 inference(modus_ponens,[status(thm)],[220, 219])).
% 35.81/22.49 tff(222,plain,
% 35.81/22.49 ((~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))))) | ordered(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.81/22.49 inference(unit_resolution,[status(thm)],[221, 218])).
% 35.81/22.49 tff(223,plain,
% 35.81/22.49 (ordered(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.81/22.49 inference(unit_resolution,[status(thm)],[222, 211])).
% 35.81/22.49 tff(224,plain,
% 35.81/22.49 (ordered(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), truth)),
% 35.81/22.49 inference(modus_ponens,[status(thm)],[223, 200])).
% 35.81/22.49 tff(225,plain,
% 35.81/22.49 (implies(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.81/22.49 inference(monotonicity,[status(thm)],[198])).
% 35.81/22.49 tff(226,plain,
% 35.81/22.49 (implies(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(not(implies(y, x)), not(implies(x, y)))),
% 35.81/22.49 inference(transitivity,[status(thm)],[225, 203, 180, 182, 27])).
% 35.81/22.49 tff(227,plain,
% 35.81/22.49 (ordered(implies(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))) <=> ordered(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.81/22.49 inference(monotonicity,[status(thm)],[226, 158])).
% 35.81/22.49 tff(228,plain,
% 35.81/22.49 (ordered(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) <=> ordered(implies(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.81/22.49 inference(symmetry,[status(thm)],[227])).
% 35.81/22.49 tff(229,plain,
% 35.81/22.49 ((~ordered(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))) <=> (~ordered(implies(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))))),
% 35.81/22.49 inference(monotonicity,[status(thm)],[228])).
% 35.81/22.49 tff(230,assumption,(~ordered(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), introduced(assumption)).
% 35.81/22.49 tff(231,plain,
% 35.81/22.49 (~ordered(implies(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.81/22.49 inference(modus_ponens,[status(thm)],[230, 229])).
% 35.81/22.49 tff(232,plain,
% 35.81/22.49 (((~![Z: $i, Y: $i, X: $i] : ((~ordered(X, Y)) | ordered(implies(X, Z), implies(Y, Z)))) | ((~ordered(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), truth)) | ordered(implies(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~ordered(X, Y)) | ordered(implies(X, Z), implies(Y, Z)))) | (~ordered(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), truth)) | ordered(implies(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))))),
% 35.81/22.49 inference(rewrite,[status(thm)],[])).
% 35.81/22.49 tff(233,plain,
% 35.81/22.49 ((~![Z: $i, Y: $i, X: $i] : ((~ordered(X, Y)) | ordered(implies(X, Z), implies(Y, Z)))) | ((~ordered(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), truth)) | ordered(implies(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))))),
% 35.81/22.49 inference(quant_inst,[status(thm)],[])).
% 35.81/22.49 tff(234,plain,
% 35.81/22.49 ((~![Z: $i, Y: $i, X: $i] : ((~ordered(X, Y)) | ordered(implies(X, Z), implies(Y, Z)))) | (~ordered(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), truth)) | ordered(implies(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.81/22.49 inference(modus_ponens,[status(thm)],[233, 232])).
% 35.81/22.49 tff(235,plain,
% 35.81/22.49 ((~ordered(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), truth)) | ordered(implies(big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.81/22.49 inference(unit_resolution,[status(thm)],[234, 75])).
% 35.81/22.49 tff(236,plain,
% 35.81/22.49 ($false),
% 35.81/22.49 inference(unit_resolution,[status(thm)],[235, 231, 224])).
% 35.81/22.49 tff(237,plain,(ordered(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))), inference(lemma,lemma(discharge,[]))).
% 35.81/22.49 tff(238,plain,
% 35.81/22.49 (((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | ((~ordered(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))) | (implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth))) <=> ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | (~ordered(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))) | (implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth))),
% 35.81/22.49 inference(rewrite,[status(thm)],[])).
% 35.81/22.49 tff(239,plain,
% 35.81/22.49 ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | ((~ordered(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))) | (implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth))),
% 35.81/22.49 inference(quant_inst,[status(thm)],[])).
% 35.81/22.50 tff(240,plain,
% 35.81/22.50 ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | (~ordered(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))) | (implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth)),
% 35.81/22.50 inference(modus_ponens,[status(thm)],[239, 238])).
% 35.81/22.50 tff(241,plain,
% 35.81/22.50 (implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth),
% 35.81/22.50 inference(unit_resolution,[status(thm)],[240, 92, 237])).
% 35.81/22.50 tff(242,plain,
% 35.81/22.50 (implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))),
% 35.81/22.50 inference(transitivity,[status(thm)],[241, 145, 84])).
% 35.81/22.50 tff(243,plain,
% 35.81/22.50 (implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.81/22.50 inference(monotonicity,[status(thm)],[242])).
% 35.81/22.50 tff(244,plain,
% 35.81/22.50 ((~![Y: $i, X: $i] : (implies(implies(X, Y), Y) = implies(implies(Y, X), X))) | (implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y)))))),
% 35.81/22.50 inference(quant_inst,[status(thm)],[])).
% 35.81/22.50 tff(245,plain,
% 35.81/22.50 (implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.81/22.50 inference(unit_resolution,[status(thm)],[244, 116])).
% 35.81/22.50 tff(246,plain,
% 35.81/22.50 (implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.81/22.50 inference(symmetry,[status(thm)],[245])).
% 35.81/22.50 tff(247,plain,
% 35.81/22.50 (truth = implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.81/22.50 inference(symmetry,[status(thm)],[178])).
% 35.81/22.50 tff(248,plain,
% 35.81/22.50 (implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))) = implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.81/22.50 inference(transitivity,[status(thm)],[142, 141, 247])).
% 35.81/22.50 tff(249,plain,
% 35.81/22.50 (implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.83/22.50 inference(monotonicity,[status(thm)],[248])).
% 35.83/22.50 tff(250,plain,
% 35.83/22.50 (truth = implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))))),
% 35.83/22.50 inference(transitivity,[status(thm)],[145, 84])).
% 35.83/22.50 tff(251,plain,
% 35.83/22.50 (implies(truth, implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.83/22.50 inference(monotonicity,[status(thm)],[250])).
% 35.83/22.50 tff(252,plain,
% 35.83/22.50 (implies(not(implies(y, x)), not(implies(x, y))) = implies(not(not(implies(x, y))), not(not(implies(y, x))))),
% 35.83/22.50 inference(transitivity,[status(thm)],[184, 251, 249, 246, 243, 144, 34])).
% 35.83/22.50 tff(253,plain,
% 35.83/22.50 (implies(implies(not(implies(y, x)), not(implies(x, y))), not(implies(x, y))) = implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), not(implies(x, y)))),
% 35.83/22.50 inference(monotonicity,[status(thm)],[252])).
% 35.83/22.50 tff(254,plain,
% 35.83/22.50 ((~![Y: $i, X: $i] : (implies(implies(X, Y), Y) = implies(implies(Y, X), X))) | (implies(implies(not(implies(y, x)), not(implies(x, y))), not(implies(x, y))) = implies(implies(not(implies(x, y)), not(implies(y, x))), not(implies(y, x))))),
% 35.83/22.50 inference(quant_inst,[status(thm)],[])).
% 35.83/22.50 tff(255,plain,
% 35.83/22.50 (implies(implies(not(implies(y, x)), not(implies(x, y))), not(implies(x, y))) = implies(implies(not(implies(x, y)), not(implies(y, x))), not(implies(y, x)))),
% 35.83/22.50 inference(unit_resolution,[status(thm)],[254, 116])).
% 35.83/22.50 tff(256,plain,
% 35.83/22.50 (implies(implies(not(implies(x, y)), not(implies(y, x))), not(implies(y, x))) = implies(implies(not(implies(y, x)), not(implies(x, y))), not(implies(x, y)))),
% 35.83/22.50 inference(symmetry,[status(thm)],[255])).
% 35.83/22.50 tff(257,plain,
% 35.83/22.50 (implies(implies(not(implies(x, y)), not(implies(y, x))), not(implies(y, x))) = implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), not(implies(x, y)))),
% 35.83/22.50 inference(transitivity,[status(thm)],[256, 253])).
% 35.83/22.50 tff(258,plain,
% 35.83/22.50 (ordered(not(implies(y, x)), implies(implies(not(implies(x, y)), not(implies(y, x))), not(implies(y, x)))) <=> ordered(not(implies(y, x)), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), not(implies(x, y))))),
% 35.83/22.50 inference(monotonicity,[status(thm)],[257])).
% 35.83/22.50 tff(259,plain,
% 35.83/22.50 (ordered(not(implies(y, x)), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), not(implies(x, y)))) <=> ordered(not(implies(y, x)), implies(implies(not(implies(x, y)), not(implies(y, x))), not(implies(y, x))))),
% 35.83/22.50 inference(symmetry,[status(thm)],[258])).
% 35.83/22.50 tff(260,plain,
% 35.83/22.50 (((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | ((~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y))))) | ordered(not(implies(y, x)), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), not(implies(x, y)))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | (~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y))))) | ordered(not(implies(y, x)), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), not(implies(x, y)))))),
% 35.83/22.50 inference(rewrite,[status(thm)],[])).
% 35.83/22.50 tff(261,plain,
% 35.83/22.50 ((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | ((~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y))))) | ordered(not(implies(y, x)), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), not(implies(x, y)))))),
% 35.83/22.50 inference(quant_inst,[status(thm)],[])).
% 35.83/22.50 tff(262,plain,
% 35.83/22.50 ((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | (~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y))))) | ordered(not(implies(y, x)), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), not(implies(x, y))))),
% 35.83/22.50 inference(modus_ponens,[status(thm)],[261, 260])).
% 35.83/22.50 tff(263,plain,
% 35.83/22.50 (ordered(not(implies(y, x)), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), not(implies(x, y))))),
% 35.83/22.50 inference(unit_resolution,[status(thm)],[262, 218, 210])).
% 35.83/22.50 tff(264,plain,
% 35.83/22.50 (ordered(not(implies(y, x)), implies(implies(not(implies(x, y)), not(implies(y, x))), not(implies(y, x))))),
% 35.83/22.50 inference(modus_ponens,[status(thm)],[263, 259])).
% 35.83/22.50 tff(265,plain,
% 35.83/22.50 (((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | ((~ordered(not(implies(y, x)), implies(implies(not(implies(x, y)), not(implies(y, x))), not(implies(y, x))))) | ordered(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x)))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | (~ordered(not(implies(y, x)), implies(implies(not(implies(x, y)), not(implies(y, x))), not(implies(y, x))))) | ordered(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x)))))),
% 35.83/22.50 inference(rewrite,[status(thm)],[])).
% 35.83/22.50 tff(266,plain,
% 35.83/22.50 ((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | ((~ordered(not(implies(y, x)), implies(implies(not(implies(x, y)), not(implies(y, x))), not(implies(y, x))))) | ordered(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x)))))),
% 35.83/22.50 inference(quant_inst,[status(thm)],[])).
% 35.83/22.50 tff(267,plain,
% 35.83/22.50 ((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | (~ordered(not(implies(y, x)), implies(implies(not(implies(x, y)), not(implies(y, x))), not(implies(y, x))))) | ordered(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x))))),
% 35.83/22.50 inference(modus_ponens,[status(thm)],[266, 265])).
% 35.83/22.50 tff(268,plain,
% 35.83/22.50 ((~ordered(not(implies(y, x)), implies(implies(not(implies(x, y)), not(implies(y, x))), not(implies(y, x))))) | ordered(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x))))),
% 35.83/22.50 inference(unit_resolution,[status(thm)],[267, 218])).
% 35.83/22.50 tff(269,plain,
% 35.83/22.50 (ordered(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x))))),
% 35.83/22.50 inference(unit_resolution,[status(thm)],[268, 264])).
% 35.83/22.50 tff(270,plain,
% 35.83/22.50 (((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | ((~ordered(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x))))) | (implies(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x)))) = truth))) <=> ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | (~ordered(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x))))) | (implies(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x)))) = truth))),
% 35.83/22.50 inference(rewrite,[status(thm)],[])).
% 35.83/22.50 tff(271,plain,
% 35.83/22.50 ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | ((~ordered(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x))))) | (implies(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x)))) = truth))),
% 35.83/22.50 inference(quant_inst,[status(thm)],[])).
% 35.83/22.50 tff(272,plain,
% 35.83/22.50 ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | (~ordered(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x))))) | (implies(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x)))) = truth)),
% 35.83/22.50 inference(modus_ponens,[status(thm)],[271, 270])).
% 35.83/22.50 tff(273,plain,
% 35.83/22.50 (implies(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x)))) = truth),
% 35.83/22.50 inference(unit_resolution,[status(thm)],[272, 92, 269])).
% 35.83/22.50 tff(274,plain,
% 35.83/22.50 (truth = implies(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x))))),
% 35.83/22.50 inference(symmetry,[status(thm)],[273])).
% 35.83/22.50 tff(275,plain,
% 35.83/22.50 (implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y)))) = implies(truth, implies(not(implies(y, x)), not(implies(x, y))))),
% 35.83/22.50 inference(symmetry,[status(thm)],[251])).
% 35.83/22.50 tff(276,plain,
% 35.83/22.50 (implies(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.83/22.50 inference(symmetry,[status(thm)],[249])).
% 35.83/22.50 tff(277,plain,
% 35.83/22.50 (implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(implies(not(implies(y, x)), not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.83/22.50 inference(symmetry,[status(thm)],[243])).
% 35.83/22.50 tff(278,plain,
% 35.83/22.50 (implies(not(not(implies(x, y))), not(not(implies(y, x)))) = implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.83/22.50 inference(symmetry,[status(thm)],[34])).
% 35.83/22.50 tff(279,plain,
% 35.83/22.50 (implies(not(not(implies(x, y))), not(not(implies(y, x)))) = implies(not(implies(y, x)), not(implies(x, y)))),
% 35.83/22.50 inference(transitivity,[status(thm)],[278, 157, 277, 245, 276, 275, 27])).
% 35.83/22.50 tff(280,plain,
% 35.83/22.50 (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth) = implies(implies(not(implies(y, x)), not(implies(x, y))), implies(implies(not(implies(x, y)), not(implies(y, x))), implies(not(implies(y, x)), not(implies(y, x)))))),
% 35.83/22.50 inference(monotonicity,[status(thm)],[279, 274])).
% 35.83/22.50 tff(281,plain,
% 35.83/22.50 (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth) = truth),
% 35.83/22.50 inference(transitivity,[status(thm)],[280, 33])).
% 35.83/22.50 tff(282,plain,
% 35.83/22.50 ((implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth) <=> (implies(not(not(implies(x, y))), not(not(implies(y, x)))) = truth)),
% 35.83/22.50 inference(monotonicity,[status(thm)],[34])).
% 35.83/22.50 tff(283,plain,
% 35.83/22.50 ((implies(not(not(implies(x, y))), not(not(implies(y, x)))) = truth) <=> (implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth)),
% 35.83/22.50 inference(symmetry,[status(thm)],[282])).
% 35.83/22.50 tff(284,plain,
% 35.83/22.50 ((~(implies(not(not(implies(x, y))), not(not(implies(y, x)))) = truth)) <=> (~(implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth))),
% 35.83/22.50 inference(monotonicity,[status(thm)],[283])).
% 35.83/22.50 tff(285,assumption,(~(implies(not(not(implies(x, y))), not(not(implies(y, x)))) = truth)), introduced(assumption)).
% 35.83/22.50 tff(286,plain,
% 35.83/22.50 (~(implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth)),
% 35.83/22.50 inference(modus_ponens,[status(thm)],[285, 284])).
% 35.83/22.50 tff(287,plain,
% 35.83/22.50 (((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | ((~ordered(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))) | (implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth))) <=> ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | (~ordered(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))) | (implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth))),
% 35.83/22.50 inference(rewrite,[status(thm)],[])).
% 35.83/22.50 tff(288,plain,
% 35.83/22.50 ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | ((~ordered(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))) | (implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth))),
% 35.83/22.50 inference(quant_inst,[status(thm)],[])).
% 35.83/22.50 tff(289,plain,
% 35.83/22.50 ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | (~ordered(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))) | (implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth)),
% 35.83/22.50 inference(modus_ponens,[status(thm)],[288, 287])).
% 35.83/22.50 tff(290,plain,
% 35.83/22.50 ((~ordered(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))) | (implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth)),
% 35.83/22.50 inference(unit_resolution,[status(thm)],[289, 92])).
% 35.83/22.50 tff(291,plain,
% 35.83/22.50 (~ordered(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.83/22.50 inference(unit_resolution,[status(thm)],[290, 286])).
% 35.83/22.50 tff(292,plain,
% 35.83/22.50 (ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y))))) <=> ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth)),
% 35.83/22.50 inference(monotonicity,[status(thm)],[178])).
% 35.83/22.50 tff(293,plain,
% 35.83/22.50 (ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), truth) <=> ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))))),
% 35.83/22.50 inference(symmetry,[status(thm)],[292])).
% 35.83/22.50 tff(294,plain,
% 35.83/22.50 (ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))))),
% 35.83/22.50 inference(modus_ponens,[status(thm)],[68, 293])).
% 35.83/22.50 tff(295,plain,
% 35.83/22.50 (((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | ((~ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))))) | ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y))))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | (~ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))))) | ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y))))))),
% 35.83/22.51 inference(rewrite,[status(thm)],[])).
% 35.83/22.51 tff(296,plain,
% 35.83/22.51 ((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | ((~ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))))) | ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y))))))),
% 35.83/22.51 inference(quant_inst,[status(thm)],[])).
% 35.83/22.51 tff(297,plain,
% 35.83/22.51 ((~![Z: $i, Y: $i, X: $i] : ((~ordered(Y, implies(X, Z))) | ordered(X, implies(Y, Z)))) | (~ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))))) | ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y)))))),
% 35.83/22.51 inference(modus_ponens,[status(thm)],[296, 295])).
% 35.83/22.51 tff(298,plain,
% 35.83/22.51 ((~ordered(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))))) | ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y)))))),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[297, 218])).
% 35.83/22.51 tff(299,plain,
% 35.83/22.51 (ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y)))))),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[298, 294])).
% 35.83/22.51 tff(300,plain,
% 35.83/22.51 (((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | ((~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y)))))) | (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y))))) = truth))) <=> ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | (~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y)))))) | (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y))))) = truth))),
% 35.83/22.51 inference(rewrite,[status(thm)],[])).
% 35.83/22.51 tff(301,plain,
% 35.83/22.51 ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | ((~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y)))))) | (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y))))) = truth))),
% 35.83/22.51 inference(quant_inst,[status(thm)],[])).
% 35.83/22.51 tff(302,plain,
% 35.83/22.51 ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | (~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y)))))) | (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y))))) = truth)),
% 35.83/22.51 inference(modus_ponens,[status(thm)],[301, 300])).
% 35.83/22.51 tff(303,plain,
% 35.83/22.51 (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y))))) = truth),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[302, 92, 299])).
% 35.83/22.51 tff(304,plain,
% 35.83/22.51 (implies(not(not(implies(x, y))), not(not(implies(y, x)))) = implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.83/22.51 inference(transitivity,[status(thm)],[159, 157, 277, 245, 276])).
% 35.83/22.51 tff(305,plain,
% 35.83/22.51 (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(implies(implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y)))))))))), implies(not(not(not(not(not(not(not(not(implies(y, x))))))))), not(not(not(not(not(not(not(not(implies(x, y))))))))))), implies(not(implies(y, x)), not(implies(x, y)))))),
% 35.83/22.51 inference(monotonicity,[status(thm)],[304])).
% 35.83/22.51 tff(306,plain,
% 35.83/22.51 (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))) = truth),
% 35.83/22.51 inference(transitivity,[status(thm)],[305, 303])).
% 35.83/22.51 tff(307,plain,
% 35.83/22.51 (ordered(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))) <=> ordered(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.83/22.51 inference(monotonicity,[status(thm)],[306, 158])).
% 35.83/22.51 tff(308,plain,
% 35.83/22.51 (ordered(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) <=> ordered(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.83/22.51 inference(symmetry,[status(thm)],[307])).
% 35.83/22.51 tff(309,plain,
% 35.83/22.51 ((~ordered(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))) <=> (~ordered(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))))),
% 35.83/22.51 inference(monotonicity,[status(thm)],[308])).
% 35.83/22.51 tff(310,assumption,(~ordered(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))), introduced(assumption)).
% 35.83/22.51 tff(311,plain,
% 35.83/22.51 (~ordered(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.83/22.51 inference(modus_ponens,[status(thm)],[310, 309])).
% 35.83/22.51 tff(312,assumption,(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth) = truth), introduced(assumption)).
% 35.83/22.51 tff(313,plain,
% 35.83/22.51 (((~![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))) | ((~(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth) = truth)) | ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth))) <=> ((~![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))) | (~(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth) = truth)) | ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth))),
% 35.83/22.51 inference(rewrite,[status(thm)],[])).
% 35.83/22.51 tff(314,plain,
% 35.83/22.51 ((~![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))) | ((~(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth) = truth)) | ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth))),
% 35.83/22.51 inference(quant_inst,[status(thm)],[])).
% 35.83/22.51 tff(315,plain,
% 35.83/22.51 ((~![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))) | (~(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth) = truth)) | ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth)),
% 35.83/22.51 inference(modus_ponens,[status(thm)],[314, 313])).
% 35.83/22.51 tff(316,plain,
% 35.83/22.51 ((~(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth) = truth)) | ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth)),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[315, 48])).
% 35.83/22.51 tff(317,plain,
% 35.83/22.51 (ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth)),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[316, 312])).
% 35.83/22.51 tff(318,plain,
% 35.83/22.51 (((~![Z: $i, Y: $i, X: $i] : ((~ordered(X, Y)) | ordered(implies(X, Z), implies(Y, Z)))) | ((~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth)) | ordered(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~ordered(X, Y)) | ordered(implies(X, Z), implies(Y, Z)))) | (~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth)) | ordered(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))))),
% 35.83/22.51 inference(rewrite,[status(thm)],[])).
% 35.83/22.51 tff(319,plain,
% 35.83/22.51 ((~![Z: $i, Y: $i, X: $i] : ((~ordered(X, Y)) | ordered(implies(X, Z), implies(Y, Z)))) | ((~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth)) | ordered(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))))),
% 35.83/22.51 inference(quant_inst,[status(thm)],[])).
% 35.83/22.51 tff(320,plain,
% 35.83/22.51 ((~![Z: $i, Y: $i, X: $i] : ((~ordered(X, Y)) | ordered(implies(X, Z), implies(Y, Z)))) | (~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth)) | ordered(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.83/22.51 inference(modus_ponens,[status(thm)],[319, 318])).
% 35.83/22.51 tff(321,plain,
% 35.83/22.51 ((~ordered(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth)) | ordered(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(not(implies(x, y))), not(not(implies(y, x))))), implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))))),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[320, 75])).
% 35.83/22.51 tff(322,plain,
% 35.83/22.51 ($false),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[321, 317, 311])).
% 35.83/22.51 tff(323,plain,((~(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth) = truth)) | ordered(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))), inference(lemma,lemma(discharge,[]))).
% 35.83/22.51 tff(324,plain,
% 35.83/22.51 (~(implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth) = truth)),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[323, 291])).
% 35.83/22.51 tff(325,plain,
% 35.83/22.51 ($false),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[324, 281])).
% 35.83/22.51 tff(326,plain,((~(implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(not(not(implies(x, y))), not(not(implies(y, x)))))) | (implies(not(not(implies(x, y))), not(not(implies(y, x)))) = truth)), inference(lemma,lemma(discharge,[]))).
% 35.83/22.51 tff(327,plain,
% 35.83/22.51 (implies(not(not(implies(x, y))), not(not(implies(y, x)))) = truth),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[326, 31])).
% 35.83/22.51 tff(328,plain,
% 35.83/22.51 (truth = implies(not(not(implies(x, y))), not(not(implies(y, x))))),
% 35.83/22.51 inference(symmetry,[status(thm)],[327])).
% 35.83/22.51 tff(329,plain,
% 35.83/22.51 (implies(truth, implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.83/22.51 inference(monotonicity,[status(thm)],[328])).
% 35.83/22.51 tff(330,plain,
% 35.83/22.51 (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), implies(not(implies(y, x)), not(implies(x, y)))) = implies(truth, implies(not(implies(y, x)), not(implies(x, y))))),
% 35.83/22.51 inference(symmetry,[status(thm)],[329])).
% 35.83/22.51 tff(331,plain,
% 35.83/22.51 (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth) = implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.83/22.51 inference(monotonicity,[status(thm)],[327, 328])).
% 35.83/22.51 tff(332,plain,
% 35.83/22.51 (implies(not(implies(y, x)), not(implies(x, y))) = truth),
% 35.83/22.51 inference(transitivity,[status(thm)],[184, 329, 178])).
% 35.83/22.51 tff(333,plain,
% 35.83/22.51 (implies(truth, implies(not(implies(y, x)), not(implies(x, y)))) = implies(not(not(implies(x, y))), not(not(implies(y, x))))),
% 35.83/22.51 inference(transitivity,[status(thm)],[329, 178, 328])).
% 35.83/22.51 tff(334,plain,
% 35.83/22.51 (implies(implies(truth, implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth)),
% 35.83/22.51 inference(monotonicity,[status(thm)],[333, 332])).
% 35.83/22.51 tff(335,plain,
% 35.83/22.51 ((~![Y: $i, X: $i] : (big_V(X, Y) = implies(implies(X, Y), Y))) | (big_V(truth, implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(truth, implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y)))))),
% 35.83/22.51 inference(quant_inst,[status(thm)],[])).
% 35.83/22.51 tff(336,plain,
% 35.83/22.51 (big_V(truth, implies(not(implies(y, x)), not(implies(x, y)))) = implies(implies(truth, implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[335, 173])).
% 35.83/22.51 tff(337,plain,
% 35.83/22.51 (truth = implies(not(implies(y, x)), not(implies(x, y)))),
% 35.83/22.51 inference(transitivity,[status(thm)],[247, 330, 27])).
% 35.83/22.51 tff(338,plain,
% 35.83/22.51 (big_V(truth, truth) = big_V(truth, implies(not(implies(y, x)), not(implies(x, y))))),
% 35.83/22.51 inference(monotonicity,[status(thm)],[337])).
% 35.83/22.51 tff(339,plain,
% 35.83/22.51 (big_V(truth, truth) = implies(not(implies(y, x)), not(implies(x, y)))),
% 35.83/22.51 inference(transitivity,[status(thm)],[338, 336, 334, 331, 31, 327, 247, 330, 27])).
% 35.83/22.51 tff(340,plain,
% 35.83/22.51 (implies(big_V(truth, truth), implies(implies(x, y), implies(y, x))) = implies(implies(not(implies(y, x)), not(implies(x, y))), implies(implies(x, y), implies(y, x)))),
% 35.83/22.51 inference(monotonicity,[status(thm)],[339])).
% 35.83/22.51 tff(341,plain,
% 35.83/22.51 ((~![Z: $i, Y: $i, X: $i] : (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z)))) | (implies(big_V(truth, truth), implies(implies(x, y), implies(y, x))) = big_hat(implies(truth, implies(implies(x, y), implies(y, x))), implies(truth, implies(implies(x, y), implies(y, x)))))),
% 35.83/22.51 inference(quant_inst,[status(thm)],[])).
% 35.83/22.51 tff(342,plain,
% 35.83/22.51 (implies(big_V(truth, truth), implies(implies(x, y), implies(y, x))) = big_hat(implies(truth, implies(implies(x, y), implies(y, x))), implies(truth, implies(implies(x, y), implies(y, x))))),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[341, 194])).
% 35.83/22.51 tff(343,plain,
% 35.83/22.51 (big_hat(implies(truth, implies(implies(x, y), implies(y, x))), implies(truth, implies(implies(x, y), implies(y, x)))) = implies(big_V(truth, truth), implies(implies(x, y), implies(y, x)))),
% 35.83/22.51 inference(symmetry,[status(thm)],[342])).
% 35.83/22.51 tff(344,plain,
% 35.83/22.51 (implies(truth, implies(implies(x, y), implies(y, x))) = implies(implies(not(implies(y, x)), not(implies(x, y))), implies(implies(x, y), implies(y, x)))),
% 35.83/22.51 inference(monotonicity,[status(thm)],[337])).
% 35.83/22.51 tff(345,plain,
% 35.83/22.51 (implies(implies(not(implies(y, x)), not(implies(x, y))), implies(implies(x, y), implies(y, x))) = implies(truth, implies(implies(x, y), implies(y, x)))),
% 35.83/22.51 inference(symmetry,[status(thm)],[344])).
% 35.83/22.51 tff(346,plain,
% 35.83/22.51 (truth = implies(implies(not(implies(y, x)), not(implies(x, y))), implies(implies(x, y), implies(y, x)))),
% 35.83/22.51 inference(symmetry,[status(thm)],[18])).
% 35.83/22.51 tff(347,plain,
% 35.83/22.51 ((~![Y: $i, X: $i] : (implies(implies(not(X), not(Y)), implies(Y, X)) = truth)) | (implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(implies(x, y)), not(implies(x, y)))) = truth)),
% 35.83/22.51 inference(quant_inst,[status(thm)],[])).
% 35.83/22.51 tff(348,plain,
% 35.83/22.51 (implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(implies(x, y)), not(implies(x, y)))) = truth),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[347, 16])).
% 35.83/22.51 tff(349,plain,
% 35.83/22.51 ((~![X: $i] : (implies(truth, X) = X)) | (implies(truth, not(implies(x, y))) = not(implies(x, y)))),
% 35.83/22.51 inference(quant_inst,[status(thm)],[])).
% 35.83/22.51 tff(350,plain,
% 35.83/22.51 (implies(truth, not(implies(x, y))) = not(implies(x, y))),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[349, 25])).
% 35.83/22.51 tff(351,plain,
% 35.83/22.51 (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), not(implies(x, y))) = implies(truth, not(implies(x, y)))),
% 35.83/22.51 inference(monotonicity,[status(thm)],[327])).
% 35.83/22.51 tff(352,plain,
% 35.83/22.51 (implies(not(implies(y, x)), not(implies(x, y))) = implies(not(not(implies(x, y))), not(not(implies(y, x))))),
% 35.83/22.51 inference(transitivity,[status(thm)],[184, 329, 178, 328])).
% 35.83/22.51 tff(353,plain,
% 35.83/22.51 (implies(implies(not(implies(y, x)), not(implies(x, y))), not(implies(x, y))) = implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), not(implies(x, y)))),
% 35.83/22.51 inference(monotonicity,[status(thm)],[352])).
% 35.83/22.51 tff(354,plain,
% 35.83/22.51 ((~![Y: $i, X: $i] : (big_V(X, Y) = implies(implies(X, Y), Y))) | (big_V(not(implies(y, x)), not(implies(x, y))) = implies(implies(not(implies(y, x)), not(implies(x, y))), not(implies(x, y))))),
% 35.83/22.51 inference(quant_inst,[status(thm)],[])).
% 35.83/22.51 tff(355,plain,
% 35.83/22.51 (big_V(not(implies(y, x)), not(implies(x, y))) = implies(implies(not(implies(y, x)), not(implies(x, y))), not(implies(x, y)))),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[354, 173])).
% 35.83/22.51 tff(356,plain,
% 35.83/22.51 (^[Y: $i, X: $i] : refl((not(big_hat(X, Y)) = big_V(not(X), not(Y))) <=> (not(big_hat(X, Y)) = big_V(not(X), not(Y))))),
% 35.83/22.51 inference(bind,[status(th)],[])).
% 35.83/22.51 tff(357,plain,
% 35.83/22.51 (![Y: $i, X: $i] : (not(big_hat(X, Y)) = big_V(not(X), not(Y))) <=> ![Y: $i, X: $i] : (not(big_hat(X, Y)) = big_V(not(X), not(Y)))),
% 35.83/22.51 inference(quant_intro,[status(thm)],[356])).
% 35.83/22.51 tff(358,plain,
% 35.83/22.51 (![Y: $i, X: $i] : (not(big_hat(X, Y)) = big_V(not(X), not(Y))) <=> ![Y: $i, X: $i] : (not(big_hat(X, Y)) = big_V(not(X), not(Y)))),
% 35.83/22.51 inference(rewrite,[status(thm)],[])).
% 35.83/22.51 tff(359,axiom,(![Y: $i, X: $i] : (not(big_hat(X, Y)) = big_V(not(X), not(Y)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','lemma_5')).
% 35.83/22.51 tff(360,plain,
% 35.83/22.51 (![Y: $i, X: $i] : (not(big_hat(X, Y)) = big_V(not(X), not(Y)))),
% 35.83/22.51 inference(modus_ponens,[status(thm)],[359, 358])).
% 35.83/22.51 tff(361,plain,(
% 35.83/22.51 ![Y: $i, X: $i] : (not(big_hat(X, Y)) = big_V(not(X), not(Y)))),
% 35.83/22.51 inference(skolemize,[status(sab)],[360])).
% 35.83/22.51 tff(362,plain,
% 35.83/22.51 (![Y: $i, X: $i] : (not(big_hat(X, Y)) = big_V(not(X), not(Y)))),
% 35.83/22.51 inference(modus_ponens,[status(thm)],[361, 357])).
% 35.83/22.51 tff(363,plain,
% 35.83/22.51 ((~![Y: $i, X: $i] : (not(big_hat(X, Y)) = big_V(not(X), not(Y)))) | (not(big_hat(implies(y, x), implies(x, y))) = big_V(not(implies(y, x)), not(implies(x, y))))),
% 35.83/22.51 inference(quant_inst,[status(thm)],[])).
% 35.83/22.51 tff(364,plain,
% 35.83/22.51 (not(big_hat(implies(y, x), implies(x, y))) = big_V(not(implies(y, x)), not(implies(x, y)))),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[363, 362])).
% 35.83/22.51 tff(365,plain,
% 35.83/22.51 (^[Y: $i, X: $i] : refl((big_hat(X, Y) = not(big_V(not(X), not(Y)))) <=> (big_hat(X, Y) = not(big_V(not(X), not(Y)))))),
% 35.83/22.51 inference(bind,[status(th)],[])).
% 35.83/22.51 tff(366,plain,
% 35.83/22.51 (![Y: $i, X: $i] : (big_hat(X, Y) = not(big_V(not(X), not(Y)))) <=> ![Y: $i, X: $i] : (big_hat(X, Y) = not(big_V(not(X), not(Y))))),
% 35.83/22.51 inference(quant_intro,[status(thm)],[365])).
% 35.83/22.51 tff(367,plain,
% 35.83/22.51 (![Y: $i, X: $i] : (big_hat(X, Y) = not(big_V(not(X), not(Y)))) <=> ![Y: $i, X: $i] : (big_hat(X, Y) = not(big_V(not(X), not(Y))))),
% 35.83/22.51 inference(rewrite,[status(thm)],[])).
% 35.83/22.51 tff(368,axiom,(![Y: $i, X: $i] : (big_hat(X, Y) = not(big_V(not(X), not(Y))))), file('/export/starexec/sandbox/benchmark/Axioms/LCL001-1.ax','big_hat_definition')).
% 35.83/22.51 tff(369,plain,
% 35.83/22.51 (![Y: $i, X: $i] : (big_hat(X, Y) = not(big_V(not(X), not(Y))))),
% 35.83/22.51 inference(modus_ponens,[status(thm)],[368, 367])).
% 35.83/22.51 tff(370,plain,(
% 35.83/22.51 ![Y: $i, X: $i] : (big_hat(X, Y) = not(big_V(not(X), not(Y))))),
% 35.83/22.51 inference(skolemize,[status(sab)],[369])).
% 35.83/22.51 tff(371,plain,
% 35.83/22.51 (![Y: $i, X: $i] : (big_hat(X, Y) = not(big_V(not(X), not(Y))))),
% 35.83/22.51 inference(modus_ponens,[status(thm)],[370, 366])).
% 35.83/22.51 tff(372,plain,
% 35.83/22.51 ((~![Y: $i, X: $i] : (big_hat(X, Y) = not(big_V(not(X), not(Y))))) | (big_hat(implies(y, x), implies(x, y)) = not(big_V(not(implies(y, x)), not(implies(x, y)))))),
% 35.83/22.51 inference(quant_inst,[status(thm)],[])).
% 35.83/22.51 tff(373,plain,
% 35.83/22.51 (big_hat(implies(y, x), implies(x, y)) = not(big_V(not(implies(y, x)), not(implies(x, y))))),
% 35.83/22.51 inference(unit_resolution,[status(thm)],[372, 371])).
% 35.83/22.51 tff(374,plain,
% 35.83/22.51 (not(big_V(not(implies(y, x)), not(implies(x, y)))) = big_hat(implies(y, x), implies(x, y))),
% 35.83/22.52 inference(symmetry,[status(thm)],[373])).
% 35.83/22.52 tff(375,plain,
% 35.83/22.52 (big_V(not(implies(y, x)), not(implies(x, y))) = not(implies(x, y))),
% 35.83/22.52 inference(transitivity,[status(thm)],[355, 353, 351, 350])).
% 35.83/22.52 tff(376,plain,
% 35.83/22.52 (not(big_V(not(implies(y, x)), not(implies(x, y)))) = not(not(implies(x, y)))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[375])).
% 35.83/22.52 tff(377,plain,
% 35.83/22.52 (not(not(implies(x, y))) = not(big_V(not(implies(y, x)), not(implies(x, y))))),
% 35.83/22.52 inference(symmetry,[status(thm)],[376])).
% 35.83/22.52 tff(378,plain,
% 35.83/22.52 (not(not(implies(x, y))) = big_hat(implies(y, x), implies(x, y))),
% 35.83/22.52 inference(transitivity,[status(thm)],[377, 374])).
% 35.83/22.52 tff(379,plain,
% 35.83/22.52 (not(not(not(implies(x, y)))) = not(big_hat(implies(y, x), implies(x, y)))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[378])).
% 35.83/22.52 tff(380,plain,
% 35.83/22.52 (not(not(not(implies(x, y)))) = not(implies(x, y))),
% 35.83/22.52 inference(transitivity,[status(thm)],[379, 364, 355, 353, 351, 350])).
% 35.83/22.52 tff(381,plain,
% 35.83/22.52 (implies(not(not(not(implies(x, y)))), not(implies(x, y))) = implies(not(implies(x, y)), not(implies(x, y)))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[380])).
% 35.83/22.52 tff(382,plain,
% 35.83/22.52 (big_V(truth, truth) = implies(not(not(implies(x, y))), not(not(implies(y, x))))),
% 35.83/22.52 inference(transitivity,[status(thm)],[338, 336, 334, 331, 31])).
% 35.83/22.52 tff(383,plain,
% 35.83/22.52 (implies(big_V(truth, truth), not(implies(x, y))) = implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), not(implies(x, y)))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[382])).
% 35.83/22.52 tff(384,plain,
% 35.83/22.52 ((~![Z: $i, Y: $i, X: $i] : (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z)))) | (implies(big_V(truth, truth), not(implies(x, y))) = big_hat(implies(truth, not(implies(x, y))), implies(truth, not(implies(x, y)))))),
% 35.83/22.52 inference(quant_inst,[status(thm)],[])).
% 35.83/22.52 tff(385,plain,
% 35.83/22.52 (implies(big_V(truth, truth), not(implies(x, y))) = big_hat(implies(truth, not(implies(x, y))), implies(truth, not(implies(x, y))))),
% 35.83/22.52 inference(unit_resolution,[status(thm)],[384, 194])).
% 35.83/22.52 tff(386,plain,
% 35.83/22.52 (big_hat(implies(truth, not(implies(x, y))), implies(truth, not(implies(x, y)))) = implies(big_V(truth, truth), not(implies(x, y)))),
% 35.83/22.52 inference(symmetry,[status(thm)],[385])).
% 35.83/22.52 tff(387,plain,
% 35.83/22.52 (not(implies(x, y)) = implies(truth, not(implies(x, y)))),
% 35.83/22.52 inference(symmetry,[status(thm)],[350])).
% 35.83/22.52 tff(388,plain,
% 35.83/22.52 (big_hat(not(implies(x, y)), not(implies(x, y))) = big_hat(implies(truth, not(implies(x, y))), implies(truth, not(implies(x, y))))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[387, 387])).
% 35.83/22.52 tff(389,plain,
% 35.83/22.52 (big_hat(not(implies(x, y)), not(implies(x, y))) = not(implies(x, y))),
% 35.83/22.52 inference(transitivity,[status(thm)],[388, 386, 383, 351, 350])).
% 35.83/22.52 tff(390,plain,
% 35.83/22.52 (implies(not(not(not(implies(x, y)))), big_hat(not(implies(x, y)), not(implies(x, y)))) = implies(not(not(not(implies(x, y)))), not(implies(x, y)))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[389])).
% 35.83/22.52 tff(391,plain,
% 35.83/22.52 (^[Z: $i, Y: $i, X: $i] : refl((implies(X, big_hat(Y, Z)) = big_hat(implies(X, Y), implies(X, Z))) <=> (implies(X, big_hat(Y, Z)) = big_hat(implies(X, Y), implies(X, Z))))),
% 35.83/22.52 inference(bind,[status(th)],[])).
% 35.83/22.52 tff(392,plain,
% 35.83/22.52 (![Z: $i, Y: $i, X: $i] : (implies(X, big_hat(Y, Z)) = big_hat(implies(X, Y), implies(X, Z))) <=> ![Z: $i, Y: $i, X: $i] : (implies(X, big_hat(Y, Z)) = big_hat(implies(X, Y), implies(X, Z)))),
% 35.83/22.52 inference(quant_intro,[status(thm)],[391])).
% 35.83/22.52 tff(393,plain,
% 35.83/22.52 (![Z: $i, Y: $i, X: $i] : (implies(X, big_hat(Y, Z)) = big_hat(implies(X, Y), implies(X, Z))) <=> ![Z: $i, Y: $i, X: $i] : (implies(X, big_hat(Y, Z)) = big_hat(implies(X, Y), implies(X, Z)))),
% 35.83/22.52 inference(rewrite,[status(thm)],[])).
% 35.83/22.52 tff(394,axiom,(![Z: $i, Y: $i, X: $i] : (implies(X, big_hat(Y, Z)) = big_hat(implies(X, Y), implies(X, Z)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','lemma_7')).
% 35.83/22.52 tff(395,plain,
% 35.83/22.52 (![Z: $i, Y: $i, X: $i] : (implies(X, big_hat(Y, Z)) = big_hat(implies(X, Y), implies(X, Z)))),
% 35.83/22.52 inference(modus_ponens,[status(thm)],[394, 393])).
% 35.83/22.52 tff(396,plain,(
% 35.83/22.52 ![Z: $i, Y: $i, X: $i] : (implies(X, big_hat(Y, Z)) = big_hat(implies(X, Y), implies(X, Z)))),
% 35.83/22.52 inference(skolemize,[status(sab)],[395])).
% 35.83/22.52 tff(397,plain,
% 35.83/22.52 (![Z: $i, Y: $i, X: $i] : (implies(X, big_hat(Y, Z)) = big_hat(implies(X, Y), implies(X, Z)))),
% 35.83/22.52 inference(modus_ponens,[status(thm)],[396, 392])).
% 35.83/22.52 tff(398,plain,
% 35.83/22.52 ((~![Z: $i, Y: $i, X: $i] : (implies(X, big_hat(Y, Z)) = big_hat(implies(X, Y), implies(X, Z)))) | (implies(not(not(not(implies(x, y)))), big_hat(not(implies(x, y)), not(implies(x, y)))) = big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y)))))),
% 35.83/22.52 inference(quant_inst,[status(thm)],[])).
% 35.83/22.52 tff(399,plain,
% 35.83/22.52 (implies(not(not(not(implies(x, y)))), big_hat(not(implies(x, y)), not(implies(x, y)))) = big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y))))),
% 35.83/22.52 inference(unit_resolution,[status(thm)],[398, 397])).
% 35.83/22.52 tff(400,plain,
% 35.83/22.52 (big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y)))) = implies(not(not(not(implies(x, y)))), big_hat(not(implies(x, y)), not(implies(x, y))))),
% 35.83/22.52 inference(symmetry,[status(thm)],[399])).
% 35.83/22.52 tff(401,plain,
% 35.83/22.52 (big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y)))) = implies(not(implies(x, y)), not(implies(x, y)))),
% 35.83/22.52 inference(transitivity,[status(thm)],[400, 390, 381])).
% 35.83/22.52 tff(402,plain,
% 35.83/22.52 (implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y))))) = implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(implies(x, y)), not(implies(x, y))))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[401])).
% 35.83/22.52 tff(403,plain,
% 35.83/22.52 (implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y))))) = implies(truth, implies(implies(x, y), implies(y, x)))),
% 35.83/22.52 inference(transitivity,[status(thm)],[402, 348, 346, 345])).
% 35.83/22.52 tff(404,plain,
% 35.83/22.52 (big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y))))), implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y)))))) = big_hat(implies(truth, implies(implies(x, y), implies(y, x))), implies(truth, implies(implies(x, y), implies(y, x))))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[403, 403])).
% 35.83/22.52 tff(405,plain,
% 35.83/22.52 ((~![Z: $i, Y: $i, X: $i] : (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z)))) | (implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y))))) = big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y))))), implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y)))))))),
% 35.83/22.52 inference(quant_inst,[status(thm)],[])).
% 35.83/22.52 tff(406,plain,
% 35.83/22.52 (implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y))))) = big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y))))), implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y))))))),
% 35.83/22.52 inference(unit_resolution,[status(thm)],[405, 194])).
% 35.83/22.52 tff(407,plain,
% 35.83/22.52 (implies(not(not(not(implies(x, y)))), not(implies(x, y))) = implies(not(not(not(implies(x, y)))), big_hat(not(implies(x, y)), not(implies(x, y))))),
% 35.83/22.52 inference(symmetry,[status(thm)],[390])).
% 35.83/22.52 tff(408,plain,
% 35.83/22.52 (implies(not(implies(x, y)), not(implies(x, y))) = implies(not(not(not(implies(x, y)))), not(implies(x, y)))),
% 35.83/22.52 inference(symmetry,[status(thm)],[381])).
% 35.83/22.52 tff(409,plain,
% 35.83/22.52 ((~![Y: $i, X: $i] : (big_V(X, Y) = implies(implies(X, Y), Y))) | (big_V(implies(x, y), implies(y, x)) = implies(implies(implies(x, y), implies(y, x)), implies(y, x)))),
% 35.83/22.52 inference(quant_inst,[status(thm)],[])).
% 35.83/22.52 tff(410,plain,
% 35.83/22.52 (big_V(implies(x, y), implies(y, x)) = implies(implies(implies(x, y), implies(y, x)), implies(y, x))),
% 35.83/22.52 inference(unit_resolution,[status(thm)],[409, 173])).
% 35.83/22.52 tff(411,plain,
% 35.83/22.52 (implies(implies(implies(x, y), implies(y, x)), implies(y, x)) = big_V(implies(x, y), implies(y, x))),
% 35.83/22.52 inference(symmetry,[status(thm)],[410])).
% 35.83/22.52 tff(412,plain,
% 35.83/22.52 ((~![X: $i] : (implies(truth, X) = X)) | (implies(truth, implies(implies(x, y), implies(y, x))) = implies(implies(x, y), implies(y, x)))),
% 35.83/22.52 inference(quant_inst,[status(thm)],[])).
% 35.83/22.52 tff(413,plain,
% 35.83/22.52 (implies(truth, implies(implies(x, y), implies(y, x))) = implies(implies(x, y), implies(y, x))),
% 35.83/22.52 inference(unit_resolution,[status(thm)],[412, 25])).
% 35.83/22.52 tff(414,plain,
% 35.83/22.52 (implies(implies(x, y), implies(y, x)) = implies(truth, implies(implies(x, y), implies(y, x)))),
% 35.83/22.52 inference(symmetry,[status(thm)],[413])).
% 35.83/22.52 tff(415,plain,
% 35.83/22.52 (implies(implies(x, y), implies(y, x)) = implies(not(implies(y, x)), not(implies(x, y)))),
% 35.83/22.52 inference(transitivity,[status(thm)],[414, 344, 18, 247, 330, 27])).
% 35.83/22.52 tff(416,plain,
% 35.83/22.52 (implies(implies(implies(x, y), implies(y, x)), implies(y, x)) = implies(implies(not(implies(y, x)), not(implies(x, y))), implies(y, x))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[415])).
% 35.83/22.52 tff(417,plain,
% 35.83/22.52 (implies(implies(not(implies(y, x)), not(implies(x, y))), implies(y, x)) = implies(implies(implies(x, y), implies(y, x)), implies(y, x))),
% 35.83/22.52 inference(symmetry,[status(thm)],[416])).
% 35.83/22.52 tff(418,plain,
% 35.83/22.52 (implies(truth, implies(y, x)) = implies(implies(not(implies(y, x)), not(implies(x, y))), implies(y, x))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[337])).
% 35.83/22.52 tff(419,plain,
% 35.83/22.52 ((~![X: $i] : (implies(truth, X) = X)) | (implies(truth, implies(y, x)) = implies(y, x))),
% 35.83/22.52 inference(quant_inst,[status(thm)],[])).
% 35.83/22.52 tff(420,plain,
% 35.83/22.52 (implies(truth, implies(y, x)) = implies(y, x)),
% 35.83/22.52 inference(unit_resolution,[status(thm)],[419, 25])).
% 35.83/22.52 tff(421,plain,
% 35.83/22.52 (implies(y, x) = implies(truth, implies(y, x))),
% 35.83/22.52 inference(symmetry,[status(thm)],[420])).
% 35.83/22.52 tff(422,plain,
% 35.83/22.52 (implies(y, x) = big_V(implies(x, y), implies(y, x))),
% 35.83/22.52 inference(transitivity,[status(thm)],[421, 418, 417, 411])).
% 35.83/22.52 tff(423,plain,
% 35.83/22.52 (not(implies(y, x)) = not(big_V(implies(x, y), implies(y, x)))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[422])).
% 35.83/22.52 tff(424,plain,
% 35.83/22.52 (not(big_V(implies(x, y), implies(y, x))) = not(implies(y, x))),
% 35.83/22.52 inference(symmetry,[status(thm)],[423])).
% 35.83/22.52 tff(425,plain,
% 35.83/22.52 (^[Y: $i, X: $i] : refl((not(big_V(X, Y)) = big_hat(not(X), not(Y))) <=> (not(big_V(X, Y)) = big_hat(not(X), not(Y))))),
% 35.83/22.52 inference(bind,[status(th)],[])).
% 35.83/22.52 tff(426,plain,
% 35.83/22.52 (![Y: $i, X: $i] : (not(big_V(X, Y)) = big_hat(not(X), not(Y))) <=> ![Y: $i, X: $i] : (not(big_V(X, Y)) = big_hat(not(X), not(Y)))),
% 35.83/22.52 inference(quant_intro,[status(thm)],[425])).
% 35.83/22.52 tff(427,plain,
% 35.83/22.52 (![Y: $i, X: $i] : (not(big_V(X, Y)) = big_hat(not(X), not(Y))) <=> ![Y: $i, X: $i] : (not(big_V(X, Y)) = big_hat(not(X), not(Y)))),
% 35.83/22.52 inference(rewrite,[status(thm)],[])).
% 35.83/22.52 tff(428,axiom,(![Y: $i, X: $i] : (not(big_V(X, Y)) = big_hat(not(X), not(Y)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','lemma_4')).
% 35.83/22.52 tff(429,plain,
% 35.83/22.52 (![Y: $i, X: $i] : (not(big_V(X, Y)) = big_hat(not(X), not(Y)))),
% 35.83/22.52 inference(modus_ponens,[status(thm)],[428, 427])).
% 35.83/22.52 tff(430,plain,(
% 35.83/22.52 ![Y: $i, X: $i] : (not(big_V(X, Y)) = big_hat(not(X), not(Y)))),
% 35.83/22.52 inference(skolemize,[status(sab)],[429])).
% 35.83/22.52 tff(431,plain,
% 35.83/22.52 (![Y: $i, X: $i] : (not(big_V(X, Y)) = big_hat(not(X), not(Y)))),
% 35.83/22.52 inference(modus_ponens,[status(thm)],[430, 426])).
% 35.83/22.52 tff(432,plain,
% 35.83/22.52 ((~![Y: $i, X: $i] : (not(big_V(X, Y)) = big_hat(not(X), not(Y)))) | (not(big_V(implies(x, y), implies(y, x))) = big_hat(not(implies(x, y)), not(implies(y, x))))),
% 35.83/22.52 inference(quant_inst,[status(thm)],[])).
% 35.83/22.52 tff(433,plain,
% 35.83/22.52 (not(big_V(implies(x, y), implies(y, x))) = big_hat(not(implies(x, y)), not(implies(y, x)))),
% 35.83/22.52 inference(unit_resolution,[status(thm)],[432, 431])).
% 35.83/22.52 tff(434,plain,
% 35.83/22.52 (big_hat(not(implies(x, y)), not(implies(y, x))) = not(big_V(implies(x, y), implies(y, x)))),
% 35.83/22.52 inference(symmetry,[status(thm)],[433])).
% 35.83/22.52 tff(435,plain,
% 35.83/22.52 ((~![X: $i] : (implies(truth, X) = X)) | (implies(truth, big_hat(not(implies(x, y)), not(implies(y, x)))) = big_hat(not(implies(x, y)), not(implies(y, x))))),
% 35.83/22.52 inference(quant_inst,[status(thm)],[])).
% 35.83/22.52 tff(436,plain,
% 35.83/22.52 (implies(truth, big_hat(not(implies(x, y)), not(implies(y, x)))) = big_hat(not(implies(x, y)), not(implies(y, x)))),
% 35.83/22.52 inference(unit_resolution,[status(thm)],[435, 25])).
% 35.83/22.52 tff(437,plain,
% 35.83/22.52 (implies(truth, big_hat(not(implies(x, y)), not(implies(y, x)))) = not(implies(y, x))),
% 35.83/22.52 inference(transitivity,[status(thm)],[436, 434, 424])).
% 35.83/22.52 tff(438,plain,
% 35.83/22.52 (not(implies(y, x)) = big_hat(not(implies(x, y)), not(implies(y, x)))),
% 35.83/22.52 inference(transitivity,[status(thm)],[423, 433])).
% 35.83/22.52 tff(439,plain,
% 35.83/22.52 (implies(not(implies(x, y)), not(implies(y, x))) = implies(not(implies(x, y)), big_hat(not(implies(x, y)), not(implies(y, x))))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[438])).
% 35.83/22.52 tff(440,plain,
% 35.83/22.52 (implies(not(implies(x, y)), big_hat(not(implies(x, y)), not(implies(y, x)))) = implies(not(implies(x, y)), not(implies(y, x)))),
% 35.83/22.52 inference(symmetry,[status(thm)],[439])).
% 35.83/22.52 tff(441,plain,
% 35.83/22.52 (implies(implies(not(implies(x, y)), big_hat(not(implies(x, y)), not(implies(y, x)))), implies(truth, big_hat(not(implies(x, y)), not(implies(y, x))))) = implies(implies(not(implies(x, y)), not(implies(y, x))), not(implies(y, x)))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[440, 437])).
% 35.83/22.52 tff(442,plain,
% 35.83/22.52 (implies(implies(not(implies(x, y)), not(implies(y, x))), not(implies(y, x))) = implies(implies(not(implies(x, y)), big_hat(not(implies(x, y)), not(implies(y, x)))), implies(truth, big_hat(not(implies(x, y)), not(implies(y, x)))))),
% 35.83/22.52 inference(symmetry,[status(thm)],[441])).
% 35.83/22.52 tff(443,plain,
% 35.83/22.52 (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), not(implies(x, y))) = implies(implies(not(implies(y, x)), not(implies(x, y))), not(implies(x, y)))),
% 35.83/22.52 inference(symmetry,[status(thm)],[353])).
% 35.83/22.52 tff(444,plain,
% 35.83/22.52 (implies(truth, not(implies(x, y))) = implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), not(implies(x, y)))),
% 35.83/22.52 inference(symmetry,[status(thm)],[351])).
% 35.83/22.52 tff(445,plain,
% 35.83/22.52 (not(implies(x, y)) = implies(implies(not(implies(x, y)), big_hat(not(implies(x, y)), not(implies(y, x)))), implies(truth, big_hat(not(implies(x, y)), not(implies(y, x)))))),
% 35.83/22.52 inference(transitivity,[status(thm)],[387, 444, 443, 255, 442])).
% 35.83/22.52 tff(446,plain,
% 35.83/22.52 (implies(not(implies(x, y)), not(implies(x, y))) = implies(implies(truth, not(implies(x, y))), implies(implies(not(implies(x, y)), big_hat(not(implies(x, y)), not(implies(y, x)))), implies(truth, big_hat(not(implies(x, y)), not(implies(y, x))))))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[387, 445])).
% 35.83/22.52 tff(447,plain,
% 35.83/22.52 (implies(implies(truth, not(implies(x, y))), implies(implies(not(implies(x, y)), big_hat(not(implies(x, y)), not(implies(y, x)))), implies(truth, big_hat(not(implies(x, y)), not(implies(y, x)))))) = implies(not(implies(x, y)), not(implies(x, y)))),
% 35.83/22.52 inference(symmetry,[status(thm)],[446])).
% 35.83/22.52 tff(448,plain,
% 35.83/22.52 ((~![Z: $i, Y: $i, X: $i] : (implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth)) | (implies(implies(truth, not(implies(x, y))), implies(implies(not(implies(x, y)), big_hat(not(implies(x, y)), not(implies(y, x)))), implies(truth, big_hat(not(implies(x, y)), not(implies(y, x)))))) = truth)),
% 35.83/22.52 inference(quant_inst,[status(thm)],[])).
% 35.83/22.52 tff(449,plain,
% 35.83/22.52 (implies(implies(truth, not(implies(x, y))), implies(implies(not(implies(x, y)), big_hat(not(implies(x, y)), not(implies(y, x)))), implies(truth, big_hat(not(implies(x, y)), not(implies(y, x)))))) = truth),
% 35.83/22.52 inference(unit_resolution,[status(thm)],[448, 7])).
% 35.83/22.52 tff(450,plain,
% 35.83/22.52 (truth = implies(implies(truth, not(implies(x, y))), implies(implies(not(implies(x, y)), big_hat(not(implies(x, y)), not(implies(y, x)))), implies(truth, big_hat(not(implies(x, y)), not(implies(y, x))))))),
% 35.83/22.52 inference(symmetry,[status(thm)],[449])).
% 35.83/22.52 tff(451,plain,
% 35.83/22.52 (truth = big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y))))),
% 35.83/22.52 inference(transitivity,[status(thm)],[450, 447, 408, 407, 399])).
% 35.83/22.52 tff(452,plain,
% 35.83/22.52 (implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), truth) = implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), big_hat(implies(not(not(not(implies(x, y)))), not(implies(x, y))), implies(not(not(not(implies(x, y)))), not(implies(x, y)))))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[451])).
% 35.83/22.52 tff(453,plain,
% 35.83/22.52 (implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), truth) = truth),
% 35.83/22.52 inference(transitivity,[status(thm)],[452, 406, 404, 343, 340, 18])).
% 35.83/22.52 tff(454,plain,
% 35.83/22.52 (implies(implies(not(not(implies(x, y))), truth), implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), truth)) = implies(implies(not(not(implies(x, y))), truth), truth)),
% 35.83/22.52 inference(monotonicity,[status(thm)],[453])).
% 35.83/22.52 tff(455,plain,
% 35.83/22.52 (implies(implies(not(not(implies(x, y))), truth), truth) = implies(implies(not(not(implies(x, y))), truth), implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), truth))),
% 35.83/22.52 inference(symmetry,[status(thm)],[454])).
% 35.83/22.52 tff(456,plain,
% 35.83/22.52 ((~![Y: $i, X: $i] : (implies(implies(X, Y), Y) = implies(implies(Y, X), X))) | (implies(implies(truth, not(not(implies(x, y)))), not(not(implies(x, y)))) = implies(implies(not(not(implies(x, y))), truth), truth))),
% 35.83/22.52 inference(quant_inst,[status(thm)],[])).
% 35.83/22.52 tff(457,plain,
% 35.83/22.52 (implies(implies(truth, not(not(implies(x, y)))), not(not(implies(x, y)))) = implies(implies(not(not(implies(x, y))), truth), truth)),
% 35.83/22.52 inference(unit_resolution,[status(thm)],[456, 116])).
% 35.83/22.52 tff(458,plain,
% 35.83/22.52 ((~![X: $i] : (implies(truth, X) = X)) | (implies(truth, not(not(implies(x, y)))) = not(not(implies(x, y))))),
% 35.83/22.52 inference(quant_inst,[status(thm)],[])).
% 35.83/22.52 tff(459,plain,
% 35.83/22.52 (implies(truth, not(not(implies(x, y)))) = not(not(implies(x, y)))),
% 35.83/22.52 inference(unit_resolution,[status(thm)],[458, 25])).
% 35.83/22.52 tff(460,plain,
% 35.83/22.52 (implies(implies(truth, not(not(implies(x, y)))), not(not(implies(x, y)))) = implies(not(not(implies(x, y))), not(not(implies(x, y))))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[459])).
% 35.83/22.52 tff(461,plain,
% 35.83/22.52 (implies(not(not(implies(x, y))), not(not(implies(x, y)))) = implies(implies(truth, not(not(implies(x, y)))), not(not(implies(x, y))))),
% 35.83/22.52 inference(symmetry,[status(thm)],[460])).
% 35.83/22.52 tff(462,plain,
% 35.83/22.52 ((~![X: $i] : (implies(truth, X) = X)) | (implies(truth, implies(not(not(implies(x, y))), not(not(implies(x, y))))) = implies(not(not(implies(x, y))), not(not(implies(x, y)))))),
% 35.83/22.52 inference(quant_inst,[status(thm)],[])).
% 35.83/22.52 tff(463,plain,
% 35.83/22.52 (implies(truth, implies(not(not(implies(x, y))), not(not(implies(x, y))))) = implies(not(not(implies(x, y))), not(not(implies(x, y))))),
% 35.83/22.52 inference(unit_resolution,[status(thm)],[462, 25])).
% 35.83/22.52 tff(464,plain,
% 35.83/22.52 (implies(implies(truth, not(not(implies(x, y)))), implies(truth, not(not(implies(x, y))))) = implies(not(not(implies(x, y))), not(not(implies(x, y))))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[459, 459])).
% 35.83/22.52 tff(465,plain,
% 35.83/22.52 (implies(not(not(implies(x, y))), not(not(implies(x, y)))) = implies(implies(truth, not(not(implies(x, y)))), implies(truth, not(not(implies(x, y)))))),
% 35.83/22.52 inference(symmetry,[status(thm)],[464])).
% 35.83/22.52 tff(466,plain,
% 35.83/22.52 (implies(truth, truth) = implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth)),
% 35.83/22.52 inference(monotonicity,[status(thm)],[328])).
% 35.83/22.52 tff(467,plain,
% 35.83/22.52 (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth) = implies(truth, truth)),
% 35.83/22.52 inference(symmetry,[status(thm)],[466])).
% 35.83/22.52 tff(468,plain,
% 35.83/22.52 (implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x))))) = implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth)),
% 35.83/22.52 inference(symmetry,[status(thm)],[331])).
% 35.83/22.52 tff(469,plain,
% 35.83/22.52 (implies(not(not(implies(x, y))), not(not(implies(y, x)))) = implies(truth, implies(not(not(implies(x, y))), not(not(implies(y, x)))))),
% 35.83/22.52 inference(symmetry,[status(thm)],[31])).
% 35.83/22.52 tff(470,plain,
% 35.83/22.52 (truth = implies(truth, truth)),
% 35.83/22.52 inference(transitivity,[status(thm)],[328, 469, 468, 467])).
% 35.83/22.52 tff(471,plain,
% 35.83/22.52 (implies(truth, implies(not(not(implies(x, y))), not(not(implies(x, y))))) = implies(implies(truth, truth), implies(implies(truth, not(not(implies(x, y)))), implies(truth, not(not(implies(x, y))))))),
% 35.83/22.52 inference(monotonicity,[status(thm)],[470, 465])).
% 35.83/22.52 tff(472,plain,
% 35.83/22.52 (implies(implies(truth, truth), implies(implies(truth, not(not(implies(x, y)))), implies(truth, not(not(implies(x, y)))))) = implies(truth, implies(not(not(implies(x, y))), not(not(implies(x, y)))))),
% 35.83/22.52 inference(symmetry,[status(thm)],[471])).
% 35.83/22.52 tff(473,plain,
% 35.83/22.52 ((~![Z: $i, Y: $i, X: $i] : (implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth)) | (implies(implies(truth, truth), implies(implies(truth, not(not(implies(x, y)))), implies(truth, not(not(implies(x, y)))))) = truth)),
% 35.83/22.52 inference(quant_inst,[status(thm)],[])).
% 35.83/22.52 tff(474,plain,
% 35.83/22.52 (implies(implies(truth, truth), implies(implies(truth, not(not(implies(x, y)))), implies(truth, not(not(implies(x, y)))))) = truth),
% 35.83/22.52 inference(unit_resolution,[status(thm)],[473, 7])).
% 35.83/22.52 tff(475,plain,
% 35.83/22.52 (truth = implies(implies(truth, truth), implies(implies(truth, not(not(implies(x, y)))), implies(truth, not(not(implies(x, y))))))),
% 35.83/22.52 inference(symmetry,[status(thm)],[474])).
% 35.83/22.52 tff(476,plain,
% 35.83/22.52 (truth = implies(implies(not(not(implies(x, y))), truth), implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), truth))),
% 35.83/22.52 inference(transitivity,[status(thm)],[475, 472, 463, 461, 457, 455])).
% 35.83/22.52 tff(477,plain,
% 35.83/22.52 ((~![Z: $i, Y: $i, X: $i] : (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z)))) | (implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), not(not(implies(x, y)))) = big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), not(not(implies(x, y)))), implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), not(not(implies(x, y))))))),
% 35.83/22.52 inference(quant_inst,[status(thm)],[])).
% 35.83/22.52 tff(478,plain,
% 35.83/22.52 (implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), not(not(implies(x, y)))) = big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), not(not(implies(x, y)))), implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), not(not(implies(x, y)))))),
% 35.83/22.52 inference(unit_resolution,[status(thm)],[477, 194])).
% 35.83/22.52 tff(479,plain,
% 35.83/22.52 (big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), not(not(implies(x, y)))), implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), not(not(implies(x, y))))) = implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), not(not(implies(x, y))))),
% 35.83/22.53 inference(symmetry,[status(thm)],[478])).
% 35.83/22.53 tff(480,plain,
% 35.83/22.53 (not(big_hat(not(implies(x, y)), not(implies(x, y)))) = not(not(implies(x, y)))),
% 35.83/22.53 inference(monotonicity,[status(thm)],[389])).
% 35.83/22.53 tff(481,plain,
% 35.83/22.53 ((~![Y: $i, X: $i] : (not(big_hat(X, Y)) = big_V(not(X), not(Y)))) | (not(big_hat(not(implies(x, y)), not(implies(x, y)))) = big_V(not(not(implies(x, y))), not(not(implies(x, y)))))),
% 35.83/22.53 inference(quant_inst,[status(thm)],[])).
% 35.83/22.53 tff(482,plain,
% 35.83/22.53 (not(big_hat(not(implies(x, y)), not(implies(x, y)))) = big_V(not(not(implies(x, y))), not(not(implies(x, y))))),
% 35.83/22.53 inference(unit_resolution,[status(thm)],[481, 362])).
% 35.83/22.53 tff(483,plain,
% 35.83/22.53 (big_V(not(not(implies(x, y))), not(not(implies(x, y)))) = not(big_hat(not(implies(x, y)), not(implies(x, y))))),
% 35.83/22.53 inference(symmetry,[status(thm)],[482])).
% 35.83/22.53 tff(484,plain,
% 35.83/22.53 ((~![Y: $i, X: $i] : (big_V(X, Y) = implies(implies(X, Y), Y))) | (big_V(not(not(implies(x, y))), not(not(implies(x, y)))) = implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), not(not(implies(x, y)))))),
% 35.83/22.53 inference(quant_inst,[status(thm)],[])).
% 35.83/22.53 tff(485,plain,
% 35.83/22.53 (big_V(not(not(implies(x, y))), not(not(implies(x, y)))) = implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), not(not(implies(x, y))))),
% 35.83/22.53 inference(unit_resolution,[status(thm)],[484, 173])).
% 35.83/22.53 tff(486,plain,
% 35.83/22.53 (implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), not(not(implies(x, y)))) = big_V(not(not(implies(x, y))), not(not(implies(x, y))))),
% 35.83/22.53 inference(symmetry,[status(thm)],[485])).
% 35.83/22.53 tff(487,plain,
% 35.83/22.53 (implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), not(not(implies(x, y)))) = not(not(implies(x, y)))),
% 35.83/22.53 inference(transitivity,[status(thm)],[486, 483, 480])).
% 35.83/22.53 tff(488,plain,
% 35.83/22.53 (big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), not(not(implies(x, y)))), implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), not(not(implies(x, y))))) = big_hat(not(not(implies(x, y))), not(not(implies(x, y))))),
% 35.83/22.53 inference(monotonicity,[status(thm)],[487, 487])).
% 35.83/22.53 tff(489,plain,
% 35.83/22.53 (big_hat(not(not(implies(x, y))), not(not(implies(x, y)))) = big_hat(implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), not(not(implies(x, y)))), implies(implies(not(not(implies(x, y))), not(not(implies(x, y)))), not(not(implies(x, y)))))),
% 35.83/22.53 inference(symmetry,[status(thm)],[488])).
% 35.83/22.53 tff(490,plain,
% 35.83/22.53 (not(not(implies(x, y))) = implies(truth, not(not(implies(x, y))))),
% 35.83/22.53 inference(symmetry,[status(thm)],[459])).
% 35.83/22.53 tff(491,plain,
% 35.83/22.53 (big_hat(not(not(implies(x, y))), not(not(implies(x, y)))) = big_hat(implies(truth, not(not(implies(x, y)))), implies(truth, not(not(implies(x, y)))))),
% 35.83/22.53 inference(monotonicity,[status(thm)],[490, 490])).
% 35.83/22.53 tff(492,plain,
% 35.83/22.53 (big_hat(implies(truth, not(not(implies(x, y)))), implies(truth, not(not(implies(x, y))))) = big_hat(not(not(implies(x, y))), not(not(implies(x, y))))),
% 35.83/22.53 inference(symmetry,[status(thm)],[491])).
% 35.83/22.53 tff(493,plain,
% 35.83/22.53 ((~![Z: $i, Y: $i, X: $i] : (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z)))) | (implies(big_V(truth, truth), not(not(implies(x, y)))) = big_hat(implies(truth, not(not(implies(x, y)))), implies(truth, not(not(implies(x, y))))))),
% 35.83/22.53 inference(quant_inst,[status(thm)],[])).
% 35.83/22.53 tff(494,plain,
% 35.83/22.53 (implies(big_V(truth, truth), not(not(implies(x, y)))) = big_hat(implies(truth, not(not(implies(x, y)))), implies(truth, not(not(implies(x, y)))))),
% 35.83/22.53 inference(unit_resolution,[status(thm)],[493, 194])).
% 35.83/22.53 tff(495,plain,
% 35.83/22.53 (big_V(truth, truth) = truth),
% 35.83/22.53 inference(transitivity,[status(thm)],[338, 336, 334, 331, 31, 327])).
% 35.83/22.53 tff(496,plain,
% 35.83/22.53 (implies(big_V(truth, truth), not(not(implies(x, y)))) = implies(truth, not(not(implies(x, y))))),
% 35.83/22.53 inference(monotonicity,[status(thm)],[495])).
% 35.83/22.53 tff(497,plain,
% 35.83/22.53 (implies(truth, not(not(implies(x, y)))) = implies(big_V(truth, truth), not(not(implies(x, y))))),
% 35.83/22.53 inference(symmetry,[status(thm)],[496])).
% 35.83/22.53 tff(498,plain,
% 35.83/22.53 (not(not(implies(x, y))) = implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), not(not(implies(x, y))))),
% 35.83/22.53 inference(transitivity,[status(thm)],[490, 497, 494, 492, 489, 479])).
% 35.83/22.53 tff(499,plain,
% 35.83/22.53 (implies(not(not(implies(x, y))), truth) = implies(implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), not(not(implies(x, y)))), implies(implies(not(not(implies(x, y))), truth), implies(big_V(implies(not(not(implies(x, y))), not(not(implies(x, y)))), implies(not(not(implies(x, y))), not(not(implies(x, y))))), truth)))),
% 35.83/22.53 inference(monotonicity,[status(thm)],[498, 476])).
% 35.83/22.53 tff(500,plain,
% 35.83/22.53 (implies(not(not(implies(x, y))), not(not(implies(x, y)))) = implies(truth, implies(not(not(implies(x, y))), not(not(implies(x, y)))))),
% 35.83/22.53 inference(symmetry,[status(thm)],[463])).
% 35.83/22.53 tff(501,plain,
% 35.83/22.53 (implies(not(not(implies(x, y))), not(not(implies(x, y)))) = truth),
% 35.83/22.53 inference(transitivity,[status(thm)],[500, 471, 474])).
% 35.83/22.53 tff(502,plain,
% 35.83/22.53 (implies(not(not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(x, y))))) = implies(not(not(implies(x, y))), truth)),
% 35.83/22.53 inference(monotonicity,[status(thm)],[501])).
% 35.83/22.53 tff(503,plain,
% 35.83/22.53 (implies(not(not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(x, y))))) = truth),
% 35.83/22.53 inference(transitivity,[status(thm)],[502, 499, 9])).
% 35.83/22.53 tff(504,plain,
% 35.83/22.53 (implies(not(not(implies(x, y))), truth) = implies(not(not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(x, y)))))),
% 35.83/22.53 inference(symmetry,[status(thm)],[502])).
% 35.83/22.53 tff(505,plain,
% 35.83/22.53 ((implies(not(not(implies(x, y))), truth) = truth) <=> (implies(not(not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(x, y))))) = truth)),
% 35.83/22.53 inference(monotonicity,[status(thm)],[504])).
% 35.83/22.53 tff(506,plain,
% 35.83/22.53 ((~(implies(not(not(implies(x, y))), truth) = truth)) <=> (~(implies(not(not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(x, y))))) = truth))),
% 35.83/22.53 inference(monotonicity,[status(thm)],[505])).
% 35.83/22.53 tff(507,plain,
% 35.83/22.53 ((~![X: $i] : (implies(truth, X) = X)) | (implies(truth, implies(x, y)) = implies(x, y))),
% 35.83/22.53 inference(quant_inst,[status(thm)],[])).
% 35.83/22.53 tff(508,plain,
% 35.83/22.53 (implies(truth, implies(x, y)) = implies(x, y)),
% 35.83/22.53 inference(unit_resolution,[status(thm)],[507, 25])).
% 35.83/22.53 tff(509,plain,
% 35.83/22.53 (big_V(truth, implies(not(implies(y, x)), not(implies(x, y)))) = big_V(truth, truth)),
% 35.83/22.53 inference(symmetry,[status(thm)],[338])).
% 35.83/22.53 tff(510,plain,
% 35.83/22.53 (implies(implies(truth, implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y)))) = big_V(truth, implies(not(implies(y, x)), not(implies(x, y))))),
% 35.83/22.53 inference(symmetry,[status(thm)],[336])).
% 35.83/22.53 tff(511,plain,
% 35.83/22.53 (implies(implies(not(not(implies(x, y))), not(not(implies(y, x)))), truth) = implies(implies(truth, implies(not(implies(y, x)), not(implies(x, y)))), implies(not(implies(y, x)), not(implies(x, y))))),
% 35.83/22.53 inference(symmetry,[status(thm)],[334])).
% 35.83/22.53 tff(512,plain,
% 35.83/22.53 (truth = big_V(truth, truth)),
% 35.83/22.53 inference(transitivity,[status(thm)],[328, 469, 468, 511, 510, 509])).
% 35.83/22.53 tff(513,plain,
% 35.83/22.53 (implies(truth, implies(x, y)) = implies(big_V(truth, truth), implies(x, y))),
% 35.83/22.53 inference(monotonicity,[status(thm)],[512])).
% 35.83/22.53 tff(514,plain,
% 35.83/22.53 (implies(big_V(truth, truth), implies(x, y)) = implies(truth, implies(x, y))),
% 35.83/22.53 inference(symmetry,[status(thm)],[513])).
% 35.83/22.53 tff(515,plain,
% 35.83/22.53 ((~![Z: $i, Y: $i, X: $i] : (implies(big_V(X, Y), Z) = big_hat(implies(X, Z), implies(Y, Z)))) | (implies(big_V(truth, truth), implies(x, y)) = big_hat(implies(truth, implies(x, y)), implies(truth, implies(x, y))))),
% 35.83/22.53 inference(quant_inst,[status(thm)],[])).
% 35.83/22.53 tff(516,plain,
% 35.83/22.53 (implies(big_V(truth, truth), implies(x, y)) = big_hat(implies(truth, implies(x, y)), implies(truth, implies(x, y)))),
% 35.83/22.53 inference(unit_resolution,[status(thm)],[515, 194])).
% 35.83/22.53 tff(517,plain,
% 35.83/22.53 (big_hat(implies(truth, implies(x, y)), implies(truth, implies(x, y))) = implies(big_V(truth, truth), implies(x, y))),
% 35.83/22.53 inference(symmetry,[status(thm)],[516])).
% 35.83/22.53 tff(518,plain,
% 35.83/22.53 (big_hat(implies(truth, implies(x, y)), implies(truth, implies(x, y))) = big_hat(implies(x, y), implies(x, y))),
% 35.83/22.53 inference(monotonicity,[status(thm)],[508, 508])).
% 35.83/22.53 tff(519,plain,
% 35.83/22.53 (big_hat(implies(x, y), implies(x, y)) = big_hat(implies(truth, implies(x, y)), implies(truth, implies(x, y)))),
% 35.83/22.53 inference(symmetry,[status(thm)],[518])).
% 35.83/22.53 tff(520,plain,
% 35.83/22.53 ((~![Y: $i, X: $i] : (big_hat(X, Y) = not(big_V(not(X), not(Y))))) | (big_hat(implies(x, y), implies(x, y)) = not(big_V(not(implies(x, y)), not(implies(x, y)))))),
% 35.83/22.53 inference(quant_inst,[status(thm)],[])).
% 35.83/22.53 tff(521,plain,
% 35.83/22.53 (big_hat(implies(x, y), implies(x, y)) = not(big_V(not(implies(x, y)), not(implies(x, y))))),
% 35.83/22.53 inference(unit_resolution,[status(thm)],[520, 371])).
% 35.83/22.53 tff(522,plain,
% 35.83/22.53 (not(big_V(not(implies(x, y)), not(implies(x, y)))) = big_hat(implies(x, y), implies(x, y))),
% 35.83/22.53 inference(symmetry,[status(thm)],[521])).
% 35.83/22.53 tff(523,plain,
% 35.83/22.53 ((~![Y: $i, X: $i] : (not(big_V(X, Y)) = big_hat(not(X), not(Y)))) | (not(big_V(not(implies(x, y)), not(implies(x, y)))) = big_hat(not(not(implies(x, y))), not(not(implies(x, y)))))),
% 35.83/22.53 inference(quant_inst,[status(thm)],[])).
% 35.83/22.53 tff(524,plain,
% 35.83/22.53 (not(big_V(not(implies(x, y)), not(implies(x, y)))) = big_hat(not(not(implies(x, y))), not(not(implies(x, y))))),
% 35.83/22.53 inference(unit_resolution,[status(thm)],[523, 431])).
% 35.83/22.53 tff(525,plain,
% 35.83/22.53 (big_hat(not(not(implies(x, y))), not(not(implies(x, y)))) = not(big_V(not(implies(x, y)), not(implies(x, y))))),
% 35.83/22.53 inference(symmetry,[status(thm)],[524])).
% 35.83/22.53 tff(526,plain,
% 35.83/22.53 (not(not(implies(x, y))) = implies(x, y)),
% 35.83/22.53 inference(transitivity,[status(thm)],[490, 497, 494, 492, 525, 522, 519, 517, 514, 508])).
% 35.83/22.53 tff(527,plain,
% 35.83/22.53 (ordered(not(not(implies(x, y))), truth) <=> ordered(implies(x, y), truth)),
% 35.83/22.53 inference(monotonicity,[status(thm)],[526])).
% 35.83/22.53 tff(528,plain,
% 35.83/22.53 (ordered(implies(x, y), truth) <=> ordered(not(not(implies(x, y))), truth)),
% 35.83/22.53 inference(symmetry,[status(thm)],[527])).
% 35.83/22.53 tff(529,plain,
% 35.83/22.53 ((~ordered(implies(x, y), truth)) <=> (~ordered(not(not(implies(x, y))), truth))),
% 35.83/22.53 inference(monotonicity,[status(thm)],[528])).
% 35.83/22.53 tff(530,plain,
% 35.83/22.53 (ordered(implies(implies(x, y), implies(y, x)), implies(truth, implies(y, x))) <=> ordered(implies(not(implies(y, x)), not(implies(x, y))), implies(y, x))),
% 35.83/22.53 inference(monotonicity,[status(thm)],[415, 420])).
% 35.83/22.53 tff(531,plain,
% 35.83/22.53 (ordered(implies(not(implies(y, x)), not(implies(x, y))), implies(y, x)) <=> ordered(implies(implies(x, y), implies(y, x)), implies(truth, implies(y, x)))),
% 35.83/22.53 inference(symmetry,[status(thm)],[530])).
% 35.83/22.53 tff(532,plain,
% 35.83/22.53 ((~ordered(implies(not(implies(y, x)), not(implies(x, y))), implies(y, x))) <=> (~ordered(implies(implies(x, y), implies(y, x)), implies(truth, implies(y, x))))),
% 35.83/22.53 inference(monotonicity,[status(thm)],[531])).
% 35.83/22.53 tff(533,plain,
% 35.83/22.53 (ordered(implies(implies(x, y), implies(y, x)), implies(y, x)) <=> ordered(implies(not(implies(y, x)), not(implies(x, y))), implies(y, x))),
% 35.83/22.53 inference(monotonicity,[status(thm)],[415])).
% 35.83/22.53 tff(534,plain,
% 35.83/22.53 ((~ordered(implies(implies(x, y), implies(y, x)), implies(y, x))) <=> (~ordered(implies(not(implies(y, x)), not(implies(x, y))), implies(y, x)))),
% 35.83/22.53 inference(monotonicity,[status(thm)],[533])).
% 35.83/22.53 tff(535,plain,
% 35.83/22.53 ((implies(implies(implies(x, y), implies(y, x)), implies(y, x)) = truth) <=> (big_V(implies(x, y), implies(y, x)) = truth)),
% 35.83/22.53 inference(monotonicity,[status(thm)],[411])).
% 35.83/22.53 tff(536,plain,
% 35.83/22.53 ((big_V(implies(x, y), implies(y, x)) = truth) <=> (implies(implies(implies(x, y), implies(y, x)), implies(y, x)) = truth)),
% 35.83/22.53 inference(symmetry,[status(thm)],[535])).
% 35.83/22.53 tff(537,plain,
% 35.83/22.53 ((~(big_V(implies(x, y), implies(y, x)) = truth)) <=> (~(implies(implies(implies(x, y), implies(y, x)), implies(y, x)) = truth))),
% 35.83/22.53 inference(monotonicity,[status(thm)],[536])).
% 35.83/22.53 tff(538,plain,
% 35.83/22.53 ((~(big_V(implies(x, y), implies(y, x)) = truth)) <=> (~(big_V(implies(x, y), implies(y, x)) = truth))),
% 35.83/22.53 inference(rewrite,[status(thm)],[])).
% 35.83/22.53 tff(539,axiom,(~(big_V(implies(x, y), implies(y, x)) = truth)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_mv_4')).
% 35.87/22.53 tff(540,plain,
% 35.87/22.53 (~(big_V(implies(x, y), implies(y, x)) = truth)),
% 35.87/22.53 inference(modus_ponens,[status(thm)],[539, 538])).
% 35.87/22.53 tff(541,plain,
% 35.87/22.53 (~(implies(implies(implies(x, y), implies(y, x)), implies(y, x)) = truth)),
% 35.87/22.53 inference(modus_ponens,[status(thm)],[540, 537])).
% 35.87/22.53 tff(542,plain,
% 35.87/22.53 (((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | ((~ordered(implies(implies(x, y), implies(y, x)), implies(y, x))) | (implies(implies(implies(x, y), implies(y, x)), implies(y, x)) = truth))) <=> ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | (~ordered(implies(implies(x, y), implies(y, x)), implies(y, x))) | (implies(implies(implies(x, y), implies(y, x)), implies(y, x)) = truth))),
% 35.87/22.53 inference(rewrite,[status(thm)],[])).
% 35.87/22.53 tff(543,plain,
% 35.87/22.53 ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | ((~ordered(implies(implies(x, y), implies(y, x)), implies(y, x))) | (implies(implies(implies(x, y), implies(y, x)), implies(y, x)) = truth))),
% 35.87/22.53 inference(quant_inst,[status(thm)],[])).
% 35.87/22.53 tff(544,plain,
% 35.87/22.53 ((~![Y: $i, X: $i] : ((~ordered(X, Y)) | (implies(X, Y) = truth))) | (~ordered(implies(implies(x, y), implies(y, x)), implies(y, x))) | (implies(implies(implies(x, y), implies(y, x)), implies(y, x)) = truth)),
% 35.87/22.53 inference(modus_ponens,[status(thm)],[543, 542])).
% 35.87/22.53 tff(545,plain,
% 35.87/22.53 ((~ordered(implies(implies(x, y), implies(y, x)), implies(y, x))) | (implies(implies(implies(x, y), implies(y, x)), implies(y, x)) = truth)),
% 35.87/22.53 inference(unit_resolution,[status(thm)],[544, 92])).
% 35.87/22.53 tff(546,plain,
% 35.87/22.53 (~ordered(implies(implies(x, y), implies(y, x)), implies(y, x))),
% 35.87/22.53 inference(unit_resolution,[status(thm)],[545, 541])).
% 35.87/22.53 tff(547,plain,
% 35.87/22.53 (~ordered(implies(not(implies(y, x)), not(implies(x, y))), implies(y, x))),
% 35.87/22.53 inference(modus_ponens,[status(thm)],[546, 534])).
% 35.87/22.53 tff(548,plain,
% 35.87/22.53 (~ordered(implies(implies(x, y), implies(y, x)), implies(truth, implies(y, x)))),
% 35.87/22.53 inference(modus_ponens,[status(thm)],[547, 532])).
% 35.87/22.53 tff(549,plain,
% 35.87/22.53 (((~![Z: $i, Y: $i, X: $i] : ((~ordered(X, Y)) | ordered(implies(X, Z), implies(Y, Z)))) | ((~ordered(implies(x, y), truth)) | ordered(implies(implies(x, y), implies(y, x)), implies(truth, implies(y, x))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~ordered(X, Y)) | ordered(implies(X, Z), implies(Y, Z)))) | (~ordered(implies(x, y), truth)) | ordered(implies(implies(x, y), implies(y, x)), implies(truth, implies(y, x))))),
% 35.87/22.53 inference(rewrite,[status(thm)],[])).
% 35.87/22.53 tff(550,plain,
% 35.87/22.53 ((~![Z: $i, Y: $i, X: $i] : ((~ordered(X, Y)) | ordered(implies(X, Z), implies(Y, Z)))) | ((~ordered(implies(x, y), truth)) | ordered(implies(implies(x, y), implies(y, x)), implies(truth, implies(y, x))))),
% 35.87/22.53 inference(quant_inst,[status(thm)],[])).
% 35.87/22.53 tff(551,plain,
% 35.87/22.53 ((~![Z: $i, Y: $i, X: $i] : ((~ordered(X, Y)) | ordered(implies(X, Z), implies(Y, Z)))) | (~ordered(implies(x, y), truth)) | ordered(implies(implies(x, y), implies(y, x)), implies(truth, implies(y, x)))),
% 35.87/22.53 inference(modus_ponens,[status(thm)],[550, 549])).
% 35.87/22.53 tff(552,plain,
% 35.87/22.53 ((~ordered(implies(x, y), truth)) | ordered(implies(implies(x, y), implies(y, x)), implies(truth, implies(y, x)))),
% 35.87/22.53 inference(unit_resolution,[status(thm)],[551, 75])).
% 35.87/22.53 tff(553,plain,
% 35.87/22.53 (~ordered(implies(x, y), truth)),
% 35.87/22.53 inference(unit_resolution,[status(thm)],[552, 548])).
% 35.87/22.53 tff(554,plain,
% 35.87/22.53 (~ordered(not(not(implies(x, y))), truth)),
% 35.87/22.53 inference(modus_ponens,[status(thm)],[553, 529])).
% 35.87/22.53 tff(555,plain,
% 35.87/22.53 (((~![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))) | ((~(implies(not(not(implies(x, y))), truth) = truth)) | ordered(not(not(implies(x, y))), truth))) <=> ((~![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))) | (~(implies(not(not(implies(x, y))), truth) = truth)) | ordered(not(not(implies(x, y))), truth))),
% 35.87/22.53 inference(rewrite,[status(thm)],[])).
% 35.87/22.53 tff(556,plain,
% 35.87/22.53 ((~![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))) | ((~(implies(not(not(implies(x, y))), truth) = truth)) | ordered(not(not(implies(x, y))), truth))),
% 35.87/22.54 inference(quant_inst,[status(thm)],[])).
% 35.87/22.54 tff(557,plain,
% 35.87/22.54 ((~![Y: $i, X: $i] : ((~(implies(X, Y) = truth)) | ordered(X, Y))) | (~(implies(not(not(implies(x, y))), truth) = truth)) | ordered(not(not(implies(x, y))), truth)),
% 35.87/22.54 inference(modus_ponens,[status(thm)],[556, 555])).
% 35.87/22.54 tff(558,plain,
% 35.87/22.54 (~(implies(not(not(implies(x, y))), truth) = truth)),
% 35.87/22.54 inference(unit_resolution,[status(thm)],[557, 48, 554])).
% 35.87/22.54 tff(559,plain,
% 35.87/22.54 (~(implies(not(not(implies(x, y))), implies(not(not(implies(x, y))), not(not(implies(x, y))))) = truth)),
% 35.87/22.54 inference(modus_ponens,[status(thm)],[558, 506])).
% 35.87/22.54 tff(560,plain,
% 35.87/22.54 ($false),
% 35.87/22.54 inference(unit_resolution,[status(thm)],[559, 503])).
% 35.87/22.54 % SZS output end Proof
%------------------------------------------------------------------------------