TSTP Solution File: LCL542+1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : LCL542+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n020.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 04:56:38 EDT 2022
% Result : Theorem 0.20s 0.41s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.11 % Problem : LCL542+1 : TPTP v8.1.0. Released v3.3.0.
% 0.04/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n020.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Sep 1 22:12:15 EDT 2022
% 0.13/0.34 % CPUTime :
% 0.13/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.34 Usage: tptp [options] [-file:]file
% 0.13/0.34 -h, -? prints this message.
% 0.13/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.34 -m, -model generate model.
% 0.13/0.34 -p, -proof generate proof.
% 0.13/0.34 -c, -core generate unsat core of named formulas.
% 0.13/0.34 -st, -statistics display statistics.
% 0.13/0.34 -t:timeout set timeout (in second).
% 0.13/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.34 -<param>:<value> configuration parameter and value.
% 0.13/0.34 -o:<output-file> file to place output in.
% 0.20/0.41 % SZS status Theorem
% 0.20/0.41 % SZS output start Proof
% 0.20/0.41 tff(is_a_theorem_type, type, (
% 0.20/0.41 is_a_theorem: $i > $o)).
% 0.20/0.41 tff(necessarily_type, type, (
% 0.20/0.41 necessarily: $i > $i)).
% 0.20/0.41 tff(implies_type, type, (
% 0.20/0.41 implies: ( $i * $i ) > $i)).
% 0.20/0.41 tff(tptp_fun_X_4_type, type, (
% 0.20/0.41 tptp_fun_X_4: $i)).
% 0.20/0.41 tff(and_type, type, (
% 0.20/0.41 and: ( $i * $i ) > $i)).
% 0.20/0.41 tff(tptp_fun_Y_3_type, type, (
% 0.20/0.41 tptp_fun_Y_3: $i)).
% 0.20/0.41 tff(strict_implies_type, type, (
% 0.20/0.41 strict_implies: ( $i * $i ) > $i)).
% 0.20/0.41 tff(op_strict_implies_type, type, (
% 0.20/0.41 op_strict_implies: $o)).
% 0.20/0.41 tff(axiom_m2_type, type, (
% 0.20/0.41 axiom_m2: $o)).
% 0.20/0.41 tff(and_1_type, type, (
% 0.20/0.41 and_1: $o)).
% 0.20/0.41 tff(necessitation_type, type, (
% 0.20/0.41 necessitation: $o)).
% 0.20/0.41 tff(1,plain,
% 0.20/0.41 (^[X: $i, Y: $i] : refl((strict_implies(X, Y) = necessarily(implies(X, Y))) <=> (strict_implies(X, Y) = necessarily(implies(X, Y))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(2,plain,
% 0.20/0.41 (![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y))) <=> ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[1])).
% 0.20/0.41 tff(3,plain,
% 0.20/0.41 (![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y))) <=> ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(4,plain,
% 0.20/0.41 (($false | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) <=> ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(5,plain,
% 0.20/0.41 ((~$true) <=> $false),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(6,axiom,(op_strict_implies), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','s1_0_op_strict_implies')).
% 0.20/0.41 tff(7,plain,
% 0.20/0.41 (op_strict_implies <=> $true),
% 0.20/0.41 inference(iff_true,[status(thm)],[6])).
% 0.20/0.41 tff(8,plain,
% 0.20/0.41 ((~op_strict_implies) <=> (~$true)),
% 0.20/0.41 inference(monotonicity,[status(thm)],[7])).
% 0.20/0.41 tff(9,plain,
% 0.20/0.41 ((~op_strict_implies) <=> $false),
% 0.20/0.41 inference(transitivity,[status(thm)],[8, 5])).
% 0.20/0.41 tff(10,plain,
% 0.20/0.41 (((~op_strict_implies) | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) <=> ($false | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[9])).
% 0.20/0.41 tff(11,plain,
% 0.20/0.41 (((~op_strict_implies) | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) <=> ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 0.20/0.41 inference(transitivity,[status(thm)],[10, 4])).
% 0.20/0.41 tff(12,plain,
% 0.20/0.41 (((~op_strict_implies) | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) <=> ((~op_strict_implies) | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(13,plain,
% 0.20/0.41 ((op_strict_implies => ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) <=> ((~op_strict_implies) | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y))))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(14,axiom,(op_strict_implies => ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+1.ax','op_strict_implies')).
% 0.20/0.41 tff(15,plain,
% 0.20/0.41 ((~op_strict_implies) | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.20/0.41 tff(16,plain,
% 0.20/0.41 ((~op_strict_implies) | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[15, 12])).
% 0.20/0.41 tff(17,plain,
% 0.20/0.41 ((~op_strict_implies) | ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.20/0.41 tff(18,plain,
% 0.20/0.41 (![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[17, 11])).
% 0.20/0.41 tff(19,plain,
% 0.20/0.41 (![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[18, 3])).
% 0.20/0.41 tff(20,plain,(
% 0.20/0.41 ![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 0.20/0.41 inference(skolemize,[status(sab)],[19])).
% 0.20/0.41 tff(21,plain,
% 0.20/0.41 (![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[20, 2])).
% 0.20/0.41 tff(22,plain,
% 0.20/0.41 ((~![X: $i, Y: $i] : (strict_implies(X, Y) = necessarily(implies(X, Y)))) | (strict_implies(and(X!4, Y!3), X!4) = necessarily(implies(and(X!4, Y!3), X!4)))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(23,plain,
% 0.20/0.41 (strict_implies(and(X!4, Y!3), X!4) = necessarily(implies(and(X!4, Y!3), X!4))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[22, 21])).
% 0.20/0.41 tff(24,plain,
% 0.20/0.41 (necessarily(implies(and(X!4, Y!3), X!4)) = strict_implies(and(X!4, Y!3), X!4)),
% 0.20/0.41 inference(symmetry,[status(thm)],[23])).
% 0.20/0.41 tff(25,plain,
% 0.20/0.41 (is_a_theorem(necessarily(implies(and(X!4, Y!3), X!4))) <=> is_a_theorem(strict_implies(and(X!4, Y!3), X!4))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[24])).
% 0.20/0.41 tff(26,plain,
% 0.20/0.41 (is_a_theorem(strict_implies(and(X!4, Y!3), X!4)) <=> is_a_theorem(necessarily(implies(and(X!4, Y!3), X!4)))),
% 0.20/0.41 inference(symmetry,[status(thm)],[25])).
% 0.20/0.41 tff(27,plain,
% 0.20/0.41 ((~is_a_theorem(strict_implies(and(X!4, Y!3), X!4))) <=> (~is_a_theorem(necessarily(implies(and(X!4, Y!3), X!4))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[26])).
% 0.20/0.41 tff(28,plain,
% 0.20/0.41 ((~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))) <=> (~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(29,plain,
% 0.20/0.41 (($false <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))) <=> (~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(30,axiom,(~axiom_m2), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','s1_0_axiom_m2')).
% 0.20/0.41 tff(31,plain,
% 0.20/0.41 (axiom_m2 <=> $false),
% 0.20/0.41 inference(iff_false,[status(thm)],[30])).
% 0.20/0.41 tff(32,plain,
% 0.20/0.41 ((axiom_m2 <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))) <=> ($false <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X)))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[31])).
% 0.20/0.41 tff(33,plain,
% 0.20/0.41 ((axiom_m2 <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))) <=> (~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X)))),
% 0.20/0.41 inference(transitivity,[status(thm)],[32, 29])).
% 0.20/0.41 tff(34,plain,
% 0.20/0.41 ((axiom_m2 <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))) <=> (axiom_m2 <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(35,axiom,(axiom_m2 <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+0.ax','axiom_m2')).
% 0.20/0.41 tff(36,plain,
% 0.20/0.41 (axiom_m2 <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.20/0.41 tff(37,plain,
% 0.20/0.41 (axiom_m2 <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[36, 34])).
% 0.20/0.41 tff(38,plain,
% 0.20/0.41 (axiom_m2 <=> ![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[37, 34])).
% 0.20/0.41 tff(39,plain,
% 0.20/0.41 (~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[38, 33])).
% 0.20/0.41 tff(40,plain,
% 0.20/0.41 (~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[39, 28])).
% 0.20/0.41 tff(41,plain,
% 0.20/0.41 (~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[40, 28])).
% 0.20/0.41 tff(42,plain,
% 0.20/0.41 (~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[41, 28])).
% 0.20/0.41 tff(43,plain,
% 0.20/0.41 (~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[42, 28])).
% 0.20/0.41 tff(44,plain,
% 0.20/0.41 (~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[43, 28])).
% 0.20/0.41 tff(45,plain,
% 0.20/0.41 (~![X: $i, Y: $i] : is_a_theorem(strict_implies(and(X, Y), X))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[44, 28])).
% 0.20/0.41 tff(46,plain,(
% 0.20/0.41 ~is_a_theorem(strict_implies(and(X!4, Y!3), X!4))),
% 0.20/0.41 inference(skolemize,[status(sab)],[45])).
% 0.20/0.41 tff(47,plain,
% 0.20/0.41 (~is_a_theorem(necessarily(implies(and(X!4, Y!3), X!4)))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[46, 27])).
% 0.20/0.41 tff(48,plain,
% 0.20/0.41 (^[X: $i, Y: $i] : refl(is_a_theorem(implies(and(X, Y), X)) <=> is_a_theorem(implies(and(X, Y), X)))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(49,plain,
% 0.20/0.41 (![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X)) <=> ![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[48])).
% 0.20/0.41 tff(50,plain,
% 0.20/0.41 (![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X)) <=> ![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(51,plain,
% 0.20/0.41 (($true <=> ![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))) <=> ![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(52,axiom,(and_1), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+2.ax','hilbert_and_1')).
% 0.20/0.41 tff(53,plain,
% 0.20/0.41 (and_1 <=> $true),
% 0.20/0.41 inference(iff_true,[status(thm)],[52])).
% 0.20/0.41 tff(54,plain,
% 0.20/0.41 ((and_1 <=> ![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))) <=> ($true <=> ![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X)))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[53])).
% 0.20/0.41 tff(55,plain,
% 0.20/0.41 ((and_1 <=> ![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))) <=> ![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))),
% 0.20/0.41 inference(transitivity,[status(thm)],[54, 51])).
% 0.20/0.41 tff(56,plain,
% 0.20/0.41 ((and_1 <=> ![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))) <=> (and_1 <=> ![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(57,axiom,(and_1 <=> ![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+0.ax','and_1')).
% 0.20/0.41 tff(58,plain,
% 0.20/0.41 (and_1 <=> ![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.20/0.41 tff(59,plain,
% 0.20/0.41 (and_1 <=> ![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[58, 56])).
% 0.20/0.41 tff(60,plain,
% 0.20/0.41 (![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[59, 55])).
% 0.20/0.41 tff(61,plain,
% 0.20/0.41 (![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[60, 50])).
% 0.20/0.41 tff(62,plain,(
% 0.20/0.41 ![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))),
% 0.20/0.41 inference(skolemize,[status(sab)],[61])).
% 0.20/0.41 tff(63,plain,
% 0.20/0.41 (![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))),
% 0.20/0.41 inference(modus_ponens,[status(thm)],[62, 49])).
% 0.20/0.41 tff(64,plain,
% 0.20/0.41 ((~![X: $i, Y: $i] : is_a_theorem(implies(and(X, Y), X))) | is_a_theorem(implies(and(X!4, Y!3), X!4))),
% 0.20/0.41 inference(quant_inst,[status(thm)],[])).
% 0.20/0.41 tff(65,plain,
% 0.20/0.41 (is_a_theorem(implies(and(X!4, Y!3), X!4))),
% 0.20/0.41 inference(unit_resolution,[status(thm)],[64, 63])).
% 0.20/0.41 tff(66,plain,
% 0.20/0.41 (^[X: $i] : refl((is_a_theorem(necessarily(X)) | (~is_a_theorem(X))) <=> (is_a_theorem(necessarily(X)) | (~is_a_theorem(X))))),
% 0.20/0.41 inference(bind,[status(th)],[])).
% 0.20/0.41 tff(67,plain,
% 0.20/0.41 (![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X))) <=> ![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))),
% 0.20/0.41 inference(quant_intro,[status(thm)],[66])).
% 0.20/0.41 tff(68,plain,
% 0.20/0.41 (![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X))) <=> ![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(69,plain,
% 0.20/0.41 (($true <=> ![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))) <=> ![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))),
% 0.20/0.41 inference(rewrite,[status(thm)],[])).
% 0.20/0.41 tff(70,axiom,(necessitation), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+3.ax','km4b_necessitation')).
% 0.20/0.41 tff(71,plain,
% 0.20/0.41 (necessitation <=> $true),
% 0.20/0.41 inference(iff_true,[status(thm)],[70])).
% 0.20/0.41 tff(72,plain,
% 0.20/0.41 ((necessitation <=> ![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))) <=> ($true <=> ![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X))))),
% 0.20/0.41 inference(monotonicity,[status(thm)],[71])).
% 0.20/0.41 tff(73,plain,
% 0.20/0.42 ((necessitation <=> ![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))) <=> ![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))),
% 0.20/0.42 inference(transitivity,[status(thm)],[72, 69])).
% 0.20/0.42 tff(74,plain,
% 0.20/0.42 ((necessitation <=> ![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))) <=> (necessitation <=> ![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(75,plain,
% 0.20/0.42 ((necessitation <=> ![X: $i] : (is_a_theorem(X) => is_a_theorem(necessarily(X)))) <=> (necessitation <=> ![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(76,axiom,(necessitation <=> ![X: $i] : (is_a_theorem(X) => is_a_theorem(necessarily(X)))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL007+0.ax','necessitation')).
% 0.20/0.42 tff(77,plain,
% 0.20/0.42 (necessitation <=> ![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[76, 75])).
% 0.20/0.42 tff(78,plain,
% 0.20/0.42 (necessitation <=> ![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[77, 74])).
% 0.20/0.42 tff(79,plain,
% 0.20/0.42 (necessitation <=> ![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[78, 74])).
% 0.20/0.42 tff(80,plain,
% 0.20/0.42 (![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[79, 73])).
% 0.20/0.42 tff(81,plain,
% 0.20/0.42 (![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[80, 68])).
% 0.20/0.42 tff(82,plain,(
% 0.20/0.42 ![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))),
% 0.20/0.42 inference(skolemize,[status(sab)],[81])).
% 0.20/0.42 tff(83,plain,
% 0.20/0.42 (![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[82, 67])).
% 0.20/0.42 tff(84,plain,
% 0.20/0.42 (((~![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))) | (is_a_theorem(necessarily(implies(and(X!4, Y!3), X!4))) | (~is_a_theorem(implies(and(X!4, Y!3), X!4))))) <=> ((~![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))) | is_a_theorem(necessarily(implies(and(X!4, Y!3), X!4))) | (~is_a_theorem(implies(and(X!4, Y!3), X!4))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(85,plain,
% 0.20/0.42 ((~![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))) | (is_a_theorem(necessarily(implies(and(X!4, Y!3), X!4))) | (~is_a_theorem(implies(and(X!4, Y!3), X!4))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(86,plain,
% 0.20/0.42 ((~![X: $i] : (is_a_theorem(necessarily(X)) | (~is_a_theorem(X)))) | is_a_theorem(necessarily(implies(and(X!4, Y!3), X!4))) | (~is_a_theorem(implies(and(X!4, Y!3), X!4)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.20/0.42 tff(87,plain,
% 0.20/0.42 (is_a_theorem(necessarily(implies(and(X!4, Y!3), X!4)))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[86, 83, 65])).
% 0.20/0.42 tff(88,plain,
% 0.20/0.42 ($false),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[87, 47])).
% 0.20/0.42 % SZS output end Proof
%------------------------------------------------------------------------------