TSTP Solution File: LCL494+1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : LCL494+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 04:56:30 EDT 2022
% Result : Theorem 0.12s 0.40s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL494+1 : TPTP v8.1.0. Released v3.3.0.
% 0.12/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n020.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Sep 1 21:54:44 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 0.12/0.40 % SZS status Theorem
% 0.12/0.40 % SZS output start Proof
% 0.12/0.40 tff(is_a_theorem_type, type, (
% 0.12/0.40 is_a_theorem: $i > $o)).
% 0.12/0.40 tff(implies_type, type, (
% 0.12/0.40 implies: ( $i * $i ) > $i)).
% 0.12/0.40 tff(or_type, type, (
% 0.12/0.40 or: ( $i * $i ) > $i)).
% 0.12/0.40 tff(not_type, type, (
% 0.12/0.40 not: $i > $i)).
% 0.12/0.40 tff(tptp_fun_X_1_type, type, (
% 0.12/0.40 tptp_fun_X_1: $i)).
% 0.12/0.40 tff(tptp_fun_Y_0_type, type, (
% 0.12/0.40 tptp_fun_Y_0: $i)).
% 0.12/0.40 tff(equiv_type, type, (
% 0.12/0.40 equiv: ( $i * $i ) > $i)).
% 0.12/0.40 tff(op_implies_or_type, type, (
% 0.12/0.40 op_implies_or: $o)).
% 0.12/0.40 tff(and_type, type, (
% 0.12/0.40 and: ( $i * $i ) > $i)).
% 0.12/0.40 tff(op_implies_and_type, type, (
% 0.12/0.40 op_implies_and: $o)).
% 0.12/0.40 tff(op_and_type, type, (
% 0.12/0.40 op_and: $o)).
% 0.12/0.40 tff(op_equiv_type, type, (
% 0.12/0.40 op_equiv: $o)).
% 0.12/0.40 tff(r3_type, type, (
% 0.12/0.40 r3: $o)).
% 0.12/0.40 tff(equivalence_2_type, type, (
% 0.12/0.40 equivalence_2: $o)).
% 0.12/0.40 tff(modus_ponens_type, type, (
% 0.12/0.40 modus_ponens: $o)).
% 0.12/0.40 tff(r2_type, type, (
% 0.12/0.40 r2: $o)).
% 0.12/0.40 tff(1,plain,
% 0.12/0.40 (^[X: $i, Y: $i] : refl((implies(X, Y) = or(not(X), Y)) <=> (implies(X, Y) = or(not(X), Y)))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(2,plain,
% 0.12/0.40 (![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y)) <=> ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.40 tff(3,plain,
% 0.12/0.40 (![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y)) <=> ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(4,plain,
% 0.12/0.40 (($false | ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))) <=> ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(5,plain,
% 0.12/0.40 ((~$true) <=> $false),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(6,axiom,(op_implies_or), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+4.ax','principia_op_implies_or')).
% 0.12/0.40 tff(7,plain,
% 0.12/0.40 (op_implies_or <=> $true),
% 0.12/0.40 inference(iff_true,[status(thm)],[6])).
% 0.12/0.40 tff(8,plain,
% 0.12/0.40 ((~op_implies_or) <=> (~$true)),
% 0.12/0.40 inference(monotonicity,[status(thm)],[7])).
% 0.12/0.40 tff(9,plain,
% 0.12/0.40 ((~op_implies_or) <=> $false),
% 0.12/0.40 inference(transitivity,[status(thm)],[8, 5])).
% 0.12/0.40 tff(10,plain,
% 0.12/0.40 (((~op_implies_or) | ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))) <=> ($false | ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y)))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[9])).
% 0.12/0.40 tff(11,plain,
% 0.12/0.40 (((~op_implies_or) | ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))) <=> ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))),
% 0.12/0.40 inference(transitivity,[status(thm)],[10, 4])).
% 0.12/0.40 tff(12,plain,
% 0.12/0.40 (((~op_implies_or) | ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))) <=> ((~op_implies_or) | ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(13,plain,
% 0.12/0.40 ((op_implies_or => ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))) <=> ((~op_implies_or) | ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(14,axiom,(op_implies_or => ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+1.ax','op_implies_or')).
% 0.12/0.40 tff(15,plain,
% 0.12/0.40 ((~op_implies_or) | ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.12/0.40 tff(16,plain,
% 0.12/0.40 ((~op_implies_or) | ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[15, 12])).
% 0.12/0.40 tff(17,plain,
% 0.12/0.40 (![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[16, 11])).
% 0.12/0.40 tff(18,plain,
% 0.12/0.40 (![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[17, 3])).
% 0.12/0.40 tff(19,plain,(
% 0.12/0.40 ![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))),
% 0.12/0.40 inference(skolemize,[status(sab)],[18])).
% 0.12/0.40 tff(20,plain,
% 0.12/0.40 (![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[19, 2])).
% 0.12/0.40 tff(21,plain,
% 0.12/0.40 ((~![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))) | (implies(Y!0, X!1) = or(not(Y!0), X!1))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(22,plain,
% 0.12/0.40 (implies(Y!0, X!1) = or(not(Y!0), X!1)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[21, 20])).
% 0.12/0.40 tff(23,plain,
% 0.12/0.40 (^[X: $i, Y: $i] : refl((implies(X, Y) = not(and(X, not(Y)))) <=> (implies(X, Y) = not(and(X, not(Y)))))),
% 0.12/0.41 inference(bind,[status(th)],[])).
% 0.12/0.41 tff(24,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y)))) <=> ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 0.12/0.41 inference(quant_intro,[status(thm)],[23])).
% 0.12/0.41 tff(25,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y)))) <=> ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(26,plain,
% 0.12/0.41 (($false | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))) <=> ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(27,axiom,(op_implies_and), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','hilbert_op_implies_and')).
% 0.12/0.41 tff(28,plain,
% 0.12/0.41 (op_implies_and <=> $true),
% 0.12/0.41 inference(iff_true,[status(thm)],[27])).
% 0.12/0.41 tff(29,plain,
% 0.12/0.41 ((~op_implies_and) <=> (~$true)),
% 0.12/0.41 inference(monotonicity,[status(thm)],[28])).
% 0.12/0.41 tff(30,plain,
% 0.12/0.41 ((~op_implies_and) <=> $false),
% 0.12/0.41 inference(transitivity,[status(thm)],[29, 5])).
% 0.12/0.41 tff(31,plain,
% 0.12/0.41 (((~op_implies_and) | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))) <=> ($false | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y)))))),
% 0.12/0.41 inference(monotonicity,[status(thm)],[30])).
% 0.12/0.41 tff(32,plain,
% 0.12/0.41 (((~op_implies_and) | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))) <=> ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 0.12/0.41 inference(transitivity,[status(thm)],[31, 26])).
% 0.12/0.41 tff(33,plain,
% 0.12/0.41 (((~op_implies_and) | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))) <=> ((~op_implies_and) | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y)))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(34,plain,
% 0.12/0.41 ((op_implies_and => ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))) <=> ((~op_implies_and) | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y)))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(35,axiom,(op_implies_and => ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+1.ax','op_implies_and')).
% 0.12/0.41 tff(36,plain,
% 0.12/0.41 ((~op_implies_and) | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.12/0.41 tff(37,plain,
% 0.12/0.41 ((~op_implies_and) | ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[36, 33])).
% 0.12/0.41 tff(38,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[37, 32])).
% 0.12/0.41 tff(39,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[38, 25])).
% 0.12/0.41 tff(40,plain,(
% 0.12/0.41 ![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 0.12/0.41 inference(skolemize,[status(sab)],[39])).
% 0.12/0.41 tff(41,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[40, 24])).
% 0.12/0.41 tff(42,plain,
% 0.12/0.41 ((~![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))) | (implies(Y!0, X!1) = not(and(Y!0, not(X!1))))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(43,plain,
% 0.12/0.41 (implies(Y!0, X!1) = not(and(Y!0, not(X!1)))),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[42, 41])).
% 0.12/0.41 tff(44,plain,
% 0.12/0.41 (not(and(Y!0, not(X!1))) = implies(Y!0, X!1)),
% 0.12/0.41 inference(symmetry,[status(thm)],[43])).
% 0.12/0.41 tff(45,plain,
% 0.12/0.41 (^[X: $i, Y: $i] : refl((and(X, Y) = not(or(not(X), not(Y)))) <=> (and(X, Y) = not(or(not(X), not(Y)))))),
% 0.12/0.41 inference(bind,[status(th)],[])).
% 0.12/0.41 tff(46,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y)))) <=> ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))),
% 0.12/0.41 inference(quant_intro,[status(thm)],[45])).
% 0.12/0.41 tff(47,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y)))) <=> ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(48,plain,
% 0.12/0.41 (($false | ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))) <=> ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(49,axiom,(op_and), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+4.ax','principia_op_and')).
% 0.12/0.41 tff(50,plain,
% 0.12/0.41 (op_and <=> $true),
% 0.12/0.41 inference(iff_true,[status(thm)],[49])).
% 0.12/0.41 tff(51,plain,
% 0.12/0.41 ((~op_and) <=> (~$true)),
% 0.12/0.41 inference(monotonicity,[status(thm)],[50])).
% 0.12/0.41 tff(52,plain,
% 0.12/0.41 ((~op_and) <=> $false),
% 0.12/0.41 inference(transitivity,[status(thm)],[51, 5])).
% 0.12/0.41 tff(53,plain,
% 0.12/0.41 (((~op_and) | ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))) <=> ($false | ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y)))))),
% 0.12/0.41 inference(monotonicity,[status(thm)],[52])).
% 0.12/0.41 tff(54,plain,
% 0.12/0.41 (((~op_and) | ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))) <=> ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))),
% 0.12/0.41 inference(transitivity,[status(thm)],[53, 48])).
% 0.12/0.41 tff(55,plain,
% 0.12/0.41 (((~op_and) | ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))) <=> ((~op_and) | ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y)))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(56,plain,
% 0.12/0.41 ((op_and => ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))) <=> ((~op_and) | ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y)))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(57,axiom,(op_and => ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+1.ax','op_and')).
% 0.12/0.41 tff(58,plain,
% 0.12/0.41 ((~op_and) | ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.12/0.41 tff(59,plain,
% 0.12/0.41 ((~op_and) | ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[58, 55])).
% 0.12/0.41 tff(60,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[59, 54])).
% 0.12/0.41 tff(61,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[60, 47])).
% 0.12/0.41 tff(62,plain,(
% 0.12/0.41 ![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))),
% 0.12/0.41 inference(skolemize,[status(sab)],[61])).
% 0.12/0.41 tff(63,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[62, 46])).
% 0.12/0.41 tff(64,plain,
% 0.12/0.41 ((~![X: $i, Y: $i] : (and(X, Y) = not(or(not(X), not(Y))))) | (and(Y!0, not(X!1)) = not(or(not(Y!0), not(not(X!1)))))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(65,plain,
% 0.12/0.41 (and(Y!0, not(X!1)) = not(or(not(Y!0), not(not(X!1))))),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[64, 63])).
% 0.12/0.41 tff(66,plain,
% 0.12/0.41 (not(or(not(Y!0), not(not(X!1)))) = and(Y!0, not(X!1))),
% 0.12/0.41 inference(symmetry,[status(thm)],[65])).
% 0.12/0.41 tff(67,plain,
% 0.12/0.41 (not(not(or(not(Y!0), not(not(X!1))))) = not(and(Y!0, not(X!1)))),
% 0.12/0.41 inference(monotonicity,[status(thm)],[66])).
% 0.12/0.41 tff(68,plain,
% 0.12/0.41 (not(not(or(not(Y!0), not(not(X!1))))) = or(not(Y!0), X!1)),
% 0.12/0.41 inference(transitivity,[status(thm)],[67, 44, 22])).
% 0.12/0.41 tff(69,plain,
% 0.12/0.41 (or(not(not(or(not(Y!0), not(not(X!1))))), not(equiv(X!1, Y!0))) = or(or(not(Y!0), X!1), not(equiv(X!1, Y!0)))),
% 0.12/0.41 inference(monotonicity,[status(thm)],[68])).
% 0.12/0.41 tff(70,plain,
% 0.12/0.41 ((~![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))) | (implies(not(or(not(Y!0), not(not(X!1)))), not(equiv(X!1, Y!0))) = or(not(not(or(not(Y!0), not(not(X!1))))), not(equiv(X!1, Y!0))))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(71,plain,
% 0.12/0.41 (implies(not(or(not(Y!0), not(not(X!1)))), not(equiv(X!1, Y!0))) = or(not(not(or(not(Y!0), not(not(X!1))))), not(equiv(X!1, Y!0)))),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[70, 20])).
% 0.12/0.41 tff(72,plain,
% 0.12/0.41 (^[X: $i, Y: $i] : refl((equiv(X, Y) = and(implies(X, Y), implies(Y, X))) <=> (equiv(X, Y) = and(implies(X, Y), implies(Y, X))))),
% 0.12/0.41 inference(bind,[status(th)],[])).
% 0.12/0.41 tff(73,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X))) <=> ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))),
% 0.12/0.41 inference(quant_intro,[status(thm)],[72])).
% 0.12/0.41 tff(74,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X))) <=> ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(75,plain,
% 0.12/0.41 (($false | ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))) <=> ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(76,axiom,(op_equiv), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+4.ax','principia_op_equiv')).
% 0.12/0.41 tff(77,plain,
% 0.12/0.41 (op_equiv <=> $true),
% 0.12/0.41 inference(iff_true,[status(thm)],[76])).
% 0.12/0.41 tff(78,plain,
% 0.12/0.41 ((~op_equiv) <=> (~$true)),
% 0.12/0.41 inference(monotonicity,[status(thm)],[77])).
% 0.12/0.41 tff(79,plain,
% 0.12/0.41 ((~op_equiv) <=> $false),
% 0.12/0.41 inference(transitivity,[status(thm)],[78, 5])).
% 0.12/0.41 tff(80,plain,
% 0.12/0.41 (((~op_equiv) | ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))) <=> ($false | ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X))))),
% 0.12/0.41 inference(monotonicity,[status(thm)],[79])).
% 0.12/0.41 tff(81,plain,
% 0.12/0.41 (((~op_equiv) | ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))) <=> ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))),
% 0.12/0.41 inference(transitivity,[status(thm)],[80, 75])).
% 0.12/0.41 tff(82,plain,
% 0.12/0.41 (((~op_equiv) | ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))) <=> ((~op_equiv) | ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(83,plain,
% 0.12/0.41 ((op_equiv => ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))) <=> ((~op_equiv) | ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X))))),
% 0.12/0.41 inference(rewrite,[status(thm)],[])).
% 0.12/0.41 tff(84,axiom,(op_equiv => ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+1.ax','op_equiv')).
% 0.12/0.41 tff(85,plain,
% 0.12/0.41 ((~op_equiv) | ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[84, 83])).
% 0.12/0.41 tff(86,plain,
% 0.12/0.41 ((~op_equiv) | ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[85, 82])).
% 0.12/0.41 tff(87,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[86, 81])).
% 0.12/0.41 tff(88,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[87, 74])).
% 0.12/0.41 tff(89,plain,(
% 0.12/0.41 ![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))),
% 0.12/0.41 inference(skolemize,[status(sab)],[88])).
% 0.12/0.41 tff(90,plain,
% 0.12/0.41 (![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))),
% 0.12/0.41 inference(modus_ponens,[status(thm)],[89, 73])).
% 0.12/0.41 tff(91,plain,
% 0.12/0.41 ((~![X: $i, Y: $i] : (equiv(X, Y) = and(implies(X, Y), implies(Y, X)))) | (equiv(X!1, Y!0) = and(implies(X!1, Y!0), implies(Y!0, X!1)))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(92,plain,
% 0.12/0.41 (equiv(X!1, Y!0) = and(implies(X!1, Y!0), implies(Y!0, X!1))),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[91, 90])).
% 0.12/0.41 tff(93,plain,
% 0.12/0.41 (and(implies(X!1, Y!0), implies(Y!0, X!1)) = equiv(X!1, Y!0)),
% 0.12/0.41 inference(symmetry,[status(thm)],[92])).
% 0.12/0.41 tff(94,plain,
% 0.12/0.41 (and(implies(X!1, Y!0), not(and(Y!0, not(X!1)))) = and(implies(X!1, Y!0), implies(Y!0, X!1))),
% 0.12/0.41 inference(monotonicity,[status(thm)],[44])).
% 0.12/0.41 tff(95,plain,
% 0.12/0.41 (and(implies(X!1, Y!0), not(and(Y!0, not(X!1)))) = equiv(X!1, Y!0)),
% 0.12/0.41 inference(transitivity,[status(thm)],[94, 93])).
% 0.12/0.41 tff(96,plain,
% 0.12/0.41 (not(and(implies(X!1, Y!0), not(and(Y!0, not(X!1))))) = not(equiv(X!1, Y!0))),
% 0.12/0.41 inference(monotonicity,[status(thm)],[95])).
% 0.12/0.41 tff(97,plain,
% 0.12/0.41 ((~![X: $i, Y: $i] : (implies(X, Y) = not(and(X, not(Y))))) | (implies(implies(X!1, Y!0), and(Y!0, not(X!1))) = not(and(implies(X!1, Y!0), not(and(Y!0, not(X!1))))))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(98,plain,
% 0.12/0.41 (implies(implies(X!1, Y!0), and(Y!0, not(X!1))) = not(and(implies(X!1, Y!0), not(and(Y!0, not(X!1)))))),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[97, 41])).
% 0.12/0.41 tff(99,plain,
% 0.12/0.41 ((~![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))) | (implies(X!1, Y!0) = or(not(X!1), Y!0))),
% 0.12/0.41 inference(quant_inst,[status(thm)],[])).
% 0.12/0.41 tff(100,plain,
% 0.12/0.41 (implies(X!1, Y!0) = or(not(X!1), Y!0)),
% 0.12/0.41 inference(unit_resolution,[status(thm)],[99, 20])).
% 0.12/0.42 tff(101,plain,
% 0.12/0.42 (or(not(X!1), Y!0) = implies(X!1, Y!0)),
% 0.12/0.42 inference(symmetry,[status(thm)],[100])).
% 0.12/0.42 tff(102,plain,
% 0.12/0.42 (implies(or(not(X!1), Y!0), not(or(not(Y!0), not(not(X!1))))) = implies(implies(X!1, Y!0), and(Y!0, not(X!1)))),
% 0.12/0.42 inference(monotonicity,[status(thm)],[101, 66])).
% 0.12/0.42 tff(103,plain,
% 0.12/0.42 ((~![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))) | (implies(or(not(X!1), Y!0), not(or(not(Y!0), not(not(X!1))))) = or(not(or(not(X!1), Y!0)), not(or(not(Y!0), not(not(X!1))))))),
% 0.12/0.42 inference(quant_inst,[status(thm)],[])).
% 0.12/0.42 tff(104,plain,
% 0.12/0.42 (implies(or(not(X!1), Y!0), not(or(not(Y!0), not(not(X!1))))) = or(not(or(not(X!1), Y!0)), not(or(not(Y!0), not(not(X!1)))))),
% 0.12/0.42 inference(unit_resolution,[status(thm)],[103, 20])).
% 0.12/0.42 tff(105,plain,
% 0.12/0.42 (or(not(or(not(X!1), Y!0)), not(or(not(Y!0), not(not(X!1))))) = implies(or(not(X!1), Y!0), not(or(not(Y!0), not(not(X!1)))))),
% 0.12/0.42 inference(symmetry,[status(thm)],[104])).
% 0.12/0.42 tff(106,plain,
% 0.12/0.42 (not(or(not(X!1), Y!0)) = not(implies(X!1, Y!0))),
% 0.12/0.42 inference(monotonicity,[status(thm)],[101])).
% 0.12/0.42 tff(107,plain,
% 0.12/0.42 (not(implies(X!1, Y!0)) = not(or(not(X!1), Y!0))),
% 0.12/0.42 inference(symmetry,[status(thm)],[106])).
% 0.12/0.42 tff(108,plain,
% 0.12/0.42 (or(not(implies(X!1, Y!0)), not(or(not(Y!0), not(not(X!1))))) = or(not(or(not(X!1), Y!0)), not(or(not(Y!0), not(not(X!1)))))),
% 0.12/0.42 inference(monotonicity,[status(thm)],[107])).
% 0.12/0.42 tff(109,plain,
% 0.12/0.42 (or(not(implies(X!1, Y!0)), not(or(not(Y!0), not(not(X!1))))) = not(equiv(X!1, Y!0))),
% 0.12/0.42 inference(transitivity,[status(thm)],[108, 105, 102, 98, 96])).
% 0.12/0.42 tff(110,plain,
% 0.12/0.42 (implies(not(or(not(Y!0), not(not(X!1)))), or(not(implies(X!1, Y!0)), not(or(not(Y!0), not(not(X!1)))))) = implies(not(or(not(Y!0), not(not(X!1)))), not(equiv(X!1, Y!0)))),
% 0.12/0.42 inference(monotonicity,[status(thm)],[109])).
% 0.12/0.42 tff(111,plain,
% 0.12/0.42 (implies(not(or(not(Y!0), not(not(X!1)))), or(not(implies(X!1, Y!0)), not(or(not(Y!0), not(not(X!1)))))) = or(or(not(Y!0), X!1), not(equiv(X!1, Y!0)))),
% 0.12/0.42 inference(transitivity,[status(thm)],[110, 71, 69])).
% 0.12/0.42 tff(112,plain,
% 0.12/0.42 (is_a_theorem(implies(not(or(not(Y!0), not(not(X!1)))), or(not(implies(X!1, Y!0)), not(or(not(Y!0), not(not(X!1))))))) <=> is_a_theorem(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))))),
% 0.12/0.42 inference(monotonicity,[status(thm)],[111])).
% 0.12/0.42 tff(113,plain,
% 0.12/0.42 (is_a_theorem(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0)))) <=> is_a_theorem(implies(not(or(not(Y!0), not(not(X!1)))), or(not(implies(X!1, Y!0)), not(or(not(Y!0), not(not(X!1)))))))),
% 0.12/0.42 inference(symmetry,[status(thm)],[112])).
% 0.12/0.42 tff(114,plain,
% 0.12/0.42 ((~is_a_theorem(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))))) <=> (~is_a_theorem(implies(not(or(not(Y!0), not(not(X!1)))), or(not(implies(X!1, Y!0)), not(or(not(Y!0), not(not(X!1))))))))),
% 0.12/0.42 inference(monotonicity,[status(thm)],[113])).
% 0.12/0.42 tff(115,plain,
% 0.12/0.42 (or(not(Y!0), X!1) = implies(Y!0, X!1)),
% 0.12/0.42 inference(symmetry,[status(thm)],[22])).
% 0.12/0.42 tff(116,plain,
% 0.12/0.42 (or(not(equiv(X!1, Y!0)), or(not(Y!0), X!1)) = or(not(equiv(X!1, Y!0)), implies(Y!0, X!1))),
% 0.12/0.42 inference(monotonicity,[status(thm)],[115])).
% 0.12/0.42 tff(117,plain,
% 0.12/0.42 (implies(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))), or(not(equiv(X!1, Y!0)), or(not(Y!0), X!1))) = implies(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))), or(not(equiv(X!1, Y!0)), implies(Y!0, X!1)))),
% 0.12/0.42 inference(monotonicity,[status(thm)],[116])).
% 0.12/0.42 tff(118,plain,
% 0.12/0.42 (is_a_theorem(implies(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))), or(not(equiv(X!1, Y!0)), or(not(Y!0), X!1)))) <=> is_a_theorem(implies(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))), or(not(equiv(X!1, Y!0)), implies(Y!0, X!1))))),
% 0.12/0.42 inference(monotonicity,[status(thm)],[117])).
% 0.12/0.42 tff(119,plain,
% 0.12/0.42 (^[P: $i, Q: $i] : refl(is_a_theorem(implies(or(P, Q), or(Q, P))) <=> is_a_theorem(implies(or(P, Q), or(Q, P))))),
% 0.12/0.42 inference(bind,[status(th)],[])).
% 0.12/0.42 tff(120,plain,
% 0.12/0.42 (![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P))) <=> ![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))),
% 0.12/0.42 inference(quant_intro,[status(thm)],[119])).
% 0.12/0.42 tff(121,plain,
% 0.12/0.42 (![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P))) <=> ![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(122,plain,
% 0.20/0.42 (($true <=> ![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))) <=> ![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(123,axiom,(r3), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+4.ax','principia_r3')).
% 0.20/0.42 tff(124,plain,
% 0.20/0.42 (r3 <=> $true),
% 0.20/0.42 inference(iff_true,[status(thm)],[123])).
% 0.20/0.42 tff(125,plain,
% 0.20/0.42 ((r3 <=> ![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))) <=> ($true <=> ![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[124])).
% 0.20/0.42 tff(126,plain,
% 0.20/0.42 ((r3 <=> ![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))) <=> ![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))),
% 0.20/0.42 inference(transitivity,[status(thm)],[125, 122])).
% 0.20/0.42 tff(127,plain,
% 0.20/0.42 ((r3 <=> ![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))) <=> (r3 <=> ![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(128,axiom,(r3 <=> ![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+0.ax','r3')).
% 0.20/0.42 tff(129,plain,
% 0.20/0.42 (r3 <=> ![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[128, 127])).
% 0.20/0.42 tff(130,plain,
% 0.20/0.42 (r3 <=> ![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[129, 127])).
% 0.20/0.42 tff(131,plain,
% 0.20/0.42 (![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[130, 126])).
% 0.20/0.42 tff(132,plain,
% 0.20/0.42 (![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[131, 121])).
% 0.20/0.42 tff(133,plain,(
% 0.20/0.42 ![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))),
% 0.20/0.42 inference(skolemize,[status(sab)],[132])).
% 0.20/0.42 tff(134,plain,
% 0.20/0.42 (![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[133, 120])).
% 0.20/0.42 tff(135,plain,
% 0.20/0.42 ((~![P: $i, Q: $i] : is_a_theorem(implies(or(P, Q), or(Q, P)))) | is_a_theorem(implies(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))), or(not(equiv(X!1, Y!0)), or(not(Y!0), X!1))))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(136,plain,
% 0.20/0.42 (is_a_theorem(implies(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))), or(not(equiv(X!1, Y!0)), or(not(Y!0), X!1))))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[135, 134])).
% 0.20/0.42 tff(137,plain,
% 0.20/0.42 (is_a_theorem(implies(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))), or(not(equiv(X!1, Y!0)), implies(Y!0, X!1))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[136, 118])).
% 0.20/0.42 tff(138,plain,
% 0.20/0.42 ((~![X: $i, Y: $i] : (implies(X, Y) = or(not(X), Y))) | (implies(equiv(X!1, Y!0), implies(Y!0, X!1)) = or(not(equiv(X!1, Y!0)), implies(Y!0, X!1)))),
% 0.20/0.42 inference(quant_inst,[status(thm)],[])).
% 0.20/0.42 tff(139,plain,
% 0.20/0.42 (implies(equiv(X!1, Y!0), implies(Y!0, X!1)) = or(not(equiv(X!1, Y!0)), implies(Y!0, X!1))),
% 0.20/0.42 inference(unit_resolution,[status(thm)],[138, 20])).
% 0.20/0.42 tff(140,plain,
% 0.20/0.42 (or(not(equiv(X!1, Y!0)), implies(Y!0, X!1)) = implies(equiv(X!1, Y!0), implies(Y!0, X!1))),
% 0.20/0.42 inference(symmetry,[status(thm)],[139])).
% 0.20/0.42 tff(141,plain,
% 0.20/0.42 (is_a_theorem(or(not(equiv(X!1, Y!0)), implies(Y!0, X!1))) <=> is_a_theorem(implies(equiv(X!1, Y!0), implies(Y!0, X!1)))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[140])).
% 0.20/0.42 tff(142,plain,
% 0.20/0.42 (is_a_theorem(implies(equiv(X!1, Y!0), implies(Y!0, X!1))) <=> is_a_theorem(or(not(equiv(X!1, Y!0)), implies(Y!0, X!1)))),
% 0.20/0.42 inference(symmetry,[status(thm)],[141])).
% 0.20/0.42 tff(143,plain,
% 0.20/0.42 ((~is_a_theorem(implies(equiv(X!1, Y!0), implies(Y!0, X!1)))) <=> (~is_a_theorem(or(not(equiv(X!1, Y!0)), implies(Y!0, X!1))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[142])).
% 0.20/0.42 tff(144,plain,
% 0.20/0.42 ((~![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X)))) <=> (~![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(145,plain,
% 0.20/0.42 (($false <=> ![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X)))) <=> (~![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(146,axiom,(~equivalence_2), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','hilbert_equivalence_2')).
% 0.20/0.42 tff(147,plain,
% 0.20/0.42 (equivalence_2 <=> $false),
% 0.20/0.42 inference(iff_false,[status(thm)],[146])).
% 0.20/0.42 tff(148,plain,
% 0.20/0.42 ((equivalence_2 <=> ![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X)))) <=> ($false <=> ![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[147])).
% 0.20/0.42 tff(149,plain,
% 0.20/0.42 ((equivalence_2 <=> ![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X)))) <=> (~![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X))))),
% 0.20/0.42 inference(transitivity,[status(thm)],[148, 145])).
% 0.20/0.42 tff(150,plain,
% 0.20/0.42 ((equivalence_2 <=> ![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X)))) <=> (equivalence_2 <=> ![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(151,axiom,(equivalence_2 <=> ![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X)))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+0.ax','equivalence_2')).
% 0.20/0.42 tff(152,plain,
% 0.20/0.42 (equivalence_2 <=> ![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[151, 150])).
% 0.20/0.42 tff(153,plain,
% 0.20/0.42 (equivalence_2 <=> ![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[152, 150])).
% 0.20/0.42 tff(154,plain,
% 0.20/0.42 (~![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[153, 149])).
% 0.20/0.42 tff(155,plain,
% 0.20/0.42 (~![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[154, 144])).
% 0.20/0.42 tff(156,plain,
% 0.20/0.42 (~![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[155, 144])).
% 0.20/0.42 tff(157,plain,
% 0.20/0.42 (~![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[156, 144])).
% 0.20/0.42 tff(158,plain,
% 0.20/0.42 (~![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[157, 144])).
% 0.20/0.42 tff(159,plain,
% 0.20/0.42 (~![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[158, 144])).
% 0.20/0.42 tff(160,plain,
% 0.20/0.42 (~![X: $i, Y: $i] : is_a_theorem(implies(equiv(X, Y), implies(Y, X)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[159, 144])).
% 0.20/0.42 tff(161,plain,(
% 0.20/0.42 ~is_a_theorem(implies(equiv(X!1, Y!0), implies(Y!0, X!1)))),
% 0.20/0.42 inference(skolemize,[status(sab)],[160])).
% 0.20/0.42 tff(162,plain,
% 0.20/0.42 (~is_a_theorem(or(not(equiv(X!1, Y!0)), implies(Y!0, X!1)))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[161, 143])).
% 0.20/0.42 tff(163,plain,
% 0.20/0.42 (^[X: $i, Y: $i] : refl((is_a_theorem(Y) | (~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y)))) <=> (is_a_theorem(Y) | (~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y)))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(164,plain,
% 0.20/0.42 (![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y)))) <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[163])).
% 0.20/0.42 tff(165,plain,
% 0.20/0.42 (^[X: $i, Y: $i] : trans(monotonicity(trans(monotonicity(rewrite((is_a_theorem(X) & is_a_theorem(implies(X, Y))) <=> (~((~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y)))))), ((~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))) <=> (~(~((~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y)))))))), rewrite((~(~((~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y)))))) <=> ((~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y))))), ((~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))) <=> ((~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y)))))), ((is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y))))) <=> (is_a_theorem(Y) | ((~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y))))))), rewrite((is_a_theorem(Y) | ((~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y))))) <=> (is_a_theorem(Y) | (~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y))))), ((is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y))))) <=> (is_a_theorem(Y) | (~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y))))))),
% 0.20/0.42 inference(bind,[status(th)],[])).
% 0.20/0.42 tff(166,plain,
% 0.20/0.42 (![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y))))) <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y))))),
% 0.20/0.42 inference(quant_intro,[status(thm)],[165])).
% 0.20/0.42 tff(167,plain,
% 0.20/0.42 (![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y))))) <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(168,plain,
% 0.20/0.42 (($true <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))) <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(169,axiom,(modus_ponens), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+4.ax','principia_modus_ponens')).
% 0.20/0.42 tff(170,plain,
% 0.20/0.42 (modus_ponens <=> $true),
% 0.20/0.42 inference(iff_true,[status(thm)],[169])).
% 0.20/0.42 tff(171,plain,
% 0.20/0.42 ((modus_ponens <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))) <=> ($true <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y))))))),
% 0.20/0.42 inference(monotonicity,[status(thm)],[170])).
% 0.20/0.42 tff(172,plain,
% 0.20/0.42 ((modus_ponens <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))) <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 0.20/0.42 inference(transitivity,[status(thm)],[171, 168])).
% 0.20/0.42 tff(173,plain,
% 0.20/0.42 ((modus_ponens <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))) <=> (modus_ponens <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y))))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(174,plain,
% 0.20/0.42 ((modus_ponens <=> ![X: $i, Y: $i] : ((is_a_theorem(X) & is_a_theorem(implies(X, Y))) => is_a_theorem(Y))) <=> (modus_ponens <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y))))))),
% 0.20/0.42 inference(rewrite,[status(thm)],[])).
% 0.20/0.42 tff(175,axiom,(modus_ponens <=> ![X: $i, Y: $i] : ((is_a_theorem(X) & is_a_theorem(implies(X, Y))) => is_a_theorem(Y))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+0.ax','modus_ponens')).
% 0.20/0.42 tff(176,plain,
% 0.20/0.42 (modus_ponens <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[175, 174])).
% 0.20/0.42 tff(177,plain,
% 0.20/0.42 (modus_ponens <=> ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[176, 173])).
% 0.20/0.42 tff(178,plain,
% 0.20/0.42 (![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[177, 172])).
% 0.20/0.42 tff(179,plain,
% 0.20/0.42 (![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[178, 167])).
% 0.20/0.42 tff(180,plain,(
% 0.20/0.42 ![X: $i, Y: $i] : (is_a_theorem(Y) | (~(is_a_theorem(X) & is_a_theorem(implies(X, Y)))))),
% 0.20/0.42 inference(skolemize,[status(sab)],[179])).
% 0.20/0.42 tff(181,plain,
% 0.20/0.42 (![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[180, 166])).
% 0.20/0.42 tff(182,plain,
% 0.20/0.42 (![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y))))),
% 0.20/0.42 inference(modus_ponens,[status(thm)],[181, 164])).
% 0.20/0.42 tff(183,plain,
% 0.20/0.42 (((~![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y))))) | (is_a_theorem(or(not(equiv(X!1, Y!0)), implies(Y!0, X!1))) | (~is_a_theorem(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))))) | (~is_a_theorem(implies(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))), or(not(equiv(X!1, Y!0)), implies(Y!0, X!1))))))) <=> ((~![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y))))) | is_a_theorem(or(not(equiv(X!1, Y!0)), implies(Y!0, X!1))) | (~is_a_theorem(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))))) | (~is_a_theorem(implies(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))), or(not(equiv(X!1, Y!0)), implies(Y!0, X!1))))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(184,plain,
% 0.20/0.43 ((~![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y))))) | (is_a_theorem(or(not(equiv(X!1, Y!0)), implies(Y!0, X!1))) | (~is_a_theorem(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))))) | (~is_a_theorem(implies(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))), or(not(equiv(X!1, Y!0)), implies(Y!0, X!1))))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(185,plain,
% 0.20/0.43 ((~![X: $i, Y: $i] : (is_a_theorem(Y) | (~is_a_theorem(X)) | (~is_a_theorem(implies(X, Y))))) | is_a_theorem(or(not(equiv(X!1, Y!0)), implies(Y!0, X!1))) | (~is_a_theorem(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))))) | (~is_a_theorem(implies(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))), or(not(equiv(X!1, Y!0)), implies(Y!0, X!1)))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[184, 183])).
% 0.20/0.43 tff(186,plain,
% 0.20/0.43 (is_a_theorem(or(not(equiv(X!1, Y!0)), implies(Y!0, X!1))) | (~is_a_theorem(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))))) | (~is_a_theorem(implies(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))), or(not(equiv(X!1, Y!0)), implies(Y!0, X!1)))))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[185, 182])).
% 0.20/0.43 tff(187,plain,
% 0.20/0.43 (~is_a_theorem(or(or(not(Y!0), X!1), not(equiv(X!1, Y!0))))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[186, 162, 137])).
% 0.20/0.43 tff(188,plain,
% 0.20/0.43 (~is_a_theorem(implies(not(or(not(Y!0), not(not(X!1)))), or(not(implies(X!1, Y!0)), not(or(not(Y!0), not(not(X!1)))))))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[187, 114])).
% 0.20/0.43 tff(189,plain,
% 0.20/0.43 (^[P: $i, Q: $i] : refl(is_a_theorem(implies(Q, or(P, Q))) <=> is_a_theorem(implies(Q, or(P, Q))))),
% 0.20/0.43 inference(bind,[status(th)],[])).
% 0.20/0.43 tff(190,plain,
% 0.20/0.43 (![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q))) <=> ![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))),
% 0.20/0.43 inference(quant_intro,[status(thm)],[189])).
% 0.20/0.43 tff(191,plain,
% 0.20/0.43 (![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q))) <=> ![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(192,plain,
% 0.20/0.43 (($true <=> ![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))) <=> ![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(193,axiom,(r2), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+4.ax','principia_r2')).
% 0.20/0.43 tff(194,plain,
% 0.20/0.43 (r2 <=> $true),
% 0.20/0.43 inference(iff_true,[status(thm)],[193])).
% 0.20/0.43 tff(195,plain,
% 0.20/0.43 ((r2 <=> ![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))) <=> ($true <=> ![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q))))),
% 0.20/0.43 inference(monotonicity,[status(thm)],[194])).
% 0.20/0.43 tff(196,plain,
% 0.20/0.43 ((r2 <=> ![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))) <=> ![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))),
% 0.20/0.43 inference(transitivity,[status(thm)],[195, 192])).
% 0.20/0.43 tff(197,plain,
% 0.20/0.43 ((r2 <=> ![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))) <=> (r2 <=> ![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q))))),
% 0.20/0.43 inference(rewrite,[status(thm)],[])).
% 0.20/0.43 tff(198,axiom,(r2 <=> ![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))), file('/export/starexec/sandbox2/benchmark/Axioms/LCL006+0.ax','r2')).
% 0.20/0.43 tff(199,plain,
% 0.20/0.43 (r2 <=> ![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[198, 197])).
% 0.20/0.43 tff(200,plain,
% 0.20/0.43 (r2 <=> ![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[199, 197])).
% 0.20/0.43 tff(201,plain,
% 0.20/0.43 (![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[200, 196])).
% 0.20/0.43 tff(202,plain,
% 0.20/0.43 (![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[201, 191])).
% 0.20/0.43 tff(203,plain,(
% 0.20/0.43 ![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))),
% 0.20/0.43 inference(skolemize,[status(sab)],[202])).
% 0.20/0.43 tff(204,plain,
% 0.20/0.43 (![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))),
% 0.20/0.43 inference(modus_ponens,[status(thm)],[203, 190])).
% 0.20/0.43 tff(205,plain,
% 0.20/0.43 ((~![P: $i, Q: $i] : is_a_theorem(implies(Q, or(P, Q)))) | is_a_theorem(implies(not(or(not(Y!0), not(not(X!1)))), or(not(implies(X!1, Y!0)), not(or(not(Y!0), not(not(X!1)))))))),
% 0.20/0.43 inference(quant_inst,[status(thm)],[])).
% 0.20/0.43 tff(206,plain,
% 0.20/0.43 (is_a_theorem(implies(not(or(not(Y!0), not(not(X!1)))), or(not(implies(X!1, Y!0)), not(or(not(Y!0), not(not(X!1)))))))),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[205, 204])).
% 0.20/0.43 tff(207,plain,
% 0.20/0.43 ($false),
% 0.20/0.43 inference(unit_resolution,[status(thm)],[206, 188])).
% 0.20/0.43 % SZS output end Proof
%------------------------------------------------------------------------------