TSTP Solution File: LCL542+1 by ePrincess---1.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : ePrincess---1.0
% Problem : LCL542+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : ePrincess-casc -timeout=%d %s
% Computer : n027.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 : 600s
% DateTime : Sun Jul 17 09:37:01 EDT 2022
% Result : Theorem 4.19s 1.67s
% Output : Proof 6.50s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : LCL542+1 : TPTP v8.1.0. Released v3.3.0.
% 0.03/0.12 % Command : ePrincess-casc -timeout=%d %s
% 0.12/0.33 % Computer : n027.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 : 600
% 0.12/0.33 % DateTime : Sat Jul 2 16:53:54 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.54/0.61 ____ _
% 0.54/0.61 ___ / __ \_____(_)___ ________ __________
% 0.54/0.61 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/
% 0.54/0.61 / __/ ____/ / / / / / / /__/ __(__ |__ )
% 0.54/0.61 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/
% 0.54/0.61
% 0.54/0.61 A Theorem Prover for First-Order Logic
% 0.54/0.61 (ePrincess v.1.0)
% 0.54/0.61
% 0.54/0.61 (c) Philipp Rümmer, 2009-2015
% 0.54/0.61 (c) Peter Backeman, 2014-2015
% 0.54/0.61 (contributions by Angelo Brillout, Peter Baumgartner)
% 0.54/0.61 Free software under GNU Lesser General Public License (LGPL).
% 0.54/0.61 Bug reports to peter@backeman.se
% 0.54/0.61
% 0.54/0.61 For more information, visit http://user.uu.se/~petba168/breu/
% 0.54/0.61
% 0.54/0.61 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 0.73/0.68 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all
% 1.96/1.06 Prover 0: Preprocessing ...
% 3.80/1.53 Prover 0: Constructing countermodel ...
% 4.19/1.67 Prover 0: proved (991ms)
% 4.19/1.67
% 4.19/1.67 No countermodel exists, formula is valid
% 4.19/1.67 % SZS status Theorem for theBenchmark
% 4.19/1.67
% 4.19/1.67 Generating proof ... found it (size 8)
% 5.81/2.06
% 5.81/2.06 % SZS output start Proof for theBenchmark
% 5.81/2.06 Assumed formulas after preprocessing and simplification:
% 5.81/2.06 | (0) ? [v0] : ? [v1] : ? [v2] : ? [v3] : ? [v4] : ? [v5] : ? [v6] : ? [v7] : ? [v8] : ? [v9] : ? [v10] : ? [v11] : ? [v12] : ? [v13] : ? [v14] : ? [v15] : ? [v16] : ? [v17] : ? [v18] : ? [v19] : ? [v20] : ? [v21] : ? [v22] : ? [v23] : ? [v24] : ? [v25] : ? [v26] : ? [v27] : ? [v28] : ? [v29] : ? [v30] : ? [v31] : ? [v32] : ? [v33] : ? [v34] : ? [v35] : ? [v36] : ? [v37] : ? [v38] : ? [v39] : ? [v40] : ? [v41] : ? [v42] : ? [v43] : ? [v44] : ? [v45] : ? [v46] : ? [v47] : ? [v48] : ? [v49] : ? [v50] : ? [v51] : ? [v52] : ? [v53] : ? [v54] : ? [v55] : ? [v56] : ? [v57] : ? [v58] : ? [v59] : ? [v60] : ? [v61] : ? [v62] : ? [v63] : ? [v64] : ? [v65] : ? [v66] : ? [v67] : ? [v68] : ? [v69] : ? [v70] : ? [v71] : ? [v72] : ? [v73] : ? [v74] : ? [v75] : ? [v76] : ? [v77] : ? [v78] : ? [v79] : ? [v80] : ? [v81] : ? [v82] : ? [v83] : ? [v84] : ? [v85] : ? [v86] : ? [v87] : ? [v88] : ? [v89] : ? [v90] : ? [v91] : ? [v92] : ? [v93] : ? [v94] : ? [v95] : ? [v96] : ? [v97] : ? [v98] : ? [v99] : ? [v100] : ? [v101] : ? [v102] : ? [v103] : ? [v104] : ? [v105] : ? [v106] : ? [v107] : ? [v108] : ? [v109] : ? [v110] : ? [v111] : ? [v112] : ? [v113] : ? [v114] : ? [v115] : ? [v116] : ? [v117] : ? [v118] : ? [v119] : ? [v120] : ? [v121] : ? [v122] : ? [v123] : ? [v124] : ? [v125] : ? [v126] : ? [v127] : ? [v128] : ? [v129] : ? [v130] : ? [v131] : ? [v132] : ? [v133] : ? [v134] : ? [v135] : ? [v136] : ? [v137] : ? [v138] : ? [v139] : ? [v140] : ? [v141] : ? [v142] : ? [v143] : ? [v144] : ? [v145] : ? [v146] : ? [v147] : ? [v148] : ? [v149] : ? [v150] : ? [v151] : ? [v152] : ? [v153] : ? [v154] : ? [v155] : ? [v156] : ? [v157] : (strict_implies(v44, v42) = v45 & and(v42, v43) = v44 & op_implies & op_strict_equiv & op_strict_implies & op_possibly & axiom_B & axiom_4 & axiom_M & axiom_K & necessitation & op_equiv & op_implies_and & op_or & equivalence_3 & equivalence_2 & equivalence_1 & or_3 & or_2 & or_1 & and_3 & and_2 & and_1 & implies_3 & implies_2 & implies_1 & modus_tollens & substitution_of_equivalents & modus_ponens & ~ axiom_m2 & ~ is_a_theorem(v45) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ! [v163] : ! [v164] : ! [v165] : ! [v166] : ( ~ (or(v158, v159) = v163) | ~ (implies(v163, v160) = v164) | ~ (implies(v162, v164) = v165) | ~ (implies(v161, v165) = v166) | ~ (implies(v159, v160) = v162) | ~ (implies(v158, v160) = v161) | is_a_theorem(v166)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ! [v163] : ! [v164] : ! [v165] : ( ~ (necessarily(v160) = v161) | ~ (necessarily(v159) = v163) | ~ (necessarily(v158) = v162) | ~ (implies(v162, v163) = v164) | ~ (implies(v161, v164) = v165) | ~ (implies(v158, v159) = v160) | is_a_theorem(v165)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ! [v163] : ! [v164] : ! [v165] : ( ~ (implies(v162, v163) = v164) | ~ (implies(v161, v164) = v165) | ~ (implies(v159, v160) = v162) | ~ (implies(v158, v160) = v163) | ~ (implies(v158, v159) = v161) | is_a_theorem(v165)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ! [v163] : ! [v164] : ( ~ (not(v159) = v160) | ~ (not(v158) = v161) | ~ (implies(v162, v163) = v164) | ~ (implies(v160, v161) = v162) | ~ (implies(v158, v159) = v163) | is_a_theorem(v164)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ! [v163] : ! [v164] : ( ~ (equiv(v158, v159) = v162) | ~ (implies(v161, v162) = v163) | ~ (implies(v160, v163) = v164) | ~ (implies(v159, v158) = v161) | ~ (implies(v158, v159) = v160) | is_a_theorem(v164)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ( ~ (strict_implies(v159, v158) = v161) | ~ (strict_implies(v158, v159) = v160) | ~ (and(v160, v161) = v162) | strict_equiv(v158, v159) = v162) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ( ~ (and(v160, v161) = v162) | ~ (not(v159) = v161) | ~ (not(v158) = v160) | ? [v163] : (or(v158, v159) = v163 & not(v162) = v163)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ( ~ (and(v160, v161) = v162) | ~ (implies(v159, v158) = v161) | ~ (implies(v158, v159) = v160) | equiv(v158, v159) = v162) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ( ~ (and(v158, v159) = v160) | ~ (implies(v159, v160) = v161) | ~ (implies(v158, v161) = v162) | is_a_theorem(v162)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ( ~ (equiv(v158, v159) = v160) | ~ (implies(v160, v161) = v162) | ~ (implies(v159, v158) = v161) | is_a_theorem(v162)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ( ~ (equiv(v158, v159) = v160) | ~ (implies(v160, v161) = v162) | ~ (implies(v158, v159) = v161) | is_a_theorem(v162)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ( ~ (implies(v161, v160) = v162) | ~ (implies(v158, v160) = v161) | ~ (implies(v158, v159) = v160) | is_a_theorem(v162)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : (v159 = v158 | ~ (strict_equiv(v161, v160) = v159) | ~ (strict_equiv(v161, v160) = v158)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : (v159 = v158 | ~ (strict_implies(v161, v160) = v159) | ~ (strict_implies(v161, v160) = v158)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : (v159 = v158 | ~ (or(v161, v160) = v159) | ~ (or(v161, v160) = v158)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : (v159 = v158 | ~ (and(v161, v160) = v159) | ~ (and(v161, v160) = v158)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : (v159 = v158 | ~ (equiv(v161, v160) = v159) | ~ (equiv(v161, v160) = v158)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : (v159 = v158 | ~ (implies(v161, v160) = v159) | ~ (implies(v161, v160) = v158)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (possibly(v158) = v159) | ~ (necessarily(v159) = v160) | ~ (implies(v158, v160) = v161) | is_a_theorem(v161)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (necessarily(v159) = v160) | ~ (necessarily(v158) = v159) | ~ (implies(v159, v160) = v161) | is_a_theorem(v161)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (or(v158, v159) = v160) | ~ (implies(v159, v160) = v161) | is_a_theorem(v161)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (or(v158, v159) = v160) | ~ (implies(v158, v160) = v161) | is_a_theorem(v161)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (and(v158, v160) = v161) | ~ (not(v159) = v160) | ? [v162] : (not(v161) = v162 & implies(v158, v159) = v162)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (and(v158, v159) = v160) | ~ (implies(v160, v159) = v161) | is_a_theorem(v161)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (and(v158, v159) = v160) | ~ (implies(v160, v158) = v161) | is_a_theorem(v161)) & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (implies(v159, v158) = v160) | ~ (implies(v158, v160) = v161) | is_a_theorem(v161)) & ! [v158] : ! [v159] : ! [v160] : (v159 = v158 | ~ (possibly(v160) = v159) | ~ (possibly(v160) = v158)) & ! [v158] : ! [v159] : ! [v160] : (v159 = v158 | ~ (necessarily(v160) = v159) | ~ (necessarily(v160) = v158)) & ! [v158] : ! [v159] : ! [v160] : (v159 = v158 | ~ (not(v160) = v159) | ~ (not(v160) = v158)) & ! [v158] : ! [v159] : ! [v160] : (v159 = v158 | ~ (equiv(v158, v159) = v160) | ~ is_a_theorem(v160)) & ! [v158] : ! [v159] : ! [v160] : ( ~ (strict_equiv(v158, v159) = v160) | ? [v161] : ? [v162] : (strict_implies(v159, v158) = v162 & strict_implies(v158, v159) = v161 & and(v161, v162) = v160)) & ! [v158] : ! [v159] : ! [v160] : ( ~ (strict_implies(v158, v159) = v160) | ? [v161] : (necessarily(v161) = v160 & implies(v158, v159) = v161)) & ! [v158] : ! [v159] : ! [v160] : ( ~ (necessarily(v158) = v159) | ~ (implies(v159, v158) = v160) | is_a_theorem(v160)) & ! [v158] : ! [v159] : ! [v160] : ( ~ (or(v158, v159) = v160) | ? [v161] : ? [v162] : ? [v163] : (and(v161, v162) = v163 & not(v163) = v160 & not(v159) = v162 & not(v158) = v161)) & ! [v158] : ! [v159] : ! [v160] : ( ~ (equiv(v158, v159) = v160) | ? [v161] : ? [v162] : (and(v161, v162) = v160 & implies(v159, v158) = v162 & implies(v158, v159) = v161)) & ! [v158] : ! [v159] : ! [v160] : ( ~ (implies(v158, v159) = v160) | ~ is_a_theorem(v160) | ~ is_a_theorem(v158) | is_a_theorem(v159)) & ! [v158] : ! [v159] : ! [v160] : ( ~ (implies(v158, v159) = v160) | ? [v161] : ? [v162] : (and(v158, v161) = v162 & not(v162) = v160 & not(v159) = v161)) & ! [v158] : ! [v159] : ! [v160] : ( ~ (implies(v158, v159) = v160) | ? [v161] : (strict_implies(v158, v159) = v161 & necessarily(v160) = v161)) & ! [v158] : ! [v159] : ( ~ (possibly(v158) = v159) | ? [v160] : ? [v161] : (necessarily(v160) = v161 & not(v161) = v159 & not(v158) = v160)) & ! [v158] : ! [v159] : ( ~ (necessarily(v158) = v159) | ~ is_a_theorem(v158) | is_a_theorem(v159)) & ! [v158] : ! [v159] : ( ~ (not(v158) = v159) | ? [v160] : ? [v161] : (possibly(v158) = v160 & necessarily(v159) = v161 & not(v161) = v160)) & ( ~ op_necessarily | ( ! [v158] : ! [v159] : ( ~ (necessarily(v158) = v159) | ? [v160] : ? [v161] : (possibly(v160) = v161 & not(v161) = v159 & not(v158) = v160)) & ! [v158] : ! [v159] : ( ~ (not(v158) = v159) | ? [v160] : ? [v161] : (possibly(v159) = v161 & necessarily(v158) = v160 & not(v161) = v160)))) & ( ~ op_implies_or | ( ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (or(v160, v159) = v161) | ~ (not(v158) = v160) | implies(v158, v159) = v161) & ! [v158] : ! [v159] : ! [v160] : ( ~ (implies(v158, v159) = v160) | ? [v161] : (or(v161, v159) = v160 & not(v158) = v161)))) & ( ~ op_and | ( ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ( ~ (or(v160, v161) = v162) | ~ (not(v159) = v161) | ~ (not(v158) = v160) | ? [v163] : (and(v158, v159) = v163 & not(v162) = v163)) & ! [v158] : ! [v159] : ! [v160] : ( ~ (and(v158, v159) = v160) | ? [v161] : ? [v162] : ? [v163] : (or(v161, v162) = v163 & not(v163) = v160 & not(v159) = v162 & not(v158) = v161)))) & (( ~ (v88 = v87) & strict_equiv(v87, v88) = v89 & is_a_theorem(v89) & ~ substitution_strict_equiv) | (substitution_strict_equiv & ! [v158] : ! [v159] : ! [v160] : (v159 = v158 | ~ (strict_equiv(v158, v159) = v160) | ~ is_a_theorem(v160)))) & ((possibly(v83) = v84 & necessarily(v84) = v85 & implies(v84, v85) = v86 & ~ axiom_5 & ~ is_a_theorem(v86)) | (axiom_5 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (possibly(v158) = v159) | ~ (necessarily(v159) = v160) | ~ (implies(v159, v160) = v161) | is_a_theorem(v161)))) & ((possibly(v66) = v67 & possibly(v65) = v69 & possibly(v64) = v68 & strict_implies(v67, v70) = v71 & and(v68, v69) = v70 & and(v64, v65) = v66 & ~ axiom_s2 & ~ is_a_theorem(v71)) | (axiom_s2 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ! [v163] : ! [v164] : ! [v165] : ( ~ (possibly(v160) = v161) | ~ (possibly(v159) = v163) | ~ (possibly(v158) = v162) | ~ (strict_implies(v161, v164) = v165) | ~ (and(v162, v163) = v164) | ~ (and(v158, v159) = v160) | is_a_theorem(v165)))) & ((possibly(v56) = v58 & possibly(v55) = v60 & strict_implies(v59, v61) = v62 & strict_implies(v57, v62) = v63 & strict_implies(v55, v56) = v57 & not(v60) = v61 & not(v58) = v59 & ~ axiom_s3 & ~ is_a_theorem(v63)) | (axiom_s3 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ! [v163] : ! [v164] : ! [v165] : ! [v166] : ( ~ (possibly(v159) = v161) | ~ (possibly(v158) = v163) | ~ (strict_implies(v162, v164) = v165) | ~ (strict_implies(v160, v165) = v166) | ~ (strict_implies(v158, v159) = v160) | ~ (not(v163) = v164) | ~ (not(v161) = v162) | is_a_theorem(v166)))) & ((possibly(v20) = v21 & strict_implies(v20, v21) = v22 & ~ axiom_m6 & ~ is_a_theorem(v22)) | (axiom_m6 & ! [v158] : ! [v159] : ! [v160] : ( ~ (possibly(v158) = v159) | ~ (strict_implies(v158, v159) = v160) | is_a_theorem(v160)))) & ((possibly(v17) = v18 & strict_implies(v18, v15) = v19 & and(v15, v16) = v17 & ~ axiom_m7 & ~ is_a_theorem(v19)) | (axiom_m7 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ( ~ (possibly(v160) = v161) | ~ (strict_implies(v161, v158) = v162) | ~ (and(v158, v159) = v160) | is_a_theorem(v162)))) & ((possibly(v9) = v12 & possibly(v8) = v11 & strict_implies(v11, v12) = v13 & strict_implies(v10, v13) = v14 & strict_implies(v8, v9) = v10 & ~ axiom_m8 & ~ is_a_theorem(v14)) | (axiom_m8 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ! [v163] : ! [v164] : ( ~ (possibly(v159) = v162) | ~ (possibly(v158) = v161) | ~ (strict_implies(v161, v162) = v163) | ~ (strict_implies(v160, v163) = v164) | ~ (strict_implies(v158, v159) = v160) | is_a_theorem(v164)))) & ((possibly(v5) = v6 & possibly(v4) = v5 & strict_implies(v6, v5) = v7 & ~ axiom_m9 & ~ is_a_theorem(v7)) | (axiom_m9 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (possibly(v159) = v160) | ~ (possibly(v158) = v159) | ~ (strict_implies(v160, v159) = v161) | is_a_theorem(v161)))) & ((possibly(v0) = v1 & strict_implies(v1, v2) = v3 & necessarily(v1) = v2 & ~ axiom_m10 & ~ is_a_theorem(v3)) | (axiom_m10 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (possibly(v158) = v159) | ~ (strict_implies(v159, v160) = v161) | ~ (necessarily(v159) = v160) | is_a_theorem(v161)))) & ((strict_implies(v93, v94) = v95 & is_a_theorem(v95) & is_a_theorem(v93) & ~ modus_ponens_strict_implies & ~ is_a_theorem(v94)) | (modus_ponens_strict_implies & ! [v158] : ! [v159] : ! [v160] : ( ~ (strict_implies(v158, v159) = v160) | ~ is_a_theorem(v160) | ~ is_a_theorem(v158) | is_a_theorem(v159)))) & ((strict_implies(v52, v53) = v54 & necessarily(v52) = v53 & necessarily(v51) = v52 & ~ axiom_s4 & ~ is_a_theorem(v54)) | (axiom_s4 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (strict_implies(v159, v160) = v161) | ~ (necessarily(v159) = v160) | ~ (necessarily(v158) = v159) | is_a_theorem(v161)))) & ((strict_implies(v48, v49) = v50 & and(v47, v46) = v49 & and(v46, v47) = v48 & ~ axiom_m1 & ~ is_a_theorem(v50)) | (axiom_m1 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ( ~ (strict_implies(v160, v161) = v162) | ~ (and(v159, v158) = v161) | ~ (and(v158, v159) = v160) | is_a_theorem(v162)))) & ((strict_implies(v38, v40) = v41 & and(v37, v36) = v38 & and(v35, v36) = v39 & and(v34, v39) = v40 & and(v34, v35) = v37 & ~ axiom_m3 & ~ is_a_theorem(v41)) | (axiom_m3 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ! [v163] : ! [v164] : ! [v165] : ( ~ (strict_implies(v162, v164) = v165) | ~ (and(v161, v160) = v162) | ~ (and(v159, v160) = v163) | ~ (and(v158, v163) = v164) | ~ (and(v158, v159) = v161) | is_a_theorem(v165)))) & ((strict_implies(v31, v32) = v33 & and(v31, v31) = v32 & ~ axiom_m4 & ~ is_a_theorem(v33)) | (axiom_m4 & ! [v158] : ! [v159] : ! [v160] : ( ~ (strict_implies(v158, v159) = v160) | ~ (and(v158, v158) = v159) | is_a_theorem(v160)))) & ((strict_implies(v28, v29) = v30 & strict_implies(v24, v25) = v27 & strict_implies(v23, v25) = v29 & strict_implies(v23, v24) = v26 & and(v26, v27) = v28 & ~ axiom_m5 & ~ is_a_theorem(v30)) | (axiom_m5 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ! [v163] : ! [v164] : ! [v165] : ( ~ (strict_implies(v163, v164) = v165) | ~ (strict_implies(v159, v160) = v162) | ~ (strict_implies(v158, v160) = v164) | ~ (strict_implies(v158, v159) = v161) | ~ (and(v161, v162) = v163) | is_a_theorem(v165)))) & ((necessarily(v80) = v81 & necessarily(v77) = v78 & necessarily(v75) = v76 & and(v76, v78) = v79 & implies(v79, v81) = v82 & implies(v73, v74) = v77 & implies(v72, v74) = v80 & implies(v72, v73) = v75 & ~ axiom_s1 & ~ is_a_theorem(v82)) | (axiom_s1 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ! [v163] : ! [v164] : ! [v165] : ! [v166] : ! [v167] : ! [v168] : ( ~ (necessarily(v166) = v167) | ~ (necessarily(v163) = v164) | ~ (necessarily(v161) = v162) | ~ (and(v162, v164) = v165) | ~ (implies(v165, v167) = v168) | ~ (implies(v159, v160) = v163) | ~ (implies(v158, v160) = v166) | ~ (implies(v158, v159) = v161) | is_a_theorem(v168)))) & ((or(v121, v121) = v122 & implies(v122, v121) = v123 & ~ r1 & ~ is_a_theorem(v123)) | (r1 & ! [v158] : ! [v159] : ! [v160] : ( ~ (or(v158, v158) = v159) | ~ (implies(v159, v158) = v160) | is_a_theorem(v160)))) & ((or(v117, v118) = v119 & implies(v118, v119) = v120 & ~ r2 & ~ is_a_theorem(v120)) | (r2 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (or(v158, v159) = v160) | ~ (implies(v159, v160) = v161) | is_a_theorem(v161)))) & ((or(v113, v112) = v115 & or(v112, v113) = v114 & implies(v114, v115) = v116 & ~ r3 & ~ is_a_theorem(v116)) | (r3 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ( ~ (or(v159, v158) = v161) | ~ (or(v158, v159) = v160) | ~ (implies(v160, v161) = v162) | is_a_theorem(v162)))) & ((or(v105, v109) = v110 & or(v105, v106) = v107 & or(v104, v107) = v108 & or(v104, v106) = v109 & implies(v108, v110) = v111 & ~ r4 & ~ is_a_theorem(v111)) | (r4 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ! [v163] : ! [v164] : ! [v165] : ( ~ (or(v159, v163) = v164) | ~ (or(v159, v160) = v161) | ~ (or(v158, v161) = v162) | ~ (or(v158, v160) = v163) | ~ (implies(v162, v164) = v165) | is_a_theorem(v165)))) & ((or(v96, v98) = v101 & or(v96, v97) = v100 & implies(v100, v101) = v102 & implies(v99, v102) = v103 & implies(v97, v98) = v99 & ~ r5 & ~ is_a_theorem(v103)) | (r5 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ! [v163] : ! [v164] : ! [v165] : ( ~ (or(v158, v160) = v163) | ~ (or(v158, v159) = v162) | ~ (implies(v162, v163) = v164) | ~ (implies(v161, v164) = v165) | ~ (implies(v159, v160) = v161) | is_a_theorem(v165)))) & ((and(v155, v155) = v156 & implies(v155, v156) = v157 & ~ kn1 & ~ is_a_theorem(v157)) | (kn1 & ! [v158] : ! [v159] : ! [v160] : ( ~ (and(v158, v158) = v159) | ~ (implies(v158, v159) = v160) | is_a_theorem(v160)))) & ((and(v151, v152) = v153 & implies(v153, v151) = v154 & ~ kn2 & ~ is_a_theorem(v154)) | (kn2 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (and(v158, v159) = v160) | ~ (implies(v160, v158) = v161) | is_a_theorem(v161)))) & ((and(v143, v141) = v147 & and(v142, v143) = v145 & not(v147) = v148 & not(v145) = v146 & implies(v146, v148) = v149 & implies(v144, v149) = v150 & implies(v141, v142) = v144 & ~ kn3 & ~ is_a_theorem(v150)) | (kn3 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ! [v163] : ! [v164] : ! [v165] : ! [v166] : ! [v167] : ( ~ (and(v160, v158) = v164) | ~ (and(v159, v160) = v162) | ~ (not(v164) = v165) | ~ (not(v162) = v163) | ~ (implies(v163, v165) = v166) | ~ (implies(v161, v166) = v167) | ~ (implies(v158, v159) = v161) | is_a_theorem(v167)))) & ((and(v90, v91) = v92 & is_a_theorem(v91) & is_a_theorem(v90) & ~ adjunction & ~ is_a_theorem(v92)) | (adjunction & ! [v158] : ! [v159] : ! [v160] : ( ~ (and(v158, v159) = v160) | ~ is_a_theorem(v159) | ~ is_a_theorem(v158) | is_a_theorem(v160)))) & ((not(v128) = v130 & implies(v130, v129) = v131 & implies(v128, v131) = v132 & ~ cn2 & ~ is_a_theorem(v132)) | (cn2 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ( ~ (not(v158) = v160) | ~ (implies(v160, v159) = v161) | ~ (implies(v158, v161) = v162) | is_a_theorem(v162)))) & ((not(v124) = v125 & implies(v126, v124) = v127 & implies(v125, v124) = v126 & ~ cn3 & ~ is_a_theorem(v127)) | (cn3 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ( ~ (not(v158) = v159) | ~ (implies(v160, v158) = v161) | ~ (implies(v159, v158) = v160) | is_a_theorem(v161)))) & ((implies(v137, v138) = v139 & implies(v136, v139) = v140 & implies(v134, v135) = v137 & implies(v133, v135) = v138 & implies(v133, v134) = v136 & ~ cn1 & ~ is_a_theorem(v140)) | (cn1 & ! [v158] : ! [v159] : ! [v160] : ! [v161] : ! [v162] : ! [v163] : ! [v164] : ! [v165] : ( ~ (implies(v162, v163) = v164) | ~ (implies(v161, v164) = v165) | ~ (implies(v159, v160) = v162) | ~ (implies(v158, v160) = v163) | ~ (implies(v158, v159) = v161) | is_a_theorem(v165)))))
% 6.37/2.13 | Instantiating (0) with all_0_0_0, all_0_1_1, all_0_2_2, all_0_3_3, all_0_4_4, all_0_5_5, all_0_6_6, all_0_7_7, all_0_8_8, all_0_9_9, all_0_10_10, all_0_11_11, all_0_12_12, all_0_13_13, all_0_14_14, all_0_15_15, all_0_16_16, all_0_17_17, all_0_18_18, all_0_19_19, all_0_20_20, all_0_21_21, all_0_22_22, all_0_23_23, all_0_24_24, all_0_25_25, all_0_26_26, all_0_27_27, all_0_28_28, all_0_29_29, all_0_30_30, all_0_31_31, all_0_32_32, all_0_33_33, all_0_34_34, all_0_35_35, all_0_36_36, all_0_37_37, all_0_38_38, all_0_39_39, all_0_40_40, all_0_41_41, all_0_42_42, all_0_43_43, all_0_44_44, all_0_45_45, all_0_46_46, all_0_47_47, all_0_48_48, all_0_49_49, all_0_50_50, all_0_51_51, all_0_52_52, all_0_53_53, all_0_54_54, all_0_55_55, all_0_56_56, all_0_57_57, all_0_58_58, all_0_59_59, all_0_60_60, all_0_61_61, all_0_62_62, all_0_63_63, all_0_64_64, all_0_65_65, all_0_66_66, all_0_67_67, all_0_68_68, all_0_69_69, all_0_70_70, all_0_71_71, all_0_72_72, all_0_73_73, all_0_74_74, all_0_75_75, all_0_76_76, all_0_77_77, all_0_78_78, all_0_79_79, all_0_80_80, all_0_81_81, all_0_82_82, all_0_83_83, all_0_84_84, all_0_85_85, all_0_86_86, all_0_87_87, all_0_88_88, all_0_89_89, all_0_90_90, all_0_91_91, all_0_92_92, all_0_93_93, all_0_94_94, all_0_95_95, all_0_96_96, all_0_97_97, all_0_98_98, all_0_99_99, all_0_100_100, all_0_101_101, all_0_102_102, all_0_103_103, all_0_104_104, all_0_105_105, all_0_106_106, all_0_107_107, all_0_108_108, all_0_109_109, all_0_110_110, all_0_111_111, all_0_112_112, all_0_113_113, all_0_114_114, all_0_115_115, all_0_116_116, all_0_117_117, all_0_118_118, all_0_119_119, all_0_120_120, all_0_121_121, all_0_122_122, all_0_123_123, all_0_124_124, all_0_125_125, all_0_126_126, all_0_127_127, all_0_128_128, all_0_129_129, all_0_130_130, all_0_131_131, all_0_132_132, all_0_133_133, all_0_134_134, all_0_135_135, all_0_136_136, all_0_137_137, all_0_138_138, all_0_139_139, all_0_140_140, all_0_141_141, all_0_142_142, all_0_143_143, all_0_144_144, all_0_145_145, all_0_146_146, all_0_147_147, all_0_148_148, all_0_149_149, all_0_150_150, all_0_151_151, all_0_152_152, all_0_153_153, all_0_154_154, all_0_155_155, all_0_156_156, all_0_157_157 yields:
% 6.37/2.13 | (1) strict_implies(all_0_113_113, all_0_115_115) = all_0_112_112 & and(all_0_115_115, all_0_114_114) = all_0_113_113 & op_implies & op_strict_equiv & op_strict_implies & op_possibly & axiom_B & axiom_4 & axiom_M & axiom_K & necessitation & op_equiv & op_implies_and & op_or & equivalence_3 & equivalence_2 & equivalence_1 & or_3 & or_2 & or_1 & and_3 & and_2 & and_1 & implies_3 & implies_2 & implies_1 & modus_tollens & substitution_of_equivalents & modus_ponens & ~ axiom_m2 & ~ is_a_theorem(all_0_112_112) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (or(v0, v1) = v5) | ~ (implies(v5, v2) = v6) | ~ (implies(v4, v6) = v7) | ~ (implies(v3, v7) = v8) | ~ (implies(v1, v2) = v4) | ~ (implies(v0, v2) = v3) | is_a_theorem(v8)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (necessarily(v2) = v3) | ~ (necessarily(v1) = v5) | ~ (necessarily(v0) = v4) | ~ (implies(v4, v5) = v6) | ~ (implies(v3, v6) = v7) | ~ (implies(v0, v1) = v2) | is_a_theorem(v7)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (implies(v4, v5) = v6) | ~ (implies(v3, v6) = v7) | ~ (implies(v1, v2) = v4) | ~ (implies(v0, v2) = v5) | ~ (implies(v0, v1) = v3) | is_a_theorem(v7)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (not(v1) = v2) | ~ (not(v0) = v3) | ~ (implies(v4, v5) = v6) | ~ (implies(v2, v3) = v4) | ~ (implies(v0, v1) = v5) | is_a_theorem(v6)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (equiv(v0, v1) = v4) | ~ (implies(v3, v4) = v5) | ~ (implies(v2, v5) = v6) | ~ (implies(v1, v0) = v3) | ~ (implies(v0, v1) = v2) | is_a_theorem(v6)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (strict_implies(v1, v0) = v3) | ~ (strict_implies(v0, v1) = v2) | ~ (and(v2, v3) = v4) | strict_equiv(v0, v1) = v4) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (and(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) = v2) | ? [v5] : (or(v0, v1) = v5 & not(v4) = v5)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (and(v2, v3) = v4) | ~ (implies(v1, v0) = v3) | ~ (implies(v0, v1) = v2) | equiv(v0, v1) = v4) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (and(v0, v1) = v2) | ~ (implies(v1, v2) = v3) | ~ (implies(v0, v3) = v4) | is_a_theorem(v4)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (equiv(v0, v1) = v2) | ~ (implies(v2, v3) = v4) | ~ (implies(v1, v0) = v3) | is_a_theorem(v4)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (equiv(v0, v1) = v2) | ~ (implies(v2, v3) = v4) | ~ (implies(v0, v1) = v3) | is_a_theorem(v4)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (implies(v3, v2) = v4) | ~ (implies(v0, v2) = v3) | ~ (implies(v0, v1) = v2) | is_a_theorem(v4)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (strict_equiv(v3, v2) = v1) | ~ (strict_equiv(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (strict_implies(v3, v2) = v1) | ~ (strict_implies(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (or(v3, v2) = v1) | ~ (or(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (and(v3, v2) = v1) | ~ (and(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equiv(v3, v2) = v1) | ~ (equiv(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (implies(v3, v2) = v1) | ~ (implies(v3, v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (possibly(v0) = v1) | ~ (necessarily(v1) = v2) | ~ (implies(v0, v2) = v3) | is_a_theorem(v3)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (necessarily(v1) = v2) | ~ (necessarily(v0) = v1) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v0, v1) = v2) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v0, v1) = v2) | ~ (implies(v0, v2) = v3) | is_a_theorem(v3)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v2) = v3) | ~ (not(v1) = v2) | ? [v4] : (not(v3) = v4 & implies(v0, v1) = v4)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v1) = v2) | ~ (implies(v2, v1) = v3) | is_a_theorem(v3)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v1) = v2) | ~ (implies(v2, v0) = v3) | is_a_theorem(v3)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (implies(v1, v0) = v2) | ~ (implies(v0, v2) = v3) | is_a_theorem(v3)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (possibly(v2) = v1) | ~ (possibly(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (necessarily(v2) = v1) | ~ (necessarily(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (not(v2) = v1) | ~ (not(v2) = v0)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (equiv(v0, v1) = v2) | ~ is_a_theorem(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (strict_equiv(v0, v1) = v2) | ? [v3] : ? [v4] : (strict_implies(v1, v0) = v4 & strict_implies(v0, v1) = v3 & and(v3, v4) = v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (strict_implies(v0, v1) = v2) | ? [v3] : (necessarily(v3) = v2 & implies(v0, v1) = v3)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (necessarily(v0) = v1) | ~ (implies(v1, v0) = v2) | is_a_theorem(v2)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (and(v3, v4) = v5 & not(v5) = v2 & not(v1) = v4 & not(v0) = v3)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (equiv(v0, v1) = v2) | ? [v3] : ? [v4] : (and(v3, v4) = v2 & implies(v1, v0) = v4 & implies(v0, v1) = v3)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ~ is_a_theorem(v2) | ~ is_a_theorem(v0) | is_a_theorem(v1)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : (and(v0, v3) = v4 & not(v4) = v2 & not(v1) = v3)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : (strict_implies(v0, v1) = v3 & necessarily(v2) = v3)) & ! [v0] : ! [v1] : ( ~ (possibly(v0) = v1) | ? [v2] : ? [v3] : (necessarily(v2) = v3 & not(v3) = v1 & not(v0) = v2)) & ! [v0] : ! [v1] : ( ~ (necessarily(v0) = v1) | ~ is_a_theorem(v0) | is_a_theorem(v1)) & ! [v0] : ! [v1] : ( ~ (not(v0) = v1) | ? [v2] : ? [v3] : (possibly(v0) = v2 & necessarily(v1) = v3 & not(v3) = v2)) & ( ~ op_necessarily | ( ! [v0] : ! [v1] : ( ~ (necessarily(v0) = v1) | ? [v2] : ? [v3] : (possibly(v2) = v3 & not(v3) = v1 & not(v0) = v2)) & ! [v0] : ! [v1] : ( ~ (not(v0) = v1) | ? [v2] : ? [v3] : (possibly(v1) = v3 & necessarily(v0) = v2 & not(v3) = v2)))) & ( ~ op_implies_or | ( ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v2, v1) = v3) | ~ (not(v0) = v2) | implies(v0, v1) = v3) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : (or(v3, v1) = v2 & not(v0) = v3)))) & ( ~ op_and | ( ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (or(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) = v2) | ? [v5] : (and(v0, v1) = v5 & not(v4) = v5)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (or(v3, v4) = v5 & not(v5) = v2 & not(v1) = v4 & not(v0) = v3)))) & (( ~ (all_0_69_69 = all_0_70_70) & strict_equiv(all_0_70_70, all_0_69_69) = all_0_68_68 & is_a_theorem(all_0_68_68) & ~ substitution_strict_equiv) | (substitution_strict_equiv & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (strict_equiv(v0, v1) = v2) | ~ is_a_theorem(v2)))) & ((possibly(all_0_74_74) = all_0_73_73 & necessarily(all_0_73_73) = all_0_72_72 & implies(all_0_73_73, all_0_72_72) = all_0_71_71 & ~ axiom_5 & ~ is_a_theorem(all_0_71_71)) | (axiom_5 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (possibly(v0) = v1) | ~ (necessarily(v1) = v2) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3)))) & ((possibly(all_0_91_91) = all_0_90_90 & possibly(all_0_92_92) = all_0_88_88 & possibly(all_0_93_93) = all_0_89_89 & strict_implies(all_0_90_90, all_0_87_87) = all_0_86_86 & and(all_0_89_89, all_0_88_88) = all_0_87_87 & and(all_0_93_93, all_0_92_92) = all_0_91_91 & ~ axiom_s2 & ~ is_a_theorem(all_0_86_86)) | (axiom_s2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (possibly(v2) = v3) | ~ (possibly(v1) = v5) | ~ (possibly(v0) = v4) | ~ (strict_implies(v3, v6) = v7) | ~ (and(v4, v5) = v6) | ~ (and(v0, v1) = v2) | is_a_theorem(v7)))) & ((possibly(all_0_101_101) = all_0_99_99 & possibly(all_0_102_102) = all_0_97_97 & strict_implies(all_0_98_98, all_0_96_96) = all_0_95_95 & strict_implies(all_0_100_100, all_0_95_95) = all_0_94_94 & strict_implies(all_0_102_102, all_0_101_101) = all_0_100_100 & not(all_0_97_97) = all_0_96_96 & not(all_0_99_99) = all_0_98_98 & ~ axiom_s3 & ~ is_a_theorem(all_0_94_94)) | (axiom_s3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (possibly(v1) = v3) | ~ (possibly(v0) = v5) | ~ (strict_implies(v4, v6) = v7) | ~ (strict_implies(v2, v7) = v8) | ~ (strict_implies(v0, v1) = v2) | ~ (not(v5) = v6) | ~ (not(v3) = v4) | is_a_theorem(v8)))) & ((possibly(all_0_137_137) = all_0_136_136 & strict_implies(all_0_137_137, all_0_136_136) = all_0_135_135 & ~ axiom_m6 & ~ is_a_theorem(all_0_135_135)) | (axiom_m6 & ! [v0] : ! [v1] : ! [v2] : ( ~ (possibly(v0) = v1) | ~ (strict_implies(v0, v1) = v2) | is_a_theorem(v2)))) & ((possibly(all_0_140_140) = all_0_139_139 & strict_implies(all_0_139_139, all_0_142_142) = all_0_138_138 & and(all_0_142_142, all_0_141_141) = all_0_140_140 & ~ axiom_m7 & ~ is_a_theorem(all_0_138_138)) | (axiom_m7 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (possibly(v2) = v3) | ~ (strict_implies(v3, v0) = v4) | ~ (and(v0, v1) = v2) | is_a_theorem(v4)))) & ((possibly(all_0_148_148) = all_0_145_145 & possibly(all_0_149_149) = all_0_146_146 & strict_implies(all_0_146_146, all_0_145_145) = all_0_144_144 & strict_implies(all_0_147_147, all_0_144_144) = all_0_143_143 & strict_implies(all_0_149_149, all_0_148_148) = all_0_147_147 & ~ axiom_m8 & ~ is_a_theorem(all_0_143_143)) | (axiom_m8 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (possibly(v1) = v4) | ~ (possibly(v0) = v3) | ~ (strict_implies(v3, v4) = v5) | ~ (strict_implies(v2, v5) = v6) | ~ (strict_implies(v0, v1) = v2) | is_a_theorem(v6)))) & ((possibly(all_0_152_152) = all_0_151_151 & possibly(all_0_153_153) = all_0_152_152 & strict_implies(all_0_151_151, all_0_152_152) = all_0_150_150 & ~ axiom_m9 & ~ is_a_theorem(all_0_150_150)) | (axiom_m9 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (possibly(v1) = v2) | ~ (possibly(v0) = v1) | ~ (strict_implies(v2, v1) = v3) | is_a_theorem(v3)))) & ((possibly(all_0_157_157) = all_0_156_156 & strict_implies(all_0_156_156, all_0_155_155) = all_0_154_154 & necessarily(all_0_156_156) = all_0_155_155 & ~ axiom_m10 & ~ is_a_theorem(all_0_154_154)) | (axiom_m10 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (possibly(v0) = v1) | ~ (strict_implies(v1, v2) = v3) | ~ (necessarily(v1) = v2) | is_a_theorem(v3)))) & ((strict_implies(all_0_64_64, all_0_63_63) = all_0_62_62 & is_a_theorem(all_0_62_62) & is_a_theorem(all_0_64_64) & ~ modus_ponens_strict_implies & ~ is_a_theorem(all_0_63_63)) | (modus_ponens_strict_implies & ! [v0] : ! [v1] : ! [v2] : ( ~ (strict_implies(v0, v1) = v2) | ~ is_a_theorem(v2) | ~ is_a_theorem(v0) | is_a_theorem(v1)))) & ((strict_implies(all_0_105_105, all_0_104_104) = all_0_103_103 & necessarily(all_0_105_105) = all_0_104_104 & necessarily(all_0_106_106) = all_0_105_105 & ~ axiom_s4 & ~ is_a_theorem(all_0_103_103)) | (axiom_s4 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (strict_implies(v1, v2) = v3) | ~ (necessarily(v1) = v2) | ~ (necessarily(v0) = v1) | is_a_theorem(v3)))) & ((strict_implies(all_0_109_109, all_0_108_108) = all_0_107_107 & and(all_0_110_110, all_0_111_111) = all_0_108_108 & and(all_0_111_111, all_0_110_110) = all_0_109_109 & ~ axiom_m1 & ~ is_a_theorem(all_0_107_107)) | (axiom_m1 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (strict_implies(v2, v3) = v4) | ~ (and(v1, v0) = v3) | ~ (and(v0, v1) = v2) | is_a_theorem(v4)))) & ((strict_implies(all_0_119_119, all_0_117_117) = all_0_116_116 & and(all_0_120_120, all_0_121_121) = all_0_119_119 & and(all_0_122_122, all_0_121_121) = all_0_118_118 & and(all_0_123_123, all_0_118_118) = all_0_117_117 & and(all_0_123_123, all_0_122_122) = all_0_120_120 & ~ axiom_m3 & ~ is_a_theorem(all_0_116_116)) | (axiom_m3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (strict_implies(v4, v6) = v7) | ~ (and(v3, v2) = v4) | ~ (and(v1, v2) = v5) | ~ (and(v0, v5) = v6) | ~ (and(v0, v1) = v3) | is_a_theorem(v7)))) & ((strict_implies(all_0_126_126, all_0_125_125) = all_0_124_124 & and(all_0_126_126, all_0_126_126) = all_0_125_125 & ~ axiom_m4 & ~ is_a_theorem(all_0_124_124)) | (axiom_m4 & ! [v0] : ! [v1] : ! [v2] : ( ~ (strict_implies(v0, v1) = v2) | ~ (and(v0, v0) = v1) | is_a_theorem(v2)))) & ((strict_implies(all_0_129_129, all_0_128_128) = all_0_127_127 & strict_implies(all_0_133_133, all_0_132_132) = all_0_130_130 & strict_implies(all_0_134_134, all_0_132_132) = all_0_128_128 & strict_implies(all_0_134_134, all_0_133_133) = all_0_131_131 & and(all_0_131_131, all_0_130_130) = all_0_129_129 & ~ axiom_m5 & ~ is_a_theorem(all_0_127_127)) | (axiom_m5 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (strict_implies(v5, v6) = v7) | ~ (strict_implies(v1, v2) = v4) | ~ (strict_implies(v0, v2) = v6) | ~ (strict_implies(v0, v1) = v3) | ~ (and(v3, v4) = v5) | is_a_theorem(v7)))) & ((necessarily(all_0_77_77) = all_0_76_76 & necessarily(all_0_80_80) = all_0_79_79 & necessarily(all_0_82_82) = all_0_81_81 & and(all_0_81_81, all_0_79_79) = all_0_78_78 & implies(all_0_78_78, all_0_76_76) = all_0_75_75 & implies(all_0_84_84, all_0_83_83) = all_0_80_80 & implies(all_0_85_85, all_0_83_83) = all_0_77_77 & implies(all_0_85_85, all_0_84_84) = all_0_82_82 & ~ axiom_s1 & ~ is_a_theorem(all_0_75_75)) | (axiom_s1 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (necessarily(v8) = v9) | ~ (necessarily(v5) = v6) | ~ (necessarily(v3) = v4) | ~ (and(v4, v6) = v7) | ~ (implies(v7, v9) = v10) | ~ (implies(v1, v2) = v5) | ~ (implies(v0, v2) = v8) | ~ (implies(v0, v1) = v3) | is_a_theorem(v10)))) & ((or(all_0_36_36, all_0_36_36) = all_0_35_35 & implies(all_0_35_35, all_0_36_36) = all_0_34_34 & ~ r1 & ~ is_a_theorem(all_0_34_34)) | (r1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v0) = v1) | ~ (implies(v1, v0) = v2) | is_a_theorem(v2)))) & ((or(all_0_40_40, all_0_39_39) = all_0_38_38 & implies(all_0_39_39, all_0_38_38) = all_0_37_37 & ~ r2 & ~ is_a_theorem(all_0_37_37)) | (r2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v0, v1) = v2) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3)))) & ((or(all_0_44_44, all_0_45_45) = all_0_42_42 & or(all_0_45_45, all_0_44_44) = all_0_43_43 & implies(all_0_43_43, all_0_42_42) = all_0_41_41 & ~ r3 & ~ is_a_theorem(all_0_41_41)) | (r3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (or(v1, v0) = v3) | ~ (or(v0, v1) = v2) | ~ (implies(v2, v3) = v4) | is_a_theorem(v4)))) & ((or(all_0_52_52, all_0_48_48) = all_0_47_47 & or(all_0_52_52, all_0_51_51) = all_0_50_50 & or(all_0_53_53, all_0_50_50) = all_0_49_49 & or(all_0_53_53, all_0_51_51) = all_0_48_48 & implies(all_0_49_49, all_0_47_47) = all_0_46_46 & ~ r4 & ~ is_a_theorem(all_0_46_46)) | (r4 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (or(v1, v5) = v6) | ~ (or(v1, v2) = v3) | ~ (or(v0, v3) = v4) | ~ (or(v0, v2) = v5) | ~ (implies(v4, v6) = v7) | is_a_theorem(v7)))) & ((or(all_0_61_61, all_0_59_59) = all_0_56_56 & or(all_0_61_61, all_0_60_60) = all_0_57_57 & implies(all_0_57_57, all_0_56_56) = all_0_55_55 & implies(all_0_58_58, all_0_55_55) = all_0_54_54 & implies(all_0_60_60, all_0_59_59) = all_0_58_58 & ~ r5 & ~ is_a_theorem(all_0_54_54)) | (r5 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (or(v0, v2) = v5) | ~ (or(v0, v1) = v4) | ~ (implies(v4, v5) = v6) | ~ (implies(v3, v6) = v7) | ~ (implies(v1, v2) = v3) | is_a_theorem(v7)))) & ((and(all_0_2_2, all_0_2_2) = all_0_1_1 & implies(all_0_2_2, all_0_1_1) = all_0_0_0 & ~ kn1 & ~ is_a_theorem(all_0_0_0)) | (kn1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v0) = v1) | ~ (implies(v0, v1) = v2) | is_a_theorem(v2)))) & ((and(all_0_6_6, all_0_5_5) = all_0_4_4 & implies(all_0_4_4, all_0_6_6) = all_0_3_3 & ~ kn2 & ~ is_a_theorem(all_0_3_3)) | (kn2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v1) = v2) | ~ (implies(v2, v0) = v3) | is_a_theorem(v3)))) & ((and(all_0_14_14, all_0_16_16) = all_0_10_10 & and(all_0_15_15, all_0_14_14) = all_0_12_12 & not(all_0_10_10) = all_0_9_9 & not(all_0_12_12) = all_0_11_11 & implies(all_0_11_11, all_0_9_9) = all_0_8_8 & implies(all_0_13_13, all_0_8_8) = all_0_7_7 & implies(all_0_16_16, all_0_15_15) = all_0_13_13 & ~ kn3 & ~ is_a_theorem(all_0_7_7)) | (kn3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (and(v2, v0) = v6) | ~ (and(v1, v2) = v4) | ~ (not(v6) = v7) | ~ (not(v4) = v5) | ~ (implies(v5, v7) = v8) | ~ (implies(v3, v8) = v9) | ~ (implies(v0, v1) = v3) | is_a_theorem(v9)))) & ((and(all_0_67_67, all_0_66_66) = all_0_65_65 & is_a_theorem(all_0_66_66) & is_a_theorem(all_0_67_67) & ~ adjunction & ~ is_a_theorem(all_0_65_65)) | (adjunction & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ~ is_a_theorem(v1) | ~ is_a_theorem(v0) | is_a_theorem(v2)))) & ((not(all_0_29_29) = all_0_27_27 & implies(all_0_27_27, all_0_28_28) = all_0_26_26 & implies(all_0_29_29, all_0_26_26) = all_0_25_25 & ~ cn2 & ~ is_a_theorem(all_0_25_25)) | (cn2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (not(v0) = v2) | ~ (implies(v2, v1) = v3) | ~ (implies(v0, v3) = v4) | is_a_theorem(v4)))) & ((not(all_0_33_33) = all_0_32_32 & implies(all_0_31_31, all_0_33_33) = all_0_30_30 & implies(all_0_32_32, all_0_33_33) = all_0_31_31 & ~ cn3 & ~ is_a_theorem(all_0_30_30)) | (cn3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (not(v0) = v1) | ~ (implies(v2, v0) = v3) | ~ (implies(v1, v0) = v2) | is_a_theorem(v3)))) & ((implies(all_0_20_20, all_0_19_19) = all_0_18_18 & implies(all_0_21_21, all_0_18_18) = all_0_17_17 & implies(all_0_23_23, all_0_22_22) = all_0_20_20 & implies(all_0_24_24, all_0_22_22) = all_0_19_19 & implies(all_0_24_24, all_0_23_23) = all_0_21_21 & ~ cn1 & ~ is_a_theorem(all_0_17_17)) | (cn1 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (implies(v4, v5) = v6) | ~ (implies(v3, v6) = v7) | ~ (implies(v1, v2) = v4) | ~ (implies(v0, v2) = v5) | ~ (implies(v0, v1) = v3) | is_a_theorem(v7))))
% 6.50/2.16 |
% 6.50/2.16 | Applying alpha-rule on (1) yields:
% 6.50/2.16 | (2) or_2
% 6.50/2.16 | (3) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (equiv(v0, v1) = v2) | ~ (implies(v2, v3) = v4) | ~ (implies(v0, v1) = v3) | is_a_theorem(v4))
% 6.50/2.16 | (4) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (not(v2) = v1) | ~ (not(v2) = v0))
% 6.50/2.16 | (5) (not(all_0_33_33) = all_0_32_32 & implies(all_0_31_31, all_0_33_33) = all_0_30_30 & implies(all_0_32_32, all_0_33_33) = all_0_31_31 & ~ cn3 & ~ is_a_theorem(all_0_30_30)) | (cn3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (not(v0) = v1) | ~ (implies(v2, v0) = v3) | ~ (implies(v1, v0) = v2) | is_a_theorem(v3)))
% 6.50/2.16 | (6) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (equiv(v0, v1) = v4) | ~ (implies(v3, v4) = v5) | ~ (implies(v2, v5) = v6) | ~ (implies(v1, v0) = v3) | ~ (implies(v0, v1) = v2) | is_a_theorem(v6))
% 6.50/2.16 | (7) op_implies_and
% 6.50/2.16 | (8) (possibly(all_0_137_137) = all_0_136_136 & strict_implies(all_0_137_137, all_0_136_136) = all_0_135_135 & ~ axiom_m6 & ~ is_a_theorem(all_0_135_135)) | (axiom_m6 & ! [v0] : ! [v1] : ! [v2] : ( ~ (possibly(v0) = v1) | ~ (strict_implies(v0, v1) = v2) | is_a_theorem(v2)))
% 6.50/2.16 | (9) op_strict_equiv
% 6.50/2.16 | (10) axiom_K
% 6.50/2.16 | (11) or_3
% 6.50/2.16 | (12) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v1) = v2) | ~ (implies(v2, v0) = v3) | is_a_theorem(v3))
% 6.50/2.16 | (13) (and(all_0_2_2, all_0_2_2) = all_0_1_1 & implies(all_0_2_2, all_0_1_1) = all_0_0_0 & ~ kn1 & ~ is_a_theorem(all_0_0_0)) | (kn1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v0) = v1) | ~ (implies(v0, v1) = v2) | is_a_theorem(v2)))
% 6.50/2.17 | (14) (or(all_0_61_61, all_0_59_59) = all_0_56_56 & or(all_0_61_61, all_0_60_60) = all_0_57_57 & implies(all_0_57_57, all_0_56_56) = all_0_55_55 & implies(all_0_58_58, all_0_55_55) = all_0_54_54 & implies(all_0_60_60, all_0_59_59) = all_0_58_58 & ~ r5 & ~ is_a_theorem(all_0_54_54)) | (r5 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (or(v0, v2) = v5) | ~ (or(v0, v1) = v4) | ~ (implies(v4, v5) = v6) | ~ (implies(v3, v6) = v7) | ~ (implies(v1, v2) = v3) | is_a_theorem(v7)))
% 6.50/2.17 | (15) necessitation
% 6.50/2.17 | (16) axiom_B
% 6.50/2.17 | (17) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (equiv(v0, v1) = v2) | ~ is_a_theorem(v2))
% 6.50/2.17 | (18) (possibly(all_0_91_91) = all_0_90_90 & possibly(all_0_92_92) = all_0_88_88 & possibly(all_0_93_93) = all_0_89_89 & strict_implies(all_0_90_90, all_0_87_87) = all_0_86_86 & and(all_0_89_89, all_0_88_88) = all_0_87_87 & and(all_0_93_93, all_0_92_92) = all_0_91_91 & ~ axiom_s2 & ~ is_a_theorem(all_0_86_86)) | (axiom_s2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (possibly(v2) = v3) | ~ (possibly(v1) = v5) | ~ (possibly(v0) = v4) | ~ (strict_implies(v3, v6) = v7) | ~ (and(v4, v5) = v6) | ~ (and(v0, v1) = v2) | is_a_theorem(v7)))
% 6.50/2.17 | (19) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (and(v3, v2) = v1) | ~ (and(v3, v2) = v0))
% 6.50/2.17 | (20) equivalence_2
% 6.50/2.17 | (21) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (and(v0, v1) = v2) | ~ (implies(v1, v2) = v3) | ~ (implies(v0, v3) = v4) | is_a_theorem(v4))
% 6.50/2.17 | (22) (or(all_0_52_52, all_0_48_48) = all_0_47_47 & or(all_0_52_52, all_0_51_51) = all_0_50_50 & or(all_0_53_53, all_0_50_50) = all_0_49_49 & or(all_0_53_53, all_0_51_51) = all_0_48_48 & implies(all_0_49_49, all_0_47_47) = all_0_46_46 & ~ r4 & ~ is_a_theorem(all_0_46_46)) | (r4 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (or(v1, v5) = v6) | ~ (or(v1, v2) = v3) | ~ (or(v0, v3) = v4) | ~ (or(v0, v2) = v5) | ~ (implies(v4, v6) = v7) | is_a_theorem(v7)))
% 6.50/2.17 | (23) (possibly(all_0_152_152) = all_0_151_151 & possibly(all_0_153_153) = all_0_152_152 & strict_implies(all_0_151_151, all_0_152_152) = all_0_150_150 & ~ axiom_m9 & ~ is_a_theorem(all_0_150_150)) | (axiom_m9 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (possibly(v1) = v2) | ~ (possibly(v0) = v1) | ~ (strict_implies(v2, v1) = v3) | is_a_theorem(v3)))
% 6.50/2.17 | (24) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (possibly(v2) = v1) | ~ (possibly(v2) = v0))
% 6.50/2.17 | (25) op_strict_implies
% 6.50/2.17 | (26) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (implies(v3, v2) = v4) | ~ (implies(v0, v2) = v3) | ~ (implies(v0, v1) = v2) | is_a_theorem(v4))
% 6.50/2.17 | (27) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (implies(v3, v2) = v1) | ~ (implies(v3, v2) = v0))
% 6.50/2.17 | (28) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (implies(v1, v0) = v2) | ~ (implies(v0, v2) = v3) | is_a_theorem(v3))
% 6.50/2.17 | (29) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (equiv(v0, v1) = v2) | ~ (implies(v2, v3) = v4) | ~ (implies(v1, v0) = v3) | is_a_theorem(v4))
% 6.50/2.17 | (30) (strict_implies(all_0_64_64, all_0_63_63) = all_0_62_62 & is_a_theorem(all_0_62_62) & is_a_theorem(all_0_64_64) & ~ modus_ponens_strict_implies & ~ is_a_theorem(all_0_63_63)) | (modus_ponens_strict_implies & ! [v0] : ! [v1] : ! [v2] : ( ~ (strict_implies(v0, v1) = v2) | ~ is_a_theorem(v2) | ~ is_a_theorem(v0) | is_a_theorem(v1)))
% 6.50/2.17 | (31) op_or
% 6.50/2.17 | (32) and(all_0_115_115, all_0_114_114) = all_0_113_113
% 6.50/2.17 | (33) (possibly(all_0_74_74) = all_0_73_73 & necessarily(all_0_73_73) = all_0_72_72 & implies(all_0_73_73, all_0_72_72) = all_0_71_71 & ~ axiom_5 & ~ is_a_theorem(all_0_71_71)) | (axiom_5 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (possibly(v0) = v1) | ~ (necessarily(v1) = v2) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3)))
% 6.50/2.17 | (34) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (necessarily(v2) = v3) | ~ (necessarily(v1) = v5) | ~ (necessarily(v0) = v4) | ~ (implies(v4, v5) = v6) | ~ (implies(v3, v6) = v7) | ~ (implies(v0, v1) = v2) | is_a_theorem(v7))
% 6.50/2.17 | (35) (and(all_0_14_14, all_0_16_16) = all_0_10_10 & and(all_0_15_15, all_0_14_14) = all_0_12_12 & not(all_0_10_10) = all_0_9_9 & not(all_0_12_12) = all_0_11_11 & implies(all_0_11_11, all_0_9_9) = all_0_8_8 & implies(all_0_13_13, all_0_8_8) = all_0_7_7 & implies(all_0_16_16, all_0_15_15) = all_0_13_13 & ~ kn3 & ~ is_a_theorem(all_0_7_7)) | (kn3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ( ~ (and(v2, v0) = v6) | ~ (and(v1, v2) = v4) | ~ (not(v6) = v7) | ~ (not(v4) = v5) | ~ (implies(v5, v7) = v8) | ~ (implies(v3, v8) = v9) | ~ (implies(v0, v1) = v3) | is_a_theorem(v9)))
% 6.50/2.17 | (36) (or(all_0_36_36, all_0_36_36) = all_0_35_35 & implies(all_0_35_35, all_0_36_36) = all_0_34_34 & ~ r1 & ~ is_a_theorem(all_0_34_34)) | (r1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v0) = v1) | ~ (implies(v1, v0) = v2) | is_a_theorem(v2)))
% 6.50/2.17 | (37) modus_ponens
% 6.50/2.17 | (38) and_3
% 6.50/2.17 | (39) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (and(v2, v3) = v4) | ~ (implies(v1, v0) = v3) | ~ (implies(v0, v1) = v2) | equiv(v0, v1) = v4)
% 6.50/2.17 | (40) ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (and(v3, v4) = v5 & not(v5) = v2 & not(v1) = v4 & not(v0) = v3))
% 6.50/2.17 | (41) ~ is_a_theorem(all_0_112_112)
% 6.50/2.17 | (42) (strict_implies(all_0_109_109, all_0_108_108) = all_0_107_107 & and(all_0_110_110, all_0_111_111) = all_0_108_108 & and(all_0_111_111, all_0_110_110) = all_0_109_109 & ~ axiom_m1 & ~ is_a_theorem(all_0_107_107)) | (axiom_m1 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (strict_implies(v2, v3) = v4) | ~ (and(v1, v0) = v3) | ~ (and(v0, v1) = v2) | is_a_theorem(v4)))
% 6.50/2.17 | (43) op_implies
% 6.50/2.17 | (44) modus_tollens
% 6.50/2.17 | (45) equivalence_1
% 6.50/2.17 | (46) ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : (strict_implies(v0, v1) = v3 & necessarily(v2) = v3))
% 6.50/2.17 | (47) (and(all_0_67_67, all_0_66_66) = all_0_65_65 & is_a_theorem(all_0_66_66) & is_a_theorem(all_0_67_67) & ~ adjunction & ~ is_a_theorem(all_0_65_65)) | (adjunction & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ~ is_a_theorem(v1) | ~ is_a_theorem(v0) | is_a_theorem(v2)))
% 6.50/2.17 | (48) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v2) = v3) | ~ (not(v1) = v2) | ? [v4] : (not(v3) = v4 & implies(v0, v1) = v4))
% 6.50/2.17 | (49) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (strict_implies(v3, v2) = v1) | ~ (strict_implies(v3, v2) = v0))
% 6.50/2.17 | (50) implies_1
% 6.50/2.17 | (51) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (implies(v4, v5) = v6) | ~ (implies(v3, v6) = v7) | ~ (implies(v1, v2) = v4) | ~ (implies(v0, v2) = v5) | ~ (implies(v0, v1) = v3) | is_a_theorem(v7))
% 6.50/2.17 | (52) (or(all_0_44_44, all_0_45_45) = all_0_42_42 & or(all_0_45_45, all_0_44_44) = all_0_43_43 & implies(all_0_43_43, all_0_42_42) = all_0_41_41 & ~ r3 & ~ is_a_theorem(all_0_41_41)) | (r3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (or(v1, v0) = v3) | ~ (or(v0, v1) = v2) | ~ (implies(v2, v3) = v4) | is_a_theorem(v4)))
% 6.50/2.17 | (53) (and(all_0_6_6, all_0_5_5) = all_0_4_4 & implies(all_0_4_4, all_0_6_6) = all_0_3_3 & ~ kn2 & ~ is_a_theorem(all_0_3_3)) | (kn2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v1) = v2) | ~ (implies(v2, v0) = v3) | is_a_theorem(v3)))
% 6.50/2.17 | (54) (strict_implies(all_0_105_105, all_0_104_104) = all_0_103_103 & necessarily(all_0_105_105) = all_0_104_104 & necessarily(all_0_106_106) = all_0_105_105 & ~ axiom_s4 & ~ is_a_theorem(all_0_103_103)) | (axiom_s4 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (strict_implies(v1, v2) = v3) | ~ (necessarily(v1) = v2) | ~ (necessarily(v0) = v1) | is_a_theorem(v3)))
% 6.50/2.17 | (55) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (or(v3, v2) = v1) | ~ (or(v3, v2) = v0))
% 6.50/2.17 | (56) (not(all_0_29_29) = all_0_27_27 & implies(all_0_27_27, all_0_28_28) = all_0_26_26 & implies(all_0_29_29, all_0_26_26) = all_0_25_25 & ~ cn2 & ~ is_a_theorem(all_0_25_25)) | (cn2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (not(v0) = v2) | ~ (implies(v2, v1) = v3) | ~ (implies(v0, v3) = v4) | is_a_theorem(v4)))
% 6.50/2.17 | (57) ! [v0] : ! [v1] : ( ~ (not(v0) = v1) | ? [v2] : ? [v3] : (possibly(v0) = v2 & necessarily(v1) = v3 & not(v3) = v2))
% 6.50/2.18 | (58) ! [v0] : ! [v1] : ! [v2] : ( ~ (strict_equiv(v0, v1) = v2) | ? [v3] : ? [v4] : (strict_implies(v1, v0) = v4 & strict_implies(v0, v1) = v3 & and(v3, v4) = v2))
% 6.50/2.18 | (59) ~ op_necessarily | ( ! [v0] : ! [v1] : ( ~ (necessarily(v0) = v1) | ? [v2] : ? [v3] : (possibly(v2) = v3 & not(v3) = v1 & not(v0) = v2)) & ! [v0] : ! [v1] : ( ~ (not(v0) = v1) | ? [v2] : ? [v3] : (possibly(v1) = v3 & necessarily(v0) = v2 & not(v3) = v2)))
% 6.50/2.18 | (60) and_1
% 6.50/2.18 | (61) axiom_4
% 6.50/2.18 | (62) (or(all_0_40_40, all_0_39_39) = all_0_38_38 & implies(all_0_39_39, all_0_38_38) = all_0_37_37 & ~ r2 & ~ is_a_theorem(all_0_37_37)) | (r2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v0, v1) = v2) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3)))
% 6.50/2.18 | (63) (possibly(all_0_157_157) = all_0_156_156 & strict_implies(all_0_156_156, all_0_155_155) = all_0_154_154 & necessarily(all_0_156_156) = all_0_155_155 & ~ axiom_m10 & ~ is_a_theorem(all_0_154_154)) | (axiom_m10 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (possibly(v0) = v1) | ~ (strict_implies(v1, v2) = v3) | ~ (necessarily(v1) = v2) | is_a_theorem(v3)))
% 6.50/2.18 | (64) (possibly(all_0_140_140) = all_0_139_139 & strict_implies(all_0_139_139, all_0_142_142) = all_0_138_138 & and(all_0_142_142, all_0_141_141) = all_0_140_140 & ~ axiom_m7 & ~ is_a_theorem(all_0_138_138)) | (axiom_m7 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (possibly(v2) = v3) | ~ (strict_implies(v3, v0) = v4) | ~ (and(v0, v1) = v2) | is_a_theorem(v4)))
% 6.50/2.18 | (65) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (strict_equiv(v3, v2) = v1) | ~ (strict_equiv(v3, v2) = v0))
% 6.50/2.18 | (66) ! [v0] : ! [v1] : ! [v2] : ( ~ (necessarily(v0) = v1) | ~ (implies(v1, v0) = v2) | is_a_theorem(v2))
% 6.50/2.18 | (67) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (possibly(v0) = v1) | ~ (necessarily(v1) = v2) | ~ (implies(v0, v2) = v3) | is_a_theorem(v3))
% 6.50/2.18 | (68) ! [v0] : ! [v1] : ! [v2] : ( ~ (strict_implies(v0, v1) = v2) | ? [v3] : (necessarily(v3) = v2 & implies(v0, v1) = v3))
% 6.50/2.18 | (69) ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ~ is_a_theorem(v2) | ~ is_a_theorem(v0) | is_a_theorem(v1))
% 6.50/2.18 | (70) op_equiv
% 6.50/2.18 | (71) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v1) = v2) | ~ (implies(v2, v1) = v3) | is_a_theorem(v3))
% 6.50/2.18 | (72) ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : (and(v0, v3) = v4 & not(v4) = v2 & not(v1) = v3))
% 6.50/2.18 | (73) (strict_implies(all_0_126_126, all_0_125_125) = all_0_124_124 & and(all_0_126_126, all_0_126_126) = all_0_125_125 & ~ axiom_m4 & ~ is_a_theorem(all_0_124_124)) | (axiom_m4 & ! [v0] : ! [v1] : ! [v2] : ( ~ (strict_implies(v0, v1) = v2) | ~ (and(v0, v0) = v1) | is_a_theorem(v2)))
% 6.50/2.18 | (74) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (strict_implies(v1, v0) = v3) | ~ (strict_implies(v0, v1) = v2) | ~ (and(v2, v3) = v4) | strict_equiv(v0, v1) = v4)
% 6.50/2.18 | (75) ! [v0] : ! [v1] : ( ~ (necessarily(v0) = v1) | ~ is_a_theorem(v0) | is_a_theorem(v1))
% 6.50/2.18 | (76) (strict_implies(all_0_119_119, all_0_117_117) = all_0_116_116 & and(all_0_120_120, all_0_121_121) = all_0_119_119 & and(all_0_122_122, all_0_121_121) = all_0_118_118 & and(all_0_123_123, all_0_118_118) = all_0_117_117 & and(all_0_123_123, all_0_122_122) = all_0_120_120 & ~ axiom_m3 & ~ is_a_theorem(all_0_116_116)) | (axiom_m3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (strict_implies(v4, v6) = v7) | ~ (and(v3, v2) = v4) | ~ (and(v1, v2) = v5) | ~ (and(v0, v5) = v6) | ~ (and(v0, v1) = v3) | is_a_theorem(v7)))
% 6.50/2.18 | (77) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (necessarily(v1) = v2) | ~ (necessarily(v0) = v1) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3))
% 6.50/2.18 | (78) (strict_implies(all_0_129_129, all_0_128_128) = all_0_127_127 & strict_implies(all_0_133_133, all_0_132_132) = all_0_130_130 & strict_implies(all_0_134_134, all_0_132_132) = all_0_128_128 & strict_implies(all_0_134_134, all_0_133_133) = all_0_131_131 & and(all_0_131_131, all_0_130_130) = all_0_129_129 & ~ axiom_m5 & ~ is_a_theorem(all_0_127_127)) | (axiom_m5 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (strict_implies(v5, v6) = v7) | ~ (strict_implies(v1, v2) = v4) | ~ (strict_implies(v0, v2) = v6) | ~ (strict_implies(v0, v1) = v3) | ~ (and(v3, v4) = v5) | is_a_theorem(v7)))
% 6.50/2.18 | (79) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (and(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) = v2) | ? [v5] : (or(v0, v1) = v5 & not(v4) = v5))
% 6.50/2.18 | (80) implies_2
% 6.50/2.18 | (81) axiom_M
% 6.50/2.18 | (82) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v0, v1) = v2) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3))
% 6.50/2.18 | (83) and_2
% 6.50/2.18 | (84) implies_3
% 6.50/2.18 | (85) (necessarily(all_0_77_77) = all_0_76_76 & necessarily(all_0_80_80) = all_0_79_79 & necessarily(all_0_82_82) = all_0_81_81 & and(all_0_81_81, all_0_79_79) = all_0_78_78 & implies(all_0_78_78, all_0_76_76) = all_0_75_75 & implies(all_0_84_84, all_0_83_83) = all_0_80_80 & implies(all_0_85_85, all_0_83_83) = all_0_77_77 & implies(all_0_85_85, all_0_84_84) = all_0_82_82 & ~ axiom_s1 & ~ is_a_theorem(all_0_75_75)) | (axiom_s1 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ! [v9] : ! [v10] : ( ~ (necessarily(v8) = v9) | ~ (necessarily(v5) = v6) | ~ (necessarily(v3) = v4) | ~ (and(v4, v6) = v7) | ~ (implies(v7, v9) = v10) | ~ (implies(v1, v2) = v5) | ~ (implies(v0, v2) = v8) | ~ (implies(v0, v1) = v3) | is_a_theorem(v10)))
% 6.50/2.18 | (86) ! [v0] : ! [v1] : ! [v2] : ( ~ (equiv(v0, v1) = v2) | ? [v3] : ? [v4] : (and(v3, v4) = v2 & implies(v1, v0) = v4 & implies(v0, v1) = v3))
% 6.50/2.18 | (87) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equiv(v3, v2) = v1) | ~ (equiv(v3, v2) = v0))
% 6.50/2.18 | (88) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (or(v0, v1) = v5) | ~ (implies(v5, v2) = v6) | ~ (implies(v4, v6) = v7) | ~ (implies(v3, v7) = v8) | ~ (implies(v1, v2) = v4) | ~ (implies(v0, v2) = v3) | is_a_theorem(v8))
% 6.50/2.18 | (89) op_possibly
% 6.50/2.18 | (90) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v0, v1) = v2) | ~ (implies(v0, v2) = v3) | is_a_theorem(v3))
% 6.50/2.18 | (91) substitution_of_equivalents
% 6.50/2.18 | (92) strict_implies(all_0_113_113, all_0_115_115) = all_0_112_112
% 6.50/2.18 | (93) ~ axiom_m2
% 6.50/2.18 | (94) or_1
% 6.50/2.18 | (95) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (not(v1) = v2) | ~ (not(v0) = v3) | ~ (implies(v4, v5) = v6) | ~ (implies(v2, v3) = v4) | ~ (implies(v0, v1) = v5) | is_a_theorem(v6))
% 6.50/2.18 | (96) equivalence_3
% 6.50/2.18 | (97) ~ op_and | ( ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (or(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) = v2) | ? [v5] : (and(v0, v1) = v5 & not(v4) = v5)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (or(v3, v4) = v5 & not(v5) = v2 & not(v1) = v4 & not(v0) = v3)))
% 6.50/2.18 | (98) ! [v0] : ! [v1] : ( ~ (possibly(v0) = v1) | ? [v2] : ? [v3] : (necessarily(v2) = v3 & not(v3) = v1 & not(v0) = v2))
% 6.50/2.18 | (99) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (necessarily(v2) = v1) | ~ (necessarily(v2) = v0))
% 6.50/2.18 | (100) (possibly(all_0_148_148) = all_0_145_145 & possibly(all_0_149_149) = all_0_146_146 & strict_implies(all_0_146_146, all_0_145_145) = all_0_144_144 & strict_implies(all_0_147_147, all_0_144_144) = all_0_143_143 & strict_implies(all_0_149_149, all_0_148_148) = all_0_147_147 & ~ axiom_m8 & ~ is_a_theorem(all_0_143_143)) | (axiom_m8 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ( ~ (possibly(v1) = v4) | ~ (possibly(v0) = v3) | ~ (strict_implies(v3, v4) = v5) | ~ (strict_implies(v2, v5) = v6) | ~ (strict_implies(v0, v1) = v2) | is_a_theorem(v6)))
% 6.50/2.18 | (101) (implies(all_0_20_20, all_0_19_19) = all_0_18_18 & implies(all_0_21_21, all_0_18_18) = all_0_17_17 & implies(all_0_23_23, all_0_22_22) = all_0_20_20 & implies(all_0_24_24, all_0_22_22) = all_0_19_19 & implies(all_0_24_24, all_0_23_23) = all_0_21_21 & ~ cn1 & ~ is_a_theorem(all_0_17_17)) | (cn1 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (implies(v4, v5) = v6) | ~ (implies(v3, v6) = v7) | ~ (implies(v1, v2) = v4) | ~ (implies(v0, v2) = v5) | ~ (implies(v0, v1) = v3) | is_a_theorem(v7)))
% 6.50/2.18 | (102) ( ~ (all_0_69_69 = all_0_70_70) & strict_equiv(all_0_70_70, all_0_69_69) = all_0_68_68 & is_a_theorem(all_0_68_68) & ~ substitution_strict_equiv) | (substitution_strict_equiv & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (strict_equiv(v0, v1) = v2) | ~ is_a_theorem(v2)))
% 6.50/2.18 | (103) (possibly(all_0_101_101) = all_0_99_99 & possibly(all_0_102_102) = all_0_97_97 & strict_implies(all_0_98_98, all_0_96_96) = all_0_95_95 & strict_implies(all_0_100_100, all_0_95_95) = all_0_94_94 & strict_implies(all_0_102_102, all_0_101_101) = all_0_100_100 & not(all_0_97_97) = all_0_96_96 & not(all_0_99_99) = all_0_98_98 & ~ axiom_s3 & ~ is_a_theorem(all_0_94_94)) | (axiom_s3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ! [v8] : ( ~ (possibly(v1) = v3) | ~ (possibly(v0) = v5) | ~ (strict_implies(v4, v6) = v7) | ~ (strict_implies(v2, v7) = v8) | ~ (strict_implies(v0, v1) = v2) | ~ (not(v5) = v6) | ~ (not(v3) = v4) | is_a_theorem(v8)))
% 6.50/2.19 | (104) ~ op_implies_or | ( ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v2, v1) = v3) | ~ (not(v0) = v2) | implies(v0, v1) = v3) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : (or(v3, v1) = v2 & not(v0) = v3)))
% 6.50/2.19 |
% 6.50/2.19 | Instantiating formula (68) with all_0_112_112, all_0_115_115, all_0_113_113 and discharging atoms strict_implies(all_0_113_113, all_0_115_115) = all_0_112_112, yields:
% 6.50/2.19 | (105) ? [v0] : (necessarily(v0) = all_0_112_112 & implies(all_0_113_113, all_0_115_115) = v0)
% 6.50/2.19 |
% 6.50/2.19 | Instantiating (105) with all_8_0_158 yields:
% 6.50/2.19 | (106) necessarily(all_8_0_158) = all_0_112_112 & implies(all_0_113_113, all_0_115_115) = all_8_0_158
% 6.50/2.19 |
% 6.50/2.19 | Applying alpha-rule on (106) yields:
% 6.50/2.19 | (107) necessarily(all_8_0_158) = all_0_112_112
% 6.50/2.19 | (108) implies(all_0_113_113, all_0_115_115) = all_8_0_158
% 6.50/2.19 |
% 6.50/2.19 | Instantiating formula (12) with all_8_0_158, all_0_113_113, all_0_114_114, all_0_115_115 and discharging atoms and(all_0_115_115, all_0_114_114) = all_0_113_113, implies(all_0_113_113, all_0_115_115) = all_8_0_158, yields:
% 6.50/2.19 | (109) is_a_theorem(all_8_0_158)
% 6.50/2.19 |
% 6.50/2.19 | Instantiating formula (75) with all_0_112_112, all_8_0_158 and discharging atoms necessarily(all_8_0_158) = all_0_112_112, is_a_theorem(all_8_0_158), ~ is_a_theorem(all_0_112_112), yields:
% 6.50/2.19 | (110) $false
% 6.50/2.19 |
% 6.50/2.19 |-The branch is then unsatisfiable
% 6.50/2.19 % SZS output end Proof for theBenchmark
% 6.50/2.19
% 6.50/2.19 1560ms
%------------------------------------------------------------------------------