TSTP Solution File: LCL538+1 by ePrincess---1.0
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- Process Solution
%------------------------------------------------------------------------------
% File : ePrincess---1.0
% Problem : LCL538+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : ePrincess-casc -timeout=%d %s
% Computer : n028.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 09:37:00 EDT 2022
% Result : Theorem 19.11s 6.31s
% Output : Proof 22.27s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : LCL538+1 : TPTP v8.1.0. Released v3.3.0.
% 0.11/0.13 % Command : ePrincess-casc -timeout=%d %s
% 0.12/0.33 % Computer : n028.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 : Mon Jul 4 20:56:20 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.61/0.60 ____ _
% 0.61/0.60 ___ / __ \_____(_)___ ________ __________
% 0.61/0.60 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/
% 0.61/0.60 / __/ ____/ / / / / / / /__/ __(__ |__ )
% 0.61/0.60 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/
% 0.61/0.60
% 0.61/0.60 A Theorem Prover for First-Order Logic
% 0.61/0.60 (ePrincess v.1.0)
% 0.61/0.60
% 0.61/0.60 (c) Philipp Rümmer, 2009-2015
% 0.61/0.60 (c) Peter Backeman, 2014-2015
% 0.61/0.60 (contributions by Angelo Brillout, Peter Baumgartner)
% 0.61/0.60 Free software under GNU Lesser General Public License (LGPL).
% 0.61/0.60 Bug reports to peter@backeman.se
% 0.61/0.60
% 0.61/0.60 For more information, visit http://user.uu.se/~petba168/breu/
% 0.61/0.60
% 0.61/0.60 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 0.70/0.66 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all
% 2.03/1.04 Prover 0: Preprocessing ...
% 3.96/1.53 Prover 0: Constructing countermodel ...
% 17.36/5.95 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all
% 17.66/6.03 Prover 1: Preprocessing ...
% 18.59/6.23 Prover 1: Constructing countermodel ...
% 19.11/6.31 Prover 1: proved (361ms)
% 19.11/6.31 Prover 0: stopped
% 19.11/6.31
% 19.11/6.31 No countermodel exists, formula is valid
% 19.11/6.31 % SZS status Theorem for theBenchmark
% 19.11/6.31
% 19.11/6.31 Generating proof ... found it (size 39)
% 21.86/6.93
% 21.86/6.93 % SZS output start Proof for theBenchmark
% 21.86/6.93 Assumed formulas after preprocessing and simplification:
% 21.86/6.93 | (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] : ? [v158] : ? [v159] : ? [v160] : ? [v161] : ? [v162] : ? [v163] : ? [v164] : ? [v165] : ? [v166] : ? [v167] : ? [v168] : ? [v169] : ? [v170] : ? [v171] : ? [v172] : ? [v173] : ? [v174] : ? [v175] : ? [v176] : ? [v177] : ? [v178] : ? [v179] : ? [v180] : ? [v181] : ? [v182] : ? [v183] : ? [v184] : ? [v185] : ? [v186] : ? [v187] : ? [v188] : ( ~ (v115 = 0) & strict_implies(v112, v113) = v114 & is_a_theorem(v114) = 0 & is_a_theorem(v113) = v115 & is_a_theorem(v112) = 0 & 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 & ~ modus_ponens_strict_implies & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ! [v194] : ! [v195] : ! [v196] : ! [v197] : ( ~ (or(v189, v190) = v194) | ~ (implies(v194, v191) = v195) | ~ (implies(v193, v195) = v196) | ~ (implies(v192, v196) = v197) | ~ (implies(v190, v191) = v193) | ~ (implies(v189, v191) = v192) | is_a_theorem(v197) = 0) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ! [v194] : ! [v195] : ! [v196] : ( ~ (necessarily(v191) = v192) | ~ (necessarily(v190) = v194) | ~ (necessarily(v189) = v193) | ~ (implies(v193, v194) = v195) | ~ (implies(v192, v195) = v196) | ~ (implies(v189, v190) = v191) | is_a_theorem(v196) = 0) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ! [v194] : ! [v195] : ! [v196] : ( ~ (implies(v193, v194) = v195) | ~ (implies(v192, v195) = v196) | ~ (implies(v190, v191) = v193) | ~ (implies(v189, v191) = v194) | ~ (implies(v189, v190) = v192) | is_a_theorem(v196) = 0) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ! [v194] : ! [v195] : ( ~ (not(v190) = v191) | ~ (not(v189) = v192) | ~ (implies(v193, v194) = v195) | ~ (implies(v191, v192) = v193) | ~ (implies(v189, v190) = v194) | is_a_theorem(v195) = 0) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ! [v194] : ! [v195] : ( ~ (equiv(v189, v190) = v193) | ~ (implies(v192, v193) = v194) | ~ (implies(v191, v194) = v195) | ~ (implies(v190, v189) = v192) | ~ (implies(v189, v190) = v191) | is_a_theorem(v195) = 0) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ( ~ (strict_implies(v190, v189) = v192) | ~ (strict_implies(v189, v190) = v191) | ~ (and(v191, v192) = v193) | strict_equiv(v189, v190) = v193) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ( ~ (and(v191, v192) = v193) | ~ (not(v190) = v192) | ~ (not(v189) = v191) | ? [v194] : (or(v189, v190) = v194 & not(v193) = v194)) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ( ~ (and(v191, v192) = v193) | ~ (implies(v190, v189) = v192) | ~ (implies(v189, v190) = v191) | equiv(v189, v190) = v193) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ( ~ (and(v189, v190) = v191) | ~ (implies(v190, v191) = v192) | ~ (implies(v189, v192) = v193) | is_a_theorem(v193) = 0) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ( ~ (equiv(v189, v190) = v191) | ~ (implies(v191, v192) = v193) | ~ (implies(v190, v189) = v192) | is_a_theorem(v193) = 0) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ( ~ (equiv(v189, v190) = v191) | ~ (implies(v191, v192) = v193) | ~ (implies(v189, v190) = v192) | is_a_theorem(v193) = 0) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ( ~ (implies(v192, v191) = v193) | ~ (implies(v189, v191) = v192) | ~ (implies(v189, v190) = v191) | is_a_theorem(v193) = 0) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : (v190 = v189 | ~ (strict_equiv(v192, v191) = v190) | ~ (strict_equiv(v192, v191) = v189)) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : (v190 = v189 | ~ (strict_implies(v192, v191) = v190) | ~ (strict_implies(v192, v191) = v189)) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : (v190 = v189 | ~ (or(v192, v191) = v190) | ~ (or(v192, v191) = v189)) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : (v190 = v189 | ~ (and(v192, v191) = v190) | ~ (and(v192, v191) = v189)) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : (v190 = v189 | ~ (equiv(v192, v191) = v190) | ~ (equiv(v192, v191) = v189)) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : (v190 = v189 | ~ (implies(v192, v191) = v190) | ~ (implies(v192, v191) = v189)) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (possibly(v189) = v190) | ~ (necessarily(v190) = v191) | ~ (implies(v189, v191) = v192) | is_a_theorem(v192) = 0) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (necessarily(v190) = v191) | ~ (necessarily(v189) = v190) | ~ (implies(v190, v191) = v192) | is_a_theorem(v192) = 0) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (or(v189, v190) = v191) | ~ (implies(v190, v191) = v192) | is_a_theorem(v192) = 0) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (or(v189, v190) = v191) | ~ (implies(v189, v191) = v192) | is_a_theorem(v192) = 0) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (and(v189, v191) = v192) | ~ (not(v190) = v191) | ? [v193] : (not(v192) = v193 & implies(v189, v190) = v193)) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (and(v189, v190) = v191) | ~ (implies(v191, v190) = v192) | is_a_theorem(v192) = 0) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (and(v189, v190) = v191) | ~ (implies(v191, v189) = v192) | is_a_theorem(v192) = 0) & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (implies(v190, v189) = v191) | ~ (implies(v189, v191) = v192) | is_a_theorem(v192) = 0) & ! [v189] : ! [v190] : ! [v191] : (v191 = 0 | ~ (is_a_theorem(v190) = v191) | ~ (is_a_theorem(v189) = 0) | ? [v192] : ? [v193] : ( ~ (v193 = 0) & implies(v189, v190) = v192 & is_a_theorem(v192) = v193)) & ! [v189] : ! [v190] : ! [v191] : (v190 = v189 | ~ (possibly(v191) = v190) | ~ (possibly(v191) = v189)) & ! [v189] : ! [v190] : ! [v191] : (v190 = v189 | ~ (necessarily(v191) = v190) | ~ (necessarily(v191) = v189)) & ! [v189] : ! [v190] : ! [v191] : (v190 = v189 | ~ (not(v191) = v190) | ~ (not(v191) = v189)) & ! [v189] : ! [v190] : ! [v191] : (v190 = v189 | ~ (equiv(v189, v190) = v191) | ? [v192] : ( ~ (v192 = 0) & is_a_theorem(v191) = v192)) & ! [v189] : ! [v190] : ! [v191] : (v190 = v189 | ~ (is_a_theorem(v191) = v190) | ~ (is_a_theorem(v191) = v189)) & ! [v189] : ! [v190] : ! [v191] : ( ~ (strict_implies(v189, v190) = v191) | ? [v192] : (necessarily(v192) = v191 & implies(v189, v190) = v192)) & ! [v189] : ! [v190] : ! [v191] : ( ~ (necessarily(v189) = v190) | ~ (implies(v190, v189) = v191) | is_a_theorem(v191) = 0) & ! [v189] : ! [v190] : ( ~ (necessarily(v189) = v190) | ? [v191] : ? [v192] : (is_a_theorem(v190) = v192 & is_a_theorem(v189) = v191 & ( ~ (v191 = 0) | v192 = 0))) & ! [v189] : ! [v190] : ( ~ (not(v189) = v190) | ? [v191] : ? [v192] : (possibly(v189) = v191 & necessarily(v190) = v192 & not(v192) = v191)) & ( ~ op_necessarily | ! [v189] : ! [v190] : ( ~ (not(v189) = v190) | ? [v191] : ? [v192] : (possibly(v190) = v192 & necessarily(v189) = v191 & not(v192) = v191))) & ( ~ op_implies_or | ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (or(v191, v190) = v192) | ~ (not(v189) = v191) | implies(v189, v190) = v192)) & ( ~ op_and | ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ( ~ (or(v191, v192) = v193) | ~ (not(v190) = v192) | ~ (not(v189) = v191) | ? [v194] : (and(v189, v190) = v194 & not(v193) = v194))) & ((v109 = 0 & v108 = 0 & ~ (v111 = 0) & and(v106, v107) = v110 & is_a_theorem(v110) = v111 & is_a_theorem(v107) = 0 & is_a_theorem(v106) = 0 & ~ adjunction) | (adjunction & ! [v189] : ! [v190] : ! [v191] : ( ~ (and(v189, v190) = v191) | ? [v192] : ? [v193] : ? [v194] : (is_a_theorem(v191) = v194 & is_a_theorem(v190) = v193 & is_a_theorem(v189) = v192 & ( ~ (v193 = 0) | ~ (v192 = 0) | v194 = 0))))) & ((v105 = 0 & ~ (v103 = v102) & strict_equiv(v102, v103) = v104 & is_a_theorem(v104) = 0 & ~ substitution_strict_equiv) | (substitution_strict_equiv & ! [v189] : ! [v190] : ! [v191] : (v190 = v189 | ~ (strict_equiv(v189, v190) = v191) | ? [v192] : ( ~ (v192 = 0) & is_a_theorem(v191) = v192)))) & (( ~ (v188 = 0) & and(v185, v185) = v186 & implies(v185, v186) = v187 & is_a_theorem(v187) = v188 & ~ kn1) | (kn1 & ! [v189] : ! [v190] : ! [v191] : ( ~ (and(v189, v189) = v190) | ~ (implies(v189, v190) = v191) | is_a_theorem(v191) = 0))) & (( ~ (v184 = 0) & and(v180, v181) = v182 & implies(v182, v180) = v183 & is_a_theorem(v183) = v184 & ~ kn2) | (kn2 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (and(v189, v190) = v191) | ~ (implies(v191, v189) = v192) | is_a_theorem(v192) = 0))) & (( ~ (v179 = 0) & and(v171, v169) = v175 & and(v170, v171) = v173 & not(v175) = v176 & not(v173) = v174 & implies(v174, v176) = v177 & implies(v172, v177) = v178 & implies(v169, v170) = v172 & is_a_theorem(v178) = v179 & ~ kn3) | (kn3 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ! [v194] : ! [v195] : ! [v196] : ! [v197] : ! [v198] : ( ~ (and(v191, v189) = v195) | ~ (and(v190, v191) = v193) | ~ (not(v195) = v196) | ~ (not(v193) = v194) | ~ (implies(v194, v196) = v197) | ~ (implies(v192, v197) = v198) | ~ (implies(v189, v190) = v192) | is_a_theorem(v198) = 0))) & (( ~ (v168 = 0) & implies(v164, v165) = v166 & implies(v163, v166) = v167 & implies(v161, v162) = v164 & implies(v160, v162) = v165 & implies(v160, v161) = v163 & is_a_theorem(v167) = v168 & ~ cn1) | (cn1 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ! [v194] : ! [v195] : ! [v196] : ( ~ (implies(v193, v194) = v195) | ~ (implies(v192, v195) = v196) | ~ (implies(v190, v191) = v193) | ~ (implies(v189, v191) = v194) | ~ (implies(v189, v190) = v192) | is_a_theorem(v196) = 0))) & (( ~ (v159 = 0) & not(v154) = v156 & implies(v156, v155) = v157 & implies(v154, v157) = v158 & is_a_theorem(v158) = v159 & ~ cn2) | (cn2 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ( ~ (not(v189) = v191) | ~ (implies(v191, v190) = v192) | ~ (implies(v189, v192) = v193) | is_a_theorem(v193) = 0))) & (( ~ (v153 = 0) & not(v149) = v150 & implies(v151, v149) = v152 & implies(v150, v149) = v151 & is_a_theorem(v152) = v153 & ~ cn3) | (cn3 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (not(v189) = v190) | ~ (implies(v191, v189) = v192) | ~ (implies(v190, v189) = v191) | is_a_theorem(v192) = 0))) & (( ~ (v148 = 0) & or(v145, v145) = v146 & implies(v146, v145) = v147 & is_a_theorem(v147) = v148 & ~ r1) | (r1 & ! [v189] : ! [v190] : ! [v191] : ( ~ (or(v189, v189) = v190) | ~ (implies(v190, v189) = v191) | is_a_theorem(v191) = 0))) & (( ~ (v144 = 0) & or(v140, v141) = v142 & implies(v141, v142) = v143 & is_a_theorem(v143) = v144 & ~ r2) | (r2 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (or(v189, v190) = v191) | ~ (implies(v190, v191) = v192) | is_a_theorem(v192) = 0))) & (( ~ (v139 = 0) & or(v135, v134) = v137 & or(v134, v135) = v136 & implies(v136, v137) = v138 & is_a_theorem(v138) = v139 & ~ r3) | (r3 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ( ~ (or(v190, v189) = v192) | ~ (or(v189, v190) = v191) | ~ (implies(v191, v192) = v193) | is_a_theorem(v193) = 0))) & (( ~ (v133 = 0) & or(v126, v130) = v131 & or(v126, v127) = v128 & or(v125, v128) = v129 & or(v125, v127) = v130 & implies(v129, v131) = v132 & is_a_theorem(v132) = v133 & ~ r4) | (r4 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ! [v194] : ! [v195] : ! [v196] : ( ~ (or(v190, v194) = v195) | ~ (or(v190, v191) = v192) | ~ (or(v189, v192) = v193) | ~ (or(v189, v191) = v194) | ~ (implies(v193, v195) = v196) | is_a_theorem(v196) = 0))) & (( ~ (v124 = 0) & or(v116, v118) = v121 & or(v116, v117) = v120 & implies(v120, v121) = v122 & implies(v119, v122) = v123 & implies(v117, v118) = v119 & is_a_theorem(v123) = v124 & ~ r5) | (r5 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ! [v194] : ! [v195] : ! [v196] : ( ~ (or(v189, v191) = v194) | ~ (or(v189, v190) = v193) | ~ (implies(v193, v194) = v195) | ~ (implies(v192, v195) = v196) | ~ (implies(v190, v191) = v192) | is_a_theorem(v196) = 0))) & (( ~ (v101 = 0) & possibly(v97) = v98 & necessarily(v98) = v99 & implies(v98, v99) = v100 & is_a_theorem(v100) = v101 & ~ axiom_5) | (axiom_5 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (possibly(v189) = v190) | ~ (necessarily(v190) = v191) | ~ (implies(v190, v191) = v192) | is_a_theorem(v192) = 0))) & (( ~ (v96 = 0) & necessarily(v93) = v94 & necessarily(v90) = v91 & necessarily(v88) = v89 & and(v89, v91) = v92 & implies(v92, v94) = v95 & implies(v86, v87) = v90 & implies(v85, v87) = v93 & implies(v85, v86) = v88 & is_a_theorem(v95) = v96 & ~ axiom_s1) | (axiom_s1 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ! [v194] : ! [v195] : ! [v196] : ! [v197] : ! [v198] : ! [v199] : ( ~ (necessarily(v197) = v198) | ~ (necessarily(v194) = v195) | ~ (necessarily(v192) = v193) | ~ (and(v193, v195) = v196) | ~ (implies(v196, v198) = v199) | ~ (implies(v190, v191) = v194) | ~ (implies(v189, v191) = v197) | ~ (implies(v189, v190) = v192) | is_a_theorem(v199) = 0))) & (( ~ (v84 = 0) & possibly(v78) = v79 & possibly(v77) = v81 & possibly(v76) = v80 & strict_implies(v79, v82) = v83 & and(v80, v81) = v82 & and(v76, v77) = v78 & is_a_theorem(v83) = v84 & ~ axiom_s2) | (axiom_s2 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ! [v194] : ! [v195] : ! [v196] : ( ~ (possibly(v191) = v192) | ~ (possibly(v190) = v194) | ~ (possibly(v189) = v193) | ~ (strict_implies(v192, v195) = v196) | ~ (and(v193, v194) = v195) | ~ (and(v189, v190) = v191) | is_a_theorem(v196) = 0))) & (( ~ (v75 = 0) & possibly(v67) = v69 & possibly(v66) = v71 & strict_implies(v70, v72) = v73 & strict_implies(v68, v73) = v74 & strict_implies(v66, v67) = v68 & not(v71) = v72 & not(v69) = v70 & is_a_theorem(v74) = v75 & ~ axiom_s3) | (axiom_s3 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ! [v194] : ! [v195] : ! [v196] : ! [v197] : ( ~ (possibly(v190) = v192) | ~ (possibly(v189) = v194) | ~ (strict_implies(v193, v195) = v196) | ~ (strict_implies(v191, v196) = v197) | ~ (strict_implies(v189, v190) = v191) | ~ (not(v194) = v195) | ~ (not(v192) = v193) | is_a_theorem(v197) = 0))) & (( ~ (v65 = 0) & strict_implies(v62, v63) = v64 & necessarily(v62) = v63 & necessarily(v61) = v62 & is_a_theorem(v64) = v65 & ~ axiom_s4) | (axiom_s4 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (strict_implies(v190, v191) = v192) | ~ (necessarily(v190) = v191) | ~ (necessarily(v189) = v190) | is_a_theorem(v192) = 0))) & (( ~ (v60 = 0) & strict_implies(v57, v58) = v59 & and(v56, v55) = v58 & and(v55, v56) = v57 & is_a_theorem(v59) = v60 & ~ axiom_m1) | (axiom_m1 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ( ~ (strict_implies(v191, v192) = v193) | ~ (and(v190, v189) = v192) | ~ (and(v189, v190) = v191) | is_a_theorem(v193) = 0))) & (( ~ (v54 = 0) & strict_implies(v52, v50) = v53 & and(v50, v51) = v52 & is_a_theorem(v53) = v54 & ~ axiom_m2) | (axiom_m2 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (strict_implies(v191, v189) = v192) | ~ (and(v189, v190) = v191) | is_a_theorem(v192) = 0))) & (( ~ (v49 = 0) & strict_implies(v45, v47) = v48 & and(v44, v43) = v45 & and(v42, v43) = v46 & and(v41, v46) = v47 & and(v41, v42) = v44 & is_a_theorem(v48) = v49 & ~ axiom_m3) | (axiom_m3 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ! [v194] : ! [v195] : ! [v196] : ( ~ (strict_implies(v193, v195) = v196) | ~ (and(v192, v191) = v193) | ~ (and(v190, v191) = v194) | ~ (and(v189, v194) = v195) | ~ (and(v189, v190) = v192) | is_a_theorem(v196) = 0))) & (( ~ (v40 = 0) & strict_implies(v37, v38) = v39 & and(v37, v37) = v38 & is_a_theorem(v39) = v40 & ~ axiom_m4) | (axiom_m4 & ! [v189] : ! [v190] : ! [v191] : ( ~ (strict_implies(v189, v190) = v191) | ~ (and(v189, v189) = v190) | is_a_theorem(v191) = 0))) & (( ~ (v36 = 0) & strict_implies(v33, v34) = v35 & strict_implies(v29, v30) = v32 & strict_implies(v28, v30) = v34 & strict_implies(v28, v29) = v31 & and(v31, v32) = v33 & is_a_theorem(v35) = v36 & ~ axiom_m5) | (axiom_m5 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ! [v194] : ! [v195] : ! [v196] : ( ~ (strict_implies(v194, v195) = v196) | ~ (strict_implies(v190, v191) = v193) | ~ (strict_implies(v189, v191) = v195) | ~ (strict_implies(v189, v190) = v192) | ~ (and(v192, v193) = v194) | is_a_theorem(v196) = 0))) & (( ~ (v27 = 0) & possibly(v24) = v25 & strict_implies(v24, v25) = v26 & is_a_theorem(v26) = v27 & ~ axiom_m6) | (axiom_m6 & ! [v189] : ! [v190] : ! [v191] : ( ~ (possibly(v189) = v190) | ~ (strict_implies(v189, v190) = v191) | is_a_theorem(v191) = 0))) & (( ~ (v23 = 0) & possibly(v20) = v21 & strict_implies(v21, v18) = v22 & and(v18, v19) = v20 & is_a_theorem(v22) = v23 & ~ axiom_m7) | (axiom_m7 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ( ~ (possibly(v191) = v192) | ~ (strict_implies(v192, v189) = v193) | ~ (and(v189, v190) = v191) | is_a_theorem(v193) = 0))) & (( ~ (v17 = 0) & possibly(v11) = v14 & possibly(v10) = v13 & strict_implies(v13, v14) = v15 & strict_implies(v12, v15) = v16 & strict_implies(v10, v11) = v12 & is_a_theorem(v16) = v17 & ~ axiom_m8) | (axiom_m8 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ! [v193] : ! [v194] : ! [v195] : ( ~ (possibly(v190) = v193) | ~ (possibly(v189) = v192) | ~ (strict_implies(v192, v193) = v194) | ~ (strict_implies(v191, v194) = v195) | ~ (strict_implies(v189, v190) = v191) | is_a_theorem(v195) = 0))) & (( ~ (v9 = 0) & possibly(v6) = v7 & possibly(v5) = v6 & strict_implies(v7, v6) = v8 & is_a_theorem(v8) = v9 & ~ axiom_m9) | (axiom_m9 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (possibly(v190) = v191) | ~ (possibly(v189) = v190) | ~ (strict_implies(v191, v190) = v192) | is_a_theorem(v192) = 0))) & (( ~ (v4 = 0) & possibly(v0) = v1 & strict_implies(v1, v2) = v3 & necessarily(v1) = v2 & is_a_theorem(v3) = v4 & ~ axiom_m10) | (axiom_m10 & ! [v189] : ! [v190] : ! [v191] : ! [v192] : ( ~ (possibly(v189) = v190) | ~ (strict_implies(v190, v191) = v192) | ~ (necessarily(v190) = v191) | is_a_theorem(v192) = 0))))
% 21.86/6.99 | 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, all_0_158_158, all_0_159_159, all_0_160_160, all_0_161_161, all_0_162_162, all_0_163_163, all_0_164_164, all_0_165_165, all_0_166_166, all_0_167_167, all_0_168_168, all_0_169_169, all_0_170_170, all_0_171_171, all_0_172_172, all_0_173_173, all_0_174_174, all_0_175_175, all_0_176_176, all_0_177_177, all_0_178_178, all_0_179_179, all_0_180_180, all_0_181_181, all_0_182_182, all_0_183_183, all_0_184_184, all_0_185_185, all_0_186_186, all_0_187_187, all_0_188_188 yields:
% 21.86/6.99 | (1) ~ (all_0_73_73 = 0) & strict_implies(all_0_76_76, all_0_75_75) = all_0_74_74 & is_a_theorem(all_0_74_74) = 0 & is_a_theorem(all_0_75_75) = all_0_73_73 & is_a_theorem(all_0_76_76) = 0 & 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 & ~ modus_ponens_strict_implies & ! [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) = 0) & ! [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) = 0) & ! [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) = 0) & ! [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) = 0) & ! [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) = 0) & ! [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) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (equiv(v0, v1) = v2) | ~ (implies(v2, v3) = v4) | ~ (implies(v1, v0) = v3) | is_a_theorem(v4) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (equiv(v0, v1) = v2) | ~ (implies(v2, v3) = v4) | ~ (implies(v0, v1) = v3) | is_a_theorem(v4) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (implies(v3, v2) = v4) | ~ (implies(v0, v2) = v3) | ~ (implies(v0, v1) = v2) | is_a_theorem(v4) = 0) & ! [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) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (necessarily(v1) = v2) | ~ (necessarily(v0) = v1) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v0, v1) = v2) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v0, v1) = v2) | ~ (implies(v0, v2) = v3) | is_a_theorem(v3) = 0) & ! [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) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v1) = v2) | ~ (implies(v2, v0) = v3) | is_a_theorem(v3) = 0) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (implies(v1, v0) = v2) | ~ (implies(v0, v2) = v3) | is_a_theorem(v3) = 0) & ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (is_a_theorem(v1) = v2) | ~ (is_a_theorem(v0) = 0) | ? [v3] : ? [v4] : ( ~ (v4 = 0) & implies(v0, v1) = v3 & is_a_theorem(v3) = v4)) & ! [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) | ? [v3] : ( ~ (v3 = 0) & is_a_theorem(v2) = v3)) & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (is_a_theorem(v2) = v1) | ~ (is_a_theorem(v2) = v0)) & ! [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) = 0) & ! [v0] : ! [v1] : ( ~ (necessarily(v0) = v1) | ? [v2] : ? [v3] : (is_a_theorem(v1) = v3 & is_a_theorem(v0) = v2 & ( ~ (v2 = 0) | v3 = 0))) & ! [v0] : ! [v1] : ( ~ (not(v0) = v1) | ? [v2] : ? [v3] : (possibly(v0) = v2 & necessarily(v1) = v3 & not(v3) = v2)) & ( ~ op_necessarily | ! [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)) & ( ~ 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))) & ((all_0_79_79 = 0 & all_0_80_80 = 0 & ~ (all_0_77_77 = 0) & and(all_0_82_82, all_0_81_81) = all_0_78_78 & is_a_theorem(all_0_78_78) = all_0_77_77 & is_a_theorem(all_0_81_81) = 0 & is_a_theorem(all_0_82_82) = 0 & ~ adjunction) | (adjunction & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (is_a_theorem(v2) = v5 & is_a_theorem(v1) = v4 & is_a_theorem(v0) = v3 & ( ~ (v4 = 0) | ~ (v3 = 0) | v5 = 0))))) & ((all_0_83_83 = 0 & ~ (all_0_85_85 = all_0_86_86) & strict_equiv(all_0_86_86, all_0_85_85) = all_0_84_84 & is_a_theorem(all_0_84_84) = 0 & ~ substitution_strict_equiv) | (substitution_strict_equiv & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (strict_equiv(v0, v1) = v2) | ? [v3] : ( ~ (v3 = 0) & is_a_theorem(v2) = v3)))) & (( ~ (all_0_0_0 = 0) & and(all_0_3_3, all_0_3_3) = all_0_2_2 & implies(all_0_3_3, all_0_2_2) = all_0_1_1 & is_a_theorem(all_0_1_1) = all_0_0_0 & ~ kn1) | (kn1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v0) = v1) | ~ (implies(v0, v1) = v2) | is_a_theorem(v2) = 0))) & (( ~ (all_0_4_4 = 0) & and(all_0_8_8, all_0_7_7) = all_0_6_6 & implies(all_0_6_6, all_0_8_8) = all_0_5_5 & is_a_theorem(all_0_5_5) = all_0_4_4 & ~ kn2) | (kn2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v1) = v2) | ~ (implies(v2, v0) = v3) | is_a_theorem(v3) = 0))) & (( ~ (all_0_9_9 = 0) & and(all_0_17_17, all_0_19_19) = all_0_13_13 & and(all_0_18_18, all_0_17_17) = all_0_15_15 & not(all_0_13_13) = all_0_12_12 & not(all_0_15_15) = all_0_14_14 & implies(all_0_14_14, all_0_12_12) = all_0_11_11 & implies(all_0_16_16, all_0_11_11) = all_0_10_10 & implies(all_0_19_19, all_0_18_18) = all_0_16_16 & is_a_theorem(all_0_10_10) = all_0_9_9 & ~ kn3) | (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) = 0))) & (( ~ (all_0_20_20 = 0) & implies(all_0_24_24, all_0_23_23) = all_0_22_22 & implies(all_0_25_25, all_0_22_22) = all_0_21_21 & implies(all_0_27_27, all_0_26_26) = all_0_24_24 & implies(all_0_28_28, all_0_26_26) = all_0_23_23 & implies(all_0_28_28, all_0_27_27) = all_0_25_25 & is_a_theorem(all_0_21_21) = all_0_20_20 & ~ cn1) | (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) = 0))) & (( ~ (all_0_29_29 = 0) & not(all_0_34_34) = all_0_32_32 & implies(all_0_32_32, all_0_33_33) = all_0_31_31 & implies(all_0_34_34, all_0_31_31) = all_0_30_30 & is_a_theorem(all_0_30_30) = all_0_29_29 & ~ cn2) | (cn2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (not(v0) = v2) | ~ (implies(v2, v1) = v3) | ~ (implies(v0, v3) = v4) | is_a_theorem(v4) = 0))) & (( ~ (all_0_35_35 = 0) & not(all_0_39_39) = all_0_38_38 & implies(all_0_37_37, all_0_39_39) = all_0_36_36 & implies(all_0_38_38, all_0_39_39) = all_0_37_37 & is_a_theorem(all_0_36_36) = all_0_35_35 & ~ cn3) | (cn3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (not(v0) = v1) | ~ (implies(v2, v0) = v3) | ~ (implies(v1, v0) = v2) | is_a_theorem(v3) = 0))) & (( ~ (all_0_40_40 = 0) & or(all_0_43_43, all_0_43_43) = all_0_42_42 & implies(all_0_42_42, all_0_43_43) = all_0_41_41 & is_a_theorem(all_0_41_41) = all_0_40_40 & ~ r1) | (r1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v0) = v1) | ~ (implies(v1, v0) = v2) | is_a_theorem(v2) = 0))) & (( ~ (all_0_44_44 = 0) & or(all_0_48_48, all_0_47_47) = all_0_46_46 & implies(all_0_47_47, all_0_46_46) = all_0_45_45 & is_a_theorem(all_0_45_45) = all_0_44_44 & ~ r2) | (r2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v0, v1) = v2) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3) = 0))) & (( ~ (all_0_49_49 = 0) & or(all_0_53_53, all_0_54_54) = all_0_51_51 & or(all_0_54_54, all_0_53_53) = all_0_52_52 & implies(all_0_52_52, all_0_51_51) = all_0_50_50 & is_a_theorem(all_0_50_50) = all_0_49_49 & ~ r3) | (r3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (or(v1, v0) = v3) | ~ (or(v0, v1) = v2) | ~ (implies(v2, v3) = v4) | is_a_theorem(v4) = 0))) & (( ~ (all_0_55_55 = 0) & or(all_0_62_62, all_0_58_58) = all_0_57_57 & or(all_0_62_62, all_0_61_61) = all_0_60_60 & or(all_0_63_63, all_0_60_60) = all_0_59_59 & or(all_0_63_63, all_0_61_61) = all_0_58_58 & implies(all_0_59_59, all_0_57_57) = all_0_56_56 & is_a_theorem(all_0_56_56) = all_0_55_55 & ~ r4) | (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) = 0))) & (( ~ (all_0_64_64 = 0) & or(all_0_72_72, all_0_70_70) = all_0_67_67 & or(all_0_72_72, all_0_71_71) = all_0_68_68 & implies(all_0_68_68, all_0_67_67) = all_0_66_66 & implies(all_0_69_69, all_0_66_66) = all_0_65_65 & implies(all_0_71_71, all_0_70_70) = all_0_69_69 & is_a_theorem(all_0_65_65) = all_0_64_64 & ~ r5) | (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) = 0))) & (( ~ (all_0_87_87 = 0) & possibly(all_0_91_91) = all_0_90_90 & necessarily(all_0_90_90) = all_0_89_89 & implies(all_0_90_90, all_0_89_89) = all_0_88_88 & is_a_theorem(all_0_88_88) = all_0_87_87 & ~ axiom_5) | (axiom_5 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (possibly(v0) = v1) | ~ (necessarily(v1) = v2) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3) = 0))) & (( ~ (all_0_92_92 = 0) & necessarily(all_0_95_95) = all_0_94_94 & necessarily(all_0_98_98) = all_0_97_97 & necessarily(all_0_100_100) = all_0_99_99 & and(all_0_99_99, all_0_97_97) = all_0_96_96 & implies(all_0_96_96, all_0_94_94) = all_0_93_93 & implies(all_0_102_102, all_0_101_101) = all_0_98_98 & implies(all_0_103_103, all_0_101_101) = all_0_95_95 & implies(all_0_103_103, all_0_102_102) = all_0_100_100 & is_a_theorem(all_0_93_93) = all_0_92_92 & ~ axiom_s1) | (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) = 0))) & (( ~ (all_0_104_104 = 0) & possibly(all_0_110_110) = all_0_109_109 & possibly(all_0_111_111) = all_0_107_107 & possibly(all_0_112_112) = all_0_108_108 & strict_implies(all_0_109_109, all_0_106_106) = all_0_105_105 & and(all_0_108_108, all_0_107_107) = all_0_106_106 & and(all_0_112_112, all_0_111_111) = all_0_110_110 & is_a_theorem(all_0_105_105) = all_0_104_104 & ~ axiom_s2) | (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) = 0))) & (( ~ (all_0_113_113 = 0) & possibly(all_0_121_121) = all_0_119_119 & possibly(all_0_122_122) = all_0_117_117 & strict_implies(all_0_118_118, all_0_116_116) = all_0_115_115 & strict_implies(all_0_120_120, all_0_115_115) = all_0_114_114 & strict_implies(all_0_122_122, all_0_121_121) = all_0_120_120 & not(all_0_117_117) = all_0_116_116 & not(all_0_119_119) = all_0_118_118 & is_a_theorem(all_0_114_114) = all_0_113_113 & ~ axiom_s3) | (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) = 0))) & (( ~ (all_0_123_123 = 0) & strict_implies(all_0_126_126, all_0_125_125) = all_0_124_124 & necessarily(all_0_126_126) = all_0_125_125 & necessarily(all_0_127_127) = all_0_126_126 & is_a_theorem(all_0_124_124) = all_0_123_123 & ~ axiom_s4) | (axiom_s4 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (strict_implies(v1, v2) = v3) | ~ (necessarily(v1) = v2) | ~ (necessarily(v0) = v1) | is_a_theorem(v3) = 0))) & (( ~ (all_0_128_128 = 0) & strict_implies(all_0_131_131, all_0_130_130) = all_0_129_129 & and(all_0_132_132, all_0_133_133) = all_0_130_130 & and(all_0_133_133, all_0_132_132) = all_0_131_131 & is_a_theorem(all_0_129_129) = all_0_128_128 & ~ axiom_m1) | (axiom_m1 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (strict_implies(v2, v3) = v4) | ~ (and(v1, v0) = v3) | ~ (and(v0, v1) = v2) | is_a_theorem(v4) = 0))) & (( ~ (all_0_134_134 = 0) & strict_implies(all_0_136_136, all_0_138_138) = all_0_135_135 & and(all_0_138_138, all_0_137_137) = all_0_136_136 & is_a_theorem(all_0_135_135) = all_0_134_134 & ~ axiom_m2) | (axiom_m2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (strict_implies(v2, v0) = v3) | ~ (and(v0, v1) = v2) | is_a_theorem(v3) = 0))) & (( ~ (all_0_139_139 = 0) & strict_implies(all_0_143_143, all_0_141_141) = all_0_140_140 & and(all_0_144_144, all_0_145_145) = all_0_143_143 & and(all_0_146_146, all_0_145_145) = all_0_142_142 & and(all_0_147_147, all_0_142_142) = all_0_141_141 & and(all_0_147_147, all_0_146_146) = all_0_144_144 & is_a_theorem(all_0_140_140) = all_0_139_139 & ~ axiom_m3) | (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) = 0))) & (( ~ (all_0_148_148 = 0) & strict_implies(all_0_151_151, all_0_150_150) = all_0_149_149 & and(all_0_151_151, all_0_151_151) = all_0_150_150 & is_a_theorem(all_0_149_149) = all_0_148_148 & ~ axiom_m4) | (axiom_m4 & ! [v0] : ! [v1] : ! [v2] : ( ~ (strict_implies(v0, v1) = v2) | ~ (and(v0, v0) = v1) | is_a_theorem(v2) = 0))) & (( ~ (all_0_152_152 = 0) & strict_implies(all_0_155_155, all_0_154_154) = all_0_153_153 & strict_implies(all_0_159_159, all_0_158_158) = all_0_156_156 & strict_implies(all_0_160_160, all_0_158_158) = all_0_154_154 & strict_implies(all_0_160_160, all_0_159_159) = all_0_157_157 & and(all_0_157_157, all_0_156_156) = all_0_155_155 & is_a_theorem(all_0_153_153) = all_0_152_152 & ~ axiom_m5) | (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) = 0))) & (( ~ (all_0_161_161 = 0) & possibly(all_0_164_164) = all_0_163_163 & strict_implies(all_0_164_164, all_0_163_163) = all_0_162_162 & is_a_theorem(all_0_162_162) = all_0_161_161 & ~ axiom_m6) | (axiom_m6 & ! [v0] : ! [v1] : ! [v2] : ( ~ (possibly(v0) = v1) | ~ (strict_implies(v0, v1) = v2) | is_a_theorem(v2) = 0))) & (( ~ (all_0_165_165 = 0) & possibly(all_0_168_168) = all_0_167_167 & strict_implies(all_0_167_167, all_0_170_170) = all_0_166_166 & and(all_0_170_170, all_0_169_169) = all_0_168_168 & is_a_theorem(all_0_166_166) = all_0_165_165 & ~ axiom_m7) | (axiom_m7 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (possibly(v2) = v3) | ~ (strict_implies(v3, v0) = v4) | ~ (and(v0, v1) = v2) | is_a_theorem(v4) = 0))) & (( ~ (all_0_171_171 = 0) & possibly(all_0_177_177) = all_0_174_174 & possibly(all_0_178_178) = all_0_175_175 & strict_implies(all_0_175_175, all_0_174_174) = all_0_173_173 & strict_implies(all_0_176_176, all_0_173_173) = all_0_172_172 & strict_implies(all_0_178_178, all_0_177_177) = all_0_176_176 & is_a_theorem(all_0_172_172) = all_0_171_171 & ~ axiom_m8) | (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) = 0))) & (( ~ (all_0_179_179 = 0) & possibly(all_0_182_182) = all_0_181_181 & possibly(all_0_183_183) = all_0_182_182 & strict_implies(all_0_181_181, all_0_182_182) = all_0_180_180 & is_a_theorem(all_0_180_180) = all_0_179_179 & ~ axiom_m9) | (axiom_m9 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (possibly(v1) = v2) | ~ (possibly(v0) = v1) | ~ (strict_implies(v2, v1) = v3) | is_a_theorem(v3) = 0))) & (( ~ (all_0_184_184 = 0) & possibly(all_0_188_188) = all_0_187_187 & strict_implies(all_0_187_187, all_0_186_186) = all_0_185_185 & necessarily(all_0_187_187) = all_0_186_186 & is_a_theorem(all_0_185_185) = all_0_184_184 & ~ axiom_m10) | (axiom_m10 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (possibly(v0) = v1) | ~ (strict_implies(v1, v2) = v3) | ~ (necessarily(v1) = v2) | is_a_theorem(v3) = 0)))
% 22.27/7.01 |
% 22.27/7.01 | Applying alpha-rule on (1) yields:
% 22.27/7.01 | (2) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (and(v0, v1) = v2) | ~ (implies(v1, v2) = v3) | ~ (implies(v0, v3) = v4) | is_a_theorem(v4) = 0)
% 22.27/7.02 | (3) ( ~ (all_0_55_55 = 0) & or(all_0_62_62, all_0_58_58) = all_0_57_57 & or(all_0_62_62, all_0_61_61) = all_0_60_60 & or(all_0_63_63, all_0_60_60) = all_0_59_59 & or(all_0_63_63, all_0_61_61) = all_0_58_58 & implies(all_0_59_59, all_0_57_57) = all_0_56_56 & is_a_theorem(all_0_56_56) = all_0_55_55 & ~ r4) | (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) = 0))
% 22.27/7.02 | (4) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (or(v3, v2) = v1) | ~ (or(v3, v2) = v0))
% 22.27/7.02 | (5) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equiv(v3, v2) = v1) | ~ (equiv(v3, v2) = v0))
% 22.27/7.02 | (6) op_implies_and
% 22.27/7.02 | (7) ( ~ (all_0_134_134 = 0) & strict_implies(all_0_136_136, all_0_138_138) = all_0_135_135 & and(all_0_138_138, all_0_137_137) = all_0_136_136 & is_a_theorem(all_0_135_135) = all_0_134_134 & ~ axiom_m2) | (axiom_m2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (strict_implies(v2, v0) = v3) | ~ (and(v0, v1) = v2) | is_a_theorem(v3) = 0))
% 22.27/7.02 | (8) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (implies(v3, v2) = v4) | ~ (implies(v0, v2) = v3) | ~ (implies(v0, v1) = v2) | is_a_theorem(v4) = 0)
% 22.27/7.02 | (9) equivalence_2
% 22.27/7.02 | (10) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v1) = v2) | ~ (implies(v2, v1) = v3) | is_a_theorem(v3) = 0)
% 22.27/7.02 | (11) ! [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) = 0)
% 22.27/7.02 | (12) is_a_theorem(all_0_76_76) = 0
% 22.27/7.02 | (13) and_2
% 22.27/7.02 | (14) ( ~ (all_0_44_44 = 0) & or(all_0_48_48, all_0_47_47) = all_0_46_46 & implies(all_0_47_47, all_0_46_46) = all_0_45_45 & is_a_theorem(all_0_45_45) = all_0_44_44 & ~ r2) | (r2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v0, v1) = v2) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3) = 0))
% 22.27/7.02 | (15) ( ~ (all_0_128_128 = 0) & strict_implies(all_0_131_131, all_0_130_130) = all_0_129_129 & and(all_0_132_132, all_0_133_133) = all_0_130_130 & and(all_0_133_133, all_0_132_132) = all_0_131_131 & is_a_theorem(all_0_129_129) = all_0_128_128 & ~ axiom_m1) | (axiom_m1 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (strict_implies(v2, v3) = v4) | ~ (and(v1, v0) = v3) | ~ (and(v0, v1) = v2) | is_a_theorem(v4) = 0))
% 22.27/7.02 | (16) ~ modus_ponens_strict_implies
% 22.27/7.02 | (17) op_implies
% 22.27/7.02 | (18) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (necessarily(v1) = v2) | ~ (necessarily(v0) = v1) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3) = 0)
% 22.27/7.02 | (19) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v2) = v3) | ~ (not(v1) = v2) | ? [v4] : (not(v3) = v4 & implies(v0, v1) = v4))
% 22.27/7.02 | (20) ( ~ (all_0_161_161 = 0) & possibly(all_0_164_164) = all_0_163_163 & strict_implies(all_0_164_164, all_0_163_163) = all_0_162_162 & is_a_theorem(all_0_162_162) = all_0_161_161 & ~ axiom_m6) | (axiom_m6 & ! [v0] : ! [v1] : ! [v2] : ( ~ (possibly(v0) = v1) | ~ (strict_implies(v0, v1) = v2) | is_a_theorem(v2) = 0))
% 22.27/7.02 | (21) op_or
% 22.27/7.02 | (22) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (equiv(v0, v1) = v2) | ? [v3] : ( ~ (v3 = 0) & is_a_theorem(v2) = v3))
% 22.27/7.02 | (23) modus_tollens
% 22.27/7.02 | (24) ! [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) = 0)
% 22.27/7.02 | (25) axiom_M
% 22.27/7.02 | (26) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (possibly(v0) = v1) | ~ (necessarily(v1) = v2) | ~ (implies(v0, v2) = v3) | is_a_theorem(v3) = 0)
% 22.27/7.02 | (27) ! [v0] : ! [v1] : ! [v2] : (v2 = 0 | ~ (is_a_theorem(v1) = v2) | ~ (is_a_theorem(v0) = 0) | ? [v3] : ? [v4] : ( ~ (v4 = 0) & implies(v0, v1) = v3 & is_a_theorem(v3) = v4))
% 22.27/7.02 | (28) implies_2
% 22.27/7.02 | (29) strict_implies(all_0_76_76, all_0_75_75) = all_0_74_74
% 22.27/7.02 | (30) ! [v0] : ! [v1] : ! [v2] : ( ~ (necessarily(v0) = v1) | ~ (implies(v1, v0) = v2) | is_a_theorem(v2) = 0)
% 22.27/7.02 | (31) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (strict_implies(v3, v2) = v1) | ~ (strict_implies(v3, v2) = v0))
% 22.27/7.02 | (32) ~ op_implies_or | ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v2, v1) = v3) | ~ (not(v0) = v2) | implies(v0, v1) = v3)
% 22.27/7.02 | (33) ( ~ (all_0_152_152 = 0) & strict_implies(all_0_155_155, all_0_154_154) = all_0_153_153 & strict_implies(all_0_159_159, all_0_158_158) = all_0_156_156 & strict_implies(all_0_160_160, all_0_158_158) = all_0_154_154 & strict_implies(all_0_160_160, all_0_159_159) = all_0_157_157 & and(all_0_157_157, all_0_156_156) = all_0_155_155 & is_a_theorem(all_0_153_153) = all_0_152_152 & ~ axiom_m5) | (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) = 0))
% 22.27/7.02 | (34) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (not(v2) = v1) | ~ (not(v2) = v0))
% 22.27/7.02 | (35) ! [v0] : ! [v1] : ! [v2] : ( ~ (strict_implies(v0, v1) = v2) | ? [v3] : (necessarily(v3) = v2 & implies(v0, v1) = v3))
% 22.27/7.02 | (36) ( ~ (all_0_87_87 = 0) & possibly(all_0_91_91) = all_0_90_90 & necessarily(all_0_90_90) = all_0_89_89 & implies(all_0_90_90, all_0_89_89) = all_0_88_88 & is_a_theorem(all_0_88_88) = all_0_87_87 & ~ axiom_5) | (axiom_5 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (possibly(v0) = v1) | ~ (necessarily(v1) = v2) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3) = 0))
% 22.27/7.02 | (37) is_a_theorem(all_0_74_74) = 0
% 22.27/7.02 | (38) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v0, v1) = v2) | ~ (implies(v0, v2) = v3) | is_a_theorem(v3) = 0)
% 22.27/7.02 | (39) is_a_theorem(all_0_75_75) = all_0_73_73
% 22.27/7.02 | (40) ( ~ (all_0_20_20 = 0) & implies(all_0_24_24, all_0_23_23) = all_0_22_22 & implies(all_0_25_25, all_0_22_22) = all_0_21_21 & implies(all_0_27_27, all_0_26_26) = all_0_24_24 & implies(all_0_28_28, all_0_26_26) = all_0_23_23 & implies(all_0_28_28, all_0_27_27) = all_0_25_25 & is_a_theorem(all_0_21_21) = all_0_20_20 & ~ cn1) | (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) = 0))
% 22.27/7.02 | (41) implies_3
% 22.27/7.02 | (42) or_1
% 22.27/7.02 | (43) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (equiv(v0, v1) = v2) | ~ (implies(v2, v3) = v4) | ~ (implies(v0, v1) = v3) | is_a_theorem(v4) = 0)
% 22.27/7.02 | (44) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (possibly(v2) = v1) | ~ (possibly(v2) = v0))
% 22.27/7.02 | (45) axiom_4
% 22.27/7.02 | (46) ( ~ (all_0_29_29 = 0) & not(all_0_34_34) = all_0_32_32 & implies(all_0_32_32, all_0_33_33) = all_0_31_31 & implies(all_0_34_34, all_0_31_31) = all_0_30_30 & is_a_theorem(all_0_30_30) = all_0_29_29 & ~ cn2) | (cn2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (not(v0) = v2) | ~ (implies(v2, v1) = v3) | ~ (implies(v0, v3) = v4) | is_a_theorem(v4) = 0))
% 22.27/7.02 | (47) ~ op_necessarily | ! [v0] : ! [v1] : ( ~ (not(v0) = v1) | ? [v2] : ? [v3] : (possibly(v1) = v3 & necessarily(v0) = v2 & not(v3) = v2))
% 22.27/7.03 | (48) ( ~ (all_0_40_40 = 0) & or(all_0_43_43, all_0_43_43) = all_0_42_42 & implies(all_0_42_42, all_0_43_43) = all_0_41_41 & is_a_theorem(all_0_41_41) = all_0_40_40 & ~ r1) | (r1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v0) = v1) | ~ (implies(v1, v0) = v2) | is_a_theorem(v2) = 0))
% 22.27/7.03 | (49) ( ~ (all_0_179_179 = 0) & possibly(all_0_182_182) = all_0_181_181 & possibly(all_0_183_183) = all_0_182_182 & strict_implies(all_0_181_181, all_0_182_182) = all_0_180_180 & is_a_theorem(all_0_180_180) = all_0_179_179 & ~ axiom_m9) | (axiom_m9 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (possibly(v1) = v2) | ~ (possibly(v0) = v1) | ~ (strict_implies(v2, v1) = v3) | is_a_theorem(v3) = 0))
% 22.27/7.03 | (50) op_equiv
% 22.27/7.03 | (51) necessitation
% 22.27/7.03 | (52) ( ~ (all_0_113_113 = 0) & possibly(all_0_121_121) = all_0_119_119 & possibly(all_0_122_122) = all_0_117_117 & strict_implies(all_0_118_118, all_0_116_116) = all_0_115_115 & strict_implies(all_0_120_120, all_0_115_115) = all_0_114_114 & strict_implies(all_0_122_122, all_0_121_121) = all_0_120_120 & not(all_0_117_117) = all_0_116_116 & not(all_0_119_119) = all_0_118_118 & is_a_theorem(all_0_114_114) = all_0_113_113 & ~ axiom_s3) | (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) = 0))
% 22.27/7.03 | (53) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (implies(v3, v2) = v1) | ~ (implies(v3, v2) = v0))
% 22.27/7.03 | (54) ( ~ (all_0_139_139 = 0) & strict_implies(all_0_143_143, all_0_141_141) = all_0_140_140 & and(all_0_144_144, all_0_145_145) = all_0_143_143 & and(all_0_146_146, all_0_145_145) = all_0_142_142 & and(all_0_147_147, all_0_142_142) = all_0_141_141 & and(all_0_147_147, all_0_146_146) = all_0_144_144 & is_a_theorem(all_0_140_140) = all_0_139_139 & ~ axiom_m3) | (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) = 0))
% 22.27/7.03 | (55) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (strict_implies(v1, v0) = v3) | ~ (strict_implies(v0, v1) = v2) | ~ (and(v2, v3) = v4) | strict_equiv(v0, v1) = v4)
% 22.27/7.03 | (56) equivalence_3
% 22.27/7.03 | (57) ( ~ (all_0_92_92 = 0) & necessarily(all_0_95_95) = all_0_94_94 & necessarily(all_0_98_98) = all_0_97_97 & necessarily(all_0_100_100) = all_0_99_99 & and(all_0_99_99, all_0_97_97) = all_0_96_96 & implies(all_0_96_96, all_0_94_94) = all_0_93_93 & implies(all_0_102_102, all_0_101_101) = all_0_98_98 & implies(all_0_103_103, all_0_101_101) = all_0_95_95 & implies(all_0_103_103, all_0_102_102) = all_0_100_100 & is_a_theorem(all_0_93_93) = all_0_92_92 & ~ axiom_s1) | (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) = 0))
% 22.27/7.03 | (58) ( ~ (all_0_64_64 = 0) & or(all_0_72_72, all_0_70_70) = all_0_67_67 & or(all_0_72_72, all_0_71_71) = all_0_68_68 & implies(all_0_68_68, all_0_67_67) = all_0_66_66 & implies(all_0_69_69, all_0_66_66) = all_0_65_65 & implies(all_0_71_71, all_0_70_70) = all_0_69_69 & is_a_theorem(all_0_65_65) = all_0_64_64 & ~ r5) | (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) = 0))
% 22.27/7.03 | (59) ( ~ (all_0_4_4 = 0) & and(all_0_8_8, all_0_7_7) = all_0_6_6 & implies(all_0_6_6, all_0_8_8) = all_0_5_5 & is_a_theorem(all_0_5_5) = all_0_4_4 & ~ kn2) | (kn2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v1) = v2) | ~ (implies(v2, v0) = v3) | is_a_theorem(v3) = 0))
% 22.27/7.03 | (60) ~ 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))
% 22.27/7.03 | (61) ! [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) = 0)
% 22.27/7.03 | (62) ! [v0] : ! [v1] : ( ~ (necessarily(v0) = v1) | ? [v2] : ? [v3] : (is_a_theorem(v1) = v3 & is_a_theorem(v0) = v2 & ( ~ (v2 = 0) | v3 = 0)))
% 22.27/7.03 | (63) modus_ponens
% 22.27/7.03 | (64) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (implies(v1, v0) = v2) | ~ (implies(v0, v2) = v3) | is_a_theorem(v3) = 0)
% 22.27/7.03 | (65) ( ~ (all_0_104_104 = 0) & possibly(all_0_110_110) = all_0_109_109 & possibly(all_0_111_111) = all_0_107_107 & possibly(all_0_112_112) = all_0_108_108 & strict_implies(all_0_109_109, all_0_106_106) = all_0_105_105 & and(all_0_108_108, all_0_107_107) = all_0_106_106 & and(all_0_112_112, all_0_111_111) = all_0_110_110 & is_a_theorem(all_0_105_105) = all_0_104_104 & ~ axiom_s2) | (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) = 0))
% 22.27/7.03 | (66) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (strict_equiv(v3, v2) = v1) | ~ (strict_equiv(v3, v2) = v0))
% 22.27/7.03 | (67) (all_0_79_79 = 0 & all_0_80_80 = 0 & ~ (all_0_77_77 = 0) & and(all_0_82_82, all_0_81_81) = all_0_78_78 & is_a_theorem(all_0_78_78) = all_0_77_77 & is_a_theorem(all_0_81_81) = 0 & is_a_theorem(all_0_82_82) = 0 & ~ adjunction) | (adjunction & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (is_a_theorem(v2) = v5 & is_a_theorem(v1) = v4 & is_a_theorem(v0) = v3 & ( ~ (v4 = 0) | ~ (v3 = 0) | v5 = 0))))
% 22.27/7.03 | (68) ( ~ (all_0_184_184 = 0) & possibly(all_0_188_188) = all_0_187_187 & strict_implies(all_0_187_187, all_0_186_186) = all_0_185_185 & necessarily(all_0_187_187) = all_0_186_186 & is_a_theorem(all_0_185_185) = all_0_184_184 & ~ axiom_m10) | (axiom_m10 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (possibly(v0) = v1) | ~ (strict_implies(v1, v2) = v3) | ~ (necessarily(v1) = v2) | is_a_theorem(v3) = 0))
% 22.27/7.03 | (69) and_3
% 22.27/7.03 | (70) ( ~ (all_0_123_123 = 0) & strict_implies(all_0_126_126, all_0_125_125) = all_0_124_124 & necessarily(all_0_126_126) = all_0_125_125 & necessarily(all_0_127_127) = all_0_126_126 & is_a_theorem(all_0_124_124) = all_0_123_123 & ~ axiom_s4) | (axiom_s4 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (strict_implies(v1, v2) = v3) | ~ (necessarily(v1) = v2) | ~ (necessarily(v0) = v1) | is_a_theorem(v3) = 0))
% 22.27/7.03 | (71) ( ~ (all_0_148_148 = 0) & strict_implies(all_0_151_151, all_0_150_150) = all_0_149_149 & and(all_0_151_151, all_0_151_151) = all_0_150_150 & is_a_theorem(all_0_149_149) = all_0_148_148 & ~ axiom_m4) | (axiom_m4 & ! [v0] : ! [v1] : ! [v2] : ( ~ (strict_implies(v0, v1) = v2) | ~ (and(v0, v0) = v1) | is_a_theorem(v2) = 0))
% 22.27/7.03 | (72) ! [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) = 0)
% 22.27/7.03 | (73) and_1
% 22.27/7.03 | (74) or_3
% 22.27/7.03 | (75) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (and(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) = v2) | ? [v5] : (or(v0, v1) = v5 & not(v4) = v5))
% 22.27/7.03 | (76) ~ (all_0_73_73 = 0)
% 22.27/7.03 | (77) axiom_B
% 22.27/7.03 | (78) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v1) = v2) | ~ (implies(v2, v0) = v3) | is_a_theorem(v3) = 0)
% 22.27/7.03 | (79) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v0, v1) = v2) | ~ (implies(v1, v2) = v3) | is_a_theorem(v3) = 0)
% 22.27/7.03 | (80) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (equiv(v0, v1) = v2) | ~ (implies(v2, v3) = v4) | ~ (implies(v1, v0) = v3) | is_a_theorem(v4) = 0)
% 22.27/7.03 | (81) ! [v0] : ! [v1] : ( ~ (not(v0) = v1) | ? [v2] : ? [v3] : (possibly(v0) = v2 & necessarily(v1) = v3 & not(v3) = v2))
% 22.27/7.03 | (82) ! [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) = 0)
% 22.27/7.03 | (83) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (and(v2, v3) = v4) | ~ (implies(v1, v0) = v3) | ~ (implies(v0, v1) = v2) | equiv(v0, v1) = v4)
% 22.27/7.03 | (84) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (and(v3, v2) = v1) | ~ (and(v3, v2) = v0))
% 22.27/7.03 | (85) ( ~ (all_0_35_35 = 0) & not(all_0_39_39) = all_0_38_38 & implies(all_0_37_37, all_0_39_39) = all_0_36_36 & implies(all_0_38_38, all_0_39_39) = all_0_37_37 & is_a_theorem(all_0_36_36) = all_0_35_35 & ~ cn3) | (cn3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (not(v0) = v1) | ~ (implies(v2, v0) = v3) | ~ (implies(v1, v0) = v2) | is_a_theorem(v3) = 0))
% 22.27/7.03 | (86) op_strict_implies
% 22.27/7.03 | (87) equivalence_1
% 22.27/7.03 | (88) op_strict_equiv
% 22.27/7.03 | (89) (all_0_83_83 = 0 & ~ (all_0_85_85 = all_0_86_86) & strict_equiv(all_0_86_86, all_0_85_85) = all_0_84_84 & is_a_theorem(all_0_84_84) = 0 & ~ substitution_strict_equiv) | (substitution_strict_equiv & ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (strict_equiv(v0, v1) = v2) | ? [v3] : ( ~ (v3 = 0) & is_a_theorem(v2) = v3)))
% 22.27/7.03 | (90) op_possibly
% 22.27/7.03 | (91) ( ~ (all_0_165_165 = 0) & possibly(all_0_168_168) = all_0_167_167 & strict_implies(all_0_167_167, all_0_170_170) = all_0_166_166 & and(all_0_170_170, all_0_169_169) = all_0_168_168 & is_a_theorem(all_0_166_166) = all_0_165_165 & ~ axiom_m7) | (axiom_m7 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (possibly(v2) = v3) | ~ (strict_implies(v3, v0) = v4) | ~ (and(v0, v1) = v2) | is_a_theorem(v4) = 0))
% 22.27/7.03 | (92) ( ~ (all_0_171_171 = 0) & possibly(all_0_177_177) = all_0_174_174 & possibly(all_0_178_178) = all_0_175_175 & strict_implies(all_0_175_175, all_0_174_174) = all_0_173_173 & strict_implies(all_0_176_176, all_0_173_173) = all_0_172_172 & strict_implies(all_0_178_178, all_0_177_177) = all_0_176_176 & is_a_theorem(all_0_172_172) = all_0_171_171 & ~ axiom_m8) | (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) = 0))
% 22.27/7.03 | (93) or_2
% 22.27/7.03 | (94) ( ~ (all_0_0_0 = 0) & and(all_0_3_3, all_0_3_3) = all_0_2_2 & implies(all_0_3_3, all_0_2_2) = all_0_1_1 & is_a_theorem(all_0_1_1) = all_0_0_0 & ~ kn1) | (kn1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v0) = v1) | ~ (implies(v0, v1) = v2) | is_a_theorem(v2) = 0))
% 22.27/7.03 | (95) ( ~ (all_0_49_49 = 0) & or(all_0_53_53, all_0_54_54) = all_0_51_51 & or(all_0_54_54, all_0_53_53) = all_0_52_52 & implies(all_0_52_52, all_0_51_51) = all_0_50_50 & is_a_theorem(all_0_50_50) = all_0_49_49 & ~ r3) | (r3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (or(v1, v0) = v3) | ~ (or(v0, v1) = v2) | ~ (implies(v2, v3) = v4) | is_a_theorem(v4) = 0))
% 22.27/7.03 | (96) axiom_K
% 22.27/7.03 | (97) implies_1
% 22.27/7.03 | (98) substitution_of_equivalents
% 22.27/7.04 | (99) ( ~ (all_0_9_9 = 0) & and(all_0_17_17, all_0_19_19) = all_0_13_13 & and(all_0_18_18, all_0_17_17) = all_0_15_15 & not(all_0_13_13) = all_0_12_12 & not(all_0_15_15) = all_0_14_14 & implies(all_0_14_14, all_0_12_12) = all_0_11_11 & implies(all_0_16_16, all_0_11_11) = all_0_10_10 & implies(all_0_19_19, all_0_18_18) = all_0_16_16 & is_a_theorem(all_0_10_10) = all_0_9_9 & ~ kn3) | (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) = 0))
% 22.27/7.04 | (100) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (is_a_theorem(v2) = v1) | ~ (is_a_theorem(v2) = v0))
% 22.27/7.04 | (101) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (necessarily(v2) = v1) | ~ (necessarily(v2) = v0))
% 22.27/7.04 |
% 22.27/7.04 | Instantiating formula (35) with all_0_74_74, all_0_75_75, all_0_76_76 and discharging atoms strict_implies(all_0_76_76, all_0_75_75) = all_0_74_74, yields:
% 22.27/7.04 | (102) ? [v0] : (necessarily(v0) = all_0_74_74 & implies(all_0_76_76, all_0_75_75) = v0)
% 22.27/7.04 |
% 22.27/7.04 | Instantiating formula (27) with all_0_73_73, all_0_75_75, all_0_74_74 and discharging atoms is_a_theorem(all_0_74_74) = 0, is_a_theorem(all_0_75_75) = all_0_73_73, yields:
% 22.27/7.04 | (103) all_0_73_73 = 0 | ? [v0] : ? [v1] : ( ~ (v1 = 0) & implies(all_0_74_74, all_0_75_75) = v0 & is_a_theorem(v0) = v1)
% 22.27/7.04 |
% 22.27/7.04 | Instantiating formula (27) with all_0_73_73, all_0_75_75, all_0_76_76 and discharging atoms is_a_theorem(all_0_75_75) = all_0_73_73, is_a_theorem(all_0_76_76) = 0, yields:
% 22.27/7.04 | (104) all_0_73_73 = 0 | ? [v0] : ? [v1] : ( ~ (v1 = 0) & implies(all_0_76_76, all_0_75_75) = v0 & is_a_theorem(v0) = v1)
% 22.27/7.04 |
% 22.27/7.04 | Instantiating (102) with all_8_0_189 yields:
% 22.27/7.04 | (105) necessarily(all_8_0_189) = all_0_74_74 & implies(all_0_76_76, all_0_75_75) = all_8_0_189
% 22.27/7.04 |
% 22.27/7.04 | Applying alpha-rule on (105) yields:
% 22.27/7.04 | (106) necessarily(all_8_0_189) = all_0_74_74
% 22.27/7.04 | (107) implies(all_0_76_76, all_0_75_75) = all_8_0_189
% 22.27/7.04 |
% 22.27/7.04 +-Applying beta-rule and splitting (103), into two cases.
% 22.27/7.04 |-Branch one:
% 22.27/7.04 | (108) all_0_73_73 = 0
% 22.27/7.04 |
% 22.27/7.04 | Equations (108) can reduce 76 to:
% 22.27/7.04 | (109) $false
% 22.27/7.04 |
% 22.27/7.04 |-The branch is then unsatisfiable
% 22.27/7.04 |-Branch two:
% 22.27/7.04 | (76) ~ (all_0_73_73 = 0)
% 22.27/7.04 | (111) ? [v0] : ? [v1] : ( ~ (v1 = 0) & implies(all_0_74_74, all_0_75_75) = v0 & is_a_theorem(v0) = v1)
% 22.27/7.04 |
% 22.27/7.04 +-Applying beta-rule and splitting (104), into two cases.
% 22.27/7.04 |-Branch one:
% 22.27/7.04 | (108) all_0_73_73 = 0
% 22.27/7.04 |
% 22.27/7.04 | Equations (108) can reduce 76 to:
% 22.27/7.04 | (109) $false
% 22.27/7.04 |
% 22.27/7.04 |-The branch is then unsatisfiable
% 22.27/7.04 |-Branch two:
% 22.27/7.04 | (76) ~ (all_0_73_73 = 0)
% 22.27/7.04 | (115) ? [v0] : ? [v1] : ( ~ (v1 = 0) & implies(all_0_76_76, all_0_75_75) = v0 & is_a_theorem(v0) = v1)
% 22.27/7.04 |
% 22.27/7.04 | Instantiating (115) with all_19_0_192, all_19_1_193 yields:
% 22.27/7.04 | (116) ~ (all_19_0_192 = 0) & implies(all_0_76_76, all_0_75_75) = all_19_1_193 & is_a_theorem(all_19_1_193) = all_19_0_192
% 22.27/7.04 |
% 22.27/7.04 | Applying alpha-rule on (116) yields:
% 22.27/7.04 | (117) ~ (all_19_0_192 = 0)
% 22.27/7.04 | (118) implies(all_0_76_76, all_0_75_75) = all_19_1_193
% 22.27/7.04 | (119) is_a_theorem(all_19_1_193) = all_19_0_192
% 22.27/7.04 |
% 22.27/7.04 | Instantiating formula (53) with all_0_76_76, all_0_75_75, all_8_0_189, all_19_1_193 and discharging atoms implies(all_0_76_76, all_0_75_75) = all_19_1_193, implies(all_0_76_76, all_0_75_75) = all_8_0_189, yields:
% 22.27/7.04 | (120) all_19_1_193 = all_8_0_189
% 22.27/7.04 |
% 22.27/7.04 | From (120) and (119) follows:
% 22.27/7.04 | (121) is_a_theorem(all_8_0_189) = all_19_0_192
% 22.27/7.04 |
% 22.27/7.04 | Instantiating formula (27) with all_19_0_192, all_8_0_189, all_0_74_74 and discharging atoms is_a_theorem(all_8_0_189) = all_19_0_192, is_a_theorem(all_0_74_74) = 0, yields:
% 22.27/7.04 | (122) all_19_0_192 = 0 | ? [v0] : ? [v1] : ( ~ (v1 = 0) & implies(all_0_74_74, all_8_0_189) = v0 & is_a_theorem(v0) = v1)
% 22.27/7.04 |
% 22.27/7.04 | Instantiating formula (27) with all_19_0_192, all_8_0_189, all_0_76_76 and discharging atoms is_a_theorem(all_8_0_189) = all_19_0_192, is_a_theorem(all_0_76_76) = 0, yields:
% 22.27/7.04 | (123) all_19_0_192 = 0 | ? [v0] : ? [v1] : ( ~ (v1 = 0) & implies(all_0_76_76, all_8_0_189) = v0 & is_a_theorem(v0) = v1)
% 22.27/7.04 |
% 22.27/7.04 +-Applying beta-rule and splitting (123), into two cases.
% 22.27/7.04 |-Branch one:
% 22.27/7.04 | (124) all_19_0_192 = 0
% 22.27/7.04 |
% 22.27/7.04 | Equations (124) can reduce 117 to:
% 22.27/7.04 | (109) $false
% 22.27/7.04 |
% 22.27/7.04 |-The branch is then unsatisfiable
% 22.27/7.04 |-Branch two:
% 22.27/7.04 | (117) ~ (all_19_0_192 = 0)
% 22.27/7.04 | (127) ? [v0] : ? [v1] : ( ~ (v1 = 0) & implies(all_0_76_76, all_8_0_189) = v0 & is_a_theorem(v0) = v1)
% 22.27/7.04 |
% 22.27/7.04 +-Applying beta-rule and splitting (122), into two cases.
% 22.27/7.04 |-Branch one:
% 22.27/7.04 | (124) all_19_0_192 = 0
% 22.27/7.04 |
% 22.27/7.04 | Equations (124) can reduce 117 to:
% 22.27/7.04 | (109) $false
% 22.27/7.04 |
% 22.27/7.04 |-The branch is then unsatisfiable
% 22.27/7.04 |-Branch two:
% 22.27/7.04 | (117) ~ (all_19_0_192 = 0)
% 22.27/7.04 | (131) ? [v0] : ? [v1] : ( ~ (v1 = 0) & implies(all_0_74_74, all_8_0_189) = v0 & is_a_theorem(v0) = v1)
% 22.27/7.04 |
% 22.27/7.04 | Instantiating (131) with all_55_0_208, all_55_1_209 yields:
% 22.27/7.04 | (132) ~ (all_55_0_208 = 0) & implies(all_0_74_74, all_8_0_189) = all_55_1_209 & is_a_theorem(all_55_1_209) = all_55_0_208
% 22.27/7.04 |
% 22.27/7.04 | Applying alpha-rule on (132) yields:
% 22.27/7.04 | (133) ~ (all_55_0_208 = 0)
% 22.27/7.04 | (134) implies(all_0_74_74, all_8_0_189) = all_55_1_209
% 22.27/7.04 | (135) is_a_theorem(all_55_1_209) = all_55_0_208
% 22.27/7.04 |
% 22.27/7.04 | Instantiating formula (30) with all_55_1_209, all_0_74_74, all_8_0_189 and discharging atoms necessarily(all_8_0_189) = all_0_74_74, implies(all_0_74_74, all_8_0_189) = all_55_1_209, yields:
% 22.27/7.04 | (136) is_a_theorem(all_55_1_209) = 0
% 22.27/7.04 |
% 22.27/7.04 | Instantiating formula (27) with all_55_0_208, all_55_1_209, all_0_74_74 and discharging atoms is_a_theorem(all_55_1_209) = all_55_0_208, is_a_theorem(all_0_74_74) = 0, yields:
% 22.27/7.04 | (137) all_55_0_208 = 0 | ? [v0] : ? [v1] : ( ~ (v1 = 0) & implies(all_0_74_74, all_55_1_209) = v0 & is_a_theorem(v0) = v1)
% 22.27/7.04 |
% 22.27/7.04 | Instantiating formula (27) with all_55_0_208, all_55_1_209, all_0_76_76 and discharging atoms is_a_theorem(all_55_1_209) = all_55_0_208, is_a_theorem(all_0_76_76) = 0, yields:
% 22.27/7.04 | (138) all_55_0_208 = 0 | ? [v0] : ? [v1] : ( ~ (v1 = 0) & implies(all_0_76_76, all_55_1_209) = v0 & is_a_theorem(v0) = v1)
% 22.27/7.04 |
% 22.27/7.04 +-Applying beta-rule and splitting (137), into two cases.
% 22.27/7.04 |-Branch one:
% 22.27/7.04 | (139) all_55_0_208 = 0
% 22.27/7.04 |
% 22.27/7.04 | Equations (139) can reduce 133 to:
% 22.27/7.04 | (109) $false
% 22.27/7.04 |
% 22.27/7.04 |-The branch is then unsatisfiable
% 22.27/7.04 |-Branch two:
% 22.27/7.04 | (133) ~ (all_55_0_208 = 0)
% 22.27/7.04 | (142) ? [v0] : ? [v1] : ( ~ (v1 = 0) & implies(all_0_74_74, all_55_1_209) = v0 & is_a_theorem(v0) = v1)
% 22.27/7.04 |
% 22.27/7.04 +-Applying beta-rule and splitting (138), into two cases.
% 22.27/7.04 |-Branch one:
% 22.27/7.04 | (139) all_55_0_208 = 0
% 22.27/7.04 |
% 22.27/7.04 | Equations (139) can reduce 133 to:
% 22.27/7.04 | (109) $false
% 22.27/7.04 |
% 22.27/7.04 |-The branch is then unsatisfiable
% 22.27/7.04 |-Branch two:
% 22.27/7.04 | (133) ~ (all_55_0_208 = 0)
% 22.27/7.04 | (146) ? [v0] : ? [v1] : ( ~ (v1 = 0) & implies(all_0_76_76, all_55_1_209) = v0 & is_a_theorem(v0) = v1)
% 22.27/7.04 |
% 22.27/7.04 | Instantiating formula (100) with all_55_1_209, 0, all_55_0_208 and discharging atoms is_a_theorem(all_55_1_209) = all_55_0_208, is_a_theorem(all_55_1_209) = 0, yields:
% 22.27/7.04 | (139) all_55_0_208 = 0
% 22.27/7.04 |
% 22.27/7.04 | Equations (139) can reduce 133 to:
% 22.27/7.04 | (109) $false
% 22.27/7.04 |
% 22.27/7.04 |-The branch is then unsatisfiable
% 22.27/7.04 % SZS output end Proof for theBenchmark
% 22.27/7.04
% 22.27/7.04 6428ms
%------------------------------------------------------------------------------