TSTP Solution File: LCL499+1 by ePrincess---1.0
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%------------------------------------------------------------------------------
% File : ePrincess---1.0
% Problem : LCL499+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : ePrincess-casc -timeout=%d %s
% Computer : n012.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 09:36:50 EDT 2022
% Result : Theorem 28.11s 8.79s
% Output : Proof 32.04s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : LCL499+1 : TPTP v8.1.0. Released v3.3.0.
% 0.00/0.12 % Command : ePrincess-casc -timeout=%d %s
% 0.13/0.33 % Computer : n012.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 600
% 0.13/0.33 % DateTime : Sat Jul 2 21:52:01 EDT 2022
% 0.13/0.33 % CPUTime :
% 0.59/0.59 ____ _
% 0.59/0.59 ___ / __ \_____(_)___ ________ __________
% 0.59/0.59 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/
% 0.59/0.59 / __/ ____/ / / / / / / /__/ __(__ |__ )
% 0.59/0.59 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/
% 0.59/0.59
% 0.59/0.59 A Theorem Prover for First-Order Logic
% 0.59/0.60 (ePrincess v.1.0)
% 0.59/0.60
% 0.59/0.60 (c) Philipp Rümmer, 2009-2015
% 0.59/0.60 (c) Peter Backeman, 2014-2015
% 0.59/0.60 (contributions by Angelo Brillout, Peter Baumgartner)
% 0.59/0.60 Free software under GNU Lesser General Public License (LGPL).
% 0.59/0.60 Bug reports to peter@backeman.se
% 0.59/0.60
% 0.59/0.60 For more information, visit http://user.uu.se/~petba168/breu/
% 0.59/0.60
% 0.59/0.60 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 0.75/0.65 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all
% 1.74/1.00 Prover 0: Preprocessing ...
% 3.05/1.36 Prover 0: Constructing countermodel ...
% 16.66/5.94 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all
% 16.99/6.00 Prover 1: Preprocessing ...
% 17.54/6.13 Prover 1: Constructing countermodel ...
% 27.25/8.53 Prover 2: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all
% 27.25/8.57 Prover 2: Preprocessing ...
% 27.64/8.68 Prover 2: Warning: ignoring some quantifiers
% 27.64/8.69 Prover 2: Constructing countermodel ...
% 28.11/8.79 Prover 2: proved (252ms)
% 28.11/8.79 Prover 0: stopped
% 28.11/8.79 Prover 1: stopped
% 28.11/8.79
% 28.11/8.79 No countermodel exists, formula is valid
% 28.11/8.79 % SZS status Theorem for theBenchmark
% 28.11/8.79
% 28.11/8.79 Generating proof ... Warning: ignoring some quantifiers
% 31.64/9.55 found it (size 59)
% 31.64/9.55
% 31.64/9.55 % SZS output start Proof for theBenchmark
% 31.64/9.55 Assumed formulas after preprocessing and simplification:
% 31.64/9.55 | (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] : ( ~ (v39 = 0) & and(v36, v36) = v37 & implies(v36, v37) = v38 & is_a_theorem(v38) = v39 & op_equiv & op_implies_or & op_implies_and & op_and & op_or & r5 & r4 & r3 & r2 & r1 & substitution_of_equivalents & modus_ponens & ~ kn1 & ! [v124] : ! [v125] : ! [v126] : ! [v127] : ! [v128] : ! [v129] : ( ~ (or(v124, v126) = v128) | ~ (or(v124, v125) = v127) | ~ (implies(v127, v128) = v129) | ? [v130] : ? [v131] : (implies(v130, v129) = v131 & implies(v125, v126) = v130 & is_a_theorem(v131) = 0)) & ! [v124] : ! [v125] : ! [v126] : ! [v127] : ! [v128] : ( ~ (or(v126, v127) = v128) | ~ (not(v125) = v127) | ~ (not(v124) = v126) | ? [v129] : (and(v124, v125) = v129 & not(v128) = v129)) & ! [v124] : ! [v125] : ! [v126] : ! [v127] : ! [v128] : ( ~ (or(v125, v127) = v128) | ~ (or(v124, v126) = v127) | ? [v129] : ? [v130] : ? [v131] : (or(v125, v126) = v129 & or(v124, v129) = v130 & implies(v130, v128) = v131 & is_a_theorem(v131) = 0)) & ! [v124] : ! [v125] : ! [v126] : ! [v127] : ! [v128] : ( ~ (or(v125, v126) = v127) | ~ (or(v124, v127) = v128) | ? [v129] : ? [v130] : ? [v131] : (or(v125, v129) = v130 & or(v124, v126) = v129 & implies(v128, v130) = v131 & is_a_theorem(v131) = 0)) & ! [v124] : ! [v125] : ! [v126] : ! [v127] : ! [v128] : ( ~ (and(v126, v127) = v128) | ~ (not(v125) = v127) | ~ (not(v124) = v126) | ? [v129] : (or(v124, v125) = v129 & not(v128) = v129)) & ! [v124] : ! [v125] : ! [v126] : ! [v127] : (v125 = v124 | ~ (or(v127, v126) = v125) | ~ (or(v127, v126) = v124)) & ! [v124] : ! [v125] : ! [v126] : ! [v127] : (v125 = v124 | ~ (and(v127, v126) = v125) | ~ (and(v127, v126) = v124)) & ! [v124] : ! [v125] : ! [v126] : ! [v127] : (v125 = v124 | ~ (equiv(v127, v126) = v125) | ~ (equiv(v127, v126) = v124)) & ! [v124] : ! [v125] : ! [v126] : ! [v127] : (v125 = v124 | ~ (implies(v127, v126) = v125) | ~ (implies(v127, v126) = v124)) & ! [v124] : ! [v125] : ! [v126] : ! [v127] : ( ~ (or(v126, v125) = v127) | ~ (not(v124) = v126) | implies(v124, v125) = v127) & ! [v124] : ! [v125] : ! [v126] : ! [v127] : ( ~ (and(v124, v126) = v127) | ~ (not(v125) = v126) | ? [v128] : (not(v127) = v128 & implies(v124, v125) = v128)) & ! [v124] : ! [v125] : ! [v126] : (v125 = v124 | ~ (not(v126) = v125) | ~ (not(v126) = v124)) & ! [v124] : ! [v125] : ! [v126] : (v125 = v124 | ~ (equiv(v124, v125) = v126) | ? [v127] : ( ~ (v127 = 0) & is_a_theorem(v126) = v127)) & ! [v124] : ! [v125] : ! [v126] : (v125 = v124 | ~ (is_a_theorem(v126) = v125) | ~ (is_a_theorem(v126) = v124)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (or(v125, v124) = v126) | ? [v127] : ? [v128] : (or(v124, v125) = v127 & implies(v127, v126) = v128 & is_a_theorem(v128) = 0)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (or(v124, v125) = v126) | ? [v127] : ? [v128] : ? [v129] : (and(v127, v128) = v129 & not(v129) = v126 & not(v125) = v128 & not(v124) = v127)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (or(v124, v125) = v126) | ? [v127] : ? [v128] : (or(v125, v124) = v127 & implies(v126, v127) = v128 & is_a_theorem(v128) = 0)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (or(v124, v125) = v126) | ? [v127] : (implies(v125, v126) = v127 & is_a_theorem(v127) = 0)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (and(v124, v125) = v126) | ? [v127] : ? [v128] : ? [v129] : (or(v127, v128) = v129 & not(v129) = v126 & not(v125) = v128 & not(v124) = v127)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (equiv(v124, v125) = v126) | ? [v127] : ? [v128] : (and(v127, v128) = v126 & implies(v125, v124) = v128 & implies(v124, v125) = v127)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (implies(v125, v124) = v126) | ? [v127] : ? [v128] : (and(v128, v126) = v127 & equiv(v124, v125) = v127 & implies(v124, v125) = v128)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (implies(v124, v125) = v126) | ? [v127] : ? [v128] : (and(v126, v128) = v127 & equiv(v124, v125) = v127 & implies(v125, v124) = v128)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (implies(v124, v125) = v126) | ? [v127] : ? [v128] : (and(v124, v127) = v128 & not(v128) = v126 & not(v125) = v127)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (implies(v124, v125) = v126) | ? [v127] : (or(v127, v125) = v126 & not(v124) = v127)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (implies(v124, v125) = v126) | ? [v127] : ((v127 = 0 & is_a_theorem(v125) = 0) | ( ~ (v127 = 0) & is_a_theorem(v126) = v127) | ( ~ (v127 = 0) & is_a_theorem(v124) = v127))) & ! [v124] : ! [v125] : ( ~ (or(v124, v124) = v125) | ? [v126] : (implies(v125, v124) = v126 & is_a_theorem(v126) = 0)) & ? [v124] : ? [v125] : ? [v126] : or(v125, v124) = v126 & ? [v124] : ? [v125] : ? [v126] : and(v125, v124) = v126 & ? [v124] : ? [v125] : ? [v126] : equiv(v125, v124) = v126 & ? [v124] : ? [v125] : ? [v126] : implies(v125, v124) = v126 & ? [v124] : ? [v125] : not(v124) = v125 & ? [v124] : ? [v125] : is_a_theorem(v124) = v125 & (( ~ (v123 = 0) & not(v117) = v118 & not(v116) = v119 & implies(v120, v121) = v122 & implies(v118, v119) = v120 & implies(v116, v117) = v121 & is_a_theorem(v122) = v123 & ~ modus_tollens) | (modus_tollens & ! [v124] : ! [v125] : ! [v126] : ! [v127] : ! [v128] : ( ~ (not(v125) = v126) | ~ (not(v124) = v127) | ~ (implies(v126, v127) = v128) | ? [v129] : ? [v130] : (implies(v128, v129) = v130 & implies(v124, v125) = v129 & is_a_theorem(v130) = 0)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (implies(v124, v125) = v126) | ? [v127] : ? [v128] : ? [v129] : ? [v130] : (not(v125) = v127 & not(v124) = v128 & implies(v129, v126) = v130 & implies(v127, v128) = v129 & is_a_theorem(v130) = 0)))) & (( ~ (v115 = 0) & implies(v112, v111) = v113 & implies(v111, v113) = v114 & is_a_theorem(v114) = v115 & ~ implies_1) | (implies_1 & ! [v124] : ! [v125] : ! [v126] : ( ~ (implies(v125, v124) = v126) | ? [v127] : (implies(v124, v126) = v127 & is_a_theorem(v127) = 0)))) & (( ~ (v110 = 0) & implies(v108, v107) = v109 & implies(v105, v107) = v108 & implies(v105, v106) = v107 & is_a_theorem(v109) = v110 & ~ implies_2) | (implies_2 & ! [v124] : ! [v125] : ! [v126] : ( ~ (implies(v124, v125) = v126) | ? [v127] : ? [v128] : (implies(v127, v126) = v128 & implies(v124, v126) = v127 & is_a_theorem(v128) = 0)))) & (( ~ (v104 = 0) & implies(v100, v101) = v102 & implies(v99, v102) = v103 & implies(v97, v98) = v100 & implies(v96, v98) = v101 & implies(v96, v97) = v99 & is_a_theorem(v103) = v104 & ~ implies_3) | (implies_3 & ! [v124] : ! [v125] : ! [v126] : ! [v127] : ! [v128] : ! [v129] : ( ~ (implies(v127, v128) = v129) | ~ (implies(v125, v126) = v127) | ~ (implies(v124, v126) = v128) | ? [v130] : ? [v131] : (implies(v130, v129) = v131 & implies(v124, v125) = v130 & is_a_theorem(v131) = 0)))) & (( ~ (v95 = 0) & and(v91, v92) = v93 & implies(v93, v91) = v94 & is_a_theorem(v94) = v95 & ~ and_1) | (and_1 & ! [v124] : ! [v125] : ! [v126] : ( ~ (and(v124, v125) = v126) | ? [v127] : (implies(v126, v124) = v127 & is_a_theorem(v127) = 0)))) & (( ~ (v90 = 0) & and(v86, v87) = v88 & implies(v88, v87) = v89 & is_a_theorem(v89) = v90 & ~ and_2) | (and_2 & ! [v124] : ! [v125] : ! [v126] : ( ~ (and(v124, v125) = v126) | ? [v127] : (implies(v126, v125) = v127 & is_a_theorem(v127) = 0)))) & (( ~ (v85 = 0) & and(v80, v81) = v82 & implies(v81, v82) = v83 & implies(v80, v83) = v84 & is_a_theorem(v84) = v85 & ~ and_3) | (and_3 & ! [v124] : ! [v125] : ! [v126] : ( ~ (and(v124, v125) = v126) | ? [v127] : ? [v128] : (implies(v125, v126) = v127 & implies(v124, v127) = v128 & is_a_theorem(v128) = 0)))) & (( ~ (v79 = 0) & or(v75, v76) = v77 & implies(v75, v77) = v78 & is_a_theorem(v78) = v79 & ~ or_1) | (or_1 & ! [v124] : ! [v125] : ! [v126] : ( ~ (or(v124, v125) = v126) | ? [v127] : (implies(v124, v126) = v127 & is_a_theorem(v127) = 0)))) & (( ~ (v74 = 0) & or(v70, v71) = v72 & implies(v71, v72) = v73 & is_a_theorem(v73) = v74 & ~ or_2) | (or_2 & ! [v124] : ! [v125] : ! [v126] : ( ~ (or(v124, v125) = v126) | ? [v127] : (implies(v125, v126) = v127 & is_a_theorem(v127) = 0)))) & (( ~ (v69 = 0) & or(v60, v61) = v65 & implies(v65, v62) = v66 & implies(v64, v66) = v67 & implies(v63, v67) = v68 & implies(v61, v62) = v64 & implies(v60, v62) = v63 & is_a_theorem(v68) = v69 & ~ or_3) | (or_3 & ! [v124] : ! [v125] : ! [v126] : ! [v127] : ! [v128] : ( ~ (or(v124, v125) = v127) | ~ (implies(v127, v126) = v128) | ? [v129] : ? [v130] : ? [v131] : ? [v132] : (implies(v130, v128) = v131 & implies(v129, v131) = v132 & implies(v125, v126) = v130 & implies(v124, v126) = v129 & is_a_theorem(v132) = 0)))) & (( ~ (v59 = 0) & equiv(v54, v55) = v56 & implies(v56, v57) = v58 & implies(v54, v55) = v57 & is_a_theorem(v58) = v59 & ~ equivalence_1) | (equivalence_1 & ! [v124] : ! [v125] : ! [v126] : ( ~ (equiv(v124, v125) = v126) | ? [v127] : ? [v128] : (implies(v126, v127) = v128 & implies(v124, v125) = v127 & is_a_theorem(v128) = 0)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (implies(v124, v125) = v126) | ? [v127] : ? [v128] : (equiv(v124, v125) = v127 & implies(v127, v126) = v128 & is_a_theorem(v128) = 0)))) & (( ~ (v53 = 0) & equiv(v48, v49) = v50 & implies(v50, v51) = v52 & implies(v49, v48) = v51 & is_a_theorem(v52) = v53 & ~ equivalence_2) | (equivalence_2 & ! [v124] : ! [v125] : ! [v126] : ( ~ (equiv(v124, v125) = v126) | ? [v127] : ? [v128] : (implies(v126, v127) = v128 & implies(v125, v124) = v127 & is_a_theorem(v128) = 0)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (implies(v125, v124) = v126) | ? [v127] : ? [v128] : (equiv(v124, v125) = v127 & implies(v127, v126) = v128 & is_a_theorem(v128) = 0)))) & (( ~ (v47 = 0) & equiv(v40, v41) = v44 & implies(v43, v44) = v45 & implies(v42, v45) = v46 & implies(v41, v40) = v43 & implies(v40, v41) = v42 & is_a_theorem(v46) = v47 & ~ equivalence_3) | (equivalence_3 & ! [v124] : ! [v125] : ! [v126] : ( ~ (equiv(v124, v125) = v126) | ? [v127] : ? [v128] : ? [v129] : ? [v130] : (implies(v128, v126) = v129 & implies(v127, v129) = v130 & implies(v125, v124) = v128 & implies(v124, v125) = v127 & is_a_theorem(v130) = 0)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (implies(v125, v124) = v126) | ? [v127] : ? [v128] : ? [v129] : ? [v130] : (equiv(v124, v125) = v128 & implies(v127, v129) = v130 & implies(v126, v128) = v129 & implies(v124, v125) = v127 & is_a_theorem(v130) = 0)) & ! [v124] : ! [v125] : ! [v126] : ( ~ (implies(v124, v125) = v126) | ? [v127] : ? [v128] : ? [v129] : ? [v130] : (equiv(v124, v125) = v128 & implies(v127, v128) = v129 & implies(v126, v129) = v130 & implies(v125, v124) = v127 & is_a_theorem(v130) = 0)))) & (( ~ (v35 = 0) & and(v31, v32) = v33 & implies(v33, v31) = v34 & is_a_theorem(v34) = v35 & ~ kn2) | (kn2 & ! [v124] : ! [v125] : ! [v126] : ( ~ (and(v124, v125) = v126) | ? [v127] : (implies(v126, v124) = v127 & is_a_theorem(v127) = 0)))) & (( ~ (v30 = 0) & and(v22, v20) = v26 & and(v21, v22) = v24 & not(v26) = v27 & not(v24) = v25 & implies(v25, v27) = v28 & implies(v23, v28) = v29 & implies(v20, v21) = v23 & is_a_theorem(v29) = v30 & ~ kn3) | (kn3 & ! [v124] : ! [v125] : ! [v126] : ! [v127] : ! [v128] : ! [v129] : ! [v130] : ! [v131] : ( ~ (and(v126, v124) = v129) | ~ (and(v125, v126) = v127) | ~ (not(v129) = v130) | ~ (not(v127) = v128) | ~ (implies(v128, v130) = v131) | ? [v132] : ? [v133] : (implies(v132, v131) = v133 & implies(v124, v125) = v132 & is_a_theorem(v133) = 0)))) & (( ~ (v19 = 0) & implies(v15, v16) = v17 & implies(v14, v17) = v18 & implies(v12, v13) = v15 & implies(v11, v13) = v16 & implies(v11, v12) = v14 & is_a_theorem(v18) = v19 & ~ cn1) | (cn1 & ! [v124] : ! [v125] : ! [v126] : ! [v127] : ! [v128] : ! [v129] : ( ~ (implies(v127, v128) = v129) | ~ (implies(v125, v126) = v127) | ~ (implies(v124, v126) = v128) | ? [v130] : ? [v131] : (implies(v130, v129) = v131 & implies(v124, v125) = v130 & is_a_theorem(v131) = 0)))) & (( ~ (v10 = 0) & not(v5) = v7 & implies(v7, v6) = v8 & implies(v5, v8) = v9 & is_a_theorem(v9) = v10 & ~ cn2) | (cn2 & ! [v124] : ! [v125] : ! [v126] : ! [v127] : ( ~ (not(v124) = v126) | ~ (implies(v126, v125) = v127) | ? [v128] : (implies(v124, v127) = v128 & is_a_theorem(v128) = 0)))) & (( ~ (v4 = 0) & not(v0) = v1 & implies(v2, v0) = v3 & implies(v1, v0) = v2 & is_a_theorem(v3) = v4 & ~ cn3) | (cn3 & ! [v124] : ! [v125] : ( ~ (not(v124) = v125) | ? [v126] : ? [v127] : (implies(v126, v124) = v127 & implies(v125, v124) = v126 & is_a_theorem(v127) = 0)))))
% 32.04/9.61 | 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 yields:
% 32.04/9.61 | (1) ~ (all_0_84_84 = 0) & and(all_0_87_87, all_0_87_87) = all_0_86_86 & implies(all_0_87_87, all_0_86_86) = all_0_85_85 & is_a_theorem(all_0_85_85) = all_0_84_84 & op_equiv & op_implies_or & op_implies_and & op_and & op_or & r5 & r4 & r3 & r2 & r1 & substitution_of_equivalents & modus_ponens & ~ kn1 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (or(v0, v2) = v4) | ~ (or(v0, v1) = v3) | ~ (implies(v3, v4) = v5) | ? [v6] : ? [v7] : (implies(v6, v5) = v7 & implies(v1, v2) = v6 & is_a_theorem(v7) = 0)) & ! [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] : ! [v3] : ! [v4] : ( ~ (or(v1, v3) = v4) | ~ (or(v0, v2) = v3) | ? [v5] : ? [v6] : ? [v7] : (or(v1, v2) = v5 & or(v0, v5) = v6 & implies(v6, v4) = v7 & is_a_theorem(v7) = 0)) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (or(v1, v2) = v3) | ~ (or(v0, v3) = v4) | ? [v5] : ? [v6] : ? [v7] : (or(v1, v5) = v6 & or(v0, v2) = v5 & implies(v4, v6) = v7 & is_a_theorem(v7) = 0)) & ! [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] : (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] : ( ~ (or(v2, v1) = v3) | ~ (not(v0) = v2) | implies(v0, v1) = v3) & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v2) = v3) | ~ (not(v1) = v2) | ? [v4] : (not(v3) = v4 & implies(v0, v1) = v4)) & ! [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] : ( ~ (or(v1, v0) = v2) | ? [v3] : ? [v4] : (or(v0, v1) = v3 & implies(v3, v2) = v4 & is_a_theorem(v4) = 0)) & ! [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] : ( ~ (or(v0, v1) = v2) | ? [v3] : ? [v4] : (or(v1, v0) = v3 & implies(v2, v3) = v4 & is_a_theorem(v4) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v1) = v2) | ? [v3] : (implies(v1, v2) = v3 & is_a_theorem(v3) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (or(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(v1, v0) = v2) | ? [v3] : ? [v4] : (and(v4, v2) = v3 & equiv(v0, v1) = v3 & implies(v0, v1) = v4)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : (and(v2, v4) = v3 & equiv(v0, v1) = v3 & implies(v1, v0) = v4)) & ! [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] : (or(v3, v1) = v2 & not(v0) = v3)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ((v3 = 0 & is_a_theorem(v1) = 0) | ( ~ (v3 = 0) & is_a_theorem(v2) = v3) | ( ~ (v3 = 0) & is_a_theorem(v0) = v3))) & ! [v0] : ! [v1] : ( ~ (or(v0, v0) = v1) | ? [v2] : (implies(v1, v0) = v2 & is_a_theorem(v2) = 0)) & ? [v0] : ? [v1] : ? [v2] : or(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : and(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : equiv(v1, v0) = v2 & ? [v0] : ? [v1] : ? [v2] : implies(v1, v0) = v2 & ? [v0] : ? [v1] : not(v0) = v1 & ? [v0] : ? [v1] : is_a_theorem(v0) = v1 & (( ~ (all_0_0_0 = 0) & not(all_0_6_6) = all_0_5_5 & not(all_0_7_7) = all_0_4_4 & implies(all_0_3_3, all_0_2_2) = all_0_1_1 & implies(all_0_5_5, all_0_4_4) = all_0_3_3 & implies(all_0_7_7, all_0_6_6) = all_0_2_2 & is_a_theorem(all_0_1_1) = all_0_0_0 & ~ modus_tollens) | (modus_tollens & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (not(v1) = v2) | ~ (not(v0) = v3) | ~ (implies(v2, v3) = v4) | ? [v5] : ? [v6] : (implies(v4, v5) = v6 & implies(v0, v1) = v5 & is_a_theorem(v6) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : (not(v1) = v3 & not(v0) = v4 & implies(v5, v2) = v6 & implies(v3, v4) = v5 & is_a_theorem(v6) = 0)))) & (( ~ (all_0_8_8 = 0) & implies(all_0_11_11, all_0_12_12) = all_0_10_10 & implies(all_0_12_12, all_0_10_10) = all_0_9_9 & is_a_theorem(all_0_9_9) = all_0_8_8 & ~ implies_1) | (implies_1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v1, v0) = v2) | ? [v3] : (implies(v0, v2) = v3 & is_a_theorem(v3) = 0)))) & (( ~ (all_0_13_13 = 0) & implies(all_0_15_15, all_0_16_16) = all_0_14_14 & implies(all_0_18_18, all_0_16_16) = all_0_15_15 & implies(all_0_18_18, all_0_17_17) = all_0_16_16 & is_a_theorem(all_0_14_14) = all_0_13_13 & ~ implies_2) | (implies_2 & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : (implies(v3, v2) = v4 & implies(v0, v2) = v3 & is_a_theorem(v4) = 0)))) & (( ~ (all_0_19_19 = 0) & implies(all_0_23_23, all_0_22_22) = all_0_21_21 & implies(all_0_24_24, all_0_21_21) = all_0_20_20 & implies(all_0_26_26, all_0_25_25) = all_0_23_23 & implies(all_0_27_27, all_0_25_25) = all_0_22_22 & implies(all_0_27_27, all_0_26_26) = all_0_24_24 & is_a_theorem(all_0_20_20) = all_0_19_19 & ~ implies_3) | (implies_3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (implies(v3, v4) = v5) | ~ (implies(v1, v2) = v3) | ~ (implies(v0, v2) = v4) | ? [v6] : ? [v7] : (implies(v6, v5) = v7 & implies(v0, v1) = v6 & is_a_theorem(v7) = 0)))) & (( ~ (all_0_28_28 = 0) & and(all_0_32_32, all_0_31_31) = all_0_30_30 & implies(all_0_30_30, all_0_32_32) = all_0_29_29 & is_a_theorem(all_0_29_29) = all_0_28_28 & ~ and_1) | (and_1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : (implies(v2, v0) = v3 & is_a_theorem(v3) = 0)))) & (( ~ (all_0_33_33 = 0) & and(all_0_37_37, all_0_36_36) = all_0_35_35 & implies(all_0_35_35, all_0_36_36) = all_0_34_34 & is_a_theorem(all_0_34_34) = all_0_33_33 & ~ and_2) | (and_2 & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : (implies(v2, v1) = v3 & is_a_theorem(v3) = 0)))) & (( ~ (all_0_38_38 = 0) & and(all_0_43_43, all_0_42_42) = all_0_41_41 & implies(all_0_42_42, all_0_41_41) = all_0_40_40 & implies(all_0_43_43, all_0_40_40) = all_0_39_39 & is_a_theorem(all_0_39_39) = all_0_38_38 & ~ and_3) | (and_3 & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : ? [v4] : (implies(v1, v2) = v3 & implies(v0, v3) = v4 & is_a_theorem(v4) = 0)))) & (( ~ (all_0_44_44 = 0) & or(all_0_48_48, all_0_47_47) = all_0_46_46 & implies(all_0_48_48, all_0_46_46) = all_0_45_45 & is_a_theorem(all_0_45_45) = all_0_44_44 & ~ or_1) | (or_1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v1) = v2) | ? [v3] : (implies(v0, v2) = v3 & is_a_theorem(v3) = 0)))) & (( ~ (all_0_49_49 = 0) & or(all_0_53_53, all_0_52_52) = all_0_51_51 & implies(all_0_52_52, all_0_51_51) = all_0_50_50 & is_a_theorem(all_0_50_50) = all_0_49_49 & ~ or_2) | (or_2 & ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v1) = v2) | ? [v3] : (implies(v1, v2) = v3 & is_a_theorem(v3) = 0)))) & (( ~ (all_0_54_54 = 0) & or(all_0_63_63, all_0_62_62) = all_0_58_58 & implies(all_0_58_58, all_0_61_61) = all_0_57_57 & implies(all_0_59_59, all_0_57_57) = all_0_56_56 & implies(all_0_60_60, all_0_56_56) = all_0_55_55 & implies(all_0_62_62, all_0_61_61) = all_0_59_59 & implies(all_0_63_63, all_0_61_61) = all_0_60_60 & is_a_theorem(all_0_55_55) = all_0_54_54 & ~ or_3) | (or_3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (or(v0, v1) = v3) | ~ (implies(v3, v2) = v4) | ? [v5] : ? [v6] : ? [v7] : ? [v8] : (implies(v6, v4) = v7 & implies(v5, v7) = v8 & implies(v1, v2) = v6 & implies(v0, v2) = v5 & is_a_theorem(v8) = 0)))) & (( ~ (all_0_64_64 = 0) & equiv(all_0_69_69, all_0_68_68) = all_0_67_67 & implies(all_0_67_67, all_0_66_66) = all_0_65_65 & implies(all_0_69_69, all_0_68_68) = all_0_66_66 & is_a_theorem(all_0_65_65) = all_0_64_64 & ~ equivalence_1) | (equivalence_1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (equiv(v0, v1) = v2) | ? [v3] : ? [v4] : (implies(v2, v3) = v4 & implies(v0, v1) = v3 & is_a_theorem(v4) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : (equiv(v0, v1) = v3 & implies(v3, v2) = v4 & is_a_theorem(v4) = 0)))) & (( ~ (all_0_70_70 = 0) & equiv(all_0_75_75, all_0_74_74) = all_0_73_73 & implies(all_0_73_73, all_0_72_72) = all_0_71_71 & implies(all_0_74_74, all_0_75_75) = all_0_72_72 & is_a_theorem(all_0_71_71) = all_0_70_70 & ~ equivalence_2) | (equivalence_2 & ! [v0] : ! [v1] : ! [v2] : ( ~ (equiv(v0, v1) = v2) | ? [v3] : ? [v4] : (implies(v2, v3) = v4 & implies(v1, v0) = v3 & is_a_theorem(v4) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v1, v0) = v2) | ? [v3] : ? [v4] : (equiv(v0, v1) = v3 & implies(v3, v2) = v4 & is_a_theorem(v4) = 0)))) & (( ~ (all_0_76_76 = 0) & equiv(all_0_83_83, all_0_82_82) = all_0_79_79 & implies(all_0_80_80, all_0_79_79) = all_0_78_78 & implies(all_0_81_81, all_0_78_78) = all_0_77_77 & implies(all_0_82_82, all_0_83_83) = all_0_80_80 & implies(all_0_83_83, all_0_82_82) = all_0_81_81 & is_a_theorem(all_0_77_77) = all_0_76_76 & ~ equivalence_3) | (equivalence_3 & ! [v0] : ! [v1] : ! [v2] : ( ~ (equiv(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : (implies(v4, v2) = v5 & implies(v3, v5) = v6 & implies(v1, v0) = v4 & implies(v0, v1) = v3 & is_a_theorem(v6) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v1, v0) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : (equiv(v0, v1) = v4 & implies(v3, v5) = v6 & implies(v2, v4) = v5 & implies(v0, v1) = v3 & is_a_theorem(v6) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : (equiv(v0, v1) = v4 & implies(v3, v4) = v5 & implies(v2, v5) = v6 & implies(v1, v0) = v3 & is_a_theorem(v6) = 0)))) & (( ~ (all_0_88_88 = 0) & and(all_0_92_92, all_0_91_91) = all_0_90_90 & implies(all_0_90_90, all_0_92_92) = all_0_89_89 & is_a_theorem(all_0_89_89) = all_0_88_88 & ~ kn2) | (kn2 & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : (implies(v2, v0) = v3 & is_a_theorem(v3) = 0)))) & (( ~ (all_0_93_93 = 0) & and(all_0_101_101, all_0_103_103) = all_0_97_97 & and(all_0_102_102, all_0_101_101) = all_0_99_99 & not(all_0_97_97) = all_0_96_96 & not(all_0_99_99) = all_0_98_98 & implies(all_0_98_98, all_0_96_96) = all_0_95_95 & implies(all_0_100_100, all_0_95_95) = all_0_94_94 & implies(all_0_103_103, all_0_102_102) = all_0_100_100 & is_a_theorem(all_0_94_94) = all_0_93_93 & ~ kn3) | (kn3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (and(v2, v0) = v5) | ~ (and(v1, v2) = v3) | ~ (not(v5) = v6) | ~ (not(v3) = v4) | ~ (implies(v4, v6) = v7) | ? [v8] : ? [v9] : (implies(v8, v7) = v9 & implies(v0, v1) = v8 & is_a_theorem(v9) = 0)))) & (( ~ (all_0_104_104 = 0) & implies(all_0_108_108, all_0_107_107) = all_0_106_106 & implies(all_0_109_109, all_0_106_106) = all_0_105_105 & implies(all_0_111_111, all_0_110_110) = all_0_108_108 & implies(all_0_112_112, all_0_110_110) = all_0_107_107 & implies(all_0_112_112, all_0_111_111) = all_0_109_109 & is_a_theorem(all_0_105_105) = all_0_104_104 & ~ cn1) | (cn1 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (implies(v3, v4) = v5) | ~ (implies(v1, v2) = v3) | ~ (implies(v0, v2) = v4) | ? [v6] : ? [v7] : (implies(v6, v5) = v7 & implies(v0, v1) = v6 & is_a_theorem(v7) = 0)))) & (( ~ (all_0_113_113 = 0) & not(all_0_118_118) = all_0_116_116 & implies(all_0_116_116, all_0_117_117) = all_0_115_115 & implies(all_0_118_118, all_0_115_115) = all_0_114_114 & is_a_theorem(all_0_114_114) = all_0_113_113 & ~ cn2) | (cn2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (not(v0) = v2) | ~ (implies(v2, v1) = v3) | ? [v4] : (implies(v0, v3) = v4 & is_a_theorem(v4) = 0)))) & (( ~ (all_0_119_119 = 0) & not(all_0_123_123) = all_0_122_122 & implies(all_0_121_121, all_0_123_123) = all_0_120_120 & implies(all_0_122_122, all_0_123_123) = all_0_121_121 & is_a_theorem(all_0_120_120) = all_0_119_119 & ~ cn3) | (cn3 & ! [v0] : ! [v1] : ( ~ (not(v0) = v1) | ? [v2] : ? [v3] : (implies(v2, v0) = v3 & implies(v1, v0) = v2 & is_a_theorem(v3) = 0))))
% 32.04/9.63 |
% 32.04/9.63 | Applying alpha-rule on (1) yields:
% 32.04/9.63 | (2) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (equiv(v0, v1) = v2) | ? [v3] : ( ~ (v3 = 0) & is_a_theorem(v2) = v3))
% 32.04/9.63 | (3) ( ~ (all_0_38_38 = 0) & and(all_0_43_43, all_0_42_42) = all_0_41_41 & implies(all_0_42_42, all_0_41_41) = all_0_40_40 & implies(all_0_43_43, all_0_40_40) = all_0_39_39 & is_a_theorem(all_0_39_39) = all_0_38_38 & ~ and_3) | (and_3 & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : ? [v4] : (implies(v1, v2) = v3 & implies(v0, v3) = v4 & is_a_theorem(v4) = 0)))
% 32.04/9.63 | (4) ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (or(v3, v4) = v5 & not(v5) = v2 & not(v1) = v4 & not(v0) = v3))
% 32.04/9.63 | (5) ~ (all_0_84_84 = 0)
% 32.04/9.63 | (6) ( ~ (all_0_119_119 = 0) & not(all_0_123_123) = all_0_122_122 & implies(all_0_121_121, all_0_123_123) = all_0_120_120 & implies(all_0_122_122, all_0_123_123) = all_0_121_121 & is_a_theorem(all_0_120_120) = all_0_119_119 & ~ cn3) | (cn3 & ! [v0] : ! [v1] : ( ~ (not(v0) = v1) | ? [v2] : ? [v3] : (implies(v2, v0) = v3 & implies(v1, v0) = v2 & is_a_theorem(v3) = 0)))
% 32.04/9.63 | (7) op_implies_or
% 32.04/9.63 | (8) ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ((v3 = 0 & is_a_theorem(v1) = 0) | ( ~ (v3 = 0) & is_a_theorem(v2) = v3) | ( ~ (v3 = 0) & is_a_theorem(v0) = v3)))
% 32.04/9.63 | (9) ? [v0] : ? [v1] : ? [v2] : equiv(v1, v0) = v2
% 32.04/9.63 | (10) modus_ponens
% 32.04/9.63 | (11) r3
% 32.04/9.63 | (12) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equiv(v3, v2) = v1) | ~ (equiv(v3, v2) = v0))
% 32.04/9.63 | (13) ( ~ (all_0_113_113 = 0) & not(all_0_118_118) = all_0_116_116 & implies(all_0_116_116, all_0_117_117) = all_0_115_115 & implies(all_0_118_118, all_0_115_115) = all_0_114_114 & is_a_theorem(all_0_114_114) = all_0_113_113 & ~ cn2) | (cn2 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (not(v0) = v2) | ~ (implies(v2, v1) = v3) | ? [v4] : (implies(v0, v3) = v4 & is_a_theorem(v4) = 0)))
% 32.04/9.63 | (14) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (not(v2) = v1) | ~ (not(v2) = v0))
% 32.04/9.63 | (15) ~ kn1
% 32.04/9.63 | (16) ( ~ (all_0_88_88 = 0) & and(all_0_92_92, all_0_91_91) = all_0_90_90 & implies(all_0_90_90, all_0_92_92) = all_0_89_89 & is_a_theorem(all_0_89_89) = all_0_88_88 & ~ kn2) | (kn2 & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : (implies(v2, v0) = v3 & is_a_theorem(v3) = 0)))
% 32.04/9.63 | (17) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v2) = v3) | ~ (not(v1) = v2) | ? [v4] : (not(v3) = v4 & implies(v0, v1) = v4))
% 32.04/9.63 | (18) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (implies(v3, v2) = v1) | ~ (implies(v3, v2) = v0))
% 32.04/9.63 | (19) ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v1, v0) = v2) | ? [v3] : ? [v4] : (or(v0, v1) = v3 & implies(v3, v2) = v4 & is_a_theorem(v4) = 0))
% 32.04/9.63 | (20) ! [v0] : ! [v1] : ! [v2] : ( ~ (equiv(v0, v1) = v2) | ? [v3] : ? [v4] : (and(v3, v4) = v2 & implies(v1, v0) = v4 & implies(v0, v1) = v3))
% 32.04/9.63 | (21) substitution_of_equivalents
% 32.04/9.63 | (22) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (is_a_theorem(v2) = v1) | ~ (is_a_theorem(v2) = v0))
% 32.04/9.63 | (23) ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v1, v0) = v2) | ? [v3] : ? [v4] : (and(v4, v2) = v3 & equiv(v0, v1) = v3 & implies(v0, v1) = v4))
% 32.04/9.63 | (24) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v2, v1) = v3) | ~ (not(v0) = v2) | implies(v0, v1) = v3)
% 32.04/9.63 | (25) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (or(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) = v2) | ? [v5] : (and(v0, v1) = v5 & not(v4) = v5))
% 32.04/9.63 | (26) implies(all_0_87_87, all_0_86_86) = all_0_85_85
% 32.04/9.63 | (27) op_equiv
% 32.04/9.63 | (28) op_and
% 32.04/9.63 | (29) r2
% 32.04/9.63 | (30) ( ~ (all_0_19_19 = 0) & implies(all_0_23_23, all_0_22_22) = all_0_21_21 & implies(all_0_24_24, all_0_21_21) = all_0_20_20 & implies(all_0_26_26, all_0_25_25) = all_0_23_23 & implies(all_0_27_27, all_0_25_25) = all_0_22_22 & implies(all_0_27_27, all_0_26_26) = all_0_24_24 & is_a_theorem(all_0_20_20) = all_0_19_19 & ~ implies_3) | (implies_3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (implies(v3, v4) = v5) | ~ (implies(v1, v2) = v3) | ~ (implies(v0, v2) = v4) | ? [v6] : ? [v7] : (implies(v6, v5) = v7 & implies(v0, v1) = v6 & is_a_theorem(v7) = 0)))
% 32.04/9.63 | (31) ( ~ (all_0_8_8 = 0) & implies(all_0_11_11, all_0_12_12) = all_0_10_10 & implies(all_0_12_12, all_0_10_10) = all_0_9_9 & is_a_theorem(all_0_9_9) = all_0_8_8 & ~ implies_1) | (implies_1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v1, v0) = v2) | ? [v3] : (implies(v0, v2) = v3 & is_a_theorem(v3) = 0)))
% 32.04/9.63 | (32) ( ~ (all_0_104_104 = 0) & implies(all_0_108_108, all_0_107_107) = all_0_106_106 & implies(all_0_109_109, all_0_106_106) = all_0_105_105 & implies(all_0_111_111, all_0_110_110) = all_0_108_108 & implies(all_0_112_112, all_0_110_110) = all_0_107_107 & implies(all_0_112_112, all_0_111_111) = all_0_109_109 & is_a_theorem(all_0_105_105) = all_0_104_104 & ~ cn1) | (cn1 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (implies(v3, v4) = v5) | ~ (implies(v1, v2) = v3) | ~ (implies(v0, v2) = v4) | ? [v6] : ? [v7] : (implies(v6, v5) = v7 & implies(v0, v1) = v6 & is_a_theorem(v7) = 0)))
% 32.04/9.64 | (33) ( ~ (all_0_0_0 = 0) & not(all_0_6_6) = all_0_5_5 & not(all_0_7_7) = all_0_4_4 & implies(all_0_3_3, all_0_2_2) = all_0_1_1 & implies(all_0_5_5, all_0_4_4) = all_0_3_3 & implies(all_0_7_7, all_0_6_6) = all_0_2_2 & is_a_theorem(all_0_1_1) = all_0_0_0 & ~ modus_tollens) | (modus_tollens & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (not(v1) = v2) | ~ (not(v0) = v3) | ~ (implies(v2, v3) = v4) | ? [v5] : ? [v6] : (implies(v4, v5) = v6 & implies(v0, v1) = v5 & is_a_theorem(v6) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : (not(v1) = v3 & not(v0) = v4 & implies(v5, v2) = v6 & implies(v3, v4) = v5 & is_a_theorem(v6) = 0)))
% 32.04/9.64 | (34) ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v1) = v2) | ? [v3] : ? [v4] : (or(v1, v0) = v3 & implies(v2, v3) = v4 & is_a_theorem(v4) = 0))
% 32.04/9.64 | (35) is_a_theorem(all_0_85_85) = all_0_84_84
% 32.04/9.64 | (36) ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : (and(v0, v3) = v4 & not(v4) = v2 & not(v1) = v3))
% 32.04/9.64 | (37) ( ~ (all_0_76_76 = 0) & equiv(all_0_83_83, all_0_82_82) = all_0_79_79 & implies(all_0_80_80, all_0_79_79) = all_0_78_78 & implies(all_0_81_81, all_0_78_78) = all_0_77_77 & implies(all_0_82_82, all_0_83_83) = all_0_80_80 & implies(all_0_83_83, all_0_82_82) = all_0_81_81 & is_a_theorem(all_0_77_77) = all_0_76_76 & ~ equivalence_3) | (equivalence_3 & ! [v0] : ! [v1] : ! [v2] : ( ~ (equiv(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : (implies(v4, v2) = v5 & implies(v3, v5) = v6 & implies(v1, v0) = v4 & implies(v0, v1) = v3 & is_a_theorem(v6) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v1, v0) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : (equiv(v0, v1) = v4 & implies(v3, v5) = v6 & implies(v2, v4) = v5 & implies(v0, v1) = v3 & is_a_theorem(v6) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : ? [v6] : (equiv(v0, v1) = v4 & implies(v3, v4) = v5 & implies(v2, v5) = v6 & implies(v1, v0) = v3 & is_a_theorem(v6) = 0)))
% 32.04/9.64 | (38) ( ~ (all_0_28_28 = 0) & and(all_0_32_32, all_0_31_31) = all_0_30_30 & implies(all_0_30_30, all_0_32_32) = all_0_29_29 & is_a_theorem(all_0_29_29) = all_0_28_28 & ~ and_1) | (and_1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : (implies(v2, v0) = v3 & is_a_theorem(v3) = 0)))
% 32.04/9.64 | (39) ! [v0] : ! [v1] : ( ~ (or(v0, v0) = v1) | ? [v2] : (implies(v1, v0) = v2 & is_a_theorem(v2) = 0))
% 32.04/9.64 | (40) ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : (or(v3, v1) = v2 & not(v0) = v3))
% 32.04/9.64 | (41) ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : (and(v2, v4) = v3 & equiv(v0, v1) = v3 & implies(v1, v0) = v4))
% 32.04/9.64 | (42) ? [v0] : ? [v1] : ? [v2] : and(v1, v0) = v2
% 32.04/9.64 | (43) ( ~ (all_0_54_54 = 0) & or(all_0_63_63, all_0_62_62) = all_0_58_58 & implies(all_0_58_58, all_0_61_61) = all_0_57_57 & implies(all_0_59_59, all_0_57_57) = all_0_56_56 & implies(all_0_60_60, all_0_56_56) = all_0_55_55 & implies(all_0_62_62, all_0_61_61) = all_0_59_59 & implies(all_0_63_63, all_0_61_61) = all_0_60_60 & is_a_theorem(all_0_55_55) = all_0_54_54 & ~ or_3) | (or_3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (or(v0, v1) = v3) | ~ (implies(v3, v2) = v4) | ? [v5] : ? [v6] : ? [v7] : ? [v8] : (implies(v6, v4) = v7 & implies(v5, v7) = v8 & implies(v1, v2) = v6 & implies(v0, v2) = v5 & is_a_theorem(v8) = 0)))
% 32.04/9.64 | (44) ? [v0] : ? [v1] : ? [v2] : or(v1, v0) = v2
% 32.04/9.64 | (45) ( ~ (all_0_33_33 = 0) & and(all_0_37_37, all_0_36_36) = all_0_35_35 & implies(all_0_35_35, all_0_36_36) = all_0_34_34 & is_a_theorem(all_0_34_34) = all_0_33_33 & ~ and_2) | (and_2 & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : (implies(v2, v1) = v3 & is_a_theorem(v3) = 0)))
% 32.04/9.64 | (46) ( ~ (all_0_13_13 = 0) & implies(all_0_15_15, all_0_16_16) = all_0_14_14 & implies(all_0_18_18, all_0_16_16) = all_0_15_15 & implies(all_0_18_18, all_0_17_17) = all_0_16_16 & is_a_theorem(all_0_14_14) = all_0_13_13 & ~ implies_2) | (implies_2 & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : (implies(v3, v2) = v4 & implies(v0, v2) = v3 & is_a_theorem(v4) = 0)))
% 32.04/9.64 | (47) ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v1) = v2) | ? [v3] : (implies(v1, v2) = v3 & is_a_theorem(v3) = 0))
% 32.04/9.64 | (48) r5
% 32.04/9.64 | (49) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ( ~ (or(v0, v2) = v4) | ~ (or(v0, v1) = v3) | ~ (implies(v3, v4) = v5) | ? [v6] : ? [v7] : (implies(v6, v5) = v7 & implies(v1, v2) = v6 & is_a_theorem(v7) = 0))
% 32.04/9.64 | (50) ? [v0] : ? [v1] : is_a_theorem(v0) = v1
% 32.04/9.64 | (51) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (and(v3, v2) = v1) | ~ (and(v3, v2) = v0))
% 32.04/9.64 | (52) ( ~ (all_0_70_70 = 0) & equiv(all_0_75_75, all_0_74_74) = all_0_73_73 & implies(all_0_73_73, all_0_72_72) = all_0_71_71 & implies(all_0_74_74, all_0_75_75) = all_0_72_72 & is_a_theorem(all_0_71_71) = all_0_70_70 & ~ equivalence_2) | (equivalence_2 & ! [v0] : ! [v1] : ! [v2] : ( ~ (equiv(v0, v1) = v2) | ? [v3] : ? [v4] : (implies(v2, v3) = v4 & implies(v1, v0) = v3 & is_a_theorem(v4) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v1, v0) = v2) | ? [v3] : ? [v4] : (equiv(v0, v1) = v3 & implies(v3, v2) = v4 & is_a_theorem(v4) = 0)))
% 32.04/9.64 | (53) op_or
% 32.04/9.64 | (54) ( ~ (all_0_64_64 = 0) & equiv(all_0_69_69, all_0_68_68) = all_0_67_67 & implies(all_0_67_67, all_0_66_66) = all_0_65_65 & implies(all_0_69_69, all_0_68_68) = all_0_66_66 & is_a_theorem(all_0_65_65) = all_0_64_64 & ~ equivalence_1) | (equivalence_1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (equiv(v0, v1) = v2) | ? [v3] : ? [v4] : (implies(v2, v3) = v4 & implies(v0, v1) = v3 & is_a_theorem(v4) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : (equiv(v0, v1) = v3 & implies(v3, v2) = v4 & is_a_theorem(v4) = 0)))
% 32.04/9.65 | (55) ? [v0] : ? [v1] : ? [v2] : implies(v1, v0) = v2
% 32.04/9.65 | (56) r1
% 32.04/9.65 | (57) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (or(v3, v2) = v1) | ~ (or(v3, v2) = v0))
% 32.04/9.65 | (58) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (and(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) = v2) | ? [v5] : (or(v0, v1) = v5 & not(v4) = v5))
% 32.04/9.65 | (59) ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (and(v3, v4) = v5 & not(v5) = v2 & not(v1) = v4 & not(v0) = v3))
% 32.04/9.65 | (60) ? [v0] : ? [v1] : not(v0) = v1
% 32.04/9.65 | (61) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (or(v1, v3) = v4) | ~ (or(v0, v2) = v3) | ? [v5] : ? [v6] : ? [v7] : (or(v1, v2) = v5 & or(v0, v5) = v6 & implies(v6, v4) = v7 & is_a_theorem(v7) = 0))
% 32.04/9.65 | (62) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (or(v1, v2) = v3) | ~ (or(v0, v3) = v4) | ? [v5] : ? [v6] : ? [v7] : (or(v1, v5) = v6 & or(v0, v2) = v5 & implies(v4, v6) = v7 & is_a_theorem(v7) = 0))
% 32.04/9.65 | (63) r4
% 32.04/9.65 | (64) ( ~ (all_0_44_44 = 0) & or(all_0_48_48, all_0_47_47) = all_0_46_46 & implies(all_0_48_48, all_0_46_46) = all_0_45_45 & is_a_theorem(all_0_45_45) = all_0_44_44 & ~ or_1) | (or_1 & ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v1) = v2) | ? [v3] : (implies(v0, v2) = v3 & is_a_theorem(v3) = 0)))
% 32.04/9.65 | (65) op_implies_and
% 32.04/9.65 | (66) ( ~ (all_0_93_93 = 0) & and(all_0_101_101, all_0_103_103) = all_0_97_97 & and(all_0_102_102, all_0_101_101) = all_0_99_99 & not(all_0_97_97) = all_0_96_96 & not(all_0_99_99) = all_0_98_98 & implies(all_0_98_98, all_0_96_96) = all_0_95_95 & implies(all_0_100_100, all_0_95_95) = all_0_94_94 & implies(all_0_103_103, all_0_102_102) = all_0_100_100 & is_a_theorem(all_0_94_94) = all_0_93_93 & ~ kn3) | (kn3 & ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ! [v5] : ! [v6] : ! [v7] : ( ~ (and(v2, v0) = v5) | ~ (and(v1, v2) = v3) | ~ (not(v5) = v6) | ~ (not(v3) = v4) | ~ (implies(v4, v6) = v7) | ? [v8] : ? [v9] : (implies(v8, v7) = v9 & implies(v0, v1) = v8 & is_a_theorem(v9) = 0)))
% 32.04/9.65 | (67) and(all_0_87_87, all_0_87_87) = all_0_86_86
% 32.04/9.65 | (68) ( ~ (all_0_49_49 = 0) & or(all_0_53_53, all_0_52_52) = all_0_51_51 & implies(all_0_52_52, all_0_51_51) = all_0_50_50 & is_a_theorem(all_0_50_50) = all_0_49_49 & ~ or_2) | (or_2 & ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v1) = v2) | ? [v3] : (implies(v1, v2) = v3 & is_a_theorem(v3) = 0)))
% 32.04/9.65 |
% 32.04/9.65 | Instantiating formula (4) with all_0_86_86, all_0_87_87, all_0_87_87 and discharging atoms and(all_0_87_87, all_0_87_87) = all_0_86_86, yields:
% 32.04/9.65 | (69) ? [v0] : ? [v1] : ? [v2] : (or(v0, v1) = v2 & not(v2) = all_0_86_86 & not(all_0_87_87) = v1 & not(all_0_87_87) = v0)
% 32.04/9.65 |
% 32.04/9.65 | Instantiating formula (40) with all_0_85_85, all_0_86_86, all_0_87_87 and discharging atoms implies(all_0_87_87, all_0_86_86) = all_0_85_85, yields:
% 32.04/9.65 | (70) ? [v0] : (or(v0, all_0_86_86) = all_0_85_85 & not(all_0_87_87) = v0)
% 32.04/9.65 |
% 32.04/9.65 | Instantiating (69) with all_25_0_145, all_25_1_146, all_25_2_147 yields:
% 32.04/9.65 | (71) or(all_25_2_147, all_25_1_146) = all_25_0_145 & not(all_25_0_145) = all_0_86_86 & not(all_0_87_87) = all_25_1_146 & not(all_0_87_87) = all_25_2_147
% 32.04/9.65 |
% 32.04/9.65 | Applying alpha-rule on (71) yields:
% 32.04/9.65 | (72) or(all_25_2_147, all_25_1_146) = all_25_0_145
% 32.04/9.65 | (73) not(all_25_0_145) = all_0_86_86
% 32.04/9.65 | (74) not(all_0_87_87) = all_25_1_146
% 32.04/9.65 | (75) not(all_0_87_87) = all_25_2_147
% 32.04/9.65 |
% 32.04/9.65 | Instantiating (70) with all_29_0_150 yields:
% 32.04/9.65 | (76) or(all_29_0_150, all_0_86_86) = all_0_85_85 & not(all_0_87_87) = all_29_0_150
% 32.04/9.65 |
% 32.04/9.65 | Applying alpha-rule on (76) yields:
% 32.04/9.65 | (77) or(all_29_0_150, all_0_86_86) = all_0_85_85
% 32.04/9.65 | (78) not(all_0_87_87) = all_29_0_150
% 32.04/9.65 |
% 32.04/9.65 | Instantiating formula (14) with all_0_87_87, all_25_1_146, all_29_0_150 and discharging atoms not(all_0_87_87) = all_29_0_150, not(all_0_87_87) = all_25_1_146, yields:
% 32.04/9.65 | (79) all_29_0_150 = all_25_1_146
% 32.04/9.65 |
% 32.04/9.65 | Instantiating formula (14) with all_0_87_87, all_25_2_147, all_29_0_150 and discharging atoms not(all_0_87_87) = all_29_0_150, not(all_0_87_87) = all_25_2_147, yields:
% 32.04/9.65 | (80) all_29_0_150 = all_25_2_147
% 32.04/9.65 |
% 32.04/9.65 | Combining equations (79,80) yields a new equation:
% 32.04/9.65 | (81) all_25_1_146 = all_25_2_147
% 32.04/9.65 |
% 32.04/9.65 | Simplifying 81 yields:
% 32.04/9.65 | (82) all_25_1_146 = all_25_2_147
% 32.04/9.65 |
% 32.04/9.65 | From (80) and (77) follows:
% 32.04/9.65 | (83) or(all_25_2_147, all_0_86_86) = all_0_85_85
% 32.04/9.65 |
% 32.04/9.65 | From (82) and (72) follows:
% 32.04/9.65 | (84) or(all_25_2_147, all_25_2_147) = all_25_0_145
% 32.04/9.65 |
% 32.04/9.66 | Instantiating formula (39) with all_25_0_145, all_25_2_147 and discharging atoms or(all_25_2_147, all_25_2_147) = all_25_0_145, yields:
% 32.04/9.66 | (85) ? [v0] : (implies(all_25_0_145, all_25_2_147) = v0 & is_a_theorem(v0) = 0)
% 32.04/9.66 |
% 32.04/9.66 | Instantiating formula (47) with all_25_0_145, all_25_2_147, all_25_2_147 and discharging atoms or(all_25_2_147, all_25_2_147) = all_25_0_145, yields:
% 32.04/9.66 | (86) ? [v0] : (implies(all_25_2_147, all_25_0_145) = v0 & is_a_theorem(v0) = 0)
% 32.04/9.66 |
% 32.04/9.66 | Instantiating formula (19) with all_0_85_85, all_25_2_147, all_0_86_86 and discharging atoms or(all_25_2_147, all_0_86_86) = all_0_85_85, yields:
% 32.04/9.66 | (87) ? [v0] : ? [v1] : (or(all_0_86_86, all_25_2_147) = v0 & implies(v0, all_0_85_85) = v1 & is_a_theorem(v1) = 0)
% 32.04/9.66 |
% 32.04/9.66 | Instantiating formula (34) with all_0_85_85, all_0_86_86, all_25_2_147 and discharging atoms or(all_25_2_147, all_0_86_86) = all_0_85_85, yields:
% 32.04/9.66 | (88) ? [v0] : ? [v1] : (or(all_0_86_86, all_25_2_147) = v0 & implies(all_0_85_85, v0) = v1 & is_a_theorem(v1) = 0)
% 32.04/9.66 |
% 32.04/9.66 | Instantiating (88) with all_43_0_152, all_43_1_153 yields:
% 32.04/9.66 | (89) or(all_0_86_86, all_25_2_147) = all_43_1_153 & implies(all_0_85_85, all_43_1_153) = all_43_0_152 & is_a_theorem(all_43_0_152) = 0
% 32.04/9.66 |
% 32.04/9.66 | Applying alpha-rule on (89) yields:
% 32.04/9.66 | (90) or(all_0_86_86, all_25_2_147) = all_43_1_153
% 32.04/9.66 | (91) implies(all_0_85_85, all_43_1_153) = all_43_0_152
% 32.04/9.66 | (92) is_a_theorem(all_43_0_152) = 0
% 32.04/9.66 |
% 32.04/9.66 | Instantiating (87) with all_47_0_157, all_47_1_158 yields:
% 32.04/9.66 | (93) or(all_0_86_86, all_25_2_147) = all_47_1_158 & implies(all_47_1_158, all_0_85_85) = all_47_0_157 & is_a_theorem(all_47_0_157) = 0
% 32.04/9.66 |
% 32.04/9.66 | Applying alpha-rule on (93) yields:
% 32.04/9.66 | (94) or(all_0_86_86, all_25_2_147) = all_47_1_158
% 32.04/9.66 | (95) implies(all_47_1_158, all_0_85_85) = all_47_0_157
% 32.04/9.66 | (96) is_a_theorem(all_47_0_157) = 0
% 32.04/9.66 |
% 32.04/9.66 | Instantiating (86) with all_49_0_159 yields:
% 32.04/9.66 | (97) implies(all_25_2_147, all_25_0_145) = all_49_0_159 & is_a_theorem(all_49_0_159) = 0
% 32.04/9.66 |
% 32.04/9.66 | Applying alpha-rule on (97) yields:
% 32.04/9.66 | (98) implies(all_25_2_147, all_25_0_145) = all_49_0_159
% 32.04/9.66 | (99) is_a_theorem(all_49_0_159) = 0
% 32.04/9.66 |
% 32.04/9.66 | Instantiating (85) with all_70_0_179 yields:
% 32.04/9.66 | (100) implies(all_25_0_145, all_25_2_147) = all_70_0_179 & is_a_theorem(all_70_0_179) = 0
% 32.04/9.66 |
% 32.04/9.66 | Applying alpha-rule on (100) yields:
% 32.04/9.66 | (101) implies(all_25_0_145, all_25_2_147) = all_70_0_179
% 32.04/9.66 | (102) is_a_theorem(all_70_0_179) = 0
% 32.04/9.66 |
% 32.04/9.66 | Instantiating formula (57) with all_0_86_86, all_25_2_147, all_43_1_153, all_47_1_158 and discharging atoms or(all_0_86_86, all_25_2_147) = all_47_1_158, or(all_0_86_86, all_25_2_147) = all_43_1_153, yields:
% 32.04/9.66 | (103) all_47_1_158 = all_43_1_153
% 32.04/9.66 |
% 32.04/9.66 | From (103) and (94) follows:
% 32.04/9.66 | (90) or(all_0_86_86, all_25_2_147) = all_43_1_153
% 32.04/9.66 |
% 32.04/9.66 | From (103) and (95) follows:
% 32.04/9.66 | (105) implies(all_43_1_153, all_0_85_85) = all_47_0_157
% 32.04/9.66 |
% 32.04/9.66 | Instantiating formula (24) with all_43_1_153, all_0_86_86, all_25_2_147, all_25_0_145 and discharging atoms or(all_0_86_86, all_25_2_147) = all_43_1_153, not(all_25_0_145) = all_0_86_86, yields:
% 32.04/9.66 | (106) implies(all_25_0_145, all_25_2_147) = all_43_1_153
% 32.04/9.66 |
% 32.04/9.66 | Instantiating formula (8) with all_47_0_157, all_0_85_85, all_43_1_153 and discharging atoms implies(all_43_1_153, all_0_85_85) = all_47_0_157, yields:
% 32.04/9.66 | (107) ? [v0] : ((v0 = 0 & is_a_theorem(all_0_85_85) = 0) | ( ~ (v0 = 0) & is_a_theorem(all_47_0_157) = v0) | ( ~ (v0 = 0) & is_a_theorem(all_43_1_153) = v0))
% 32.04/9.66 |
% 32.04/9.66 | Instantiating formula (23) with all_49_0_159, all_25_2_147, all_25_0_145 and discharging atoms implies(all_25_2_147, all_25_0_145) = all_49_0_159, yields:
% 32.04/9.66 | (108) ? [v0] : ? [v1] : (and(v1, all_49_0_159) = v0 & equiv(all_25_0_145, all_25_2_147) = v0 & implies(all_25_0_145, all_25_2_147) = v1)
% 32.04/9.66 |
% 32.04/9.66 | Instantiating formula (41) with all_49_0_159, all_25_0_145, all_25_2_147 and discharging atoms implies(all_25_2_147, all_25_0_145) = all_49_0_159, yields:
% 32.04/9.66 | (109) ? [v0] : ? [v1] : (and(all_49_0_159, v1) = v0 & equiv(all_25_2_147, all_25_0_145) = v0 & implies(all_25_0_145, all_25_2_147) = v1)
% 32.04/9.66 |
% 32.04/9.66 | Instantiating (109) with all_147_0_234, all_147_1_235 yields:
% 32.04/9.66 | (110) and(all_49_0_159, all_147_0_234) = all_147_1_235 & equiv(all_25_2_147, all_25_0_145) = all_147_1_235 & implies(all_25_0_145, all_25_2_147) = all_147_0_234
% 32.04/9.66 |
% 32.04/9.66 | Applying alpha-rule on (110) yields:
% 32.04/9.66 | (111) and(all_49_0_159, all_147_0_234) = all_147_1_235
% 32.04/9.66 | (112) equiv(all_25_2_147, all_25_0_145) = all_147_1_235
% 32.04/9.66 | (113) implies(all_25_0_145, all_25_2_147) = all_147_0_234
% 32.04/9.66 |
% 32.04/9.66 | Instantiating (108) with all_149_0_236, all_149_1_237 yields:
% 32.04/9.66 | (114) and(all_149_0_236, all_49_0_159) = all_149_1_237 & equiv(all_25_0_145, all_25_2_147) = all_149_1_237 & implies(all_25_0_145, all_25_2_147) = all_149_0_236
% 32.04/9.66 |
% 32.04/9.66 | Applying alpha-rule on (114) yields:
% 32.04/9.66 | (115) and(all_149_0_236, all_49_0_159) = all_149_1_237
% 32.04/9.66 | (116) equiv(all_25_0_145, all_25_2_147) = all_149_1_237
% 32.04/9.66 | (117) implies(all_25_0_145, all_25_2_147) = all_149_0_236
% 32.04/9.66 |
% 32.04/9.66 | Instantiating (107) with all_209_0_292 yields:
% 32.04/9.66 | (118) (all_209_0_292 = 0 & is_a_theorem(all_0_85_85) = 0) | ( ~ (all_209_0_292 = 0) & is_a_theorem(all_47_0_157) = all_209_0_292) | ( ~ (all_209_0_292 = 0) & is_a_theorem(all_43_1_153) = all_209_0_292)
% 32.04/9.66 |
% 32.04/9.66 +-Applying beta-rule and splitting (118), into two cases.
% 32.04/9.66 |-Branch one:
% 32.04/9.66 | (119) (all_209_0_292 = 0 & is_a_theorem(all_0_85_85) = 0) | ( ~ (all_209_0_292 = 0) & is_a_theorem(all_47_0_157) = all_209_0_292)
% 32.04/9.66 |
% 32.04/9.66 +-Applying beta-rule and splitting (119), into two cases.
% 32.04/9.66 |-Branch one:
% 32.04/9.66 | (120) all_209_0_292 = 0 & is_a_theorem(all_0_85_85) = 0
% 32.04/9.66 |
% 32.04/9.66 | Applying alpha-rule on (120) yields:
% 32.04/9.66 | (121) all_209_0_292 = 0
% 32.04/9.66 | (122) is_a_theorem(all_0_85_85) = 0
% 32.04/9.66 |
% 32.04/9.66 | Instantiating formula (22) with all_0_85_85, 0, all_0_84_84 and discharging atoms is_a_theorem(all_0_85_85) = all_0_84_84, is_a_theorem(all_0_85_85) = 0, yields:
% 32.04/9.66 | (123) all_0_84_84 = 0
% 32.04/9.66 |
% 32.04/9.66 | Equations (123) can reduce 5 to:
% 32.04/9.66 | (124) $false
% 32.04/9.66 |
% 32.04/9.66 |-The branch is then unsatisfiable
% 32.04/9.66 |-Branch two:
% 32.04/9.66 | (125) ~ (all_209_0_292 = 0) & is_a_theorem(all_47_0_157) = all_209_0_292
% 32.04/9.66 |
% 32.04/9.66 | Applying alpha-rule on (125) yields:
% 32.04/9.66 | (126) ~ (all_209_0_292 = 0)
% 32.04/9.66 | (127) is_a_theorem(all_47_0_157) = all_209_0_292
% 32.04/9.66 |
% 32.04/9.66 | Instantiating formula (22) with all_47_0_157, all_209_0_292, 0 and discharging atoms is_a_theorem(all_47_0_157) = all_209_0_292, is_a_theorem(all_47_0_157) = 0, yields:
% 32.04/9.66 | (121) all_209_0_292 = 0
% 32.04/9.66 |
% 32.04/9.66 | Equations (121) can reduce 126 to:
% 32.04/9.66 | (124) $false
% 32.04/9.66 |
% 32.04/9.66 |-The branch is then unsatisfiable
% 32.04/9.66 |-Branch two:
% 32.04/9.66 | (130) ~ (all_209_0_292 = 0) & is_a_theorem(all_43_1_153) = all_209_0_292
% 32.04/9.66 |
% 32.04/9.66 | Applying alpha-rule on (130) yields:
% 32.04/9.66 | (126) ~ (all_209_0_292 = 0)
% 32.04/9.66 | (132) is_a_theorem(all_43_1_153) = all_209_0_292
% 32.04/9.66 |
% 32.04/9.66 | Instantiating formula (18) with all_25_0_145, all_25_2_147, all_149_0_236, all_70_0_179 and discharging atoms implies(all_25_0_145, all_25_2_147) = all_149_0_236, implies(all_25_0_145, all_25_2_147) = all_70_0_179, yields:
% 32.04/9.66 | (133) all_149_0_236 = all_70_0_179
% 32.04/9.66 |
% 32.04/9.66 | Instantiating formula (18) with all_25_0_145, all_25_2_147, all_147_0_234, all_149_0_236 and discharging atoms implies(all_25_0_145, all_25_2_147) = all_149_0_236, implies(all_25_0_145, all_25_2_147) = all_147_0_234, yields:
% 32.04/9.66 | (134) all_149_0_236 = all_147_0_234
% 32.04/9.66 |
% 32.04/9.66 | Instantiating formula (18) with all_25_0_145, all_25_2_147, all_43_1_153, all_149_0_236 and discharging atoms implies(all_25_0_145, all_25_2_147) = all_149_0_236, implies(all_25_0_145, all_25_2_147) = all_43_1_153, yields:
% 32.04/9.66 | (135) all_149_0_236 = all_43_1_153
% 32.04/9.66 |
% 32.04/9.66 | Combining equations (133,134) yields a new equation:
% 32.04/9.66 | (136) all_147_0_234 = all_70_0_179
% 32.04/9.66 |
% 32.04/9.66 | Combining equations (135,134) yields a new equation:
% 32.04/9.66 | (137) all_147_0_234 = all_43_1_153
% 32.04/9.66 |
% 32.04/9.66 | Combining equations (137,136) yields a new equation:
% 32.04/9.66 | (138) all_70_0_179 = all_43_1_153
% 32.04/9.66 |
% 32.04/9.66 | From (138) and (102) follows:
% 32.04/9.66 | (139) is_a_theorem(all_43_1_153) = 0
% 32.04/9.66 |
% 32.04/9.66 | Instantiating formula (22) with all_43_1_153, 0, all_209_0_292 and discharging atoms is_a_theorem(all_43_1_153) = all_209_0_292, is_a_theorem(all_43_1_153) = 0, yields:
% 32.04/9.67 | (121) all_209_0_292 = 0
% 32.04/9.67 |
% 32.04/9.67 | Equations (121) can reduce 126 to:
% 32.04/9.67 | (124) $false
% 32.04/9.67 |
% 32.04/9.67 |-The branch is then unsatisfiable
% 32.04/9.67 % SZS output end Proof for theBenchmark
% 32.04/9.67
% 32.04/9.67 9058ms
%------------------------------------------------------------------------------