TSTP Solution File: LCL459+1 by ePrincess---1.0
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%------------------------------------------------------------------------------
% File : ePrincess---1.0
% Problem : LCL459+1 : TPTP v8.1.0. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : ePrincess-casc -timeout=%d %s
% Computer : n005.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 09:36:39 EDT 2022
% Result : Theorem 26.11s 8.81s
% Output : Proof 35.33s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : LCL459+1 : TPTP v8.1.0. Released v3.3.0.
% 0.04/0.12 % Command : ePrincess-casc -timeout=%d %s
% 0.13/0.33 % Computer : n005.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 16:40:22 EDT 2022
% 0.13/0.33 % CPUTime :
% 0.67/0.63 ____ _
% 0.67/0.63 ___ / __ \_____(_)___ ________ __________
% 0.67/0.63 / _ \/ /_/ / ___/ / __ \/ ___/ _ \/ ___/ ___/
% 0.67/0.63 / __/ ____/ / / / / / / /__/ __(__ |__ )
% 0.67/0.63 \___/_/ /_/ /_/_/ /_/\___/\___/____/____/
% 0.67/0.63
% 0.67/0.63 A Theorem Prover for First-Order Logic
% 0.67/0.63 (ePrincess v.1.0)
% 0.67/0.63
% 0.67/0.63 (c) Philipp Rümmer, 2009-2015
% 0.67/0.63 (c) Peter Backeman, 2014-2015
% 0.67/0.63 (contributions by Angelo Brillout, Peter Baumgartner)
% 0.67/0.63 Free software under GNU Lesser General Public License (LGPL).
% 0.67/0.63 Bug reports to peter@backeman.se
% 0.67/0.63
% 0.67/0.63 For more information, visit http://user.uu.se/~petba168/breu/
% 0.67/0.63
% 0.67/0.63 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 0.78/0.68 Prover 0: Options: -triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all
% 1.80/1.01 Prover 0: Preprocessing ...
% 3.24/1.37 Prover 0: Constructing countermodel ...
% 14.21/5.97 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -resolutionMethod=normal +ignoreQuantifiers -generateTriggers=all
% 14.37/6.02 Prover 1: Preprocessing ...
% 15.11/6.17 Prover 1: Constructing countermodel ...
% 25.04/8.57 Prover 2: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -resolutionMethod=nonUnifying +ignoreQuantifiers -generateTriggers=all
% 25.25/8.61 Prover 2: Preprocessing ...
% 25.44/8.72 Prover 2: Warning: ignoring some quantifiers
% 25.44/8.72 Prover 2: Constructing countermodel ...
% 26.11/8.80 Prover 2: proved (236ms)
% 26.11/8.81 Prover 0: stopped
% 26.11/8.81 Prover 1: stopped
% 26.11/8.81
% 26.11/8.81 No countermodel exists, formula is valid
% 26.11/8.81 % SZS status Theorem for theBenchmark
% 26.11/8.81
% 26.11/8.81 Generating proof ... Warning: ignoring some quantifiers
% 34.95/11.70 found it (size 28)
% 34.95/11.70
% 34.95/11.70 % SZS output start Proof for theBenchmark
% 34.95/11.70 Assumed formulas after preprocessing and simplification:
% 34.95/11.70 | (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] : ( ~ (v72 = 0) & and(v69, v69) = v70 & implies(v69, v70) = v71 & is_a_theorem(v71) = v72 & 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 & ~ kn1 & ! [v73] : ! [v74] : ! [v75] : ! [v76] : ! [v77] : ! [v78] : ( ~ (implies(v76, v77) = v78) | ~ (implies(v74, v75) = v76) | ~ (implies(v73, v75) = v77) | ? [v79] : ? [v80] : (implies(v79, v78) = v80 & implies(v73, v74) = v79 & is_a_theorem(v80) = 0)) & ! [v73] : ! [v74] : ! [v75] : ! [v76] : ! [v77] : ( ~ (or(v73, v74) = v76) | ~ (implies(v76, v75) = v77) | ? [v78] : ? [v79] : ? [v80] : ? [v81] : (implies(v79, v77) = v80 & implies(v78, v80) = v81 & implies(v74, v75) = v79 & implies(v73, v75) = v78 & is_a_theorem(v81) = 0)) & ! [v73] : ! [v74] : ! [v75] : ! [v76] : ! [v77] : ( ~ (and(v75, v76) = v77) | ~ (not(v74) = v76) | ~ (not(v73) = v75) | ? [v78] : (or(v73, v74) = v78 & not(v77) = v78)) & ! [v73] : ! [v74] : ! [v75] : ! [v76] : ! [v77] : ( ~ (not(v74) = v75) | ~ (not(v73) = v76) | ~ (implies(v75, v76) = v77) | ? [v78] : ? [v79] : (implies(v77, v78) = v79 & implies(v73, v74) = v78 & is_a_theorem(v79) = 0)) & ! [v73] : ! [v74] : ! [v75] : ! [v76] : (v74 = v73 | ~ (or(v76, v75) = v74) | ~ (or(v76, v75) = v73)) & ! [v73] : ! [v74] : ! [v75] : ! [v76] : (v74 = v73 | ~ (and(v76, v75) = v74) | ~ (and(v76, v75) = v73)) & ! [v73] : ! [v74] : ! [v75] : ! [v76] : (v74 = v73 | ~ (equiv(v76, v75) = v74) | ~ (equiv(v76, v75) = v73)) & ! [v73] : ! [v74] : ! [v75] : ! [v76] : (v74 = v73 | ~ (implies(v76, v75) = v74) | ~ (implies(v76, v75) = v73)) & ! [v73] : ! [v74] : ! [v75] : ! [v76] : ( ~ (and(v73, v75) = v76) | ~ (not(v74) = v75) | ? [v77] : (not(v76) = v77 & implies(v73, v74) = v77)) & ! [v73] : ! [v74] : ! [v75] : (v74 = v73 | ~ (not(v75) = v74) | ~ (not(v75) = v73)) & ! [v73] : ! [v74] : ! [v75] : (v74 = v73 | ~ (equiv(v73, v74) = v75) | ? [v76] : ( ~ (v76 = 0) & is_a_theorem(v75) = v76)) & ! [v73] : ! [v74] : ! [v75] : (v74 = v73 | ~ (is_a_theorem(v75) = v74) | ~ (is_a_theorem(v75) = v73)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (or(v73, v74) = v75) | ? [v76] : ? [v77] : ? [v78] : (and(v76, v77) = v78 & not(v78) = v75 & not(v74) = v77 & not(v73) = v76)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (or(v73, v74) = v75) | ? [v76] : (implies(v74, v75) = v76 & is_a_theorem(v76) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (or(v73, v74) = v75) | ? [v76] : (implies(v73, v75) = v76 & is_a_theorem(v76) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (and(v73, v74) = v75) | ? [v76] : ? [v77] : (implies(v74, v75) = v76 & implies(v73, v76) = v77 & is_a_theorem(v77) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (and(v73, v74) = v75) | ? [v76] : (implies(v75, v74) = v76 & is_a_theorem(v76) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (and(v73, v74) = v75) | ? [v76] : (implies(v75, v73) = v76 & is_a_theorem(v76) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (equiv(v73, v74) = v75) | ? [v76] : ? [v77] : ? [v78] : ? [v79] : (implies(v77, v75) = v78 & implies(v76, v78) = v79 & implies(v74, v73) = v77 & implies(v73, v74) = v76 & is_a_theorem(v79) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (equiv(v73, v74) = v75) | ? [v76] : ? [v77] : (and(v76, v77) = v75 & implies(v74, v73) = v77 & implies(v73, v74) = v76)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (equiv(v73, v74) = v75) | ? [v76] : ? [v77] : (implies(v75, v76) = v77 & implies(v74, v73) = v76 & is_a_theorem(v77) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (equiv(v73, v74) = v75) | ? [v76] : ? [v77] : (implies(v75, v76) = v77 & implies(v73, v74) = v76 & is_a_theorem(v77) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (implies(v74, v73) = v75) | ? [v76] : ? [v77] : ? [v78] : ? [v79] : (equiv(v73, v74) = v77 & implies(v76, v78) = v79 & implies(v75, v77) = v78 & implies(v73, v74) = v76 & is_a_theorem(v79) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (implies(v74, v73) = v75) | ? [v76] : ? [v77] : (and(v77, v75) = v76 & equiv(v73, v74) = v76 & implies(v73, v74) = v77)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (implies(v74, v73) = v75) | ? [v76] : ? [v77] : (equiv(v73, v74) = v76 & implies(v76, v75) = v77 & is_a_theorem(v77) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (implies(v74, v73) = v75) | ? [v76] : (implies(v73, v75) = v76 & is_a_theorem(v76) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (implies(v73, v74) = v75) | ? [v76] : ? [v77] : ? [v78] : ? [v79] : (not(v74) = v76 & not(v73) = v77 & implies(v78, v75) = v79 & implies(v76, v77) = v78 & is_a_theorem(v79) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (implies(v73, v74) = v75) | ? [v76] : ? [v77] : ? [v78] : ? [v79] : (equiv(v73, v74) = v77 & implies(v76, v77) = v78 & implies(v75, v78) = v79 & implies(v74, v73) = v76 & is_a_theorem(v79) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (implies(v73, v74) = v75) | ? [v76] : ? [v77] : (and(v75, v77) = v76 & equiv(v73, v74) = v76 & implies(v74, v73) = v77)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (implies(v73, v74) = v75) | ? [v76] : ? [v77] : (and(v73, v76) = v77 & not(v77) = v75 & not(v74) = v76)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (implies(v73, v74) = v75) | ? [v76] : ? [v77] : (equiv(v73, v74) = v76 & implies(v76, v75) = v77 & is_a_theorem(v77) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (implies(v73, v74) = v75) | ? [v76] : ? [v77] : (implies(v76, v75) = v77 & implies(v73, v75) = v76 & is_a_theorem(v77) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (implies(v73, v74) = v75) | ? [v76] : ((v76 = 0 & is_a_theorem(v74) = 0) | ( ~ (v76 = 0) & is_a_theorem(v75) = v76) | ( ~ (v76 = 0) & is_a_theorem(v73) = v76))) & ? [v73] : ? [v74] : ? [v75] : or(v74, v73) = v75 & ? [v73] : ? [v74] : ? [v75] : and(v74, v73) = v75 & ? [v73] : ? [v74] : ? [v75] : equiv(v74, v73) = v75 & ? [v73] : ? [v74] : ? [v75] : implies(v74, v73) = v75 & ? [v73] : ? [v74] : not(v73) = v74 & ? [v73] : ? [v74] : is_a_theorem(v73) = v74 & ( ~ op_implies_or | ( ! [v73] : ! [v74] : ! [v75] : ! [v76] : ( ~ (or(v75, v74) = v76) | ~ (not(v73) = v75) | implies(v73, v74) = v76) & ! [v73] : ! [v74] : ! [v75] : ( ~ (implies(v73, v74) = v75) | ? [v76] : (or(v76, v74) = v75 & not(v73) = v76)))) & ( ~ op_and | ( ! [v73] : ! [v74] : ! [v75] : ! [v76] : ! [v77] : ( ~ (or(v75, v76) = v77) | ~ (not(v74) = v76) | ~ (not(v73) = v75) | ? [v78] : (and(v73, v74) = v78 & not(v77) = v78)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (and(v73, v74) = v75) | ? [v76] : ? [v77] : ? [v78] : (or(v76, v77) = v78 & not(v78) = v75 & not(v74) = v77 & not(v73) = v76)))) & (( ~ (v68 = 0) & and(v64, v65) = v66 & implies(v66, v64) = v67 & is_a_theorem(v67) = v68 & ~ kn2) | (kn2 & ! [v73] : ! [v74] : ! [v75] : ( ~ (and(v73, v74) = v75) | ? [v76] : (implies(v75, v73) = v76 & is_a_theorem(v76) = 0)))) & (( ~ (v63 = 0) & and(v55, v53) = v59 & and(v54, v55) = v57 & not(v59) = v60 & not(v57) = v58 & implies(v58, v60) = v61 & implies(v56, v61) = v62 & implies(v53, v54) = v56 & is_a_theorem(v62) = v63 & ~ kn3) | (kn3 & ! [v73] : ! [v74] : ! [v75] : ! [v76] : ! [v77] : ! [v78] : ! [v79] : ! [v80] : ( ~ (and(v75, v73) = v78) | ~ (and(v74, v75) = v76) | ~ (not(v78) = v79) | ~ (not(v76) = v77) | ~ (implies(v77, v79) = v80) | ? [v81] : ? [v82] : (implies(v81, v80) = v82 & implies(v73, v74) = v81 & is_a_theorem(v82) = 0)))) & (( ~ (v52 = 0) & implies(v48, v49) = v50 & implies(v47, v50) = v51 & implies(v45, v46) = v48 & implies(v44, v46) = v49 & implies(v44, v45) = v47 & is_a_theorem(v51) = v52 & ~ cn1) | (cn1 & ! [v73] : ! [v74] : ! [v75] : ! [v76] : ! [v77] : ! [v78] : ( ~ (implies(v76, v77) = v78) | ~ (implies(v74, v75) = v76) | ~ (implies(v73, v75) = v77) | ? [v79] : ? [v80] : (implies(v79, v78) = v80 & implies(v73, v74) = v79 & is_a_theorem(v80) = 0)))) & (( ~ (v43 = 0) & not(v38) = v40 & implies(v40, v39) = v41 & implies(v38, v41) = v42 & is_a_theorem(v42) = v43 & ~ cn2) | (cn2 & ! [v73] : ! [v74] : ! [v75] : ! [v76] : ( ~ (not(v73) = v75) | ~ (implies(v75, v74) = v76) | ? [v77] : (implies(v73, v76) = v77 & is_a_theorem(v77) = 0)))) & (( ~ (v37 = 0) & not(v33) = v34 & implies(v35, v33) = v36 & implies(v34, v33) = v35 & is_a_theorem(v36) = v37 & ~ cn3) | (cn3 & ! [v73] : ! [v74] : ( ~ (not(v73) = v74) | ? [v75] : ? [v76] : (implies(v75, v73) = v76 & implies(v74, v73) = v75 & is_a_theorem(v76) = 0)))) & (( ~ (v32 = 0) & or(v29, v29) = v30 & implies(v30, v29) = v31 & is_a_theorem(v31) = v32 & ~ r1) | (r1 & ! [v73] : ! [v74] : ( ~ (or(v73, v73) = v74) | ? [v75] : (implies(v74, v73) = v75 & is_a_theorem(v75) = 0)))) & (( ~ (v28 = 0) & or(v24, v25) = v26 & implies(v25, v26) = v27 & is_a_theorem(v27) = v28 & ~ r2) | (r2 & ! [v73] : ! [v74] : ! [v75] : ( ~ (or(v73, v74) = v75) | ? [v76] : (implies(v74, v75) = v76 & is_a_theorem(v76) = 0)))) & (( ~ (v23 = 0) & or(v19, v18) = v21 & or(v18, v19) = v20 & implies(v20, v21) = v22 & is_a_theorem(v22) = v23 & ~ r3) | (r3 & ! [v73] : ! [v74] : ! [v75] : ( ~ (or(v74, v73) = v75) | ? [v76] : ? [v77] : (or(v73, v74) = v76 & implies(v76, v75) = v77 & is_a_theorem(v77) = 0)) & ! [v73] : ! [v74] : ! [v75] : ( ~ (or(v73, v74) = v75) | ? [v76] : ? [v77] : (or(v74, v73) = v76 & implies(v75, v76) = v77 & is_a_theorem(v77) = 0)))) & (( ~ (v17 = 0) & or(v10, v14) = v15 & or(v10, v11) = v12 & or(v9, v12) = v13 & or(v9, v11) = v14 & implies(v13, v15) = v16 & is_a_theorem(v16) = v17 & ~ r4) | (r4 & ! [v73] : ! [v74] : ! [v75] : ! [v76] : ! [v77] : ( ~ (or(v74, v76) = v77) | ~ (or(v73, v75) = v76) | ? [v78] : ? [v79] : ? [v80] : (or(v74, v75) = v78 & or(v73, v78) = v79 & implies(v79, v77) = v80 & is_a_theorem(v80) = 0)) & ! [v73] : ! [v74] : ! [v75] : ! [v76] : ! [v77] : ( ~ (or(v74, v75) = v76) | ~ (or(v73, v76) = v77) | ? [v78] : ? [v79] : ? [v80] : (or(v74, v78) = v79 & or(v73, v75) = v78 & implies(v77, v79) = v80 & is_a_theorem(v80) = 0)))) & (( ~ (v8 = 0) & or(v0, v2) = v5 & or(v0, v1) = v4 & implies(v4, v5) = v6 & implies(v3, v6) = v7 & implies(v1, v2) = v3 & is_a_theorem(v7) = v8 & ~ r5) | (r5 & ! [v73] : ! [v74] : ! [v75] : ! [v76] : ! [v77] : ! [v78] : ( ~ (or(v73, v75) = v77) | ~ (or(v73, v74) = v76) | ~ (implies(v76, v77) = v78) | ? [v79] : ? [v80] : (implies(v79, v78) = v80 & implies(v74, v75) = v79 & is_a_theorem(v80) = 0)))))
% 35.33/11.77 | 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 yields:
% 35.33/11.77 | (1) ~ (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 & 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 & ~ kn1 & ! [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)) & ! [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)) & ! [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] : ( ~ (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] : ! [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] : ( ~ (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(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] : (implies(v1, v2) = v3 & is_a_theorem(v3) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v1) = v2) | ? [v3] : (implies(v0, v2) = v3 & is_a_theorem(v3) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : ? [v4] : (implies(v1, v2) = v3 & implies(v0, v3) = v4 & is_a_theorem(v4) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : (implies(v2, v1) = v3 & is_a_theorem(v3) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : (implies(v2, v0) = v3 & is_a_theorem(v3) = 0)) & ! [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] : ( ~ (equiv(v0, v1) = v2) | ? [v3] : ? [v4] : (and(v3, v4) = v2 & implies(v1, v0) = v4 & implies(v0, v1) = v3)) & ! [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] : ( ~ (equiv(v0, v1) = v2) | ? [v3] : ? [v4] : (implies(v2, v3) = v4 & implies(v0, v1) = v3 & is_a_theorem(v4) = 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(v1, v0) = v2) | ? [v3] : ? [v4] : (and(v4, v2) = v3 & equiv(v0, v1) = v3 & implies(v0, v1) = v4)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v1, v0) = v2) | ? [v3] : ? [v4] : (equiv(v0, v1) = v3 & implies(v3, v2) = v4 & is_a_theorem(v4) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v1, v0) = v2) | ? [v3] : (implies(v0, v2) = v3 & is_a_theorem(v3) = 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)) & ! [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)) & ! [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] : ? [v4] : (equiv(v0, v1) = v3 & implies(v3, v2) = v4 & is_a_theorem(v4) = 0)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : (implies(v3, v2) = v4 & implies(v0, v2) = v3 & is_a_theorem(v4) = 0)) & ! [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] : ? [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 & ( ~ op_implies_or | ( ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v2, v1) = v3) | ~ (not(v0) = v2) | implies(v0, v1) = v3) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : (or(v3, v1) = v2 & not(v0) = v3)))) & ( ~ op_and | ( ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (or(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) = v2) | ? [v5] : (and(v0, v1) = v5 & not(v4) = v5)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (or(v3, v4) = v5 & not(v5) = v2 & not(v1) = v4 & not(v0) = v3)))) & (( ~ (all_0_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] : ( ~ (and(v0, v1) = v2) | ? [v3] : (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] : ( ~ (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_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] : ( ~ (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_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] : ( ~ (not(v0) = v2) | ~ (implies(v2, v1) = v3) | ? [v4] : (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] : ( ~ (not(v0) = v1) | ? [v2] : ? [v3] : (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] : ( ~ (or(v0, v0) = v1) | ? [v2] : (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] : ( ~ (or(v0, v1) = v2) | ? [v3] : (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] : ( ~ (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] : (or(v1, v0) = v3 & 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] : ( ~ (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)))) & (( ~ (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] : ( ~ (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))))
% 35.33/11.79 |
% 35.33/11.79 | Applying alpha-rule on (1) yields:
% 35.33/11.79 | (2) ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v1, v0) = v2) | ? [v3] : ? [v4] : (and(v4, v2) = v3 & equiv(v0, v1) = v3 & implies(v0, v1) = v4))
% 35.33/11.79 | (3) ( ~ (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] : ( ~ (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)))
% 35.33/11.79 | (4) ? [v0] : ? [v1] : ? [v2] : equiv(v1, v0) = v2
% 35.33/11.79 | (5) and(all_0_3_3, all_0_3_3) = all_0_2_2
% 35.33/11.79 | (6) ! [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)))
% 35.33/11.79 | (7) equivalence_1
% 35.33/11.79 | (8) ? [v0] : ? [v1] : ? [v2] : or(v1, v0) = v2
% 35.33/11.79 | (9) modus_ponens
% 35.33/11.79 | (10) ( ~ (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] : ( ~ (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)))
% 35.33/11.79 | (11) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (and(v3, v2) = v1) | ~ (and(v3, v2) = v0))
% 35.33/11.79 | (12) implies_3
% 35.33/11.79 | (13) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (implies(v3, v2) = v1) | ~ (implies(v3, v2) = v0))
% 35.33/11.79 | (14) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (and(v0, v2) = v3) | ~ (not(v1) = v2) | ? [v4] : (not(v3) = v4 & implies(v0, v1) = v4))
% 35.33/11.79 | (15) ~ kn1
% 35.33/11.79 | (16) ! [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))
% 35.33/11.79 | (17) substitution_of_equivalents
% 35.33/11.79 | (18) ! [v0] : ! [v1] : ! [v2] : ( ~ (equiv(v0, v1) = v2) | ? [v3] : ? [v4] : (implies(v2, v3) = v4 & implies(v0, v1) = v3 & is_a_theorem(v4) = 0))
% 35.33/11.79 | (19) implies_1
% 35.33/11.79 | (20) ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v1, v0) = v2) | ? [v3] : ? [v4] : (equiv(v0, v1) = v3 & implies(v3, v2) = v4 & is_a_theorem(v4) = 0))
% 35.33/11.79 | (21) modus_tollens
% 35.33/11.79 | (22) ! [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))
% 35.33/11.79 | (23) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (or(v3, v2) = v1) | ~ (or(v3, v2) = v0))
% 35.33/11.79 | (24) ( ~ (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] : ( ~ (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)))
% 35.33/11.79 | (25) ! [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))
% 35.33/11.80 | (26) op_equiv
% 35.33/11.80 | (27) ( ~ (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] : ( ~ (or(v0, v0) = v1) | ? [v2] : (implies(v1, v0) = v2 & is_a_theorem(v2) = 0)))
% 35.33/11.80 | (28) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (equiv(v0, v1) = v2) | ? [v3] : ( ~ (v3 = 0) & is_a_theorem(v2) = v3))
% 35.33/11.80 | (29) ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v1) = v2) | ? [v3] : (implies(v1, v2) = v3 & is_a_theorem(v3) = 0))
% 35.33/11.80 | (30) ? [v0] : ? [v1] : not(v0) = v1
% 35.33/11.80 | (31) is_a_theorem(all_0_1_1) = all_0_0_0
% 35.33/11.80 | (32) ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (and(v3, v4) = v5 & not(v5) = v2 & not(v1) = v4 & not(v0) = v3))
% 35.33/11.80 | (33) ( ~ (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] : ( ~ (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] : (or(v1, v0) = v3 & implies(v2, v3) = v4 & is_a_theorem(v4) = 0)))
% 35.33/11.80 | (34) ! [v0] : ! [v1] : ! [v2] : ( ~ (equiv(v0, v1) = v2) | ? [v3] : ? [v4] : (implies(v2, v3) = v4 & implies(v1, v0) = v3 & is_a_theorem(v4) = 0))
% 35.33/11.80 | (35) or_1
% 35.33/11.80 | (36) ? [v0] : ? [v1] : ? [v2] : implies(v1, v0) = v2
% 35.33/11.80 | (37) and_2
% 35.33/11.80 | (38) ? [v0] : ? [v1] : ? [v2] : and(v1, v0) = v2
% 35.33/11.80 | (39) and_1
% 35.33/11.80 | (40) implies(all_0_3_3, all_0_2_2) = all_0_1_1
% 35.33/11.80 | (41) ! [v0] : ! [v1] : ! [v2] : ( ~ (or(v0, v1) = v2) | ? [v3] : (implies(v0, v2) = v3 & is_a_theorem(v3) = 0))
% 35.33/11.80 | (42) ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : (and(v0, v3) = v4 & not(v4) = v2 & not(v1) = v3))
% 35.33/11.80 | (43) ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : (equiv(v0, v1) = v3 & implies(v3, v2) = v4 & is_a_theorem(v4) = 0))
% 35.33/11.80 | (44) ( ~ (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] : ( ~ (not(v0) = v2) | ~ (implies(v2, v1) = v3) | ? [v4] : (implies(v0, v3) = v4 & is_a_theorem(v4) = 0)))
% 35.33/11.80 | (45) ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v1, v0) = v2) | ? [v3] : (implies(v0, v2) = v3 & is_a_theorem(v3) = 0))
% 35.33/11.80 | (46) ! [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))
% 35.33/11.80 | (47) ! [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))
% 35.33/11.80 | (48) ~ op_implies_or | ( ! [v0] : ! [v1] : ! [v2] : ! [v3] : ( ~ (or(v2, v1) = v3) | ~ (not(v0) = v2) | implies(v0, v1) = v3) & ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : (or(v3, v1) = v2 & not(v0) = v3)))
% 35.33/11.80 | (49) ! [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))
% 35.33/11.80 | (50) ( ~ (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] : ( ~ (not(v0) = v1) | ? [v2] : ? [v3] : (implies(v2, v0) = v3 & implies(v1, v0) = v2 & is_a_theorem(v3) = 0)))
% 35.33/11.80 | (51) ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : (implies(v3, v2) = v4 & implies(v0, v2) = v3 & is_a_theorem(v4) = 0))
% 35.33/11.80 | (52) ! [v0] : ! [v1] : ! [v2] : ( ~ (equiv(v0, v1) = v2) | ? [v3] : ? [v4] : (and(v3, v4) = v2 & implies(v1, v0) = v4 & implies(v0, v1) = v3))
% 35.33/11.80 | (53) equivalence_3
% 35.33/11.80 | (54) ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : ? [v4] : (implies(v1, v2) = v3 & implies(v0, v3) = v4 & is_a_theorem(v4) = 0))
% 35.33/11.80 | (55) ( ~ (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] : ( ~ (or(v0, v1) = v2) | ? [v3] : (implies(v1, v2) = v3 & is_a_theorem(v3) = 0)))
% 35.33/11.80 | (56) ! [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))
% 35.33/11.81 | (57) op_or
% 35.33/11.81 | (58) ( ~ (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] : ( ~ (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)))
% 35.33/11.81 | (59) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (is_a_theorem(v2) = v1) | ~ (is_a_theorem(v2) = v0))
% 35.33/11.81 | (60) implies_2
% 35.33/11.81 | (61) ( ~ (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] : ( ~ (and(v0, v1) = v2) | ? [v3] : (implies(v2, v0) = v3 & is_a_theorem(v3) = 0)))
% 35.33/11.81 | (62) ! [v0] : ! [v1] : ! [v2] : ( ~ (implies(v0, v1) = v2) | ? [v3] : ? [v4] : (and(v2, v4) = v3 & equiv(v0, v1) = v3 & implies(v1, v0) = v4))
% 35.33/11.81 | (63) and_3
% 35.33/11.81 | (64) ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : (implies(v2, v0) = v3 & is_a_theorem(v3) = 0))
% 35.33/11.81 | (65) ~ (all_0_0_0 = 0)
% 35.33/11.81 | (66) ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (and(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) = v2) | ? [v5] : (or(v0, v1) = v5 & not(v4) = v5))
% 35.33/11.81 | (67) ! [v0] : ! [v1] : ! [v2] : ! [v3] : (v1 = v0 | ~ (equiv(v3, v2) = v1) | ~ (equiv(v3, v2) = v0))
% 35.33/11.81 | (68) or_3
% 35.33/11.81 | (69) ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : (implies(v2, v1) = v3 & is_a_theorem(v3) = 0))
% 35.33/11.81 | (70) or_2
% 35.33/11.81 | (71) equivalence_2
% 35.33/11.81 | (72) ! [v0] : ! [v1] : ! [v2] : (v1 = v0 | ~ (not(v2) = v1) | ~ (not(v2) = v0))
% 35.33/11.81 | (73) ~ op_and | ( ! [v0] : ! [v1] : ! [v2] : ! [v3] : ! [v4] : ( ~ (or(v2, v3) = v4) | ~ (not(v1) = v3) | ~ (not(v0) = v2) | ? [v5] : (and(v0, v1) = v5 & not(v4) = v5)) & ! [v0] : ! [v1] : ! [v2] : ( ~ (and(v0, v1) = v2) | ? [v3] : ? [v4] : ? [v5] : (or(v3, v4) = v5 & not(v5) = v2 & not(v1) = v4 & not(v0) = v3)))
% 35.33/11.81 | (74) op_implies_and
% 35.33/11.81 | (75) ? [v0] : ? [v1] : is_a_theorem(v0) = v1
% 35.33/11.81 |
% 35.33/11.81 | Instantiating formula (54) with all_0_2_2, all_0_3_3, all_0_3_3 and discharging atoms and(all_0_3_3, all_0_3_3) = all_0_2_2, yields:
% 35.33/11.81 | (76) ? [v0] : ? [v1] : (implies(all_0_3_3, v0) = v1 & implies(all_0_3_3, all_0_2_2) = v0 & is_a_theorem(v1) = 0)
% 35.33/11.81 |
% 35.33/11.81 | Instantiating formula (51) with all_0_1_1, all_0_2_2, all_0_3_3 and discharging atoms implies(all_0_3_3, all_0_2_2) = all_0_1_1, yields:
% 35.33/11.81 | (77) ? [v0] : ? [v1] : (implies(v0, all_0_1_1) = v1 & implies(all_0_3_3, all_0_1_1) = v0 & is_a_theorem(v1) = 0)
% 35.33/11.81 |
% 35.33/11.81 | Instantiating (77) with all_21_0_90, all_21_1_91 yields:
% 35.33/11.81 | (78) implies(all_21_1_91, all_0_1_1) = all_21_0_90 & implies(all_0_3_3, all_0_1_1) = all_21_1_91 & is_a_theorem(all_21_0_90) = 0
% 35.33/11.81 |
% 35.33/11.81 | Applying alpha-rule on (78) yields:
% 35.33/11.81 | (79) implies(all_21_1_91, all_0_1_1) = all_21_0_90
% 35.33/11.81 | (80) implies(all_0_3_3, all_0_1_1) = all_21_1_91
% 35.33/11.81 | (81) is_a_theorem(all_21_0_90) = 0
% 35.33/11.81 |
% 35.33/11.81 | Instantiating (76) with all_39_0_110, all_39_1_111 yields:
% 35.33/11.81 | (82) implies(all_0_3_3, all_39_1_111) = all_39_0_110 & implies(all_0_3_3, all_0_2_2) = all_39_1_111 & is_a_theorem(all_39_0_110) = 0
% 35.33/11.81 |
% 35.33/11.81 | Applying alpha-rule on (82) yields:
% 35.33/11.81 | (83) implies(all_0_3_3, all_39_1_111) = all_39_0_110
% 35.33/11.81 | (84) implies(all_0_3_3, all_0_2_2) = all_39_1_111
% 35.33/11.81 | (85) is_a_theorem(all_39_0_110) = 0
% 35.33/11.81 |
% 35.33/11.81 | Instantiating formula (13) with all_0_3_3, all_0_2_2, all_39_1_111, all_0_1_1 and discharging atoms implies(all_0_3_3, all_0_2_2) = all_39_1_111, implies(all_0_3_3, all_0_2_2) = all_0_1_1, yields:
% 35.33/11.81 | (86) all_39_1_111 = all_0_1_1
% 35.33/11.81 |
% 35.33/11.81 | From (86) and (83) follows:
% 35.33/11.81 | (87) implies(all_0_3_3, all_0_1_1) = all_39_0_110
% 35.33/11.81 |
% 35.33/11.81 | Instantiating formula (13) with all_0_3_3, all_0_1_1, all_39_0_110, all_21_1_91 and discharging atoms implies(all_0_3_3, all_0_1_1) = all_39_0_110, implies(all_0_3_3, all_0_1_1) = all_21_1_91, yields:
% 35.33/11.81 | (88) all_39_0_110 = all_21_1_91
% 35.33/11.81 |
% 35.33/11.81 | From (88) and (85) follows:
% 35.33/11.81 | (89) is_a_theorem(all_21_1_91) = 0
% 35.33/11.81 |
% 35.33/11.81 | Instantiating formula (6) with all_21_0_90, all_0_1_1, all_21_1_91 and discharging atoms implies(all_21_1_91, all_0_1_1) = all_21_0_90, yields:
% 35.33/11.81 | (90) ? [v0] : ((v0 = 0 & is_a_theorem(all_0_1_1) = 0) | ( ~ (v0 = 0) & is_a_theorem(all_21_0_90) = v0) | ( ~ (v0 = 0) & is_a_theorem(all_21_1_91) = v0))
% 35.33/11.82 |
% 35.33/11.82 | Instantiating (90) with all_107_0_179 yields:
% 35.33/11.82 | (91) (all_107_0_179 = 0 & is_a_theorem(all_0_1_1) = 0) | ( ~ (all_107_0_179 = 0) & is_a_theorem(all_21_0_90) = all_107_0_179) | ( ~ (all_107_0_179 = 0) & is_a_theorem(all_21_1_91) = all_107_0_179)
% 35.33/11.82 |
% 35.33/11.82 +-Applying beta-rule and splitting (91), into two cases.
% 35.33/11.82 |-Branch one:
% 35.33/11.82 | (92) (all_107_0_179 = 0 & is_a_theorem(all_0_1_1) = 0) | ( ~ (all_107_0_179 = 0) & is_a_theorem(all_21_0_90) = all_107_0_179)
% 35.33/11.82 |
% 35.33/11.82 +-Applying beta-rule and splitting (92), into two cases.
% 35.33/11.82 |-Branch one:
% 35.33/11.82 | (93) all_107_0_179 = 0 & is_a_theorem(all_0_1_1) = 0
% 35.33/11.82 |
% 35.33/11.82 | Applying alpha-rule on (93) yields:
% 35.33/11.82 | (94) all_107_0_179 = 0
% 35.33/11.82 | (95) is_a_theorem(all_0_1_1) = 0
% 35.33/11.82 |
% 35.33/11.82 | Instantiating formula (59) with all_0_1_1, 0, all_0_0_0 and discharging atoms is_a_theorem(all_0_1_1) = all_0_0_0, is_a_theorem(all_0_1_1) = 0, yields:
% 35.33/11.82 | (96) all_0_0_0 = 0
% 35.33/11.82 |
% 35.33/11.82 | Equations (96) can reduce 65 to:
% 35.33/11.82 | (97) $false
% 35.33/11.82 |
% 35.33/11.82 |-The branch is then unsatisfiable
% 35.33/11.82 |-Branch two:
% 35.33/11.82 | (98) ~ (all_107_0_179 = 0) & is_a_theorem(all_21_0_90) = all_107_0_179
% 35.33/11.82 |
% 35.33/11.82 | Applying alpha-rule on (98) yields:
% 35.33/11.82 | (99) ~ (all_107_0_179 = 0)
% 35.33/11.82 | (100) is_a_theorem(all_21_0_90) = all_107_0_179
% 35.33/11.82 |
% 35.33/11.82 | Instantiating formula (59) with all_21_0_90, all_107_0_179, 0 and discharging atoms is_a_theorem(all_21_0_90) = all_107_0_179, is_a_theorem(all_21_0_90) = 0, yields:
% 35.33/11.82 | (94) all_107_0_179 = 0
% 35.33/11.82 |
% 35.33/11.82 | Equations (94) can reduce 99 to:
% 35.33/11.82 | (97) $false
% 35.33/11.82 |
% 35.33/11.82 |-The branch is then unsatisfiable
% 35.33/11.82 |-Branch two:
% 35.33/11.82 | (103) ~ (all_107_0_179 = 0) & is_a_theorem(all_21_1_91) = all_107_0_179
% 35.33/11.82 |
% 35.33/11.82 | Applying alpha-rule on (103) yields:
% 35.33/11.82 | (99) ~ (all_107_0_179 = 0)
% 35.33/11.82 | (105) is_a_theorem(all_21_1_91) = all_107_0_179
% 35.33/11.82 |
% 35.33/11.82 | Instantiating formula (59) with all_21_1_91, all_107_0_179, 0 and discharging atoms is_a_theorem(all_21_1_91) = all_107_0_179, is_a_theorem(all_21_1_91) = 0, yields:
% 35.33/11.82 | (94) all_107_0_179 = 0
% 35.33/11.82 |
% 35.33/11.82 | Equations (94) can reduce 99 to:
% 35.33/11.82 | (97) $false
% 35.33/11.82 |
% 35.33/11.82 |-The branch is then unsatisfiable
% 35.33/11.82 % SZS output end Proof for theBenchmark
% 35.33/11.82
% 35.33/11.82 11182ms
%------------------------------------------------------------------------------