%------------------------------------------------------------------------------ % File : Princess---230619 % Problem : MGT027+1 : TPTP v9.3.0. Released v2.0.0. % Transfm : none % Format : tptp % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s % Computer : n004.cluster.edu % Model : x86_64 x86_64 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz % Memory : 8046.5625MB % OS : Linux 6.8.0-71-generic % CPULimit : 300s % WCLimit : 300s % DateTime : Tue Sep 8 06:17:22 AM UTC 2026 % Result : Theorem 22.97s 3.70s % Output : Proof 23.93s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.00/0.02 % Problem : MGT027+1 : TPTP v9.3.0. Released v2.0.0. % 0.00/0.03 % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s % 0.09/0.36 % Computer : n004.cluster.edu % 0.09/0.36 % Model : x86_64 x86_64 % 0.09/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.09/0.36 % Memory : 8046.5625MB % 0.09/0.36 % OS : Linux 6.8.0-71-generic % 0.02/0.36 % CPULimit : 300 % 0.02/0.36 % WCLimit : 300 % 0.02/0.36 % DateTime : Mon Sep 7 12:13:30 UTC 2026 % 0.02/0.36 % CPUTime : % 0.02/0.37 OpenJDK 64-Bit Server VM warning: Options -Xverify:none and -noverify were deprecated in JDK 13 and will likely be removed in a future release. % 0.54/0.57 ________ _____ % 0.54/0.57 ___ __ \_________(_)________________________________ % 0.54/0.57 __ /_/ /_ ___/_ /__ __ \ ___/ _ \_ ___/_ ___/ % 0.54/0.57 _ ____/_ / _ / _ / / / /__ / __/(__ )_(__ ) % 0.54/0.57 /_/ /_/ /_/ /_/ /_/\___/ \___//____/ /____/ % 0.54/0.57 % 0.54/0.57 A Theorem Prover for First-Order Logic modulo Linear Integer Arithmetic % 0.54/0.57 (2023-06-19) % 0.54/0.57 % 0.54/0.57 (c) Philipp Rümmer, 2009-2023 % 0.54/0.57 Contributors: Peter Backeman, Peter Baumgartner, Angelo Brillout, Zafer Esen, % 0.54/0.57 Amanda Stjerna. % 0.54/0.57 Free software under BSD-3-Clause. % 0.54/0.57 % 0.54/0.57 For more information, visit http://www.philipp.ruemmer.org/princess.shtml % 0.54/0.57 % 0.54/0.57 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ... % 0.54/0.58 Running up to 7 provers in parallel. % 0.54/0.59 Prover 2: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMinimalAndEmpty -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1065072994 % 0.54/0.59 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1571432423 % 0.54/0.59 Prover 4: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=1868514696 % 0.54/0.59 Prover 3: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1922548996 % 0.54/0.59 Prover 0: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1042961893 % 0.54/0.59 Prover 5: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=complete -randomSeed=1259561288 % 0.54/0.59 Prover 6: Options: -triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximalOutermost -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1399714365 % 2.91/1.20 Prover 1: Preprocessing ... % 3.29/1.23 Prover 4: Preprocessing ... % 3.29/1.26 Prover 3: Preprocessing ... % 3.29/1.26 Prover 0: Preprocessing ... % 3.29/1.26 Prover 2: Preprocessing ... % 3.29/1.26 Prover 5: Preprocessing ... % 3.29/1.26 Prover 6: Preprocessing ... % 7.22/1.73 Prover 5: Proving ... % 7.22/1.74 Prover 2: Proving ... % 7.22/1.75 Prover 1: Constructing countermodel ... % 7.22/1.75 Prover 3: Constructing countermodel ... % 7.22/1.79 Prover 6: Proving ... % 8.81/2.01 Prover 4: Constructing countermodel ... % 9.91/2.09 Prover 3: gave up % 9.91/2.09 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-236303470 % 9.91/2.11 Prover 0: Proving ... % 9.91/2.14 Prover 7: Preprocessing ... % 11.12/2.27 Prover 7: Warning: ignoring some quantifiers % 11.12/2.29 Prover 7: Constructing countermodel ... % 11.90/2.34 Prover 1: gave up % 11.90/2.35 Prover 8: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-200781089 % 11.90/2.38 Prover 8: Preprocessing ... % 13.48/2.58 Prover 8: Warning: ignoring some quantifiers % 14.29/2.60 Prover 8: Constructing countermodel ... % 16.69/2.90 Prover 8: gave up % 16.69/2.90 Prover 9: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allMinimal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1423531889 % 16.69/2.99 Prover 9: Preprocessing ... % 18.99/3.29 Prover 9: Constructing countermodel ... % 22.20/3.69 Prover 7: Found proof (size 77) % 22.20/3.69 Prover 7: proved (1601ms) % 22.20/3.70 Prover 4: stopped % 22.20/3.70 Prover 6: stopped % 22.20/3.70 Prover 0: stopped % 22.20/3.70 Prover 2: stopped % 22.97/3.70 Prover 5: stopped % 22.97/3.70 Prover 9: stopped % 22.97/3.70 % 22.97/3.70 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % 22.97/3.70 % 22.97/3.72 % SZS output start Proof for theBenchmark % 22.97/3.72 Assumptions after simplification: % 22.97/3.72 --------------------------------- % 22.97/3.72 % 22.97/3.72 (l10) % 22.97/3.77 $i(efficient_producers) & $i(zero) & $i(first_movers) & ! [v0: $i] : ! [v1: % 22.97/3.77 $i] : ( ~ (appear(efficient_producers, v0) = v1) | ~ $i(v0) | ~ stable(v0) % 22.97/3.77 | ~ environment(v0) | ? [v2: $i] : ($i(v2) & greater(v2, v1) & ! [v3: $i] % 22.97/3.77 : ! [v4: $i] : ( ~ (growth_rate(first_movers, v3) = v4) | ~ $i(v3) | ~ % 22.97/3.77 subpopulations(first_movers, efficient_producers, v0, v3) | ~ % 22.97/3.77 greater_or_equal(v3, v2) | greater(zero, v4)))) % 22.97/3.77 % 22.97/3.77 (mp_EP_in_stable_environments) % 22.97/3.78 $i(efficient_producers) & ! [v0: $i] : ! [v1: $i] : ( ~ % 22.97/3.78 (appear(efficient_producers, v0) = v1) | ~ $i(v0) | ~ stable(v0) | ~ % 22.97/3.78 environment(v0) | in_environment(v0, v1)) % 22.97/3.78 % 22.97/3.78 (mp_contracts_from) % 22.97/3.78 $i(zero) & $i(first_movers) & ! [v0: $i] : ! [v1: $i] : ( ~ $i(v1) | ~ % 22.97/3.78 $i(v0) | ~ in_environment(v0, v1) | ~ stable(v0) | ~ environment(v0) | % 22.97/3.78 contracts_from(v1, first_movers) | ? [v2: $i] : ? [v3: $i] : ? [v4: $i] : % 22.97/3.78 (cardinality_at_time(first_movers, v2) = v3 & growth_rate(first_movers, v2) % 22.97/3.78 = v4 & $i(v4) & $i(v3) & $i(v2) & greater_or_equal(v2, v1) & greater(v3, % 22.97/3.78 zero) & ~ greater(zero, v4))) % 22.97/3.78 % 22.97/3.78 (mp_greater_or_equal) % 22.97/3.78 ! [v0: $i] : ! [v1: $i] : (v1 = v0 | ~ $i(v1) | ~ $i(v0) | ~ % 22.97/3.78 greater_or_equal(v0, v1) | greater(v0, v1)) & ! [v0: $i] : ! [v1: $i] : ( % 22.97/3.79 ~ $i(v1) | ~ $i(v0) | ~ greater(v0, v1) | greater_or_equal(v0, v1)) & ? % 22.97/3.79 [v0: $i] : ( ~ $i(v0) | greater_or_equal(v0, v0)) % 22.97/3.79 % 22.97/3.79 (mp_greater_transitivity) % 22.97/3.79 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | % 22.97/3.79 ~ greater(v1, v2) | ~ greater(v0, v1) | greater(v0, v2)) % 22.97/3.79 % 22.97/3.79 (mp_long_stable_environments) % 22.97/3.79 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ $i(v2) | ~ $i(v1) | ~ $i(v0) | % 22.97/3.79 ~ greater(v2, v1) | ~ in_environment(v0, v1) | ~ stable(v0) | ~ % 22.97/3.79 environment(v0) | in_environment(v0, v2)) % 22.97/3.79 % 22.97/3.79 (mp_non_empty_fm_and_ep) % 22.97/3.80 $i(efficient_producers) & $i(zero) & $i(first_movers) & ! [v0: $i] : ! [v1: % 22.97/3.80 $i] : ! [v2: $i] : ( ~ (cardinality_at_time(efficient_producers, v1) = v2) % 22.97/3.80 | ~ $i(v1) | ~ $i(v0) | ~ greater(v2, zero) | ~ in_environment(v0, v1) | % 22.97/3.80 ~ environment(v0) | subpopulations(first_movers, efficient_producers, v0, % 22.97/3.80 v1) | ? [v3: $i] : (cardinality_at_time(first_movers, v1) = v3 & $i(v3) & % 22.97/3.80 ~ greater(v3, zero))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ % 22.97/3.80 (cardinality_at_time(first_movers, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ~ % 22.97/3.80 greater(v2, zero) | ~ in_environment(v0, v1) | ~ environment(v0) | % 22.97/3.80 subpopulations(first_movers, efficient_producers, v0, v1) | ? [v3: $i] : % 22.97/3.80 (cardinality_at_time(efficient_producers, v1) = v3 & $i(v3) & ~ greater(v3, % 22.97/3.80 zero))) % 22.97/3.80 % 22.97/3.80 (prove_l9) % 23.44/3.80 $i(efficient_producers) & $i(first_movers) & ? [v0: $i] : ? [v1: $i] : % 23.44/3.80 (appear(efficient_producers, v0) = v1 & $i(v1) & $i(v0) & stable(v0) & % 23.44/3.80 environment(v0) & ! [v2: $i] : ( ~ $i(v2) | ~ contracts_from(v2, % 23.44/3.80 first_movers) | ~ greater(v2, v1))) % 23.44/3.80 % 23.44/3.80 (t6) % 23.44/3.80 $i(efficient_producers) & $i(zero) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : % 23.44/3.80 ! [v3: $i] : ( ~ (appear(efficient_producers, v0) = v2) | ~ % 23.44/3.80 (cardinality_at_time(efficient_producers, v1) = v3) | ~ $i(v1) | ~ $i(v0) % 23.44/3.80 | ~ greater_or_equal(v1, v2) | ~ in_environment(v0, v1) | ~ % 23.44/3.80 environment(v0) | greater(v3, zero)) % 23.44/3.80 % 23.44/3.80 Those formulas are unsatisfiable: % 23.44/3.80 --------------------------------- % 23.44/3.80 % 23.44/3.80 Begin of proof % 23.44/3.81 | % 23.44/3.81 | ALPHA: (mp_contracts_from) implies: % 23.44/3.81 | (1) ! [v0: $i] : ! [v1: $i] : ( ~ $i(v1) | ~ $i(v0) | ~ % 23.44/3.81 | in_environment(v0, v1) | ~ stable(v0) | ~ environment(v0) | % 23.44/3.81 | contracts_from(v1, first_movers) | ? [v2: $i] : ? [v3: $i] : ? % 23.44/3.81 | [v4: $i] : (cardinality_at_time(first_movers, v2) = v3 & % 23.44/3.81 | growth_rate(first_movers, v2) = v4 & $i(v4) & $i(v3) & $i(v2) & % 23.44/3.81 | greater_or_equal(v2, v1) & greater(v3, zero) & ~ greater(zero, % 23.44/3.81 | v4))) % 23.44/3.81 | % 23.44/3.81 | ALPHA: (mp_non_empty_fm_and_ep) implies: % 23.44/3.81 | (2) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ % 23.44/3.81 | (cardinality_at_time(first_movers, v1) = v2) | ~ $i(v1) | ~ $i(v0) % 23.44/3.81 | | ~ greater(v2, zero) | ~ in_environment(v0, v1) | ~ % 23.44/3.81 | environment(v0) | subpopulations(first_movers, efficient_producers, % 23.44/3.81 | v0, v1) | ? [v3: $i] : (cardinality_at_time(efficient_producers, % 23.44/3.81 | v1) = v3 & $i(v3) & ~ greater(v3, zero))) % 23.44/3.81 | % 23.44/3.81 | ALPHA: (mp_EP_in_stable_environments) implies: % 23.44/3.82 | (3) ! [v0: $i] : ! [v1: $i] : ( ~ (appear(efficient_producers, v0) = v1) % 23.44/3.82 | | ~ $i(v0) | ~ stable(v0) | ~ environment(v0) | in_environment(v0, % 23.44/3.82 | v1)) % 23.44/3.82 | % 23.44/3.82 | ALPHA: (mp_greater_or_equal) implies: % 23.44/3.82 | (4) ! [v0: $i] : ! [v1: $i] : ( ~ $i(v1) | ~ $i(v0) | ~ greater(v0, v1) % 23.44/3.82 | | greater_or_equal(v0, v1)) % 23.44/3.82 | (5) ! [v0: $i] : ! [v1: $i] : (v1 = v0 | ~ $i(v1) | ~ $i(v0) | ~ % 23.44/3.82 | greater_or_equal(v0, v1) | greater(v0, v1)) % 23.44/3.82 | % 23.44/3.82 | ALPHA: (t6) implies: % 23.44/3.82 | (6) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~ % 23.44/3.82 | (appear(efficient_producers, v0) = v2) | ~ % 23.44/3.82 | (cardinality_at_time(efficient_producers, v1) = v3) | ~ $i(v1) | ~ % 23.44/3.82 | $i(v0) | ~ greater_or_equal(v1, v2) | ~ in_environment(v0, v1) | ~ % 23.44/3.82 | environment(v0) | greater(v3, zero)) % 23.44/3.82 | % 23.44/3.82 | ALPHA: (l10) implies: % 23.44/3.82 | (7) ! [v0: $i] : ! [v1: $i] : ( ~ (appear(efficient_producers, v0) = v1) % 23.44/3.82 | | ~ $i(v0) | ~ stable(v0) | ~ environment(v0) | ? [v2: $i] : % 23.44/3.82 | ($i(v2) & greater(v2, v1) & ! [v3: $i] : ! [v4: $i] : ( ~ % 23.44/3.82 | (growth_rate(first_movers, v3) = v4) | ~ $i(v3) | ~ % 23.44/3.82 | subpopulations(first_movers, efficient_producers, v0, v3) | ~ % 23.44/3.82 | greater_or_equal(v3, v2) | greater(zero, v4)))) % 23.44/3.82 | % 23.44/3.82 | ALPHA: (prove_l9) implies: % 23.44/3.82 | (8) ? [v0: $i] : ? [v1: $i] : (appear(efficient_producers, v0) = v1 & % 23.44/3.82 | $i(v1) & $i(v0) & stable(v0) & environment(v0) & ! [v2: $i] : ( ~ % 23.44/3.82 | $i(v2) | ~ contracts_from(v2, first_movers) | ~ greater(v2, v1))) % 23.44/3.82 | % 23.44/3.83 | DELTA: instantiating (8) with fresh symbols all_13_0, all_13_1 gives: % 23.44/3.83 | (9) appear(efficient_producers, all_13_1) = all_13_0 & $i(all_13_0) & % 23.44/3.83 | $i(all_13_1) & stable(all_13_1) & environment(all_13_1) & ! [v0: $i] : % 23.44/3.83 | ( ~ $i(v0) | ~ contracts_from(v0, first_movers) | ~ greater(v0, % 23.44/3.83 | all_13_0)) % 23.44/3.83 | % 23.44/3.83 | ALPHA: (9) implies: % 23.44/3.83 | (10) environment(all_13_1) % 23.44/3.83 | (11) stable(all_13_1) % 23.44/3.83 | (12) $i(all_13_1) % 23.44/3.83 | (13) $i(all_13_0) % 23.44/3.83 | (14) appear(efficient_producers, all_13_1) = all_13_0 % 23.44/3.83 | (15) ! [v0: $i] : ( ~ $i(v0) | ~ contracts_from(v0, first_movers) | ~ % 23.44/3.83 | greater(v0, all_13_0)) % 23.44/3.83 | % 23.44/3.83 | GROUND_INST: instantiating (3) with all_13_1, all_13_0, simplifying with (10), % 23.44/3.83 | (11), (12), (14) gives: % 23.44/3.83 | (16) in_environment(all_13_1, all_13_0) % 23.44/3.83 | % 23.44/3.83 | GROUND_INST: instantiating (7) with all_13_1, all_13_0, simplifying with (10), % 23.44/3.83 | (11), (12), (14) gives: % 23.44/3.83 | (17) ? [v0: $i] : ($i(v0) & greater(v0, all_13_0) & ! [v1: $i] : ! [v2: % 23.44/3.83 | $i] : ( ~ (growth_rate(first_movers, v1) = v2) | ~ $i(v1) | ~ % 23.44/3.84 | subpopulations(first_movers, efficient_producers, all_13_1, v1) | % 23.44/3.84 | ~ greater_or_equal(v1, v0) | greater(zero, v2))) % 23.44/3.84 | % 23.44/3.84 | DELTA: instantiating (17) with fresh symbol all_22_0 gives: % 23.44/3.84 | (18) $i(all_22_0) & greater(all_22_0, all_13_0) & ! [v0: $i] : ! [v1: $i] % 23.44/3.84 | : ( ~ (growth_rate(first_movers, v0) = v1) | ~ $i(v0) | ~ % 23.44/3.84 | subpopulations(first_movers, efficient_producers, all_13_1, v0) | ~ % 23.44/3.84 | greater_or_equal(v0, all_22_0) | greater(zero, v1)) % 23.44/3.84 | % 23.44/3.84 | ALPHA: (18) implies: % 23.44/3.84 | (19) greater(all_22_0, all_13_0) % 23.44/3.84 | (20) $i(all_22_0) % 23.44/3.84 | (21) ! [v0: $i] : ! [v1: $i] : ( ~ (growth_rate(first_movers, v0) = v1) | % 23.44/3.84 | ~ $i(v0) | ~ subpopulations(first_movers, efficient_producers, % 23.44/3.84 | all_13_1, v0) | ~ greater_or_equal(v0, all_22_0) | greater(zero, % 23.44/3.84 | v1)) % 23.44/3.84 | % 23.44/3.84 | GROUND_INST: instantiating (mp_long_stable_environments) with all_13_1, % 23.44/3.84 | all_13_0, all_22_0, simplifying with (10), (11), (12), (13), % 23.44/3.84 | (16), (19), (20) gives: % 23.44/3.84 | (22) in_environment(all_13_1, all_22_0) % 23.44/3.84 | % 23.44/3.84 | GROUND_INST: instantiating (4) with all_22_0, all_13_0, simplifying with (13), % 23.44/3.84 | (19), (20) gives: % 23.44/3.84 | (23) greater_or_equal(all_22_0, all_13_0) % 23.44/3.84 | % 23.44/3.84 | GROUND_INST: instantiating (1) with all_13_1, all_22_0, simplifying with (10), % 23.44/3.84 | (11), (12), (20), (22) gives: % 23.44/3.84 | (24) contracts_from(all_22_0, first_movers) | ? [v0: $i] : ? [v1: $i] : % 23.44/3.85 | ? [v2: $i] : (cardinality_at_time(first_movers, v0) = v1 & % 23.44/3.85 | growth_rate(first_movers, v0) = v2 & $i(v2) & $i(v1) & $i(v0) & % 23.44/3.85 | greater_or_equal(v0, all_22_0) & greater(v1, zero) & ~ % 23.44/3.85 | greater(zero, v2)) % 23.44/3.85 | % 23.44/3.85 | BETA: splitting (24) gives: % 23.44/3.85 | % 23.44/3.85 | Case 1: % 23.44/3.85 | | % 23.44/3.85 | | (25) contracts_from(all_22_0, first_movers) % 23.44/3.85 | | % 23.44/3.85 | | GROUND_INST: instantiating (15) with all_22_0, simplifying with (19), (20), % 23.44/3.85 | | (25) gives: % 23.44/3.85 | | (26) $false % 23.44/3.85 | | % 23.44/3.85 | | CLOSE: (26) is inconsistent. % 23.44/3.85 | | % 23.44/3.85 | Case 2: % 23.44/3.85 | | % 23.44/3.85 | | (27) ? [v0: $i] : ? [v1: $i] : ? [v2: $i] : % 23.44/3.85 | | (cardinality_at_time(first_movers, v0) = v1 & % 23.44/3.85 | | growth_rate(first_movers, v0) = v2 & $i(v2) & $i(v1) & $i(v0) & % 23.44/3.85 | | greater_or_equal(v0, all_22_0) & greater(v1, zero) & ~ % 23.44/3.85 | | greater(zero, v2)) % 23.44/3.85 | | % 23.44/3.85 | | DELTA: instantiating (27) with fresh symbols all_66_0, all_66_1, all_66_2 % 23.44/3.85 | | gives: % 23.44/3.85 | | (28) cardinality_at_time(first_movers, all_66_2) = all_66_1 & % 23.44/3.85 | | growth_rate(first_movers, all_66_2) = all_66_0 & $i(all_66_0) & % 23.44/3.85 | | $i(all_66_1) & $i(all_66_2) & greater_or_equal(all_66_2, all_22_0) & % 23.44/3.85 | | greater(all_66_1, zero) & ~ greater(zero, all_66_0) % 23.44/3.85 | | % 23.44/3.85 | | ALPHA: (28) implies: % 23.44/3.85 | | (29) ~ greater(zero, all_66_0) % 23.44/3.85 | | (30) greater(all_66_1, zero) % 23.44/3.85 | | (31) greater_or_equal(all_66_2, all_22_0) % 23.44/3.85 | | (32) $i(all_66_2) % 23.44/3.85 | | (33) growth_rate(first_movers, all_66_2) = all_66_0 % 23.44/3.85 | | (34) cardinality_at_time(first_movers, all_66_2) = all_66_1 % 23.44/3.85 | | % 23.44/3.86 | | GROUND_INST: instantiating (5) with all_66_2, all_22_0, simplifying with % 23.44/3.86 | | (20), (31), (32) gives: % 23.44/3.86 | | (35) all_66_2 = all_22_0 | greater(all_66_2, all_22_0) % 23.44/3.86 | | % 23.44/3.86 | | GROUND_INST: instantiating (2) with all_13_1, all_22_0, all_66_1, % 23.44/3.86 | | simplifying with (10), (12), (20), (22), (30) gives: % 23.44/3.86 | | (36) ~ (cardinality_at_time(first_movers, all_22_0) = all_66_1) | % 23.44/3.86 | | subpopulations(first_movers, efficient_producers, all_13_1, % 23.44/3.86 | | all_22_0) | ? [v0: $i] : % 23.44/3.86 | | (cardinality_at_time(efficient_producers, all_22_0) = v0 & $i(v0) & % 23.44/3.86 | | ~ greater(v0, zero)) % 23.44/3.86 | | % 23.44/3.86 | | GROUND_INST: instantiating (2) with all_13_1, all_66_2, all_66_2, % 23.44/3.86 | | simplifying with (10), (12), (32) gives: % 23.44/3.86 | | (37) ~ (cardinality_at_time(first_movers, all_66_2) = all_66_2) | ~ % 23.44/3.86 | | greater(all_66_2, zero) | ~ in_environment(all_13_1, all_66_2) | % 23.44/3.86 | | subpopulations(first_movers, efficient_producers, all_13_1, % 23.44/3.86 | | all_66_2) | ? [v0: $i] : % 23.44/3.86 | | (cardinality_at_time(efficient_producers, all_66_2) = v0 & $i(v0) & % 23.44/3.86 | | ~ greater(v0, zero)) % 23.44/3.86 | | % 23.44/3.86 | | BETA: splitting (36) gives: % 23.44/3.86 | | % 23.44/3.86 | | Case 1: % 23.44/3.86 | | | % 23.44/3.86 | | | (38) subpopulations(first_movers, efficient_producers, all_13_1, % 23.44/3.86 | | | all_22_0) % 23.44/3.86 | | | % 23.44/3.86 | | | BETA: splitting (35) gives: % 23.44/3.86 | | | % 23.44/3.86 | | | Case 1: % 23.44/3.86 | | | | % 23.44/3.86 | | | | (39) greater(all_66_2, all_22_0) % 23.44/3.86 | | | | % 23.44/3.87 | | | | REF_CLOSE: (2), (4), (6), (10), (11), (12), (13), (14), (19), (20), % 23.44/3.87 | | | | (21), (22), (29), (30), (31), (32), (33), (34), (37), (39), % 23.44/3.87 | | | | (mp_greater_transitivity), (mp_long_stable_environments) are % 23.44/3.87 | | | | inconsistent by sub-proof #1. % 23.44/3.87 | | | | % 23.44/3.87 | | | Case 2: % 23.44/3.87 | | | | % 23.44/3.87 | | | | (40) all_66_2 = all_22_0 % 23.44/3.87 | | | | % 23.44/3.87 | | | | REDUCE: (33), (40) imply: % 23.44/3.87 | | | | (41) growth_rate(first_movers, all_22_0) = all_66_0 % 23.44/3.87 | | | | % 23.44/3.87 | | | | REDUCE: (31), (40) imply: % 23.44/3.87 | | | | (42) greater_or_equal(all_22_0, all_22_0) % 23.44/3.87 | | | | % 23.44/3.87 | | | | GROUND_INST: instantiating (21) with all_22_0, all_66_0, simplifying % 23.44/3.87 | | | | with (20), (29), (38), (41), (42) gives: % 23.44/3.87 | | | | (43) $false % 23.44/3.87 | | | | % 23.44/3.87 | | | | CLOSE: (43) is inconsistent. % 23.44/3.87 | | | | % 23.44/3.87 | | | End of split % 23.44/3.87 | | | % 23.44/3.87 | | Case 2: % 23.44/3.87 | | | % 23.44/3.87 | | | (44) ~ (cardinality_at_time(first_movers, all_22_0) = all_66_1) | ? % 23.44/3.87 | | | [v0: $i] : (cardinality_at_time(efficient_producers, all_22_0) = % 23.44/3.87 | | | v0 & $i(v0) & ~ greater(v0, zero)) % 23.44/3.87 | | | % 23.44/3.87 | | | BETA: splitting (35) gives: % 23.44/3.87 | | | % 23.44/3.87 | | | Case 1: % 23.44/3.87 | | | | % 23.44/3.87 | | | | (45) greater(all_66_2, all_22_0) % 23.44/3.87 | | | | % 23.44/3.88 | | | | REF_CLOSE: (2), (4), (6), (10), (11), (12), (13), (14), (19), (20), % 23.44/3.88 | | | | (21), (22), (29), (30), (31), (32), (33), (34), (37), (45), % 23.44/3.88 | | | | (mp_greater_transitivity), (mp_long_stable_environments) are % 23.44/3.88 | | | | inconsistent by sub-proof #1. % 23.44/3.88 | | | | % 23.44/3.88 | | | Case 2: % 23.44/3.88 | | | | % 23.44/3.88 | | | | (46) all_66_2 = all_22_0 % 23.44/3.88 | | | | % 23.44/3.88 | | | | REDUCE: (34), (46) imply: % 23.44/3.88 | | | | (47) cardinality_at_time(first_movers, all_22_0) = all_66_1 % 23.44/3.88 | | | | % 23.44/3.88 | | | | BETA: splitting (44) gives: % 23.44/3.88 | | | | % 23.44/3.88 | | | | Case 1: % 23.44/3.88 | | | | | % 23.44/3.88 | | | | | (48) ~ (cardinality_at_time(first_movers, all_22_0) = all_66_1) % 23.44/3.88 | | | | | % 23.44/3.88 | | | | | PRED_UNIFY: (47), (48) imply: % 23.44/3.88 | | | | | (49) $false % 23.44/3.88 | | | | | % 23.44/3.88 | | | | | CLOSE: (49) is inconsistent. % 23.44/3.88 | | | | | % 23.44/3.88 | | | | Case 2: % 23.44/3.88 | | | | | % 23.44/3.88 | | | | | (50) ? [v0: $i] : (cardinality_at_time(efficient_producers, % 23.44/3.88 | | | | | all_22_0) = v0 & $i(v0) & ~ greater(v0, zero)) % 23.44/3.88 | | | | | % 23.44/3.88 | | | | | DELTA: instantiating (50) with fresh symbol all_127_0 gives: % 23.44/3.88 | | | | | (51) cardinality_at_time(efficient_producers, all_22_0) = all_127_0 % 23.44/3.88 | | | | | & $i(all_127_0) & ~ greater(all_127_0, zero) % 23.44/3.88 | | | | | % 23.44/3.88 | | | | | ALPHA: (51) implies: % 23.44/3.88 | | | | | (52) ~ greater(all_127_0, zero) % 23.44/3.88 | | | | | (53) cardinality_at_time(efficient_producers, all_22_0) = all_127_0 % 23.44/3.88 | | | | | % 23.44/3.89 | | | | | GROUND_INST: instantiating (6) with all_13_1, all_22_0, all_13_0, % 23.44/3.89 | | | | | all_127_0, simplifying with (10), (12), (14), (20), (22), % 23.44/3.89 | | | | | (23), (52), (53) gives: % 23.44/3.89 | | | | | (54) $false % 23.44/3.89 | | | | | % 23.44/3.89 | | | | | CLOSE: (54) is inconsistent. % 23.44/3.89 | | | | | % 23.44/3.89 | | | | End of split % 23.44/3.89 | | | | % 23.44/3.89 | | | End of split % 23.44/3.89 | | | % 23.44/3.89 | | End of split % 23.44/3.89 | | % 23.44/3.89 | End of split % 23.44/3.89 | % 23.44/3.89 End of proof % 23.44/3.89 % 23.44/3.89 Sub-proof #1 shows that the following formulas are inconsistent: % 23.44/3.89 ---------------------------------------------------------------- % 23.44/3.89 (1) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~ % 23.44/3.89 (appear(efficient_producers, v0) = v2) | ~ % 23.44/3.89 (cardinality_at_time(efficient_producers, v1) = v3) | ~ $i(v1) | ~ % 23.44/3.89 $i(v0) | ~ greater_or_equal(v1, v2) | ~ in_environment(v0, v1) | ~ % 23.44/3.89 environment(v0) | greater(v3, zero)) % 23.44/3.89 (2) in_environment(all_13_1, all_22_0) % 23.44/3.89 (3) greater(all_66_1, zero) % 23.44/3.89 (4) environment(all_13_1) % 23.44/3.89 (5) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ % 23.44/3.89 (cardinality_at_time(first_movers, v1) = v2) | ~ $i(v1) | ~ $i(v0) | % 23.44/3.89 ~ greater(v2, zero) | ~ in_environment(v0, v1) | ~ environment(v0) | % 23.44/3.89 subpopulations(first_movers, efficient_producers, v0, v1) | ? [v3: $i] % 23.44/3.89 : (cardinality_at_time(efficient_producers, v1) = v3 & $i(v3) & ~ % 23.44/3.89 greater(v3, zero))) % 23.44/3.89 (6) stable(all_13_1) % 23.44/3.89 (7) ~ greater(zero, all_66_0) % 23.44/3.89 (8) cardinality_at_time(first_movers, all_66_2) = all_66_1 % 23.44/3.90 (9) ~ (cardinality_at_time(first_movers, all_66_2) = all_66_2) | ~ % 23.44/3.90 greater(all_66_2, zero) | ~ in_environment(all_13_1, all_66_2) | % 23.44/3.90 subpopulations(first_movers, efficient_producers, all_13_1, all_66_2) | % 23.44/3.90 ? [v0: $i] : (cardinality_at_time(efficient_producers, all_66_2) = v0 & % 23.44/3.90 $i(v0) & ~ greater(v0, zero)) % 23.93/3.90 (10) ! [v0: $i] : ! [v1: $i] : ( ~ $i(v1) | ~ $i(v0) | ~ greater(v0, v1) % 23.93/3.90 | greater_or_equal(v0, v1)) % 23.93/3.90 (11) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ $i(v2) | ~ $i(v1) | ~ % 23.93/3.90 $i(v0) | ~ greater(v1, v2) | ~ greater(v0, v1) | greater(v0, v2)) % 23.93/3.90 (12) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ $i(v2) | ~ $i(v1) | ~ % 23.93/3.90 $i(v0) | ~ greater(v2, v1) | ~ in_environment(v0, v1) | ~ % 23.93/3.90 stable(v0) | ~ environment(v0) | in_environment(v0, v2)) % 23.93/3.90 (13) appear(efficient_producers, all_13_1) = all_13_0 % 23.93/3.90 (14) ! [v0: $i] : ! [v1: $i] : ( ~ (growth_rate(first_movers, v0) = v1) | % 23.93/3.90 ~ $i(v0) | ~ subpopulations(first_movers, efficient_producers, % 23.93/3.90 all_13_1, v0) | ~ greater_or_equal(v0, all_22_0) | greater(zero, % 23.93/3.90 v1)) % 23.93/3.90 (15) $i(all_22_0) % 23.93/3.90 (16) greater(all_22_0, all_13_0) % 23.93/3.90 (17) greater_or_equal(all_66_2, all_22_0) % 23.93/3.90 (18) $i(all_66_2) % 23.93/3.90 (19) $i(all_13_1) % 23.93/3.90 (20) growth_rate(first_movers, all_66_2) = all_66_0 % 23.93/3.90 (21) greater(all_66_2, all_22_0) % 23.93/3.90 (22) $i(all_13_0) % 23.93/3.90 % 23.93/3.90 Begin of proof % 23.93/3.90 | % 23.93/3.91 | GROUND_INST: instantiating (12) with all_13_1, all_22_0, all_66_2, simplifying % 23.93/3.91 | with (2), (4), (6), (15), (18), (19), (21) gives: % 23.93/3.91 | (23) in_environment(all_13_1, all_66_2) % 23.93/3.91 | % 23.93/3.91 | GROUND_INST: instantiating (11) with all_66_2, all_22_0, all_13_0, simplifying % 23.93/3.91 | with (15), (16), (18), (21), (22) gives: % 23.93/3.91 | (24) greater(all_66_2, all_13_0) % 23.93/3.91 | % 23.93/3.91 | GROUND_INST: instantiating (5) with all_13_1, all_66_2, all_66_1, simplifying % 23.93/3.91 | with (3), (4), (8), (18), (19), (23) gives: % 23.93/3.91 | (25) subpopulations(first_movers, efficient_producers, all_13_1, all_66_2) % 23.93/3.91 | | ? [v0: $i] : (cardinality_at_time(efficient_producers, all_66_2) = % 23.93/3.91 | v0 & $i(v0) & ~ greater(v0, zero)) % 23.93/3.91 | % 23.93/3.91 | GROUND_INST: instantiating (10) with all_66_2, all_13_0, simplifying with % 23.93/3.91 | (18), (22), (24) gives: % 23.93/3.91 | (26) greater_or_equal(all_66_2, all_13_0) % 23.93/3.91 | % 23.93/3.91 | BETA: splitting (9) gives: % 23.93/3.91 | % 23.93/3.91 | Case 1: % 23.93/3.91 | | % 23.93/3.91 | | (27) subpopulations(first_movers, efficient_producers, all_13_1, % 23.93/3.91 | | all_66_2) % 23.93/3.91 | | % 23.93/3.91 | | GROUND_INST: instantiating (14) with all_66_2, all_66_0, simplifying with % 23.93/3.91 | | (7), (17), (18), (20), (27) gives: % 23.93/3.91 | | (28) $false % 23.93/3.91 | | % 23.93/3.91 | | CLOSE: (28) is inconsistent. % 23.93/3.91 | | % 23.93/3.91 | Case 2: % 23.93/3.91 | | % 23.93/3.91 | | (29) ~ subpopulations(first_movers, efficient_producers, all_13_1, % 23.93/3.91 | | all_66_2) % 23.93/3.91 | | % 23.93/3.91 | | BETA: splitting (25) gives: % 23.93/3.91 | | % 23.93/3.91 | | Case 1: % 23.93/3.91 | | | % 23.93/3.91 | | | (30) subpopulations(first_movers, efficient_producers, all_13_1, % 23.93/3.91 | | | all_66_2) % 23.93/3.91 | | | % 23.93/3.91 | | | PRED_UNIFY: (29), (30) imply: % 23.93/3.91 | | | (31) $false % 23.93/3.91 | | | % 23.93/3.91 | | | CLOSE: (31) is inconsistent. % 23.93/3.92 | | | % 23.93/3.92 | | Case 2: % 23.93/3.92 | | | % 23.93/3.92 | | | (32) ? [v0: $i] : (cardinality_at_time(efficient_producers, all_66_2) % 23.93/3.92 | | | = v0 & $i(v0) & ~ greater(v0, zero)) % 23.93/3.92 | | | % 23.93/3.92 | | | DELTA: instantiating (32) with fresh symbol all_211_0 gives: % 23.93/3.92 | | | (33) cardinality_at_time(efficient_producers, all_66_2) = all_211_0 & % 23.93/3.92 | | | $i(all_211_0) & ~ greater(all_211_0, zero) % 23.93/3.92 | | | % 23.93/3.92 | | | ALPHA: (33) implies: % 23.93/3.92 | | | (34) ~ greater(all_211_0, zero) % 23.93/3.92 | | | (35) cardinality_at_time(efficient_producers, all_66_2) = all_211_0 % 23.93/3.92 | | | % 23.93/3.92 | | | GROUND_INST: instantiating (1) with all_13_1, all_66_2, all_13_0, % 23.93/3.92 | | | all_211_0, simplifying with (4), (13), (18), (19), (23), % 23.93/3.92 | | | (26), (34), (35) gives: % 23.93/3.92 | | | (36) $false % 23.93/3.92 | | | % 23.93/3.92 | | | CLOSE: (36) is inconsistent. % 23.93/3.92 | | | % 23.93/3.92 | | End of split % 23.93/3.92 | | % 23.93/3.92 | End of split % 23.93/3.92 | % 23.93/3.92 End of proof % 23.93/3.92 % SZS output end Proof for theBenchmark % 23.93/3.92 % 23.93/3.92 3351ms %------------------------------------------------------------------------------