TSTP Solution File: MGT027+1 by Princess---230619
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%------------------------------------------------------------------------------
% File : Princess---230619
% Problem : MGT027+1 : TPTP v8.1.2. Released v2.0.0.
% Transfm : none
% Format : tptp
% Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s
% Computer : n007.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 : 300s
% DateTime : Thu Aug 31 09:16:17 EDT 2023
% Result : Theorem 12.91s 2.44s
% Output : Proof 13.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : MGT027+1 : TPTP v8.1.2. Released v2.0.0.
% 0.07/0.13 % Command : princess -inputFormat=tptp +threads -portfolio=casc +printProof -timeoutSec=%d %s
% 0.13/0.33 % Computer : n007.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 : 300
% 0.13/0.33 % DateTime : Mon Aug 28 06:22:27 EDT 2023
% 0.13/0.33 % CPUTime :
% 0.20/0.59 ________ _____
% 0.20/0.59 ___ __ \_________(_)________________________________
% 0.20/0.59 __ /_/ /_ ___/_ /__ __ \ ___/ _ \_ ___/_ ___/
% 0.20/0.59 _ ____/_ / _ / _ / / / /__ / __/(__ )_(__ )
% 0.20/0.59 /_/ /_/ /_/ /_/ /_/\___/ \___//____/ /____/
% 0.20/0.59
% 0.20/0.59 A Theorem Prover for First-Order Logic modulo Linear Integer Arithmetic
% 0.20/0.59 (2023-06-19)
% 0.20/0.59
% 0.20/0.59 (c) Philipp Rümmer, 2009-2023
% 0.20/0.59 Contributors: Peter Backeman, Peter Baumgartner, Angelo Brillout, Zafer Esen,
% 0.20/0.59 Amanda Stjerna.
% 0.20/0.59 Free software under BSD-3-Clause.
% 0.20/0.59
% 0.20/0.59 For more information, visit http://www.philipp.ruemmer.org/princess.shtml
% 0.20/0.59
% 0.20/0.59 Loading /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 0.20/0.60 Running up to 7 provers in parallel.
% 0.20/0.62 Prover 0: Options: +triggersInConjecture +genTotalityAxioms +tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1042961893
% 0.20/0.62 Prover 1: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-1571432423
% 0.20/0.62 Prover 2: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMinimalAndEmpty -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1065072994
% 0.20/0.62 Prover 3: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1922548996
% 0.20/0.62 Prover 4: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=simple -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=1868514696
% 0.20/0.62 Prover 5: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allMaximal -realRatSaturationRounds=1 -ignoreQuantifiers -constructProofs=never -generateTriggers=complete -randomSeed=1259561288
% 0.20/0.62 Prover 6: Options: -triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=none +reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximalOutermost -realRatSaturationRounds=0 -ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=-1399714365
% 2.23/0.99 Prover 4: Preprocessing ...
% 2.23/0.99 Prover 1: Preprocessing ...
% 2.53/1.03 Prover 6: Preprocessing ...
% 2.53/1.03 Prover 0: Preprocessing ...
% 2.53/1.03 Prover 5: Preprocessing ...
% 2.53/1.03 Prover 3: Preprocessing ...
% 2.53/1.03 Prover 2: Preprocessing ...
% 4.38/1.29 Prover 6: Proving ...
% 4.38/1.29 Prover 5: Proving ...
% 4.38/1.31 Prover 2: Proving ...
% 4.38/1.31 Prover 1: Constructing countermodel ...
% 4.38/1.31 Prover 3: Constructing countermodel ...
% 5.46/1.44 Prover 4: Constructing countermodel ...
% 5.64/1.48 Prover 0: Proving ...
% 6.75/1.59 Prover 3: gave up
% 6.75/1.60 Prover 7: Options: +triggersInConjecture -genTotalityAxioms +tightFunctionScopes -clausifier=simple +reverseFunctionalityPropagation +boolFunsAsPreds -triggerStrategy=allUni -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-236303470
% 7.33/1.66 Prover 7: Preprocessing ...
% 7.46/1.69 Prover 1: gave up
% 7.46/1.69 Prover 8: Options: +triggersInConjecture +genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=maximal -realRatSaturationRounds=0 +ignoreQuantifiers -constructProofs=always -generateTriggers=all -randomSeed=-200781089
% 7.71/1.72 Prover 8: Preprocessing ...
% 7.71/1.73 Prover 7: Warning: ignoring some quantifiers
% 7.71/1.74 Prover 7: Constructing countermodel ...
% 7.86/1.80 Prover 8: Warning: ignoring some quantifiers
% 7.86/1.81 Prover 8: Constructing countermodel ...
% 9.32/2.00 Prover 8: gave up
% 9.32/2.00 Prover 9: Options: +triggersInConjecture -genTotalityAxioms -tightFunctionScopes -clausifier=none -reverseFunctionalityPropagation -boolFunsAsPreds -triggerStrategy=allMinimal -realRatSaturationRounds=1 +ignoreQuantifiers -constructProofs=never -generateTriggers=all -randomSeed=1423531889
% 10.00/2.04 Prover 9: Preprocessing ...
% 10.94/2.16 Prover 9: Constructing countermodel ...
% 12.91/2.44 Prover 7: Found proof (size 77)
% 12.91/2.44 Prover 7: proved (833ms)
% 12.91/2.44 Prover 4: stopped
% 12.91/2.44 Prover 0: stopped
% 12.91/2.44 Prover 5: stopped
% 12.91/2.44 Prover 9: stopped
% 12.91/2.44 Prover 6: stopped
% 12.91/2.44 Prover 2: stopped
% 12.91/2.44
% 12.91/2.44 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% 12.91/2.44
% 12.91/2.45 % SZS output start Proof for theBenchmark
% 12.91/2.46 Assumptions after simplification:
% 12.91/2.46 ---------------------------------
% 12.91/2.46
% 12.91/2.46 (l10)
% 13.20/2.48 $i(efficient_producers) & $i(zero) & $i(first_movers) & ! [v0: $i] : ! [v1:
% 13.20/2.48 $i] : ( ~ (appear(efficient_producers, v0) = v1) | ~ $i(v0) | ~ stable(v0)
% 13.20/2.48 | ~ environment(v0) | ? [v2: $i] : ($i(v2) & greater(v2, v1) & ! [v3: $i]
% 13.20/2.48 : ! [v4: $i] : ( ~ (growth_rate(first_movers, v3) = v4) | ~ $i(v3) | ~
% 13.20/2.48 subpopulations(first_movers, efficient_producers, v0, v3) | ~
% 13.20/2.48 greater_or_equal(v3, v2) | greater(zero, v4))))
% 13.20/2.48
% 13.20/2.48 (mp_EP_in_stable_environments)
% 13.20/2.48 $i(efficient_producers) & ! [v0: $i] : ! [v1: $i] : ( ~
% 13.20/2.48 (appear(efficient_producers, v0) = v1) | ~ $i(v0) | ~ stable(v0) | ~
% 13.20/2.48 environment(v0) | in_environment(v0, v1))
% 13.20/2.48
% 13.20/2.48 (mp_contracts_from)
% 13.20/2.49 $i(zero) & $i(first_movers) & ! [v0: $i] : ! [v1: $i] : ( ~ $i(v1) | ~
% 13.20/2.49 $i(v0) | ~ in_environment(v0, v1) | ~ stable(v0) | ~ environment(v0) |
% 13.20/2.49 contracts_from(v1, first_movers) | ? [v2: $i] : ? [v3: $i] : ? [v4: $i] :
% 13.20/2.49 (cardinality_at_time(first_movers, v2) = v3 & growth_rate(first_movers, v2)
% 13.20/2.49 = v4 & $i(v4) & $i(v3) & $i(v2) & greater_or_equal(v2, v1) & greater(v3,
% 13.20/2.49 zero) & ~ greater(zero, v4)))
% 13.20/2.49
% 13.20/2.49 (mp_greater_or_equal)
% 13.20/2.49 ! [v0: $i] : ! [v1: $i] : (v1 = v0 | ~ $i(v1) | ~ $i(v0) | ~
% 13.20/2.49 greater_or_equal(v0, v1) | greater(v0, v1)) & ! [v0: $i] : ! [v1: $i] : (
% 13.20/2.49 ~ $i(v1) | ~ $i(v0) | ~ greater(v0, v1) | greater_or_equal(v0, v1)) & ?
% 13.20/2.49 [v0: $i] : ( ~ $i(v0) | greater_or_equal(v0, v0))
% 13.20/2.49
% 13.20/2.49 (mp_greater_transitivity)
% 13.20/2.49 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ $i(v2) | ~ $i(v1) | ~ $i(v0) |
% 13.20/2.49 ~ greater(v1, v2) | ~ greater(v0, v1) | greater(v0, v2))
% 13.20/2.49
% 13.20/2.49 (mp_long_stable_environments)
% 13.20/2.49 ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ $i(v2) | ~ $i(v1) | ~ $i(v0) |
% 13.20/2.49 ~ greater(v2, v1) | ~ in_environment(v0, v1) | ~ stable(v0) | ~
% 13.20/2.49 environment(v0) | in_environment(v0, v2))
% 13.20/2.49
% 13.20/2.49 (mp_non_empty_fm_and_ep)
% 13.20/2.49 $i(efficient_producers) & $i(zero) & $i(first_movers) & ! [v0: $i] : ! [v1:
% 13.20/2.49 $i] : ! [v2: $i] : ( ~ (cardinality_at_time(efficient_producers, v1) = v2)
% 13.20/2.49 | ~ $i(v1) | ~ $i(v0) | ~ greater(v2, zero) | ~ in_environment(v0, v1) |
% 13.20/2.49 ~ environment(v0) | subpopulations(first_movers, efficient_producers, v0,
% 13.20/2.49 v1) | ? [v3: $i] : (cardinality_at_time(first_movers, v1) = v3 & $i(v3) &
% 13.20/2.49 ~ greater(v3, zero))) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 13.20/2.49 (cardinality_at_time(first_movers, v1) = v2) | ~ $i(v1) | ~ $i(v0) | ~
% 13.20/2.49 greater(v2, zero) | ~ in_environment(v0, v1) | ~ environment(v0) |
% 13.20/2.49 subpopulations(first_movers, efficient_producers, v0, v1) | ? [v3: $i] :
% 13.20/2.49 (cardinality_at_time(efficient_producers, v1) = v3 & $i(v3) & ~ greater(v3,
% 13.20/2.49 zero)))
% 13.20/2.49
% 13.20/2.49 (prove_l9)
% 13.20/2.50 $i(efficient_producers) & $i(first_movers) & ? [v0: $i] : ? [v1: $i] :
% 13.20/2.50 (appear(efficient_producers, v0) = v1 & $i(v1) & $i(v0) & stable(v0) &
% 13.20/2.50 environment(v0) & ! [v2: $i] : ( ~ $i(v2) | ~ contracts_from(v2,
% 13.20/2.50 first_movers) | ~ greater(v2, v1)))
% 13.20/2.50
% 13.20/2.50 (t6)
% 13.20/2.50 $i(efficient_producers) & $i(zero) & ! [v0: $i] : ! [v1: $i] : ! [v2: $i] :
% 13.20/2.50 ! [v3: $i] : ( ~ (appear(efficient_producers, v0) = v2) | ~
% 13.20/2.50 (cardinality_at_time(efficient_producers, v1) = v3) | ~ $i(v1) | ~ $i(v0)
% 13.20/2.50 | ~ greater_or_equal(v1, v2) | ~ in_environment(v0, v1) | ~
% 13.20/2.50 environment(v0) | greater(v3, zero))
% 13.20/2.50
% 13.20/2.50 Those formulas are unsatisfiable:
% 13.20/2.50 ---------------------------------
% 13.20/2.50
% 13.20/2.50 Begin of proof
% 13.20/2.50 |
% 13.20/2.50 | ALPHA: (mp_contracts_from) implies:
% 13.20/2.50 | (1) ! [v0: $i] : ! [v1: $i] : ( ~ $i(v1) | ~ $i(v0) | ~
% 13.20/2.50 | in_environment(v0, v1) | ~ stable(v0) | ~ environment(v0) |
% 13.20/2.50 | contracts_from(v1, first_movers) | ? [v2: $i] : ? [v3: $i] : ?
% 13.20/2.50 | [v4: $i] : (cardinality_at_time(first_movers, v2) = v3 &
% 13.20/2.50 | growth_rate(first_movers, v2) = v4 & $i(v4) & $i(v3) & $i(v2) &
% 13.20/2.50 | greater_or_equal(v2, v1) & greater(v3, zero) & ~ greater(zero,
% 13.20/2.50 | v4)))
% 13.20/2.50 |
% 13.20/2.50 | ALPHA: (mp_non_empty_fm_and_ep) implies:
% 13.20/2.50 | (2) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 13.20/2.50 | (cardinality_at_time(first_movers, v1) = v2) | ~ $i(v1) | ~ $i(v0)
% 13.20/2.50 | | ~ greater(v2, zero) | ~ in_environment(v0, v1) | ~
% 13.20/2.50 | environment(v0) | subpopulations(first_movers, efficient_producers,
% 13.20/2.50 | v0, v1) | ? [v3: $i] : (cardinality_at_time(efficient_producers,
% 13.20/2.50 | v1) = v3 & $i(v3) & ~ greater(v3, zero)))
% 13.20/2.50 |
% 13.20/2.50 | ALPHA: (mp_EP_in_stable_environments) implies:
% 13.20/2.50 | (3) ! [v0: $i] : ! [v1: $i] : ( ~ (appear(efficient_producers, v0) = v1)
% 13.20/2.50 | | ~ $i(v0) | ~ stable(v0) | ~ environment(v0) | in_environment(v0,
% 13.20/2.50 | v1))
% 13.20/2.50 |
% 13.20/2.50 | ALPHA: (mp_greater_or_equal) implies:
% 13.20/2.50 | (4) ! [v0: $i] : ! [v1: $i] : ( ~ $i(v1) | ~ $i(v0) | ~ greater(v0, v1)
% 13.20/2.50 | | greater_or_equal(v0, v1))
% 13.20/2.50 | (5) ! [v0: $i] : ! [v1: $i] : (v1 = v0 | ~ $i(v1) | ~ $i(v0) | ~
% 13.20/2.50 | greater_or_equal(v0, v1) | greater(v0, v1))
% 13.20/2.50 |
% 13.20/2.50 | ALPHA: (t6) implies:
% 13.20/2.50 | (6) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~
% 13.20/2.50 | (appear(efficient_producers, v0) = v2) | ~
% 13.20/2.50 | (cardinality_at_time(efficient_producers, v1) = v3) | ~ $i(v1) | ~
% 13.20/2.50 | $i(v0) | ~ greater_or_equal(v1, v2) | ~ in_environment(v0, v1) | ~
% 13.20/2.51 | environment(v0) | greater(v3, zero))
% 13.20/2.51 |
% 13.20/2.51 | ALPHA: (l10) implies:
% 13.20/2.51 | (7) ! [v0: $i] : ! [v1: $i] : ( ~ (appear(efficient_producers, v0) = v1)
% 13.20/2.51 | | ~ $i(v0) | ~ stable(v0) | ~ environment(v0) | ? [v2: $i] :
% 13.20/2.51 | ($i(v2) & greater(v2, v1) & ! [v3: $i] : ! [v4: $i] : ( ~
% 13.20/2.51 | (growth_rate(first_movers, v3) = v4) | ~ $i(v3) | ~
% 13.20/2.51 | subpopulations(first_movers, efficient_producers, v0, v3) | ~
% 13.20/2.51 | greater_or_equal(v3, v2) | greater(zero, v4))))
% 13.20/2.51 |
% 13.20/2.51 | ALPHA: (prove_l9) implies:
% 13.20/2.51 | (8) ? [v0: $i] : ? [v1: $i] : (appear(efficient_producers, v0) = v1 &
% 13.20/2.51 | $i(v1) & $i(v0) & stable(v0) & environment(v0) & ! [v2: $i] : ( ~
% 13.20/2.51 | $i(v2) | ~ contracts_from(v2, first_movers) | ~ greater(v2, v1)))
% 13.20/2.51 |
% 13.20/2.51 | DELTA: instantiating (8) with fresh symbols all_13_0, all_13_1 gives:
% 13.20/2.51 | (9) appear(efficient_producers, all_13_1) = all_13_0 & $i(all_13_0) &
% 13.20/2.51 | $i(all_13_1) & stable(all_13_1) & environment(all_13_1) & ! [v0: $i] :
% 13.20/2.51 | ( ~ $i(v0) | ~ contracts_from(v0, first_movers) | ~ greater(v0,
% 13.20/2.51 | all_13_0))
% 13.20/2.51 |
% 13.20/2.51 | ALPHA: (9) implies:
% 13.20/2.51 | (10) environment(all_13_1)
% 13.20/2.51 | (11) stable(all_13_1)
% 13.20/2.51 | (12) $i(all_13_1)
% 13.20/2.51 | (13) $i(all_13_0)
% 13.20/2.51 | (14) appear(efficient_producers, all_13_1) = all_13_0
% 13.20/2.51 | (15) ! [v0: $i] : ( ~ $i(v0) | ~ contracts_from(v0, first_movers) | ~
% 13.20/2.51 | greater(v0, all_13_0))
% 13.20/2.51 |
% 13.20/2.51 | GROUND_INST: instantiating (3) with all_13_1, all_13_0, simplifying with (10),
% 13.20/2.51 | (11), (12), (14) gives:
% 13.20/2.51 | (16) in_environment(all_13_1, all_13_0)
% 13.20/2.51 |
% 13.20/2.51 | GROUND_INST: instantiating (7) with all_13_1, all_13_0, simplifying with (10),
% 13.20/2.51 | (11), (12), (14) gives:
% 13.20/2.51 | (17) ? [v0: $i] : ($i(v0) & greater(v0, all_13_0) & ! [v1: $i] : ! [v2:
% 13.20/2.51 | $i] : ( ~ (growth_rate(first_movers, v1) = v2) | ~ $i(v1) | ~
% 13.20/2.51 | subpopulations(first_movers, efficient_producers, all_13_1, v1) |
% 13.20/2.51 | ~ greater_or_equal(v1, v0) | greater(zero, v2)))
% 13.20/2.51 |
% 13.20/2.51 | DELTA: instantiating (17) with fresh symbol all_22_0 gives:
% 13.20/2.51 | (18) $i(all_22_0) & greater(all_22_0, all_13_0) & ! [v0: $i] : ! [v1: $i]
% 13.20/2.51 | : ( ~ (growth_rate(first_movers, v0) = v1) | ~ $i(v0) | ~
% 13.20/2.51 | subpopulations(first_movers, efficient_producers, all_13_1, v0) | ~
% 13.20/2.51 | greater_or_equal(v0, all_22_0) | greater(zero, v1))
% 13.20/2.51 |
% 13.20/2.51 | ALPHA: (18) implies:
% 13.20/2.51 | (19) greater(all_22_0, all_13_0)
% 13.20/2.51 | (20) $i(all_22_0)
% 13.20/2.51 | (21) ! [v0: $i] : ! [v1: $i] : ( ~ (growth_rate(first_movers, v0) = v1) |
% 13.20/2.51 | ~ $i(v0) | ~ subpopulations(first_movers, efficient_producers,
% 13.20/2.51 | all_13_1, v0) | ~ greater_or_equal(v0, all_22_0) | greater(zero,
% 13.20/2.51 | v1))
% 13.20/2.51 |
% 13.20/2.51 | GROUND_INST: instantiating (mp_long_stable_environments) with all_13_1,
% 13.20/2.51 | all_13_0, all_22_0, simplifying with (10), (11), (12), (13),
% 13.20/2.51 | (16), (19), (20) gives:
% 13.20/2.51 | (22) in_environment(all_13_1, all_22_0)
% 13.20/2.51 |
% 13.20/2.51 | GROUND_INST: instantiating (4) with all_22_0, all_13_0, simplifying with (13),
% 13.20/2.51 | (19), (20) gives:
% 13.20/2.51 | (23) greater_or_equal(all_22_0, all_13_0)
% 13.20/2.51 |
% 13.20/2.52 | GROUND_INST: instantiating (1) with all_13_1, all_22_0, simplifying with (10),
% 13.20/2.52 | (11), (12), (20), (22) gives:
% 13.20/2.52 | (24) contracts_from(all_22_0, first_movers) | ? [v0: $i] : ? [v1: $i] :
% 13.20/2.52 | ? [v2: $i] : (cardinality_at_time(first_movers, v0) = v1 &
% 13.20/2.52 | growth_rate(first_movers, v0) = v2 & $i(v2) & $i(v1) & $i(v0) &
% 13.20/2.52 | greater_or_equal(v0, all_22_0) & greater(v1, zero) & ~
% 13.20/2.52 | greater(zero, v2))
% 13.20/2.52 |
% 13.20/2.52 | BETA: splitting (24) gives:
% 13.20/2.52 |
% 13.20/2.52 | Case 1:
% 13.20/2.52 | |
% 13.20/2.52 | | (25) contracts_from(all_22_0, first_movers)
% 13.20/2.52 | |
% 13.20/2.52 | | GROUND_INST: instantiating (15) with all_22_0, simplifying with (19), (20),
% 13.20/2.52 | | (25) gives:
% 13.20/2.52 | | (26) $false
% 13.20/2.52 | |
% 13.20/2.52 | | CLOSE: (26) is inconsistent.
% 13.20/2.52 | |
% 13.20/2.52 | Case 2:
% 13.20/2.52 | |
% 13.20/2.52 | | (27) ? [v0: $i] : ? [v1: $i] : ? [v2: $i] :
% 13.20/2.52 | | (cardinality_at_time(first_movers, v0) = v1 &
% 13.20/2.52 | | growth_rate(first_movers, v0) = v2 & $i(v2) & $i(v1) & $i(v0) &
% 13.20/2.52 | | greater_or_equal(v0, all_22_0) & greater(v1, zero) & ~
% 13.20/2.52 | | greater(zero, v2))
% 13.20/2.52 | |
% 13.20/2.52 | | DELTA: instantiating (27) with fresh symbols all_66_0, all_66_1, all_66_2
% 13.20/2.52 | | gives:
% 13.20/2.52 | | (28) cardinality_at_time(first_movers, all_66_2) = all_66_1 &
% 13.20/2.52 | | growth_rate(first_movers, all_66_2) = all_66_0 & $i(all_66_0) &
% 13.20/2.52 | | $i(all_66_1) & $i(all_66_2) & greater_or_equal(all_66_2, all_22_0) &
% 13.20/2.52 | | greater(all_66_1, zero) & ~ greater(zero, all_66_0)
% 13.20/2.52 | |
% 13.20/2.52 | | ALPHA: (28) implies:
% 13.20/2.52 | | (29) ~ greater(zero, all_66_0)
% 13.20/2.52 | | (30) greater(all_66_1, zero)
% 13.20/2.52 | | (31) greater_or_equal(all_66_2, all_22_0)
% 13.20/2.52 | | (32) $i(all_66_2)
% 13.20/2.52 | | (33) growth_rate(first_movers, all_66_2) = all_66_0
% 13.20/2.52 | | (34) cardinality_at_time(first_movers, all_66_2) = all_66_1
% 13.20/2.52 | |
% 13.20/2.52 | | GROUND_INST: instantiating (5) with all_66_2, all_22_0, simplifying with
% 13.20/2.52 | | (20), (31), (32) gives:
% 13.20/2.52 | | (35) all_66_2 = all_22_0 | greater(all_66_2, all_22_0)
% 13.20/2.52 | |
% 13.20/2.52 | | GROUND_INST: instantiating (2) with all_13_1, all_22_0, all_66_1,
% 13.20/2.52 | | simplifying with (10), (12), (20), (22), (30) gives:
% 13.20/2.52 | | (36) ~ (cardinality_at_time(first_movers, all_22_0) = all_66_1) |
% 13.20/2.52 | | subpopulations(first_movers, efficient_producers, all_13_1,
% 13.20/2.52 | | all_22_0) | ? [v0: $i] :
% 13.20/2.52 | | (cardinality_at_time(efficient_producers, all_22_0) = v0 & $i(v0) &
% 13.20/2.52 | | ~ greater(v0, zero))
% 13.20/2.52 | |
% 13.20/2.52 | | GROUND_INST: instantiating (2) with all_13_1, all_66_2, all_66_2,
% 13.20/2.52 | | simplifying with (10), (12), (32) gives:
% 13.20/2.52 | | (37) ~ (cardinality_at_time(first_movers, all_66_2) = all_66_2) | ~
% 13.20/2.52 | | greater(all_66_2, zero) | ~ in_environment(all_13_1, all_66_2) |
% 13.20/2.52 | | subpopulations(first_movers, efficient_producers, all_13_1,
% 13.20/2.52 | | all_66_2) | ? [v0: $i] :
% 13.20/2.52 | | (cardinality_at_time(efficient_producers, all_66_2) = v0 & $i(v0) &
% 13.20/2.52 | | ~ greater(v0, zero))
% 13.20/2.52 | |
% 13.20/2.52 | | BETA: splitting (36) gives:
% 13.20/2.52 | |
% 13.20/2.52 | | Case 1:
% 13.20/2.52 | | |
% 13.20/2.52 | | | (38) subpopulations(first_movers, efficient_producers, all_13_1,
% 13.20/2.52 | | | all_22_0)
% 13.20/2.52 | | |
% 13.20/2.52 | | | BETA: splitting (35) gives:
% 13.20/2.52 | | |
% 13.20/2.52 | | | Case 1:
% 13.20/2.52 | | | |
% 13.20/2.52 | | | | (39) greater(all_66_2, all_22_0)
% 13.20/2.52 | | | |
% 13.20/2.53 | | | | REF_CLOSE: (2), (4), (6), (10), (11), (12), (13), (14), (19), (20),
% 13.20/2.53 | | | | (21), (22), (29), (30), (31), (32), (33), (34), (37), (39),
% 13.20/2.53 | | | | (mp_greater_transitivity), (mp_long_stable_environments) are
% 13.20/2.53 | | | | inconsistent by sub-proof #1.
% 13.20/2.53 | | | |
% 13.20/2.53 | | | Case 2:
% 13.20/2.53 | | | |
% 13.20/2.53 | | | | (40) all_66_2 = all_22_0
% 13.20/2.53 | | | |
% 13.20/2.53 | | | | REDUCE: (33), (40) imply:
% 13.20/2.53 | | | | (41) growth_rate(first_movers, all_22_0) = all_66_0
% 13.20/2.53 | | | |
% 13.20/2.53 | | | | REDUCE: (31), (40) imply:
% 13.20/2.53 | | | | (42) greater_or_equal(all_22_0, all_22_0)
% 13.20/2.53 | | | |
% 13.20/2.53 | | | | GROUND_INST: instantiating (21) with all_22_0, all_66_0, simplifying
% 13.20/2.53 | | | | with (20), (29), (38), (41), (42) gives:
% 13.20/2.53 | | | | (43) $false
% 13.20/2.53 | | | |
% 13.20/2.53 | | | | CLOSE: (43) is inconsistent.
% 13.20/2.53 | | | |
% 13.20/2.53 | | | End of split
% 13.20/2.53 | | |
% 13.20/2.53 | | Case 2:
% 13.20/2.53 | | |
% 13.20/2.53 | | | (44) ~ (cardinality_at_time(first_movers, all_22_0) = all_66_1) | ?
% 13.20/2.53 | | | [v0: $i] : (cardinality_at_time(efficient_producers, all_22_0) =
% 13.20/2.53 | | | v0 & $i(v0) & ~ greater(v0, zero))
% 13.20/2.53 | | |
% 13.20/2.53 | | | BETA: splitting (35) gives:
% 13.20/2.53 | | |
% 13.20/2.53 | | | Case 1:
% 13.20/2.53 | | | |
% 13.20/2.53 | | | | (45) greater(all_66_2, all_22_0)
% 13.20/2.53 | | | |
% 13.20/2.53 | | | | REF_CLOSE: (2), (4), (6), (10), (11), (12), (13), (14), (19), (20),
% 13.20/2.53 | | | | (21), (22), (29), (30), (31), (32), (33), (34), (37), (45),
% 13.20/2.53 | | | | (mp_greater_transitivity), (mp_long_stable_environments) are
% 13.20/2.53 | | | | inconsistent by sub-proof #1.
% 13.20/2.53 | | | |
% 13.20/2.53 | | | Case 2:
% 13.20/2.53 | | | |
% 13.20/2.53 | | | | (46) all_66_2 = all_22_0
% 13.20/2.53 | | | |
% 13.20/2.53 | | | | REDUCE: (34), (46) imply:
% 13.20/2.53 | | | | (47) cardinality_at_time(first_movers, all_22_0) = all_66_1
% 13.20/2.53 | | | |
% 13.20/2.53 | | | | BETA: splitting (44) gives:
% 13.20/2.53 | | | |
% 13.20/2.53 | | | | Case 1:
% 13.20/2.53 | | | | |
% 13.20/2.53 | | | | | (48) ~ (cardinality_at_time(first_movers, all_22_0) = all_66_1)
% 13.20/2.53 | | | | |
% 13.20/2.53 | | | | | PRED_UNIFY: (47), (48) imply:
% 13.20/2.53 | | | | | (49) $false
% 13.20/2.53 | | | | |
% 13.20/2.53 | | | | | CLOSE: (49) is inconsistent.
% 13.20/2.53 | | | | |
% 13.20/2.53 | | | | Case 2:
% 13.20/2.53 | | | | |
% 13.20/2.53 | | | | | (50) ? [v0: $i] : (cardinality_at_time(efficient_producers,
% 13.20/2.53 | | | | | all_22_0) = v0 & $i(v0) & ~ greater(v0, zero))
% 13.20/2.53 | | | | |
% 13.20/2.53 | | | | | DELTA: instantiating (50) with fresh symbol all_127_0 gives:
% 13.20/2.53 | | | | | (51) cardinality_at_time(efficient_producers, all_22_0) = all_127_0
% 13.20/2.53 | | | | | & $i(all_127_0) & ~ greater(all_127_0, zero)
% 13.20/2.53 | | | | |
% 13.20/2.53 | | | | | ALPHA: (51) implies:
% 13.20/2.53 | | | | | (52) ~ greater(all_127_0, zero)
% 13.20/2.53 | | | | | (53) cardinality_at_time(efficient_producers, all_22_0) = all_127_0
% 13.20/2.53 | | | | |
% 13.20/2.53 | | | | | GROUND_INST: instantiating (6) with all_13_1, all_22_0, all_13_0,
% 13.20/2.53 | | | | | all_127_0, simplifying with (10), (12), (14), (20), (22),
% 13.20/2.53 | | | | | (23), (52), (53) gives:
% 13.20/2.53 | | | | | (54) $false
% 13.20/2.53 | | | | |
% 13.20/2.53 | | | | | CLOSE: (54) is inconsistent.
% 13.20/2.53 | | | | |
% 13.20/2.53 | | | | End of split
% 13.20/2.53 | | | |
% 13.20/2.53 | | | End of split
% 13.20/2.53 | | |
% 13.20/2.53 | | End of split
% 13.20/2.53 | |
% 13.20/2.53 | End of split
% 13.20/2.53 |
% 13.20/2.53 End of proof
% 13.20/2.54
% 13.20/2.54 Sub-proof #1 shows that the following formulas are inconsistent:
% 13.20/2.54 ----------------------------------------------------------------
% 13.20/2.54 (1) $i(all_22_0)
% 13.20/2.54 (2) ! [v0: $i] : ! [v1: $i] : ( ~ (growth_rate(first_movers, v0) = v1) | ~
% 13.20/2.54 $i(v0) | ~ subpopulations(first_movers, efficient_producers, all_13_1,
% 13.20/2.54 v0) | ~ greater_or_equal(v0, all_22_0) | greater(zero, v1))
% 13.20/2.54 (3) greater(all_22_0, all_13_0)
% 13.20/2.54 (4) cardinality_at_time(first_movers, all_66_2) = all_66_1
% 13.20/2.54 (5) greater(all_66_2, all_22_0)
% 13.20/2.54 (6) greater_or_equal(all_66_2, all_22_0)
% 13.20/2.54 (7) appear(efficient_producers, all_13_1) = all_13_0
% 13.20/2.54 (8) environment(all_13_1)
% 13.20/2.54 (9) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~
% 13.20/2.54 (cardinality_at_time(first_movers, v1) = v2) | ~ $i(v1) | ~ $i(v0) |
% 13.20/2.54 ~ greater(v2, zero) | ~ in_environment(v0, v1) | ~ environment(v0) |
% 13.20/2.54 subpopulations(first_movers, efficient_producers, v0, v1) | ? [v3: $i]
% 13.20/2.54 : (cardinality_at_time(efficient_producers, v1) = v3 & $i(v3) & ~
% 13.20/2.54 greater(v3, zero)))
% 13.20/2.54 (10) ~ (cardinality_at_time(first_movers, all_66_2) = all_66_2) | ~
% 13.20/2.54 greater(all_66_2, zero) | ~ in_environment(all_13_1, all_66_2) |
% 13.20/2.54 subpopulations(first_movers, efficient_producers, all_13_1, all_66_2) |
% 13.20/2.54 ? [v0: $i] : (cardinality_at_time(efficient_producers, all_66_2) = v0 &
% 13.20/2.54 $i(v0) & ~ greater(v0, zero))
% 13.20/2.54 (11) $i(all_13_1)
% 13.20/2.54 (12) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ! [v3: $i] : ( ~
% 13.20/2.54 (appear(efficient_producers, v0) = v2) | ~
% 13.20/2.54 (cardinality_at_time(efficient_producers, v1) = v3) | ~ $i(v1) | ~
% 13.20/2.54 $i(v0) | ~ greater_or_equal(v1, v2) | ~ in_environment(v0, v1) | ~
% 13.20/2.54 environment(v0) | greater(v3, zero))
% 13.20/2.54 (13) ! [v0: $i] : ! [v1: $i] : ( ~ $i(v1) | ~ $i(v0) | ~ greater(v0, v1)
% 13.20/2.54 | greater_or_equal(v0, v1))
% 13.20/2.54 (14) greater(all_66_1, zero)
% 13.20/2.54 (15) growth_rate(first_movers, all_66_2) = all_66_0
% 13.20/2.54 (16) stable(all_13_1)
% 13.20/2.54 (17) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ $i(v2) | ~ $i(v1) | ~
% 13.20/2.54 $i(v0) | ~ greater(v2, v1) | ~ in_environment(v0, v1) | ~
% 13.20/2.54 stable(v0) | ~ environment(v0) | in_environment(v0, v2))
% 13.20/2.54 (18) in_environment(all_13_1, all_22_0)
% 13.20/2.54 (19) ~ greater(zero, all_66_0)
% 13.20/2.54 (20) $i(all_66_2)
% 13.20/2.54 (21) $i(all_13_0)
% 13.20/2.54 (22) ! [v0: $i] : ! [v1: $i] : ! [v2: $i] : ( ~ $i(v2) | ~ $i(v1) | ~
% 13.20/2.54 $i(v0) | ~ greater(v1, v2) | ~ greater(v0, v1) | greater(v0, v2))
% 13.20/2.54
% 13.20/2.54 Begin of proof
% 13.20/2.54 |
% 13.20/2.54 | GROUND_INST: instantiating (17) with all_13_1, all_22_0, all_66_2, simplifying
% 13.20/2.54 | with (1), (5), (8), (11), (16), (18), (20) gives:
% 13.20/2.54 | (23) in_environment(all_13_1, all_66_2)
% 13.20/2.54 |
% 13.20/2.54 | GROUND_INST: instantiating (22) with all_66_2, all_22_0, all_13_0, simplifying
% 13.20/2.54 | with (1), (3), (5), (20), (21) gives:
% 13.20/2.54 | (24) greater(all_66_2, all_13_0)
% 13.20/2.54 |
% 13.20/2.54 | GROUND_INST: instantiating (9) with all_13_1, all_66_2, all_66_1, simplifying
% 13.20/2.54 | with (4), (8), (11), (14), (20), (23) gives:
% 13.20/2.54 | (25) subpopulations(first_movers, efficient_producers, all_13_1, all_66_2)
% 13.20/2.54 | | ? [v0: $i] : (cardinality_at_time(efficient_producers, all_66_2) =
% 13.20/2.54 | v0 & $i(v0) & ~ greater(v0, zero))
% 13.20/2.54 |
% 13.20/2.54 | GROUND_INST: instantiating (13) with all_66_2, all_13_0, simplifying with
% 13.20/2.54 | (20), (21), (24) gives:
% 13.20/2.55 | (26) greater_or_equal(all_66_2, all_13_0)
% 13.20/2.55 |
% 13.20/2.55 | BETA: splitting (10) gives:
% 13.20/2.55 |
% 13.20/2.55 | Case 1:
% 13.20/2.55 | |
% 13.20/2.55 | | (27) subpopulations(first_movers, efficient_producers, all_13_1,
% 13.20/2.55 | | all_66_2)
% 13.20/2.55 | |
% 13.20/2.55 | | GROUND_INST: instantiating (2) with all_66_2, all_66_0, simplifying with
% 13.20/2.55 | | (6), (15), (19), (20), (27) gives:
% 13.20/2.55 | | (28) $false
% 13.20/2.55 | |
% 13.20/2.55 | | CLOSE: (28) is inconsistent.
% 13.20/2.55 | |
% 13.20/2.55 | Case 2:
% 13.20/2.55 | |
% 13.20/2.55 | | (29) ~ subpopulations(first_movers, efficient_producers, all_13_1,
% 13.20/2.55 | | all_66_2)
% 13.20/2.55 | |
% 13.20/2.55 | | BETA: splitting (25) gives:
% 13.20/2.55 | |
% 13.20/2.55 | | Case 1:
% 13.20/2.55 | | |
% 13.20/2.55 | | | (30) subpopulations(first_movers, efficient_producers, all_13_1,
% 13.20/2.55 | | | all_66_2)
% 13.20/2.55 | | |
% 13.20/2.55 | | | PRED_UNIFY: (29), (30) imply:
% 13.20/2.55 | | | (31) $false
% 13.20/2.55 | | |
% 13.20/2.55 | | | CLOSE: (31) is inconsistent.
% 13.20/2.55 | | |
% 13.20/2.55 | | Case 2:
% 13.20/2.55 | | |
% 13.20/2.55 | | | (32) ? [v0: $i] : (cardinality_at_time(efficient_producers, all_66_2)
% 13.20/2.55 | | | = v0 & $i(v0) & ~ greater(v0, zero))
% 13.20/2.55 | | |
% 13.20/2.55 | | | DELTA: instantiating (32) with fresh symbol all_211_0 gives:
% 13.20/2.55 | | | (33) cardinality_at_time(efficient_producers, all_66_2) = all_211_0 &
% 13.20/2.55 | | | $i(all_211_0) & ~ greater(all_211_0, zero)
% 13.20/2.55 | | |
% 13.20/2.55 | | | ALPHA: (33) implies:
% 13.20/2.55 | | | (34) ~ greater(all_211_0, zero)
% 13.20/2.55 | | | (35) cardinality_at_time(efficient_producers, all_66_2) = all_211_0
% 13.20/2.55 | | |
% 13.20/2.55 | | | GROUND_INST: instantiating (12) with all_13_1, all_66_2, all_13_0,
% 13.20/2.55 | | | all_211_0, simplifying with (7), (8), (11), (20), (23), (26),
% 13.20/2.55 | | | (34), (35) gives:
% 13.20/2.55 | | | (36) $false
% 13.20/2.55 | | |
% 13.20/2.55 | | | CLOSE: (36) is inconsistent.
% 13.20/2.55 | | |
% 13.20/2.55 | | End of split
% 13.20/2.55 | |
% 13.20/2.55 | End of split
% 13.20/2.55 |
% 13.20/2.55 End of proof
% 13.20/2.55 % SZS output end Proof for theBenchmark
% 13.20/2.55
% 13.20/2.55 1955ms
%------------------------------------------------------------------------------