TSTP Solution File: MGT036+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : MGT036+1 : TPTP v8.1.2. Released v2.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n019.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 09:17:07 EDT 2023
% Result : Theorem 0.21s 0.42s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : MGT036+1 : TPTP v8.1.2. Released v2.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.36 % Computer : n019.cluster.edu
% 0.13/0.36 % Model : x86_64 x86_64
% 0.13/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.36 % Memory : 8042.1875MB
% 0.13/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.36 % CPULimit : 300
% 0.13/0.36 % WCLimit : 300
% 0.13/0.36 % DateTime : Mon Aug 28 06:22:58 EDT 2023
% 0.13/0.36 % CPUTime :
% 0.21/0.42 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.21/0.42
% 0.21/0.42 % SZS status Theorem
% 0.21/0.42
% 0.21/0.43 % SZS output start Proof
% 0.21/0.43 Take the following subset of the input axioms:
% 0.21/0.44 fof(a12, hypothesis, ![E, T, S1, S2]: ((environment(E) & (in_environment(E, T) & (~greater(zero, growth_rate(S1, T)) & greater(resilience(S2), resilience(S1))))) => ~greater(zero, growth_rate(S2, T)))).
% 0.21/0.44 fof(a2, hypothesis, greater(resilience(efficient_producers), resilience(first_movers))).
% 0.21/0.44 fof(d2, hypothesis, ![E2, T2, S1_2, S2_2]: ((environment(E2) & subpopulations(S1_2, S2_2, E2, T2)) => ((greater_or_equal(growth_rate(S2_2, T2), zero) & greater(zero, growth_rate(S1_2, T2))) <=> outcompetes(S2_2, S1_2, T2)))).
% 0.21/0.44 fof(mp_growth_rate_relationships, axiom, ![E2, T2, S1_2, S2_2]: (((environment(E2) & subpopulations(S1_2, S2_2, E2, T2)) => greater_or_equal(growth_rate(S1_2, T2), zero)) <=> ~greater(zero, growth_rate(S1_2, T2)))).
% 0.21/0.44 fof(mp_symmetry_of_FM_and_EP, axiom, ![E2, T2]: ((environment(E2) & subpopulations(first_movers, efficient_producers, E2, T2)) => subpopulations(efficient_producers, first_movers, E2, T2))).
% 0.21/0.44 fof(mp_time_point_occur, axiom, ![E2, T2]: ((environment(E2) & subpopulations(first_movers, efficient_producers, E2, T2)) => in_environment(E2, T2))).
% 0.21/0.44 fof(prove_t5, conjecture, ![E2, T2]: ((environment(E2) & subpopulations(first_movers, efficient_producers, E2, T2)) => ~outcompetes(first_movers, efficient_producers, T2))).
% 0.21/0.44
% 0.21/0.44 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.44 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.44 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.44 fresh(y, y, x1...xn) = u
% 0.21/0.44 C => fresh(s, t, x1...xn) = v
% 0.21/0.44 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.44 variables of u and v.
% 0.21/0.44 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.44 input problem has no model of domain size 1).
% 0.21/0.44
% 0.21/0.44 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.44
% 0.21/0.44 Axiom 1 (prove_t5): environment(e) = true2.
% 0.21/0.44 Axiom 2 (prove_t5_2): outcompetes(first_movers, efficient_producers, t) = true2.
% 0.21/0.44 Axiom 3 (mp_growth_rate_relationships_2): fresh6(X, X, Y) = true2.
% 0.21/0.44 Axiom 4 (a2): greater(resilience(efficient_producers), resilience(first_movers)) = true2.
% 0.21/0.44 Axiom 5 (prove_t5_1): subpopulations(first_movers, efficient_producers, e, t) = true2.
% 0.21/0.44 Axiom 6 (mp_time_point_occur): fresh(X, X, Y, Z) = true2.
% 0.21/0.44 Axiom 7 (d2_2): fresh20(X, X, Y, Z) = true2.
% 0.21/0.44 Axiom 8 (d2_1): fresh18(X, X, Y, Z) = true2.
% 0.21/0.44 Axiom 9 (a12): fresh12(X, X, Y, Z) = true2.
% 0.21/0.44 Axiom 10 (mp_symmetry_of_FM_and_EP): fresh4(X, X, Y, Z) = subpopulations(efficient_producers, first_movers, Y, Z).
% 0.21/0.44 Axiom 11 (mp_symmetry_of_FM_and_EP): fresh3(X, X, Y, Z) = true2.
% 0.21/0.44 Axiom 12 (mp_time_point_occur): fresh2(X, X, Y, Z) = in_environment(Y, Z).
% 0.21/0.44 Axiom 13 (a12): fresh11(X, X, Y, Z, W) = fresh12(environment(Y), true2, Z, W).
% 0.21/0.44 Axiom 14 (a12): fresh10(X, X, Y, Z, W) = greater(zero, growth_rate(Z, W)).
% 0.21/0.44 Axiom 15 (d2_1): fresh8(X, X, Y, Z, W) = greater_or_equal(growth_rate(Z, W), zero).
% 0.21/0.44 Axiom 16 (d2_2): fresh7(X, X, Y, Z, W) = greater(zero, growth_rate(Z, W)).
% 0.21/0.44 Axiom 17 (d2_2): fresh19(X, X, Y, Z, W, V) = fresh20(environment(Y), true2, Z, V).
% 0.21/0.44 Axiom 18 (d2_1): fresh17(X, X, Y, Z, W, V) = fresh18(environment(Y), true2, W, V).
% 0.21/0.44 Axiom 19 (a12): fresh9(X, X, Y, Z, W, V) = fresh10(in_environment(Y, V), true2, Y, Z, V).
% 0.21/0.44 Axiom 20 (mp_growth_rate_relationships_2): fresh6(greater(zero, growth_rate(X, Y)), true2, Z) = environment(Z).
% 0.21/0.44 Axiom 21 (mp_symmetry_of_FM_and_EP): fresh4(subpopulations(first_movers, efficient_producers, X, Y), true2, X, Y) = fresh3(environment(X), true2, X, Y).
% 0.21/0.44 Axiom 22 (mp_time_point_occur): fresh2(subpopulations(first_movers, efficient_producers, X, Y), true2, X, Y) = fresh(environment(X), true2, X, Y).
% 0.21/0.44 Axiom 23 (d2_1): fresh17(outcompetes(X, Y, Z), true2, W, Y, X, Z) = fresh8(subpopulations(Y, X, W, Z), true2, W, X, Z).
% 0.21/0.44 Axiom 24 (d2_2): fresh19(outcompetes(X, Y, Z), true2, W, Y, X, Z) = fresh7(subpopulations(Y, X, W, Z), true2, W, Y, Z).
% 0.21/0.44 Axiom 25 (a12): fresh9(greater(resilience(X), resilience(Y)), true2, Z, Y, X, W) = fresh11(greater(zero, growth_rate(X, W)), true2, Z, Y, W).
% 0.21/0.44
% 0.21/0.44 Lemma 26: subpopulations(efficient_producers, first_movers, e, t) = true2.
% 0.21/0.44 Proof:
% 0.21/0.44 subpopulations(efficient_producers, first_movers, e, t)
% 0.21/0.44 = { by axiom 10 (mp_symmetry_of_FM_and_EP) R->L }
% 0.21/0.44 fresh4(true2, true2, e, t)
% 0.21/0.44 = { by axiom 5 (prove_t5_1) R->L }
% 0.21/0.44 fresh4(subpopulations(first_movers, efficient_producers, e, t), true2, e, t)
% 0.21/0.44 = { by axiom 21 (mp_symmetry_of_FM_and_EP) }
% 0.21/0.44 fresh3(environment(e), true2, e, t)
% 0.21/0.44 = { by axiom 1 (prove_t5) }
% 0.21/0.44 fresh3(true2, true2, e, t)
% 0.21/0.44 = { by axiom 11 (mp_symmetry_of_FM_and_EP) }
% 0.21/0.44 true2
% 0.21/0.44
% 0.21/0.44 Lemma 27: greater(zero, growth_rate(efficient_producers, t)) = true2.
% 0.21/0.44 Proof:
% 0.21/0.44 greater(zero, growth_rate(efficient_producers, t))
% 0.21/0.44 = { by axiom 16 (d2_2) R->L }
% 0.21/0.44 fresh7(true2, true2, e, efficient_producers, t)
% 0.21/0.44 = { by lemma 26 R->L }
% 0.21/0.44 fresh7(subpopulations(efficient_producers, first_movers, e, t), true2, e, efficient_producers, t)
% 0.21/0.44 = { by axiom 24 (d2_2) R->L }
% 0.21/0.44 fresh19(outcompetes(first_movers, efficient_producers, t), true2, e, efficient_producers, first_movers, t)
% 0.21/0.44 = { by axiom 2 (prove_t5_2) }
% 0.21/0.44 fresh19(true2, true2, e, efficient_producers, first_movers, t)
% 0.21/0.44 = { by axiom 17 (d2_2) }
% 0.21/0.44 fresh20(environment(e), true2, efficient_producers, t)
% 0.21/0.44 = { by axiom 1 (prove_t5) }
% 0.21/0.44 fresh20(true2, true2, efficient_producers, t)
% 0.21/0.44 = { by axiom 7 (d2_2) }
% 0.21/0.44 true2
% 0.21/0.44
% 0.21/0.44 Goal 1 (mp_growth_rate_relationships_1): tuple(greater_or_equal(growth_rate(X, Y), zero), greater(zero, growth_rate(X, Y))) = tuple(true2, true2).
% 0.21/0.44 The goal is true when:
% 0.21/0.44 X = first_movers
% 0.21/0.44 Y = t
% 0.21/0.44
% 0.21/0.44 Proof:
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), greater(zero, growth_rate(first_movers, t)))
% 0.21/0.44 = { by axiom 14 (a12) R->L }
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), fresh10(true2, true2, e, first_movers, t))
% 0.21/0.44 = { by axiom 6 (mp_time_point_occur) R->L }
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), fresh10(fresh(true2, true2, e, t), true2, e, first_movers, t))
% 0.21/0.44 = { by axiom 1 (prove_t5) R->L }
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), fresh10(fresh(environment(e), true2, e, t), true2, e, first_movers, t))
% 0.21/0.44 = { by axiom 22 (mp_time_point_occur) R->L }
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), fresh10(fresh2(subpopulations(first_movers, efficient_producers, e, t), true2, e, t), true2, e, first_movers, t))
% 0.21/0.44 = { by axiom 5 (prove_t5_1) }
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), fresh10(fresh2(true2, true2, e, t), true2, e, first_movers, t))
% 0.21/0.44 = { by axiom 12 (mp_time_point_occur) }
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), fresh10(in_environment(e, t), true2, e, first_movers, t))
% 0.21/0.44 = { by axiom 19 (a12) R->L }
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), fresh9(true2, true2, e, first_movers, efficient_producers, t))
% 0.21/0.44 = { by axiom 4 (a2) R->L }
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), fresh9(greater(resilience(efficient_producers), resilience(first_movers)), true2, e, first_movers, efficient_producers, t))
% 0.21/0.44 = { by axiom 25 (a12) }
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), fresh11(greater(zero, growth_rate(efficient_producers, t)), true2, e, first_movers, t))
% 0.21/0.44 = { by lemma 27 }
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), fresh11(true2, true2, e, first_movers, t))
% 0.21/0.44 = { by axiom 13 (a12) }
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), fresh12(environment(e), true2, first_movers, t))
% 0.21/0.44 = { by axiom 20 (mp_growth_rate_relationships_2) R->L }
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), fresh12(fresh6(greater(zero, growth_rate(efficient_producers, t)), true2, e), true2, first_movers, t))
% 0.21/0.44 = { by lemma 27 }
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), fresh12(fresh6(true2, true2, e), true2, first_movers, t))
% 0.21/0.44 = { by axiom 3 (mp_growth_rate_relationships_2) }
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), fresh12(true2, true2, first_movers, t))
% 0.21/0.44 = { by axiom 9 (a12) }
% 0.21/0.44 tuple(greater_or_equal(growth_rate(first_movers, t), zero), true2)
% 0.21/0.44 = { by axiom 15 (d2_1) R->L }
% 0.21/0.44 tuple(fresh8(true2, true2, e, first_movers, t), true2)
% 0.21/0.44 = { by lemma 26 R->L }
% 0.21/0.44 tuple(fresh8(subpopulations(efficient_producers, first_movers, e, t), true2, e, first_movers, t), true2)
% 0.21/0.44 = { by axiom 23 (d2_1) R->L }
% 0.21/0.44 tuple(fresh17(outcompetes(first_movers, efficient_producers, t), true2, e, efficient_producers, first_movers, t), true2)
% 0.21/0.44 = { by axiom 2 (prove_t5_2) }
% 0.21/0.44 tuple(fresh17(true2, true2, e, efficient_producers, first_movers, t), true2)
% 0.21/0.44 = { by axiom 18 (d2_1) }
% 0.21/0.44 tuple(fresh18(environment(e), true2, first_movers, t), true2)
% 0.21/0.44 = { by axiom 1 (prove_t5) }
% 0.21/0.44 tuple(fresh18(true2, true2, first_movers, t), true2)
% 0.21/0.44 = { by axiom 8 (d2_1) }
% 0.21/0.44 tuple(true2, true2)
% 0.21/0.44 % SZS output end Proof
% 0.21/0.44
% 0.21/0.44 RESULT: Theorem (the conjecture is true).
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