TSTP Solution File: MGT032+2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : MGT032+2 : 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 : n028.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 09:17:05 EDT 2023
% Result : Theorem 0.21s 0.40s
% 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 : MGT032+2 : 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
% 0.14/0.35 % Computer : n028.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Mon Aug 28 06:50:54 EDT 2023
% 0.14/0.36 % CPUTime :
% 0.21/0.40 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 0.21/0.40
% 0.21/0.40 % SZS status Theorem
% 0.21/0.40
% 0.21/0.41 % SZS output start Proof
% 0.21/0.41 Take the following subset of the input axioms:
% 0.21/0.41 fof(l1, hypothesis, ![E]: ((environment(E) & stable(E)) => ?[To]: (in_environment(E, To) & ![T]: ((subpopulations(first_movers, efficient_producers, E, T) & greater_or_equal(T, To)) => greater(growth_rate(efficient_producers, T), growth_rate(first_movers, T)))))).
% 0.21/0.41 fof(mp1_high_growth_rates, axiom, ![S1, S2, T2, E2]: ((environment(E2) & (subpopulations(S1, S2, E2, T2) & greater(growth_rate(S2, T2), growth_rate(S1, T2)))) => selection_favors(S2, S1, T2))).
% 0.21/0.41 fof(prove_t1, conjecture, ![E2]: ((environment(E2) & stable(E2)) => ?[To2]: (in_environment(E2, To2) & ![T2]: ((subpopulations(first_movers, efficient_producers, E2, T2) & greater_or_equal(T2, To2)) => selection_favors(efficient_producers, first_movers, T2))))).
% 0.21/0.41
% 0.21/0.41 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.41 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.41 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.41 fresh(y, y, x1...xn) = u
% 0.21/0.41 C => fresh(s, t, x1...xn) = v
% 0.21/0.41 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.41 variables of u and v.
% 0.21/0.41 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.41 input problem has no model of domain size 1).
% 0.21/0.41
% 0.21/0.41 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.41
% 0.21/0.41 Axiom 1 (prove_t1): environment(e) = true2.
% 0.21/0.41 Axiom 2 (prove_t1_1): stable(e) = true2.
% 0.21/0.41 Axiom 3 (prove_t1_4): fresh(X, X, Y) = true2.
% 0.21/0.41 Axiom 4 (l1): fresh9(X, X, Y) = true2.
% 0.21/0.41 Axiom 5 (l1_1): fresh5(X, X, Y) = in_environment(Y, to(Y)).
% 0.21/0.41 Axiom 6 (l1_1): fresh4(X, X, Y) = true2.
% 0.21/0.41 Axiom 7 (prove_t1_3): fresh2(X, X, Y) = true2.
% 0.21/0.41 Axiom 8 (prove_t1_4): fresh(in_environment(e, X), true2, X) = greater_or_equal(t(X), X).
% 0.21/0.41 Axiom 9 (l1): fresh8(X, X, Y, Z) = fresh9(environment(Y), true2, Z).
% 0.21/0.41 Axiom 10 (l1): fresh7(X, X, Y, Z) = greater(growth_rate(efficient_producers, Z), growth_rate(first_movers, Z)).
% 0.21/0.41 Axiom 11 (l1_1): fresh5(stable(X), true2, X) = fresh4(environment(X), true2, X).
% 0.21/0.41 Axiom 12 (prove_t1_3): fresh2(in_environment(e, X), true2, X) = subpopulations(first_movers, efficient_producers, e, t(X)).
% 0.21/0.41 Axiom 13 (mp1_high_growth_rates): fresh11(X, X, Y, Z, W) = true2.
% 0.21/0.41 Axiom 14 (l1): fresh6(X, X, Y, Z) = fresh7(stable(Y), true2, Y, Z).
% 0.21/0.41 Axiom 15 (mp1_high_growth_rates): fresh10(X, X, Y, Z, W, V) = fresh11(environment(Y), true2, Z, W, V).
% 0.21/0.41 Axiom 16 (mp1_high_growth_rates): fresh3(X, X, Y, Z, W, V) = selection_favors(W, Z, V).
% 0.21/0.41 Axiom 17 (l1): fresh6(greater_or_equal(X, to(Y)), true2, Y, X) = fresh8(subpopulations(first_movers, efficient_producers, Y, X), true2, Y, X).
% 0.21/0.41 Axiom 18 (mp1_high_growth_rates): fresh10(greater(growth_rate(X, Y), growth_rate(Z, Y)), true2, W, Z, X, Y) = fresh3(subpopulations(Z, X, W, Y), true2, W, Z, X, Y).
% 0.21/0.41
% 0.21/0.41 Lemma 19: in_environment(e, to(e)) = true2.
% 0.21/0.41 Proof:
% 0.21/0.41 in_environment(e, to(e))
% 0.21/0.41 = { by axiom 5 (l1_1) R->L }
% 0.21/0.41 fresh5(true2, true2, e)
% 0.21/0.41 = { by axiom 2 (prove_t1_1) R->L }
% 0.21/0.41 fresh5(stable(e), true2, e)
% 0.21/0.41 = { by axiom 11 (l1_1) }
% 0.21/0.41 fresh4(environment(e), true2, e)
% 0.21/0.41 = { by axiom 1 (prove_t1) }
% 0.21/0.41 fresh4(true2, true2, e)
% 0.21/0.41 = { by axiom 6 (l1_1) }
% 0.21/0.41 true2
% 0.21/0.41
% 0.21/0.41 Lemma 20: subpopulations(first_movers, efficient_producers, e, t(to(e))) = true2.
% 0.21/0.41 Proof:
% 0.21/0.41 subpopulations(first_movers, efficient_producers, e, t(to(e)))
% 0.21/0.41 = { by axiom 12 (prove_t1_3) R->L }
% 0.21/0.41 fresh2(in_environment(e, to(e)), true2, to(e))
% 0.21/0.41 = { by lemma 19 }
% 0.21/0.41 fresh2(true2, true2, to(e))
% 0.21/0.41 = { by axiom 7 (prove_t1_3) }
% 0.21/0.41 true2
% 0.21/0.41
% 0.21/0.41 Goal 1 (prove_t1_2): tuple(selection_favors(efficient_producers, first_movers, t(X)), in_environment(e, X)) = tuple(true2, true2).
% 0.21/0.41 The goal is true when:
% 0.21/0.41 X = to(e)
% 0.21/0.41
% 0.21/0.41 Proof:
% 0.21/0.41 tuple(selection_favors(efficient_producers, first_movers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by axiom 16 (mp1_high_growth_rates) R->L }
% 0.21/0.41 tuple(fresh3(true2, true2, e, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by lemma 20 R->L }
% 0.21/0.41 tuple(fresh3(subpopulations(first_movers, efficient_producers, e, t(to(e))), true2, e, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by axiom 18 (mp1_high_growth_rates) R->L }
% 0.21/0.41 tuple(fresh10(greater(growth_rate(efficient_producers, t(to(e))), growth_rate(first_movers, t(to(e)))), true2, e, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by axiom 10 (l1) R->L }
% 0.21/0.41 tuple(fresh10(fresh7(true2, true2, e, t(to(e))), true2, e, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by axiom 2 (prove_t1_1) R->L }
% 0.21/0.41 tuple(fresh10(fresh7(stable(e), true2, e, t(to(e))), true2, e, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by axiom 14 (l1) R->L }
% 0.21/0.41 tuple(fresh10(fresh6(true2, true2, e, t(to(e))), true2, e, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by axiom 3 (prove_t1_4) R->L }
% 0.21/0.41 tuple(fresh10(fresh6(fresh(true2, true2, to(e)), true2, e, t(to(e))), true2, e, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by lemma 19 R->L }
% 0.21/0.41 tuple(fresh10(fresh6(fresh(in_environment(e, to(e)), true2, to(e)), true2, e, t(to(e))), true2, e, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by axiom 8 (prove_t1_4) }
% 0.21/0.41 tuple(fresh10(fresh6(greater_or_equal(t(to(e)), to(e)), true2, e, t(to(e))), true2, e, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by axiom 17 (l1) }
% 0.21/0.41 tuple(fresh10(fresh8(subpopulations(first_movers, efficient_producers, e, t(to(e))), true2, e, t(to(e))), true2, e, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by lemma 20 }
% 0.21/0.41 tuple(fresh10(fresh8(true2, true2, e, t(to(e))), true2, e, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by axiom 9 (l1) }
% 0.21/0.41 tuple(fresh10(fresh9(environment(e), true2, t(to(e))), true2, e, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by axiom 1 (prove_t1) }
% 0.21/0.41 tuple(fresh10(fresh9(true2, true2, t(to(e))), true2, e, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by axiom 4 (l1) }
% 0.21/0.41 tuple(fresh10(true2, true2, e, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by axiom 15 (mp1_high_growth_rates) }
% 0.21/0.41 tuple(fresh11(environment(e), true2, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by axiom 1 (prove_t1) }
% 0.21/0.41 tuple(fresh11(true2, true2, first_movers, efficient_producers, t(to(e))), in_environment(e, to(e)))
% 0.21/0.41 = { by axiom 13 (mp1_high_growth_rates) }
% 0.21/0.41 tuple(true2, in_environment(e, to(e)))
% 0.21/0.41 = { by lemma 19 }
% 0.21/0.41 tuple(true2, true2)
% 0.21/0.41 % SZS output end Proof
% 0.21/0.41
% 0.21/0.41 RESULT: Theorem (the conjecture is true).
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