TSTP Solution File: MGT036+3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : MGT036+3 : 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 : n011.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 09:17:08 EDT 2023
% Result : Theorem 0.13s 0.38s
% Output : Proof 0.13s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.12 % Problem : MGT036+3 : TPTP v8.1.2. Released v2.0.0.
% 0.10/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n011.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Mon Aug 28 06:10:25 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.13/0.38 Command-line arguments: --flatten
% 0.13/0.38
% 0.13/0.38 % SZS status Theorem
% 0.13/0.38
% 0.13/0.39 % SZS output start Proof
% 0.13/0.39 Take the following subset of the input axioms:
% 0.13/0.39 fof(a13_star, hypothesis, ?[E, T]: (environment(E) & (subpopulations(first_movers, efficient_producers, E, T) & (greater_or_equal(growth_rate(first_movers, T), zero) & greater(zero, growth_rate(efficient_producers, T)))))).
% 0.13/0.39 fof(d2, hypothesis, ![S1, S2, E2, T2]: ((environment(E2) & subpopulations(S1, S2, E2, T2)) => ((greater_or_equal(growth_rate(S2, T2), zero) & greater(zero, growth_rate(S1, T2))) <=> outcompetes(S2, S1, T2)))).
% 0.13/0.39 fof(mp_symmetry_of_subpopulations, axiom, ![E2, T2, S1_2, S2_2]: ((environment(E2) & subpopulations(S1_2, S2_2, E2, T2)) => subpopulations(S2_2, S1_2, E2, T2))).
% 0.13/0.39 fof(prove_t5_star, conjecture, ?[E2, T2]: (environment(E2) & (subpopulations(first_movers, efficient_producers, E2, T2) & outcompetes(first_movers, efficient_producers, T2)))).
% 0.13/0.39
% 0.13/0.39 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.13/0.39 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.13/0.39 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.13/0.39 fresh(y, y, x1...xn) = u
% 0.13/0.39 C => fresh(s, t, x1...xn) = v
% 0.13/0.39 where fresh is a fresh function symbol and x1..xn are the free
% 0.13/0.39 variables of u and v.
% 0.13/0.39 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.13/0.39 input problem has no model of domain size 1).
% 0.13/0.39
% 0.13/0.39 The encoding turns the above axioms into the following unit equations and goals:
% 0.13/0.39
% 0.13/0.39 Axiom 1 (a13_star): environment(e) = true2.
% 0.13/0.39 Axiom 2 (a13_star_2): greater_or_equal(growth_rate(first_movers, t), zero) = true2.
% 0.13/0.39 Axiom 3 (a13_star_3): greater(zero, growth_rate(efficient_producers, t)) = true2.
% 0.13/0.39 Axiom 4 (a13_star_1): subpopulations(first_movers, efficient_producers, e, t) = true2.
% 0.13/0.39 Axiom 5 (d2): fresh8(X, X, Y, Z, W) = true2.
% 0.13/0.39 Axiom 6 (mp_symmetry_of_subpopulations): fresh(X, X, Y, Z, W, V) = true2.
% 0.13/0.39 Axiom 7 (d2): fresh7(X, X, Y, Z, W, V) = fresh8(environment(Y), true2, Z, W, V).
% 0.13/0.39 Axiom 8 (d2): fresh6(X, X, Y, Z, W, V) = outcompetes(W, Z, V).
% 0.13/0.39 Axiom 9 (mp_symmetry_of_subpopulations): fresh2(X, X, Y, Z, W, V) = subpopulations(W, Z, Y, V).
% 0.13/0.39 Axiom 10 (d2): fresh5(X, X, Y, Z, W, V) = fresh6(subpopulations(Z, W, Y, V), true2, Y, Z, W, V).
% 0.13/0.39 Axiom 11 (d2): fresh5(greater(zero, growth_rate(X, Y)), true2, Z, X, W, Y) = fresh7(greater_or_equal(growth_rate(W, Y), zero), true2, Z, X, W, Y).
% 0.13/0.39 Axiom 12 (mp_symmetry_of_subpopulations): fresh2(subpopulations(X, Y, Z, W), true2, Z, X, Y, W) = fresh(environment(Z), true2, Z, X, Y, W).
% 0.13/0.39
% 0.13/0.39 Lemma 13: subpopulations(first_movers, efficient_producers, e, t) = environment(e).
% 0.13/0.39 Proof:
% 0.13/0.39 subpopulations(first_movers, efficient_producers, e, t)
% 0.13/0.39 = { by axiom 4 (a13_star_1) }
% 0.13/0.39 true2
% 0.13/0.39 = { by axiom 1 (a13_star) R->L }
% 0.13/0.39 environment(e)
% 0.13/0.39
% 0.13/0.39 Goal 1 (prove_t5_star): tuple(environment(X), subpopulations(first_movers, efficient_producers, X, Y), outcompetes(first_movers, efficient_producers, Y)) = tuple(true2, true2, true2).
% 0.13/0.39 The goal is true when:
% 0.13/0.39 X = e
% 0.13/0.39 Y = t
% 0.13/0.39
% 0.13/0.39 Proof:
% 0.13/0.39 tuple(environment(e), subpopulations(first_movers, efficient_producers, e, t), outcompetes(first_movers, efficient_producers, t))
% 0.13/0.39 = { by lemma 13 }
% 0.13/0.39 tuple(environment(e), environment(e), outcompetes(first_movers, efficient_producers, t))
% 0.13/0.39 = { by axiom 8 (d2) R->L }
% 0.13/0.39 tuple(environment(e), environment(e), fresh6(environment(e), environment(e), e, efficient_producers, first_movers, t))
% 0.13/0.39 = { by axiom 1 (a13_star) }
% 0.13/0.39 tuple(environment(e), environment(e), fresh6(true2, environment(e), e, efficient_producers, first_movers, t))
% 0.13/0.39 = { by axiom 6 (mp_symmetry_of_subpopulations) R->L }
% 0.13/0.39 tuple(environment(e), environment(e), fresh6(fresh(environment(e), environment(e), e, first_movers, efficient_producers, t), environment(e), e, efficient_producers, first_movers, t))
% 0.13/0.39 = { by axiom 1 (a13_star) }
% 0.13/0.39 tuple(environment(e), environment(e), fresh6(fresh(environment(e), true2, e, first_movers, efficient_producers, t), environment(e), e, efficient_producers, first_movers, t))
% 0.13/0.39 = { by axiom 12 (mp_symmetry_of_subpopulations) R->L }
% 0.13/0.40 tuple(environment(e), environment(e), fresh6(fresh2(subpopulations(first_movers, efficient_producers, e, t), true2, e, first_movers, efficient_producers, t), environment(e), e, efficient_producers, first_movers, t))
% 0.13/0.40 = { by axiom 1 (a13_star) R->L }
% 0.13/0.40 tuple(environment(e), environment(e), fresh6(fresh2(subpopulations(first_movers, efficient_producers, e, t), environment(e), e, first_movers, efficient_producers, t), environment(e), e, efficient_producers, first_movers, t))
% 0.13/0.40 = { by lemma 13 }
% 0.13/0.40 tuple(environment(e), environment(e), fresh6(fresh2(environment(e), environment(e), e, first_movers, efficient_producers, t), environment(e), e, efficient_producers, first_movers, t))
% 0.13/0.40 = { by axiom 9 (mp_symmetry_of_subpopulations) }
% 0.13/0.40 tuple(environment(e), environment(e), fresh6(subpopulations(efficient_producers, first_movers, e, t), environment(e), e, efficient_producers, first_movers, t))
% 0.13/0.40 = { by axiom 1 (a13_star) }
% 0.13/0.40 tuple(environment(e), environment(e), fresh6(subpopulations(efficient_producers, first_movers, e, t), true2, e, efficient_producers, first_movers, t))
% 0.13/0.40 = { by axiom 10 (d2) R->L }
% 0.13/0.40 tuple(environment(e), environment(e), fresh5(environment(e), environment(e), e, efficient_producers, first_movers, t))
% 0.13/0.40 = { by axiom 1 (a13_star) }
% 0.13/0.40 tuple(environment(e), environment(e), fresh5(true2, environment(e), e, efficient_producers, first_movers, t))
% 0.13/0.40 = { by axiom 3 (a13_star_3) R->L }
% 0.13/0.40 tuple(environment(e), environment(e), fresh5(greater(zero, growth_rate(efficient_producers, t)), environment(e), e, efficient_producers, first_movers, t))
% 0.13/0.40 = { by axiom 1 (a13_star) }
% 0.13/0.40 tuple(environment(e), environment(e), fresh5(greater(zero, growth_rate(efficient_producers, t)), true2, e, efficient_producers, first_movers, t))
% 0.13/0.40 = { by axiom 11 (d2) }
% 0.13/0.40 tuple(environment(e), environment(e), fresh7(greater_or_equal(growth_rate(first_movers, t), zero), true2, e, efficient_producers, first_movers, t))
% 0.13/0.40 = { by axiom 1 (a13_star) R->L }
% 0.13/0.40 tuple(environment(e), environment(e), fresh7(greater_or_equal(growth_rate(first_movers, t), zero), environment(e), e, efficient_producers, first_movers, t))
% 0.13/0.40 = { by axiom 2 (a13_star_2) }
% 0.13/0.40 tuple(environment(e), environment(e), fresh7(true2, environment(e), e, efficient_producers, first_movers, t))
% 0.13/0.40 = { by axiom 1 (a13_star) R->L }
% 0.13/0.40 tuple(environment(e), environment(e), fresh7(environment(e), environment(e), e, efficient_producers, first_movers, t))
% 0.13/0.40 = { by axiom 7 (d2) }
% 0.13/0.40 tuple(environment(e), environment(e), fresh8(environment(e), true2, efficient_producers, first_movers, t))
% 0.13/0.40 = { by axiom 1 (a13_star) R->L }
% 0.13/0.40 tuple(environment(e), environment(e), fresh8(environment(e), environment(e), efficient_producers, first_movers, t))
% 0.13/0.40 = { by axiom 5 (d2) }
% 0.13/0.40 tuple(environment(e), environment(e), true2)
% 0.13/0.40 = { by axiom 1 (a13_star) R->L }
% 0.13/0.40 tuple(environment(e), environment(e), environment(e))
% 0.13/0.40 = { by axiom 1 (a13_star) }
% 0.13/0.40 tuple(true2, environment(e), environment(e))
% 0.13/0.40 = { by axiom 1 (a13_star) }
% 0.13/0.40 tuple(true2, true2, environment(e))
% 0.13/0.40 = { by axiom 1 (a13_star) }
% 0.13/0.40 tuple(true2, true2, true2)
% 0.13/0.40 % SZS output end Proof
% 0.13/0.40
% 0.13/0.40 RESULT: Theorem (the conjecture is true).
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