TSTP Solution File: SWV417+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SWV417+1 : TPTP v8.1.2. Released v3.3.0.
% 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 : n010.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 23:04:10 EDT 2023
% Result : Theorem 0.19s 0.49s
% Output : Proof 0.19s
% 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 : SWV417+1 : TPTP v8.1.2. Released v3.3.0.
% 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 : n010.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 : Tue Aug 29 06:24:19 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.19/0.49 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 0.19/0.49
% 0.19/0.49 % SZS status Theorem
% 0.19/0.49
% 0.19/0.49 % SZS output start Proof
% 0.19/0.49 Take the following subset of the input axioms:
% 0.19/0.49 fof(ax14, axiom, ![U, V]: ((contains_pq(U, V) & issmallestelement_pq(U, V)) => findmin_pq_res(U, V)=V)).
% 0.19/0.49 fof(ax20, axiom, ![U2]: ~contains_slb(create_slb, U2)).
% 0.19/0.49 fof(ax22, axiom, ![U2, V2]: ~pair_in_list(create_slb, U2, V2)).
% 0.19/0.49 fof(ax38, axiom, ![W, X, Y, U2, V2]: (strictly_less_than(X, Y) => (check_cpq(triple(U2, insert_slb(V2, pair(X, Y)), W)) <=> $false))).
% 0.19/0.49 fof(ax40, axiom, ![U2, V2]: (ok(triple(U2, V2, bad)) <=> $false)).
% 0.19/0.49 fof(ax58, axiom, ![U2, V2]: (pi_sharp_find_min(U2, V2) <=> (contains_pq(U2, V2) & issmallestelement_pq(U2, V2)))).
% 0.19/0.49 fof(ax8, axiom, ![U2]: ~contains_pq(create_pq, U2)).
% 0.19/0.49 fof(co4, conjecture, ![U2, V2, W2]: (pi_find_min(triple(U2, V2, W2)) => (phi(findmin_cpq_eff(triple(U2, V2, W2))) => ?[X2]: (pi_sharp_find_min(i(triple(U2, V2, W2)), X2) & findmin_cpq_res(triple(U2, V2, W2))=findmin_pq_res(i(triple(U2, V2, W2)), X2))))).
% 0.19/0.49 fof(main4_l7, lemma, ![U2, V2, W2]: (phi(findmin_cpq_eff(triple(U2, V2, W2))) => pi_sharp_find_min(i(triple(U2, V2, W2)), findmin_cpq_res(triple(U2, V2, W2))))).
% 0.19/0.49 fof(stricly_smaller_definition, axiom, ![U2, V2]: (strictly_less_than(U2, V2) <=> (less_than(U2, V2) & ~less_than(V2, U2)))).
% 0.19/0.49
% 0.19/0.49 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.49 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.49 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.49 fresh(y, y, x1...xn) = u
% 0.19/0.49 C => fresh(s, t, x1...xn) = v
% 0.19/0.49 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.49 variables of u and v.
% 0.19/0.49 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.49 input problem has no model of domain size 1).
% 0.19/0.49
% 0.19/0.49 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.49
% 0.19/0.49 Axiom 1 (ax14): fresh50(X, X, Y, Z) = findmin_pq_res(Y, Z).
% 0.19/0.49 Axiom 2 (ax58_1): fresh22(X, X, Y, Z) = true2.
% 0.19/0.49 Axiom 3 (ax58_2): fresh21(X, X, Y, Z) = true2.
% 0.19/0.49 Axiom 4 (ax14): fresh2(X, X, Y, Z) = Z.
% 0.19/0.49 Axiom 5 (co4_1): phi(findmin_cpq_eff(triple(u, v, w))) = true2.
% 0.19/0.49 Axiom 6 (ax58_1): fresh22(pi_sharp_find_min(X, Y), true2, X, Y) = contains_pq(X, Y).
% 0.19/0.49 Axiom 7 (ax58_2): fresh21(pi_sharp_find_min(X, Y), true2, X, Y) = issmallestelement_pq(X, Y).
% 0.19/0.49 Axiom 8 (main4_l7): fresh7(X, X, Y, Z, W) = true2.
% 0.19/0.49 Axiom 9 (ax14): fresh50(issmallestelement_pq(X, Y), true2, X, Y) = fresh2(contains_pq(X, Y), true2, X, Y).
% 0.19/0.49 Axiom 10 (main4_l7): fresh7(phi(findmin_cpq_eff(triple(X, Y, Z))), true2, X, Y, Z) = pi_sharp_find_min(i(triple(X, Y, Z)), findmin_cpq_res(triple(X, Y, Z))).
% 0.19/0.49
% 0.19/0.49 Lemma 11: pi_sharp_find_min(i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))) = true2.
% 0.19/0.49 Proof:
% 0.19/0.49 pi_sharp_find_min(i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w)))
% 0.19/0.49 = { by axiom 10 (main4_l7) R->L }
% 0.19/0.49 fresh7(phi(findmin_cpq_eff(triple(u, v, w))), true2, u, v, w)
% 0.19/0.49 = { by axiom 5 (co4_1) }
% 0.19/0.49 fresh7(true2, true2, u, v, w)
% 0.19/0.49 = { by axiom 8 (main4_l7) }
% 0.19/0.50 true2
% 0.19/0.50
% 0.19/0.50 Goal 1 (co4_2): tuple2(findmin_cpq_res(triple(u, v, w)), pi_sharp_find_min(i(triple(u, v, w)), X)) = tuple2(findmin_pq_res(i(triple(u, v, w)), X), true2).
% 0.19/0.50 The goal is true when:
% 0.19/0.50 X = findmin_cpq_res(triple(u, v, w))
% 0.19/0.50
% 0.19/0.50 Proof:
% 0.19/0.50 tuple2(findmin_cpq_res(triple(u, v, w)), pi_sharp_find_min(i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))))
% 0.19/0.50 = { by lemma 11 }
% 0.19/0.50 tuple2(findmin_cpq_res(triple(u, v, w)), true2)
% 0.19/0.50 = { by axiom 4 (ax14) R->L }
% 0.19/0.50 tuple2(fresh2(true2, true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2)
% 0.19/0.50 = { by axiom 2 (ax58_1) R->L }
% 0.19/0.50 tuple2(fresh2(fresh22(true2, true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2)
% 0.19/0.50 = { by lemma 11 R->L }
% 0.19/0.50 tuple2(fresh2(fresh22(pi_sharp_find_min(i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2)
% 0.19/0.50 = { by axiom 6 (ax58_1) }
% 0.19/0.50 tuple2(fresh2(contains_pq(i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2)
% 0.19/0.50 = { by axiom 9 (ax14) R->L }
% 0.19/0.50 tuple2(fresh50(issmallestelement_pq(i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2)
% 0.19/0.50 = { by axiom 7 (ax58_2) R->L }
% 0.19/0.50 tuple2(fresh50(fresh21(pi_sharp_find_min(i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2)
% 0.19/0.50 = { by lemma 11 }
% 0.19/0.50 tuple2(fresh50(fresh21(true2, true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2)
% 0.19/0.50 = { by axiom 3 (ax58_2) }
% 0.19/0.50 tuple2(fresh50(true2, true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2)
% 0.19/0.50 = { by axiom 1 (ax14) }
% 0.19/0.50 tuple2(findmin_pq_res(i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))), true2)
% 0.19/0.50 % SZS output end Proof
% 0.19/0.50
% 0.19/0.50 RESULT: Theorem (the conjecture is true).
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