TSTP Solution File: SWV371+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SWV371+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 : n029.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 23:03:59 EDT 2023
% Result : Theorem 0.18s 0.50s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SWV371+1 : TPTP v8.1.2. Released v3.3.0.
% 0.11/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.33 % Computer : n029.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 : Tue Aug 29 03:19:11 EDT 2023
% 0.13/0.33 % CPUTime :
% 0.18/0.50 Command-line arguments: --ground-connectedness --complete-subsets
% 0.18/0.50
% 0.18/0.50 % SZS status Theorem
% 0.18/0.50
% 0.18/0.50 % SZS output start Proof
% 0.18/0.50 Take the following subset of the input axioms:
% 0.18/0.50 fof(ax58, axiom, ![U, V]: (pi_sharp_find_min(U, V) <=> (contains_pq(U, V) & issmallestelement_pq(U, V)))).
% 0.18/0.50 fof(l7_co, conjecture, ![W, U2, V2]: (phi(findmin_cpq_eff(triple(U2, V2, W))) => pi_sharp_find_min(i(triple(U2, V2, W)), findmin_cpq_res(triple(U2, V2, W))))).
% 0.18/0.50 fof(l7_l8, lemma, ![U2, V2, W2]: (phi(findmin_cpq_eff(triple(U2, V2, W2))) => contains_pq(i(triple(U2, V2, W2)), findmin_cpq_res(triple(U2, V2, W2))))).
% 0.18/0.50 fof(l7_lX, lemma, ![U2, V2, W2]: (phi(findmin_cpq_eff(triple(U2, V2, W2))) => issmallestelement_pq(i(triple(U2, V2, W2)), findmin_cpq_res(triple(U2, V2, W2))))).
% 0.18/0.50
% 0.18/0.50 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.18/0.50 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.18/0.50 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.18/0.50 fresh(y, y, x1...xn) = u
% 0.18/0.50 C => fresh(s, t, x1...xn) = v
% 0.18/0.50 where fresh is a fresh function symbol and x1..xn are the free
% 0.18/0.50 variables of u and v.
% 0.18/0.50 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.18/0.50 input problem has no model of domain size 1).
% 0.18/0.50
% 0.18/0.50 The encoding turns the above axioms into the following unit equations and goals:
% 0.18/0.50
% 0.18/0.50 Axiom 1 (ax58): fresh25(X, X, Y, Z) = pi_sharp_find_min(Y, Z).
% 0.18/0.50 Axiom 2 (ax58): fresh24(X, X, Y, Z) = true2.
% 0.18/0.50 Axiom 3 (l7_l8): fresh8(X, X, Y, Z, W) = true2.
% 0.18/0.50 Axiom 4 (l7_lX): fresh7(X, X, Y, Z, W) = true2.
% 0.18/0.50 Axiom 5 (l7_co): phi(findmin_cpq_eff(triple(u, v, w))) = true2.
% 0.18/0.50 Axiom 6 (ax58): fresh25(issmallestelement_pq(X, Y), true2, X, Y) = fresh24(contains_pq(X, Y), true2, X, Y).
% 0.18/0.50 Axiom 7 (l7_lX): fresh7(phi(findmin_cpq_eff(triple(X, Y, Z))), true2, X, Y, Z) = issmallestelement_pq(i(triple(X, Y, Z)), findmin_cpq_res(triple(X, Y, Z))).
% 0.18/0.50 Axiom 8 (l7_l8): fresh8(phi(findmin_cpq_eff(triple(X, Y, Z))), true2, X, Y, Z) = contains_pq(i(triple(X, Y, Z)), findmin_cpq_res(triple(X, Y, Z))).
% 0.18/0.50
% 0.18/0.50 Goal 1 (l7_co_1): pi_sharp_find_min(i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w))) = true2.
% 0.18/0.50 Proof:
% 0.18/0.50 pi_sharp_find_min(i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w)))
% 0.18/0.50 = { by axiom 1 (ax58) R->L }
% 0.18/0.50 fresh25(true2, true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w)))
% 0.18/0.51 = { by axiom 4 (l7_lX) R->L }
% 0.18/0.51 fresh25(fresh7(true2, true2, u, v, w), true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w)))
% 0.18/0.51 = { by axiom 5 (l7_co) R->L }
% 0.18/0.51 fresh25(fresh7(phi(findmin_cpq_eff(triple(u, v, w))), true2, u, v, w), true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w)))
% 0.18/0.51 = { by axiom 7 (l7_lX) }
% 0.18/0.51 fresh25(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)))
% 0.18/0.51 = { by axiom 6 (ax58) }
% 0.18/0.51 fresh24(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)))
% 0.18/0.51 = { by axiom 8 (l7_l8) R->L }
% 0.18/0.51 fresh24(fresh8(phi(findmin_cpq_eff(triple(u, v, w))), true2, u, v, w), true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w)))
% 0.18/0.51 = { by axiom 5 (l7_co) }
% 0.18/0.51 fresh24(fresh8(true2, true2, u, v, w), true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w)))
% 0.18/0.51 = { by axiom 3 (l7_l8) }
% 0.18/0.51 fresh24(true2, true2, i(triple(u, v, w)), findmin_cpq_res(triple(u, v, w)))
% 0.18/0.51 = { by axiom 2 (ax58) }
% 0.18/0.51 true2
% 0.18/0.51 % SZS output end Proof
% 0.18/0.51
% 0.18/0.51 RESULT: Theorem (the conjecture is true).
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