TSTP Solution File: LCL512+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL512+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 : n032.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 08:19:16 EDT 2023
% Result : Theorem 0.13s 0.37s
% 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
% 0.00/0.09 % Problem : LCL512+1 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.10 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.08/0.28 % Computer : n032.cluster.edu
% 0.08/0.28 % Model : x86_64 x86_64
% 0.08/0.28 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.28 % Memory : 8042.1875MB
% 0.08/0.28 % OS : Linux 3.10.0-693.el7.x86_64
% 0.08/0.28 % CPULimit : 300
% 0.08/0.28 % WCLimit : 300
% 0.08/0.28 % DateTime : Fri Aug 25 05:15:57 EDT 2023
% 0.08/0.28 % CPUTime :
% 0.13/0.37 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.13/0.37
% 0.13/0.37 % SZS status Theorem
% 0.13/0.37
% 0.13/0.37 % SZS output start Proof
% 0.13/0.37 Take the following subset of the input axioms:
% 0.13/0.37 fof(equivalence_1, axiom, equivalence_1 <=> ![X, Y]: is_a_theorem(implies(equiv(X, Y), implies(X, Y)))).
% 0.13/0.37 fof(hilbert_equivalence_1, conjecture, equivalence_1).
% 0.13/0.37 fof(kn2, axiom, kn2 <=> ![P, Q]: is_a_theorem(implies(and(P, Q), P))).
% 0.13/0.37 fof(op_equiv, axiom, op_equiv => ![X2, Y2]: equiv(X2, Y2)=and(implies(X2, Y2), implies(Y2, X2))).
% 0.13/0.37 fof(rosser_kn2, axiom, kn2).
% 0.13/0.37 fof(rosser_op_equiv, axiom, op_equiv).
% 0.13/0.37
% 0.13/0.37 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.13/0.37 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.13/0.37 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.13/0.37 fresh(y, y, x1...xn) = u
% 0.13/0.37 C => fresh(s, t, x1...xn) = v
% 0.13/0.37 where fresh is a fresh function symbol and x1..xn are the free
% 0.13/0.37 variables of u and v.
% 0.13/0.37 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.13/0.37 input problem has no model of domain size 1).
% 0.13/0.37
% 0.13/0.37 The encoding turns the above axioms into the following unit equations and goals:
% 0.13/0.37
% 0.13/0.37 Axiom 1 (rosser_kn2): kn2 = true.
% 0.13/0.37 Axiom 2 (rosser_op_equiv): op_equiv = true.
% 0.13/0.37 Axiom 3 (equivalence_1): fresh46(X, X) = true.
% 0.13/0.37 Axiom 4 (kn2_1): fresh31(X, X, Y, Z) = true.
% 0.13/0.37 Axiom 5 (op_equiv): fresh23(X, X, Y, Z) = equiv(Y, Z).
% 0.13/0.37 Axiom 6 (kn2_1): fresh31(kn2, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 0.13/0.37 Axiom 7 (op_equiv): fresh23(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
% 0.13/0.37 Axiom 8 (equivalence_1): fresh46(is_a_theorem(implies(equiv(x3, y3), implies(x3, y3))), true) = equivalence_1.
% 0.13/0.37
% 0.13/0.37 Goal 1 (hilbert_equivalence_1): equivalence_1 = true.
% 0.13/0.37 Proof:
% 0.13/0.37 equivalence_1
% 0.13/0.37 = { by axiom 8 (equivalence_1) R->L }
% 0.13/0.37 fresh46(is_a_theorem(implies(equiv(x3, y3), implies(x3, y3))), true)
% 0.13/0.37 = { by axiom 5 (op_equiv) R->L }
% 0.13/0.37 fresh46(is_a_theorem(implies(fresh23(true, true, x3, y3), implies(x3, y3))), true)
% 0.13/0.37 = { by axiom 2 (rosser_op_equiv) R->L }
% 0.13/0.37 fresh46(is_a_theorem(implies(fresh23(op_equiv, true, x3, y3), implies(x3, y3))), true)
% 0.13/0.38 = { by axiom 7 (op_equiv) }
% 0.13/0.38 fresh46(is_a_theorem(implies(and(implies(x3, y3), implies(y3, x3)), implies(x3, y3))), true)
% 0.13/0.38 = { by axiom 6 (kn2_1) R->L }
% 0.13/0.38 fresh46(fresh31(kn2, true, implies(x3, y3), implies(y3, x3)), true)
% 0.13/0.38 = { by axiom 1 (rosser_kn2) }
% 0.13/0.38 fresh46(fresh31(true, true, implies(x3, y3), implies(y3, x3)), true)
% 0.13/0.38 = { by axiom 4 (kn2_1) }
% 0.13/0.38 fresh46(true, true)
% 0.13/0.38 = { by axiom 3 (equivalence_1) }
% 0.13/0.38 true
% 0.13/0.38 % SZS output end Proof
% 0.13/0.38
% 0.13/0.38 RESULT: Theorem (the conjecture is true).
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