TSTP Solution File: LCL542+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL542+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 : n031.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 08:19:22 EDT 2023
% Result : Theorem 0.19s 0.57s
% 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
% 0.12/0.13 % Problem : LCL542+1 : TPTP v8.1.2. Released v3.3.0.
% 0.12/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n031.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.35 % CPULimit : 300
% 0.12/0.35 % WCLimit : 300
% 0.12/0.35 % DateTime : Thu Aug 24 18:14:51 EDT 2023
% 0.12/0.35 % CPUTime :
% 0.19/0.57 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.19/0.57
% 0.19/0.57 % SZS status Theorem
% 0.19/0.57
% 0.19/0.57 % SZS output start Proof
% 0.19/0.57 Take the following subset of the input axioms:
% 0.19/0.57 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 0.19/0.57 fof(axiom_m2, axiom, axiom_m2 <=> ![X2, Y2]: is_a_theorem(strict_implies(and(X2, Y2), X2))).
% 0.19/0.57 fof(hilbert_and_1, axiom, and_1).
% 0.19/0.57 fof(km4b_necessitation, axiom, necessitation).
% 0.19/0.57 fof(necessitation, axiom, necessitation <=> ![X2]: (is_a_theorem(X2) => is_a_theorem(necessarily(X2)))).
% 0.19/0.57 fof(op_strict_implies, axiom, op_strict_implies => ![X2, Y2]: strict_implies(X2, Y2)=necessarily(implies(X2, Y2))).
% 0.19/0.57 fof(s1_0_axiom_m2, conjecture, axiom_m2).
% 0.19/0.57 fof(s1_0_op_strict_implies, axiom, op_strict_implies).
% 0.19/0.57
% 0.19/0.57 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.57 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.57 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.57 fresh(y, y, x1...xn) = u
% 0.19/0.57 C => fresh(s, t, x1...xn) = v
% 0.19/0.57 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.57 variables of u and v.
% 0.19/0.57 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.57 input problem has no model of domain size 1).
% 0.19/0.57
% 0.19/0.57 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.57
% 0.19/0.57 Axiom 1 (hilbert_and_1): and_1 = true.
% 0.19/0.57 Axiom 2 (km4b_necessitation): necessitation = true.
% 0.19/0.57 Axiom 3 (s1_0_op_strict_implies): op_strict_implies = true.
% 0.19/0.57 Axiom 4 (axiom_m2): fresh88(X, X) = true.
% 0.19/0.57 Axiom 5 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)).
% 0.19/0.57 Axiom 6 (necessitation_1): fresh33(X, X, Y) = true.
% 0.19/0.57 Axiom 7 (and_1_1): fresh107(X, X, Y, Z) = true.
% 0.19/0.57 Axiom 8 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X).
% 0.19/0.57 Axiom 9 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z).
% 0.19/0.57 Axiom 10 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)).
% 0.19/0.57 Axiom 11 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 0.19/0.57 Axiom 12 (axiom_m2): fresh88(is_a_theorem(strict_implies(and(x7, y3), x7)), true) = axiom_m2.
% 0.19/0.57
% 0.19/0.57 Goal 1 (s1_0_axiom_m2): axiom_m2 = true.
% 0.19/0.57 Proof:
% 0.19/0.57 axiom_m2
% 0.19/0.57 = { by axiom 12 (axiom_m2) R->L }
% 0.19/0.57 fresh88(is_a_theorem(strict_implies(and(x7, y3), x7)), true)
% 0.19/0.57 = { by axiom 9 (op_strict_implies) R->L }
% 0.19/0.57 fresh88(is_a_theorem(fresh23(true, true, and(x7, y3), x7)), true)
% 0.19/0.57 = { by axiom 3 (s1_0_op_strict_implies) R->L }
% 0.19/0.57 fresh88(is_a_theorem(fresh23(op_strict_implies, true, and(x7, y3), x7)), true)
% 0.19/0.57 = { by axiom 10 (op_strict_implies) }
% 0.19/0.57 fresh88(is_a_theorem(necessarily(implies(and(x7, y3), x7))), true)
% 0.19/0.57 = { by axiom 5 (necessitation_1) R->L }
% 0.19/0.57 fresh88(fresh34(true, true, implies(and(x7, y3), x7)), true)
% 0.19/0.57 = { by axiom 2 (km4b_necessitation) R->L }
% 0.19/0.57 fresh88(fresh34(necessitation, true, implies(and(x7, y3), x7)), true)
% 0.19/0.57 = { by axiom 8 (necessitation_1) }
% 0.19/0.57 fresh88(fresh33(is_a_theorem(implies(and(x7, y3), x7)), true, implies(and(x7, y3), x7)), true)
% 0.19/0.57 = { by axiom 11 (and_1_1) R->L }
% 0.19/0.57 fresh88(fresh33(fresh107(and_1, true, x7, y3), true, implies(and(x7, y3), x7)), true)
% 0.19/0.57 = { by axiom 1 (hilbert_and_1) }
% 0.19/0.57 fresh88(fresh33(fresh107(true, true, x7, y3), true, implies(and(x7, y3), x7)), true)
% 0.19/0.57 = { by axiom 7 (and_1_1) }
% 0.19/0.57 fresh88(fresh33(true, true, implies(and(x7, y3), x7)), true)
% 0.19/0.57 = { by axiom 6 (necessitation_1) }
% 0.19/0.57 fresh88(true, true)
% 0.19/0.57 = { by axiom 4 (axiom_m2) }
% 0.19/0.57 true
% 0.19/0.57 % SZS output end Proof
% 0.19/0.57
% 0.19/0.57 RESULT: Theorem (the conjecture is true).
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