TSTP Solution File: LCL499+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL499+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 : n004.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 08:19:14 EDT 2023
% Result : Theorem 0.21s 0.51s
% Output : Proof 0.21s
% 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 : LCL499+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
% 0.14/0.35 % Computer : n004.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 : Thu Aug 24 18:28:38 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.21/0.51 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.21/0.51
% 0.21/0.51 % SZS status Theorem
% 0.21/0.51
% 0.21/0.51 % SZS output start Proof
% 0.21/0.51 Take the following subset of the input axioms:
% 0.21/0.51 fof(kn1, axiom, kn1 <=> ![P]: is_a_theorem(implies(P, and(P, P)))).
% 0.21/0.51 fof(modus_ponens, axiom, modus_ponens <=> ![X, Y]: ((is_a_theorem(X) & is_a_theorem(implies(X, Y))) => is_a_theorem(Y))).
% 0.21/0.51 fof(op_and, axiom, op_and => ![X2, Y2]: and(X2, Y2)=not(or(not(X2), not(Y2)))).
% 0.21/0.51 fof(op_implies_or, axiom, op_implies_or => ![X2, Y2]: implies(X2, Y2)=or(not(X2), Y2)).
% 0.21/0.51 fof(principia_modus_ponens, axiom, modus_ponens).
% 0.21/0.51 fof(principia_op_and, axiom, op_and).
% 0.21/0.51 fof(principia_op_implies_or, axiom, op_implies_or).
% 0.21/0.51 fof(principia_r1, axiom, r1).
% 0.21/0.51 fof(principia_r3, axiom, r3).
% 0.21/0.51 fof(r1, axiom, r1 <=> ![P2]: is_a_theorem(implies(or(P2, P2), P2))).
% 0.21/0.51 fof(r3, axiom, r3 <=> ![Q, P2]: is_a_theorem(implies(or(P2, Q), or(Q, P2)))).
% 0.21/0.51 fof(rosser_kn1, conjecture, kn1).
% 0.21/0.51
% 0.21/0.51 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.51 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.51 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.51 fresh(y, y, x1...xn) = u
% 0.21/0.51 C => fresh(s, t, x1...xn) = v
% 0.21/0.51 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.51 variables of u and v.
% 0.21/0.51 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.51 input problem has no model of domain size 1).
% 0.21/0.51
% 0.21/0.51 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.51
% 0.21/0.51 Axiom 1 (principia_modus_ponens): modus_ponens = true.
% 0.21/0.51 Axiom 2 (principia_r1): r1 = true.
% 0.21/0.51 Axiom 3 (principia_r3): r3 = true.
% 0.21/0.51 Axiom 4 (principia_op_and): op_and = true.
% 0.21/0.51 Axiom 5 (principia_op_implies_or): op_implies_or = true.
% 0.21/0.51 Axiom 6 (kn1): fresh34(X, X) = true.
% 0.21/0.51 Axiom 7 (modus_ponens_2): fresh60(X, X, Y) = true.
% 0.21/0.51 Axiom 8 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 0.21/0.51 Axiom 9 (r1_1): fresh12(X, X, Y) = true.
% 0.21/0.51 Axiom 10 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 0.21/0.51 Axiom 11 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
% 0.21/0.51 Axiom 12 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
% 0.21/0.51 Axiom 13 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
% 0.21/0.51 Axiom 14 (r3_1): fresh8(X, X, Y, Z) = true.
% 0.21/0.51 Axiom 15 (r1_1): fresh12(r1, true, X) = is_a_theorem(implies(or(X, X), X)).
% 0.21/0.51 Axiom 16 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
% 0.21/0.51 Axiom 17 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
% 0.21/0.51 Axiom 18 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 0.21/0.51 Axiom 19 (kn1): fresh34(is_a_theorem(implies(p11, and(p11, p11))), true) = kn1.
% 0.21/0.51
% 0.21/0.51 Lemma 20: or(not(X), Y) = implies(X, Y).
% 0.21/0.51 Proof:
% 0.21/0.51 or(not(X), Y)
% 0.21/0.51 = { by axiom 13 (op_implies_or) R->L }
% 0.21/0.51 fresh21(op_implies_or, true, X, Y)
% 0.21/0.51 = { by axiom 5 (principia_op_implies_or) }
% 0.21/0.51 fresh21(true, true, X, Y)
% 0.21/0.51 = { by axiom 12 (op_implies_or) }
% 0.21/0.51 implies(X, Y)
% 0.21/0.51
% 0.21/0.51 Goal 1 (rosser_kn1): kn1 = true.
% 0.21/0.51 Proof:
% 0.21/0.51 kn1
% 0.21/0.51 = { by axiom 19 (kn1) R->L }
% 0.21/0.51 fresh34(is_a_theorem(implies(p11, and(p11, p11))), true)
% 0.21/0.51 = { by lemma 20 R->L }
% 0.21/0.51 fresh34(is_a_theorem(or(not(p11), and(p11, p11))), true)
% 0.21/0.51 = { by axiom 8 (modus_ponens_2) R->L }
% 0.21/0.51 fresh34(fresh28(true, true, or(not(p11), and(p11, p11))), true)
% 0.21/0.51 = { by axiom 9 (r1_1) R->L }
% 0.21/0.51 fresh34(fresh28(fresh12(true, true, not(p11)), true, or(not(p11), and(p11, p11))), true)
% 0.21/0.51 = { by axiom 2 (principia_r1) R->L }
% 0.21/0.51 fresh34(fresh28(fresh12(r1, true, not(p11)), true, or(not(p11), and(p11, p11))), true)
% 0.21/0.51 = { by axiom 15 (r1_1) }
% 0.21/0.51 fresh34(fresh28(is_a_theorem(implies(or(not(p11), not(p11)), not(p11))), true, or(not(p11), and(p11, p11))), true)
% 0.21/0.51 = { by lemma 20 }
% 0.21/0.52 fresh34(fresh28(is_a_theorem(implies(implies(p11, not(p11)), not(p11))), true, or(not(p11), and(p11, p11))), true)
% 0.21/0.52 = { by lemma 20 R->L }
% 0.21/0.52 fresh34(fresh28(is_a_theorem(or(not(implies(p11, not(p11))), not(p11))), true, or(not(p11), and(p11, p11))), true)
% 0.21/0.52 = { by lemma 20 R->L }
% 0.21/0.52 fresh34(fresh28(is_a_theorem(or(not(or(not(p11), not(p11))), not(p11))), true, or(not(p11), and(p11, p11))), true)
% 0.21/0.52 = { by axiom 16 (op_and) R->L }
% 0.21/0.52 fresh34(fresh28(is_a_theorem(or(fresh24(op_and, true, p11, p11), not(p11))), true, or(not(p11), and(p11, p11))), true)
% 0.21/0.52 = { by axiom 4 (principia_op_and) }
% 0.21/0.52 fresh34(fresh28(is_a_theorem(or(fresh24(true, true, p11, p11), not(p11))), true, or(not(p11), and(p11, p11))), true)
% 0.21/0.52 = { by axiom 11 (op_and) }
% 0.21/0.52 fresh34(fresh28(is_a_theorem(or(and(p11, p11), not(p11))), true, or(not(p11), and(p11, p11))), true)
% 0.21/0.52 = { by axiom 18 (modus_ponens_2) R->L }
% 0.21/0.52 fresh34(fresh59(is_a_theorem(implies(or(and(p11, p11), not(p11)), or(not(p11), and(p11, p11)))), true, or(and(p11, p11), not(p11)), or(not(p11), and(p11, p11))), true)
% 0.21/0.52 = { by axiom 17 (r3_1) R->L }
% 0.21/0.52 fresh34(fresh59(fresh8(r3, true, and(p11, p11), not(p11)), true, or(and(p11, p11), not(p11)), or(not(p11), and(p11, p11))), true)
% 0.21/0.52 = { by axiom 3 (principia_r3) }
% 0.21/0.52 fresh34(fresh59(fresh8(true, true, and(p11, p11), not(p11)), true, or(and(p11, p11), not(p11)), or(not(p11), and(p11, p11))), true)
% 0.21/0.52 = { by axiom 14 (r3_1) }
% 0.21/0.52 fresh34(fresh59(true, true, or(and(p11, p11), not(p11)), or(not(p11), and(p11, p11))), true)
% 0.21/0.52 = { by axiom 10 (modus_ponens_2) }
% 0.21/0.52 fresh34(fresh60(modus_ponens, true, or(not(p11), and(p11, p11))), true)
% 0.21/0.52 = { by axiom 1 (principia_modus_ponens) }
% 0.21/0.52 fresh34(fresh60(true, true, or(not(p11), and(p11, p11))), true)
% 0.21/0.52 = { by axiom 7 (modus_ponens_2) }
% 0.21/0.52 fresh34(true, true)
% 0.21/0.52 = { by axiom 6 (kn1) }
% 0.21/0.52 true
% 0.21/0.52 % SZS output end Proof
% 0.21/0.52
% 0.21/0.52 RESULT: Theorem (the conjecture is true).
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