TSTP Solution File: LCL459+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL459+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 : n001.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 08:19:05 EDT 2023
% Result : Theorem 0.20s 0.47s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LCL459+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 : n001.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 18:23:11 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.20/0.47 Command-line arguments: --ground-connectedness --complete-subsets
% 0.20/0.47
% 0.20/0.47 % SZS status Theorem
% 0.20/0.47
% 0.20/0.48 % SZS output start Proof
% 0.20/0.48 Take the following subset of the input axioms:
% 0.20/0.48 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 0.20/0.48 fof(and_3, axiom, and_3 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, and(X2, Y2))))).
% 0.20/0.48 fof(hilbert_and_1, axiom, and_1).
% 0.20/0.48 fof(hilbert_and_3, axiom, and_3).
% 0.20/0.48 fof(hilbert_implies_2, axiom, implies_2).
% 0.20/0.48 fof(hilbert_modus_ponens, axiom, modus_ponens).
% 0.20/0.48 fof(implies_2, axiom, implies_2 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, implies(X2, Y2)), implies(X2, Y2)))).
% 0.20/0.48 fof(kn1, axiom, kn1 <=> ![P]: is_a_theorem(implies(P, and(P, P)))).
% 0.20/0.48 fof(kn2, axiom, kn2 <=> ![Q, P2]: is_a_theorem(implies(and(P2, Q), P2))).
% 0.20/0.48 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 0.20/0.48 fof(rosser_kn1, conjecture, kn1).
% 0.20/0.48
% 0.20/0.48 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.48 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.48 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.48 fresh(y, y, x1...xn) = u
% 0.20/0.48 C => fresh(s, t, x1...xn) = v
% 0.20/0.48 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.48 variables of u and v.
% 0.20/0.48 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.48 input problem has no model of domain size 1).
% 0.20/0.48
% 0.20/0.48 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.48
% 0.20/0.48 Axiom 1 (hilbert_modus_ponens): modus_ponens = true.
% 0.20/0.48 Axiom 2 (hilbert_implies_2): implies_2 = true.
% 0.20/0.48 Axiom 3 (hilbert_and_1): and_1 = true.
% 0.20/0.48 Axiom 4 (hilbert_and_3): and_3 = true.
% 0.20/0.48 Axiom 5 (kn1): fresh34(X, X) = true.
% 0.20/0.48 Axiom 6 (kn2): fresh32(X, X) = true.
% 0.20/0.48 Axiom 7 (modus_ponens_2): fresh60(X, X, Y) = true.
% 0.20/0.48 Axiom 8 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 0.20/0.48 Axiom 9 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 0.20/0.48 Axiom 10 (and_1_1): fresh58(X, X, Y, Z) = true.
% 0.20/0.48 Axiom 11 (and_3_1): fresh53(X, X, Y, Z) = true.
% 0.20/0.48 Axiom 12 (implies_2_1): fresh37(X, X, Y, Z) = true.
% 0.20/0.48 Axiom 13 (kn1_1): fresh33(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
% 0.20/0.48 Axiom 14 (and_1_1): fresh58(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 0.20/0.48 Axiom 15 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 0.20/0.48 Axiom 16 (kn1): fresh34(is_a_theorem(implies(p11, and(p11, p11))), true) = kn1.
% 0.20/0.48 Axiom 17 (kn2): fresh32(is_a_theorem(implies(and(p10, q8), p10)), true) = kn2.
% 0.20/0.48 Axiom 18 (and_3_1): fresh53(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
% 0.20/0.48 Axiom 19 (implies_2_1): fresh37(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
% 0.20/0.48
% 0.20/0.48 Lemma 20: modus_ponens = kn2.
% 0.20/0.48 Proof:
% 0.20/0.48 modus_ponens
% 0.20/0.48 = { by axiom 1 (hilbert_modus_ponens) }
% 0.20/0.48 true
% 0.20/0.48 = { by axiom 6 (kn2) R->L }
% 0.20/0.48 fresh32(modus_ponens, modus_ponens)
% 0.20/0.48 = { by axiom 1 (hilbert_modus_ponens) }
% 0.20/0.48 fresh32(modus_ponens, true)
% 0.20/0.48 = { by axiom 1 (hilbert_modus_ponens) }
% 0.20/0.48 fresh32(true, true)
% 0.20/0.48 = { by axiom 10 (and_1_1) R->L }
% 0.20/0.48 fresh32(fresh58(modus_ponens, modus_ponens, p10, q8), true)
% 0.20/0.48 = { by axiom 1 (hilbert_modus_ponens) }
% 0.20/0.48 fresh32(fresh58(modus_ponens, true, p10, q8), true)
% 0.20/0.48 = { by axiom 1 (hilbert_modus_ponens) }
% 0.20/0.48 fresh32(fresh58(true, true, p10, q8), true)
% 0.20/0.48 = { by axiom 3 (hilbert_and_1) R->L }
% 0.20/0.48 fresh32(fresh58(and_1, true, p10, q8), true)
% 0.20/0.48 = { by axiom 14 (and_1_1) }
% 0.20/0.48 fresh32(is_a_theorem(implies(and(p10, q8), p10)), true)
% 0.20/0.48 = { by axiom 17 (kn2) }
% 0.20/0.48 kn2
% 0.20/0.48
% 0.20/0.48 Lemma 21: is_a_theorem(implies(X, and(X, X))) = fresh33(kn1, modus_ponens, X).
% 0.20/0.48 Proof:
% 0.20/0.48 is_a_theorem(implies(X, and(X, X)))
% 0.20/0.48 = { by axiom 13 (kn1_1) R->L }
% 0.20/0.48 fresh33(kn1, true, X)
% 0.20/0.48 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 0.20/0.49 fresh33(kn1, modus_ponens, X)
% 0.20/0.49
% 0.20/0.49 Goal 1 (rosser_kn1): kn1 = true.
% 0.20/0.49 Proof:
% 0.20/0.49 kn1
% 0.20/0.49 = { by axiom 16 (kn1) R->L }
% 0.20/0.49 fresh34(is_a_theorem(implies(p11, and(p11, p11))), true)
% 0.20/0.49 = { by lemma 21 }
% 0.20/0.49 fresh34(fresh33(kn1, modus_ponens, p11), true)
% 0.20/0.49 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 0.20/0.49 fresh34(fresh33(kn1, modus_ponens, p11), modus_ponens)
% 0.20/0.49 = { by lemma 20 }
% 0.20/0.49 fresh34(fresh33(kn1, kn2, p11), modus_ponens)
% 0.20/0.49 = { by lemma 20 }
% 0.20/0.49 fresh34(fresh33(kn1, kn2, p11), kn2)
% 0.20/0.49 = { by lemma 20 R->L }
% 0.20/0.49 fresh34(fresh33(kn1, modus_ponens, p11), kn2)
% 0.20/0.49 = { by lemma 21 R->L }
% 0.20/0.49 fresh34(is_a_theorem(implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 8 (modus_ponens_2) R->L }
% 0.20/0.49 fresh34(fresh28(kn2, kn2, implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by lemma 20 R->L }
% 0.20/0.49 fresh34(fresh28(modus_ponens, kn2, implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 1 (hilbert_modus_ponens) }
% 0.20/0.49 fresh34(fresh28(true, kn2, implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 11 (and_3_1) R->L }
% 0.20/0.49 fresh34(fresh28(fresh53(kn2, kn2, p11, p11), kn2, implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by lemma 20 R->L }
% 0.20/0.49 fresh34(fresh28(fresh53(kn2, modus_ponens, p11, p11), kn2, implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 1 (hilbert_modus_ponens) }
% 0.20/0.49 fresh34(fresh28(fresh53(kn2, true, p11, p11), kn2, implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by lemma 20 R->L }
% 0.20/0.49 fresh34(fresh28(fresh53(modus_ponens, true, p11, p11), kn2, implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 1 (hilbert_modus_ponens) }
% 0.20/0.49 fresh34(fresh28(fresh53(true, true, p11, p11), kn2, implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 4 (hilbert_and_3) R->L }
% 0.20/0.49 fresh34(fresh28(fresh53(and_3, true, p11, p11), kn2, implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 18 (and_3_1) }
% 0.20/0.49 fresh34(fresh28(is_a_theorem(implies(p11, implies(p11, and(p11, p11)))), kn2, implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by lemma 20 R->L }
% 0.20/0.49 fresh34(fresh28(is_a_theorem(implies(p11, implies(p11, and(p11, p11)))), modus_ponens, implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 1 (hilbert_modus_ponens) }
% 0.20/0.49 fresh34(fresh28(is_a_theorem(implies(p11, implies(p11, and(p11, p11)))), true, implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 15 (modus_ponens_2) R->L }
% 0.20/0.49 fresh34(fresh59(is_a_theorem(implies(implies(p11, implies(p11, and(p11, p11))), implies(p11, and(p11, p11)))), true, implies(p11, implies(p11, and(p11, p11))), implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 0.20/0.49 fresh34(fresh59(is_a_theorem(implies(implies(p11, implies(p11, and(p11, p11))), implies(p11, and(p11, p11)))), modus_ponens, implies(p11, implies(p11, and(p11, p11))), implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 19 (implies_2_1) R->L }
% 0.20/0.49 fresh34(fresh59(fresh37(implies_2, true, p11, and(p11, p11)), modus_ponens, implies(p11, implies(p11, and(p11, p11))), implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 2 (hilbert_implies_2) }
% 0.20/0.49 fresh34(fresh59(fresh37(true, true, p11, and(p11, p11)), modus_ponens, implies(p11, implies(p11, and(p11, p11))), implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 0.20/0.49 fresh34(fresh59(fresh37(modus_ponens, true, p11, and(p11, p11)), modus_ponens, implies(p11, implies(p11, and(p11, p11))), implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by lemma 20 }
% 0.20/0.49 fresh34(fresh59(fresh37(kn2, true, p11, and(p11, p11)), modus_ponens, implies(p11, implies(p11, and(p11, p11))), implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 0.20/0.49 fresh34(fresh59(fresh37(kn2, modus_ponens, p11, and(p11, p11)), modus_ponens, implies(p11, implies(p11, and(p11, p11))), implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by lemma 20 }
% 0.20/0.49 fresh34(fresh59(fresh37(kn2, kn2, p11, and(p11, p11)), modus_ponens, implies(p11, implies(p11, and(p11, p11))), implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 12 (implies_2_1) }
% 0.20/0.49 fresh34(fresh59(true, modus_ponens, implies(p11, implies(p11, and(p11, p11))), implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 0.20/0.49 fresh34(fresh59(modus_ponens, modus_ponens, implies(p11, implies(p11, and(p11, p11))), implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by lemma 20 }
% 0.20/0.49 fresh34(fresh59(kn2, modus_ponens, implies(p11, implies(p11, and(p11, p11))), implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by lemma 20 }
% 0.20/0.49 fresh34(fresh59(kn2, kn2, implies(p11, implies(p11, and(p11, p11))), implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 9 (modus_ponens_2) }
% 0.20/0.49 fresh34(fresh60(modus_ponens, true, implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 0.20/0.49 fresh34(fresh60(modus_ponens, modus_ponens, implies(p11, and(p11, p11))), kn2)
% 0.20/0.49 = { by axiom 7 (modus_ponens_2) }
% 0.20/0.49 fresh34(true, kn2)
% 0.20/0.49 = { by axiom 1 (hilbert_modus_ponens) R->L }
% 0.20/0.49 fresh34(modus_ponens, kn2)
% 0.20/0.49 = { by lemma 20 }
% 0.20/0.49 fresh34(kn2, kn2)
% 0.20/0.49 = { by axiom 5 (kn1) }
% 0.20/0.49 true
% 0.20/0.49 % SZS output end Proof
% 0.20/0.49
% 0.20/0.49 RESULT: Theorem (the conjecture is true).
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