TSTP Solution File: LCL483+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL483+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 08:19:10 EDT 2023
% Result : Theorem 0.18s 0.48s
% 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
% 0.00/0.12 % Problem : LCL483+1 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.16/0.34 % Computer : n001.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Thu Aug 24 19:35:11 EDT 2023
% 0.16/0.34 % CPUTime :
% 0.18/0.48 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.18/0.48
% 0.18/0.48 % SZS status Theorem
% 0.18/0.48
% 0.18/0.49 % SZS output start Proof
% 0.18/0.49 Take the following subset of the input axioms:
% 0.18/0.49 fof(hilbert_modus_tollens, conjecture, modus_tollens).
% 0.18/0.49 fof(hilbert_op_implies_and, axiom, op_implies_and).
% 0.18/0.49 fof(hilbert_op_or, axiom, op_or).
% 0.18/0.49 fof(modus_tollens, axiom, modus_tollens <=> ![X, Y]: is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y)))).
% 0.18/0.49 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 0.18/0.49 fof(op_implies_or, axiom, op_implies_or => ![X2, Y2]: implies(X2, Y2)=or(not(X2), Y2)).
% 0.18/0.49 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 0.18/0.49 fof(principia_op_implies_or, axiom, op_implies_or).
% 0.18/0.49 fof(principia_r3, axiom, r3).
% 0.18/0.49 fof(r3, axiom, r3 <=> ![P, Q]: is_a_theorem(implies(or(P, Q), or(Q, P)))).
% 0.18/0.49
% 0.18/0.49 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.18/0.49 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.18/0.49 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.18/0.49 fresh(y, y, x1...xn) = u
% 0.18/0.49 C => fresh(s, t, x1...xn) = v
% 0.18/0.49 where fresh is a fresh function symbol and x1..xn are the free
% 0.18/0.49 variables of u and v.
% 0.18/0.49 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.18/0.49 input problem has no model of domain size 1).
% 0.18/0.49
% 0.18/0.49 The encoding turns the above axioms into the following unit equations and goals:
% 0.18/0.49
% 0.18/0.49 Axiom 1 (principia_r3): r3 = true.
% 0.18/0.49 Axiom 2 (hilbert_op_or): op_or = true.
% 0.18/0.49 Axiom 3 (hilbert_op_implies_and): op_implies_and = true.
% 0.18/0.49 Axiom 4 (principia_op_implies_or): op_implies_or = true.
% 0.18/0.49 Axiom 5 (modus_tollens): fresh26(X, X) = true.
% 0.18/0.49 Axiom 6 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 0.18/0.49 Axiom 7 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 0.18/0.49 Axiom 8 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
% 0.18/0.49 Axiom 9 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
% 0.18/0.49 Axiom 10 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 0.18/0.49 Axiom 11 (r3_1): fresh8(X, X, Y, Z) = true.
% 0.18/0.49 Axiom 12 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 0.18/0.49 Axiom 13 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
% 0.18/0.49 Axiom 14 (modus_tollens): fresh26(is_a_theorem(implies(implies(not(y13), not(x13)), implies(x13, y13))), true) = modus_tollens.
% 0.18/0.49
% 0.18/0.49 Goal 1 (hilbert_modus_tollens): modus_tollens = true.
% 0.18/0.49 Proof:
% 0.18/0.49 modus_tollens
% 0.18/0.49 = { by axiom 14 (modus_tollens) R->L }
% 0.18/0.49 fresh26(is_a_theorem(implies(implies(not(y13), not(x13)), implies(x13, y13))), true)
% 0.18/0.49 = { by axiom 6 (op_implies_and) R->L }
% 0.18/0.49 fresh26(is_a_theorem(implies(fresh22(true, true, not(y13), not(x13)), implies(x13, y13))), true)
% 0.18/0.49 = { by axiom 3 (hilbert_op_implies_and) R->L }
% 0.18/0.49 fresh26(is_a_theorem(implies(fresh22(op_implies_and, true, not(y13), not(x13)), implies(x13, y13))), true)
% 0.18/0.49 = { by axiom 7 (op_implies_and) }
% 0.18/0.49 fresh26(is_a_theorem(implies(not(and(not(y13), not(not(x13)))), implies(x13, y13))), true)
% 0.18/0.49 = { by axiom 12 (op_or) R->L }
% 0.18/0.49 fresh26(is_a_theorem(implies(fresh20(op_or, true, y13, not(x13)), implies(x13, y13))), true)
% 0.18/0.49 = { by axiom 2 (hilbert_op_or) }
% 0.18/0.49 fresh26(is_a_theorem(implies(fresh20(true, true, y13, not(x13)), implies(x13, y13))), true)
% 0.18/0.49 = { by axiom 10 (op_or) }
% 0.18/0.49 fresh26(is_a_theorem(implies(or(y13, not(x13)), implies(x13, y13))), true)
% 0.18/0.49 = { by axiom 8 (op_implies_or) R->L }
% 0.18/0.49 fresh26(is_a_theorem(implies(or(y13, not(x13)), fresh21(true, true, x13, y13))), true)
% 0.18/0.49 = { by axiom 4 (principia_op_implies_or) R->L }
% 0.18/0.49 fresh26(is_a_theorem(implies(or(y13, not(x13)), fresh21(op_implies_or, true, x13, y13))), true)
% 0.18/0.49 = { by axiom 9 (op_implies_or) }
% 0.18/0.49 fresh26(is_a_theorem(implies(or(y13, not(x13)), or(not(x13), y13))), true)
% 0.18/0.49 = { by axiom 13 (r3_1) R->L }
% 0.18/0.49 fresh26(fresh8(r3, true, y13, not(x13)), true)
% 0.18/0.49 = { by axiom 1 (principia_r3) }
% 0.18/0.49 fresh26(fresh8(true, true, y13, not(x13)), true)
% 0.18/0.49 = { by axiom 11 (r3_1) }
% 0.18/0.49 fresh26(true, true)
% 0.18/0.49 = { by axiom 5 (modus_tollens) }
% 0.18/0.49 true
% 0.18/0.49 % SZS output end Proof
% 0.18/0.49
% 0.18/0.49 RESULT: Theorem (the conjecture is true).
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