TSTP Solution File: LCL494+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL494+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 : n012.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 08:19:13 EDT 2023
% Result : Theorem 0.20s 0.53s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LCL494+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
% 0.14/0.36 % Computer : n012.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Fri Aug 25 04:54:56 EDT 2023
% 0.14/0.36 % CPUTime :
% 0.20/0.53 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 0.20/0.53
% 0.20/0.53 % SZS status Theorem
% 0.20/0.53
% 0.20/0.54 % SZS output start Proof
% 0.20/0.54 Take the following subset of the input axioms:
% 0.20/0.54 fof(equivalence_2, axiom, equivalence_2 <=> ![X, Y]: is_a_theorem(implies(equiv(X, Y), implies(Y, X)))).
% 0.20/0.54 fof(hilbert_equivalence_2, conjecture, equivalence_2).
% 0.20/0.54 fof(hilbert_op_implies_and, axiom, op_implies_and).
% 0.20/0.54 fof(hilbert_op_or, axiom, op_or).
% 0.20/0.54 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 0.20/0.54 fof(op_and, axiom, op_and => ![X2, Y2]: and(X2, Y2)=not(or(not(X2), not(Y2)))).
% 0.20/0.54 fof(op_equiv, axiom, op_equiv => ![X2, Y2]: equiv(X2, Y2)=and(implies(X2, Y2), implies(Y2, X2))).
% 0.20/0.54 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 0.20/0.54 fof(op_implies_or, axiom, op_implies_or => ![X2, Y2]: implies(X2, Y2)=or(not(X2), Y2)).
% 0.20/0.54 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 0.20/0.54 fof(principia_modus_ponens, axiom, modus_ponens).
% 0.20/0.54 fof(principia_op_and, axiom, op_and).
% 0.20/0.54 fof(principia_op_equiv, axiom, op_equiv).
% 0.20/0.54 fof(principia_op_implies_or, axiom, op_implies_or).
% 0.20/0.54 fof(principia_r2, axiom, r2).
% 0.20/0.54 fof(principia_r3, axiom, r3).
% 0.20/0.54 fof(r2, axiom, r2 <=> ![P, Q]: is_a_theorem(implies(Q, or(P, Q)))).
% 0.20/0.54 fof(r3, axiom, r3 <=> ![P2, Q2]: is_a_theorem(implies(or(P2, Q2), or(Q2, P2)))).
% 0.20/0.54
% 0.20/0.54 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.54 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.54 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.54 fresh(y, y, x1...xn) = u
% 0.20/0.54 C => fresh(s, t, x1...xn) = v
% 0.20/0.54 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.54 variables of u and v.
% 0.20/0.54 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.54 input problem has no model of domain size 1).
% 0.20/0.54
% 0.20/0.54 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.54
% 0.20/0.54 Axiom 1 (principia_modus_ponens): modus_ponens = true.
% 0.20/0.54 Axiom 2 (principia_r2): r2 = true.
% 0.20/0.54 Axiom 3 (principia_r3): r3 = true.
% 0.20/0.54 Axiom 4 (principia_op_equiv): op_equiv = true.
% 0.20/0.54 Axiom 5 (hilbert_op_or): op_or = true.
% 0.20/0.54 Axiom 6 (principia_op_and): op_and = true.
% 0.20/0.54 Axiom 7 (hilbert_op_implies_and): op_implies_and = true.
% 0.20/0.54 Axiom 8 (principia_op_implies_or): op_implies_or = true.
% 0.20/0.54 Axiom 9 (equivalence_2): fresh44(X, X) = true.
% 0.20/0.54 Axiom 10 (modus_ponens_2): fresh60(X, X, Y) = true.
% 0.20/0.54 Axiom 11 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 0.20/0.54 Axiom 12 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 0.20/0.54 Axiom 13 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
% 0.20/0.54 Axiom 14 (op_equiv): fresh23(X, X, Y, Z) = equiv(Y, Z).
% 0.20/0.54 Axiom 15 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 0.20/0.54 Axiom 16 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 0.20/0.54 Axiom 17 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
% 0.20/0.54 Axiom 18 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
% 0.20/0.54 Axiom 19 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 0.20/0.54 Axiom 20 (r2_1): fresh10(X, X, Y, Z) = true.
% 0.20/0.54 Axiom 21 (r3_1): fresh8(X, X, Y, Z) = true.
% 0.20/0.54 Axiom 22 (r2_1): fresh10(r2, true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
% 0.20/0.54 Axiom 23 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
% 0.20/0.54 Axiom 24 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 0.20/0.54 Axiom 25 (op_equiv): fresh23(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
% 0.20/0.54 Axiom 26 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
% 0.20/0.54 Axiom 27 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 0.20/0.54 Axiom 28 (equivalence_2): fresh44(is_a_theorem(implies(equiv(x2, y2), implies(y2, x2))), true) = equivalence_2.
% 0.20/0.54
% 0.20/0.54 Lemma 29: implies(not(X), Y) = or(X, Y).
% 0.20/0.54 Proof:
% 0.20/0.54 implies(not(X), Y)
% 0.20/0.54 = { by axiom 15 (op_implies_and) R->L }
% 0.20/0.54 fresh22(true, true, not(X), Y)
% 0.20/0.54 = { by axiom 7 (hilbert_op_implies_and) R->L }
% 0.20/0.54 fresh22(op_implies_and, true, not(X), Y)
% 0.20/0.54 = { by axiom 16 (op_implies_and) }
% 0.20/0.54 not(and(not(X), not(Y)))
% 0.20/0.54 = { by axiom 24 (op_or) R->L }
% 0.20/0.54 fresh20(op_or, true, X, Y)
% 0.20/0.54 = { by axiom 5 (hilbert_op_or) }
% 0.20/0.54 fresh20(true, true, X, Y)
% 0.20/0.54 = { by axiom 19 (op_or) }
% 0.20/0.54 or(X, Y)
% 0.20/0.54
% 0.20/0.54 Lemma 30: or(not(X), Y) = implies(X, Y).
% 0.20/0.54 Proof:
% 0.20/0.54 or(not(X), Y)
% 0.20/0.54 = { by axiom 18 (op_implies_or) R->L }
% 0.20/0.54 fresh21(op_implies_or, true, X, Y)
% 0.20/0.54 = { by axiom 8 (principia_op_implies_or) }
% 0.20/0.54 fresh21(true, true, X, Y)
% 0.20/0.54 = { by axiom 17 (op_implies_or) }
% 0.20/0.54 implies(X, Y)
% 0.20/0.54
% 0.20/0.54 Goal 1 (hilbert_equivalence_2): equivalence_2 = true.
% 0.20/0.54 Proof:
% 0.20/0.54 equivalence_2
% 0.20/0.54 = { by axiom 28 (equivalence_2) R->L }
% 0.20/0.54 fresh44(is_a_theorem(implies(equiv(x2, y2), implies(y2, x2))), true)
% 0.20/0.54 = { by axiom 14 (op_equiv) R->L }
% 0.20/0.54 fresh44(is_a_theorem(implies(fresh23(true, true, x2, y2), implies(y2, x2))), true)
% 0.20/0.54 = { by axiom 4 (principia_op_equiv) R->L }
% 0.20/0.54 fresh44(is_a_theorem(implies(fresh23(op_equiv, true, x2, y2), implies(y2, x2))), true)
% 0.20/0.54 = { by axiom 25 (op_equiv) }
% 0.20/0.54 fresh44(is_a_theorem(implies(and(implies(x2, y2), implies(y2, x2)), implies(y2, x2))), true)
% 0.20/0.54 = { by axiom 13 (op_and) R->L }
% 0.20/0.54 fresh44(is_a_theorem(implies(fresh24(true, true, implies(x2, y2), implies(y2, x2)), implies(y2, x2))), true)
% 0.20/0.55 = { by axiom 6 (principia_op_and) R->L }
% 0.20/0.55 fresh44(is_a_theorem(implies(fresh24(op_and, true, implies(x2, y2), implies(y2, x2)), implies(y2, x2))), true)
% 0.20/0.55 = { by axiom 23 (op_and) }
% 0.20/0.55 fresh44(is_a_theorem(implies(not(or(not(implies(x2, y2)), not(implies(y2, x2)))), implies(y2, x2))), true)
% 0.20/0.55 = { by lemma 30 }
% 0.20/0.55 fresh44(is_a_theorem(implies(not(implies(implies(x2, y2), not(implies(y2, x2)))), implies(y2, x2))), true)
% 0.20/0.55 = { by lemma 29 }
% 0.20/0.55 fresh44(is_a_theorem(or(implies(implies(x2, y2), not(implies(y2, x2))), implies(y2, x2))), true)
% 0.20/0.55 = { by axiom 11 (modus_ponens_2) R->L }
% 0.20/0.55 fresh44(fresh28(true, true, or(implies(implies(x2, y2), not(implies(y2, x2))), implies(y2, x2))), true)
% 0.20/0.55 = { by axiom 20 (r2_1) R->L }
% 0.20/0.55 fresh44(fresh28(fresh10(true, true, not(implies(x2, y2)), not(implies(y2, x2))), true, or(implies(implies(x2, y2), not(implies(y2, x2))), implies(y2, x2))), true)
% 0.20/0.55 = { by axiom 2 (principia_r2) R->L }
% 0.20/0.55 fresh44(fresh28(fresh10(r2, true, not(implies(x2, y2)), not(implies(y2, x2))), true, or(implies(implies(x2, y2), not(implies(y2, x2))), implies(y2, x2))), true)
% 0.20/0.55 = { by axiom 22 (r2_1) }
% 0.20/0.55 fresh44(fresh28(is_a_theorem(implies(not(implies(y2, x2)), or(not(implies(x2, y2)), not(implies(y2, x2))))), true, or(implies(implies(x2, y2), not(implies(y2, x2))), implies(y2, x2))), true)
% 0.20/0.55 = { by lemma 30 }
% 0.20/0.55 fresh44(fresh28(is_a_theorem(implies(not(implies(y2, x2)), implies(implies(x2, y2), not(implies(y2, x2))))), true, or(implies(implies(x2, y2), not(implies(y2, x2))), implies(y2, x2))), true)
% 0.20/0.55 = { by lemma 29 }
% 0.20/0.55 fresh44(fresh28(is_a_theorem(or(implies(y2, x2), implies(implies(x2, y2), not(implies(y2, x2))))), true, or(implies(implies(x2, y2), not(implies(y2, x2))), implies(y2, x2))), true)
% 0.20/0.55 = { by axiom 27 (modus_ponens_2) R->L }
% 0.20/0.55 fresh44(fresh59(is_a_theorem(implies(or(implies(y2, x2), implies(implies(x2, y2), not(implies(y2, x2)))), or(implies(implies(x2, y2), not(implies(y2, x2))), implies(y2, x2)))), true, or(implies(y2, x2), implies(implies(x2, y2), not(implies(y2, x2)))), or(implies(implies(x2, y2), not(implies(y2, x2))), implies(y2, x2))), true)
% 0.20/0.55 = { by axiom 26 (r3_1) R->L }
% 0.20/0.55 fresh44(fresh59(fresh8(r3, true, implies(y2, x2), implies(implies(x2, y2), not(implies(y2, x2)))), true, or(implies(y2, x2), implies(implies(x2, y2), not(implies(y2, x2)))), or(implies(implies(x2, y2), not(implies(y2, x2))), implies(y2, x2))), true)
% 0.20/0.55 = { by axiom 3 (principia_r3) }
% 0.20/0.55 fresh44(fresh59(fresh8(true, true, implies(y2, x2), implies(implies(x2, y2), not(implies(y2, x2)))), true, or(implies(y2, x2), implies(implies(x2, y2), not(implies(y2, x2)))), or(implies(implies(x2, y2), not(implies(y2, x2))), implies(y2, x2))), true)
% 0.20/0.55 = { by axiom 21 (r3_1) }
% 0.20/0.55 fresh44(fresh59(true, true, or(implies(y2, x2), implies(implies(x2, y2), not(implies(y2, x2)))), or(implies(implies(x2, y2), not(implies(y2, x2))), implies(y2, x2))), true)
% 0.20/0.55 = { by axiom 12 (modus_ponens_2) }
% 0.20/0.55 fresh44(fresh60(modus_ponens, true, or(implies(implies(x2, y2), not(implies(y2, x2))), implies(y2, x2))), true)
% 0.20/0.55 = { by axiom 1 (principia_modus_ponens) }
% 0.20/0.55 fresh44(fresh60(true, true, or(implies(implies(x2, y2), not(implies(y2, x2))), implies(y2, x2))), true)
% 0.20/0.55 = { by axiom 10 (modus_ponens_2) }
% 0.20/0.55 fresh44(true, true)
% 0.20/0.55 = { by axiom 9 (equivalence_2) }
% 0.20/0.55 true
% 0.20/0.55 % SZS output end Proof
% 0.20/0.55
% 0.20/0.55 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------