TSTP Solution File: LCL500+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL500+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 : n017.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 3.03s 0.78s
% Output : Proof 3.03s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.09/0.14 % Problem : LCL500+1 : TPTP v8.1.2. Released v3.3.0.
% 0.15/0.15 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.37 % Computer : n017.cluster.edu
% 0.15/0.37 % Model : x86_64 x86_64
% 0.15/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.37 % Memory : 8042.1875MB
% 0.15/0.37 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.37 % CPULimit : 300
% 0.15/0.37 % WCLimit : 300
% 0.15/0.37 % DateTime : Fri Aug 25 04:14:54 EDT 2023
% 0.15/0.37 % CPUTime :
% 3.03/0.78 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 3.03/0.78
% 3.03/0.78 % SZS status Theorem
% 3.03/0.78
% 3.03/0.80 % SZS output start Proof
% 3.03/0.80 Take the following subset of the input axioms:
% 3.03/0.80 fof(kn2, axiom, kn2 <=> ![P, Q]: is_a_theorem(implies(and(P, Q), P))).
% 3.03/0.80 fof(modus_ponens, axiom, modus_ponens <=> ![X, Y]: ((is_a_theorem(X) & is_a_theorem(implies(X, Y))) => is_a_theorem(Y))).
% 3.03/0.80 fof(op_and, axiom, op_and => ![X2, Y2]: and(X2, Y2)=not(or(not(X2), not(Y2)))).
% 3.03/0.80 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 3.03/0.80 fof(op_implies_or, axiom, op_implies_or => ![X2, Y2]: implies(X2, Y2)=or(not(X2), Y2)).
% 3.03/0.80 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 3.03/0.80 fof(principia_modus_ponens, axiom, modus_ponens).
% 3.03/0.80 fof(principia_op_and, axiom, op_and).
% 3.03/0.80 fof(principia_op_implies_or, axiom, op_implies_or).
% 3.03/0.80 fof(principia_r2, axiom, r2).
% 3.03/0.80 fof(principia_r3, axiom, r3).
% 3.03/0.80 fof(principia_r5, axiom, r5).
% 3.03/0.80 fof(r2, axiom, r2 <=> ![P2, Q2]: is_a_theorem(implies(Q2, or(P2, Q2)))).
% 3.03/0.80 fof(r3, axiom, r3 <=> ![P2, Q2]: is_a_theorem(implies(or(P2, Q2), or(Q2, P2)))).
% 3.03/0.80 fof(r5, axiom, r5 <=> ![R, P2, Q2]: is_a_theorem(implies(implies(Q2, R), implies(or(P2, Q2), or(P2, R))))).
% 3.03/0.80 fof(rosser_kn2, conjecture, kn2).
% 3.03/0.80 fof(rosser_op_implies_and, axiom, op_implies_and).
% 3.03/0.80 fof(rosser_op_or, axiom, op_or).
% 3.03/0.80
% 3.03/0.80 Now clausify the problem and encode Horn clauses using encoding 3 of
% 3.03/0.80 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 3.03/0.80 We repeatedly replace C & s=t => u=v by the two clauses:
% 3.03/0.80 fresh(y, y, x1...xn) = u
% 3.03/0.80 C => fresh(s, t, x1...xn) = v
% 3.03/0.80 where fresh is a fresh function symbol and x1..xn are the free
% 3.03/0.80 variables of u and v.
% 3.03/0.80 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 3.03/0.80 input problem has no model of domain size 1).
% 3.03/0.80
% 3.03/0.80 The encoding turns the above axioms into the following unit equations and goals:
% 3.03/0.80
% 3.03/0.80 Axiom 1 (principia_modus_ponens): modus_ponens = true.
% 3.03/0.80 Axiom 2 (principia_r2): r2 = true.
% 3.03/0.80 Axiom 3 (principia_r3): r3 = true.
% 3.03/0.80 Axiom 4 (principia_r5): r5 = true.
% 3.03/0.80 Axiom 5 (rosser_op_or): op_or = true.
% 3.03/0.80 Axiom 6 (principia_op_and): op_and = true.
% 3.03/0.80 Axiom 7 (rosser_op_implies_and): op_implies_and = true.
% 3.03/0.80 Axiom 8 (principia_op_implies_or): op_implies_or = true.
% 3.03/0.80 Axiom 9 (kn2): fresh32(X, X) = true.
% 3.03/0.80 Axiom 10 (modus_ponens_2): fresh60(X, X, Y) = true.
% 3.03/0.80 Axiom 11 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 3.03/0.80 Axiom 12 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 3.03/0.80 Axiom 13 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
% 3.03/0.80 Axiom 14 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 3.03/0.80 Axiom 15 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 3.03/0.80 Axiom 16 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
% 3.03/0.81 Axiom 17 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
% 3.03/0.81 Axiom 18 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 3.03/0.81 Axiom 19 (r2_1): fresh10(X, X, Y, Z) = true.
% 3.03/0.81 Axiom 20 (r3_1): fresh8(X, X, Y, Z) = true.
% 3.03/0.81 Axiom 21 (r2_1): fresh10(r2, true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
% 3.03/0.81 Axiom 22 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
% 3.03/0.81 Axiom 23 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 3.03/0.81 Axiom 24 (r5_1): fresh4(X, X, Y, Z, W) = true.
% 3.03/0.81 Axiom 25 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
% 3.03/0.81 Axiom 26 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 3.03/0.81 Axiom 27 (kn2): fresh32(is_a_theorem(implies(and(p10, q8), p10)), true) = kn2.
% 3.03/0.81 Axiom 28 (r5_1): fresh4(r5, true, X, Y, Z) = is_a_theorem(implies(implies(Y, Z), implies(or(X, Y), or(X, Z)))).
% 3.03/0.81
% 3.03/0.81 Lemma 29: implies(not(X), Y) = or(X, Y).
% 3.03/0.81 Proof:
% 3.03/0.81 implies(not(X), Y)
% 3.03/0.81 = { by axiom 14 (op_implies_and) R->L }
% 3.03/0.81 fresh22(true, true, not(X), Y)
% 3.03/0.81 = { by axiom 7 (rosser_op_implies_and) R->L }
% 3.03/0.81 fresh22(op_implies_and, true, not(X), Y)
% 3.03/0.81 = { by axiom 15 (op_implies_and) }
% 3.03/0.81 not(and(not(X), not(Y)))
% 3.03/0.81 = { by axiom 23 (op_or) R->L }
% 3.03/0.81 fresh20(op_or, true, X, Y)
% 3.03/0.81 = { by axiom 5 (rosser_op_or) }
% 3.03/0.81 fresh20(true, true, X, Y)
% 3.03/0.81 = { by axiom 18 (op_or) }
% 3.03/0.81 or(X, Y)
% 3.03/0.81
% 3.03/0.81 Lemma 30: or(not(X), Y) = implies(X, Y).
% 3.03/0.81 Proof:
% 3.03/0.81 or(not(X), Y)
% 3.03/0.81 = { by axiom 17 (op_implies_or) R->L }
% 3.03/0.81 fresh21(op_implies_or, true, X, Y)
% 3.03/0.81 = { by axiom 8 (principia_op_implies_or) }
% 3.03/0.81 fresh21(true, true, X, Y)
% 3.03/0.81 = { by axiom 16 (op_implies_or) }
% 3.03/0.81 implies(X, Y)
% 3.03/0.81
% 3.03/0.81 Lemma 31: fresh59(X, X, Y, Z) = true.
% 3.03/0.81 Proof:
% 3.03/0.81 fresh59(X, X, Y, Z)
% 3.03/0.81 = { by axiom 12 (modus_ponens_2) }
% 3.03/0.81 fresh60(modus_ponens, true, Z)
% 3.03/0.81 = { by axiom 1 (principia_modus_ponens) }
% 3.03/0.81 fresh60(true, true, Z)
% 3.03/0.81 = { by axiom 10 (modus_ponens_2) }
% 3.03/0.81 true
% 3.03/0.81
% 3.03/0.81 Lemma 32: not(implies(X, not(Y))) = and(X, Y).
% 3.03/0.81 Proof:
% 3.03/0.81 not(implies(X, not(Y)))
% 3.03/0.81 = { by lemma 30 R->L }
% 3.03/0.81 not(or(not(X), not(Y)))
% 3.03/0.81 = { by axiom 22 (op_and) R->L }
% 3.03/0.81 fresh24(op_and, true, X, Y)
% 3.03/0.81 = { by axiom 6 (principia_op_and) }
% 3.03/0.81 fresh24(true, true, X, Y)
% 3.03/0.81 = { by axiom 13 (op_and) }
% 3.03/0.81 and(X, Y)
% 3.03/0.81
% 3.03/0.81 Lemma 33: or(implies(X, not(Y)), Z) = implies(and(X, Y), Z).
% 3.03/0.81 Proof:
% 3.03/0.81 or(implies(X, not(Y)), Z)
% 3.03/0.81 = { by lemma 29 R->L }
% 3.03/0.81 implies(not(implies(X, not(Y))), Z)
% 3.03/0.81 = { by lemma 32 }
% 3.03/0.81 implies(and(X, Y), Z)
% 3.03/0.81
% 3.03/0.81 Lemma 34: is_a_theorem(implies(or(X, Y), or(Y, X))) = true.
% 3.03/0.81 Proof:
% 3.03/0.81 is_a_theorem(implies(or(X, Y), or(Y, X)))
% 3.03/0.81 = { by axiom 25 (r3_1) R->L }
% 3.03/0.81 fresh8(r3, true, X, Y)
% 3.03/0.81 = { by axiom 3 (principia_r3) }
% 3.03/0.81 fresh8(true, true, X, Y)
% 3.03/0.81 = { by axiom 20 (r3_1) }
% 3.03/0.81 true
% 3.03/0.81
% 3.03/0.81 Goal 1 (rosser_kn2): kn2 = true.
% 3.03/0.81 Proof:
% 3.03/0.81 kn2
% 3.03/0.81 = { by axiom 27 (kn2) R->L }
% 3.03/0.81 fresh32(is_a_theorem(implies(and(p10, q8), p10)), true)
% 3.03/0.81 = { by lemma 33 R->L }
% 3.03/0.81 fresh32(is_a_theorem(or(implies(p10, not(q8)), p10)), true)
% 3.03/0.81 = { by axiom 11 (modus_ponens_2) R->L }
% 3.03/0.81 fresh32(fresh28(true, true, or(implies(p10, not(q8)), p10)), true)
% 3.03/0.81 = { by lemma 31 R->L }
% 3.03/0.81 fresh32(fresh28(fresh59(true, true, implies(or(not(q8), not(p10)), implies(p10, not(q8))), or(implies(p10, not(q8)), not(or(not(q8), not(p10))))), true, or(implies(p10, not(q8)), p10)), true)
% 3.03/0.81 = { by lemma 34 R->L }
% 3.03/0.81 fresh32(fresh28(fresh59(is_a_theorem(implies(or(not(or(not(q8), not(p10))), implies(p10, not(q8))), or(implies(p10, not(q8)), not(or(not(q8), not(p10)))))), true, implies(or(not(q8), not(p10)), implies(p10, not(q8))), or(implies(p10, not(q8)), not(or(not(q8), not(p10))))), true, or(implies(p10, not(q8)), p10)), true)
% 3.03/0.81 = { by lemma 30 }
% 3.03/0.81 fresh32(fresh28(fresh59(is_a_theorem(implies(implies(or(not(q8), not(p10)), implies(p10, not(q8))), or(implies(p10, not(q8)), not(or(not(q8), not(p10)))))), true, implies(or(not(q8), not(p10)), implies(p10, not(q8))), or(implies(p10, not(q8)), not(or(not(q8), not(p10))))), true, or(implies(p10, not(q8)), p10)), true)
% 3.03/0.81 = { by axiom 26 (modus_ponens_2) }
% 3.03/0.81 fresh32(fresh28(fresh28(is_a_theorem(implies(or(not(q8), not(p10)), implies(p10, not(q8)))), true, or(implies(p10, not(q8)), not(or(not(q8), not(p10))))), true, or(implies(p10, not(q8)), p10)), true)
% 3.03/0.81 = { by lemma 30 R->L }
% 3.03/0.81 fresh32(fresh28(fresh28(is_a_theorem(implies(or(not(q8), not(p10)), or(not(p10), not(q8)))), true, or(implies(p10, not(q8)), not(or(not(q8), not(p10))))), true, or(implies(p10, not(q8)), p10)), true)
% 3.03/0.81 = { by lemma 34 }
% 3.03/0.81 fresh32(fresh28(fresh28(true, true, or(implies(p10, not(q8)), not(or(not(q8), not(p10))))), true, or(implies(p10, not(q8)), p10)), true)
% 3.03/0.81 = { by axiom 11 (modus_ponens_2) }
% 3.03/0.81 fresh32(fresh28(is_a_theorem(or(implies(p10, not(q8)), not(or(not(q8), not(p10))))), true, or(implies(p10, not(q8)), p10)), true)
% 3.03/0.81 = { by lemma 29 R->L }
% 3.03/0.81 fresh32(fresh28(is_a_theorem(or(implies(p10, not(q8)), not(implies(not(not(q8)), not(p10))))), true, or(implies(p10, not(q8)), p10)), true)
% 3.03/0.81 = { by lemma 32 }
% 3.03/0.81 fresh32(fresh28(is_a_theorem(or(implies(p10, not(q8)), and(not(not(q8)), p10))), true, or(implies(p10, not(q8)), p10)), true)
% 3.03/0.81 = { by axiom 26 (modus_ponens_2) R->L }
% 3.03/0.81 fresh32(fresh59(is_a_theorem(implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.81 = { by axiom 11 (modus_ponens_2) R->L }
% 3.03/0.81 fresh32(fresh59(fresh28(true, true, implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.81 = { by lemma 31 R->L }
% 3.03/0.81 fresh32(fresh59(fresh28(fresh59(true, true, or(p10, implies(not(not(q8)), not(p10))), or(implies(not(not(q8)), not(p10)), p10)), true, implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.81 = { by lemma 34 R->L }
% 3.03/0.82 fresh32(fresh59(fresh28(fresh59(is_a_theorem(implies(or(p10, implies(not(not(q8)), not(p10))), or(implies(not(not(q8)), not(p10)), p10))), true, or(p10, implies(not(not(q8)), not(p10))), or(implies(not(not(q8)), not(p10)), p10)), true, implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.82 = { by axiom 26 (modus_ponens_2) }
% 3.03/0.82 fresh32(fresh59(fresh28(fresh28(is_a_theorem(or(p10, implies(not(not(q8)), not(p10)))), true, or(implies(not(not(q8)), not(p10)), p10)), true, implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.82 = { by lemma 30 R->L }
% 3.03/0.82 fresh32(fresh59(fresh28(fresh28(is_a_theorem(or(p10, or(not(not(not(q8))), not(p10)))), true, or(implies(not(not(q8)), not(p10)), p10)), true, implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.82 = { by lemma 29 R->L }
% 3.03/0.82 fresh32(fresh59(fresh28(fresh28(is_a_theorem(implies(not(p10), or(not(not(not(q8))), not(p10)))), true, or(implies(not(not(q8)), not(p10)), p10)), true, implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.82 = { by axiom 21 (r2_1) R->L }
% 3.03/0.82 fresh32(fresh59(fresh28(fresh28(fresh10(r2, true, not(not(not(q8))), not(p10)), true, or(implies(not(not(q8)), not(p10)), p10)), true, implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.82 = { by axiom 2 (principia_r2) }
% 3.03/0.82 fresh32(fresh59(fresh28(fresh28(fresh10(true, true, not(not(not(q8))), not(p10)), true, or(implies(not(not(q8)), not(p10)), p10)), true, implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.82 = { by axiom 19 (r2_1) }
% 3.03/0.82 fresh32(fresh59(fresh28(fresh28(true, true, or(implies(not(not(q8)), not(p10)), p10)), true, implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.82 = { by axiom 11 (modus_ponens_2) }
% 3.03/0.82 fresh32(fresh59(fresh28(is_a_theorem(or(implies(not(not(q8)), not(p10)), p10)), true, implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.82 = { by lemma 33 }
% 3.03/0.82 fresh32(fresh59(fresh28(is_a_theorem(implies(and(not(not(q8)), p10), p10)), true, implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.82 = { by axiom 26 (modus_ponens_2) R->L }
% 3.03/0.82 fresh32(fresh59(fresh59(is_a_theorem(implies(implies(and(not(not(q8)), p10), p10), implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)))), true, implies(and(not(not(q8)), p10), p10), implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.82 = { by axiom 28 (r5_1) R->L }
% 3.03/0.82 fresh32(fresh59(fresh59(fresh4(r5, true, implies(p10, not(q8)), and(not(not(q8)), p10), p10), true, implies(and(not(not(q8)), p10), p10), implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.82 = { by axiom 4 (principia_r5) }
% 3.03/0.82 fresh32(fresh59(fresh59(fresh4(true, true, implies(p10, not(q8)), and(not(not(q8)), p10), p10), true, implies(and(not(not(q8)), p10), p10), implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.82 = { by axiom 24 (r5_1) }
% 3.03/0.82 fresh32(fresh59(fresh59(true, true, implies(and(not(not(q8)), p10), p10), implies(or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10))), true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.82 = { by lemma 31 }
% 3.03/0.82 fresh32(fresh59(true, true, or(implies(p10, not(q8)), and(not(not(q8)), p10)), or(implies(p10, not(q8)), p10)), true)
% 3.03/0.82 = { by lemma 31 }
% 3.03/0.82 fresh32(true, true)
% 3.03/0.82 = { by axiom 9 (kn2) }
% 3.03/0.82 true
% 3.03/0.82 % SZS output end Proof
% 3.03/0.82
% 3.03/0.82 RESULT: Theorem (the conjecture is true).
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