TSTP Solution File: LCL495+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL495+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 : n014.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 63.79s 8.77s
% Output : Proof 65.26s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.14 % Problem : LCL495+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
% 0.15/0.36 % Computer : n014.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 300
% 0.15/0.36 % DateTime : Thu Aug 24 17:44:18 EDT 2023
% 0.15/0.36 % CPUTime :
% 63.79/8.77 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 63.79/8.77
% 63.79/8.77 % SZS status Theorem
% 63.79/8.77
% 65.26/8.82 % SZS output start Proof
% 65.26/8.82 Take the following subset of the input axioms:
% 65.26/8.82 fof(equivalence_3, axiom, equivalence_3 <=> ![X, Y]: is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y))))).
% 65.26/8.82 fof(hilbert_equivalence_3, conjecture, equivalence_3).
% 65.26/8.82 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 65.26/8.82 fof(op_and, axiom, op_and => ![X2, Y2]: and(X2, Y2)=not(or(not(X2), not(Y2)))).
% 65.26/8.82 fof(op_equiv, axiom, op_equiv => ![X2, Y2]: equiv(X2, Y2)=and(implies(X2, Y2), implies(Y2, X2))).
% 65.26/8.82 fof(op_implies_or, axiom, op_implies_or => ![X2, Y2]: implies(X2, Y2)=or(not(X2), Y2)).
% 65.26/8.82 fof(principia_modus_ponens, axiom, modus_ponens).
% 65.26/8.82 fof(principia_op_and, axiom, op_and).
% 65.26/8.82 fof(principia_op_equiv, axiom, op_equiv).
% 65.26/8.82 fof(principia_op_implies_or, axiom, op_implies_or).
% 65.26/8.82 fof(principia_r3, axiom, r3).
% 65.26/8.82 fof(principia_r4, axiom, r4).
% 65.26/8.82 fof(principia_r5, axiom, r5).
% 65.26/8.82 fof(r3, axiom, r3 <=> ![P, Q]: is_a_theorem(implies(or(P, Q), or(Q, P)))).
% 65.26/8.82 fof(r4, axiom, r4 <=> ![R, P2, Q2]: is_a_theorem(implies(or(P2, or(Q2, R)), or(Q2, or(P2, R))))).
% 65.26/8.83 fof(r5, axiom, r5 <=> ![P2, Q2, R2]: is_a_theorem(implies(implies(Q2, R2), implies(or(P2, Q2), or(P2, R2))))).
% 65.26/8.83
% 65.26/8.83 Now clausify the problem and encode Horn clauses using encoding 3 of
% 65.26/8.83 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 65.26/8.83 We repeatedly replace C & s=t => u=v by the two clauses:
% 65.26/8.83 fresh(y, y, x1...xn) = u
% 65.26/8.83 C => fresh(s, t, x1...xn) = v
% 65.26/8.83 where fresh is a fresh function symbol and x1..xn are the free
% 65.26/8.83 variables of u and v.
% 65.26/8.83 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 65.26/8.83 input problem has no model of domain size 1).
% 65.26/8.83
% 65.26/8.83 The encoding turns the above axioms into the following unit equations and goals:
% 65.26/8.83
% 65.26/8.83 Axiom 1 (principia_modus_ponens): modus_ponens = true.
% 65.26/8.83 Axiom 2 (principia_r3): r3 = true.
% 65.26/8.83 Axiom 3 (principia_r4): r4 = true.
% 65.26/8.83 Axiom 4 (principia_r5): r5 = true.
% 65.26/8.83 Axiom 5 (principia_op_equiv): op_equiv = true.
% 65.26/8.83 Axiom 6 (principia_op_and): op_and = true.
% 65.26/8.83 Axiom 7 (principia_op_implies_or): op_implies_or = true.
% 65.26/8.83 Axiom 8 (equivalence_3): fresh42(X, X) = true.
% 65.26/8.83 Axiom 9 (modus_ponens_2): fresh60(X, X, Y) = true.
% 65.26/8.83 Axiom 10 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 65.26/8.83 Axiom 11 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 65.26/8.83 Axiom 12 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
% 65.26/8.83 Axiom 13 (op_equiv): fresh23(X, X, Y, Z) = equiv(Y, Z).
% 65.26/8.83 Axiom 14 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
% 65.26/8.83 Axiom 15 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
% 65.26/8.83 Axiom 16 (r3_1): fresh8(X, X, Y, Z) = true.
% 65.26/8.83 Axiom 17 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
% 65.26/8.83 Axiom 18 (r4_1): fresh6(X, X, Y, Z, W) = true.
% 65.26/8.83 Axiom 19 (r5_1): fresh4(X, X, Y, Z, W) = true.
% 65.26/8.83 Axiom 20 (op_equiv): fresh23(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
% 65.26/8.83 Axiom 21 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
% 65.26/8.83 Axiom 22 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 65.26/8.83 Axiom 23 (r5_1): fresh4(r5, true, X, Y, Z) = is_a_theorem(implies(implies(Y, Z), implies(or(X, Y), or(X, Z)))).
% 65.26/8.83 Axiom 24 (r4_1): fresh6(r4, true, X, Y, Z) = is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))).
% 65.26/8.83 Axiom 25 (equivalence_3): fresh42(is_a_theorem(implies(implies(x, y), implies(implies(y, x), equiv(x, y)))), true) = equivalence_3.
% 65.26/8.83
% 65.26/8.83 Lemma 26: or(not(X), Y) = implies(X, Y).
% 65.26/8.83 Proof:
% 65.26/8.83 or(not(X), Y)
% 65.26/8.83 = { by axiom 15 (op_implies_or) R->L }
% 65.26/8.83 fresh21(op_implies_or, true, X, Y)
% 65.26/8.83 = { by axiom 7 (principia_op_implies_or) }
% 65.26/8.83 fresh21(true, true, X, Y)
% 65.26/8.83 = { by axiom 14 (op_implies_or) }
% 65.26/8.83 implies(X, Y)
% 65.26/8.83
% 65.26/8.83 Lemma 27: fresh59(X, X, Y, Z) = true.
% 65.26/8.83 Proof:
% 65.26/8.83 fresh59(X, X, Y, Z)
% 65.26/8.83 = { by axiom 11 (modus_ponens_2) }
% 65.26/8.83 fresh60(modus_ponens, true, Z)
% 65.26/8.83 = { by axiom 1 (principia_modus_ponens) }
% 65.26/8.83 fresh60(true, true, Z)
% 65.26/8.83 = { by axiom 9 (modus_ponens_2) }
% 65.26/8.83 true
% 65.26/8.83
% 65.26/8.83 Lemma 28: is_a_theorem(implies(implies(X, Y), or(Y, not(X)))) = true.
% 65.26/8.83 Proof:
% 65.26/8.83 is_a_theorem(implies(implies(X, Y), or(Y, not(X))))
% 65.26/8.83 = { by lemma 26 R->L }
% 65.26/8.83 is_a_theorem(implies(or(not(X), Y), or(Y, not(X))))
% 65.26/8.83 = { by axiom 21 (r3_1) R->L }
% 65.26/8.83 fresh8(r3, true, not(X), Y)
% 65.26/8.83 = { by axiom 2 (principia_r3) }
% 65.26/8.83 fresh8(true, true, not(X), Y)
% 65.26/8.83 = { by axiom 16 (r3_1) }
% 65.26/8.83 true
% 65.26/8.83
% 65.26/8.83 Lemma 29: is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))) = true.
% 65.26/8.83 Proof:
% 65.26/8.83 is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z))))
% 65.26/8.83 = { by axiom 24 (r4_1) R->L }
% 65.26/8.83 fresh6(r4, true, X, Y, Z)
% 65.26/8.83 = { by axiom 3 (principia_r4) }
% 65.26/8.83 fresh6(true, true, X, Y, Z)
% 65.26/8.83 = { by axiom 18 (r4_1) }
% 65.26/8.83 true
% 65.26/8.83
% 65.26/8.83 Goal 1 (hilbert_equivalence_3): equivalence_3 = true.
% 65.26/8.83 Proof:
% 65.26/8.83 equivalence_3
% 65.26/8.83 = { by axiom 25 (equivalence_3) R->L }
% 65.26/8.83 fresh42(is_a_theorem(implies(implies(x, y), implies(implies(y, x), equiv(x, y)))), true)
% 65.26/8.83 = { by axiom 13 (op_equiv) R->L }
% 65.26/8.83 fresh42(is_a_theorem(implies(implies(x, y), implies(implies(y, x), fresh23(true, true, x, y)))), true)
% 65.26/8.83 = { by axiom 5 (principia_op_equiv) R->L }
% 65.26/8.83 fresh42(is_a_theorem(implies(implies(x, y), implies(implies(y, x), fresh23(op_equiv, true, x, y)))), true)
% 65.26/8.83 = { by axiom 20 (op_equiv) }
% 65.26/8.83 fresh42(is_a_theorem(implies(implies(x, y), implies(implies(y, x), and(implies(x, y), implies(y, x))))), true)
% 65.26/8.83 = { by axiom 12 (op_and) R->L }
% 65.26/8.83 fresh42(is_a_theorem(implies(implies(x, y), implies(implies(y, x), fresh24(true, true, implies(x, y), implies(y, x))))), true)
% 65.26/8.83 = { by axiom 6 (principia_op_and) R->L }
% 65.26/8.83 fresh42(is_a_theorem(implies(implies(x, y), implies(implies(y, x), fresh24(op_and, true, implies(x, y), implies(y, x))))), true)
% 65.26/8.83 = { by axiom 17 (op_and) }
% 65.26/8.83 fresh42(is_a_theorem(implies(implies(x, y), implies(implies(y, x), not(or(not(implies(x, y)), not(implies(y, x))))))), true)
% 65.26/8.83 = { by lemma 26 }
% 65.26/8.83 fresh42(is_a_theorem(implies(implies(x, y), implies(implies(y, x), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.83 = { by lemma 26 R->L }
% 65.26/8.83 fresh42(is_a_theorem(implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.83 = { by axiom 10 (modus_ponens_2) R->L }
% 65.26/8.83 fresh42(fresh28(true, true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.83 = { by lemma 27 R->L }
% 65.26/8.83 fresh42(fresh28(fresh59(true, true, or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.84 = { by lemma 27 R->L }
% 65.26/8.84 fresh42(fresh28(fresh59(fresh59(true, true, implies(implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y))), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x)))))), implies(or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x)))))))), true, or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.84 = { by axiom 19 (r5_1) R->L }
% 65.26/8.84 fresh42(fresh28(fresh59(fresh59(fresh4(true, true, not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y))), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x)))))), true, implies(implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y))), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x)))))), implies(or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x)))))))), true, or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.84 = { by axiom 4 (principia_r5) R->L }
% 65.26/8.84 fresh42(fresh28(fresh59(fresh59(fresh4(r5, true, not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y))), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x)))))), true, implies(implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y))), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x)))))), implies(or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x)))))))), true, or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.84 = { by axiom 23 (r5_1) }
% 65.26/8.84 fresh42(fresh28(fresh59(fresh59(is_a_theorem(implies(implies(implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y))), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x)))))), implies(or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))))), true, implies(implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y))), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x)))))), implies(or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x)))))))), true, or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.84 = { by axiom 22 (modus_ponens_2) }
% 65.26/8.84 fresh42(fresh28(fresh59(fresh28(is_a_theorem(implies(implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y))), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x)))))))), true, or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.84 = { by lemma 28 }
% 65.26/8.84 fresh42(fresh28(fresh59(fresh28(true, true, implies(or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x)))))))), true, or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.84 = { by axiom 10 (modus_ponens_2) }
% 65.26/8.84 fresh42(fresh28(fresh59(is_a_theorem(implies(or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x)))))))), true, or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.84 = { by axiom 22 (modus_ponens_2) }
% 65.26/8.84 fresh42(fresh28(fresh28(is_a_theorem(or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y))))), true, or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.84 = { by axiom 10 (modus_ponens_2) R->L }
% 65.26/8.84 fresh42(fresh28(fresh28(fresh28(true, true, or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y))))), true, or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.84 = { by lemma 28 R->L }
% 65.26/8.84 fresh42(fresh28(fresh28(fresh28(is_a_theorem(implies(implies(implies(x, y), not(implies(y, x))), or(not(implies(y, x)), not(implies(x, y))))), true, or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y))))), true, or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.84 = { by axiom 22 (modus_ponens_2) R->L }
% 65.26/8.84 fresh42(fresh28(fresh28(fresh59(is_a_theorem(implies(implies(implies(implies(x, y), not(implies(y, x))), or(not(implies(y, x)), not(implies(x, y)))), or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y)))))), true, implies(implies(implies(x, y), not(implies(y, x))), or(not(implies(y, x)), not(implies(x, y)))), or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y))))), true, or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.84 = { by lemma 26 R->L }
% 65.26/8.84 fresh42(fresh28(fresh28(fresh59(is_a_theorem(implies(implies(implies(implies(x, y), not(implies(y, x))), or(not(implies(y, x)), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(implies(x, y), not(implies(y, x)))), not(implies(x, y)))))), true, implies(implies(implies(x, y), not(implies(y, x))), or(not(implies(y, x)), not(implies(x, y)))), or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y))))), true, or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.84 = { by lemma 26 R->L }
% 65.26/8.84 fresh42(fresh28(fresh28(fresh59(is_a_theorem(implies(or(not(implies(implies(x, y), not(implies(y, x)))), or(not(implies(y, x)), not(implies(x, y)))), or(not(implies(y, x)), or(not(implies(implies(x, y), not(implies(y, x)))), not(implies(x, y)))))), true, implies(implies(implies(x, y), not(implies(y, x))), or(not(implies(y, x)), not(implies(x, y)))), or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y))))), true, or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.85 = { by lemma 29 }
% 65.26/8.85 fresh42(fresh28(fresh28(fresh59(true, true, implies(implies(implies(x, y), not(implies(y, x))), or(not(implies(y, x)), not(implies(x, y)))), or(not(implies(y, x)), implies(implies(implies(x, y), not(implies(y, x))), not(implies(x, y))))), true, or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.85 = { by lemma 27 }
% 65.26/8.85 fresh42(fresh28(fresh28(true, true, or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.85 = { by axiom 10 (modus_ponens_2) }
% 65.26/8.85 fresh42(fresh28(is_a_theorem(or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.85 = { by lemma 26 }
% 65.26/8.85 fresh42(fresh28(is_a_theorem(or(not(implies(y, x)), implies(implies(x, y), not(implies(implies(x, y), not(implies(y, x))))))), true, implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.85 = { by axiom 22 (modus_ponens_2) R->L }
% 65.26/8.85 fresh42(fresh59(is_a_theorem(implies(or(not(implies(y, x)), implies(implies(x, y), not(implies(implies(x, y), not(implies(y, x)))))), implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x)))))))), true, or(not(implies(y, x)), implies(implies(x, y), not(implies(implies(x, y), not(implies(y, x)))))), implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.85 = { by lemma 26 R->L }
% 65.26/8.85 fresh42(fresh59(is_a_theorem(implies(or(not(implies(y, x)), implies(implies(x, y), not(implies(implies(x, y), not(implies(y, x)))))), or(not(implies(x, y)), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x)))))))), true, or(not(implies(y, x)), implies(implies(x, y), not(implies(implies(x, y), not(implies(y, x)))))), implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.85 = { by lemma 26 R->L }
% 65.26/8.85 fresh42(fresh59(is_a_theorem(implies(or(not(implies(y, x)), or(not(implies(x, y)), not(implies(implies(x, y), not(implies(y, x)))))), or(not(implies(x, y)), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x)))))))), true, or(not(implies(y, x)), implies(implies(x, y), not(implies(implies(x, y), not(implies(y, x)))))), implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.85 = { by lemma 29 }
% 65.26/8.85 fresh42(fresh59(true, true, or(not(implies(y, x)), implies(implies(x, y), not(implies(implies(x, y), not(implies(y, x)))))), implies(implies(x, y), or(not(implies(y, x)), not(implies(implies(x, y), not(implies(y, x))))))), true)
% 65.26/8.85 = { by lemma 27 }
% 65.26/8.85 fresh42(true, true)
% 65.26/8.85 = { by axiom 8 (equivalence_3) }
% 65.26/8.85 true
% 65.26/8.85 % SZS output end Proof
% 65.26/8.85
% 65.26/8.85 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------