TSTP Solution File: LCL486+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL486+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 : n007.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 08:19:11 EDT 2023
% Result : Theorem 5.86s 1.61s
% Output : Proof 6.42s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.05/0.12 % Problem : LCL486+1 : TPTP v8.1.2. Released v3.3.0.
% 0.05/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n007.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Aug 24 17:53:40 EDT 2023
% 0.12/0.34 % CPUTime :
% 5.86/1.61 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 5.86/1.61
% 5.86/1.61 % SZS status Theorem
% 5.86/1.62
% 6.19/1.68 % SZS output start Proof
% 6.19/1.68 Take the following subset of the input axioms:
% 6.19/1.68 fof(cn1, axiom, cn1 <=> ![P, Q, R]: is_a_theorem(implies(implies(P, Q), implies(implies(Q, R), implies(P, R))))).
% 6.19/1.68 fof(hilbert_implies_3, conjecture, implies_3).
% 6.19/1.69 fof(hilbert_op_or, axiom, op_or).
% 6.19/1.69 fof(implies_1, axiom, implies_1 <=> ![X, Y]: is_a_theorem(implies(X, implies(Y, X)))).
% 6.19/1.69 fof(implies_3, axiom, implies_3 <=> ![Z, X2, Y2]: is_a_theorem(implies(implies(X2, Y2), implies(implies(Y2, Z), implies(X2, Z))))).
% 6.19/1.69 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 6.19/1.69 fof(op_implies_or, axiom, op_implies_or => ![X2, Y2]: implies(X2, Y2)=or(not(X2), Y2)).
% 6.19/1.69 fof(or_2, axiom, or_2 <=> ![X2, Y2]: is_a_theorem(implies(Y2, or(X2, Y2)))).
% 6.19/1.69 fof(principia_modus_ponens, axiom, modus_ponens).
% 6.19/1.69 fof(principia_op_implies_or, axiom, op_implies_or).
% 6.19/1.69 fof(principia_r2, axiom, r2).
% 6.19/1.69 fof(principia_r4, axiom, r4).
% 6.19/1.69 fof(principia_r5, axiom, r5).
% 6.19/1.69 fof(r2, axiom, r2 <=> ![P2, Q2]: is_a_theorem(implies(Q2, or(P2, Q2)))).
% 6.19/1.69 fof(r4, axiom, r4 <=> ![P2, Q2, R2]: is_a_theorem(implies(or(P2, or(Q2, R2)), or(Q2, or(P2, R2))))).
% 6.19/1.69 fof(r5, axiom, r5 <=> ![P2, Q2, R2]: is_a_theorem(implies(implies(Q2, R2), implies(or(P2, Q2), or(P2, R2))))).
% 6.19/1.69
% 6.19/1.69 Now clausify the problem and encode Horn clauses using encoding 3 of
% 6.19/1.69 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 6.19/1.69 We repeatedly replace C & s=t => u=v by the two clauses:
% 6.19/1.69 fresh(y, y, x1...xn) = u
% 6.19/1.69 C => fresh(s, t, x1...xn) = v
% 6.19/1.69 where fresh is a fresh function symbol and x1..xn are the free
% 6.19/1.69 variables of u and v.
% 6.19/1.69 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 6.19/1.69 input problem has no model of domain size 1).
% 6.19/1.69
% 6.19/1.69 The encoding turns the above axioms into the following unit equations and goals:
% 6.19/1.69
% 6.19/1.70 Axiom 1 (hilbert_op_or): op_or = true.
% 6.19/1.70 Axiom 2 (principia_op_implies_or): op_implies_or = true.
% 6.19/1.70 Axiom 3 (principia_modus_ponens): modus_ponens = true.
% 6.19/1.70 Axiom 4 (principia_r2): r2 = true.
% 6.19/1.70 Axiom 5 (principia_r4): r4 = true.
% 6.19/1.70 Axiom 6 (principia_r5): r5 = true.
% 6.19/1.70 Axiom 7 (implies_1): fresh40(X, X) = true.
% 6.19/1.70 Axiom 8 (implies_3): fresh36(X, X) = true.
% 6.19/1.70 Axiom 9 (or_2): fresh17(X, X) = true.
% 6.19/1.70 Axiom 10 (modus_ponens_2): fresh60(X, X, Y) = true.
% 6.19/1.70 Axiom 11 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 6.19/1.70 Axiom 12 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 6.19/1.70 Axiom 13 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
% 6.19/1.70 Axiom 14 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
% 6.19/1.70 Axiom 15 (r2_1): fresh10(X, X, Y, Z) = true.
% 6.19/1.70 Axiom 16 (r4_1): fresh6(X, X, Y, Z, W) = true.
% 6.19/1.70 Axiom 17 (r5_1): fresh4(X, X, Y, Z, W) = true.
% 6.19/1.70 Axiom 18 (implies_1_1): fresh39(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
% 6.19/1.70 Axiom 19 (or_2_1): fresh16(or_2, true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
% 6.19/1.70 Axiom 20 (r2_1): fresh10(r2, true, X, Y) = is_a_theorem(implies(Y, or(X, Y))).
% 6.19/1.70 Axiom 21 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 6.19/1.70 Axiom 22 (implies_1): fresh40(is_a_theorem(implies(x12, implies(y12, x12))), true) = implies_1.
% 6.19/1.70 Axiom 23 (or_2): fresh17(is_a_theorem(implies(y5, or(x5, y5))), true) = or_2.
% 6.19/1.70 Axiom 24 (cn1_1): fresh51(cn1, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))).
% 6.19/1.70 Axiom 25 (r5_1): fresh4(r5, true, X, Y, Z) = is_a_theorem(implies(implies(Y, Z), implies(or(X, Y), or(X, Z)))).
% 6.19/1.70 Axiom 26 (r4_1): fresh6(r4, true, X, Y, Z) = is_a_theorem(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))).
% 6.19/1.70 Axiom 27 (implies_3): fresh36(is_a_theorem(implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), true) = implies_3.
% 6.19/1.70
% 6.19/1.70 Lemma 28: is_a_theorem(implies(X, or(Y, X))) = fresh16(or_2, op_or, Y, X).
% 6.19/1.70 Proof:
% 6.19/1.70 is_a_theorem(implies(X, or(Y, X)))
% 6.19/1.70 = { by axiom 19 (or_2_1) R->L }
% 6.19/1.70 fresh16(or_2, true, Y, X)
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) R->L }
% 6.19/1.70 fresh16(or_2, op_or, Y, X)
% 6.19/1.70
% 6.19/1.70 Lemma 29: fresh16(or_2, op_or, X, Y) = op_or.
% 6.19/1.70 Proof:
% 6.19/1.70 fresh16(or_2, op_or, X, Y)
% 6.19/1.70 = { by lemma 28 R->L }
% 6.19/1.70 is_a_theorem(implies(Y, or(X, Y)))
% 6.19/1.70 = { by axiom 20 (r2_1) R->L }
% 6.19/1.70 fresh10(r2, true, X, Y)
% 6.19/1.70 = { by axiom 4 (principia_r2) }
% 6.19/1.70 fresh10(true, true, X, Y)
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) R->L }
% 6.19/1.70 fresh10(op_or, true, X, Y)
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) R->L }
% 6.19/1.70 fresh10(op_or, op_or, X, Y)
% 6.19/1.70 = { by axiom 15 (r2_1) }
% 6.19/1.70 true
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) R->L }
% 6.19/1.70 op_or
% 6.19/1.70
% 6.19/1.70 Lemma 30: op_or = or_2.
% 6.19/1.70 Proof:
% 6.19/1.70 op_or
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) }
% 6.19/1.70 true
% 6.19/1.70 = { by axiom 9 (or_2) R->L }
% 6.19/1.70 fresh17(op_or, op_or)
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) }
% 6.19/1.70 fresh17(op_or, true)
% 6.19/1.70 = { by lemma 29 R->L }
% 6.19/1.70 fresh17(fresh16(or_2, op_or, x5, y5), true)
% 6.19/1.70 = { by lemma 28 R->L }
% 6.19/1.70 fresh17(is_a_theorem(implies(y5, or(x5, y5))), true)
% 6.19/1.70 = { by axiom 23 (or_2) }
% 6.19/1.70 or_2
% 6.19/1.70
% 6.19/1.70 Lemma 31: or(not(X), Y) = implies(X, Y).
% 6.19/1.70 Proof:
% 6.19/1.70 or(not(X), Y)
% 6.19/1.70 = { by axiom 14 (op_implies_or) R->L }
% 6.19/1.70 fresh21(op_implies_or, true, X, Y)
% 6.19/1.70 = { by axiom 2 (principia_op_implies_or) }
% 6.19/1.70 fresh21(true, true, X, Y)
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) R->L }
% 6.19/1.70 fresh21(op_or, true, X, Y)
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) R->L }
% 6.19/1.70 fresh21(op_or, op_or, X, Y)
% 6.19/1.70 = { by axiom 13 (op_implies_or) }
% 6.19/1.70 implies(X, Y)
% 6.19/1.70
% 6.19/1.70 Lemma 32: is_a_theorem(implies(X, implies(Y, X))) = fresh39(implies_1, op_or, X, Y).
% 6.19/1.70 Proof:
% 6.19/1.70 is_a_theorem(implies(X, implies(Y, X)))
% 6.19/1.70 = { by axiom 18 (implies_1_1) R->L }
% 6.19/1.70 fresh39(implies_1, true, X, Y)
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) R->L }
% 6.19/1.70 fresh39(implies_1, op_or, X, Y)
% 6.19/1.70
% 6.19/1.70 Lemma 33: or_2 = implies_1.
% 6.19/1.70 Proof:
% 6.19/1.70 or_2
% 6.19/1.70 = { by lemma 30 R->L }
% 6.19/1.70 op_or
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) }
% 6.19/1.70 true
% 6.19/1.70 = { by axiom 7 (implies_1) R->L }
% 6.19/1.70 fresh40(or_2, or_2)
% 6.19/1.70 = { by lemma 30 R->L }
% 6.19/1.70 fresh40(op_or, or_2)
% 6.19/1.70 = { by lemma 29 R->L }
% 6.19/1.70 fresh40(fresh16(or_2, op_or, not(y12), x12), or_2)
% 6.19/1.70 = { by lemma 28 R->L }
% 6.19/1.70 fresh40(is_a_theorem(implies(x12, or(not(y12), x12))), or_2)
% 6.19/1.70 = { by lemma 31 }
% 6.19/1.70 fresh40(is_a_theorem(implies(x12, implies(y12, x12))), or_2)
% 6.19/1.70 = { by lemma 32 }
% 6.19/1.70 fresh40(fresh39(implies_1, op_or, x12, y12), or_2)
% 6.19/1.70 = { by lemma 30 }
% 6.19/1.70 fresh40(fresh39(implies_1, or_2, x12, y12), or_2)
% 6.19/1.70 = { by lemma 30 R->L }
% 6.19/1.70 fresh40(fresh39(implies_1, or_2, x12, y12), op_or)
% 6.19/1.70 = { by lemma 30 R->L }
% 6.19/1.70 fresh40(fresh39(implies_1, op_or, x12, y12), op_or)
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) }
% 6.19/1.70 fresh40(fresh39(implies_1, op_or, x12, y12), true)
% 6.19/1.70 = { by lemma 32 R->L }
% 6.19/1.70 fresh40(is_a_theorem(implies(x12, implies(y12, x12))), true)
% 6.19/1.70 = { by axiom 22 (implies_1) }
% 6.19/1.70 implies_1
% 6.19/1.70
% 6.19/1.70 Lemma 34: is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = fresh51(cn1, or_2, X, Y, Z).
% 6.19/1.70 Proof:
% 6.19/1.70 is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))))
% 6.19/1.70 = { by axiom 24 (cn1_1) R->L }
% 6.19/1.70 fresh51(cn1, true, X, Y, Z)
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) R->L }
% 6.19/1.70 fresh51(cn1, op_or, X, Y, Z)
% 6.19/1.70 = { by lemma 30 }
% 6.19/1.70 fresh51(cn1, or_2, X, Y, Z)
% 6.19/1.70
% 6.19/1.70 Goal 1 (hilbert_implies_3): implies_3 = true.
% 6.19/1.70 Proof:
% 6.19/1.70 implies_3
% 6.19/1.70 = { by axiom 27 (implies_3) R->L }
% 6.19/1.70 fresh36(is_a_theorem(implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), true)
% 6.19/1.70 = { by lemma 34 }
% 6.19/1.70 fresh36(fresh51(cn1, or_2, x10, y10, z2), true)
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) R->L }
% 6.19/1.70 fresh36(fresh51(cn1, or_2, x10, y10, z2), op_or)
% 6.19/1.70 = { by lemma 30 }
% 6.19/1.70 fresh36(fresh51(cn1, or_2, x10, y10, z2), or_2)
% 6.19/1.70 = { by lemma 33 }
% 6.19/1.70 fresh36(fresh51(cn1, implies_1, x10, y10, z2), or_2)
% 6.19/1.70 = { by lemma 33 }
% 6.19/1.70 fresh36(fresh51(cn1, implies_1, x10, y10, z2), implies_1)
% 6.19/1.70 = { by lemma 33 R->L }
% 6.19/1.70 fresh36(fresh51(cn1, or_2, x10, y10, z2), implies_1)
% 6.19/1.70 = { by lemma 34 R->L }
% 6.19/1.70 fresh36(is_a_theorem(implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.19/1.70 = { by axiom 11 (modus_ponens_2) R->L }
% 6.19/1.70 fresh36(fresh28(implies_1, implies_1, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.19/1.70 = { by lemma 33 R->L }
% 6.19/1.70 fresh36(fresh28(or_2, implies_1, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.19/1.70 = { by lemma 30 R->L }
% 6.19/1.70 fresh36(fresh28(op_or, implies_1, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) }
% 6.19/1.70 fresh36(fresh28(true, implies_1, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.19/1.70 = { by axiom 17 (r5_1) R->L }
% 6.19/1.70 fresh36(fresh28(fresh4(or_2, or_2, not(x10), y10, z2), implies_1, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.19/1.70 = { by lemma 30 R->L }
% 6.19/1.70 fresh36(fresh28(fresh4(or_2, op_or, not(x10), y10, z2), implies_1, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) }
% 6.19/1.70 fresh36(fresh28(fresh4(or_2, true, not(x10), y10, z2), implies_1, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.19/1.70 = { by lemma 30 R->L }
% 6.19/1.70 fresh36(fresh28(fresh4(op_or, true, not(x10), y10, z2), implies_1, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.19/1.70 = { by axiom 1 (hilbert_op_or) }
% 6.19/1.70 fresh36(fresh28(fresh4(true, true, not(x10), y10, z2), implies_1, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.19/1.70 = { by axiom 6 (principia_r5) R->L }
% 6.19/1.70 fresh36(fresh28(fresh4(r5, true, not(x10), y10, z2), implies_1, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.19/1.70 = { by axiom 25 (r5_1) }
% 6.19/1.70 fresh36(fresh28(is_a_theorem(implies(implies(y10, z2), implies(or(not(x10), y10), or(not(x10), z2)))), implies_1, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.70 = { by lemma 31 }
% 6.42/1.70 fresh36(fresh28(is_a_theorem(implies(implies(y10, z2), implies(implies(x10, y10), or(not(x10), z2)))), implies_1, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.70 = { by lemma 31 }
% 6.42/1.70 fresh36(fresh28(is_a_theorem(implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2)))), implies_1, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.70 = { by lemma 33 R->L }
% 6.42/1.70 fresh36(fresh28(is_a_theorem(implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2)))), or_2, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.70 = { by lemma 30 R->L }
% 6.42/1.70 fresh36(fresh28(is_a_theorem(implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2)))), op_or, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.70 = { by axiom 1 (hilbert_op_or) }
% 6.42/1.70 fresh36(fresh28(is_a_theorem(implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2)))), true, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.70 = { by axiom 21 (modus_ponens_2) R->L }
% 6.42/1.71 fresh36(fresh59(is_a_theorem(implies(implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2))))), true, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by axiom 1 (hilbert_op_or) R->L }
% 6.42/1.71 fresh36(fresh59(is_a_theorem(implies(implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2))))), op_or, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by lemma 31 R->L }
% 6.42/1.71 fresh36(fresh59(is_a_theorem(implies(implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), or(not(implies(x10, y10)), implies(implies(y10, z2), implies(x10, z2))))), op_or, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by lemma 31 R->L }
% 6.42/1.71 fresh36(fresh59(is_a_theorem(implies(implies(implies(y10, z2), or(not(implies(x10, y10)), implies(x10, z2))), or(not(implies(x10, y10)), implies(implies(y10, z2), implies(x10, z2))))), op_or, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by lemma 31 R->L }
% 6.42/1.71 fresh36(fresh59(is_a_theorem(implies(implies(implies(y10, z2), or(not(implies(x10, y10)), implies(x10, z2))), or(not(implies(x10, y10)), or(not(implies(y10, z2)), implies(x10, z2))))), op_or, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by lemma 31 R->L }
% 6.42/1.71 fresh36(fresh59(is_a_theorem(implies(or(not(implies(y10, z2)), or(not(implies(x10, y10)), implies(x10, z2))), or(not(implies(x10, y10)), or(not(implies(y10, z2)), implies(x10, z2))))), op_or, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by axiom 26 (r4_1) R->L }
% 6.42/1.71 fresh36(fresh59(fresh6(r4, true, not(implies(y10, z2)), not(implies(x10, y10)), implies(x10, z2)), op_or, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by axiom 5 (principia_r4) }
% 6.42/1.71 fresh36(fresh59(fresh6(true, true, not(implies(y10, z2)), not(implies(x10, y10)), implies(x10, z2)), op_or, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by axiom 1 (hilbert_op_or) R->L }
% 6.42/1.71 fresh36(fresh59(fresh6(op_or, true, not(implies(y10, z2)), not(implies(x10, y10)), implies(x10, z2)), op_or, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by lemma 30 }
% 6.42/1.71 fresh36(fresh59(fresh6(or_2, true, not(implies(y10, z2)), not(implies(x10, y10)), implies(x10, z2)), op_or, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by axiom 1 (hilbert_op_or) R->L }
% 6.42/1.71 fresh36(fresh59(fresh6(or_2, op_or, not(implies(y10, z2)), not(implies(x10, y10)), implies(x10, z2)), op_or, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by lemma 30 }
% 6.42/1.71 fresh36(fresh59(fresh6(or_2, or_2, not(implies(y10, z2)), not(implies(x10, y10)), implies(x10, z2)), op_or, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by axiom 16 (r4_1) }
% 6.42/1.71 fresh36(fresh59(true, op_or, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by axiom 1 (hilbert_op_or) R->L }
% 6.42/1.71 fresh36(fresh59(op_or, op_or, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by lemma 30 }
% 6.42/1.71 fresh36(fresh59(or_2, op_or, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by lemma 33 }
% 6.42/1.71 fresh36(fresh59(implies_1, op_or, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by lemma 30 }
% 6.42/1.71 fresh36(fresh59(implies_1, or_2, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by lemma 33 }
% 6.42/1.71 fresh36(fresh59(implies_1, implies_1, implies(implies(y10, z2), implies(implies(x10, y10), implies(x10, z2))), implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by axiom 12 (modus_ponens_2) }
% 6.42/1.71 fresh36(fresh60(modus_ponens, true, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.71 = { by axiom 3 (principia_modus_ponens) }
% 6.42/1.72 fresh36(fresh60(true, true, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.72 = { by axiom 1 (hilbert_op_or) R->L }
% 6.42/1.72 fresh36(fresh60(op_or, true, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.72 = { by axiom 1 (hilbert_op_or) R->L }
% 6.42/1.72 fresh36(fresh60(op_or, op_or, implies(implies(x10, y10), implies(implies(y10, z2), implies(x10, z2)))), implies_1)
% 6.42/1.72 = { by axiom 10 (modus_ponens_2) }
% 6.42/1.72 fresh36(true, implies_1)
% 6.42/1.72 = { by axiom 1 (hilbert_op_or) R->L }
% 6.42/1.72 fresh36(op_or, implies_1)
% 6.42/1.72 = { by lemma 30 }
% 6.42/1.72 fresh36(or_2, implies_1)
% 6.42/1.72 = { by lemma 33 }
% 6.42/1.72 fresh36(implies_1, implies_1)
% 6.42/1.72 = { by axiom 8 (implies_3) }
% 6.42/1.72 true
% 6.42/1.72 % SZS output end Proof
% 6.42/1.72
% 6.42/1.72 RESULT: Theorem (the conjecture is true).
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