TSTP Solution File: LCL518+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL518+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 : n021.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 08:19:17 EDT 2023
% Result : Theorem 0.20s 0.66s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL518+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
% 0.13/0.34 % Computer : n021.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Aug 24 19:37:10 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.66 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.66
% 0.20/0.66 % SZS status Theorem
% 0.20/0.66
% 0.20/0.69 % SZS output start Proof
% 0.20/0.69 Take the following subset of the input axioms:
% 0.20/0.69 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 0.20/0.69 fof(cn3, axiom, cn3 <=> ![P]: is_a_theorem(implies(implies(not(P), P), P))).
% 0.20/0.69 fof(kn1, axiom, kn1 <=> ![P2]: is_a_theorem(implies(P2, and(P2, P2)))).
% 0.20/0.69 fof(kn2, axiom, kn2 <=> ![Q, P2]: is_a_theorem(implies(and(P2, Q), P2))).
% 0.20/0.69 fof(kn3, axiom, kn3 <=> ![R, P2, Q2]: is_a_theorem(implies(implies(P2, Q2), implies(not(and(Q2, R)), not(and(R, P2)))))).
% 0.20/0.69 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 0.20/0.69 fof(op_and, axiom, op_and => ![X2, Y2]: and(X2, Y2)=not(or(not(X2), not(Y2)))).
% 0.20/0.69 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 0.20/0.69 fof(op_implies_or, axiom, op_implies_or => ![X2, Y2]: implies(X2, Y2)=or(not(X2), Y2)).
% 0.20/0.69 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 0.20/0.69 fof(principia_op_and, axiom, op_and).
% 0.20/0.69 fof(principia_op_implies_or, axiom, op_implies_or).
% 0.20/0.69 fof(principia_r1, conjecture, r1).
% 0.20/0.69 fof(r1, axiom, r1 <=> ![P2]: is_a_theorem(implies(or(P2, P2), P2))).
% 0.20/0.69 fof(rosser_kn1, axiom, kn1).
% 0.20/0.69 fof(rosser_kn2, axiom, kn2).
% 0.20/0.69 fof(rosser_kn3, axiom, kn3).
% 0.20/0.69 fof(rosser_modus_ponens, axiom, modus_ponens).
% 0.20/0.69 fof(rosser_op_implies_and, axiom, op_implies_and).
% 0.20/0.69 fof(rosser_op_or, axiom, op_or).
% 0.20/0.69
% 0.20/0.69 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.69 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.69 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.69 fresh(y, y, x1...xn) = u
% 0.20/0.69 C => fresh(s, t, x1...xn) = v
% 0.20/0.69 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.69 variables of u and v.
% 0.20/0.69 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.69 input problem has no model of domain size 1).
% 0.20/0.69
% 0.20/0.69 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.69
% 0.20/0.69 Axiom 1 (rosser_op_or): op_or = true.
% 0.20/0.69 Axiom 2 (principia_op_and): op_and = true.
% 0.20/0.69 Axiom 3 (rosser_op_implies_and): op_implies_and = true.
% 0.20/0.69 Axiom 4 (principia_op_implies_or): op_implies_or = true.
% 0.20/0.69 Axiom 5 (rosser_modus_ponens): modus_ponens = true.
% 0.20/0.69 Axiom 6 (rosser_kn1): kn1 = true.
% 0.20/0.69 Axiom 7 (rosser_kn2): kn2 = true.
% 0.20/0.69 Axiom 8 (rosser_kn3): kn3 = true.
% 0.20/0.69 Axiom 9 (and_1): fresh57(X, X) = true.
% 0.20/0.69 Axiom 10 (r1): fresh13(X, X) = true.
% 0.20/0.69 Axiom 11 (modus_ponens_2): fresh60(X, X, Y) = true.
% 0.20/0.69 Axiom 12 (kn1_1): fresh33(X, X, Y) = true.
% 0.20/0.69 Axiom 13 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 0.20/0.69 Axiom 14 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 0.20/0.69 Axiom 15 (kn2_1): fresh31(X, X, Y, Z) = true.
% 0.20/0.69 Axiom 16 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
% 0.20/0.69 Axiom 17 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 0.20/0.69 Axiom 18 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
% 0.20/0.69 Axiom 19 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
% 0.20/0.69 Axiom 20 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 0.20/0.69 Axiom 21 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 0.20/0.69 Axiom 22 (kn3_1): fresh29(X, X, Y, Z, W) = true.
% 0.20/0.69 Axiom 23 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 0.20/0.69 Axiom 24 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
% 0.20/0.69 Axiom 25 (kn1_1): fresh33(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
% 0.20/0.69 Axiom 26 (and_1_1): fresh58(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 0.20/0.69 Axiom 27 (kn2_1): fresh31(kn2, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 0.20/0.69 Axiom 28 (r1_1): fresh12(r1, true, X) = is_a_theorem(implies(or(X, X), X)).
% 0.20/0.69 Axiom 29 (cn3_1): fresh47(cn3, true, X) = is_a_theorem(implies(implies(not(X), X), X)).
% 0.20/0.69 Axiom 30 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 0.20/0.69 Axiom 31 (and_1): fresh57(is_a_theorem(implies(and(x9, y9), x9)), true) = and_1.
% 0.20/0.69 Axiom 32 (r1): fresh13(is_a_theorem(implies(or(p5, p5), p5)), true) = r1.
% 0.20/0.69 Axiom 33 (kn3_1): fresh29(kn3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X))))).
% 0.20/0.69
% 0.20/0.69 Lemma 34: is_a_theorem(implies(and(X, Y), X)) = fresh58(and_1, op_or, X, Y).
% 0.20/0.69 Proof:
% 0.20/0.69 is_a_theorem(implies(and(X, Y), X))
% 0.20/0.69 = { by axiom 26 (and_1_1) R->L }
% 0.20/0.69 fresh58(and_1, true, X, Y)
% 0.20/0.69 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.69 fresh58(and_1, op_or, X, Y)
% 0.20/0.69
% 0.20/0.69 Lemma 35: fresh58(and_1, op_or, X, Y) = op_or.
% 0.20/0.69 Proof:
% 0.20/0.69 fresh58(and_1, op_or, X, Y)
% 0.20/0.69 = { by lemma 34 R->L }
% 0.20/0.69 is_a_theorem(implies(and(X, Y), X))
% 0.20/0.69 = { by axiom 27 (kn2_1) R->L }
% 0.20/0.69 fresh31(kn2, true, X, Y)
% 0.20/0.69 = { by axiom 7 (rosser_kn2) }
% 0.20/0.69 fresh31(true, true, X, Y)
% 0.20/0.69 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.69 fresh31(op_or, true, X, Y)
% 0.20/0.69 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.69 fresh31(op_or, op_or, X, Y)
% 0.20/0.69 = { by axiom 15 (kn2_1) }
% 0.20/0.69 true
% 0.20/0.69 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.69 op_or
% 0.20/0.69
% 0.20/0.69 Lemma 36: op_or = and_1.
% 0.20/0.69 Proof:
% 0.20/0.69 op_or
% 0.20/0.69 = { by axiom 1 (rosser_op_or) }
% 0.20/0.69 true
% 0.20/0.69 = { by axiom 9 (and_1) R->L }
% 0.20/0.69 fresh57(op_or, op_or)
% 0.20/0.69 = { by axiom 1 (rosser_op_or) }
% 0.20/0.69 fresh57(op_or, true)
% 0.20/0.69 = { by lemma 35 R->L }
% 0.20/0.69 fresh57(fresh58(and_1, op_or, x9, y9), true)
% 0.20/0.69 = { by lemma 34 R->L }
% 0.20/0.69 fresh57(is_a_theorem(implies(and(x9, y9), x9)), true)
% 0.20/0.69 = { by axiom 31 (and_1) }
% 0.20/0.69 and_1
% 0.20/0.69
% 0.20/0.69 Lemma 37: not(and(X, not(Y))) = implies(X, Y).
% 0.20/0.69 Proof:
% 0.20/0.69 not(and(X, not(Y)))
% 0.20/0.69 = { by axiom 21 (op_implies_and) R->L }
% 0.20/0.69 fresh22(op_implies_and, true, X, Y)
% 0.20/0.69 = { by axiom 3 (rosser_op_implies_and) }
% 0.20/0.69 fresh22(true, true, X, Y)
% 0.20/0.69 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.69 fresh22(op_or, true, X, Y)
% 0.20/0.69 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.69 fresh22(op_or, op_or, X, Y)
% 0.20/0.69 = { by axiom 17 (op_implies_and) }
% 0.20/0.69 implies(X, Y)
% 0.20/0.69
% 0.20/0.69 Lemma 38: implies(not(X), Y) = or(X, Y).
% 0.20/0.69 Proof:
% 0.20/0.69 implies(not(X), Y)
% 0.20/0.69 = { by lemma 37 R->L }
% 0.20/0.69 not(and(not(X), not(Y)))
% 0.20/0.69 = { by axiom 23 (op_or) R->L }
% 0.20/0.69 fresh20(op_or, true, X, Y)
% 0.20/0.69 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.69 fresh20(op_or, op_or, X, Y)
% 0.20/0.69 = { by axiom 20 (op_or) }
% 0.20/0.69 or(X, Y)
% 0.20/0.69
% 0.20/0.69 Lemma 39: or(not(X), Y) = implies(X, Y).
% 0.20/0.69 Proof:
% 0.20/0.69 or(not(X), Y)
% 0.20/0.69 = { by axiom 19 (op_implies_or) R->L }
% 0.20/0.70 fresh21(op_implies_or, true, X, Y)
% 0.20/0.70 = { by axiom 4 (principia_op_implies_or) }
% 0.20/0.70 fresh21(true, true, X, Y)
% 0.20/0.70 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.70 fresh21(op_or, true, X, Y)
% 0.20/0.70 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.70 fresh21(op_or, op_or, X, Y)
% 0.20/0.70 = { by axiom 18 (op_implies_or) }
% 0.20/0.70 implies(X, Y)
% 0.20/0.70
% 0.20/0.70 Lemma 40: is_a_theorem(implies(or(X, X), X)) = fresh12(r1, op_or, X).
% 0.20/0.70 Proof:
% 0.20/0.70 is_a_theorem(implies(or(X, X), X))
% 0.20/0.70 = { by axiom 28 (r1_1) R->L }
% 0.20/0.70 fresh12(r1, true, X)
% 0.20/0.70 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.70 fresh12(r1, op_or, X)
% 0.20/0.70
% 0.20/0.70 Lemma 41: fresh12(r1, op_or, X) = fresh47(cn3, op_or, X).
% 0.20/0.70 Proof:
% 0.20/0.70 fresh12(r1, op_or, X)
% 0.20/0.70 = { by lemma 40 R->L }
% 0.20/0.70 is_a_theorem(implies(or(X, X), X))
% 0.20/0.70 = { by lemma 38 R->L }
% 0.20/0.70 is_a_theorem(implies(implies(not(X), X), X))
% 0.20/0.70 = { by axiom 29 (cn3_1) R->L }
% 0.20/0.70 fresh47(cn3, true, X)
% 0.20/0.70 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.70 fresh47(cn3, op_or, X)
% 0.20/0.70
% 0.20/0.70 Lemma 42: fresh59(X, X, Y, Z) = op_or.
% 0.20/0.70 Proof:
% 0.20/0.70 fresh59(X, X, Y, Z)
% 0.20/0.70 = { by axiom 14 (modus_ponens_2) }
% 0.20/0.70 fresh60(modus_ponens, true, Z)
% 0.20/0.70 = { by axiom 5 (rosser_modus_ponens) }
% 0.20/0.70 fresh60(true, true, Z)
% 0.20/0.70 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.70 fresh60(op_or, true, Z)
% 0.20/0.70 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.70 fresh60(op_or, op_or, Z)
% 0.20/0.70 = { by axiom 11 (modus_ponens_2) }
% 0.20/0.70 true
% 0.20/0.70 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.70 op_or
% 0.20/0.70
% 0.20/0.70 Lemma 43: fresh59(is_a_theorem(implies(X, Y)), op_or, X, Y) = fresh28(is_a_theorem(X), op_or, Y).
% 0.20/0.70 Proof:
% 0.20/0.70 fresh59(is_a_theorem(implies(X, Y)), op_or, X, Y)
% 0.20/0.70 = { by axiom 1 (rosser_op_or) }
% 0.20/0.70 fresh59(is_a_theorem(implies(X, Y)), true, X, Y)
% 0.20/0.70 = { by axiom 30 (modus_ponens_2) }
% 0.20/0.70 fresh28(is_a_theorem(X), true, Y)
% 0.20/0.70 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.70 fresh28(is_a_theorem(X), op_or, Y)
% 0.20/0.70
% 0.20/0.70 Lemma 44: fresh28(is_a_theorem(or(or(X, X), Y)), and_1, or(Y, X)) = and_1.
% 0.20/0.70 Proof:
% 0.20/0.70 fresh28(is_a_theorem(or(or(X, X), Y)), and_1, or(Y, X))
% 0.20/0.70 = { by lemma 36 R->L }
% 0.20/0.70 fresh28(is_a_theorem(or(or(X, X), Y)), op_or, or(Y, X))
% 0.20/0.70 = { by lemma 43 R->L }
% 0.20/0.70 fresh59(is_a_theorem(implies(or(or(X, X), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 38 R->L }
% 0.20/0.70 fresh59(is_a_theorem(implies(or(implies(not(X), X), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 37 R->L }
% 0.20/0.70 fresh59(is_a_theorem(implies(or(not(and(not(X), not(X))), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 39 }
% 0.20/0.70 fresh59(is_a_theorem(implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by axiom 13 (modus_ponens_2) R->L }
% 0.20/0.70 fresh59(fresh28(and_1, and_1, implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 36 R->L }
% 0.20/0.70 fresh59(fresh28(op_or, and_1, implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by axiom 1 (rosser_op_or) }
% 0.20/0.70 fresh59(fresh28(true, and_1, implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by axiom 12 (kn1_1) R->L }
% 0.20/0.70 fresh59(fresh28(fresh33(op_or, op_or, not(X)), and_1, implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by axiom 1 (rosser_op_or) }
% 0.20/0.70 fresh59(fresh28(fresh33(op_or, true, not(X)), and_1, implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by axiom 1 (rosser_op_or) }
% 0.20/0.70 fresh59(fresh28(fresh33(true, true, not(X)), and_1, implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by axiom 6 (rosser_kn1) R->L }
% 0.20/0.70 fresh59(fresh28(fresh33(kn1, true, not(X)), and_1, implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by axiom 25 (kn1_1) }
% 0.20/0.70 fresh59(fresh28(is_a_theorem(implies(not(X), and(not(X), not(X)))), and_1, implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 38 }
% 0.20/0.70 fresh59(fresh28(is_a_theorem(or(X, and(not(X), not(X)))), and_1, implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 36 R->L }
% 0.20/0.70 fresh59(fresh28(is_a_theorem(or(X, and(not(X), not(X)))), op_or, implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 43 R->L }
% 0.20/0.70 fresh59(fresh59(is_a_theorem(implies(or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X)))), op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 37 R->L }
% 0.20/0.70 fresh59(fresh59(is_a_theorem(implies(or(X, and(not(X), not(X))), implies(not(and(and(not(X), not(X)), not(Y))), or(Y, X)))), op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 38 }
% 0.20/0.70 fresh59(fresh59(is_a_theorem(implies(or(X, and(not(X), not(X))), or(and(and(not(X), not(X)), not(Y)), or(Y, X)))), op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 38 R->L }
% 0.20/0.70 fresh59(fresh59(is_a_theorem(implies(or(X, and(not(X), not(X))), or(and(and(not(X), not(X)), not(Y)), implies(not(Y), X)))), op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 38 R->L }
% 0.20/0.70 fresh59(fresh59(is_a_theorem(implies(implies(not(X), and(not(X), not(X))), or(and(and(not(X), not(X)), not(Y)), implies(not(Y), X)))), op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 37 R->L }
% 0.20/0.70 fresh59(fresh59(is_a_theorem(implies(implies(not(X), and(not(X), not(X))), or(and(and(not(X), not(X)), not(Y)), not(and(not(Y), not(X)))))), op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 38 R->L }
% 0.20/0.70 fresh59(fresh59(is_a_theorem(implies(implies(not(X), and(not(X), not(X))), implies(not(and(and(not(X), not(X)), not(Y))), not(and(not(Y), not(X)))))), op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by axiom 33 (kn3_1) R->L }
% 0.20/0.70 fresh59(fresh59(fresh29(kn3, true, not(X), and(not(X), not(X)), not(Y)), op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by axiom 8 (rosser_kn3) }
% 0.20/0.70 fresh59(fresh59(fresh29(true, true, not(X), and(not(X), not(X)), not(Y)), op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.70 fresh59(fresh59(fresh29(op_or, true, not(X), and(not(X), not(X)), not(Y)), op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 36 }
% 0.20/0.70 fresh59(fresh59(fresh29(and_1, true, not(X), and(not(X), not(X)), not(Y)), op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.70 fresh59(fresh59(fresh29(and_1, op_or, not(X), and(not(X), not(X)), not(Y)), op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 36 }
% 0.20/0.70 fresh59(fresh59(fresh29(and_1, and_1, not(X), and(not(X), not(X)), not(Y)), op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by axiom 22 (kn3_1) }
% 0.20/0.70 fresh59(fresh59(true, op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.70 fresh59(fresh59(op_or, op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 36 }
% 0.20/0.70 fresh59(fresh59(and_1, op_or, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 36 }
% 0.20/0.70 fresh59(fresh59(and_1, and_1, or(X, and(not(X), not(X))), implies(implies(and(not(X), not(X)), Y), or(Y, X))), op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 42 }
% 0.20/0.70 fresh59(op_or, op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 36 }
% 0.20/0.70 fresh59(and_1, op_or, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 36 }
% 0.20/0.70 fresh59(and_1, and_1, or(or(X, X), Y), or(Y, X))
% 0.20/0.70 = { by lemma 42 }
% 0.20/0.70 op_or
% 0.20/0.70 = { by lemma 36 }
% 0.20/0.70 and_1
% 0.20/0.70
% 0.20/0.70 Goal 1 (principia_r1): r1 = true.
% 0.20/0.70 Proof:
% 0.20/0.70 r1
% 0.20/0.70 = { by axiom 32 (r1) R->L }
% 0.20/0.70 fresh13(is_a_theorem(implies(or(p5, p5), p5)), true)
% 0.20/0.70 = { by lemma 40 }
% 0.20/0.70 fresh13(fresh12(r1, op_or, p5), true)
% 0.20/0.70 = { by lemma 41 }
% 0.20/0.70 fresh13(fresh47(cn3, op_or, p5), true)
% 0.20/0.70 = { by lemma 36 }
% 0.20/0.70 fresh13(fresh47(cn3, and_1, p5), true)
% 0.20/0.70 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.70 fresh13(fresh47(cn3, and_1, p5), op_or)
% 0.20/0.70 = { by lemma 36 }
% 0.20/0.70 fresh13(fresh47(cn3, and_1, p5), and_1)
% 0.20/0.70 = { by lemma 36 R->L }
% 0.20/0.70 fresh13(fresh47(cn3, op_or, p5), and_1)
% 0.20/0.70 = { by lemma 41 R->L }
% 0.20/0.70 fresh13(fresh12(r1, op_or, p5), and_1)
% 0.20/0.70 = { by lemma 40 R->L }
% 0.20/0.70 fresh13(is_a_theorem(implies(or(p5, p5), p5)), and_1)
% 0.20/0.70 = { by lemma 39 R->L }
% 0.20/0.70 fresh13(is_a_theorem(or(not(or(p5, p5)), p5)), and_1)
% 0.20/0.70 = { by axiom 13 (modus_ponens_2) R->L }
% 0.20/0.70 fresh13(fresh28(and_1, and_1, or(not(or(p5, p5)), p5)), and_1)
% 0.20/0.70 = { by lemma 44 R->L }
% 0.20/0.70 fresh13(fresh28(fresh28(is_a_theorem(or(or(not(or(p5, p5)), not(or(p5, p5))), or(p5, p5))), and_1, or(or(p5, p5), not(or(p5, p5)))), and_1, or(not(or(p5, p5)), p5)), and_1)
% 0.20/0.70 = { by lemma 38 R->L }
% 0.20/0.70 fresh13(fresh28(fresh28(is_a_theorem(implies(not(or(not(or(p5, p5)), not(or(p5, p5)))), or(p5, p5))), and_1, or(or(p5, p5), not(or(p5, p5)))), and_1, or(not(or(p5, p5)), p5)), and_1)
% 0.20/0.70 = { by axiom 24 (op_and) R->L }
% 0.20/0.70 fresh13(fresh28(fresh28(is_a_theorem(implies(fresh24(op_and, true, or(p5, p5), or(p5, p5)), or(p5, p5))), and_1, or(or(p5, p5), not(or(p5, p5)))), and_1, or(not(or(p5, p5)), p5)), and_1)
% 0.20/0.70 = { by axiom 2 (principia_op_and) }
% 0.20/0.70 fresh13(fresh28(fresh28(is_a_theorem(implies(fresh24(true, true, or(p5, p5), or(p5, p5)), or(p5, p5))), and_1, or(or(p5, p5), not(or(p5, p5)))), and_1, or(not(or(p5, p5)), p5)), and_1)
% 0.20/0.70 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.70 fresh13(fresh28(fresh28(is_a_theorem(implies(fresh24(op_or, true, or(p5, p5), or(p5, p5)), or(p5, p5))), and_1, or(or(p5, p5), not(or(p5, p5)))), and_1, or(not(or(p5, p5)), p5)), and_1)
% 0.20/0.70 = { by axiom 1 (rosser_op_or) R->L }
% 0.20/0.70 fresh13(fresh28(fresh28(is_a_theorem(implies(fresh24(op_or, op_or, or(p5, p5), or(p5, p5)), or(p5, p5))), and_1, or(or(p5, p5), not(or(p5, p5)))), and_1, or(not(or(p5, p5)), p5)), and_1)
% 0.20/0.70 = { by axiom 16 (op_and) }
% 0.20/0.70 fresh13(fresh28(fresh28(is_a_theorem(implies(and(or(p5, p5), or(p5, p5)), or(p5, p5))), and_1, or(or(p5, p5), not(or(p5, p5)))), and_1, or(not(or(p5, p5)), p5)), and_1)
% 0.20/0.71 = { by lemma 34 }
% 0.20/0.71 fresh13(fresh28(fresh28(fresh58(and_1, op_or, or(p5, p5), or(p5, p5)), and_1, or(or(p5, p5), not(or(p5, p5)))), and_1, or(not(or(p5, p5)), p5)), and_1)
% 0.20/0.71 = { by lemma 35 }
% 0.20/0.71 fresh13(fresh28(fresh28(op_or, and_1, or(or(p5, p5), not(or(p5, p5)))), and_1, or(not(or(p5, p5)), p5)), and_1)
% 0.20/0.71 = { by lemma 36 }
% 0.20/0.71 fresh13(fresh28(fresh28(and_1, and_1, or(or(p5, p5), not(or(p5, p5)))), and_1, or(not(or(p5, p5)), p5)), and_1)
% 0.20/0.71 = { by axiom 13 (modus_ponens_2) }
% 0.20/0.71 fresh13(fresh28(is_a_theorem(or(or(p5, p5), not(or(p5, p5)))), and_1, or(not(or(p5, p5)), p5)), and_1)
% 0.20/0.71 = { by lemma 44 }
% 0.20/0.71 fresh13(and_1, and_1)
% 0.20/0.71 = { by axiom 10 (r1) }
% 0.20/0.71 true
% 0.20/0.71 % SZS output end Proof
% 0.20/0.71
% 0.20/0.71 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------