TSTP Solution File: LCL516+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL516+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 : n005.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 59.83s 8.07s
% Output : Proof 60.43s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.11 % Problem : LCL516+1 : TPTP v8.1.2. Released v3.3.0.
% 0.10/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.33 % Computer : n005.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Fri Aug 25 01:54:09 EDT 2023
% 0.13/0.33 % CPUTime :
% 59.83/8.07 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 59.83/8.07
% 59.83/8.07 % SZS status Theorem
% 59.83/8.07
% 59.83/8.11 % SZS output start Proof
% 59.83/8.11 Take the following subset of the input axioms:
% 59.83/8.12 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 59.83/8.12 fof(cn2, axiom, cn2 <=> ![P, Q]: is_a_theorem(implies(P, implies(not(P), Q)))).
% 59.83/8.12 fof(kn1, axiom, kn1 <=> ![P2]: is_a_theorem(implies(P2, and(P2, P2)))).
% 59.83/8.12 fof(kn2, axiom, kn2 <=> ![P2, Q2]: is_a_theorem(implies(and(P2, Q2), P2))).
% 59.83/8.12 fof(kn3, axiom, kn3 <=> ![R, P2, Q2]: is_a_theorem(implies(implies(P2, Q2), implies(not(and(Q2, R)), not(and(R, P2)))))).
% 59.83/8.12 fof(luka_cn2, conjecture, cn2).
% 59.83/8.12 fof(luka_op_implies, axiom, op_implies).
% 59.83/8.12 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 59.83/8.12 fof(op_equiv, axiom, op_equiv => ![X2, Y2]: equiv(X2, Y2)=and(implies(X2, Y2), implies(Y2, X2))).
% 59.83/8.12 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 59.83/8.12 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 59.83/8.12 fof(or_1, axiom, or_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, or(X2, Y2)))).
% 59.83/8.12 fof(rosser_kn1, axiom, kn1).
% 59.83/8.12 fof(rosser_kn2, axiom, kn2).
% 59.83/8.12 fof(rosser_kn3, axiom, kn3).
% 59.83/8.12 fof(rosser_modus_ponens, axiom, modus_ponens).
% 59.83/8.12 fof(rosser_op_equiv, axiom, op_equiv).
% 59.83/8.12 fof(rosser_op_implies_and, axiom, op_implies_and).
% 59.83/8.12 fof(rosser_op_or, axiom, op_or).
% 59.83/8.12
% 59.83/8.12 Now clausify the problem and encode Horn clauses using encoding 3 of
% 59.83/8.12 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 59.83/8.12 We repeatedly replace C & s=t => u=v by the two clauses:
% 59.83/8.12 fresh(y, y, x1...xn) = u
% 59.83/8.12 C => fresh(s, t, x1...xn) = v
% 59.83/8.12 where fresh is a fresh function symbol and x1..xn are the free
% 59.83/8.12 variables of u and v.
% 59.83/8.12 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 59.83/8.12 input problem has no model of domain size 1).
% 59.83/8.12
% 59.83/8.12 The encoding turns the above axioms into the following unit equations and goals:
% 59.83/8.12
% 59.83/8.12 Axiom 1 (luka_op_implies): op_implies = true.
% 59.83/8.12 Axiom 2 (rosser_op_implies_and): op_implies_and = true.
% 59.83/8.12 Axiom 3 (rosser_modus_ponens): modus_ponens = true.
% 59.83/8.12 Axiom 4 (rosser_kn1): kn1 = true.
% 59.83/8.12 Axiom 5 (rosser_kn2): kn2 = true.
% 59.83/8.12 Axiom 6 (rosser_kn3): kn3 = true.
% 59.83/8.12 Axiom 7 (rosser_op_or): op_or = true.
% 59.83/8.12 Axiom 8 (rosser_op_equiv): op_equiv = true.
% 59.83/8.12 Axiom 9 (cn2): fresh50(X, X) = true.
% 59.83/8.12 Axiom 10 (modus_ponens_2): fresh60(X, X, Y) = true.
% 59.83/8.12 Axiom 11 (kn1_1): fresh33(X, X, Y) = true.
% 59.83/8.12 Axiom 12 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 59.83/8.12 Axiom 13 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 59.83/8.12 Axiom 14 (kn2_1): fresh31(X, X, Y, Z) = true.
% 59.83/8.12 Axiom 15 (op_equiv): fresh23(X, X, Y, Z) = equiv(Y, Z).
% 59.83/8.12 Axiom 16 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 59.83/8.12 Axiom 17 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 59.83/8.12 Axiom 18 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 59.83/8.12 Axiom 19 (kn3_1): fresh29(X, X, Y, Z, W) = true.
% 59.83/8.12 Axiom 20 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 59.83/8.12 Axiom 21 (kn1_1): fresh33(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
% 59.83/8.12 Axiom 22 (or_1_1): fresh18(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
% 59.83/8.12 Axiom 23 (and_1_1): fresh58(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 59.83/8.12 Axiom 24 (kn2_1): fresh31(kn2, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 59.83/8.12 Axiom 25 (op_equiv): fresh23(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
% 59.83/8.12 Axiom 26 (cn2_1): fresh49(cn2, true, X, Y) = is_a_theorem(implies(X, implies(not(X), Y))).
% 59.83/8.12 Axiom 27 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 59.83/8.12 Axiom 28 (cn2): fresh50(is_a_theorem(implies(p7, implies(not(p7), q5))), true) = cn2.
% 59.83/8.12 Axiom 29 (kn3_1): fresh29(kn3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X))))).
% 59.83/8.12
% 59.83/8.12 Lemma 30: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
% 59.83/8.12 Proof:
% 59.83/8.12 and(implies(X, Y), implies(Y, X))
% 59.83/8.12 = { by axiom 25 (op_equiv) R->L }
% 59.83/8.12 fresh23(op_equiv, true, X, Y)
% 59.83/8.12 = { by axiom 8 (rosser_op_equiv) }
% 59.83/8.12 fresh23(true, true, X, Y)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh23(op_implies, true, X, Y)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh23(op_implies, op_implies, X, Y)
% 59.83/8.12 = { by axiom 15 (op_equiv) }
% 59.83/8.12 equiv(X, Y)
% 59.83/8.12
% 59.83/8.12 Lemma 31: fresh59(is_a_theorem(implies(X, Y)), op_implies, X, Y) = fresh28(is_a_theorem(X), op_implies, Y).
% 59.83/8.12 Proof:
% 59.83/8.12 fresh59(is_a_theorem(implies(X, Y)), op_implies, X, Y)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) }
% 59.83/8.12 fresh59(is_a_theorem(implies(X, Y)), true, X, Y)
% 59.83/8.12 = { by axiom 27 (modus_ponens_2) }
% 59.83/8.12 fresh28(is_a_theorem(X), true, Y)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh28(is_a_theorem(X), op_implies, Y)
% 59.83/8.12
% 59.83/8.12 Lemma 32: not(and(X, not(Y))) = implies(X, Y).
% 59.83/8.12 Proof:
% 59.83/8.12 not(and(X, not(Y)))
% 59.83/8.12 = { by axiom 18 (op_implies_and) R->L }
% 59.83/8.12 fresh22(op_implies_and, true, X, Y)
% 59.83/8.12 = { by axiom 2 (rosser_op_implies_and) }
% 59.83/8.12 fresh22(true, true, X, Y)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh22(op_implies, true, X, Y)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh22(op_implies, op_implies, X, Y)
% 59.83/8.12 = { by axiom 16 (op_implies_and) }
% 59.83/8.12 implies(X, Y)
% 59.83/8.12
% 59.83/8.12 Lemma 33: implies(not(X), Y) = or(X, Y).
% 59.83/8.12 Proof:
% 59.83/8.12 implies(not(X), Y)
% 59.83/8.12 = { by lemma 32 R->L }
% 59.83/8.12 not(and(not(X), not(Y)))
% 59.83/8.12 = { by axiom 20 (op_or) R->L }
% 59.83/8.12 fresh20(op_or, true, X, Y)
% 59.83/8.12 = { by axiom 7 (rosser_op_or) }
% 59.83/8.12 fresh20(true, true, X, Y)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh20(op_implies, true, X, Y)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh20(op_implies, op_implies, X, Y)
% 59.83/8.12 = { by axiom 17 (op_or) }
% 59.83/8.12 or(X, Y)
% 59.83/8.12
% 59.83/8.12 Lemma 34: is_a_theorem(implies(and(X, Y), X)) = fresh58(and_1, op_implies, X, Y).
% 59.83/8.12 Proof:
% 59.83/8.12 is_a_theorem(implies(and(X, Y), X))
% 59.83/8.12 = { by axiom 23 (and_1_1) R->L }
% 59.83/8.12 fresh58(and_1, true, X, Y)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh58(and_1, op_implies, X, Y)
% 59.83/8.12
% 59.83/8.12 Lemma 35: fresh58(and_1, op_implies, X, Y) = op_implies.
% 59.83/8.12 Proof:
% 59.83/8.12 fresh58(and_1, op_implies, X, Y)
% 59.83/8.12 = { by lemma 34 R->L }
% 59.83/8.12 is_a_theorem(implies(and(X, Y), X))
% 59.83/8.12 = { by axiom 24 (kn2_1) R->L }
% 59.83/8.12 fresh31(kn2, true, X, Y)
% 59.83/8.12 = { by axiom 5 (rosser_kn2) }
% 59.83/8.12 fresh31(true, true, X, Y)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh31(op_implies, true, X, Y)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh31(op_implies, op_implies, X, Y)
% 59.83/8.12 = { by axiom 14 (kn2_1) }
% 59.83/8.12 true
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 op_implies
% 59.83/8.12
% 59.83/8.12 Lemma 36: fresh59(X, X, Y, Z) = op_implies.
% 59.83/8.12 Proof:
% 59.83/8.12 fresh59(X, X, Y, Z)
% 59.83/8.12 = { by axiom 13 (modus_ponens_2) }
% 59.83/8.12 fresh60(modus_ponens, true, Z)
% 59.83/8.12 = { by axiom 3 (rosser_modus_ponens) }
% 59.83/8.12 fresh60(true, true, Z)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh60(op_implies, true, Z)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh60(op_implies, op_implies, Z)
% 59.83/8.12 = { by axiom 10 (modus_ponens_2) }
% 59.83/8.12 true
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 op_implies
% 59.83/8.12
% 59.83/8.12 Lemma 37: is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X))))) = op_implies.
% 59.83/8.12 Proof:
% 59.83/8.12 is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X)))))
% 59.83/8.12 = { by lemma 33 R->L }
% 59.83/8.12 is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X)))))
% 59.83/8.12 = { by axiom 29 (kn3_1) R->L }
% 59.83/8.12 fresh29(kn3, true, X, Y, Z)
% 59.83/8.12 = { by axiom 6 (rosser_kn3) }
% 59.83/8.12 fresh29(true, true, X, Y, Z)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh29(op_implies, true, X, Y, Z)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh29(op_implies, op_implies, X, Y, Z)
% 59.83/8.12 = { by axiom 19 (kn3_1) }
% 59.83/8.12 true
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 op_implies
% 59.83/8.12
% 59.83/8.12 Lemma 38: fresh28(is_a_theorem(implies(X, Y)), op_implies, or(and(Y, Z), not(and(Z, X)))) = op_implies.
% 59.83/8.12 Proof:
% 59.83/8.12 fresh28(is_a_theorem(implies(X, Y)), op_implies, or(and(Y, Z), not(and(Z, X))))
% 59.83/8.12 = { by lemma 31 R->L }
% 59.83/8.12 fresh59(is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X))))), op_implies, implies(X, Y), or(and(Y, Z), not(and(Z, X))))
% 59.83/8.12 = { by lemma 37 }
% 59.83/8.12 fresh59(op_implies, op_implies, implies(X, Y), or(and(Y, Z), not(and(Z, X))))
% 59.83/8.12 = { by lemma 36 }
% 59.83/8.12 op_implies
% 59.83/8.12
% 59.83/8.12 Lemma 39: is_a_theorem(implies(X, and(X, X))) = op_implies.
% 59.83/8.12 Proof:
% 59.83/8.12 is_a_theorem(implies(X, and(X, X)))
% 59.83/8.12 = { by axiom 21 (kn1_1) R->L }
% 59.83/8.12 fresh33(kn1, true, X)
% 59.83/8.12 = { by axiom 4 (rosser_kn1) }
% 59.83/8.12 fresh33(true, true, X)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh33(op_implies, true, X)
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 fresh33(op_implies, op_implies, X)
% 59.83/8.12 = { by axiom 11 (kn1_1) }
% 59.83/8.12 true
% 59.83/8.12 = { by axiom 1 (luka_op_implies) R->L }
% 59.83/8.12 op_implies
% 59.83/8.12
% 59.83/8.12 Lemma 40: or(and(X, not(Y)), Z) = implies(implies(X, Y), Z).
% 59.83/8.12 Proof:
% 59.83/8.12 or(and(X, not(Y)), Z)
% 59.83/8.12 = { by lemma 33 R->L }
% 59.83/8.12 implies(not(and(X, not(Y))), Z)
% 59.83/8.12 = { by lemma 32 }
% 59.83/8.12 implies(implies(X, Y), Z)
% 59.83/8.12
% 59.83/8.12 Lemma 41: is_a_theorem(implies(or(X, Y), or(and(Y, Z), implies(Z, X)))) = op_implies.
% 59.83/8.12 Proof:
% 59.83/8.12 is_a_theorem(implies(or(X, Y), or(and(Y, Z), implies(Z, X))))
% 59.83/8.12 = { by lemma 33 R->L }
% 59.83/8.12 is_a_theorem(implies(implies(not(X), Y), or(and(Y, Z), implies(Z, X))))
% 59.83/8.12 = { by lemma 32 R->L }
% 59.83/8.12 is_a_theorem(implies(implies(not(X), Y), or(and(Y, Z), not(and(Z, not(X))))))
% 59.83/8.12 = { by lemma 37 }
% 59.83/8.12 op_implies
% 59.83/8.12
% 59.83/8.12 Lemma 42: fresh28(is_a_theorem(or(X, Y)), op_implies, implies(implies(Y, Z), or(Z, X))) = op_implies.
% 59.83/8.12 Proof:
% 59.83/8.12 fresh28(is_a_theorem(or(X, Y)), op_implies, implies(implies(Y, Z), or(Z, X)))
% 59.83/8.12 = { by lemma 31 R->L }
% 59.83/8.12 fresh59(is_a_theorem(implies(or(X, Y), implies(implies(Y, Z), or(Z, X)))), op_implies, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 59.83/8.12 = { by lemma 40 R->L }
% 59.83/8.12 fresh59(is_a_theorem(implies(or(X, Y), or(and(Y, not(Z)), or(Z, X)))), op_implies, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 59.83/8.12 = { by lemma 33 R->L }
% 59.83/8.12 fresh59(is_a_theorem(implies(or(X, Y), or(and(Y, not(Z)), implies(not(Z), X)))), op_implies, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 59.83/8.12 = { by lemma 41 }
% 59.83/8.12 fresh59(op_implies, op_implies, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 59.83/8.12 = { by lemma 36 }
% 59.83/8.12 op_implies
% 59.83/8.12
% 59.83/8.12 Lemma 43: fresh28(is_a_theorem(implies(X, Y)), op_implies, or(Y, not(X))) = op_implies.
% 59.83/8.12 Proof:
% 59.83/8.12 fresh28(is_a_theorem(implies(X, Y)), op_implies, or(Y, not(X)))
% 59.83/8.12 = { by lemma 31 R->L }
% 59.83/8.12 fresh59(is_a_theorem(implies(implies(X, Y), or(Y, not(X)))), op_implies, implies(X, Y), or(Y, not(X)))
% 59.83/8.12 = { by axiom 12 (modus_ponens_2) R->L }
% 59.83/8.12 fresh59(fresh28(op_implies, op_implies, implies(implies(X, Y), or(Y, not(X)))), op_implies, implies(X, Y), or(Y, not(X)))
% 59.83/8.12 = { by lemma 36 R->L }
% 59.83/8.12 fresh59(fresh28(fresh59(op_implies, op_implies, not(and(and(not(X), not(X)), not(not(X)))), not(and(not(not(X)), not(X)))), op_implies, implies(implies(X, Y), or(Y, not(X)))), op_implies, implies(X, Y), or(Y, not(X)))
% 59.83/8.12 = { by lemma 38 R->L }
% 60.43/8.12 fresh59(fresh28(fresh59(fresh28(is_a_theorem(implies(not(X), and(not(X), not(X)))), op_implies, or(and(and(not(X), not(X)), not(not(X))), not(and(not(not(X)), not(X))))), op_implies, not(and(and(not(X), not(X)), not(not(X)))), not(and(not(not(X)), not(X)))), op_implies, implies(implies(X, Y), or(Y, not(X)))), op_implies, implies(X, Y), or(Y, not(X)))
% 60.43/8.12 = { by lemma 39 }
% 60.43/8.12 fresh59(fresh28(fresh59(fresh28(op_implies, op_implies, or(and(and(not(X), not(X)), not(not(X))), not(and(not(not(X)), not(X))))), op_implies, not(and(and(not(X), not(X)), not(not(X)))), not(and(not(not(X)), not(X)))), op_implies, implies(implies(X, Y), or(Y, not(X)))), op_implies, implies(X, Y), or(Y, not(X)))
% 60.43/8.12 = { by axiom 12 (modus_ponens_2) }
% 60.43/8.12 fresh59(fresh28(fresh59(is_a_theorem(or(and(and(not(X), not(X)), not(not(X))), not(and(not(not(X)), not(X))))), op_implies, not(and(and(not(X), not(X)), not(not(X)))), not(and(not(not(X)), not(X)))), op_implies, implies(implies(X, Y), or(Y, not(X)))), op_implies, implies(X, Y), or(Y, not(X)))
% 60.43/8.12 = { by lemma 33 R->L }
% 60.43/8.12 fresh59(fresh28(fresh59(is_a_theorem(implies(not(and(and(not(X), not(X)), not(not(X)))), not(and(not(not(X)), not(X))))), op_implies, not(and(and(not(X), not(X)), not(not(X)))), not(and(not(not(X)), not(X)))), op_implies, implies(implies(X, Y), or(Y, not(X)))), op_implies, implies(X, Y), or(Y, not(X)))
% 60.43/8.12 = { by lemma 31 }
% 60.43/8.12 fresh59(fresh28(fresh28(is_a_theorem(not(and(and(not(X), not(X)), not(not(X))))), op_implies, not(and(not(not(X)), not(X)))), op_implies, implies(implies(X, Y), or(Y, not(X)))), op_implies, implies(X, Y), or(Y, not(X)))
% 60.43/8.12 = { by lemma 32 }
% 60.43/8.12 fresh59(fresh28(fresh28(is_a_theorem(implies(and(not(X), not(X)), not(X))), op_implies, not(and(not(not(X)), not(X)))), op_implies, implies(implies(X, Y), or(Y, not(X)))), op_implies, implies(X, Y), or(Y, not(X)))
% 60.43/8.12 = { by lemma 34 }
% 60.43/8.12 fresh59(fresh28(fresh28(fresh58(and_1, op_implies, not(X), not(X)), op_implies, not(and(not(not(X)), not(X)))), op_implies, implies(implies(X, Y), or(Y, not(X)))), op_implies, implies(X, Y), or(Y, not(X)))
% 60.43/8.12 = { by lemma 35 }
% 60.43/8.12 fresh59(fresh28(fresh28(op_implies, op_implies, not(and(not(not(X)), not(X)))), op_implies, implies(implies(X, Y), or(Y, not(X)))), op_implies, implies(X, Y), or(Y, not(X)))
% 60.43/8.12 = { by axiom 12 (modus_ponens_2) }
% 60.43/8.12 fresh59(fresh28(is_a_theorem(not(and(not(not(X)), not(X)))), op_implies, implies(implies(X, Y), or(Y, not(X)))), op_implies, implies(X, Y), or(Y, not(X)))
% 60.43/8.12 = { by lemma 32 }
% 60.43/8.12 fresh59(fresh28(is_a_theorem(implies(not(not(X)), X)), op_implies, implies(implies(X, Y), or(Y, not(X)))), op_implies, implies(X, Y), or(Y, not(X)))
% 60.43/8.12 = { by lemma 33 }
% 60.43/8.12 fresh59(fresh28(is_a_theorem(or(not(X), X)), op_implies, implies(implies(X, Y), or(Y, not(X)))), op_implies, implies(X, Y), or(Y, not(X)))
% 60.43/8.12 = { by lemma 42 }
% 60.43/8.12 fresh59(op_implies, op_implies, implies(X, Y), or(Y, not(X)))
% 60.43/8.12 = { by lemma 36 }
% 60.43/8.12 op_implies
% 60.43/8.12
% 60.43/8.12 Lemma 44: is_a_theorem(or(X, not(and(X, Y)))) = op_implies.
% 60.43/8.12 Proof:
% 60.43/8.12 is_a_theorem(or(X, not(and(X, Y))))
% 60.43/8.12 = { by axiom 12 (modus_ponens_2) R->L }
% 60.43/8.12 fresh28(op_implies, op_implies, or(X, not(and(X, Y))))
% 60.43/8.12 = { by lemma 35 R->L }
% 60.43/8.12 fresh28(fresh58(and_1, op_implies, X, Y), op_implies, or(X, not(and(X, Y))))
% 60.43/8.12 = { by lemma 34 R->L }
% 60.43/8.12 fresh28(is_a_theorem(implies(and(X, Y), X)), op_implies, or(X, not(and(X, Y))))
% 60.43/8.12 = { by lemma 43 }
% 60.43/8.12 op_implies
% 60.43/8.12
% 60.43/8.12 Lemma 45: fresh28(is_a_theorem(or(and(X, Y), Z)), op_implies, or(Z, X)) = op_implies.
% 60.43/8.12 Proof:
% 60.43/8.12 fresh28(is_a_theorem(or(and(X, Y), Z)), op_implies, or(Z, X))
% 60.43/8.12 = { by lemma 31 R->L }
% 60.43/8.12 fresh59(is_a_theorem(implies(or(and(X, Y), Z), or(Z, X))), op_implies, or(and(X, Y), Z), or(Z, X))
% 60.43/8.13 = { by lemma 33 R->L }
% 60.43/8.13 fresh59(is_a_theorem(implies(implies(not(and(X, Y)), Z), or(Z, X))), op_implies, or(and(X, Y), Z), or(Z, X))
% 60.43/8.13 = { by axiom 12 (modus_ponens_2) R->L }
% 60.43/8.13 fresh59(fresh28(op_implies, op_implies, implies(implies(not(and(X, Y)), Z), or(Z, X))), op_implies, or(and(X, Y), Z), or(Z, X))
% 60.43/8.13 = { by lemma 44 R->L }
% 60.43/8.13 fresh59(fresh28(is_a_theorem(or(X, not(and(X, Y)))), op_implies, implies(implies(not(and(X, Y)), Z), or(Z, X))), op_implies, or(and(X, Y), Z), or(Z, X))
% 60.43/8.13 = { by lemma 42 }
% 60.43/8.13 fresh59(op_implies, op_implies, or(and(X, Y), Z), or(Z, X))
% 60.43/8.13 = { by lemma 36 }
% 60.43/8.13 op_implies
% 60.43/8.13
% 60.43/8.13 Lemma 46: is_a_theorem(or(X, implies(X, Y))) = op_implies.
% 60.43/8.13 Proof:
% 60.43/8.13 is_a_theorem(or(X, implies(X, Y)))
% 60.43/8.13 = { by lemma 32 R->L }
% 60.43/8.13 is_a_theorem(or(X, not(and(X, not(Y)))))
% 60.43/8.13 = { by lemma 44 }
% 60.43/8.13 op_implies
% 60.43/8.13
% 60.43/8.13 Lemma 47: is_a_theorem(or(X, and(not(X), not(X)))) = op_implies.
% 60.43/8.13 Proof:
% 60.43/8.13 is_a_theorem(or(X, and(not(X), not(X))))
% 60.43/8.13 = { by lemma 33 R->L }
% 60.43/8.13 is_a_theorem(implies(not(X), and(not(X), not(X))))
% 60.43/8.13 = { by lemma 39 }
% 60.43/8.13 op_implies
% 60.43/8.13
% 60.43/8.13 Lemma 48: is_a_theorem(equiv(X, X)) = op_implies.
% 60.43/8.13 Proof:
% 60.43/8.13 is_a_theorem(equiv(X, X))
% 60.43/8.13 = { by lemma 30 R->L }
% 60.43/8.13 is_a_theorem(and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by axiom 12 (modus_ponens_2) R->L }
% 60.43/8.13 fresh28(op_implies, op_implies, and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 45 R->L }
% 60.43/8.13 fresh28(fresh28(is_a_theorem(or(and(implies(X, X), X), implies(X, X))), op_implies, or(implies(X, X), implies(X, X))), op_implies, and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by axiom 12 (modus_ponens_2) R->L }
% 60.43/8.13 fresh28(fresh28(fresh28(op_implies, op_implies, or(and(implies(X, X), X), implies(X, X))), op_implies, or(implies(X, X), implies(X, X))), op_implies, and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 46 R->L }
% 60.43/8.13 fresh28(fresh28(fresh28(is_a_theorem(or(X, implies(X, X))), op_implies, or(and(implies(X, X), X), implies(X, X))), op_implies, or(implies(X, X), implies(X, X))), op_implies, and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 31 R->L }
% 60.43/8.13 fresh28(fresh28(fresh59(is_a_theorem(implies(or(X, implies(X, X)), or(and(implies(X, X), X), implies(X, X)))), op_implies, or(X, implies(X, X)), or(and(implies(X, X), X), implies(X, X))), op_implies, or(implies(X, X), implies(X, X))), op_implies, and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 41 }
% 60.43/8.13 fresh28(fresh28(fresh59(op_implies, op_implies, or(X, implies(X, X)), or(and(implies(X, X), X), implies(X, X))), op_implies, or(implies(X, X), implies(X, X))), op_implies, and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 36 }
% 60.43/8.13 fresh28(fresh28(op_implies, op_implies, or(implies(X, X), implies(X, X))), op_implies, and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by axiom 12 (modus_ponens_2) }
% 60.43/8.13 fresh28(is_a_theorem(or(implies(X, X), implies(X, X))), op_implies, and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 31 R->L }
% 60.43/8.13 fresh59(is_a_theorem(implies(or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))), op_implies, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 33 R->L }
% 60.43/8.13 fresh59(is_a_theorem(implies(implies(not(implies(X, X)), implies(X, X)), and(implies(X, X), implies(X, X)))), op_implies, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 40 R->L }
% 60.43/8.13 fresh59(is_a_theorem(or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), op_implies, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by axiom 12 (modus_ponens_2) R->L }
% 60.43/8.13 fresh59(fresh28(op_implies, op_implies, or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), op_implies, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 47 R->L }
% 60.43/8.13 fresh59(fresh28(is_a_theorem(or(implies(X, X), and(not(implies(X, X)), not(implies(X, X))))), op_implies, or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), op_implies, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 31 R->L }
% 60.43/8.13 fresh59(fresh59(is_a_theorem(implies(or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X))))), op_implies, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), op_implies, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 33 R->L }
% 60.43/8.13 fresh59(fresh59(is_a_theorem(implies(implies(not(implies(X, X)), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X))))), op_implies, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), op_implies, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by axiom 12 (modus_ponens_2) R->L }
% 60.43/8.13 fresh59(fresh59(fresh28(op_implies, op_implies, implies(implies(not(implies(X, X)), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X))))), op_implies, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), op_implies, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 43 R->L }
% 60.43/8.13 fresh59(fresh59(fresh28(fresh28(is_a_theorem(implies(implies(X, X), and(implies(X, X), implies(X, X)))), op_implies, or(and(implies(X, X), implies(X, X)), not(implies(X, X)))), op_implies, implies(implies(not(implies(X, X)), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X))))), op_implies, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), op_implies, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 39 }
% 60.43/8.13 fresh59(fresh59(fresh28(fresh28(op_implies, op_implies, or(and(implies(X, X), implies(X, X)), not(implies(X, X)))), op_implies, implies(implies(not(implies(X, X)), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X))))), op_implies, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), op_implies, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by axiom 12 (modus_ponens_2) }
% 60.43/8.13 fresh59(fresh59(fresh28(is_a_theorem(or(and(implies(X, X), implies(X, X)), not(implies(X, X)))), op_implies, implies(implies(not(implies(X, X)), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X))))), op_implies, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), op_implies, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 42 }
% 60.43/8.13 fresh59(fresh59(op_implies, op_implies, or(implies(X, X), and(not(implies(X, X)), not(implies(X, X)))), or(and(not(implies(X, X)), not(implies(X, X))), and(implies(X, X), implies(X, X)))), op_implies, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 36 }
% 60.43/8.13 fresh59(op_implies, op_implies, or(implies(X, X), implies(X, X)), and(implies(X, X), implies(X, X)))
% 60.43/8.13 = { by lemma 36 }
% 60.43/8.13 op_implies
% 60.43/8.13
% 60.43/8.13 Lemma 49: is_a_theorem(implies(X, or(X, Y))) = fresh18(or_1, op_implies, X, Y).
% 60.43/8.13 Proof:
% 60.43/8.13 is_a_theorem(implies(X, or(X, Y)))
% 60.43/8.13 = { by axiom 22 (or_1_1) R->L }
% 60.43/8.13 fresh18(or_1, true, X, Y)
% 60.43/8.13 = { by axiom 1 (luka_op_implies) R->L }
% 60.43/8.13 fresh18(or_1, op_implies, X, Y)
% 60.43/8.13
% 60.43/8.13 Lemma 50: fresh18(or_1, op_implies, X, Y) = fresh49(cn2, op_implies, X, Y).
% 60.43/8.13 Proof:
% 60.43/8.13 fresh18(or_1, op_implies, X, Y)
% 60.43/8.13 = { by lemma 49 R->L }
% 60.43/8.13 is_a_theorem(implies(X, or(X, Y)))
% 60.43/8.13 = { by lemma 33 R->L }
% 60.43/8.13 is_a_theorem(implies(X, implies(not(X), Y)))
% 60.43/8.13 = { by axiom 26 (cn2_1) R->L }
% 60.43/8.13 fresh49(cn2, true, X, Y)
% 60.43/8.13 = { by axiom 1 (luka_op_implies) R->L }
% 60.43/8.13 fresh49(cn2, op_implies, X, Y)
% 60.43/8.13
% 60.43/8.13 Lemma 51: fresh28(is_a_theorem(equiv(X, Y)), op_implies, implies(X, Y)) = op_implies.
% 60.43/8.13 Proof:
% 60.43/8.13 fresh28(is_a_theorem(equiv(X, Y)), op_implies, implies(X, Y))
% 60.43/8.13 = { by lemma 30 R->L }
% 60.43/8.13 fresh28(is_a_theorem(and(implies(X, Y), implies(Y, X))), op_implies, implies(X, Y))
% 60.43/8.13 = { by lemma 31 R->L }
% 60.43/8.13 fresh59(is_a_theorem(implies(and(implies(X, Y), implies(Y, X)), implies(X, Y))), op_implies, and(implies(X, Y), implies(Y, X)), implies(X, Y))
% 60.43/8.13 = { by lemma 34 }
% 60.43/8.13 fresh59(fresh58(and_1, op_implies, implies(X, Y), implies(Y, X)), op_implies, and(implies(X, Y), implies(Y, X)), implies(X, Y))
% 60.43/8.13 = { by lemma 35 }
% 60.43/8.13 fresh59(op_implies, op_implies, and(implies(X, Y), implies(Y, X)), implies(X, Y))
% 60.43/8.13 = { by lemma 36 }
% 60.43/8.13 op_implies
% 60.43/8.13
% 60.43/8.13 Lemma 52: fresh28(is_a_theorem(or(X, Y)), op_implies, or(Y, X)) = op_implies.
% 60.43/8.13 Proof:
% 60.43/8.13 fresh28(is_a_theorem(or(X, Y)), op_implies, or(Y, X))
% 60.43/8.13 = { by lemma 31 R->L }
% 60.43/8.13 fresh59(is_a_theorem(implies(or(X, Y), or(Y, X))), op_implies, or(X, Y), or(Y, X))
% 60.43/8.13 = { by lemma 33 R->L }
% 60.43/8.13 fresh59(is_a_theorem(implies(implies(not(X), Y), or(Y, X))), op_implies, or(X, Y), or(Y, X))
% 60.43/8.13 = { by axiom 12 (modus_ponens_2) R->L }
% 60.43/8.13 fresh59(fresh28(op_implies, op_implies, implies(implies(not(X), Y), or(Y, X))), op_implies, or(X, Y), or(Y, X))
% 60.43/8.13 = { by lemma 51 R->L }
% 60.43/8.13 fresh59(fresh28(fresh28(is_a_theorem(equiv(not(X), not(X))), op_implies, implies(not(X), not(X))), op_implies, implies(implies(not(X), Y), or(Y, X))), op_implies, or(X, Y), or(Y, X))
% 60.43/8.13 = { by lemma 33 }
% 60.43/8.13 fresh59(fresh28(fresh28(is_a_theorem(equiv(not(X), not(X))), op_implies, or(X, not(X))), op_implies, implies(implies(not(X), Y), or(Y, X))), op_implies, or(X, Y), or(Y, X))
% 60.43/8.13 = { by lemma 48 }
% 60.43/8.13 fresh59(fresh28(fresh28(op_implies, op_implies, or(X, not(X))), op_implies, implies(implies(not(X), Y), or(Y, X))), op_implies, or(X, Y), or(Y, X))
% 60.43/8.13 = { by axiom 12 (modus_ponens_2) }
% 60.43/8.13 fresh59(fresh28(is_a_theorem(or(X, not(X))), op_implies, implies(implies(not(X), Y), or(Y, X))), op_implies, or(X, Y), or(Y, X))
% 60.43/8.13 = { by lemma 42 }
% 60.43/8.13 fresh59(op_implies, op_implies, or(X, Y), or(Y, X))
% 60.43/8.13 = { by lemma 36 }
% 60.43/8.13 op_implies
% 60.43/8.13
% 60.43/8.13 Lemma 53: fresh28(is_a_theorem(or(X, not(Y))), op_implies, implies(or(Y, Z), or(Z, X))) = op_implies.
% 60.43/8.13 Proof:
% 60.43/8.13 fresh28(is_a_theorem(or(X, not(Y))), op_implies, implies(or(Y, Z), or(Z, X)))
% 60.43/8.13 = { by lemma 33 R->L }
% 60.43/8.13 fresh28(is_a_theorem(or(X, not(Y))), op_implies, implies(implies(not(Y), Z), or(Z, X)))
% 60.43/8.13 = { by lemma 42 }
% 60.43/8.13 op_implies
% 60.43/8.13
% 60.43/8.13 Goal 1 (luka_cn2): cn2 = true.
% 60.43/8.13 Proof:
% 60.43/8.13 cn2
% 60.43/8.13 = { by axiom 28 (cn2) R->L }
% 60.43/8.13 fresh50(is_a_theorem(implies(p7, implies(not(p7), q5))), true)
% 60.43/8.13 = { by lemma 33 }
% 60.43/8.13 fresh50(is_a_theorem(implies(p7, or(p7, q5))), true)
% 60.43/8.13 = { by lemma 49 }
% 60.43/8.13 fresh50(fresh18(or_1, op_implies, p7, q5), true)
% 60.43/8.13 = { by lemma 50 }
% 60.43/8.13 fresh50(fresh49(cn2, op_implies, p7, q5), true)
% 60.43/8.13 = { by axiom 1 (luka_op_implies) R->L }
% 60.43/8.13 fresh50(fresh49(cn2, op_implies, p7, q5), op_implies)
% 60.43/8.13 = { by lemma 50 R->L }
% 60.43/8.13 fresh50(fresh18(or_1, op_implies, p7, q5), op_implies)
% 60.43/8.13 = { by axiom 1 (luka_op_implies) }
% 60.43/8.13 fresh50(fresh18(or_1, true, p7, q5), op_implies)
% 60.43/8.13 = { by axiom 22 (or_1_1) }
% 60.43/8.13 fresh50(is_a_theorem(implies(p7, or(p7, q5))), op_implies)
% 60.43/8.13 = { by axiom 12 (modus_ponens_2) R->L }
% 60.43/8.13 fresh50(fresh28(op_implies, op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.13 = { by lemma 36 R->L }
% 60.43/8.13 fresh50(fresh28(fresh59(op_implies, op_implies, or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 53 R->L }
% 60.43/8.14 fresh50(fresh28(fresh59(fresh28(is_a_theorem(or(not(and(p7, not(or(p7, q5)))), not(or(p7, q5)))), op_implies, implies(or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), not(and(p7, not(or(p7, q5))))))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by axiom 12 (modus_ponens_2) R->L }
% 60.43/8.14 fresh50(fresh28(fresh59(fresh28(fresh28(op_implies, op_implies, or(not(and(p7, not(or(p7, q5)))), not(or(p7, q5)))), op_implies, implies(or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), not(and(p7, not(or(p7, q5))))))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 38 R->L }
% 60.43/8.14 fresh50(fresh28(fresh59(fresh28(fresh28(fresh28(is_a_theorem(implies(not(or(p7, q5)), not(or(p7, q5)))), op_implies, or(and(not(or(p7, q5)), p7), not(and(p7, not(or(p7, q5)))))), op_implies, or(not(and(p7, not(or(p7, q5)))), not(or(p7, q5)))), op_implies, implies(or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), not(and(p7, not(or(p7, q5))))))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by axiom 12 (modus_ponens_2) R->L }
% 60.43/8.14 fresh50(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(op_implies, op_implies, implies(not(or(p7, q5)), not(or(p7, q5)))), op_implies, or(and(not(or(p7, q5)), p7), not(and(p7, not(or(p7, q5)))))), op_implies, or(not(and(p7, not(or(p7, q5)))), not(or(p7, q5)))), op_implies, implies(or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), not(and(p7, not(or(p7, q5))))))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 48 R->L }
% 60.43/8.14 fresh50(fresh28(fresh59(fresh28(fresh28(fresh28(fresh28(is_a_theorem(equiv(not(or(p7, q5)), not(or(p7, q5)))), op_implies, implies(not(or(p7, q5)), not(or(p7, q5)))), op_implies, or(and(not(or(p7, q5)), p7), not(and(p7, not(or(p7, q5)))))), op_implies, or(not(and(p7, not(or(p7, q5)))), not(or(p7, q5)))), op_implies, implies(or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), not(and(p7, not(or(p7, q5))))))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 51 }
% 60.43/8.14 fresh50(fresh28(fresh59(fresh28(fresh28(fresh28(op_implies, op_implies, or(and(not(or(p7, q5)), p7), not(and(p7, not(or(p7, q5)))))), op_implies, or(not(and(p7, not(or(p7, q5)))), not(or(p7, q5)))), op_implies, implies(or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), not(and(p7, not(or(p7, q5))))))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by axiom 12 (modus_ponens_2) }
% 60.43/8.14 fresh50(fresh28(fresh59(fresh28(fresh28(is_a_theorem(or(and(not(or(p7, q5)), p7), not(and(p7, not(or(p7, q5)))))), op_implies, or(not(and(p7, not(or(p7, q5)))), not(or(p7, q5)))), op_implies, implies(or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), not(and(p7, not(or(p7, q5))))))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 45 }
% 60.43/8.14 fresh50(fresh28(fresh59(fresh28(op_implies, op_implies, implies(or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), not(and(p7, not(or(p7, q5))))))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by axiom 12 (modus_ponens_2) }
% 60.43/8.14 fresh50(fresh28(fresh59(is_a_theorem(implies(or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), not(and(p7, not(or(p7, q5))))))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 32 }
% 60.43/8.14 fresh50(fresh28(fresh59(is_a_theorem(implies(or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5))))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 31 }
% 60.43/8.14 fresh50(fresh28(fresh28(is_a_theorem(or(or(p7, q5), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by axiom 12 (modus_ponens_2) R->L }
% 60.43/8.14 fresh50(fresh28(fresh28(fresh28(op_implies, op_implies, or(or(p7, q5), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 36 R->L }
% 60.43/8.14 fresh50(fresh28(fresh28(fresh28(fresh59(op_implies, op_implies, or(p7, implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), or(p7, q5))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 53 R->L }
% 60.43/8.14 fresh50(fresh28(fresh28(fresh28(fresh59(fresh28(is_a_theorem(or(implies(not(p7), q5), not(p7))), op_implies, implies(or(p7, implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(not(p7), q5)))), op_implies, or(p7, implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), or(p7, q5))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by axiom 12 (modus_ponens_2) R->L }
% 60.43/8.14 fresh50(fresh28(fresh28(fresh28(fresh59(fresh28(fresh28(op_implies, op_implies, or(implies(not(p7), q5), not(p7))), op_implies, implies(or(p7, implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(not(p7), q5)))), op_implies, or(p7, implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), or(p7, q5))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 46 R->L }
% 60.43/8.14 fresh50(fresh28(fresh28(fresh28(fresh59(fresh28(fresh28(is_a_theorem(or(not(p7), implies(not(p7), q5))), op_implies, or(implies(not(p7), q5), not(p7))), op_implies, implies(or(p7, implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(not(p7), q5)))), op_implies, or(p7, implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), or(p7, q5))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 52 }
% 60.43/8.14 fresh50(fresh28(fresh28(fresh28(fresh59(fresh28(op_implies, op_implies, implies(or(p7, implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(not(p7), q5)))), op_implies, or(p7, implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), or(p7, q5))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by axiom 12 (modus_ponens_2) }
% 60.43/8.14 fresh50(fresh28(fresh28(fresh28(fresh59(is_a_theorem(implies(or(p7, implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), implies(not(p7), q5)))), op_implies, or(p7, implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), or(p7, q5))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 33 }
% 60.43/8.14 fresh50(fresh28(fresh28(fresh28(fresh59(is_a_theorem(implies(or(p7, implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), or(p7, q5)))), op_implies, or(p7, implies(p7, or(p7, q5))), or(implies(p7, or(p7, q5)), or(p7, q5))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 31 }
% 60.43/8.14 fresh50(fresh28(fresh28(fresh28(fresh28(is_a_theorem(or(p7, implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), or(p7, q5))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 46 }
% 60.43/8.14 fresh50(fresh28(fresh28(fresh28(fresh28(op_implies, op_implies, or(implies(p7, or(p7, q5)), or(p7, q5))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by axiom 12 (modus_ponens_2) }
% 60.43/8.14 fresh50(fresh28(fresh28(fresh28(is_a_theorem(or(implies(p7, or(p7, q5)), or(p7, q5))), op_implies, or(or(p7, q5), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 52 }
% 60.43/8.14 fresh50(fresh28(fresh28(op_implies, op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by axiom 12 (modus_ponens_2) }
% 60.43/8.14 fresh50(fresh28(is_a_theorem(or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5)))), op_implies, implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 31 R->L }
% 60.43/8.14 fresh50(fresh59(is_a_theorem(implies(or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5))), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5))), implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 33 R->L }
% 60.43/8.14 fresh50(fresh59(is_a_theorem(implies(implies(not(implies(p7, or(p7, q5))), implies(p7, or(p7, q5))), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5))), implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 40 R->L }
% 60.43/8.14 fresh50(fresh59(is_a_theorem(or(and(not(implies(p7, or(p7, q5))), not(implies(p7, or(p7, q5)))), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5))), implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by axiom 12 (modus_ponens_2) R->L }
% 60.43/8.14 fresh50(fresh59(fresh28(op_implies, op_implies, or(and(not(implies(p7, or(p7, q5))), not(implies(p7, or(p7, q5)))), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5))), implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 47 R->L }
% 60.43/8.14 fresh50(fresh59(fresh28(is_a_theorem(or(implies(p7, or(p7, q5)), and(not(implies(p7, or(p7, q5))), not(implies(p7, or(p7, q5)))))), op_implies, or(and(not(implies(p7, or(p7, q5))), not(implies(p7, or(p7, q5)))), implies(p7, or(p7, q5)))), op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5))), implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 52 }
% 60.43/8.14 fresh50(fresh59(op_implies, op_implies, or(implies(p7, or(p7, q5)), implies(p7, or(p7, q5))), implies(p7, or(p7, q5))), op_implies)
% 60.43/8.14 = { by lemma 36 }
% 60.43/8.14 fresh50(op_implies, op_implies)
% 60.43/8.14 = { by axiom 9 (cn2) }
% 60.43/8.14 true
% 60.43/8.14 % SZS output end Proof
% 60.43/8.14
% 60.43/8.14 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------