0.11/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.11/0.13	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.14/0.35	% Computer   : n011.cluster.edu
0.14/0.35	% Model      : x86_64 x86_64
0.14/0.35	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.14/0.35	% Memory     : 8042.1875MB
0.14/0.35	% OS         : Linux 3.10.0-693.el7.x86_64
0.14/0.35	% CPULimit   : 1200
0.14/0.35	% WCLimit    : 120
0.14/0.35	% DateTime   : Tue Jul 13 16:12:10 EDT 2021
0.14/0.35	% CPUTime    : 
80.55/10.61	% SZS status Theorem
80.55/10.61	
80.55/10.65	% SZS output start Proof
80.55/10.65	Take the following subset of the input axioms:
80.55/10.65	  fof(and_1, axiom, ![X, Y]: is_a_theorem(implies(and(X, Y), X)) <=> and_1).
80.55/10.65	  fof(cn2, axiom, ![P, Q]: is_a_theorem(implies(P, implies(not(P), Q))) <=> cn2).
80.55/10.65	  fof(kn1, axiom, kn1 <=> ![P]: is_a_theorem(implies(P, and(P, P)))).
80.55/10.65	  fof(kn2, axiom, ![P, Q]: is_a_theorem(implies(and(P, Q), P)) <=> kn2).
80.55/10.65	  fof(kn3, axiom, kn3 <=> ![P, Q, R]: is_a_theorem(implies(implies(P, Q), implies(not(and(Q, R)), not(and(R, P)))))).
80.55/10.65	  fof(luka_cn2, conjecture, cn2).
80.55/10.65	  fof(modus_ponens, axiom, ![X, Y]: (is_a_theorem(Y) <= (is_a_theorem(implies(X, Y)) & is_a_theorem(X))) <=> modus_ponens).
80.55/10.65	  fof(op_implies_and, axiom, op_implies_and => ![X, Y]: implies(X, Y)=not(and(X, not(Y)))).
80.55/10.65	  fof(op_or, axiom, op_or => ![X, Y]: or(X, Y)=not(and(not(X), not(Y)))).
80.55/10.65	  fof(rosser_kn1, axiom, kn1).
80.55/10.65	  fof(rosser_kn2, axiom, kn2).
80.55/10.65	  fof(rosser_kn3, axiom, kn3).
80.55/10.65	  fof(rosser_modus_ponens, axiom, modus_ponens).
80.55/10.65	  fof(rosser_op_implies_and, axiom, op_implies_and).
80.55/10.65	  fof(rosser_op_or, axiom, op_or).
80.55/10.65	
80.55/10.65	Now clausify the problem and encode Horn clauses using encoding 3 of
80.55/10.65	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
80.55/10.65	We repeatedly replace C & s=t => u=v by the two clauses:
80.55/10.65	  fresh(y, y, x1...xn) = u
80.55/10.65	  C => fresh(s, t, x1...xn) = v
80.55/10.65	where fresh is a fresh function symbol and x1..xn are the free
80.55/10.65	variables of u and v.
80.55/10.65	A predicate p(X) is encoded as p(X)=true (this is sound, because the
80.55/10.65	input problem has no model of domain size 1).
80.55/10.65	
80.55/10.65	The encoding turns the above axioms into the following unit equations and goals:
80.55/10.65	
80.55/10.65	Axiom 1 (rosser_modus_ponens): modus_ponens = true.
80.55/10.65	Axiom 2 (rosser_kn3): kn3 = true.
80.55/10.65	Axiom 3 (rosser_kn1): kn1 = true.
80.55/10.65	Axiom 4 (rosser_kn2): kn2 = true.
80.55/10.65	Axiom 5 (rosser_op_or): op_or = true.
80.55/10.65	Axiom 6 (rosser_op_implies_and): op_implies_and = true.
80.55/10.65	Axiom 7 (cn2): fresh50(X, X) = true.
80.55/10.65	Axiom 8 (modus_ponens_2): fresh60(X, X, Y) = true.
80.55/10.65	Axiom 9 (kn1_1): fresh33(X, X, Y) = true.
80.55/10.65	Axiom 10 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(is_a_theorem(Y), true, Z).
80.55/10.65	Axiom 11 (kn2_1): fresh31(X, X, Y, Z) = true.
80.55/10.65	Axiom 12 (modus_ponens_2): fresh28(X, X, Y, Z) = is_a_theorem(Z).
80.55/10.65	Axiom 13 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
80.55/10.65	Axiom 14 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
80.55/10.65	Axiom 15 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
80.55/10.65	Axiom 16 (kn1_1): fresh33(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
80.55/10.65	Axiom 17 (and_1_1): fresh58(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
80.55/10.65	Axiom 18 (kn2_1): fresh31(kn2, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
80.55/10.65	Axiom 19 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
80.55/10.65	Axiom 20 (kn3_1): fresh29(X, X, Y, Z, W) = true.
80.55/10.65	Axiom 21 (modus_ponens_2): fresh59(modus_ponens, true, X, Y) = fresh28(is_a_theorem(implies(X, Y)), true, X, Y).
80.55/10.65	Axiom 22 (cn2): fresh50(is_a_theorem(implies(p10, implies(not(p10), q7))), true) = cn2.
80.55/10.65	Axiom 23 (kn3_1): fresh29(kn3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X))))).
80.55/10.65	
80.55/10.65	Lemma 24: not(and(X, not(Y))) = implies(X, Y).
80.55/10.65	Proof:
80.55/10.65	  not(and(X, not(Y)))
80.55/10.65	= { by axiom 14 (op_implies_and) R->L }
80.55/10.65	  fresh22(op_implies_and, true, X, Y)
80.55/10.65	= { by axiom 6 (rosser_op_implies_and) }
80.55/10.65	  fresh22(true, true, X, Y)
80.55/10.65	= { by axiom 13 (op_implies_and) }
80.55/10.65	  implies(X, Y)
80.55/10.65	
80.55/10.65	Lemma 25: implies(not(X), Y) = or(X, Y).
80.55/10.65	Proof:
80.55/10.65	  implies(not(X), Y)
80.55/10.65	= { by lemma 24 R->L }
80.55/10.65	  not(and(not(X), not(Y)))
80.55/10.65	= { by axiom 19 (op_or) R->L }
80.55/10.65	  fresh20(op_or, true, X, Y)
80.55/10.65	= { by axiom 5 (rosser_op_or) }
80.55/10.65	  fresh20(true, true, X, Y)
80.55/10.65	= { by axiom 15 (op_or) }
80.55/10.65	  or(X, Y)
80.55/10.65	
80.55/10.65	Lemma 26: fresh58(and_1, true, X, Y) = true.
80.55/10.65	Proof:
80.55/10.65	  fresh58(and_1, true, X, Y)
80.55/10.65	= { by axiom 17 (and_1_1) }
80.55/10.65	  is_a_theorem(implies(and(X, Y), X))
80.55/10.65	= { by axiom 18 (kn2_1) R->L }
80.55/10.65	  fresh31(kn2, true, X, Y)
80.55/10.65	= { by axiom 4 (rosser_kn2) }
80.55/10.65	  fresh31(true, true, X, Y)
80.55/10.65	= { by axiom 11 (kn2_1) }
80.55/10.65	  true
80.55/10.65	
80.55/10.65	Lemma 27: is_a_theorem(implies(X, and(X, X))) = true.
80.55/10.65	Proof:
80.55/10.65	  is_a_theorem(implies(X, and(X, X)))
80.55/10.65	= { by axiom 16 (kn1_1) R->L }
80.55/10.65	  fresh33(kn1, true, X)
80.55/10.65	= { by axiom 3 (rosser_kn1) }
80.55/10.65	  fresh33(true, true, X)
80.55/10.65	= { by axiom 9 (kn1_1) }
80.55/10.65	  true
80.55/10.65	
80.55/10.65	Lemma 28: fresh28(is_a_theorem(implies(X, Y)), true, X, Y) = fresh60(is_a_theorem(X), true, Y).
80.55/10.65	Proof:
80.55/10.65	  fresh28(is_a_theorem(implies(X, Y)), true, X, Y)
80.55/10.65	= { by axiom 21 (modus_ponens_2) R->L }
80.55/10.65	  fresh59(modus_ponens, true, X, Y)
80.55/10.65	= { by axiom 1 (rosser_modus_ponens) }
80.55/10.65	  fresh59(true, true, X, Y)
80.55/10.65	= { by axiom 10 (modus_ponens_2) }
80.55/10.65	  fresh60(is_a_theorem(X), true, Y)
80.55/10.65	
80.55/10.65	Lemma 29: is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X))))) = true.
80.55/10.65	Proof:
80.55/10.65	  is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X)))))
80.55/10.65	= { by lemma 25 R->L }
80.55/10.65	  is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X)))))
80.55/10.65	= { by axiom 23 (kn3_1) R->L }
80.55/10.65	  fresh29(kn3, true, X, Y, Z)
80.55/10.65	= { by axiom 2 (rosser_kn3) }
80.55/10.65	  fresh29(true, true, X, Y, Z)
80.55/10.65	= { by axiom 20 (kn3_1) }
80.55/10.65	  true
80.55/10.65	
80.55/10.65	Lemma 30: fresh60(is_a_theorem(or(X, Y)), true, or(and(Y, Z), implies(Z, X))) = is_a_theorem(or(and(Y, Z), implies(Z, X))).
80.55/10.65	Proof:
80.55/10.65	  fresh60(is_a_theorem(or(X, Y)), true, or(and(Y, Z), implies(Z, X)))
80.55/10.65	= { by lemma 28 R->L }
80.55/10.65	  fresh28(is_a_theorem(implies(or(X, Y), or(and(Y, Z), implies(Z, X)))), true, or(X, Y), or(and(Y, Z), implies(Z, X)))
80.55/10.65	= { by lemma 25 R->L }
80.55/10.65	  fresh28(is_a_theorem(implies(implies(not(X), Y), or(and(Y, Z), implies(Z, X)))), true, or(X, Y), or(and(Y, Z), implies(Z, X)))
80.55/10.65	= { by lemma 24 R->L }
80.55/10.65	  fresh28(is_a_theorem(implies(implies(not(X), Y), or(and(Y, Z), not(and(Z, not(X)))))), true, or(X, Y), or(and(Y, Z), implies(Z, X)))
80.55/10.65	= { by lemma 29 }
80.55/10.65	  fresh28(true, true, or(X, Y), or(and(Y, Z), implies(Z, X)))
80.55/10.65	= { by axiom 12 (modus_ponens_2) }
80.55/10.65	  is_a_theorem(or(and(Y, Z), implies(Z, X)))
80.55/10.65	
80.55/10.65	Lemma 31: or(and(X, not(Y)), Z) = implies(implies(X, Y), Z).
80.55/10.65	Proof:
80.55/10.65	  or(and(X, not(Y)), Z)
80.55/10.65	= { by lemma 25 R->L }
80.55/10.65	  implies(not(and(X, not(Y))), Z)
80.55/10.65	= { by lemma 24 }
80.55/10.65	  implies(implies(X, Y), Z)
80.55/10.65	
80.55/10.65	Lemma 32: is_a_theorem(or(not(X), X)) = true.
80.55/10.65	Proof:
80.55/10.65	  is_a_theorem(or(not(X), X))
80.55/10.65	= { by axiom 12 (modus_ponens_2) R->L }
80.55/10.65	  fresh28(true, true, implies(and(not(X), not(X)), not(X)), or(not(X), X))
80.55/10.65	= { by axiom 8 (modus_ponens_2) R->L }
80.55/10.65	  fresh28(fresh60(true, true, or(and(and(not(X), not(X)), not(not(X))), implies(not(not(X)), X))), true, implies(and(not(X), not(X)), not(X)), or(not(X), X))
80.55/10.65	= { by lemma 27 R->L }
80.55/10.65	  fresh28(fresh60(is_a_theorem(implies(not(X), and(not(X), not(X)))), true, or(and(and(not(X), not(X)), not(not(X))), implies(not(not(X)), X))), true, implies(and(not(X), not(X)), not(X)), or(not(X), X))
80.55/10.65	= { by lemma 25 }
80.55/10.65	  fresh28(fresh60(is_a_theorem(or(X, and(not(X), not(X)))), true, or(and(and(not(X), not(X)), not(not(X))), implies(not(not(X)), X))), true, implies(and(not(X), not(X)), not(X)), or(not(X), X))
80.55/10.65	= { by lemma 30 }
80.55/10.65	  fresh28(is_a_theorem(or(and(and(not(X), not(X)), not(not(X))), implies(not(not(X)), X))), true, implies(and(not(X), not(X)), not(X)), or(not(X), X))
80.55/10.65	= { by lemma 25 }
80.55/10.65	  fresh28(is_a_theorem(or(and(and(not(X), not(X)), not(not(X))), or(not(X), X))), true, implies(and(not(X), not(X)), not(X)), or(not(X), X))
80.55/10.65	= { by lemma 31 }
80.55/10.65	  fresh28(is_a_theorem(implies(implies(and(not(X), not(X)), not(X)), or(not(X), X))), true, implies(and(not(X), not(X)), not(X)), or(not(X), X))
80.55/10.65	= { by lemma 28 }
80.55/10.65	  fresh60(is_a_theorem(implies(and(not(X), not(X)), not(X))), true, or(not(X), X))
80.55/10.65	= { by axiom 17 (and_1_1) R->L }
80.55/10.65	  fresh60(fresh58(and_1, true, not(X), not(X)), true, or(not(X), X))
80.55/10.65	= { by lemma 26 }
80.55/10.65	  fresh60(true, true, or(not(X), X))
80.55/10.65	= { by axiom 8 (modus_ponens_2) }
80.55/10.65	  true
80.55/10.65	
80.55/10.65	Lemma 33: is_a_theorem(or(and(X, Y), implies(Y, not(X)))) = true.
80.55/10.65	Proof:
80.55/10.65	  is_a_theorem(or(and(X, Y), implies(Y, not(X))))
80.55/10.65	= { by lemma 30 R->L }
80.55/10.65	  fresh60(is_a_theorem(or(not(X), X)), true, or(and(X, Y), implies(Y, not(X))))
80.55/10.65	= { by lemma 32 }
80.55/10.65	  fresh60(true, true, or(and(X, Y), implies(Y, not(X))))
80.55/10.65	= { by axiom 8 (modus_ponens_2) }
80.55/10.65	  true
80.55/10.65	
80.55/10.65	Lemma 34: fresh60(is_a_theorem(implies(X, Y)), true, or(Y, not(X))) = is_a_theorem(or(Y, not(X))).
80.55/10.65	Proof:
80.55/10.65	  fresh60(is_a_theorem(implies(X, Y)), true, or(Y, not(X)))
80.55/10.65	= { by lemma 28 R->L }
80.55/10.65	  fresh28(is_a_theorem(implies(implies(X, Y), or(Y, not(X)))), true, implies(X, Y), or(Y, not(X)))
80.55/10.65	= { by lemma 31 R->L }
80.55/10.65	  fresh28(is_a_theorem(or(and(X, not(Y)), or(Y, not(X)))), true, implies(X, Y), or(Y, not(X)))
80.55/10.65	= { by lemma 25 R->L }
80.55/10.65	  fresh28(is_a_theorem(or(and(X, not(Y)), implies(not(Y), not(X)))), true, implies(X, Y), or(Y, not(X)))
80.55/10.65	= { by lemma 33 }
80.55/10.65	  fresh28(true, true, implies(X, Y), or(Y, not(X)))
80.55/10.65	= { by axiom 12 (modus_ponens_2) }
80.55/10.65	  is_a_theorem(or(Y, not(X)))
80.55/10.65	
80.55/10.65	Lemma 35: is_a_theorem(or(X, not(and(X, Y)))) = true.
80.55/10.65	Proof:
80.55/10.65	  is_a_theorem(or(X, not(and(X, Y))))
80.55/10.65	= { by lemma 34 R->L }
80.55/10.65	  fresh60(is_a_theorem(implies(and(X, Y), X)), true, or(X, not(and(X, Y))))
80.55/10.65	= { by axiom 17 (and_1_1) R->L }
80.55/10.65	  fresh60(fresh58(and_1, true, X, Y), true, or(X, not(and(X, Y))))
80.55/10.65	= { by lemma 26 }
80.55/10.65	  fresh60(true, true, or(X, not(and(X, Y))))
80.55/10.65	= { by axiom 8 (modus_ponens_2) }
80.55/10.65	  true
80.55/10.65	
80.55/10.65	Lemma 36: fresh28(is_a_theorem(or(X, Y)), true, not(X), Y) = fresh60(is_a_theorem(not(X)), true, Y).
80.55/10.65	Proof:
80.55/10.65	  fresh28(is_a_theorem(or(X, Y)), true, not(X), Y)
80.55/10.65	= { by lemma 25 R->L }
80.55/10.65	  fresh28(is_a_theorem(implies(not(X), Y)), true, not(X), Y)
80.55/10.65	= { by lemma 28 }
80.55/10.65	  fresh60(is_a_theorem(not(X)), true, Y)
80.55/10.65	
80.55/10.65	Lemma 37: fresh60(is_a_theorem(implies(X, Y)), true, or(and(Y, Z), not(and(Z, X)))) = is_a_theorem(or(and(Y, Z), not(and(Z, X)))).
80.55/10.65	Proof:
80.55/10.65	  fresh60(is_a_theorem(implies(X, Y)), true, or(and(Y, Z), not(and(Z, X))))
80.55/10.65	= { by lemma 28 R->L }
80.55/10.65	  fresh28(is_a_theorem(implies(implies(X, Y), or(and(Y, Z), not(and(Z, X))))), true, implies(X, Y), or(and(Y, Z), not(and(Z, X))))
80.55/10.65	= { by lemma 29 }
80.55/10.65	  fresh28(true, true, implies(X, Y), or(and(Y, Z), not(and(Z, X))))
80.55/10.65	= { by axiom 12 (modus_ponens_2) }
80.55/10.66	  is_a_theorem(or(and(Y, Z), not(and(Z, X))))
80.55/10.66	
80.55/10.66	Lemma 38: is_a_theorem(or(and(X, Y), not(and(Y, and(X, Z))))) = true.
80.55/10.66	Proof:
80.55/10.66	  is_a_theorem(or(and(X, Y), not(and(Y, and(X, Z)))))
80.55/10.66	= { by lemma 37 R->L }
80.55/10.66	  fresh60(is_a_theorem(implies(and(X, Z), X)), true, or(and(X, Y), not(and(Y, and(X, Z)))))
80.55/10.66	= { by axiom 17 (and_1_1) R->L }
80.55/10.66	  fresh60(fresh58(and_1, true, X, Z), true, or(and(X, Y), not(and(Y, and(X, Z)))))
80.55/10.66	= { by lemma 26 }
80.55/10.66	  fresh60(true, true, or(and(X, Y), not(and(Y, and(X, Z)))))
80.55/10.66	= { by axiom 8 (modus_ponens_2) }
80.55/10.66	  true
80.55/10.66	
80.55/10.66	Goal 1 (luka_cn2): cn2 = true.
80.55/10.66	Proof:
80.55/10.66	  cn2
80.55/10.66	= { by axiom 22 (cn2) R->L }
80.55/10.66	  fresh50(is_a_theorem(implies(p10, implies(not(p10), q7))), true)
80.55/10.66	= { by lemma 24 R->L }
80.55/10.66	  fresh50(is_a_theorem(implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.66	= { by axiom 12 (modus_ponens_2) R->L }
80.55/10.66	  fresh50(fresh28(true, true, not(and(and(not(p10), not(q7)), p10)), implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.66	= { by lemma 33 R->L }
80.55/10.66	  fresh50(fresh28(is_a_theorem(or(and(and(not(p10), not(q7)), p10), implies(p10, not(and(not(p10), not(q7)))))), true, not(and(and(not(p10), not(q7)), p10)), implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.66	= { by lemma 36 }
80.55/10.66	  fresh50(fresh60(is_a_theorem(not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.66	= { by axiom 12 (modus_ponens_2) R->L }
80.55/10.66	  fresh50(fresh60(fresh28(true, true, not(and(and(p10, p10), and(not(p10), not(q7)))), not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.66	= { by axiom 8 (modus_ponens_2) R->L }
80.55/10.66	  fresh50(fresh60(fresh28(fresh60(true, true, or(and(and(p10, p10), and(not(p10), not(q7))), not(and(and(not(p10), not(q7)), p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7)))), not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.66	= { by lemma 27 R->L }
80.55/10.66	  fresh50(fresh60(fresh28(fresh60(is_a_theorem(implies(p10, and(p10, p10))), true, or(and(and(p10, p10), and(not(p10), not(q7))), not(and(and(not(p10), not(q7)), p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7)))), not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.66	= { by lemma 37 }
80.55/10.66	  fresh50(fresh60(fresh28(is_a_theorem(or(and(and(p10, p10), and(not(p10), not(q7))), not(and(and(not(p10), not(q7)), p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7)))), not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.66	= { by lemma 36 }
80.55/10.66	  fresh50(fresh60(fresh60(is_a_theorem(not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.66	= { by axiom 12 (modus_ponens_2) R->L }
80.55/10.66	  fresh50(fresh60(fresh60(fresh28(true, true, not(and(not(p10), and(p10, p10))), not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.66	= { by lemma 38 R->L }
80.55/10.66	  fresh50(fresh60(fresh60(fresh28(is_a_theorem(or(and(not(p10), and(p10, p10)), not(and(and(p10, p10), and(not(p10), not(q7)))))), true, not(and(not(p10), and(p10, p10))), not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.66	= { by lemma 36 }
80.55/10.66	  fresh50(fresh60(fresh60(fresh60(is_a_theorem(not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.66	= { by axiom 12 (modus_ponens_2) R->L }
80.55/10.66	  fresh50(fresh60(fresh60(fresh60(fresh28(true, true, implies(p10, p10), not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.66	= { by lemma 38 R->L }
80.55/10.66	  fresh50(fresh60(fresh60(fresh60(fresh28(is_a_theorem(or(and(p10, not(p10)), not(and(not(p10), and(p10, p10))))), true, implies(p10, p10), not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.66	= { by lemma 31 }
80.55/10.66	  fresh50(fresh60(fresh60(fresh60(fresh28(is_a_theorem(implies(implies(p10, p10), not(and(not(p10), and(p10, p10))))), true, implies(p10, p10), not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.66	= { by lemma 28 }
80.55/10.66	  fresh50(fresh60(fresh60(fresh60(fresh60(is_a_theorem(implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.67	= { by axiom 12 (modus_ponens_2) R->L }
80.55/10.67	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh28(true, true, not(not(implies(p10, p10))), implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.67	= { by lemma 32 R->L }
80.55/10.67	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh28(is_a_theorem(or(not(implies(p10, p10)), implies(p10, p10))), true, not(not(implies(p10, p10))), implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.67	= { by lemma 36 }
80.55/10.67	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(is_a_theorem(not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
80.55/10.67	= { by axiom 12 (modus_ponens_2) R->L }
80.55/10.67	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh28(true, true, or(implies(p10, p10), implies(p10, p10)), not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.67	= { by axiom 8 (modus_ponens_2) R->L }
81.02/10.67	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh28(fresh60(true, true, or(and(not(implies(p10, p10)), not(implies(p10, p10))), not(not(implies(p10, p10))))), true, or(implies(p10, p10), implies(p10, p10)), not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.67	= { by lemma 27 R->L }
81.02/10.67	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh28(fresh60(is_a_theorem(implies(not(implies(p10, p10)), and(not(implies(p10, p10)), not(implies(p10, p10))))), true, or(and(not(implies(p10, p10)), not(implies(p10, p10))), not(not(implies(p10, p10))))), true, or(implies(p10, p10), implies(p10, p10)), not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.67	= { by lemma 34 }
81.02/10.67	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh28(is_a_theorem(or(and(not(implies(p10, p10)), not(implies(p10, p10))), not(not(implies(p10, p10))))), true, or(implies(p10, p10), implies(p10, p10)), not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.67	= { by lemma 31 }
81.02/10.67	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh28(is_a_theorem(implies(implies(not(implies(p10, p10)), implies(p10, p10)), not(not(implies(p10, p10))))), true, or(implies(p10, p10), implies(p10, p10)), not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.67	= { by lemma 25 }
81.02/10.67	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh28(is_a_theorem(implies(or(implies(p10, p10), implies(p10, p10)), not(not(implies(p10, p10))))), true, or(implies(p10, p10), implies(p10, p10)), not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.67	= { by lemma 28 }
81.02/10.67	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(is_a_theorem(or(implies(p10, p10), implies(p10, p10))), true, not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.67	= { by axiom 12 (modus_ponens_2) R->L }
81.02/10.67	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(fresh28(true, true, or(and(implies(p10, p10), p10), implies(p10, p10)), or(implies(p10, p10), implies(p10, p10))), true, not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.67	= { by axiom 8 (modus_ponens_2) R->L }
81.02/10.67	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(fresh28(fresh60(true, true, or(and(not(and(implies(p10, p10), p10)), not(implies(p10, p10))), implies(not(implies(p10, p10)), implies(p10, p10)))), true, or(and(implies(p10, p10), p10), implies(p10, p10)), or(implies(p10, p10), implies(p10, p10))), true, not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.67	= { by lemma 35 R->L }
81.02/10.67	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(fresh28(fresh60(is_a_theorem(or(implies(p10, p10), not(and(implies(p10, p10), p10)))), true, or(and(not(and(implies(p10, p10), p10)), not(implies(p10, p10))), implies(not(implies(p10, p10)), implies(p10, p10)))), true, or(and(implies(p10, p10), p10), implies(p10, p10)), or(implies(p10, p10), implies(p10, p10))), true, not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by lemma 30 }
81.02/10.68	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(fresh28(is_a_theorem(or(and(not(and(implies(p10, p10), p10)), not(implies(p10, p10))), implies(not(implies(p10, p10)), implies(p10, p10)))), true, or(and(implies(p10, p10), p10), implies(p10, p10)), or(implies(p10, p10), implies(p10, p10))), true, not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by lemma 25 }
81.02/10.68	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(fresh28(is_a_theorem(or(and(not(and(implies(p10, p10), p10)), not(implies(p10, p10))), or(implies(p10, p10), implies(p10, p10)))), true, or(and(implies(p10, p10), p10), implies(p10, p10)), or(implies(p10, p10), implies(p10, p10))), true, not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by lemma 31 }
81.02/10.68	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(fresh28(is_a_theorem(implies(implies(not(and(implies(p10, p10), p10)), implies(p10, p10)), or(implies(p10, p10), implies(p10, p10)))), true, or(and(implies(p10, p10), p10), implies(p10, p10)), or(implies(p10, p10), implies(p10, p10))), true, not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by lemma 25 }
81.02/10.68	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(fresh28(is_a_theorem(implies(or(and(implies(p10, p10), p10), implies(p10, p10)), or(implies(p10, p10), implies(p10, p10)))), true, or(and(implies(p10, p10), p10), implies(p10, p10)), or(implies(p10, p10), implies(p10, p10))), true, not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by lemma 28 }
81.02/10.68	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(is_a_theorem(or(and(implies(p10, p10), p10), implies(p10, p10))), true, or(implies(p10, p10), implies(p10, p10))), true, not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by lemma 30 R->L }
81.02/10.68	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(is_a_theorem(or(p10, implies(p10, p10))), true, or(and(implies(p10, p10), p10), implies(p10, p10))), true, or(implies(p10, p10), implies(p10, p10))), true, not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by lemma 24 R->L }
81.02/10.68	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(is_a_theorem(or(p10, not(and(p10, not(p10))))), true, or(and(implies(p10, p10), p10), implies(p10, p10))), true, or(implies(p10, p10), implies(p10, p10))), true, not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by lemma 35 }
81.02/10.68	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(true, true, or(and(implies(p10, p10), p10), implies(p10, p10))), true, or(implies(p10, p10), implies(p10, p10))), true, not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by axiom 8 (modus_ponens_2) }
81.02/10.68	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(true, true, or(implies(p10, p10), implies(p10, p10))), true, not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by axiom 8 (modus_ponens_2) }
81.02/10.68	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(fresh60(true, true, not(not(implies(p10, p10)))), true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by axiom 8 (modus_ponens_2) }
81.02/10.68	  fresh50(fresh60(fresh60(fresh60(fresh60(fresh60(true, true, implies(p10, p10)), true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by axiom 8 (modus_ponens_2) }
81.02/10.68	  fresh50(fresh60(fresh60(fresh60(fresh60(true, true, not(and(not(p10), and(p10, p10)))), true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by axiom 8 (modus_ponens_2) }
81.02/10.68	  fresh50(fresh60(fresh60(fresh60(true, true, not(and(and(p10, p10), and(not(p10), not(q7))))), true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by axiom 8 (modus_ponens_2) }
81.02/10.68	  fresh50(fresh60(fresh60(true, true, not(and(and(not(p10), not(q7)), p10))), true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by axiom 8 (modus_ponens_2) }
81.02/10.68	  fresh50(fresh60(true, true, implies(p10, not(and(not(p10), not(q7))))), true)
81.02/10.68	= { by axiom 8 (modus_ponens_2) }
81.02/10.68	  fresh50(true, true)
81.02/10.68	= { by axiom 7 (cn2) }
81.02/10.68	  true
81.02/10.68	% SZS output end Proof
81.02/10.68	
81.02/10.68	RESULT: Theorem (the conjecture is true).
81.02/10.70	EOF