0.06/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.06/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.34	% Computer   : n014.cluster.edu
0.12/0.34	% Model      : x86_64 x86_64
0.12/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.34	% Memory     : 8042.1875MB
0.12/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.34	% CPULimit   : 960
0.12/0.34	% WCLimit    : 120
0.12/0.34	% DateTime   : Thu Jul  2 08:21:40 EDT 2020
0.12/0.34	% CPUTime    : 
17.74/2.62	% SZS status Theorem
17.74/2.62	
17.74/2.62	% SZS output start Proof
17.74/2.62	Take the following subset of the input axioms:
18.10/2.67	  fof(and_1, axiom, ![X, Y]: is_a_theorem(implies(and(X, Y), X)) <=> and_1).
18.10/2.67	  fof(and_3, axiom, ![X, Y]: is_a_theorem(implies(X, implies(Y, and(X, Y)))) <=> and_3).
18.10/2.67	  fof(axiom_5, axiom, ![X]: is_a_theorem(implies(possibly(X), necessarily(possibly(X)))) <=> axiom_5).
18.10/2.67	  fof(axiom_M, axiom, ![X]: is_a_theorem(implies(necessarily(X), X)) <=> axiom_M).
18.10/2.67	  fof(axiom_m9, axiom, ![X]: is_a_theorem(strict_implies(possibly(possibly(X)), possibly(X))) <=> axiom_m9).
18.10/2.67	  fof(cn3, axiom, ![P]: is_a_theorem(implies(implies(not(P), P), P)) <=> cn3).
18.10/2.67	  fof(hilbert_and_1, axiom, and_1).
18.10/2.67	  fof(hilbert_and_3, axiom, and_3).
18.10/2.67	  fof(hilbert_implies_1, axiom, implies_1).
18.10/2.67	  fof(hilbert_implies_2, axiom, implies_2).
18.10/2.67	  fof(hilbert_modus_ponens, axiom, modus_ponens).
18.10/2.67	  fof(hilbert_modus_tollens, axiom, modus_tollens).
18.10/2.67	  fof(hilbert_op_equiv, axiom, op_equiv).
18.10/2.67	  fof(hilbert_op_implies_and, axiom, op_implies_and).
18.10/2.67	  fof(hilbert_op_or, axiom, op_or).
18.10/2.67	  fof(implies_1, axiom, implies_1 <=> ![X, Y]: is_a_theorem(implies(X, implies(Y, X)))).
18.10/2.67	  fof(implies_2, axiom, implies_2 <=> ![X, Y]: is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y)))).
18.10/2.67	  fof(km5_axiom_5, axiom, axiom_5).
18.10/2.67	  fof(km5_axiom_M, axiom, axiom_M).
18.10/2.67	  fof(km5_necessitation, axiom, necessitation).
18.10/2.67	  fof(km5_op_possibly, axiom, op_possibly).
18.10/2.67	  fof(kn1, axiom, kn1 <=> ![P]: is_a_theorem(implies(P, and(P, P)))).
18.10/2.67	  fof(modus_ponens, axiom, ![X, Y]: (is_a_theorem(Y) <= (is_a_theorem(X) & is_a_theorem(implies(X, Y)))) <=> modus_ponens).
18.10/2.67	  fof(modus_tollens, axiom, ![X, Y]: is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))) <=> modus_tollens).
18.10/2.67	  fof(necessitation, axiom, necessitation <=> ![X]: (is_a_theorem(X) => is_a_theorem(necessarily(X)))).
18.10/2.67	  fof(op_equiv, axiom, ![X, Y]: and(implies(X, Y), implies(Y, X))=equiv(X, Y) <= op_equiv).
18.10/2.67	  fof(op_implies_and, axiom, op_implies_and => ![X, Y]: not(and(X, not(Y)))=implies(X, Y)).
18.10/2.67	  fof(op_or, axiom, op_or => ![X, Y]: or(X, Y)=not(and(not(X), not(Y)))).
18.10/2.67	  fof(op_possibly, axiom, op_possibly => ![X]: possibly(X)=not(necessarily(not(X)))).
18.10/2.67	  fof(op_strict_implies, axiom, op_strict_implies => ![X, Y]: necessarily(implies(X, Y))=strict_implies(X, Y)).
18.10/2.67	  fof(s1_0_m6s3m9b_axiom_m9, conjecture, axiom_m9).
18.10/2.67	  fof(s1_0_op_strict_implies, axiom, op_strict_implies).
18.10/2.67	  fof(substitution_of_equivalents, axiom, ![X, Y]: (X=Y <= is_a_theorem(equiv(X, Y))) <=> substitution_of_equivalents).
18.10/2.67	  fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
18.10/2.67	
18.10/2.67	Now clausify the problem and encode Horn clauses using encoding 3 of
18.10/2.67	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
18.10/2.67	We repeatedly replace C & s=t => u=v by the two clauses:
18.10/2.67	  fresh(y, y, x1...xn) = u
18.10/2.67	  C => fresh(s, t, x1...xn) = v
18.10/2.67	where fresh is a fresh function symbol and x1..xn are the free
18.10/2.67	variables of u and v.
18.10/2.67	A predicate p(X) is encoded as p(X)=true (this is sound, because the
18.10/2.67	input problem has no model of domain size 1).
18.10/2.67	
18.10/2.67	The encoding turns the above axioms into the following unit equations and goals:
18.10/2.67	
18.10/2.67	Axiom 1 (and_1_1): fresh107(X, X, Y, Z) = true.
18.10/2.67	Axiom 2 (and_3_1): fresh103(X, X, Y, Z) = true.
18.10/2.67	Axiom 3 (axiom_5_1): fresh99(X, X, Y) = true.
18.10/2.67	Axiom 4 (axiom_M_1): fresh93(X, X, Y) = true.
18.10/2.67	Axiom 5 (axiom_m9): fresh74(X, X) = true.
18.10/2.67	Axiom 6 (implies_1_1): fresh51(X, X, Y, Z) = true.
18.10/2.67	Axiom 7 (implies_2_1): fresh49(X, X, Y, Z) = true.
18.10/2.67	Axiom 8 (modus_ponens_2): fresh40(X, X, Y, Z) = is_a_theorem(Z).
18.10/2.67	Axiom 9 (modus_ponens_2): fresh116(X, X, Y) = true.
18.10/2.67	Axiom 10 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(is_a_theorem(Y), true, Z).
18.10/2.67	Axiom 11 (modus_tollens_1): fresh35(X, X, Y, Z) = true.
18.10/2.67	Axiom 12 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)).
18.10/2.67	Axiom 13 (necessitation_1): fresh33(X, X, Y) = true.
18.10/2.67	Axiom 14 (op_equiv): fresh30(X, X, Y, Z) = equiv(Y, Z).
18.10/2.67	Axiom 15 (op_implies_and): fresh29(X, X, Y, Z) = implies(Y, Z).
18.10/2.67	Axiom 16 (op_or): fresh26(X, X, Y, Z) = or(Y, Z).
18.10/2.67	Axiom 17 (op_possibly): fresh25(X, X, Y) = possibly(Y).
18.10/2.67	Axiom 18 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z).
18.10/2.67	Axiom 19 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y.
18.10/2.67	Axiom 20 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z.
18.10/2.67	Axiom 21 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
18.10/2.67	Axiom 22 (modus_tollens_1): fresh35(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
18.10/2.67	Axiom 23 (kn1_1): fresh45(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
18.10/2.67	Axiom 24 (implies_1_1): fresh51(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
18.10/2.67	Axiom 25 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y).
18.10/2.67	Axiom 26 (cn3_1): fresh59(cn3, true, X) = is_a_theorem(implies(implies(not(X), X), X)).
18.10/2.67	Axiom 27 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
18.10/2.67	Axiom 28 (modus_ponens_2): fresh115(modus_ponens, true, X, Y) = fresh40(is_a_theorem(implies(X, Y)), true, X, Y).
18.10/2.67	Axiom 29 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
18.10/2.67	Axiom 30 (op_equiv): fresh30(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
18.10/2.67	Axiom 31 (op_or): fresh26(op_or, true, X, Y) = not(and(not(X), not(Y))).
18.10/2.67	Axiom 32 (op_implies_and): fresh29(op_implies_and, true, X, Y) = not(and(X, not(Y))).
18.10/2.67	Axiom 33 (hilbert_op_implies_and): op_implies_and = true.
18.10/2.67	Axiom 34 (hilbert_modus_tollens): modus_tollens = true.
18.10/2.67	Axiom 35 (hilbert_implies_2): implies_2 = true.
18.10/2.67	Axiom 36 (hilbert_implies_1): implies_1 = true.
18.10/2.67	Axiom 37 (hilbert_and_1): and_1 = true.
18.10/2.67	Axiom 38 (hilbert_and_3): and_3 = true.
18.10/2.67	Axiom 39 (substitution_of_equivalents): substitution_of_equivalents = true.
18.10/2.67	Axiom 40 (hilbert_modus_ponens): modus_ponens = true.
18.10/2.67	Axiom 41 (hilbert_op_or): op_or = true.
18.10/2.67	Axiom 42 (hilbert_op_equiv): op_equiv = true.
18.10/2.67	Axiom 43 (axiom_M_1): fresh93(axiom_M, true, X) = is_a_theorem(implies(necessarily(X), X)).
18.10/2.67	Axiom 44 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X).
18.10/2.67	Axiom 45 (axiom_m9_1): fresh73(axiom_m9, true, X) = is_a_theorem(strict_implies(possibly(possibly(X)), possibly(X))).
18.10/2.67	Axiom 46 (axiom_m9): fresh74(is_a_theorem(strict_implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) = axiom_m9.
18.10/2.67	Axiom 47 (axiom_5_1): fresh99(axiom_5, true, X) = is_a_theorem(implies(possibly(X), necessarily(possibly(X)))).
18.10/2.67	Axiom 48 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)).
18.10/2.67	Axiom 49 (op_possibly): fresh25(op_possibly, true, X) = not(necessarily(not(X))).
18.10/2.67	Axiom 50 (km5_axiom_M): axiom_M = true.
18.10/2.67	Axiom 51 (km5_necessitation): necessitation = true.
18.10/2.67	Axiom 52 (km5_op_possibly): op_possibly = true.
18.10/2.67	Axiom 53 (km5_axiom_5): axiom_5 = true.
18.10/2.69	Axiom 54 (s1_0_op_strict_implies): op_strict_implies = true.
18.10/2.69	
18.10/2.69	Lemma 55: not(and(Y, not(X))) = implies(Y, X).
18.10/2.69	Proof:
18.10/2.69	  not(and(Y, not(X)))
18.10/2.69	= { by axiom 32 (op_implies_and) }
18.10/2.69	  fresh29(op_implies_and, true, Y, X)
18.10/2.69	= { by axiom 33 (hilbert_op_implies_and) }
18.10/2.69	  fresh29(true, true, Y, X)
18.10/2.69	= { by axiom 15 (op_implies_and) }
18.10/2.69	  implies(Y, X)
18.10/2.69	
18.10/2.69	Lemma 56: implies(not(Y), X) = or(Y, X).
18.10/2.69	Proof:
18.10/2.69	  implies(not(Y), X)
18.10/2.69	= { by lemma 55 }
18.10/2.69	  not(and(not(Y), not(X)))
18.10/2.69	= { by axiom 31 (op_or) }
18.10/2.69	  fresh26(op_or, true, Y, X)
18.10/2.69	= { by axiom 41 (hilbert_op_or) }
18.10/2.69	  fresh26(true, true, Y, X)
18.10/2.69	= { by axiom 16 (op_or) }
18.10/2.69	  or(Y, X)
18.10/2.69	
18.10/2.69	Lemma 57: not(necessarily(not(X))) = possibly(X).
18.10/2.69	Proof:
18.10/2.69	  not(necessarily(not(X)))
18.10/2.69	= { by axiom 49 (op_possibly) }
18.10/2.69	  fresh25(op_possibly, true, X)
18.10/2.69	= { by axiom 52 (km5_op_possibly) }
18.10/2.69	  fresh25(true, true, X)
18.10/2.69	= { by axiom 17 (op_possibly) }
18.10/2.69	  possibly(X)
18.10/2.69	
18.10/2.69	Lemma 58: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
18.10/2.69	Proof:
18.10/2.69	  and(implies(X, Y), implies(Y, X))
18.10/2.69	= { by axiom 30 (op_equiv) }
18.10/2.69	  fresh30(op_equiv, true, X, Y)
18.10/2.69	= { by axiom 42 (hilbert_op_equiv) }
18.10/2.69	  fresh30(true, true, X, Y)
18.10/2.69	= { by axiom 14 (op_equiv) }
18.10/2.69	  equiv(X, Y)
18.10/2.69	
18.10/2.69	Lemma 59: fresh40(is_a_theorem(implies(X, Y)), true, X, Y) = fresh116(is_a_theorem(X), true, Y).
18.10/2.69	Proof:
18.10/2.69	  fresh40(is_a_theorem(implies(X, Y)), true, X, Y)
18.10/2.69	= { by axiom 28 (modus_ponens_2) }
18.10/2.69	  fresh115(modus_ponens, true, X, Y)
18.10/2.69	= { by axiom 40 (hilbert_modus_ponens) }
18.10/2.69	  fresh115(true, true, X, Y)
18.10/2.69	= { by axiom 10 (modus_ponens_2) }
18.10/2.69	  fresh116(is_a_theorem(X), true, Y)
18.10/2.69	
18.10/2.69	Lemma 60: is_a_theorem(implies(X, implies(Y, and(X, Y)))) = true.
18.10/2.69	Proof:
18.10/2.69	  is_a_theorem(implies(X, implies(Y, and(X, Y))))
18.10/2.69	= { by axiom 21 (and_3_1) }
18.10/2.69	  fresh103(and_3, true, X, Y)
18.10/2.69	= { by axiom 38 (hilbert_and_3) }
18.10/2.69	  fresh103(true, true, X, Y)
18.10/2.69	= { by axiom 2 (and_3_1) }
18.10/2.69	  true
18.10/2.69	
18.10/2.69	Lemma 61: fresh116(is_a_theorem(Y), true, implies(X, and(Y, X))) = is_a_theorem(implies(X, and(Y, X))).
18.10/2.69	Proof:
18.10/2.69	  fresh116(is_a_theorem(Y), true, implies(X, and(Y, X)))
18.10/2.69	= { by lemma 59 }
18.10/2.69	  fresh40(is_a_theorem(implies(Y, implies(X, and(Y, X)))), true, Y, implies(X, and(Y, X)))
18.10/2.69	= { by lemma 60 }
18.10/2.69	  fresh40(true, true, Y, implies(X, and(Y, X)))
18.10/2.69	= { by axiom 8 (modus_ponens_2) }
18.10/2.69	  is_a_theorem(implies(X, and(Y, X)))
18.10/2.69	
18.10/2.69	Lemma 62: fresh3(is_a_theorem(equiv(X, Y)), true, X, Y) = X.
18.10/2.69	Proof:
18.10/2.69	  fresh3(is_a_theorem(equiv(X, Y)), true, X, Y)
18.10/2.69	= { by axiom 25 (substitution_of_equivalents_2) }
18.10/2.69	  fresh4(substitution_of_equivalents, true, X, Y)
18.10/2.69	= { by axiom 39 (substitution_of_equivalents) }
18.10/2.69	  fresh4(true, true, X, Y)
18.10/2.69	= { by axiom 19 (substitution_of_equivalents_2) }
18.10/2.70	  X
18.10/2.70	
18.10/2.70	Lemma 63: necessarily(possibly(X)) = possibly(X).
18.10/2.70	Proof:
18.10/2.70	  necessarily(possibly(X))
18.10/2.70	= { by axiom 20 (substitution_of_equivalents_2) }
18.10/2.70	  fresh3(true, true, possibly(X), necessarily(possibly(X)))
18.10/2.70	= { by axiom 9 (modus_ponens_2) }
18.10/2.70	  fresh3(fresh116(true, true, equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
18.10/2.70	= { by axiom 4 (axiom_M_1) }
18.10/2.70	  fresh3(fresh116(fresh93(true, true, possibly(X)), true, equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
18.10/2.70	= { by axiom 50 (km5_axiom_M) }
18.10/2.70	  fresh3(fresh116(fresh93(axiom_M, true, possibly(X)), true, equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
18.10/2.70	= { by axiom 43 (axiom_M_1) }
18.10/2.70	  fresh3(fresh116(is_a_theorem(implies(necessarily(possibly(X)), possibly(X))), true, equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
18.10/2.70	= { by lemma 59 }
18.10/2.70	  fresh3(fresh40(is_a_theorem(implies(implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
18.10/2.70	= { by lemma 58 }
18.10/2.70	  fresh3(fresh40(is_a_theorem(implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
18.10/2.70	= { by lemma 61 }
18.10/2.70	  fresh3(fresh40(fresh116(is_a_theorem(implies(possibly(X), necessarily(possibly(X)))), true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
18.10/2.70	= { by axiom 47 (axiom_5_1) }
18.10/2.70	  fresh3(fresh40(fresh116(fresh99(axiom_5, true, X), true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
18.10/2.70	= { by axiom 53 (km5_axiom_5) }
18.10/2.70	  fresh3(fresh40(fresh116(fresh99(true, true, X), true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
18.10/2.70	= { by axiom 3 (axiom_5_1) }
18.10/2.70	  fresh3(fresh40(fresh116(true, true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
18.10/2.70	= { by axiom 9 (modus_ponens_2) }
18.10/2.70	  fresh3(fresh40(true, true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
18.10/2.70	= { by axiom 8 (modus_ponens_2) }
18.10/2.70	  fresh3(is_a_theorem(equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
18.10/2.70	= { by lemma 62 }
18.10/2.70	  possibly(X)
18.10/2.70	
18.10/2.70	Lemma 64: possibly(necessarily(not(X))) = not(possibly(X)).
18.10/2.70	Proof:
18.10/2.70	  possibly(necessarily(not(X)))
18.10/2.70	= { by lemma 57 }
18.10/2.70	  not(necessarily(not(necessarily(not(X)))))
18.10/2.70	= { by lemma 57 }
18.10/2.70	  not(necessarily(possibly(X)))
18.10/2.70	= { by lemma 63 }
18.70/2.72	  not(possibly(X))
18.70/2.72	
18.70/2.72	Lemma 65: implies(not(X), X) = not(not(X)).
18.70/2.72	Proof:
18.70/2.72	  implies(not(X), X)
18.70/2.72	= { by lemma 55 }
18.70/2.72	  not(and(not(X), not(X)))
18.70/2.72	= { by axiom 20 (substitution_of_equivalents_2) }
18.70/2.72	  not(fresh3(true, true, not(X), and(not(X), not(X))))
18.70/2.72	= { by axiom 9 (modus_ponens_2) }
18.70/2.72	  not(fresh3(fresh116(true, true, equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.72	= { by axiom 1 (and_1_1) }
18.70/2.72	  not(fresh3(fresh116(fresh107(true, true, not(X), not(X)), true, equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.72	= { by axiom 37 (hilbert_and_1) }
18.70/2.72	  not(fresh3(fresh116(fresh107(and_1, true, not(X), not(X)), true, equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.72	= { by axiom 27 (and_1_1) }
18.70/2.72	  not(fresh3(fresh116(is_a_theorem(implies(and(not(X), not(X)), not(X))), true, equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.72	= { by lemma 59 }
18.70/2.74	  not(fresh3(fresh40(is_a_theorem(implies(implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.74	= { by lemma 58 }
18.70/2.74	  not(fresh3(fresh40(is_a_theorem(implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.74	= { by lemma 61 }
18.70/2.74	  not(fresh3(fresh40(fresh116(is_a_theorem(implies(not(X), and(not(X), not(X)))), true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.74	= { by axiom 8 (modus_ponens_2) }
18.70/2.74	  not(fresh3(fresh40(fresh116(fresh40(true, true, implies(not(X), implies(not(X), and(not(X), not(X)))), implies(not(X), and(not(X), not(X)))), true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.74	= { by axiom 7 (implies_2_1) }
18.70/2.74	  not(fresh3(fresh40(fresh116(fresh40(fresh49(true, true, not(X), and(not(X), not(X))), true, implies(not(X), implies(not(X), and(not(X), not(X)))), implies(not(X), and(not(X), not(X)))), true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.74	= { by axiom 35 (hilbert_implies_2) }
18.70/2.74	  not(fresh3(fresh40(fresh116(fresh40(fresh49(implies_2, true, not(X), and(not(X), not(X))), true, implies(not(X), implies(not(X), and(not(X), not(X)))), implies(not(X), and(not(X), not(X)))), true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.74	= { by axiom 29 (implies_2_1) }
18.70/2.74	  not(fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(not(X), implies(not(X), and(not(X), not(X)))), implies(not(X), and(not(X), not(X))))), true, implies(not(X), implies(not(X), and(not(X), not(X)))), implies(not(X), and(not(X), not(X)))), true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.74	= { by lemma 59 }
18.70/2.74	  not(fresh3(fresh40(fresh116(fresh116(is_a_theorem(implies(not(X), implies(not(X), and(not(X), not(X))))), true, implies(not(X), and(not(X), not(X)))), true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.74	= { by lemma 60 }
18.70/2.74	  not(fresh3(fresh40(fresh116(fresh116(true, true, implies(not(X), and(not(X), not(X)))), true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.74	= { by axiom 9 (modus_ponens_2) }
18.70/2.74	  not(fresh3(fresh40(fresh116(true, true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.74	= { by axiom 9 (modus_ponens_2) }
18.70/2.74	  not(fresh3(fresh40(true, true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.74	= { by axiom 8 (modus_ponens_2) }
18.70/2.74	  not(fresh3(is_a_theorem(equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X))))
18.70/2.74	= { by lemma 62 }
18.70/2.74	  not(not(X))
18.70/2.74	
18.70/2.74	Lemma 66: is_a_theorem(implies(or(X, X), X)) = fresh59(cn3, true, X).
18.70/2.74	Proof:
18.70/2.74	  is_a_theorem(implies(or(X, X), X))
18.70/2.74	= { by lemma 56 }
18.70/2.74	  is_a_theorem(implies(implies(not(X), X), X))
18.70/2.74	= { by axiom 26 (cn3_1) }
19.14/2.78	  fresh59(cn3, true, X)
19.14/2.78	
19.14/2.78	Goal 1 (s1_0_m6s3m9b_axiom_m9): axiom_m9 = true.
19.14/2.78	Proof:
19.14/2.78	  axiom_m9
19.14/2.78	= { by axiom 46 (axiom_m9) }
19.14/2.78	  fresh74(is_a_theorem(strict_implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.78	= { by axiom 18 (op_strict_implies) }
19.14/2.78	  fresh74(is_a_theorem(fresh23(true, true, possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.78	= { by axiom 54 (s1_0_op_strict_implies) }
19.14/2.78	  fresh74(is_a_theorem(fresh23(op_strict_implies, true, possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.78	= { by axiom 48 (op_strict_implies) }
19.14/2.78	  fresh74(is_a_theorem(necessarily(implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X)))), true)
19.14/2.78	= { by axiom 12 (necessitation_1) }
19.14/2.78	  fresh74(fresh34(true, true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.78	= { by axiom 51 (km5_necessitation) }
19.14/2.78	  fresh74(fresh34(necessitation, true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.78	= { by axiom 44 (necessitation_1) }
19.14/2.78	  fresh74(fresh33(is_a_theorem(implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.78	= { by lemma 63 }
19.14/2.78	  fresh74(fresh33(is_a_theorem(implies(possibly(necessarily(possibly(sK17_axiom_m9_X))), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.78	= { by lemma 57 }
19.14/2.78	  fresh74(fresh33(is_a_theorem(implies(possibly(necessarily(not(necessarily(not(sK17_axiom_m9_X))))), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.78	= { by lemma 64 }
19.14/2.78	  fresh74(fresh33(is_a_theorem(implies(not(possibly(necessarily(not(sK17_axiom_m9_X)))), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.78	= { by lemma 64 }
19.14/2.78	  fresh74(fresh33(is_a_theorem(implies(not(not(possibly(sK17_axiom_m9_X))), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.78	= { by lemma 65 }
19.14/2.78	  fresh74(fresh33(is_a_theorem(implies(implies(not(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.78	= { by lemma 56 }
19.14/2.78	  fresh74(fresh33(is_a_theorem(implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.78	= { by axiom 8 (modus_ponens_2) }
19.14/2.79	  fresh74(fresh33(fresh40(true, true, or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by axiom 11 (modus_tollens_1) }
19.14/2.79	  fresh74(fresh33(fresh40(fresh35(true, true, or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X)), true, or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by axiom 34 (hilbert_modus_tollens) }
19.14/2.79	  fresh74(fresh33(fresh40(fresh35(modus_tollens, true, or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X)), true, or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by axiom 22 (modus_tollens_1) }
19.14/2.79	  fresh74(fresh33(fresh40(is_a_theorem(implies(implies(not(possibly(sK17_axiom_m9_X)), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X)))), true, or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by lemma 56 }
19.14/2.79	  fresh74(fresh33(fresh40(is_a_theorem(implies(or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X)))), true, or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by axiom 28 (modus_ponens_2) }
19.14/2.79	  fresh74(fresh33(fresh115(modus_ponens, true, or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by axiom 40 (hilbert_modus_ponens) }
19.14/2.79	  fresh74(fresh33(fresh115(true, true, or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by axiom 10 (modus_ponens_2) }
19.14/2.79	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X))))), true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by lemma 56 }
19.14/2.79	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(sK17_axiom_m9_X), not(implies(not(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))))), true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by lemma 65 }
19.14/2.79	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(sK17_axiom_m9_X), not(not(not(possibly(sK17_axiom_m9_X)))))), true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by lemma 65 }
19.14/2.79	  fresh74(fresh33(fresh116(is_a_theorem(or(possibly(sK17_axiom_m9_X), implies(not(not(possibly(sK17_axiom_m9_X))), not(possibly(sK17_axiom_m9_X))))), true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by lemma 56 }
19.14/2.79	  fresh74(fresh33(fresh116(is_a_theorem(implies(not(possibly(sK17_axiom_m9_X)), implies(not(not(possibly(sK17_axiom_m9_X))), not(possibly(sK17_axiom_m9_X))))), true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by axiom 24 (implies_1_1) }
19.14/2.79	  fresh74(fresh33(fresh116(fresh51(implies_1, true, not(possibly(sK17_axiom_m9_X)), not(not(possibly(sK17_axiom_m9_X)))), true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by axiom 36 (hilbert_implies_1) }
19.14/2.79	  fresh74(fresh33(fresh116(fresh51(true, true, not(possibly(sK17_axiom_m9_X)), not(not(possibly(sK17_axiom_m9_X)))), true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by axiom 6 (implies_1_1) }
19.14/2.79	  fresh74(fresh33(fresh116(true, true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by axiom 9 (modus_ponens_2) }
19.14/2.79	  fresh74(fresh33(true, true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true)
19.14/2.79	= { by axiom 13 (necessitation_1) }
19.14/2.79	  fresh74(true, true)
19.14/2.79	= { by axiom 5 (axiom_m9) }
19.14/2.79	  true
19.14/2.79	% SZS output end Proof
19.14/2.79	
19.14/2.79	RESULT: Theorem (the conjecture is true).
19.14/2.80	EOF