0.11/0.12	% 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   : n013.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 16:14:48 EDT 2021
0.14/0.35	% CPUTime    : 
49.89/6.67	% SZS status Theorem
49.89/6.67	
49.89/6.75	% SZS output start Proof
49.89/6.75	Take the following subset of the input axioms:
50.51/6.75	  fof(and_1, axiom, ![X, Y]: is_a_theorem(implies(and(X, Y), X)) <=> and_1).
50.51/6.75	  fof(and_3, axiom, and_3 <=> ![X, Y]: is_a_theorem(implies(X, implies(Y, and(X, Y))))).
50.51/6.75	  fof(axiom_4, axiom, axiom_4 <=> ![X]: is_a_theorem(implies(necessarily(X), necessarily(necessarily(X))))).
50.51/6.75	  fof(axiom_5, axiom, ![X]: is_a_theorem(implies(possibly(X), necessarily(possibly(X)))) <=> axiom_5).
50.51/6.75	  fof(axiom_B, axiom, ![X]: is_a_theorem(implies(X, necessarily(possibly(X)))) <=> axiom_B).
50.51/6.75	  fof(axiom_M, axiom, axiom_M <=> ![X]: is_a_theorem(implies(necessarily(X), X))).
50.51/6.75	  fof(hilbert_and_1, axiom, and_1).
50.51/6.75	  fof(hilbert_and_3, axiom, and_3).
50.51/6.75	  fof(hilbert_implies_2, axiom, implies_2).
50.51/6.75	  fof(hilbert_modus_ponens, axiom, modus_ponens).
50.51/6.75	  fof(hilbert_modus_tollens, axiom, modus_tollens).
50.51/6.75	  fof(hilbert_op_equiv, axiom, op_equiv).
50.51/6.75	  fof(hilbert_op_implies_and, axiom, op_implies_and).
50.51/6.75	  fof(hilbert_op_or, axiom, op_or).
50.51/6.75	  fof(hilbert_or_1, axiom, or_1).
50.51/6.75	  fof(implies_2, axiom, ![X, Y]: is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))) <=> implies_2).
50.51/6.75	  fof(km4b_axiom_4, axiom, axiom_4).
50.51/6.75	  fof(km4b_axiom_B, axiom, axiom_B).
50.51/6.75	  fof(km4b_axiom_M, axiom, axiom_M).
50.51/6.75	  fof(km4b_op_possibly, axiom, op_possibly).
50.51/6.75	  fof(km5_axiom_5, conjecture, axiom_5).
50.51/6.75	  fof(modus_ponens, axiom, ![X, Y]: (is_a_theorem(Y) <= (is_a_theorem(implies(X, Y)) & is_a_theorem(X))) <=> modus_ponens).
50.51/6.75	  fof(modus_tollens, axiom, ![X, Y]: is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))) <=> modus_tollens).
50.51/6.75	  fof(op_equiv, axiom, op_equiv => ![X, Y]: and(implies(X, Y), implies(Y, X))=equiv(X, Y)).
50.51/6.75	  fof(op_implies_and, axiom, op_implies_and => ![X, Y]: implies(X, Y)=not(and(X, not(Y)))).
50.51/6.75	  fof(op_or, axiom, op_or => ![X, Y]: or(X, Y)=not(and(not(X), not(Y)))).
50.51/6.75	  fof(op_possibly, axiom, ![X]: possibly(X)=not(necessarily(not(X))) <= op_possibly).
50.51/6.75	  fof(or_1, axiom, or_1 <=> ![X, Y]: is_a_theorem(implies(X, or(X, Y)))).
50.51/6.75	  fof(substitution_of_equivalents, axiom, ![X, Y]: (is_a_theorem(equiv(X, Y)) => X=Y) <=> substitution_of_equivalents).
50.51/6.75	  fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
50.51/6.75	
50.51/6.75	Now clausify the problem and encode Horn clauses using encoding 3 of
50.51/6.75	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
50.51/6.75	We repeatedly replace C & s=t => u=v by the two clauses:
50.51/6.75	  fresh(y, y, x1...xn) = u
50.51/6.75	  C => fresh(s, t, x1...xn) = v
50.51/6.75	where fresh is a fresh function symbol and x1..xn are the free
50.51/6.75	variables of u and v.
50.51/6.75	A predicate p(X) is encoded as p(X)=true (this is sound, because the
50.51/6.75	input problem has no model of domain size 1).
50.51/6.75	
50.51/6.75	The encoding turns the above axioms into the following unit equations and goals:
50.51/6.75	
50.51/6.75	Axiom 1 (hilbert_and_1): and_1 = true.
50.51/6.75	Axiom 2 (hilbert_implies_2): implies_2 = true.
50.51/6.75	Axiom 3 (hilbert_modus_ponens): modus_ponens = true.
50.51/6.75	Axiom 4 (hilbert_and_3): and_3 = true.
50.51/6.75	Axiom 5 (hilbert_or_1): or_1 = true.
50.51/6.75	Axiom 6 (hilbert_modus_tollens): modus_tollens = true.
50.51/6.75	Axiom 7 (substitution_of_equivalents): substitution_of_equivalents = true.
50.51/6.75	Axiom 8 (km4b_axiom_B): axiom_B = true.
50.51/6.75	Axiom 9 (km4b_axiom_4): axiom_4 = true.
50.51/6.75	Axiom 10 (km4b_axiom_M): axiom_M = true.
50.51/6.75	Axiom 11 (hilbert_op_implies_and): op_implies_and = true.
50.51/6.75	Axiom 12 (hilbert_op_or): op_or = true.
50.51/6.75	Axiom 13 (hilbert_op_equiv): op_equiv = true.
50.51/6.75	Axiom 14 (km4b_op_possibly): op_possibly = true.
50.51/6.75	Axiom 15 (axiom_5): fresh100(X, X) = true.
50.51/6.75	Axiom 16 (modus_ponens_2): fresh116(X, X, Y) = true.
50.51/6.75	Axiom 17 (axiom_4_1): fresh101(X, X, Y) = true.
50.51/6.75	Axiom 18 (axiom_B_1): fresh97(X, X, Y) = true.
50.51/6.75	Axiom 19 (axiom_M_1): fresh93(X, X, Y) = true.
50.51/6.75	Axiom 20 (op_possibly): fresh25(X, X, Y) = possibly(Y).
50.51/6.75	Axiom 21 (axiom_M_1): fresh93(axiom_M, true, X) = is_a_theorem(implies(necessarily(X), X)).
50.51/6.75	Axiom 22 (op_possibly): fresh25(op_possibly, true, X) = not(necessarily(not(X))).
50.51/6.75	Axiom 23 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(is_a_theorem(Y), true, Z).
50.51/6.75	Axiom 24 (and_1_1): fresh107(X, X, Y, Z) = true.
50.51/6.75	Axiom 25 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
50.51/6.75	Axiom 26 (and_3_1): fresh103(X, X, Y, Z) = true.
50.51/6.75	Axiom 27 (implies_2_1): fresh49(X, X, Y, Z) = true.
50.51/6.75	Axiom 28 (modus_ponens_2): fresh40(X, X, Y, Z) = is_a_theorem(Z).
50.51/6.75	Axiom 29 (modus_tollens_1): fresh35(X, X, Y, Z) = true.
50.51/6.75	Axiom 30 (op_equiv): fresh30(X, X, Y, Z) = equiv(Y, Z).
50.51/6.75	Axiom 31 (op_equiv): fresh30(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
50.51/6.75	Axiom 32 (op_implies_and): fresh29(X, X, Y, Z) = implies(Y, Z).
50.51/6.75	Axiom 33 (op_implies_and): fresh29(op_implies_and, true, X, Y) = not(and(X, not(Y))).
50.51/6.75	Axiom 34 (op_or): fresh26(X, X, Y, Z) = or(Y, Z).
50.51/6.75	Axiom 35 (or_1_1): fresh21(X, X, Y, Z) = true.
50.51/6.75	Axiom 36 (or_1_1): fresh21(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
50.51/6.75	Axiom 37 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y.
50.51/6.75	Axiom 38 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z.
50.51/6.75	Axiom 39 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
50.51/6.75	Axiom 40 (axiom_B_1): fresh97(axiom_B, true, X) = is_a_theorem(implies(X, necessarily(possibly(X)))).
50.51/6.75	Axiom 41 (op_or): fresh26(op_or, true, X, Y) = not(and(not(X), not(Y))).
50.51/6.75	Axiom 42 (axiom_4_1): fresh101(axiom_4, true, X) = is_a_theorem(implies(necessarily(X), necessarily(necessarily(X)))).
50.51/6.75	Axiom 43 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
50.51/6.75	Axiom 44 (modus_ponens_2): fresh115(modus_ponens, true, X, Y) = fresh40(is_a_theorem(implies(X, Y)), true, X, Y).
50.51/6.75	Axiom 45 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y).
50.51/6.75	Axiom 46 (modus_tollens_1): fresh35(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
50.51/6.75	Axiom 47 (axiom_5): fresh100(is_a_theorem(implies(possibly(x16), necessarily(possibly(x16)))), true) = axiom_5.
50.51/6.75	
50.51/6.75	Lemma 48: fresh40(is_a_theorem(implies(X, Y)), true, X, Y) = fresh116(is_a_theorem(X), true, Y).
50.51/6.75	Proof:
50.51/6.75	  fresh40(is_a_theorem(implies(X, Y)), true, X, Y)
50.51/6.75	= { by axiom 44 (modus_ponens_2) R->L }
50.51/6.75	  fresh115(modus_ponens, true, X, Y)
50.51/6.75	= { by axiom 3 (hilbert_modus_ponens) }
50.51/6.75	  fresh115(true, true, X, Y)
50.51/6.75	= { by axiom 23 (modus_ponens_2) }
50.51/6.75	  fresh116(is_a_theorem(X), true, Y)
50.51/6.75	
50.51/6.75	Lemma 49: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
50.51/6.75	Proof:
50.51/6.75	  and(implies(X, Y), implies(Y, X))
50.51/6.75	= { by axiom 31 (op_equiv) R->L }
50.51/6.75	  fresh30(op_equiv, true, X, Y)
50.51/6.75	= { by axiom 13 (hilbert_op_equiv) }
50.51/6.75	  fresh30(true, true, X, Y)
50.51/6.75	= { by axiom 30 (op_equiv) }
50.51/6.75	  equiv(X, Y)
50.51/6.75	
50.51/6.75	Lemma 50: is_a_theorem(implies(X, implies(Y, and(X, Y)))) = true.
50.51/6.75	Proof:
50.51/6.75	  is_a_theorem(implies(X, implies(Y, and(X, Y))))
50.51/6.75	= { by axiom 39 (and_3_1) R->L }
50.51/6.75	  fresh103(and_3, true, X, Y)
50.51/6.75	= { by axiom 4 (hilbert_and_3) }
50.51/6.75	  fresh103(true, true, X, Y)
50.51/6.75	= { by axiom 26 (and_3_1) }
50.51/6.75	  true
50.51/6.75	
50.51/6.75	Lemma 51: fresh116(is_a_theorem(X), true, implies(Y, and(X, Y))) = is_a_theorem(implies(Y, and(X, Y))).
50.51/6.75	Proof:
50.51/6.75	  fresh116(is_a_theorem(X), true, implies(Y, and(X, Y)))
50.51/6.75	= { by lemma 48 R->L }
50.51/6.75	  fresh40(is_a_theorem(implies(X, implies(Y, and(X, Y)))), true, X, implies(Y, and(X, Y)))
50.51/6.75	= { by lemma 50 }
50.51/6.75	  fresh40(true, true, X, implies(Y, and(X, Y)))
50.51/6.75	= { by axiom 28 (modus_ponens_2) }
50.51/6.75	  is_a_theorem(implies(Y, and(X, Y)))
50.51/6.75	
50.51/6.75	Lemma 52: fresh3(is_a_theorem(equiv(X, Y)), true, X, Y) = X.
50.51/6.75	Proof:
50.51/6.75	  fresh3(is_a_theorem(equiv(X, Y)), true, X, Y)
50.51/6.75	= { by axiom 45 (substitution_of_equivalents_2) R->L }
50.51/6.75	  fresh4(substitution_of_equivalents, true, X, Y)
50.51/6.75	= { by axiom 7 (substitution_of_equivalents) }
50.51/6.75	  fresh4(true, true, X, Y)
50.51/6.75	= { by axiom 37 (substitution_of_equivalents_2) }
50.51/6.75	  X
50.51/6.75	
50.51/6.75	Lemma 53: and(X, X) = X.
50.51/6.75	Proof:
50.51/6.75	  and(X, X)
50.51/6.75	= { by axiom 38 (substitution_of_equivalents_2) R->L }
50.51/6.75	  fresh3(true, true, X, and(X, X))
50.51/6.75	= { by axiom 16 (modus_ponens_2) R->L }
50.51/6.75	  fresh3(fresh116(true, true, equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by axiom 24 (and_1_1) R->L }
50.51/6.76	  fresh3(fresh116(fresh107(true, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by axiom 1 (hilbert_and_1) R->L }
50.51/6.76	  fresh3(fresh116(fresh107(and_1, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by axiom 25 (and_1_1) }
50.51/6.76	  fresh3(fresh116(is_a_theorem(implies(and(X, X), X)), true, equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by lemma 48 R->L }
50.51/6.76	  fresh3(fresh40(is_a_theorem(implies(implies(and(X, X), X), equiv(X, and(X, X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by lemma 49 R->L }
50.51/6.76	  fresh3(fresh40(is_a_theorem(implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by lemma 51 R->L }
50.51/6.76	  fresh3(fresh40(fresh116(is_a_theorem(implies(X, and(X, X))), true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by axiom 28 (modus_ponens_2) R->L }
50.51/6.76	  fresh3(fresh40(fresh116(fresh40(true, true, implies(X, implies(X, and(X, X))), implies(X, and(X, X))), true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by axiom 27 (implies_2_1) R->L }
50.51/6.76	  fresh3(fresh40(fresh116(fresh40(fresh49(true, true, X, and(X, X)), true, implies(X, implies(X, and(X, X))), implies(X, and(X, X))), true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by axiom 2 (hilbert_implies_2) R->L }
50.51/6.76	  fresh3(fresh40(fresh116(fresh40(fresh49(implies_2, true, X, and(X, X)), true, implies(X, implies(X, and(X, X))), implies(X, and(X, X))), true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by axiom 43 (implies_2_1) }
50.51/6.76	  fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(X, implies(X, and(X, X))), implies(X, and(X, X)))), true, implies(X, implies(X, and(X, X))), implies(X, and(X, X))), true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by lemma 48 }
50.51/6.76	  fresh3(fresh40(fresh116(fresh116(is_a_theorem(implies(X, implies(X, and(X, X)))), true, implies(X, and(X, X))), true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by lemma 50 }
50.51/6.76	  fresh3(fresh40(fresh116(fresh116(true, true, implies(X, and(X, X))), true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by axiom 16 (modus_ponens_2) }
50.51/6.76	  fresh3(fresh40(fresh116(true, true, implies(implies(and(X, X), X), and(implies(X, and(X, X)), implies(and(X, X), X)))), true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by axiom 16 (modus_ponens_2) }
50.51/6.76	  fresh3(fresh40(true, true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by axiom 28 (modus_ponens_2) }
50.51/6.76	  fresh3(is_a_theorem(equiv(X, and(X, X))), true, X, and(X, X))
50.51/6.76	= { by lemma 52 }
50.51/6.76	  X
50.51/6.76	
50.51/6.76	Lemma 54: not(and(X, not(Y))) = implies(X, Y).
50.51/6.76	Proof:
50.51/6.76	  not(and(X, not(Y)))
50.51/6.76	= { by axiom 33 (op_implies_and) R->L }
50.51/6.76	  fresh29(op_implies_and, true, X, Y)
50.51/6.76	= { by axiom 11 (hilbert_op_implies_and) }
50.51/6.76	  fresh29(true, true, X, Y)
50.51/6.76	= { by axiom 32 (op_implies_and) }
50.51/6.76	  implies(X, Y)
50.51/6.76	
50.51/6.76	Lemma 55: implies(not(X), Y) = or(X, Y).
50.51/6.76	Proof:
50.51/6.76	  implies(not(X), Y)
50.51/6.76	= { by lemma 54 R->L }
50.51/6.76	  not(and(not(X), not(Y)))
50.51/6.76	= { by axiom 41 (op_or) R->L }
50.51/6.76	  fresh26(op_or, true, X, Y)
50.51/6.76	= { by axiom 12 (hilbert_op_or) }
50.51/6.76	  fresh26(true, true, X, Y)
50.51/6.76	= { by axiom 34 (op_or) }
50.51/6.76	  or(X, Y)
50.51/6.76	
50.51/6.76	Lemma 56: not(not(X)) = or(X, X).
50.51/6.76	Proof:
50.51/6.76	  not(not(X))
50.51/6.76	= { by lemma 53 R->L }
50.51/6.76	  not(and(not(X), not(X)))
50.51/6.76	= { by lemma 54 }
50.51/6.76	  implies(not(X), X)
50.51/6.76	= { by lemma 55 }
50.51/6.76	  or(X, X)
50.51/6.76	
50.51/6.76	Lemma 57: not(necessarily(not(X))) = possibly(X).
50.51/6.76	Proof:
50.51/6.76	  not(necessarily(not(X)))
50.51/6.76	= { by axiom 22 (op_possibly) R->L }
50.51/6.76	  fresh25(op_possibly, true, X)
50.51/6.76	= { by axiom 14 (km4b_op_possibly) }
50.51/6.76	  fresh25(true, true, X)
50.51/6.76	= { by axiom 20 (op_possibly) }
50.51/6.76	  possibly(X)
50.51/6.76	
50.51/6.76	Lemma 58: is_a_theorem(implies(X, or(X, Y))) = true.
50.51/6.76	Proof:
50.51/6.76	  is_a_theorem(implies(X, or(X, Y)))
50.51/6.76	= { by axiom 36 (or_1_1) R->L }
50.51/6.76	  fresh21(or_1, true, X, Y)
50.51/6.76	= { by axiom 5 (hilbert_or_1) }
50.51/6.76	  fresh21(true, true, X, Y)
50.51/6.76	= { by axiom 35 (or_1_1) }
50.51/6.76	  true
50.51/6.76	
50.51/6.76	Goal 1 (km5_axiom_5): axiom_5 = true.
50.51/6.76	Proof:
50.51/6.76	  axiom_5
50.51/6.76	= { by axiom 47 (axiom_5) R->L }
50.51/6.76	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(possibly(x16)))), true)
50.51/6.76	= { by lemma 57 R->L }
50.51/6.76	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(not(x16)))))), true)
50.51/6.76	= { by lemma 52 R->L }
50.51/6.77	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(fresh3(is_a_theorem(equiv(necessarily(not(x16)), necessarily(necessarily(not(x16))))), true, necessarily(not(x16)), necessarily(necessarily(not(x16)))))))), true)
50.51/6.77	= { by axiom 28 (modus_ponens_2) R->L }
50.51/6.77	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(fresh3(fresh40(true, true, implies(necessarily(necessarily(not(x16))), necessarily(not(x16))), equiv(necessarily(not(x16)), necessarily(necessarily(not(x16))))), true, necessarily(not(x16)), necessarily(necessarily(not(x16)))))))), true)
50.51/6.77	= { by axiom 16 (modus_ponens_2) R->L }
50.51/6.77	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(fresh3(fresh40(fresh116(true, true, implies(implies(necessarily(necessarily(not(x16))), necessarily(not(x16))), and(implies(necessarily(not(x16)), necessarily(necessarily(not(x16)))), implies(necessarily(necessarily(not(x16))), necessarily(not(x16)))))), true, implies(necessarily(necessarily(not(x16))), necessarily(not(x16))), equiv(necessarily(not(x16)), necessarily(necessarily(not(x16))))), true, necessarily(not(x16)), necessarily(necessarily(not(x16)))))))), true)
50.51/6.77	= { by axiom 17 (axiom_4_1) R->L }
50.51/6.77	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(fresh3(fresh40(fresh116(fresh101(true, true, not(x16)), true, implies(implies(necessarily(necessarily(not(x16))), necessarily(not(x16))), and(implies(necessarily(not(x16)), necessarily(necessarily(not(x16)))), implies(necessarily(necessarily(not(x16))), necessarily(not(x16)))))), true, implies(necessarily(necessarily(not(x16))), necessarily(not(x16))), equiv(necessarily(not(x16)), necessarily(necessarily(not(x16))))), true, necessarily(not(x16)), necessarily(necessarily(not(x16)))))))), true)
50.51/6.77	= { by axiom 9 (km4b_axiom_4) R->L }
50.51/6.77	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(fresh3(fresh40(fresh116(fresh101(axiom_4, true, not(x16)), true, implies(implies(necessarily(necessarily(not(x16))), necessarily(not(x16))), and(implies(necessarily(not(x16)), necessarily(necessarily(not(x16)))), implies(necessarily(necessarily(not(x16))), necessarily(not(x16)))))), true, implies(necessarily(necessarily(not(x16))), necessarily(not(x16))), equiv(necessarily(not(x16)), necessarily(necessarily(not(x16))))), true, necessarily(not(x16)), necessarily(necessarily(not(x16)))))))), true)
50.51/6.77	= { by axiom 42 (axiom_4_1) }
50.51/6.77	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(fresh3(fresh40(fresh116(is_a_theorem(implies(necessarily(not(x16)), necessarily(necessarily(not(x16))))), true, implies(implies(necessarily(necessarily(not(x16))), necessarily(not(x16))), and(implies(necessarily(not(x16)), necessarily(necessarily(not(x16)))), implies(necessarily(necessarily(not(x16))), necessarily(not(x16)))))), true, implies(necessarily(necessarily(not(x16))), necessarily(not(x16))), equiv(necessarily(not(x16)), necessarily(necessarily(not(x16))))), true, necessarily(not(x16)), necessarily(necessarily(not(x16)))))))), true)
50.51/6.77	= { by lemma 51 }
50.51/6.77	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(fresh3(fresh40(is_a_theorem(implies(implies(necessarily(necessarily(not(x16))), necessarily(not(x16))), and(implies(necessarily(not(x16)), necessarily(necessarily(not(x16)))), implies(necessarily(necessarily(not(x16))), necessarily(not(x16)))))), true, implies(necessarily(necessarily(not(x16))), necessarily(not(x16))), equiv(necessarily(not(x16)), necessarily(necessarily(not(x16))))), true, necessarily(not(x16)), necessarily(necessarily(not(x16)))))))), true)
50.51/6.77	= { by lemma 49 }
50.73/6.78	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(fresh3(fresh40(is_a_theorem(implies(implies(necessarily(necessarily(not(x16))), necessarily(not(x16))), equiv(necessarily(not(x16)), necessarily(necessarily(not(x16)))))), true, implies(necessarily(necessarily(not(x16))), necessarily(not(x16))), equiv(necessarily(not(x16)), necessarily(necessarily(not(x16))))), true, necessarily(not(x16)), necessarily(necessarily(not(x16)))))))), true)
50.73/6.78	= { by lemma 48 }
50.73/6.78	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(fresh3(fresh116(is_a_theorem(implies(necessarily(necessarily(not(x16))), necessarily(not(x16)))), true, equiv(necessarily(not(x16)), necessarily(necessarily(not(x16))))), true, necessarily(not(x16)), necessarily(necessarily(not(x16)))))))), true)
50.73/6.78	= { by axiom 21 (axiom_M_1) R->L }
50.73/6.78	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(fresh3(fresh116(fresh93(axiom_M, true, necessarily(not(x16))), true, equiv(necessarily(not(x16)), necessarily(necessarily(not(x16))))), true, necessarily(not(x16)), necessarily(necessarily(not(x16)))))))), true)
50.73/6.78	= { by axiom 10 (km4b_axiom_M) }
50.73/6.78	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(fresh3(fresh116(fresh93(true, true, necessarily(not(x16))), true, equiv(necessarily(not(x16)), necessarily(necessarily(not(x16))))), true, necessarily(not(x16)), necessarily(necessarily(not(x16)))))))), true)
50.73/6.78	= { by axiom 19 (axiom_M_1) }
50.73/6.78	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(fresh3(fresh116(true, true, equiv(necessarily(not(x16)), necessarily(necessarily(not(x16))))), true, necessarily(not(x16)), necessarily(necessarily(not(x16)))))))), true)
50.73/6.78	= { by axiom 16 (modus_ponens_2) }
50.73/6.78	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(fresh3(true, true, necessarily(not(x16)), necessarily(necessarily(not(x16)))))))), true)
50.73/6.78	= { by axiom 38 (substitution_of_equivalents_2) }
50.73/6.78	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(necessarily(not(x16))))))), true)
50.73/6.78	= { by axiom 38 (substitution_of_equivalents_2) R->L }
50.73/6.78	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(true, true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.78	= { by axiom 16 (modus_ponens_2) R->L }
50.73/6.78	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh116(true, true, equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.78	= { by lemma 58 R->L }
50.73/6.78	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh116(is_a_theorem(implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16))))), true, equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.78	= { by lemma 48 R->L }
50.73/6.78	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh40(is_a_theorem(implies(implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))))), true, implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.78	= { by lemma 49 R->L }
50.73/6.78	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh40(is_a_theorem(implies(implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), and(implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))), implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16))))))), true, implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.78	= { by lemma 51 R->L }
50.73/6.78	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh40(fresh116(is_a_theorem(implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, implies(implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), and(implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))), implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16))))))), true, implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.78	= { by axiom 28 (modus_ponens_2) R->L }
50.73/6.78	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh40(fresh116(fresh40(true, true, or(necessarily(not(x16)), not(or(necessarily(not(x16)), necessarily(not(x16))))), implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, implies(implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), and(implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))), implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16))))))), true, implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.78	= { by axiom 29 (modus_tollens_1) R->L }
50.73/6.79	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh40(fresh116(fresh40(fresh35(true, true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))), true, or(necessarily(not(x16)), not(or(necessarily(not(x16)), necessarily(not(x16))))), implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, implies(implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), and(implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))), implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16))))))), true, implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.79	= { by axiom 6 (hilbert_modus_tollens) R->L }
50.73/6.79	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh40(fresh116(fresh40(fresh35(modus_tollens, true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))), true, or(necessarily(not(x16)), not(or(necessarily(not(x16)), necessarily(not(x16))))), implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, implies(implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), and(implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))), implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16))))))), true, implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.79	= { by axiom 46 (modus_tollens_1) }
50.73/6.79	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(not(necessarily(not(x16))), not(or(necessarily(not(x16)), necessarily(not(x16))))), implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))))), true, or(necessarily(not(x16)), not(or(necessarily(not(x16)), necessarily(not(x16))))), implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, implies(implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), and(implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))), implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16))))))), true, implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.79	= { by lemma 55 }
50.73/6.79	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(or(necessarily(not(x16)), not(or(necessarily(not(x16)), necessarily(not(x16))))), implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))))), true, or(necessarily(not(x16)), not(or(necessarily(not(x16)), necessarily(not(x16))))), implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, implies(implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), and(implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))), implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16))))))), true, implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.79	= { by lemma 48 }
50.73/6.79	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh40(fresh116(fresh116(is_a_theorem(or(necessarily(not(x16)), not(or(necessarily(not(x16)), necessarily(not(x16)))))), true, implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, implies(implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), and(implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))), implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16))))))), true, implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.79	= { by lemma 56 R->L }
50.73/6.79	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh40(fresh116(fresh116(is_a_theorem(or(necessarily(not(x16)), not(not(not(necessarily(not(x16))))))), true, implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, implies(implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), and(implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))), implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16))))))), true, implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.79	= { by lemma 56 }
50.73/6.79	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh40(fresh116(fresh116(is_a_theorem(or(necessarily(not(x16)), or(not(necessarily(not(x16))), not(necessarily(not(x16)))))), true, implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, implies(implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), and(implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))), implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16))))))), true, implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.79	= { by lemma 55 R->L }
50.73/6.79	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh40(fresh116(fresh116(is_a_theorem(implies(not(necessarily(not(x16))), or(not(necessarily(not(x16))), not(necessarily(not(x16)))))), true, implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, implies(implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), and(implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))), implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16))))))), true, implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.79	= { by lemma 58 }
50.73/6.79	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh40(fresh116(fresh116(true, true, implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, implies(implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), and(implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))), implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16))))))), true, implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.79	= { by axiom 16 (modus_ponens_2) }
50.73/6.80	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh40(fresh116(true, true, implies(implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), and(implies(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16))), implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16))))))), true, implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.80	= { by axiom 16 (modus_ponens_2) }
50.73/6.80	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(fresh40(true, true, implies(necessarily(not(x16)), or(necessarily(not(x16)), necessarily(not(x16)))), equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.80	= { by axiom 28 (modus_ponens_2) }
50.73/6.80	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(fresh3(is_a_theorem(equiv(or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))), true, or(necessarily(not(x16)), necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.80	= { by lemma 52 }
50.73/6.80	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(or(necessarily(not(x16)), necessarily(not(x16)))))))), true)
50.73/6.80	= { by lemma 55 R->L }
50.73/6.80	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(implies(not(necessarily(not(x16))), necessarily(not(x16)))))))), true)
50.73/6.80	= { by lemma 54 R->L }
50.73/6.80	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(not(necessarily(not(and(not(necessarily(not(x16))), not(necessarily(not(x16)))))))))), true)
50.73/6.80	= { by lemma 57 }
50.73/6.80	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(possibly(and(not(necessarily(not(x16))), not(necessarily(not(x16)))))))), true)
50.73/6.80	= { by lemma 57 }
50.73/6.80	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(possibly(and(not(necessarily(not(x16))), possibly(x16)))))), true)
50.73/6.80	= { by lemma 57 }
50.73/6.80	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(possibly(and(possibly(x16), possibly(x16)))))), true)
50.73/6.80	= { by lemma 53 }
50.73/6.80	  fresh100(is_a_theorem(implies(possibly(x16), necessarily(possibly(possibly(x16))))), true)
50.73/6.80	= { by axiom 40 (axiom_B_1) R->L }
50.73/6.80	  fresh100(fresh97(axiom_B, true, possibly(x16)), true)
50.73/6.80	= { by axiom 8 (km4b_axiom_B) }
50.73/6.80	  fresh100(fresh97(true, true, possibly(x16)), true)
50.73/6.80	= { by axiom 18 (axiom_B_1) }
50.73/6.80	  fresh100(true, true)
50.73/6.80	= { by axiom 15 (axiom_5) }
50.73/6.80	  true
50.73/6.80	% SZS output end Proof
50.73/6.80	
50.73/6.80	RESULT: Theorem (the conjecture is true).
50.73/6.81	EOF