0.07/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.07/0.14	% 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 16:12:40 EDT 2021
0.14/0.36	% CPUTime    : 
14.76/2.51	% SZS status Theorem
14.76/2.51	
14.76/2.54	% SZS output start Proof
14.76/2.54	Take the following subset of the input axioms:
14.76/2.54	  fof(and_1, axiom, ![X, Y]: is_a_theorem(implies(and(X, Y), X)) <=> and_1).
14.76/2.54	  fof(and_3, axiom, and_3 <=> ![X, Y]: is_a_theorem(implies(X, implies(Y, and(X, Y))))).
14.76/2.54	  fof(axiom_5, axiom, ![X]: is_a_theorem(implies(possibly(X), necessarily(possibly(X)))) <=> axiom_5).
14.76/2.54	  fof(axiom_B, axiom, ![X]: is_a_theorem(implies(X, necessarily(possibly(X)))) <=> axiom_B).
14.76/2.54	  fof(axiom_M, axiom, axiom_M <=> ![X]: is_a_theorem(implies(necessarily(X), X))).
14.76/2.54	  fof(hilbert_and_1, axiom, and_1).
14.76/2.54	  fof(hilbert_and_3, axiom, and_3).
14.76/2.54	  fof(hilbert_implies_2, axiom, implies_2).
14.76/2.54	  fof(hilbert_modus_ponens, axiom, modus_ponens).
14.76/2.54	  fof(hilbert_modus_tollens, axiom, modus_tollens).
14.76/2.54	  fof(hilbert_op_equiv, axiom, op_equiv).
14.76/2.54	  fof(hilbert_op_implies_and, axiom, op_implies_and).
14.76/2.54	  fof(hilbert_op_or, axiom, op_or).
14.76/2.54	  fof(hilbert_or_1, axiom, or_1).
14.76/2.54	  fof(implies_2, axiom, ![X, Y]: is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))) <=> implies_2).
14.76/2.54	  fof(km4b_axiom_B, conjecture, axiom_B).
14.76/2.54	  fof(km5_axiom_5, axiom, axiom_5).
14.76/2.54	  fof(km5_axiom_M, axiom, axiom_M).
14.76/2.54	  fof(km5_op_possibly, axiom, op_possibly).
14.76/2.54	  fof(modus_ponens, axiom, ![X, Y]: (is_a_theorem(Y) <= (is_a_theorem(implies(X, Y)) & is_a_theorem(X))) <=> modus_ponens).
14.76/2.54	  fof(modus_tollens, axiom, ![X, Y]: is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))) <=> modus_tollens).
14.76/2.54	  fof(op_equiv, axiom, op_equiv => ![X, Y]: and(implies(X, Y), implies(Y, X))=equiv(X, Y)).
14.76/2.54	  fof(op_implies_and, axiom, op_implies_and => ![X, Y]: implies(X, Y)=not(and(X, not(Y)))).
14.76/2.54	  fof(op_or, axiom, op_or => ![X, Y]: or(X, Y)=not(and(not(X), not(Y)))).
14.76/2.54	  fof(op_possibly, axiom, ![X]: possibly(X)=not(necessarily(not(X))) <= op_possibly).
14.76/2.54	  fof(or_1, axiom, or_1 <=> ![X, Y]: is_a_theorem(implies(X, or(X, Y)))).
14.76/2.54	  fof(substitution_of_equivalents, axiom, ![X, Y]: (is_a_theorem(equiv(X, Y)) => X=Y) <=> substitution_of_equivalents).
14.76/2.54	  fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
14.76/2.54	
14.76/2.54	Now clausify the problem and encode Horn clauses using encoding 3 of
14.76/2.54	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
14.76/2.54	We repeatedly replace C & s=t => u=v by the two clauses:
14.76/2.54	  fresh(y, y, x1...xn) = u
14.76/2.54	  C => fresh(s, t, x1...xn) = v
14.76/2.54	where fresh is a fresh function symbol and x1..xn are the free
14.76/2.54	variables of u and v.
14.76/2.54	A predicate p(X) is encoded as p(X)=true (this is sound, because the
14.76/2.54	input problem has no model of domain size 1).
14.76/2.54	
14.76/2.54	The encoding turns the above axioms into the following unit equations and goals:
14.76/2.54	
14.76/2.54	Axiom 1 (hilbert_and_1): and_1 = true.
14.76/2.54	Axiom 2 (hilbert_implies_2): implies_2 = true.
14.76/2.54	Axiom 3 (hilbert_modus_ponens): modus_ponens = true.
14.76/2.54	Axiom 4 (hilbert_and_3): and_3 = true.
14.76/2.54	Axiom 5 (hilbert_or_1): or_1 = true.
14.76/2.54	Axiom 6 (hilbert_modus_tollens): modus_tollens = true.
14.76/2.54	Axiom 7 (substitution_of_equivalents): substitution_of_equivalents = true.
14.76/2.54	Axiom 8 (km5_axiom_5): axiom_5 = true.
14.76/2.54	Axiom 9 (km5_axiom_M): axiom_M = true.
14.76/2.54	Axiom 10 (hilbert_op_implies_and): op_implies_and = true.
14.76/2.54	Axiom 11 (hilbert_op_or): op_or = true.
14.76/2.54	Axiom 12 (hilbert_op_equiv): op_equiv = true.
14.76/2.54	Axiom 13 (km5_op_possibly): op_possibly = true.
14.76/2.54	Axiom 14 (axiom_B): fresh98(X, X) = true.
14.76/2.54	Axiom 15 (modus_ponens_2): fresh116(X, X, Y) = true.
14.76/2.54	Axiom 16 (axiom_5_1): fresh99(X, X, Y) = true.
14.76/2.54	Axiom 17 (axiom_M_1): fresh93(X, X, Y) = true.
14.76/2.54	Axiom 18 (op_possibly): fresh25(X, X, Y) = possibly(Y).
14.76/2.54	Axiom 19 (axiom_M_1): fresh93(axiom_M, true, X) = is_a_theorem(implies(necessarily(X), X)).
14.76/2.54	Axiom 20 (op_possibly): fresh25(op_possibly, true, X) = not(necessarily(not(X))).
14.76/2.54	Axiom 21 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(is_a_theorem(Y), true, Z).
14.76/2.54	Axiom 22 (and_1_1): fresh107(X, X, Y, Z) = true.
14.76/2.54	Axiom 23 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
14.76/2.54	Axiom 24 (and_3_1): fresh103(X, X, Y, Z) = true.
14.76/2.54	Axiom 25 (implies_2_1): fresh49(X, X, Y, Z) = true.
14.76/2.54	Axiom 26 (modus_ponens_2): fresh40(X, X, Y, Z) = is_a_theorem(Z).
14.76/2.54	Axiom 27 (modus_tollens_1): fresh35(X, X, Y, Z) = true.
14.76/2.54	Axiom 28 (op_equiv): fresh30(X, X, Y, Z) = equiv(Y, Z).
14.76/2.54	Axiom 29 (op_equiv): fresh30(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
14.76/2.54	Axiom 30 (op_implies_and): fresh29(X, X, Y, Z) = implies(Y, Z).
14.76/2.54	Axiom 31 (op_implies_and): fresh29(op_implies_and, true, X, Y) = not(and(X, not(Y))).
14.76/2.54	Axiom 32 (op_or): fresh26(X, X, Y, Z) = or(Y, Z).
14.76/2.54	Axiom 33 (or_1_1): fresh21(X, X, Y, Z) = true.
14.76/2.54	Axiom 34 (or_1_1): fresh21(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
14.76/2.54	Axiom 35 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y.
14.76/2.54	Axiom 36 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z.
14.76/2.54	Axiom 37 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
14.76/2.54	Axiom 38 (op_or): fresh26(op_or, true, X, Y) = not(and(not(X), not(Y))).
14.76/2.54	Axiom 39 (axiom_5_1): fresh99(axiom_5, true, X) = is_a_theorem(implies(possibly(X), necessarily(possibly(X)))).
14.76/2.54	Axiom 40 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
14.76/2.54	Axiom 41 (axiom_B): fresh98(is_a_theorem(implies(x18, necessarily(possibly(x18)))), true) = axiom_B.
14.76/2.54	Axiom 42 (modus_ponens_2): fresh115(modus_ponens, true, X, Y) = fresh40(is_a_theorem(implies(X, Y)), true, X, Y).
14.76/2.54	Axiom 43 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y).
14.76/2.54	Axiom 44 (modus_tollens_1): fresh35(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
14.76/2.54	
14.76/2.54	Lemma 45: fresh40(is_a_theorem(implies(X, Y)), true, X, Y) = fresh116(is_a_theorem(X), true, Y).
14.76/2.54	Proof:
14.76/2.54	  fresh40(is_a_theorem(implies(X, Y)), true, X, Y)
14.76/2.54	= { by axiom 42 (modus_ponens_2) R->L }
14.76/2.54	  fresh115(modus_ponens, true, X, Y)
14.76/2.54	= { by axiom 3 (hilbert_modus_ponens) }
14.76/2.54	  fresh115(true, true, X, Y)
14.76/2.54	= { by axiom 21 (modus_ponens_2) }
14.76/2.54	  fresh116(is_a_theorem(X), true, Y)
14.76/2.54	
14.76/2.54	Lemma 46: and(implies(X, Y), implies(Y, X)) = equiv(X, Y).
14.76/2.54	Proof:
14.76/2.54	  and(implies(X, Y), implies(Y, X))
14.76/2.54	= { by axiom 29 (op_equiv) R->L }
14.76/2.54	  fresh30(op_equiv, true, X, Y)
14.76/2.54	= { by axiom 12 (hilbert_op_equiv) }
14.76/2.54	  fresh30(true, true, X, Y)
14.76/2.54	= { by axiom 28 (op_equiv) }
14.76/2.54	  equiv(X, Y)
14.76/2.54	
14.76/2.54	Lemma 47: is_a_theorem(implies(X, implies(Y, and(X, Y)))) = true.
14.76/2.54	Proof:
14.76/2.54	  is_a_theorem(implies(X, implies(Y, and(X, Y))))
14.76/2.54	= { by axiom 37 (and_3_1) R->L }
14.76/2.54	  fresh103(and_3, true, X, Y)
14.76/2.54	= { by axiom 4 (hilbert_and_3) }
14.76/2.54	  fresh103(true, true, X, Y)
14.76/2.54	= { by axiom 24 (and_3_1) }
14.76/2.54	  true
14.76/2.54	
14.76/2.54	Lemma 48: fresh116(is_a_theorem(X), true, implies(Y, and(X, Y))) = is_a_theorem(implies(Y, and(X, Y))).
14.76/2.54	Proof:
14.76/2.54	  fresh116(is_a_theorem(X), true, implies(Y, and(X, Y)))
14.76/2.54	= { by lemma 45 R->L }
14.76/2.54	  fresh40(is_a_theorem(implies(X, implies(Y, and(X, Y)))), true, X, implies(Y, and(X, Y)))
14.76/2.54	= { by lemma 47 }
14.76/2.54	  fresh40(true, true, X, implies(Y, and(X, Y)))
14.76/2.54	= { by axiom 26 (modus_ponens_2) }
14.76/2.54	  is_a_theorem(implies(Y, and(X, Y)))
14.76/2.54	
14.76/2.54	Lemma 49: fresh3(is_a_theorem(equiv(X, Y)), true, X, Y) = X.
14.76/2.54	Proof:
14.76/2.54	  fresh3(is_a_theorem(equiv(X, Y)), true, X, Y)
14.76/2.54	= { by axiom 43 (substitution_of_equivalents_2) R->L }
14.76/2.54	  fresh4(substitution_of_equivalents, true, X, Y)
14.76/2.54	= { by axiom 7 (substitution_of_equivalents) }
14.76/2.54	  fresh4(true, true, X, Y)
14.76/2.54	= { by axiom 35 (substitution_of_equivalents_2) }
14.76/2.54	  X
14.76/2.54	
14.76/2.54	Lemma 50: and(X, X) = X.
14.76/2.54	Proof:
14.76/2.54	  and(X, X)
14.76/2.54	= { by axiom 36 (substitution_of_equivalents_2) R->L }
14.76/2.54	  fresh3(true, true, X, and(X, X))
14.76/2.54	= { by axiom 15 (modus_ponens_2) R->L }
14.76/2.54	  fresh3(fresh116(true, true, equiv(X, and(X, X))), true, X, and(X, X))
14.76/2.54	= { by axiom 22 (and_1_1) R->L }
14.76/2.54	  fresh3(fresh116(fresh107(true, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X))
14.76/2.54	= { by axiom 1 (hilbert_and_1) R->L }
14.76/2.54	  fresh3(fresh116(fresh107(and_1, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X))
14.76/2.54	= { by axiom 23 (and_1_1) }
14.76/2.54	  fresh3(fresh116(is_a_theorem(implies(and(X, X), X)), true, equiv(X, and(X, X))), true, X, and(X, X))
14.76/2.54	= { by lemma 45 R->L }
14.76/2.54	  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))
14.76/2.54	= { by lemma 46 R->L }
14.76/2.54	  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))
14.76/2.54	= { by lemma 48 R->L }
14.76/2.54	  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))
14.76/2.54	= { by axiom 26 (modus_ponens_2) R->L }
14.76/2.55	  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))
14.76/2.55	= { by axiom 25 (implies_2_1) R->L }
14.76/2.55	  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))
14.76/2.55	= { by axiom 2 (hilbert_implies_2) R->L }
14.76/2.55	  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))
14.76/2.55	= { by axiom 40 (implies_2_1) }
14.76/2.55	  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))
14.76/2.55	= { by lemma 45 }
14.76/2.55	  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))
14.76/2.55	= { by lemma 47 }
14.76/2.55	  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))
14.76/2.55	= { by axiom 15 (modus_ponens_2) }
14.76/2.55	  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))
14.76/2.55	= { by axiom 15 (modus_ponens_2) }
14.76/2.55	  fresh3(fresh40(true, true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
14.76/2.55	= { by axiom 26 (modus_ponens_2) }
14.76/2.55	  fresh3(is_a_theorem(equiv(X, and(X, X))), true, X, and(X, X))
14.76/2.55	= { by lemma 49 }
14.76/2.55	  X
14.76/2.55	
14.76/2.55	Lemma 51: not(and(X, not(Y))) = implies(X, Y).
14.76/2.55	Proof:
14.76/2.55	  not(and(X, not(Y)))
14.76/2.55	= { by axiom 31 (op_implies_and) R->L }
14.76/2.55	  fresh29(op_implies_and, true, X, Y)
14.76/2.55	= { by axiom 10 (hilbert_op_implies_and) }
14.76/2.55	  fresh29(true, true, X, Y)
14.76/2.55	= { by axiom 30 (op_implies_and) }
14.76/2.55	  implies(X, Y)
14.76/2.55	
14.76/2.55	Lemma 52: implies(not(X), Y) = or(X, Y).
14.76/2.55	Proof:
14.76/2.55	  implies(not(X), Y)
14.76/2.55	= { by lemma 51 R->L }
14.76/2.55	  not(and(not(X), not(Y)))
14.76/2.55	= { by axiom 38 (op_or) R->L }
14.76/2.55	  fresh26(op_or, true, X, Y)
14.76/2.55	= { by axiom 11 (hilbert_op_or) }
14.76/2.55	  fresh26(true, true, X, Y)
14.76/2.55	= { by axiom 32 (op_or) }
14.76/2.55	  or(X, Y)
14.76/2.55	
14.76/2.55	Lemma 53: not(not(X)) = or(X, X).
14.76/2.55	Proof:
14.76/2.55	  not(not(X))
14.76/2.55	= { by lemma 50 R->L }
14.76/2.55	  not(and(not(X), not(X)))
14.76/2.55	= { by lemma 51 }
14.76/2.55	  implies(not(X), X)
14.76/2.55	= { by lemma 52 }
14.76/2.55	  or(X, X)
14.76/2.55	
14.76/2.55	Lemma 54: not(necessarily(not(X))) = possibly(X).
14.76/2.55	Proof:
14.76/2.55	  not(necessarily(not(X)))
14.76/2.55	= { by axiom 20 (op_possibly) R->L }
14.76/2.55	  fresh25(op_possibly, true, X)
14.76/2.55	= { by axiom 13 (km5_op_possibly) }
14.76/2.55	  fresh25(true, true, X)
14.76/2.55	= { by axiom 18 (op_possibly) }
14.76/2.55	  possibly(X)
14.76/2.55	
14.76/2.55	Lemma 55: is_a_theorem(implies(necessarily(X), X)) = true.
14.76/2.55	Proof:
14.76/2.55	  is_a_theorem(implies(necessarily(X), X))
14.76/2.55	= { by axiom 19 (axiom_M_1) R->L }
14.76/2.55	  fresh93(axiom_M, true, X)
14.76/2.55	= { by axiom 9 (km5_axiom_M) }
14.76/2.55	  fresh93(true, true, X)
14.76/2.55	= { by axiom 17 (axiom_M_1) }
14.76/2.55	  true
14.76/2.55	
14.76/2.55	Lemma 56: is_a_theorem(implies(X, or(X, Y))) = true.
14.76/2.55	Proof:
14.76/2.55	  is_a_theorem(implies(X, or(X, Y)))
14.76/2.55	= { by axiom 34 (or_1_1) R->L }
14.76/2.55	  fresh21(or_1, true, X, Y)
14.76/2.55	= { by axiom 5 (hilbert_or_1) }
14.76/2.55	  fresh21(true, true, X, Y)
14.76/2.55	= { by axiom 33 (or_1_1) }
14.76/2.55	  true
14.76/2.55	
14.76/2.55	Lemma 57: fresh116(is_a_theorem(or(X, not(Y))), true, implies(Y, X)) = is_a_theorem(implies(Y, X)).
14.76/2.55	Proof:
14.76/2.55	  fresh116(is_a_theorem(or(X, not(Y))), true, implies(Y, X))
14.76/2.55	= { by lemma 45 R->L }
14.76/2.55	  fresh40(is_a_theorem(implies(or(X, not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
14.76/2.55	= { by lemma 52 R->L }
14.76/2.55	  fresh40(is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X))), true, or(X, not(Y)), implies(Y, X))
14.76/2.55	= { by axiom 44 (modus_tollens_1) R->L }
14.76/2.55	  fresh40(fresh35(modus_tollens, true, Y, X), true, or(X, not(Y)), implies(Y, X))
14.76/2.55	= { by axiom 6 (hilbert_modus_tollens) }
14.76/2.55	  fresh40(fresh35(true, true, Y, X), true, or(X, not(Y)), implies(Y, X))
14.76/2.55	= { by axiom 27 (modus_tollens_1) }
14.76/2.55	  fresh40(true, true, or(X, not(Y)), implies(Y, X))
14.76/2.55	= { by axiom 26 (modus_ponens_2) }
14.76/2.55	  is_a_theorem(implies(Y, X))
14.76/2.55	
14.76/2.55	Goal 1 (km4b_axiom_B): axiom_B = true.
14.76/2.55	Proof:
14.76/2.55	  axiom_B
14.76/2.55	= { by axiom 41 (axiom_B) R->L }
14.76/2.55	  fresh98(is_a_theorem(implies(x18, necessarily(possibly(x18)))), true)
14.76/2.55	= { by axiom 36 (substitution_of_equivalents_2) R->L }
14.76/2.55	  fresh98(is_a_theorem(implies(x18, fresh3(true, true, possibly(x18), necessarily(possibly(x18))))), true)
14.76/2.55	= { by axiom 15 (modus_ponens_2) R->L }
14.76/2.55	  fresh98(is_a_theorem(implies(x18, fresh3(fresh116(true, true, equiv(possibly(x18), necessarily(possibly(x18)))), true, possibly(x18), necessarily(possibly(x18))))), true)
14.76/2.55	= { by lemma 55 R->L }
14.76/2.55	  fresh98(is_a_theorem(implies(x18, fresh3(fresh116(is_a_theorem(implies(necessarily(possibly(x18)), possibly(x18))), true, equiv(possibly(x18), necessarily(possibly(x18)))), true, possibly(x18), necessarily(possibly(x18))))), true)
14.76/2.55	= { by lemma 45 R->L }
14.76/2.55	  fresh98(is_a_theorem(implies(x18, fresh3(fresh40(is_a_theorem(implies(implies(necessarily(possibly(x18)), possibly(x18)), equiv(possibly(x18), necessarily(possibly(x18))))), true, implies(necessarily(possibly(x18)), possibly(x18)), equiv(possibly(x18), necessarily(possibly(x18)))), true, possibly(x18), necessarily(possibly(x18))))), true)
14.76/2.55	= { by lemma 46 R->L }
14.76/2.55	  fresh98(is_a_theorem(implies(x18, fresh3(fresh40(is_a_theorem(implies(implies(necessarily(possibly(x18)), possibly(x18)), and(implies(possibly(x18), necessarily(possibly(x18))), implies(necessarily(possibly(x18)), possibly(x18))))), true, implies(necessarily(possibly(x18)), possibly(x18)), equiv(possibly(x18), necessarily(possibly(x18)))), true, possibly(x18), necessarily(possibly(x18))))), true)
14.76/2.55	= { by lemma 48 R->L }
14.76/2.55	  fresh98(is_a_theorem(implies(x18, fresh3(fresh40(fresh116(is_a_theorem(implies(possibly(x18), necessarily(possibly(x18)))), true, implies(implies(necessarily(possibly(x18)), possibly(x18)), and(implies(possibly(x18), necessarily(possibly(x18))), implies(necessarily(possibly(x18)), possibly(x18))))), true, implies(necessarily(possibly(x18)), possibly(x18)), equiv(possibly(x18), necessarily(possibly(x18)))), true, possibly(x18), necessarily(possibly(x18))))), true)
14.76/2.55	= { by axiom 39 (axiom_5_1) R->L }
14.76/2.55	  fresh98(is_a_theorem(implies(x18, fresh3(fresh40(fresh116(fresh99(axiom_5, true, x18), true, implies(implies(necessarily(possibly(x18)), possibly(x18)), and(implies(possibly(x18), necessarily(possibly(x18))), implies(necessarily(possibly(x18)), possibly(x18))))), true, implies(necessarily(possibly(x18)), possibly(x18)), equiv(possibly(x18), necessarily(possibly(x18)))), true, possibly(x18), necessarily(possibly(x18))))), true)
14.76/2.55	= { by axiom 8 (km5_axiom_5) }
14.76/2.55	  fresh98(is_a_theorem(implies(x18, fresh3(fresh40(fresh116(fresh99(true, true, x18), true, implies(implies(necessarily(possibly(x18)), possibly(x18)), and(implies(possibly(x18), necessarily(possibly(x18))), implies(necessarily(possibly(x18)), possibly(x18))))), true, implies(necessarily(possibly(x18)), possibly(x18)), equiv(possibly(x18), necessarily(possibly(x18)))), true, possibly(x18), necessarily(possibly(x18))))), true)
14.76/2.55	= { by axiom 16 (axiom_5_1) }
14.76/2.55	  fresh98(is_a_theorem(implies(x18, fresh3(fresh40(fresh116(true, true, implies(implies(necessarily(possibly(x18)), possibly(x18)), and(implies(possibly(x18), necessarily(possibly(x18))), implies(necessarily(possibly(x18)), possibly(x18))))), true, implies(necessarily(possibly(x18)), possibly(x18)), equiv(possibly(x18), necessarily(possibly(x18)))), true, possibly(x18), necessarily(possibly(x18))))), true)
14.76/2.55	= { by axiom 15 (modus_ponens_2) }
14.76/2.55	  fresh98(is_a_theorem(implies(x18, fresh3(fresh40(true, true, implies(necessarily(possibly(x18)), possibly(x18)), equiv(possibly(x18), necessarily(possibly(x18)))), true, possibly(x18), necessarily(possibly(x18))))), true)
14.76/2.55	= { by axiom 26 (modus_ponens_2) }
14.76/2.55	  fresh98(is_a_theorem(implies(x18, fresh3(is_a_theorem(equiv(possibly(x18), necessarily(possibly(x18)))), true, possibly(x18), necessarily(possibly(x18))))), true)
14.76/2.55	= { by lemma 49 }
14.76/2.55	  fresh98(is_a_theorem(implies(x18, possibly(x18))), true)
14.76/2.55	= { by lemma 57 R->L }
14.76/2.55	  fresh98(fresh116(is_a_theorem(or(possibly(x18), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.55	= { by lemma 50 R->L }
14.76/2.55	  fresh98(fresh116(is_a_theorem(or(and(possibly(x18), possibly(x18)), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.55	= { by lemma 52 R->L }
14.76/2.55	  fresh98(fresh116(is_a_theorem(implies(not(and(possibly(x18), possibly(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.55	= { by lemma 54 R->L }
14.76/2.55	  fresh98(fresh116(is_a_theorem(implies(not(and(possibly(x18), not(necessarily(not(x18))))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.55	= { by lemma 51 }
14.76/2.55	  fresh98(fresh116(is_a_theorem(implies(implies(possibly(x18), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.55	= { by lemma 54 R->L }
14.76/2.55	  fresh98(fresh116(is_a_theorem(implies(implies(not(necessarily(not(x18))), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.55	= { by lemma 52 }
14.76/2.55	  fresh98(fresh116(is_a_theorem(implies(or(necessarily(not(x18)), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.55	= { by lemma 49 R->L }
14.76/2.55	  fresh98(fresh116(is_a_theorem(implies(fresh3(is_a_theorem(equiv(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.55	= { by axiom 26 (modus_ponens_2) R->L }
14.76/2.55	  fresh98(fresh116(is_a_theorem(implies(fresh3(fresh40(true, true, implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), equiv(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.55	= { by axiom 15 (modus_ponens_2) R->L }
14.76/2.55	  fresh98(fresh116(is_a_theorem(implies(fresh3(fresh40(fresh116(true, true, implies(implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), and(implies(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18))))))), true, implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), equiv(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.55	= { by axiom 15 (modus_ponens_2) R->L }
14.76/2.56	  fresh98(fresh116(is_a_theorem(implies(fresh3(fresh40(fresh116(fresh116(true, true, implies(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, implies(implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), and(implies(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18))))))), true, implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), equiv(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.56	= { by lemma 56 R->L }
14.76/2.56	  fresh98(fresh116(is_a_theorem(implies(fresh3(fresh40(fresh116(fresh116(is_a_theorem(implies(not(necessarily(not(x18))), or(not(necessarily(not(x18))), not(necessarily(not(x18)))))), true, implies(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, implies(implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), and(implies(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18))))))), true, implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), equiv(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.56	= { by lemma 52 }
14.76/2.56	  fresh98(fresh116(is_a_theorem(implies(fresh3(fresh40(fresh116(fresh116(is_a_theorem(or(necessarily(not(x18)), or(not(necessarily(not(x18))), not(necessarily(not(x18)))))), true, implies(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, implies(implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), and(implies(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18))))))), true, implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), equiv(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.56	= { by lemma 53 R->L }
14.76/2.56	  fresh98(fresh116(is_a_theorem(implies(fresh3(fresh40(fresh116(fresh116(is_a_theorem(or(necessarily(not(x18)), not(not(not(necessarily(not(x18))))))), true, implies(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, implies(implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), and(implies(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18))))))), true, implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), equiv(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.56	= { by lemma 53 }
14.76/2.56	  fresh98(fresh116(is_a_theorem(implies(fresh3(fresh40(fresh116(fresh116(is_a_theorem(or(necessarily(not(x18)), not(or(necessarily(not(x18)), necessarily(not(x18)))))), true, implies(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, implies(implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), and(implies(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18))))))), true, implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), equiv(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.56	= { by lemma 57 }
14.76/2.56	  fresh98(fresh116(is_a_theorem(implies(fresh3(fresh40(fresh116(is_a_theorem(implies(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, implies(implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), and(implies(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18))))))), true, implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), equiv(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.56	= { by lemma 48 }
14.76/2.56	  fresh98(fresh116(is_a_theorem(implies(fresh3(fresh40(is_a_theorem(implies(implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), and(implies(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18))))))), true, implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), equiv(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.56	= { by lemma 46 }
14.76/2.56	  fresh98(fresh116(is_a_theorem(implies(fresh3(fresh40(is_a_theorem(implies(implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), equiv(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))))), true, implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18)))), equiv(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.56	= { by lemma 45 }
14.76/2.56	  fresh98(fresh116(is_a_theorem(implies(fresh3(fresh116(is_a_theorem(implies(necessarily(not(x18)), or(necessarily(not(x18)), necessarily(not(x18))))), true, equiv(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.56	= { by lemma 56 }
14.76/2.56	  fresh98(fresh116(is_a_theorem(implies(fresh3(fresh116(true, true, equiv(or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18)))), true, or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.56	= { by axiom 15 (modus_ponens_2) }
14.76/2.56	  fresh98(fresh116(is_a_theorem(implies(fresh3(true, true, or(necessarily(not(x18)), necessarily(not(x18))), necessarily(not(x18))), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.56	= { by axiom 36 (substitution_of_equivalents_2) }
14.76/2.56	  fresh98(fresh116(is_a_theorem(implies(necessarily(not(x18)), not(x18))), true, implies(x18, possibly(x18))), true)
14.76/2.56	= { by lemma 55 }
14.76/2.56	  fresh98(fresh116(true, true, implies(x18, possibly(x18))), true)
14.76/2.56	= { by axiom 15 (modus_ponens_2) }
14.76/2.56	  fresh98(true, true)
14.76/2.56	= { by axiom 14 (axiom_B) }
14.76/2.56	  true
14.76/2.56	% SZS output end Proof
14.76/2.56	
14.76/2.56	RESULT: Theorem (the conjecture is true).
14.76/2.57	EOF