0.03/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.03/0.13 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.13/0.34 % Computer : n008.cluster.edu 0.13/0.34 % Model : x86_64 x86_64 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.34 % Memory : 8042.1875MB 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.34 % CPULimit : 180 0.13/0.34 % DateTime : Thu Aug 29 13:56:08 EDT 2019 0.13/0.34 % CPUTime : 54.25/54.47 % SZS status Theorem 54.25/54.47 54.25/54.47 % SZS output start Proof 54.25/54.47 Take the following subset of the input axioms: 55.83/55.99 fof(and_2, axiom, and_2 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), Y))). 55.83/55.99 fof(and_3, axiom, ![X, Y]: is_a_theorem(implies(X, implies(Y, and(X, Y)))) <=> and_3). 55.83/55.99 fof(axiom_4, axiom, ![X]: is_a_theorem(implies(necessarily(X), necessarily(necessarily(X)))) <=> axiom_4). 55.83/55.99 fof(axiom_M, axiom, axiom_M <=> ![X]: is_a_theorem(implies(necessarily(X), X))). 55.83/55.99 fof(axiom_m9, axiom, ![X]: is_a_theorem(strict_implies(possibly(possibly(X)), possibly(X))) <=> axiom_m9). 55.83/55.99 fof(cn3, axiom, ![P]: is_a_theorem(implies(implies(not(P), P), P)) <=> cn3). 55.83/55.99 fof(hilbert_and_2, axiom, and_2). 55.83/55.99 fof(hilbert_and_3, axiom, and_3). 55.83/55.99 fof(hilbert_implies_1, axiom, implies_1). 55.83/55.99 fof(hilbert_implies_2, axiom, implies_2). 55.83/55.99 fof(hilbert_modus_ponens, axiom, modus_ponens). 55.83/55.99 fof(hilbert_op_equiv, axiom, op_equiv). 55.83/55.99 fof(hilbert_op_implies_and, axiom, op_implies_and). 55.83/55.99 fof(hilbert_op_or, axiom, op_or). 55.83/55.99 fof(hilbert_or_1, axiom, or_1). 55.83/55.99 fof(hilbert_or_3, axiom, or_3). 55.83/55.99 fof(implies_1, axiom, implies_1 <=> ![X, Y]: is_a_theorem(implies(X, implies(Y, X)))). 55.83/55.99 fof(implies_2, axiom, implies_2 <=> ![X, Y]: is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y)))). 55.83/55.99 fof(km4b_axiom_4, axiom, axiom_4). 55.83/55.99 fof(km4b_axiom_M, axiom, axiom_M). 55.83/55.99 fof(km4b_necessitation, axiom, necessitation). 55.83/55.99 fof(km4b_op_possibly, axiom, op_possibly). 55.83/55.99 fof(kn1, axiom, ![P]: is_a_theorem(implies(P, and(P, P))) <=> kn1). 55.83/55.99 fof(modus_ponens, axiom, modus_ponens <=> ![X, Y]: ((is_a_theorem(X) & is_a_theorem(implies(X, Y))) => is_a_theorem(Y))). 55.83/55.99 fof(necessitation, axiom, necessitation <=> ![X]: (is_a_theorem(necessarily(X)) <= is_a_theorem(X))). 55.83/55.99 fof(op_equiv, axiom, op_equiv => ![X, Y]: equiv(X, Y)=and(implies(X, Y), implies(Y, X))). 55.83/55.99 fof(op_implies_and, axiom, op_implies_and => ![X, Y]: implies(X, Y)=not(and(X, not(Y)))). 55.83/55.99 fof(op_or, axiom, op_or => ![X, Y]: or(X, Y)=not(and(not(X), not(Y)))). 55.83/55.99 fof(op_possibly, axiom, ![X]: not(necessarily(not(X)))=possibly(X) <= op_possibly). 55.83/55.99 fof(op_strict_implies, axiom, ![X, Y]: necessarily(implies(X, Y))=strict_implies(X, Y) <= op_strict_implies). 55.83/55.99 fof(or_1, axiom, or_1 <=> ![X, Y]: is_a_theorem(implies(X, or(X, Y)))). 55.83/55.99 fof(or_3, axiom, ![X, Y, Z]: is_a_theorem(implies(implies(X, Z), implies(implies(Y, Z), implies(or(X, Y), Z)))) <=> or_3). 55.83/55.99 fof(r1, axiom, r1 <=> ![P]: is_a_theorem(implies(or(P, P), P))). 55.83/55.99 fof(s1_0_m6s3m9b_axiom_m9, conjecture, axiom_m9). 55.83/55.99 fof(s1_0_op_strict_implies, axiom, op_strict_implies). 55.83/55.99 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X, Y]: (is_a_theorem(equiv(X, Y)) => X=Y)). 55.83/55.99 fof(substitution_of_equivalents, axiom, substitution_of_equivalents). 55.83/55.99 55.83/55.99 Now clausify the problem and encode Horn clauses using encoding 3 of 55.83/55.99 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 55.83/55.99 We repeatedly replace C & s=t => u=v by the two clauses: 55.83/55.99 fresh(y, y, x1...xn) = u 55.83/55.99 C => fresh(s, t, x1...xn) = v 55.83/55.99 where fresh is a fresh function symbol and x1..xn are the free 55.83/55.99 variables of u and v. 55.83/55.99 A predicate p(X) is encoded as p(X)=true (this is sound, because the 55.83/55.99 input problem has no model of domain size 1). 55.83/55.99 55.83/55.99 The encoding turns the above axioms into the following unit equations and goals: 55.83/55.99 55.83/55.99 Axiom 1 (and_2_1): fresh105(X, X, Y, Z) = true. 55.83/55.99 Axiom 2 (and_3_1): fresh103(X, X, Y, Z) = true. 55.83/55.99 Axiom 3 (axiom_4_1): fresh101(X, X, Y) = true. 55.83/55.99 Axiom 4 (axiom_M_1): fresh93(X, X, Y) = true. 55.83/55.99 Axiom 5 (axiom_m9): fresh74(X, X) = true. 55.83/55.99 Axiom 6 (implies_1_1): fresh51(X, X, Y, Z) = true. 55.83/55.99 Axiom 7 (implies_2_1): fresh49(X, X, Y, Z) = true. 55.83/55.99 Axiom 8 (modus_ponens_2): fresh40(X, X, Y, Z) = is_a_theorem(Z). 55.83/55.99 Axiom 9 (modus_ponens_2): fresh116(X, X, Y) = true. 55.83/55.99 Axiom 10 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(is_a_theorem(Y), true, Z). 55.83/55.99 Axiom 11 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)). 55.83/55.99 Axiom 12 (necessitation_1): fresh33(X, X, Y) = true. 55.83/55.99 Axiom 13 (op_equiv): fresh30(X, X, Y, Z) = equiv(Y, Z). 55.83/55.99 Axiom 14 (op_implies_and): fresh29(X, X, Y, Z) = implies(Y, Z). 55.83/55.99 Axiom 15 (op_or): fresh26(X, X, Y, Z) = or(Y, Z). 55.83/55.99 Axiom 16 (op_possibly): fresh25(X, X, Y) = possibly(Y). 55.83/55.99 Axiom 17 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z). 55.83/55.99 Axiom 18 (or_1_1): fresh21(X, X, Y, Z) = true. 55.83/55.99 Axiom 19 (or_3_1): fresh17(X, X, Y, Z, W) = true. 55.83/55.99 Axiom 20 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y. 55.83/55.99 Axiom 21 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z. 55.83/55.99 Axiom 22 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))). 55.83/55.99 Axiom 23 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y). 55.83/55.99 Axiom 24 (modus_ponens_2): fresh115(modus_ponens, true, X, Y) = fresh40(is_a_theorem(implies(X, Y)), true, X, Y). 55.83/55.99 Axiom 25 (kn1_1): fresh45(kn1, true, X) = is_a_theorem(implies(X, and(X, X))). 55.83/55.99 Axiom 26 (and_2_1): fresh105(and_2, true, X, Y) = is_a_theorem(implies(and(X, Y), Y)). 55.83/55.99 Axiom 27 (cn3_1): fresh59(cn3, true, X) = is_a_theorem(implies(implies(not(X), X), X)). 55.83/55.99 Axiom 28 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))). 55.83/55.99 Axiom 29 (or_3_1): fresh17(or_3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Z), implies(implies(Y, Z), implies(or(X, Y), Z)))). 55.83/55.99 Axiom 30 (implies_1_1): fresh51(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))). 55.83/55.99 Axiom 31 (or_1_1): fresh21(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))). 55.83/55.99 Axiom 32 (r1_1): fresh15(r1, true, X) = is_a_theorem(implies(or(X, X), X)). 55.83/55.99 Axiom 33 (op_or): fresh26(op_or, true, X, Y) = not(and(not(X), not(Y))). 55.83/55.99 Axiom 34 (op_equiv): fresh30(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)). 55.83/55.99 Axiom 35 (op_implies_and): fresh29(op_implies_and, true, X, Y) = not(and(X, not(Y))). 55.83/55.99 Axiom 36 (hilbert_and_2): and_2 = true. 55.83/55.99 Axiom 37 (substitution_of_equivalents): substitution_of_equivalents = true. 55.83/55.99 Axiom 38 (hilbert_and_3): and_3 = true. 55.83/55.99 Axiom 39 (hilbert_implies_1): implies_1 = true. 55.83/55.99 Axiom 40 (hilbert_or_1): or_1 = true. 55.83/55.99 Axiom 41 (hilbert_modus_ponens): modus_ponens = true. 55.83/55.99 Axiom 42 (hilbert_or_3): or_3 = true. 55.83/55.99 Axiom 43 (hilbert_implies_2): implies_2 = true. 55.83/55.99 Axiom 44 (hilbert_op_equiv): op_equiv = true. 55.83/55.99 Axiom 45 (hilbert_op_implies_and): op_implies_and = true. 55.83/55.99 Axiom 46 (hilbert_op_or): op_or = true. 55.83/55.99 Axiom 47 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X). 55.83/55.99 Axiom 48 (axiom_4_1): fresh101(axiom_4, true, X) = is_a_theorem(implies(necessarily(X), necessarily(necessarily(X)))). 55.83/55.99 Axiom 49 (axiom_m9_1): fresh73(axiom_m9, true, X) = is_a_theorem(strict_implies(possibly(possibly(X)), possibly(X))). 55.83/55.99 Axiom 50 (axiom_m9): fresh74(is_a_theorem(strict_implies(possibly(possibly(sK11_axiom_m9_X)), possibly(sK11_axiom_m9_X))), true) = axiom_m9. 55.83/55.99 Axiom 51 (axiom_M_1): fresh93(axiom_M, true, X) = is_a_theorem(implies(necessarily(X), X)). 55.83/55.99 Axiom 52 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)). 55.83/55.99 Axiom 53 (op_possibly): fresh25(op_possibly, true, X) = not(necessarily(not(X))). 55.83/55.99 Axiom 54 (km4b_axiom_M): axiom_M = true. 55.83/55.99 Axiom 55 (km4b_necessitation): necessitation = true. 55.83/55.99 Axiom 56 (km4b_op_possibly): op_possibly = true. 55.83/55.99 Axiom 57 (km4b_axiom_4): axiom_4 = true. 55.83/55.99 Axiom 58 (s1_0_op_strict_implies): op_strict_implies = true. 55.83/55.99 55.83/55.99 Lemma 59: not(necessarily(not(X))) = possibly(X). 55.83/55.99 Proof: 55.83/55.99 not(necessarily(not(X))) 55.83/55.99 = { by axiom 53 (op_possibly) } 55.83/55.99 fresh25(op_possibly, true, X) 55.83/55.99 = { by axiom 56 (km4b_op_possibly) } 55.83/55.99 fresh25(true, true, X) 55.83/55.99 = { by axiom 16 (op_possibly) } 55.83/55.99 possibly(X) 55.83/55.99 55.83/55.99 Lemma 60: fresh3(is_a_theorem(equiv(X, Y)), true, X, Y) = X. 55.83/55.99 Proof: 55.83/55.99 fresh3(is_a_theorem(equiv(X, Y)), true, X, Y) 55.83/55.99 = { by axiom 23 (substitution_of_equivalents_2) } 55.83/55.99 fresh4(substitution_of_equivalents, true, X, Y) 55.83/55.99 = { by axiom 37 (substitution_of_equivalents) } 55.83/55.99 fresh4(true, true, X, Y) 55.83/55.99 = { by axiom 20 (substitution_of_equivalents_2) } 55.83/55.99 X 55.83/55.99 55.83/55.99 Lemma 61: not(and(X, not(Y))) = implies(X, Y). 55.83/55.99 Proof: 55.83/55.99 not(and(X, not(Y))) 55.83/55.99 = { by axiom 35 (op_implies_and) } 55.83/55.99 fresh29(op_implies_and, true, X, Y) 55.83/55.99 = { by axiom 45 (hilbert_op_implies_and) } 55.83/55.99 fresh29(true, true, X, Y) 55.83/55.99 = { by axiom 14 (op_implies_and) } 55.83/55.99 implies(X, Y) 55.83/55.99 55.83/55.99 Lemma 62: implies(not(X), Y) = or(X, Y). 55.83/55.99 Proof: 55.83/55.99 implies(not(X), Y) 55.83/55.99 = { by lemma 61 } 55.83/55.99 not(and(not(X), not(Y))) 55.83/55.99 = { by axiom 33 (op_or) } 55.83/55.99 fresh26(op_or, true, X, Y) 55.83/55.99 = { by axiom 46 (hilbert_op_or) } 55.83/55.99 fresh26(true, true, X, Y) 55.83/55.99 = { by axiom 15 (op_or) } 55.83/55.99 or(X, Y) 55.83/55.99 55.83/55.99 Lemma 63: fresh40(is_a_theorem(implies(X, Y)), true, X, Y) = fresh116(is_a_theorem(X), true, Y). 55.83/55.99 Proof: 55.83/55.99 fresh40(is_a_theorem(implies(X, Y)), true, X, Y) 55.83/55.99 = { by axiom 24 (modus_ponens_2) } 55.83/55.99 fresh115(modus_ponens, true, X, Y) 55.83/55.99 = { by axiom 41 (hilbert_modus_ponens) } 55.83/55.99 fresh115(true, true, X, Y) 55.83/55.99 = { by axiom 10 (modus_ponens_2) } 55.83/55.99 fresh116(is_a_theorem(X), true, Y) 55.83/55.99 55.83/55.99 Lemma 64: is_a_theorem(implies(X, implies(Y, and(X, Y)))) = true. 55.83/55.99 Proof: 55.83/55.99 is_a_theorem(implies(X, implies(Y, and(X, Y)))) 55.83/55.99 = { by axiom 22 (and_3_1) } 55.83/56.00 fresh103(and_3, true, X, Y) 55.83/56.00 = { by axiom 38 (hilbert_and_3) } 55.83/56.00 fresh103(true, true, X, Y) 55.83/56.00 = { by axiom 2 (and_3_1) } 55.83/56.00 true 55.83/56.00 55.83/56.00 Lemma 65: fresh116(is_a_theorem(X), true, implies(Y, and(X, Y))) = is_a_theorem(implies(Y, and(X, Y))). 55.83/56.00 Proof: 55.83/56.00 fresh116(is_a_theorem(X), true, implies(Y, and(X, Y))) 55.83/56.00 = { by lemma 63 } 55.83/56.00 fresh40(is_a_theorem(implies(X, implies(Y, and(X, Y)))), true, X, implies(Y, and(X, Y))) 55.83/56.00 = { by lemma 64 } 55.83/56.00 fresh40(true, true, X, implies(Y, and(X, Y))) 55.83/56.00 = { by axiom 8 (modus_ponens_2) } 55.83/56.00 is_a_theorem(implies(Y, and(X, Y))) 55.83/56.00 55.83/56.00 Lemma 66: and(implies(X, Y), implies(Y, X)) = equiv(X, Y). 55.83/56.00 Proof: 55.83/56.00 and(implies(X, Y), implies(Y, X)) 55.83/56.00 = { by axiom 34 (op_equiv) } 55.83/56.00 fresh30(op_equiv, true, X, Y) 55.83/56.00 = { by axiom 44 (hilbert_op_equiv) } 55.83/56.00 fresh30(true, true, X, Y) 55.83/56.00 = { by axiom 13 (op_equiv) } 55.83/56.00 equiv(X, Y) 55.83/56.00 55.83/56.00 Lemma 67: fresh116(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y)) = is_a_theorem(implies(X, Y)). 55.83/56.00 Proof: 55.83/56.00 fresh116(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y)) 55.83/56.00 = { by lemma 63 } 55.83/56.00 fresh40(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y)) 55.83/56.00 = { by axiom 28 (implies_2_1) } 55.83/56.00 fresh40(fresh49(implies_2, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y)) 55.83/56.00 = { by axiom 43 (hilbert_implies_2) } 55.83/56.00 fresh40(fresh49(true, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y)) 55.83/56.00 = { by axiom 7 (implies_2_1) } 55.83/56.00 fresh40(true, true, implies(X, implies(X, Y)), implies(X, Y)) 55.83/56.00 = { by axiom 8 (modus_ponens_2) } 55.83/56.00 is_a_theorem(implies(X, Y)) 55.83/56.00 55.83/56.00 Lemma 68: and(X, X) = X. 55.83/56.00 Proof: 55.83/56.00 and(X, X) 55.83/56.00 = { by axiom 21 (substitution_of_equivalents_2) } 55.83/56.00 fresh3(true, true, X, and(X, X)) 55.83/56.00 = { by axiom 9 (modus_ponens_2) } 55.83/56.00 fresh3(fresh116(true, true, equiv(X, and(X, X))), true, X, and(X, X)) 55.83/56.00 = { by axiom 1 (and_2_1) } 55.83/56.00 fresh3(fresh116(fresh105(true, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X)) 55.83/56.00 = { by axiom 36 (hilbert_and_2) } 55.83/56.00 fresh3(fresh116(fresh105(and_2, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X)) 55.83/56.00 = { by axiom 26 (and_2_1) } 55.83/56.00 fresh3(fresh116(is_a_theorem(implies(and(X, X), X)), true, equiv(X, and(X, X))), true, X, and(X, X)) 55.83/56.00 = { by lemma 63 } 55.83/56.00 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)) 55.83/56.00 = { by lemma 66 } 55.83/56.00 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)) 55.83/56.00 = { by lemma 65 } 55.83/56.00 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)) 55.83/56.00 = { by lemma 67 } 55.83/56.00 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)) 55.83/56.00 = { by lemma 64 } 55.83/56.00 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)) 55.83/56.00 = { by axiom 9 (modus_ponens_2) } 55.83/56.00 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)) 55.83/56.00 = { by axiom 9 (modus_ponens_2) } 55.83/56.00 fresh3(fresh40(true, true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X)) 55.83/56.00 = { by axiom 8 (modus_ponens_2) } 55.83/56.00 fresh3(is_a_theorem(equiv(X, and(X, X))), true, X, and(X, X)) 55.83/56.00 = { by lemma 60 } 55.83/56.00 X 55.83/56.00 55.83/56.00 Lemma 69: not(and(X, possibly(Y))) = implies(X, necessarily(not(Y))). 55.83/56.00 Proof: 55.83/56.00 not(and(X, possibly(Y))) 55.83/56.00 = { by lemma 59 } 55.83/56.00 not(and(X, not(necessarily(not(Y))))) 55.83/56.00 = { by lemma 61 } 55.83/56.00 implies(X, necessarily(not(Y))) 55.83/56.00 55.83/56.00 Lemma 70: is_a_theorem(implies(X, X)) = true. 55.83/56.00 Proof: 55.83/56.00 is_a_theorem(implies(X, X)) 55.83/56.00 = { by lemma 67 } 55.83/56.00 fresh116(is_a_theorem(implies(X, implies(X, X))), true, implies(X, X)) 55.83/56.00 = { by axiom 30 (implies_1_1) } 55.83/56.00 fresh116(fresh51(implies_1, true, X, X), true, implies(X, X)) 55.83/56.00 = { by axiom 39 (hilbert_implies_1) } 55.83/56.00 fresh116(fresh51(true, true, X, X), true, implies(X, X)) 55.83/56.00 = { by axiom 6 (implies_1_1) } 55.83/56.00 fresh116(true, true, implies(X, X)) 55.83/56.00 = { by axiom 9 (modus_ponens_2) } 57.25/57.46 true 57.25/57.46 57.25/57.46 Goal 1 (s1_0_m6s3m9b_axiom_m9): axiom_m9 = true. 57.25/57.46 Proof: 57.25/57.46 axiom_m9 57.25/57.46 = { by axiom 50 (axiom_m9) } 57.25/57.46 fresh74(is_a_theorem(strict_implies(possibly(possibly(sK11_axiom_m9_X)), possibly(sK11_axiom_m9_X))), true) 57.25/57.46 = { by lemma 59 } 57.25/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(not(possibly(sK11_axiom_m9_X)))), possibly(sK11_axiom_m9_X))), true) 57.25/57.46 = { by lemma 60 } 57.25/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(is_a_theorem(equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.25/57.46 = { by axiom 8 (modus_ponens_2) } 57.25/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(true, true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.25/57.46 = { by axiom 9 (modus_ponens_2) } 57.25/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(true, true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.25/57.46 = { by axiom 9 (modus_ponens_2) } 57.25/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(fresh116(true, true, or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.25/57.46 = { by lemma 70 } 57.25/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(fresh116(is_a_theorem(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.25/57.46 = { by lemma 63 } 57.25/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))))), true, implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.25/57.46 = { by lemma 68 } 57.25/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), or(and(not(necessarily(not(sK11_axiom_m9_X))), not(necessarily(not(sK11_axiom_m9_X)))), necessarily(not(sK11_axiom_m9_X))))), true, implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.25/57.46 = { by axiom 15 (op_or) } 57.25/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), fresh26(true, true, and(not(necessarily(not(sK11_axiom_m9_X))), not(necessarily(not(sK11_axiom_m9_X)))), necessarily(not(sK11_axiom_m9_X))))), true, implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.25/57.46 = { by axiom 46 (hilbert_op_or) } 57.25/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), fresh26(op_or, true, and(not(necessarily(not(sK11_axiom_m9_X))), not(necessarily(not(sK11_axiom_m9_X)))), necessarily(not(sK11_axiom_m9_X))))), true, implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.25/57.46 = { by axiom 33 (op_or) } 57.25/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), not(and(not(and(not(necessarily(not(sK11_axiom_m9_X))), not(necessarily(not(sK11_axiom_m9_X))))), not(necessarily(not(sK11_axiom_m9_X))))))), true, implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.25/57.46 = { by lemma 61 } 57.25/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), not(and(implies(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), not(necessarily(not(sK11_axiom_m9_X))))))), true, implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.25/57.46 = { by lemma 61 } 57.25/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), implies(implies(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))))), true, implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.25/57.46 = { by lemma 62 } 57.25/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), implies(or(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))))), true, implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.25/57.46 = { by lemma 67 } 57.32/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(fresh40(fresh116(is_a_theorem(implies(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), implies(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), implies(or(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), implies(or(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))))), true, implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.46 = { by axiom 29 (or_3_1) } 57.32/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(fresh40(fresh116(fresh17(or_3, true, necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), implies(or(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))))), true, implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.46 = { by axiom 42 (hilbert_or_3) } 57.32/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(fresh40(fresh116(fresh17(true, true, necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), implies(or(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))))), true, implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.46 = { by axiom 19 (or_3_1) } 57.32/57.46 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(fresh40(fresh116(true, true, implies(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), implies(or(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))))), true, implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.46 = { by axiom 9 (modus_ponens_2) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(fresh40(true, true, implies(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 8 (modus_ponens_2) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(is_a_theorem(or(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 62 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(is_a_theorem(implies(not(not(necessarily(not(sK11_axiom_m9_X)))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 68 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(is_a_theorem(implies(not(and(not(necessarily(not(sK11_axiom_m9_X))), not(necessarily(not(sK11_axiom_m9_X))))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 61 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(is_a_theorem(implies(implies(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 59 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(is_a_theorem(implies(implies(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 61 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(is_a_theorem(not(and(implies(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), not(necessarily(not(sK11_axiom_m9_X)))))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 69 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(is_a_theorem(not(and(not(and(possibly(sK11_axiom_m9_X), possibly(sK11_axiom_m9_X))), not(necessarily(not(sK11_axiom_m9_X)))))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 33 (op_or) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(is_a_theorem(fresh26(op_or, true, and(possibly(sK11_axiom_m9_X), possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 46 (hilbert_op_or) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(is_a_theorem(fresh26(true, true, and(possibly(sK11_axiom_m9_X), possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 15 (op_or) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(is_a_theorem(or(and(possibly(sK11_axiom_m9_X), possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 68 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(fresh116(is_a_theorem(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X)))), true, implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 65 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(is_a_theorem(implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(or(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 62 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(is_a_theorem(implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), and(implies(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 34 (op_equiv) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(is_a_theorem(implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), fresh30(op_equiv, true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 44 (hilbert_op_equiv) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(is_a_theorem(implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), fresh30(true, true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 13 (op_equiv) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh40(is_a_theorem(implies(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), true, implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X))), equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 63 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh116(is_a_theorem(implies(necessarily(not(sK11_axiom_m9_X)), not(possibly(sK11_axiom_m9_X)))), true, equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 68 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh116(is_a_theorem(implies(necessarily(not(sK11_axiom_m9_X)), not(and(possibly(sK11_axiom_m9_X), possibly(sK11_axiom_m9_X))))), true, equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 69 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh116(is_a_theorem(implies(necessarily(not(sK11_axiom_m9_X)), implies(possibly(sK11_axiom_m9_X), necessarily(not(sK11_axiom_m9_X))))), true, equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 59 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh116(is_a_theorem(implies(necessarily(not(sK11_axiom_m9_X)), implies(not(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))))), true, equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 62 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh116(is_a_theorem(implies(necessarily(not(sK11_axiom_m9_X)), or(necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), true, equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 31 (or_1_1) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh116(fresh21(or_1, true, necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), true, equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 40 (hilbert_or_1) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh116(fresh21(true, true, necessarily(not(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))), true, equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 18 (or_1_1) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(fresh116(true, true, equiv(not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X)))), true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 9 (modus_ponens_2) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(fresh3(true, true, not(possibly(sK11_axiom_m9_X)), necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 21 (substitution_of_equivalents_2) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(necessarily(not(sK11_axiom_m9_X)))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 21 (substitution_of_equivalents_2) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(fresh3(true, true, necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 9 (modus_ponens_2) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(fresh3(fresh116(true, true, equiv(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), true, necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 4 (axiom_M_1) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(fresh3(fresh116(fresh93(true, true, necessarily(not(sK11_axiom_m9_X))), true, equiv(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), true, necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 54 (km4b_axiom_M) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(fresh3(fresh116(fresh93(axiom_M, true, necessarily(not(sK11_axiom_m9_X))), true, equiv(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), true, necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 51 (axiom_M_1) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(fresh3(fresh116(is_a_theorem(implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))), true, equiv(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), true, necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 63 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(fresh3(fresh40(is_a_theorem(implies(implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), equiv(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X)))))), true, implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), equiv(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), true, necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 66 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(fresh3(fresh40(is_a_theorem(implies(implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), and(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X)))), implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))))), true, implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), equiv(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), true, necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 65 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(fresh3(fresh40(fresh116(is_a_theorem(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), true, implies(implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), and(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X)))), implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))))), true, implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), equiv(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), true, necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 48 (axiom_4_1) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(fresh3(fresh40(fresh116(fresh101(axiom_4, true, not(sK11_axiom_m9_X)), true, implies(implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), and(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X)))), implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))))), true, implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), equiv(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), true, necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 57 (km4b_axiom_4) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(fresh3(fresh40(fresh116(fresh101(true, true, not(sK11_axiom_m9_X)), true, implies(implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), and(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X)))), implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))))), true, implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), equiv(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), true, necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 3 (axiom_4_1) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(fresh3(fresh40(fresh116(true, true, implies(implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), and(implies(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X)))), implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X)))))), true, implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), equiv(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), true, necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 9 (modus_ponens_2) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(fresh3(fresh40(true, true, implies(necessarily(necessarily(not(sK11_axiom_m9_X))), necessarily(not(sK11_axiom_m9_X))), equiv(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), true, necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 8 (modus_ponens_2) } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(fresh3(is_a_theorem(equiv(necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), true, necessarily(not(sK11_axiom_m9_X)), necessarily(necessarily(not(sK11_axiom_m9_X))))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 60 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(not(necessarily(not(sK11_axiom_m9_X))), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 59 } 57.32/57.47 fresh74(is_a_theorem(strict_implies(possibly(sK11_axiom_m9_X), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 17 (op_strict_implies) } 57.32/57.47 fresh74(is_a_theorem(fresh23(true, true, possibly(sK11_axiom_m9_X), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 58 (s1_0_op_strict_implies) } 57.32/57.47 fresh74(is_a_theorem(fresh23(op_strict_implies, true, possibly(sK11_axiom_m9_X), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 52 (op_strict_implies) } 57.32/57.47 fresh74(is_a_theorem(necessarily(implies(possibly(sK11_axiom_m9_X), possibly(sK11_axiom_m9_X)))), true) 57.32/57.47 = { by axiom 11 (necessitation_1) } 57.32/57.47 fresh74(fresh34(true, true, implies(possibly(sK11_axiom_m9_X), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 55 (km4b_necessitation) } 57.32/57.47 fresh74(fresh34(necessitation, true, implies(possibly(sK11_axiom_m9_X), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 47 (necessitation_1) } 57.32/57.47 fresh74(fresh33(is_a_theorem(implies(possibly(sK11_axiom_m9_X), possibly(sK11_axiom_m9_X))), true, implies(possibly(sK11_axiom_m9_X), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by lemma 70 } 57.32/57.47 fresh74(fresh33(true, true, implies(possibly(sK11_axiom_m9_X), possibly(sK11_axiom_m9_X))), true) 57.32/57.47 = { by axiom 12 (necessitation_1) } 57.32/57.47 fresh74(true, true) 57.32/57.47 = { by axiom 5 (axiom_m9) } 57.32/57.47 true 57.32/57.47 % SZS output end Proof 57.32/57.47 57.32/57.47 RESULT: Theorem (the conjecture is true). 57.34/57.49 EOF