TSTP Solution File: LCL534+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL534+1 : TPTP v8.1.2. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n012.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 08:19:20 EDT 2023
% Result : Theorem 137.73s 18.02s
% Output : Proof 137.73s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LCL534+1 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n012.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 : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 01:56:12 EDT 2023
% 0.13/0.34 % CPUTime :
% 137.73/18.02 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 137.73/18.02
% 137.73/18.02 % SZS status Theorem
% 137.73/18.02
% 137.73/18.08 % SZS output start Proof
% 137.73/18.08 Take the following subset of the input axioms:
% 137.73/18.08 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 137.73/18.08 fof(and_3, axiom, and_3 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, and(X2, Y2))))).
% 137.73/18.08 fof(axiom_5, axiom, axiom_5 <=> ![X2]: is_a_theorem(implies(possibly(X2), necessarily(possibly(X2))))).
% 137.73/18.08 fof(axiom_K, axiom, axiom_K <=> ![X2, Y2]: is_a_theorem(implies(necessarily(implies(X2, Y2)), implies(necessarily(X2), necessarily(Y2))))).
% 137.73/18.08 fof(axiom_M, axiom, axiom_M <=> ![X2]: is_a_theorem(implies(necessarily(X2), X2))).
% 137.73/18.08 fof(axiom_s3, axiom, axiom_s3 <=> ![X2, Y2]: is_a_theorem(strict_implies(strict_implies(X2, Y2), strict_implies(not(possibly(Y2)), not(possibly(X2)))))).
% 137.73/18.08 fof(cn3, axiom, cn3 <=> ![P]: is_a_theorem(implies(implies(not(P), P), P))).
% 137.73/18.08 fof(equivalence_3, axiom, equivalence_3 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, Y2), implies(implies(Y2, X2), equiv(X2, Y2))))).
% 137.73/18.08 fof(hilbert_and_1, axiom, and_1).
% 137.73/18.08 fof(hilbert_and_3, axiom, and_3).
% 137.73/18.08 fof(hilbert_equivalence_3, axiom, equivalence_3).
% 137.73/18.08 fof(hilbert_implies_1, axiom, implies_1).
% 137.73/18.08 fof(hilbert_implies_2, axiom, implies_2).
% 137.73/18.08 fof(hilbert_modus_ponens, axiom, modus_ponens).
% 137.73/18.08 fof(hilbert_modus_tollens, axiom, modus_tollens).
% 137.73/18.08 fof(hilbert_op_implies_and, axiom, op_implies_and).
% 137.73/18.08 fof(hilbert_op_or, axiom, op_or).
% 137.73/18.08 fof(hilbert_or_1, axiom, or_1).
% 137.73/18.08 fof(hilbert_or_3, axiom, or_3).
% 137.73/18.08 fof(implies_1, axiom, implies_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, X2)))).
% 137.73/18.08 fof(implies_2, axiom, implies_2 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, implies(X2, Y2)), implies(X2, Y2)))).
% 137.73/18.08 fof(km5_axiom_5, axiom, axiom_5).
% 137.73/18.08 fof(km5_axiom_K, axiom, axiom_K).
% 137.73/18.08 fof(km5_axiom_M, axiom, axiom_M).
% 137.73/18.08 fof(km5_necessitation, axiom, necessitation).
% 137.73/18.08 fof(km5_op_possibly, axiom, op_possibly).
% 137.73/18.08 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 137.73/18.08 fof(modus_tollens, axiom, modus_tollens <=> ![X2, Y2]: is_a_theorem(implies(implies(not(Y2), not(X2)), implies(X2, Y2)))).
% 137.73/18.08 fof(necessitation, axiom, necessitation <=> ![X2]: (is_a_theorem(X2) => is_a_theorem(necessarily(X2)))).
% 137.73/18.08 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 137.73/18.08 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 137.73/18.08 fof(op_possibly, axiom, op_possibly => ![X2]: possibly(X2)=not(necessarily(not(X2)))).
% 137.73/18.08 fof(op_strict_implies, axiom, op_strict_implies => ![X2, Y2]: strict_implies(X2, Y2)=necessarily(implies(X2, Y2))).
% 137.73/18.08 fof(or_1, axiom, or_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, or(X2, Y2)))).
% 137.73/18.08 fof(or_3, axiom, or_3 <=> ![Z, X2, Y2]: is_a_theorem(implies(implies(X2, Z), implies(implies(Y2, Z), implies(or(X2, Y2), Z))))).
% 137.73/18.08 fof(s1_0_m6s3m9b_axiom_s3, conjecture, axiom_s3).
% 137.73/18.08 fof(s1_0_op_strict_implies, axiom, op_strict_implies).
% 137.73/18.08 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X2, Y2]: (is_a_theorem(equiv(X2, Y2)) => X2=Y2)).
% 137.73/18.08 fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
% 137.73/18.08
% 137.73/18.08 Now clausify the problem and encode Horn clauses using encoding 3 of
% 137.73/18.08 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 137.73/18.08 We repeatedly replace C & s=t => u=v by the two clauses:
% 137.73/18.08 fresh(y, y, x1...xn) = u
% 137.73/18.08 C => fresh(s, t, x1...xn) = v
% 137.73/18.08 where fresh is a fresh function symbol and x1..xn are the free
% 137.73/18.08 variables of u and v.
% 137.73/18.08 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 137.73/18.08 input problem has no model of domain size 1).
% 137.73/18.08
% 137.73/18.08 The encoding turns the above axioms into the following unit equations and goals:
% 137.73/18.08
% 137.73/18.08 Axiom 1 (hilbert_modus_ponens): modus_ponens = true.
% 137.73/18.09 Axiom 2 (substitution_of_equivalents): substitution_of_equivalents = true.
% 137.73/18.09 Axiom 3 (hilbert_modus_tollens): modus_tollens = true.
% 137.73/18.09 Axiom 4 (hilbert_implies_1): implies_1 = true.
% 137.73/18.09 Axiom 5 (hilbert_implies_2): implies_2 = true.
% 137.73/18.09 Axiom 6 (hilbert_and_1): and_1 = true.
% 137.73/18.09 Axiom 7 (hilbert_and_3): and_3 = true.
% 137.73/18.09 Axiom 8 (hilbert_or_1): or_1 = true.
% 137.73/18.09 Axiom 9 (hilbert_or_3): or_3 = true.
% 137.73/18.09 Axiom 10 (hilbert_equivalence_3): equivalence_3 = true.
% 137.73/18.09 Axiom 11 (hilbert_op_or): op_or = true.
% 137.73/18.09 Axiom 12 (km5_necessitation): necessitation = true.
% 137.73/18.09 Axiom 13 (km5_axiom_K): axiom_K = true.
% 137.73/18.09 Axiom 14 (km5_axiom_M): axiom_M = true.
% 137.73/18.09 Axiom 15 (km5_axiom_5): axiom_5 = true.
% 137.73/18.09 Axiom 16 (km5_op_possibly): op_possibly = true.
% 137.73/18.09 Axiom 17 (hilbert_op_implies_and): op_implies_and = true.
% 137.73/18.09 Axiom 18 (s1_0_op_strict_implies): op_strict_implies = true.
% 137.73/18.09 Axiom 19 (axiom_s3): fresh68(X, X) = true.
% 137.73/18.09 Axiom 20 (modus_ponens_2): fresh116(X, X, Y) = true.
% 137.73/18.09 Axiom 21 (axiom_5_1): fresh99(X, X, Y) = true.
% 137.73/18.09 Axiom 22 (axiom_M_1): fresh93(X, X, Y) = true.
% 137.73/18.09 Axiom 23 (modus_ponens_2): fresh40(X, X, Y) = is_a_theorem(Y).
% 137.73/18.09 Axiom 24 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)).
% 137.73/18.09 Axiom 25 (necessitation_1): fresh33(X, X, Y) = true.
% 137.73/18.09 Axiom 26 (op_possibly): fresh25(X, X, Y) = possibly(Y).
% 137.73/18.09 Axiom 27 (op_possibly): fresh25(op_possibly, true, X) = not(necessarily(not(X))).
% 137.73/18.09 Axiom 28 (axiom_M_1): fresh93(axiom_M, true, X) = is_a_theorem(implies(necessarily(X), X)).
% 137.73/18.09 Axiom 29 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(modus_ponens, true, Z).
% 137.73/18.09 Axiom 30 (and_1_1): fresh107(X, X, Y, Z) = true.
% 137.73/18.09 Axiom 31 (and_3_1): fresh103(X, X, Y, Z) = true.
% 137.73/18.09 Axiom 32 (axiom_K_1): fresh95(X, X, Y, Z) = true.
% 137.73/18.09 Axiom 33 (equivalence_3_1): fresh53(X, X, Y, Z) = true.
% 137.73/18.09 Axiom 34 (implies_1_1): fresh51(X, X, Y, Z) = true.
% 137.73/18.09 Axiom 35 (implies_2_1): fresh49(X, X, Y, Z) = true.
% 137.73/18.09 Axiom 36 (modus_tollens_1): fresh35(X, X, Y, Z) = true.
% 137.73/18.09 Axiom 37 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X).
% 137.73/18.09 Axiom 38 (op_implies_and): fresh29(X, X, Y, Z) = implies(Y, Z).
% 137.73/18.09 Axiom 39 (op_implies_and): fresh29(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 137.73/18.09 Axiom 40 (op_or): fresh26(X, X, Y, Z) = or(Y, Z).
% 137.73/18.09 Axiom 41 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z).
% 137.73/18.09 Axiom 42 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)).
% 137.73/18.09 Axiom 43 (or_1_1): fresh21(X, X, Y, Z) = true.
% 137.73/18.09 Axiom 44 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y.
% 137.73/18.09 Axiom 45 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z.
% 137.73/18.09 Axiom 46 (implies_1_1): fresh51(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
% 137.73/18.09 Axiom 47 (or_1_1): fresh21(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
% 137.73/18.09 Axiom 48 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 137.73/18.09 Axiom 49 (op_or): fresh26(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 137.73/18.09 Axiom 50 (or_3_1): fresh17(X, X, Y, Z, W) = true.
% 137.73/18.09 Axiom 51 (axiom_5_1): fresh99(axiom_5, true, X) = is_a_theorem(implies(possibly(X), necessarily(possibly(X)))).
% 137.73/18.09 Axiom 52 (cn3_1): fresh59(cn3, true, X) = is_a_theorem(implies(implies(not(X), X), X)).
% 137.73/18.09 Axiom 53 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
% 137.73/18.09 Axiom 54 (modus_ponens_2): fresh115(is_a_theorem(implies(X, Y)), true, X, Y) = fresh40(is_a_theorem(X), true, Y).
% 137.73/18.09 Axiom 55 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y).
% 137.73/18.09 Axiom 56 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
% 137.73/18.09 Axiom 57 (modus_tollens_1): fresh35(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
% 137.73/18.09 Axiom 58 (axiom_K_1): fresh95(axiom_K, true, X, Y) = is_a_theorem(implies(necessarily(implies(X, Y)), implies(necessarily(X), necessarily(Y)))).
% 137.73/18.09 Axiom 59 (equivalence_3_1): fresh53(equivalence_3, true, X, Y) = is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y)))).
% 137.73/18.09 Axiom 60 (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)))).
% 137.73/18.09 Axiom 61 (axiom_s3): fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), strict_implies(not(possibly(y5)), not(possibly(x10))))), true) = axiom_s3.
% 137.73/18.09
% 137.73/18.09 Lemma 62: fresh115(X, X, Y, Z) = true.
% 137.73/18.09 Proof:
% 137.73/18.09 fresh115(X, X, Y, Z)
% 137.73/18.09 = { by axiom 29 (modus_ponens_2) }
% 137.73/18.09 fresh116(modus_ponens, true, Z)
% 137.73/18.09 = { by axiom 1 (hilbert_modus_ponens) }
% 137.73/18.09 fresh116(true, true, Z)
% 137.73/18.09 = { by axiom 20 (modus_ponens_2) }
% 137.73/18.09 true
% 137.73/18.09
% 137.73/18.09 Lemma 63: fresh40(is_a_theorem(implies(X, Y)), true, implies(implies(Y, X), equiv(X, Y))) = true.
% 137.73/18.09 Proof:
% 137.73/18.09 fresh40(is_a_theorem(implies(X, Y)), true, implies(implies(Y, X), equiv(X, Y)))
% 137.73/18.09 = { by axiom 54 (modus_ponens_2) R->L }
% 137.73/18.09 fresh115(is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y)))), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 137.73/18.09 = { by axiom 59 (equivalence_3_1) R->L }
% 137.73/18.09 fresh115(fresh53(equivalence_3, true, X, Y), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 137.73/18.09 = { by axiom 10 (hilbert_equivalence_3) }
% 137.73/18.09 fresh115(fresh53(true, true, X, Y), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 137.73/18.09 = { by axiom 33 (equivalence_3_1) }
% 137.73/18.09 fresh115(true, true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 137.73/18.09 = { by lemma 62 }
% 137.73/18.09 true
% 137.73/18.09
% 137.73/18.09 Lemma 64: fresh40(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y)) = true.
% 137.73/18.09 Proof:
% 137.73/18.09 fresh40(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y))
% 137.73/18.09 = { by axiom 54 (modus_ponens_2) R->L }
% 137.73/18.09 fresh115(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 137.73/18.09 = { by axiom 56 (implies_2_1) R->L }
% 137.73/18.09 fresh115(fresh49(implies_2, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
% 137.73/18.09 = { by axiom 5 (hilbert_implies_2) }
% 137.73/18.09 fresh115(fresh49(true, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
% 137.73/18.09 = { by axiom 35 (implies_2_1) }
% 137.73/18.09 fresh115(true, true, implies(X, implies(X, Y)), implies(X, Y))
% 137.73/18.09 = { by lemma 62 }
% 137.73/18.09 true
% 137.73/18.09
% 137.73/18.09 Lemma 65: fresh3(is_a_theorem(equiv(X, Y)), true, X, Y) = X.
% 137.73/18.09 Proof:
% 137.73/18.09 fresh3(is_a_theorem(equiv(X, Y)), true, X, Y)
% 137.73/18.09 = { by axiom 55 (substitution_of_equivalents_2) R->L }
% 137.73/18.09 fresh4(substitution_of_equivalents, true, X, Y)
% 137.73/18.09 = { by axiom 2 (substitution_of_equivalents) }
% 137.73/18.09 fresh4(true, true, X, Y)
% 137.73/18.09 = { by axiom 44 (substitution_of_equivalents_2) }
% 137.73/18.09 X
% 137.73/18.09
% 137.73/18.09 Lemma 66: and(X, X) = X.
% 137.73/18.09 Proof:
% 137.73/18.09 and(X, X)
% 137.73/18.09 = { by axiom 45 (substitution_of_equivalents_2) R->L }
% 137.73/18.09 fresh3(true, true, X, and(X, X))
% 137.73/18.09 = { by lemma 62 R->L }
% 137.73/18.09 fresh3(fresh115(true, true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
% 137.73/18.09 = { by lemma 63 R->L }
% 137.73/18.09 fresh3(fresh115(fresh40(is_a_theorem(implies(X, and(X, X))), true, 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))
% 137.73/18.09 = { by axiom 23 (modus_ponens_2) R->L }
% 137.73/18.09 fresh3(fresh115(fresh40(fresh40(true, true, implies(X, and(X, X))), true, 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))
% 137.73/18.09 = { by axiom 31 (and_3_1) R->L }
% 137.73/18.09 fresh3(fresh115(fresh40(fresh40(fresh103(true, true, X, X), true, implies(X, and(X, X))), true, 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))
% 137.73/18.09 = { by axiom 7 (hilbert_and_3) R->L }
% 137.73/18.09 fresh3(fresh115(fresh40(fresh40(fresh103(and_3, true, X, X), true, implies(X, and(X, X))), true, 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))
% 137.73/18.09 = { by axiom 53 (and_3_1) }
% 137.73/18.09 fresh3(fresh115(fresh40(fresh40(is_a_theorem(implies(X, implies(X, and(X, X)))), true, implies(X, and(X, X))), true, 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))
% 137.73/18.09 = { by lemma 64 }
% 137.73/18.09 fresh3(fresh115(fresh40(true, true, 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))
% 137.73/18.09 = { by axiom 23 (modus_ponens_2) }
% 137.73/18.09 fresh3(fresh115(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))
% 137.73/18.09 = { by axiom 54 (modus_ponens_2) }
% 137.73/18.09 fresh3(fresh40(is_a_theorem(implies(and(X, X), X)), true, equiv(X, and(X, X))), true, X, and(X, X))
% 137.73/18.09 = { by axiom 48 (and_1_1) R->L }
% 137.73/18.09 fresh3(fresh40(fresh107(and_1, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X))
% 137.73/18.09 = { by axiom 6 (hilbert_and_1) }
% 137.73/18.09 fresh3(fresh40(fresh107(true, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X))
% 137.73/18.09 = { by axiom 30 (and_1_1) }
% 137.73/18.09 fresh3(fresh40(true, true, equiv(X, and(X, X))), true, X, and(X, X))
% 137.73/18.09 = { by axiom 23 (modus_ponens_2) }
% 137.73/18.09 fresh3(is_a_theorem(equiv(X, and(X, X))), true, X, and(X, X))
% 137.73/18.09 = { by lemma 65 }
% 137.73/18.09 X
% 137.73/18.09
% 137.73/18.09 Lemma 67: is_a_theorem(implies(necessarily(X), X)) = true.
% 137.73/18.09 Proof:
% 137.73/18.09 is_a_theorem(implies(necessarily(X), X))
% 137.73/18.09 = { by axiom 28 (axiom_M_1) R->L }
% 137.73/18.09 fresh93(axiom_M, true, X)
% 137.73/18.09 = { by axiom 14 (km5_axiom_M) }
% 137.73/18.09 fresh93(true, true, X)
% 137.73/18.09 = { by axiom 22 (axiom_M_1) }
% 137.73/18.09 true
% 137.73/18.09
% 137.73/18.09 Lemma 68: necessarily(possibly(X)) = possibly(X).
% 137.73/18.09 Proof:
% 137.73/18.09 necessarily(possibly(X))
% 137.73/18.09 = { by axiom 45 (substitution_of_equivalents_2) R->L }
% 137.73/18.09 fresh3(true, true, possibly(X), necessarily(possibly(X)))
% 137.73/18.09 = { by lemma 62 R->L }
% 137.73/18.09 fresh3(fresh115(true, true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
% 137.73/18.09 = { by lemma 63 R->L }
% 137.73/18.09 fresh3(fresh115(fresh40(is_a_theorem(implies(possibly(X), necessarily(possibly(X)))), true, implies(implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
% 137.73/18.09 = { by axiom 51 (axiom_5_1) R->L }
% 137.73/18.09 fresh3(fresh115(fresh40(fresh99(axiom_5, true, X), true, implies(implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
% 137.73/18.09 = { by axiom 15 (km5_axiom_5) }
% 137.73/18.09 fresh3(fresh115(fresh40(fresh99(true, true, X), true, implies(implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
% 137.73/18.09 = { by axiom 21 (axiom_5_1) }
% 137.73/18.09 fresh3(fresh115(fresh40(true, true, implies(implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
% 137.73/18.09 = { by axiom 23 (modus_ponens_2) }
% 137.73/18.09 fresh3(fresh115(is_a_theorem(implies(implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
% 137.73/18.09 = { by axiom 54 (modus_ponens_2) }
% 137.73/18.09 fresh3(fresh40(is_a_theorem(implies(necessarily(possibly(X)), possibly(X))), true, equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
% 137.73/18.09 = { by lemma 67 }
% 137.73/18.09 fresh3(fresh40(true, true, equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
% 137.73/18.09 = { by axiom 23 (modus_ponens_2) }
% 137.73/18.09 fresh3(is_a_theorem(equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X)))
% 137.73/18.09 = { by lemma 65 }
% 137.73/18.09 possibly(X)
% 137.73/18.09
% 137.73/18.09 Lemma 69: not(and(X, not(Y))) = implies(X, Y).
% 137.73/18.09 Proof:
% 137.73/18.09 not(and(X, not(Y)))
% 137.73/18.09 = { by axiom 39 (op_implies_and) R->L }
% 137.73/18.09 fresh29(op_implies_and, true, X, Y)
% 137.73/18.09 = { by axiom 17 (hilbert_op_implies_and) }
% 137.73/18.09 fresh29(true, true, X, Y)
% 137.73/18.09 = { by axiom 38 (op_implies_and) }
% 137.73/18.09 implies(X, Y)
% 137.73/18.09
% 137.73/18.09 Lemma 70: implies(not(X), Y) = or(X, Y).
% 137.73/18.09 Proof:
% 137.73/18.09 implies(not(X), Y)
% 137.73/18.09 = { by lemma 69 R->L }
% 137.73/18.09 not(and(not(X), not(Y)))
% 137.73/18.09 = { by axiom 49 (op_or) R->L }
% 137.73/18.10 fresh26(op_or, true, X, Y)
% 137.73/18.10 = { by axiom 11 (hilbert_op_or) }
% 137.73/18.10 fresh26(true, true, X, Y)
% 137.73/18.10 = { by axiom 40 (op_or) }
% 137.73/18.10 or(X, Y)
% 137.73/18.10
% 137.73/18.10 Lemma 71: is_a_theorem(implies(or(X, X), X)) = fresh59(cn3, true, X).
% 137.73/18.10 Proof:
% 137.73/18.10 is_a_theorem(implies(or(X, X), X))
% 137.73/18.10 = { by lemma 70 R->L }
% 137.73/18.10 is_a_theorem(implies(implies(not(X), X), X))
% 137.73/18.10 = { by axiom 52 (cn3_1) R->L }
% 137.73/18.10 fresh59(cn3, true, X)
% 137.73/18.10
% 137.73/18.10 Lemma 72: fresh40(is_a_theorem(implies(X, Y)), true, implies(or(X, X), Y)) = true.
% 137.73/18.10 Proof:
% 137.73/18.10 fresh40(is_a_theorem(implies(X, Y)), true, implies(or(X, X), Y))
% 137.73/18.10 = { by axiom 54 (modus_ponens_2) R->L }
% 137.73/18.10 fresh115(is_a_theorem(implies(implies(X, Y), implies(or(X, X), Y))), true, implies(X, Y), implies(or(X, X), Y))
% 137.73/18.10 = { by axiom 23 (modus_ponens_2) R->L }
% 137.73/18.10 fresh115(fresh40(true, true, implies(implies(X, Y), implies(or(X, X), Y))), true, implies(X, Y), implies(or(X, X), Y))
% 137.73/18.10 = { by axiom 50 (or_3_1) R->L }
% 137.73/18.10 fresh115(fresh40(fresh17(true, true, X, X, Y), true, implies(implies(X, Y), implies(or(X, X), Y))), true, implies(X, Y), implies(or(X, X), Y))
% 137.73/18.10 = { by axiom 9 (hilbert_or_3) R->L }
% 137.73/18.10 fresh115(fresh40(fresh17(or_3, true, X, X, Y), true, implies(implies(X, Y), implies(or(X, X), Y))), true, implies(X, Y), implies(or(X, X), Y))
% 137.73/18.10 = { by axiom 60 (or_3_1) }
% 137.73/18.10 fresh115(fresh40(is_a_theorem(implies(implies(X, Y), implies(implies(X, Y), implies(or(X, X), Y)))), true, implies(implies(X, Y), implies(or(X, X), Y))), true, implies(X, Y), implies(or(X, X), Y))
% 137.73/18.10 = { by lemma 64 }
% 137.73/18.10 fresh115(true, true, implies(X, Y), implies(or(X, X), Y))
% 137.73/18.10 = { by lemma 62 }
% 137.73/18.10 true
% 137.73/18.10
% 137.73/18.10 Lemma 73: fresh59(cn3, true, X) = true.
% 137.73/18.10 Proof:
% 137.73/18.10 fresh59(cn3, true, X)
% 137.73/18.10 = { by lemma 71 R->L }
% 137.73/18.10 is_a_theorem(implies(or(X, X), X))
% 137.73/18.10 = { by axiom 23 (modus_ponens_2) R->L }
% 137.73/18.10 fresh40(true, true, implies(or(X, X), X))
% 137.73/18.10 = { by lemma 64 R->L }
% 137.73/18.10 fresh40(fresh40(is_a_theorem(implies(X, implies(X, X))), true, implies(X, X)), true, implies(or(X, X), X))
% 137.73/18.10 = { by axiom 46 (implies_1_1) R->L }
% 137.73/18.10 fresh40(fresh40(fresh51(implies_1, true, X, X), true, implies(X, X)), true, implies(or(X, X), X))
% 137.73/18.10 = { by axiom 4 (hilbert_implies_1) }
% 137.73/18.10 fresh40(fresh40(fresh51(true, true, X, X), true, implies(X, X)), true, implies(or(X, X), X))
% 137.73/18.10 = { by axiom 34 (implies_1_1) }
% 137.73/18.10 fresh40(fresh40(true, true, implies(X, X)), true, implies(or(X, X), X))
% 137.73/18.10 = { by axiom 23 (modus_ponens_2) }
% 137.73/18.10 fresh40(is_a_theorem(implies(X, X)), true, implies(or(X, X), X))
% 137.73/18.10 = { by lemma 72 }
% 137.73/18.10 true
% 137.73/18.10
% 137.73/18.10 Lemma 74: or(X, X) = X.
% 137.73/18.10 Proof:
% 137.73/18.10 or(X, X)
% 137.73/18.10 = { by axiom 45 (substitution_of_equivalents_2) R->L }
% 137.73/18.10 fresh3(true, true, X, or(X, X))
% 137.73/18.10 = { by lemma 62 R->L }
% 137.73/18.10 fresh3(fresh115(true, true, implies(or(X, X), X), equiv(X, or(X, X))), true, X, or(X, X))
% 137.73/18.10 = { by lemma 63 R->L }
% 137.73/18.10 fresh3(fresh115(fresh40(is_a_theorem(implies(X, or(X, X))), true, implies(implies(or(X, X), X), equiv(X, or(X, X)))), true, implies(or(X, X), X), equiv(X, or(X, X))), true, X, or(X, X))
% 137.73/18.10 = { by axiom 47 (or_1_1) R->L }
% 137.73/18.10 fresh3(fresh115(fresh40(fresh21(or_1, true, X, X), true, implies(implies(or(X, X), X), equiv(X, or(X, X)))), true, implies(or(X, X), X), equiv(X, or(X, X))), true, X, or(X, X))
% 137.73/18.10 = { by axiom 8 (hilbert_or_1) }
% 137.73/18.10 fresh3(fresh115(fresh40(fresh21(true, true, X, X), true, implies(implies(or(X, X), X), equiv(X, or(X, X)))), true, implies(or(X, X), X), equiv(X, or(X, X))), true, X, or(X, X))
% 137.73/18.10 = { by axiom 43 (or_1_1) }
% 137.73/18.10 fresh3(fresh115(fresh40(true, true, implies(implies(or(X, X), X), equiv(X, or(X, X)))), true, implies(or(X, X), X), equiv(X, or(X, X))), true, X, or(X, X))
% 137.73/18.10 = { by axiom 23 (modus_ponens_2) }
% 137.73/18.10 fresh3(fresh115(is_a_theorem(implies(implies(or(X, X), X), equiv(X, or(X, X)))), true, implies(or(X, X), X), equiv(X, or(X, X))), true, X, or(X, X))
% 137.73/18.10 = { by axiom 54 (modus_ponens_2) }
% 137.73/18.10 fresh3(fresh40(is_a_theorem(implies(or(X, X), X)), true, equiv(X, or(X, X))), true, X, or(X, X))
% 137.73/18.10 = { by lemma 71 }
% 137.73/18.10 fresh3(fresh40(fresh59(cn3, true, X), true, equiv(X, or(X, X))), true, X, or(X, X))
% 137.73/18.10 = { by lemma 73 }
% 137.73/18.10 fresh3(fresh40(true, true, equiv(X, or(X, X))), true, X, or(X, X))
% 137.73/18.10 = { by axiom 23 (modus_ponens_2) }
% 137.73/18.10 fresh3(is_a_theorem(equiv(X, or(X, X))), true, X, or(X, X))
% 137.73/18.10 = { by lemma 65 }
% 137.73/18.10 X
% 137.73/18.10
% 137.73/18.10 Lemma 75: is_a_theorem(implies(or(X, not(Y)), implies(Y, X))) = true.
% 137.73/18.10 Proof:
% 137.73/18.10 is_a_theorem(implies(or(X, not(Y)), implies(Y, X)))
% 137.73/18.10 = { by lemma 70 R->L }
% 137.73/18.10 is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X)))
% 137.73/18.10 = { by axiom 57 (modus_tollens_1) R->L }
% 137.73/18.10 fresh35(modus_tollens, true, Y, X)
% 137.73/18.10 = { by axiom 3 (hilbert_modus_tollens) }
% 137.73/18.10 fresh35(true, true, Y, X)
% 137.73/18.10 = { by axiom 36 (modus_tollens_1) }
% 137.73/18.10 true
% 137.73/18.10
% 137.73/18.10 Lemma 76: not(necessarily(not(X))) = possibly(X).
% 137.73/18.10 Proof:
% 137.73/18.10 not(necessarily(not(X)))
% 137.73/18.10 = { by axiom 27 (op_possibly) R->L }
% 137.73/18.10 fresh25(op_possibly, true, X)
% 137.73/18.10 = { by axiom 16 (km5_op_possibly) }
% 137.73/18.10 fresh25(true, true, X)
% 137.73/18.10 = { by axiom 26 (op_possibly) }
% 137.73/18.10 possibly(X)
% 137.73/18.10
% 137.73/18.10 Lemma 77: implies(implies(X, necessarily(not(Y))), Z) = or(and(X, possibly(Y)), Z).
% 137.73/18.10 Proof:
% 137.73/18.10 implies(implies(X, necessarily(not(Y))), Z)
% 137.73/18.10 = { by lemma 69 R->L }
% 137.73/18.10 implies(not(and(X, not(necessarily(not(Y))))), Z)
% 137.73/18.10 = { by lemma 76 }
% 137.73/18.10 implies(not(and(X, possibly(Y))), Z)
% 137.73/18.10 = { by lemma 70 }
% 137.73/18.10 or(and(X, possibly(Y)), Z)
% 137.73/18.10
% 137.73/18.10 Lemma 78: or(necessarily(not(X)), Y) = implies(possibly(X), Y).
% 137.73/18.10 Proof:
% 137.73/18.10 or(necessarily(not(X)), Y)
% 137.73/18.10 = { by lemma 70 R->L }
% 137.73/18.10 implies(not(necessarily(not(X))), Y)
% 137.73/18.10 = { by lemma 76 }
% 137.73/18.11 implies(possibly(X), Y)
% 137.73/18.11
% 137.73/18.11 Lemma 79: fresh40(is_a_theorem(implies(possibly(X), X)), true, equiv(X, possibly(X))) = true.
% 137.73/18.11 Proof:
% 137.73/18.11 fresh40(is_a_theorem(implies(possibly(X), X)), true, equiv(X, possibly(X)))
% 137.73/18.11 = { by axiom 54 (modus_ponens_2) R->L }
% 137.73/18.11 fresh115(is_a_theorem(implies(implies(possibly(X), X), equiv(X, possibly(X)))), true, implies(possibly(X), X), equiv(X, possibly(X)))
% 137.73/18.11 = { by axiom 23 (modus_ponens_2) R->L }
% 137.73/18.11 fresh115(fresh40(true, true, implies(implies(possibly(X), X), equiv(X, possibly(X)))), true, implies(possibly(X), X), equiv(X, possibly(X)))
% 137.73/18.11 = { by lemma 62 R->L }
% 137.73/18.11 fresh115(fresh40(fresh115(true, true, or(possibly(X), not(X)), implies(X, possibly(X))), true, implies(implies(possibly(X), X), equiv(X, possibly(X)))), true, implies(possibly(X), X), equiv(X, possibly(X)))
% 137.73/18.11 = { by lemma 75 R->L }
% 137.73/18.11 fresh115(fresh40(fresh115(is_a_theorem(implies(or(possibly(X), not(X)), implies(X, possibly(X)))), true, or(possibly(X), not(X)), implies(X, possibly(X))), true, implies(implies(possibly(X), X), equiv(X, possibly(X)))), true, implies(possibly(X), X), equiv(X, possibly(X)))
% 137.73/18.11 = { by axiom 54 (modus_ponens_2) }
% 137.73/18.11 fresh115(fresh40(fresh40(is_a_theorem(or(possibly(X), not(X))), true, implies(X, possibly(X))), true, implies(implies(possibly(X), X), equiv(X, possibly(X)))), true, implies(possibly(X), X), equiv(X, possibly(X)))
% 137.73/18.11 = { by lemma 66 R->L }
% 137.73/18.11 fresh115(fresh40(fresh40(is_a_theorem(or(and(possibly(X), possibly(X)), not(X))), true, implies(X, possibly(X))), true, implies(implies(possibly(X), X), equiv(X, possibly(X)))), true, implies(possibly(X), X), equiv(X, possibly(X)))
% 137.73/18.11 = { by lemma 77 R->L }
% 137.73/18.11 fresh115(fresh40(fresh40(is_a_theorem(implies(implies(possibly(X), necessarily(not(X))), not(X))), true, implies(X, possibly(X))), true, implies(implies(possibly(X), X), equiv(X, possibly(X)))), true, implies(possibly(X), X), equiv(X, possibly(X)))
% 137.73/18.11 = { by lemma 78 R->L }
% 137.73/18.11 fresh115(fresh40(fresh40(is_a_theorem(implies(or(necessarily(not(X)), necessarily(not(X))), not(X))), true, implies(X, possibly(X))), true, implies(implies(possibly(X), X), equiv(X, possibly(X)))), true, implies(possibly(X), X), equiv(X, possibly(X)))
% 137.73/18.11 = { by axiom 23 (modus_ponens_2) R->L }
% 137.73/18.11 fresh115(fresh40(fresh40(fresh40(true, true, implies(or(necessarily(not(X)), necessarily(not(X))), not(X))), true, implies(X, possibly(X))), true, implies(implies(possibly(X), X), equiv(X, possibly(X)))), true, implies(possibly(X), X), equiv(X, possibly(X)))
% 137.73/18.11 = { by lemma 67 R->L }
% 137.73/18.11 fresh115(fresh40(fresh40(fresh40(is_a_theorem(implies(necessarily(not(X)), not(X))), true, implies(or(necessarily(not(X)), necessarily(not(X))), not(X))), true, implies(X, possibly(X))), true, implies(implies(possibly(X), X), equiv(X, possibly(X)))), true, implies(possibly(X), X), equiv(X, possibly(X)))
% 137.73/18.11 = { by lemma 72 }
% 137.73/18.11 fresh115(fresh40(fresh40(true, true, implies(X, possibly(X))), true, implies(implies(possibly(X), X), equiv(X, possibly(X)))), true, implies(possibly(X), X), equiv(X, possibly(X)))
% 137.73/18.11 = { by axiom 23 (modus_ponens_2) }
% 137.73/18.11 fresh115(fresh40(is_a_theorem(implies(X, possibly(X))), true, implies(implies(possibly(X), X), equiv(X, possibly(X)))), true, implies(possibly(X), X), equiv(X, possibly(X)))
% 137.73/18.11 = { by lemma 63 }
% 137.73/18.11 fresh115(true, true, implies(possibly(X), X), equiv(X, possibly(X)))
% 137.73/18.11 = { by lemma 62 }
% 137.73/18.11 true
% 137.73/18.11
% 137.73/18.11 Lemma 80: possibly(necessarily(not(X))) = not(possibly(X)).
% 137.73/18.11 Proof:
% 137.73/18.11 possibly(necessarily(not(X)))
% 137.73/18.11 = { by lemma 76 R->L }
% 137.73/18.11 not(necessarily(not(necessarily(not(X)))))
% 137.73/18.11 = { by lemma 76 }
% 137.73/18.11 not(necessarily(possibly(X)))
% 137.73/18.11 = { by lemma 68 }
% 137.73/18.11 not(possibly(X))
% 137.73/18.11
% 137.73/18.11 Lemma 81: is_a_theorem(or(possibly(X), necessarily(not(X)))) = true.
% 137.73/18.11 Proof:
% 137.73/18.11 is_a_theorem(or(possibly(X), necessarily(not(X))))
% 137.73/18.11 = { by lemma 66 R->L }
% 137.73/18.11 is_a_theorem(or(and(possibly(X), possibly(X)), necessarily(not(X))))
% 137.73/18.11 = { by lemma 77 R->L }
% 137.73/18.11 is_a_theorem(implies(implies(possibly(X), necessarily(not(X))), necessarily(not(X))))
% 137.73/18.11 = { by lemma 78 R->L }
% 137.73/18.11 is_a_theorem(implies(or(necessarily(not(X)), necessarily(not(X))), necessarily(not(X))))
% 137.73/18.11 = { by lemma 71 }
% 137.73/18.11 fresh59(cn3, true, necessarily(not(X)))
% 137.73/18.11 = { by lemma 73 }
% 137.73/18.11 true
% 137.73/18.11
% 137.73/18.11 Lemma 82: not(possibly(X)) = necessarily(not(X)).
% 137.73/18.11 Proof:
% 137.73/18.11 not(possibly(X))
% 137.73/18.11 = { by axiom 45 (substitution_of_equivalents_2) R->L }
% 137.73/18.11 fresh3(true, true, necessarily(not(X)), not(possibly(X)))
% 137.73/18.11 = { by lemma 79 R->L }
% 137.73/18.11 fresh3(fresh40(is_a_theorem(implies(possibly(necessarily(not(X))), necessarily(not(X)))), true, equiv(necessarily(not(X)), possibly(necessarily(not(X))))), true, necessarily(not(X)), not(possibly(X)))
% 137.73/18.11 = { by lemma 80 }
% 137.73/18.11 fresh3(fresh40(is_a_theorem(implies(not(possibly(X)), necessarily(not(X)))), true, equiv(necessarily(not(X)), possibly(necessarily(not(X))))), true, necessarily(not(X)), not(possibly(X)))
% 137.73/18.11 = { by lemma 70 }
% 137.73/18.11 fresh3(fresh40(is_a_theorem(or(possibly(X), necessarily(not(X)))), true, equiv(necessarily(not(X)), possibly(necessarily(not(X))))), true, necessarily(not(X)), not(possibly(X)))
% 137.73/18.11 = { by lemma 81 }
% 137.73/18.11 fresh3(fresh40(true, true, equiv(necessarily(not(X)), possibly(necessarily(not(X))))), true, necessarily(not(X)), not(possibly(X)))
% 137.73/18.11 = { by axiom 23 (modus_ponens_2) }
% 137.73/18.11 fresh3(is_a_theorem(equiv(necessarily(not(X)), possibly(necessarily(not(X))))), true, necessarily(not(X)), not(possibly(X)))
% 137.73/18.11 = { by lemma 80 }
% 137.73/18.11 fresh3(is_a_theorem(equiv(necessarily(not(X)), not(possibly(X)))), true, necessarily(not(X)), not(possibly(X)))
% 137.73/18.11 = { by lemma 65 }
% 137.73/18.11 necessarily(not(X))
% 137.73/18.11
% 137.73/18.11 Lemma 83: not(not(X)) = X.
% 137.73/18.11 Proof:
% 137.73/18.11 not(not(X))
% 137.73/18.11 = { by lemma 66 R->L }
% 137.73/18.11 not(and(not(X), not(X)))
% 137.73/18.11 = { by lemma 69 }
% 137.73/18.11 implies(not(X), X)
% 137.73/18.11 = { by lemma 70 }
% 137.73/18.11 or(X, X)
% 137.73/18.11 = { by lemma 74 }
% 137.73/18.11 X
% 137.73/18.11
% 137.73/18.11 Lemma 84: necessarily(implies(X, Y)) = strict_implies(X, Y).
% 137.73/18.11 Proof:
% 137.73/18.11 necessarily(implies(X, Y))
% 137.73/18.11 = { by axiom 42 (op_strict_implies) R->L }
% 137.73/18.11 fresh23(op_strict_implies, true, X, Y)
% 137.73/18.11 = { by axiom 18 (s1_0_op_strict_implies) }
% 137.73/18.11 fresh23(true, true, X, Y)
% 137.73/18.11 = { by axiom 41 (op_strict_implies) }
% 137.73/18.11 strict_implies(X, Y)
% 137.73/18.11
% 137.73/18.11 Lemma 85: fresh33(is_a_theorem(X), true, X) = is_a_theorem(necessarily(X)).
% 137.73/18.11 Proof:
% 137.73/18.11 fresh33(is_a_theorem(X), true, X)
% 137.73/18.11 = { by axiom 37 (necessitation_1) R->L }
% 137.73/18.11 fresh34(necessitation, true, X)
% 137.73/18.11 = { by axiom 12 (km5_necessitation) }
% 137.73/18.11 fresh34(true, true, X)
% 137.73/18.11 = { by axiom 24 (necessitation_1) }
% 137.73/18.11 is_a_theorem(necessarily(X))
% 137.73/18.11
% 137.73/18.11 Lemma 86: possibly(and(X, not(Y))) = not(strict_implies(X, Y)).
% 137.73/18.11 Proof:
% 137.73/18.11 possibly(and(X, not(Y)))
% 137.73/18.11 = { by lemma 76 R->L }
% 137.73/18.11 not(necessarily(not(and(X, not(Y)))))
% 137.73/18.11 = { by lemma 69 }
% 137.73/18.11 not(necessarily(implies(X, Y)))
% 137.73/18.11 = { by lemma 84 }
% 137.73/18.11 not(strict_implies(X, Y))
% 137.73/18.11
% 137.73/18.11 Lemma 87: not(not(strict_implies(X, Y))) = possibly(strict_implies(X, Y)).
% 137.73/18.11 Proof:
% 137.73/18.12 not(not(strict_implies(X, Y)))
% 137.73/18.12 = { by lemma 86 R->L }
% 137.73/18.12 not(possibly(and(X, not(Y))))
% 137.73/18.12 = { by lemma 80 R->L }
% 137.73/18.12 possibly(necessarily(not(and(X, not(Y)))))
% 137.73/18.12 = { by lemma 69 }
% 137.73/18.12 possibly(necessarily(implies(X, Y)))
% 137.73/18.12 = { by lemma 84 }
% 137.73/18.12 possibly(strict_implies(X, Y))
% 137.73/18.12
% 137.73/18.12 Lemma 88: is_a_theorem(implies(or(X, Y), or(Y, X))) = true.
% 137.73/18.12 Proof:
% 137.73/18.12 is_a_theorem(implies(or(X, Y), or(Y, X)))
% 137.73/18.12 = { by lemma 70 R->L }
% 137.73/18.12 is_a_theorem(implies(or(X, Y), implies(not(Y), X)))
% 137.73/18.12 = { by lemma 74 R->L }
% 137.73/18.12 is_a_theorem(implies(or(X, or(Y, Y)), implies(not(Y), X)))
% 137.73/18.12 = { by lemma 66 R->L }
% 137.73/18.12 is_a_theorem(implies(or(X, or(Y, Y)), implies(and(not(Y), not(Y)), X)))
% 137.73/18.12 = { by lemma 70 R->L }
% 137.73/18.12 is_a_theorem(implies(or(X, implies(not(Y), Y)), implies(and(not(Y), not(Y)), X)))
% 137.73/18.12 = { by lemma 69 R->L }
% 137.73/18.12 is_a_theorem(implies(or(X, not(and(not(Y), not(Y)))), implies(and(not(Y), not(Y)), X)))
% 137.73/18.12 = { by lemma 75 }
% 137.73/18.12 true
% 137.73/18.12
% 137.73/18.12 Lemma 89: is_a_theorem(implies(strict_implies(X, Y), implies(necessarily(X), necessarily(Y)))) = true.
% 137.73/18.12 Proof:
% 137.73/18.12 is_a_theorem(implies(strict_implies(X, Y), implies(necessarily(X), necessarily(Y))))
% 137.73/18.12 = { by lemma 84 R->L }
% 137.73/18.12 is_a_theorem(implies(necessarily(implies(X, Y)), implies(necessarily(X), necessarily(Y))))
% 137.73/18.12 = { by axiom 58 (axiom_K_1) R->L }
% 137.73/18.12 fresh95(axiom_K, true, X, Y)
% 137.73/18.12 = { by axiom 13 (km5_axiom_K) }
% 137.73/18.12 fresh95(true, true, X, Y)
% 137.73/18.12 = { by axiom 32 (axiom_K_1) }
% 137.73/18.13 true
% 137.73/18.13
% 137.73/18.13 Goal 1 (s1_0_m6s3m9b_axiom_s3): axiom_s3 = true.
% 137.73/18.13 Proof:
% 137.73/18.13 axiom_s3
% 137.73/18.13 = { by axiom 61 (axiom_s3) R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), strict_implies(not(possibly(y5)), not(possibly(x10))))), true)
% 137.73/18.13 = { by lemma 82 }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), strict_implies(necessarily(not(y5)), not(possibly(x10))))), true)
% 137.73/18.13 = { by lemma 82 }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(strict_implies(x10, y5), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 83 R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(strict_implies(not(not(x10)), y5), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 66 R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(strict_implies(not(and(not(x10), not(x10))), y5), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 69 }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(strict_implies(implies(not(x10), x10), y5), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 84 R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(necessarily(implies(implies(not(x10), x10), y5)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 69 R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(necessarily(implies(not(and(not(x10), not(x10))), y5)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 70 }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(necessarily(or(and(not(x10), not(x10)), y5)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 65 R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(necessarily(fresh3(is_a_theorem(equiv(or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), true, or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by axiom 23 (modus_ponens_2) R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(necessarily(fresh3(fresh40(true, true, equiv(or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), true, or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 88 R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(necessarily(fresh3(fresh40(is_a_theorem(implies(or(y5, and(not(x10), not(x10))), or(and(not(x10), not(x10)), y5))), true, equiv(or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), true, or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by axiom 54 (modus_ponens_2) R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(necessarily(fresh3(fresh115(is_a_theorem(implies(implies(or(y5, and(not(x10), not(x10))), or(and(not(x10), not(x10)), y5)), equiv(or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10)))))), true, implies(or(y5, and(not(x10), not(x10))), or(and(not(x10), not(x10)), y5)), equiv(or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), true, or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by axiom 23 (modus_ponens_2) R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(necessarily(fresh3(fresh115(fresh40(true, true, implies(implies(or(y5, and(not(x10), not(x10))), or(and(not(x10), not(x10)), y5)), equiv(or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10)))))), true, implies(or(y5, and(not(x10), not(x10))), or(and(not(x10), not(x10)), y5)), equiv(or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), true, or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 88 R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(necessarily(fresh3(fresh115(fresh40(is_a_theorem(implies(or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), true, implies(implies(or(y5, and(not(x10), not(x10))), or(and(not(x10), not(x10)), y5)), equiv(or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10)))))), true, implies(or(y5, and(not(x10), not(x10))), or(and(not(x10), not(x10)), y5)), equiv(or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), true, or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 63 }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(necessarily(fresh3(fresh115(true, true, implies(or(y5, and(not(x10), not(x10))), or(and(not(x10), not(x10)), y5)), equiv(or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), true, or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 62 }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(necessarily(fresh3(true, true, or(and(not(x10), not(x10)), y5), or(y5, and(not(x10), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by axiom 45 (substitution_of_equivalents_2) }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(necessarily(or(y5, and(not(x10), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 70 R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(necessarily(implies(not(y5), and(not(x10), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 84 }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), and(not(x10), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 66 }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 65 R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(fresh3(is_a_theorem(equiv(strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by axiom 23 (modus_ponens_2) R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(true, true, equiv(strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 81 R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(or(possibly(and(not(y5), not(not(x10)))), necessarily(not(and(not(y5), not(not(x10))))))), true, equiv(strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 69 }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(or(possibly(and(not(y5), not(not(x10)))), necessarily(implies(not(y5), not(x10))))), true, equiv(strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 86 }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(or(not(strict_implies(not(y5), not(x10))), necessarily(implies(not(y5), not(x10))))), true, equiv(strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 70 R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(implies(not(not(strict_implies(not(y5), not(x10)))), necessarily(implies(not(y5), not(x10))))), true, equiv(strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 87 }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(implies(possibly(strict_implies(not(y5), not(x10))), necessarily(implies(not(y5), not(x10))))), true, equiv(strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 84 }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(implies(possibly(strict_implies(not(y5), not(x10))), strict_implies(not(y5), not(x10)))), true, equiv(strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10))))), true, strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 79 }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(fresh3(true, true, strict_implies(not(y5), not(x10)), possibly(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by axiom 45 (substitution_of_equivalents_2) }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(possibly(strict_implies(not(y5), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 65 R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(fresh3(is_a_theorem(equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by axiom 23 (modus_ponens_2) R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(true, true, equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 73 R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(fresh59(cn3, true, possibly(strict_implies(not(y5), not(x10)))), true, equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 68 R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(fresh59(cn3, true, necessarily(possibly(strict_implies(not(y5), not(x10))))), true, equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.13 = { by lemma 76 R->L }
% 137.73/18.13 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(fresh59(cn3, true, necessarily(not(necessarily(not(strict_implies(not(y5), not(x10))))))), true, equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 71 R->L }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(implies(or(necessarily(not(necessarily(not(strict_implies(not(y5), not(x10)))))), necessarily(not(necessarily(not(strict_implies(not(y5), not(x10))))))), necessarily(not(necessarily(not(strict_implies(not(y5), not(x10)))))))), true, equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 78 }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(implies(implies(possibly(necessarily(not(strict_implies(not(y5), not(x10))))), necessarily(not(necessarily(not(strict_implies(not(y5), not(x10))))))), necessarily(not(necessarily(not(strict_implies(not(y5), not(x10)))))))), true, equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 77 }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(or(and(possibly(necessarily(not(strict_implies(not(y5), not(x10))))), possibly(necessarily(not(strict_implies(not(y5), not(x10)))))), necessarily(not(necessarily(not(strict_implies(not(y5), not(x10)))))))), true, equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 66 }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(or(possibly(necessarily(not(strict_implies(not(y5), not(x10))))), necessarily(not(necessarily(not(strict_implies(not(y5), not(x10)))))))), true, equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 76 }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(or(possibly(necessarily(not(strict_implies(not(y5), not(x10))))), necessarily(possibly(strict_implies(not(y5), not(x10)))))), true, equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 68 R->L }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(or(necessarily(possibly(necessarily(not(strict_implies(not(y5), not(x10)))))), necessarily(possibly(strict_implies(not(y5), not(x10)))))), true, equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 76 R->L }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(or(necessarily(not(necessarily(not(necessarily(not(strict_implies(not(y5), not(x10)))))))), necessarily(possibly(strict_implies(not(y5), not(x10)))))), true, equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 76 }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(or(necessarily(not(necessarily(possibly(strict_implies(not(y5), not(x10)))))), necessarily(possibly(strict_implies(not(y5), not(x10)))))), true, equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 78 }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(implies(possibly(necessarily(possibly(strict_implies(not(y5), not(x10))))), necessarily(possibly(strict_implies(not(y5), not(x10)))))), true, equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 68 }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(implies(possibly(possibly(strict_implies(not(y5), not(x10)))), necessarily(possibly(strict_implies(not(y5), not(x10)))))), true, equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 68 }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(implies(possibly(possibly(strict_implies(not(y5), not(x10)))), possibly(strict_implies(not(y5), not(x10))))), true, equiv(possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10)))))), true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 79 }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(fresh3(true, true, possibly(strict_implies(not(y5), not(x10))), possibly(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by axiom 45 (substitution_of_equivalents_2) }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(possibly(possibly(strict_implies(not(y5), not(x10)))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 68 R->L }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(possibly(necessarily(possibly(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 87 R->L }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(possibly(necessarily(not(not(strict_implies(not(y5), not(x10)))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 80 }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(not(possibly(not(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 82 }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(necessarily(not(not(strict_implies(not(y5), not(x10))))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 83 }
% 137.73/18.14 fresh68(is_a_theorem(strict_implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 84 R->L }
% 137.73/18.14 fresh68(is_a_theorem(necessarily(implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10)))))), true)
% 137.73/18.14 = { by lemma 85 R->L }
% 137.73/18.14 fresh68(fresh33(is_a_theorem(implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 84 R->L }
% 137.73/18.14 fresh68(fresh33(is_a_theorem(implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by axiom 23 (modus_ponens_2) R->L }
% 137.73/18.14 fresh68(fresh33(fresh40(true, true, implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by axiom 25 (necessitation_1) R->L }
% 137.73/18.14 fresh68(fresh33(fresh40(fresh33(true, true, implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 89 R->L }
% 137.73/18.14 fresh68(fresh33(fresh40(fresh33(is_a_theorem(implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 85 }
% 137.73/18.14 fresh68(fresh33(fresh40(is_a_theorem(necessarily(implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 84 }
% 137.73/18.14 fresh68(fresh33(fresh40(is_a_theorem(strict_implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by axiom 54 (modus_ponens_2) R->L }
% 137.73/18.14 fresh68(fresh33(fresh115(is_a_theorem(implies(strict_implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10)))), implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10))))))), true, strict_implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10)))), implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 89 }
% 137.73/18.14 fresh68(fresh33(fresh115(true, true, strict_implies(strict_implies(not(y5), not(x10)), implies(necessarily(not(y5)), necessarily(not(x10)))), implies(necessarily(strict_implies(not(y5), not(x10))), necessarily(implies(necessarily(not(y5)), necessarily(not(x10)))))), true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.14 = { by lemma 62 }
% 137.73/18.14 fresh68(fresh33(true, true, implies(necessarily(strict_implies(not(y5), not(x10))), strict_implies(necessarily(not(y5)), necessarily(not(x10))))), true)
% 137.73/18.15 = { by axiom 25 (necessitation_1) }
% 137.73/18.15 fresh68(true, true)
% 137.73/18.15 = { by axiom 19 (axiom_s3) }
% 137.73/18.15 true
% 137.73/18.15 % SZS output end Proof
% 137.73/18.15
% 137.73/18.15 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------