TSTP Solution File: LCL549+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL549+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 : n002.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 08:19:23 EDT 2023
% Result : Theorem 51.41s 7.00s
% Output : Proof 53.00s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.11 % Problem : LCL549+1 : TPTP v8.1.2. Released v3.3.0.
% 0.10/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.11/0.33 % Computer : n002.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Fri Aug 25 00:56:02 EDT 2023
% 0.11/0.33 % CPUTime :
% 51.41/7.00 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 51.41/7.00
% 51.41/7.00 % SZS status Theorem
% 51.41/7.00
% 52.56/7.04 % SZS output start Proof
% 52.56/7.04 Take the following subset of the input axioms:
% 52.56/7.05 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 52.56/7.05 fof(and_3, axiom, and_3 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, and(X2, Y2))))).
% 52.56/7.05 fof(axiom_4, axiom, axiom_4 <=> ![X2]: is_a_theorem(implies(necessarily(X2), necessarily(necessarily(X2))))).
% 52.56/7.05 fof(axiom_B, axiom, axiom_B <=> ![X2]: is_a_theorem(implies(X2, necessarily(possibly(X2))))).
% 52.56/7.05 fof(axiom_M, axiom, axiom_M <=> ![X2]: is_a_theorem(implies(necessarily(X2), X2))).
% 52.56/7.05 fof(axiom_m10, axiom, axiom_m10 <=> ![X2]: is_a_theorem(strict_implies(possibly(X2), necessarily(possibly(X2))))).
% 52.56/7.05 fof(equivalence_3, axiom, equivalence_3 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, Y2), implies(implies(Y2, X2), equiv(X2, Y2))))).
% 52.56/7.05 fof(hilbert_and_1, axiom, and_1).
% 52.56/7.05 fof(hilbert_and_3, axiom, and_3).
% 52.56/7.05 fof(hilbert_equivalence_3, axiom, equivalence_3).
% 52.56/7.05 fof(hilbert_implies_1, axiom, implies_1).
% 52.56/7.05 fof(hilbert_implies_2, axiom, implies_2).
% 52.56/7.05 fof(hilbert_modus_ponens, axiom, modus_ponens).
% 52.56/7.05 fof(hilbert_modus_tollens, axiom, modus_tollens).
% 52.56/7.05 fof(hilbert_op_implies_and, axiom, op_implies_and).
% 52.56/7.05 fof(hilbert_op_or, axiom, op_or).
% 52.56/7.05 fof(hilbert_or_1, axiom, or_1).
% 52.56/7.05 fof(hilbert_or_3, axiom, or_3).
% 52.56/7.05 fof(implies_1, axiom, implies_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, X2)))).
% 52.56/7.05 fof(implies_2, axiom, implies_2 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, implies(X2, Y2)), implies(X2, Y2)))).
% 52.56/7.05 fof(km4b_axiom_4, axiom, axiom_4).
% 52.56/7.05 fof(km4b_axiom_B, axiom, axiom_B).
% 52.56/7.05 fof(km4b_axiom_M, axiom, axiom_M).
% 52.56/7.05 fof(km4b_necessitation, axiom, necessitation).
% 52.56/7.05 fof(km4b_op_possibly, axiom, op_possibly).
% 52.56/7.05 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 52.56/7.05 fof(modus_tollens, axiom, modus_tollens <=> ![X2, Y2]: is_a_theorem(implies(implies(not(Y2), not(X2)), implies(X2, Y2)))).
% 52.56/7.05 fof(necessitation, axiom, necessitation <=> ![X2]: (is_a_theorem(X2) => is_a_theorem(necessarily(X2)))).
% 52.56/7.05 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 52.56/7.05 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 52.56/7.05 fof(op_possibly, axiom, op_possibly => ![X2]: possibly(X2)=not(necessarily(not(X2)))).
% 52.56/7.05 fof(op_strict_implies, axiom, op_strict_implies => ![X2, Y2]: strict_implies(X2, Y2)=necessarily(implies(X2, Y2))).
% 52.56/7.05 fof(or_1, axiom, or_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, or(X2, Y2)))).
% 52.56/7.05 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))))).
% 52.56/7.05 fof(s1_0_m10_axiom_m10, conjecture, axiom_m10).
% 52.56/7.05 fof(s1_0_op_strict_implies, axiom, op_strict_implies).
% 52.56/7.05 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X2, Y2]: (is_a_theorem(equiv(X2, Y2)) => X2=Y2)).
% 52.56/7.05 fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
% 52.56/7.05
% 52.56/7.05 Now clausify the problem and encode Horn clauses using encoding 3 of
% 52.56/7.05 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 52.56/7.05 We repeatedly replace C & s=t => u=v by the two clauses:
% 52.56/7.05 fresh(y, y, x1...xn) = u
% 52.56/7.05 C => fresh(s, t, x1...xn) = v
% 52.56/7.05 where fresh is a fresh function symbol and x1..xn are the free
% 52.56/7.05 variables of u and v.
% 52.56/7.05 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 52.56/7.05 input problem has no model of domain size 1).
% 52.56/7.05
% 52.56/7.05 The encoding turns the above axioms into the following unit equations and goals:
% 52.56/7.05
% 52.56/7.05 Axiom 1 (hilbert_modus_ponens): modus_ponens = true.
% 52.56/7.05 Axiom 2 (substitution_of_equivalents): substitution_of_equivalents = true.
% 52.56/7.05 Axiom 3 (hilbert_modus_tollens): modus_tollens = true.
% 52.56/7.05 Axiom 4 (hilbert_implies_1): implies_1 = true.
% 52.56/7.05 Axiom 5 (hilbert_implies_2): implies_2 = true.
% 52.56/7.05 Axiom 6 (hilbert_and_1): and_1 = true.
% 52.56/7.05 Axiom 7 (hilbert_and_3): and_3 = true.
% 52.56/7.05 Axiom 8 (hilbert_or_1): or_1 = true.
% 52.56/7.05 Axiom 9 (hilbert_or_3): or_3 = true.
% 52.56/7.05 Axiom 10 (hilbert_equivalence_3): equivalence_3 = true.
% 52.56/7.05 Axiom 11 (hilbert_op_or): op_or = true.
% 52.56/7.05 Axiom 12 (km4b_necessitation): necessitation = true.
% 52.56/7.05 Axiom 13 (km4b_axiom_M): axiom_M = true.
% 52.56/7.05 Axiom 14 (km4b_axiom_4): axiom_4 = true.
% 52.56/7.05 Axiom 15 (km4b_axiom_B): axiom_B = true.
% 52.56/7.05 Axiom 16 (km4b_op_possibly): op_possibly = true.
% 52.56/7.05 Axiom 17 (hilbert_op_implies_and): op_implies_and = true.
% 52.56/7.05 Axiom 18 (s1_0_op_strict_implies): op_strict_implies = true.
% 52.56/7.05 Axiom 19 (axiom_m10): fresh91(X, X) = true.
% 52.56/7.05 Axiom 20 (modus_ponens_2): fresh116(X, X, Y) = true.
% 52.56/7.05 Axiom 21 (axiom_4_1): fresh101(X, X, Y) = true.
% 52.56/7.05 Axiom 22 (axiom_B_1): fresh97(X, X, Y) = true.
% 52.56/7.05 Axiom 23 (axiom_M_1): fresh93(X, X, Y) = true.
% 52.56/7.05 Axiom 24 (modus_ponens_2): fresh40(X, X, Y) = is_a_theorem(Y).
% 52.56/7.05 Axiom 25 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)).
% 52.56/7.05 Axiom 26 (necessitation_1): fresh33(X, X, Y) = true.
% 52.56/7.05 Axiom 27 (op_possibly): fresh25(X, X, Y) = possibly(Y).
% 52.56/7.05 Axiom 28 (op_possibly): fresh25(op_possibly, true, X) = not(necessarily(not(X))).
% 52.56/7.05 Axiom 29 (axiom_M_1): fresh93(axiom_M, true, X) = is_a_theorem(implies(necessarily(X), X)).
% 52.56/7.05 Axiom 30 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(modus_ponens, true, Z).
% 52.56/7.05 Axiom 31 (and_1_1): fresh107(X, X, Y, Z) = true.
% 52.56/7.05 Axiom 32 (and_3_1): fresh103(X, X, Y, Z) = true.
% 52.56/7.05 Axiom 33 (equivalence_3_1): fresh53(X, X, Y, Z) = true.
% 52.56/7.05 Axiom 34 (implies_1_1): fresh51(X, X, Y, Z) = true.
% 52.56/7.05 Axiom 35 (implies_2_1): fresh49(X, X, Y, Z) = true.
% 52.56/7.05 Axiom 36 (modus_tollens_1): fresh35(X, X, Y, Z) = true.
% 52.56/7.05 Axiom 37 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X).
% 52.56/7.05 Axiom 38 (op_implies_and): fresh29(X, X, Y, Z) = implies(Y, Z).
% 52.56/7.05 Axiom 39 (op_implies_and): fresh29(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 52.56/7.05 Axiom 40 (op_or): fresh26(X, X, Y, Z) = or(Y, Z).
% 52.56/7.05 Axiom 41 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z).
% 52.56/7.05 Axiom 42 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)).
% 52.56/7.05 Axiom 43 (or_1_1): fresh21(X, X, Y, Z) = true.
% 52.56/7.05 Axiom 44 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y.
% 52.56/7.05 Axiom 45 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z.
% 52.56/7.05 Axiom 46 (implies_1_1): fresh51(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))).
% 52.56/7.05 Axiom 47 (axiom_B_1): fresh97(axiom_B, true, X) = is_a_theorem(implies(X, necessarily(possibly(X)))).
% 52.56/7.05 Axiom 48 (or_1_1): fresh21(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
% 52.56/7.05 Axiom 49 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 52.56/7.05 Axiom 50 (op_or): fresh26(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 52.56/7.05 Axiom 51 (or_3_1): fresh17(X, X, Y, Z, W) = true.
% 52.56/7.05 Axiom 52 (axiom_4_1): fresh101(axiom_4, true, X) = is_a_theorem(implies(necessarily(X), necessarily(necessarily(X)))).
% 52.56/7.05 Axiom 53 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
% 52.56/7.05 Axiom 54 (modus_ponens_2): fresh115(is_a_theorem(implies(X, Y)), true, X, Y) = fresh40(is_a_theorem(X), true, Y).
% 52.56/7.05 Axiom 55 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y).
% 52.56/7.05 Axiom 56 (axiom_m10): fresh91(is_a_theorem(strict_implies(possibly(x), necessarily(possibly(x)))), true) = axiom_m10.
% 52.56/7.05 Axiom 57 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
% 52.56/7.05 Axiom 58 (modus_tollens_1): fresh35(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
% 52.56/7.05 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)))).
% 52.56/7.05 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)))).
% 52.56/7.05
% 52.56/7.05 Lemma 61: fresh115(X, X, Y, Z) = true.
% 52.56/7.05 Proof:
% 52.56/7.05 fresh115(X, X, Y, Z)
% 52.56/7.05 = { by axiom 30 (modus_ponens_2) }
% 52.56/7.05 fresh116(modus_ponens, true, Z)
% 52.56/7.05 = { by axiom 1 (hilbert_modus_ponens) }
% 52.56/7.05 fresh116(true, true, Z)
% 52.56/7.05 = { by axiom 20 (modus_ponens_2) }
% 52.56/7.05 true
% 52.56/7.05
% 52.56/7.05 Lemma 62: fresh40(is_a_theorem(implies(X, Y)), true, implies(implies(Y, X), equiv(X, Y))) = true.
% 52.56/7.05 Proof:
% 52.56/7.05 fresh40(is_a_theorem(implies(X, Y)), true, implies(implies(Y, X), equiv(X, Y)))
% 52.56/7.05 = { by axiom 54 (modus_ponens_2) R->L }
% 52.56/7.05 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)))
% 52.56/7.05 = { by axiom 59 (equivalence_3_1) R->L }
% 52.56/7.05 fresh115(fresh53(equivalence_3, true, X, Y), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 52.56/7.05 = { by axiom 10 (hilbert_equivalence_3) }
% 52.56/7.05 fresh115(fresh53(true, true, X, Y), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 52.56/7.05 = { by axiom 33 (equivalence_3_1) }
% 52.56/7.05 fresh115(true, true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 52.56/7.05 = { by lemma 61 }
% 52.56/7.05 true
% 52.56/7.05
% 52.56/7.05 Lemma 63: fresh40(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y)) = true.
% 52.56/7.05 Proof:
% 52.56/7.05 fresh40(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y))
% 52.56/7.05 = { by axiom 54 (modus_ponens_2) R->L }
% 52.56/7.05 fresh115(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 52.56/7.05 = { by axiom 57 (implies_2_1) R->L }
% 52.56/7.05 fresh115(fresh49(implies_2, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
% 52.56/7.05 = { by axiom 5 (hilbert_implies_2) }
% 52.56/7.05 fresh115(fresh49(true, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
% 52.56/7.05 = { by axiom 35 (implies_2_1) }
% 52.56/7.05 fresh115(true, true, implies(X, implies(X, Y)), implies(X, Y))
% 52.56/7.05 = { by lemma 61 }
% 52.56/7.05 true
% 52.56/7.05
% 52.56/7.05 Lemma 64: fresh3(is_a_theorem(equiv(X, Y)), true, X, Y) = X.
% 52.56/7.05 Proof:
% 52.56/7.05 fresh3(is_a_theorem(equiv(X, Y)), true, X, Y)
% 52.56/7.05 = { by axiom 55 (substitution_of_equivalents_2) R->L }
% 52.56/7.05 fresh4(substitution_of_equivalents, true, X, Y)
% 52.56/7.05 = { by axiom 2 (substitution_of_equivalents) }
% 52.56/7.05 fresh4(true, true, X, Y)
% 52.56/7.05 = { by axiom 44 (substitution_of_equivalents_2) }
% 52.56/7.05 X
% 52.56/7.05
% 52.56/7.05 Lemma 65: and(X, X) = X.
% 52.56/7.05 Proof:
% 52.56/7.05 and(X, X)
% 52.56/7.05 = { by axiom 45 (substitution_of_equivalents_2) R->L }
% 52.56/7.05 fresh3(true, true, X, and(X, X))
% 52.56/7.05 = { by lemma 61 R->L }
% 52.56/7.05 fresh3(fresh115(true, true, implies(and(X, X), X), equiv(X, and(X, X))), true, X, and(X, X))
% 52.56/7.05 = { by lemma 62 R->L }
% 52.56/7.05 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))
% 52.56/7.05 = { by axiom 24 (modus_ponens_2) R->L }
% 52.56/7.05 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))
% 52.56/7.05 = { by axiom 32 (and_3_1) R->L }
% 52.56/7.05 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))
% 52.56/7.05 = { by axiom 7 (hilbert_and_3) R->L }
% 52.56/7.05 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))
% 52.56/7.05 = { by axiom 53 (and_3_1) }
% 52.56/7.05 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))
% 52.56/7.05 = { by lemma 63 }
% 52.56/7.05 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))
% 52.56/7.05 = { by axiom 24 (modus_ponens_2) }
% 52.56/7.05 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))
% 52.56/7.05 = { by axiom 54 (modus_ponens_2) }
% 52.56/7.05 fresh3(fresh40(is_a_theorem(implies(and(X, X), X)), true, equiv(X, and(X, X))), true, X, and(X, X))
% 52.56/7.05 = { by axiom 49 (and_1_1) R->L }
% 52.56/7.05 fresh3(fresh40(fresh107(and_1, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X))
% 52.56/7.05 = { by axiom 6 (hilbert_and_1) }
% 52.56/7.05 fresh3(fresh40(fresh107(true, true, X, X), true, equiv(X, and(X, X))), true, X, and(X, X))
% 52.56/7.05 = { by axiom 31 (and_1_1) }
% 52.56/7.05 fresh3(fresh40(true, true, equiv(X, and(X, X))), true, X, and(X, X))
% 52.56/7.05 = { by axiom 24 (modus_ponens_2) }
% 52.56/7.05 fresh3(is_a_theorem(equiv(X, and(X, X))), true, X, and(X, X))
% 52.56/7.05 = { by lemma 64 }
% 52.56/7.05 X
% 52.56/7.05
% 52.56/7.05 Lemma 66: or(X, X) = X.
% 52.56/7.05 Proof:
% 52.56/7.05 or(X, X)
% 52.56/7.05 = { by lemma 64 R->L }
% 52.56/7.05 fresh3(is_a_theorem(equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.05 = { by axiom 24 (modus_ponens_2) R->L }
% 52.56/7.05 fresh3(fresh40(true, true, equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.05 = { by axiom 43 (or_1_1) R->L }
% 52.56/7.05 fresh3(fresh40(fresh21(true, true, X, X), true, equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.05 = { by axiom 8 (hilbert_or_1) R->L }
% 52.56/7.05 fresh3(fresh40(fresh21(or_1, true, X, X), true, equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.05 = { by axiom 48 (or_1_1) }
% 52.56/7.05 fresh3(fresh40(is_a_theorem(implies(X, or(X, X))), true, equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.05 = { by axiom 54 (modus_ponens_2) R->L }
% 52.56/7.05 fresh3(fresh115(is_a_theorem(implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.05 = { by axiom 24 (modus_ponens_2) R->L }
% 52.56/7.05 fresh3(fresh115(fresh40(true, true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.05 = { by lemma 61 R->L }
% 52.56/7.05 fresh3(fresh115(fresh40(fresh115(true, true, implies(X, X), implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.05 = { by lemma 63 R->L }
% 52.56/7.05 fresh3(fresh115(fresh40(fresh115(fresh40(is_a_theorem(implies(implies(X, X), implies(implies(X, X), implies(or(X, X), X)))), true, implies(implies(X, X), implies(or(X, X), X))), true, implies(X, X), implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.06 = { by axiom 60 (or_3_1) R->L }
% 52.56/7.06 fresh3(fresh115(fresh40(fresh115(fresh40(fresh17(or_3, true, X, X, X), true, implies(implies(X, X), implies(or(X, X), X))), true, implies(X, X), implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.06 = { by axiom 9 (hilbert_or_3) }
% 52.56/7.06 fresh3(fresh115(fresh40(fresh115(fresh40(fresh17(true, true, X, X, X), true, implies(implies(X, X), implies(or(X, X), X))), true, implies(X, X), implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.06 = { by axiom 51 (or_3_1) }
% 52.56/7.06 fresh3(fresh115(fresh40(fresh115(fresh40(true, true, implies(implies(X, X), implies(or(X, X), X))), true, implies(X, X), implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.06 = { by axiom 24 (modus_ponens_2) }
% 52.56/7.06 fresh3(fresh115(fresh40(fresh115(is_a_theorem(implies(implies(X, X), implies(or(X, X), X))), true, implies(X, X), implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.06 = { by axiom 54 (modus_ponens_2) }
% 52.56/7.06 fresh3(fresh115(fresh40(fresh40(is_a_theorem(implies(X, X)), true, implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.06 = { by axiom 24 (modus_ponens_2) R->L }
% 52.56/7.06 fresh3(fresh115(fresh40(fresh40(fresh40(true, true, implies(X, X)), true, implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.06 = { by axiom 34 (implies_1_1) R->L }
% 52.56/7.06 fresh3(fresh115(fresh40(fresh40(fresh40(fresh51(true, true, X, X), true, implies(X, X)), true, implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.06 = { by axiom 4 (hilbert_implies_1) R->L }
% 52.56/7.06 fresh3(fresh115(fresh40(fresh40(fresh40(fresh51(implies_1, true, X, X), true, implies(X, X)), true, implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.06 = { by axiom 46 (implies_1_1) }
% 52.56/7.06 fresh3(fresh115(fresh40(fresh40(fresh40(is_a_theorem(implies(X, implies(X, X))), true, implies(X, X)), true, implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.06 = { by lemma 63 }
% 52.56/7.06 fresh3(fresh115(fresh40(fresh40(true, true, implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.06 = { by axiom 24 (modus_ponens_2) }
% 52.56/7.06 fresh3(fresh115(fresh40(is_a_theorem(implies(or(X, X), X)), true, implies(implies(X, or(X, X)), equiv(or(X, X), X))), true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.06 = { by lemma 62 }
% 52.56/7.06 fresh3(fresh115(true, true, implies(X, or(X, X)), equiv(or(X, X), X)), true, or(X, X), X)
% 52.56/7.06 = { by lemma 61 }
% 52.56/7.06 fresh3(true, true, or(X, X), X)
% 52.56/7.06 = { by axiom 45 (substitution_of_equivalents_2) }
% 52.56/7.06 X
% 52.56/7.06
% 52.56/7.06 Lemma 67: not(and(X, not(Y))) = implies(X, Y).
% 52.56/7.06 Proof:
% 52.56/7.06 not(and(X, not(Y)))
% 52.56/7.06 = { by axiom 39 (op_implies_and) R->L }
% 52.56/7.06 fresh29(op_implies_and, true, X, Y)
% 52.56/7.06 = { by axiom 17 (hilbert_op_implies_and) }
% 52.56/7.06 fresh29(true, true, X, Y)
% 52.56/7.06 = { by axiom 38 (op_implies_and) }
% 52.56/7.06 implies(X, Y)
% 52.56/7.06
% 52.56/7.06 Lemma 68: implies(not(X), Y) = or(X, Y).
% 52.56/7.06 Proof:
% 52.56/7.06 implies(not(X), Y)
% 52.56/7.06 = { by lemma 67 R->L }
% 52.56/7.06 not(and(not(X), not(Y)))
% 52.56/7.06 = { by axiom 50 (op_or) R->L }
% 52.56/7.06 fresh26(op_or, true, X, Y)
% 52.56/7.06 = { by axiom 11 (hilbert_op_or) }
% 52.56/7.06 fresh26(true, true, X, Y)
% 52.56/7.06 = { by axiom 40 (op_or) }
% 52.56/7.06 or(X, Y)
% 52.56/7.06
% 52.56/7.06 Lemma 69: not(necessarily(not(X))) = possibly(X).
% 52.56/7.06 Proof:
% 52.56/7.06 not(necessarily(not(X)))
% 52.56/7.06 = { by axiom 28 (op_possibly) R->L }
% 52.56/7.06 fresh25(op_possibly, true, X)
% 52.56/7.06 = { by axiom 16 (km4b_op_possibly) }
% 52.56/7.06 fresh25(true, true, X)
% 52.56/7.06 = { by axiom 27 (op_possibly) }
% 52.56/7.06 possibly(X)
% 52.56/7.06
% 52.56/7.06 Lemma 70: or(necessarily(not(X)), Y) = implies(possibly(X), Y).
% 52.56/7.06 Proof:
% 52.56/7.06 or(necessarily(not(X)), Y)
% 52.56/7.06 = { by lemma 68 R->L }
% 52.56/7.06 implies(not(necessarily(not(X))), Y)
% 52.56/7.06 = { by lemma 69 }
% 52.56/7.06 implies(possibly(X), Y)
% 52.56/7.06
% 52.56/7.06 Lemma 71: fresh40(is_a_theorem(or(X, Y)), true, or(Y, X)) = true.
% 52.56/7.06 Proof:
% 52.56/7.06 fresh40(is_a_theorem(or(X, Y)), true, or(Y, X))
% 52.56/7.06 = { by axiom 54 (modus_ponens_2) R->L }
% 52.56/7.06 fresh115(is_a_theorem(implies(or(X, Y), or(Y, X))), true, or(X, Y), or(Y, X))
% 52.56/7.06 = { by lemma 68 R->L }
% 52.56/7.06 fresh115(is_a_theorem(implies(or(X, Y), implies(not(Y), X))), true, or(X, Y), or(Y, X))
% 52.56/7.06 = { by lemma 66 R->L }
% 52.56/7.06 fresh115(is_a_theorem(implies(or(X, or(Y, Y)), implies(not(Y), X))), true, or(X, Y), or(Y, X))
% 52.56/7.06 = { by lemma 65 R->L }
% 52.56/7.06 fresh115(is_a_theorem(implies(or(X, or(Y, Y)), implies(and(not(Y), not(Y)), X))), true, or(X, Y), or(Y, X))
% 52.56/7.06 = { by lemma 68 R->L }
% 52.56/7.06 fresh115(is_a_theorem(implies(or(X, implies(not(Y), Y)), implies(and(not(Y), not(Y)), X))), true, or(X, Y), or(Y, X))
% 52.56/7.06 = { by lemma 67 R->L }
% 52.56/7.06 fresh115(is_a_theorem(implies(or(X, not(and(not(Y), not(Y)))), implies(and(not(Y), not(Y)), X))), true, or(X, Y), or(Y, X))
% 52.56/7.06 = { by lemma 68 R->L }
% 52.56/7.06 fresh115(is_a_theorem(implies(implies(not(X), not(and(not(Y), not(Y)))), implies(and(not(Y), not(Y)), X))), true, or(X, Y), or(Y, X))
% 52.56/7.06 = { by axiom 58 (modus_tollens_1) R->L }
% 52.56/7.06 fresh115(fresh35(modus_tollens, true, and(not(Y), not(Y)), X), true, or(X, Y), or(Y, X))
% 52.56/7.06 = { by axiom 3 (hilbert_modus_tollens) }
% 52.56/7.06 fresh115(fresh35(true, true, and(not(Y), not(Y)), X), true, or(X, Y), or(Y, X))
% 52.56/7.06 = { by axiom 36 (modus_tollens_1) }
% 52.56/7.06 fresh115(true, true, or(X, Y), or(Y, X))
% 52.56/7.06 = { by lemma 61 }
% 52.56/7.06 true
% 52.56/7.06
% 52.56/7.06 Goal 1 (s1_0_m10_axiom_m10): axiom_m10 = true.
% 52.56/7.06 Proof:
% 52.56/7.06 axiom_m10
% 52.56/7.06 = { by axiom 56 (axiom_m10) R->L }
% 52.56/7.06 fresh91(is_a_theorem(strict_implies(possibly(x), necessarily(possibly(x)))), true)
% 52.56/7.06 = { by axiom 41 (op_strict_implies) R->L }
% 52.56/7.06 fresh91(is_a_theorem(fresh23(true, true, possibly(x), necessarily(possibly(x)))), true)
% 52.56/7.06 = { by axiom 18 (s1_0_op_strict_implies) R->L }
% 52.56/7.06 fresh91(is_a_theorem(fresh23(op_strict_implies, true, possibly(x), necessarily(possibly(x)))), true)
% 52.56/7.06 = { by axiom 42 (op_strict_implies) }
% 52.56/7.06 fresh91(is_a_theorem(necessarily(implies(possibly(x), necessarily(possibly(x))))), true)
% 52.56/7.06 = { by axiom 25 (necessitation_1) R->L }
% 52.56/7.06 fresh91(fresh34(true, true, implies(possibly(x), necessarily(possibly(x)))), true)
% 52.56/7.06 = { by axiom 12 (km4b_necessitation) R->L }
% 52.56/7.06 fresh91(fresh34(necessitation, true, implies(possibly(x), necessarily(possibly(x)))), true)
% 52.56/7.06 = { by axiom 37 (necessitation_1) }
% 52.56/7.06 fresh91(fresh33(is_a_theorem(implies(possibly(x), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 52.56/7.06 = { by lemma 70 R->L }
% 52.56/7.06 fresh91(fresh33(is_a_theorem(or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 52.56/7.06 = { by axiom 24 (modus_ponens_2) R->L }
% 52.56/7.06 fresh91(fresh33(fresh40(true, true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 52.56/7.06 = { by lemma 71 R->L }
% 52.56/7.06 fresh91(fresh33(fresh40(fresh40(is_a_theorem(or(necessarily(not(x)), necessarily(possibly(possibly(x))))), true, or(necessarily(possibly(possibly(x))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 52.56/7.06 = { by lemma 70 }
% 52.56/7.06 fresh91(fresh33(fresh40(fresh40(is_a_theorem(implies(possibly(x), necessarily(possibly(possibly(x))))), true, or(necessarily(possibly(possibly(x))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 52.56/7.06 = { by axiom 47 (axiom_B_1) R->L }
% 52.56/7.06 fresh91(fresh33(fresh40(fresh40(fresh97(axiom_B, true, possibly(x)), true, or(necessarily(possibly(possibly(x))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 52.56/7.06 = { by axiom 15 (km4b_axiom_B) }
% 53.00/7.06 fresh91(fresh33(fresh40(fresh40(fresh97(true, true, possibly(x)), true, or(necessarily(possibly(possibly(x))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.06 = { by axiom 22 (axiom_B_1) }
% 53.00/7.06 fresh91(fresh33(fresh40(fresh40(true, true, or(necessarily(possibly(possibly(x))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.06 = { by axiom 24 (modus_ponens_2) }
% 53.00/7.06 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(possibly(possibly(x))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.06 = { by lemma 65 R->L }
% 53.00/7.06 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(possibly(and(possibly(x), possibly(x)))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.06 = { by lemma 69 R->L }
% 53.00/7.06 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(possibly(and(not(necessarily(not(x))), possibly(x)))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.06 = { by lemma 69 R->L }
% 53.00/7.06 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(possibly(and(not(necessarily(not(x))), not(necessarily(not(x)))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.06 = { by lemma 69 R->L }
% 53.00/7.06 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(necessarily(not(and(not(necessarily(not(x))), not(necessarily(not(x)))))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.06 = { by lemma 67 }
% 53.00/7.06 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(necessarily(implies(not(necessarily(not(x))), necessarily(not(x)))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.06 = { by lemma 68 }
% 53.00/7.06 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(necessarily(or(necessarily(not(x)), necessarily(not(x)))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.06 = { by lemma 66 }
% 53.00/7.06 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(necessarily(necessarily(not(x))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.06 = { by axiom 45 (substitution_of_equivalents_2) R->L }
% 53.00/7.06 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(fresh3(true, true, necessarily(not(x)), necessarily(necessarily(not(x)))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.06 = { by lemma 61 R->L }
% 53.00/7.07 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(fresh3(fresh115(true, true, implies(necessarily(necessarily(not(x))), necessarily(not(x))), equiv(necessarily(not(x)), necessarily(necessarily(not(x))))), true, necessarily(not(x)), necessarily(necessarily(not(x)))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.07 = { by lemma 62 R->L }
% 53.00/7.07 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(fresh3(fresh115(fresh40(is_a_theorem(implies(necessarily(not(x)), necessarily(necessarily(not(x))))), true, implies(implies(necessarily(necessarily(not(x))), necessarily(not(x))), equiv(necessarily(not(x)), necessarily(necessarily(not(x)))))), true, implies(necessarily(necessarily(not(x))), necessarily(not(x))), equiv(necessarily(not(x)), necessarily(necessarily(not(x))))), true, necessarily(not(x)), necessarily(necessarily(not(x)))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.07 = { by axiom 52 (axiom_4_1) R->L }
% 53.00/7.07 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(fresh3(fresh115(fresh40(fresh101(axiom_4, true, not(x)), true, implies(implies(necessarily(necessarily(not(x))), necessarily(not(x))), equiv(necessarily(not(x)), necessarily(necessarily(not(x)))))), true, implies(necessarily(necessarily(not(x))), necessarily(not(x))), equiv(necessarily(not(x)), necessarily(necessarily(not(x))))), true, necessarily(not(x)), necessarily(necessarily(not(x)))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.07 = { by axiom 14 (km4b_axiom_4) }
% 53.00/7.07 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(fresh3(fresh115(fresh40(fresh101(true, true, not(x)), true, implies(implies(necessarily(necessarily(not(x))), necessarily(not(x))), equiv(necessarily(not(x)), necessarily(necessarily(not(x)))))), true, implies(necessarily(necessarily(not(x))), necessarily(not(x))), equiv(necessarily(not(x)), necessarily(necessarily(not(x))))), true, necessarily(not(x)), necessarily(necessarily(not(x)))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.07 = { by axiom 21 (axiom_4_1) }
% 53.00/7.07 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(fresh3(fresh115(fresh40(true, true, implies(implies(necessarily(necessarily(not(x))), necessarily(not(x))), equiv(necessarily(not(x)), necessarily(necessarily(not(x)))))), true, implies(necessarily(necessarily(not(x))), necessarily(not(x))), equiv(necessarily(not(x)), necessarily(necessarily(not(x))))), true, necessarily(not(x)), necessarily(necessarily(not(x)))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.07 = { by axiom 24 (modus_ponens_2) }
% 53.00/7.07 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(fresh3(fresh115(is_a_theorem(implies(implies(necessarily(necessarily(not(x))), necessarily(not(x))), equiv(necessarily(not(x)), necessarily(necessarily(not(x)))))), true, implies(necessarily(necessarily(not(x))), necessarily(not(x))), equiv(necessarily(not(x)), necessarily(necessarily(not(x))))), true, necessarily(not(x)), necessarily(necessarily(not(x)))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.07 = { by axiom 54 (modus_ponens_2) }
% 53.00/7.07 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(fresh3(fresh40(is_a_theorem(implies(necessarily(necessarily(not(x))), necessarily(not(x)))), true, equiv(necessarily(not(x)), necessarily(necessarily(not(x))))), true, necessarily(not(x)), necessarily(necessarily(not(x)))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.07 = { by axiom 29 (axiom_M_1) R->L }
% 53.00/7.07 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(fresh3(fresh40(fresh93(axiom_M, true, necessarily(not(x))), true, equiv(necessarily(not(x)), necessarily(necessarily(not(x))))), true, necessarily(not(x)), necessarily(necessarily(not(x)))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.07 = { by axiom 13 (km4b_axiom_M) }
% 53.00/7.07 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(fresh3(fresh40(fresh93(true, true, necessarily(not(x))), true, equiv(necessarily(not(x)), necessarily(necessarily(not(x))))), true, necessarily(not(x)), necessarily(necessarily(not(x)))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.07 = { by axiom 23 (axiom_M_1) }
% 53.00/7.07 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(fresh3(fresh40(true, true, equiv(necessarily(not(x)), necessarily(necessarily(not(x))))), true, necessarily(not(x)), necessarily(necessarily(not(x)))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.07 = { by axiom 24 (modus_ponens_2) }
% 53.00/7.07 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(fresh3(is_a_theorem(equiv(necessarily(not(x)), necessarily(necessarily(not(x))))), true, necessarily(not(x)), necessarily(necessarily(not(x)))))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.07 = { by lemma 64 }
% 53.00/7.07 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(not(necessarily(not(x)))), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.07 = { by lemma 69 }
% 53.00/7.07 fresh91(fresh33(fresh40(is_a_theorem(or(necessarily(possibly(x)), necessarily(not(x)))), true, or(necessarily(not(x)), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.07 = { by lemma 71 }
% 53.00/7.07 fresh91(fresh33(true, true, implies(possibly(x), necessarily(possibly(x)))), true)
% 53.00/7.07 = { by axiom 26 (necessitation_1) }
% 53.00/7.07 fresh91(true, true)
% 53.00/7.07 = { by axiom 19 (axiom_m10) }
% 53.00/7.07 true
% 53.00/7.07 % SZS output end Proof
% 53.00/7.07
% 53.00/7.07 RESULT: Theorem (the conjecture is true).
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