TSTP Solution File: LCL524+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL524+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 : n026.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 08:19:19 EDT 2023
% Result : Theorem 15.49s 2.36s
% Output : Proof 15.49s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL524+1 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.17/0.35 % Computer : n026.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Thu Aug 24 19:43:02 EDT 2023
% 0.17/0.35 % CPUTime :
% 15.49/2.36 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 15.49/2.36
% 15.49/2.36 % SZS status Theorem
% 15.49/2.36
% 15.49/2.39 % SZS output start Proof
% 15.49/2.39 Take the following subset of the input axioms:
% 15.49/2.39 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 15.49/2.39 fof(and_3, axiom, and_3 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, and(X2, Y2))))).
% 15.49/2.39 fof(axiom_5, axiom, axiom_5 <=> ![X2]: is_a_theorem(implies(possibly(X2), necessarily(possibly(X2))))).
% 15.49/2.39 fof(axiom_B, axiom, axiom_B <=> ![X2]: is_a_theorem(implies(X2, necessarily(possibly(X2))))).
% 15.49/2.39 fof(axiom_M, axiom, axiom_M <=> ![X2]: is_a_theorem(implies(necessarily(X2), X2))).
% 15.49/2.39 fof(equivalence_3, axiom, equivalence_3 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, Y2), implies(implies(Y2, X2), equiv(X2, Y2))))).
% 15.49/2.39 fof(hilbert_and_1, axiom, and_1).
% 15.49/2.39 fof(hilbert_and_3, axiom, and_3).
% 15.49/2.39 fof(hilbert_equivalence_3, axiom, equivalence_3).
% 15.49/2.39 fof(hilbert_implies_2, axiom, implies_2).
% 15.49/2.39 fof(hilbert_modus_ponens, axiom, modus_ponens).
% 15.49/2.39 fof(hilbert_modus_tollens, axiom, modus_tollens).
% 15.49/2.39 fof(hilbert_op_implies_and, axiom, op_implies_and).
% 15.49/2.39 fof(hilbert_op_or, axiom, op_or).
% 15.49/2.39 fof(hilbert_or_3, axiom, or_3).
% 15.49/2.39 fof(implies_2, axiom, implies_2 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, implies(X2, Y2)), implies(X2, Y2)))).
% 15.49/2.39 fof(km4b_axiom_B, conjecture, axiom_B).
% 15.49/2.39 fof(km5_axiom_5, axiom, axiom_5).
% 15.49/2.39 fof(km5_axiom_M, axiom, axiom_M).
% 15.49/2.39 fof(km5_op_possibly, axiom, op_possibly).
% 15.49/2.39 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 15.49/2.39 fof(modus_tollens, axiom, modus_tollens <=> ![X2, Y2]: is_a_theorem(implies(implies(not(Y2), not(X2)), implies(X2, Y2)))).
% 15.49/2.39 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 15.49/2.39 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 15.49/2.39 fof(op_possibly, axiom, op_possibly => ![X2]: possibly(X2)=not(necessarily(not(X2)))).
% 15.49/2.39 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))))).
% 15.49/2.39 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X2, Y2]: (is_a_theorem(equiv(X2, Y2)) => X2=Y2)).
% 15.49/2.39 fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
% 15.49/2.39
% 15.49/2.39 Now clausify the problem and encode Horn clauses using encoding 3 of
% 15.49/2.39 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 15.49/2.39 We repeatedly replace C & s=t => u=v by the two clauses:
% 15.49/2.39 fresh(y, y, x1...xn) = u
% 15.49/2.39 C => fresh(s, t, x1...xn) = v
% 15.49/2.39 where fresh is a fresh function symbol and x1..xn are the free
% 15.49/2.39 variables of u and v.
% 15.49/2.39 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 15.49/2.39 input problem has no model of domain size 1).
% 15.49/2.39
% 15.49/2.39 The encoding turns the above axioms into the following unit equations and goals:
% 15.49/2.39
% 15.49/2.39 Axiom 1 (hilbert_modus_ponens): modus_ponens = true.
% 15.49/2.39 Axiom 2 (substitution_of_equivalents): substitution_of_equivalents = true.
% 15.49/2.39 Axiom 3 (hilbert_modus_tollens): modus_tollens = true.
% 15.49/2.39 Axiom 4 (hilbert_implies_2): implies_2 = true.
% 15.49/2.39 Axiom 5 (hilbert_and_1): and_1 = true.
% 15.49/2.39 Axiom 6 (hilbert_and_3): and_3 = true.
% 15.49/2.39 Axiom 7 (hilbert_or_3): or_3 = true.
% 15.49/2.39 Axiom 8 (hilbert_equivalence_3): equivalence_3 = true.
% 15.49/2.39 Axiom 9 (km5_axiom_M): axiom_M = true.
% 15.49/2.39 Axiom 10 (km5_axiom_5): axiom_5 = true.
% 15.49/2.39 Axiom 11 (hilbert_op_or): op_or = true.
% 15.49/2.39 Axiom 12 (hilbert_op_implies_and): op_implies_and = true.
% 15.49/2.39 Axiom 13 (km5_op_possibly): op_possibly = true.
% 15.49/2.39 Axiom 14 (axiom_B): fresh98(X, X) = true.
% 15.49/2.39 Axiom 15 (modus_ponens_2): fresh116(X, X, Y) = true.
% 15.49/2.39 Axiom 16 (axiom_5_1): fresh99(X, X, Y) = true.
% 15.49/2.39 Axiom 17 (axiom_M_1): fresh93(X, X, Y) = true.
% 15.49/2.39 Axiom 18 (modus_ponens_2): fresh40(X, X, Y) = is_a_theorem(Y).
% 15.49/2.39 Axiom 19 (op_possibly): fresh25(X, X, Y) = possibly(Y).
% 15.49/2.39 Axiom 20 (op_possibly): fresh25(op_possibly, true, X) = not(necessarily(not(X))).
% 15.49/2.39 Axiom 21 (axiom_M_1): fresh93(axiom_M, true, X) = is_a_theorem(implies(necessarily(X), X)).
% 15.49/2.39 Axiom 22 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(modus_ponens, true, Z).
% 15.49/2.39 Axiom 23 (and_1_1): fresh107(X, X, Y, Z) = true.
% 15.49/2.39 Axiom 24 (and_3_1): fresh103(X, X, Y, Z) = true.
% 15.49/2.39 Axiom 25 (equivalence_3_1): fresh53(X, X, Y, Z) = true.
% 15.49/2.39 Axiom 26 (implies_2_1): fresh49(X, X, Y, Z) = true.
% 15.49/2.39 Axiom 27 (modus_tollens_1): fresh35(X, X, Y, Z) = true.
% 15.49/2.39 Axiom 28 (op_implies_and): fresh29(X, X, Y, Z) = implies(Y, Z).
% 15.49/2.39 Axiom 29 (op_implies_and): fresh29(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 15.49/2.39 Axiom 30 (op_or): fresh26(X, X, Y, Z) = or(Y, Z).
% 15.49/2.39 Axiom 31 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y.
% 15.49/2.39 Axiom 32 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z.
% 15.49/2.39 Axiom 33 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 15.49/2.39 Axiom 34 (op_or): fresh26(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 15.49/2.39 Axiom 35 (or_3_1): fresh17(X, X, Y, Z, W) = true.
% 15.49/2.39 Axiom 36 (axiom_5_1): fresh99(axiom_5, true, X) = is_a_theorem(implies(possibly(X), necessarily(possibly(X)))).
% 15.49/2.39 Axiom 37 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
% 15.49/2.39 Axiom 38 (modus_ponens_2): fresh115(is_a_theorem(implies(X, Y)), true, X, Y) = fresh40(is_a_theorem(X), true, Y).
% 15.49/2.39 Axiom 39 (axiom_B): fresh98(is_a_theorem(implies(x13, necessarily(possibly(x13)))), true) = axiom_B.
% 15.49/2.39 Axiom 40 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y).
% 15.49/2.39 Axiom 41 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
% 15.49/2.39 Axiom 42 (modus_tollens_1): fresh35(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
% 15.49/2.39 Axiom 43 (equivalence_3_1): fresh53(equivalence_3, true, X, Y) = is_a_theorem(implies(implies(X, Y), implies(implies(Y, X), equiv(X, Y)))).
% 15.49/2.39 Axiom 44 (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)))).
% 15.49/2.39
% 15.49/2.39 Lemma 45: not(and(X, not(Y))) = implies(X, Y).
% 15.49/2.39 Proof:
% 15.49/2.39 not(and(X, not(Y)))
% 15.49/2.39 = { by axiom 29 (op_implies_and) R->L }
% 15.49/2.39 fresh29(op_implies_and, true, X, Y)
% 15.49/2.39 = { by axiom 12 (hilbert_op_implies_and) }
% 15.49/2.39 fresh29(true, true, X, Y)
% 15.49/2.39 = { by axiom 28 (op_implies_and) }
% 15.49/2.39 implies(X, Y)
% 15.49/2.39
% 15.49/2.39 Lemma 46: implies(not(X), Y) = or(X, Y).
% 15.49/2.39 Proof:
% 15.49/2.39 implies(not(X), Y)
% 15.49/2.39 = { by lemma 45 R->L }
% 15.49/2.39 not(and(not(X), not(Y)))
% 15.49/2.39 = { by axiom 34 (op_or) R->L }
% 15.49/2.39 fresh26(op_or, true, X, Y)
% 15.49/2.39 = { by axiom 11 (hilbert_op_or) }
% 15.49/2.39 fresh26(true, true, X, Y)
% 15.49/2.39 = { by axiom 30 (op_or) }
% 15.49/2.39 or(X, Y)
% 15.49/2.39
% 15.49/2.39 Lemma 47: not(necessarily(not(X))) = possibly(X).
% 15.49/2.39 Proof:
% 15.49/2.39 not(necessarily(not(X)))
% 15.49/2.39 = { by axiom 20 (op_possibly) R->L }
% 15.49/2.39 fresh25(op_possibly, true, X)
% 15.49/2.39 = { by axiom 13 (km5_op_possibly) }
% 15.49/2.39 fresh25(true, true, X)
% 15.49/2.39 = { by axiom 19 (op_possibly) }
% 15.49/2.39 possibly(X)
% 15.49/2.39
% 15.49/2.39 Lemma 48: is_a_theorem(implies(necessarily(X), X)) = true.
% 15.49/2.39 Proof:
% 15.49/2.39 is_a_theorem(implies(necessarily(X), X))
% 15.49/2.39 = { by axiom 21 (axiom_M_1) R->L }
% 15.49/2.39 fresh93(axiom_M, true, X)
% 15.49/2.39 = { by axiom 9 (km5_axiom_M) }
% 15.49/2.39 fresh93(true, true, X)
% 15.49/2.39 = { by axiom 17 (axiom_M_1) }
% 15.49/2.39 true
% 15.49/2.39
% 15.49/2.39 Lemma 49: fresh115(X, X, Y, Z) = true.
% 15.49/2.39 Proof:
% 15.49/2.39 fresh115(X, X, Y, Z)
% 15.49/2.39 = { by axiom 22 (modus_ponens_2) }
% 15.49/2.39 fresh116(modus_ponens, true, Z)
% 15.49/2.39 = { by axiom 1 (hilbert_modus_ponens) }
% 15.49/2.39 fresh116(true, true, Z)
% 15.49/2.39 = { by axiom 15 (modus_ponens_2) }
% 15.49/2.39 true
% 15.49/2.39
% 15.49/2.39 Lemma 50: fresh3(is_a_theorem(equiv(X, Y)), true, X, Y) = X.
% 15.49/2.39 Proof:
% 15.49/2.39 fresh3(is_a_theorem(equiv(X, Y)), true, X, Y)
% 15.49/2.39 = { by axiom 40 (substitution_of_equivalents_2) R->L }
% 15.49/2.39 fresh4(substitution_of_equivalents, true, X, Y)
% 15.49/2.39 = { by axiom 2 (substitution_of_equivalents) }
% 15.49/2.39 fresh4(true, true, X, Y)
% 15.49/2.39 = { by axiom 31 (substitution_of_equivalents_2) }
% 15.49/2.39 X
% 15.49/2.39
% 15.49/2.39 Lemma 51: fresh40(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y)) = true.
% 15.49/2.39 Proof:
% 15.49/2.39 fresh40(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y))
% 15.49/2.39 = { by axiom 38 (modus_ponens_2) R->L }
% 15.49/2.39 fresh115(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y))
% 15.49/2.39 = { by axiom 41 (implies_2_1) R->L }
% 15.49/2.39 fresh115(fresh49(implies_2, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
% 15.49/2.39 = { by axiom 4 (hilbert_implies_2) }
% 15.49/2.39 fresh115(fresh49(true, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y))
% 15.49/2.39 = { by axiom 26 (implies_2_1) }
% 15.49/2.39 fresh115(true, true, implies(X, implies(X, Y)), implies(X, Y))
% 15.49/2.39 = { by lemma 49 }
% 15.49/2.39 true
% 15.49/2.39
% 15.49/2.39 Lemma 52: fresh40(is_a_theorem(implies(X, Y)), true, implies(implies(Y, X), equiv(X, Y))) = true.
% 15.49/2.39 Proof:
% 15.49/2.39 fresh40(is_a_theorem(implies(X, Y)), true, implies(implies(Y, X), equiv(X, Y)))
% 15.49/2.39 = { by axiom 38 (modus_ponens_2) R->L }
% 15.49/2.39 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)))
% 15.49/2.39 = { by axiom 43 (equivalence_3_1) R->L }
% 15.49/2.39 fresh115(fresh53(equivalence_3, true, X, Y), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 15.49/2.39 = { by axiom 8 (hilbert_equivalence_3) }
% 15.49/2.39 fresh115(fresh53(true, true, X, Y), true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 15.49/2.39 = { by axiom 25 (equivalence_3_1) }
% 15.49/2.39 fresh115(true, true, implies(X, Y), implies(implies(Y, X), equiv(X, Y)))
% 15.49/2.39 = { by lemma 49 }
% 15.49/2.39 true
% 15.49/2.39
% 15.49/2.39 Goal 1 (km4b_axiom_B): axiom_B = true.
% 15.49/2.39 Proof:
% 15.49/2.39 axiom_B
% 15.49/2.40 = { by axiom 39 (axiom_B) R->L }
% 15.49/2.40 fresh98(is_a_theorem(implies(x13, necessarily(possibly(x13)))), true)
% 15.49/2.40 = { by axiom 32 (substitution_of_equivalents_2) R->L }
% 15.49/2.40 fresh98(is_a_theorem(implies(x13, fresh3(true, true, possibly(x13), necessarily(possibly(x13))))), true)
% 15.49/2.40 = { by lemma 49 R->L }
% 15.49/2.40 fresh98(is_a_theorem(implies(x13, fresh3(fresh115(true, true, implies(necessarily(possibly(x13)), possibly(x13)), equiv(possibly(x13), necessarily(possibly(x13)))), true, possibly(x13), necessarily(possibly(x13))))), true)
% 15.49/2.40 = { by lemma 52 R->L }
% 15.49/2.40 fresh98(is_a_theorem(implies(x13, fresh3(fresh115(fresh40(is_a_theorem(implies(possibly(x13), necessarily(possibly(x13)))), true, implies(implies(necessarily(possibly(x13)), possibly(x13)), equiv(possibly(x13), necessarily(possibly(x13))))), true, implies(necessarily(possibly(x13)), possibly(x13)), equiv(possibly(x13), necessarily(possibly(x13)))), true, possibly(x13), necessarily(possibly(x13))))), true)
% 15.49/2.40 = { by axiom 36 (axiom_5_1) R->L }
% 15.49/2.40 fresh98(is_a_theorem(implies(x13, fresh3(fresh115(fresh40(fresh99(axiom_5, true, x13), true, implies(implies(necessarily(possibly(x13)), possibly(x13)), equiv(possibly(x13), necessarily(possibly(x13))))), true, implies(necessarily(possibly(x13)), possibly(x13)), equiv(possibly(x13), necessarily(possibly(x13)))), true, possibly(x13), necessarily(possibly(x13))))), true)
% 15.49/2.40 = { by axiom 10 (km5_axiom_5) }
% 15.49/2.40 fresh98(is_a_theorem(implies(x13, fresh3(fresh115(fresh40(fresh99(true, true, x13), true, implies(implies(necessarily(possibly(x13)), possibly(x13)), equiv(possibly(x13), necessarily(possibly(x13))))), true, implies(necessarily(possibly(x13)), possibly(x13)), equiv(possibly(x13), necessarily(possibly(x13)))), true, possibly(x13), necessarily(possibly(x13))))), true)
% 15.49/2.40 = { by axiom 16 (axiom_5_1) }
% 15.49/2.40 fresh98(is_a_theorem(implies(x13, fresh3(fresh115(fresh40(true, true, implies(implies(necessarily(possibly(x13)), possibly(x13)), equiv(possibly(x13), necessarily(possibly(x13))))), true, implies(necessarily(possibly(x13)), possibly(x13)), equiv(possibly(x13), necessarily(possibly(x13)))), true, possibly(x13), necessarily(possibly(x13))))), true)
% 15.49/2.40 = { by axiom 18 (modus_ponens_2) }
% 15.49/2.40 fresh98(is_a_theorem(implies(x13, fresh3(fresh115(is_a_theorem(implies(implies(necessarily(possibly(x13)), possibly(x13)), equiv(possibly(x13), necessarily(possibly(x13))))), true, implies(necessarily(possibly(x13)), possibly(x13)), equiv(possibly(x13), necessarily(possibly(x13)))), true, possibly(x13), necessarily(possibly(x13))))), true)
% 15.49/2.40 = { by axiom 38 (modus_ponens_2) }
% 15.49/2.40 fresh98(is_a_theorem(implies(x13, fresh3(fresh40(is_a_theorem(implies(necessarily(possibly(x13)), possibly(x13))), true, equiv(possibly(x13), necessarily(possibly(x13)))), true, possibly(x13), necessarily(possibly(x13))))), true)
% 15.49/2.40 = { by lemma 48 }
% 15.49/2.40 fresh98(is_a_theorem(implies(x13, fresh3(fresh40(true, true, equiv(possibly(x13), necessarily(possibly(x13)))), true, possibly(x13), necessarily(possibly(x13))))), true)
% 15.49/2.40 = { by axiom 18 (modus_ponens_2) }
% 15.49/2.40 fresh98(is_a_theorem(implies(x13, fresh3(is_a_theorem(equiv(possibly(x13), necessarily(possibly(x13)))), true, possibly(x13), necessarily(possibly(x13))))), true)
% 15.49/2.40 = { by lemma 50 }
% 15.49/2.40 fresh98(is_a_theorem(implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 18 (modus_ponens_2) R->L }
% 15.49/2.40 fresh98(fresh40(true, true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by lemma 49 R->L }
% 15.49/2.40 fresh98(fresh40(fresh115(true, true, implies(necessarily(not(x13)), not(x13)), implies(or(necessarily(not(x13)), necessarily(not(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by lemma 51 R->L }
% 15.49/2.40 fresh98(fresh40(fresh115(fresh40(is_a_theorem(implies(implies(necessarily(not(x13)), not(x13)), implies(implies(necessarily(not(x13)), not(x13)), implies(or(necessarily(not(x13)), necessarily(not(x13))), not(x13))))), true, implies(implies(necessarily(not(x13)), not(x13)), implies(or(necessarily(not(x13)), necessarily(not(x13))), not(x13)))), true, implies(necessarily(not(x13)), not(x13)), implies(or(necessarily(not(x13)), necessarily(not(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 44 (or_3_1) R->L }
% 15.49/2.40 fresh98(fresh40(fresh115(fresh40(fresh17(or_3, true, necessarily(not(x13)), necessarily(not(x13)), not(x13)), true, implies(implies(necessarily(not(x13)), not(x13)), implies(or(necessarily(not(x13)), necessarily(not(x13))), not(x13)))), true, implies(necessarily(not(x13)), not(x13)), implies(or(necessarily(not(x13)), necessarily(not(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 7 (hilbert_or_3) }
% 15.49/2.40 fresh98(fresh40(fresh115(fresh40(fresh17(true, true, necessarily(not(x13)), necessarily(not(x13)), not(x13)), true, implies(implies(necessarily(not(x13)), not(x13)), implies(or(necessarily(not(x13)), necessarily(not(x13))), not(x13)))), true, implies(necessarily(not(x13)), not(x13)), implies(or(necessarily(not(x13)), necessarily(not(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 35 (or_3_1) }
% 15.49/2.40 fresh98(fresh40(fresh115(fresh40(true, true, implies(implies(necessarily(not(x13)), not(x13)), implies(or(necessarily(not(x13)), necessarily(not(x13))), not(x13)))), true, implies(necessarily(not(x13)), not(x13)), implies(or(necessarily(not(x13)), necessarily(not(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 18 (modus_ponens_2) }
% 15.49/2.40 fresh98(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x13)), not(x13)), implies(or(necessarily(not(x13)), necessarily(not(x13))), not(x13)))), true, implies(necessarily(not(x13)), not(x13)), implies(or(necessarily(not(x13)), necessarily(not(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 38 (modus_ponens_2) }
% 15.49/2.40 fresh98(fresh40(fresh40(is_a_theorem(implies(necessarily(not(x13)), not(x13))), true, implies(or(necessarily(not(x13)), necessarily(not(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by lemma 48 }
% 15.49/2.40 fresh98(fresh40(fresh40(true, true, implies(or(necessarily(not(x13)), necessarily(not(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 18 (modus_ponens_2) }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(implies(or(necessarily(not(x13)), necessarily(not(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by lemma 46 R->L }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(implies(implies(not(necessarily(not(x13))), necessarily(not(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by lemma 47 }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(implies(implies(possibly(x13), necessarily(not(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by lemma 45 R->L }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(implies(not(and(possibly(x13), not(necessarily(not(x13))))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by lemma 47 }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(implies(not(and(possibly(x13), possibly(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by lemma 46 }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(and(possibly(x13), possibly(x13)), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 32 (substitution_of_equivalents_2) R->L }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(fresh3(true, true, possibly(x13), and(possibly(x13), possibly(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by lemma 49 R->L }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(fresh3(fresh115(true, true, implies(and(possibly(x13), possibly(x13)), possibly(x13)), equiv(possibly(x13), and(possibly(x13), possibly(x13)))), true, possibly(x13), and(possibly(x13), possibly(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by lemma 52 R->L }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(fresh3(fresh115(fresh40(is_a_theorem(implies(possibly(x13), and(possibly(x13), possibly(x13)))), true, implies(implies(and(possibly(x13), possibly(x13)), possibly(x13)), equiv(possibly(x13), and(possibly(x13), possibly(x13))))), true, implies(and(possibly(x13), possibly(x13)), possibly(x13)), equiv(possibly(x13), and(possibly(x13), possibly(x13)))), true, possibly(x13), and(possibly(x13), possibly(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 18 (modus_ponens_2) R->L }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(fresh3(fresh115(fresh40(fresh40(true, true, implies(possibly(x13), and(possibly(x13), possibly(x13)))), true, implies(implies(and(possibly(x13), possibly(x13)), possibly(x13)), equiv(possibly(x13), and(possibly(x13), possibly(x13))))), true, implies(and(possibly(x13), possibly(x13)), possibly(x13)), equiv(possibly(x13), and(possibly(x13), possibly(x13)))), true, possibly(x13), and(possibly(x13), possibly(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 24 (and_3_1) R->L }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(fresh3(fresh115(fresh40(fresh40(fresh103(true, true, possibly(x13), possibly(x13)), true, implies(possibly(x13), and(possibly(x13), possibly(x13)))), true, implies(implies(and(possibly(x13), possibly(x13)), possibly(x13)), equiv(possibly(x13), and(possibly(x13), possibly(x13))))), true, implies(and(possibly(x13), possibly(x13)), possibly(x13)), equiv(possibly(x13), and(possibly(x13), possibly(x13)))), true, possibly(x13), and(possibly(x13), possibly(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 6 (hilbert_and_3) R->L }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(fresh3(fresh115(fresh40(fresh40(fresh103(and_3, true, possibly(x13), possibly(x13)), true, implies(possibly(x13), and(possibly(x13), possibly(x13)))), true, implies(implies(and(possibly(x13), possibly(x13)), possibly(x13)), equiv(possibly(x13), and(possibly(x13), possibly(x13))))), true, implies(and(possibly(x13), possibly(x13)), possibly(x13)), equiv(possibly(x13), and(possibly(x13), possibly(x13)))), true, possibly(x13), and(possibly(x13), possibly(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 37 (and_3_1) }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(fresh3(fresh115(fresh40(fresh40(is_a_theorem(implies(possibly(x13), implies(possibly(x13), and(possibly(x13), possibly(x13))))), true, implies(possibly(x13), and(possibly(x13), possibly(x13)))), true, implies(implies(and(possibly(x13), possibly(x13)), possibly(x13)), equiv(possibly(x13), and(possibly(x13), possibly(x13))))), true, implies(and(possibly(x13), possibly(x13)), possibly(x13)), equiv(possibly(x13), and(possibly(x13), possibly(x13)))), true, possibly(x13), and(possibly(x13), possibly(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by lemma 51 }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(fresh3(fresh115(fresh40(true, true, implies(implies(and(possibly(x13), possibly(x13)), possibly(x13)), equiv(possibly(x13), and(possibly(x13), possibly(x13))))), true, implies(and(possibly(x13), possibly(x13)), possibly(x13)), equiv(possibly(x13), and(possibly(x13), possibly(x13)))), true, possibly(x13), and(possibly(x13), possibly(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 18 (modus_ponens_2) }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(fresh3(fresh115(is_a_theorem(implies(implies(and(possibly(x13), possibly(x13)), possibly(x13)), equiv(possibly(x13), and(possibly(x13), possibly(x13))))), true, implies(and(possibly(x13), possibly(x13)), possibly(x13)), equiv(possibly(x13), and(possibly(x13), possibly(x13)))), true, possibly(x13), and(possibly(x13), possibly(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 38 (modus_ponens_2) }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(fresh3(fresh40(is_a_theorem(implies(and(possibly(x13), possibly(x13)), possibly(x13))), true, equiv(possibly(x13), and(possibly(x13), possibly(x13)))), true, possibly(x13), and(possibly(x13), possibly(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 33 (and_1_1) R->L }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(fresh3(fresh40(fresh107(and_1, true, possibly(x13), possibly(x13)), true, equiv(possibly(x13), and(possibly(x13), possibly(x13)))), true, possibly(x13), and(possibly(x13), possibly(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 5 (hilbert_and_1) }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(fresh3(fresh40(fresh107(true, true, possibly(x13), possibly(x13)), true, equiv(possibly(x13), and(possibly(x13), possibly(x13)))), true, possibly(x13), and(possibly(x13), possibly(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 23 (and_1_1) }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(fresh3(fresh40(true, true, equiv(possibly(x13), and(possibly(x13), possibly(x13)))), true, possibly(x13), and(possibly(x13), possibly(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 18 (modus_ponens_2) }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(fresh3(is_a_theorem(equiv(possibly(x13), and(possibly(x13), possibly(x13)))), true, possibly(x13), and(possibly(x13), possibly(x13))), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by lemma 50 }
% 15.49/2.40 fresh98(fresh40(is_a_theorem(or(possibly(x13), not(x13))), true, implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 38 (modus_ponens_2) R->L }
% 15.49/2.40 fresh98(fresh115(is_a_theorem(implies(or(possibly(x13), not(x13)), implies(x13, possibly(x13)))), true, or(possibly(x13), not(x13)), implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by lemma 46 R->L }
% 15.49/2.40 fresh98(fresh115(is_a_theorem(implies(implies(not(possibly(x13)), not(x13)), implies(x13, possibly(x13)))), true, or(possibly(x13), not(x13)), implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 42 (modus_tollens_1) R->L }
% 15.49/2.40 fresh98(fresh115(fresh35(modus_tollens, true, x13, possibly(x13)), true, or(possibly(x13), not(x13)), implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 3 (hilbert_modus_tollens) }
% 15.49/2.40 fresh98(fresh115(fresh35(true, true, x13, possibly(x13)), true, or(possibly(x13), not(x13)), implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by axiom 27 (modus_tollens_1) }
% 15.49/2.40 fresh98(fresh115(true, true, or(possibly(x13), not(x13)), implies(x13, possibly(x13))), true)
% 15.49/2.40 = { by lemma 49 }
% 15.49/2.40 fresh98(true, true)
% 15.49/2.40 = { by axiom 14 (axiom_B) }
% 15.49/2.40 true
% 15.49/2.40 % SZS output end Proof
% 15.49/2.40
% 15.49/2.40 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------