TSTP Solution File: LCL533+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL533+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 : n013.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 20.33s 3.03s
% Output : Proof 21.51s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.13 % Problem : LCL533+1 : TPTP v8.1.2. Released v3.3.0.
% 0.11/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.37 % Computer : n013.cluster.edu
% 0.14/0.37 % Model : x86_64 x86_64
% 0.14/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.37 % Memory : 8042.1875MB
% 0.14/0.37 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.37 % CPULimit : 300
% 0.14/0.37 % WCLimit : 300
% 0.14/0.37 % DateTime : Thu Aug 24 18:16:47 EDT 2023
% 0.14/0.38 % CPUTime :
% 20.33/3.03 Command-line arguments: --no-flatten-goal
% 20.33/3.03
% 20.33/3.03 % SZS status Theorem
% 20.33/3.03
% 21.12/3.08 % SZS output start Proof
% 21.12/3.08 Take the following subset of the input axioms:
% 21.12/3.08 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 21.12/3.08 fof(and_3, axiom, and_3 <=> ![X2, Y2]: is_a_theorem(implies(X2, implies(Y2, and(X2, Y2))))).
% 21.12/3.08 fof(axiom_M, axiom, axiom_M <=> ![X2]: is_a_theorem(implies(necessarily(X2), X2))).
% 21.12/3.08 fof(axiom_m6, axiom, axiom_m6 <=> ![X2]: is_a_theorem(strict_implies(X2, possibly(X2)))).
% 21.12/3.08 fof(hilbert_and_1, axiom, and_1).
% 21.12/3.08 fof(hilbert_and_3, axiom, and_3).
% 21.12/3.08 fof(hilbert_implies_2, axiom, implies_2).
% 21.12/3.08 fof(hilbert_modus_ponens, axiom, modus_ponens).
% 21.12/3.08 fof(hilbert_modus_tollens, axiom, modus_tollens).
% 21.12/3.08 fof(hilbert_op_equiv, axiom, op_equiv).
% 21.12/3.08 fof(hilbert_op_implies_and, axiom, op_implies_and).
% 21.12/3.08 fof(hilbert_op_or, axiom, op_or).
% 21.12/3.08 fof(hilbert_or_3, axiom, or_3).
% 21.12/3.08 fof(implies_2, axiom, implies_2 <=> ![X2, Y2]: is_a_theorem(implies(implies(X2, implies(X2, Y2)), implies(X2, Y2)))).
% 21.12/3.08 fof(km5_axiom_M, axiom, axiom_M).
% 21.12/3.08 fof(km5_necessitation, axiom, necessitation).
% 21.12/3.08 fof(km5_op_possibly, axiom, op_possibly).
% 21.12/3.08 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 21.12/3.08 fof(modus_tollens, axiom, modus_tollens <=> ![X2, Y2]: is_a_theorem(implies(implies(not(Y2), not(X2)), implies(X2, Y2)))).
% 21.12/3.08 fof(necessitation, axiom, necessitation <=> ![X2]: (is_a_theorem(X2) => is_a_theorem(necessarily(X2)))).
% 21.12/3.08 fof(op_equiv, axiom, op_equiv => ![X2, Y2]: equiv(X2, Y2)=and(implies(X2, Y2), implies(Y2, X2))).
% 21.12/3.08 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 21.12/3.08 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 21.12/3.08 fof(op_possibly, axiom, op_possibly => ![X2]: possibly(X2)=not(necessarily(not(X2)))).
% 21.12/3.08 fof(op_strict_implies, axiom, op_strict_implies => ![X2, Y2]: strict_implies(X2, Y2)=necessarily(implies(X2, Y2))).
% 21.12/3.08 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))))).
% 21.12/3.08 fof(s1_0_m6s3m9b_axiom_m6, conjecture, axiom_m6).
% 21.12/3.08 fof(s1_0_op_implies, axiom, op_implies).
% 21.12/3.08 fof(s1_0_op_strict_implies, axiom, op_strict_implies).
% 21.12/3.08 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X2, Y2]: (is_a_theorem(equiv(X2, Y2)) => X2=Y2)).
% 21.12/3.08 fof(substitution_of_equivalents, axiom, substitution_of_equivalents).
% 21.12/3.08
% 21.12/3.08 Now clausify the problem and encode Horn clauses using encoding 3 of
% 21.12/3.08 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 21.12/3.08 We repeatedly replace C & s=t => u=v by the two clauses:
% 21.12/3.08 fresh(y, y, x1...xn) = u
% 21.12/3.08 C => fresh(s, t, x1...xn) = v
% 21.12/3.08 where fresh is a fresh function symbol and x1..xn are the free
% 21.12/3.08 variables of u and v.
% 21.12/3.08 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 21.12/3.08 input problem has no model of domain size 1).
% 21.12/3.08
% 21.12/3.08 The encoding turns the above axioms into the following unit equations and goals:
% 21.12/3.08
% 21.12/3.08 Axiom 1 (s1_0_op_implies): op_implies = true.
% 21.12/3.08 Axiom 2 (hilbert_op_implies_and): op_implies_and = true.
% 21.12/3.08 Axiom 3 (s1_0_op_strict_implies): op_strict_implies = true.
% 21.12/3.08 Axiom 4 (hilbert_modus_ponens): modus_ponens = true.
% 21.12/3.08 Axiom 5 (substitution_of_equivalents): substitution_of_equivalents = true.
% 21.12/3.08 Axiom 6 (hilbert_modus_tollens): modus_tollens = true.
% 21.12/3.08 Axiom 7 (hilbert_implies_2): implies_2 = true.
% 21.12/3.08 Axiom 8 (hilbert_and_1): and_1 = true.
% 21.12/3.08 Axiom 9 (hilbert_and_3): and_3 = true.
% 21.12/3.08 Axiom 10 (hilbert_or_3): or_3 = true.
% 21.12/3.08 Axiom 11 (hilbert_op_or): op_or = true.
% 21.12/3.08 Axiom 12 (hilbert_op_equiv): op_equiv = true.
% 21.12/3.08 Axiom 13 (km5_necessitation): necessitation = true.
% 21.12/3.08 Axiom 14 (km5_axiom_M): axiom_M = true.
% 21.12/3.08 Axiom 15 (km5_op_possibly): op_possibly = true.
% 21.12/3.08 Axiom 16 (axiom_m6): fresh80(X, X) = true.
% 21.12/3.08 Axiom 17 (modus_ponens_2): fresh116(X, X, Y) = true.
% 21.12/3.08 Axiom 18 (axiom_M_1): fresh93(X, X, Y) = true.
% 21.12/3.08 Axiom 19 (modus_ponens_2): fresh40(X, X, Y) = is_a_theorem(Y).
% 21.12/3.08 Axiom 20 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)).
% 21.12/3.08 Axiom 21 (necessitation_1): fresh33(X, X, Y) = true.
% 21.12/3.08 Axiom 22 (op_possibly): fresh25(X, X, Y) = possibly(Y).
% 21.12/3.08 Axiom 23 (op_possibly): fresh25(op_possibly, true, X) = not(necessarily(not(X))).
% 21.12/3.08 Axiom 24 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(modus_ponens, true, Z).
% 21.12/3.08 Axiom 25 (and_1_1): fresh107(X, X, Y, Z) = true.
% 21.12/3.08 Axiom 26 (and_3_1): fresh103(X, X, Y, Z) = true.
% 21.12/3.08 Axiom 27 (implies_2_1): fresh49(X, X, Y, Z) = true.
% 21.12/3.08 Axiom 28 (modus_tollens_1): fresh35(X, X, Y, Z) = true.
% 21.12/3.08 Axiom 29 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X).
% 21.12/3.08 Axiom 30 (op_equiv): fresh30(X, X, Y, Z) = equiv(Y, Z).
% 21.12/3.08 Axiom 31 (op_implies_and): fresh29(X, X, Y, Z) = implies(Y, Z).
% 21.12/3.08 Axiom 32 (op_or): fresh26(X, X, Y, Z) = or(Y, Z).
% 21.12/3.08 Axiom 33 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z).
% 21.12/3.08 Axiom 34 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)).
% 21.12/3.08 Axiom 35 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y.
% 21.12/3.08 Axiom 36 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z.
% 21.12/3.08 Axiom 37 (op_implies_and): fresh29(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 21.12/3.08 Axiom 38 (axiom_m6_1): fresh79(axiom_m6, true, X) = is_a_theorem(strict_implies(X, possibly(X))).
% 21.12/3.08 Axiom 39 (axiom_M_1): fresh93(axiom_M, true, X) = is_a_theorem(implies(necessarily(X), X)).
% 21.12/3.08 Axiom 40 (or_3_1): fresh17(X, X, Y, Z, W) = true.
% 21.12/3.08 Axiom 41 (op_or): fresh26(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 21.12/3.08 Axiom 42 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 21.12/3.08 Axiom 43 (axiom_m6): fresh80(is_a_theorem(strict_implies(x3, possibly(x3))), true) = axiom_m6.
% 21.12/3.08 Axiom 44 (op_equiv): fresh30(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)).
% 21.12/3.08 Axiom 45 (modus_ponens_2): fresh115(is_a_theorem(implies(X, Y)), true, X, Y) = fresh40(is_a_theorem(X), true, Y).
% 21.12/3.08 Axiom 46 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y).
% 21.12/3.08 Axiom 47 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))).
% 21.12/3.08 Axiom 48 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))).
% 21.12/3.08 Axiom 49 (modus_tollens_1): fresh35(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))).
% 21.12/3.08 Axiom 50 (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)))).
% 21.12/3.08
% 21.12/3.08 Lemma 51: not(and(X, not(Y))) = implies(X, Y).
% 21.12/3.08 Proof:
% 21.12/3.08 not(and(X, not(Y)))
% 21.12/3.08 = { by axiom 37 (op_implies_and) R->L }
% 21.12/3.08 fresh29(op_implies_and, true, X, Y)
% 21.12/3.08 = { by axiom 2 (hilbert_op_implies_and) }
% 21.12/3.08 fresh29(true, true, X, Y)
% 21.12/3.08 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.08 fresh29(op_implies, true, X, Y)
% 21.12/3.08 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.08 fresh29(op_implies, op_implies, X, Y)
% 21.12/3.08 = { by axiom 31 (op_implies_and) }
% 21.12/3.08 implies(X, Y)
% 21.12/3.08
% 21.12/3.08 Lemma 52: implies(not(X), Y) = or(X, Y).
% 21.12/3.08 Proof:
% 21.12/3.08 implies(not(X), Y)
% 21.12/3.08 = { by lemma 51 R->L }
% 21.12/3.08 not(and(not(X), not(Y)))
% 21.12/3.08 = { by axiom 41 (op_or) R->L }
% 21.12/3.08 fresh26(op_or, true, X, Y)
% 21.12/3.08 = { by axiom 11 (hilbert_op_or) }
% 21.12/3.08 fresh26(true, true, X, Y)
% 21.12/3.08 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.08 fresh26(op_implies, true, X, Y)
% 21.12/3.08 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.08 fresh26(op_implies, op_implies, X, Y)
% 21.12/3.08 = { by axiom 32 (op_or) }
% 21.12/3.08 or(X, Y)
% 21.12/3.08
% 21.12/3.08 Lemma 53: is_a_theorem(strict_implies(X, possibly(X))) = fresh79(axiom_m6, op_implies, X).
% 21.12/3.08 Proof:
% 21.12/3.08 is_a_theorem(strict_implies(X, possibly(X)))
% 21.12/3.08 = { by axiom 38 (axiom_m6_1) R->L }
% 21.12/3.08 fresh79(axiom_m6, true, X)
% 21.12/3.08 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.08 fresh79(axiom_m6, op_implies, X)
% 21.12/3.08
% 21.12/3.08 Lemma 54: fresh115(X, X, Y, Z) = op_implies.
% 21.12/3.08 Proof:
% 21.12/3.08 fresh115(X, X, Y, Z)
% 21.12/3.08 = { by axiom 24 (modus_ponens_2) }
% 21.12/3.08 fresh116(modus_ponens, true, Z)
% 21.12/3.08 = { by axiom 4 (hilbert_modus_ponens) }
% 21.12/3.08 fresh116(true, true, Z)
% 21.12/3.08 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.08 fresh116(op_implies, true, Z)
% 21.12/3.08 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.08 fresh116(op_implies, op_implies, Z)
% 21.12/3.08 = { by axiom 17 (modus_ponens_2) }
% 21.12/3.08 true
% 21.12/3.08 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.08 op_implies
% 21.12/3.08
% 21.12/3.08 Lemma 55: is_a_theorem(implies(X, implies(Y, and(X, Y)))) = op_implies.
% 21.12/3.08 Proof:
% 21.12/3.08 is_a_theorem(implies(X, implies(Y, and(X, Y))))
% 21.12/3.08 = { by axiom 47 (and_3_1) R->L }
% 21.12/3.08 fresh103(and_3, true, X, Y)
% 21.12/3.08 = { by axiom 9 (hilbert_and_3) }
% 21.12/3.08 fresh103(true, true, X, Y)
% 21.12/3.08 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.08 fresh103(op_implies, true, X, Y)
% 21.12/3.08 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.08 fresh103(op_implies, op_implies, X, Y)
% 21.12/3.08 = { by axiom 26 (and_3_1) }
% 21.12/3.08 true
% 21.12/3.08 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.08 op_implies
% 21.12/3.08
% 21.12/3.08 Lemma 56: fresh115(is_a_theorem(implies(X, Y)), op_implies, X, Y) = fresh40(is_a_theorem(X), op_implies, Y).
% 21.12/3.08 Proof:
% 21.12/3.08 fresh115(is_a_theorem(implies(X, Y)), op_implies, X, Y)
% 21.12/3.08 = { by axiom 1 (s1_0_op_implies) }
% 21.12/3.08 fresh115(is_a_theorem(implies(X, Y)), true, X, Y)
% 21.12/3.08 = { by axiom 45 (modus_ponens_2) }
% 21.12/3.08 fresh40(is_a_theorem(X), true, Y)
% 21.12/3.08 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.08 fresh40(is_a_theorem(X), op_implies, Y)
% 21.12/3.08
% 21.12/3.08 Lemma 57: fresh40(is_a_theorem(implies(X, implies(X, Y))), op_implies, implies(X, Y)) = op_implies.
% 21.12/3.08 Proof:
% 21.12/3.08 fresh40(is_a_theorem(implies(X, implies(X, Y))), op_implies, implies(X, Y))
% 21.12/3.08 = { by lemma 56 R->L }
% 21.12/3.08 fresh115(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), op_implies, implies(X, implies(X, Y)), implies(X, Y))
% 21.12/3.08 = { by axiom 48 (implies_2_1) R->L }
% 21.12/3.08 fresh115(fresh49(implies_2, true, X, Y), op_implies, implies(X, implies(X, Y)), implies(X, Y))
% 21.12/3.08 = { by axiom 7 (hilbert_implies_2) }
% 21.12/3.08 fresh115(fresh49(true, true, X, Y), op_implies, implies(X, implies(X, Y)), implies(X, Y))
% 21.12/3.08 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.08 fresh115(fresh49(op_implies, true, X, Y), op_implies, implies(X, implies(X, Y)), implies(X, Y))
% 21.12/3.08 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.09 fresh115(fresh49(op_implies, op_implies, X, Y), op_implies, implies(X, implies(X, Y)), implies(X, Y))
% 21.12/3.09 = { by axiom 27 (implies_2_1) }
% 21.12/3.09 fresh115(true, op_implies, implies(X, implies(X, Y)), implies(X, Y))
% 21.12/3.09 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.09 fresh115(op_implies, op_implies, implies(X, implies(X, Y)), implies(X, Y))
% 21.12/3.09 = { by lemma 54 }
% 21.12/3.09 op_implies
% 21.12/3.09
% 21.12/3.09 Goal 1 (s1_0_m6s3m9b_axiom_m6): axiom_m6 = true.
% 21.12/3.09 Proof:
% 21.12/3.09 axiom_m6
% 21.12/3.09 = { by axiom 43 (axiom_m6) R->L }
% 21.12/3.09 fresh80(is_a_theorem(strict_implies(x3, possibly(x3))), true)
% 21.12/3.09 = { by lemma 53 }
% 21.12/3.09 fresh80(fresh79(axiom_m6, op_implies, x3), true)
% 21.12/3.09 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.09 fresh80(fresh79(axiom_m6, op_implies, x3), op_implies)
% 21.12/3.09 = { by lemma 53 R->L }
% 21.12/3.09 fresh80(is_a_theorem(strict_implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 33 (op_strict_implies) R->L }
% 21.12/3.09 fresh80(is_a_theorem(fresh23(op_implies, op_implies, x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 1 (s1_0_op_implies) }
% 21.12/3.09 fresh80(is_a_theorem(fresh23(op_implies, true, x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 1 (s1_0_op_implies) }
% 21.12/3.09 fresh80(is_a_theorem(fresh23(true, true, x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 3 (s1_0_op_strict_implies) R->L }
% 21.12/3.09 fresh80(is_a_theorem(fresh23(op_strict_implies, true, x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 34 (op_strict_implies) }
% 21.12/3.09 fresh80(is_a_theorem(necessarily(implies(x3, possibly(x3)))), op_implies)
% 21.12/3.09 = { by axiom 20 (necessitation_1) R->L }
% 21.12/3.09 fresh80(fresh34(op_implies, op_implies, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 1 (s1_0_op_implies) }
% 21.12/3.09 fresh80(fresh34(op_implies, true, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 1 (s1_0_op_implies) }
% 21.12/3.09 fresh80(fresh34(true, true, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 13 (km5_necessitation) R->L }
% 21.12/3.09 fresh80(fresh34(necessitation, true, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 29 (necessitation_1) }
% 21.12/3.09 fresh80(fresh33(is_a_theorem(implies(x3, possibly(x3))), true, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.09 fresh80(fresh33(is_a_theorem(implies(x3, possibly(x3))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 22 (op_possibly) R->L }
% 21.12/3.09 fresh80(fresh33(is_a_theorem(implies(x3, fresh25(op_implies, op_implies, x3))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 1 (s1_0_op_implies) }
% 21.12/3.09 fresh80(fresh33(is_a_theorem(implies(x3, fresh25(op_implies, true, x3))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 1 (s1_0_op_implies) }
% 21.12/3.09 fresh80(fresh33(is_a_theorem(implies(x3, fresh25(true, true, x3))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 15 (km5_op_possibly) R->L }
% 21.12/3.09 fresh80(fresh33(is_a_theorem(implies(x3, fresh25(op_possibly, true, x3))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 23 (op_possibly) }
% 21.12/3.09 fresh80(fresh33(is_a_theorem(implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 19 (modus_ponens_2) R->L }
% 21.12/3.09 fresh80(fresh33(fresh40(op_implies, op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by lemma 54 R->L }
% 21.12/3.09 fresh80(fresh33(fresh40(fresh115(op_implies, op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by lemma 57 R->L }
% 21.12/3.09 fresh80(fresh33(fresh40(fresh115(fresh40(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(implies(necessarily(not(x3)), not(x3)), implies(or(necessarily(not(x3)), necessarily(not(x3))), not(x3))))), op_implies, implies(implies(necessarily(not(x3)), not(x3)), implies(or(necessarily(not(x3)), necessarily(not(x3))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 50 (or_3_1) R->L }
% 21.12/3.09 fresh80(fresh33(fresh40(fresh115(fresh40(fresh17(or_3, true, necessarily(not(x3)), necessarily(not(x3)), not(x3)), op_implies, implies(implies(necessarily(not(x3)), not(x3)), implies(or(necessarily(not(x3)), necessarily(not(x3))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 10 (hilbert_or_3) }
% 21.12/3.09 fresh80(fresh33(fresh40(fresh115(fresh40(fresh17(true, true, necessarily(not(x3)), necessarily(not(x3)), not(x3)), op_implies, implies(implies(necessarily(not(x3)), not(x3)), implies(or(necessarily(not(x3)), necessarily(not(x3))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.09 fresh80(fresh33(fresh40(fresh115(fresh40(fresh17(op_implies, true, necessarily(not(x3)), necessarily(not(x3)), not(x3)), op_implies, implies(implies(necessarily(not(x3)), not(x3)), implies(or(necessarily(not(x3)), necessarily(not(x3))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.09 fresh80(fresh33(fresh40(fresh115(fresh40(fresh17(op_implies, op_implies, necessarily(not(x3)), necessarily(not(x3)), not(x3)), op_implies, implies(implies(necessarily(not(x3)), not(x3)), implies(or(necessarily(not(x3)), necessarily(not(x3))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 40 (or_3_1) }
% 21.12/3.09 fresh80(fresh33(fresh40(fresh115(fresh40(true, op_implies, implies(implies(necessarily(not(x3)), not(x3)), implies(or(necessarily(not(x3)), necessarily(not(x3))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.12/3.09 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.12/3.09 fresh80(fresh33(fresh40(fresh115(fresh40(op_implies, op_implies, implies(implies(necessarily(not(x3)), not(x3)), implies(or(necessarily(not(x3)), necessarily(not(x3))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.09 = { by axiom 19 (modus_ponens_2) }
% 21.29/3.09 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(or(necessarily(not(x3)), necessarily(not(x3))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.10 = { by lemma 52 R->L }
% 21.29/3.10 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(implies(not(necessarily(not(x3))), necessarily(not(x3))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.10 = { by lemma 51 R->L }
% 21.29/3.10 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(and(not(necessarily(not(x3))), not(necessarily(not(x3))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.10 = { by axiom 36 (substitution_of_equivalents_2) R->L }
% 21.29/3.10 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(op_implies, op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.10 = { by lemma 54 R->L }
% 21.29/3.10 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh115(op_implies, op_implies, implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.10 = { by lemma 54 R->L }
% 21.29/3.10 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh115(fresh115(op_implies, op_implies, implies(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))), implies(implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), and(implies(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))), implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3))))))), op_implies, implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.10 = { by lemma 55 R->L }
% 21.29/3.10 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh115(fresh115(is_a_theorem(implies(implies(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))), implies(implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), and(implies(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))), implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))))))), op_implies, implies(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))), implies(implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), and(implies(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))), implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3))))))), op_implies, implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.10 = { by lemma 56 }
% 21.29/3.10 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh115(fresh40(is_a_theorem(implies(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, implies(implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), and(implies(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))), implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3))))))), op_implies, implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.10 = { by axiom 19 (modus_ponens_2) R->L }
% 21.29/3.10 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh115(fresh40(fresh40(op_implies, op_implies, implies(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, implies(implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), and(implies(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))), implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3))))))), op_implies, implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.10 = { by lemma 55 R->L }
% 21.29/3.11 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh115(fresh40(fresh40(is_a_theorem(implies(not(necessarily(not(x3))), implies(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))))), op_implies, implies(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, implies(implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), and(implies(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))), implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3))))))), op_implies, implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.11 = { by lemma 57 }
% 21.29/3.11 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh115(fresh40(op_implies, op_implies, implies(implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), and(implies(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))), implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3))))))), op_implies, implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.11 = { by axiom 19 (modus_ponens_2) }
% 21.29/3.11 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh115(is_a_theorem(implies(implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), and(implies(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))), implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3))))))), op_implies, implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.11 = { by axiom 44 (op_equiv) R->L }
% 21.29/3.11 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh115(is_a_theorem(implies(implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), fresh30(op_equiv, true, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))))), op_implies, implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.11 = { by axiom 12 (hilbert_op_equiv) }
% 21.29/3.11 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh115(is_a_theorem(implies(implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), fresh30(true, true, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))))), op_implies, implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.11 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.29/3.11 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh115(is_a_theorem(implies(implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), fresh30(op_implies, true, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))))), op_implies, implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.11 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.29/3.11 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh115(is_a_theorem(implies(implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), fresh30(op_implies, op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))))), op_implies, implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.11 = { by axiom 30 (op_equiv) }
% 21.29/3.11 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh115(is_a_theorem(implies(implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3))))))), op_implies, implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3)))), equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.11 = { by lemma 56 }
% 21.29/3.11 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh40(is_a_theorem(implies(and(not(necessarily(not(x3))), not(necessarily(not(x3)))), not(necessarily(not(x3))))), op_implies, equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.11 = { by axiom 42 (and_1_1) R->L }
% 21.29/3.11 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh40(fresh107(and_1, true, not(necessarily(not(x3))), not(necessarily(not(x3)))), op_implies, equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.11 = { by axiom 8 (hilbert_and_1) }
% 21.29/3.11 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh40(fresh107(true, true, not(necessarily(not(x3))), not(necessarily(not(x3)))), op_implies, equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.12 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.29/3.12 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh40(fresh107(op_implies, true, not(necessarily(not(x3))), not(necessarily(not(x3)))), op_implies, equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.12 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.29/3.12 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh40(fresh107(op_implies, op_implies, not(necessarily(not(x3))), not(necessarily(not(x3)))), op_implies, equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.12 = { by axiom 25 (and_1_1) }
% 21.29/3.12 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh40(true, op_implies, equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.12 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.29/3.12 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(fresh40(op_implies, op_implies, equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.12 = { by axiom 19 (modus_ponens_2) }
% 21.29/3.12 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(is_a_theorem(equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.12 = { by axiom 1 (s1_0_op_implies) }
% 21.29/3.12 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh3(is_a_theorem(equiv(not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), true, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.29/3.12 = { by axiom 46 (substitution_of_equivalents_2) R->L }
% 21.29/3.12 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh4(substitution_of_equivalents, true, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.50/3.12 = { by axiom 5 (substitution_of_equivalents) }
% 21.50/3.12 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh4(true, true, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.50/3.12 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.50/3.12 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh4(op_implies, true, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.50/3.12 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.50/3.12 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(fresh4(op_implies, op_implies, not(necessarily(not(x3))), and(not(necessarily(not(x3))), not(necessarily(not(x3)))))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.50/3.12 = { by axiom 35 (substitution_of_equivalents_2) }
% 21.50/3.12 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), implies(not(not(necessarily(not(x3)))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by lemma 52 }
% 21.51/3.12 fresh80(fresh33(fresh40(fresh115(is_a_theorem(implies(implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3)))), op_implies, implies(necessarily(not(x3)), not(x3)), or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by lemma 56 }
% 21.51/3.12 fresh80(fresh33(fresh40(fresh40(is_a_theorem(implies(necessarily(not(x3)), not(x3))), op_implies, or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by axiom 39 (axiom_M_1) R->L }
% 21.51/3.12 fresh80(fresh33(fresh40(fresh40(fresh93(axiom_M, true, not(x3)), op_implies, or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by axiom 14 (km5_axiom_M) }
% 21.51/3.12 fresh80(fresh33(fresh40(fresh40(fresh93(true, true, not(x3)), op_implies, or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.51/3.12 fresh80(fresh33(fresh40(fresh40(fresh93(op_implies, true, not(x3)), op_implies, or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.51/3.12 fresh80(fresh33(fresh40(fresh40(fresh93(op_implies, op_implies, not(x3)), op_implies, or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by axiom 18 (axiom_M_1) }
% 21.51/3.12 fresh80(fresh33(fresh40(fresh40(true, op_implies, or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.51/3.12 fresh80(fresh33(fresh40(fresh40(op_implies, op_implies, or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by axiom 19 (modus_ponens_2) }
% 21.51/3.12 fresh80(fresh33(fresh40(is_a_theorem(or(not(necessarily(not(x3))), not(x3))), op_implies, implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by lemma 56 R->L }
% 21.51/3.12 fresh80(fresh33(fresh115(is_a_theorem(implies(or(not(necessarily(not(x3))), not(x3)), implies(x3, not(necessarily(not(x3)))))), op_implies, or(not(necessarily(not(x3))), not(x3)), implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by lemma 52 R->L }
% 21.51/3.12 fresh80(fresh33(fresh115(is_a_theorem(implies(implies(not(not(necessarily(not(x3)))), not(x3)), implies(x3, not(necessarily(not(x3)))))), op_implies, or(not(necessarily(not(x3))), not(x3)), implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by axiom 49 (modus_tollens_1) R->L }
% 21.51/3.12 fresh80(fresh33(fresh115(fresh35(modus_tollens, true, x3, not(necessarily(not(x3)))), op_implies, or(not(necessarily(not(x3))), not(x3)), implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by axiom 6 (hilbert_modus_tollens) }
% 21.51/3.12 fresh80(fresh33(fresh115(fresh35(true, true, x3, not(necessarily(not(x3)))), op_implies, or(not(necessarily(not(x3))), not(x3)), implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.51/3.12 fresh80(fresh33(fresh115(fresh35(op_implies, true, x3, not(necessarily(not(x3)))), op_implies, or(not(necessarily(not(x3))), not(x3)), implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.51/3.12 fresh80(fresh33(fresh115(fresh35(op_implies, op_implies, x3, not(necessarily(not(x3)))), op_implies, or(not(necessarily(not(x3))), not(x3)), implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by axiom 28 (modus_tollens_1) }
% 21.51/3.12 fresh80(fresh33(fresh115(true, op_implies, or(not(necessarily(not(x3))), not(x3)), implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.51/3.12 fresh80(fresh33(fresh115(op_implies, op_implies, or(not(necessarily(not(x3))), not(x3)), implies(x3, not(necessarily(not(x3))))), op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by lemma 54 }
% 21.51/3.12 fresh80(fresh33(op_implies, op_implies, implies(x3, possibly(x3))), op_implies)
% 21.51/3.12 = { by axiom 21 (necessitation_1) }
% 21.51/3.12 fresh80(true, op_implies)
% 21.51/3.12 = { by axiom 1 (s1_0_op_implies) R->L }
% 21.51/3.12 fresh80(op_implies, op_implies)
% 21.51/3.12 = { by axiom 16 (axiom_m6) }
% 21.51/3.12 true
% 21.51/3.12 % SZS output end Proof
% 21.51/3.12
% 21.51/3.12 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------