0.06/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.06/0.13 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.12/0.34 % Computer : n014.cluster.edu 0.12/0.34 % Model : x86_64 x86_64 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.12/0.34 % Memory : 8042.1875MB 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.12/0.34 % CPULimit : 960 0.12/0.34 % WCLimit : 120 0.12/0.34 % DateTime : Thu Jul 2 08:21:40 EDT 2020 0.12/0.34 % CPUTime : 17.74/2.62 % SZS status Theorem 17.74/2.62 17.74/2.62 % SZS output start Proof 17.74/2.62 Take the following subset of the input axioms: 18.10/2.67 fof(and_1, axiom, ![X, Y]: is_a_theorem(implies(and(X, Y), X)) <=> and_1). 18.10/2.67 fof(and_3, axiom, ![X, Y]: is_a_theorem(implies(X, implies(Y, and(X, Y)))) <=> and_3). 18.10/2.67 fof(axiom_5, axiom, ![X]: is_a_theorem(implies(possibly(X), necessarily(possibly(X)))) <=> axiom_5). 18.10/2.67 fof(axiom_M, axiom, ![X]: is_a_theorem(implies(necessarily(X), X)) <=> axiom_M). 18.10/2.67 fof(axiom_m9, axiom, ![X]: is_a_theorem(strict_implies(possibly(possibly(X)), possibly(X))) <=> axiom_m9). 18.10/2.67 fof(cn3, axiom, ![P]: is_a_theorem(implies(implies(not(P), P), P)) <=> cn3). 18.10/2.67 fof(hilbert_and_1, axiom, and_1). 18.10/2.67 fof(hilbert_and_3, axiom, and_3). 18.10/2.67 fof(hilbert_implies_1, axiom, implies_1). 18.10/2.67 fof(hilbert_implies_2, axiom, implies_2). 18.10/2.67 fof(hilbert_modus_ponens, axiom, modus_ponens). 18.10/2.67 fof(hilbert_modus_tollens, axiom, modus_tollens). 18.10/2.67 fof(hilbert_op_equiv, axiom, op_equiv). 18.10/2.67 fof(hilbert_op_implies_and, axiom, op_implies_and). 18.10/2.67 fof(hilbert_op_or, axiom, op_or). 18.10/2.67 fof(implies_1, axiom, implies_1 <=> ![X, Y]: is_a_theorem(implies(X, implies(Y, X)))). 18.10/2.67 fof(implies_2, axiom, implies_2 <=> ![X, Y]: is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y)))). 18.10/2.67 fof(km5_axiom_5, axiom, axiom_5). 18.10/2.67 fof(km5_axiom_M, axiom, axiom_M). 18.10/2.67 fof(km5_necessitation, axiom, necessitation). 18.10/2.67 fof(km5_op_possibly, axiom, op_possibly). 18.10/2.67 fof(kn1, axiom, kn1 <=> ![P]: is_a_theorem(implies(P, and(P, P)))). 18.10/2.67 fof(modus_ponens, axiom, ![X, Y]: (is_a_theorem(Y) <= (is_a_theorem(X) & is_a_theorem(implies(X, Y)))) <=> modus_ponens). 18.10/2.67 fof(modus_tollens, axiom, ![X, Y]: is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))) <=> modus_tollens). 18.10/2.67 fof(necessitation, axiom, necessitation <=> ![X]: (is_a_theorem(X) => is_a_theorem(necessarily(X)))). 18.10/2.67 fof(op_equiv, axiom, ![X, Y]: and(implies(X, Y), implies(Y, X))=equiv(X, Y) <= op_equiv). 18.10/2.67 fof(op_implies_and, axiom, op_implies_and => ![X, Y]: not(and(X, not(Y)))=implies(X, Y)). 18.10/2.67 fof(op_or, axiom, op_or => ![X, Y]: or(X, Y)=not(and(not(X), not(Y)))). 18.10/2.67 fof(op_possibly, axiom, op_possibly => ![X]: possibly(X)=not(necessarily(not(X)))). 18.10/2.67 fof(op_strict_implies, axiom, op_strict_implies => ![X, Y]: necessarily(implies(X, Y))=strict_implies(X, Y)). 18.10/2.67 fof(s1_0_m6s3m9b_axiom_m9, conjecture, axiom_m9). 18.10/2.67 fof(s1_0_op_strict_implies, axiom, op_strict_implies). 18.10/2.67 fof(substitution_of_equivalents, axiom, ![X, Y]: (X=Y <= is_a_theorem(equiv(X, Y))) <=> substitution_of_equivalents). 18.10/2.67 fof(substitution_of_equivalents, axiom, substitution_of_equivalents). 18.10/2.67 18.10/2.67 Now clausify the problem and encode Horn clauses using encoding 3 of 18.10/2.67 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 18.10/2.67 We repeatedly replace C & s=t => u=v by the two clauses: 18.10/2.67 fresh(y, y, x1...xn) = u 18.10/2.67 C => fresh(s, t, x1...xn) = v 18.10/2.67 where fresh is a fresh function symbol and x1..xn are the free 18.10/2.67 variables of u and v. 18.10/2.67 A predicate p(X) is encoded as p(X)=true (this is sound, because the 18.10/2.67 input problem has no model of domain size 1). 18.10/2.67 18.10/2.67 The encoding turns the above axioms into the following unit equations and goals: 18.10/2.67 18.10/2.67 Axiom 1 (and_1_1): fresh107(X, X, Y, Z) = true. 18.10/2.67 Axiom 2 (and_3_1): fresh103(X, X, Y, Z) = true. 18.10/2.67 Axiom 3 (axiom_5_1): fresh99(X, X, Y) = true. 18.10/2.67 Axiom 4 (axiom_M_1): fresh93(X, X, Y) = true. 18.10/2.67 Axiom 5 (axiom_m9): fresh74(X, X) = true. 18.10/2.67 Axiom 6 (implies_1_1): fresh51(X, X, Y, Z) = true. 18.10/2.67 Axiom 7 (implies_2_1): fresh49(X, X, Y, Z) = true. 18.10/2.67 Axiom 8 (modus_ponens_2): fresh40(X, X, Y, Z) = is_a_theorem(Z). 18.10/2.67 Axiom 9 (modus_ponens_2): fresh116(X, X, Y) = true. 18.10/2.67 Axiom 10 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(is_a_theorem(Y), true, Z). 18.10/2.67 Axiom 11 (modus_tollens_1): fresh35(X, X, Y, Z) = true. 18.10/2.67 Axiom 12 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)). 18.10/2.67 Axiom 13 (necessitation_1): fresh33(X, X, Y) = true. 18.10/2.67 Axiom 14 (op_equiv): fresh30(X, X, Y, Z) = equiv(Y, Z). 18.10/2.67 Axiom 15 (op_implies_and): fresh29(X, X, Y, Z) = implies(Y, Z). 18.10/2.67 Axiom 16 (op_or): fresh26(X, X, Y, Z) = or(Y, Z). 18.10/2.67 Axiom 17 (op_possibly): fresh25(X, X, Y) = possibly(Y). 18.10/2.67 Axiom 18 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z). 18.10/2.67 Axiom 19 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y. 18.10/2.67 Axiom 20 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z. 18.10/2.67 Axiom 21 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))). 18.10/2.67 Axiom 22 (modus_tollens_1): fresh35(modus_tollens, true, X, Y) = is_a_theorem(implies(implies(not(Y), not(X)), implies(X, Y))). 18.10/2.67 Axiom 23 (kn1_1): fresh45(kn1, true, X) = is_a_theorem(implies(X, and(X, X))). 18.10/2.67 Axiom 24 (implies_1_1): fresh51(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))). 18.10/2.67 Axiom 25 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y). 18.10/2.67 Axiom 26 (cn3_1): fresh59(cn3, true, X) = is_a_theorem(implies(implies(not(X), X), X)). 18.10/2.67 Axiom 27 (and_1_1): fresh107(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)). 18.10/2.67 Axiom 28 (modus_ponens_2): fresh115(modus_ponens, true, X, Y) = fresh40(is_a_theorem(implies(X, Y)), true, X, Y). 18.10/2.67 Axiom 29 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))). 18.10/2.67 Axiom 30 (op_equiv): fresh30(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)). 18.10/2.67 Axiom 31 (op_or): fresh26(op_or, true, X, Y) = not(and(not(X), not(Y))). 18.10/2.67 Axiom 32 (op_implies_and): fresh29(op_implies_and, true, X, Y) = not(and(X, not(Y))). 18.10/2.67 Axiom 33 (hilbert_op_implies_and): op_implies_and = true. 18.10/2.67 Axiom 34 (hilbert_modus_tollens): modus_tollens = true. 18.10/2.67 Axiom 35 (hilbert_implies_2): implies_2 = true. 18.10/2.67 Axiom 36 (hilbert_implies_1): implies_1 = true. 18.10/2.67 Axiom 37 (hilbert_and_1): and_1 = true. 18.10/2.67 Axiom 38 (hilbert_and_3): and_3 = true. 18.10/2.67 Axiom 39 (substitution_of_equivalents): substitution_of_equivalents = true. 18.10/2.67 Axiom 40 (hilbert_modus_ponens): modus_ponens = true. 18.10/2.67 Axiom 41 (hilbert_op_or): op_or = true. 18.10/2.67 Axiom 42 (hilbert_op_equiv): op_equiv = true. 18.10/2.67 Axiom 43 (axiom_M_1): fresh93(axiom_M, true, X) = is_a_theorem(implies(necessarily(X), X)). 18.10/2.67 Axiom 44 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X). 18.10/2.67 Axiom 45 (axiom_m9_1): fresh73(axiom_m9, true, X) = is_a_theorem(strict_implies(possibly(possibly(X)), possibly(X))). 18.10/2.67 Axiom 46 (axiom_m9): fresh74(is_a_theorem(strict_implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) = axiom_m9. 18.10/2.67 Axiom 47 (axiom_5_1): fresh99(axiom_5, true, X) = is_a_theorem(implies(possibly(X), necessarily(possibly(X)))). 18.10/2.67 Axiom 48 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)). 18.10/2.67 Axiom 49 (op_possibly): fresh25(op_possibly, true, X) = not(necessarily(not(X))). 18.10/2.67 Axiom 50 (km5_axiom_M): axiom_M = true. 18.10/2.67 Axiom 51 (km5_necessitation): necessitation = true. 18.10/2.67 Axiom 52 (km5_op_possibly): op_possibly = true. 18.10/2.67 Axiom 53 (km5_axiom_5): axiom_5 = true. 18.10/2.69 Axiom 54 (s1_0_op_strict_implies): op_strict_implies = true. 18.10/2.69 18.10/2.69 Lemma 55: not(and(Y, not(X))) = implies(Y, X). 18.10/2.69 Proof: 18.10/2.69 not(and(Y, not(X))) 18.10/2.69 = { by axiom 32 (op_implies_and) } 18.10/2.69 fresh29(op_implies_and, true, Y, X) 18.10/2.69 = { by axiom 33 (hilbert_op_implies_and) } 18.10/2.69 fresh29(true, true, Y, X) 18.10/2.69 = { by axiom 15 (op_implies_and) } 18.10/2.69 implies(Y, X) 18.10/2.69 18.10/2.69 Lemma 56: implies(not(Y), X) = or(Y, X). 18.10/2.69 Proof: 18.10/2.69 implies(not(Y), X) 18.10/2.69 = { by lemma 55 } 18.10/2.69 not(and(not(Y), not(X))) 18.10/2.69 = { by axiom 31 (op_or) } 18.10/2.69 fresh26(op_or, true, Y, X) 18.10/2.69 = { by axiom 41 (hilbert_op_or) } 18.10/2.69 fresh26(true, true, Y, X) 18.10/2.69 = { by axiom 16 (op_or) } 18.10/2.69 or(Y, X) 18.10/2.69 18.10/2.69 Lemma 57: not(necessarily(not(X))) = possibly(X). 18.10/2.69 Proof: 18.10/2.69 not(necessarily(not(X))) 18.10/2.69 = { by axiom 49 (op_possibly) } 18.10/2.69 fresh25(op_possibly, true, X) 18.10/2.69 = { by axiom 52 (km5_op_possibly) } 18.10/2.69 fresh25(true, true, X) 18.10/2.69 = { by axiom 17 (op_possibly) } 18.10/2.69 possibly(X) 18.10/2.69 18.10/2.69 Lemma 58: and(implies(X, Y), implies(Y, X)) = equiv(X, Y). 18.10/2.69 Proof: 18.10/2.69 and(implies(X, Y), implies(Y, X)) 18.10/2.69 = { by axiom 30 (op_equiv) } 18.10/2.69 fresh30(op_equiv, true, X, Y) 18.10/2.69 = { by axiom 42 (hilbert_op_equiv) } 18.10/2.69 fresh30(true, true, X, Y) 18.10/2.69 = { by axiom 14 (op_equiv) } 18.10/2.69 equiv(X, Y) 18.10/2.69 18.10/2.69 Lemma 59: fresh40(is_a_theorem(implies(X, Y)), true, X, Y) = fresh116(is_a_theorem(X), true, Y). 18.10/2.69 Proof: 18.10/2.69 fresh40(is_a_theorem(implies(X, Y)), true, X, Y) 18.10/2.69 = { by axiom 28 (modus_ponens_2) } 18.10/2.69 fresh115(modus_ponens, true, X, Y) 18.10/2.69 = { by axiom 40 (hilbert_modus_ponens) } 18.10/2.69 fresh115(true, true, X, Y) 18.10/2.69 = { by axiom 10 (modus_ponens_2) } 18.10/2.69 fresh116(is_a_theorem(X), true, Y) 18.10/2.69 18.10/2.69 Lemma 60: is_a_theorem(implies(X, implies(Y, and(X, Y)))) = true. 18.10/2.69 Proof: 18.10/2.69 is_a_theorem(implies(X, implies(Y, and(X, Y)))) 18.10/2.69 = { by axiom 21 (and_3_1) } 18.10/2.69 fresh103(and_3, true, X, Y) 18.10/2.69 = { by axiom 38 (hilbert_and_3) } 18.10/2.69 fresh103(true, true, X, Y) 18.10/2.69 = { by axiom 2 (and_3_1) } 18.10/2.69 true 18.10/2.69 18.10/2.69 Lemma 61: fresh116(is_a_theorem(Y), true, implies(X, and(Y, X))) = is_a_theorem(implies(X, and(Y, X))). 18.10/2.69 Proof: 18.10/2.69 fresh116(is_a_theorem(Y), true, implies(X, and(Y, X))) 18.10/2.69 = { by lemma 59 } 18.10/2.69 fresh40(is_a_theorem(implies(Y, implies(X, and(Y, X)))), true, Y, implies(X, and(Y, X))) 18.10/2.69 = { by lemma 60 } 18.10/2.69 fresh40(true, true, Y, implies(X, and(Y, X))) 18.10/2.69 = { by axiom 8 (modus_ponens_2) } 18.10/2.69 is_a_theorem(implies(X, and(Y, X))) 18.10/2.69 18.10/2.69 Lemma 62: fresh3(is_a_theorem(equiv(X, Y)), true, X, Y) = X. 18.10/2.69 Proof: 18.10/2.69 fresh3(is_a_theorem(equiv(X, Y)), true, X, Y) 18.10/2.69 = { by axiom 25 (substitution_of_equivalents_2) } 18.10/2.69 fresh4(substitution_of_equivalents, true, X, Y) 18.10/2.69 = { by axiom 39 (substitution_of_equivalents) } 18.10/2.69 fresh4(true, true, X, Y) 18.10/2.69 = { by axiom 19 (substitution_of_equivalents_2) } 18.10/2.70 X 18.10/2.70 18.10/2.70 Lemma 63: necessarily(possibly(X)) = possibly(X). 18.10/2.70 Proof: 18.10/2.70 necessarily(possibly(X)) 18.10/2.70 = { by axiom 20 (substitution_of_equivalents_2) } 18.10/2.70 fresh3(true, true, possibly(X), necessarily(possibly(X))) 18.10/2.70 = { by axiom 9 (modus_ponens_2) } 18.10/2.70 fresh3(fresh116(true, true, equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X))) 18.10/2.70 = { by axiom 4 (axiom_M_1) } 18.10/2.70 fresh3(fresh116(fresh93(true, true, possibly(X)), true, equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X))) 18.10/2.70 = { by axiom 50 (km5_axiom_M) } 18.10/2.70 fresh3(fresh116(fresh93(axiom_M, true, possibly(X)), true, equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X))) 18.10/2.70 = { by axiom 43 (axiom_M_1) } 18.10/2.70 fresh3(fresh116(is_a_theorem(implies(necessarily(possibly(X)), possibly(X))), true, equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X))) 18.10/2.70 = { by lemma 59 } 18.10/2.70 fresh3(fresh40(is_a_theorem(implies(implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X))) 18.10/2.70 = { by lemma 58 } 18.10/2.70 fresh3(fresh40(is_a_theorem(implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X))) 18.10/2.70 = { by lemma 61 } 18.10/2.70 fresh3(fresh40(fresh116(is_a_theorem(implies(possibly(X), necessarily(possibly(X)))), true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X))) 18.10/2.70 = { by axiom 47 (axiom_5_1) } 18.10/2.70 fresh3(fresh40(fresh116(fresh99(axiom_5, true, X), true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X))) 18.10/2.70 = { by axiom 53 (km5_axiom_5) } 18.10/2.70 fresh3(fresh40(fresh116(fresh99(true, true, X), true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X))) 18.10/2.70 = { by axiom 3 (axiom_5_1) } 18.10/2.70 fresh3(fresh40(fresh116(true, true, implies(implies(necessarily(possibly(X)), possibly(X)), and(implies(possibly(X), necessarily(possibly(X))), implies(necessarily(possibly(X)), possibly(X))))), true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X))) 18.10/2.70 = { by axiom 9 (modus_ponens_2) } 18.10/2.70 fresh3(fresh40(true, true, implies(necessarily(possibly(X)), possibly(X)), equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X))) 18.10/2.70 = { by axiom 8 (modus_ponens_2) } 18.10/2.70 fresh3(is_a_theorem(equiv(possibly(X), necessarily(possibly(X)))), true, possibly(X), necessarily(possibly(X))) 18.10/2.70 = { by lemma 62 } 18.10/2.70 possibly(X) 18.10/2.70 18.10/2.70 Lemma 64: possibly(necessarily(not(X))) = not(possibly(X)). 18.10/2.70 Proof: 18.10/2.70 possibly(necessarily(not(X))) 18.10/2.70 = { by lemma 57 } 18.10/2.70 not(necessarily(not(necessarily(not(X))))) 18.10/2.70 = { by lemma 57 } 18.10/2.70 not(necessarily(possibly(X))) 18.10/2.70 = { by lemma 63 } 18.70/2.72 not(possibly(X)) 18.70/2.72 18.70/2.72 Lemma 65: implies(not(X), X) = not(not(X)). 18.70/2.72 Proof: 18.70/2.72 implies(not(X), X) 18.70/2.72 = { by lemma 55 } 18.70/2.72 not(and(not(X), not(X))) 18.70/2.72 = { by axiom 20 (substitution_of_equivalents_2) } 18.70/2.72 not(fresh3(true, true, not(X), and(not(X), not(X)))) 18.70/2.72 = { by axiom 9 (modus_ponens_2) } 18.70/2.72 not(fresh3(fresh116(true, true, equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.72 = { by axiom 1 (and_1_1) } 18.70/2.72 not(fresh3(fresh116(fresh107(true, true, not(X), not(X)), true, equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.72 = { by axiom 37 (hilbert_and_1) } 18.70/2.72 not(fresh3(fresh116(fresh107(and_1, true, not(X), not(X)), true, equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.72 = { by axiom 27 (and_1_1) } 18.70/2.72 not(fresh3(fresh116(is_a_theorem(implies(and(not(X), not(X)), not(X))), true, equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.72 = { by lemma 59 } 18.70/2.74 not(fresh3(fresh40(is_a_theorem(implies(implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.74 = { by lemma 58 } 18.70/2.74 not(fresh3(fresh40(is_a_theorem(implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.74 = { by lemma 61 } 18.70/2.74 not(fresh3(fresh40(fresh116(is_a_theorem(implies(not(X), and(not(X), not(X)))), true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.74 = { by axiom 8 (modus_ponens_2) } 18.70/2.74 not(fresh3(fresh40(fresh116(fresh40(true, true, implies(not(X), implies(not(X), and(not(X), not(X)))), implies(not(X), and(not(X), not(X)))), true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.74 = { by axiom 7 (implies_2_1) } 18.70/2.74 not(fresh3(fresh40(fresh116(fresh40(fresh49(true, true, not(X), and(not(X), not(X))), true, implies(not(X), implies(not(X), and(not(X), not(X)))), implies(not(X), and(not(X), not(X)))), true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.74 = { by axiom 35 (hilbert_implies_2) } 18.70/2.74 not(fresh3(fresh40(fresh116(fresh40(fresh49(implies_2, true, not(X), and(not(X), not(X))), true, implies(not(X), implies(not(X), and(not(X), not(X)))), implies(not(X), and(not(X), not(X)))), true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.74 = { by axiom 29 (implies_2_1) } 18.70/2.74 not(fresh3(fresh40(fresh116(fresh40(is_a_theorem(implies(implies(not(X), implies(not(X), and(not(X), not(X)))), implies(not(X), and(not(X), not(X))))), true, implies(not(X), implies(not(X), and(not(X), not(X)))), implies(not(X), and(not(X), not(X)))), true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.74 = { by lemma 59 } 18.70/2.74 not(fresh3(fresh40(fresh116(fresh116(is_a_theorem(implies(not(X), implies(not(X), and(not(X), not(X))))), true, implies(not(X), and(not(X), not(X)))), true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.74 = { by lemma 60 } 18.70/2.74 not(fresh3(fresh40(fresh116(fresh116(true, true, implies(not(X), and(not(X), not(X)))), true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.74 = { by axiom 9 (modus_ponens_2) } 18.70/2.74 not(fresh3(fresh40(fresh116(true, true, implies(implies(and(not(X), not(X)), not(X)), and(implies(not(X), and(not(X), not(X))), implies(and(not(X), not(X)), not(X))))), true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.74 = { by axiom 9 (modus_ponens_2) } 18.70/2.74 not(fresh3(fresh40(true, true, implies(and(not(X), not(X)), not(X)), equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.74 = { by axiom 8 (modus_ponens_2) } 18.70/2.74 not(fresh3(is_a_theorem(equiv(not(X), and(not(X), not(X)))), true, not(X), and(not(X), not(X)))) 18.70/2.74 = { by lemma 62 } 18.70/2.74 not(not(X)) 18.70/2.74 18.70/2.74 Lemma 66: is_a_theorem(implies(or(X, X), X)) = fresh59(cn3, true, X). 18.70/2.74 Proof: 18.70/2.74 is_a_theorem(implies(or(X, X), X)) 18.70/2.74 = { by lemma 56 } 18.70/2.74 is_a_theorem(implies(implies(not(X), X), X)) 18.70/2.74 = { by axiom 26 (cn3_1) } 19.14/2.78 fresh59(cn3, true, X) 19.14/2.78 19.14/2.78 Goal 1 (s1_0_m6s3m9b_axiom_m9): axiom_m9 = true. 19.14/2.78 Proof: 19.14/2.78 axiom_m9 19.14/2.78 = { by axiom 46 (axiom_m9) } 19.14/2.78 fresh74(is_a_theorem(strict_implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.78 = { by axiom 18 (op_strict_implies) } 19.14/2.78 fresh74(is_a_theorem(fresh23(true, true, possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.78 = { by axiom 54 (s1_0_op_strict_implies) } 19.14/2.78 fresh74(is_a_theorem(fresh23(op_strict_implies, true, possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.78 = { by axiom 48 (op_strict_implies) } 19.14/2.78 fresh74(is_a_theorem(necessarily(implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X)))), true) 19.14/2.78 = { by axiom 12 (necessitation_1) } 19.14/2.78 fresh74(fresh34(true, true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.78 = { by axiom 51 (km5_necessitation) } 19.14/2.78 fresh74(fresh34(necessitation, true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.78 = { by axiom 44 (necessitation_1) } 19.14/2.78 fresh74(fresh33(is_a_theorem(implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.78 = { by lemma 63 } 19.14/2.78 fresh74(fresh33(is_a_theorem(implies(possibly(necessarily(possibly(sK17_axiom_m9_X))), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.78 = { by lemma 57 } 19.14/2.78 fresh74(fresh33(is_a_theorem(implies(possibly(necessarily(not(necessarily(not(sK17_axiom_m9_X))))), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.78 = { by lemma 64 } 19.14/2.78 fresh74(fresh33(is_a_theorem(implies(not(possibly(necessarily(not(sK17_axiom_m9_X)))), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.78 = { by lemma 64 } 19.14/2.78 fresh74(fresh33(is_a_theorem(implies(not(not(possibly(sK17_axiom_m9_X))), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.78 = { by lemma 65 } 19.14/2.78 fresh74(fresh33(is_a_theorem(implies(implies(not(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.78 = { by lemma 56 } 19.14/2.78 fresh74(fresh33(is_a_theorem(implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.78 = { by axiom 8 (modus_ponens_2) } 19.14/2.79 fresh74(fresh33(fresh40(true, true, or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by axiom 11 (modus_tollens_1) } 19.14/2.79 fresh74(fresh33(fresh40(fresh35(true, true, or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X)), true, or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by axiom 34 (hilbert_modus_tollens) } 19.14/2.79 fresh74(fresh33(fresh40(fresh35(modus_tollens, true, or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X)), true, or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by axiom 22 (modus_tollens_1) } 19.14/2.79 fresh74(fresh33(fresh40(is_a_theorem(implies(implies(not(possibly(sK17_axiom_m9_X)), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X)))), true, or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by lemma 56 } 19.14/2.79 fresh74(fresh33(fresh40(is_a_theorem(implies(or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X)))), true, or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by axiom 28 (modus_ponens_2) } 19.14/2.79 fresh74(fresh33(fresh115(modus_ponens, true, or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by axiom 40 (hilbert_modus_ponens) } 19.14/2.79 fresh74(fresh33(fresh115(true, true, or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)))), implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by axiom 10 (modus_ponens_2) } 19.14/2.79 fresh74(fresh33(fresh116(is_a_theorem(or(possibly(sK17_axiom_m9_X), not(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X))))), true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by lemma 56 } 19.14/2.79 fresh74(fresh33(fresh116(is_a_theorem(or(possibly(sK17_axiom_m9_X), not(implies(not(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))))), true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by lemma 65 } 19.14/2.79 fresh74(fresh33(fresh116(is_a_theorem(or(possibly(sK17_axiom_m9_X), not(not(not(possibly(sK17_axiom_m9_X)))))), true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by lemma 65 } 19.14/2.79 fresh74(fresh33(fresh116(is_a_theorem(or(possibly(sK17_axiom_m9_X), implies(not(not(possibly(sK17_axiom_m9_X))), not(possibly(sK17_axiom_m9_X))))), true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by lemma 56 } 19.14/2.79 fresh74(fresh33(fresh116(is_a_theorem(implies(not(possibly(sK17_axiom_m9_X)), implies(not(not(possibly(sK17_axiom_m9_X))), not(possibly(sK17_axiom_m9_X))))), true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by axiom 24 (implies_1_1) } 19.14/2.79 fresh74(fresh33(fresh116(fresh51(implies_1, true, not(possibly(sK17_axiom_m9_X)), not(not(possibly(sK17_axiom_m9_X)))), true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by axiom 36 (hilbert_implies_1) } 19.14/2.79 fresh74(fresh33(fresh116(fresh51(true, true, not(possibly(sK17_axiom_m9_X)), not(not(possibly(sK17_axiom_m9_X)))), true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by axiom 6 (implies_1_1) } 19.14/2.79 fresh74(fresh33(fresh116(true, true, implies(or(possibly(sK17_axiom_m9_X), possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by axiom 9 (modus_ponens_2) } 19.14/2.79 fresh74(fresh33(true, true, implies(possibly(possibly(sK17_axiom_m9_X)), possibly(sK17_axiom_m9_X))), true) 19.14/2.79 = { by axiom 13 (necessitation_1) } 19.14/2.79 fresh74(true, true) 19.14/2.79 = { by axiom 5 (axiom_m9) } 19.14/2.79 true 19.14/2.79 % SZS output end Proof 19.14/2.79 19.14/2.79 RESULT: Theorem (the conjecture is true). 19.14/2.80 EOF