0.11/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.11/0.13 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.13/0.34 % Computer : n003.cluster.edu 0.13/0.34 % Model : x86_64 x86_64 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.34 % Memory : 8042.1875MB 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.34 % CPULimit : 180 0.13/0.34 % DateTime : Thu Aug 29 09:10:17 EDT 2019 0.13/0.34 % CPUTime : 1.75/1.93 % SZS status Theorem 1.75/1.93 1.75/1.93 % SZS output start Proof 1.75/1.93 Take the following subset of the input axioms: 1.75/1.98 fof(and_2, axiom, and_2 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), Y))). 1.75/1.98 fof(and_3, axiom, ![X, Y]: is_a_theorem(implies(X, implies(Y, and(X, Y)))) <=> and_3). 1.75/1.98 fof(axiom_m1, axiom, axiom_m1 <=> ![X, Y]: is_a_theorem(strict_implies(and(X, Y), and(Y, X)))). 1.75/1.98 fof(axiom_m4, axiom, ![X]: is_a_theorem(strict_implies(X, and(X, X))) <=> axiom_m4). 1.75/1.98 fof(hilbert_and_2, axiom, and_2). 1.75/1.98 fof(hilbert_and_3, axiom, and_3). 1.75/1.98 fof(hilbert_implies_1, axiom, implies_1). 1.75/1.98 fof(hilbert_implies_2, axiom, implies_2). 1.75/1.98 fof(hilbert_modus_ponens, axiom, modus_ponens). 1.75/1.98 fof(hilbert_op_equiv, axiom, op_equiv). 1.75/1.98 fof(implies_1, axiom, implies_1 <=> ![X, Y]: is_a_theorem(implies(X, implies(Y, X)))). 1.75/1.98 fof(implies_2, axiom, implies_2 <=> ![X, Y]: is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y)))). 1.75/1.98 fof(km5_necessitation, axiom, necessitation). 1.75/1.98 fof(kn1, axiom, ![P]: is_a_theorem(implies(P, and(P, P))) <=> kn1). 1.75/1.98 fof(modus_ponens, axiom, modus_ponens <=> ![X, Y]: ((is_a_theorem(X) & is_a_theorem(implies(X, Y))) => is_a_theorem(Y))). 1.75/1.98 fof(necessitation, axiom, necessitation <=> ![X]: (is_a_theorem(necessarily(X)) <= is_a_theorem(X))). 1.75/1.98 fof(op_equiv, axiom, op_equiv => ![X, Y]: equiv(X, Y)=and(implies(X, Y), implies(Y, X))). 1.75/1.98 fof(op_strict_implies, axiom, ![X, Y]: necessarily(implies(X, Y))=strict_implies(X, Y) <= op_strict_implies). 1.75/1.98 fof(s1_0_axiom_m4, conjecture, axiom_m4). 1.75/1.98 fof(s1_0_op_strict_implies, axiom, op_strict_implies). 1.75/1.98 fof(substitution_of_equivalents, axiom, substitution_of_equivalents <=> ![X, Y]: (is_a_theorem(equiv(X, Y)) => X=Y)). 1.75/1.98 fof(substitution_of_equivalents, axiom, substitution_of_equivalents). 1.75/1.98 1.75/1.98 Now clausify the problem and encode Horn clauses using encoding 3 of 1.75/1.98 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 1.75/1.98 We repeatedly replace C & s=t => u=v by the two clauses: 1.75/1.98 fresh(y, y, x1...xn) = u 1.75/1.98 C => fresh(s, t, x1...xn) = v 1.75/1.98 where fresh is a fresh function symbol and x1..xn are the free 1.75/1.98 variables of u and v. 1.75/1.98 A predicate p(X) is encoded as p(X)=true (this is sound, because the 1.75/1.98 input problem has no model of domain size 1). 1.75/1.98 1.75/1.98 The encoding turns the above axioms into the following unit equations and goals: 1.75/1.98 1.75/1.98 Axiom 1 (and_2_1): fresh105(X, X, Y, Z) = true. 1.75/1.98 Axiom 2 (and_3_1): fresh103(X, X, Y, Z) = true. 1.75/1.98 Axiom 3 (axiom_m4): fresh84(X, X) = true. 1.75/1.98 Axiom 4 (implies_1_1): fresh51(X, X, Y, Z) = true. 1.75/1.98 Axiom 5 (implies_2_1): fresh49(X, X, Y, Z) = true. 1.75/1.98 Axiom 6 (modus_ponens_2): fresh40(X, X, Y, Z) = is_a_theorem(Z). 1.75/1.98 Axiom 7 (modus_ponens_2): fresh116(X, X, Y) = true. 1.75/1.98 Axiom 8 (modus_ponens_2): fresh115(X, X, Y, Z) = fresh116(is_a_theorem(Y), true, Z). 1.75/1.98 Axiom 9 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)). 1.75/1.98 Axiom 10 (necessitation_1): fresh33(X, X, Y) = true. 1.75/1.98 Axiom 11 (op_equiv): fresh30(X, X, Y, Z) = equiv(Y, Z). 1.75/1.98 Axiom 12 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z). 1.75/1.98 Axiom 13 (substitution_of_equivalents_2): fresh4(X, X, Y, Z) = Y. 1.75/1.98 Axiom 14 (substitution_of_equivalents_2): fresh3(X, X, Y, Z) = Z. 1.75/1.98 Axiom 15 (and_3_1): fresh103(and_3, true, X, Y) = is_a_theorem(implies(X, implies(Y, and(X, Y)))). 1.75/1.98 Axiom 16 (substitution_of_equivalents_2): fresh4(substitution_of_equivalents, true, X, Y) = fresh3(is_a_theorem(equiv(X, Y)), true, X, Y). 1.75/1.98 Axiom 17 (modus_ponens_2): fresh115(modus_ponens, true, X, Y) = fresh40(is_a_theorem(implies(X, Y)), true, X, Y). 1.75/1.98 Axiom 18 (kn1_1): fresh45(kn1, true, X) = is_a_theorem(implies(X, and(X, X))). 1.75/1.98 Axiom 19 (and_2_1): fresh105(and_2, true, X, Y) = is_a_theorem(implies(and(X, Y), Y)). 1.75/1.98 Axiom 20 (implies_2_1): fresh49(implies_2, true, X, Y) = is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))). 1.75/1.98 Axiom 21 (implies_1_1): fresh51(implies_1, true, X, Y) = is_a_theorem(implies(X, implies(Y, X))). 1.75/1.98 Axiom 22 (op_equiv): fresh30(op_equiv, true, X, Y) = and(implies(X, Y), implies(Y, X)). 1.75/1.98 Axiom 23 (hilbert_and_2): and_2 = true. 1.75/1.98 Axiom 24 (substitution_of_equivalents): substitution_of_equivalents = true. 1.75/1.98 Axiom 25 (hilbert_and_3): and_3 = true. 1.75/1.98 Axiom 26 (hilbert_implies_1): implies_1 = true. 1.75/1.98 Axiom 27 (hilbert_modus_ponens): modus_ponens = true. 1.75/1.98 Axiom 28 (hilbert_implies_2): implies_2 = true. 1.75/1.98 Axiom 29 (hilbert_op_equiv): op_equiv = true. 1.75/1.98 Axiom 30 (axiom_m1_1): fresh89(axiom_m1, true, X, Y) = is_a_theorem(strict_implies(and(X, Y), and(Y, X))). 1.75/1.98 Axiom 31 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X). 1.75/1.98 Axiom 32 (axiom_m4_1): fresh83(axiom_m4, true, X) = is_a_theorem(strict_implies(X, and(X, X))). 1.75/1.98 Axiom 33 (axiom_m4): fresh84(is_a_theorem(strict_implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) = axiom_m4. 1.75/1.98 Axiom 34 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)). 1.75/1.98 Axiom 35 (km5_necessitation): necessitation = true. 1.75/1.98 Axiom 36 (s1_0_op_strict_implies): op_strict_implies = true. 1.75/1.98 1.75/1.98 Lemma 37: fresh40(is_a_theorem(implies(X, Y)), true, X, Y) = fresh116(is_a_theorem(X), true, Y). 1.75/1.98 Proof: 1.75/1.98 fresh40(is_a_theorem(implies(X, Y)), true, X, Y) 1.75/1.98 = { by axiom 17 (modus_ponens_2) } 1.75/1.98 fresh115(modus_ponens, true, X, Y) 1.75/1.98 = { by axiom 27 (hilbert_modus_ponens) } 1.75/1.98 fresh115(true, true, X, Y) 1.75/1.98 = { by axiom 8 (modus_ponens_2) } 1.75/1.98 fresh116(is_a_theorem(X), true, Y) 1.75/1.98 1.75/1.98 Lemma 38: is_a_theorem(implies(X, implies(Y, and(X, Y)))) = true. 1.75/1.98 Proof: 1.75/1.98 is_a_theorem(implies(X, implies(Y, and(X, Y)))) 1.75/1.98 = { by axiom 15 (and_3_1) } 1.75/1.98 fresh103(and_3, true, X, Y) 1.75/1.98 = { by axiom 25 (hilbert_and_3) } 1.75/1.98 fresh103(true, true, X, Y) 1.75/1.98 = { by axiom 2 (and_3_1) } 1.75/1.98 true 1.75/1.98 1.75/1.98 Lemma 39: fresh116(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y)) = is_a_theorem(implies(X, Y)). 1.75/1.98 Proof: 1.75/1.98 fresh116(is_a_theorem(implies(X, implies(X, Y))), true, implies(X, Y)) 1.75/1.98 = { by lemma 37 } 1.75/1.98 fresh40(is_a_theorem(implies(implies(X, implies(X, Y)), implies(X, Y))), true, implies(X, implies(X, Y)), implies(X, Y)) 1.75/1.98 = { by axiom 20 (implies_2_1) } 1.75/1.98 fresh40(fresh49(implies_2, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y)) 1.75/1.98 = { by axiom 28 (hilbert_implies_2) } 1.75/1.98 fresh40(fresh49(true, true, X, Y), true, implies(X, implies(X, Y)), implies(X, Y)) 1.75/1.98 = { by axiom 5 (implies_2_1) } 1.75/1.98 fresh40(true, true, implies(X, implies(X, Y)), implies(X, Y)) 1.75/1.98 = { by axiom 6 (modus_ponens_2) } 1.83/2.03 is_a_theorem(implies(X, Y)) 1.83/2.03 1.83/2.03 Goal 1 (s1_0_axiom_m4): axiom_m4 = true. 1.83/2.03 Proof: 1.83/2.03 axiom_m4 1.83/2.03 = { by axiom 33 (axiom_m4) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 13 (substitution_of_equivalents_2) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh4(true, true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 24 (substitution_of_equivalents) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh4(substitution_of_equivalents, true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 16 (substitution_of_equivalents_2) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(is_a_theorem(equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 6 (modus_ponens_2) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(fresh40(true, true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 7 (modus_ponens_2) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(fresh40(fresh116(true, true, implies(implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), and(implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X)))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 7 (modus_ponens_2) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(fresh40(fresh116(fresh116(true, true, implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, implies(implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), and(implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X)))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by lemma 38 } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(fresh40(fresh116(fresh116(is_a_theorem(implies(sK10_axiom_m4_X, implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)))), true, implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, implies(implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), and(implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X)))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by lemma 39 } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(fresh40(fresh116(is_a_theorem(implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, implies(implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), and(implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X)))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by lemma 37 } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(fresh40(fresh40(is_a_theorem(implies(implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), implies(implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), and(implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X))))), true, implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), implies(implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), and(implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X)))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by lemma 38 } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(fresh40(fresh40(true, true, implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), implies(implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), and(implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X)))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 6 (modus_ponens_2) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(implies(implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), and(implies(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X)))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 22 (op_equiv) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(implies(implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), fresh30(op_equiv, true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 29 (hilbert_op_equiv) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(implies(implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), fresh30(true, true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 11 (op_equiv) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(fresh40(is_a_theorem(implies(implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X), equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by lemma 37 } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(fresh116(is_a_theorem(implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), sK10_axiom_m4_X)), true, equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 19 (and_2_1) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(fresh116(fresh105(and_2, true, sK10_axiom_m4_X, sK10_axiom_m4_X), true, equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 23 (hilbert_and_2) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(fresh116(fresh105(true, true, sK10_axiom_m4_X, sK10_axiom_m4_X), true, equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 1 (and_2_1) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(fresh116(true, true, equiv(sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 7 (modus_ponens_2) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(fresh3(true, true, sK10_axiom_m4_X, and(sK10_axiom_m4_X, sK10_axiom_m4_X)), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 14 (substitution_of_equivalents_2) } 1.83/2.03 fresh84(is_a_theorem(strict_implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 12 (op_strict_implies) } 1.83/2.03 fresh84(is_a_theorem(fresh23(true, true, and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 36 (s1_0_op_strict_implies) } 1.83/2.03 fresh84(is_a_theorem(fresh23(op_strict_implies, true, and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 34 (op_strict_implies) } 1.83/2.03 fresh84(is_a_theorem(necessarily(implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X)))), true) 1.83/2.03 = { by axiom 9 (necessitation_1) } 1.83/2.03 fresh84(fresh34(true, true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 35 (km5_necessitation) } 1.83/2.03 fresh84(fresh34(necessitation, true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 31 (necessitation_1) } 1.83/2.03 fresh84(fresh33(is_a_theorem(implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by lemma 39 } 1.83/2.03 fresh84(fresh33(fresh116(is_a_theorem(implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X)))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 21 (implies_1_1) } 1.83/2.03 fresh84(fresh33(fresh116(fresh51(implies_1, true, and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X)), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 26 (hilbert_implies_1) } 1.83/2.03 fresh84(fresh33(fresh116(fresh51(true, true, and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X)), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 4 (implies_1_1) } 1.83/2.03 fresh84(fresh33(fresh116(true, true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 7 (modus_ponens_2) } 1.83/2.03 fresh84(fresh33(true, true, implies(and(sK10_axiom_m4_X, sK10_axiom_m4_X), and(sK10_axiom_m4_X, sK10_axiom_m4_X))), true) 1.83/2.03 = { by axiom 10 (necessitation_1) } 1.83/2.03 fresh84(true, true) 1.83/2.03 = { by axiom 3 (axiom_m4) } 1.83/2.03 true 1.83/2.03 % SZS output end Proof 1.83/2.03 1.83/2.03 RESULT: Theorem (the conjecture is true). 1.83/2.03 EOF