0.02/0.10 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.02/0.11 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.10/0.31 % Computer : n029.cluster.edu 0.10/0.31 % Model : x86_64 x86_64 0.10/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.10/0.31 % Memory : 8042.1875MB 0.10/0.31 % OS : Linux 3.10.0-693.el7.x86_64 0.10/0.31 % CPULimit : 960 0.10/0.31 % WCLimit : 120 0.10/0.31 % DateTime : Thu Jul 2 06:50:15 EDT 2020 0.16/0.31 % CPUTime : 0.17/0.51 % SZS status Theorem 0.17/0.51 0.17/0.51 % SZS output start Proof 0.17/0.51 Take the following subset of the input axioms: 0.17/0.53 fof(axiom_5, axiom, ![X]: is_a_theorem(implies(possibly(X), necessarily(possibly(X)))) <=> axiom_5). 0.17/0.53 fof(axiom_m10, axiom, axiom_m10 <=> ![X]: is_a_theorem(strict_implies(possibly(X), necessarily(possibly(X))))). 0.17/0.53 fof(hilbert_op_equiv, axiom, op_equiv). 0.17/0.53 fof(hilbert_op_implies_and, axiom, op_implies_and). 0.17/0.53 fof(km5_axiom_5, axiom, axiom_5). 0.17/0.53 fof(km5_necessitation, axiom, necessitation). 0.17/0.53 fof(necessitation, axiom, necessitation <=> ![X]: (is_a_theorem(X) => is_a_theorem(necessarily(X)))). 0.17/0.53 fof(op_strict_implies, axiom, op_strict_implies => ![X, Y]: necessarily(implies(X, Y))=strict_implies(X, Y)). 0.17/0.53 fof(s1_0_m10_axiom_m10, conjecture, axiom_m10). 0.17/0.53 fof(s1_0_op_equiv, axiom, op_equiv). 0.17/0.53 fof(s1_0_op_implies, axiom, op_implies). 0.17/0.53 fof(s1_0_op_strict_implies, axiom, op_strict_implies). 0.17/0.53 0.17/0.53 Now clausify the problem and encode Horn clauses using encoding 3 of 0.17/0.53 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 0.17/0.53 We repeatedly replace C & s=t => u=v by the two clauses: 0.17/0.53 fresh(y, y, x1...xn) = u 0.17/0.53 C => fresh(s, t, x1...xn) = v 0.17/0.53 where fresh is a fresh function symbol and x1..xn are the free 0.17/0.53 variables of u and v. 0.17/0.53 A predicate p(X) is encoded as p(X)=true (this is sound, because the 0.17/0.53 input problem has no model of domain size 1). 0.17/0.53 0.17/0.53 The encoding turns the above axioms into the following unit equations and goals: 0.17/0.53 0.17/0.53 Axiom 1 (axiom_5_1): fresh99(X, X, Y) = true. 0.17/0.53 Axiom 2 (axiom_m10): fresh91(X, X) = true. 0.17/0.53 Axiom 3 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)). 0.17/0.53 Axiom 4 (necessitation_1): fresh33(X, X, Y) = true. 0.17/0.53 Axiom 5 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z). 0.17/0.53 Axiom 6 (hilbert_op_implies_and): op_implies_and = true. 0.17/0.53 Axiom 7 (hilbert_op_equiv): op_equiv = true. 0.17/0.53 Axiom 8 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X). 0.17/0.53 Axiom 9 (axiom_m10_1): fresh90(axiom_m10, true, X) = is_a_theorem(strict_implies(possibly(X), necessarily(possibly(X)))). 0.17/0.53 Axiom 10 (axiom_m10): fresh91(is_a_theorem(strict_implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) = axiom_m10. 0.17/0.53 Axiom 11 (axiom_5_1): fresh99(axiom_5, true, X) = is_a_theorem(implies(possibly(X), necessarily(possibly(X)))). 0.17/0.53 Axiom 12 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)). 0.17/0.53 Axiom 13 (km5_necessitation): necessitation = true. 0.17/0.53 Axiom 14 (km5_axiom_5): axiom_5 = true. 0.17/0.53 Axiom 15 (s1_0_op_strict_implies): op_strict_implies = true. 0.17/0.53 Axiom 16 (s1_0_op_implies): op_implies = true. 0.17/0.53 Axiom 17 (s1_0_op_equiv): op_equiv = true. 0.17/0.53 0.17/0.53 Lemma 18: op_implies_and = op_implies. 0.17/0.53 Proof: 0.17/0.53 op_implies_and 0.17/0.53 = { by axiom 6 (hilbert_op_implies_and) } 0.17/0.53 true 0.17/0.53 = { by axiom 16 (s1_0_op_implies) } 0.17/0.53 op_implies 0.17/0.53 0.17/0.53 Lemma 19: is_a_theorem(strict_implies(possibly(X), necessarily(possibly(X)))) = fresh90(axiom_m10, op_implies_and, X). 0.17/0.54 Proof: 0.17/0.54 is_a_theorem(strict_implies(possibly(X), necessarily(possibly(X)))) 0.17/0.54 = { by axiom 9 (axiom_m10_1) } 0.17/0.54 fresh90(axiom_m10, true, X) 0.17/0.54 = { by axiom 6 (hilbert_op_implies_and) } 0.17/0.56 fresh90(axiom_m10, op_implies_and, X) 0.17/0.56 0.17/0.56 Goal 1 (s1_0_m10_axiom_m10): axiom_m10 = true. 0.17/0.56 Proof: 0.17/0.56 axiom_m10 0.17/0.56 = { by axiom 10 (axiom_m10) } 0.17/0.56 fresh91(is_a_theorem(strict_implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 5 (op_strict_implies) } 0.17/0.56 fresh91(is_a_theorem(fresh23(op_implies_and, op_implies_and, possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 6 (hilbert_op_implies_and) } 0.17/0.56 fresh91(is_a_theorem(fresh23(true, op_implies_and, possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 15 (s1_0_op_strict_implies) } 0.17/0.56 fresh91(is_a_theorem(fresh23(op_strict_implies, op_implies_and, possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 6 (hilbert_op_implies_and) } 0.17/0.56 fresh91(is_a_theorem(fresh23(op_strict_implies, true, possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 12 (op_strict_implies) } 0.17/0.56 fresh91(is_a_theorem(necessarily(implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X))))), true) 0.17/0.56 = { by axiom 3 (necessitation_1) } 0.17/0.56 fresh91(fresh34(op_implies, op_implies, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by lemma 18 } 0.17/0.56 fresh91(fresh34(op_implies_and, op_implies, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 6 (hilbert_op_implies_and) } 0.17/0.56 fresh91(fresh34(true, op_implies, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 13 (km5_necessitation) } 0.17/0.56 fresh91(fresh34(necessitation, op_implies, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by lemma 18 } 0.17/0.56 fresh91(fresh34(necessitation, op_implies_and, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 6 (hilbert_op_implies_and) } 0.17/0.56 fresh91(fresh34(necessitation, true, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 8 (necessitation_1) } 0.17/0.56 fresh91(fresh33(is_a_theorem(implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 11 (axiom_5_1) } 0.17/0.56 fresh91(fresh33(fresh99(axiom_5, true, sK9_axiom_m10_X), true, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 14 (km5_axiom_5) } 0.17/0.56 fresh91(fresh33(fresh99(true, true, sK9_axiom_m10_X), true, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 6 (hilbert_op_implies_and) } 0.17/0.56 fresh91(fresh33(fresh99(op_implies_and, true, sK9_axiom_m10_X), true, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by lemma 18 } 0.17/0.56 fresh91(fresh33(fresh99(op_implies, true, sK9_axiom_m10_X), true, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 6 (hilbert_op_implies_and) } 0.17/0.56 fresh91(fresh33(fresh99(op_implies, op_implies_and, sK9_axiom_m10_X), true, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by lemma 18 } 0.17/0.56 fresh91(fresh33(fresh99(op_implies, op_implies, sK9_axiom_m10_X), true, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 1 (axiom_5_1) } 0.17/0.56 fresh91(fresh33(true, true, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 16 (s1_0_op_implies) } 0.17/0.56 fresh91(fresh33(op_implies, true, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 6 (hilbert_op_implies_and) } 0.17/0.56 fresh91(fresh33(op_implies, op_implies_and, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by lemma 18 } 0.17/0.56 fresh91(fresh33(op_implies, op_implies, implies(possibly(sK9_axiom_m10_X), necessarily(possibly(sK9_axiom_m10_X)))), true) 0.17/0.56 = { by axiom 4 (necessitation_1) } 0.17/0.56 fresh91(true, true) 0.17/0.56 = { by axiom 16 (s1_0_op_implies) } 0.17/0.56 fresh91(op_implies, true) 0.17/0.56 = { by axiom 6 (hilbert_op_implies_and) } 0.17/0.56 fresh91(op_implies, op_implies_and) 0.17/0.56 = { by lemma 18 } 0.17/0.56 fresh91(op_implies, op_implies) 0.17/0.56 = { by axiom 2 (axiom_m10) } 0.17/0.56 true 0.17/0.56 % SZS output end Proof 0.17/0.56 0.17/0.56 RESULT: Theorem (the conjecture is true). 0.17/0.56 EOF