TSTP Solution File: LCL536+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL536+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 : n003.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 08:19:21 EDT 2023
% Result : Theorem 0.19s 0.56s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL536+1 : TPTP v8.1.2. Released v3.3.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.33 % Computer : n003.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Fri Aug 25 06:12:38 EDT 2023
% 0.13/0.33 % CPUTime :
% 0.19/0.56 Command-line arguments: --ground-connectedness --complete-subsets
% 0.19/0.56
% 0.19/0.56 % SZS status Theorem
% 0.19/0.56
% 0.19/0.57 % SZS output start Proof
% 0.19/0.57 Take the following subset of the input axioms:
% 0.19/0.57 fof(axiom_5, axiom, axiom_5 <=> ![X]: is_a_theorem(implies(possibly(X), necessarily(possibly(X))))).
% 0.19/0.57 fof(axiom_m10, axiom, axiom_m10 <=> ![X2]: is_a_theorem(strict_implies(possibly(X2), necessarily(possibly(X2))))).
% 0.19/0.57 fof(km5_axiom_5, axiom, axiom_5).
% 0.19/0.57 fof(km5_necessitation, axiom, necessitation).
% 0.19/0.57 fof(necessitation, axiom, necessitation <=> ![X2]: (is_a_theorem(X2) => is_a_theorem(necessarily(X2)))).
% 0.19/0.57 fof(op_strict_implies, axiom, op_strict_implies => ![Y, X2]: strict_implies(X2, Y)=necessarily(implies(X2, Y))).
% 0.19/0.57 fof(s1_0_m10_axiom_m10, conjecture, axiom_m10).
% 0.19/0.57 fof(s1_0_op_implies, axiom, op_implies).
% 0.19/0.57 fof(s1_0_op_strict_implies, axiom, op_strict_implies).
% 0.19/0.57
% 0.19/0.57 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.57 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.57 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.57 fresh(y, y, x1...xn) = u
% 0.19/0.57 C => fresh(s, t, x1...xn) = v
% 0.19/0.57 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.57 variables of u and v.
% 0.19/0.57 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.57 input problem has no model of domain size 1).
% 0.19/0.57
% 0.19/0.57 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.57
% 0.19/0.57 Axiom 1 (s1_0_op_implies): op_implies = true.
% 0.19/0.57 Axiom 2 (s1_0_op_strict_implies): op_strict_implies = true.
% 0.19/0.57 Axiom 3 (km5_necessitation): necessitation = true.
% 0.19/0.57 Axiom 4 (km5_axiom_5): axiom_5 = true.
% 0.19/0.57 Axiom 5 (axiom_m10): fresh91(X, X) = true.
% 0.19/0.57 Axiom 6 (axiom_5_1): fresh99(X, X, Y) = true.
% 0.19/0.57 Axiom 7 (necessitation_1): fresh34(X, X, Y) = is_a_theorem(necessarily(Y)).
% 0.19/0.57 Axiom 8 (necessitation_1): fresh33(X, X, Y) = true.
% 0.19/0.57 Axiom 9 (necessitation_1): fresh34(necessitation, true, X) = fresh33(is_a_theorem(X), true, X).
% 0.19/0.57 Axiom 10 (op_strict_implies): fresh23(X, X, Y, Z) = strict_implies(Y, Z).
% 0.19/0.57 Axiom 11 (op_strict_implies): fresh23(op_strict_implies, true, X, Y) = necessarily(implies(X, Y)).
% 0.19/0.57 Axiom 12 (axiom_m10_1): fresh90(axiom_m10, true, X) = is_a_theorem(strict_implies(possibly(X), necessarily(possibly(X)))).
% 0.19/0.57 Axiom 13 (axiom_5_1): fresh99(axiom_5, true, X) = is_a_theorem(implies(possibly(X), necessarily(possibly(X)))).
% 0.19/0.57 Axiom 14 (axiom_m10): fresh91(is_a_theorem(strict_implies(possibly(x), necessarily(possibly(x)))), true) = axiom_m10.
% 0.19/0.57
% 0.19/0.57 Lemma 15: is_a_theorem(strict_implies(possibly(X), necessarily(possibly(X)))) = fresh90(axiom_m10, op_implies, X).
% 0.19/0.57 Proof:
% 0.19/0.57 is_a_theorem(strict_implies(possibly(X), necessarily(possibly(X))))
% 0.19/0.57 = { by axiom 12 (axiom_m10_1) R->L }
% 0.19/0.57 fresh90(axiom_m10, true, X)
% 0.19/0.57 = { by axiom 1 (s1_0_op_implies) R->L }
% 0.19/0.57 fresh90(axiom_m10, op_implies, X)
% 0.19/0.57
% 0.19/0.57 Goal 1 (s1_0_m10_axiom_m10): axiom_m10 = true.
% 0.19/0.57 Proof:
% 0.19/0.57 axiom_m10
% 0.19/0.57 = { by axiom 14 (axiom_m10) R->L }
% 0.19/0.57 fresh91(is_a_theorem(strict_implies(possibly(x), necessarily(possibly(x)))), true)
% 0.19/0.57 = { by lemma 15 }
% 0.19/0.57 fresh91(fresh90(axiom_m10, op_implies, x), true)
% 0.19/0.57 = { by axiom 1 (s1_0_op_implies) R->L }
% 0.19/0.57 fresh91(fresh90(axiom_m10, op_implies, x), op_implies)
% 0.19/0.57 = { by lemma 15 R->L }
% 0.19/0.57 fresh91(is_a_theorem(strict_implies(possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 10 (op_strict_implies) R->L }
% 0.19/0.57 fresh91(is_a_theorem(fresh23(op_implies, op_implies, possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 1 (s1_0_op_implies) }
% 0.19/0.57 fresh91(is_a_theorem(fresh23(op_implies, true, possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 1 (s1_0_op_implies) }
% 0.19/0.57 fresh91(is_a_theorem(fresh23(true, true, possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 2 (s1_0_op_strict_implies) R->L }
% 0.19/0.57 fresh91(is_a_theorem(fresh23(op_strict_implies, true, possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 11 (op_strict_implies) }
% 0.19/0.57 fresh91(is_a_theorem(necessarily(implies(possibly(x), necessarily(possibly(x))))), op_implies)
% 0.19/0.57 = { by axiom 7 (necessitation_1) R->L }
% 0.19/0.57 fresh91(fresh34(op_implies, op_implies, implies(possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 1 (s1_0_op_implies) }
% 0.19/0.57 fresh91(fresh34(op_implies, true, implies(possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 1 (s1_0_op_implies) }
% 0.19/0.57 fresh91(fresh34(true, true, implies(possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 3 (km5_necessitation) R->L }
% 0.19/0.57 fresh91(fresh34(necessitation, true, implies(possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 9 (necessitation_1) }
% 0.19/0.57 fresh91(fresh33(is_a_theorem(implies(possibly(x), necessarily(possibly(x)))), true, implies(possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 1 (s1_0_op_implies) R->L }
% 0.19/0.57 fresh91(fresh33(is_a_theorem(implies(possibly(x), necessarily(possibly(x)))), op_implies, implies(possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 13 (axiom_5_1) R->L }
% 0.19/0.57 fresh91(fresh33(fresh99(axiom_5, true, x), op_implies, implies(possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 4 (km5_axiom_5) }
% 0.19/0.57 fresh91(fresh33(fresh99(true, true, x), op_implies, implies(possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 1 (s1_0_op_implies) R->L }
% 0.19/0.57 fresh91(fresh33(fresh99(op_implies, true, x), op_implies, implies(possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 1 (s1_0_op_implies) R->L }
% 0.19/0.57 fresh91(fresh33(fresh99(op_implies, op_implies, x), op_implies, implies(possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 6 (axiom_5_1) }
% 0.19/0.57 fresh91(fresh33(true, op_implies, implies(possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 1 (s1_0_op_implies) R->L }
% 0.19/0.57 fresh91(fresh33(op_implies, op_implies, implies(possibly(x), necessarily(possibly(x)))), op_implies)
% 0.19/0.57 = { by axiom 8 (necessitation_1) }
% 0.19/0.57 fresh91(true, op_implies)
% 0.19/0.57 = { by axiom 1 (s1_0_op_implies) R->L }
% 0.19/0.57 fresh91(op_implies, op_implies)
% 0.19/0.57 = { by axiom 5 (axiom_m10) }
% 0.19/0.57 true
% 0.19/0.57 % SZS output end Proof
% 0.19/0.57
% 0.19/0.57 RESULT: Theorem (the conjecture is true).
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