TSTP Solution File: LCL452+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL452+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 : n023.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 08:19:03 EDT 2023
% Result : Theorem 0.19s 0.46s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : LCL452+1 : TPTP v8.1.2. Released v3.3.0.
% 0.11/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n023.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 : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 07:34:42 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.46 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.19/0.46
% 0.19/0.46 % SZS status Theorem
% 0.19/0.46
% 0.19/0.47 % SZS output start Proof
% 0.19/0.47 Take the following subset of the input axioms:
% 0.19/0.47 fof(cn2, axiom, cn2 <=> ![P, Q]: is_a_theorem(implies(P, implies(not(P), Q)))).
% 0.19/0.47 fof(hilbert_op_implies_and, axiom, op_implies_and).
% 0.19/0.47 fof(hilbert_op_or, axiom, op_or).
% 0.19/0.47 fof(hilbert_or_1, axiom, or_1).
% 0.19/0.47 fof(luka_cn2, conjecture, cn2).
% 0.19/0.47 fof(op_implies_and, axiom, op_implies_and => ![X, Y]: implies(X, Y)=not(and(X, not(Y)))).
% 0.19/0.47 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 0.19/0.47 fof(or_1, axiom, or_1 <=> ![X2, Y2]: is_a_theorem(implies(X2, or(X2, Y2)))).
% 0.19/0.47
% 0.19/0.47 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.47 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.47 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.47 fresh(y, y, x1...xn) = u
% 0.19/0.47 C => fresh(s, t, x1...xn) = v
% 0.19/0.47 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.47 variables of u and v.
% 0.19/0.47 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.47 input problem has no model of domain size 1).
% 0.19/0.47
% 0.19/0.47 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.47
% 0.19/0.47 Axiom 1 (hilbert_or_1): or_1 = true.
% 0.19/0.47 Axiom 2 (hilbert_op_or): op_or = true.
% 0.19/0.47 Axiom 3 (hilbert_op_implies_and): op_implies_and = true.
% 0.19/0.47 Axiom 4 (cn2): fresh50(X, X) = true.
% 0.19/0.47 Axiom 5 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 0.19/0.47 Axiom 6 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 0.19/0.47 Axiom 7 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 0.19/0.47 Axiom 8 (or_1_1): fresh18(X, X, Y, Z) = true.
% 0.19/0.47 Axiom 9 (or_1_1): fresh18(or_1, true, X, Y) = is_a_theorem(implies(X, or(X, Y))).
% 0.19/0.47 Axiom 10 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 0.19/0.47 Axiom 11 (cn2): fresh50(is_a_theorem(implies(p7, implies(not(p7), q5))), true) = cn2.
% 0.19/0.47
% 0.19/0.47 Goal 1 (luka_cn2): cn2 = true.
% 0.19/0.47 Proof:
% 0.19/0.47 cn2
% 0.19/0.47 = { by axiom 11 (cn2) R->L }
% 0.19/0.47 fresh50(is_a_theorem(implies(p7, implies(not(p7), q5))), true)
% 0.19/0.47 = { by axiom 5 (op_implies_and) R->L }
% 0.19/0.47 fresh50(is_a_theorem(implies(p7, fresh22(true, true, not(p7), q5))), true)
% 0.19/0.47 = { by axiom 3 (hilbert_op_implies_and) R->L }
% 0.19/0.47 fresh50(is_a_theorem(implies(p7, fresh22(op_implies_and, true, not(p7), q5))), true)
% 0.19/0.47 = { by axiom 6 (op_implies_and) }
% 0.19/0.47 fresh50(is_a_theorem(implies(p7, not(and(not(p7), not(q5))))), true)
% 0.19/0.47 = { by axiom 10 (op_or) R->L }
% 0.19/0.47 fresh50(is_a_theorem(implies(p7, fresh20(op_or, true, p7, q5))), true)
% 0.19/0.47 = { by axiom 2 (hilbert_op_or) }
% 0.19/0.47 fresh50(is_a_theorem(implies(p7, fresh20(true, true, p7, q5))), true)
% 0.19/0.47 = { by axiom 7 (op_or) }
% 0.19/0.47 fresh50(is_a_theorem(implies(p7, or(p7, q5))), true)
% 0.19/0.47 = { by axiom 9 (or_1_1) R->L }
% 0.19/0.47 fresh50(fresh18(or_1, true, p7, q5), true)
% 0.19/0.47 = { by axiom 1 (hilbert_or_1) }
% 0.19/0.47 fresh50(fresh18(true, true, p7, q5), true)
% 0.19/0.47 = { by axiom 8 (or_1_1) }
% 0.19/0.47 fresh50(true, true)
% 0.19/0.47 = { by axiom 4 (cn2) }
% 0.19/0.47 true
% 0.19/0.47 % SZS output end Proof
% 0.19/0.47
% 0.19/0.47 RESULT: Theorem (the conjecture is true).
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