TSTP Solution File: LCL258-3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL258-3 : TPTP v8.1.2. Released v2.3.0.
% 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 : n025.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 08:18:19 EDT 2023
% Result : Unsatisfiable 0.19s 0.67s
% 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
% 0.00/0.12 % Problem : LCL258-3 : TPTP v8.1.2. Released v2.3.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n025.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 : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Aug 25 05:16:06 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.19/0.67 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.19/0.67
% 0.19/0.67 % SZS status Unsatisfiable
% 0.19/0.67
% 0.19/0.68 % SZS output start Proof
% 0.19/0.68 Take the following subset of the input axioms:
% 0.19/0.68 fof(axiom_1_3, axiom, ![A, B]: axiom(implies(A, or(B, A)))).
% 0.19/0.68 fof(axiom_1_5, axiom, ![C, A2, B2]: axiom(implies(or(A2, or(B2, C)), or(B2, or(A2, C))))).
% 0.19/0.68 fof(implies_definition, axiom, ![X, Y]: implies(X, Y)=or(not(X), Y)).
% 0.19/0.68 fof(prove_this, negated_conjecture, ~theorem(implies(p, implies(implies(p, q), q)))).
% 0.19/0.68 fof(rule_1, axiom, ![X2]: (theorem(X2) | ~axiom(X2))).
% 0.19/0.68 fof(rule_2, axiom, ![X2, Y2]: (theorem(X2) | (~theorem(implies(Y2, X2)) | ~theorem(Y2)))).
% 0.19/0.68
% 0.19/0.68 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.68 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.68 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.68 fresh(y, y, x1...xn) = u
% 0.19/0.68 C => fresh(s, t, x1...xn) = v
% 0.19/0.68 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.68 variables of u and v.
% 0.19/0.68 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.68 input problem has no model of domain size 1).
% 0.19/0.68
% 0.19/0.68 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.68
% 0.19/0.68 Axiom 1 (rule_2): fresh(X, X, Y) = true.
% 0.19/0.68 Axiom 2 (rule_1): fresh2(X, X, Y) = true.
% 0.19/0.68 Axiom 3 (implies_definition): implies(X, Y) = or(not(X), Y).
% 0.19/0.68 Axiom 4 (rule_2): fresh3(X, X, Y, Z) = theorem(Y).
% 0.19/0.68 Axiom 5 (rule_1): fresh2(axiom(X), true, X) = theorem(X).
% 0.19/0.68 Axiom 6 (axiom_1_3): axiom(implies(X, or(Y, X))) = true.
% 0.19/0.68 Axiom 7 (rule_2): fresh3(theorem(implies(X, Y)), true, Y, X) = fresh(theorem(X), true, Y).
% 0.19/0.68 Axiom 8 (axiom_1_5): axiom(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))) = true.
% 0.19/0.68
% 0.19/0.68 Lemma 9: theorem(implies(X, implies(Y, X))) = true.
% 0.19/0.68 Proof:
% 0.19/0.68 theorem(implies(X, implies(Y, X)))
% 0.19/0.68 = { by axiom 5 (rule_1) R->L }
% 0.19/0.68 fresh2(axiom(implies(X, implies(Y, X))), true, implies(X, implies(Y, X)))
% 0.19/0.68 = { by axiom 3 (implies_definition) }
% 0.19/0.68 fresh2(axiom(implies(X, or(not(Y), X))), true, implies(X, implies(Y, X)))
% 0.19/0.68 = { by axiom 6 (axiom_1_3) }
% 0.19/0.68 fresh2(true, true, implies(X, implies(Y, X)))
% 0.19/0.68 = { by axiom 2 (rule_1) }
% 0.19/0.68 true
% 0.19/0.68
% 0.19/0.68 Lemma 10: fresh(theorem(implies(X, implies(Y, Z))), true, implies(Y, implies(X, Z))) = theorem(implies(Y, implies(X, Z))).
% 0.19/0.68 Proof:
% 0.19/0.68 fresh(theorem(implies(X, implies(Y, Z))), true, implies(Y, implies(X, Z)))
% 0.19/0.68 = { by axiom 7 (rule_2) R->L }
% 0.19/0.68 fresh3(theorem(implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))), true, implies(Y, implies(X, Z)), implies(X, implies(Y, Z)))
% 0.19/0.68 = { by axiom 3 (implies_definition) }
% 0.19/0.68 fresh3(theorem(implies(implies(X, implies(Y, Z)), or(not(Y), implies(X, Z)))), true, implies(Y, implies(X, Z)), implies(X, implies(Y, Z)))
% 0.19/0.68 = { by axiom 3 (implies_definition) }
% 0.19/0.68 fresh3(theorem(implies(implies(X, or(not(Y), Z)), or(not(Y), implies(X, Z)))), true, implies(Y, implies(X, Z)), implies(X, implies(Y, Z)))
% 0.19/0.68 = { by axiom 3 (implies_definition) }
% 0.19/0.68 fresh3(theorem(implies(implies(X, or(not(Y), Z)), or(not(Y), or(not(X), Z)))), true, implies(Y, implies(X, Z)), implies(X, implies(Y, Z)))
% 0.19/0.68 = { by axiom 3 (implies_definition) }
% 0.19/0.68 fresh3(theorem(implies(or(not(X), or(not(Y), Z)), or(not(Y), or(not(X), Z)))), true, implies(Y, implies(X, Z)), implies(X, implies(Y, Z)))
% 0.19/0.68 = { by axiom 5 (rule_1) R->L }
% 0.19/0.68 fresh3(fresh2(axiom(implies(or(not(X), or(not(Y), Z)), or(not(Y), or(not(X), Z)))), true, implies(or(not(X), or(not(Y), Z)), or(not(Y), or(not(X), Z)))), true, implies(Y, implies(X, Z)), implies(X, implies(Y, Z)))
% 0.19/0.68 = { by axiom 8 (axiom_1_5) }
% 0.19/0.68 fresh3(fresh2(true, true, implies(or(not(X), or(not(Y), Z)), or(not(Y), or(not(X), Z)))), true, implies(Y, implies(X, Z)), implies(X, implies(Y, Z)))
% 0.19/0.68 = { by axiom 2 (rule_1) }
% 0.19/0.68 fresh3(true, true, implies(Y, implies(X, Z)), implies(X, implies(Y, Z)))
% 0.19/0.68 = { by axiom 4 (rule_2) }
% 0.19/0.68 theorem(implies(Y, implies(X, Z)))
% 0.19/0.68
% 0.19/0.68 Goal 1 (prove_this): theorem(implies(p, implies(implies(p, q), q))) = true.
% 0.19/0.68 Proof:
% 0.19/0.68 theorem(implies(p, implies(implies(p, q), q)))
% 0.19/0.68 = { by lemma 10 R->L }
% 0.19/0.68 fresh(theorem(implies(implies(p, q), implies(p, q))), true, implies(p, implies(implies(p, q), q)))
% 0.19/0.68 = { by axiom 4 (rule_2) R->L }
% 0.19/0.68 fresh(fresh3(true, true, implies(implies(p, q), implies(p, q)), implies(X, implies(Y, X))), true, implies(p, implies(implies(p, q), q)))
% 0.19/0.68 = { by axiom 1 (rule_2) R->L }
% 0.19/0.68 fresh(fresh3(fresh(true, true, implies(implies(X, implies(Y, X)), implies(implies(p, q), implies(p, q)))), true, implies(implies(p, q), implies(p, q)), implies(X, implies(Y, X))), true, implies(p, implies(implies(p, q), q)))
% 0.19/0.68 = { by lemma 9 R->L }
% 0.19/0.68 fresh(fresh3(fresh(theorem(implies(implies(p, q), implies(implies(X, implies(Y, X)), implies(p, q)))), true, implies(implies(X, implies(Y, X)), implies(implies(p, q), implies(p, q)))), true, implies(implies(p, q), implies(p, q)), implies(X, implies(Y, X))), true, implies(p, implies(implies(p, q), q)))
% 0.19/0.68 = { by lemma 10 }
% 0.19/0.68 fresh(fresh3(theorem(implies(implies(X, implies(Y, X)), implies(implies(p, q), implies(p, q)))), true, implies(implies(p, q), implies(p, q)), implies(X, implies(Y, X))), true, implies(p, implies(implies(p, q), q)))
% 0.19/0.68 = { by axiom 7 (rule_2) }
% 0.19/0.68 fresh(fresh(theorem(implies(X, implies(Y, X))), true, implies(implies(p, q), implies(p, q))), true, implies(p, implies(implies(p, q), q)))
% 0.19/0.68 = { by lemma 9 }
% 0.19/0.68 fresh(fresh(true, true, implies(implies(p, q), implies(p, q))), true, implies(p, implies(implies(p, q), q)))
% 0.19/0.68 = { by axiom 1 (rule_2) }
% 0.19/0.68 fresh(true, true, implies(p, implies(implies(p, q), q)))
% 0.19/0.68 = { by axiom 1 (rule_2) }
% 0.19/0.68 true
% 0.19/0.68 % SZS output end Proof
% 0.19/0.68
% 0.19/0.68 RESULT: Unsatisfiable (the axioms are contradictory).
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