TSTP Solution File: LCL212-3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL212-3 : 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 : n017.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 08:18:02 EDT 2023
% Result : Unsatisfiable 0.13s 0.39s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : LCL212-3 : TPTP v8.1.2. Released v2.3.0.
% 0.06/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n017.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Aug 25 04:13:39 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.13/0.39 Command-line arguments: --flatten
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% 0.13/0.39 % SZS status Unsatisfiable
% 0.13/0.39
% 0.20/0.40 % SZS output start Proof
% 0.20/0.40 Take the following subset of the input axioms:
% 0.20/0.40 fof(axiom_1_3, axiom, ![A, B]: axiom(implies(A, or(B, A)))).
% 0.20/0.40 fof(implies_definition, axiom, ![X, Y]: implies(X, Y)=or(not(X), Y)).
% 0.20/0.40 fof(prove_this, negated_conjecture, ~theorem(implies(not(p), implies(q, implies(implies(p, q), q))))).
% 0.20/0.40 fof(rule_1, axiom, ![X2]: (theorem(X2) | ~axiom(X2))).
% 0.20/0.40 fof(rule_2, axiom, ![X2, Y2]: (theorem(X2) | (~theorem(implies(Y2, X2)) | ~theorem(Y2)))).
% 0.20/0.40
% 0.20/0.40 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.40 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.40 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.40 fresh(y, y, x1...xn) = u
% 0.20/0.40 C => fresh(s, t, x1...xn) = v
% 0.20/0.40 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.40 variables of u and v.
% 0.20/0.40 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.40 input problem has no model of domain size 1).
% 0.20/0.40
% 0.20/0.40 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.40
% 0.20/0.40 Axiom 1 (rule_2): fresh(X, X, Y) = true.
% 0.20/0.40 Axiom 2 (rule_1): fresh2(X, X, Y) = true.
% 0.20/0.40 Axiom 3 (implies_definition): implies(X, Y) = or(not(X), Y).
% 0.20/0.40 Axiom 4 (rule_2): fresh3(X, X, Y, Z) = theorem(Y).
% 0.20/0.40 Axiom 5 (rule_1): fresh2(axiom(X), true, X) = theorem(X).
% 0.20/0.40 Axiom 6 (axiom_1_3): axiom(implies(X, or(Y, X))) = true.
% 0.20/0.40 Axiom 7 (rule_2): fresh3(theorem(implies(X, Y)), true, Y, X) = fresh(theorem(X), true, Y).
% 0.20/0.40
% 0.20/0.40 Lemma 8: theorem(implies(X, implies(Y, X))) = true.
% 0.20/0.40 Proof:
% 0.20/0.40 theorem(implies(X, implies(Y, X)))
% 0.20/0.40 = { by axiom 5 (rule_1) R->L }
% 0.20/0.40 fresh2(axiom(implies(X, implies(Y, X))), true, implies(X, implies(Y, X)))
% 0.20/0.40 = { by axiom 3 (implies_definition) }
% 0.20/0.40 fresh2(axiom(implies(X, or(not(Y), X))), true, implies(X, implies(Y, X)))
% 0.20/0.40 = { by axiom 6 (axiom_1_3) }
% 0.20/0.40 fresh2(true, true, implies(X, implies(Y, X)))
% 0.20/0.40 = { by axiom 2 (rule_1) }
% 0.20/0.40 true
% 0.20/0.40
% 0.20/0.40 Goal 1 (prove_this): theorem(implies(not(p), implies(q, implies(implies(p, q), q)))) = true.
% 0.20/0.40 Proof:
% 0.20/0.40 theorem(implies(not(p), implies(q, implies(implies(p, q), q))))
% 0.20/0.40 = { by axiom 4 (rule_2) R->L }
% 0.20/0.40 fresh3(true, true, implies(not(p), implies(q, implies(implies(p, q), q))), implies(q, implies(implies(p, q), q)))
% 0.20/0.40 = { by lemma 8 R->L }
% 0.20/0.40 fresh3(theorem(implies(implies(q, implies(implies(p, q), q)), implies(not(p), implies(q, implies(implies(p, q), q))))), true, implies(not(p), implies(q, implies(implies(p, q), q))), implies(q, implies(implies(p, q), q)))
% 0.20/0.40 = { by axiom 7 (rule_2) }
% 0.20/0.40 fresh(theorem(implies(q, implies(implies(p, q), q))), true, implies(not(p), implies(q, implies(implies(p, q), q))))
% 0.20/0.40 = { by lemma 8 }
% 0.20/0.40 fresh(true, true, implies(not(p), implies(q, implies(implies(p, q), q))))
% 0.20/0.40 = { by axiom 1 (rule_2) }
% 0.20/0.40 true
% 0.20/0.40 % SZS output end Proof
% 0.20/0.40
% 0.20/0.40 RESULT: Unsatisfiable (the axioms are contradictory).
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