TSTP Solution File: LCL008-1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL008-1 : TPTP v8.1.2. Released v1.0.0.
% 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 08:17:01 EDT 2023
% Result : Unsatisfiable 0.15s 0.37s
% 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
% 0.08/0.14 % Problem : LCL008-1 : TPTP v8.1.2. Released v1.0.0.
% 0.08/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.34 % Computer : n003.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Fri Aug 25 05:25:08 EDT 2023
% 0.15/0.34 % CPUTime :
% 0.15/0.37 Command-line arguments: --no-flatten-goal
% 0.15/0.37
% 0.15/0.37 % SZS status Unsatisfiable
% 0.15/0.37
% 0.19/0.38 % SZS output start Proof
% 0.19/0.38 Take the following subset of the input axioms:
% 0.19/0.38 fof(condensed_detachment, axiom, ![X, Y]: (~is_a_theorem(equivalent(X, Y)) | (~is_a_theorem(X) | is_a_theorem(Y)))).
% 0.19/0.38 fof(prove_ec_4, negated_conjecture, ~is_a_theorem(equivalent(equivalent(a, b), equivalent(b, a)))).
% 0.19/0.38 fof(yql, axiom, ![Z, X2, Y2]: is_a_theorem(equivalent(equivalent(X2, Y2), equivalent(equivalent(Z, Y2), equivalent(X2, Z))))).
% 0.19/0.38
% 0.19/0.38 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.38 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.38 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.38 fresh(y, y, x1...xn) = u
% 0.19/0.38 C => fresh(s, t, x1...xn) = v
% 0.19/0.38 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.38 variables of u and v.
% 0.19/0.38 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.38 input problem has no model of domain size 1).
% 0.19/0.38
% 0.19/0.38 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.38
% 0.19/0.38 Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 0.19/0.38 Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 0.19/0.38 Axiom 3 (condensed_detachment): fresh(is_a_theorem(equivalent(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 0.19/0.38 Axiom 4 (yql): is_a_theorem(equivalent(equivalent(X, Y), equivalent(equivalent(Z, Y), equivalent(X, Z)))) = true.
% 0.19/0.38
% 0.19/0.38 Lemma 5: fresh2(is_a_theorem(equivalent(X, Y)), true, equivalent(equivalent(Z, Y), equivalent(X, Z))) = is_a_theorem(equivalent(equivalent(Z, Y), equivalent(X, Z))).
% 0.19/0.38 Proof:
% 0.19/0.38 fresh2(is_a_theorem(equivalent(X, Y)), true, equivalent(equivalent(Z, Y), equivalent(X, Z)))
% 0.19/0.38 = { by axiom 3 (condensed_detachment) R->L }
% 0.19/0.38 fresh(is_a_theorem(equivalent(equivalent(X, Y), equivalent(equivalent(Z, Y), equivalent(X, Z)))), true, equivalent(X, Y), equivalent(equivalent(Z, Y), equivalent(X, Z)))
% 0.19/0.38 = { by axiom 4 (yql) }
% 0.19/0.38 fresh(true, true, equivalent(X, Y), equivalent(equivalent(Z, Y), equivalent(X, Z)))
% 0.19/0.38 = { by axiom 2 (condensed_detachment) }
% 0.19/0.38 is_a_theorem(equivalent(equivalent(Z, Y), equivalent(X, Z)))
% 0.19/0.38
% 0.19/0.38 Lemma 6: fresh2(is_a_theorem(equivalent(X, equivalent(equivalent(Y, Z), equivalent(W, Y)))), true, equivalent(equivalent(W, Z), X)) = is_a_theorem(equivalent(equivalent(W, Z), X)).
% 0.19/0.38 Proof:
% 0.19/0.38 fresh2(is_a_theorem(equivalent(X, equivalent(equivalent(Y, Z), equivalent(W, Y)))), true, equivalent(equivalent(W, Z), X))
% 0.19/0.38 = { by axiom 3 (condensed_detachment) R->L }
% 0.19/0.38 fresh(is_a_theorem(equivalent(equivalent(X, equivalent(equivalent(Y, Z), equivalent(W, Y))), equivalent(equivalent(W, Z), X))), true, equivalent(X, equivalent(equivalent(Y, Z), equivalent(W, Y))), equivalent(equivalent(W, Z), X))
% 0.19/0.38 = { by lemma 5 R->L }
% 0.19/0.38 fresh(fresh2(is_a_theorem(equivalent(equivalent(W, Z), equivalent(equivalent(Y, Z), equivalent(W, Y)))), true, equivalent(equivalent(X, equivalent(equivalent(Y, Z), equivalent(W, Y))), equivalent(equivalent(W, Z), X))), true, equivalent(X, equivalent(equivalent(Y, Z), equivalent(W, Y))), equivalent(equivalent(W, Z), X))
% 0.19/0.38 = { by axiom 4 (yql) }
% 0.19/0.38 fresh(fresh2(true, true, equivalent(equivalent(X, equivalent(equivalent(Y, Z), equivalent(W, Y))), equivalent(equivalent(W, Z), X))), true, equivalent(X, equivalent(equivalent(Y, Z), equivalent(W, Y))), equivalent(equivalent(W, Z), X))
% 0.19/0.38 = { by axiom 1 (condensed_detachment) }
% 0.19/0.38 fresh(true, true, equivalent(X, equivalent(equivalent(Y, Z), equivalent(W, Y))), equivalent(equivalent(W, Z), X))
% 0.19/0.38 = { by axiom 2 (condensed_detachment) }
% 0.19/0.38 is_a_theorem(equivalent(equivalent(W, Z), X))
% 0.19/0.38
% 0.19/0.38 Lemma 7: is_a_theorem(equivalent(equivalent(X, Y), equivalent(X, Y))) = true.
% 0.19/0.38 Proof:
% 0.19/0.38 is_a_theorem(equivalent(equivalent(X, Y), equivalent(X, Y)))
% 0.19/0.38 = { by lemma 6 R->L }
% 0.19/0.38 fresh2(is_a_theorem(equivalent(equivalent(X, Y), equivalent(equivalent(Z, Y), equivalent(X, Z)))), true, equivalent(equivalent(X, Y), equivalent(X, Y)))
% 0.19/0.38 = { by axiom 4 (yql) }
% 0.19/0.38 fresh2(true, true, equivalent(equivalent(X, Y), equivalent(X, Y)))
% 0.19/0.38 = { by axiom 1 (condensed_detachment) }
% 0.19/0.38 true
% 0.19/0.38
% 0.19/0.38 Goal 1 (prove_ec_4): is_a_theorem(equivalent(equivalent(a, b), equivalent(b, a))) = true.
% 0.19/0.38 Proof:
% 0.19/0.38 is_a_theorem(equivalent(equivalent(a, b), equivalent(b, a)))
% 0.19/0.38 = { by lemma 5 R->L }
% 0.19/0.38 fresh2(is_a_theorem(equivalent(b, b)), true, equivalent(equivalent(a, b), equivalent(b, a)))
% 0.19/0.38 = { by axiom 2 (condensed_detachment) R->L }
% 0.19/0.38 fresh2(fresh(true, true, equivalent(equivalent(X, Y), equivalent(X, Y)), equivalent(b, b)), true, equivalent(equivalent(a, b), equivalent(b, a)))
% 0.19/0.38 = { by axiom 1 (condensed_detachment) R->L }
% 0.19/0.38 fresh2(fresh(fresh2(true, true, equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), equivalent(b, b))), true, equivalent(equivalent(X, Y), equivalent(X, Y)), equivalent(b, b)), true, equivalent(equivalent(a, b), equivalent(b, a)))
% 0.19/0.38 = { by axiom 1 (condensed_detachment) R->L }
% 0.19/0.38 fresh2(fresh(fresh2(fresh2(true, true, equivalent(equivalent(b, b), equivalent(equivalent(b, equivalent(X, Y)), equivalent(equivalent(X, Y), b)))), true, equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), equivalent(b, b))), true, equivalent(equivalent(X, Y), equivalent(X, Y)), equivalent(b, b)), true, equivalent(equivalent(a, b), equivalent(b, a)))
% 0.19/0.38 = { by axiom 1 (condensed_detachment) R->L }
% 0.19/0.38 fresh2(fresh(fresh2(fresh2(fresh2(true, true, equivalent(equivalent(equivalent(b, equivalent(X, Y)), equivalent(equivalent(X, Y), b)), equivalent(equivalent(equivalent(X, Y), b), equivalent(b, equivalent(X, Y))))), true, equivalent(equivalent(b, b), equivalent(equivalent(b, equivalent(X, Y)), equivalent(equivalent(X, Y), b)))), true, equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), equivalent(b, b))), true, equivalent(equivalent(X, Y), equivalent(X, Y)), equivalent(b, b)), true, equivalent(equivalent(a, b), equivalent(b, a)))
% 0.19/0.38 = { by lemma 7 R->L }
% 0.19/0.38 fresh2(fresh(fresh2(fresh2(fresh2(is_a_theorem(equivalent(equivalent(equivalent(X, Y), b), equivalent(equivalent(X, Y), b))), true, equivalent(equivalent(equivalent(b, equivalent(X, Y)), equivalent(equivalent(X, Y), b)), equivalent(equivalent(equivalent(X, Y), b), equivalent(b, equivalent(X, Y))))), true, equivalent(equivalent(b, b), equivalent(equivalent(b, equivalent(X, Y)), equivalent(equivalent(X, Y), b)))), true, equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), equivalent(b, b))), true, equivalent(equivalent(X, Y), equivalent(X, Y)), equivalent(b, b)), true, equivalent(equivalent(a, b), equivalent(b, a)))
% 0.19/0.38 = { by lemma 5 }
% 0.19/0.39 fresh2(fresh(fresh2(fresh2(is_a_theorem(equivalent(equivalent(equivalent(b, equivalent(X, Y)), equivalent(equivalent(X, Y), b)), equivalent(equivalent(equivalent(X, Y), b), equivalent(b, equivalent(X, Y))))), true, equivalent(equivalent(b, b), equivalent(equivalent(b, equivalent(X, Y)), equivalent(equivalent(X, Y), b)))), true, equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), equivalent(b, b))), true, equivalent(equivalent(X, Y), equivalent(X, Y)), equivalent(b, b)), true, equivalent(equivalent(a, b), equivalent(b, a)))
% 0.19/0.39 = { by lemma 6 }
% 0.19/0.39 fresh2(fresh(fresh2(is_a_theorem(equivalent(equivalent(b, b), equivalent(equivalent(b, equivalent(X, Y)), equivalent(equivalent(X, Y), b)))), true, equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), equivalent(b, b))), true, equivalent(equivalent(X, Y), equivalent(X, Y)), equivalent(b, b)), true, equivalent(equivalent(a, b), equivalent(b, a)))
% 0.19/0.39 = { by lemma 6 }
% 0.19/0.39 fresh2(fresh(is_a_theorem(equivalent(equivalent(equivalent(X, Y), equivalent(X, Y)), equivalent(b, b))), true, equivalent(equivalent(X, Y), equivalent(X, Y)), equivalent(b, b)), true, equivalent(equivalent(a, b), equivalent(b, a)))
% 0.19/0.39 = { by axiom 3 (condensed_detachment) }
% 0.19/0.39 fresh2(fresh2(is_a_theorem(equivalent(equivalent(X, Y), equivalent(X, Y))), true, equivalent(b, b)), true, equivalent(equivalent(a, b), equivalent(b, a)))
% 0.19/0.39 = { by lemma 7 }
% 0.19/0.39 fresh2(fresh2(true, true, equivalent(b, b)), true, equivalent(equivalent(a, b), equivalent(b, a)))
% 0.19/0.39 = { by axiom 1 (condensed_detachment) }
% 0.19/0.39 fresh2(true, true, equivalent(equivalent(a, b), equivalent(b, a)))
% 0.19/0.39 = { by axiom 1 (condensed_detachment) }
% 0.19/0.39 true
% 0.19/0.39 % SZS output end Proof
% 0.19/0.39
% 0.19/0.39 RESULT: Unsatisfiable (the axioms are contradictory).
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