TSTP Solution File: LCL077-2 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL077-2 : 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 : n022.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 08:17:18 EDT 2023
% Result : Unsatisfiable 0.20s 0.49s
% 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
% 0.00/0.12 % Problem : LCL077-2 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n022.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Fri Aug 25 05:26:25 EDT 2023
% 0.14/0.34 % CPUTime :
% 0.20/0.49 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 0.20/0.49
% 0.20/0.49 % SZS status Unsatisfiable
% 0.20/0.49
% 0.20/0.49 % SZS output start Proof
% 0.20/0.49 Take the following subset of the input axioms:
% 0.20/0.49 fof(cn_18, axiom, ![X, Y]: is_a_theorem(implies(X, implies(Y, X)))).
% 0.20/0.49 fof(cn_35, axiom, ![Z, X4, Y2]: is_a_theorem(implies(implies(X4, implies(Y2, Z)), implies(implies(X4, Y2), implies(X4, Z))))).
% 0.20/0.49 fof(cn_49, axiom, ![X4, Y2]: is_a_theorem(implies(implies(not(X4), not(Y2)), implies(Y2, X4)))).
% 0.20/0.49 fof(condensed_detachment, axiom, ![X4, Y2]: (~is_a_theorem(implies(X4, Y2)) | (~is_a_theorem(X4) | is_a_theorem(Y2)))).
% 0.20/0.49 fof(prove_cn_39, negated_conjecture, ~is_a_theorem(implies(not(not(a)), a))).
% 0.20/0.49 fof(transitivity, axiom, ![X1, X2, X3]: (~is_a_theorem(implies(X1, X2)) | (~is_a_theorem(implies(X2, X3)) | is_a_theorem(implies(X1, X3))))).
% 0.20/0.49
% 0.20/0.49 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.49 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.49 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.49 fresh(y, y, x1...xn) = u
% 0.20/0.49 C => fresh(s, t, x1...xn) = v
% 0.20/0.49 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.49 variables of u and v.
% 0.20/0.49 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.49 input problem has no model of domain size 1).
% 0.20/0.49
% 0.20/0.49 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.49
% 0.20/0.49 Axiom 1 (condensed_detachment): fresh4(X, X, Y) = true.
% 0.20/0.49 Axiom 2 (transitivity): fresh(X, X, Y, Z) = true.
% 0.20/0.49 Axiom 3 (condensed_detachment): fresh3(X, X, Y, Z) = is_a_theorem(Z).
% 0.20/0.49 Axiom 4 (cn_18): is_a_theorem(implies(X, implies(Y, X))) = true.
% 0.20/0.49 Axiom 5 (transitivity): fresh2(X, X, Y, Z, W) = is_a_theorem(implies(Y, W)).
% 0.20/0.49 Axiom 6 (condensed_detachment): fresh3(is_a_theorem(implies(X, Y)), true, X, Y) = fresh4(is_a_theorem(X), true, Y).
% 0.20/0.49 Axiom 7 (transitivity): fresh2(is_a_theorem(implies(X, Y)), true, Z, X, Y) = fresh(is_a_theorem(implies(Z, X)), true, Z, Y).
% 0.20/0.49 Axiom 8 (cn_49): is_a_theorem(implies(implies(not(X), not(Y)), implies(Y, X))) = true.
% 0.20/0.49 Axiom 9 (cn_35): is_a_theorem(implies(implies(X, implies(Y, Z)), implies(implies(X, Y), implies(X, Z)))) = true.
% 0.20/0.49
% 0.20/0.49 Lemma 10: fresh(is_a_theorem(implies(X, implies(not(Y), not(Z)))), true, X, implies(Z, Y)) = is_a_theorem(implies(X, implies(Z, Y))).
% 0.20/0.49 Proof:
% 0.20/0.49 fresh(is_a_theorem(implies(X, implies(not(Y), not(Z)))), true, X, implies(Z, Y))
% 0.20/0.49 = { by axiom 7 (transitivity) R->L }
% 0.20/0.49 fresh2(is_a_theorem(implies(implies(not(Y), not(Z)), implies(Z, Y))), true, X, implies(not(Y), not(Z)), implies(Z, Y))
% 0.20/0.49 = { by axiom 8 (cn_49) }
% 0.20/0.49 fresh2(true, true, X, implies(not(Y), not(Z)), implies(Z, Y))
% 0.20/0.49 = { by axiom 5 (transitivity) }
% 0.20/0.49 is_a_theorem(implies(X, implies(Z, Y)))
% 0.20/0.49
% 0.20/0.49 Goal 1 (prove_cn_39): is_a_theorem(implies(not(not(a)), a)) = true.
% 0.20/0.49 Proof:
% 0.20/0.49 is_a_theorem(implies(not(not(a)), a))
% 0.20/0.49 = { by axiom 3 (condensed_detachment) R->L }
% 0.20/0.49 fresh3(true, true, implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))
% 0.20/0.49 = { by axiom 1 (condensed_detachment) R->L }
% 0.20/0.49 fresh3(fresh4(true, true, implies(implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))), true, implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))
% 0.20/0.49 = { by axiom 2 (transitivity) R->L }
% 0.20/0.49 fresh3(fresh4(fresh(true, true, not(not(a)), implies(implies(X, not(not(a))), a)), true, implies(implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))), true, implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))
% 0.20/0.49 = { by axiom 2 (transitivity) R->L }
% 0.20/0.49 fresh3(fresh4(fresh(fresh(true, true, not(not(a)), implies(not(a), not(implies(X, not(not(a)))))), true, not(not(a)), implies(implies(X, not(not(a))), a)), true, implies(implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))), true, implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))
% 0.20/0.49 = { by axiom 4 (cn_18) R->L }
% 0.20/0.49 fresh3(fresh4(fresh(fresh(is_a_theorem(implies(not(not(a)), implies(not(not(implies(X, not(not(a))))), not(not(a))))), true, not(not(a)), implies(not(a), not(implies(X, not(not(a)))))), true, not(not(a)), implies(implies(X, not(not(a))), a)), true, implies(implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))), true, implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))
% 0.20/0.49 = { by lemma 10 }
% 0.20/0.49 fresh3(fresh4(fresh(is_a_theorem(implies(not(not(a)), implies(not(a), not(implies(X, not(not(a))))))), true, not(not(a)), implies(implies(X, not(not(a))), a)), true, implies(implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))), true, implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))
% 0.20/0.49 = { by lemma 10 }
% 0.20/0.49 fresh3(fresh4(is_a_theorem(implies(not(not(a)), implies(implies(X, not(not(a))), a))), true, implies(implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))), true, implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))
% 0.20/0.49 = { by axiom 6 (condensed_detachment) R->L }
% 0.20/0.49 fresh3(fresh3(is_a_theorem(implies(implies(not(not(a)), implies(implies(X, not(not(a))), a)), implies(implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a)))), true, implies(not(not(a)), implies(implies(X, not(not(a))), a)), implies(implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))), true, implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))
% 0.20/0.49 = { by axiom 9 (cn_35) }
% 0.20/0.49 fresh3(fresh3(true, true, implies(not(not(a)), implies(implies(X, not(not(a))), a)), implies(implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))), true, implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))
% 0.20/0.49 = { by axiom 3 (condensed_detachment) }
% 0.20/0.49 fresh3(is_a_theorem(implies(implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))), true, implies(not(not(a)), implies(X, not(not(a)))), implies(not(not(a)), a))
% 0.20/0.49 = { by axiom 6 (condensed_detachment) }
% 0.20/0.49 fresh4(is_a_theorem(implies(not(not(a)), implies(X, not(not(a))))), true, implies(not(not(a)), a))
% 0.20/0.49 = { by axiom 4 (cn_18) }
% 0.20/0.49 fresh4(true, true, implies(not(not(a)), a))
% 0.20/0.49 = { by axiom 1 (condensed_detachment) }
% 0.20/0.49 true
% 0.20/0.49 % SZS output end Proof
% 0.20/0.49
% 0.20/0.49 RESULT: Unsatisfiable (the axioms are contradictory).
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