TSTP Solution File: LCL358-1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL358-1 : 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 : n031.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 08:18:38 EDT 2023
% Result : Unsatisfiable 0.20s 0.43s
% 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 : LCL358-1 : TPTP v8.1.2. Released v2.3.0.
% 0.12/0.13 % 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 : n031.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 02:42:06 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.43 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.43
% 0.20/0.43 % SZS status Unsatisfiable
% 0.20/0.43
% 0.20/0.43 % SZS output start Proof
% 0.20/0.43 Take the following subset of the input axioms:
% 0.20/0.43 fof(cn_1, axiom, ![X, Y, Z]: is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))))).
% 0.20/0.43 fof(condensed_detachment, axiom, ![X2, Y2]: (~is_a_theorem(implies(X2, Y2)) | (~is_a_theorem(X2) | is_a_theorem(Y2)))).
% 0.20/0.43 fof(prove_cn_07, negated_conjecture, ~is_a_theorem(implies(implies(x, implies(implies(y, z), u)), implies(implies(y, v), implies(x, implies(implies(v, z), u)))))).
% 0.20/0.43
% 0.20/0.43 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.43 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.43 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.43 fresh(y, y, x1...xn) = u
% 0.20/0.43 C => fresh(s, t, x1...xn) = v
% 0.20/0.43 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.43 variables of u and v.
% 0.20/0.43 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.43 input problem has no model of domain size 1).
% 0.20/0.43
% 0.20/0.43 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.43
% 0.20/0.43 Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 0.20/0.43 Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 0.20/0.43 Axiom 3 (condensed_detachment): fresh(is_a_theorem(implies(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 0.20/0.43 Axiom 4 (cn_1): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
% 0.20/0.43
% 0.20/0.43 Lemma 5: fresh2(is_a_theorem(implies(X, Y)), true, implies(implies(Y, Z), implies(X, Z))) = is_a_theorem(implies(implies(Y, Z), implies(X, Z))).
% 0.20/0.43 Proof:
% 0.20/0.43 fresh2(is_a_theorem(implies(X, Y)), true, implies(implies(Y, Z), implies(X, Z)))
% 0.20/0.43 = { by axiom 3 (condensed_detachment) R->L }
% 0.20/0.43 fresh(is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))), true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 0.20/0.43 = { by axiom 4 (cn_1) }
% 0.20/0.43 fresh(true, true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 0.20/0.43 = { by axiom 2 (condensed_detachment) }
% 0.20/0.43 is_a_theorem(implies(implies(Y, Z), implies(X, Z)))
% 0.20/0.43
% 0.20/0.43 Lemma 6: fresh2(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, implies(implies(Z, X), W)) = is_a_theorem(implies(implies(Z, X), W)).
% 0.20/0.43 Proof:
% 0.20/0.43 fresh2(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, implies(implies(Z, X), W))
% 0.20/0.43 = { by axiom 3 (condensed_detachment) R->L }
% 0.20/0.43 fresh(is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 0.20/0.43 = { by lemma 5 R->L }
% 0.20/0.43 fresh(fresh2(is_a_theorem(implies(implies(Z, X), implies(implies(X, Y), implies(Z, Y)))), true, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 0.20/0.43 = { by axiom 4 (cn_1) }
% 0.20/0.43 fresh(fresh2(true, true, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 0.20/0.43 = { by axiom 1 (condensed_detachment) }
% 0.20/0.43 fresh(true, true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 0.20/0.43 = { by axiom 2 (condensed_detachment) }
% 0.20/0.43 is_a_theorem(implies(implies(Z, X), W))
% 0.20/0.43
% 0.20/0.43 Goal 1 (prove_cn_07): is_a_theorem(implies(implies(x, implies(implies(y, z), u)), implies(implies(y, v), implies(x, implies(implies(v, z), u))))) = true.
% 0.20/0.43 Proof:
% 0.20/0.43 is_a_theorem(implies(implies(x, implies(implies(y, z), u)), implies(implies(y, v), implies(x, implies(implies(v, z), u)))))
% 0.20/0.43 = { by lemma 6 R->L }
% 0.20/0.44 fresh2(is_a_theorem(implies(implies(implies(implies(implies(y, z), u), implies(implies(v, z), u)), implies(x, implies(implies(v, z), u))), implies(implies(y, v), implies(x, implies(implies(v, z), u))))), true, implies(implies(x, implies(implies(y, z), u)), implies(implies(y, v), implies(x, implies(implies(v, z), u)))))
% 0.20/0.44 = { by lemma 5 R->L }
% 0.20/0.44 fresh2(fresh2(is_a_theorem(implies(implies(y, v), implies(implies(implies(y, z), u), implies(implies(v, z), u)))), true, implies(implies(implies(implies(implies(y, z), u), implies(implies(v, z), u)), implies(x, implies(implies(v, z), u))), implies(implies(y, v), implies(x, implies(implies(v, z), u))))), true, implies(implies(x, implies(implies(y, z), u)), implies(implies(y, v), implies(x, implies(implies(v, z), u)))))
% 0.20/0.44 = { by lemma 6 R->L }
% 0.20/0.44 fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(v, z), implies(y, z)), implies(implies(implies(y, z), u), implies(implies(v, z), u)))), true, implies(implies(y, v), implies(implies(implies(y, z), u), implies(implies(v, z), u)))), true, implies(implies(implies(implies(implies(y, z), u), implies(implies(v, z), u)), implies(x, implies(implies(v, z), u))), implies(implies(y, v), implies(x, implies(implies(v, z), u))))), true, implies(implies(x, implies(implies(y, z), u)), implies(implies(y, v), implies(x, implies(implies(v, z), u)))))
% 0.20/0.44 = { by axiom 4 (cn_1) }
% 0.20/0.44 fresh2(fresh2(fresh2(true, true, implies(implies(y, v), implies(implies(implies(y, z), u), implies(implies(v, z), u)))), true, implies(implies(implies(implies(implies(y, z), u), implies(implies(v, z), u)), implies(x, implies(implies(v, z), u))), implies(implies(y, v), implies(x, implies(implies(v, z), u))))), true, implies(implies(x, implies(implies(y, z), u)), implies(implies(y, v), implies(x, implies(implies(v, z), u)))))
% 0.20/0.44 = { by axiom 1 (condensed_detachment) }
% 0.20/0.44 fresh2(fresh2(true, true, implies(implies(implies(implies(implies(y, z), u), implies(implies(v, z), u)), implies(x, implies(implies(v, z), u))), implies(implies(y, v), implies(x, implies(implies(v, z), u))))), true, implies(implies(x, implies(implies(y, z), u)), implies(implies(y, v), implies(x, implies(implies(v, z), u)))))
% 0.20/0.44 = { by axiom 1 (condensed_detachment) }
% 0.20/0.44 fresh2(true, true, implies(implies(x, implies(implies(y, z), u)), implies(implies(y, v), implies(x, implies(implies(v, z), u)))))
% 0.20/0.44 = { by axiom 1 (condensed_detachment) }
% 0.20/0.44 true
% 0.20/0.44 % SZS output end Proof
% 0.20/0.44
% 0.20/0.44 RESULT: Unsatisfiable (the axioms are contradictory).
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