TSTP Solution File: LCL359-1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL359-1 : 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 : n032.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 0.11s 0.32s
% Output : Proof 0.11s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : LCL359-1 : TPTP v8.1.2. Released v2.3.0.
% 0.00/0.09 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.11/0.28 % Computer : n032.cluster.edu
% 0.11/0.28 % Model : x86_64 x86_64
% 0.11/0.28 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.28 % Memory : 8042.1875MB
% 0.11/0.28 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.28 % CPULimit : 300
% 0.11/0.28 % WCLimit : 300
% 0.11/0.28 % DateTime : Thu Aug 24 19:38:42 EDT 2023
% 0.11/0.29 % CPUTime :
% 0.11/0.32 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.11/0.32
% 0.11/0.32 % SZS status Unsatisfiable
% 0.11/0.32
% 0.11/0.32 % SZS output start Proof
% 0.11/0.32 Take the following subset of the input axioms:
% 0.11/0.32 fof(cn_1, axiom, ![X, Y, Z]: is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))))).
% 0.11/0.32 fof(condensed_detachment, axiom, ![X2, Y2]: (~is_a_theorem(implies(X2, Y2)) | (~is_a_theorem(X2) | is_a_theorem(Y2)))).
% 0.11/0.32 fof(prove_cn_08, negated_conjecture, ~is_a_theorem(implies(implies(x, y), implies(implies(z, x), implies(implies(y, u), implies(z, u)))))).
% 0.11/0.32
% 0.11/0.32 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.11/0.32 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.11/0.32 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.11/0.32 fresh(y, y, x1...xn) = u
% 0.11/0.32 C => fresh(s, t, x1...xn) = v
% 0.11/0.32 where fresh is a fresh function symbol and x1..xn are the free
% 0.11/0.32 variables of u and v.
% 0.11/0.32 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.11/0.32 input problem has no model of domain size 1).
% 0.11/0.32
% 0.11/0.32 The encoding turns the above axioms into the following unit equations and goals:
% 0.11/0.32
% 0.11/0.32 Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 0.11/0.32 Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 0.11/0.32 Axiom 3 (condensed_detachment): fresh(is_a_theorem(implies(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 0.11/0.32 Axiom 4 (cn_1): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
% 0.11/0.32
% 0.11/0.32 Lemma 5: is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))) = true.
% 0.11/0.32 Proof:
% 0.11/0.32 is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W)))
% 0.11/0.32 = { by axiom 2 (condensed_detachment) R->L }
% 0.11/0.33 fresh(true, true, implies(implies(Z, X), implies(implies(X, Y), implies(Z, Y))), implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W)))
% 0.11/0.33 = { by axiom 4 (cn_1) R->L }
% 0.11/0.33 fresh(is_a_theorem(implies(implies(implies(Z, X), implies(implies(X, Y), implies(Z, Y))), implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W)))), true, implies(implies(Z, X), implies(implies(X, Y), implies(Z, Y))), implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W)))
% 0.11/0.33 = { by axiom 3 (condensed_detachment) }
% 0.11/0.33 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)))
% 0.11/0.33 = { by axiom 4 (cn_1) }
% 0.11/0.33 fresh2(true, true, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W)))
% 0.11/0.33 = { by axiom 1 (condensed_detachment) }
% 0.11/0.33 true
% 0.11/0.33
% 0.11/0.33 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)).
% 0.11/0.33 Proof:
% 0.11/0.33 fresh2(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, implies(implies(Z, X), W))
% 0.11/0.33 = { by axiom 3 (condensed_detachment) R->L }
% 0.11/0.33 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))
% 0.11/0.33 = { by lemma 5 }
% 0.11/0.33 fresh(true, true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 0.11/0.33 = { by axiom 2 (condensed_detachment) }
% 0.11/0.33 is_a_theorem(implies(implies(Z, X), W))
% 0.11/0.33
% 0.11/0.33 Goal 1 (prove_cn_08): is_a_theorem(implies(implies(x, y), implies(implies(z, x), implies(implies(y, u), implies(z, u))))) = true.
% 0.11/0.33 Proof:
% 0.11/0.33 is_a_theorem(implies(implies(x, y), implies(implies(z, x), implies(implies(y, u), implies(z, u)))))
% 0.11/0.33 = { by lemma 6 R->L }
% 0.11/0.33 fresh2(is_a_theorem(implies(implies(implies(y, u), implies(x, u)), implies(implies(z, x), implies(implies(y, u), implies(z, u))))), true, implies(implies(x, y), implies(implies(z, x), implies(implies(y, u), implies(z, u)))))
% 0.11/0.33 = { by lemma 6 R->L }
% 0.11/0.33 fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(x, u), implies(z, u)), implies(implies(y, u), implies(z, u))), implies(implies(z, x), implies(implies(y, u), implies(z, u))))), true, implies(implies(implies(y, u), implies(x, u)), implies(implies(z, x), implies(implies(y, u), implies(z, u))))), true, implies(implies(x, y), implies(implies(z, x), implies(implies(y, u), implies(z, u)))))
% 0.11/0.33 = { by lemma 5 }
% 0.11/0.33 fresh2(fresh2(true, true, implies(implies(implies(y, u), implies(x, u)), implies(implies(z, x), implies(implies(y, u), implies(z, u))))), true, implies(implies(x, y), implies(implies(z, x), implies(implies(y, u), implies(z, u)))))
% 0.11/0.33 = { by axiom 1 (condensed_detachment) }
% 0.11/0.33 fresh2(true, true, implies(implies(x, y), implies(implies(z, x), implies(implies(y, u), implies(z, u)))))
% 0.11/0.33 = { by axiom 1 (condensed_detachment) }
% 0.11/0.33 true
% 0.11/0.33 % SZS output end Proof
% 0.11/0.33
% 0.11/0.33 RESULT: Unsatisfiable (the axioms are contradictory).
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