TSTP Solution File: LCL109-5 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL109-5 : 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 : n005.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 08:17:28 EDT 2023
% Result : Unsatisfiable 0.19s 0.40s
% 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
% 0.11/0.12 % Problem : LCL109-5 : TPTP v8.1.2. Released v1.0.0.
% 0.11/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n005.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Aug 24 16:48:23 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.19/0.40 Command-line arguments: --no-flatten-goal
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% 0.19/0.40 % SZS status Unsatisfiable
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% 0.19/0.41 % SZS output start Proof
% 0.19/0.41 Take the following subset of the input axioms:
% 0.19/0.41 fof(big_V_definition, axiom, ![X, Y]: big_V(X, Y)=implies(implies(X, Y), Y)).
% 0.19/0.41 fof(lemma_1, axiom, ![Z, X2, Y2]: (~ordered(X2, Y2) | ordered(implies(X2, Z), implies(Y2, Z)))).
% 0.19/0.41 fof(partial_order_definition1, axiom, ![X2, Y2]: (~ordered(X2, Y2) | implies(X2, Y2)=truth)).
% 0.19/0.41 fof(partial_order_definition2, axiom, ![X2, Y2]: (implies(X2, Y2)!=truth | ordered(X2, Y2))).
% 0.19/0.41 fof(prove_mv_4, negated_conjecture, big_V(implies(x, y), implies(y, x))!=truth).
% 0.19/0.41 fof(wajsberg_1, axiom, ![X2]: implies(truth, X2)=X2).
% 0.19/0.41 fof(wajsberg_2, axiom, ![X2, Y2, Z2]: implies(implies(X2, Y2), implies(implies(Y2, Z2), implies(X2, Z2)))=truth).
% 0.19/0.41 fof(wajsberg_3, axiom, ![X2, Y2]: implies(implies(X2, Y2), Y2)=implies(implies(Y2, X2), X2)).
% 0.19/0.41
% 0.19/0.41 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.41 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.41 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.41 fresh(y, y, x1...xn) = u
% 0.19/0.41 C => fresh(s, t, x1...xn) = v
% 0.19/0.41 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.41 variables of u and v.
% 0.19/0.41 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.41 input problem has no model of domain size 1).
% 0.19/0.41
% 0.19/0.41 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.41
% 0.19/0.41 Axiom 1 (wajsberg_1): implies(truth, X) = X.
% 0.19/0.41 Axiom 2 (partial_order_definition2): fresh(X, X, Y, Z) = true.
% 0.19/0.41 Axiom 3 (partial_order_definition1): fresh2(X, X, Y, Z) = truth.
% 0.19/0.41 Axiom 4 (big_V_definition): big_V(X, Y) = implies(implies(X, Y), Y).
% 0.19/0.41 Axiom 5 (wajsberg_3): implies(implies(X, Y), Y) = implies(implies(Y, X), X).
% 0.19/0.41 Axiom 6 (lemma_1): fresh5(X, X, Y, Z, W) = true.
% 0.19/0.41 Axiom 7 (partial_order_definition2): fresh(implies(X, Y), truth, X, Y) = ordered(X, Y).
% 0.19/0.41 Axiom 8 (partial_order_definition1): fresh2(ordered(X, Y), true, X, Y) = implies(X, Y).
% 0.19/0.41 Axiom 9 (lemma_1): fresh5(ordered(X, Y), true, X, Y, Z) = ordered(implies(X, Z), implies(Y, Z)).
% 0.19/0.41 Axiom 10 (wajsberg_2): implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth.
% 0.19/0.41
% 0.19/0.41 Lemma 11: implies(X, big_V(X, Y)) = truth.
% 0.19/0.41 Proof:
% 0.19/0.41 implies(X, big_V(X, Y))
% 0.19/0.41 = { by axiom 4 (big_V_definition) }
% 0.19/0.41 implies(X, implies(implies(X, Y), Y))
% 0.19/0.41 = { by axiom 1 (wajsberg_1) R->L }
% 0.19/0.41 implies(X, implies(implies(X, Y), implies(truth, Y)))
% 0.19/0.41 = { by axiom 1 (wajsberg_1) R->L }
% 0.19/0.41 implies(implies(truth, X), implies(implies(X, Y), implies(truth, Y)))
% 0.19/0.41 = { by axiom 10 (wajsberg_2) }
% 0.19/0.41 truth
% 0.19/0.41
% 0.19/0.41 Goal 1 (prove_mv_4): big_V(implies(x, y), implies(y, x)) = truth.
% 0.19/0.41 Proof:
% 0.19/0.41 big_V(implies(x, y), implies(y, x))
% 0.19/0.41 = { by axiom 4 (big_V_definition) }
% 0.19/0.41 implies(implies(implies(x, y), implies(y, x)), implies(y, x))
% 0.19/0.41 = { by axiom 8 (partial_order_definition1) R->L }
% 0.19/0.41 fresh2(ordered(implies(implies(x, y), implies(y, x)), implies(y, x)), true, implies(implies(x, y), implies(y, x)), implies(y, x))
% 0.19/0.41 = { by axiom 1 (wajsberg_1) R->L }
% 0.19/0.41 fresh2(ordered(implies(implies(x, y), implies(y, x)), implies(truth, implies(y, x))), true, implies(implies(x, y), implies(y, x)), implies(y, x))
% 0.19/0.41 = { by axiom 9 (lemma_1) R->L }
% 0.19/0.41 fresh2(fresh5(ordered(implies(x, y), truth), true, implies(x, y), truth, implies(y, x)), true, implies(implies(x, y), implies(y, x)), implies(y, x))
% 0.19/0.41 = { by axiom 7 (partial_order_definition2) R->L }
% 0.19/0.41 fresh2(fresh5(fresh(implies(implies(x, y), truth), truth, implies(x, y), truth), true, implies(x, y), truth, implies(y, x)), true, implies(implies(x, y), implies(y, x)), implies(y, x))
% 0.19/0.41 = { by lemma 11 R->L }
% 0.19/0.41 fresh2(fresh5(fresh(implies(implies(x, y), implies(truth, big_V(truth, implies(x, y)))), truth, implies(x, y), truth), true, implies(x, y), truth, implies(y, x)), true, implies(implies(x, y), implies(y, x)), implies(y, x))
% 0.19/0.41 = { by axiom 1 (wajsberg_1) }
% 0.19/0.42 fresh2(fresh5(fresh(implies(implies(x, y), big_V(truth, implies(x, y))), truth, implies(x, y), truth), true, implies(x, y), truth, implies(y, x)), true, implies(implies(x, y), implies(y, x)), implies(y, x))
% 0.19/0.42 = { by axiom 4 (big_V_definition) }
% 0.19/0.42 fresh2(fresh5(fresh(implies(implies(x, y), implies(implies(truth, implies(x, y)), implies(x, y))), truth, implies(x, y), truth), true, implies(x, y), truth, implies(y, x)), true, implies(implies(x, y), implies(y, x)), implies(y, x))
% 0.19/0.42 = { by axiom 5 (wajsberg_3) }
% 0.19/0.42 fresh2(fresh5(fresh(implies(implies(x, y), implies(implies(implies(x, y), truth), truth)), truth, implies(x, y), truth), true, implies(x, y), truth, implies(y, x)), true, implies(implies(x, y), implies(y, x)), implies(y, x))
% 0.19/0.42 = { by axiom 4 (big_V_definition) R->L }
% 0.19/0.42 fresh2(fresh5(fresh(implies(implies(x, y), big_V(implies(x, y), truth)), truth, implies(x, y), truth), true, implies(x, y), truth, implies(y, x)), true, implies(implies(x, y), implies(y, x)), implies(y, x))
% 0.19/0.42 = { by lemma 11 }
% 0.19/0.42 fresh2(fresh5(fresh(truth, truth, implies(x, y), truth), true, implies(x, y), truth, implies(y, x)), true, implies(implies(x, y), implies(y, x)), implies(y, x))
% 0.19/0.42 = { by axiom 2 (partial_order_definition2) }
% 0.19/0.42 fresh2(fresh5(true, true, implies(x, y), truth, implies(y, x)), true, implies(implies(x, y), implies(y, x)), implies(y, x))
% 0.19/0.42 = { by axiom 6 (lemma_1) }
% 0.19/0.42 fresh2(true, true, implies(implies(x, y), implies(y, x)), implies(y, x))
% 0.19/0.42 = { by axiom 3 (partial_order_definition1) }
% 0.19/0.42 truth
% 0.19/0.42 % SZS output end Proof
% 0.19/0.42
% 0.19/0.42 RESULT: Unsatisfiable (the axioms are contradictory).
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