TSTP Solution File: LCL109-3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL109-3 : 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 : n002.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 0.43s
% 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
% 0.00/0.12 % Problem : LCL109-3 : 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
% 0.12/0.34 % Computer : n002.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Aug 25 02:35:47 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.19/0.43 Command-line arguments: --no-flatten-goal
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% 0.19/0.43 % SZS status Unsatisfiable
% 0.19/0.43
% 0.19/0.44 % SZS output start Proof
% 0.19/0.44 Take the following subset of the input axioms:
% 0.19/0.44 fof(lemma_1, axiom, ![X]: implies(X, X)=truth).
% 0.19/0.44 fof(lemma_10, axiom, ![Y, X2]: implies(not(X2), not(Y))=implies(Y, X2)).
% 0.19/0.44 fof(lemma_4, axiom, ![X2, Y2]: implies(X2, implies(Y2, X2))=truth).
% 0.19/0.44 fof(lemma_7, axiom, ![Z, X2, Y2]: implies(X2, implies(Y2, Z))=implies(Y2, implies(X2, Z))).
% 0.19/0.44 fof(lemma_8, axiom, ![X2]: implies(X2, not(truth))=not(X2)).
% 0.19/0.44 fof(prove_wajsberg_mv_4, negated_conjecture, implies(implies(implies(a, b), implies(b, a)), implies(b, a))!=truth).
% 0.19/0.44 fof(wajsberg_1, axiom, ![X2]: implies(truth, X2)=X2).
% 0.19/0.44 fof(wajsberg_3, axiom, ![X2, Y2]: implies(implies(X2, Y2), Y2)=implies(implies(Y2, X2), X2)).
% 0.19/0.44
% 0.19/0.44 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.44 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.44 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.44 fresh(y, y, x1...xn) = u
% 0.19/0.44 C => fresh(s, t, x1...xn) = v
% 0.19/0.44 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.44 variables of u and v.
% 0.19/0.44 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.44 input problem has no model of domain size 1).
% 0.19/0.44
% 0.19/0.44 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.44
% 0.19/0.44 Axiom 1 (lemma_1): implies(X, X) = truth.
% 0.19/0.44 Axiom 2 (wajsberg_1): implies(truth, X) = X.
% 0.19/0.44 Axiom 3 (lemma_8): implies(X, not(truth)) = not(X).
% 0.19/0.44 Axiom 4 (lemma_4): implies(X, implies(Y, X)) = truth.
% 0.19/0.44 Axiom 5 (lemma_7): implies(X, implies(Y, Z)) = implies(Y, implies(X, Z)).
% 0.19/0.44 Axiom 6 (lemma_10): implies(not(X), not(Y)) = implies(Y, X).
% 0.19/0.44 Axiom 7 (wajsberg_3): implies(implies(X, Y), Y) = implies(implies(Y, X), X).
% 0.19/0.44
% 0.19/0.44 Lemma 8: implies(Y, not(X)) = implies(X, not(Y)).
% 0.19/0.44 Proof:
% 0.19/0.44 implies(Y, not(X))
% 0.19/0.44 = { by axiom 3 (lemma_8) R->L }
% 0.19/0.44 implies(Y, implies(X, not(truth)))
% 0.19/0.44 = { by axiom 5 (lemma_7) }
% 0.19/0.44 implies(X, implies(Y, not(truth)))
% 0.19/0.44 = { by axiom 3 (lemma_8) }
% 0.19/0.44 implies(X, not(Y))
% 0.19/0.44
% 0.19/0.44 Lemma 9: implies(not(X), implies(Y, not(Z))) = implies(Y, implies(Z, X)).
% 0.19/0.44 Proof:
% 0.19/0.44 implies(not(X), implies(Y, not(Z)))
% 0.19/0.44 = { by axiom 5 (lemma_7) R->L }
% 0.19/0.44 implies(Y, implies(not(X), not(Z)))
% 0.19/0.44 = { by axiom 6 (lemma_10) }
% 0.19/0.44 implies(Y, implies(Z, X))
% 0.19/0.44
% 0.19/0.44 Goal 1 (prove_wajsberg_mv_4): implies(implies(implies(a, b), implies(b, a)), implies(b, a)) = truth.
% 0.19/0.44 Proof:
% 0.19/0.44 implies(implies(implies(a, b), implies(b, a)), implies(b, a))
% 0.19/0.44 = { by axiom 5 (lemma_7) }
% 0.19/0.44 implies(b, implies(implies(implies(a, b), implies(b, a)), a))
% 0.19/0.44 = { by axiom 2 (wajsberg_1) R->L }
% 0.19/0.44 implies(b, implies(implies(implies(a, b), implies(truth, implies(b, a))), a))
% 0.19/0.44 = { by axiom 4 (lemma_4) R->L }
% 0.19/0.44 implies(b, implies(implies(implies(a, b), implies(implies(a, implies(b, a)), implies(b, a))), a))
% 0.19/0.44 = { by axiom 7 (wajsberg_3) }
% 0.19/0.44 implies(b, implies(implies(implies(a, b), implies(implies(implies(b, a), a), a)), a))
% 0.19/0.44 = { by axiom 7 (wajsberg_3) }
% 0.19/0.44 implies(b, implies(implies(implies(a, b), implies(implies(implies(a, b), b), a)), a))
% 0.19/0.44 = { by lemma 9 R->L }
% 0.19/0.44 implies(b, implies(implies(not(a), implies(implies(a, b), not(implies(implies(a, b), b)))), a))
% 0.19/0.44 = { by lemma 8 }
% 0.19/0.44 implies(b, implies(implies(not(a), implies(implies(implies(a, b), b), not(implies(a, b)))), a))
% 0.19/0.44 = { by axiom 6 (lemma_10) R->L }
% 0.19/0.44 implies(b, implies(implies(not(a), implies(implies(not(b), not(implies(a, b))), not(implies(a, b)))), a))
% 0.19/0.44 = { by axiom 7 (wajsberg_3) R->L }
% 0.19/0.44 implies(b, implies(implies(not(a), implies(implies(not(implies(a, b)), not(b)), not(b))), a))
% 0.19/0.45 = { by axiom 6 (lemma_10) }
% 0.19/0.45 implies(b, implies(implies(not(a), implies(implies(b, implies(a, b)), not(b))), a))
% 0.19/0.45 = { by lemma 8 R->L }
% 0.19/0.45 implies(b, implies(implies(not(a), implies(b, not(implies(b, implies(a, b))))), a))
% 0.19/0.45 = { by lemma 9 }
% 0.19/0.45 implies(b, implies(implies(b, implies(implies(b, implies(a, b)), a)), a))
% 0.19/0.45 = { by axiom 4 (lemma_4) }
% 0.19/0.45 implies(b, implies(implies(b, implies(truth, a)), a))
% 0.19/0.45 = { by axiom 2 (wajsberg_1) }
% 0.19/0.45 implies(b, implies(implies(b, a), a))
% 0.19/0.45 = { by axiom 5 (lemma_7) R->L }
% 0.19/0.45 implies(implies(b, a), implies(b, a))
% 0.19/0.45 = { by axiom 1 (lemma_1) }
% 0.19/0.45 truth
% 0.19/0.45 % SZS output end Proof
% 0.19/0.45
% 0.19/0.45 RESULT: Unsatisfiable (the axioms are contradictory).
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