TSTP Solution File: GEO059-2 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : GEO059-2 : 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 : n020.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 : Wed Aug 30 23:27:08 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
% 0.06/0.12 % Problem : GEO059-2 : TPTP v8.1.2. Released v1.0.0.
% 0.06/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n020.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Tue Aug 29 20:29:13 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.43 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.19/0.43
% 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(prove_congruence, negated_conjecture, ~equidistant(v, u, v, reflection(reflection(u, v), v))).
% 0.19/0.44 fof(reflection, axiom, ![V, U]: reflection(U, V)=extension(U, V, U, V)).
% 0.19/0.44 fof(reflexivity_for_equidistance, axiom, ![X, Y]: equidistant(X, Y, Y, X)).
% 0.19/0.44 fof(segment_construction2, axiom, ![W, X2, Y2, V3]: equidistant(Y2, extension(X2, Y2, W, V3), W, V3)).
% 0.19/0.44 fof(transitivity_for_equidistance, axiom, ![Z, V2, X2, Y2, V3, W2]: (~equidistant(X2, Y2, Z, V3) | (~equidistant(X2, Y2, V2, W2) | equidistant(Z, V3, V2, W2)))).
% 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 (reflection): reflection(X, Y) = extension(X, Y, X, Y).
% 0.19/0.44 Axiom 2 (reflexivity_for_equidistance): equidistant(X, Y, Y, X) = true.
% 0.19/0.44 Axiom 3 (transitivity_for_equidistance): fresh3(X, X, Y, Z, W, V) = true.
% 0.19/0.44 Axiom 4 (transitivity_for_equidistance): fresh4(X, X, Y, Z, W, V, U, T) = equidistant(W, V, U, T).
% 0.19/0.44 Axiom 5 (segment_construction2): equidistant(X, extension(Y, X, Z, W), Z, W) = true.
% 0.19/0.44 Axiom 6 (transitivity_for_equidistance): fresh4(equidistant(X, Y, Z, W), true, X, Y, V, U, Z, W) = fresh3(equidistant(X, Y, V, U), true, V, U, Z, W).
% 0.19/0.44
% 0.19/0.44 Lemma 7: fresh3(equidistant(X, Y, Z, W), true, Z, W, Y, X) = equidistant(Z, W, Y, X).
% 0.19/0.44 Proof:
% 0.19/0.44 fresh3(equidistant(X, Y, Z, W), true, Z, W, Y, X)
% 0.19/0.44 = { by axiom 6 (transitivity_for_equidistance) R->L }
% 0.19/0.44 fresh4(equidistant(X, Y, Y, X), true, X, Y, Z, W, Y, X)
% 0.19/0.44 = { by axiom 2 (reflexivity_for_equidistance) }
% 0.19/0.44 fresh4(true, true, X, Y, Z, W, Y, X)
% 0.19/0.44 = { by axiom 4 (transitivity_for_equidistance) }
% 0.19/0.44 equidistant(Z, W, Y, X)
% 0.19/0.44
% 0.19/0.44 Lemma 8: equidistant(X, Y, reflection(X, Y), Y) = true.
% 0.19/0.44 Proof:
% 0.19/0.44 equidistant(X, Y, reflection(X, Y), Y)
% 0.19/0.44 = { by lemma 7 R->L }
% 0.19/0.44 fresh3(equidistant(Y, reflection(X, Y), X, Y), true, X, Y, reflection(X, Y), Y)
% 0.19/0.44 = { by axiom 1 (reflection) }
% 0.19/0.44 fresh3(equidistant(Y, extension(X, Y, X, Y), X, Y), true, X, Y, reflection(X, Y), Y)
% 0.19/0.44 = { by axiom 5 (segment_construction2) }
% 0.19/0.44 fresh3(true, true, X, Y, reflection(X, Y), Y)
% 0.19/0.44 = { by axiom 3 (transitivity_for_equidistance) }
% 0.19/0.45 true
% 0.19/0.45
% 0.19/0.45 Goal 1 (prove_congruence): equidistant(v, u, v, reflection(reflection(u, v), v)) = true.
% 0.19/0.45 Proof:
% 0.19/0.45 equidistant(v, u, v, reflection(reflection(u, v), v))
% 0.19/0.45 = { by lemma 7 R->L }
% 0.19/0.45 fresh3(equidistant(reflection(reflection(u, v), v), v, v, u), true, v, u, v, reflection(reflection(u, v), v))
% 0.19/0.45 = { by axiom 4 (transitivity_for_equidistance) R->L }
% 0.19/0.45 fresh3(fresh4(true, true, reflection(u, v), v, reflection(reflection(u, v), v), v, v, u), true, v, u, v, reflection(reflection(u, v), v))
% 0.19/0.45 = { by axiom 3 (transitivity_for_equidistance) R->L }
% 0.19/0.45 fresh3(fresh4(fresh3(true, true, reflection(u, v), v, v, u), true, reflection(u, v), v, reflection(reflection(u, v), v), v, v, u), true, v, u, v, reflection(reflection(u, v), v))
% 0.19/0.45 = { by lemma 8 R->L }
% 0.19/0.45 fresh3(fresh4(fresh3(equidistant(u, v, reflection(u, v), v), true, reflection(u, v), v, v, u), true, reflection(u, v), v, reflection(reflection(u, v), v), v, v, u), true, v, u, v, reflection(reflection(u, v), v))
% 0.19/0.45 = { by lemma 7 }
% 0.19/0.45 fresh3(fresh4(equidistant(reflection(u, v), v, v, u), true, reflection(u, v), v, reflection(reflection(u, v), v), v, v, u), true, v, u, v, reflection(reflection(u, v), v))
% 0.19/0.45 = { by axiom 6 (transitivity_for_equidistance) }
% 0.19/0.45 fresh3(fresh3(equidistant(reflection(u, v), v, reflection(reflection(u, v), v), v), true, reflection(reflection(u, v), v), v, v, u), true, v, u, v, reflection(reflection(u, v), v))
% 0.19/0.45 = { by lemma 8 }
% 0.19/0.45 fresh3(fresh3(true, true, reflection(reflection(u, v), v), v, v, u), true, v, u, v, reflection(reflection(u, v), v))
% 0.19/0.45 = { by axiom 3 (transitivity_for_equidistance) }
% 0.19/0.45 fresh3(true, true, v, u, v, reflection(reflection(u, v), v))
% 0.19/0.45 = { by axiom 3 (transitivity_for_equidistance) }
% 0.19/0.45 true
% 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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