TSTP Solution File: COL006-3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : COL006-3 : 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 : 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 : Wed Aug 30 18:31:36 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 : COL006-3 : 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
% 0.13/0.34 % Computer : n005.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 : Sun Aug 27 05:14:38 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.43 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 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(k_definition, axiom, ![X, Y]: apply(apply(k, X), Y)=X).
% 0.19/0.44 fof(prove_strong_fixed_point, negated_conjecture, ~fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))))).
% 0.19/0.44 fof(s_definition, axiom, ![Z, X2, Y2]: apply(apply(apply(s, X2), Y2), Z)=apply(apply(X2, Z), apply(Y2, Z))).
% 0.19/0.44 fof(strong_fixed_point, axiom, ![Strong_fixed_point]: (apply(Strong_fixed_point, fixed_pt)!=apply(fixed_pt, apply(Strong_fixed_point, fixed_pt)) | fixed_point(Strong_fixed_point))).
% 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 (strong_fixed_point): fresh(X, X, Y) = true.
% 0.19/0.44 Axiom 2 (k_definition): apply(apply(k, X), Y) = X.
% 0.19/0.44 Axiom 3 (s_definition): apply(apply(apply(s, X), Y), Z) = apply(apply(X, Z), apply(Y, Z)).
% 0.19/0.44 Axiom 4 (strong_fixed_point): fresh(apply(X, fixed_pt), apply(fixed_pt, apply(X, fixed_pt)), X) = fixed_point(X).
% 0.19/0.44
% 0.19/0.44 Lemma 5: apply(apply(apply(s, k), X), Y) = Y.
% 0.19/0.44 Proof:
% 0.19/0.44 apply(apply(apply(s, k), X), Y)
% 0.19/0.44 = { by axiom 3 (s_definition) }
% 0.19/0.44 apply(apply(k, Y), apply(X, Y))
% 0.19/0.44 = { by axiom 2 (k_definition) }
% 0.19/0.44 Y
% 0.19/0.44
% 0.19/0.45 Lemma 6: apply(apply(apply(s, apply(k, X)), Y), Z) = apply(X, apply(Y, Z)).
% 0.19/0.45 Proof:
% 0.19/0.45 apply(apply(apply(s, apply(k, X)), Y), Z)
% 0.19/0.45 = { by axiom 3 (s_definition) }
% 0.19/0.45 apply(apply(apply(k, X), Z), apply(Y, Z))
% 0.19/0.45 = { by axiom 2 (k_definition) }
% 0.19/0.45 apply(X, apply(Y, Z))
% 0.19/0.45
% 0.19/0.45 Goal 1 (prove_strong_fixed_point): fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))))) = true.
% 0.19/0.45 Proof:
% 0.19/0.45 fixed_point(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))))
% 0.19/0.45 = { by axiom 4 (strong_fixed_point) R->L }
% 0.19/0.45 fresh(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(fixed_pt, apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt)), apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))))
% 0.19/0.45 = { by lemma 6 }
% 0.19/0.45 fresh(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(fixed_pt, apply(apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt))), apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))))
% 0.19/0.45 = { by axiom 2 (k_definition) R->L }
% 0.19/0.45 fresh(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(k, fixed_pt), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt)), apply(apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt))), apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))))
% 0.19/0.45 = { by axiom 3 (s_definition) R->L }
% 0.19/0.45 fresh(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(s, apply(k, fixed_pt)), apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt)), apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))))
% 0.19/0.45 = { by lemma 6 R->L }
% 0.19/0.45 fresh(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(apply(apply(s, apply(k, s)), k), fixed_pt), apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt)), apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))))
% 0.19/0.45 = { by axiom 2 (k_definition) R->L }
% 0.19/0.45 fresh(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(apply(apply(s, apply(k, s)), k), fixed_pt), apply(apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))), fixed_pt)), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt)), apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))))
% 0.19/0.45 = { by axiom 3 (s_definition) R->L }
% 0.19/0.45 fresh(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt)), apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))))
% 0.19/0.45 = { by lemma 5 R->L }
% 0.19/0.45 fresh(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt), apply(apply(apply(s, k), k), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt))), apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))))
% 0.19/0.46 = { by lemma 5 R->L }
% 0.19/0.46 fresh(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(apply(s, k), k), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt)), apply(apply(apply(s, k), k), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt))), apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))))
% 0.19/0.46 = { by axiom 3 (s_definition) R->L }
% 0.19/0.46 fresh(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)), apply(apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), fixed_pt)), apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))))
% 0.19/0.46 = { by lemma 6 R->L }
% 0.19/0.46 fresh(apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))), fixed_pt), apply(apply(s, apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k)))), apply(apply(s, apply(apply(s, apply(k, s)), k)), apply(k, apply(apply(s, apply(apply(s, k), k)), apply(apply(s, k), k))))))
% 0.19/0.46 = { by axiom 1 (strong_fixed_point) }
% 0.19/0.46 true
% 0.19/0.46 % SZS output end Proof
% 0.19/0.46
% 0.19/0.46 RESULT: Unsatisfiable (the axioms are contradictory).
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