TSTP Solution File: COL079-2 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : COL079-2 : TPTP v8.1.2. Released v1.2.0.
% 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 : Wed Aug 30 18:32:02 EDT 2023
% Result : Unsatisfiable 0.18s 0.45s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : COL079-2 : TPTP v8.1.2. Released v1.2.0.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.33 % Computer : n002.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Sun Aug 27 05:24:18 EDT 2023
% 0.13/0.33 % CPUTime :
% 0.18/0.45 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.18/0.45
% 0.18/0.45 % SZS status Unsatisfiable
% 0.18/0.45
% 0.18/0.45 % SZS output start Proof
% 0.18/0.45 Take the following subset of the input axioms:
% 0.18/0.45 fof(abstraction, axiom, ![X, Y, Z]: apply(apply(apply(abstraction, X), Y), Z)=apply(apply(X, k(Z)), apply(Y, Z))).
% 0.18/0.45 fof(extensionality2, axiom, ![X2, Y2]: (X2=Y2 | apply(X2, n(X2, Y2))!=apply(Y2, n(X2, Y2)))).
% 0.18/0.45 fof(k_definition, axiom, ![X2, Y2]: apply(k(X2), Y2)=X2).
% 0.18/0.45 fof(prove_TRC2a, negated_conjecture, apply(abstraction, apply(abstraction, apply(abstraction, b)))!=apply(abstraction, b)).
% 0.18/0.45
% 0.18/0.45 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.18/0.45 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.18/0.45 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.18/0.45 fresh(y, y, x1...xn) = u
% 0.18/0.45 C => fresh(s, t, x1...xn) = v
% 0.18/0.45 where fresh is a fresh function symbol and x1..xn are the free
% 0.18/0.45 variables of u and v.
% 0.18/0.45 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.18/0.45 input problem has no model of domain size 1).
% 0.18/0.45
% 0.18/0.45 The encoding turns the above axioms into the following unit equations and goals:
% 0.18/0.45
% 0.18/0.45 Axiom 1 (k_definition): apply(k(X), Y) = X.
% 0.18/0.46 Axiom 2 (extensionality2): fresh(X, X, Y, Z) = Z.
% 0.18/0.46 Axiom 3 (abstraction): apply(apply(apply(abstraction, X), Y), Z) = apply(apply(X, k(Z)), apply(Y, Z)).
% 0.18/0.46 Axiom 4 (extensionality2): fresh(apply(X, n(X, Y)), apply(Y, n(X, Y)), X, Y) = X.
% 0.18/0.46
% 0.18/0.46 Lemma 5: apply(apply(apply(abstraction, X), k(Y)), Z) = apply(apply(X, k(Z)), Y).
% 0.18/0.46 Proof:
% 0.18/0.46 apply(apply(apply(abstraction, X), k(Y)), Z)
% 0.18/0.46 = { by axiom 3 (abstraction) }
% 0.18/0.46 apply(apply(X, k(Z)), apply(k(Y), Z))
% 0.18/0.46 = { by axiom 1 (k_definition) }
% 0.18/0.46 apply(apply(X, k(Z)), Y)
% 0.18/0.46
% 0.18/0.46 Goal 1 (prove_TRC2a): apply(abstraction, apply(abstraction, apply(abstraction, b))) = apply(abstraction, b).
% 0.18/0.46 Proof:
% 0.18/0.46 apply(abstraction, apply(abstraction, apply(abstraction, b)))
% 0.18/0.46 = { by axiom 4 (extensionality2) R->L }
% 0.18/0.46 fresh(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))
% 0.18/0.46 = { by axiom 2 (extensionality2) R->L }
% 0.18/0.46 fresh(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), fresh(apply(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), n(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))))), apply(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), n(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))))), apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b)))), apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))
% 0.18/0.46 = { by axiom 3 (abstraction) }
% 0.18/0.46 fresh(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), fresh(apply(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), n(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))))), apply(apply(apply(abstraction, apply(abstraction, b)), k(n(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b)))))), apply(n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b)), n(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b)))))), apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b)))), apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))
% 0.18/0.46 = { by lemma 5 }
% 0.18/0.46 fresh(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), fresh(apply(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), n(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))))), apply(apply(apply(abstraction, b), k(apply(n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b)), n(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))))))), n(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))))), apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b)))), apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))
% 0.18/0.46 = { by lemma 5 }
% 0.18/0.46 fresh(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), fresh(apply(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), n(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))))), apply(apply(b, k(n(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b)))))), apply(n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b)), n(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b)))))), apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b)))), apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))
% 0.18/0.46 = { by axiom 3 (abstraction) R->L }
% 0.18/0.46 fresh(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), fresh(apply(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), n(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))))), apply(apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), n(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))))), apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, b), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b)))), apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))
% 0.18/0.46 = { by axiom 4 (extensionality2) }
% 0.18/0.47 fresh(apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(apply(abstraction, apply(abstraction, apply(abstraction, b))), n(apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))), apply(abstraction, apply(abstraction, apply(abstraction, b))), apply(abstraction, b))
% 0.18/0.47 = { by axiom 2 (extensionality2) }
% 0.18/0.47 apply(abstraction, b)
% 0.18/0.47 % SZS output end Proof
% 0.18/0.47
% 0.18/0.47 RESULT: Unsatisfiable (the axioms are contradictory).
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