TSTP Solution File: COL076-2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : COL076-2 : 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 : n025.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 18:32:01 EDT 2023
% Result : Unsatisfiable 25.01s 3.67s
% Output : Proof 25.01s
% 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 : COL076-2 : TPTP v8.1.2. Released v1.2.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n025.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Sun Aug 27 04:57:23 EDT 2023
% 0.14/0.34 % CPUTime :
% 25.01/3.67 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 25.01/3.67
% 25.01/3.67 % SZS status Unsatisfiable
% 25.01/3.67
% 25.01/3.68 % SZS output start Proof
% 25.01/3.68 Take the following subset of the input axioms:
% 25.01/3.68 fof(abstraction, axiom, ![X, Y, Z]: apply(apply(apply(abstraction, X), Y), Z)=apply(apply(X, apply(k, Z)), apply(Y, Z))).
% 25.01/3.68 fof(diagonal_combinator, axiom, ![Y2, X2]: apply(apply(f, X2), Y2)=apply(X2, X2)).
% 25.01/3.68 fof(extensionality2, axiom, ![Y2, X2]: (X2=Y2 | apply(X2, n(X2, Y2))!=apply(Y2, n(X2, Y2)))).
% 25.01/3.68 fof(k_definition, axiom, ![Y2, X2]: apply(apply(k, X2), Y2)=X2).
% 25.01/3.68 fof(pairwise_application, axiom, ![Y2, X2, Z2]: apply(pair(X2, Y2), Z2)=pair(apply(X2, Z2), apply(Y2, Z2))).
% 25.01/3.68 fof(prove_self_referential, negated_conjecture, ![Y2]: Y2!=apply(eq, pair(apply(k, Y2), apply(k, projection2)))).
% 25.01/3.68
% 25.01/3.68 Now clausify the problem and encode Horn clauses using encoding 3 of
% 25.01/3.68 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 25.01/3.68 We repeatedly replace C & s=t => u=v by the two clauses:
% 25.01/3.68 fresh(y, y, x1...xn) = u
% 25.01/3.68 C => fresh(s, t, x1...xn) = v
% 25.01/3.68 where fresh is a fresh function symbol and x1..xn are the free
% 25.01/3.68 variables of u and v.
% 25.01/3.68 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 25.01/3.68 input problem has no model of domain size 1).
% 25.01/3.68
% 25.01/3.68 The encoding turns the above axioms into the following unit equations and goals:
% 25.01/3.68
% 25.01/3.68 Axiom 1 (k_definition): apply(apply(k, X), Y) = X.
% 25.01/3.68 Axiom 2 (diagonal_combinator): apply(apply(f, X), Y) = apply(X, X).
% 25.01/3.68 Axiom 3 (extensionality2): fresh(X, X, Y, Z) = Z.
% 25.01/3.68 Axiom 4 (pairwise_application): apply(pair(X, Y), Z) = pair(apply(X, Z), apply(Y, Z)).
% 25.01/3.68 Axiom 5 (abstraction): apply(apply(apply(abstraction, X), Y), Z) = apply(apply(X, apply(k, Z)), apply(Y, Z)).
% 25.01/3.68 Axiom 6 (extensionality2): fresh(apply(X, n(X, Y)), apply(Y, n(X, Y)), X, Y) = X.
% 25.01/3.68
% 25.01/3.68 Lemma 7: apply(k, apply(X, X)) = apply(f, X).
% 25.01/3.68 Proof:
% 25.01/3.68 apply(k, apply(X, X))
% 25.01/3.68 = { by axiom 6 (extensionality2) R->L }
% 25.01/3.68 fresh(apply(apply(k, apply(X, X)), n(apply(k, apply(X, X)), apply(f, X))), apply(apply(f, X), n(apply(k, apply(X, X)), apply(f, X))), apply(k, apply(X, X)), apply(f, X))
% 25.01/3.68 = { by axiom 2 (diagonal_combinator) }
% 25.01/3.68 fresh(apply(apply(k, apply(X, X)), n(apply(k, apply(X, X)), apply(f, X))), apply(X, X), apply(k, apply(X, X)), apply(f, X))
% 25.01/3.68 = { by axiom 1 (k_definition) }
% 25.01/3.68 fresh(apply(X, X), apply(X, X), apply(k, apply(X, X)), apply(f, X))
% 25.01/3.68 = { by axiom 3 (extensionality2) }
% 25.01/3.68 apply(f, X)
% 25.01/3.68
% 25.01/3.68 Lemma 8: apply(pair(X, apply(k, Y)), Z) = pair(apply(X, Z), Y).
% 25.01/3.68 Proof:
% 25.01/3.68 apply(pair(X, apply(k, Y)), Z)
% 25.01/3.68 = { by axiom 4 (pairwise_application) }
% 25.01/3.68 pair(apply(X, Z), apply(apply(k, Y), Z))
% 25.01/3.68 = { by axiom 1 (k_definition) }
% 25.01/3.68 pair(apply(X, Z), Y)
% 25.01/3.68
% 25.01/3.68 Goal 1 (prove_self_referential): X = apply(eq, pair(apply(k, X), apply(k, projection2))).
% 25.01/3.68 The goal is true when:
% 25.01/3.68 X = apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))
% 25.01/3.68
% 25.01/3.68 Proof:
% 25.01/3.68 apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))
% 25.01/3.68 = { by axiom 5 (abstraction) }
% 25.01/3.68 apply(apply(apply(k, eq), apply(k, pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(f, pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))
% 25.01/3.68 = { by axiom 1 (k_definition) }
% 25.01/3.68 apply(eq, apply(f, pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))
% 25.01/3.68 = { by lemma 7 R->L }
% 25.01/3.68 apply(eq, apply(k, apply(pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))
% 25.01/3.68 = { by lemma 8 }
% 25.01/3.68 apply(eq, apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), projection2)))
% 25.01/3.68 = { by axiom 1 (k_definition) R->L }
% 25.01/3.68 apply(eq, apply(k, pair(apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))), apply(apply(k, projection2), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))))))
% 25.01/3.68 = { by axiom 1 (k_definition) R->L }
% 25.01/3.68 apply(eq, apply(k, pair(apply(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(apply(k, projection2), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))))))
% 25.01/3.68 = { by axiom 4 (pairwise_application) R->L }
% 25.01/3.68 apply(eq, apply(k, apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))))
% 25.01/3.69 = { by axiom 6 (extensionality2) R->L }
% 25.01/3.69 apply(eq, fresh(apply(apply(k, apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))), n(apply(k, apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))), pair(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(k, projection2)))), apply(pair(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(k, projection2)), n(apply(k, apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))), pair(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(k, projection2)))), apply(k, apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))), pair(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(k, projection2))))
% 25.01/3.69 = { by axiom 1 (k_definition) }
% 25.01/3.69 apply(eq, fresh(apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(pair(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(k, projection2)), n(apply(k, apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))), pair(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(k, projection2)))), apply(k, apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))), pair(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(k, projection2))))
% 25.01/3.69 = { by lemma 8 }
% 25.01/3.69 apply(eq, fresh(apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), pair(apply(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), n(apply(k, apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))), pair(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(k, projection2)))), projection2), apply(k, apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))), pair(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(k, projection2))))
% 25.01/3.69 = { by axiom 1 (k_definition) R->L }
% 25.01/3.69 apply(eq, fresh(apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), pair(apply(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), n(apply(k, apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))), pair(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(k, projection2)))), apply(apply(k, projection2), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))), apply(k, apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))), pair(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(k, projection2))))
% 25.01/3.69 = { by axiom 2 (diagonal_combinator) }
% 25.01/3.69 apply(eq, fresh(apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), pair(apply(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(apply(k, projection2), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))), apply(k, apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))), pair(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(k, projection2))))
% 25.01/3.69 = { by axiom 4 (pairwise_application) R->L }
% 25.01/3.69 apply(eq, fresh(apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(k, apply(pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))), pair(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(k, projection2))))
% 25.01/3.69 = { by axiom 3 (extensionality2) }
% 25.01/3.69 apply(eq, pair(apply(f, apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2))))), apply(k, projection2)))
% 25.01/3.69 = { by lemma 7 R->L }
% 25.01/3.69 apply(eq, pair(apply(k, apply(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))))), apply(k, projection2)))
% 25.01/3.69 = { by axiom 1 (k_definition) }
% 25.01/3.69 apply(eq, pair(apply(k, apply(apply(apply(abstraction, apply(k, eq)), f), pair(apply(apply(abstraction, apply(k, eq)), f), apply(k, projection2)))), apply(k, projection2)))
% 25.01/3.69 % SZS output end Proof
% 25.01/3.69
% 25.01/3.69 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------