TSTP Solution File: COL077-1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : COL077-1 : 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 18:32:01 EDT 2023
% Result : Unsatisfiable 0.21s 0.46s
% Output : Proof 0.21s
% 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 : COL077-1 : 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
% 0.14/0.35 % Computer : n002.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Sun Aug 27 05:28:33 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.21/0.46 Command-line arguments: --flatten
% 0.21/0.46
% 0.21/0.46 % SZS status Unsatisfiable
% 0.21/0.46
% 0.21/0.46 % SZS output start Proof
% 0.21/0.46 Take the following subset of the input axioms:
% 0.21/0.46 fof(abstraction, axiom, ![X, Y, Z]: apply(apply(apply(abstraction, X), Y), Z)=apply(apply(X, k(Z)), apply(Y, Z))).
% 0.21/0.46 fof(extensionality2, axiom, ![X2, Y2]: (X2=Y2 | apply(X2, n(X2, Y2))!=apply(Y2, n(X2, Y2)))).
% 0.21/0.46 fof(identity_definition, axiom, ![X2]: apply(identity, X2)=X2).
% 0.21/0.46 fof(k_definition, axiom, ![X2, Y2]: apply(k(X2), Y2)=X2).
% 0.21/0.46 fof(prove_TRC1a, negated_conjecture, apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction)!=identity).
% 0.21/0.46
% 0.21/0.46 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.46 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.46 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.46 fresh(y, y, x1...xn) = u
% 0.21/0.46 C => fresh(s, t, x1...xn) = v
% 0.21/0.46 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.46 variables of u and v.
% 0.21/0.46 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.46 input problem has no model of domain size 1).
% 0.21/0.46
% 0.21/0.46 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.46
% 0.21/0.46 Axiom 1 (identity_definition): apply(identity, X) = X.
% 0.21/0.46 Axiom 2 (k_definition): apply(k(X), Y) = X.
% 0.21/0.47 Axiom 3 (extensionality2): fresh(X, X, Y, Z) = Z.
% 0.21/0.47 Axiom 4 (abstraction): apply(apply(apply(abstraction, X), Y), Z) = apply(apply(X, k(Z)), apply(Y, Z)).
% 0.21/0.47 Axiom 5 (extensionality2): fresh(apply(X, n(X, Y)), apply(Y, n(X, Y)), X, Y) = X.
% 0.21/0.47
% 0.21/0.47 Lemma 6: apply(apply(apply(apply(abstraction, abstraction), Y), X), Z) = apply(X, apply(apply(Y, X), Z)).
% 0.21/0.47 Proof:
% 0.21/0.47 apply(apply(apply(apply(abstraction, abstraction), Y), X), Z)
% 0.21/0.47 = { by axiom 4 (abstraction) }
% 0.21/0.47 apply(apply(apply(abstraction, k(X)), apply(Y, X)), Z)
% 0.21/0.47 = { by axiom 4 (abstraction) }
% 0.21/0.47 apply(apply(k(X), k(Z)), apply(apply(Y, X), Z))
% 0.21/0.47 = { by axiom 2 (k_definition) }
% 0.21/0.47 apply(X, apply(apply(Y, X), Z))
% 0.21/0.47
% 0.21/0.47 Goal 1 (prove_TRC1a): apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction) = identity.
% 0.21/0.47 Proof:
% 0.21/0.47 apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction)
% 0.21/0.47 = { by axiom 5 (extensionality2) R->L }
% 0.21/0.47 fresh(apply(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), n(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity)), apply(identity, n(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity)), apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity)
% 0.21/0.47 = { by axiom 1 (identity_definition) }
% 0.21/0.47 fresh(apply(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), n(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity)), n(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity), apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity)
% 0.21/0.47 = { by lemma 6 }
% 0.21/0.47 fresh(apply(apply(apply(abstraction, apply(apply(abstraction, abstraction), abstraction)), abstraction), n(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity)), n(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity), apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity)
% 0.21/0.47 = { by axiom 4 (abstraction) }
% 0.21/0.47 fresh(apply(apply(apply(apply(abstraction, abstraction), abstraction), k(n(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity))), apply(abstraction, n(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity))), n(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity), apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity)
% 0.21/0.47 = { by lemma 6 }
% 0.21/0.47 fresh(apply(k(n(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity)), apply(apply(abstraction, k(n(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity))), apply(abstraction, n(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity)))), n(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity), apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity)
% 0.21/0.47 = { by axiom 2 (k_definition) }
% 0.21/0.47 fresh(n(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity), n(apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity), apply(apply(apply(apply(apply(abstraction, abstraction), abstraction), abstraction), abstraction), abstraction), identity)
% 0.21/0.47 = { by axiom 3 (extensionality2) }
% 0.21/0.47 identity
% 0.21/0.47 % SZS output end Proof
% 0.21/0.47
% 0.21/0.47 RESULT: Unsatisfiable (the axioms are contradictory).
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