TSTP Solution File: CAT009-4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : CAT009-4 : 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 : 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 18:18:53 EDT 2023
% Result : Unsatisfiable 0.21s 0.41s
% 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
% 0.07/0.13 % Problem : CAT009-4 : TPTP v8.1.2. Released v1.0.0.
% 0.13/0.14 % 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 00:44:33 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.21/0.41 Command-line arguments: --flatten
% 0.21/0.41
% 0.21/0.41 % SZS status Unsatisfiable
% 0.21/0.41
% 0.21/0.42 % SZS output start Proof
% 0.21/0.42 Take the following subset of the input axioms:
% 0.21/0.42 fof(ab_exists, hypothesis, there_exists(compose(a, b))).
% 0.21/0.42 fof(associativity_of_compose, axiom, ![X, Y, Z]: compose(X, compose(Y, Z))=compose(compose(X, Y), Z)).
% 0.21/0.42 fof(codomain_has_elements, axiom, ![X2]: (~there_exists(codomain(X2)) | there_exists(X2))).
% 0.21/0.42 fof(compose_codomain, axiom, ![X2]: compose(codomain(X2), X2)=X2).
% 0.21/0.42 fof(compose_domain, axiom, ![X2]: compose(X2, domain(X2))=X2).
% 0.21/0.42 fof(composition_implies_domain, axiom, ![X2, Y2]: (~there_exists(compose(X2, Y2)) | there_exists(domain(X2)))).
% 0.21/0.42 fof(domain_codomain_composition1, axiom, ![X2, Y2]: (~there_exists(compose(X2, Y2)) | domain(X2)=codomain(Y2))).
% 0.21/0.42 fof(prove_domain_of_ab_equals_domain_of_b, negated_conjecture, domain(compose(a, b))!=domain(b)).
% 0.21/0.42
% 0.21/0.42 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.42 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.42 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.42 fresh(y, y, x1...xn) = u
% 0.21/0.42 C => fresh(s, t, x1...xn) = v
% 0.21/0.42 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.42 variables of u and v.
% 0.21/0.42 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.42 input problem has no model of domain size 1).
% 0.21/0.42
% 0.21/0.42 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.42
% 0.21/0.42 Axiom 1 (composition_implies_domain): fresh9(X, X, Y) = true.
% 0.21/0.42 Axiom 2 (codomain_has_elements): fresh8(X, X, Y) = true.
% 0.21/0.42 Axiom 3 (ab_exists): there_exists(compose(a, b)) = true.
% 0.21/0.42 Axiom 4 (compose_domain): compose(X, domain(X)) = X.
% 0.21/0.42 Axiom 5 (compose_codomain): compose(codomain(X), X) = X.
% 0.21/0.42 Axiom 6 (domain_codomain_composition1): fresh7(X, X, Y, Z) = codomain(Z).
% 0.21/0.42 Axiom 7 (associativity_of_compose): compose(X, compose(Y, Z)) = compose(compose(X, Y), Z).
% 0.21/0.42 Axiom 8 (codomain_has_elements): fresh8(there_exists(codomain(X)), true, X) = there_exists(X).
% 0.21/0.42 Axiom 9 (composition_implies_domain): fresh9(there_exists(compose(X, Y)), true, X) = there_exists(domain(X)).
% 0.21/0.42 Axiom 10 (domain_codomain_composition1): fresh7(there_exists(compose(X, Y)), true, X, Y) = domain(X).
% 0.21/0.42
% 0.21/0.42 Lemma 11: fresh7(there_exists(compose(X, Y)), there_exists(compose(a, b)), X, Y) = domain(X).
% 0.21/0.42 Proof:
% 0.21/0.42 fresh7(there_exists(compose(X, Y)), there_exists(compose(a, b)), X, Y)
% 0.21/0.42 = { by axiom 3 (ab_exists) }
% 0.21/0.42 fresh7(there_exists(compose(X, Y)), true, X, Y)
% 0.21/0.42 = { by axiom 10 (domain_codomain_composition1) }
% 0.21/0.42 domain(X)
% 0.21/0.42
% 0.21/0.42 Lemma 12: codomain(domain(domain(b))) = domain(b).
% 0.21/0.42 Proof:
% 0.21/0.42 codomain(domain(domain(b)))
% 0.21/0.42 = { by axiom 6 (domain_codomain_composition1) R->L }
% 0.21/0.42 fresh7(there_exists(compose(a, b)), there_exists(compose(a, b)), b, domain(domain(b)))
% 0.21/0.42 = { by axiom 3 (ab_exists) }
% 0.21/0.42 fresh7(true, there_exists(compose(a, b)), b, domain(domain(b)))
% 0.21/0.42 = { by axiom 2 (codomain_has_elements) R->L }
% 0.21/0.42 fresh7(fresh8(there_exists(compose(a, b)), there_exists(compose(a, b)), b), there_exists(compose(a, b)), b, domain(domain(b)))
% 0.21/0.42 = { by axiom 3 (ab_exists) }
% 0.21/0.42 fresh7(fresh8(true, there_exists(compose(a, b)), b), there_exists(compose(a, b)), b, domain(domain(b)))
% 0.21/0.42 = { by axiom 1 (composition_implies_domain) R->L }
% 0.21/0.42 fresh7(fresh8(fresh9(there_exists(compose(a, b)), there_exists(compose(a, b)), a), there_exists(compose(a, b)), b), there_exists(compose(a, b)), b, domain(domain(b)))
% 0.21/0.42 = { by axiom 3 (ab_exists) }
% 0.21/0.42 fresh7(fresh8(fresh9(there_exists(compose(a, b)), true, a), there_exists(compose(a, b)), b), there_exists(compose(a, b)), b, domain(domain(b)))
% 0.21/0.42 = { by axiom 9 (composition_implies_domain) }
% 0.21/0.42 fresh7(fresh8(there_exists(domain(a)), there_exists(compose(a, b)), b), there_exists(compose(a, b)), b, domain(domain(b)))
% 0.21/0.42 = { by lemma 11 R->L }
% 0.21/0.42 fresh7(fresh8(there_exists(fresh7(there_exists(compose(a, b)), there_exists(compose(a, b)), a, b)), there_exists(compose(a, b)), b), there_exists(compose(a, b)), b, domain(domain(b)))
% 0.21/0.42 = { by axiom 6 (domain_codomain_composition1) }
% 0.21/0.42 fresh7(fresh8(there_exists(codomain(b)), there_exists(compose(a, b)), b), there_exists(compose(a, b)), b, domain(domain(b)))
% 0.21/0.42 = { by axiom 3 (ab_exists) }
% 0.21/0.43 fresh7(fresh8(there_exists(codomain(b)), true, b), there_exists(compose(a, b)), b, domain(domain(b)))
% 0.21/0.43 = { by axiom 8 (codomain_has_elements) }
% 0.21/0.43 fresh7(there_exists(b), there_exists(compose(a, b)), b, domain(domain(b)))
% 0.21/0.43 = { by axiom 4 (compose_domain) R->L }
% 0.21/0.43 fresh7(there_exists(compose(b, domain(b))), there_exists(compose(a, b)), b, domain(domain(b)))
% 0.21/0.43 = { by axiom 4 (compose_domain) R->L }
% 0.21/0.43 fresh7(there_exists(compose(b, compose(domain(b), domain(domain(b))))), there_exists(compose(a, b)), b, domain(domain(b)))
% 0.21/0.43 = { by axiom 7 (associativity_of_compose) }
% 0.21/0.43 fresh7(there_exists(compose(compose(b, domain(b)), domain(domain(b)))), there_exists(compose(a, b)), b, domain(domain(b)))
% 0.21/0.43 = { by axiom 4 (compose_domain) }
% 0.21/0.43 fresh7(there_exists(compose(b, domain(domain(b)))), there_exists(compose(a, b)), b, domain(domain(b)))
% 0.21/0.43 = { by lemma 11 }
% 0.21/0.43 domain(b)
% 0.21/0.43
% 0.21/0.43 Goal 1 (prove_domain_of_ab_equals_domain_of_b): domain(compose(a, b)) = domain(b).
% 0.21/0.43 Proof:
% 0.21/0.43 domain(compose(a, b))
% 0.21/0.43 = { by lemma 11 R->L }
% 0.21/0.43 fresh7(there_exists(compose(compose(a, b), domain(b))), there_exists(compose(a, b)), compose(a, b), domain(b))
% 0.21/0.43 = { by axiom 7 (associativity_of_compose) R->L }
% 0.21/0.43 fresh7(there_exists(compose(a, compose(b, domain(b)))), there_exists(compose(a, b)), compose(a, b), domain(b))
% 0.21/0.43 = { by axiom 4 (compose_domain) }
% 0.21/0.43 fresh7(there_exists(compose(a, b)), there_exists(compose(a, b)), compose(a, b), domain(b))
% 0.21/0.43 = { by axiom 6 (domain_codomain_composition1) }
% 0.21/0.43 codomain(domain(b))
% 0.21/0.43 = { by axiom 4 (compose_domain) R->L }
% 0.21/0.43 codomain(compose(domain(b), domain(domain(b))))
% 0.21/0.43 = { by lemma 12 R->L }
% 0.21/0.43 codomain(compose(codomain(domain(domain(b))), domain(domain(b))))
% 0.21/0.43 = { by axiom 5 (compose_codomain) }
% 0.21/0.43 codomain(domain(domain(b)))
% 0.21/0.43 = { by lemma 12 }
% 0.21/0.43 domain(b)
% 0.21/0.43 % SZS output end Proof
% 0.21/0.43
% 0.21/0.43 RESULT: Unsatisfiable (the axioms are contradictory).
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