TSTP Solution File: CAT018-4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : CAT018-4 : 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 : n020.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 18:18:59 EDT 2023
% Result : Unsatisfiable 0.21s 0.45s
% 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
% 0.00/0.13 % Problem : CAT018-4 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.14 % 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 : n020.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 00:12:14 EDT 2023
% 0.14/0.36 % CPUTime :
% 0.21/0.45 Command-line arguments: --no-flatten-goal
% 0.21/0.45
% 0.21/0.45 % SZS status Unsatisfiable
% 0.21/0.45
% 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(associativity_of_compose, axiom, ![X, Y, Z]: compose(X, compose(Y, Z))=compose(compose(X, Y), Z)).
% 0.21/0.46 fof(assume_ab_exists, hypothesis, there_exists(compose(a, b))).
% 0.21/0.46 fof(assume_bc_exists, hypothesis, there_exists(compose(b, c))).
% 0.21/0.46 fof(compose_codomain, axiom, ![X2]: compose(codomain(X2), X2)=X2).
% 0.21/0.46 fof(compose_domain, axiom, ![X2]: compose(X2, domain(X2))=X2).
% 0.21/0.46 fof(composition_implies_domain, axiom, ![X2, Y2]: (~there_exists(compose(X2, Y2)) | there_exists(domain(X2)))).
% 0.21/0.46 fof(domain_codomain_composition1, axiom, ![X2, Y2]: (~there_exists(compose(X2, Y2)) | domain(X2)=codomain(Y2))).
% 0.21/0.46 fof(domain_codomain_composition2, axiom, ![X2, Y2]: (~there_exists(domain(X2)) | (domain(X2)!=codomain(Y2) | there_exists(compose(X2, Y2))))).
% 0.21/0.46 fof(domain_has_elements, axiom, ![X2]: (~there_exists(domain(X2)) | there_exists(X2))).
% 0.21/0.46 fof(prove_a_bc_exists, negated_conjecture, ~there_exists(compose(a, compose(b, c)))).
% 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 (composition_implies_domain): fresh9(X, X, Y) = true.
% 0.21/0.46 Axiom 2 (domain_has_elements): fresh4(X, X, Y) = true.
% 0.21/0.46 Axiom 3 (compose_domain): compose(X, domain(X)) = X.
% 0.21/0.46 Axiom 4 (compose_codomain): compose(codomain(X), X) = X.
% 0.21/0.46 Axiom 5 (assume_ab_exists): there_exists(compose(a, b)) = true.
% 0.21/0.46 Axiom 6 (assume_bc_exists): there_exists(compose(b, c)) = true.
% 0.21/0.46 Axiom 7 (domain_codomain_composition1): fresh7(X, X, Y, Z) = codomain(Z).
% 0.21/0.46 Axiom 8 (domain_codomain_composition2): fresh6(X, X, Y, Z) = there_exists(compose(Y, Z)).
% 0.21/0.46 Axiom 9 (domain_codomain_composition2): fresh5(X, X, Y, Z) = true.
% 0.21/0.46 Axiom 10 (associativity_of_compose): compose(X, compose(Y, Z)) = compose(compose(X, Y), Z).
% 0.21/0.46 Axiom 11 (domain_has_elements): fresh4(there_exists(domain(X)), true, X) = there_exists(X).
% 0.21/0.46 Axiom 12 (composition_implies_domain): fresh9(there_exists(compose(X, Y)), true, X) = there_exists(domain(X)).
% 0.21/0.46 Axiom 13 (domain_codomain_composition2): fresh6(there_exists(domain(X)), true, X, Y) = fresh5(domain(X), codomain(Y), X, Y).
% 0.21/0.46 Axiom 14 (domain_codomain_composition1): fresh7(there_exists(compose(X, Y)), true, X, Y) = domain(X).
% 0.21/0.46
% 0.21/0.46 Lemma 15: domain(a) = codomain(b).
% 0.21/0.46 Proof:
% 0.21/0.46 domain(a)
% 0.21/0.46 = { by axiom 14 (domain_codomain_composition1) R->L }
% 0.21/0.46 fresh7(there_exists(compose(a, b)), true, a, b)
% 0.21/0.46 = { by axiom 5 (assume_ab_exists) }
% 0.21/0.46 fresh7(true, true, a, b)
% 0.21/0.46 = { by axiom 7 (domain_codomain_composition1) }
% 0.21/0.46 codomain(b)
% 0.21/0.46
% 0.21/0.46 Lemma 16: there_exists(codomain(b)) = true.
% 0.21/0.46 Proof:
% 0.21/0.46 there_exists(codomain(b))
% 0.21/0.46 = { by lemma 15 R->L }
% 0.21/0.46 there_exists(domain(a))
% 0.21/0.46 = { by axiom 12 (composition_implies_domain) R->L }
% 0.21/0.46 fresh9(there_exists(compose(a, b)), true, a)
% 0.21/0.46 = { by axiom 5 (assume_ab_exists) }
% 0.21/0.46 fresh9(true, true, a)
% 0.21/0.46 = { by axiom 1 (composition_implies_domain) }
% 0.21/0.46 true
% 0.21/0.46
% 0.21/0.46 Lemma 17: codomain(codomain(b)) = codomain(b).
% 0.21/0.46 Proof:
% 0.21/0.46 codomain(codomain(b))
% 0.21/0.46 = { by lemma 15 R->L }
% 0.21/0.46 codomain(domain(a))
% 0.21/0.46 = { by axiom 7 (domain_codomain_composition1) R->L }
% 0.21/0.46 fresh7(true, true, a, domain(a))
% 0.21/0.46 = { by axiom 2 (domain_has_elements) R->L }
% 0.21/0.46 fresh7(fresh4(true, true, a), true, a, domain(a))
% 0.21/0.46 = { by lemma 16 R->L }
% 0.21/0.46 fresh7(fresh4(there_exists(codomain(b)), true, a), true, a, domain(a))
% 0.21/0.46 = { by lemma 15 R->L }
% 0.21/0.46 fresh7(fresh4(there_exists(domain(a)), true, a), true, a, domain(a))
% 0.21/0.46 = { by axiom 11 (domain_has_elements) }
% 0.21/0.46 fresh7(there_exists(a), true, a, domain(a))
% 0.21/0.46 = { by axiom 3 (compose_domain) R->L }
% 0.21/0.46 fresh7(there_exists(compose(a, domain(a))), true, a, domain(a))
% 0.21/0.46 = { by axiom 14 (domain_codomain_composition1) }
% 0.21/0.46 domain(a)
% 0.21/0.46 = { by lemma 15 }
% 0.21/0.46 codomain(b)
% 0.21/0.46
% 0.21/0.46 Goal 1 (prove_a_bc_exists): there_exists(compose(a, compose(b, c))) = true.
% 0.21/0.46 Proof:
% 0.21/0.46 there_exists(compose(a, compose(b, c)))
% 0.21/0.46 = { by axiom 8 (domain_codomain_composition2) R->L }
% 0.21/0.46 fresh6(true, true, a, compose(b, c))
% 0.21/0.46 = { by lemma 16 R->L }
% 0.21/0.46 fresh6(there_exists(codomain(b)), true, a, compose(b, c))
% 0.21/0.46 = { by lemma 15 R->L }
% 0.21/0.46 fresh6(there_exists(domain(a)), true, a, compose(b, c))
% 0.21/0.46 = { by axiom 13 (domain_codomain_composition2) }
% 0.21/0.46 fresh5(domain(a), codomain(compose(b, c)), a, compose(b, c))
% 0.21/0.46 = { by lemma 15 }
% 0.21/0.46 fresh5(codomain(b), codomain(compose(b, c)), a, compose(b, c))
% 0.21/0.46 = { by axiom 7 (domain_codomain_composition1) R->L }
% 0.21/0.46 fresh5(codomain(b), fresh7(true, true, codomain(b), compose(b, c)), a, compose(b, c))
% 0.21/0.46 = { by axiom 6 (assume_bc_exists) R->L }
% 0.21/0.46 fresh5(codomain(b), fresh7(there_exists(compose(b, c)), true, codomain(b), compose(b, c)), a, compose(b, c))
% 0.21/0.46 = { by axiom 4 (compose_codomain) R->L }
% 0.21/0.46 fresh5(codomain(b), fresh7(there_exists(compose(compose(codomain(b), b), c)), true, codomain(b), compose(b, c)), a, compose(b, c))
% 0.21/0.46 = { by axiom 10 (associativity_of_compose) R->L }
% 0.21/0.46 fresh5(codomain(b), fresh7(there_exists(compose(codomain(b), compose(b, c))), true, codomain(b), compose(b, c)), a, compose(b, c))
% 0.21/0.46 = { by axiom 14 (domain_codomain_composition1) }
% 0.21/0.46 fresh5(codomain(b), domain(codomain(b)), a, compose(b, c))
% 0.21/0.46 = { by axiom 14 (domain_codomain_composition1) R->L }
% 0.21/0.46 fresh5(codomain(b), fresh7(there_exists(compose(codomain(b), codomain(b))), true, codomain(b), codomain(b)), a, compose(b, c))
% 0.21/0.46 = { by lemma 17 R->L }
% 0.21/0.46 fresh5(codomain(b), fresh7(there_exists(compose(codomain(codomain(b)), codomain(b))), true, codomain(b), codomain(b)), a, compose(b, c))
% 0.21/0.47 = { by axiom 4 (compose_codomain) }
% 0.21/0.47 fresh5(codomain(b), fresh7(there_exists(codomain(b)), true, codomain(b), codomain(b)), a, compose(b, c))
% 0.21/0.47 = { by lemma 16 }
% 0.21/0.47 fresh5(codomain(b), fresh7(true, true, codomain(b), codomain(b)), a, compose(b, c))
% 0.21/0.47 = { by axiom 7 (domain_codomain_composition1) }
% 0.21/0.47 fresh5(codomain(b), codomain(codomain(b)), a, compose(b, c))
% 0.21/0.47 = { by lemma 17 }
% 0.21/0.47 fresh5(codomain(b), codomain(b), a, compose(b, c))
% 0.21/0.47 = { by axiom 9 (domain_codomain_composition2) }
% 0.21/0.47 true
% 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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