TSTP Solution File: CAT006-3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : CAT006-3 : 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 : n023.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 18:18:51 EDT 2023
% Result : Unsatisfiable 0.14s 0.39s
% Output : Proof 0.14s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : CAT006-3 : TPTP v8.1.2. Released v1.0.0.
% 0.11/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.33 % Computer : n023.cluster.edu
% 0.14/0.33 % Model : x86_64 x86_64
% 0.14/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.33 % Memory : 8042.1875MB
% 0.14/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.33 % CPULimit : 300
% 0.14/0.33 % WCLimit : 300
% 0.14/0.33 % DateTime : Sun Aug 27 00:39:11 EDT 2023
% 0.14/0.34 % CPUTime :
% 0.14/0.39 Command-line arguments: --ground-connectedness --complete-subsets
% 0.14/0.39
% 0.14/0.39 % SZS status Unsatisfiable
% 0.14/0.39
% 0.14/0.39 % SZS output start Proof
% 0.14/0.39 Take the following subset of the input axioms:
% 0.14/0.39 fof(codomain_has_elements, axiom, ![X]: (~there_exists(codomain(X)) | there_exists(X))).
% 0.14/0.39 fof(compose_domain, axiom, ![X2]: compose(X2, domain(X2))=X2).
% 0.14/0.39 fof(composition_implies_codomain, axiom, ![Y, X2]: (~there_exists(compose(X2, Y)) | there_exists(codomain(X2)))).
% 0.14/0.39 fof(da_exists, hypothesis, there_exists(compose(d, a))).
% 0.14/0.39 fof(domain_codomain_composition1, axiom, ![X2, Y2]: (~there_exists(compose(X2, Y2)) | domain(X2)=codomain(Y2))).
% 0.14/0.39 fof(dx_equals_x, hypothesis, ![X2]: (~there_exists(compose(d, X2)) | compose(d, X2)=X2)).
% 0.14/0.39 fof(prove_codomain_of_a_is_d, negated_conjecture, codomain(a)!=d).
% 0.14/0.39
% 0.14/0.39 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.14/0.39 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.14/0.39 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.14/0.39 fresh(y, y, x1...xn) = u
% 0.14/0.39 C => fresh(s, t, x1...xn) = v
% 0.14/0.39 where fresh is a fresh function symbol and x1..xn are the free
% 0.14/0.39 variables of u and v.
% 0.14/0.39 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.14/0.39 input problem has no model of domain size 1).
% 0.14/0.39
% 0.14/0.39 The encoding turns the above axioms into the following unit equations and goals:
% 0.14/0.39
% 0.14/0.39 Axiom 1 (composition_implies_codomain): fresh16(X, X, Y) = true.
% 0.14/0.39 Axiom 2 (codomain_has_elements): fresh15(X, X, Y) = true.
% 0.14/0.39 Axiom 3 (dx_equals_x): fresh2(X, X, Y) = Y.
% 0.14/0.39 Axiom 4 (compose_domain): compose(X, domain(X)) = X.
% 0.14/0.39 Axiom 5 (da_exists): there_exists(compose(d, a)) = true.
% 0.14/0.40 Axiom 6 (domain_codomain_composition1): fresh13(X, X, Y, Z) = codomain(Z).
% 0.14/0.40 Axiom 7 (codomain_has_elements): fresh15(there_exists(codomain(X)), true, X) = there_exists(X).
% 0.14/0.40 Axiom 8 (composition_implies_codomain): fresh16(there_exists(compose(X, Y)), true, X) = there_exists(codomain(X)).
% 0.14/0.40 Axiom 9 (dx_equals_x): fresh2(there_exists(compose(d, X)), true, X) = compose(d, X).
% 0.14/0.40 Axiom 10 (domain_codomain_composition1): fresh13(there_exists(compose(X, Y)), true, X, Y) = domain(X).
% 0.14/0.40
% 0.14/0.40 Lemma 11: compose(d, codomain(a)) = d.
% 0.14/0.40 Proof:
% 0.14/0.40 compose(d, codomain(a))
% 0.14/0.40 = { by axiom 6 (domain_codomain_composition1) R->L }
% 0.14/0.40 compose(d, fresh13(true, true, d, a))
% 0.14/0.40 = { by axiom 5 (da_exists) R->L }
% 0.14/0.40 compose(d, fresh13(there_exists(compose(d, a)), true, d, a))
% 0.14/0.40 = { by axiom 10 (domain_codomain_composition1) }
% 0.14/0.40 compose(d, domain(d))
% 0.14/0.40 = { by axiom 4 (compose_domain) }
% 0.14/0.40 d
% 0.14/0.40
% 0.14/0.40 Goal 1 (prove_codomain_of_a_is_d): codomain(a) = d.
% 0.14/0.40 Proof:
% 0.14/0.40 codomain(a)
% 0.14/0.40 = { by axiom 3 (dx_equals_x) R->L }
% 0.14/0.40 fresh2(true, true, codomain(a))
% 0.14/0.40 = { by axiom 2 (codomain_has_elements) R->L }
% 0.14/0.40 fresh2(fresh15(true, true, d), true, codomain(a))
% 0.14/0.40 = { by axiom 1 (composition_implies_codomain) R->L }
% 0.14/0.40 fresh2(fresh15(fresh16(true, true, d), true, d), true, codomain(a))
% 0.14/0.40 = { by axiom 5 (da_exists) R->L }
% 0.14/0.40 fresh2(fresh15(fresh16(there_exists(compose(d, a)), true, d), true, d), true, codomain(a))
% 0.14/0.40 = { by axiom 8 (composition_implies_codomain) }
% 0.14/0.40 fresh2(fresh15(there_exists(codomain(d)), true, d), true, codomain(a))
% 0.14/0.40 = { by axiom 7 (codomain_has_elements) }
% 0.14/0.40 fresh2(there_exists(d), true, codomain(a))
% 0.14/0.40 = { by lemma 11 R->L }
% 0.14/0.40 fresh2(there_exists(compose(d, codomain(a))), true, codomain(a))
% 0.14/0.40 = { by axiom 9 (dx_equals_x) }
% 0.14/0.40 compose(d, codomain(a))
% 0.14/0.40 = { by lemma 11 }
% 0.14/0.40 d
% 0.14/0.40 % SZS output end Proof
% 0.14/0.40
% 0.14/0.40 RESULT: Unsatisfiable (the axioms are contradictory).
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