TSTP Solution File: CAT010-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : CAT010-1 : 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 : n012.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 0.19s 0.51s
% Output : Proof 0.19s
% 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 : CAT010-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n012.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Sun Aug 27 00:07:40 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.51 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.19/0.51
% 0.19/0.51 % SZS status Unsatisfiable
% 0.19/0.51
% 0.19/0.52 % SZS output start Proof
% 0.19/0.52 Take the following subset of the input axioms:
% 0.19/0.52 fof(ba_defined, hypothesis, defined(b, a)).
% 0.19/0.52 fof(category_theory_axiom3, axiom, ![X, Y, Z, Yz]: (~product(Y, Z, Yz) | (~defined(X, Yz) | defined(X, Y)))).
% 0.19/0.52 fof(closure_of_composition, axiom, ![X2, Y2]: (~defined(X2, Y2) | product(X2, Y2, compose(X2, Y2)))).
% 0.19/0.52 fof(codomain_is_an_identity_map, axiom, ![X2]: identity_map(codomain(X2))).
% 0.19/0.52 fof(composition_is_well_defined, axiom, ![W, X2, Y2, Z2]: (~product(X2, Y2, Z2) | (~product(X2, Y2, W) | Z2=W))).
% 0.19/0.52 fof(identity1, axiom, ![X2, Y2]: (~defined(X2, Y2) | (~identity_map(X2) | product(X2, Y2, Y2)))).
% 0.19/0.52 fof(identity2, axiom, ![X2, Y2]: (~defined(X2, Y2) | (~identity_map(Y2) | product(X2, Y2, X2)))).
% 0.19/0.52 fof(mapping_from_codomain_of_x_to_x, axiom, ![X2]: defined(codomain(X2), X2)).
% 0.19/0.52 fof(product_on_codomain, axiom, ![X2]: product(codomain(X2), X2, X2)).
% 0.19/0.52 fof(prove_codomain_of_ba_equals_codomain_of_b, negated_conjecture, codomain(compose(b, a))!=codomain(b)).
% 0.19/0.52
% 0.19/0.52 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.52 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.52 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.52 fresh(y, y, x1...xn) = u
% 0.19/0.52 C => fresh(s, t, x1...xn) = v
% 0.19/0.52 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.52 variables of u and v.
% 0.19/0.52 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.52 input problem has no model of domain size 1).
% 0.19/0.52
% 0.19/0.52 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.52
% 0.19/0.52 Axiom 1 (ba_defined): defined(b, a) = true.
% 0.19/0.52 Axiom 2 (codomain_is_an_identity_map): identity_map(codomain(X)) = true.
% 0.19/0.52 Axiom 3 (mapping_from_codomain_of_x_to_x): defined(codomain(X), X) = true.
% 0.19/0.52 Axiom 4 (product_on_codomain): product(codomain(X), X, X) = true.
% 0.19/0.52 Axiom 5 (composition_is_well_defined): fresh(X, X, Y, Z) = Z.
% 0.19/0.52 Axiom 6 (category_theory_axiom3): fresh11(X, X, Y, Z) = true.
% 0.19/0.52 Axiom 7 (closure_of_composition): fresh7(X, X, Y, Z) = true.
% 0.19/0.52 Axiom 8 (identity1): fresh6(X, X, Y, Z) = product(Y, Z, Z).
% 0.19/0.52 Axiom 9 (identity1): fresh5(X, X, Y, Z) = true.
% 0.19/0.52 Axiom 10 (identity2): fresh4(X, X, Y, Z) = product(Y, Z, Y).
% 0.19/0.52 Axiom 11 (identity2): fresh3(X, X, Y, Z) = true.
% 0.19/0.52 Axiom 12 (category_theory_axiom3): fresh12(X, X, Y, Z, W) = defined(W, Y).
% 0.19/0.52 Axiom 13 (closure_of_composition): fresh7(defined(X, Y), true, X, Y) = product(X, Y, compose(X, Y)).
% 0.19/0.52 Axiom 14 (identity1): fresh6(identity_map(X), true, X, Y) = fresh5(defined(X, Y), true, X, Y).
% 0.19/0.52 Axiom 15 (identity2): fresh4(identity_map(X), true, Y, X) = fresh3(defined(Y, X), true, Y, X).
% 0.19/0.52 Axiom 16 (composition_is_well_defined): fresh2(X, X, Y, Z, W, V) = W.
% 0.19/0.52 Axiom 17 (category_theory_axiom3): fresh12(product(X, Y, Z), true, X, Z, W) = fresh11(defined(W, Z), true, X, W).
% 0.19/0.52 Axiom 18 (composition_is_well_defined): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 0.19/0.52
% 0.19/0.52 Lemma 19: defined(codomain(compose(b, a)), codomain(b)) = true.
% 0.19/0.52 Proof:
% 0.19/0.52 defined(codomain(compose(b, a)), codomain(b))
% 0.19/0.52 = { by axiom 12 (category_theory_axiom3) R->L }
% 0.19/0.52 fresh12(true, true, codomain(b), b, codomain(compose(b, a)))
% 0.19/0.52 = { by axiom 4 (product_on_codomain) R->L }
% 0.19/0.52 fresh12(product(codomain(b), b, b), true, codomain(b), b, codomain(compose(b, a)))
% 0.19/0.52 = { by axiom 17 (category_theory_axiom3) }
% 0.19/0.52 fresh11(defined(codomain(compose(b, a)), b), true, codomain(b), codomain(compose(b, a)))
% 0.19/0.52 = { by axiom 12 (category_theory_axiom3) R->L }
% 0.19/0.52 fresh11(fresh12(true, true, b, compose(b, a), codomain(compose(b, a))), true, codomain(b), codomain(compose(b, a)))
% 0.19/0.52 = { by axiom 7 (closure_of_composition) R->L }
% 0.19/0.52 fresh11(fresh12(fresh7(true, true, b, a), true, b, compose(b, a), codomain(compose(b, a))), true, codomain(b), codomain(compose(b, a)))
% 0.19/0.52 = { by axiom 1 (ba_defined) R->L }
% 0.19/0.52 fresh11(fresh12(fresh7(defined(b, a), true, b, a), true, b, compose(b, a), codomain(compose(b, a))), true, codomain(b), codomain(compose(b, a)))
% 0.19/0.52 = { by axiom 13 (closure_of_composition) }
% 0.19/0.52 fresh11(fresh12(product(b, a, compose(b, a)), true, b, compose(b, a), codomain(compose(b, a))), true, codomain(b), codomain(compose(b, a)))
% 0.19/0.52 = { by axiom 17 (category_theory_axiom3) }
% 0.19/0.52 fresh11(fresh11(defined(codomain(compose(b, a)), compose(b, a)), true, b, codomain(compose(b, a))), true, codomain(b), codomain(compose(b, a)))
% 0.19/0.52 = { by axiom 3 (mapping_from_codomain_of_x_to_x) }
% 0.19/0.52 fresh11(fresh11(true, true, b, codomain(compose(b, a))), true, codomain(b), codomain(compose(b, a)))
% 0.19/0.52 = { by axiom 6 (category_theory_axiom3) }
% 0.19/0.52 fresh11(true, true, codomain(b), codomain(compose(b, a)))
% 0.19/0.52 = { by axiom 6 (category_theory_axiom3) }
% 0.19/0.52 true
% 0.19/0.52
% 0.19/0.52 Goal 1 (prove_codomain_of_ba_equals_codomain_of_b): codomain(compose(b, a)) = codomain(b).
% 0.19/0.52 Proof:
% 0.19/0.52 codomain(compose(b, a))
% 0.19/0.52 = { by axiom 16 (composition_is_well_defined) R->L }
% 0.19/0.52 fresh2(true, true, codomain(compose(b, a)), codomain(b), codomain(compose(b, a)), codomain(b))
% 0.19/0.52 = { by axiom 9 (identity1) R->L }
% 0.19/0.52 fresh2(fresh5(true, true, codomain(compose(b, a)), codomain(b)), true, codomain(compose(b, a)), codomain(b), codomain(compose(b, a)), codomain(b))
% 0.19/0.52 = { by lemma 19 R->L }
% 0.19/0.52 fresh2(fresh5(defined(codomain(compose(b, a)), codomain(b)), true, codomain(compose(b, a)), codomain(b)), true, codomain(compose(b, a)), codomain(b), codomain(compose(b, a)), codomain(b))
% 0.19/0.52 = { by axiom 14 (identity1) R->L }
% 0.19/0.52 fresh2(fresh6(identity_map(codomain(compose(b, a))), true, codomain(compose(b, a)), codomain(b)), true, codomain(compose(b, a)), codomain(b), codomain(compose(b, a)), codomain(b))
% 0.19/0.52 = { by axiom 2 (codomain_is_an_identity_map) }
% 0.19/0.52 fresh2(fresh6(true, true, codomain(compose(b, a)), codomain(b)), true, codomain(compose(b, a)), codomain(b), codomain(compose(b, a)), codomain(b))
% 0.19/0.52 = { by axiom 8 (identity1) }
% 0.19/0.52 fresh2(product(codomain(compose(b, a)), codomain(b), codomain(b)), true, codomain(compose(b, a)), codomain(b), codomain(compose(b, a)), codomain(b))
% 0.19/0.52 = { by axiom 18 (composition_is_well_defined) }
% 0.19/0.52 fresh(product(codomain(compose(b, a)), codomain(b), codomain(compose(b, a))), true, codomain(compose(b, a)), codomain(b))
% 0.19/0.52 = { by axiom 10 (identity2) R->L }
% 0.19/0.52 fresh(fresh4(true, true, codomain(compose(b, a)), codomain(b)), true, codomain(compose(b, a)), codomain(b))
% 0.19/0.52 = { by axiom 2 (codomain_is_an_identity_map) R->L }
% 0.19/0.52 fresh(fresh4(identity_map(codomain(b)), true, codomain(compose(b, a)), codomain(b)), true, codomain(compose(b, a)), codomain(b))
% 0.19/0.52 = { by axiom 15 (identity2) }
% 0.19/0.52 fresh(fresh3(defined(codomain(compose(b, a)), codomain(b)), true, codomain(compose(b, a)), codomain(b)), true, codomain(compose(b, a)), codomain(b))
% 0.19/0.52 = { by lemma 19 }
% 0.19/0.52 fresh(fresh3(true, true, codomain(compose(b, a)), codomain(b)), true, codomain(compose(b, a)), codomain(b))
% 0.19/0.52 = { by axiom 11 (identity2) }
% 0.19/0.52 fresh(true, true, codomain(compose(b, a)), codomain(b))
% 0.19/0.52 = { by axiom 5 (composition_is_well_defined) }
% 0.19/0.52 codomain(b)
% 0.19/0.52 % SZS output end Proof
% 0.19/0.52
% 0.19/0.52 RESULT: Unsatisfiable (the axioms are contradictory).
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