TSTP Solution File: CAT006-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : CAT006-1 : 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 : n013.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 0.21s 0.50s
% 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
% 0.00/0.12 % Problem : CAT006-1 : 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
% 0.18/0.35 % Computer : n013.cluster.edu
% 0.18/0.35 % Model : x86_64 x86_64
% 0.18/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.35 % Memory : 8042.1875MB
% 0.18/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.35 % CPULimit : 300
% 0.18/0.35 % WCLimit : 300
% 0.18/0.35 % DateTime : Sun Aug 27 00:20:17 EDT 2023
% 0.18/0.35 % CPUTime :
% 0.21/0.50 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.21/0.50
% 0.21/0.50 % SZS status Unsatisfiable
% 0.21/0.50
% 0.21/0.51 % SZS output start Proof
% 0.21/0.51 Take the following subset of the input axioms:
% 0.21/0.51 fof(category_theory_axiom3, axiom, ![X, Y, Z, Yz]: (~product(Y, Z, Yz) | (~defined(X, Yz) | defined(X, Y)))).
% 0.21/0.51 fof(codomain_is_an_identity_map, axiom, ![X2]: identity_map(codomain(X2))).
% 0.21/0.51 fof(composition_is_well_defined, axiom, ![W, X2, Y2, Z2]: (~product(X2, Y2, Z2) | (~product(X2, Y2, W) | Z2=W))).
% 0.21/0.51 fof(h_is_the_identity_map, hypothesis, identity_map(h)).
% 0.21/0.51 fof(ha_defined, hypothesis, defined(h, a)).
% 0.21/0.51 fof(identity1, axiom, ![X2, Y2]: (~defined(X2, Y2) | (~identity_map(X2) | product(X2, Y2, Y2)))).
% 0.21/0.51 fof(identity2, axiom, ![X2, Y2]: (~defined(X2, Y2) | (~identity_map(Y2) | product(X2, Y2, X2)))).
% 0.21/0.51 fof(mapping_from_codomain_of_x_to_x, axiom, ![X2]: defined(codomain(X2), X2)).
% 0.21/0.51 fof(prove_codomain_of_a_is_h, negated_conjecture, codomain(a)!=h).
% 0.21/0.51
% 0.21/0.51 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.51 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.51 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.51 fresh(y, y, x1...xn) = u
% 0.21/0.51 C => fresh(s, t, x1...xn) = v
% 0.21/0.51 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.51 variables of u and v.
% 0.21/0.51 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.51 input problem has no model of domain size 1).
% 0.21/0.51
% 0.21/0.51 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.51
% 0.21/0.51 Axiom 1 (h_is_the_identity_map): identity_map(h) = true.
% 0.21/0.51 Axiom 2 (codomain_is_an_identity_map): identity_map(codomain(X)) = true.
% 0.21/0.51 Axiom 3 (ha_defined): defined(h, a) = true.
% 0.21/0.51 Axiom 4 (mapping_from_codomain_of_x_to_x): defined(codomain(X), X) = true.
% 0.21/0.51 Axiom 5 (composition_is_well_defined): fresh(X, X, Y, Z) = Z.
% 0.21/0.51 Axiom 6 (category_theory_axiom3): fresh11(X, X, Y, Z) = true.
% 0.21/0.51 Axiom 7 (identity1): fresh6(X, X, Y, Z) = product(Y, Z, Z).
% 0.21/0.51 Axiom 8 (identity1): fresh5(X, X, Y, Z) = true.
% 0.21/0.51 Axiom 9 (identity2): fresh4(X, X, Y, Z) = product(Y, Z, Y).
% 0.21/0.51 Axiom 10 (identity2): fresh3(X, X, Y, Z) = true.
% 0.21/0.51 Axiom 11 (category_theory_axiom3): fresh12(X, X, Y, Z, W) = defined(W, Y).
% 0.21/0.51 Axiom 12 (identity1): fresh6(identity_map(X), true, X, Y) = fresh5(defined(X, Y), true, X, Y).
% 0.21/0.51 Axiom 13 (identity2): fresh4(identity_map(X), true, Y, X) = fresh3(defined(Y, X), true, Y, X).
% 0.21/0.51 Axiom 14 (composition_is_well_defined): fresh2(X, X, Y, Z, W, V) = W.
% 0.21/0.51 Axiom 15 (category_theory_axiom3): fresh12(product(X, Y, Z), true, X, Z, W) = fresh11(defined(W, Z), true, X, W).
% 0.21/0.51 Axiom 16 (composition_is_well_defined): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 0.21/0.51
% 0.21/0.51 Lemma 17: defined(codomain(a), h) = true.
% 0.21/0.51 Proof:
% 0.21/0.51 defined(codomain(a), h)
% 0.21/0.51 = { by axiom 11 (category_theory_axiom3) R->L }
% 0.21/0.51 fresh12(true, true, h, a, codomain(a))
% 0.21/0.51 = { by axiom 8 (identity1) R->L }
% 0.21/0.51 fresh12(fresh5(true, true, h, a), true, h, a, codomain(a))
% 0.21/0.51 = { by axiom 3 (ha_defined) R->L }
% 0.21/0.51 fresh12(fresh5(defined(h, a), true, h, a), true, h, a, codomain(a))
% 0.21/0.51 = { by axiom 12 (identity1) R->L }
% 0.21/0.51 fresh12(fresh6(identity_map(h), true, h, a), true, h, a, codomain(a))
% 0.21/0.51 = { by axiom 1 (h_is_the_identity_map) }
% 0.21/0.51 fresh12(fresh6(true, true, h, a), true, h, a, codomain(a))
% 0.21/0.51 = { by axiom 7 (identity1) }
% 0.21/0.51 fresh12(product(h, a, a), true, h, a, codomain(a))
% 0.21/0.51 = { by axiom 15 (category_theory_axiom3) }
% 0.21/0.51 fresh11(defined(codomain(a), a), true, h, codomain(a))
% 0.21/0.51 = { by axiom 4 (mapping_from_codomain_of_x_to_x) }
% 0.21/0.51 fresh11(true, true, h, codomain(a))
% 0.21/0.51 = { by axiom 6 (category_theory_axiom3) }
% 0.21/0.51 true
% 0.21/0.51
% 0.21/0.51 Goal 1 (prove_codomain_of_a_is_h): codomain(a) = h.
% 0.21/0.51 Proof:
% 0.21/0.51 codomain(a)
% 0.21/0.51 = { by axiom 14 (composition_is_well_defined) R->L }
% 0.21/0.51 fresh2(true, true, codomain(a), h, codomain(a), h)
% 0.21/0.51 = { by axiom 8 (identity1) R->L }
% 0.21/0.51 fresh2(fresh5(true, true, codomain(a), h), true, codomain(a), h, codomain(a), h)
% 0.21/0.51 = { by lemma 17 R->L }
% 0.21/0.51 fresh2(fresh5(defined(codomain(a), h), true, codomain(a), h), true, codomain(a), h, codomain(a), h)
% 0.21/0.51 = { by axiom 12 (identity1) R->L }
% 0.21/0.51 fresh2(fresh6(identity_map(codomain(a)), true, codomain(a), h), true, codomain(a), h, codomain(a), h)
% 0.21/0.51 = { by axiom 2 (codomain_is_an_identity_map) }
% 0.21/0.51 fresh2(fresh6(true, true, codomain(a), h), true, codomain(a), h, codomain(a), h)
% 0.21/0.51 = { by axiom 7 (identity1) }
% 0.21/0.51 fresh2(product(codomain(a), h, h), true, codomain(a), h, codomain(a), h)
% 0.21/0.51 = { by axiom 16 (composition_is_well_defined) }
% 0.21/0.51 fresh(product(codomain(a), h, codomain(a)), true, codomain(a), h)
% 0.21/0.51 = { by axiom 9 (identity2) R->L }
% 0.21/0.51 fresh(fresh4(true, true, codomain(a), h), true, codomain(a), h)
% 0.21/0.51 = { by axiom 1 (h_is_the_identity_map) R->L }
% 0.21/0.51 fresh(fresh4(identity_map(h), true, codomain(a), h), true, codomain(a), h)
% 0.21/0.51 = { by axiom 13 (identity2) }
% 0.21/0.51 fresh(fresh3(defined(codomain(a), h), true, codomain(a), h), true, codomain(a), h)
% 0.21/0.51 = { by lemma 17 }
% 0.21/0.51 fresh(fresh3(true, true, codomain(a), h), true, codomain(a), h)
% 0.21/0.51 = { by axiom 10 (identity2) }
% 0.21/0.51 fresh(true, true, codomain(a), h)
% 0.21/0.51 = { by axiom 5 (composition_is_well_defined) }
% 0.21/0.51 h
% 0.21/0.51 % SZS output end Proof
% 0.21/0.51
% 0.21/0.51 RESULT: Unsatisfiable (the axioms are contradictory).
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