TSTP Solution File: CAT018-1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : CAT018-1 : 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 : n009.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 18:18:58 EDT 2023
% Result : Unsatisfiable 0.20s 0.50s
% Output : Proof 0.20s
% 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 : CAT018-1 : TPTP v8.1.2. Released v1.0.0.
% 0.12/0.13 % 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 : n009.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 00:08:05 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.50 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.20/0.50
% 0.20/0.50 % SZS status Unsatisfiable
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% 0.20/0.51 % SZS output start Proof
% 0.20/0.51 Take the following subset of the input axioms:
% 0.20/0.51 fof(assume_ab_exists, hypothesis, defined(a, b)).
% 0.20/0.51 fof(assume_bc_exists, hypothesis, defined(b, c)).
% 0.20/0.51 fof(category_theory_axiom1, axiom, ![X, Y, Z, Xy, Yz]: (~product(X, Y, Xy) | (~product(Y, Z, Yz) | (~defined(Xy, Z) | defined(X, Yz))))).
% 0.20/0.51 fof(category_theory_axiom3, axiom, ![X2, Y2, Z2, Yz2]: (~product(Y2, Z2, Yz2) | (~defined(X2, Yz2) | defined(X2, Y2)))).
% 0.20/0.51 fof(category_theory_axiom6, axiom, ![X2, Y2, Z2]: (~defined(X2, Y2) | (~defined(Y2, Z2) | (~identity_map(Y2) | defined(X2, Z2))))).
% 0.20/0.51 fof(closure_of_composition, axiom, ![X2, Y2]: (~defined(X2, Y2) | product(X2, Y2, compose(X2, Y2)))).
% 0.20/0.51 fof(codomain_is_an_identity_map, axiom, ![X2]: identity_map(codomain(X2))).
% 0.20/0.51 fof(product_on_codomain, axiom, ![X2]: product(codomain(X2), X2, X2)).
% 0.20/0.51 fof(prove_a_bc_exists, negated_conjecture, ~defined(a, compose(b, c))).
% 0.20/0.51
% 0.20/0.51 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.51 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.51 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.51 fresh(y, y, x1...xn) = u
% 0.20/0.51 C => fresh(s, t, x1...xn) = v
% 0.20/0.51 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.51 variables of u and v.
% 0.20/0.51 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.51 input problem has no model of domain size 1).
% 0.20/0.51
% 0.20/0.51 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.51
% 0.20/0.51 Axiom 1 (assume_bc_exists): defined(b, c) = true.
% 0.20/0.51 Axiom 2 (assume_ab_exists): defined(a, b) = true.
% 0.20/0.51 Axiom 3 (codomain_is_an_identity_map): identity_map(codomain(X)) = true.
% 0.20/0.51 Axiom 4 (product_on_codomain): product(codomain(X), X, X) = true.
% 0.20/0.51 Axiom 5 (category_theory_axiom1): fresh27(X, X, Y, Z) = true.
% 0.20/0.51 Axiom 6 (category_theory_axiom6): fresh19(X, X, Y, Z) = true.
% 0.20/0.51 Axiom 7 (category_theory_axiom3): fresh11(X, X, Y, Z) = true.
% 0.20/0.51 Axiom 8 (closure_of_composition): fresh7(X, X, Y, Z) = true.
% 0.20/0.51 Axiom 9 (category_theory_axiom3): fresh12(X, X, Y, Z, W) = defined(W, Y).
% 0.20/0.51 Axiom 10 (category_theory_axiom6): fresh8(X, X, Y, Z, W) = defined(Y, W).
% 0.20/0.51 Axiom 11 (category_theory_axiom6): fresh18(X, X, Y, Z, W) = fresh19(defined(Y, Z), true, Y, W).
% 0.20/0.51 Axiom 12 (category_theory_axiom1): fresh14(X, X, Y, Z, W, V) = defined(Y, V).
% 0.20/0.51 Axiom 13 (closure_of_composition): fresh7(defined(X, Y), true, X, Y) = product(X, Y, compose(X, Y)).
% 0.20/0.51 Axiom 14 (category_theory_axiom1): fresh26(X, X, Y, Z, W, V, U) = fresh27(defined(W, V), true, Y, U).
% 0.20/0.51 Axiom 15 (category_theory_axiom6): fresh18(identity_map(X), true, Y, X, Z) = fresh8(defined(X, Z), true, Y, X, Z).
% 0.20/0.51 Axiom 16 (category_theory_axiom3): fresh12(product(X, Y, Z), true, X, Z, W) = fresh11(defined(W, Z), true, X, W).
% 0.20/0.51 Axiom 17 (category_theory_axiom1): fresh26(product(X, Y, Z), true, W, X, V, Y, Z) = fresh14(product(W, X, V), true, W, V, Y, Z).
% 0.20/0.51
% 0.20/0.51 Goal 1 (prove_a_bc_exists): defined(a, compose(b, c)) = true.
% 0.20/0.51 Proof:
% 0.20/0.51 defined(a, compose(b, c))
% 0.20/0.51 = { by axiom 10 (category_theory_axiom6) R->L }
% 0.20/0.51 fresh8(true, true, a, codomain(b), compose(b, c))
% 0.20/0.51 = { by axiom 5 (category_theory_axiom1) R->L }
% 0.20/0.51 fresh8(fresh27(true, true, codomain(b), compose(b, c)), true, a, codomain(b), compose(b, c))
% 0.20/0.51 = { by axiom 1 (assume_bc_exists) R->L }
% 0.20/0.51 fresh8(fresh27(defined(b, c), true, codomain(b), compose(b, c)), true, a, codomain(b), compose(b, c))
% 0.20/0.51 = { by axiom 14 (category_theory_axiom1) R->L }
% 0.20/0.51 fresh8(fresh26(true, true, codomain(b), b, b, c, compose(b, c)), true, a, codomain(b), compose(b, c))
% 0.20/0.51 = { by axiom 8 (closure_of_composition) R->L }
% 0.20/0.51 fresh8(fresh26(fresh7(true, true, b, c), true, codomain(b), b, b, c, compose(b, c)), true, a, codomain(b), compose(b, c))
% 0.20/0.51 = { by axiom 1 (assume_bc_exists) R->L }
% 0.20/0.51 fresh8(fresh26(fresh7(defined(b, c), true, b, c), true, codomain(b), b, b, c, compose(b, c)), true, a, codomain(b), compose(b, c))
% 0.20/0.51 = { by axiom 13 (closure_of_composition) }
% 0.20/0.51 fresh8(fresh26(product(b, c, compose(b, c)), true, codomain(b), b, b, c, compose(b, c)), true, a, codomain(b), compose(b, c))
% 0.20/0.51 = { by axiom 17 (category_theory_axiom1) }
% 0.20/0.51 fresh8(fresh14(product(codomain(b), b, b), true, codomain(b), b, c, compose(b, c)), true, a, codomain(b), compose(b, c))
% 0.20/0.51 = { by axiom 4 (product_on_codomain) }
% 0.20/0.51 fresh8(fresh14(true, true, codomain(b), b, c, compose(b, c)), true, a, codomain(b), compose(b, c))
% 0.20/0.51 = { by axiom 12 (category_theory_axiom1) }
% 0.20/0.51 fresh8(defined(codomain(b), compose(b, c)), true, a, codomain(b), compose(b, c))
% 0.20/0.51 = { by axiom 15 (category_theory_axiom6) R->L }
% 0.20/0.51 fresh18(identity_map(codomain(b)), true, a, codomain(b), compose(b, c))
% 0.20/0.51 = { by axiom 3 (codomain_is_an_identity_map) }
% 0.20/0.51 fresh18(true, true, a, codomain(b), compose(b, c))
% 0.20/0.51 = { by axiom 11 (category_theory_axiom6) }
% 0.20/0.51 fresh19(defined(a, codomain(b)), true, a, compose(b, c))
% 0.20/0.51 = { by axiom 9 (category_theory_axiom3) R->L }
% 0.20/0.51 fresh19(fresh12(true, true, codomain(b), b, a), true, a, compose(b, c))
% 0.20/0.51 = { by axiom 4 (product_on_codomain) R->L }
% 0.20/0.51 fresh19(fresh12(product(codomain(b), b, b), true, codomain(b), b, a), true, a, compose(b, c))
% 0.20/0.51 = { by axiom 16 (category_theory_axiom3) }
% 0.20/0.51 fresh19(fresh11(defined(a, b), true, codomain(b), a), true, a, compose(b, c))
% 0.20/0.51 = { by axiom 2 (assume_ab_exists) }
% 0.20/0.51 fresh19(fresh11(true, true, codomain(b), a), true, a, compose(b, c))
% 0.20/0.51 = { by axiom 7 (category_theory_axiom3) }
% 0.20/0.51 fresh19(true, true, a, compose(b, c))
% 0.20/0.51 = { by axiom 6 (category_theory_axiom6) }
% 0.20/0.51 true
% 0.20/0.51 % SZS output end Proof
% 0.20/0.51
% 0.20/0.51 RESULT: Unsatisfiable (the axioms are contradictory).
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