TSTP Solution File: CAT003-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : CAT003-1 : 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 : n011.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 18:18:48 EDT 2023
% Result : Unsatisfiable 14.15s 2.27s
% Output : Proof 14.99s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.14 % Problem : CAT003-1 : TPTP v8.1.2. Released v1.0.0.
% 0.07/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.36 % Computer : n011.cluster.edu
% 0.13/0.36 % Model : x86_64 x86_64
% 0.13/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.36 % Memory : 8042.1875MB
% 0.13/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.36 % CPULimit : 300
% 0.13/0.36 % WCLimit : 300
% 0.13/0.36 % DateTime : Sun Aug 27 00:28:09 EDT 2023
% 0.13/0.36 % CPUTime :
% 14.15/2.27 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 14.15/2.27
% 14.15/2.27 % SZS status Unsatisfiable
% 14.15/2.27
% 14.99/2.29 % SZS output start Proof
% 14.99/2.29 Take the following subset of the input axioms:
% 14.99/2.29 fof(ab_equals_c, hypothesis, product(a, b, c)).
% 14.99/2.29 fof(associative_property1, axiom, ![X, Y, Z]: (~product(X, Y, Z) | defined(X, Y))).
% 14.99/2.29 fof(cancellation_for_product, hypothesis, ![W, X2, Y2]: (~product(X2, c, W) | (~product(Y2, c, W) | X2=Y2))).
% 14.99/2.29 fof(category_theory_axiom1, axiom, ![Xy, Yz, X2, Y2, Z2]: (~product(X2, Y2, Xy) | (~product(Y2, Z2, Yz) | (~defined(Xy, Z2) | defined(X2, Yz))))).
% 14.99/2.29 fof(category_theory_axiom2, axiom, ![Xyz, X2, Y2, Z2, Xy2, Yz2]: (~product(X2, Y2, Xy2) | (~product(Xy2, Z2, Xyz) | (~product(Y2, Z2, Yz2) | product(X2, Yz2, Xyz))))).
% 14.99/2.29 fof(category_theory_axiom3, axiom, ![X2, Y2, Z2, Yz2]: (~product(Y2, Z2, Yz2) | (~defined(X2, Yz2) | defined(X2, Y2)))).
% 14.99/2.29 fof(category_theory_axiom4, axiom, ![X2, Y2, Z2, Xy2, Yz2]: (~product(Y2, Z2, Yz2) | (~product(X2, Y2, Xy2) | (~defined(X2, Yz2) | defined(Xy2, Z2))))).
% 14.99/2.29 fof(category_theory_axiom6, axiom, ![X2, Y2, Z2]: (~defined(X2, Y2) | (~defined(Y2, Z2) | (~identity_map(Y2) | defined(X2, Z2))))).
% 14.99/2.29 fof(closure_of_composition, axiom, ![X2, Y2]: (~defined(X2, Y2) | product(X2, Y2, compose(X2, Y2)))).
% 14.99/2.29 fof(codomain_is_an_identity_map, axiom, ![X2]: identity_map(codomain(X2))).
% 14.99/2.29 fof(composition_is_well_defined, axiom, ![X2, Y2, Z2, W2]: (~product(X2, Y2, Z2) | (~product(X2, Y2, W2) | Z2=W2))).
% 14.99/2.29 fof(ga_equals_d, hypothesis, product(g, a, d)).
% 14.99/2.29 fof(ha_equals_d, hypothesis, product(h, a, d)).
% 14.99/2.29 fof(product_on_codomain, axiom, ![X2]: product(codomain(X2), X2, X2)).
% 14.99/2.29 fof(prove_h_equals_g, negated_conjecture, h!=g).
% 14.99/2.29
% 14.99/2.29 Now clausify the problem and encode Horn clauses using encoding 3 of
% 14.99/2.29 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 14.99/2.29 We repeatedly replace C & s=t => u=v by the two clauses:
% 14.99/2.29 fresh(y, y, x1...xn) = u
% 14.99/2.29 C => fresh(s, t, x1...xn) = v
% 14.99/2.30 where fresh is a fresh function symbol and x1..xn are the free
% 14.99/2.30 variables of u and v.
% 14.99/2.30 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 14.99/2.30 input problem has no model of domain size 1).
% 14.99/2.30
% 14.99/2.30 The encoding turns the above axioms into the following unit equations and goals:
% 14.99/2.30
% 14.99/2.30 Axiom 1 (codomain_is_an_identity_map): identity_map(codomain(X)) = true.
% 14.99/2.30 Axiom 2 (ab_equals_c): product(a, b, c) = true.
% 14.99/2.30 Axiom 3 (ha_equals_d): product(h, a, d) = true.
% 14.99/2.30 Axiom 4 (ga_equals_d): product(g, a, d) = true.
% 14.99/2.30 Axiom 5 (product_on_codomain): product(codomain(X), X, X) = true.
% 14.99/2.30 Axiom 6 (cancellation_for_product): fresh(X, X, Y, Z) = Z.
% 14.99/2.30 Axiom 7 (category_theory_axiom1): fresh29(X, X, Y, Z) = true.
% 14.99/2.30 Axiom 8 (category_theory_axiom4): fresh25(X, X, Y, Z) = true.
% 14.99/2.30 Axiom 9 (category_theory_axiom6): fresh21(X, X, Y, Z) = true.
% 14.99/2.30 Axiom 10 (associative_property1): fresh18(X, X, Y, Z) = true.
% 14.99/2.30 Axiom 11 (category_theory_axiom3): fresh13(X, X, Y, Z) = true.
% 14.99/2.30 Axiom 12 (closure_of_composition): fresh9(X, X, Y, Z) = true.
% 14.99/2.30 Axiom 13 (composition_is_well_defined): fresh3(X, X, Y, Z) = Z.
% 14.99/2.30 Axiom 14 (category_theory_axiom2): fresh27(X, X, Y, Z, W) = true.
% 14.99/2.30 Axiom 15 (category_theory_axiom3): fresh14(X, X, Y, Z, W) = defined(W, Y).
% 14.99/2.30 Axiom 16 (category_theory_axiom6): fresh10(X, X, Y, Z, W) = defined(Y, W).
% 14.99/2.30 Axiom 17 (cancellation_for_product): fresh2(X, X, Y, Z, W) = Y.
% 14.99/2.30 Axiom 18 (category_theory_axiom6): fresh20(X, X, Y, Z, W) = fresh21(defined(Y, Z), true, Y, W).
% 14.99/2.30 Axiom 19 (category_theory_axiom1): fresh16(X, X, Y, Z, W, V) = defined(Y, V).
% 14.99/2.30 Axiom 20 (category_theory_axiom4): fresh12(X, X, Y, Z, W, V) = defined(V, Y).
% 14.99/2.30 Axiom 21 (closure_of_composition): fresh9(defined(X, Y), true, X, Y) = product(X, Y, compose(X, Y)).
% 14.99/2.30 Axiom 22 (composition_is_well_defined): fresh4(X, X, Y, Z, W, V) = W.
% 14.99/2.30 Axiom 23 (category_theory_axiom1): fresh28(X, X, Y, Z, W, V, U) = fresh29(defined(W, V), true, Y, U).
% 14.99/2.30 Axiom 24 (category_theory_axiom4): fresh24(X, X, Y, Z, W, V, U) = fresh25(defined(V, W), true, Z, U).
% 14.99/2.30 Axiom 25 (associative_property1): fresh18(product(X, Y, Z), true, X, Y) = defined(X, Y).
% 14.99/2.30 Axiom 26 (category_theory_axiom2): fresh15(X, X, Y, Z, W, V, U) = product(Y, U, V).
% 14.99/2.30 Axiom 27 (category_theory_axiom6): fresh20(identity_map(X), true, Y, X, Z) = fresh10(defined(X, Z), true, Y, X, Z).
% 14.99/2.30 Axiom 28 (category_theory_axiom2): fresh26(X, X, Y, Z, W, V, U, T) = fresh27(product(Y, Z, W), true, Y, U, T).
% 14.99/2.30 Axiom 29 (category_theory_axiom3): fresh14(product(X, Y, Z), true, X, Z, W) = fresh13(defined(W, Z), true, X, W).
% 14.99/2.30 Axiom 30 (cancellation_for_product): fresh2(product(X, c, Y), true, Z, Y, X) = fresh(product(Z, c, Y), true, Z, X).
% 14.99/2.30 Axiom 31 (composition_is_well_defined): fresh4(product(X, Y, Z), true, X, Y, W, Z) = fresh3(product(X, Y, W), true, W, Z).
% 14.99/2.30 Axiom 32 (category_theory_axiom1): fresh28(product(X, Y, Z), true, W, X, V, Y, Z) = fresh16(product(W, X, V), true, W, V, Y, Z).
% 14.99/2.30 Axiom 33 (category_theory_axiom4): fresh24(product(X, Y, Z), true, Y, W, V, X, Z) = fresh12(product(Y, W, V), true, W, V, X, Z).
% 14.99/2.30 Axiom 34 (category_theory_axiom2): fresh26(product(X, Y, Z), true, W, V, X, Y, Z, U) = fresh15(product(V, Y, U), true, W, V, X, Z, U).
% 14.99/2.30
% 14.99/2.30 Lemma 35: defined(h, c) = true.
% 14.99/2.30 Proof:
% 14.99/2.30 defined(h, c)
% 14.99/2.30 = { by axiom 16 (category_theory_axiom6) R->L }
% 14.99/2.30 fresh10(true, true, h, codomain(a), c)
% 14.99/2.30 = { by axiom 7 (category_theory_axiom1) R->L }
% 14.99/2.30 fresh10(fresh29(true, true, codomain(a), c), true, h, codomain(a), c)
% 14.99/2.30 = { by axiom 10 (associative_property1) R->L }
% 14.99/2.30 fresh10(fresh29(fresh18(true, true, a, b), true, codomain(a), c), true, h, codomain(a), c)
% 14.99/2.30 = { by axiom 2 (ab_equals_c) R->L }
% 14.99/2.30 fresh10(fresh29(fresh18(product(a, b, c), true, a, b), true, codomain(a), c), true, h, codomain(a), c)
% 14.99/2.30 = { by axiom 25 (associative_property1) }
% 14.99/2.30 fresh10(fresh29(defined(a, b), true, codomain(a), c), true, h, codomain(a), c)
% 14.99/2.30 = { by axiom 23 (category_theory_axiom1) R->L }
% 14.99/2.30 fresh10(fresh28(true, true, codomain(a), a, a, b, c), true, h, codomain(a), c)
% 14.99/2.30 = { by axiom 2 (ab_equals_c) R->L }
% 14.99/2.30 fresh10(fresh28(product(a, b, c), true, codomain(a), a, a, b, c), true, h, codomain(a), c)
% 14.99/2.30 = { by axiom 32 (category_theory_axiom1) }
% 14.99/2.30 fresh10(fresh16(product(codomain(a), a, a), true, codomain(a), a, b, c), true, h, codomain(a), c)
% 14.99/2.30 = { by axiom 5 (product_on_codomain) }
% 14.99/2.30 fresh10(fresh16(true, true, codomain(a), a, b, c), true, h, codomain(a), c)
% 14.99/2.30 = { by axiom 19 (category_theory_axiom1) }
% 14.99/2.30 fresh10(defined(codomain(a), c), true, h, codomain(a), c)
% 14.99/2.30 = { by axiom 27 (category_theory_axiom6) R->L }
% 14.99/2.30 fresh20(identity_map(codomain(a)), true, h, codomain(a), c)
% 14.99/2.30 = { by axiom 1 (codomain_is_an_identity_map) }
% 14.99/2.30 fresh20(true, true, h, codomain(a), c)
% 14.99/2.30 = { by axiom 18 (category_theory_axiom6) }
% 14.99/2.30 fresh21(defined(h, codomain(a)), true, h, c)
% 14.99/2.30 = { by axiom 15 (category_theory_axiom3) R->L }
% 14.99/2.30 fresh21(fresh14(true, true, codomain(a), a, h), true, h, c)
% 14.99/2.30 = { by axiom 5 (product_on_codomain) R->L }
% 14.99/2.30 fresh21(fresh14(product(codomain(a), a, a), true, codomain(a), a, h), true, h, c)
% 14.99/2.30 = { by axiom 29 (category_theory_axiom3) }
% 14.99/2.30 fresh21(fresh13(defined(h, a), true, codomain(a), h), true, h, c)
% 14.99/2.30 = { by axiom 25 (associative_property1) R->L }
% 14.99/2.30 fresh21(fresh13(fresh18(product(h, a, d), true, h, a), true, codomain(a), h), true, h, c)
% 14.99/2.30 = { by axiom 3 (ha_equals_d) }
% 14.99/2.30 fresh21(fresh13(fresh18(true, true, h, a), true, codomain(a), h), true, h, c)
% 14.99/2.30 = { by axiom 10 (associative_property1) }
% 14.99/2.30 fresh21(fresh13(true, true, codomain(a), h), true, h, c)
% 14.99/2.30 = { by axiom 11 (category_theory_axiom3) }
% 14.99/2.30 fresh21(true, true, h, c)
% 14.99/2.30 = { by axiom 9 (category_theory_axiom6) }
% 14.99/2.30 true
% 14.99/2.30
% 14.99/2.30 Lemma 36: product(h, c, compose(h, c)) = true.
% 14.99/2.30 Proof:
% 14.99/2.30 product(h, c, compose(h, c))
% 14.99/2.30 = { by axiom 21 (closure_of_composition) R->L }
% 14.99/2.30 fresh9(defined(h, c), true, h, c)
% 14.99/2.30 = { by lemma 35 }
% 14.99/2.30 fresh9(true, true, h, c)
% 14.99/2.30 = { by axiom 12 (closure_of_composition) }
% 14.99/2.30 true
% 14.99/2.30
% 14.99/2.30 Lemma 37: fresh27(product(X, a, d), true, X, compose(d, b), c) = product(X, c, compose(d, b)).
% 14.99/2.30 Proof:
% 14.99/2.30 fresh27(product(X, a, d), true, X, compose(d, b), c)
% 14.99/2.30 = { by axiom 28 (category_theory_axiom2) R->L }
% 14.99/2.30 fresh26(true, true, X, a, d, b, compose(d, b), c)
% 14.99/2.30 = { by axiom 12 (closure_of_composition) R->L }
% 14.99/2.30 fresh26(fresh9(true, true, d, b), true, X, a, d, b, compose(d, b), c)
% 14.99/2.30 = { by axiom 8 (category_theory_axiom4) R->L }
% 14.99/2.30 fresh26(fresh9(fresh25(true, true, b, d), true, d, b), true, X, a, d, b, compose(d, b), c)
% 14.99/2.30 = { by lemma 35 R->L }
% 14.99/2.30 fresh26(fresh9(fresh25(defined(h, c), true, b, d), true, d, b), true, X, a, d, b, compose(d, b), c)
% 14.99/2.30 = { by axiom 24 (category_theory_axiom4) R->L }
% 14.99/2.30 fresh26(fresh9(fresh24(true, true, a, b, c, h, d), true, d, b), true, X, a, d, b, compose(d, b), c)
% 14.99/2.30 = { by axiom 3 (ha_equals_d) R->L }
% 14.99/2.30 fresh26(fresh9(fresh24(product(h, a, d), true, a, b, c, h, d), true, d, b), true, X, a, d, b, compose(d, b), c)
% 14.99/2.30 = { by axiom 33 (category_theory_axiom4) }
% 14.99/2.30 fresh26(fresh9(fresh12(product(a, b, c), true, b, c, h, d), true, d, b), true, X, a, d, b, compose(d, b), c)
% 14.99/2.30 = { by axiom 2 (ab_equals_c) }
% 14.99/2.30 fresh26(fresh9(fresh12(true, true, b, c, h, d), true, d, b), true, X, a, d, b, compose(d, b), c)
% 14.99/2.30 = { by axiom 20 (category_theory_axiom4) }
% 14.99/2.30 fresh26(fresh9(defined(d, b), true, d, b), true, X, a, d, b, compose(d, b), c)
% 14.99/2.30 = { by axiom 21 (closure_of_composition) }
% 14.99/2.30 fresh26(product(d, b, compose(d, b)), true, X, a, d, b, compose(d, b), c)
% 14.99/2.30 = { by axiom 34 (category_theory_axiom2) }
% 14.99/2.30 fresh15(product(a, b, c), true, X, a, d, compose(d, b), c)
% 14.99/2.30 = { by axiom 2 (ab_equals_c) }
% 14.99/2.30 fresh15(true, true, X, a, d, compose(d, b), c)
% 14.99/2.30 = { by axiom 26 (category_theory_axiom2) }
% 14.99/2.30 product(X, c, compose(d, b))
% 14.99/2.30
% 14.99/2.30 Goal 1 (prove_h_equals_g): h = g.
% 14.99/2.30 Proof:
% 14.99/2.30 h
% 14.99/2.31 = { by axiom 6 (cancellation_for_product) R->L }
% 14.99/2.31 fresh(true, true, g, h)
% 14.99/2.31 = { by axiom 14 (category_theory_axiom2) R->L }
% 14.99/2.31 fresh(fresh27(true, true, g, compose(d, b), c), true, g, h)
% 14.99/2.31 = { by axiom 4 (ga_equals_d) R->L }
% 14.99/2.31 fresh(fresh27(product(g, a, d), true, g, compose(d, b), c), true, g, h)
% 14.99/2.31 = { by lemma 37 }
% 14.99/2.31 fresh(product(g, c, compose(d, b)), true, g, h)
% 14.99/2.31 = { by axiom 22 (composition_is_well_defined) R->L }
% 14.99/2.31 fresh(product(g, c, fresh4(true, true, h, c, compose(d, b), compose(h, c))), true, g, h)
% 14.99/2.31 = { by lemma 36 R->L }
% 14.99/2.31 fresh(product(g, c, fresh4(product(h, c, compose(h, c)), true, h, c, compose(d, b), compose(h, c))), true, g, h)
% 14.99/2.31 = { by axiom 31 (composition_is_well_defined) }
% 14.99/2.31 fresh(product(g, c, fresh3(product(h, c, compose(d, b)), true, compose(d, b), compose(h, c))), true, g, h)
% 14.99/2.31 = { by lemma 37 R->L }
% 14.99/2.31 fresh(product(g, c, fresh3(fresh27(product(h, a, d), true, h, compose(d, b), c), true, compose(d, b), compose(h, c))), true, g, h)
% 14.99/2.31 = { by axiom 3 (ha_equals_d) }
% 14.99/2.31 fresh(product(g, c, fresh3(fresh27(true, true, h, compose(d, b), c), true, compose(d, b), compose(h, c))), true, g, h)
% 14.99/2.31 = { by axiom 14 (category_theory_axiom2) }
% 14.99/2.31 fresh(product(g, c, fresh3(true, true, compose(d, b), compose(h, c))), true, g, h)
% 14.99/2.31 = { by axiom 13 (composition_is_well_defined) }
% 14.99/2.31 fresh(product(g, c, compose(h, c)), true, g, h)
% 14.99/2.31 = { by axiom 30 (cancellation_for_product) R->L }
% 14.99/2.31 fresh2(product(h, c, compose(h, c)), true, g, compose(h, c), h)
% 14.99/2.31 = { by lemma 36 }
% 14.99/2.31 fresh2(true, true, g, compose(h, c), h)
% 14.99/2.31 = { by axiom 17 (cancellation_for_product) }
% 14.99/2.31 g
% 14.99/2.31 % SZS output end Proof
% 14.99/2.31
% 14.99/2.31 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------