TSTP Solution File: CAT001-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : CAT001-1 : 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 18:18:46 EDT 2023
% Result : Unsatisfiable 13.24s 2.06s
% Output : Proof 13.24s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : CAT001-1 : 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
% 0.13/0.35 % Computer : n011.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Sun Aug 27 00:51:54 EDT 2023
% 0.13/0.35 % CPUTime :
% 13.24/2.06 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 13.24/2.06
% 13.24/2.06 % SZS status Unsatisfiable
% 13.24/2.06
% 13.24/2.08 % SZS output start Proof
% 13.24/2.08 Take the following subset of the input axioms:
% 13.24/2.08 fof(ab_equals_c, hypothesis, product(a, b, c)).
% 13.24/2.08 fof(associative_property1, axiom, ![X, Y, Z]: (~product(X, Y, Z) | defined(X, Y))).
% 13.24/2.08 fof(bg_equals_d, hypothesis, product(b, g, d)).
% 13.24/2.08 fof(bh_equals_d, hypothesis, product(b, h, d)).
% 13.24/2.08 fof(cancellation_for_product, hypothesis, ![X1, X2, X3]: (~product(c, X1, X2) | (~product(c, X3, X2) | X1=X3))).
% 13.24/2.08 fof(category_theory_axiom1, axiom, ![Xy, Yz, X4, Y2, Z2]: (~product(X4, Y2, Xy) | (~product(Y2, Z2, Yz) | (~defined(Xy, Z2) | defined(X4, Yz))))).
% 13.24/2.08 fof(category_theory_axiom2, axiom, ![Xyz, X4, Y2, Z2, Xy2, Yz2]: (~product(X4, Y2, Xy2) | (~product(Xy2, Z2, Xyz) | (~product(Y2, Z2, Yz2) | product(X4, Yz2, Xyz))))).
% 13.24/2.08 fof(category_theory_axiom3, axiom, ![X4, Y2, Z2, Yz2]: (~product(Y2, Z2, Yz2) | (~defined(X4, Yz2) | defined(X4, Y2)))).
% 13.24/2.08 fof(category_theory_axiom4, axiom, ![X4, Y2, Z2, Xy2, Yz2]: (~product(Y2, Z2, Yz2) | (~product(X4, Y2, Xy2) | (~defined(X4, Yz2) | defined(Xy2, Z2))))).
% 13.24/2.08 fof(category_theory_axiom6, axiom, ![X4, Y2, Z2]: (~defined(X4, Y2) | (~defined(Y2, Z2) | (~identity_map(Y2) | defined(X4, Z2))))).
% 13.24/2.08 fof(closure_of_composition, axiom, ![X4, Y2]: (~defined(X4, Y2) | product(X4, Y2, compose(X4, Y2)))).
% 13.24/2.08 fof(codomain_is_an_identity_map, axiom, ![X4]: identity_map(codomain(X4))).
% 13.24/2.08 fof(composition_is_well_defined, axiom, ![W, X4, Y2, Z2]: (~product(X4, Y2, Z2) | (~product(X4, Y2, W) | Z2=W))).
% 13.24/2.08 fof(product_on_codomain, axiom, ![X4]: product(codomain(X4), X4, X4)).
% 13.24/2.08 fof(prove_h_equals_g, negated_conjecture, h!=g).
% 13.24/2.08
% 13.24/2.08 Now clausify the problem and encode Horn clauses using encoding 3 of
% 13.24/2.08 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 13.24/2.08 We repeatedly replace C & s=t => u=v by the two clauses:
% 13.24/2.08 fresh(y, y, x1...xn) = u
% 13.24/2.08 C => fresh(s, t, x1...xn) = v
% 13.24/2.08 where fresh is a fresh function symbol and x1..xn are the free
% 13.24/2.08 variables of u and v.
% 13.24/2.08 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 13.24/2.08 input problem has no model of domain size 1).
% 13.24/2.08
% 13.24/2.08 The encoding turns the above axioms into the following unit equations and goals:
% 13.24/2.08
% 13.24/2.08 Axiom 1 (codomain_is_an_identity_map): identity_map(codomain(X)) = true.
% 13.24/2.08 Axiom 2 (bh_equals_d): product(b, h, d) = true.
% 13.24/2.08 Axiom 3 (bg_equals_d): product(b, g, d) = true.
% 13.24/2.08 Axiom 4 (ab_equals_c): product(a, b, c) = true.
% 13.24/2.08 Axiom 5 (product_on_codomain): product(codomain(X), X, X) = true.
% 13.24/2.08 Axiom 6 (cancellation_for_product): fresh(X, X, Y, Z) = Z.
% 13.24/2.08 Axiom 7 (category_theory_axiom1): fresh29(X, X, Y, Z) = true.
% 13.24/2.08 Axiom 8 (category_theory_axiom4): fresh25(X, X, Y, Z) = true.
% 13.24/2.08 Axiom 9 (category_theory_axiom6): fresh21(X, X, Y, Z) = true.
% 13.24/2.08 Axiom 10 (associative_property1): fresh18(X, X, Y, Z) = true.
% 13.24/2.08 Axiom 11 (category_theory_axiom3): fresh13(X, X, Y, Z) = true.
% 13.24/2.08 Axiom 12 (closure_of_composition): fresh9(X, X, Y, Z) = true.
% 13.24/2.08 Axiom 13 (composition_is_well_defined): fresh3(X, X, Y, Z) = Z.
% 13.24/2.08 Axiom 14 (category_theory_axiom2): fresh27(X, X, Y, Z, W) = true.
% 13.24/2.08 Axiom 15 (category_theory_axiom3): fresh14(X, X, Y, Z, W) = defined(W, Y).
% 13.24/2.08 Axiom 16 (category_theory_axiom6): fresh10(X, X, Y, Z, W) = defined(Y, W).
% 13.24/2.08 Axiom 17 (cancellation_for_product): fresh2(X, X, Y, Z, W) = Y.
% 13.24/2.08 Axiom 18 (category_theory_axiom6): fresh20(X, X, Y, Z, W) = fresh21(defined(Y, Z), true, Y, W).
% 13.24/2.08 Axiom 19 (category_theory_axiom1): fresh16(X, X, Y, Z, W, V) = defined(Y, V).
% 13.24/2.08 Axiom 20 (category_theory_axiom4): fresh12(X, X, Y, Z, W, V) = defined(V, Y).
% 13.24/2.08 Axiom 21 (closure_of_composition): fresh9(defined(X, Y), true, X, Y) = product(X, Y, compose(X, Y)).
% 13.24/2.08 Axiom 22 (composition_is_well_defined): fresh4(X, X, Y, Z, W, V) = W.
% 13.24/2.08 Axiom 23 (category_theory_axiom1): fresh28(X, X, Y, Z, W, V, U) = fresh29(defined(W, V), true, Y, U).
% 13.24/2.08 Axiom 24 (category_theory_axiom4): fresh24(X, X, Y, Z, W, V, U) = fresh25(defined(V, W), true, Z, U).
% 13.24/2.08 Axiom 25 (associative_property1): fresh18(product(X, Y, Z), true, X, Y) = defined(X, Y).
% 13.24/2.08 Axiom 26 (category_theory_axiom2): fresh15(X, X, Y, Z, W, V, U) = product(Y, U, V).
% 13.24/2.08 Axiom 27 (category_theory_axiom6): fresh20(identity_map(X), true, Y, X, Z) = fresh10(defined(X, Z), true, Y, X, Z).
% 13.24/2.08 Axiom 28 (category_theory_axiom2): fresh26(X, X, Y, Z, W, V, U, T) = fresh27(product(Y, Z, W), true, Y, U, T).
% 13.24/2.08 Axiom 29 (category_theory_axiom3): fresh14(product(X, Y, Z), true, X, Z, W) = fresh13(defined(W, Z), true, X, W).
% 13.24/2.08 Axiom 30 (cancellation_for_product): fresh2(product(c, X, Y), true, Z, Y, X) = fresh(product(c, Z, Y), true, Z, X).
% 13.24/2.08 Axiom 31 (composition_is_well_defined): fresh4(product(X, Y, Z), true, X, Y, W, Z) = fresh3(product(X, Y, W), true, W, Z).
% 13.24/2.08 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).
% 13.24/2.08 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).
% 13.24/2.08 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).
% 13.24/2.08
% 13.24/2.08 Lemma 35: defined(a, d) = true.
% 13.24/2.08 Proof:
% 13.24/2.08 defined(a, d)
% 13.24/2.08 = { by axiom 16 (category_theory_axiom6) R->L }
% 13.24/2.08 fresh10(true, true, a, codomain(b), d)
% 13.24/2.08 = { by axiom 7 (category_theory_axiom1) R->L }
% 13.24/2.08 fresh10(fresh29(true, true, codomain(b), d), true, a, codomain(b), d)
% 13.24/2.08 = { by axiom 10 (associative_property1) R->L }
% 13.24/2.08 fresh10(fresh29(fresh18(true, true, b, h), true, codomain(b), d), true, a, codomain(b), d)
% 13.24/2.08 = { by axiom 2 (bh_equals_d) R->L }
% 13.24/2.08 fresh10(fresh29(fresh18(product(b, h, d), true, b, h), true, codomain(b), d), true, a, codomain(b), d)
% 13.24/2.08 = { by axiom 25 (associative_property1) }
% 13.24/2.08 fresh10(fresh29(defined(b, h), true, codomain(b), d), true, a, codomain(b), d)
% 13.24/2.08 = { by axiom 23 (category_theory_axiom1) R->L }
% 13.24/2.08 fresh10(fresh28(true, true, codomain(b), b, b, h, d), true, a, codomain(b), d)
% 13.24/2.08 = { by axiom 2 (bh_equals_d) R->L }
% 13.24/2.08 fresh10(fresh28(product(b, h, d), true, codomain(b), b, b, h, d), true, a, codomain(b), d)
% 13.24/2.08 = { by axiom 32 (category_theory_axiom1) }
% 13.24/2.08 fresh10(fresh16(product(codomain(b), b, b), true, codomain(b), b, h, d), true, a, codomain(b), d)
% 13.24/2.08 = { by axiom 5 (product_on_codomain) }
% 13.24/2.08 fresh10(fresh16(true, true, codomain(b), b, h, d), true, a, codomain(b), d)
% 13.24/2.08 = { by axiom 19 (category_theory_axiom1) }
% 13.24/2.08 fresh10(defined(codomain(b), d), true, a, codomain(b), d)
% 13.24/2.08 = { by axiom 27 (category_theory_axiom6) R->L }
% 13.24/2.08 fresh20(identity_map(codomain(b)), true, a, codomain(b), d)
% 13.24/2.08 = { by axiom 1 (codomain_is_an_identity_map) }
% 13.24/2.08 fresh20(true, true, a, codomain(b), d)
% 13.24/2.08 = { by axiom 18 (category_theory_axiom6) }
% 13.24/2.08 fresh21(defined(a, codomain(b)), true, a, d)
% 13.24/2.08 = { by axiom 15 (category_theory_axiom3) R->L }
% 13.24/2.08 fresh21(fresh14(true, true, codomain(b), b, a), true, a, d)
% 13.24/2.08 = { by axiom 5 (product_on_codomain) R->L }
% 13.24/2.08 fresh21(fresh14(product(codomain(b), b, b), true, codomain(b), b, a), true, a, d)
% 13.24/2.08 = { by axiom 29 (category_theory_axiom3) }
% 13.24/2.08 fresh21(fresh13(defined(a, b), true, codomain(b), a), true, a, d)
% 13.24/2.08 = { by axiom 25 (associative_property1) R->L }
% 13.24/2.08 fresh21(fresh13(fresh18(product(a, b, c), true, a, b), true, codomain(b), a), true, a, d)
% 13.24/2.08 = { by axiom 4 (ab_equals_c) }
% 13.24/2.08 fresh21(fresh13(fresh18(true, true, a, b), true, codomain(b), a), true, a, d)
% 13.24/2.08 = { by axiom 10 (associative_property1) }
% 13.24/2.08 fresh21(fresh13(true, true, codomain(b), a), true, a, d)
% 13.24/2.08 = { by axiom 11 (category_theory_axiom3) }
% 13.24/2.08 fresh21(true, true, a, d)
% 13.24/2.08 = { by axiom 9 (category_theory_axiom6) }
% 13.24/2.08 true
% 13.24/2.08
% 13.24/2.08 Lemma 36: fresh12(product(b, X, Y), true, X, Y, a, c) = fresh25(defined(a, Y), true, X, c).
% 13.24/2.08 Proof:
% 13.24/2.08 fresh12(product(b, X, Y), true, X, Y, a, c)
% 13.24/2.08 = { by axiom 33 (category_theory_axiom4) R->L }
% 13.24/2.08 fresh24(product(a, b, c), true, b, X, Y, a, c)
% 13.24/2.08 = { by axiom 4 (ab_equals_c) }
% 13.24/2.08 fresh24(true, true, b, X, Y, a, c)
% 13.24/2.08 = { by axiom 24 (category_theory_axiom4) }
% 13.24/2.08 fresh25(defined(a, Y), true, X, c)
% 13.24/2.08
% 13.24/2.08 Lemma 37: product(c, h, compose(c, h)) = true.
% 13.24/2.08 Proof:
% 13.24/2.08 product(c, h, compose(c, h))
% 13.24/2.08 = { by axiom 21 (closure_of_composition) R->L }
% 13.24/2.08 fresh9(defined(c, h), true, c, h)
% 13.24/2.08 = { by axiom 20 (category_theory_axiom4) R->L }
% 13.24/2.08 fresh9(fresh12(true, true, h, d, a, c), true, c, h)
% 13.24/2.08 = { by axiom 2 (bh_equals_d) R->L }
% 13.24/2.09 fresh9(fresh12(product(b, h, d), true, h, d, a, c), true, c, h)
% 13.24/2.09 = { by lemma 36 }
% 13.24/2.09 fresh9(fresh25(defined(a, d), true, h, c), true, c, h)
% 13.24/2.09 = { by lemma 35 }
% 13.24/2.09 fresh9(fresh25(true, true, h, c), true, c, h)
% 13.24/2.09 = { by axiom 8 (category_theory_axiom4) }
% 13.24/2.09 fresh9(true, true, c, h)
% 13.24/2.09 = { by axiom 12 (closure_of_composition) }
% 13.24/2.09 true
% 13.24/2.09
% 13.24/2.09 Lemma 38: product(c, g, compose(c, g)) = true.
% 13.24/2.09 Proof:
% 13.24/2.09 product(c, g, compose(c, g))
% 13.24/2.09 = { by axiom 21 (closure_of_composition) R->L }
% 13.24/2.09 fresh9(defined(c, g), true, c, g)
% 13.24/2.09 = { by axiom 20 (category_theory_axiom4) R->L }
% 13.24/2.09 fresh9(fresh12(true, true, g, d, a, c), true, c, g)
% 13.24/2.09 = { by axiom 3 (bg_equals_d) R->L }
% 13.24/2.09 fresh9(fresh12(product(b, g, d), true, g, d, a, c), true, c, g)
% 13.24/2.09 = { by lemma 36 }
% 13.24/2.09 fresh9(fresh25(defined(a, d), true, g, c), true, c, g)
% 13.24/2.09 = { by lemma 35 }
% 13.24/2.09 fresh9(fresh25(true, true, g, c), true, c, g)
% 13.24/2.09 = { by axiom 8 (category_theory_axiom4) }
% 13.24/2.09 fresh9(true, true, c, g)
% 13.24/2.09 = { by axiom 12 (closure_of_composition) }
% 13.24/2.09 true
% 13.24/2.09
% 13.24/2.09 Lemma 39: fresh3(product(a, d, X), true, X, compose(a, d)) = X.
% 13.24/2.09 Proof:
% 13.24/2.09 fresh3(product(a, d, X), true, X, compose(a, d))
% 13.24/2.09 = { by axiom 31 (composition_is_well_defined) R->L }
% 13.24/2.09 fresh4(product(a, d, compose(a, d)), true, a, d, X, compose(a, d))
% 13.24/2.09 = { by axiom 21 (closure_of_composition) R->L }
% 13.24/2.09 fresh4(fresh9(defined(a, d), true, a, d), true, a, d, X, compose(a, d))
% 13.24/2.09 = { by lemma 35 }
% 13.24/2.09 fresh4(fresh9(true, true, a, d), true, a, d, X, compose(a, d))
% 13.24/2.09 = { by axiom 12 (closure_of_composition) }
% 13.24/2.09 fresh4(true, true, a, d, X, compose(a, d))
% 13.24/2.09 = { by axiom 22 (composition_is_well_defined) }
% 13.24/2.09 X
% 13.24/2.09
% 13.24/2.09 Goal 1 (prove_h_equals_g): h = g.
% 13.24/2.09 Proof:
% 13.24/2.09 h
% 13.24/2.09 = { by axiom 6 (cancellation_for_product) R->L }
% 13.24/2.09 fresh(true, true, g, h)
% 13.24/2.09 = { by lemma 38 R->L }
% 13.24/2.09 fresh(product(c, g, compose(c, g)), true, g, h)
% 13.24/2.09 = { by lemma 39 R->L }
% 13.24/2.09 fresh(product(c, g, fresh3(product(a, d, compose(c, g)), true, compose(c, g), compose(a, d))), true, g, h)
% 13.24/2.09 = { by axiom 26 (category_theory_axiom2) R->L }
% 13.24/2.09 fresh(product(c, g, fresh3(fresh15(true, true, a, b, c, compose(c, g), d), true, compose(c, g), compose(a, d))), true, g, h)
% 13.24/2.09 = { by axiom 3 (bg_equals_d) R->L }
% 13.24/2.09 fresh(product(c, g, fresh3(fresh15(product(b, g, d), true, a, b, c, compose(c, g), d), true, compose(c, g), compose(a, d))), true, g, h)
% 13.24/2.09 = { by axiom 34 (category_theory_axiom2) R->L }
% 13.24/2.09 fresh(product(c, g, fresh3(fresh26(product(c, g, compose(c, g)), true, a, b, c, g, compose(c, g), d), true, compose(c, g), compose(a, d))), true, g, h)
% 13.24/2.09 = { by lemma 38 }
% 13.24/2.09 fresh(product(c, g, fresh3(fresh26(true, true, a, b, c, g, compose(c, g), d), true, compose(c, g), compose(a, d))), true, g, h)
% 13.24/2.09 = { by axiom 28 (category_theory_axiom2) }
% 13.24/2.09 fresh(product(c, g, fresh3(fresh27(product(a, b, c), true, a, compose(c, g), d), true, compose(c, g), compose(a, d))), true, g, h)
% 13.24/2.09 = { by axiom 4 (ab_equals_c) }
% 13.24/2.09 fresh(product(c, g, fresh3(fresh27(true, true, a, compose(c, g), d), true, compose(c, g), compose(a, d))), true, g, h)
% 13.24/2.09 = { by axiom 14 (category_theory_axiom2) }
% 13.24/2.09 fresh(product(c, g, fresh3(true, true, compose(c, g), compose(a, d))), true, g, h)
% 13.24/2.09 = { by axiom 13 (composition_is_well_defined) }
% 13.24/2.09 fresh(product(c, g, compose(a, d)), true, g, h)
% 13.24/2.09 = { by axiom 13 (composition_is_well_defined) R->L }
% 13.24/2.09 fresh(product(c, g, fresh3(true, true, compose(c, h), compose(a, d))), true, g, h)
% 13.24/2.09 = { by axiom 14 (category_theory_axiom2) R->L }
% 13.24/2.09 fresh(product(c, g, fresh3(fresh27(true, true, a, compose(c, h), d), true, compose(c, h), compose(a, d))), true, g, h)
% 13.24/2.09 = { by axiom 4 (ab_equals_c) R->L }
% 13.24/2.09 fresh(product(c, g, fresh3(fresh27(product(a, b, c), true, a, compose(c, h), d), true, compose(c, h), compose(a, d))), true, g, h)
% 13.24/2.09 = { by axiom 28 (category_theory_axiom2) R->L }
% 13.24/2.09 fresh(product(c, g, fresh3(fresh26(true, true, a, b, c, h, compose(c, h), d), true, compose(c, h), compose(a, d))), true, g, h)
% 13.24/2.09 = { by lemma 37 R->L }
% 13.24/2.09 fresh(product(c, g, fresh3(fresh26(product(c, h, compose(c, h)), true, a, b, c, h, compose(c, h), d), true, compose(c, h), compose(a, d))), true, g, h)
% 13.24/2.09 = { by axiom 34 (category_theory_axiom2) }
% 13.24/2.09 fresh(product(c, g, fresh3(fresh15(product(b, h, d), true, a, b, c, compose(c, h), d), true, compose(c, h), compose(a, d))), true, g, h)
% 13.24/2.09 = { by axiom 2 (bh_equals_d) }
% 13.24/2.09 fresh(product(c, g, fresh3(fresh15(true, true, a, b, c, compose(c, h), d), true, compose(c, h), compose(a, d))), true, g, h)
% 13.24/2.09 = { by axiom 26 (category_theory_axiom2) }
% 13.24/2.09 fresh(product(c, g, fresh3(product(a, d, compose(c, h)), true, compose(c, h), compose(a, d))), true, g, h)
% 13.24/2.09 = { by lemma 39 }
% 13.24/2.09 fresh(product(c, g, compose(c, h)), true, g, h)
% 13.24/2.09 = { by axiom 30 (cancellation_for_product) R->L }
% 13.24/2.09 fresh2(product(c, h, compose(c, h)), true, g, compose(c, h), h)
% 13.24/2.09 = { by lemma 37 }
% 13.24/2.09 fresh2(true, true, g, compose(c, h), h)
% 13.24/2.09 = { by axiom 17 (cancellation_for_product) }
% 13.24/2.09 g
% 13.24/2.09 % SZS output end Proof
% 13.24/2.09
% 13.24/2.09 RESULT: Unsatisfiable (the axioms are contradictory).
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