TSTP Solution File: CAT002-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : CAT002-1 : 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 : n019.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 18:18:47 EDT 2023
% Result : Unsatisfiable 4.08s 0.94s
% Output : Proof 4.24s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : CAT002-1 : TPTP v8.1.2. Released v1.0.0.
% 0.13/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.36 % Computer : n019.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Sun Aug 27 00:08:43 EDT 2023
% 0.14/0.36 % CPUTime :
% 4.08/0.94 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 4.08/0.94
% 4.08/0.94 % SZS status Unsatisfiable
% 4.08/0.94
% 4.24/0.95 % SZS output start Proof
% 4.24/0.95 Take the following subset of the input axioms:
% 4.24/0.95 fof(ab_equals_c, hypothesis, product(a, b, c)).
% 4.24/0.95 fof(associative_property1, axiom, ![X, Y, Z]: (~product(X, Y, Z) | defined(X, Y))).
% 4.24/0.95 fof(associative_property2, axiom, ![Xy, X2, Y2, Z2]: (~product(X2, Y2, Xy) | (~defined(Xy, Z2) | defined(Y2, Z2)))).
% 4.24/0.95 fof(cancellation_for_product1, hypothesis, ![W, X2, Y2]: (~product(a, X2, W) | (~product(a, Y2, W) | X2=Y2))).
% 4.24/0.95 fof(cancellation_for_product2, hypothesis, ![X2, Y2, W2]: (~product(b, X2, W2) | (~product(b, Y2, W2) | X2=Y2))).
% 4.24/0.95 fof(category_theory_axiom2, axiom, ![Yz, Xyz, X2, Y2, Z2, Xy2]: (~product(X2, Y2, Xy2) | (~product(Xy2, Z2, Xyz) | (~product(Y2, Z2, Yz) | product(X2, Yz, Xyz))))).
% 4.24/0.95 fof(cg_equals_d, hypothesis, product(c, g, d)).
% 4.24/0.95 fof(ch_equals_d, hypothesis, product(c, h, d)).
% 4.24/0.95 fof(closure_of_composition, axiom, ![X2, Y2]: (~defined(X2, Y2) | product(X2, Y2, compose(X2, Y2)))).
% 4.24/0.95 fof(prove_h_equals_g, negated_conjecture, h!=g).
% 4.24/0.95
% 4.24/0.96 Now clausify the problem and encode Horn clauses using encoding 3 of
% 4.24/0.96 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 4.24/0.96 We repeatedly replace C & s=t => u=v by the two clauses:
% 4.24/0.96 fresh(y, y, x1...xn) = u
% 4.24/0.96 C => fresh(s, t, x1...xn) = v
% 4.24/0.96 where fresh is a fresh function symbol and x1..xn are the free
% 4.24/0.96 variables of u and v.
% 4.24/0.96 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 4.24/0.96 input problem has no model of domain size 1).
% 4.24/0.96
% 4.24/0.96 The encoding turns the above axioms into the following unit equations and goals:
% 4.24/0.96
% 4.24/0.96 Axiom 1 (ab_equals_c): product(a, b, c) = true.
% 4.24/0.96 Axiom 2 (ch_equals_d): product(c, h, d) = true.
% 4.24/0.96 Axiom 3 (cg_equals_d): product(c, g, d) = true.
% 4.24/0.96 Axiom 4 (cancellation_for_product2): fresh(X, X, Y, Z) = Z.
% 4.24/0.96 Axiom 5 (associative_property1): fresh20(X, X, Y, Z) = true.
% 4.24/0.96 Axiom 6 (associative_property2): fresh19(X, X, Y, Z) = true.
% 4.24/0.96 Axiom 7 (closure_of_composition): fresh11(X, X, Y, Z) = true.
% 4.24/0.96 Axiom 8 (cancellation_for_product1): fresh3(X, X, Y, Z) = Z.
% 4.24/0.96 Axiom 9 (category_theory_axiom2): fresh29(X, X, Y, Z, W) = true.
% 4.24/0.96 Axiom 10 (associative_property2): fresh21(X, X, Y, Z, W) = defined(Y, W).
% 4.24/0.96 Axiom 11 (closure_of_composition): fresh11(defined(X, Y), true, X, Y) = product(X, Y, compose(X, Y)).
% 4.24/0.96 Axiom 12 (cancellation_for_product1): fresh4(X, X, Y, Z, W) = Y.
% 4.24/0.96 Axiom 13 (cancellation_for_product2): fresh2(X, X, Y, Z, W) = Y.
% 4.24/0.96 Axiom 14 (associative_property1): fresh20(product(X, Y, Z), true, X, Y) = defined(X, Y).
% 4.24/0.96 Axiom 15 (associative_property2): fresh21(product(X, Y, Z), true, Y, Z, W) = fresh19(defined(Z, W), true, Y, W).
% 4.24/0.96 Axiom 16 (category_theory_axiom2): fresh17(X, X, Y, Z, W, V, U) = product(Y, U, V).
% 4.24/0.96 Axiom 17 (cancellation_for_product1): fresh4(product(a, X, Y), true, Z, Y, X) = fresh3(product(a, Z, Y), true, Z, X).
% 4.24/0.96 Axiom 18 (cancellation_for_product2): fresh2(product(b, X, Y), true, Z, Y, X) = fresh(product(b, Z, Y), true, Z, X).
% 4.24/0.96 Axiom 19 (category_theory_axiom2): fresh28(X, X, Y, Z, W, V, U, T) = fresh29(product(Y, Z, W), true, Y, U, T).
% 4.24/0.96 Axiom 20 (category_theory_axiom2): fresh28(product(X, Y, Z), true, W, V, X, Y, Z, U) = fresh17(product(V, Y, U), true, W, V, X, Z, U).
% 4.24/0.96
% 4.24/0.96 Lemma 21: fresh19(defined(c, X), true, b, X) = defined(b, X).
% 4.24/0.96 Proof:
% 4.24/0.96 fresh19(defined(c, X), true, b, X)
% 4.24/0.96 = { by axiom 15 (associative_property2) R->L }
% 4.24/0.96 fresh21(product(a, b, c), true, b, c, X)
% 4.24/0.96 = { by axiom 1 (ab_equals_c) }
% 4.24/0.96 fresh21(true, true, b, c, X)
% 4.24/0.96 = { by axiom 10 (associative_property2) }
% 4.24/0.96 defined(b, X)
% 4.24/0.96
% 4.24/0.96 Lemma 22: product(b, h, compose(b, h)) = true.
% 4.24/0.96 Proof:
% 4.24/0.96 product(b, h, compose(b, h))
% 4.24/0.96 = { by axiom 11 (closure_of_composition) R->L }
% 4.24/0.96 fresh11(defined(b, h), true, b, h)
% 4.24/0.96 = { by lemma 21 R->L }
% 4.24/0.96 fresh11(fresh19(defined(c, h), true, b, h), true, b, h)
% 4.24/0.96 = { by axiom 14 (associative_property1) R->L }
% 4.24/0.96 fresh11(fresh19(fresh20(product(c, h, d), true, c, h), true, b, h), true, b, h)
% 4.24/0.96 = { by axiom 2 (ch_equals_d) }
% 4.24/0.96 fresh11(fresh19(fresh20(true, true, c, h), true, b, h), true, b, h)
% 4.24/0.96 = { by axiom 5 (associative_property1) }
% 4.24/0.96 fresh11(fresh19(true, true, b, h), true, b, h)
% 4.24/0.96 = { by axiom 6 (associative_property2) }
% 4.24/0.96 fresh11(true, true, b, h)
% 4.24/0.96 = { by axiom 7 (closure_of_composition) }
% 4.24/0.96 true
% 4.24/0.96
% 4.24/0.96 Lemma 23: product(b, g, compose(b, g)) = true.
% 4.24/0.96 Proof:
% 4.24/0.96 product(b, g, compose(b, g))
% 4.24/0.96 = { by axiom 11 (closure_of_composition) R->L }
% 4.24/0.96 fresh11(defined(b, g), true, b, g)
% 4.24/0.96 = { by lemma 21 R->L }
% 4.24/0.96 fresh11(fresh19(defined(c, g), true, b, g), true, b, g)
% 4.24/0.96 = { by axiom 14 (associative_property1) R->L }
% 4.24/0.96 fresh11(fresh19(fresh20(product(c, g, d), true, c, g), true, b, g), true, b, g)
% 4.24/0.96 = { by axiom 3 (cg_equals_d) }
% 4.24/0.96 fresh11(fresh19(fresh20(true, true, c, g), true, b, g), true, b, g)
% 4.24/0.96 = { by axiom 5 (associative_property1) }
% 4.24/0.96 fresh11(fresh19(true, true, b, g), true, b, g)
% 4.24/0.96 = { by axiom 6 (associative_property2) }
% 4.24/0.96 fresh11(true, true, b, g)
% 4.24/0.96 = { by axiom 7 (closure_of_composition) }
% 4.24/0.96 true
% 4.24/0.96
% 4.24/0.96 Goal 1 (prove_h_equals_g): h = g.
% 4.24/0.96 Proof:
% 4.24/0.96 h
% 4.24/0.96 = { by axiom 4 (cancellation_for_product2) R->L }
% 4.24/0.96 fresh(true, true, g, h)
% 4.24/0.96 = { by lemma 23 R->L }
% 4.24/0.96 fresh(product(b, g, compose(b, g)), true, g, h)
% 4.24/0.96 = { by axiom 12 (cancellation_for_product1) R->L }
% 4.24/0.96 fresh(product(b, g, fresh4(true, true, compose(b, g), d, compose(b, h))), true, g, h)
% 4.24/0.96 = { by axiom 9 (category_theory_axiom2) R->L }
% 4.24/0.96 fresh(product(b, g, fresh4(fresh29(true, true, a, d, compose(b, h)), true, compose(b, g), d, compose(b, h))), true, g, h)
% 4.24/0.96 = { by axiom 1 (ab_equals_c) R->L }
% 4.24/0.96 fresh(product(b, g, fresh4(fresh29(product(a, b, c), true, a, d, compose(b, h)), true, compose(b, g), d, compose(b, h))), true, g, h)
% 4.24/0.96 = { by axiom 19 (category_theory_axiom2) R->L }
% 4.24/0.96 fresh(product(b, g, fresh4(fresh28(true, true, a, b, c, h, d, compose(b, h)), true, compose(b, g), d, compose(b, h))), true, g, h)
% 4.24/0.96 = { by axiom 2 (ch_equals_d) R->L }
% 4.24/0.96 fresh(product(b, g, fresh4(fresh28(product(c, h, d), true, a, b, c, h, d, compose(b, h)), true, compose(b, g), d, compose(b, h))), true, g, h)
% 4.24/0.96 = { by axiom 20 (category_theory_axiom2) }
% 4.24/0.96 fresh(product(b, g, fresh4(fresh17(product(b, h, compose(b, h)), true, a, b, c, d, compose(b, h)), true, compose(b, g), d, compose(b, h))), true, g, h)
% 4.24/0.96 = { by lemma 22 }
% 4.24/0.96 fresh(product(b, g, fresh4(fresh17(true, true, a, b, c, d, compose(b, h)), true, compose(b, g), d, compose(b, h))), true, g, h)
% 4.24/0.96 = { by axiom 16 (category_theory_axiom2) }
% 4.24/0.96 fresh(product(b, g, fresh4(product(a, compose(b, h), d), true, compose(b, g), d, compose(b, h))), true, g, h)
% 4.24/0.96 = { by axiom 17 (cancellation_for_product1) }
% 4.24/0.96 fresh(product(b, g, fresh3(product(a, compose(b, g), d), true, compose(b, g), compose(b, h))), true, g, h)
% 4.24/0.96 = { by axiom 16 (category_theory_axiom2) R->L }
% 4.24/0.96 fresh(product(b, g, fresh3(fresh17(true, true, a, b, c, d, compose(b, g)), true, compose(b, g), compose(b, h))), true, g, h)
% 4.24/0.96 = { by lemma 23 R->L }
% 4.24/0.96 fresh(product(b, g, fresh3(fresh17(product(b, g, compose(b, g)), true, a, b, c, d, compose(b, g)), true, compose(b, g), compose(b, h))), true, g, h)
% 4.24/0.96 = { by axiom 20 (category_theory_axiom2) R->L }
% 4.24/0.96 fresh(product(b, g, fresh3(fresh28(product(c, g, d), true, a, b, c, g, d, compose(b, g)), true, compose(b, g), compose(b, h))), true, g, h)
% 4.24/0.96 = { by axiom 3 (cg_equals_d) }
% 4.24/0.96 fresh(product(b, g, fresh3(fresh28(true, true, a, b, c, g, d, compose(b, g)), true, compose(b, g), compose(b, h))), true, g, h)
% 4.24/0.96 = { by axiom 19 (category_theory_axiom2) }
% 4.24/0.96 fresh(product(b, g, fresh3(fresh29(product(a, b, c), true, a, d, compose(b, g)), true, compose(b, g), compose(b, h))), true, g, h)
% 4.24/0.96 = { by axiom 1 (ab_equals_c) }
% 4.24/0.96 fresh(product(b, g, fresh3(fresh29(true, true, a, d, compose(b, g)), true, compose(b, g), compose(b, h))), true, g, h)
% 4.24/0.96 = { by axiom 9 (category_theory_axiom2) }
% 4.24/0.96 fresh(product(b, g, fresh3(true, true, compose(b, g), compose(b, h))), true, g, h)
% 4.24/0.96 = { by axiom 8 (cancellation_for_product1) }
% 4.24/0.96 fresh(product(b, g, compose(b, h)), true, g, h)
% 4.24/0.96 = { by axiom 18 (cancellation_for_product2) R->L }
% 4.24/0.96 fresh2(product(b, h, compose(b, h)), true, g, compose(b, h), h)
% 4.24/0.96 = { by lemma 22 }
% 4.24/0.96 fresh2(true, true, g, compose(b, h), h)
% 4.24/0.96 = { by axiom 13 (cancellation_for_product2) }
% 4.24/0.96 g
% 4.24/0.96 % SZS output end Proof
% 4.24/0.96
% 4.24/0.96 RESULT: Unsatisfiable (the axioms are contradictory).
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