TSTP Solution File: CAT004-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : CAT004-1 : 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 : n012.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 18:18:49 EDT 2023
% Result : Unsatisfiable 3.93s 0.86s
% Output : Proof 3.93s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : CAT004-1 : 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
% 0.14/0.33 % Computer : n012.cluster.edu
% 0.14/0.33 % Model : x86_64 x86_64
% 0.14/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.33 % Memory : 8042.1875MB
% 0.14/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Sun Aug 27 00:08:40 EDT 2023
% 0.14/0.34 % CPUTime :
% 3.07/0.86 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 3.93/0.86
% 3.93/0.86 % SZS status Unsatisfiable
% 3.93/0.86
% 3.93/0.87 % SZS output start Proof
% 3.93/0.87 Take the following subset of the input axioms:
% 3.93/0.87 fof(ab_equals_c, hypothesis, product(a, b, c)).
% 3.93/0.87 fof(associative_property1, axiom, ![X, Y, Z]: (~product(X, Y, Z) | defined(X, Y))).
% 3.93/0.87 fof(cancellation_for_product1, hypothesis, ![W, X2, Y2]: (~product(X2, a, W) | (~product(Y2, a, W) | X2=Y2))).
% 3.93/0.87 fof(cancellation_for_product2, hypothesis, ![X2, Y2, W2]: (~product(X2, b, W2) | (~product(Y2, b, W2) | X2=Y2))).
% 3.93/0.87 fof(category_theory_axiom3, axiom, ![Yz, X2, Y2, Z2]: (~product(Y2, Z2, Yz) | (~defined(X2, Yz) | defined(X2, Y2)))).
% 3.93/0.87 fof(category_theory_axiom5, axiom, ![Xy, Xyz, X2, Y2, Z2, Yz2]: (~product(Y2, Z2, Yz2) | (~product(X2, Yz2, Xyz) | (~product(X2, Y2, Xy) | product(Xy, Z2, Xyz))))).
% 3.93/0.87 fof(closure_of_composition, axiom, ![X2, Y2]: (~defined(X2, Y2) | product(X2, Y2, compose(X2, Y2)))).
% 3.93/0.87 fof(gc_equals_d, hypothesis, product(g, c, d)).
% 3.93/0.87 fof(hc_equals_d, hypothesis, product(h, c, d)).
% 3.93/0.87 fof(prove_h_equals_g, negated_conjecture, h!=g).
% 3.93/0.87
% 3.93/0.87 Now clausify the problem and encode Horn clauses using encoding 3 of
% 3.93/0.87 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 3.93/0.87 We repeatedly replace C & s=t => u=v by the two clauses:
% 3.93/0.87 fresh(y, y, x1...xn) = u
% 3.93/0.87 C => fresh(s, t, x1...xn) = v
% 3.93/0.87 where fresh is a fresh function symbol and x1..xn are the free
% 3.93/0.87 variables of u and v.
% 3.93/0.87 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 3.93/0.87 input problem has no model of domain size 1).
% 3.93/0.87
% 3.93/0.87 The encoding turns the above axioms into the following unit equations and goals:
% 3.93/0.87
% 3.93/0.87 Axiom 1 (hc_equals_d): product(h, c, d) = true.
% 3.93/0.87 Axiom 2 (gc_equals_d): product(g, c, d) = true.
% 3.93/0.87 Axiom 3 (ab_equals_c): product(a, b, c) = true.
% 3.93/0.87 Axiom 4 (cancellation_for_product2): fresh(X, X, Y, Z) = Z.
% 3.93/0.87 Axiom 5 (associative_property1): fresh20(X, X, Y, Z) = true.
% 3.93/0.87 Axiom 6 (category_theory_axiom3): fresh15(X, X, Y, Z) = true.
% 3.93/0.87 Axiom 7 (closure_of_composition): fresh11(X, X, Y, Z) = true.
% 3.93/0.87 Axiom 8 (cancellation_for_product1): fresh3(X, X, Y, Z) = Z.
% 3.93/0.87 Axiom 9 (category_theory_axiom5): fresh25(X, X, Y, Z, W) = true.
% 3.93/0.87 Axiom 10 (category_theory_axiom3): fresh16(X, X, Y, Z, W) = defined(W, Y).
% 3.93/0.87 Axiom 11 (cancellation_for_product1): fresh4(X, X, Y, Z, W) = Y.
% 3.93/0.87 Axiom 12 (cancellation_for_product2): fresh2(X, X, Y, Z, W) = Y.
% 3.93/0.87 Axiom 13 (closure_of_composition): fresh11(defined(X, Y), true, X, Y) = product(X, Y, compose(X, Y)).
% 3.93/0.87 Axiom 14 (associative_property1): fresh20(product(X, Y, Z), true, X, Y) = defined(X, Y).
% 3.93/0.88 Axiom 15 (category_theory_axiom5): fresh13(X, X, Y, Z, W, V, U) = product(U, Z, V).
% 3.93/0.88 Axiom 16 (category_theory_axiom5): fresh24(X, X, Y, Z, W, V, U, T) = fresh25(product(Y, Z, W), true, Z, U, T).
% 3.93/0.88 Axiom 17 (category_theory_axiom3): fresh16(product(X, Y, Z), true, X, Z, W) = fresh15(defined(W, Z), true, X, W).
% 3.93/0.88 Axiom 18 (cancellation_for_product1): fresh4(product(X, a, Y), true, Z, Y, X) = fresh3(product(Z, a, Y), true, Z, X).
% 3.93/0.88 Axiom 19 (cancellation_for_product2): fresh2(product(X, b, Y), true, Z, Y, X) = fresh(product(Z, b, Y), true, Z, X).
% 3.93/0.88 Axiom 20 (category_theory_axiom5): fresh24(product(X, Y, Z), true, W, V, Y, X, Z, U) = fresh13(product(X, W, U), true, W, V, Y, Z, U).
% 3.93/0.88
% 3.93/0.88 Lemma 21: fresh15(defined(X, c), true, a, X) = defined(X, a).
% 3.93/0.88 Proof:
% 3.93/0.88 fresh15(defined(X, c), true, a, X)
% 3.93/0.88 = { by axiom 17 (category_theory_axiom3) R->L }
% 3.93/0.88 fresh16(product(a, b, c), true, a, c, X)
% 3.93/0.88 = { by axiom 3 (ab_equals_c) }
% 3.93/0.88 fresh16(true, true, a, c, X)
% 3.93/0.88 = { by axiom 10 (category_theory_axiom3) }
% 3.93/0.88 defined(X, a)
% 3.93/0.88
% 3.93/0.88 Lemma 22: product(h, a, compose(h, a)) = true.
% 3.93/0.88 Proof:
% 3.93/0.88 product(h, a, compose(h, a))
% 3.93/0.88 = { by axiom 13 (closure_of_composition) R->L }
% 3.93/0.88 fresh11(defined(h, a), true, h, a)
% 3.93/0.88 = { by lemma 21 R->L }
% 3.93/0.88 fresh11(fresh15(defined(h, c), true, a, h), true, h, a)
% 3.93/0.88 = { by axiom 14 (associative_property1) R->L }
% 3.93/0.88 fresh11(fresh15(fresh20(product(h, c, d), true, h, c), true, a, h), true, h, a)
% 3.93/0.88 = { by axiom 1 (hc_equals_d) }
% 3.93/0.88 fresh11(fresh15(fresh20(true, true, h, c), true, a, h), true, h, a)
% 3.93/0.88 = { by axiom 5 (associative_property1) }
% 3.93/0.88 fresh11(fresh15(true, true, a, h), true, h, a)
% 3.93/0.88 = { by axiom 6 (category_theory_axiom3) }
% 3.93/0.88 fresh11(true, true, h, a)
% 3.93/0.88 = { by axiom 7 (closure_of_composition) }
% 3.93/0.88 true
% 3.93/0.88
% 3.93/0.88 Lemma 23: product(g, a, compose(g, a)) = true.
% 3.93/0.88 Proof:
% 3.93/0.88 product(g, a, compose(g, a))
% 3.93/0.88 = { by axiom 13 (closure_of_composition) R->L }
% 3.93/0.88 fresh11(defined(g, a), true, g, a)
% 3.93/0.88 = { by lemma 21 R->L }
% 3.93/0.88 fresh11(fresh15(defined(g, c), true, a, g), true, g, a)
% 3.93/0.88 = { by axiom 14 (associative_property1) R->L }
% 3.93/0.88 fresh11(fresh15(fresh20(product(g, c, d), true, g, c), true, a, g), true, g, a)
% 3.93/0.88 = { by axiom 2 (gc_equals_d) }
% 3.93/0.88 fresh11(fresh15(fresh20(true, true, g, c), true, a, g), true, g, a)
% 3.93/0.88 = { by axiom 5 (associative_property1) }
% 3.93/0.88 fresh11(fresh15(true, true, a, g), true, g, a)
% 3.93/0.88 = { by axiom 6 (category_theory_axiom3) }
% 3.93/0.88 fresh11(true, true, g, a)
% 3.93/0.88 = { by axiom 7 (closure_of_composition) }
% 3.93/0.88 true
% 3.93/0.88
% 3.93/0.88 Goal 1 (prove_h_equals_g): h = g.
% 3.93/0.88 Proof:
% 3.93/0.88 h
% 3.93/0.88 = { by axiom 8 (cancellation_for_product1) R->L }
% 3.93/0.88 fresh3(true, true, g, h)
% 3.93/0.88 = { by lemma 23 R->L }
% 3.93/0.88 fresh3(product(g, a, compose(g, a)), true, g, h)
% 3.93/0.88 = { by axiom 12 (cancellation_for_product2) R->L }
% 3.93/0.88 fresh3(product(g, a, fresh2(true, true, compose(g, a), d, compose(h, a))), true, g, h)
% 3.93/0.88 = { by axiom 9 (category_theory_axiom5) R->L }
% 3.93/0.88 fresh3(product(g, a, fresh2(fresh25(true, true, b, d, compose(h, a)), true, compose(g, a), d, compose(h, a))), true, g, h)
% 3.93/0.88 = { by axiom 3 (ab_equals_c) R->L }
% 3.93/0.88 fresh3(product(g, a, fresh2(fresh25(product(a, b, c), true, b, d, compose(h, a)), true, compose(g, a), d, compose(h, a))), true, g, h)
% 3.93/0.88 = { by axiom 16 (category_theory_axiom5) R->L }
% 3.93/0.88 fresh3(product(g, a, fresh2(fresh24(true, true, a, b, c, h, d, compose(h, a)), true, compose(g, a), d, compose(h, a))), true, g, h)
% 3.93/0.88 = { by axiom 1 (hc_equals_d) R->L }
% 3.93/0.88 fresh3(product(g, a, fresh2(fresh24(product(h, c, d), true, a, b, c, h, d, compose(h, a)), true, compose(g, a), d, compose(h, a))), true, g, h)
% 3.93/0.88 = { by axiom 20 (category_theory_axiom5) }
% 3.93/0.88 fresh3(product(g, a, fresh2(fresh13(product(h, a, compose(h, a)), true, a, b, c, d, compose(h, a)), true, compose(g, a), d, compose(h, a))), true, g, h)
% 3.93/0.88 = { by lemma 22 }
% 3.93/0.88 fresh3(product(g, a, fresh2(fresh13(true, true, a, b, c, d, compose(h, a)), true, compose(g, a), d, compose(h, a))), true, g, h)
% 3.93/0.88 = { by axiom 15 (category_theory_axiom5) }
% 3.93/0.88 fresh3(product(g, a, fresh2(product(compose(h, a), b, d), true, compose(g, a), d, compose(h, a))), true, g, h)
% 3.93/0.88 = { by axiom 19 (cancellation_for_product2) }
% 3.93/0.88 fresh3(product(g, a, fresh(product(compose(g, a), b, d), true, compose(g, a), compose(h, a))), true, g, h)
% 3.93/0.88 = { by axiom 15 (category_theory_axiom5) R->L }
% 3.93/0.88 fresh3(product(g, a, fresh(fresh13(true, true, a, b, c, d, compose(g, a)), true, compose(g, a), compose(h, a))), true, g, h)
% 3.93/0.88 = { by lemma 23 R->L }
% 3.93/0.88 fresh3(product(g, a, fresh(fresh13(product(g, a, compose(g, a)), true, a, b, c, d, compose(g, a)), true, compose(g, a), compose(h, a))), true, g, h)
% 3.93/0.88 = { by axiom 20 (category_theory_axiom5) R->L }
% 3.93/0.88 fresh3(product(g, a, fresh(fresh24(product(g, c, d), true, a, b, c, g, d, compose(g, a)), true, compose(g, a), compose(h, a))), true, g, h)
% 3.93/0.88 = { by axiom 2 (gc_equals_d) }
% 3.93/0.88 fresh3(product(g, a, fresh(fresh24(true, true, a, b, c, g, d, compose(g, a)), true, compose(g, a), compose(h, a))), true, g, h)
% 3.93/0.88 = { by axiom 16 (category_theory_axiom5) }
% 3.93/0.88 fresh3(product(g, a, fresh(fresh25(product(a, b, c), true, b, d, compose(g, a)), true, compose(g, a), compose(h, a))), true, g, h)
% 3.93/0.88 = { by axiom 3 (ab_equals_c) }
% 3.93/0.88 fresh3(product(g, a, fresh(fresh25(true, true, b, d, compose(g, a)), true, compose(g, a), compose(h, a))), true, g, h)
% 3.93/0.88 = { by axiom 9 (category_theory_axiom5) }
% 3.93/0.88 fresh3(product(g, a, fresh(true, true, compose(g, a), compose(h, a))), true, g, h)
% 3.93/0.88 = { by axiom 4 (cancellation_for_product2) }
% 3.93/0.88 fresh3(product(g, a, compose(h, a)), true, g, h)
% 3.93/0.88 = { by axiom 18 (cancellation_for_product1) R->L }
% 3.93/0.88 fresh4(product(h, a, compose(h, a)), true, g, compose(h, a), h)
% 3.93/0.88 = { by lemma 22 }
% 3.93/0.88 fresh4(true, true, g, compose(h, a), h)
% 3.93/0.88 = { by axiom 11 (cancellation_for_product1) }
% 3.93/0.88 g
% 3.93/0.88 % SZS output end Proof
% 3.93/0.88
% 3.93/0.88 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------