TSTP Solution File: HAL006+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : HAL006+1 : TPTP v8.1.2. Released v2.6.0.
% 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 : n014.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 : Thu Aug 31 01:53:52 EDT 2023
% Result : Theorem 121.88s 16.19s
% Output : Proof 121.88s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : HAL006+1 : TPTP v8.1.2. Released v2.6.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.16/0.34 % Computer : n014.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Mon Aug 28 02:28:16 EDT 2023
% 0.20/0.34 % CPUTime :
% 121.88/16.19 Command-line arguments: --no-flatten-goal
% 121.88/16.19
% 121.88/16.19 % SZS status Theorem
% 121.88/16.19
% 121.88/16.20 % SZS output start Proof
% 121.88/16.20 Take the following subset of the input axioms:
% 121.88/16.20 fof(alpha_morphism, axiom, morphism(alpha, a, b)).
% 121.88/16.20 fof(g_morphism, axiom, morphism(g, b, e)).
% 121.88/16.20 fof(lemma12, conjecture, ![E]: (element(E, e) => ?[B1, B2]: (element(B1, b) & (element(B2, b) & apply(g, subtract(b, B1, B2))=E)))).
% 121.88/16.20 fof(lemma8, axiom, ![E2]: (element(E2, e) => ?[E1, A, B1_2]: (element(B1_2, b) & (element(E1, e) & (subtract(e, apply(g, B1_2), E2)=E1 & (element(A, a) & (apply(gamma, apply(f, A))=E1 & apply(g, apply(alpha, A))=E1))))))).
% 121.88/16.20 fof(morphism, axiom, ![Morphism, Dom, Cod]: (morphism(Morphism, Dom, Cod) => (![El]: (element(El, Dom) => element(apply(Morphism, El), Cod)) & apply(Morphism, zero(Dom))=zero(Cod)))).
% 121.88/16.20 fof(subtract_cancellation, axiom, ![El1, El2, Dom2]: ((element(El1, Dom2) & element(El2, Dom2)) => subtract(Dom2, El1, subtract(Dom2, El1, El2))=El2)).
% 121.88/16.20 fof(subtract_distribution, axiom, ![Morphism2, Dom2, Cod2]: (morphism(Morphism2, Dom2, Cod2) => ![El1_2, El2_2]: ((element(El1_2, Dom2) & element(El2_2, Dom2)) => apply(Morphism2, subtract(Dom2, El1_2, El2_2))=subtract(Cod2, apply(Morphism2, El1_2), apply(Morphism2, El2_2))))).
% 121.88/16.20
% 121.88/16.20 Now clausify the problem and encode Horn clauses using encoding 3 of
% 121.88/16.20 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 121.88/16.20 We repeatedly replace C & s=t => u=v by the two clauses:
% 121.88/16.20 fresh(y, y, x1...xn) = u
% 121.88/16.20 C => fresh(s, t, x1...xn) = v
% 121.88/16.20 where fresh is a fresh function symbol and x1..xn are the free
% 121.88/16.20 variables of u and v.
% 121.88/16.20 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 121.88/16.20 input problem has no model of domain size 1).
% 121.88/16.20
% 121.88/16.20 The encoding turns the above axioms into the following unit equations and goals:
% 121.88/16.20
% 121.88/16.20 Axiom 1 (lemma12): element(e2, e) = true2.
% 121.88/16.20 Axiom 2 (lemma8_1): fresh21(X, X, Y) = e1(Y).
% 121.88/16.20 Axiom 3 (lemma8_2): fresh20(X, X, Y) = e1(Y).
% 121.88/16.20 Axiom 4 (lemma8_4): fresh18(X, X, Y) = true2.
% 121.88/16.20 Axiom 5 (lemma8_5): fresh17(X, X, Y) = true2.
% 121.88/16.20 Axiom 6 (alpha_morphism): morphism(alpha, a, b) = true2.
% 121.88/16.20 Axiom 7 (g_morphism): morphism(g, b, e) = true2.
% 121.88/16.20 Axiom 8 (subtract_cancellation): fresh(X, X, Y, Z, W) = W.
% 121.88/16.20 Axiom 9 (lemma8_4): fresh18(element(X, e), true2, X) = element(b1(X), b).
% 121.88/16.20 Axiom 10 (lemma8_5): fresh17(element(X, e), true2, X) = element(a2(X), a).
% 121.88/16.20 Axiom 11 (morphism_1): fresh14(X, X, Y, Z, W) = true2.
% 121.88/16.20 Axiom 12 (lemma8_1): fresh21(element(X, e), true2, X) = apply(g, apply(alpha, a2(X))).
% 121.88/16.20 Axiom 13 (morphism_1): fresh15(X, X, Y, Z, W, V) = element(apply(Y, V), W).
% 121.88/16.20 Axiom 14 (subtract_cancellation): fresh8(X, X, Y, Z, W) = subtract(Y, Z, subtract(Y, Z, W)).
% 121.88/16.20 Axiom 15 (lemma8_2): fresh20(element(X, e), true2, X) = subtract(e, apply(g, b1(X)), X).
% 121.88/16.20 Axiom 16 (subtract_cancellation): fresh8(element(X, Y), true2, Y, Z, X) = fresh(element(Z, Y), true2, Y, Z, X).
% 121.88/16.20 Axiom 17 (subtract_distribution): fresh7(X, X, Y, Z, W, V, U) = apply(Y, subtract(Z, V, U)).
% 121.88/16.20 Axiom 18 (subtract_distribution): fresh29(X, X, Y, Z, W, V, U) = subtract(W, apply(Y, V), apply(Y, U)).
% 121.88/16.20 Axiom 19 (morphism_1): fresh15(element(X, Y), true2, Z, Y, W, X) = fresh14(morphism(Z, Y, W), true2, Z, W, X).
% 121.88/16.20 Axiom 20 (subtract_distribution): fresh28(X, X, Y, Z, W, V, U) = fresh29(element(V, Z), true2, Y, Z, W, V, U).
% 121.88/16.20 Axiom 21 (subtract_distribution): fresh28(element(X, Y), true2, Z, Y, W, V, X) = fresh7(morphism(Z, Y, W), true2, Z, Y, W, V, X).
% 121.88/16.20
% 121.88/16.20 Lemma 22: element(b1(e2), b) = true2.
% 121.88/16.20 Proof:
% 121.88/16.20 element(b1(e2), b)
% 121.88/16.20 = { by axiom 9 (lemma8_4) R->L }
% 121.88/16.20 fresh18(element(e2, e), true2, e2)
% 121.88/16.20 = { by axiom 1 (lemma12) }
% 121.88/16.20 fresh18(true2, true2, e2)
% 121.88/16.20 = { by axiom 4 (lemma8_4) }
% 121.88/16.20 true2
% 121.88/16.20
% 121.88/16.20 Lemma 23: element(apply(alpha, a2(e2)), b) = true2.
% 121.88/16.20 Proof:
% 121.88/16.20 element(apply(alpha, a2(e2)), b)
% 121.88/16.20 = { by axiom 13 (morphism_1) R->L }
% 121.88/16.20 fresh15(true2, true2, alpha, a, b, a2(e2))
% 121.88/16.20 = { by axiom 5 (lemma8_5) R->L }
% 121.88/16.20 fresh15(fresh17(true2, true2, e2), true2, alpha, a, b, a2(e2))
% 121.88/16.20 = { by axiom 1 (lemma12) R->L }
% 121.88/16.20 fresh15(fresh17(element(e2, e), true2, e2), true2, alpha, a, b, a2(e2))
% 121.88/16.20 = { by axiom 10 (lemma8_5) }
% 121.88/16.20 fresh15(element(a2(e2), a), true2, alpha, a, b, a2(e2))
% 121.88/16.20 = { by axiom 19 (morphism_1) }
% 121.88/16.20 fresh14(morphism(alpha, a, b), true2, alpha, b, a2(e2))
% 121.88/16.20 = { by axiom 6 (alpha_morphism) }
% 121.88/16.20 fresh14(true2, true2, alpha, b, a2(e2))
% 121.88/16.20 = { by axiom 11 (morphism_1) }
% 121.88/16.20 true2
% 121.88/16.20
% 121.88/16.20 Goal 1 (lemma12_1): tuple(apply(g, subtract(b, X, Y)), element(X, b), element(Y, b)) = tuple(e2, true2, true2).
% 121.88/16.20 The goal is true when:
% 121.88/16.20 X = b1(e2)
% 121.88/16.20 Y = apply(alpha, a2(e2))
% 121.88/16.20
% 121.88/16.20 Proof:
% 121.88/16.20 tuple(apply(g, subtract(b, b1(e2), apply(alpha, a2(e2)))), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.20 = { by axiom 17 (subtract_distribution) R->L }
% 121.88/16.21 tuple(fresh7(true2, true2, g, b, e, b1(e2), apply(alpha, a2(e2))), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 7 (g_morphism) R->L }
% 121.88/16.21 tuple(fresh7(morphism(g, b, e), true2, g, b, e, b1(e2), apply(alpha, a2(e2))), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 21 (subtract_distribution) R->L }
% 121.88/16.21 tuple(fresh28(element(apply(alpha, a2(e2)), b), true2, g, b, e, b1(e2), apply(alpha, a2(e2))), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by lemma 23 }
% 121.88/16.21 tuple(fresh28(true2, true2, g, b, e, b1(e2), apply(alpha, a2(e2))), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 20 (subtract_distribution) }
% 121.88/16.21 tuple(fresh29(element(b1(e2), b), true2, g, b, e, b1(e2), apply(alpha, a2(e2))), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by lemma 22 }
% 121.88/16.21 tuple(fresh29(true2, true2, g, b, e, b1(e2), apply(alpha, a2(e2))), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 18 (subtract_distribution) }
% 121.88/16.21 tuple(subtract(e, apply(g, b1(e2)), apply(g, apply(alpha, a2(e2)))), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 12 (lemma8_1) R->L }
% 121.88/16.21 tuple(subtract(e, apply(g, b1(e2)), fresh21(element(e2, e), true2, e2)), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 1 (lemma12) }
% 121.88/16.21 tuple(subtract(e, apply(g, b1(e2)), fresh21(true2, true2, e2)), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 2 (lemma8_1) }
% 121.88/16.21 tuple(subtract(e, apply(g, b1(e2)), e1(e2)), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 3 (lemma8_2) R->L }
% 121.88/16.21 tuple(subtract(e, apply(g, b1(e2)), fresh20(true2, true2, e2)), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 1 (lemma12) R->L }
% 121.88/16.21 tuple(subtract(e, apply(g, b1(e2)), fresh20(element(e2, e), true2, e2)), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 15 (lemma8_2) }
% 121.88/16.21 tuple(subtract(e, apply(g, b1(e2)), subtract(e, apply(g, b1(e2)), e2)), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 14 (subtract_cancellation) R->L }
% 121.88/16.21 tuple(fresh8(true2, true2, e, apply(g, b1(e2)), e2), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 1 (lemma12) R->L }
% 121.88/16.21 tuple(fresh8(element(e2, e), true2, e, apply(g, b1(e2)), e2), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 16 (subtract_cancellation) }
% 121.88/16.21 tuple(fresh(element(apply(g, b1(e2)), e), true2, e, apply(g, b1(e2)), e2), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 13 (morphism_1) R->L }
% 121.88/16.21 tuple(fresh(fresh15(true2, true2, g, b, e, b1(e2)), true2, e, apply(g, b1(e2)), e2), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by lemma 22 R->L }
% 121.88/16.21 tuple(fresh(fresh15(element(b1(e2), b), true2, g, b, e, b1(e2)), true2, e, apply(g, b1(e2)), e2), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 19 (morphism_1) }
% 121.88/16.21 tuple(fresh(fresh14(morphism(g, b, e), true2, g, e, b1(e2)), true2, e, apply(g, b1(e2)), e2), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 7 (g_morphism) }
% 121.88/16.21 tuple(fresh(fresh14(true2, true2, g, e, b1(e2)), true2, e, apply(g, b1(e2)), e2), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 11 (morphism_1) }
% 121.88/16.21 tuple(fresh(true2, true2, e, apply(g, b1(e2)), e2), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by axiom 8 (subtract_cancellation) }
% 121.88/16.21 tuple(e2, element(b1(e2), b), element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by lemma 22 }
% 121.88/16.21 tuple(e2, true2, element(apply(alpha, a2(e2)), b))
% 121.88/16.21 = { by lemma 23 }
% 121.88/16.21 tuple(e2, true2, true2)
% 121.88/16.21 % SZS output end Proof
% 121.88/16.21
% 121.88/16.21 RESULT: Theorem (the conjecture is true).
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