0.09/0.09	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.09/0.10	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.09/0.30	% Computer   : n015.cluster.edu
0.09/0.30	% Model      : x86_64 x86_64
0.09/0.30	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.09/0.30	% Memory     : 8042.1875MB
0.09/0.30	% OS         : Linux 3.10.0-693.el7.x86_64
0.09/0.30	% CPULimit   : 960
0.09/0.30	% WCLimit    : 120
0.09/0.30	% DateTime   : Thu Jul  2 08:24:16 EDT 2020
0.09/0.30	% CPUTime    : 
196.95/25.40	% SZS status Theorem
196.95/25.40	
197.98/25.43	% SZS output start Proof
197.98/25.43	Take the following subset of the input axioms:
197.98/25.45	  fof(alpha_morphism, axiom, morphism(alpha, a, b)).
197.98/25.45	  fof(g_morphism, axiom, morphism(g, b, e)).
197.98/25.45	  fof(lemma12, conjecture, ![E]: (element(E, e) => ?[B1, B2]: (apply(g, subtract(b, B1, B2))=E & (element(B2, b) & element(B1, b))))).
197.98/25.45	  fof(lemma8, axiom, ![E]: (?[B1, A, E1]: (element(E1, e) & (E1=apply(g, apply(alpha, A)) & (E1=apply(gamma, apply(f, A)) & (element(A, a) & (subtract(e, apply(g, B1), E)=E1 & element(B1, b)))))) <= element(E, e))).
197.98/25.45	  fof(morphism, axiom, ![Dom, Cod, Morphism]: ((zero(Cod)=apply(Morphism, zero(Dom)) & ![El]: (element(El, Dom) => element(apply(Morphism, El), Cod))) <= morphism(Morphism, Dom, Cod))).
197.98/25.45	  fof(subtract_cancellation, axiom, ![Dom, El1, El2]: ((element(El1, Dom) & element(El2, Dom)) => El2=subtract(Dom, El1, subtract(Dom, El1, El2)))).
197.98/25.45	  fof(subtract_distribution, axiom, ![Dom, Cod, Morphism]: (![El1, El2]: (subtract(Cod, apply(Morphism, El1), apply(Morphism, El2))=apply(Morphism, subtract(Dom, El1, El2)) <= (element(El1, Dom) & element(El2, Dom))) <= morphism(Morphism, Dom, Cod))).
197.98/25.45	
197.98/25.45	Now clausify the problem and encode Horn clauses using encoding 3 of
197.98/25.45	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
197.98/25.45	We repeatedly replace C & s=t => u=v by the two clauses:
197.98/25.45	  fresh(y, y, x1...xn) = u
197.98/25.45	  C => fresh(s, t, x1...xn) = v
197.98/25.45	where fresh is a fresh function symbol and x1..xn are the free
197.98/25.45	variables of u and v.
197.98/25.45	A predicate p(X) is encoded as p(X)=true (this is sound, because the
197.98/25.45	input problem has no model of domain size 1).
197.98/25.45	
197.98/25.45	The encoding turns the above axioms into the following unit equations and goals:
197.98/25.45	
197.98/25.45	Axiom 1 (lemma8): fresh22(X, X, Y) = sK4_lemma8_E1(Y).
197.98/25.45	Axiom 2 (lemma8_2): fresh20(X, X, Y) = sK4_lemma8_E1(Y).
197.98/25.45	Axiom 3 (lemma8_4): fresh18(X, X, Y) = true2.
197.98/25.45	Axiom 4 (lemma8_5): fresh17(X, X, Y) = true2.
197.98/25.45	Axiom 5 (morphism_1): fresh15(X, X, Y, Z, W, V) = element(apply(Y, V), W).
197.98/25.45	Axiom 6 (morphism_1): fresh14(X, X, Y, Z, W) = true2.
197.98/25.45	Axiom 7 (subtract_cancellation): fresh(X, X, Y, Z, W) = W.
197.98/25.45	Axiom 8 (subtract_cancellation): fresh8(X, X, Y, Z, W) = subtract(Y, Z, subtract(Y, Z, W)).
197.98/25.45	Axiom 9 (subtract_distribution): fresh7(X, X, Y, Z, W, V, U) = apply(Y, subtract(Z, V, U)).
197.98/25.45	Axiom 10 (subtract_distribution): fresh33(X, X, Y, Z, W, V, U) = subtract(W, apply(Y, V), apply(Y, U)).
197.98/25.45	Axiom 11 (subtract_distribution): fresh32(X, X, Y, Z, W, V, U) = fresh33(element(V, Z), true2, Y, Z, W, V, U).
197.98/25.45	Axiom 12 (morphism_1): fresh15(element(X, Y), true2, Z, Y, W, X) = fresh14(morphism(Z, Y, W), true2, Z, W, X).
197.98/25.45	Axiom 13 (subtract_cancellation): fresh8(element(X, Y), true2, Y, Z, X) = fresh(element(Z, Y), true2, Y, Z, X).
197.98/25.45	Axiom 14 (subtract_distribution): fresh32(element(X, Y), true2, Z, Y, W, V, X) = fresh7(morphism(Z, Y, W), true2, Z, Y, W, V, X).
197.98/25.45	Axiom 15 (alpha_morphism): morphism(alpha, a, b) = true2.
197.98/25.45	Axiom 16 (lemma8_5): fresh17(element(X, e), true2, X) = element(sK2_lemma8_A(X), a).
197.98/25.45	Axiom 17 (lemma8_4): fresh18(element(X, e), true2, X) = element(sK3_lemma8_B1(X), b).
197.98/25.45	Axiom 18 (lemma8_2): fresh20(element(X, e), true2, X) = subtract(e, apply(g, sK3_lemma8_B1(X)), X).
197.98/25.45	Axiom 19 (lemma8): fresh22(element(X, e), true2, X) = apply(g, apply(alpha, sK2_lemma8_A(X))).
197.98/25.45	Axiom 20 (g_morphism): morphism(g, b, e) = true2.
197.98/25.45	Axiom 21 (lemma12): element(sK1_lemma12_E, e) = true2.
197.98/25.45	
197.98/25.45	Lemma 22: element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b) = true2.
197.98/25.45	Proof:
197.98/25.45	  element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b)
197.98/25.45	= { by axiom 5 (morphism_1) }
197.98/25.45	  fresh15(true2, true2, alpha, a, b, sK2_lemma8_A(sK1_lemma12_E))
197.98/25.45	= { by axiom 4 (lemma8_5) }
197.98/25.45	  fresh15(fresh17(true2, true2, sK1_lemma12_E), true2, alpha, a, b, sK2_lemma8_A(sK1_lemma12_E))
197.98/25.45	= { by axiom 21 (lemma12) }
197.98/25.45	  fresh15(fresh17(element(sK1_lemma12_E, e), true2, sK1_lemma12_E), true2, alpha, a, b, sK2_lemma8_A(sK1_lemma12_E))
197.98/25.45	= { by axiom 16 (lemma8_5) }
197.98/25.45	  fresh15(element(sK2_lemma8_A(sK1_lemma12_E), a), true2, alpha, a, b, sK2_lemma8_A(sK1_lemma12_E))
197.98/25.45	= { by axiom 12 (morphism_1) }
197.98/25.45	  fresh14(morphism(alpha, a, b), true2, alpha, b, sK2_lemma8_A(sK1_lemma12_E))
197.98/25.46	= { by axiom 15 (alpha_morphism) }
197.98/25.46	  fresh14(true2, true2, alpha, b, sK2_lemma8_A(sK1_lemma12_E))
197.98/25.46	= { by axiom 6 (morphism_1) }
197.98/25.46	  true2
197.98/25.46	
197.98/25.46	Lemma 23: element(sK3_lemma8_B1(sK1_lemma12_E), b) = true2.
197.98/25.46	Proof:
197.98/25.46	  element(sK3_lemma8_B1(sK1_lemma12_E), b)
197.98/25.46	= { by axiom 17 (lemma8_4) }
197.98/25.46	  fresh18(element(sK1_lemma12_E, e), true2, sK1_lemma12_E)
197.98/25.46	= { by axiom 21 (lemma12) }
197.98/25.46	  fresh18(true2, true2, sK1_lemma12_E)
197.98/25.46	= { by axiom 3 (lemma8_4) }
197.98/25.47	  true2
197.98/25.47	
197.98/25.47	Goal 1 (lemma12_1): tuple(apply(g, subtract(b, X, Y)), element(X, b), element(Y, b)) = tuple(sK1_lemma12_E, true2, true2).
197.98/25.47	The goal is true when:
197.98/25.47	  X = sK3_lemma8_B1(sK1_lemma12_E)
197.98/25.47	  Y = apply(alpha, sK2_lemma8_A(sK1_lemma12_E))
197.98/25.47	
197.98/25.47	Proof:
197.98/25.47	  tuple(apply(g, subtract(b, sK3_lemma8_B1(sK1_lemma12_E), apply(alpha, sK2_lemma8_A(sK1_lemma12_E)))), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.47	= { by axiom 9 (subtract_distribution) }
197.98/25.47	  tuple(fresh7(true2, true2, g, b, e, sK3_lemma8_B1(sK1_lemma12_E), apply(alpha, sK2_lemma8_A(sK1_lemma12_E))), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.47	= { by axiom 20 (g_morphism) }
197.98/25.47	  tuple(fresh7(morphism(g, b, e), true2, g, b, e, sK3_lemma8_B1(sK1_lemma12_E), apply(alpha, sK2_lemma8_A(sK1_lemma12_E))), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.47	= { by axiom 14 (subtract_distribution) }
197.98/25.47	  tuple(fresh32(element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b), true2, g, b, e, sK3_lemma8_B1(sK1_lemma12_E), apply(alpha, sK2_lemma8_A(sK1_lemma12_E))), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.47	= { by lemma 22 }
197.98/25.47	  tuple(fresh32(true2, true2, g, b, e, sK3_lemma8_B1(sK1_lemma12_E), apply(alpha, sK2_lemma8_A(sK1_lemma12_E))), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.47	= { by axiom 11 (subtract_distribution) }
197.98/25.47	  tuple(fresh33(element(sK3_lemma8_B1(sK1_lemma12_E), b), true2, g, b, e, sK3_lemma8_B1(sK1_lemma12_E), apply(alpha, sK2_lemma8_A(sK1_lemma12_E))), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.47	= { by lemma 23 }
197.98/25.47	  tuple(fresh33(true2, true2, g, b, e, sK3_lemma8_B1(sK1_lemma12_E), apply(alpha, sK2_lemma8_A(sK1_lemma12_E))), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.47	= { by axiom 10 (subtract_distribution) }
197.98/25.47	  tuple(subtract(e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), apply(g, apply(alpha, sK2_lemma8_A(sK1_lemma12_E)))), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.47	= { by axiom 19 (lemma8) }
197.98/25.47	  tuple(subtract(e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), fresh22(element(sK1_lemma12_E, e), true2, sK1_lemma12_E)), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.47	= { by axiom 21 (lemma12) }
197.98/25.47	  tuple(subtract(e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), fresh22(true2, true2, sK1_lemma12_E)), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.47	= { by axiom 1 (lemma8) }
197.98/25.47	  tuple(subtract(e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), sK4_lemma8_E1(sK1_lemma12_E)), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.47	= { by axiom 2 (lemma8_2) }
197.98/25.47	  tuple(subtract(e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), fresh20(true2, true2, sK1_lemma12_E)), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.47	= { by axiom 21 (lemma12) }
197.98/25.47	  tuple(subtract(e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), fresh20(element(sK1_lemma12_E, e), true2, sK1_lemma12_E)), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.47	= { by axiom 18 (lemma8_2) }
197.98/25.47	  tuple(subtract(e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), subtract(e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), sK1_lemma12_E)), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.47	= { by axiom 8 (subtract_cancellation) }
197.98/25.48	  tuple(fresh8(true2, true2, e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), sK1_lemma12_E), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.48	= { by axiom 21 (lemma12) }
197.98/25.48	  tuple(fresh8(element(sK1_lemma12_E, e), true2, e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), sK1_lemma12_E), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.48	= { by axiom 13 (subtract_cancellation) }
197.98/25.48	  tuple(fresh(element(apply(g, sK3_lemma8_B1(sK1_lemma12_E)), e), true2, e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), sK1_lemma12_E), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.48	= { by axiom 5 (morphism_1) }
197.98/25.48	  tuple(fresh(fresh15(true2, true2, g, b, e, sK3_lemma8_B1(sK1_lemma12_E)), true2, e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), sK1_lemma12_E), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.48	= { by lemma 23 }
197.98/25.48	  tuple(fresh(fresh15(element(sK3_lemma8_B1(sK1_lemma12_E), b), true2, g, b, e, sK3_lemma8_B1(sK1_lemma12_E)), true2, e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), sK1_lemma12_E), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.48	= { by axiom 12 (morphism_1) }
197.98/25.48	  tuple(fresh(fresh14(morphism(g, b, e), true2, g, e, sK3_lemma8_B1(sK1_lemma12_E)), true2, e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), sK1_lemma12_E), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.48	= { by axiom 20 (g_morphism) }
197.98/25.48	  tuple(fresh(fresh14(true2, true2, g, e, sK3_lemma8_B1(sK1_lemma12_E)), true2, e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), sK1_lemma12_E), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.48	= { by axiom 6 (morphism_1) }
197.98/25.48	  tuple(fresh(true2, true2, e, apply(g, sK3_lemma8_B1(sK1_lemma12_E)), sK1_lemma12_E), element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.48	= { by axiom 7 (subtract_cancellation) }
197.98/25.48	  tuple(sK1_lemma12_E, element(sK3_lemma8_B1(sK1_lemma12_E), b), element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.48	= { by lemma 23 }
197.98/25.48	  tuple(sK1_lemma12_E, true2, element(apply(alpha, sK2_lemma8_A(sK1_lemma12_E)), b))
197.98/25.48	= { by lemma 22 }
197.98/25.48	  tuple(sK1_lemma12_E, true2, true2)
197.98/25.48	% SZS output end Proof
197.98/25.48	
197.98/25.48	RESULT: Theorem (the conjecture is true).
197.98/25.51	EOF