0.00/0.04	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.04	% Command    : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn
0.02/0.24	% Computer   : n010.star.cs.uiowa.edu
0.02/0.24	% Model      : x86_64 x86_64
0.02/0.24	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.02/0.24	% Memory     : 32218.625MB
0.02/0.24	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.02/0.24	% CPULimit   : 300
0.02/0.24	% DateTime   : Sat Jul 14 05:28:40 CDT 2018
0.02/0.24	% CPUTime    : 
32.45/32.70	% SZS status Theorem
32.45/32.70	
32.51/32.72	% SZS output start Proof
32.51/32.72	Take the following subset of the input axioms:
32.51/32.73	  fof(alpha_morphism, axiom, morphism(alpha, a, b)).
32.51/32.73	  fof(g_morphism, axiom, morphism(g, b, e)).
32.51/32.73	  fof(lemma12, conjecture,
32.51/32.73	      ![E]:
32.51/32.73	        (element(E, e)
32.51/32.73	           => ?[B1, B2]:
32.51/32.73	                (element(B1, b)
32.51/32.73	                   & (element(B2, b) & E=apply(g, subtract(b, B1, B2)))))).
32.51/32.73	  fof(lemma8, axiom,
32.51/32.73	      ![E]:
32.51/32.73	        (element(E, e)
32.51/32.73	           => ?[B1, A, E1]:
32.51/32.73	                (element(E1, e)
32.51/32.73	                   & (E1=subtract(e, apply(g, B1), E)
32.51/32.73	                        & (element(A, a)
32.51/32.73	                             & (apply(g, apply(alpha, A))=E1
32.51/32.73	                                  & (apply(gamma, apply(f, A))=E1 & element(B1, b)))))))).
32.51/32.73	  fof(morphism, axiom,
32.51/32.73	      ![Dom, Cod, Morphism]:
32.51/32.73	        ((zero(Cod)=apply(Morphism, zero(Dom))
32.51/32.73	            & ![El]: (element(El, Dom) => element(apply(Morphism, El), Cod)))
32.51/32.73	           <= morphism(Morphism, Dom, Cod))).
32.51/32.73	  fof(subtract_cancellation, axiom,
32.51/32.73	      ![Dom, El1, El2]:
32.51/32.73	        ((element(El2, Dom) & element(El1, Dom))
32.51/32.73	           => subtract(Dom, El1, subtract(Dom, El1, El2))=El2)).
32.51/32.73	  fof(subtract_distribution, axiom,
32.51/32.73	      ![Dom, Cod, Morphism]:
32.51/32.73	        (![El1, El2]:
32.51/32.73	           (subtract(Cod, apply(Morphism, El1),
32.51/32.73	                     apply(Morphism, El2))=apply(Morphism, subtract(Dom, El1, El2))
32.51/32.73	              <= (element(El1, Dom) & element(El2, Dom)))
32.51/32.73	           <= morphism(Morphism, Dom, Cod))).
32.51/32.73	
32.51/32.73	Now clausify the problem and encode Horn clauses using encoding 3 of
32.51/32.73	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
32.51/32.73	We repeatedly replace C & s=t => u=v by the two clauses:
32.51/32.73	  $$fresh(y, y, x1...xn) = u
32.51/32.73	  C => $$fresh(s, t, x1...xn) = v
32.51/32.73	where $$fresh is a fresh function symbol and x1..xn are the free
32.51/32.73	variables of u and v.
32.51/32.73	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
32.51/32.73	input problem has no model of domain size 1).
32.51/32.73	
32.51/32.73	The encoding turns the above axioms into the following unit equations and goals:
32.51/32.73	
32.51/32.73	Axiom 37 (lemma8): $$fresh21(X, X, Y) = sK6_lemma8_E1(Y).
32.51/32.73	Axiom 39 (lemma8_2): $$fresh19(X, X, Y) = sK6_lemma8_E1(Y).
32.51/32.73	Axiom 41 (lemma8_4): $$fresh17(X, X, Y) = $$true2.
32.51/32.73	Axiom 42 (lemma8_5): $$fresh16(X, X, Y) = $$true2.
32.51/32.73	Axiom 43 (morphism): $$fresh15(X, X, Y, Z, W, V) = element(apply(Y, V), W).
32.51/32.73	Axiom 44 (morphism): $$fresh14(X, X, Y, Z, W) = $$true2.
32.51/32.73	Axiom 62 (subtract_cancellation): $$fresh(X, X, Y, Z, W) = W.
32.51/32.73	Axiom 63 (subtract_cancellation): $$fresh8(X, X, Y, Z, W) = subtract(Y, Z, subtract(Y, Z, W)).
32.51/32.73	Axiom 64 (subtract_distribution): $$fresh37(X, X, Y, Z, W, V, U) = apply(Y, subtract(Z, V, U)).
32.51/32.73	Axiom 65 (subtract_distribution): $$fresh7(X, X, Y, Z, W, V, U) = subtract(W, apply(Y, V), apply(Y, U)).
32.51/32.73	Axiom 66 (subtract_distribution): $$fresh38(X, X, Y, Z, W, V, U) = $$fresh37(element(V, Z), $$true2, Y, Z, W, V, U).
32.51/32.73	Axiom 77 (morphism): $$fresh15(morphism(X, Y, Z), $$true2, X, Y, Z, W) = $$fresh14(element(W, Y), $$true2, X, Z, W).
32.51/32.73	Axiom 86 (subtract_cancellation): $$fresh8(element(X, Y), $$true2, Y, Z, X) = $$fresh(element(Z, Y), $$true2, Y, Z, X).
32.51/32.73	Axiom 89 (subtract_distribution): $$fresh38(morphism(X, Y, Z), $$true2, X, Y, Z, W, V) = $$fresh7(element(V, Y), $$true2, X, Y, Z, W, V).
32.51/32.73	Axiom 94 (g_morphism): morphism(g, b, e) = $$true2.
32.51/32.73	Axiom 96 (lemma8_5): $$fresh16(element(X, e), $$true2, X) = element(sK4_lemma8_A(X), a).
32.51/32.73	Axiom 97 (lemma8_4): $$fresh17(element(X, e), $$true2, X) = element(sK5_lemma8_B1(X), b).
32.51/32.73	Axiom 99 (lemma8_2): $$fresh19(element(X, e), $$true2, X) = subtract(e, apply(g, sK5_lemma8_B1(X)), X).
32.51/32.73	Axiom 101 (lemma8): $$fresh21(element(X, e), $$true2, X) = apply(g, apply(alpha, sK4_lemma8_A(X))).
32.51/32.73	Axiom 111 (alpha_morphism): morphism(alpha, a, b) = $$true2.
32.51/32.73	Axiom 122 (lemma12): element(sK3_lemma12_E, e) = $$true2.
32.51/32.73	
32.51/32.73	Lemma 123: element(sK5_lemma8_B1(sK3_lemma12_E), b) = $$true2.
32.51/32.73	Proof:
32.51/32.73	  element(sK5_lemma8_B1(sK3_lemma12_E), b)
32.51/32.73	= { by axiom 97 (lemma8_4) }
32.51/32.73	  $$fresh17(element(sK3_lemma12_E, e), $$true2, sK3_lemma12_E)
32.51/32.73	= { by axiom 122 (lemma12) }
32.51/32.73	  $$fresh17($$true2, $$true2, sK3_lemma12_E)
32.51/32.73	= { by axiom 41 (lemma8_4) }
32.51/32.73	  $$true2
32.51/32.73	
32.51/32.73	Lemma 124: element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b) = $$true2.
32.51/32.73	Proof:
32.51/32.73	  element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b)
32.51/32.73	= { by axiom 43 (morphism) }
32.51/32.73	  $$fresh15($$true2, $$true2, alpha, a, b, sK4_lemma8_A(sK3_lemma12_E))
32.51/32.73	= { by axiom 111 (alpha_morphism) }
32.51/32.73	  $$fresh15(morphism(alpha, a, b), $$true2, alpha, a, b, sK4_lemma8_A(sK3_lemma12_E))
32.51/32.73	= { by axiom 77 (morphism) }
32.51/32.73	  $$fresh14(element(sK4_lemma8_A(sK3_lemma12_E), a), $$true2, alpha, b, sK4_lemma8_A(sK3_lemma12_E))
32.51/32.73	= { by axiom 96 (lemma8_5) }
32.51/32.73	  $$fresh14($$fresh16(element(sK3_lemma12_E, e), $$true2, sK3_lemma12_E), $$true2, alpha, b, sK4_lemma8_A(sK3_lemma12_E))
32.51/32.73	= { by axiom 122 (lemma12) }
32.51/32.73	  $$fresh14($$fresh16($$true2, $$true2, sK3_lemma12_E), $$true2, alpha, b, sK4_lemma8_A(sK3_lemma12_E))
32.51/32.73	= { by axiom 42 (lemma8_5) }
32.51/32.73	  $$fresh14($$true2, $$true2, alpha, b, sK4_lemma8_A(sK3_lemma12_E))
32.51/32.73	= { by axiom 44 (morphism) }
32.51/32.74	  $$true2
32.51/32.74	
32.51/32.74	Goal 1 (lemma12_1): tuple(sK3_lemma12_E, element(X, b), element(Y, b)) = tuple(apply(g, subtract(b, X, Y)), $$true2, $$true2).
32.51/32.74	The goal is true when:
32.51/32.74	  X = sK5_lemma8_B1(sK3_lemma12_E)
32.51/32.74	  Y = apply(alpha, sK4_lemma8_A(sK3_lemma12_E))
32.51/32.74	
32.51/32.74	Proof:
32.51/32.74	  tuple(sK3_lemma12_E, element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 62 (subtract_cancellation) }
32.51/32.74	  tuple($$fresh($$true2, $$true2, e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), sK3_lemma12_E), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 44 (morphism) }
32.51/32.74	  tuple($$fresh($$fresh14($$true2, $$true2, g, e, sK5_lemma8_B1(sK3_lemma12_E)), $$true2, e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), sK3_lemma12_E), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by lemma 123 }
32.51/32.74	  tuple($$fresh($$fresh14(element(sK5_lemma8_B1(sK3_lemma12_E), b), $$true2, g, e, sK5_lemma8_B1(sK3_lemma12_E)), $$true2, e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), sK3_lemma12_E), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 77 (morphism) }
32.51/32.74	  tuple($$fresh($$fresh15(morphism(g, b, e), $$true2, g, b, e, sK5_lemma8_B1(sK3_lemma12_E)), $$true2, e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), sK3_lemma12_E), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 94 (g_morphism) }
32.51/32.74	  tuple($$fresh($$fresh15($$true2, $$true2, g, b, e, sK5_lemma8_B1(sK3_lemma12_E)), $$true2, e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), sK3_lemma12_E), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 43 (morphism) }
32.51/32.74	  tuple($$fresh(element(apply(g, sK5_lemma8_B1(sK3_lemma12_E)), e), $$true2, e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), sK3_lemma12_E), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 86 (subtract_cancellation) }
32.51/32.74	  tuple($$fresh8(element(sK3_lemma12_E, e), $$true2, e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), sK3_lemma12_E), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 122 (lemma12) }
32.51/32.74	  tuple($$fresh8($$true2, $$true2, e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), sK3_lemma12_E), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 63 (subtract_cancellation) }
32.51/32.74	  tuple(subtract(e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), subtract(e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), sK3_lemma12_E)), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 99 (lemma8_2) }
32.51/32.74	  tuple(subtract(e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), $$fresh19(element(sK3_lemma12_E, e), $$true2, sK3_lemma12_E)), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 122 (lemma12) }
32.51/32.74	  tuple(subtract(e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), $$fresh19($$true2, $$true2, sK3_lemma12_E)), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 39 (lemma8_2) }
32.51/32.74	  tuple(subtract(e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), sK6_lemma8_E1(sK3_lemma12_E)), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 37 (lemma8) }
32.51/32.74	  tuple(subtract(e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), $$fresh21($$true2, $$true2, sK3_lemma12_E)), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 122 (lemma12) }
32.51/32.74	  tuple(subtract(e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), $$fresh21(element(sK3_lemma12_E, e), $$true2, sK3_lemma12_E)), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 101 (lemma8) }
32.51/32.74	  tuple(subtract(e, apply(g, sK5_lemma8_B1(sK3_lemma12_E)), apply(g, apply(alpha, sK4_lemma8_A(sK3_lemma12_E)))), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 65 (subtract_distribution) }
32.51/32.74	  tuple($$fresh7($$true2, $$true2, g, b, e, sK5_lemma8_B1(sK3_lemma12_E), apply(alpha, sK4_lemma8_A(sK3_lemma12_E))), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by lemma 124 }
32.51/32.74	  tuple($$fresh7(element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b), $$true2, g, b, e, sK5_lemma8_B1(sK3_lemma12_E), apply(alpha, sK4_lemma8_A(sK3_lemma12_E))), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 89 (subtract_distribution) }
32.51/32.74	  tuple($$fresh38(morphism(g, b, e), $$true2, g, b, e, sK5_lemma8_B1(sK3_lemma12_E), apply(alpha, sK4_lemma8_A(sK3_lemma12_E))), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 94 (g_morphism) }
32.51/32.74	  tuple($$fresh38($$true2, $$true2, g, b, e, sK5_lemma8_B1(sK3_lemma12_E), apply(alpha, sK4_lemma8_A(sK3_lemma12_E))), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 66 (subtract_distribution) }
32.51/32.74	  tuple($$fresh37(element(sK5_lemma8_B1(sK3_lemma12_E), b), $$true2, g, b, e, sK5_lemma8_B1(sK3_lemma12_E), apply(alpha, sK4_lemma8_A(sK3_lemma12_E))), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by lemma 123 }
32.51/32.74	  tuple($$fresh37($$true2, $$true2, g, b, e, sK5_lemma8_B1(sK3_lemma12_E), apply(alpha, sK4_lemma8_A(sK3_lemma12_E))), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by axiom 64 (subtract_distribution) }
32.51/32.74	  tuple(apply(g, subtract(b, sK5_lemma8_B1(sK3_lemma12_E), apply(alpha, sK4_lemma8_A(sK3_lemma12_E)))), element(sK5_lemma8_B1(sK3_lemma12_E), b), element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by lemma 123 }
32.51/32.74	  tuple(apply(g, subtract(b, sK5_lemma8_B1(sK3_lemma12_E), apply(alpha, sK4_lemma8_A(sK3_lemma12_E)))), $$true2, element(apply(alpha, sK4_lemma8_A(sK3_lemma12_E)), b))
32.51/32.74	= { by lemma 124 }
32.51/32.74	  tuple(apply(g, subtract(b, sK5_lemma8_B1(sK3_lemma12_E), apply(alpha, sK4_lemma8_A(sK3_lemma12_E)))), $$true2, $$true2)
32.51/32.74	% SZS output end Proof
32.51/32.74	
32.51/32.74	RESULT: Theorem (the conjecture is true).
32.56/32.77	EOF