0.12/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.12/0.14	% 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   : n005.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   : 1200
0.14/0.36	% WCLimit    : 120
0.14/0.36	% DateTime   : Tue Jul 13 15:14:03 EDT 2021
0.14/0.36	% CPUTime    : 
99.43/12.98	% SZS status Theorem
99.43/12.98	
99.43/12.99	% SZS output start Proof
99.43/12.99	Take the following subset of the input axioms:
99.43/12.99	  fof(alpha_morphism, axiom, morphism(alpha, a, b)).
99.43/12.99	  fof(g_morphism, axiom, morphism(g, b, e)).
99.43/12.99	  fof(lemma12, conjecture, ![E]: (?[B1, B2]: (element(B1, b) & (element(B2, b) & apply(g, subtract(b, B1, B2))=E)) <= element(E, e))).
99.43/12.99	  fof(lemma8, axiom, ![E]: (element(E, e) => ?[B1, E1, A]: (element(E1, e) & (E1=apply(g, apply(alpha, A)) & (apply(gamma, apply(f, A))=E1 & (element(A, a) & (E1=subtract(e, apply(g, B1), E) & element(B1, b)))))))).
99.43/12.99	  fof(morphism, axiom, ![Morphism, Dom, Cod]: (morphism(Morphism, Dom, Cod) => (![El]: (element(El, Dom) => element(apply(Morphism, El), Cod)) & zero(Cod)=apply(Morphism, zero(Dom))))).
99.43/12.99	  fof(subtract_cancellation, axiom, ![Dom, El1, El2]: (El2=subtract(Dom, El1, subtract(Dom, El1, El2)) <= (element(El1, Dom) & element(El2, Dom)))).
99.43/12.99	  fof(subtract_distribution, axiom, ![Morphism, Dom, Cod]: (![El1, El2]: ((element(El2, Dom) & element(El1, Dom)) => apply(Morphism, subtract(Dom, El1, El2))=subtract(Cod, apply(Morphism, El1), apply(Morphism, El2))) <= morphism(Morphism, Dom, Cod))).
99.43/12.99	
99.43/12.99	Now clausify the problem and encode Horn clauses using encoding 3 of
99.43/12.99	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
99.43/12.99	We repeatedly replace C & s=t => u=v by the two clauses:
99.43/12.99	  fresh(y, y, x1...xn) = u
99.43/12.99	  C => fresh(s, t, x1...xn) = v
99.43/12.99	where fresh is a fresh function symbol and x1..xn are the free
99.43/12.99	variables of u and v.
99.43/12.99	A predicate p(X) is encoded as p(X)=true (this is sound, because the
99.43/12.99	input problem has no model of domain size 1).
99.43/12.99	
99.43/12.99	The encoding turns the above axioms into the following unit equations and goals:
99.43/12.99	
99.43/12.99	Axiom 1 (lemma12): element(e2, e) = true2.
99.43/12.99	Axiom 2 (lemma8): fresh22(X, X, Y) = e1(Y).
99.43/12.99	Axiom 3 (lemma8_2): fresh20(X, X, Y) = e1(Y).
99.43/12.99	Axiom 4 (lemma8_4): fresh18(X, X, Y) = true2.
99.43/12.99	Axiom 5 (lemma8_5): fresh17(X, X, Y) = true2.
99.43/12.99	Axiom 6 (alpha_morphism): morphism(alpha, a, b) = true2.
99.43/12.99	Axiom 7 (g_morphism): morphism(g, b, e) = true2.
99.43/12.99	Axiom 8 (subtract_cancellation): fresh(X, X, Y, Z, W) = W.
99.43/12.99	Axiom 9 (lemma8_4): fresh18(element(X, e), true2, X) = element(b1(X), b).
99.43/12.99	Axiom 10 (lemma8_5): fresh17(element(X, e), true2, X) = element(a2(X), a).
99.43/12.99	Axiom 11 (morphism_1): fresh14(X, X, Y, Z, W) = true2.
99.43/12.99	Axiom 12 (lemma8): fresh22(element(X, e), true2, X) = apply(g, apply(alpha, a2(X))).
99.43/12.99	Axiom 13 (morphism_1): fresh15(X, X, Y, Z, W, V) = element(apply(Y, V), W).
99.43/12.99	Axiom 14 (subtract_cancellation): fresh8(X, X, Y, Z, W) = subtract(Y, Z, subtract(Y, Z, W)).
99.43/12.99	Axiom 15 (lemma8_2): fresh20(element(X, e), true2, X) = subtract(e, apply(g, b1(X)), X).
99.43/12.99	Axiom 16 (subtract_cancellation): fresh8(element(X, Y), true2, Y, Z, X) = fresh(element(Z, Y), true2, Y, Z, X).
99.43/12.99	Axiom 17 (subtract_distribution): fresh7(X, X, Y, Z, W, V, U) = apply(Y, subtract(Z, V, U)).
99.43/12.99	Axiom 18 (subtract_distribution): fresh71(X, X, Y, Z, W, V, U) = subtract(W, apply(Y, V), apply(Y, U)).
99.43/12.99	Axiom 19 (morphism_1): fresh15(element(X, Y), true2, Z, Y, W, X) = fresh14(morphism(Z, Y, W), true2, Z, W, X).
99.43/12.99	Axiom 20 (subtract_distribution): fresh70(X, X, Y, Z, W, V, U) = fresh71(element(V, Z), true2, Y, Z, W, V, U).
99.43/12.99	Axiom 21 (subtract_distribution): fresh70(element(X, Y), true2, Z, Y, W, V, X) = fresh7(morphism(Z, Y, W), true2, Z, Y, W, V, X).
99.43/12.99	
99.43/12.99	Lemma 22: element(b1(e2), b) = true2.
99.43/12.99	Proof:
99.43/12.99	  element(b1(e2), b)
99.43/12.99	= { by axiom 9 (lemma8_4) R->L }
99.43/12.99	  fresh18(element(e2, e), true2, e2)
99.43/12.99	= { by axiom 1 (lemma12) }
99.43/12.99	  fresh18(true2, true2, e2)
99.43/12.99	= { by axiom 4 (lemma8_4) }
99.43/12.99	  true2
99.43/12.99	
99.43/12.99	Lemma 23: element(apply(alpha, a2(e2)), b) = true2.
99.43/12.99	Proof:
99.43/12.99	  element(apply(alpha, a2(e2)), b)
99.43/12.99	= { by axiom 13 (morphism_1) R->L }
99.43/12.99	  fresh15(true2, true2, alpha, a, b, a2(e2))
99.43/12.99	= { by axiom 5 (lemma8_5) R->L }
99.43/12.99	  fresh15(fresh17(true2, true2, e2), true2, alpha, a, b, a2(e2))
99.43/12.99	= { by axiom 1 (lemma12) R->L }
99.43/12.99	  fresh15(fresh17(element(e2, e), true2, e2), true2, alpha, a, b, a2(e2))
99.43/12.99	= { by axiom 10 (lemma8_5) }
99.43/12.99	  fresh15(element(a2(e2), a), true2, alpha, a, b, a2(e2))
99.43/12.99	= { by axiom 19 (morphism_1) }
99.43/12.99	  fresh14(morphism(alpha, a, b), true2, alpha, b, a2(e2))
99.43/12.99	= { by axiom 6 (alpha_morphism) }
99.43/12.99	  fresh14(true2, true2, alpha, b, a2(e2))
99.43/12.99	= { by axiom 11 (morphism_1) }
99.43/12.99	  true2
99.43/12.99	
99.43/12.99	Goal 1 (lemma12_1): tuple(apply(g, subtract(b, X, Y)), element(X, b), element(Y, b)) = tuple(e2, true2, true2).
99.43/12.99	The goal is true when:
99.43/12.99	  X = b1(e2)
99.43/12.99	  Y = apply(alpha, a2(e2))
99.43/12.99	
99.43/12.99	Proof:
99.43/12.99	  tuple(apply(g, subtract(b, b1(e2), apply(alpha, a2(e2)))), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/12.99	= { by axiom 17 (subtract_distribution) R->L }
99.43/12.99	  tuple(fresh7(true2, true2, g, b, e, b1(e2), apply(alpha, a2(e2))), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/12.99	= { by axiom 7 (g_morphism) R->L }
99.43/12.99	  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))
99.43/12.99	= { by axiom 21 (subtract_distribution) R->L }
99.43/12.99	  tuple(fresh70(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))
99.43/12.99	= { by lemma 23 }
99.43/12.99	  tuple(fresh70(true2, true2, g, b, e, b1(e2), apply(alpha, a2(e2))), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/12.99	= { by axiom 20 (subtract_distribution) }
99.43/12.99	  tuple(fresh71(element(b1(e2), b), true2, g, b, e, b1(e2), apply(alpha, a2(e2))), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/12.99	= { by lemma 22 }
99.43/12.99	  tuple(fresh71(true2, true2, g, b, e, b1(e2), apply(alpha, a2(e2))), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/12.99	= { by axiom 18 (subtract_distribution) }
99.43/12.99	  tuple(subtract(e, apply(g, b1(e2)), apply(g, apply(alpha, a2(e2)))), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/12.99	= { by axiom 12 (lemma8) R->L }
99.43/12.99	  tuple(subtract(e, apply(g, b1(e2)), fresh22(element(e2, e), true2, e2)), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/13.00	= { by axiom 1 (lemma12) }
99.43/13.00	  tuple(subtract(e, apply(g, b1(e2)), fresh22(true2, true2, e2)), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/13.00	= { by axiom 2 (lemma8) }
99.43/13.00	  tuple(subtract(e, apply(g, b1(e2)), e1(e2)), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/13.00	= { by axiom 3 (lemma8_2) R->L }
99.43/13.00	  tuple(subtract(e, apply(g, b1(e2)), fresh20(true2, true2, e2)), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/13.00	= { by axiom 1 (lemma12) R->L }
99.43/13.00	  tuple(subtract(e, apply(g, b1(e2)), fresh20(element(e2, e), true2, e2)), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/13.00	= { by axiom 15 (lemma8_2) }
99.43/13.00	  tuple(subtract(e, apply(g, b1(e2)), subtract(e, apply(g, b1(e2)), e2)), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/13.00	= { by axiom 14 (subtract_cancellation) R->L }
99.43/13.00	  tuple(fresh8(true2, true2, e, apply(g, b1(e2)), e2), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/13.00	= { by axiom 1 (lemma12) R->L }
99.43/13.00	  tuple(fresh8(element(e2, e), true2, e, apply(g, b1(e2)), e2), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/13.00	= { by axiom 16 (subtract_cancellation) }
99.43/13.00	  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))
99.43/13.00	= { by axiom 13 (morphism_1) R->L }
99.43/13.00	  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))
99.43/13.00	= { by lemma 22 R->L }
99.43/13.00	  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))
99.43/13.00	= { by axiom 19 (morphism_1) }
99.43/13.00	  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))
99.43/13.00	= { by axiom 7 (g_morphism) }
99.43/13.00	  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))
99.43/13.00	= { by axiom 11 (morphism_1) }
99.43/13.00	  tuple(fresh(true2, true2, e, apply(g, b1(e2)), e2), element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/13.00	= { by axiom 8 (subtract_cancellation) }
99.43/13.00	  tuple(e2, element(b1(e2), b), element(apply(alpha, a2(e2)), b))
99.43/13.00	= { by lemma 22 }
99.43/13.00	  tuple(e2, true2, element(apply(alpha, a2(e2)), b))
99.43/13.00	= { by lemma 23 }
99.43/13.00	  tuple(e2, true2, true2)
99.43/13.00	% SZS output end Proof
99.43/13.00	
99.43/13.00	RESULT: Theorem (the conjecture is true).
99.43/13.02	EOF