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
0.03/0.24	% Computer   : n148.star.cs.uiowa.edu
0.03/0.24	% Model      : x86_64 x86_64
0.03/0.24	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.03/0.24	% Memory     : 32218.625MB
0.03/0.24	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.03/0.24	% CPULimit   : 300
0.03/0.24	% DateTime   : Sat Jul 14 04:23:40 CDT 2018
0.03/0.24	% CPUTime    : 
0.07/0.27	% SZS status Theorem
0.07/0.27	
0.07/0.28	% SZS output start Proof
0.07/0.28	Take the following subset of the input axioms:
0.07/0.29	  fof(ax2, axiom,
0.07/0.29	      ![U]: (sorti2(U) => ![V]: (sorti2(V) => sorti2(op2(U, V))))).
0.07/0.29	  fof(ax3, axiom,
0.07/0.29	      ?[U]: (![V]: (op1(V, V)=U <= sorti1(V)) & sorti1(U))).
0.07/0.29	  fof(ax4, axiom,
0.07/0.29	      ~?[U]: (sorti2(U) & ![V]: (sorti2(V) => op2(V, V)=U))).
0.07/0.29	  fof(co1, conjecture,
0.07/0.29	      (![U]: (sorti2(h(U)) <= sorti1(U))
0.07/0.29	         & ![V]: (sorti2(V) => sorti1(j(V))))
0.07/0.29	        => ~(![W]:
0.07/0.29	               (sorti1(W) => ![X]: (op2(h(W), h(X))=h(op1(W, X)) <= sorti1(X)))
0.07/0.29	               & (![Y]:
0.07/0.29	                    (![Z]: (op1(j(Y), j(Z))=j(op2(Y, Z)) <= sorti2(Z)) <= sorti2(Y))
0.07/0.29	                    & (![X1]: (X1=h(j(X1)) <= sorti2(X1))
0.07/0.29	                         & ![X2]: (sorti1(X2) => j(h(X2))=X2))))).
0.07/0.29	
0.07/0.29	Now clausify the problem and encode Horn clauses using encoding 3 of
0.07/0.29	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.07/0.29	We repeatedly replace C & s=t => u=v by the two clauses:
0.07/0.29	  $$fresh(y, y, x1...xn) = u
0.07/0.29	  C => $$fresh(s, t, x1...xn) = v
0.07/0.29	where $$fresh is a fresh function symbol and x1..xn are the free
0.07/0.29	variables of u and v.
0.07/0.29	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
0.07/0.29	input problem has no model of domain size 1).
0.07/0.29	
0.07/0.29	The encoding turns the above axioms into the following unit equations and goals:
0.07/0.29	
0.07/0.29	Axiom 3 (ax2): $$fresh12(X, X, Y, Z) = sorti2(op2(Y, Z)).
0.07/0.29	Axiom 4 (ax2): $$fresh11(X, X, Y, Z) = $$true2.
0.07/0.29	Axiom 5 (ax3_1): $$fresh10(X, X, Y) = sK2_ax3_U.
0.07/0.29	Axiom 6 (ax4_1): $$fresh9(X, X, Y) = $$true2.
0.07/0.29	Axiom 7 (co1): $$fresh8(X, X, Y) = $$true2.
0.07/0.29	Axiom 11 (co1_3): $$fresh5(X, X, Y) = $$true2.
0.07/0.29	Axiom 12 (co1_4): $$fresh4(X, X, Y, Z) = op1(j(Y), j(Z)).
0.07/0.29	Axiom 13 (co1_4): $$fresh3(X, X, Y, Z) = j(op2(Y, Z)).
0.07/0.29	Axiom 14 (co1_5): $$fresh(X, X, Y) = Y.
0.07/0.29	Axiom 15 (ax3_1): $$fresh10(sorti1(X), $$true2, X) = op1(X, X).
0.07/0.29	Axiom 16 (ax3): sorti1(sK2_ax3_U) = $$true2.
0.07/0.29	Axiom 17 (ax2): $$fresh12(sorti2(X), $$true2, Y, X) = $$fresh11(sorti2(Y), $$true2, Y, X).
0.07/0.29	Axiom 18 (ax4_1): $$fresh9(sorti2(X), $$true2, X) = sorti2(sK1_ax4_V(X)).
0.07/0.29	Axiom 20 (co1): $$fresh8(sorti1(X), $$true2, X) = sorti2(h(X)).
0.07/0.29	Axiom 23 (co1_3): $$fresh5(sorti2(X), $$true2, X) = sorti1(j(X)).
0.07/0.29	Axiom 24 (co1_4): $$fresh4(sorti2(X), $$true2, Y, X) = $$fresh3(sorti2(Y), $$true2, Y, X).
0.07/0.29	Axiom 25 (co1_5): $$fresh(sorti2(X), $$true2, X) = h(j(X)).
0.07/0.29	
0.07/0.29	Lemma 26: sorti2(h(sK2_ax3_U)) = $$true2.
0.07/0.29	Proof:
0.07/0.29	  sorti2(h(sK2_ax3_U))
0.07/0.29	= { by axiom 20 (co1) }
0.07/0.29	  $$fresh8(sorti1(sK2_ax3_U), $$true2, sK2_ax3_U)
0.07/0.29	= { by axiom 16 (ax3) }
0.07/0.29	  $$fresh8($$true2, $$true2, sK2_ax3_U)
0.07/0.29	= { by axiom 7 (co1) }
0.07/0.29	  $$true2
0.07/0.29	
0.07/0.29	Lemma 27: sorti2(sK1_ax4_V(h(sK2_ax3_U))) = $$true2.
0.07/0.29	Proof:
0.07/0.29	  sorti2(sK1_ax4_V(h(sK2_ax3_U)))
0.07/0.29	= { by axiom 18 (ax4_1) }
0.07/0.29	  $$fresh9(sorti2(h(sK2_ax3_U)), $$true2, h(sK2_ax3_U))
0.07/0.29	= { by lemma 26 }
0.07/0.29	  $$fresh9($$true2, $$true2, h(sK2_ax3_U))
0.07/0.29	= { by axiom 6 (ax4_1) }
0.07/0.29	  $$true2
0.07/0.29	
0.07/0.29	Goal 1 (ax4): tuple(op2(sK1_ax4_V(X), sK1_ax4_V(X)), sorti2(X)) = tuple(X, $$true2).
0.07/0.29	The goal is true when:
0.07/0.29	  X = h(sK2_ax3_U)
0.07/0.29	
0.07/0.29	Proof:
0.07/0.29	  tuple(op2(sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by axiom 14 (co1_5) }
0.07/0.29	  tuple($$fresh($$true2, $$true2, op2(sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U)))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by axiom 4 (ax2) }
0.07/0.29	  tuple($$fresh($$fresh11($$true2, $$true2, sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U))), $$true2, op2(sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U)))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by lemma 27 }
0.07/0.29	  tuple($$fresh($$fresh11(sorti2(sK1_ax4_V(h(sK2_ax3_U))), $$true2, sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U))), $$true2, op2(sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U)))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by axiom 17 (ax2) }
0.07/0.29	  tuple($$fresh($$fresh12(sorti2(sK1_ax4_V(h(sK2_ax3_U))), $$true2, sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U))), $$true2, op2(sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U)))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by lemma 27 }
0.07/0.29	  tuple($$fresh($$fresh12($$true2, $$true2, sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U))), $$true2, op2(sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U)))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by axiom 3 (ax2) }
0.07/0.29	  tuple($$fresh(sorti2(op2(sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U)))), $$true2, op2(sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U)))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by axiom 25 (co1_5) }
0.07/0.29	  tuple(h(j(op2(sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U))))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by axiom 13 (co1_4) }
0.07/0.29	  tuple(h($$fresh3($$true2, $$true2, sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U)))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by lemma 27 }
0.07/0.29	  tuple(h($$fresh3(sorti2(sK1_ax4_V(h(sK2_ax3_U))), $$true2, sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U)))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by axiom 24 (co1_4) }
0.07/0.29	  tuple(h($$fresh4(sorti2(sK1_ax4_V(h(sK2_ax3_U))), $$true2, sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U)))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by lemma 27 }
0.07/0.29	  tuple(h($$fresh4($$true2, $$true2, sK1_ax4_V(h(sK2_ax3_U)), sK1_ax4_V(h(sK2_ax3_U)))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by axiom 12 (co1_4) }
0.07/0.29	  tuple(h(op1(j(sK1_ax4_V(h(sK2_ax3_U))), j(sK1_ax4_V(h(sK2_ax3_U))))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by axiom 15 (ax3_1) }
0.07/0.29	  tuple(h($$fresh10(sorti1(j(sK1_ax4_V(h(sK2_ax3_U)))), $$true2, j(sK1_ax4_V(h(sK2_ax3_U))))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by axiom 23 (co1_3) }
0.07/0.29	  tuple(h($$fresh10($$fresh5(sorti2(sK1_ax4_V(h(sK2_ax3_U))), $$true2, sK1_ax4_V(h(sK2_ax3_U))), $$true2, j(sK1_ax4_V(h(sK2_ax3_U))))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by lemma 27 }
0.07/0.29	  tuple(h($$fresh10($$fresh5($$true2, $$true2, sK1_ax4_V(h(sK2_ax3_U))), $$true2, j(sK1_ax4_V(h(sK2_ax3_U))))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by axiom 11 (co1_3) }
0.07/0.29	  tuple(h($$fresh10($$true2, $$true2, j(sK1_ax4_V(h(sK2_ax3_U))))), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by axiom 5 (ax3_1) }
0.07/0.29	  tuple(h(sK2_ax3_U), sorti2(h(sK2_ax3_U)))
0.07/0.29	= { by lemma 26 }
0.07/0.29	  tuple(h(sK2_ax3_U), $$true2)
0.07/0.29	% SZS output end Proof
0.07/0.29	
0.07/0.29	RESULT: Theorem (the conjecture is true).
0.07/0.30	EOF