WORST_CASE(?,O(n^1)) * Step 1: Sum. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: E(Cons(A(),Cons(z0,z1)),z2) -> c51() E(Cons(A(),Nil()),z0) -> c47() E(Cons(B(),Cons(z0,z1)),z2) -> c50() E(Cons(B(),Nil()),z0) -> c46() E(Cons(F(),Cons(z0,z1)),z2) -> c48() E(Cons(F(),Nil()),z0) -> c44() E(Cons(T(),Cons(z0,z1)),z2) -> c49() E(Cons(T(),Nil()),z0) -> c45() E(Nil(),z0) -> c52() EQUAL(A(),A()) -> c68() EQUAL(A(),B()) -> c67() EQUAL(A(),F()) -> c65() EQUAL(A(),T()) -> c66() EQUAL(B(),A()) -> c64() EQUAL(B(),B()) -> c63() EQUAL(B(),F()) -> c61() EQUAL(B(),T()) -> c62() EQUAL(F(),A()) -> c56() EQUAL(F(),B()) -> c55() EQUAL(F(),F()) -> c53() EQUAL(F(),T()) -> c54() EQUAL(T(),A()) -> c60() EQUAL(T(),B()) -> c59() EQUAL(T(),F()) -> c57() EQUAL(T(),T()) -> c58() GOAL(z0,z1) -> c75(Q(z0,z1)) HEAD(Cons(z0,z1)) -> c43() NOTEMPTY(Cons(z0,z1)) -> c69() NOTEMPTY(Nil()) -> c70() P(z0,z1) -> c74(P[ITE](e(z0,z1),z0,z1),E(z0,z1)) Q(z0,z1) -> c73(Q[ITE](e(z0,z1),z0,z1),E(z0,z1)) R(z0,z1) -> c72(R[ITE](e(z0,z1),z0,z1),E(z0,z1)) T'(z0,z1) -> c71(T[ITE](e(z0,z1),z0,z1),E(z0,z1)) - Weak TRS: AND(False(),False()) -> c() AND(False(),True()) -> c2() AND(True(),False()) -> c1() AND(True(),True()) -> c3() P[ITE](False(),z0,Cons(A(),z1)) -> c36() P[ITE](False(),z0,Cons(B(),z1)) -> c35() P[ITE](False(),z0,Cons(F(),z1)) -> c32(AND(r(z0,Cons(F(),z1)),p(z0,z1)),R(z0,Cons(F(),z1))) P[ITE](False(),z0,Cons(F(),z1)) -> c33(AND(r(z0,Cons(F(),z1)),p(z0,z1)),P(z0,z1)) P[ITE](False(),z0,Cons(T(),z1)) -> c34() P[ITE](True(),z0,z1) -> c37() Q[ITE](False(),z0,Cons(A(),Cons(A(),z1))) -> c20() Q[ITE](False(),z0,Cons(A(),Cons(B(),z1))) -> c19() Q[ITE](False(),z0,Cons(A(),Cons(F(),z1))) -> c17() Q[ITE](False(),z0,Cons(A(),Cons(T(),z1))) -> c18() Q[ITE](False(),z0,Cons(A(),Nil())) -> c24(Q[ITE][FALSE][ITE](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,AND(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,AND(False(),and(False(),equal(head(Nil()),F()))) ,AND(False(),equal(head(Nil()),F())) ,EQUAL(head(Nil()),F()) ,HEAD(Nil())) Q[ITE](False(),z0,Cons(B(),Cons(A(),z1))) -> c16() Q[ITE](False(),z0,Cons(B(),Cons(B(),z1))) -> c15() Q[ITE](False(),z0,Cons(B(),Cons(F(),z1))) -> c13() Q[ITE](False(),z0,Cons(B(),Cons(T(),z1))) -> c14() Q[ITE](False(),z0,Cons(B(),Nil())) -> c23(Q[ITE][FALSE][ITE](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,AND(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,AND(False(),and(False(),equal(head(Nil()),F()))) ,AND(False(),equal(head(Nil()),F())) ,EQUAL(head(Nil()),F()) ,HEAD(Nil())) Q[ITE](False(),z0,Cons(F(),Cons(A(),z1))) -> c8() Q[ITE](False(),z0,Cons(F(),Cons(B(),z1))) -> c7() Q[ITE](False(),z0,Cons(F(),Cons(F(),z1))) -> c4(AND(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,P(z0,Cons(F(),Cons(F(),z1)))) Q[ITE](False(),z0,Cons(F(),Cons(F(),z1))) -> c5(AND(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,Q(z0,Cons(F(),z1))) Q[ITE](False(),z0,Cons(F(),Cons(T(),z1))) -> c6() Q[ITE](False(),z0,Cons(F(),Nil())) -> c21(Q[ITE][FALSE][ITE](and(True() ,and(True() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,AND(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,AND(True(),and(False(),equal(head(Nil()),F()))) ,AND(False(),equal(head(Nil()),F())) ,EQUAL(head(Nil()),F()) ,HEAD(Nil())) Q[ITE](False(),z0,Cons(T(),Cons(A(),z1))) -> c12() Q[ITE](False(),z0,Cons(T(),Cons(B(),z1))) -> c11() Q[ITE](False(),z0,Cons(T(),Cons(F(),z1))) -> c9() Q[ITE](False(),z0,Cons(T(),Cons(T(),z1))) -> c10() Q[ITE](False(),z0,Cons(T(),Nil())) -> c22(Q[ITE][FALSE][ITE](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,AND(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,AND(False(),and(False(),equal(head(Nil()),F()))) ,AND(False(),equal(head(Nil()),F())) ,EQUAL(head(Nil()),F()) ,HEAD(Nil())) Q[ITE](True(),z0,z1) -> c25() Q[ITE][FALSE][ITE](False(),z0,z1) -> c40() Q[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c38(AND(p(z0,Cons(z1,z2)),q(z0,z2)),P(z0,Cons(z1,z2))) Q[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c39(AND(p(z0,Cons(z1,z2)),q(z0,z2)),Q(z0,z2)) R[ITE](False(),z0,Cons(A(),z1)) -> c30() R[ITE](False(),z0,Cons(B(),z1)) -> c29() R[ITE](False(),z0,Cons(F(),z1)) -> c26(AND(q(z0,z1),r(z0,z1)),Q(z0,z1)) R[ITE](False(),z0,Cons(F(),z1)) -> c27(AND(q(z0,z1),r(z0,z1)),R(z0,z1)) R[ITE](False(),z0,Cons(T(),z1)) -> c28() R[ITE](True(),z0,z1) -> c31() T[ITE](False(),z0,z1) -> c41() T[ITE](True(),z0,z1) -> c42() and(False(),False()) -> False() and(False(),True()) -> False() and(True(),False()) -> False() and(True(),True()) -> True() e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() equal(A(),A()) -> True() equal(A(),B()) -> False() equal(A(),F()) -> False() equal(A(),T()) -> False() equal(B(),A()) -> False() equal(B(),B()) -> True() equal(B(),F()) -> False() equal(B(),T()) -> False() equal(F(),A()) -> False() equal(F(),B()) -> False() equal(F(),F()) -> True() equal(F(),T()) -> False() equal(T(),A()) -> False() equal(T(),B()) -> False() equal(T(),F()) -> False() equal(T(),T()) -> True() goal(z0,z1) -> q(z0,z1) head(Cons(z0,z1)) -> z0 notEmpty(Cons(z0,z1)) -> True() notEmpty(Nil()) -> False() p(z0,z1) -> p[Ite](e(z0,z1),z0,z1) p[Ite](False(),z0,Cons(A(),z1)) -> False() p[Ite](False(),z0,Cons(B(),z1)) -> False() p[Ite](False(),z0,Cons(F(),z1)) -> and(r(z0,Cons(F(),z1)),p(z0,z1)) p[Ite](False(),z0,Cons(T(),z1)) -> False() p[Ite](True(),z0,z1) -> True() q(z0,z1) -> q[Ite](e(z0,z1),z0,z1) q[Ite](False(),z0,Cons(A(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) q[Ite](False(),z0,Cons(B(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) q[Ite](False(),z0,Cons(F(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(F(),z1))) -> q[Ite][False][Ite][True][Ite](and(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,z0 ,Cons(F(),Cons(F(),z1))) q[Ite](False(),z0,Cons(F(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Nil())) -> q[Ite][False][Ite](and(True() ,and(True() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) q[Ite](False(),z0,Cons(T(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) q[Ite](True(),z0,z1) -> True() q[Ite][False][Ite](False(),z0,z1) -> False() q[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> q[Ite][False][Ite][True][Ite](and(p(z0,Cons(z1,z2)),q(z0,z2)) ,z0 ,Cons(z1,z2)) r(z0,z1) -> r[Ite](e(z0,z1),z0,z1) r[Ite](False(),z0,Cons(A(),z1)) -> False() r[Ite](False(),z0,Cons(B(),z1)) -> False() r[Ite](False(),z0,Cons(F(),z1)) -> r[Ite][False][Ite][True][Ite](and(q(z0,z1),r(z0,z1)),z0,Cons(F(),z1)) r[Ite](False(),z0,Cons(T(),z1)) -> False() r[Ite](True(),z0,z1) -> True() t(z0,z1) -> t[Ite](e(z0,z1),z0,z1) t[Ite](False(),z0,z1) -> True() t[Ite](True(),z0,z1) -> True() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3} / {A/0,B/0,Cons/2,F/0,False/0,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0 ,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0 ,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0 ,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0 ,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0 ,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3,r[Ite][False][Ite][True][Ite]/3} - Obligation: innermost runtime complexity wrt. defined symbols {AND,E,EQUAL,GOAL,HEAD,NOTEMPTY,P,P[ITE],Q,Q[ITE] ,Q[ITE][FALSE][ITE],R,R[ITE],T',T[ITE],and,e,equal,goal,head,notEmpty,p,p[Ite],q,q[Ite],q[Ite][False][Ite],r ,r[Ite],t,t[Ite]} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19 ,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35,c36,c37,c38,c39,c4,c40,c41,c42,c43 ,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6,c60,c61,c62,c63,c64,c65,c66,c67,c68 ,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match],q[Ite][False][Ite][True][Ite] ,r[Ite][False][Ite][True][Ite]} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: E(Cons(A(),Cons(z0,z1)),z2) -> c51() E(Cons(A(),Nil()),z0) -> c47() E(Cons(B(),Cons(z0,z1)),z2) -> c50() E(Cons(B(),Nil()),z0) -> c46() E(Cons(F(),Cons(z0,z1)),z2) -> c48() E(Cons(F(),Nil()),z0) -> c44() E(Cons(T(),Cons(z0,z1)),z2) -> c49() E(Cons(T(),Nil()),z0) -> c45() E(Nil(),z0) -> c52() EQUAL(A(),A()) -> c68() EQUAL(A(),B()) -> c67() EQUAL(A(),F()) -> c65() EQUAL(A(),T()) -> c66() EQUAL(B(),A()) -> c64() EQUAL(B(),B()) -> c63() EQUAL(B(),F()) -> c61() EQUAL(B(),T()) -> c62() EQUAL(F(),A()) -> c56() EQUAL(F(),B()) -> c55() EQUAL(F(),F()) -> c53() EQUAL(F(),T()) -> c54() EQUAL(T(),A()) -> c60() EQUAL(T(),B()) -> c59() EQUAL(T(),F()) -> c57() EQUAL(T(),T()) -> c58() GOAL(z0,z1) -> c75(Q(z0,z1)) HEAD(Cons(z0,z1)) -> c43() NOTEMPTY(Cons(z0,z1)) -> c69() NOTEMPTY(Nil()) -> c70() P(z0,z1) -> c74(P[ITE](e(z0,z1),z0,z1),E(z0,z1)) Q(z0,z1) -> c73(Q[ITE](e(z0,z1),z0,z1),E(z0,z1)) R(z0,z1) -> c72(R[ITE](e(z0,z1),z0,z1),E(z0,z1)) T'(z0,z1) -> c71(T[ITE](e(z0,z1),z0,z1),E(z0,z1)) - Weak TRS: AND(False(),False()) -> c() AND(False(),True()) -> c2() AND(True(),False()) -> c1() AND(True(),True()) -> c3() P[ITE](False(),z0,Cons(A(),z1)) -> c36() P[ITE](False(),z0,Cons(B(),z1)) -> c35() P[ITE](False(),z0,Cons(F(),z1)) -> c32(AND(r(z0,Cons(F(),z1)),p(z0,z1)),R(z0,Cons(F(),z1))) P[ITE](False(),z0,Cons(F(),z1)) -> c33(AND(r(z0,Cons(F(),z1)),p(z0,z1)),P(z0,z1)) P[ITE](False(),z0,Cons(T(),z1)) -> c34() P[ITE](True(),z0,z1) -> c37() Q[ITE](False(),z0,Cons(A(),Cons(A(),z1))) -> c20() Q[ITE](False(),z0,Cons(A(),Cons(B(),z1))) -> c19() Q[ITE](False(),z0,Cons(A(),Cons(F(),z1))) -> c17() Q[ITE](False(),z0,Cons(A(),Cons(T(),z1))) -> c18() Q[ITE](False(),z0,Cons(A(),Nil())) -> c24(Q[ITE][FALSE][ITE](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,AND(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,AND(False(),and(False(),equal(head(Nil()),F()))) ,AND(False(),equal(head(Nil()),F())) ,EQUAL(head(Nil()),F()) ,HEAD(Nil())) Q[ITE](False(),z0,Cons(B(),Cons(A(),z1))) -> c16() Q[ITE](False(),z0,Cons(B(),Cons(B(),z1))) -> c15() Q[ITE](False(),z0,Cons(B(),Cons(F(),z1))) -> c13() Q[ITE](False(),z0,Cons(B(),Cons(T(),z1))) -> c14() Q[ITE](False(),z0,Cons(B(),Nil())) -> c23(Q[ITE][FALSE][ITE](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,AND(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,AND(False(),and(False(),equal(head(Nil()),F()))) ,AND(False(),equal(head(Nil()),F())) ,EQUAL(head(Nil()),F()) ,HEAD(Nil())) Q[ITE](False(),z0,Cons(F(),Cons(A(),z1))) -> c8() Q[ITE](False(),z0,Cons(F(),Cons(B(),z1))) -> c7() Q[ITE](False(),z0,Cons(F(),Cons(F(),z1))) -> c4(AND(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,P(z0,Cons(F(),Cons(F(),z1)))) Q[ITE](False(),z0,Cons(F(),Cons(F(),z1))) -> c5(AND(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,Q(z0,Cons(F(),z1))) Q[ITE](False(),z0,Cons(F(),Cons(T(),z1))) -> c6() Q[ITE](False(),z0,Cons(F(),Nil())) -> c21(Q[ITE][FALSE][ITE](and(True() ,and(True() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,AND(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,AND(True(),and(False(),equal(head(Nil()),F()))) ,AND(False(),equal(head(Nil()),F())) ,EQUAL(head(Nil()),F()) ,HEAD(Nil())) Q[ITE](False(),z0,Cons(T(),Cons(A(),z1))) -> c12() Q[ITE](False(),z0,Cons(T(),Cons(B(),z1))) -> c11() Q[ITE](False(),z0,Cons(T(),Cons(F(),z1))) -> c9() Q[ITE](False(),z0,Cons(T(),Cons(T(),z1))) -> c10() Q[ITE](False(),z0,Cons(T(),Nil())) -> c22(Q[ITE][FALSE][ITE](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,AND(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,AND(False(),and(False(),equal(head(Nil()),F()))) ,AND(False(),equal(head(Nil()),F())) ,EQUAL(head(Nil()),F()) ,HEAD(Nil())) Q[ITE](True(),z0,z1) -> c25() Q[ITE][FALSE][ITE](False(),z0,z1) -> c40() Q[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c38(AND(p(z0,Cons(z1,z2)),q(z0,z2)),P(z0,Cons(z1,z2))) Q[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c39(AND(p(z0,Cons(z1,z2)),q(z0,z2)),Q(z0,z2)) R[ITE](False(),z0,Cons(A(),z1)) -> c30() R[ITE](False(),z0,Cons(B(),z1)) -> c29() R[ITE](False(),z0,Cons(F(),z1)) -> c26(AND(q(z0,z1),r(z0,z1)),Q(z0,z1)) R[ITE](False(),z0,Cons(F(),z1)) -> c27(AND(q(z0,z1),r(z0,z1)),R(z0,z1)) R[ITE](False(),z0,Cons(T(),z1)) -> c28() R[ITE](True(),z0,z1) -> c31() T[ITE](False(),z0,z1) -> c41() T[ITE](True(),z0,z1) -> c42() and(False(),False()) -> False() and(False(),True()) -> False() and(True(),False()) -> False() and(True(),True()) -> True() e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() equal(A(),A()) -> True() equal(A(),B()) -> False() equal(A(),F()) -> False() equal(A(),T()) -> False() equal(B(),A()) -> False() equal(B(),B()) -> True() equal(B(),F()) -> False() equal(B(),T()) -> False() equal(F(),A()) -> False() equal(F(),B()) -> False() equal(F(),F()) -> True() equal(F(),T()) -> False() equal(T(),A()) -> False() equal(T(),B()) -> False() equal(T(),F()) -> False() equal(T(),T()) -> True() goal(z0,z1) -> q(z0,z1) head(Cons(z0,z1)) -> z0 notEmpty(Cons(z0,z1)) -> True() notEmpty(Nil()) -> False() p(z0,z1) -> p[Ite](e(z0,z1),z0,z1) p[Ite](False(),z0,Cons(A(),z1)) -> False() p[Ite](False(),z0,Cons(B(),z1)) -> False() p[Ite](False(),z0,Cons(F(),z1)) -> and(r(z0,Cons(F(),z1)),p(z0,z1)) p[Ite](False(),z0,Cons(T(),z1)) -> False() p[Ite](True(),z0,z1) -> True() q(z0,z1) -> q[Ite](e(z0,z1),z0,z1) q[Ite](False(),z0,Cons(A(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) q[Ite](False(),z0,Cons(B(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) q[Ite](False(),z0,Cons(F(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(F(),z1))) -> q[Ite][False][Ite][True][Ite](and(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,z0 ,Cons(F(),Cons(F(),z1))) q[Ite](False(),z0,Cons(F(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Nil())) -> q[Ite][False][Ite](and(True() ,and(True() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) q[Ite](False(),z0,Cons(T(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) q[Ite](True(),z0,z1) -> True() q[Ite][False][Ite](False(),z0,z1) -> False() q[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> q[Ite][False][Ite][True][Ite](and(p(z0,Cons(z1,z2)),q(z0,z2)) ,z0 ,Cons(z1,z2)) r(z0,z1) -> r[Ite](e(z0,z1),z0,z1) r[Ite](False(),z0,Cons(A(),z1)) -> False() r[Ite](False(),z0,Cons(B(),z1)) -> False() r[Ite](False(),z0,Cons(F(),z1)) -> r[Ite][False][Ite][True][Ite](and(q(z0,z1),r(z0,z1)),z0,Cons(F(),z1)) r[Ite](False(),z0,Cons(T(),z1)) -> False() r[Ite](True(),z0,z1) -> True() t(z0,z1) -> t[Ite](e(z0,z1),z0,z1) t[Ite](False(),z0,z1) -> True() t[Ite](True(),z0,z1) -> True() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3} / {A/0,B/0,Cons/2,F/0,False/0,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0 ,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0 ,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0 ,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0 ,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0 ,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3,r[Ite][False][Ite][True][Ite]/3} - Obligation: innermost runtime complexity wrt. defined symbols {AND,E,EQUAL,GOAL,HEAD,NOTEMPTY,P,P[ITE],Q,Q[ITE] ,Q[ITE][FALSE][ITE],R,R[ITE],T',T[ITE],and,e,equal,goal,head,notEmpty,p,p[Ite],q,q[Ite],q[Ite][False][Ite],r ,r[Ite],t,t[Ite]} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19 ,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35,c36,c37,c38,c39,c4,c40,c41,c42,c43 ,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6,c60,c61,c62,c63,c64,c65,c66,c67,c68 ,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match],q[Ite][False][Ite][True][Ite] ,r[Ite][False][Ite][True][Ite]} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs E#(Cons(A(),Cons(z0,z1)),z2) -> c_1() E#(Cons(A(),Nil()),z0) -> c_2() E#(Cons(B(),Cons(z0,z1)),z2) -> c_3() E#(Cons(B(),Nil()),z0) -> c_4() E#(Cons(F(),Cons(z0,z1)),z2) -> c_5() E#(Cons(F(),Nil()),z0) -> c_6() E#(Cons(T(),Cons(z0,z1)),z2) -> c_7() E#(Cons(T(),Nil()),z0) -> c_8() E#(Nil(),z0) -> c_9() EQUAL#(A(),A()) -> c_10() EQUAL#(A(),B()) -> c_11() EQUAL#(A(),F()) -> c_12() EQUAL#(A(),T()) -> c_13() EQUAL#(B(),A()) -> c_14() EQUAL#(B(),B()) -> c_15() EQUAL#(B(),F()) -> c_16() EQUAL#(B(),T()) -> c_17() EQUAL#(F(),A()) -> c_18() EQUAL#(F(),B()) -> c_19() EQUAL#(F(),F()) -> c_20() EQUAL#(F(),T()) -> c_21() EQUAL#(T(),A()) -> c_22() EQUAL#(T(),B()) -> c_23() EQUAL#(T(),F()) -> c_24() EQUAL#(T(),T()) -> c_25() GOAL#(z0,z1) -> c_26(Q#(z0,z1)) HEAD#(Cons(z0,z1)) -> c_27() NOTEMPTY#(Cons(z0,z1)) -> c_28() NOTEMPTY#(Nil()) -> c_29() P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) T'#(z0,z1) -> c_33(T[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) Weak DPs AND#(False(),False()) -> c_34() AND#(False(),True()) -> c_35() AND#(True(),False()) -> c_36() AND#(True(),True()) -> c_37() P[ITE]#(False(),z0,Cons(A(),z1)) -> c_38() P[ITE]#(False(),z0,Cons(B(),z1)) -> c_39() P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,P#(z0,z1)) P[ITE]#(False(),z0,Cons(T(),z1)) -> c_42() P[ITE]#(True(),z0,z1) -> c_43() Q[ITE]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_44() Q[ITE]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_45() Q[ITE]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_46() Q[ITE]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_47() Q[ITE]#(False(),z0,Cons(A(),Nil())) -> c_48(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_49() Q[ITE]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_50() Q[ITE]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_51() Q[ITE]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_52() Q[ITE]#(False(),z0,Cons(B(),Nil())) -> c_53(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_54() Q[ITE]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_55() Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,Q#(z0,Cons(F(),z1))) Q[ITE]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_58() Q[ITE]#(False(),z0,Cons(F(),Nil())) -> c_59(Q[ITE][FALSE][ITE]#(and(True() ,and(True() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_60() Q[ITE]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_61() Q[ITE]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_62() Q[ITE]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_63() Q[ITE]#(False(),z0,Cons(T(),Nil())) -> c_64(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(True(),z0,z1) -> c_65() Q[ITE][FALSE][ITE]#(False(),z0,z1) -> c_66() Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,P#(z0,Cons(z1,z2))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,Q#(z0,z2)) R[ITE]#(False(),z0,Cons(A(),z1)) -> c_69() R[ITE]#(False(),z0,Cons(B(),z1)) -> c_70() R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),R#(z0,z1)) R[ITE]#(False(),z0,Cons(T(),z1)) -> c_73() R[ITE]#(True(),z0,z1) -> c_74() T[ITE]#(False(),z0,z1) -> c_75() T[ITE]#(True(),z0,z1) -> c_76() and#(False(),False()) -> c_77() and#(False(),True()) -> c_78() and#(True(),False()) -> c_79() and#(True(),True()) -> c_80() e#(Cons(A(),Cons(z0,z1)),z2) -> c_81() e#(Cons(A(),Nil()),z0) -> c_82() e#(Cons(B(),Cons(z0,z1)),z2) -> c_83() e#(Cons(B(),Nil()),z0) -> c_84() e#(Cons(F(),Cons(z0,z1)),z2) -> c_85() e#(Cons(F(),Nil()),z0) -> c_86() e#(Cons(T(),Cons(z0,z1)),z2) -> c_87() e#(Cons(T(),Nil()),z0) -> c_88() e#(Nil(),z0) -> c_89() equal#(A(),A()) -> c_90() equal#(A(),B()) -> c_91() equal#(A(),F()) -> c_92() equal#(A(),T()) -> c_93() equal#(B(),A()) -> c_94() equal#(B(),B()) -> c_95() equal#(B(),F()) -> c_96() equal#(B(),T()) -> c_97() equal#(F(),A()) -> c_98() equal#(F(),B()) -> c_99() equal#(F(),F()) -> c_100() equal#(F(),T()) -> c_101() equal#(T(),A()) -> c_102() equal#(T(),B()) -> c_103() equal#(T(),F()) -> c_104() equal#(T(),T()) -> c_105() goal#(z0,z1) -> c_106(q#(z0,z1)) head#(Cons(z0,z1)) -> c_107() notEmpty#(Cons(z0,z1)) -> c_108() notEmpty#(Nil()) -> c_109() p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) p[Ite]#(False(),z0,Cons(A(),z1)) -> c_111() p[Ite]#(False(),z0,Cons(B(),z1)) -> c_112() p[Ite]#(False(),z0,Cons(F(),z1)) -> c_113(and#(r(z0,Cons(F(),z1)),p(z0,z1)),r#(z0,Cons(F(),z1)),p#(z0,z1)) p[Ite]#(False(),z0,Cons(T(),z1)) -> c_114() p[Ite]#(True(),z0,z1) -> c_115() q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) q[Ite]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_117() q[Ite]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_118() q[Ite]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_119() q[Ite]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_120() q[Ite]#(False(),z0,Cons(A(),Nil())) -> c_121(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_122() q[Ite]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_123() q[Ite]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_124() q[Ite]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_125() q[Ite]#(False(),z0,Cons(B(),Nil())) -> c_126(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_127() q[Ite]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_128() q[Ite]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_129(and#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1))) q[Ite]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_130() q[Ite]#(False(),z0,Cons(F(),Nil())) -> c_131(q[Ite][False][Ite]#(and(True() ,and(True() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_132() q[Ite]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_133() q[Ite]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_134() q[Ite]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_135() q[Ite]#(False(),z0,Cons(T(),Nil())) -> c_136(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(True(),z0,z1) -> c_137() q[Ite][False][Ite]#(False(),z0,z1) -> c_138() q[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_139(and#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2)) r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) r[Ite]#(False(),z0,Cons(A(),z1)) -> c_141() r[Ite]#(False(),z0,Cons(B(),z1)) -> c_142() r[Ite]#(False(),z0,Cons(F(),z1)) -> c_143(and#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1)) r[Ite]#(False(),z0,Cons(T(),z1)) -> c_144() r[Ite]#(True(),z0,z1) -> c_145() t#(z0,z1) -> c_146(t[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) t[Ite]#(False(),z0,z1) -> c_147() t[Ite]#(True(),z0,z1) -> c_148() and mark the set of starting terms. * Step 3: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: E#(Cons(A(),Cons(z0,z1)),z2) -> c_1() E#(Cons(A(),Nil()),z0) -> c_2() E#(Cons(B(),Cons(z0,z1)),z2) -> c_3() E#(Cons(B(),Nil()),z0) -> c_4() E#(Cons(F(),Cons(z0,z1)),z2) -> c_5() E#(Cons(F(),Nil()),z0) -> c_6() E#(Cons(T(),Cons(z0,z1)),z2) -> c_7() E#(Cons(T(),Nil()),z0) -> c_8() E#(Nil(),z0) -> c_9() EQUAL#(A(),A()) -> c_10() EQUAL#(A(),B()) -> c_11() EQUAL#(A(),F()) -> c_12() EQUAL#(A(),T()) -> c_13() EQUAL#(B(),A()) -> c_14() EQUAL#(B(),B()) -> c_15() EQUAL#(B(),F()) -> c_16() EQUAL#(B(),T()) -> c_17() EQUAL#(F(),A()) -> c_18() EQUAL#(F(),B()) -> c_19() EQUAL#(F(),F()) -> c_20() EQUAL#(F(),T()) -> c_21() EQUAL#(T(),A()) -> c_22() EQUAL#(T(),B()) -> c_23() EQUAL#(T(),F()) -> c_24() EQUAL#(T(),T()) -> c_25() GOAL#(z0,z1) -> c_26(Q#(z0,z1)) HEAD#(Cons(z0,z1)) -> c_27() NOTEMPTY#(Cons(z0,z1)) -> c_28() NOTEMPTY#(Nil()) -> c_29() P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) T'#(z0,z1) -> c_33(T[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) - Weak DPs: AND#(False(),False()) -> c_34() AND#(False(),True()) -> c_35() AND#(True(),False()) -> c_36() AND#(True(),True()) -> c_37() P[ITE]#(False(),z0,Cons(A(),z1)) -> c_38() P[ITE]#(False(),z0,Cons(B(),z1)) -> c_39() P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,P#(z0,z1)) P[ITE]#(False(),z0,Cons(T(),z1)) -> c_42() P[ITE]#(True(),z0,z1) -> c_43() Q[ITE]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_44() Q[ITE]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_45() Q[ITE]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_46() Q[ITE]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_47() Q[ITE]#(False(),z0,Cons(A(),Nil())) -> c_48(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_49() Q[ITE]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_50() Q[ITE]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_51() Q[ITE]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_52() Q[ITE]#(False(),z0,Cons(B(),Nil())) -> c_53(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_54() Q[ITE]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_55() Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,Q#(z0,Cons(F(),z1))) Q[ITE]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_58() Q[ITE]#(False(),z0,Cons(F(),Nil())) -> c_59(Q[ITE][FALSE][ITE]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_60() Q[ITE]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_61() Q[ITE]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_62() Q[ITE]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_63() Q[ITE]#(False(),z0,Cons(T(),Nil())) -> c_64(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(True(),z0,z1) -> c_65() Q[ITE][FALSE][ITE]#(False(),z0,z1) -> c_66() Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,P#(z0,Cons(z1,z2))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,Q#(z0,z2)) R[ITE]#(False(),z0,Cons(A(),z1)) -> c_69() R[ITE]#(False(),z0,Cons(B(),z1)) -> c_70() R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),R#(z0,z1)) R[ITE]#(False(),z0,Cons(T(),z1)) -> c_73() R[ITE]#(True(),z0,z1) -> c_74() T[ITE]#(False(),z0,z1) -> c_75() T[ITE]#(True(),z0,z1) -> c_76() and#(False(),False()) -> c_77() and#(False(),True()) -> c_78() and#(True(),False()) -> c_79() and#(True(),True()) -> c_80() e#(Cons(A(),Cons(z0,z1)),z2) -> c_81() e#(Cons(A(),Nil()),z0) -> c_82() e#(Cons(B(),Cons(z0,z1)),z2) -> c_83() e#(Cons(B(),Nil()),z0) -> c_84() e#(Cons(F(),Cons(z0,z1)),z2) -> c_85() e#(Cons(F(),Nil()),z0) -> c_86() e#(Cons(T(),Cons(z0,z1)),z2) -> c_87() e#(Cons(T(),Nil()),z0) -> c_88() e#(Nil(),z0) -> c_89() equal#(A(),A()) -> c_90() equal#(A(),B()) -> c_91() equal#(A(),F()) -> c_92() equal#(A(),T()) -> c_93() equal#(B(),A()) -> c_94() equal#(B(),B()) -> c_95() equal#(B(),F()) -> c_96() equal#(B(),T()) -> c_97() equal#(F(),A()) -> c_98() equal#(F(),B()) -> c_99() equal#(F(),F()) -> c_100() equal#(F(),T()) -> c_101() equal#(T(),A()) -> c_102() equal#(T(),B()) -> c_103() equal#(T(),F()) -> c_104() equal#(T(),T()) -> c_105() goal#(z0,z1) -> c_106(q#(z0,z1)) head#(Cons(z0,z1)) -> c_107() notEmpty#(Cons(z0,z1)) -> c_108() notEmpty#(Nil()) -> c_109() p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) p[Ite]#(False(),z0,Cons(A(),z1)) -> c_111() p[Ite]#(False(),z0,Cons(B(),z1)) -> c_112() p[Ite]#(False(),z0,Cons(F(),z1)) -> c_113(and#(r(z0,Cons(F(),z1)),p(z0,z1)),r#(z0,Cons(F(),z1)),p#(z0,z1)) p[Ite]#(False(),z0,Cons(T(),z1)) -> c_114() p[Ite]#(True(),z0,z1) -> c_115() q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) q[Ite]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_117() q[Ite]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_118() q[Ite]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_119() q[Ite]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_120() q[Ite]#(False(),z0,Cons(A(),Nil())) -> c_121(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_122() q[Ite]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_123() q[Ite]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_124() q[Ite]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_125() q[Ite]#(False(),z0,Cons(B(),Nil())) -> c_126(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_127() q[Ite]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_128() q[Ite]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_129(and#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1))) q[Ite]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_130() q[Ite]#(False(),z0,Cons(F(),Nil())) -> c_131(q[Ite][False][Ite]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_132() q[Ite]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_133() q[Ite]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_134() q[Ite]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_135() q[Ite]#(False(),z0,Cons(T(),Nil())) -> c_136(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(True(),z0,z1) -> c_137() q[Ite][False][Ite]#(False(),z0,z1) -> c_138() q[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_139(and#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2)) r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) r[Ite]#(False(),z0,Cons(A(),z1)) -> c_141() r[Ite]#(False(),z0,Cons(B(),z1)) -> c_142() r[Ite]#(False(),z0,Cons(F(),z1)) -> c_143(and#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1)) r[Ite]#(False(),z0,Cons(T(),z1)) -> c_144() r[Ite]#(True(),z0,z1) -> c_145() t#(z0,z1) -> c_146(t[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) t[Ite]#(False(),z0,z1) -> c_147() t[Ite]#(True(),z0,z1) -> c_148() - Weak TRS: AND(False(),False()) -> c() AND(False(),True()) -> c2() AND(True(),False()) -> c1() AND(True(),True()) -> c3() E(Cons(A(),Cons(z0,z1)),z2) -> c51() E(Cons(A(),Nil()),z0) -> c47() E(Cons(B(),Cons(z0,z1)),z2) -> c50() E(Cons(B(),Nil()),z0) -> c46() E(Cons(F(),Cons(z0,z1)),z2) -> c48() E(Cons(F(),Nil()),z0) -> c44() E(Cons(T(),Cons(z0,z1)),z2) -> c49() E(Cons(T(),Nil()),z0) -> c45() E(Nil(),z0) -> c52() EQUAL(A(),A()) -> c68() EQUAL(A(),B()) -> c67() EQUAL(A(),F()) -> c65() EQUAL(A(),T()) -> c66() EQUAL(B(),A()) -> c64() EQUAL(B(),B()) -> c63() EQUAL(B(),F()) -> c61() EQUAL(B(),T()) -> c62() EQUAL(F(),A()) -> c56() EQUAL(F(),B()) -> c55() EQUAL(F(),F()) -> c53() EQUAL(F(),T()) -> c54() EQUAL(T(),A()) -> c60() EQUAL(T(),B()) -> c59() EQUAL(T(),F()) -> c57() EQUAL(T(),T()) -> c58() GOAL(z0,z1) -> c75(Q(z0,z1)) HEAD(Cons(z0,z1)) -> c43() NOTEMPTY(Cons(z0,z1)) -> c69() NOTEMPTY(Nil()) -> c70() P(z0,z1) -> c74(P[ITE](e(z0,z1),z0,z1),E(z0,z1)) P[ITE](False(),z0,Cons(A(),z1)) -> c36() P[ITE](False(),z0,Cons(B(),z1)) -> c35() P[ITE](False(),z0,Cons(F(),z1)) -> c32(AND(r(z0,Cons(F(),z1)),p(z0,z1)),R(z0,Cons(F(),z1))) P[ITE](False(),z0,Cons(F(),z1)) -> c33(AND(r(z0,Cons(F(),z1)),p(z0,z1)),P(z0,z1)) P[ITE](False(),z0,Cons(T(),z1)) -> c34() P[ITE](True(),z0,z1) -> c37() Q(z0,z1) -> c73(Q[ITE](e(z0,z1),z0,z1),E(z0,z1)) Q[ITE](False(),z0,Cons(A(),Cons(A(),z1))) -> c20() Q[ITE](False(),z0,Cons(A(),Cons(B(),z1))) -> c19() Q[ITE](False(),z0,Cons(A(),Cons(F(),z1))) -> c17() Q[ITE](False(),z0,Cons(A(),Cons(T(),z1))) -> c18() Q[ITE](False(),z0,Cons(A(),Nil())) -> c24(Q[ITE][FALSE][ITE](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,AND(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,AND(False(),and(False(),equal(head(Nil()),F()))) ,AND(False(),equal(head(Nil()),F())) ,EQUAL(head(Nil()),F()) ,HEAD(Nil())) Q[ITE](False(),z0,Cons(B(),Cons(A(),z1))) -> c16() Q[ITE](False(),z0,Cons(B(),Cons(B(),z1))) -> c15() Q[ITE](False(),z0,Cons(B(),Cons(F(),z1))) -> c13() Q[ITE](False(),z0,Cons(B(),Cons(T(),z1))) -> c14() Q[ITE](False(),z0,Cons(B(),Nil())) -> c23(Q[ITE][FALSE][ITE](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,AND(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,AND(False(),and(False(),equal(head(Nil()),F()))) ,AND(False(),equal(head(Nil()),F())) ,EQUAL(head(Nil()),F()) ,HEAD(Nil())) Q[ITE](False(),z0,Cons(F(),Cons(A(),z1))) -> c8() Q[ITE](False(),z0,Cons(F(),Cons(B(),z1))) -> c7() Q[ITE](False(),z0,Cons(F(),Cons(F(),z1))) -> c4(AND(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,P(z0,Cons(F(),Cons(F(),z1)))) Q[ITE](False(),z0,Cons(F(),Cons(F(),z1))) -> c5(AND(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,Q(z0,Cons(F(),z1))) Q[ITE](False(),z0,Cons(F(),Cons(T(),z1))) -> c6() Q[ITE](False(),z0,Cons(F(),Nil())) -> c21(Q[ITE][FALSE][ITE](and(True() ,and(True() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,AND(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,AND(True(),and(False(),equal(head(Nil()),F()))) ,AND(False(),equal(head(Nil()),F())) ,EQUAL(head(Nil()),F()) ,HEAD(Nil())) Q[ITE](False(),z0,Cons(T(),Cons(A(),z1))) -> c12() Q[ITE](False(),z0,Cons(T(),Cons(B(),z1))) -> c11() Q[ITE](False(),z0,Cons(T(),Cons(F(),z1))) -> c9() Q[ITE](False(),z0,Cons(T(),Cons(T(),z1))) -> c10() Q[ITE](False(),z0,Cons(T(),Nil())) -> c22(Q[ITE][FALSE][ITE](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,AND(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,AND(False(),and(False(),equal(head(Nil()),F()))) ,AND(False(),equal(head(Nil()),F())) ,EQUAL(head(Nil()),F()) ,HEAD(Nil())) Q[ITE](True(),z0,z1) -> c25() Q[ITE][FALSE][ITE](False(),z0,z1) -> c40() Q[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c38(AND(p(z0,Cons(z1,z2)),q(z0,z2)),P(z0,Cons(z1,z2))) Q[ITE][FALSE][ITE](True(),z0,Cons(z1,z2)) -> c39(AND(p(z0,Cons(z1,z2)),q(z0,z2)),Q(z0,z2)) R(z0,z1) -> c72(R[ITE](e(z0,z1),z0,z1),E(z0,z1)) R[ITE](False(),z0,Cons(A(),z1)) -> c30() R[ITE](False(),z0,Cons(B(),z1)) -> c29() R[ITE](False(),z0,Cons(F(),z1)) -> c26(AND(q(z0,z1),r(z0,z1)),Q(z0,z1)) R[ITE](False(),z0,Cons(F(),z1)) -> c27(AND(q(z0,z1),r(z0,z1)),R(z0,z1)) R[ITE](False(),z0,Cons(T(),z1)) -> c28() R[ITE](True(),z0,z1) -> c31() T'(z0,z1) -> c71(T[ITE](e(z0,z1),z0,z1),E(z0,z1)) T[ITE](False(),z0,z1) -> c41() T[ITE](True(),z0,z1) -> c42() and(False(),False()) -> False() and(False(),True()) -> False() and(True(),False()) -> False() and(True(),True()) -> True() e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() equal(A(),A()) -> True() equal(A(),B()) -> False() equal(A(),F()) -> False() equal(A(),T()) -> False() equal(B(),A()) -> False() equal(B(),B()) -> True() equal(B(),F()) -> False() equal(B(),T()) -> False() equal(F(),A()) -> False() equal(F(),B()) -> False() equal(F(),F()) -> True() equal(F(),T()) -> False() equal(T(),A()) -> False() equal(T(),B()) -> False() equal(T(),F()) -> False() equal(T(),T()) -> True() goal(z0,z1) -> q(z0,z1) head(Cons(z0,z1)) -> z0 notEmpty(Cons(z0,z1)) -> True() notEmpty(Nil()) -> False() p(z0,z1) -> p[Ite](e(z0,z1),z0,z1) p[Ite](False(),z0,Cons(A(),z1)) -> False() p[Ite](False(),z0,Cons(B(),z1)) -> False() p[Ite](False(),z0,Cons(F(),z1)) -> and(r(z0,Cons(F(),z1)),p(z0,z1)) p[Ite](False(),z0,Cons(T(),z1)) -> False() p[Ite](True(),z0,z1) -> True() q(z0,z1) -> q[Ite](e(z0,z1),z0,z1) q[Ite](False(),z0,Cons(A(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) q[Ite](False(),z0,Cons(B(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) q[Ite](False(),z0,Cons(F(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(F(),z1))) -> q[Ite][False][Ite][True][Ite](and(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,z0 ,Cons(F(),Cons(F(),z1))) q[Ite](False(),z0,Cons(F(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Nil())) -> q[Ite][False][Ite](and(True() ,and(True() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) q[Ite](False(),z0,Cons(T(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) q[Ite](True(),z0,z1) -> True() q[Ite][False][Ite](False(),z0,z1) -> False() q[Ite][False][Ite](True(),z0,Cons(z1,z2)) -> q[Ite][False][Ite][True][Ite](and(p(z0,Cons(z1,z2)),q(z0,z2)) ,z0 ,Cons(z1,z2)) r(z0,z1) -> r[Ite](e(z0,z1),z0,z1) r[Ite](False(),z0,Cons(A(),z1)) -> False() r[Ite](False(),z0,Cons(B(),z1)) -> False() r[Ite](False(),z0,Cons(F(),z1)) -> r[Ite][False][Ite][True][Ite](and(q(z0,z1),r(z0,z1)),z0,Cons(F(),z1)) r[Ite](False(),z0,Cons(T(),z1)) -> False() r[Ite](True(),z0,z1) -> True() t(z0,z1) -> t[Ite](e(z0,z1),z0,z1) t[Ite](False(),z0,z1) -> True() t[Ite](True(),z0,z1) -> True() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/3,c_31/3,c_32/3,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/4,c_41/4,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/4,c_57/4 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/4,c_68/4,c_69/0,c_70/0,c_71/4,c_72/4 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: and(False(),False()) -> False() and(False(),True()) -> False() and(True(),False()) -> False() and(True(),True()) -> True() e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() p(z0,z1) -> p[Ite](e(z0,z1),z0,z1) p[Ite](False(),z0,Cons(A(),z1)) -> False() p[Ite](False(),z0,Cons(B(),z1)) -> False() p[Ite](False(),z0,Cons(F(),z1)) -> and(r(z0,Cons(F(),z1)),p(z0,z1)) p[Ite](False(),z0,Cons(T(),z1)) -> False() p[Ite](True(),z0,z1) -> True() q(z0,z1) -> q[Ite](e(z0,z1),z0,z1) q[Ite](False(),z0,Cons(A(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) q[Ite](False(),z0,Cons(B(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) q[Ite](False(),z0,Cons(F(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(F(),z1))) -> q[Ite][False][Ite][True][Ite](and(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,z0 ,Cons(F(),Cons(F(),z1))) q[Ite](False(),z0,Cons(F(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Nil())) -> q[Ite][False][Ite](and(True() ,and(True(),and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) q[Ite](False(),z0,Cons(T(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) q[Ite](True(),z0,z1) -> True() r(z0,z1) -> r[Ite](e(z0,z1),z0,z1) r[Ite](False(),z0,Cons(A(),z1)) -> False() r[Ite](False(),z0,Cons(B(),z1)) -> False() r[Ite](False(),z0,Cons(F(),z1)) -> r[Ite][False][Ite][True][Ite](and(q(z0,z1),r(z0,z1)),z0,Cons(F(),z1)) r[Ite](False(),z0,Cons(T(),z1)) -> False() r[Ite](True(),z0,z1) -> True() AND#(False(),False()) -> c_34() AND#(False(),True()) -> c_35() AND#(True(),False()) -> c_36() AND#(True(),True()) -> c_37() E#(Cons(A(),Cons(z0,z1)),z2) -> c_1() E#(Cons(A(),Nil()),z0) -> c_2() E#(Cons(B(),Cons(z0,z1)),z2) -> c_3() E#(Cons(B(),Nil()),z0) -> c_4() E#(Cons(F(),Cons(z0,z1)),z2) -> c_5() E#(Cons(F(),Nil()),z0) -> c_6() E#(Cons(T(),Cons(z0,z1)),z2) -> c_7() E#(Cons(T(),Nil()),z0) -> c_8() E#(Nil(),z0) -> c_9() EQUAL#(A(),A()) -> c_10() EQUAL#(A(),B()) -> c_11() EQUAL#(A(),F()) -> c_12() EQUAL#(A(),T()) -> c_13() EQUAL#(B(),A()) -> c_14() EQUAL#(B(),B()) -> c_15() EQUAL#(B(),F()) -> c_16() EQUAL#(B(),T()) -> c_17() EQUAL#(F(),A()) -> c_18() EQUAL#(F(),B()) -> c_19() EQUAL#(F(),F()) -> c_20() EQUAL#(F(),T()) -> c_21() EQUAL#(T(),A()) -> c_22() EQUAL#(T(),B()) -> c_23() EQUAL#(T(),F()) -> c_24() EQUAL#(T(),T()) -> c_25() GOAL#(z0,z1) -> c_26(Q#(z0,z1)) HEAD#(Cons(z0,z1)) -> c_27() NOTEMPTY#(Cons(z0,z1)) -> c_28() NOTEMPTY#(Nil()) -> c_29() P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) P[ITE]#(False(),z0,Cons(A(),z1)) -> c_38() P[ITE]#(False(),z0,Cons(B(),z1)) -> c_39() P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,P#(z0,z1)) P[ITE]#(False(),z0,Cons(T(),z1)) -> c_42() P[ITE]#(True(),z0,z1) -> c_43() Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) Q[ITE]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_44() Q[ITE]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_45() Q[ITE]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_46() Q[ITE]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_47() Q[ITE]#(False(),z0,Cons(A(),Nil())) -> c_48(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_49() Q[ITE]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_50() Q[ITE]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_51() Q[ITE]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_52() Q[ITE]#(False(),z0,Cons(B(),Nil())) -> c_53(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_54() Q[ITE]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_55() Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,Q#(z0,Cons(F(),z1))) Q[ITE]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_58() Q[ITE]#(False(),z0,Cons(F(),Nil())) -> c_59(Q[ITE][FALSE][ITE]#(and(True() ,and(True() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_60() Q[ITE]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_61() Q[ITE]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_62() Q[ITE]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_63() Q[ITE]#(False(),z0,Cons(T(),Nil())) -> c_64(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(True(),z0,z1) -> c_65() Q[ITE][FALSE][ITE]#(False(),z0,z1) -> c_66() Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,P#(z0,Cons(z1,z2))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,Q#(z0,z2)) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) R[ITE]#(False(),z0,Cons(A(),z1)) -> c_69() R[ITE]#(False(),z0,Cons(B(),z1)) -> c_70() R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),R#(z0,z1)) R[ITE]#(False(),z0,Cons(T(),z1)) -> c_73() R[ITE]#(True(),z0,z1) -> c_74() T'#(z0,z1) -> c_33(T[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) T[ITE]#(False(),z0,z1) -> c_75() T[ITE]#(True(),z0,z1) -> c_76() and#(False(),False()) -> c_77() and#(False(),True()) -> c_78() and#(True(),False()) -> c_79() and#(True(),True()) -> c_80() e#(Cons(A(),Cons(z0,z1)),z2) -> c_81() e#(Cons(A(),Nil()),z0) -> c_82() e#(Cons(B(),Cons(z0,z1)),z2) -> c_83() e#(Cons(B(),Nil()),z0) -> c_84() e#(Cons(F(),Cons(z0,z1)),z2) -> c_85() e#(Cons(F(),Nil()),z0) -> c_86() e#(Cons(T(),Cons(z0,z1)),z2) -> c_87() e#(Cons(T(),Nil()),z0) -> c_88() e#(Nil(),z0) -> c_89() equal#(A(),A()) -> c_90() equal#(A(),B()) -> c_91() equal#(A(),F()) -> c_92() equal#(A(),T()) -> c_93() equal#(B(),A()) -> c_94() equal#(B(),B()) -> c_95() equal#(B(),F()) -> c_96() equal#(B(),T()) -> c_97() equal#(F(),A()) -> c_98() equal#(F(),B()) -> c_99() equal#(F(),F()) -> c_100() equal#(F(),T()) -> c_101() equal#(T(),A()) -> c_102() equal#(T(),B()) -> c_103() equal#(T(),F()) -> c_104() equal#(T(),T()) -> c_105() goal#(z0,z1) -> c_106(q#(z0,z1)) head#(Cons(z0,z1)) -> c_107() notEmpty#(Cons(z0,z1)) -> c_108() notEmpty#(Nil()) -> c_109() p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) p[Ite]#(False(),z0,Cons(A(),z1)) -> c_111() p[Ite]#(False(),z0,Cons(B(),z1)) -> c_112() p[Ite]#(False(),z0,Cons(F(),z1)) -> c_113(and#(r(z0,Cons(F(),z1)),p(z0,z1)),r#(z0,Cons(F(),z1)),p#(z0,z1)) p[Ite]#(False(),z0,Cons(T(),z1)) -> c_114() p[Ite]#(True(),z0,z1) -> c_115() q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) q[Ite]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_117() q[Ite]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_118() q[Ite]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_119() q[Ite]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_120() q[Ite]#(False(),z0,Cons(A(),Nil())) -> c_121(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_122() q[Ite]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_123() q[Ite]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_124() q[Ite]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_125() q[Ite]#(False(),z0,Cons(B(),Nil())) -> c_126(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_127() q[Ite]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_128() q[Ite]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_129(and#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1))) q[Ite]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_130() q[Ite]#(False(),z0,Cons(F(),Nil())) -> c_131(q[Ite][False][Ite]#(and(True() ,and(True() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_132() q[Ite]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_133() q[Ite]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_134() q[Ite]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_135() q[Ite]#(False(),z0,Cons(T(),Nil())) -> c_136(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(True(),z0,z1) -> c_137() q[Ite][False][Ite]#(False(),z0,z1) -> c_138() q[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_139(and#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2)) r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) r[Ite]#(False(),z0,Cons(A(),z1)) -> c_141() r[Ite]#(False(),z0,Cons(B(),z1)) -> c_142() r[Ite]#(False(),z0,Cons(F(),z1)) -> c_143(and#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1)) r[Ite]#(False(),z0,Cons(T(),z1)) -> c_144() r[Ite]#(True(),z0,z1) -> c_145() t#(z0,z1) -> c_146(t[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) t[Ite]#(False(),z0,z1) -> c_147() t[Ite]#(True(),z0,z1) -> c_148() * Step 4: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: E#(Cons(A(),Cons(z0,z1)),z2) -> c_1() E#(Cons(A(),Nil()),z0) -> c_2() E#(Cons(B(),Cons(z0,z1)),z2) -> c_3() E#(Cons(B(),Nil()),z0) -> c_4() E#(Cons(F(),Cons(z0,z1)),z2) -> c_5() E#(Cons(F(),Nil()),z0) -> c_6() E#(Cons(T(),Cons(z0,z1)),z2) -> c_7() E#(Cons(T(),Nil()),z0) -> c_8() E#(Nil(),z0) -> c_9() EQUAL#(A(),A()) -> c_10() EQUAL#(A(),B()) -> c_11() EQUAL#(A(),F()) -> c_12() EQUAL#(A(),T()) -> c_13() EQUAL#(B(),A()) -> c_14() EQUAL#(B(),B()) -> c_15() EQUAL#(B(),F()) -> c_16() EQUAL#(B(),T()) -> c_17() EQUAL#(F(),A()) -> c_18() EQUAL#(F(),B()) -> c_19() EQUAL#(F(),F()) -> c_20() EQUAL#(F(),T()) -> c_21() EQUAL#(T(),A()) -> c_22() EQUAL#(T(),B()) -> c_23() EQUAL#(T(),F()) -> c_24() EQUAL#(T(),T()) -> c_25() GOAL#(z0,z1) -> c_26(Q#(z0,z1)) HEAD#(Cons(z0,z1)) -> c_27() NOTEMPTY#(Cons(z0,z1)) -> c_28() NOTEMPTY#(Nil()) -> c_29() P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) T'#(z0,z1) -> c_33(T[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) - Weak DPs: AND#(False(),False()) -> c_34() AND#(False(),True()) -> c_35() AND#(True(),False()) -> c_36() AND#(True(),True()) -> c_37() P[ITE]#(False(),z0,Cons(A(),z1)) -> c_38() P[ITE]#(False(),z0,Cons(B(),z1)) -> c_39() P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,P#(z0,z1)) P[ITE]#(False(),z0,Cons(T(),z1)) -> c_42() P[ITE]#(True(),z0,z1) -> c_43() Q[ITE]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_44() Q[ITE]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_45() Q[ITE]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_46() Q[ITE]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_47() Q[ITE]#(False(),z0,Cons(A(),Nil())) -> c_48(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_49() Q[ITE]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_50() Q[ITE]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_51() Q[ITE]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_52() Q[ITE]#(False(),z0,Cons(B(),Nil())) -> c_53(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_54() Q[ITE]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_55() Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,Q#(z0,Cons(F(),z1))) Q[ITE]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_58() Q[ITE]#(False(),z0,Cons(F(),Nil())) -> c_59(Q[ITE][FALSE][ITE]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_60() Q[ITE]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_61() Q[ITE]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_62() Q[ITE]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_63() Q[ITE]#(False(),z0,Cons(T(),Nil())) -> c_64(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(True(),z0,z1) -> c_65() Q[ITE][FALSE][ITE]#(False(),z0,z1) -> c_66() Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,P#(z0,Cons(z1,z2))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,Q#(z0,z2)) R[ITE]#(False(),z0,Cons(A(),z1)) -> c_69() R[ITE]#(False(),z0,Cons(B(),z1)) -> c_70() R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),R#(z0,z1)) R[ITE]#(False(),z0,Cons(T(),z1)) -> c_73() R[ITE]#(True(),z0,z1) -> c_74() T[ITE]#(False(),z0,z1) -> c_75() T[ITE]#(True(),z0,z1) -> c_76() and#(False(),False()) -> c_77() and#(False(),True()) -> c_78() and#(True(),False()) -> c_79() and#(True(),True()) -> c_80() e#(Cons(A(),Cons(z0,z1)),z2) -> c_81() e#(Cons(A(),Nil()),z0) -> c_82() e#(Cons(B(),Cons(z0,z1)),z2) -> c_83() e#(Cons(B(),Nil()),z0) -> c_84() e#(Cons(F(),Cons(z0,z1)),z2) -> c_85() e#(Cons(F(),Nil()),z0) -> c_86() e#(Cons(T(),Cons(z0,z1)),z2) -> c_87() e#(Cons(T(),Nil()),z0) -> c_88() e#(Nil(),z0) -> c_89() equal#(A(),A()) -> c_90() equal#(A(),B()) -> c_91() equal#(A(),F()) -> c_92() equal#(A(),T()) -> c_93() equal#(B(),A()) -> c_94() equal#(B(),B()) -> c_95() equal#(B(),F()) -> c_96() equal#(B(),T()) -> c_97() equal#(F(),A()) -> c_98() equal#(F(),B()) -> c_99() equal#(F(),F()) -> c_100() equal#(F(),T()) -> c_101() equal#(T(),A()) -> c_102() equal#(T(),B()) -> c_103() equal#(T(),F()) -> c_104() equal#(T(),T()) -> c_105() goal#(z0,z1) -> c_106(q#(z0,z1)) head#(Cons(z0,z1)) -> c_107() notEmpty#(Cons(z0,z1)) -> c_108() notEmpty#(Nil()) -> c_109() p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) p[Ite]#(False(),z0,Cons(A(),z1)) -> c_111() p[Ite]#(False(),z0,Cons(B(),z1)) -> c_112() p[Ite]#(False(),z0,Cons(F(),z1)) -> c_113(and#(r(z0,Cons(F(),z1)),p(z0,z1)),r#(z0,Cons(F(),z1)),p#(z0,z1)) p[Ite]#(False(),z0,Cons(T(),z1)) -> c_114() p[Ite]#(True(),z0,z1) -> c_115() q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) q[Ite]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_117() q[Ite]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_118() q[Ite]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_119() q[Ite]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_120() q[Ite]#(False(),z0,Cons(A(),Nil())) -> c_121(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_122() q[Ite]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_123() q[Ite]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_124() q[Ite]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_125() q[Ite]#(False(),z0,Cons(B(),Nil())) -> c_126(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_127() q[Ite]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_128() q[Ite]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_129(and#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1))) q[Ite]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_130() q[Ite]#(False(),z0,Cons(F(),Nil())) -> c_131(q[Ite][False][Ite]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_132() q[Ite]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_133() q[Ite]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_134() q[Ite]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_135() q[Ite]#(False(),z0,Cons(T(),Nil())) -> c_136(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(True(),z0,z1) -> c_137() q[Ite][False][Ite]#(False(),z0,z1) -> c_138() q[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_139(and#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2)) r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) r[Ite]#(False(),z0,Cons(A(),z1)) -> c_141() r[Ite]#(False(),z0,Cons(B(),z1)) -> c_142() r[Ite]#(False(),z0,Cons(F(),z1)) -> c_143(and#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1)) r[Ite]#(False(),z0,Cons(T(),z1)) -> c_144() r[Ite]#(True(),z0,z1) -> c_145() t#(z0,z1) -> c_146(t[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) t[Ite]#(False(),z0,z1) -> c_147() t[Ite]#(True(),z0,z1) -> c_148() - Weak TRS: and(False(),False()) -> False() and(False(),True()) -> False() and(True(),False()) -> False() and(True(),True()) -> True() e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() p(z0,z1) -> p[Ite](e(z0,z1),z0,z1) p[Ite](False(),z0,Cons(A(),z1)) -> False() p[Ite](False(),z0,Cons(B(),z1)) -> False() p[Ite](False(),z0,Cons(F(),z1)) -> and(r(z0,Cons(F(),z1)),p(z0,z1)) p[Ite](False(),z0,Cons(T(),z1)) -> False() p[Ite](True(),z0,z1) -> True() q(z0,z1) -> q[Ite](e(z0,z1),z0,z1) q[Ite](False(),z0,Cons(A(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) q[Ite](False(),z0,Cons(B(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) q[Ite](False(),z0,Cons(F(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(F(),z1))) -> q[Ite][False][Ite][True][Ite](and(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,z0 ,Cons(F(),Cons(F(),z1))) q[Ite](False(),z0,Cons(F(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Nil())) -> q[Ite][False][Ite](and(True() ,and(True() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) q[Ite](False(),z0,Cons(T(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) q[Ite](True(),z0,z1) -> True() r(z0,z1) -> r[Ite](e(z0,z1),z0,z1) r[Ite](False(),z0,Cons(A(),z1)) -> False() r[Ite](False(),z0,Cons(B(),z1)) -> False() r[Ite](False(),z0,Cons(F(),z1)) -> r[Ite][False][Ite][True][Ite](and(q(z0,z1),r(z0,z1)),z0,Cons(F(),z1)) r[Ite](False(),z0,Cons(T(),z1)) -> False() r[Ite](True(),z0,z1) -> True() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/3,c_31/3,c_32/3,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/4,c_41/4,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/4,c_57/4 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/4,c_68/4,c_69/0,c_70/0,c_71/4,c_72/4 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,27,28,29} by application of Pre({1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,27,28,29}) = {30,31,32,33}. Here rules are labelled as follows: 1: E#(Cons(A(),Cons(z0,z1)),z2) -> c_1() 2: E#(Cons(A(),Nil()),z0) -> c_2() 3: E#(Cons(B(),Cons(z0,z1)),z2) -> c_3() 4: E#(Cons(B(),Nil()),z0) -> c_4() 5: E#(Cons(F(),Cons(z0,z1)),z2) -> c_5() 6: E#(Cons(F(),Nil()),z0) -> c_6() 7: E#(Cons(T(),Cons(z0,z1)),z2) -> c_7() 8: E#(Cons(T(),Nil()),z0) -> c_8() 9: E#(Nil(),z0) -> c_9() 10: EQUAL#(A(),A()) -> c_10() 11: EQUAL#(A(),B()) -> c_11() 12: EQUAL#(A(),F()) -> c_12() 13: EQUAL#(A(),T()) -> c_13() 14: EQUAL#(B(),A()) -> c_14() 15: EQUAL#(B(),B()) -> c_15() 16: EQUAL#(B(),F()) -> c_16() 17: EQUAL#(B(),T()) -> c_17() 18: EQUAL#(F(),A()) -> c_18() 19: EQUAL#(F(),B()) -> c_19() 20: EQUAL#(F(),F()) -> c_20() 21: EQUAL#(F(),T()) -> c_21() 22: EQUAL#(T(),A()) -> c_22() 23: EQUAL#(T(),B()) -> c_23() 24: EQUAL#(T(),F()) -> c_24() 25: EQUAL#(T(),T()) -> c_25() 26: GOAL#(z0,z1) -> c_26(Q#(z0,z1)) 27: HEAD#(Cons(z0,z1)) -> c_27() 28: NOTEMPTY#(Cons(z0,z1)) -> c_28() 29: NOTEMPTY#(Nil()) -> c_29() 30: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) 31: Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) 32: R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) 33: T'#(z0,z1) -> c_33(T[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) 34: AND#(False(),False()) -> c_34() 35: AND#(False(),True()) -> c_35() 36: AND#(True(),False()) -> c_36() 37: AND#(True(),True()) -> c_37() 38: P[ITE]#(False(),z0,Cons(A(),z1)) -> c_38() 39: P[ITE]#(False(),z0,Cons(B(),z1)) -> c_39() 40: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,R#(z0,Cons(F(),z1))) 41: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,P#(z0,z1)) 42: P[ITE]#(False(),z0,Cons(T(),z1)) -> c_42() 43: P[ITE]#(True(),z0,z1) -> c_43() 44: Q[ITE]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_44() 45: Q[ITE]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_45() 46: Q[ITE]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_46() 47: Q[ITE]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_47() 48: Q[ITE]#(False(),z0,Cons(A(),Nil())) -> c_48(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 49: Q[ITE]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_49() 50: Q[ITE]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_50() 51: Q[ITE]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_51() 52: Q[ITE]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_52() 53: Q[ITE]#(False(),z0,Cons(B(),Nil())) -> c_53(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 54: Q[ITE]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_54() 55: Q[ITE]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_55() 56: Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,P#(z0,Cons(F(),Cons(F(),z1)))) 57: Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,Q#(z0,Cons(F(),z1))) 58: Q[ITE]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_58() 59: Q[ITE]#(False(),z0,Cons(F(),Nil())) -> c_59(Q[ITE][FALSE][ITE]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 60: Q[ITE]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_60() 61: Q[ITE]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_61() 62: Q[ITE]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_62() 63: Q[ITE]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_63() 64: Q[ITE]#(False(),z0,Cons(T(),Nil())) -> c_64(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 65: Q[ITE]#(True(),z0,z1) -> c_65() 66: Q[ITE][FALSE][ITE]#(False(),z0,z1) -> c_66() 67: Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,P#(z0,Cons(z1,z2))) 68: Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,Q#(z0,z2)) 69: R[ITE]#(False(),z0,Cons(A(),z1)) -> c_69() 70: R[ITE]#(False(),z0,Cons(B(),z1)) -> c_70() 71: R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),Q#(z0,z1)) 72: R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),R#(z0,z1)) 73: R[ITE]#(False(),z0,Cons(T(),z1)) -> c_73() 74: R[ITE]#(True(),z0,z1) -> c_74() 75: T[ITE]#(False(),z0,z1) -> c_75() 76: T[ITE]#(True(),z0,z1) -> c_76() 77: and#(False(),False()) -> c_77() 78: and#(False(),True()) -> c_78() 79: and#(True(),False()) -> c_79() 80: and#(True(),True()) -> c_80() 81: e#(Cons(A(),Cons(z0,z1)),z2) -> c_81() 82: e#(Cons(A(),Nil()),z0) -> c_82() 83: e#(Cons(B(),Cons(z0,z1)),z2) -> c_83() 84: e#(Cons(B(),Nil()),z0) -> c_84() 85: e#(Cons(F(),Cons(z0,z1)),z2) -> c_85() 86: e#(Cons(F(),Nil()),z0) -> c_86() 87: e#(Cons(T(),Cons(z0,z1)),z2) -> c_87() 88: e#(Cons(T(),Nil()),z0) -> c_88() 89: e#(Nil(),z0) -> c_89() 90: equal#(A(),A()) -> c_90() 91: equal#(A(),B()) -> c_91() 92: equal#(A(),F()) -> c_92() 93: equal#(A(),T()) -> c_93() 94: equal#(B(),A()) -> c_94() 95: equal#(B(),B()) -> c_95() 96: equal#(B(),F()) -> c_96() 97: equal#(B(),T()) -> c_97() 98: equal#(F(),A()) -> c_98() 99: equal#(F(),B()) -> c_99() 100: equal#(F(),F()) -> c_100() 101: equal#(F(),T()) -> c_101() 102: equal#(T(),A()) -> c_102() 103: equal#(T(),B()) -> c_103() 104: equal#(T(),F()) -> c_104() 105: equal#(T(),T()) -> c_105() 106: goal#(z0,z1) -> c_106(q#(z0,z1)) 107: head#(Cons(z0,z1)) -> c_107() 108: notEmpty#(Cons(z0,z1)) -> c_108() 109: notEmpty#(Nil()) -> c_109() 110: p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) 111: p[Ite]#(False(),z0,Cons(A(),z1)) -> c_111() 112: p[Ite]#(False(),z0,Cons(B(),z1)) -> c_112() 113: p[Ite]#(False(),z0,Cons(F(),z1)) -> c_113(and#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1)) 114: p[Ite]#(False(),z0,Cons(T(),z1)) -> c_114() 115: p[Ite]#(True(),z0,z1) -> c_115() 116: q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) 117: q[Ite]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_117() 118: q[Ite]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_118() 119: q[Ite]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_119() 120: q[Ite]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_120() 121: q[Ite]#(False(),z0,Cons(A(),Nil())) -> c_121(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 122: q[Ite]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_122() 123: q[Ite]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_123() 124: q[Ite]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_124() 125: q[Ite]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_125() 126: q[Ite]#(False(),z0,Cons(B(),Nil())) -> c_126(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 127: q[Ite]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_127() 128: q[Ite]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_128() 129: q[Ite]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_129(and#(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1))) 130: q[Ite]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_130() 131: q[Ite]#(False(),z0,Cons(F(),Nil())) -> c_131(q[Ite][False][Ite]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 132: q[Ite]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_132() 133: q[Ite]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_133() 134: q[Ite]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_134() 135: q[Ite]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_135() 136: q[Ite]#(False(),z0,Cons(T(),Nil())) -> c_136(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 137: q[Ite]#(True(),z0,z1) -> c_137() 138: q[Ite][False][Ite]#(False(),z0,z1) -> c_138() 139: q[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_139(and#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2)) 140: r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) 141: r[Ite]#(False(),z0,Cons(A(),z1)) -> c_141() 142: r[Ite]#(False(),z0,Cons(B(),z1)) -> c_142() 143: r[Ite]#(False(),z0,Cons(F(),z1)) -> c_143(and#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1)) 144: r[Ite]#(False(),z0,Cons(T(),z1)) -> c_144() 145: r[Ite]#(True(),z0,z1) -> c_145() 146: t#(z0,z1) -> c_146(t[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) 147: t[Ite]#(False(),z0,z1) -> c_147() 148: t[Ite]#(True(),z0,z1) -> c_148() * Step 5: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: GOAL#(z0,z1) -> c_26(Q#(z0,z1)) P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) T'#(z0,z1) -> c_33(T[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) - Weak DPs: AND#(False(),False()) -> c_34() AND#(False(),True()) -> c_35() AND#(True(),False()) -> c_36() AND#(True(),True()) -> c_37() E#(Cons(A(),Cons(z0,z1)),z2) -> c_1() E#(Cons(A(),Nil()),z0) -> c_2() E#(Cons(B(),Cons(z0,z1)),z2) -> c_3() E#(Cons(B(),Nil()),z0) -> c_4() E#(Cons(F(),Cons(z0,z1)),z2) -> c_5() E#(Cons(F(),Nil()),z0) -> c_6() E#(Cons(T(),Cons(z0,z1)),z2) -> c_7() E#(Cons(T(),Nil()),z0) -> c_8() E#(Nil(),z0) -> c_9() EQUAL#(A(),A()) -> c_10() EQUAL#(A(),B()) -> c_11() EQUAL#(A(),F()) -> c_12() EQUAL#(A(),T()) -> c_13() EQUAL#(B(),A()) -> c_14() EQUAL#(B(),B()) -> c_15() EQUAL#(B(),F()) -> c_16() EQUAL#(B(),T()) -> c_17() EQUAL#(F(),A()) -> c_18() EQUAL#(F(),B()) -> c_19() EQUAL#(F(),F()) -> c_20() EQUAL#(F(),T()) -> c_21() EQUAL#(T(),A()) -> c_22() EQUAL#(T(),B()) -> c_23() EQUAL#(T(),F()) -> c_24() EQUAL#(T(),T()) -> c_25() HEAD#(Cons(z0,z1)) -> c_27() NOTEMPTY#(Cons(z0,z1)) -> c_28() NOTEMPTY#(Nil()) -> c_29() P[ITE]#(False(),z0,Cons(A(),z1)) -> c_38() P[ITE]#(False(),z0,Cons(B(),z1)) -> c_39() P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,P#(z0,z1)) P[ITE]#(False(),z0,Cons(T(),z1)) -> c_42() P[ITE]#(True(),z0,z1) -> c_43() Q[ITE]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_44() Q[ITE]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_45() Q[ITE]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_46() Q[ITE]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_47() Q[ITE]#(False(),z0,Cons(A(),Nil())) -> c_48(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_49() Q[ITE]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_50() Q[ITE]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_51() Q[ITE]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_52() Q[ITE]#(False(),z0,Cons(B(),Nil())) -> c_53(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_54() Q[ITE]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_55() Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,Q#(z0,Cons(F(),z1))) Q[ITE]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_58() Q[ITE]#(False(),z0,Cons(F(),Nil())) -> c_59(Q[ITE][FALSE][ITE]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_60() Q[ITE]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_61() Q[ITE]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_62() Q[ITE]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_63() Q[ITE]#(False(),z0,Cons(T(),Nil())) -> c_64(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(True(),z0,z1) -> c_65() Q[ITE][FALSE][ITE]#(False(),z0,z1) -> c_66() Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,P#(z0,Cons(z1,z2))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,Q#(z0,z2)) R[ITE]#(False(),z0,Cons(A(),z1)) -> c_69() R[ITE]#(False(),z0,Cons(B(),z1)) -> c_70() R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),R#(z0,z1)) R[ITE]#(False(),z0,Cons(T(),z1)) -> c_73() R[ITE]#(True(),z0,z1) -> c_74() T[ITE]#(False(),z0,z1) -> c_75() T[ITE]#(True(),z0,z1) -> c_76() and#(False(),False()) -> c_77() and#(False(),True()) -> c_78() and#(True(),False()) -> c_79() and#(True(),True()) -> c_80() e#(Cons(A(),Cons(z0,z1)),z2) -> c_81() e#(Cons(A(),Nil()),z0) -> c_82() e#(Cons(B(),Cons(z0,z1)),z2) -> c_83() e#(Cons(B(),Nil()),z0) -> c_84() e#(Cons(F(),Cons(z0,z1)),z2) -> c_85() e#(Cons(F(),Nil()),z0) -> c_86() e#(Cons(T(),Cons(z0,z1)),z2) -> c_87() e#(Cons(T(),Nil()),z0) -> c_88() e#(Nil(),z0) -> c_89() equal#(A(),A()) -> c_90() equal#(A(),B()) -> c_91() equal#(A(),F()) -> c_92() equal#(A(),T()) -> c_93() equal#(B(),A()) -> c_94() equal#(B(),B()) -> c_95() equal#(B(),F()) -> c_96() equal#(B(),T()) -> c_97() equal#(F(),A()) -> c_98() equal#(F(),B()) -> c_99() equal#(F(),F()) -> c_100() equal#(F(),T()) -> c_101() equal#(T(),A()) -> c_102() equal#(T(),B()) -> c_103() equal#(T(),F()) -> c_104() equal#(T(),T()) -> c_105() goal#(z0,z1) -> c_106(q#(z0,z1)) head#(Cons(z0,z1)) -> c_107() notEmpty#(Cons(z0,z1)) -> c_108() notEmpty#(Nil()) -> c_109() p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) p[Ite]#(False(),z0,Cons(A(),z1)) -> c_111() p[Ite]#(False(),z0,Cons(B(),z1)) -> c_112() p[Ite]#(False(),z0,Cons(F(),z1)) -> c_113(and#(r(z0,Cons(F(),z1)),p(z0,z1)),r#(z0,Cons(F(),z1)),p#(z0,z1)) p[Ite]#(False(),z0,Cons(T(),z1)) -> c_114() p[Ite]#(True(),z0,z1) -> c_115() q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) q[Ite]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_117() q[Ite]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_118() q[Ite]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_119() q[Ite]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_120() q[Ite]#(False(),z0,Cons(A(),Nil())) -> c_121(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_122() q[Ite]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_123() q[Ite]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_124() q[Ite]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_125() q[Ite]#(False(),z0,Cons(B(),Nil())) -> c_126(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_127() q[Ite]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_128() q[Ite]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_129(and#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1))) q[Ite]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_130() q[Ite]#(False(),z0,Cons(F(),Nil())) -> c_131(q[Ite][False][Ite]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_132() q[Ite]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_133() q[Ite]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_134() q[Ite]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_135() q[Ite]#(False(),z0,Cons(T(),Nil())) -> c_136(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(True(),z0,z1) -> c_137() q[Ite][False][Ite]#(False(),z0,z1) -> c_138() q[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_139(and#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2)) r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) r[Ite]#(False(),z0,Cons(A(),z1)) -> c_141() r[Ite]#(False(),z0,Cons(B(),z1)) -> c_142() r[Ite]#(False(),z0,Cons(F(),z1)) -> c_143(and#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1)) r[Ite]#(False(),z0,Cons(T(),z1)) -> c_144() r[Ite]#(True(),z0,z1) -> c_145() t#(z0,z1) -> c_146(t[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) t[Ite]#(False(),z0,z1) -> c_147() t[Ite]#(True(),z0,z1) -> c_148() - Weak TRS: and(False(),False()) -> False() and(False(),True()) -> False() and(True(),False()) -> False() and(True(),True()) -> True() e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() p(z0,z1) -> p[Ite](e(z0,z1),z0,z1) p[Ite](False(),z0,Cons(A(),z1)) -> False() p[Ite](False(),z0,Cons(B(),z1)) -> False() p[Ite](False(),z0,Cons(F(),z1)) -> and(r(z0,Cons(F(),z1)),p(z0,z1)) p[Ite](False(),z0,Cons(T(),z1)) -> False() p[Ite](True(),z0,z1) -> True() q(z0,z1) -> q[Ite](e(z0,z1),z0,z1) q[Ite](False(),z0,Cons(A(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) q[Ite](False(),z0,Cons(B(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) q[Ite](False(),z0,Cons(F(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(F(),z1))) -> q[Ite][False][Ite][True][Ite](and(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,z0 ,Cons(F(),Cons(F(),z1))) q[Ite](False(),z0,Cons(F(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Nil())) -> q[Ite][False][Ite](and(True() ,and(True() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) q[Ite](False(),z0,Cons(T(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) q[Ite](True(),z0,z1) -> True() r(z0,z1) -> r[Ite](e(z0,z1),z0,z1) r[Ite](False(),z0,Cons(A(),z1)) -> False() r[Ite](False(),z0,Cons(B(),z1)) -> False() r[Ite](False(),z0,Cons(F(),z1)) -> r[Ite][False][Ite][True][Ite](and(q(z0,z1),r(z0,z1)),z0,Cons(F(),z1)) r[Ite](False(),z0,Cons(T(),z1)) -> False() r[Ite](True(),z0,z1) -> True() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/3,c_31/3,c_32/3,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/4,c_41/4,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/4,c_57/4 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/4,c_68/4,c_69/0,c_70/0,c_71/4,c_72/4 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {5} by application of Pre({5}) = {}. Here rules are labelled as follows: 1: GOAL#(z0,z1) -> c_26(Q#(z0,z1)) 2: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) 3: Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) 4: R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) 5: T'#(z0,z1) -> c_33(T[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) 6: AND#(False(),False()) -> c_34() 7: AND#(False(),True()) -> c_35() 8: AND#(True(),False()) -> c_36() 9: AND#(True(),True()) -> c_37() 10: E#(Cons(A(),Cons(z0,z1)),z2) -> c_1() 11: E#(Cons(A(),Nil()),z0) -> c_2() 12: E#(Cons(B(),Cons(z0,z1)),z2) -> c_3() 13: E#(Cons(B(),Nil()),z0) -> c_4() 14: E#(Cons(F(),Cons(z0,z1)),z2) -> c_5() 15: E#(Cons(F(),Nil()),z0) -> c_6() 16: E#(Cons(T(),Cons(z0,z1)),z2) -> c_7() 17: E#(Cons(T(),Nil()),z0) -> c_8() 18: E#(Nil(),z0) -> c_9() 19: EQUAL#(A(),A()) -> c_10() 20: EQUAL#(A(),B()) -> c_11() 21: EQUAL#(A(),F()) -> c_12() 22: EQUAL#(A(),T()) -> c_13() 23: EQUAL#(B(),A()) -> c_14() 24: EQUAL#(B(),B()) -> c_15() 25: EQUAL#(B(),F()) -> c_16() 26: EQUAL#(B(),T()) -> c_17() 27: EQUAL#(F(),A()) -> c_18() 28: EQUAL#(F(),B()) -> c_19() 29: EQUAL#(F(),F()) -> c_20() 30: EQUAL#(F(),T()) -> c_21() 31: EQUAL#(T(),A()) -> c_22() 32: EQUAL#(T(),B()) -> c_23() 33: EQUAL#(T(),F()) -> c_24() 34: EQUAL#(T(),T()) -> c_25() 35: HEAD#(Cons(z0,z1)) -> c_27() 36: NOTEMPTY#(Cons(z0,z1)) -> c_28() 37: NOTEMPTY#(Nil()) -> c_29() 38: P[ITE]#(False(),z0,Cons(A(),z1)) -> c_38() 39: P[ITE]#(False(),z0,Cons(B(),z1)) -> c_39() 40: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,R#(z0,Cons(F(),z1))) 41: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,P#(z0,z1)) 42: P[ITE]#(False(),z0,Cons(T(),z1)) -> c_42() 43: P[ITE]#(True(),z0,z1) -> c_43() 44: Q[ITE]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_44() 45: Q[ITE]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_45() 46: Q[ITE]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_46() 47: Q[ITE]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_47() 48: Q[ITE]#(False(),z0,Cons(A(),Nil())) -> c_48(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 49: Q[ITE]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_49() 50: Q[ITE]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_50() 51: Q[ITE]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_51() 52: Q[ITE]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_52() 53: Q[ITE]#(False(),z0,Cons(B(),Nil())) -> c_53(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 54: Q[ITE]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_54() 55: Q[ITE]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_55() 56: Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,P#(z0,Cons(F(),Cons(F(),z1)))) 57: Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,Q#(z0,Cons(F(),z1))) 58: Q[ITE]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_58() 59: Q[ITE]#(False(),z0,Cons(F(),Nil())) -> c_59(Q[ITE][FALSE][ITE]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 60: Q[ITE]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_60() 61: Q[ITE]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_61() 62: Q[ITE]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_62() 63: Q[ITE]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_63() 64: Q[ITE]#(False(),z0,Cons(T(),Nil())) -> c_64(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 65: Q[ITE]#(True(),z0,z1) -> c_65() 66: Q[ITE][FALSE][ITE]#(False(),z0,z1) -> c_66() 67: Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,P#(z0,Cons(z1,z2))) 68: Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,Q#(z0,z2)) 69: R[ITE]#(False(),z0,Cons(A(),z1)) -> c_69() 70: R[ITE]#(False(),z0,Cons(B(),z1)) -> c_70() 71: R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),Q#(z0,z1)) 72: R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),R#(z0,z1)) 73: R[ITE]#(False(),z0,Cons(T(),z1)) -> c_73() 74: R[ITE]#(True(),z0,z1) -> c_74() 75: T[ITE]#(False(),z0,z1) -> c_75() 76: T[ITE]#(True(),z0,z1) -> c_76() 77: and#(False(),False()) -> c_77() 78: and#(False(),True()) -> c_78() 79: and#(True(),False()) -> c_79() 80: and#(True(),True()) -> c_80() 81: e#(Cons(A(),Cons(z0,z1)),z2) -> c_81() 82: e#(Cons(A(),Nil()),z0) -> c_82() 83: e#(Cons(B(),Cons(z0,z1)),z2) -> c_83() 84: e#(Cons(B(),Nil()),z0) -> c_84() 85: e#(Cons(F(),Cons(z0,z1)),z2) -> c_85() 86: e#(Cons(F(),Nil()),z0) -> c_86() 87: e#(Cons(T(),Cons(z0,z1)),z2) -> c_87() 88: e#(Cons(T(),Nil()),z0) -> c_88() 89: e#(Nil(),z0) -> c_89() 90: equal#(A(),A()) -> c_90() 91: equal#(A(),B()) -> c_91() 92: equal#(A(),F()) -> c_92() 93: equal#(A(),T()) -> c_93() 94: equal#(B(),A()) -> c_94() 95: equal#(B(),B()) -> c_95() 96: equal#(B(),F()) -> c_96() 97: equal#(B(),T()) -> c_97() 98: equal#(F(),A()) -> c_98() 99: equal#(F(),B()) -> c_99() 100: equal#(F(),F()) -> c_100() 101: equal#(F(),T()) -> c_101() 102: equal#(T(),A()) -> c_102() 103: equal#(T(),B()) -> c_103() 104: equal#(T(),F()) -> c_104() 105: equal#(T(),T()) -> c_105() 106: goal#(z0,z1) -> c_106(q#(z0,z1)) 107: head#(Cons(z0,z1)) -> c_107() 108: notEmpty#(Cons(z0,z1)) -> c_108() 109: notEmpty#(Nil()) -> c_109() 110: p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) 111: p[Ite]#(False(),z0,Cons(A(),z1)) -> c_111() 112: p[Ite]#(False(),z0,Cons(B(),z1)) -> c_112() 113: p[Ite]#(False(),z0,Cons(F(),z1)) -> c_113(and#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1)) 114: p[Ite]#(False(),z0,Cons(T(),z1)) -> c_114() 115: p[Ite]#(True(),z0,z1) -> c_115() 116: q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) 117: q[Ite]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_117() 118: q[Ite]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_118() 119: q[Ite]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_119() 120: q[Ite]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_120() 121: q[Ite]#(False(),z0,Cons(A(),Nil())) -> c_121(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 122: q[Ite]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_122() 123: q[Ite]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_123() 124: q[Ite]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_124() 125: q[Ite]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_125() 126: q[Ite]#(False(),z0,Cons(B(),Nil())) -> c_126(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 127: q[Ite]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_127() 128: q[Ite]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_128() 129: q[Ite]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_129(and#(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1))) 130: q[Ite]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_130() 131: q[Ite]#(False(),z0,Cons(F(),Nil())) -> c_131(q[Ite][False][Ite]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 132: q[Ite]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_132() 133: q[Ite]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_133() 134: q[Ite]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_134() 135: q[Ite]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_135() 136: q[Ite]#(False(),z0,Cons(T(),Nil())) -> c_136(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 137: q[Ite]#(True(),z0,z1) -> c_137() 138: q[Ite][False][Ite]#(False(),z0,z1) -> c_138() 139: q[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_139(and#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2)) 140: r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) 141: r[Ite]#(False(),z0,Cons(A(),z1)) -> c_141() 142: r[Ite]#(False(),z0,Cons(B(),z1)) -> c_142() 143: r[Ite]#(False(),z0,Cons(F(),z1)) -> c_143(and#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1)) 144: r[Ite]#(False(),z0,Cons(T(),z1)) -> c_144() 145: r[Ite]#(True(),z0,z1) -> c_145() 146: t#(z0,z1) -> c_146(t[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) 147: t[Ite]#(False(),z0,z1) -> c_147() 148: t[Ite]#(True(),z0,z1) -> c_148() * Step 6: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: GOAL#(z0,z1) -> c_26(Q#(z0,z1)) P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) - Weak DPs: AND#(False(),False()) -> c_34() AND#(False(),True()) -> c_35() AND#(True(),False()) -> c_36() AND#(True(),True()) -> c_37() E#(Cons(A(),Cons(z0,z1)),z2) -> c_1() E#(Cons(A(),Nil()),z0) -> c_2() E#(Cons(B(),Cons(z0,z1)),z2) -> c_3() E#(Cons(B(),Nil()),z0) -> c_4() E#(Cons(F(),Cons(z0,z1)),z2) -> c_5() E#(Cons(F(),Nil()),z0) -> c_6() E#(Cons(T(),Cons(z0,z1)),z2) -> c_7() E#(Cons(T(),Nil()),z0) -> c_8() E#(Nil(),z0) -> c_9() EQUAL#(A(),A()) -> c_10() EQUAL#(A(),B()) -> c_11() EQUAL#(A(),F()) -> c_12() EQUAL#(A(),T()) -> c_13() EQUAL#(B(),A()) -> c_14() EQUAL#(B(),B()) -> c_15() EQUAL#(B(),F()) -> c_16() EQUAL#(B(),T()) -> c_17() EQUAL#(F(),A()) -> c_18() EQUAL#(F(),B()) -> c_19() EQUAL#(F(),F()) -> c_20() EQUAL#(F(),T()) -> c_21() EQUAL#(T(),A()) -> c_22() EQUAL#(T(),B()) -> c_23() EQUAL#(T(),F()) -> c_24() EQUAL#(T(),T()) -> c_25() HEAD#(Cons(z0,z1)) -> c_27() NOTEMPTY#(Cons(z0,z1)) -> c_28() NOTEMPTY#(Nil()) -> c_29() P[ITE]#(False(),z0,Cons(A(),z1)) -> c_38() P[ITE]#(False(),z0,Cons(B(),z1)) -> c_39() P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,P#(z0,z1)) P[ITE]#(False(),z0,Cons(T(),z1)) -> c_42() P[ITE]#(True(),z0,z1) -> c_43() Q[ITE]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_44() Q[ITE]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_45() Q[ITE]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_46() Q[ITE]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_47() Q[ITE]#(False(),z0,Cons(A(),Nil())) -> c_48(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_49() Q[ITE]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_50() Q[ITE]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_51() Q[ITE]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_52() Q[ITE]#(False(),z0,Cons(B(),Nil())) -> c_53(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_54() Q[ITE]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_55() Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,Q#(z0,Cons(F(),z1))) Q[ITE]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_58() Q[ITE]#(False(),z0,Cons(F(),Nil())) -> c_59(Q[ITE][FALSE][ITE]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_60() Q[ITE]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_61() Q[ITE]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_62() Q[ITE]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_63() Q[ITE]#(False(),z0,Cons(T(),Nil())) -> c_64(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) Q[ITE]#(True(),z0,z1) -> c_65() Q[ITE][FALSE][ITE]#(False(),z0,z1) -> c_66() Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,P#(z0,Cons(z1,z2))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,Q#(z0,z2)) R[ITE]#(False(),z0,Cons(A(),z1)) -> c_69() R[ITE]#(False(),z0,Cons(B(),z1)) -> c_70() R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),R#(z0,z1)) R[ITE]#(False(),z0,Cons(T(),z1)) -> c_73() R[ITE]#(True(),z0,z1) -> c_74() T'#(z0,z1) -> c_33(T[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) T[ITE]#(False(),z0,z1) -> c_75() T[ITE]#(True(),z0,z1) -> c_76() and#(False(),False()) -> c_77() and#(False(),True()) -> c_78() and#(True(),False()) -> c_79() and#(True(),True()) -> c_80() e#(Cons(A(),Cons(z0,z1)),z2) -> c_81() e#(Cons(A(),Nil()),z0) -> c_82() e#(Cons(B(),Cons(z0,z1)),z2) -> c_83() e#(Cons(B(),Nil()),z0) -> c_84() e#(Cons(F(),Cons(z0,z1)),z2) -> c_85() e#(Cons(F(),Nil()),z0) -> c_86() e#(Cons(T(),Cons(z0,z1)),z2) -> c_87() e#(Cons(T(),Nil()),z0) -> c_88() e#(Nil(),z0) -> c_89() equal#(A(),A()) -> c_90() equal#(A(),B()) -> c_91() equal#(A(),F()) -> c_92() equal#(A(),T()) -> c_93() equal#(B(),A()) -> c_94() equal#(B(),B()) -> c_95() equal#(B(),F()) -> c_96() equal#(B(),T()) -> c_97() equal#(F(),A()) -> c_98() equal#(F(),B()) -> c_99() equal#(F(),F()) -> c_100() equal#(F(),T()) -> c_101() equal#(T(),A()) -> c_102() equal#(T(),B()) -> c_103() equal#(T(),F()) -> c_104() equal#(T(),T()) -> c_105() goal#(z0,z1) -> c_106(q#(z0,z1)) head#(Cons(z0,z1)) -> c_107() notEmpty#(Cons(z0,z1)) -> c_108() notEmpty#(Nil()) -> c_109() p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) p[Ite]#(False(),z0,Cons(A(),z1)) -> c_111() p[Ite]#(False(),z0,Cons(B(),z1)) -> c_112() p[Ite]#(False(),z0,Cons(F(),z1)) -> c_113(and#(r(z0,Cons(F(),z1)),p(z0,z1)),r#(z0,Cons(F(),z1)),p#(z0,z1)) p[Ite]#(False(),z0,Cons(T(),z1)) -> c_114() p[Ite]#(True(),z0,z1) -> c_115() q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) q[Ite]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_117() q[Ite]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_118() q[Ite]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_119() q[Ite]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_120() q[Ite]#(False(),z0,Cons(A(),Nil())) -> c_121(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_122() q[Ite]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_123() q[Ite]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_124() q[Ite]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_125() q[Ite]#(False(),z0,Cons(B(),Nil())) -> c_126(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_127() q[Ite]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_128() q[Ite]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_129(and#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1))) q[Ite]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_130() q[Ite]#(False(),z0,Cons(F(),Nil())) -> c_131(q[Ite][False][Ite]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_132() q[Ite]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_133() q[Ite]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_134() q[Ite]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_135() q[Ite]#(False(),z0,Cons(T(),Nil())) -> c_136(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) q[Ite]#(True(),z0,z1) -> c_137() q[Ite][False][Ite]#(False(),z0,z1) -> c_138() q[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_139(and#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2)) r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) r[Ite]#(False(),z0,Cons(A(),z1)) -> c_141() r[Ite]#(False(),z0,Cons(B(),z1)) -> c_142() r[Ite]#(False(),z0,Cons(F(),z1)) -> c_143(and#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1)) r[Ite]#(False(),z0,Cons(T(),z1)) -> c_144() r[Ite]#(True(),z0,z1) -> c_145() t#(z0,z1) -> c_146(t[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) t[Ite]#(False(),z0,z1) -> c_147() t[Ite]#(True(),z0,z1) -> c_148() - Weak TRS: and(False(),False()) -> False() and(False(),True()) -> False() and(True(),False()) -> False() and(True(),True()) -> True() e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() p(z0,z1) -> p[Ite](e(z0,z1),z0,z1) p[Ite](False(),z0,Cons(A(),z1)) -> False() p[Ite](False(),z0,Cons(B(),z1)) -> False() p[Ite](False(),z0,Cons(F(),z1)) -> and(r(z0,Cons(F(),z1)),p(z0,z1)) p[Ite](False(),z0,Cons(T(),z1)) -> False() p[Ite](True(),z0,z1) -> True() q(z0,z1) -> q[Ite](e(z0,z1),z0,z1) q[Ite](False(),z0,Cons(A(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) q[Ite](False(),z0,Cons(B(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) q[Ite](False(),z0,Cons(F(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(F(),z1))) -> q[Ite][False][Ite][True][Ite](and(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,z0 ,Cons(F(),Cons(F(),z1))) q[Ite](False(),z0,Cons(F(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Nil())) -> q[Ite][False][Ite](and(True() ,and(True() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) q[Ite](False(),z0,Cons(T(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) q[Ite](True(),z0,z1) -> True() r(z0,z1) -> r[Ite](e(z0,z1),z0,z1) r[Ite](False(),z0,Cons(A(),z1)) -> False() r[Ite](False(),z0,Cons(B(),z1)) -> False() r[Ite](False(),z0,Cons(F(),z1)) -> r[Ite][False][Ite][True][Ite](and(q(z0,z1),r(z0,z1)),z0,Cons(F(),z1)) r[Ite](False(),z0,Cons(T(),z1)) -> False() r[Ite](True(),z0,z1) -> True() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/3,c_31/3,c_32/3,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/4,c_41/4,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/4,c_57/4 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/4,c_68/4,c_69/0,c_70/0,c_71/4,c_72/4 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:GOAL#(z0,z1) -> c_26(Q#(z0,z1)) -->_1 Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):3 2:S:P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) -->_1 P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,P#(z0,z1)):40 -->_1 P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,R#(z0,Cons(F(),z1))):39 -->_2 e#(Nil(),z0) -> c_89():89 -->_2 e#(Cons(T(),Nil()),z0) -> c_88():88 -->_2 e#(Cons(T(),Cons(z0,z1)),z2) -> c_87():87 -->_2 e#(Cons(F(),Nil()),z0) -> c_86():86 -->_2 e#(Cons(F(),Cons(z0,z1)),z2) -> c_85():85 -->_2 e#(Cons(B(),Nil()),z0) -> c_84():84 -->_2 e#(Cons(B(),Cons(z0,z1)),z2) -> c_83():83 -->_2 e#(Cons(A(),Nil()),z0) -> c_82():82 -->_2 e#(Cons(A(),Cons(z0,z1)),z2) -> c_81():81 -->_1 P[ITE]#(True(),z0,z1) -> c_43():42 -->_1 P[ITE]#(False(),z0,Cons(T(),z1)) -> c_42():41 -->_1 P[ITE]#(False(),z0,Cons(B(),z1)) -> c_39():38 -->_1 P[ITE]#(False(),z0,Cons(A(),z1)) -> c_38():37 -->_3 E#(Nil(),z0) -> c_9():17 -->_3 E#(Cons(T(),Nil()),z0) -> c_8():16 -->_3 E#(Cons(T(),Cons(z0,z1)),z2) -> c_7():15 -->_3 E#(Cons(F(),Nil()),z0) -> c_6():14 -->_3 E#(Cons(F(),Cons(z0,z1)),z2) -> c_5():13 -->_3 E#(Cons(B(),Nil()),z0) -> c_4():12 -->_3 E#(Cons(B(),Cons(z0,z1)),z2) -> c_3():11 -->_3 E#(Cons(A(),Nil()),z0) -> c_2():10 -->_3 E#(Cons(A(),Cons(z0,z1)),z2) -> c_1():9 3:S:Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) -->_1 Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(AND#(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,Q#(z0,Cons(F(),z1))):56 -->_1 Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(AND#(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,P#(z0,Cons(F(),Cons(F(),z1)))):55 -->_2 e#(Nil(),z0) -> c_89():89 -->_2 e#(Cons(T(),Nil()),z0) -> c_88():88 -->_2 e#(Cons(T(),Cons(z0,z1)),z2) -> c_87():87 -->_2 e#(Cons(F(),Nil()),z0) -> c_86():86 -->_2 e#(Cons(F(),Cons(z0,z1)),z2) -> c_85():85 -->_2 e#(Cons(B(),Nil()),z0) -> c_84():84 -->_2 e#(Cons(B(),Cons(z0,z1)),z2) -> c_83():83 -->_2 e#(Cons(A(),Nil()),z0) -> c_82():82 -->_2 e#(Cons(A(),Cons(z0,z1)),z2) -> c_81():81 -->_1 Q[ITE]#(True(),z0,z1) -> c_65():64 -->_1 Q[ITE]#(False(),z0,Cons(T(),Nil())) -> c_64(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())):63 -->_1 Q[ITE]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_63():62 -->_1 Q[ITE]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_62():61 -->_1 Q[ITE]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_61():60 -->_1 Q[ITE]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_60():59 -->_1 Q[ITE]#(False(),z0,Cons(F(),Nil())) -> c_59(Q[ITE][FALSE][ITE]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True() ,and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True() ,and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())):58 -->_1 Q[ITE]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_58():57 -->_1 Q[ITE]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_55():54 -->_1 Q[ITE]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_54():53 -->_1 Q[ITE]#(False(),z0,Cons(B(),Nil())) -> c_53(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())):52 -->_1 Q[ITE]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_52():51 -->_1 Q[ITE]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_51():50 -->_1 Q[ITE]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_50():49 -->_1 Q[ITE]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_49():48 -->_1 Q[ITE]#(False(),z0,Cons(A(),Nil())) -> c_48(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())):47 -->_1 Q[ITE]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_47():46 -->_1 Q[ITE]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_46():45 -->_1 Q[ITE]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_45():44 -->_1 Q[ITE]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_44():43 -->_3 E#(Nil(),z0) -> c_9():17 -->_3 E#(Cons(T(),Nil()),z0) -> c_8():16 -->_3 E#(Cons(T(),Cons(z0,z1)),z2) -> c_7():15 -->_3 E#(Cons(F(),Nil()),z0) -> c_6():14 -->_3 E#(Cons(F(),Cons(z0,z1)),z2) -> c_5():13 -->_3 E#(Cons(B(),Nil()),z0) -> c_4():12 -->_3 E#(Cons(B(),Cons(z0,z1)),z2) -> c_3():11 -->_3 E#(Cons(A(),Nil()),z0) -> c_2():10 -->_3 E#(Cons(A(),Cons(z0,z1)),z2) -> c_1():9 4:S:R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) -->_1 R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),R#(z0,z1)):71 -->_1 R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),Q#(z0,z1)):70 -->_2 e#(Nil(),z0) -> c_89():89 -->_2 e#(Cons(T(),Nil()),z0) -> c_88():88 -->_2 e#(Cons(T(),Cons(z0,z1)),z2) -> c_87():87 -->_2 e#(Cons(F(),Nil()),z0) -> c_86():86 -->_2 e#(Cons(F(),Cons(z0,z1)),z2) -> c_85():85 -->_2 e#(Cons(B(),Nil()),z0) -> c_84():84 -->_2 e#(Cons(B(),Cons(z0,z1)),z2) -> c_83():83 -->_2 e#(Cons(A(),Nil()),z0) -> c_82():82 -->_2 e#(Cons(A(),Cons(z0,z1)),z2) -> c_81():81 -->_1 R[ITE]#(True(),z0,z1) -> c_74():73 -->_1 R[ITE]#(False(),z0,Cons(T(),z1)) -> c_73():72 -->_1 R[ITE]#(False(),z0,Cons(B(),z1)) -> c_70():69 -->_1 R[ITE]#(False(),z0,Cons(A(),z1)) -> c_69():68 -->_3 E#(Nil(),z0) -> c_9():17 -->_3 E#(Cons(T(),Nil()),z0) -> c_8():16 -->_3 E#(Cons(T(),Cons(z0,z1)),z2) -> c_7():15 -->_3 E#(Cons(F(),Nil()),z0) -> c_6():14 -->_3 E#(Cons(F(),Cons(z0,z1)),z2) -> c_5():13 -->_3 E#(Cons(B(),Nil()),z0) -> c_4():12 -->_3 E#(Cons(B(),Cons(z0,z1)),z2) -> c_3():11 -->_3 E#(Cons(A(),Nil()),z0) -> c_2():10 -->_3 E#(Cons(A(),Cons(z0,z1)),z2) -> c_1():9 5:W:AND#(False(),False()) -> c_34() 6:W:AND#(False(),True()) -> c_35() 7:W:AND#(True(),False()) -> c_36() 8:W:AND#(True(),True()) -> c_37() 9:W:E#(Cons(A(),Cons(z0,z1)),z2) -> c_1() 10:W:E#(Cons(A(),Nil()),z0) -> c_2() 11:W:E#(Cons(B(),Cons(z0,z1)),z2) -> c_3() 12:W:E#(Cons(B(),Nil()),z0) -> c_4() 13:W:E#(Cons(F(),Cons(z0,z1)),z2) -> c_5() 14:W:E#(Cons(F(),Nil()),z0) -> c_6() 15:W:E#(Cons(T(),Cons(z0,z1)),z2) -> c_7() 16:W:E#(Cons(T(),Nil()),z0) -> c_8() 17:W:E#(Nil(),z0) -> c_9() 18:W:EQUAL#(A(),A()) -> c_10() 19:W:EQUAL#(A(),B()) -> c_11() 20:W:EQUAL#(A(),F()) -> c_12() 21:W:EQUAL#(A(),T()) -> c_13() 22:W:EQUAL#(B(),A()) -> c_14() 23:W:EQUAL#(B(),B()) -> c_15() 24:W:EQUAL#(B(),F()) -> c_16() 25:W:EQUAL#(B(),T()) -> c_17() 26:W:EQUAL#(F(),A()) -> c_18() 27:W:EQUAL#(F(),B()) -> c_19() 28:W:EQUAL#(F(),F()) -> c_20() 29:W:EQUAL#(F(),T()) -> c_21() 30:W:EQUAL#(T(),A()) -> c_22() 31:W:EQUAL#(T(),B()) -> c_23() 32:W:EQUAL#(T(),F()) -> c_24() 33:W:EQUAL#(T(),T()) -> c_25() 34:W:HEAD#(Cons(z0,z1)) -> c_27() 35:W:NOTEMPTY#(Cons(z0,z1)) -> c_28() 36:W:NOTEMPTY#(Nil()) -> c_29() 37:W:P[ITE]#(False(),z0,Cons(A(),z1)) -> c_38() 38:W:P[ITE]#(False(),z0,Cons(B(),z1)) -> c_39() 39:W:P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,R#(z0,Cons(F(),z1))) -->_2 r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):140 -->_3 p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):110 -->_1 AND#(True(),True()) -> c_37():8 -->_1 AND#(True(),False()) -> c_36():7 -->_1 AND#(False(),True()) -> c_35():6 -->_1 AND#(False(),False()) -> c_34():5 -->_4 R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):4 40:W:P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,P#(z0,z1)) -->_2 r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):140 -->_3 p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):110 -->_1 AND#(True(),True()) -> c_37():8 -->_1 AND#(True(),False()) -> c_36():7 -->_1 AND#(False(),True()) -> c_35():6 -->_1 AND#(False(),False()) -> c_34():5 -->_4 P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):2 41:W:P[ITE]#(False(),z0,Cons(T(),z1)) -> c_42() 42:W:P[ITE]#(True(),z0,z1) -> c_43() 43:W:Q[ITE]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_44() 44:W:Q[ITE]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_45() 45:W:Q[ITE]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_46() 46:W:Q[ITE]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_47() 47:W:Q[ITE]#(False(),z0,Cons(A(),Nil())) -> c_48(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 48:W:Q[ITE]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_49() 49:W:Q[ITE]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_50() 50:W:Q[ITE]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_51() 51:W:Q[ITE]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_52() 52:W:Q[ITE]#(False(),z0,Cons(B(),Nil())) -> c_53(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 53:W:Q[ITE]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_54() 54:W:Q[ITE]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_55() 55:W:Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(AND#(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,P#(z0,Cons(F(),Cons(F(),z1)))) -->_3 q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):116 -->_2 p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):110 -->_1 AND#(True(),True()) -> c_37():8 -->_1 AND#(True(),False()) -> c_36():7 -->_1 AND#(False(),True()) -> c_35():6 -->_1 AND#(False(),False()) -> c_34():5 -->_4 P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):2 56:W:Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(AND#(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,Q#(z0,Cons(F(),z1))) -->_3 q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):116 -->_2 p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):110 -->_1 AND#(True(),True()) -> c_37():8 -->_1 AND#(True(),False()) -> c_36():7 -->_1 AND#(False(),True()) -> c_35():6 -->_1 AND#(False(),False()) -> c_34():5 -->_4 Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):3 57:W:Q[ITE]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_58() 58:W:Q[ITE]#(False(),z0,Cons(F(),Nil())) -> c_59(Q[ITE][FALSE][ITE]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 59:W:Q[ITE]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_60() 60:W:Q[ITE]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_61() 61:W:Q[ITE]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_62() 62:W:Q[ITE]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_63() 63:W:Q[ITE]#(False(),z0,Cons(T(),Nil())) -> c_64(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 64:W:Q[ITE]#(True(),z0,z1) -> c_65() 65:W:Q[ITE][FALSE][ITE]#(False(),z0,z1) -> c_66() 66:W:Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,P#(z0,Cons(z1,z2))) -->_3 q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):116 -->_2 p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):110 -->_1 AND#(True(),True()) -> c_37():8 -->_1 AND#(True(),False()) -> c_36():7 -->_1 AND#(False(),True()) -> c_35():6 -->_1 AND#(False(),False()) -> c_34():5 -->_4 P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):2 67:W:Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,Q#(z0,z2)) -->_3 q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):116 -->_2 p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):110 -->_1 AND#(True(),True()) -> c_37():8 -->_1 AND#(True(),False()) -> c_36():7 -->_1 AND#(False(),True()) -> c_35():6 -->_1 AND#(False(),False()) -> c_34():5 -->_4 Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):3 68:W:R[ITE]#(False(),z0,Cons(A(),z1)) -> c_69() 69:W:R[ITE]#(False(),z0,Cons(B(),z1)) -> c_70() 70:W:R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),Q#(z0,z1)) -->_3 r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):140 -->_2 q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):116 -->_1 AND#(True(),True()) -> c_37():8 -->_1 AND#(True(),False()) -> c_36():7 -->_1 AND#(False(),True()) -> c_35():6 -->_1 AND#(False(),False()) -> c_34():5 -->_4 Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):3 71:W:R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),R#(z0,z1)) -->_3 r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):140 -->_2 q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):116 -->_1 AND#(True(),True()) -> c_37():8 -->_1 AND#(True(),False()) -> c_36():7 -->_1 AND#(False(),True()) -> c_35():6 -->_1 AND#(False(),False()) -> c_34():5 -->_4 R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):4 72:W:R[ITE]#(False(),z0,Cons(T(),z1)) -> c_73() 73:W:R[ITE]#(True(),z0,z1) -> c_74() 74:W:T'#(z0,z1) -> c_33(T[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) -->_2 e#(Nil(),z0) -> c_89():89 -->_2 e#(Cons(T(),Nil()),z0) -> c_88():88 -->_2 e#(Cons(T(),Cons(z0,z1)),z2) -> c_87():87 -->_2 e#(Cons(F(),Nil()),z0) -> c_86():86 -->_2 e#(Cons(F(),Cons(z0,z1)),z2) -> c_85():85 -->_2 e#(Cons(B(),Nil()),z0) -> c_84():84 -->_2 e#(Cons(B(),Cons(z0,z1)),z2) -> c_83():83 -->_2 e#(Cons(A(),Nil()),z0) -> c_82():82 -->_2 e#(Cons(A(),Cons(z0,z1)),z2) -> c_81():81 -->_1 T[ITE]#(True(),z0,z1) -> c_76():76 -->_1 T[ITE]#(False(),z0,z1) -> c_75():75 -->_3 E#(Nil(),z0) -> c_9():17 -->_3 E#(Cons(T(),Nil()),z0) -> c_8():16 -->_3 E#(Cons(T(),Cons(z0,z1)),z2) -> c_7():15 -->_3 E#(Cons(F(),Nil()),z0) -> c_6():14 -->_3 E#(Cons(F(),Cons(z0,z1)),z2) -> c_5():13 -->_3 E#(Cons(B(),Nil()),z0) -> c_4():12 -->_3 E#(Cons(B(),Cons(z0,z1)),z2) -> c_3():11 -->_3 E#(Cons(A(),Nil()),z0) -> c_2():10 -->_3 E#(Cons(A(),Cons(z0,z1)),z2) -> c_1():9 75:W:T[ITE]#(False(),z0,z1) -> c_75() 76:W:T[ITE]#(True(),z0,z1) -> c_76() 77:W:and#(False(),False()) -> c_77() 78:W:and#(False(),True()) -> c_78() 79:W:and#(True(),False()) -> c_79() 80:W:and#(True(),True()) -> c_80() 81:W:e#(Cons(A(),Cons(z0,z1)),z2) -> c_81() 82:W:e#(Cons(A(),Nil()),z0) -> c_82() 83:W:e#(Cons(B(),Cons(z0,z1)),z2) -> c_83() 84:W:e#(Cons(B(),Nil()),z0) -> c_84() 85:W:e#(Cons(F(),Cons(z0,z1)),z2) -> c_85() 86:W:e#(Cons(F(),Nil()),z0) -> c_86() 87:W:e#(Cons(T(),Cons(z0,z1)),z2) -> c_87() 88:W:e#(Cons(T(),Nil()),z0) -> c_88() 89:W:e#(Nil(),z0) -> c_89() 90:W:equal#(A(),A()) -> c_90() 91:W:equal#(A(),B()) -> c_91() 92:W:equal#(A(),F()) -> c_92() 93:W:equal#(A(),T()) -> c_93() 94:W:equal#(B(),A()) -> c_94() 95:W:equal#(B(),B()) -> c_95() 96:W:equal#(B(),F()) -> c_96() 97:W:equal#(B(),T()) -> c_97() 98:W:equal#(F(),A()) -> c_98() 99:W:equal#(F(),B()) -> c_99() 100:W:equal#(F(),F()) -> c_100() 101:W:equal#(F(),T()) -> c_101() 102:W:equal#(T(),A()) -> c_102() 103:W:equal#(T(),B()) -> c_103() 104:W:equal#(T(),F()) -> c_104() 105:W:equal#(T(),T()) -> c_105() 106:W:goal#(z0,z1) -> c_106(q#(z0,z1)) -->_1 q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):116 107:W:head#(Cons(z0,z1)) -> c_107() 108:W:notEmpty#(Cons(z0,z1)) -> c_108() 109:W:notEmpty#(Nil()) -> c_109() 110:W:p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) -->_1 p[Ite]#(False(),z0,Cons(F(),z1)) -> c_113(and#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1)):113 -->_1 p[Ite]#(True(),z0,z1) -> c_115():115 -->_1 p[Ite]#(False(),z0,Cons(T(),z1)) -> c_114():114 -->_1 p[Ite]#(False(),z0,Cons(B(),z1)) -> c_112():112 -->_1 p[Ite]#(False(),z0,Cons(A(),z1)) -> c_111():111 -->_2 e#(Nil(),z0) -> c_89():89 -->_2 e#(Cons(T(),Nil()),z0) -> c_88():88 -->_2 e#(Cons(T(),Cons(z0,z1)),z2) -> c_87():87 -->_2 e#(Cons(F(),Nil()),z0) -> c_86():86 -->_2 e#(Cons(F(),Cons(z0,z1)),z2) -> c_85():85 -->_2 e#(Cons(B(),Nil()),z0) -> c_84():84 -->_2 e#(Cons(B(),Cons(z0,z1)),z2) -> c_83():83 -->_2 e#(Cons(A(),Nil()),z0) -> c_82():82 -->_2 e#(Cons(A(),Cons(z0,z1)),z2) -> c_81():81 111:W:p[Ite]#(False(),z0,Cons(A(),z1)) -> c_111() 112:W:p[Ite]#(False(),z0,Cons(B(),z1)) -> c_112() 113:W:p[Ite]#(False(),z0,Cons(F(),z1)) -> c_113(and#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1)) -->_2 r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):140 -->_3 p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):110 -->_1 and#(True(),True()) -> c_80():80 -->_1 and#(True(),False()) -> c_79():79 -->_1 and#(False(),True()) -> c_78():78 -->_1 and#(False(),False()) -> c_77():77 114:W:p[Ite]#(False(),z0,Cons(T(),z1)) -> c_114() 115:W:p[Ite]#(True(),z0,z1) -> c_115() 116:W:q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) -->_1 q[Ite]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_129(and#(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1))):129 -->_1 q[Ite]#(True(),z0,z1) -> c_137():137 -->_1 q[Ite]#(False(),z0,Cons(T(),Nil())) -> c_136(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()) ,F())))) ,z0 ,Cons(T(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())):136 -->_1 q[Ite]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_135():135 -->_1 q[Ite]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_134():134 -->_1 q[Ite]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_133():133 -->_1 q[Ite]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_132():132 -->_1 q[Ite]#(False(),z0,Cons(F(),Nil())) -> c_131(q[Ite][False][Ite]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()) ,F())))) ,z0 ,Cons(F(),Nil())) ,and#(True() ,and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())):131 -->_1 q[Ite]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_130():130 -->_1 q[Ite]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_128():128 -->_1 q[Ite]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_127():127 -->_1 q[Ite]#(False(),z0,Cons(B(),Nil())) -> c_126(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()) ,F())))) ,z0 ,Cons(B(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())):126 -->_1 q[Ite]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_125():125 -->_1 q[Ite]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_124():124 -->_1 q[Ite]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_123():123 -->_1 q[Ite]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_122():122 -->_1 q[Ite]#(False(),z0,Cons(A(),Nil())) -> c_121(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()) ,F())))) ,z0 ,Cons(A(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())):121 -->_1 q[Ite]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_120():120 -->_1 q[Ite]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_119():119 -->_1 q[Ite]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_118():118 -->_1 q[Ite]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_117():117 -->_2 e#(Nil(),z0) -> c_89():89 -->_2 e#(Cons(T(),Nil()),z0) -> c_88():88 -->_2 e#(Cons(T(),Cons(z0,z1)),z2) -> c_87():87 -->_2 e#(Cons(F(),Nil()),z0) -> c_86():86 -->_2 e#(Cons(F(),Cons(z0,z1)),z2) -> c_85():85 -->_2 e#(Cons(B(),Nil()),z0) -> c_84():84 -->_2 e#(Cons(B(),Cons(z0,z1)),z2) -> c_83():83 -->_2 e#(Cons(A(),Nil()),z0) -> c_82():82 -->_2 e#(Cons(A(),Cons(z0,z1)),z2) -> c_81():81 117:W:q[Ite]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_117() 118:W:q[Ite]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_118() 119:W:q[Ite]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_119() 120:W:q[Ite]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_120() 121:W:q[Ite]#(False(),z0,Cons(A(),Nil())) -> c_121(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 122:W:q[Ite]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_122() 123:W:q[Ite]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_123() 124:W:q[Ite]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_124() 125:W:q[Ite]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_125() 126:W:q[Ite]#(False(),z0,Cons(B(),Nil())) -> c_126(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 127:W:q[Ite]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_127() 128:W:q[Ite]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_128() 129:W:q[Ite]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_129(and#(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1))) -->_3 q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):116 -->_2 p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):110 -->_1 and#(True(),True()) -> c_80():80 -->_1 and#(True(),False()) -> c_79():79 -->_1 and#(False(),True()) -> c_78():78 -->_1 and#(False(),False()) -> c_77():77 130:W:q[Ite]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_130() 131:W:q[Ite]#(False(),z0,Cons(F(),Nil())) -> c_131(q[Ite][False][Ite]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True() ,and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 132:W:q[Ite]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_132() 133:W:q[Ite]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_133() 134:W:q[Ite]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_134() 135:W:q[Ite]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_135() 136:W:q[Ite]#(False(),z0,Cons(T(),Nil())) -> c_136(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 137:W:q[Ite]#(True(),z0,z1) -> c_137() 138:W:q[Ite][False][Ite]#(False(),z0,z1) -> c_138() 139:W:q[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_139(and#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2)) -->_3 q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):116 -->_2 p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):110 -->_1 and#(True(),True()) -> c_80():80 -->_1 and#(True(),False()) -> c_79():79 -->_1 and#(False(),True()) -> c_78():78 -->_1 and#(False(),False()) -> c_77():77 140:W:r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) -->_1 r[Ite]#(False(),z0,Cons(F(),z1)) -> c_143(and#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1)):143 -->_1 r[Ite]#(True(),z0,z1) -> c_145():145 -->_1 r[Ite]#(False(),z0,Cons(T(),z1)) -> c_144():144 -->_1 r[Ite]#(False(),z0,Cons(B(),z1)) -> c_142():142 -->_1 r[Ite]#(False(),z0,Cons(A(),z1)) -> c_141():141 -->_2 e#(Nil(),z0) -> c_89():89 -->_2 e#(Cons(T(),Nil()),z0) -> c_88():88 -->_2 e#(Cons(T(),Cons(z0,z1)),z2) -> c_87():87 -->_2 e#(Cons(F(),Nil()),z0) -> c_86():86 -->_2 e#(Cons(F(),Cons(z0,z1)),z2) -> c_85():85 -->_2 e#(Cons(B(),Nil()),z0) -> c_84():84 -->_2 e#(Cons(B(),Cons(z0,z1)),z2) -> c_83():83 -->_2 e#(Cons(A(),Nil()),z0) -> c_82():82 -->_2 e#(Cons(A(),Cons(z0,z1)),z2) -> c_81():81 141:W:r[Ite]#(False(),z0,Cons(A(),z1)) -> c_141() 142:W:r[Ite]#(False(),z0,Cons(B(),z1)) -> c_142() 143:W:r[Ite]#(False(),z0,Cons(F(),z1)) -> c_143(and#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1)) -->_3 r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):140 -->_2 q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)):116 -->_1 and#(True(),True()) -> c_80():80 -->_1 and#(True(),False()) -> c_79():79 -->_1 and#(False(),True()) -> c_78():78 -->_1 and#(False(),False()) -> c_77():77 144:W:r[Ite]#(False(),z0,Cons(T(),z1)) -> c_144() 145:W:r[Ite]#(True(),z0,z1) -> c_145() 146:W:t#(z0,z1) -> c_146(t[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) -->_1 t[Ite]#(True(),z0,z1) -> c_148():148 -->_1 t[Ite]#(False(),z0,z1) -> c_147():147 -->_2 e#(Nil(),z0) -> c_89():89 -->_2 e#(Cons(T(),Nil()),z0) -> c_88():88 -->_2 e#(Cons(T(),Cons(z0,z1)),z2) -> c_87():87 -->_2 e#(Cons(F(),Nil()),z0) -> c_86():86 -->_2 e#(Cons(F(),Cons(z0,z1)),z2) -> c_85():85 -->_2 e#(Cons(B(),Nil()),z0) -> c_84():84 -->_2 e#(Cons(B(),Cons(z0,z1)),z2) -> c_83():83 -->_2 e#(Cons(A(),Nil()),z0) -> c_82():82 -->_2 e#(Cons(A(),Cons(z0,z1)),z2) -> c_81():81 147:W:t[Ite]#(False(),z0,z1) -> c_147() 148:W:t[Ite]#(True(),z0,z1) -> c_148() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 146: t#(z0,z1) -> c_146(t[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) 147: t[Ite]#(False(),z0,z1) -> c_147() 148: t[Ite]#(True(),z0,z1) -> c_148() 139: q[Ite][False][Ite]#(True(),z0,Cons(z1,z2)) -> c_139(and#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2)) 138: q[Ite][False][Ite]#(False(),z0,z1) -> c_138() 109: notEmpty#(Nil()) -> c_109() 108: notEmpty#(Cons(z0,z1)) -> c_108() 107: head#(Cons(z0,z1)) -> c_107() 106: goal#(z0,z1) -> c_106(q#(z0,z1)) 105: equal#(T(),T()) -> c_105() 104: equal#(T(),F()) -> c_104() 103: equal#(T(),B()) -> c_103() 102: equal#(T(),A()) -> c_102() 101: equal#(F(),T()) -> c_101() 100: equal#(F(),F()) -> c_100() 99: equal#(F(),B()) -> c_99() 98: equal#(F(),A()) -> c_98() 97: equal#(B(),T()) -> c_97() 96: equal#(B(),F()) -> c_96() 95: equal#(B(),B()) -> c_95() 94: equal#(B(),A()) -> c_94() 93: equal#(A(),T()) -> c_93() 92: equal#(A(),F()) -> c_92() 91: equal#(A(),B()) -> c_91() 90: equal#(A(),A()) -> c_90() 74: T'#(z0,z1) -> c_33(T[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) 75: T[ITE]#(False(),z0,z1) -> c_75() 76: T[ITE]#(True(),z0,z1) -> c_76() 65: Q[ITE][FALSE][ITE]#(False(),z0,z1) -> c_66() 36: NOTEMPTY#(Nil()) -> c_29() 35: NOTEMPTY#(Cons(z0,z1)) -> c_28() 34: HEAD#(Cons(z0,z1)) -> c_27() 33: EQUAL#(T(),T()) -> c_25() 32: EQUAL#(T(),F()) -> c_24() 31: EQUAL#(T(),B()) -> c_23() 30: EQUAL#(T(),A()) -> c_22() 29: EQUAL#(F(),T()) -> c_21() 28: EQUAL#(F(),F()) -> c_20() 27: EQUAL#(F(),B()) -> c_19() 26: EQUAL#(F(),A()) -> c_18() 25: EQUAL#(B(),T()) -> c_17() 24: EQUAL#(B(),F()) -> c_16() 23: EQUAL#(B(),B()) -> c_15() 22: EQUAL#(B(),A()) -> c_14() 21: EQUAL#(A(),T()) -> c_13() 20: EQUAL#(A(),F()) -> c_12() 19: EQUAL#(A(),B()) -> c_11() 18: EQUAL#(A(),A()) -> c_10() 43: Q[ITE]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_44() 44: Q[ITE]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_45() 45: Q[ITE]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_46() 46: Q[ITE]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_47() 47: Q[ITE]#(False(),z0,Cons(A(),Nil())) -> c_48(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 48: Q[ITE]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_49() 49: Q[ITE]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_50() 50: Q[ITE]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_51() 51: Q[ITE]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_52() 52: Q[ITE]#(False(),z0,Cons(B(),Nil())) -> c_53(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 53: Q[ITE]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_54() 54: Q[ITE]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_55() 57: Q[ITE]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_58() 58: Q[ITE]#(False(),z0,Cons(F(),Nil())) -> c_59(Q[ITE][FALSE][ITE]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 59: Q[ITE]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_60() 60: Q[ITE]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_61() 61: Q[ITE]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_62() 62: Q[ITE]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_63() 63: Q[ITE]#(False(),z0,Cons(T(),Nil())) -> c_64(Q[ITE][FALSE][ITE]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(True(),and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,AND#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil()) ,EQUAL#(head(Nil()),F()) ,head#(Nil()) ,HEAD#(Nil())) 64: Q[ITE]#(True(),z0,z1) -> c_65() 37: P[ITE]#(False(),z0,Cons(A(),z1)) -> c_38() 38: P[ITE]#(False(),z0,Cons(B(),z1)) -> c_39() 41: P[ITE]#(False(),z0,Cons(T(),z1)) -> c_42() 42: P[ITE]#(True(),z0,z1) -> c_43() 9: E#(Cons(A(),Cons(z0,z1)),z2) -> c_1() 10: E#(Cons(A(),Nil()),z0) -> c_2() 11: E#(Cons(B(),Cons(z0,z1)),z2) -> c_3() 12: E#(Cons(B(),Nil()),z0) -> c_4() 13: E#(Cons(F(),Cons(z0,z1)),z2) -> c_5() 14: E#(Cons(F(),Nil()),z0) -> c_6() 15: E#(Cons(T(),Cons(z0,z1)),z2) -> c_7() 16: E#(Cons(T(),Nil()),z0) -> c_8() 17: E#(Nil(),z0) -> c_9() 68: R[ITE]#(False(),z0,Cons(A(),z1)) -> c_69() 69: R[ITE]#(False(),z0,Cons(B(),z1)) -> c_70() 72: R[ITE]#(False(),z0,Cons(T(),z1)) -> c_73() 73: R[ITE]#(True(),z0,z1) -> c_74() 5: AND#(False(),False()) -> c_34() 6: AND#(False(),True()) -> c_35() 7: AND#(True(),False()) -> c_36() 8: AND#(True(),True()) -> c_37() 116: q#(z0,z1) -> c_116(q[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) 143: r[Ite]#(False(),z0,Cons(F(),z1)) -> c_143(and#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1)) 140: r#(z0,z1) -> c_140(r[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) 113: p[Ite]#(False(),z0,Cons(F(),z1)) -> c_113(and#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1)) 110: p#(z0,z1) -> c_110(p[Ite]#(e(z0,z1),z0,z1),e#(z0,z1)) 129: q[Ite]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_129(and#(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1))) 117: q[Ite]#(False(),z0,Cons(A(),Cons(A(),z1))) -> c_117() 118: q[Ite]#(False(),z0,Cons(A(),Cons(B(),z1))) -> c_118() 119: q[Ite]#(False(),z0,Cons(A(),Cons(F(),z1))) -> c_119() 120: q[Ite]#(False(),z0,Cons(A(),Cons(T(),z1))) -> c_120() 121: q[Ite]#(False(),z0,Cons(A(),Nil())) -> c_121(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 122: q[Ite]#(False(),z0,Cons(B(),Cons(A(),z1))) -> c_122() 123: q[Ite]#(False(),z0,Cons(B(),Cons(B(),z1))) -> c_123() 124: q[Ite]#(False(),z0,Cons(B(),Cons(F(),z1))) -> c_124() 125: q[Ite]#(False(),z0,Cons(B(),Cons(T(),z1))) -> c_125() 126: q[Ite]#(False(),z0,Cons(B(),Nil())) -> c_126(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 127: q[Ite]#(False(),z0,Cons(F(),Cons(A(),z1))) -> c_127() 128: q[Ite]#(False(),z0,Cons(F(),Cons(B(),z1))) -> c_128() 130: q[Ite]#(False(),z0,Cons(F(),Cons(T(),z1))) -> c_130() 131: q[Ite]#(False(),z0,Cons(F(),Nil())) -> c_131(q[Ite][False][Ite]#(and(True() ,and(True() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) ,and#(True(),and(True(),and(False(),equal(head(Nil()),F())))) ,and#(True(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 132: q[Ite]#(False(),z0,Cons(T(),Cons(A(),z1))) -> c_132() 133: q[Ite]#(False(),z0,Cons(T(),Cons(B(),z1))) -> c_133() 134: q[Ite]#(False(),z0,Cons(T(),Cons(F(),z1))) -> c_134() 135: q[Ite]#(False(),z0,Cons(T(),Cons(T(),z1))) -> c_135() 136: q[Ite]#(False(),z0,Cons(T(),Nil())) -> c_136(q[Ite][False][Ite]#(and(True() ,and(False() ,and(False() ,equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) ,and#(True() ,and(False(),and(False(),equal(head(Nil()),F())))) ,and#(False(),and(False(),equal(head(Nil()),F()))) ,and#(False(),equal(head(Nil()),F())) ,equal#(head(Nil()),F()) ,head#(Nil())) 137: q[Ite]#(True(),z0,z1) -> c_137() 111: p[Ite]#(False(),z0,Cons(A(),z1)) -> c_111() 112: p[Ite]#(False(),z0,Cons(B(),z1)) -> c_112() 114: p[Ite]#(False(),z0,Cons(T(),z1)) -> c_114() 115: p[Ite]#(True(),z0,z1) -> c_115() 81: e#(Cons(A(),Cons(z0,z1)),z2) -> c_81() 82: e#(Cons(A(),Nil()),z0) -> c_82() 83: e#(Cons(B(),Cons(z0,z1)),z2) -> c_83() 84: e#(Cons(B(),Nil()),z0) -> c_84() 85: e#(Cons(F(),Cons(z0,z1)),z2) -> c_85() 86: e#(Cons(F(),Nil()),z0) -> c_86() 87: e#(Cons(T(),Cons(z0,z1)),z2) -> c_87() 88: e#(Cons(T(),Nil()),z0) -> c_88() 89: e#(Nil(),z0) -> c_89() 141: r[Ite]#(False(),z0,Cons(A(),z1)) -> c_141() 142: r[Ite]#(False(),z0,Cons(B(),z1)) -> c_142() 144: r[Ite]#(False(),z0,Cons(T(),z1)) -> c_144() 145: r[Ite]#(True(),z0,z1) -> c_145() 77: and#(False(),False()) -> c_77() 78: and#(False(),True()) -> c_78() 79: and#(True(),False()) -> c_79() 80: and#(True(),True()) -> c_80() * Step 7: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: GOAL#(z0,z1) -> c_26(Q#(z0,z1)) P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) - Weak DPs: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,P#(z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(AND#(p(z0,Cons(F(),Cons(F(),z1))),q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,Q#(z0,Cons(F(),z1))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,P#(z0,Cons(z1,z2))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,Q#(z0,z2)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),R#(z0,z1)) - Weak TRS: and(False(),False()) -> False() and(False(),True()) -> False() and(True(),False()) -> False() and(True(),True()) -> True() e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() p(z0,z1) -> p[Ite](e(z0,z1),z0,z1) p[Ite](False(),z0,Cons(A(),z1)) -> False() p[Ite](False(),z0,Cons(B(),z1)) -> False() p[Ite](False(),z0,Cons(F(),z1)) -> and(r(z0,Cons(F(),z1)),p(z0,z1)) p[Ite](False(),z0,Cons(T(),z1)) -> False() p[Ite](True(),z0,z1) -> True() q(z0,z1) -> q[Ite](e(z0,z1),z0,z1) q[Ite](False(),z0,Cons(A(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) q[Ite](False(),z0,Cons(B(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) q[Ite](False(),z0,Cons(F(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(F(),z1))) -> q[Ite][False][Ite][True][Ite](and(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,z0 ,Cons(F(),Cons(F(),z1))) q[Ite](False(),z0,Cons(F(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Nil())) -> q[Ite][False][Ite](and(True() ,and(True() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) q[Ite](False(),z0,Cons(T(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) q[Ite](True(),z0,z1) -> True() r(z0,z1) -> r[Ite](e(z0,z1),z0,z1) r[Ite](False(),z0,Cons(A(),z1)) -> False() r[Ite](False(),z0,Cons(B(),z1)) -> False() r[Ite](False(),z0,Cons(F(),z1)) -> r[Ite][False][Ite][True][Ite](and(q(z0,z1),r(z0,z1)),z0,Cons(F(),z1)) r[Ite](False(),z0,Cons(T(),z1)) -> False() r[Ite](True(),z0,z1) -> True() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/3,c_31/3,c_32/3,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/4,c_41/4,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/4,c_57/4 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/4,c_68/4,c_69/0,c_70/0,c_71/4,c_72/4 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:GOAL#(z0,z1) -> c_26(Q#(z0,z1)) -->_1 Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):3 2:S:P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) -->_1 P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,P#(z0,z1)):40 -->_1 P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,R#(z0,Cons(F(),z1))):39 3:S:Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) -->_1 Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(AND#(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,Q#(z0,Cons(F(),z1))):56 -->_1 Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(AND#(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,P#(z0,Cons(F(),Cons(F(),z1)))):55 4:S:R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)) -->_1 R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),R#(z0,z1)):71 -->_1 R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),Q#(z0,z1)):70 39:W:P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,R#(z0,Cons(F(),z1))) -->_4 R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):4 40:W:P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(AND#(r(z0,Cons(F(),z1)),p(z0,z1)) ,r#(z0,Cons(F(),z1)) ,p#(z0,z1) ,P#(z0,z1)) -->_4 P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):2 55:W:Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(AND#(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,P#(z0,Cons(F(),Cons(F(),z1)))) -->_4 P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):2 56:W:Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(AND#(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,p#(z0,Cons(F(),Cons(F(),z1))) ,q#(z0,Cons(F(),z1)) ,Q#(z0,Cons(F(),z1))) -->_4 Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):3 66:W:Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,P#(z0,Cons(z1,z2))) -->_4 P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):2 67:W:Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(AND#(p(z0,Cons(z1,z2)),q(z0,z2)) ,p#(z0,Cons(z1,z2)) ,q#(z0,z2) ,Q#(z0,z2)) -->_4 Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):3 70:W:R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),Q#(z0,z1)) -->_4 Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):3 71:W:R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(AND#(q(z0,z1),r(z0,z1)),q#(z0,z1),r#(z0,z1),R#(z0,z1)) -->_4 R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1),e#(z0,z1),E#(z0,z1)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(P#(z0,Cons(z1,z2))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(Q#(z0,z2)) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) * Step 8: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: GOAL#(z0,z1) -> c_26(Q#(z0,z1)) P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) - Weak DPs: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(P#(z0,Cons(z1,z2))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(Q#(z0,z2)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) - Weak TRS: and(False(),False()) -> False() and(False(),True()) -> False() and(True(),False()) -> False() and(True(),True()) -> True() e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() p(z0,z1) -> p[Ite](e(z0,z1),z0,z1) p[Ite](False(),z0,Cons(A(),z1)) -> False() p[Ite](False(),z0,Cons(B(),z1)) -> False() p[Ite](False(),z0,Cons(F(),z1)) -> and(r(z0,Cons(F(),z1)),p(z0,z1)) p[Ite](False(),z0,Cons(T(),z1)) -> False() p[Ite](True(),z0,z1) -> True() q(z0,z1) -> q[Ite](e(z0,z1),z0,z1) q[Ite](False(),z0,Cons(A(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(A(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(A(),Nil())) q[Ite](False(),z0,Cons(B(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(B(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(B(),Nil())) q[Ite](False(),z0,Cons(F(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Cons(F(),z1))) -> q[Ite][False][Ite][True][Ite](and(p(z0,Cons(F(),Cons(F(),z1))) ,q(z0,Cons(F(),z1))) ,z0 ,Cons(F(),Cons(F(),z1))) q[Ite](False(),z0,Cons(F(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(F(),Nil())) -> q[Ite][False][Ite](and(True() ,and(True() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(F(),Nil())) q[Ite](False(),z0,Cons(T(),Cons(A(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(B(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(F(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Cons(T(),z1))) -> False() q[Ite](False(),z0,Cons(T(),Nil())) -> q[Ite][False][Ite](and(True() ,and(False() ,and(False(),equal(head(Nil()),F())))) ,z0 ,Cons(T(),Nil())) q[Ite](True(),z0,z1) -> True() r(z0,z1) -> r[Ite](e(z0,z1),z0,z1) r[Ite](False(),z0,Cons(A(),z1)) -> False() r[Ite](False(),z0,Cons(B(),z1)) -> False() r[Ite](False(),z0,Cons(F(),z1)) -> r[Ite][False][Ite][True][Ite](and(q(z0,z1),r(z0,z1)),z0,Cons(F(),z1)) r[Ite](False(),z0,Cons(T(),z1)) -> False() r[Ite](True(),z0,z1) -> True() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/1,c_31/1,c_32/1,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/1,c_41/1,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/1,c_57/1 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/1,c_68/1,c_69/0,c_70/0,c_71/1,c_72/1 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() GOAL#(z0,z1) -> c_26(Q#(z0,z1)) P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(P#(z0,Cons(z1,z2))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(Q#(z0,z2)) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) * Step 9: RemoveHeads. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: GOAL#(z0,z1) -> c_26(Q#(z0,z1)) P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) - Weak DPs: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(P#(z0,Cons(z1,z2))) Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(Q#(z0,z2)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) - Weak TRS: e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/1,c_31/1,c_32/1,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/1,c_41/1,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/1,c_57/1 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/1,c_68/1,c_69/0,c_70/0,c_71/1,c_72/1 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:GOAL#(z0,z1) -> c_26(Q#(z0,z1)) -->_1 Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)):3 2:S:P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) -->_1 P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)):6 -->_1 P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))):5 3:S:Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) -->_1 Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))):8 -->_1 Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))):7 4:S:R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) -->_1 R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)):12 -->_1 R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)):11 5:W:P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) -->_1 R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)):4 6:W:P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) -->_1 P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)):2 7:W:Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) -->_1 P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)):2 8:W:Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) -->_1 Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)):3 9:W:Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(P#(z0,Cons(z1,z2))) -->_1 P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)):2 10:W:Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(Q#(z0,z2)) -->_1 Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)):3 11:W:R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) -->_1 Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)):3 12:W:R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) -->_1 R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)):4 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(1,GOAL#(z0,z1) -> c_26(Q#(z0,z1))) ,(9,Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_67(P#(z0,Cons(z1,z2)))) ,(10,Q[ITE][FALSE][ITE]#(True(),z0,Cons(z1,z2)) -> c_68(Q#(z0,z2)))] * Step 10: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) - Weak DPs: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) - Weak TRS: e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/1,c_31/1,c_32/1,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/1,c_41/1,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/1,c_57/1 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/1,c_68/1,c_69/0,c_70/0,c_71/1,c_72/1 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 4: R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) Consider the set of all dependency pairs 2: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) 3: Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) 4: R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) 5: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) 6: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) 7: Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) 8: Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) 11: R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) 12: R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) BEST_CASE TIME (?,?) SPACE(?,?)on application of the dependency pairs {4} These cover all (indirect) predecessors of dependency pairs {4,11,12} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ** Step 10.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) - Weak DPs: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) - Weak TRS: e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/1,c_31/1,c_32/1,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/1,c_41/1,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/1,c_57/1 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/1,c_68/1,c_69/0,c_70/0,c_71/1,c_72/1 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_30) = {1}, uargs(c_31) = {1}, uargs(c_32) = {1}, uargs(c_40) = {1}, uargs(c_41) = {1}, uargs(c_56) = {1}, uargs(c_57) = {1}, uargs(c_71) = {1}, uargs(c_72) = {1} Following symbols are considered usable: {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q#,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]# ,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q#,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} TcT has computed the following interpretation: p(A) = [2] p(AND) = [4] p(B) = [1] p(Cons) = [1] x1 + [1] x2 + [2] p(E) = [1] x2 + [4] p(EQUAL) = [2] x1 + [4] x2 + [0] p(F) = [0] p(False) = [0] p(GOAL) = [1] x2 + [0] p(HEAD) = [4] x1 + [1] p(NOTEMPTY) = [2] p(Nil) = [0] p(P) = [4] x1 + [1] x2 + [0] p(P[ITE]) = [1] x2 + [4] x3 + [1] p(Q) = [0] p(Q[ITE]) = [0] p(Q[ITE][FALSE][ITE]) = [0] p(R) = [0] p(R[ITE]) = [0] p(T) = [0] p(T') = [0] p(T[ITE]) = [0] p(True) = [0] p(and) = [0] p(c) = [0] p(c1) = [0] p(c10) = [0] p(c11) = [0] p(c12) = [0] p(c13) = [0] p(c14) = [0] p(c15) = [0] p(c16) = [0] p(c17) = [0] p(c18) = [0] p(c19) = [0] p(c2) = [0] p(c20) = [0] p(c21) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [0] p(c22) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [0] p(c23) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [0] p(c24) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [0] p(c25) = [0] p(c26) = [1] x1 + [1] x2 + [0] p(c27) = [1] x1 + [1] x2 + [0] p(c28) = [0] p(c29) = [0] p(c3) = [0] p(c30) = [0] p(c31) = [0] p(c32) = [1] x1 + [1] x2 + [0] p(c33) = [0] p(c34) = [1] p(c35) = [0] p(c36) = [1] p(c37) = [4] p(c38) = [0] p(c39) = [1] x2 + [0] p(c4) = [1] p(c40) = [1] p(c41) = [2] p(c42) = [4] p(c43) = [4] p(c44) = [1] p(c45) = [0] p(c46) = [1] p(c47) = [0] p(c48) = [0] p(c49) = [4] p(c5) = [1] p(c50) = [1] p(c51) = [0] p(c52) = [1] p(c53) = [0] p(c54) = [1] p(c55) = [4] p(c56) = [0] p(c57) = [1] p(c58) = [4] p(c59) = [1] p(c6) = [1] p(c60) = [1] p(c61) = [0] p(c62) = [4] p(c63) = [2] p(c64) = [4] p(c65) = [0] p(c66) = [1] p(c67) = [0] p(c68) = [4] p(c69) = [2] p(c7) = [0] p(c70) = [1] p(c71) = [0] p(c72) = [1] p(c73) = [0] p(c74) = [1] p(c75) = [0] p(c8) = [0] p(c9) = [2] p(e) = [0] p(e[Match][Cons][Ite][True][Match]) = [7] p(equal) = [2] x2 + [1] p(goal) = [4] x2 + [1] p(head) = [1] x1 + [1] p(notEmpty) = [0] p(p) = [4] x1 + [4] p(p[Ite]) = [1] x2 + [4] p(q) = [2] x1 + [1] p(q[Ite]) = [1] p(q[Ite][False][Ite]) = [4] x3 + [0] p(q[Ite][False][Ite][True][Ite]) = [4] p(r) = [4] x1 + [1] p(r[Ite]) = [1] x3 + [2] p(r[Ite][False][Ite][True][Ite]) = [1] x1 + [4] p(t) = [1] x2 + [0] p(t[Ite]) = [1] p(AND#) = [1] x1 + [4] x2 + [0] p(E#) = [2] x1 + [1] p(EQUAL#) = [1] p(GOAL#) = [2] p(HEAD#) = [2] x1 + [1] p(NOTEMPTY#) = [1] x1 + [0] p(P#) = [1] x1 + [1] x2 + [2] p(P[ITE]#) = [1] x2 + [1] x3 + [2] p(Q#) = [1] x1 + [1] x2 + [2] p(Q[ITE]#) = [1] x2 + [1] x3 + [2] p(Q[ITE][FALSE][ITE]#) = [1] x1 + [4] x2 + [2] p(R#) = [1] x1 + [1] x2 + [2] p(R[ITE]#) = [1] x2 + [1] x3 + [0] p(T'#) = [4] x2 + [0] p(T[ITE]#) = [1] x2 + [0] p(and#) = [1] x1 + [4] p(e#) = [2] x1 + [4] x2 + [2] p(equal#) = [2] x1 + [2] x2 + [4] p(goal#) = [0] p(head#) = [2] p(notEmpty#) = [4] p(p#) = [0] p(p[Ite]#) = [0] p(q#) = [1] x1 + [0] p(q[Ite]#) = [2] p(q[Ite][False][Ite]#) = [1] x1 + [4] x2 + [2] x3 + [1] p(r#) = [1] x1 + [0] p(r[Ite]#) = [1] p(t#) = [2] x1 + [1] x2 + [4] p(t[Ite]#) = [1] x2 + [4] p(c_1) = [4] p(c_2) = [1] p(c_3) = [2] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] p(c_7) = [1] p(c_8) = [1] p(c_9) = [0] p(c_10) = [1] p(c_11) = [0] p(c_12) = [1] p(c_13) = [1] p(c_14) = [2] p(c_15) = [2] p(c_16) = [1] p(c_17) = [1] p(c_18) = [1] p(c_19) = [0] p(c_20) = [1] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [4] p(c_25) = [1] p(c_26) = [1] p(c_27) = [2] p(c_28) = [0] p(c_29) = [0] p(c_30) = [1] x1 + [0] p(c_31) = [1] x1 + [0] p(c_32) = [1] x1 + [1] p(c_33) = [1] x1 + [1] x2 + [4] x3 + [1] p(c_34) = [2] p(c_35) = [2] p(c_36) = [4] p(c_37) = [0] p(c_38) = [1] p(c_39) = [0] p(c_40) = [1] x1 + [0] p(c_41) = [1] x1 + [2] p(c_42) = [0] p(c_43) = [1] p(c_44) = [2] p(c_45) = [0] p(c_46) = [0] p(c_47) = [0] p(c_48) = [4] x1 + [1] x2 + [1] x3 + [2] x4 + [1] x5 + [1] x6 + [1] x9 + [1] x12 + [1] x13 + [2] x14 + [2] x16 + [1] x19 + [4] x21 + [4] p(c_49) = [0] p(c_50) = [1] p(c_51) = [4] p(c_52) = [0] p(c_53) = [1] x1 + [1] x2 + [1] x3 + [1] x6 + [1] x8 + [2] x9 + [1] x10 + [1] x14 + [1] x16 + [1] x19 + [1] x21 + [0] p(c_54) = [0] p(c_55) = [1] p(c_56) = [1] x1 + [0] p(c_57) = [1] x1 + [2] p(c_58) = [0] p(c_59) = [2] x1 + [1] x2 + [1] x4 + [1] x6 + [1] x7 + [1] x8 + [1] x9 + [2] x10 + [1] x11 + [2] x12 + [1] x17 + [1] x21 + [1] p(c_60) = [0] p(c_61) = [1] p(c_62) = [1] p(c_63) = [0] p(c_64) = [1] x1 + [4] x3 + [4] x5 + [2] x6 + [2] x7 + [1] x10 + [4] x11 + [2] x15 + [4] x16 + [4] x17 + [1] x18 + [1] p(c_65) = [1] p(c_66) = [4] p(c_67) = [1] x1 + [4] p(c_68) = [1] x1 + [1] p(c_69) = [0] p(c_70) = [0] p(c_71) = [1] x1 + [0] p(c_72) = [1] x1 + [0] p(c_73) = [0] p(c_74) = [1] p(c_75) = [1] p(c_76) = [0] p(c_77) = [0] p(c_78) = [0] p(c_79) = [4] p(c_80) = [1] p(c_81) = [0] p(c_82) = [1] p(c_83) = [0] p(c_84) = [0] p(c_85) = [0] p(c_86) = [4] p(c_87) = [4] p(c_88) = [4] p(c_89) = [0] p(c_90) = [4] p(c_91) = [1] p(c_92) = [0] p(c_93) = [2] p(c_94) = [1] p(c_95) = [0] p(c_96) = [1] p(c_97) = [0] p(c_98) = [0] p(c_99) = [1] p(c_100) = [0] p(c_101) = [0] p(c_102) = [1] p(c_103) = [0] p(c_104) = [0] p(c_105) = [0] p(c_106) = [1] p(c_107) = [1] p(c_108) = [1] p(c_109) = [0] p(c_110) = [2] x1 + [0] p(c_111) = [0] p(c_112) = [0] p(c_113) = [1] x1 + [2] p(c_114) = [0] p(c_115) = [1] p(c_116) = [1] x2 + [1] p(c_117) = [0] p(c_118) = [0] p(c_119) = [0] p(c_120) = [1] p(c_121) = [0] p(c_122) = [2] p(c_123) = [0] p(c_124) = [1] p(c_125) = [4] p(c_126) = [1] x1 + [1] x2 + [0] p(c_127) = [2] p(c_128) = [4] p(c_129) = [1] x2 + [0] p(c_130) = [0] p(c_131) = [2] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [2] x6 + [1] p(c_132) = [0] p(c_133) = [2] p(c_134) = [1] p(c_135) = [2] p(c_136) = [1] x3 + [1] x4 + [2] x6 + [0] p(c_137) = [4] p(c_138) = [1] p(c_139) = [2] x2 + [0] p(c_140) = [1] x2 + [0] p(c_141) = [0] p(c_142) = [1] p(c_143) = [1] x3 + [0] p(c_144) = [0] p(c_145) = [1] p(c_146) = [2] x1 + [1] x2 + [0] p(c_147) = [1] p(c_148) = [4] Following rules are strictly oriented: R#(z0,z1) = [1] z0 + [1] z1 + [2] > [1] z0 + [1] z1 + [1] = c_32(R[ITE]#(e(z0,z1),z0,z1)) Following rules are (at-least) weakly oriented: P#(z0,z1) = [1] z0 + [1] z1 + [2] >= [1] z0 + [1] z1 + [2] = c_30(P[ITE]#(e(z0,z1),z0,z1)) P[ITE]#(False(),z0,Cons(F(),z1)) = [1] z0 + [1] z1 + [4] >= [1] z0 + [1] z1 + [4] = c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) = [1] z0 + [1] z1 + [4] >= [1] z0 + [1] z1 + [4] = c_41(P#(z0,z1)) Q#(z0,z1) = [1] z0 + [1] z1 + [2] >= [1] z0 + [1] z1 + [2] = c_31(Q[ITE]#(e(z0,z1),z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) = [1] z0 + [1] z1 + [6] >= [1] z0 + [1] z1 + [6] = c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) = [1] z0 + [1] z1 + [6] >= [1] z0 + [1] z1 + [6] = c_57(Q#(z0,Cons(F(),z1))) R[ITE]#(False(),z0,Cons(F(),z1)) = [1] z0 + [1] z1 + [2] >= [1] z0 + [1] z1 + [2] = c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) = [1] z0 + [1] z1 + [2] >= [1] z0 + [1] z1 + [2] = c_72(R#(z0,z1)) ** Step 10.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) - Weak DPs: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) - Weak TRS: e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/1,c_31/1,c_32/1,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/1,c_41/1,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/1,c_57/1 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/1,c_68/1,c_69/0,c_70/0,c_71/1,c_72/1 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () ** Step 10.b:1: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) - Weak DPs: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) - Weak TRS: e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/1,c_31/1,c_32/1,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/1,c_41/1,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/1,c_57/1 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/1,c_68/1,c_69/0,c_70/0,c_71/1,c_72/1 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) Consider the set of all dependency pairs 1: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) 2: Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) 3: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) 4: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) 5: Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) 6: Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) 7: R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) 8: R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) 9: R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) BEST_CASE TIME (?,?) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,3,4} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** Step 10.b:1.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) - Weak DPs: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) - Weak TRS: e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/1,c_31/1,c_32/1,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/1,c_41/1,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/1,c_57/1 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/1,c_68/1,c_69/0,c_70/0,c_71/1,c_72/1 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_30) = {1}, uargs(c_31) = {1}, uargs(c_32) = {1}, uargs(c_40) = {1}, uargs(c_41) = {1}, uargs(c_56) = {1}, uargs(c_57) = {1}, uargs(c_71) = {1}, uargs(c_72) = {1} Following symbols are considered usable: {e,AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q#,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]# ,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q#,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} TcT has computed the following interpretation: p(A) = [0] p(AND) = [0] p(B) = [0] p(Cons) = [1] x1 + [1] x2 + [1] p(E) = [0] p(EQUAL) = [0] p(F) = [0] p(False) = [1] p(GOAL) = [0] p(HEAD) = [0] p(NOTEMPTY) = [0] p(Nil) = [0] p(P) = [0] p(P[ITE]) = [0] p(Q) = [0] p(Q[ITE]) = [0] p(Q[ITE][FALSE][ITE]) = [0] p(R) = [0] p(R[ITE]) = [0] p(T) = [0] p(T') = [0] p(T[ITE]) = [0] p(True) = [0] p(and) = [0] p(c) = [0] p(c1) = [0] p(c10) = [0] p(c11) = [0] p(c12) = [0] p(c13) = [0] p(c14) = [0] p(c15) = [0] p(c16) = [0] p(c17) = [0] p(c18) = [0] p(c19) = [0] p(c2) = [0] p(c20) = [0] p(c21) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [0] p(c22) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [0] p(c23) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [0] p(c24) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [0] p(c25) = [0] p(c26) = [1] x1 + [1] x2 + [0] p(c27) = [1] x1 + [1] x2 + [0] p(c28) = [0] p(c29) = [0] p(c3) = [0] p(c30) = [0] p(c31) = [0] p(c32) = [1] x1 + [1] x2 + [0] p(c33) = [1] x1 + [1] x2 + [0] p(c34) = [0] p(c35) = [0] p(c36) = [0] p(c37) = [0] p(c38) = [1] x1 + [1] x2 + [0] p(c39) = [1] x1 + [1] x2 + [0] p(c4) = [1] x1 + [1] x2 + [0] p(c40) = [0] p(c41) = [0] p(c42) = [0] p(c43) = [0] p(c44) = [0] p(c45) = [0] p(c46) = [0] p(c47) = [0] p(c48) = [0] p(c49) = [0] p(c5) = [1] x1 + [1] x2 + [0] p(c50) = [0] p(c51) = [0] p(c52) = [0] p(c53) = [0] p(c54) = [0] p(c55) = [0] p(c56) = [0] p(c57) = [0] p(c58) = [0] p(c59) = [0] p(c6) = [0] p(c60) = [0] p(c61) = [0] p(c62) = [0] p(c63) = [0] p(c64) = [0] p(c65) = [0] p(c66) = [0] p(c67) = [0] p(c68) = [0] p(c69) = [0] p(c7) = [0] p(c70) = [0] p(c71) = [1] x1 + [1] x2 + [0] p(c72) = [1] x1 + [1] x2 + [0] p(c73) = [1] x1 + [1] x2 + [0] p(c74) = [1] x1 + [1] x2 + [0] p(c75) = [1] x1 + [0] p(c8) = [0] p(c9) = [0] p(e) = [1] p(e[Match][Cons][Ite][True][Match]) = [1] x1 + [1] x2 + [0] p(equal) = [0] p(goal) = [0] p(head) = [0] p(notEmpty) = [0] p(p) = [1] x2 + [0] p(p[Ite]) = [2] x1 + [2] x2 + [0] p(q) = [1] x1 + [1] p(q[Ite]) = [0] p(q[Ite][False][Ite]) = [2] x1 + [2] x3 + [1] p(q[Ite][False][Ite][True][Ite]) = [1] x1 + [1] x2 + [2] p(r) = [1] p(r[Ite]) = [1] x2 + [2] x3 + [1] p(r[Ite][False][Ite][True][Ite]) = [2] p(t) = [4] x2 + [4] p(t[Ite]) = [4] x3 + [4] p(AND#) = [4] x1 + [4] p(E#) = [0] p(EQUAL#) = [2] x1 + [1] x2 + [0] p(GOAL#) = [4] x1 + [1] p(HEAD#) = [0] p(NOTEMPTY#) = [4] x1 + [1] p(P#) = [4] x2 + [3] p(P[ITE]#) = [4] x3 + [0] p(Q#) = [4] x2 + [4] p(Q[ITE]#) = [4] x1 + [4] x3 + [0] p(Q[ITE][FALSE][ITE]#) = [2] p(R#) = [4] x2 + [0] p(R[ITE]#) = [4] x3 + [0] p(T'#) = [0] p(T[ITE]#) = [1] x2 + [1] x3 + [1] p(and#) = [1] p(e#) = [4] p(equal#) = [1] x1 + [0] p(goal#) = [2] p(head#) = [4] p(notEmpty#) = [2] x1 + [1] p(p#) = [1] x2 + [2] p(p[Ite]#) = [2] x2 + [1] x3 + [0] p(q#) = [1] x1 + [0] p(q[Ite]#) = [1] x2 + [2] p(q[Ite][False][Ite]#) = [0] p(r#) = [1] x1 + [1] x2 + [0] p(r[Ite]#) = [4] x3 + [0] p(t#) = [0] p(t[Ite]#) = [1] x1 + [2] x3 + [1] p(c_1) = [2] p(c_2) = [4] p(c_3) = [0] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] p(c_7) = [2] p(c_8) = [1] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [1] p(c_13) = [2] p(c_14) = [0] p(c_15) = [1] p(c_16) = [1] p(c_17) = [1] p(c_18) = [0] p(c_19) = [1] p(c_20) = [1] p(c_21) = [0] p(c_22) = [0] p(c_23) = [1] p(c_24) = [2] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [0] p(c_30) = [1] x1 + [0] p(c_31) = [1] x1 + [0] p(c_32) = [1] x1 + [0] p(c_33) = [1] x3 + [0] p(c_34) = [0] p(c_35) = [2] p(c_36) = [0] p(c_37) = [4] p(c_38) = [0] p(c_39) = [0] p(c_40) = [1] x1 + [0] p(c_41) = [1] x1 + [1] p(c_42) = [2] p(c_43) = [0] p(c_44) = [0] p(c_45) = [1] p(c_46) = [1] p(c_47) = [1] p(c_48) = [1] x3 + [1] x5 + [1] x6 + [2] x10 + [1] x12 + [1] x15 + [4] x21 + [1] p(c_49) = [4] p(c_50) = [0] p(c_51) = [2] p(c_52) = [0] p(c_53) = [1] x1 + [1] x2 + [4] x3 + [2] x4 + [1] x5 + [1] x6 + [2] x8 + [1] x13 + [1] x15 + [1] x16 + [1] x17 + [4] x20 + [0] p(c_54) = [1] p(c_55) = [1] p(c_56) = [1] x1 + [1] p(c_57) = [1] x1 + [4] p(c_58) = [1] p(c_59) = [2] x1 + [2] x3 + [2] x4 + [2] x6 + [1] x8 + [4] x10 + [1] x12 + [1] x13 + [1] x15 + [1] x19 + [1] x20 + [2] p(c_60) = [0] p(c_61) = [0] p(c_62) = [0] p(c_63) = [1] p(c_64) = [1] x2 + [1] x3 + [2] x6 + [2] x9 + [1] x10 + [1] x12 + [1] x13 + [1] x15 + [1] x16 + [1] p(c_65) = [0] p(c_66) = [0] p(c_67) = [1] x1 + [1] p(c_68) = [1] x1 + [1] p(c_69) = [0] p(c_70) = [0] p(c_71) = [1] x1 + [0] p(c_72) = [1] x1 + [0] p(c_73) = [0] p(c_74) = [2] p(c_75) = [0] p(c_76) = [2] p(c_77) = [1] p(c_78) = [0] p(c_79) = [1] p(c_80) = [0] p(c_81) = [0] p(c_82) = [0] p(c_83) = [1] p(c_84) = [1] p(c_85) = [0] p(c_86) = [1] p(c_87) = [1] p(c_88) = [4] p(c_89) = [0] p(c_90) = [2] p(c_91) = [0] p(c_92) = [1] p(c_93) = [1] p(c_94) = [1] p(c_95) = [0] p(c_96) = [1] p(c_97) = [2] p(c_98) = [1] p(c_99) = [0] p(c_100) = [4] p(c_101) = [0] p(c_102) = [0] p(c_103) = [4] p(c_104) = [1] p(c_105) = [0] p(c_106) = [4] p(c_107) = [0] p(c_108) = [0] p(c_109) = [0] p(c_110) = [4] x1 + [1] x2 + [0] p(c_111) = [1] p(c_112) = [0] p(c_113) = [1] x1 + [2] x2 + [4] p(c_114) = [1] p(c_115) = [1] p(c_116) = [1] x2 + [1] p(c_117) = [4] p(c_118) = [1] p(c_119) = [1] p(c_120) = [0] p(c_121) = [1] x1 + [4] x3 + [4] x4 + [1] x5 + [1] x6 + [1] p(c_122) = [0] p(c_123) = [0] p(c_124) = [2] p(c_125) = [2] p(c_126) = [1] x2 + [1] x4 + [4] p(c_127) = [1] p(c_128) = [1] p(c_129) = [1] x2 + [1] x3 + [0] p(c_130) = [0] p(c_131) = [1] x2 + [2] x4 + [2] x5 + [1] x6 + [0] p(c_132) = [0] p(c_133) = [0] p(c_134) = [1] p(c_135) = [2] p(c_136) = [4] x3 + [1] x5 + [0] p(c_137) = [1] p(c_138) = [1] p(c_139) = [1] x1 + [4] x2 + [1] p(c_140) = [1] p(c_141) = [4] p(c_142) = [0] p(c_143) = [1] x1 + [1] x3 + [0] p(c_144) = [1] p(c_145) = [2] p(c_146) = [1] p(c_147) = [0] p(c_148) = [1] Following rules are strictly oriented: P#(z0,z1) = [4] z1 + [3] > [4] z1 + [0] = c_30(P[ITE]#(e(z0,z1),z0,z1)) Following rules are (at-least) weakly oriented: P[ITE]#(False(),z0,Cons(F(),z1)) = [4] z1 + [4] >= [4] z1 + [4] = c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) = [4] z1 + [4] >= [4] z1 + [4] = c_41(P#(z0,z1)) Q#(z0,z1) = [4] z1 + [4] >= [4] z1 + [4] = c_31(Q[ITE]#(e(z0,z1),z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) = [4] z1 + [12] >= [4] z1 + [12] = c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) = [4] z1 + [12] >= [4] z1 + [12] = c_57(Q#(z0,Cons(F(),z1))) R#(z0,z1) = [4] z1 + [0] >= [4] z1 + [0] = c_32(R[ITE]#(e(z0,z1),z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) = [4] z1 + [4] >= [4] z1 + [4] = c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) = [4] z1 + [4] >= [4] z1 + [0] = c_72(R#(z0,z1)) e(Cons(A(),Cons(z0,z1)),z2) = [1] >= [1] = False() e(Cons(A(),Nil()),z0) = [1] >= [0] = e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) = [1] >= [1] = False() e(Cons(B(),Nil()),z0) = [1] >= [1] = False() e(Cons(F(),Cons(z0,z1)),z2) = [1] >= [1] = False() e(Cons(F(),Nil()),z0) = [1] >= [1] = False() e(Cons(T(),Cons(z0,z1)),z2) = [1] >= [1] = False() e(Cons(T(),Nil()),z0) = [1] >= [1] = False() e(Nil(),z0) = [1] >= [1] = False() *** Step 10.b:1.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) - Weak DPs: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) - Weak TRS: e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/1,c_31/1,c_32/1,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/1,c_41/1,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/1,c_57/1 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/1,c_68/1,c_69/0,c_70/0,c_71/1,c_72/1 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () *** Step 10.b:1.b:1: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) - Weak DPs: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) - Weak TRS: e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/1,c_31/1,c_32/1,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/1,c_41/1,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/1,c_57/1 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/1,c_68/1,c_69/0,c_70/0,c_71/1,c_72/1 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) Consider the set of all dependency pairs 1: Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) 2: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) 3: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) 4: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) 5: Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) 6: Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) 7: R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) 8: R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) 9: R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) BEST_CASE TIME (?,?) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,5,6} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 10.b:1.b:1.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) - Weak DPs: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) - Weak TRS: e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/1,c_31/1,c_32/1,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/1,c_41/1,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/1,c_57/1 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/1,c_68/1,c_69/0,c_70/0,c_71/1,c_72/1 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_30) = {1}, uargs(c_31) = {1}, uargs(c_32) = {1}, uargs(c_40) = {1}, uargs(c_41) = {1}, uargs(c_56) = {1}, uargs(c_57) = {1}, uargs(c_71) = {1}, uargs(c_72) = {1} Following symbols are considered usable: {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q#,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]# ,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q#,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} TcT has computed the following interpretation: p(A) = [0] p(AND) = [0] p(B) = [0] p(Cons) = [1] x1 + [1] x2 + [0] p(E) = [0] p(EQUAL) = [0] p(F) = [1] p(False) = [0] p(GOAL) = [0] p(HEAD) = [0] p(NOTEMPTY) = [0] p(Nil) = [1] p(P) = [0] p(P[ITE]) = [0] p(Q) = [0] p(Q[ITE]) = [0] p(Q[ITE][FALSE][ITE]) = [0] p(R) = [0] p(R[ITE]) = [0] p(T) = [0] p(T') = [0] p(T[ITE]) = [0] p(True) = [0] p(and) = [0] p(c) = [0] p(c1) = [0] p(c10) = [0] p(c11) = [0] p(c12) = [0] p(c13) = [0] p(c14) = [0] p(c15) = [0] p(c16) = [0] p(c17) = [0] p(c18) = [0] p(c19) = [0] p(c2) = [0] p(c20) = [0] p(c21) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [0] p(c22) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [0] p(c23) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [0] p(c24) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [0] p(c25) = [0] p(c26) = [1] x1 + [1] x2 + [0] p(c27) = [1] x1 + [1] x2 + [0] p(c28) = [0] p(c29) = [0] p(c3) = [0] p(c30) = [0] p(c31) = [0] p(c32) = [1] x1 + [1] x2 + [0] p(c33) = [1] x1 + [1] x2 + [0] p(c34) = [0] p(c35) = [0] p(c36) = [0] p(c37) = [0] p(c38) = [1] x1 + [1] x2 + [0] p(c39) = [1] x1 + [1] x2 + [0] p(c4) = [1] x1 + [1] x2 + [0] p(c40) = [0] p(c41) = [0] p(c42) = [0] p(c43) = [0] p(c44) = [0] p(c45) = [0] p(c46) = [0] p(c47) = [0] p(c48) = [0] p(c49) = [0] p(c5) = [1] x1 + [1] x2 + [0] p(c50) = [0] p(c51) = [0] p(c52) = [0] p(c53) = [0] p(c54) = [0] p(c55) = [0] p(c56) = [0] p(c57) = [0] p(c58) = [0] p(c59) = [0] p(c6) = [0] p(c60) = [0] p(c61) = [0] p(c62) = [0] p(c63) = [0] p(c64) = [0] p(c65) = [0] p(c66) = [0] p(c67) = [0] p(c68) = [0] p(c69) = [0] p(c7) = [0] p(c70) = [0] p(c71) = [1] x1 + [1] x2 + [0] p(c72) = [1] x1 + [1] x2 + [0] p(c73) = [2] p(c74) = [0] p(c75) = [0] p(c8) = [0] p(c9) = [4] p(e) = [0] p(e[Match][Cons][Ite][True][Match]) = [2] p(equal) = [4] x2 + [2] p(goal) = [0] p(head) = [0] p(notEmpty) = [4] p(p) = [1] x2 + [2] p(p[Ite]) = [0] p(q) = [1] p(q[Ite]) = [0] p(q[Ite][False][Ite]) = [4] x2 + [1] x3 + [1] p(q[Ite][False][Ite][True][Ite]) = [1] p(r) = [0] p(r[Ite]) = [1] x2 + [1] x3 + [4] p(r[Ite][False][Ite][True][Ite]) = [1] p(t) = [2] x1 + [1] p(t[Ite]) = [2] x1 + [2] x2 + [1] x3 + [0] p(AND#) = [1] x1 + [1] x2 + [1] p(E#) = [1] p(EQUAL#) = [2] x2 + [0] p(GOAL#) = [4] p(HEAD#) = [2] p(NOTEMPTY#) = [1] x1 + [1] p(P#) = [2] x2 + [2] p(P[ITE]#) = [2] x3 + [2] p(Q#) = [2] x2 + [3] p(Q[ITE]#) = [2] x3 + [2] p(Q[ITE][FALSE][ITE]#) = [4] x1 + [1] p(R#) = [2] x2 + [2] p(R[ITE]#) = [2] x3 + [1] p(T'#) = [1] x1 + [1] x2 + [0] p(T[ITE]#) = [1] x1 + [1] x2 + [2] p(and#) = [1] x2 + [1] p(e#) = [4] p(equal#) = [4] x2 + [0] p(goal#) = [1] x1 + [1] x2 + [0] p(head#) = [0] p(notEmpty#) = [0] p(p#) = [2] x1 + [2] x2 + [2] p(p[Ite]#) = [1] x1 + [1] x3 + [2] p(q#) = [1] x1 + [4] x2 + [0] p(q[Ite]#) = [4] x2 + [4] p(q[Ite][False][Ite]#) = [2] x3 + [1] p(r#) = [2] x2 + [0] p(r[Ite]#) = [1] x2 + [1] p(t#) = [1] x1 + [4] x2 + [1] p(t[Ite]#) = [1] x1 + [1] x3 + [4] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] p(c_7) = [1] p(c_8) = [1] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] p(c_12) = [4] p(c_13) = [1] p(c_14) = [1] p(c_15) = [1] p(c_16) = [1] p(c_17) = [1] p(c_18) = [0] p(c_19) = [0] p(c_20) = [4] p(c_21) = [1] p(c_22) = [4] p(c_23) = [0] p(c_24) = [1] p(c_25) = [2] p(c_26) = [4] x1 + [2] p(c_27) = [2] p(c_28) = [1] p(c_29) = [1] p(c_30) = [1] x1 + [0] p(c_31) = [1] x1 + [0] p(c_32) = [1] x1 + [1] p(c_33) = [2] p(c_34) = [4] p(c_35) = [1] p(c_36) = [0] p(c_37) = [0] p(c_38) = [0] p(c_39) = [1] p(c_40) = [1] x1 + [0] p(c_41) = [1] x1 + [0] p(c_42) = [1] p(c_43) = [1] p(c_44) = [0] p(c_45) = [1] p(c_46) = [1] p(c_47) = [0] p(c_48) = [1] x1 + [1] x4 + [4] x9 + [1] x13 + [4] x15 + [1] x16 + [1] x18 + [1] x20 + [0] p(c_49) = [4] p(c_50) = [1] p(c_51) = [4] p(c_52) = [0] p(c_53) = [4] x3 + [2] x4 + [4] x8 + [1] x9 + [4] x10 + [1] x12 + [1] x14 + [2] x15 + [2] x21 + [0] p(c_54) = [0] p(c_55) = [4] p(c_56) = [1] x1 + [0] p(c_57) = [1] x1 + [1] p(c_58) = [4] p(c_59) = [1] x3 + [1] x4 + [1] x6 + [1] x10 + [1] x11 + [1] x12 + [4] x13 + [1] x16 + [1] x18 + [1] x19 + [1] x20 + [1] p(c_60) = [1] p(c_61) = [1] p(c_62) = [1] p(c_63) = [4] p(c_64) = [4] x5 + [1] x6 + [1] x7 + [2] x8 + [2] x9 + [2] x10 + [4] x11 + [1] x12 + [4] x15 + [1] x18 + [1] x19 + [1] x20 + [2] x21 + [1] p(c_65) = [1] p(c_66) = [2] p(c_67) = [4] x1 + [1] p(c_68) = [4] x1 + [4] p(c_69) = [0] p(c_70) = [4] p(c_71) = [1] x1 + [0] p(c_72) = [1] x1 + [1] p(c_73) = [1] p(c_74) = [2] p(c_75) = [2] p(c_76) = [0] p(c_77) = [2] p(c_78) = [0] p(c_79) = [1] p(c_80) = [0] p(c_81) = [0] p(c_82) = [0] p(c_83) = [0] p(c_84) = [1] p(c_85) = [1] p(c_86) = [2] p(c_87) = [4] p(c_88) = [0] p(c_89) = [0] p(c_90) = [0] p(c_91) = [0] p(c_92) = [0] p(c_93) = [1] p(c_94) = [2] p(c_95) = [2] p(c_96) = [0] p(c_97) = [0] p(c_98) = [4] p(c_99) = [0] p(c_100) = [2] p(c_101) = [2] p(c_102) = [1] p(c_103) = [1] p(c_104) = [2] p(c_105) = [0] p(c_106) = [1] p(c_107) = [0] p(c_108) = [1] p(c_109) = [2] p(c_110) = [1] x2 + [1] p(c_111) = [0] p(c_112) = [1] p(c_113) = [1] x1 + [1] x2 + [1] p(c_114) = [1] p(c_115) = [2] p(c_116) = [0] p(c_117) = [4] p(c_118) = [1] p(c_119) = [0] p(c_120) = [1] p(c_121) = [4] x1 + [1] x4 + [4] x5 + [1] x6 + [0] p(c_122) = [2] p(c_123) = [0] p(c_124) = [1] p(c_125) = [0] p(c_126) = [4] x1 + [4] x2 + [4] x3 + [1] x4 + [1] x6 + [1] p(c_127) = [4] p(c_128) = [2] p(c_129) = [2] x3 + [0] p(c_130) = [0] p(c_131) = [4] x2 + [2] x3 + [2] p(c_132) = [4] p(c_133) = [4] p(c_134) = [1] p(c_135) = [0] p(c_136) = [1] x4 + [1] p(c_137) = [0] p(c_138) = [4] p(c_139) = [2] x3 + [0] p(c_140) = [1] x2 + [1] p(c_141) = [4] p(c_142) = [0] p(c_143) = [2] x3 + [2] p(c_144) = [0] p(c_145) = [1] p(c_146) = [2] x1 + [1] x2 + [0] p(c_147) = [1] p(c_148) = [1] Following rules are strictly oriented: Q#(z0,z1) = [2] z1 + [3] > [2] z1 + [2] = c_31(Q[ITE]#(e(z0,z1),z0,z1)) Following rules are (at-least) weakly oriented: P#(z0,z1) = [2] z1 + [2] >= [2] z1 + [2] = c_30(P[ITE]#(e(z0,z1),z0,z1)) P[ITE]#(False(),z0,Cons(F(),z1)) = [2] z1 + [4] >= [2] z1 + [4] = c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) = [2] z1 + [4] >= [2] z1 + [2] = c_41(P#(z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) = [2] z1 + [6] >= [2] z1 + [6] = c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) = [2] z1 + [6] >= [2] z1 + [6] = c_57(Q#(z0,Cons(F(),z1))) R#(z0,z1) = [2] z1 + [2] >= [2] z1 + [2] = c_32(R[ITE]#(e(z0,z1),z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) = [2] z1 + [3] >= [2] z1 + [3] = c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) = [2] z1 + [3] >= [2] z1 + [3] = c_72(R#(z0,z1)) **** Step 10.b:1.b:1.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) - Weak TRS: e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/1,c_31/1,c_32/1,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/1,c_41/1,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/1,c_57/1 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/1,c_68/1,c_69/0,c_70/0,c_71/1,c_72/1 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () **** Step 10.b:1.b:1.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) - Weak TRS: e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/1,c_31/1,c_32/1,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/1,c_41/1,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/1,c_57/1 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/1,c_68/1,c_69/0,c_70/0,c_71/1,c_72/1 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) -->_1 P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)):3 -->_1 P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))):2 2:W:P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) -->_1 R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)):7 3:W:P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) -->_1 P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)):1 4:W:Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) -->_1 Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))):6 -->_1 Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))):5 5:W:Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) -->_1 P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)):1 6:W:Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) -->_1 Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)):4 7:W:R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) -->_1 R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)):9 -->_1 R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)):8 8:W:R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) -->_1 Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)):4 9:W:R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) -->_1 R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: P#(z0,z1) -> c_30(P[ITE]#(e(z0,z1),z0,z1)) 5: Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_56(P#(z0,Cons(F(),Cons(F(),z1)))) 4: Q#(z0,z1) -> c_31(Q[ITE]#(e(z0,z1),z0,z1)) 8: R[ITE]#(False(),z0,Cons(F(),z1)) -> c_71(Q#(z0,z1)) 7: R#(z0,z1) -> c_32(R[ITE]#(e(z0,z1),z0,z1)) 9: R[ITE]#(False(),z0,Cons(F(),z1)) -> c_72(R#(z0,z1)) 2: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_40(R#(z0,Cons(F(),z1))) 6: Q[ITE]#(False(),z0,Cons(F(),Cons(F(),z1))) -> c_57(Q#(z0,Cons(F(),z1))) 3: P[ITE]#(False(),z0,Cons(F(),z1)) -> c_41(P#(z0,z1)) **** Step 10.b:1.b:1.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: e(Cons(A(),Cons(z0,z1)),z2) -> False() e(Cons(A(),Nil()),z0) -> e[Match][Cons][Ite][True][Match](A(),Nil(),z0) e(Cons(B(),Cons(z0,z1)),z2) -> False() e(Cons(B(),Nil()),z0) -> False() e(Cons(F(),Cons(z0,z1)),z2) -> False() e(Cons(F(),Nil()),z0) -> False() e(Cons(T(),Cons(z0,z1)),z2) -> False() e(Cons(T(),Nil()),z0) -> False() e(Nil(),z0) -> False() - Signature: {AND/2,E/2,EQUAL/2,GOAL/2,HEAD/1,NOTEMPTY/1,P/2,P[ITE]/3,Q/2,Q[ITE]/3,Q[ITE][FALSE][ITE]/3,R/2,R[ITE]/3,T'/2 ,T[ITE]/3,and/2,e/2,equal/2,goal/2,head/1,notEmpty/1,p/2,p[Ite]/3,q/2,q[Ite]/3,q[Ite][False][Ite]/3,r/2 ,r[Ite]/3,t/2,t[Ite]/3,AND#/2,E#/2,EQUAL#/2,GOAL#/2,HEAD#/1,NOTEMPTY#/1,P#/2,P[ITE]#/3,Q#/2,Q[ITE]#/3 ,Q[ITE][FALSE][ITE]#/3,R#/2,R[ITE]#/3,T'#/2,T[ITE]#/3,and#/2,e#/2,equal#/2,goal#/2,head#/1,notEmpty#/1,p#/2 ,p[Ite]#/3,q#/2,q[Ite]#/3,q[Ite][False][Ite]#/3,r#/2,r[Ite]#/3,t#/2,t[Ite]#/3} / {A/0,B/0,Cons/2,F/0,False/0 ,Nil/0,T/0,True/0,c/0,c1/0,c10/0,c11/0,c12/0,c13/0,c14/0,c15/0,c16/0,c17/0,c18/0,c19/0,c2/0,c20/0,c21/6 ,c22/6,c23/6,c24/6,c25/0,c26/2,c27/2,c28/0,c29/0,c3/0,c30/0,c31/0,c32/2,c33/2,c34/0,c35/0,c36/0,c37/0,c38/2 ,c39/2,c4/2,c40/0,c41/0,c42/0,c43/0,c44/0,c45/0,c46/0,c47/0,c48/0,c49/0,c5/2,c50/0,c51/0,c52/0,c53/0,c54/0 ,c55/0,c56/0,c57/0,c58/0,c59/0,c6/0,c60/0,c61/0,c62/0,c63/0,c64/0,c65/0,c66/0,c67/0,c68/0,c69/0,c7/0,c70/0 ,c71/2,c72/2,c73/2,c74/2,c75/1,c8/0,c9/0,e[Match][Cons][Ite][True][Match]/3,q[Ite][False][Ite][True][Ite]/3 ,r[Ite][False][Ite][True][Ite]/3,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0 ,c_13/0,c_14/0,c_15/0,c_16/0,c_17/0,c_18/0,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/1,c_27/0 ,c_28/0,c_29/0,c_30/1,c_31/1,c_32/1,c_33/3,c_34/0,c_35/0,c_36/0,c_37/0,c_38/0,c_39/0,c_40/1,c_41/1,c_42/0 ,c_43/0,c_44/0,c_45/0,c_46/0,c_47/0,c_48/21,c_49/0,c_50/0,c_51/0,c_52/0,c_53/21,c_54/0,c_55/0,c_56/1,c_57/1 ,c_58/0,c_59/21,c_60/0,c_61/0,c_62/0,c_63/0,c_64/21,c_65/0,c_66/0,c_67/1,c_68/1,c_69/0,c_70/0,c_71/1,c_72/1 ,c_73/0,c_74/0,c_75/0,c_76/0,c_77/0,c_78/0,c_79/0,c_80/0,c_81/0,c_82/0,c_83/0,c_84/0,c_85/0,c_86/0,c_87/0 ,c_88/0,c_89/0,c_90/0,c_91/0,c_92/0,c_93/0,c_94/0,c_95/0,c_96/0,c_97/0,c_98/0,c_99/0,c_100/0,c_101/0,c_102/0 ,c_103/0,c_104/0,c_105/0,c_106/1,c_107/0,c_108/0,c_109/0,c_110/2,c_111/0,c_112/0,c_113/3,c_114/0,c_115/0 ,c_116/2,c_117/0,c_118/0,c_119/0,c_120/0,c_121/6,c_122/0,c_123/0,c_124/0,c_125/0,c_126/6,c_127/0,c_128/0 ,c_129/3,c_130/0,c_131/6,c_132/0,c_133/0,c_134/0,c_135/0,c_136/6,c_137/0,c_138/0,c_139/3,c_140/2,c_141/0 ,c_142/0,c_143/3,c_144/0,c_145/0,c_146/2,c_147/0,c_148/0} - Obligation: innermost runtime complexity wrt. defined symbols {AND#,E#,EQUAL#,GOAL#,HEAD#,NOTEMPTY#,P#,P[ITE]#,Q# ,Q[ITE]#,Q[ITE][FALSE][ITE]#,R#,R[ITE]#,T'#,T[ITE]#,and#,e#,equal#,goal#,head#,notEmpty#,p#,p[Ite]#,q# ,q[Ite]#,q[Ite][False][Ite]#,r#,r[Ite]#,t#,t[Ite]#} and constructors {A,B,Cons,F,False,Nil,T,True,c,c1,c10 ,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3,c30,c31,c32,c33,c34,c35 ,c36,c37,c38,c39,c4,c40,c41,c42,c43,c44,c45,c46,c47,c48,c49,c5,c50,c51,c52,c53,c54,c55,c56,c57,c58,c59,c6 ,c60,c61,c62,c63,c64,c65,c66,c67,c68,c69,c7,c70,c71,c72,c73,c74,c75,c8,c9,e[Match][Cons][Ite][True][Match] ,q[Ite][False][Ite][True][Ite],r[Ite][False][Ite][True][Ite]} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))