YES We show the termination of the TRS R: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) active(__(X,nil())) -> mark(X) active(__(nil(),X)) -> mark(X) active(and(tt(),X)) -> mark(X) active(isList(V)) -> mark(isNeList(V)) active(isList(nil())) -> mark(tt()) active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) active(isNeList(V)) -> mark(isQid(V)) active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) active(isNePal(V)) -> mark(isQid(V)) active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) active(isPal(V)) -> mark(isNePal(V)) active(isPal(nil())) -> mark(tt()) active(isQid(a())) -> mark(tt()) active(isQid(e())) -> mark(tt()) active(isQid(i())) -> mark(tt()) active(isQid(o())) -> mark(tt()) active(isQid(u())) -> mark(tt()) active(__(X1,X2)) -> __(active(X1),X2) active(__(X1,X2)) -> __(X1,active(X2)) active(and(X1,X2)) -> and(active(X1),X2) __(mark(X1),X2) -> mark(__(X1,X2)) __(X1,mark(X2)) -> mark(__(X1,X2)) and(mark(X1),X2) -> mark(and(X1,X2)) proper(__(X1,X2)) -> __(proper(X1),proper(X2)) proper(nil()) -> ok(nil()) proper(and(X1,X2)) -> and(proper(X1),proper(X2)) proper(tt()) -> ok(tt()) proper(isList(X)) -> isList(proper(X)) proper(isNeList(X)) -> isNeList(proper(X)) proper(isQid(X)) -> isQid(proper(X)) proper(isNePal(X)) -> isNePal(proper(X)) proper(isPal(X)) -> isPal(proper(X)) proper(a()) -> ok(a()) proper(e()) -> ok(e()) proper(i()) -> ok(i()) proper(o()) -> ok(o()) proper(u()) -> ok(u()) __(ok(X1),ok(X2)) -> ok(__(X1,X2)) and(ok(X1),ok(X2)) -> ok(and(X1,X2)) isList(ok(X)) -> ok(isList(X)) isNeList(ok(X)) -> ok(isNeList(X)) isQid(ok(X)) -> ok(isQid(X)) isNePal(ok(X)) -> ok(isNePal(X)) isPal(ok(X)) -> ok(isPal(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: active#(__(__(X,Y),Z)) -> __#(X,__(Y,Z)) p2: active#(__(__(X,Y),Z)) -> __#(Y,Z) p3: active#(isList(V)) -> isNeList#(V) p4: active#(isList(__(V1,V2))) -> and#(isList(V1),isList(V2)) p5: active#(isList(__(V1,V2))) -> isList#(V1) p6: active#(isList(__(V1,V2))) -> isList#(V2) p7: active#(isNeList(V)) -> isQid#(V) p8: active#(isNeList(__(V1,V2))) -> and#(isList(V1),isNeList(V2)) p9: active#(isNeList(__(V1,V2))) -> isList#(V1) p10: active#(isNeList(__(V1,V2))) -> isNeList#(V2) p11: active#(isNeList(__(V1,V2))) -> and#(isNeList(V1),isList(V2)) p12: active#(isNeList(__(V1,V2))) -> isNeList#(V1) p13: active#(isNeList(__(V1,V2))) -> isList#(V2) p14: active#(isNePal(V)) -> isQid#(V) p15: active#(isNePal(__(I,__(P,I)))) -> and#(isQid(I),isPal(P)) p16: active#(isNePal(__(I,__(P,I)))) -> isQid#(I) p17: active#(isNePal(__(I,__(P,I)))) -> isPal#(P) p18: active#(isPal(V)) -> isNePal#(V) p19: active#(__(X1,X2)) -> __#(active(X1),X2) p20: active#(__(X1,X2)) -> active#(X1) p21: active#(__(X1,X2)) -> __#(X1,active(X2)) p22: active#(__(X1,X2)) -> active#(X2) p23: active#(and(X1,X2)) -> and#(active(X1),X2) p24: active#(and(X1,X2)) -> active#(X1) p25: __#(mark(X1),X2) -> __#(X1,X2) p26: __#(X1,mark(X2)) -> __#(X1,X2) p27: and#(mark(X1),X2) -> and#(X1,X2) p28: proper#(__(X1,X2)) -> __#(proper(X1),proper(X2)) p29: proper#(__(X1,X2)) -> proper#(X1) p30: proper#(__(X1,X2)) -> proper#(X2) p31: proper#(and(X1,X2)) -> and#(proper(X1),proper(X2)) p32: proper#(and(X1,X2)) -> proper#(X1) p33: proper#(and(X1,X2)) -> proper#(X2) p34: proper#(isList(X)) -> isList#(proper(X)) p35: proper#(isList(X)) -> proper#(X) p36: proper#(isNeList(X)) -> isNeList#(proper(X)) p37: proper#(isNeList(X)) -> proper#(X) p38: proper#(isQid(X)) -> isQid#(proper(X)) p39: proper#(isQid(X)) -> proper#(X) p40: proper#(isNePal(X)) -> isNePal#(proper(X)) p41: proper#(isNePal(X)) -> proper#(X) p42: proper#(isPal(X)) -> isPal#(proper(X)) p43: proper#(isPal(X)) -> proper#(X) p44: __#(ok(X1),ok(X2)) -> __#(X1,X2) p45: and#(ok(X1),ok(X2)) -> and#(X1,X2) p46: isList#(ok(X)) -> isList#(X) p47: isNeList#(ok(X)) -> isNeList#(X) p48: isQid#(ok(X)) -> isQid#(X) p49: isNePal#(ok(X)) -> isNePal#(X) p50: isPal#(ok(X)) -> isPal#(X) p51: top#(mark(X)) -> top#(proper(X)) p52: top#(mark(X)) -> proper#(X) p53: top#(ok(X)) -> top#(active(X)) p54: top#(ok(X)) -> active#(X) and R consists of: r1: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) r2: active(__(X,nil())) -> mark(X) r3: active(__(nil(),X)) -> mark(X) r4: active(and(tt(),X)) -> mark(X) r5: active(isList(V)) -> mark(isNeList(V)) r6: active(isList(nil())) -> mark(tt()) r7: active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) r8: active(isNeList(V)) -> mark(isQid(V)) r9: active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) r10: active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) r11: active(isNePal(V)) -> mark(isQid(V)) r12: active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) r13: active(isPal(V)) -> mark(isNePal(V)) r14: active(isPal(nil())) -> mark(tt()) r15: active(isQid(a())) -> mark(tt()) r16: active(isQid(e())) -> mark(tt()) r17: active(isQid(i())) -> mark(tt()) r18: active(isQid(o())) -> mark(tt()) r19: active(isQid(u())) -> mark(tt()) r20: active(__(X1,X2)) -> __(active(X1),X2) r21: active(__(X1,X2)) -> __(X1,active(X2)) r22: active(and(X1,X2)) -> and(active(X1),X2) r23: __(mark(X1),X2) -> mark(__(X1,X2)) r24: __(X1,mark(X2)) -> mark(__(X1,X2)) r25: and(mark(X1),X2) -> mark(and(X1,X2)) r26: proper(__(X1,X2)) -> __(proper(X1),proper(X2)) r27: proper(nil()) -> ok(nil()) r28: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) r29: proper(tt()) -> ok(tt()) r30: proper(isList(X)) -> isList(proper(X)) r31: proper(isNeList(X)) -> isNeList(proper(X)) r32: proper(isQid(X)) -> isQid(proper(X)) r33: proper(isNePal(X)) -> isNePal(proper(X)) r34: proper(isPal(X)) -> isPal(proper(X)) r35: proper(a()) -> ok(a()) r36: proper(e()) -> ok(e()) r37: proper(i()) -> ok(i()) r38: proper(o()) -> ok(o()) r39: proper(u()) -> ok(u()) r40: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) r41: and(ok(X1),ok(X2)) -> ok(and(X1,X2)) r42: isList(ok(X)) -> ok(isList(X)) r43: isNeList(ok(X)) -> ok(isNeList(X)) r44: isQid(ok(X)) -> ok(isQid(X)) r45: isNePal(ok(X)) -> ok(isNePal(X)) r46: isPal(ok(X)) -> ok(isPal(X)) r47: top(mark(X)) -> top(proper(X)) r48: top(ok(X)) -> top(active(X)) The estimated dependency graph contains the following SCCs: {p51, p53} {p20, p22, p24} {p29, p30, p32, p33, p35, p37, p39, p41, p43} {p25, p26, p44} {p47} {p27, p45} {p46} {p48} {p50} {p49} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: top#(ok(X)) -> top#(active(X)) p2: top#(mark(X)) -> top#(proper(X)) and R consists of: r1: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) r2: active(__(X,nil())) -> mark(X) r3: active(__(nil(),X)) -> mark(X) r4: active(and(tt(),X)) -> mark(X) r5: active(isList(V)) -> mark(isNeList(V)) r6: active(isList(nil())) -> mark(tt()) r7: active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) r8: active(isNeList(V)) -> mark(isQid(V)) r9: active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) r10: active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) r11: active(isNePal(V)) -> mark(isQid(V)) r12: active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) r13: active(isPal(V)) -> mark(isNePal(V)) r14: active(isPal(nil())) -> mark(tt()) r15: active(isQid(a())) -> mark(tt()) r16: active(isQid(e())) -> mark(tt()) r17: active(isQid(i())) -> mark(tt()) r18: active(isQid(o())) -> mark(tt()) r19: active(isQid(u())) -> mark(tt()) r20: active(__(X1,X2)) -> __(active(X1),X2) r21: active(__(X1,X2)) -> __(X1,active(X2)) r22: active(and(X1,X2)) -> and(active(X1),X2) r23: __(mark(X1),X2) -> mark(__(X1,X2)) r24: __(X1,mark(X2)) -> mark(__(X1,X2)) r25: and(mark(X1),X2) -> mark(and(X1,X2)) r26: proper(__(X1,X2)) -> __(proper(X1),proper(X2)) r27: proper(nil()) -> ok(nil()) r28: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) r29: proper(tt()) -> ok(tt()) r30: proper(isList(X)) -> isList(proper(X)) r31: proper(isNeList(X)) -> isNeList(proper(X)) r32: proper(isQid(X)) -> isQid(proper(X)) r33: proper(isNePal(X)) -> isNePal(proper(X)) r34: proper(isPal(X)) -> isPal(proper(X)) r35: proper(a()) -> ok(a()) r36: proper(e()) -> ok(e()) r37: proper(i()) -> ok(i()) r38: proper(o()) -> ok(o()) r39: proper(u()) -> ok(u()) r40: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) r41: and(ok(X1),ok(X2)) -> ok(and(X1,X2)) r42: isList(ok(X)) -> ok(isList(X)) r43: isNeList(ok(X)) -> ok(isNeList(X)) r44: isQid(ok(X)) -> ok(isQid(X)) r45: isNePal(ok(X)) -> ok(isNePal(X)) r46: isPal(ok(X)) -> ok(isPal(X)) r47: top(mark(X)) -> top(proper(X)) r48: top(ok(X)) -> top(active(X)) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46 Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: top#_A(x1) = ((1,0),(0,0)) x1 ok_A(x1) = x1 active_A(x1) = x1 mark_A(x1) = x1 + (5,3) proper_A(x1) = x1 ___A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (49,23) and_A(x1,x2) = ((1,1),(0,1)) x1 + x2 + (1,1) isList_A(x1) = x1 + (21,11) isNeList_A(x1) = x1 + (15,8) isQid_A(x1) = ((1,0),(0,0)) x1 + (9,5) isNePal_A(x1) = ((1,0),(1,1)) x1 + (25,8) isPal_A(x1) = ((1,0),(1,1)) x1 + (30,14) nil_A() = (4,1) tt_A() = (3,2) a_A() = (0,0) e_A() = (8,1) i_A() = (3,1) o_A() = (9,1) u_A() = (4,1) precedence: top# = ok = active = mark = proper = __ = and = isList = isNeList = isQid = isNePal = isPal = nil = tt = a = e = i = o = u partial status: pi(top#) = [] pi(ok) = [] pi(active) = [] pi(mark) = [] pi(proper) = [] pi(__) = [] pi(and) = [] pi(isList) = [] pi(isNeList) = [] pi(isQid) = [] pi(isNePal) = [] pi(isPal) = [] pi(nil) = [] pi(tt) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] 2. weighted path order base order: matrix interpretations: carrier: N^2 order: standard order interpretations: top#_A(x1) = ((1,1),(1,1)) x1 + (1,0) ok_A(x1) = ((0,1),(1,0)) x1 + (20,0) active_A(x1) = x1 + (9,9) mark_A(x1) = ((1,0),(0,0)) x1 + (2,27) proper_A(x1) = ((1,1),(1,1)) x1 + (20,10) ___A(x1,x2) = ((0,1),(1,0)) x1 + x2 + (24,24) and_A(x1,x2) = ((0,1),(1,0)) x1 + ((1,1),(1,1)) x2 + (40,21) isList_A(x1) = ((0,1),(1,0)) x1 + (23,3) isNeList_A(x1) = x1 + (18,18) isQid_A(x1) = x1 + (19,19) isNePal_A(x1) = ((1,1),(1,1)) x1 + (28,27) isPal_A(x1) = ((1,1),(1,1)) x1 + (134,133) nil_A() = (24,9) tt_A() = (23,9) a_A() = (28,27) e_A() = (0,8) i_A() = (28,27) o_A() = (28,27) u_A() = (0,0) precedence: isNeList > proper = e > isNePal = isPal > isList = tt = a = u > nil > top# = ok = active = mark = __ = and = isQid = i = o partial status: pi(top#) = [] pi(ok) = [] pi(active) = [1] pi(mark) = [] pi(proper) = [1] pi(__) = [2] pi(and) = [2] pi(isList) = [] pi(isNeList) = [] pi(isQid) = [1] pi(isNePal) = [] pi(isPal) = [] pi(nil) = [] pi(tt) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] The next rules are strictly ordered: p2 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: top#(ok(X)) -> top#(active(X)) and R consists of: r1: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) r2: active(__(X,nil())) -> mark(X) r3: active(__(nil(),X)) -> mark(X) r4: active(and(tt(),X)) -> mark(X) r5: active(isList(V)) -> mark(isNeList(V)) r6: active(isList(nil())) -> mark(tt()) r7: active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) r8: active(isNeList(V)) -> mark(isQid(V)) r9: active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) r10: active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) r11: active(isNePal(V)) -> mark(isQid(V)) r12: active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) r13: active(isPal(V)) -> mark(isNePal(V)) r14: active(isPal(nil())) -> mark(tt()) r15: active(isQid(a())) -> mark(tt()) r16: active(isQid(e())) -> mark(tt()) r17: active(isQid(i())) -> mark(tt()) r18: active(isQid(o())) -> mark(tt()) r19: active(isQid(u())) -> mark(tt()) r20: active(__(X1,X2)) -> __(active(X1),X2) r21: active(__(X1,X2)) -> __(X1,active(X2)) r22: active(and(X1,X2)) -> and(active(X1),X2) r23: __(mark(X1),X2) -> mark(__(X1,X2)) r24: __(X1,mark(X2)) -> mark(__(X1,X2)) r25: and(mark(X1),X2) -> mark(and(X1,X2)) r26: proper(__(X1,X2)) -> __(proper(X1),proper(X2)) r27: proper(nil()) -> ok(nil()) r28: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) r29: proper(tt()) -> ok(tt()) r30: proper(isList(X)) -> isList(proper(X)) r31: proper(isNeList(X)) -> isNeList(proper(X)) r32: proper(isQid(X)) -> isQid(proper(X)) r33: proper(isNePal(X)) -> isNePal(proper(X)) r34: proper(isPal(X)) -> isPal(proper(X)) r35: proper(a()) -> ok(a()) r36: proper(e()) -> ok(e()) r37: proper(i()) -> ok(i()) r38: proper(o()) -> ok(o()) r39: proper(u()) -> ok(u()) r40: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) r41: and(ok(X1),ok(X2)) -> ok(and(X1,X2)) r42: isList(ok(X)) -> ok(isList(X)) r43: isNeList(ok(X)) -> ok(isNeList(X)) r44: isQid(ok(X)) -> ok(isQid(X)) r45: isNePal(ok(X)) -> ok(isNePal(X)) r46: isPal(ok(X)) -> ok(isPal(X)) r47: top(mark(X)) -> top(proper(X)) r48: top(ok(X)) -> top(active(X)) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: top#(ok(X)) -> top#(active(X)) and R consists of: r1: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) r2: active(__(X,nil())) -> mark(X) r3: active(__(nil(),X)) -> mark(X) r4: active(and(tt(),X)) -> mark(X) r5: active(isList(V)) -> mark(isNeList(V)) r6: active(isList(nil())) -> mark(tt()) r7: active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) r8: active(isNeList(V)) -> mark(isQid(V)) r9: active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) r10: active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) r11: active(isNePal(V)) -> mark(isQid(V)) r12: active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) r13: active(isPal(V)) -> mark(isNePal(V)) r14: active(isPal(nil())) -> mark(tt()) r15: active(isQid(a())) -> mark(tt()) r16: active(isQid(e())) -> mark(tt()) r17: active(isQid(i())) -> mark(tt()) r18: active(isQid(o())) -> mark(tt()) r19: active(isQid(u())) -> mark(tt()) r20: active(__(X1,X2)) -> __(active(X1),X2) r21: active(__(X1,X2)) -> __(X1,active(X2)) r22: active(and(X1,X2)) -> and(active(X1),X2) r23: __(mark(X1),X2) -> mark(__(X1,X2)) r24: __(X1,mark(X2)) -> mark(__(X1,X2)) r25: and(mark(X1),X2) -> mark(and(X1,X2)) r26: proper(__(X1,X2)) -> __(proper(X1),proper(X2)) r27: proper(nil()) -> ok(nil()) r28: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) r29: proper(tt()) -> ok(tt()) r30: proper(isList(X)) -> isList(proper(X)) r31: proper(isNeList(X)) -> isNeList(proper(X)) r32: proper(isQid(X)) -> isQid(proper(X)) r33: proper(isNePal(X)) -> isNePal(proper(X)) r34: proper(isPal(X)) -> isPal(proper(X)) r35: proper(a()) -> ok(a()) r36: proper(e()) -> ok(e()) r37: proper(i()) -> ok(i()) r38: proper(o()) -> ok(o()) r39: proper(u()) -> ok(u()) r40: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) r41: and(ok(X1),ok(X2)) -> ok(and(X1,X2)) r42: isList(ok(X)) -> ok(isList(X)) r43: isNeList(ok(X)) -> ok(isNeList(X)) r44: isQid(ok(X)) -> ok(isQid(X)) r45: isNePal(ok(X)) -> ok(isNePal(X)) r46: isPal(ok(X)) -> ok(isPal(X)) r47: top(mark(X)) -> top(proper(X)) r48: top(ok(X)) -> top(active(X)) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r40, r41, r42, r43, r44, r45, r46 Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: max/plus interpretations on natural numbers: top#_A(x1) = max{226, x1 + 196} ok_A(x1) = max{225, x1 + 195} active_A(x1) = x1 + 69 ___A(x1,x2) = max{x1 + 40, x2 + 71} mark_A(x1) = max{132, x1 - 2} and_A(x1,x2) = max{x1 + 50, x2 + 81} isList_A(x1) = x1 + 194 isNeList_A(x1) = x1 + 133 isQid_A(x1) = max{63, x1 + 39} isNePal_A(x1) = max{63, x1 + 36} isPal_A(x1) = max{63, x1 + 33} nil_A = 30 tt_A = 3 a_A = 24 e_A = 24 i_A = 133 o_A = 2 u_A = 0 precedence: top# = active > __ = mark = and > ok = isList = isNeList = isQid = isNePal = isPal = nil = tt = a = e = i = o = u partial status: pi(top#) = [1] pi(ok) = [1] pi(active) = [1] pi(__) = [1, 2] pi(mark) = [] pi(and) = [1, 2] pi(isList) = [1] pi(isNeList) = [1] pi(isQid) = [1] pi(isNePal) = [1] pi(isPal) = [1] pi(nil) = [] pi(tt) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] 2. weighted path order base order: max/plus interpretations on natural numbers: top#_A(x1) = max{414, x1 + 361} ok_A(x1) = x1 + 313 active_A(x1) = max{54, x1 - 47} ___A(x1,x2) = max{x1 + 117, x2 + 63} mark_A(x1) = 413 and_A(x1,x2) = max{x1 - 50, x2 + 53} isList_A(x1) = x1 + 46 isNeList_A(x1) = max{314, x1 + 249} isQid_A(x1) = x1 + 312 isNePal_A(x1) = max{277, x1 + 232} isPal_A(x1) = max{72, x1 + 30} nil_A = 295 tt_A = 277 a_A = 13 e_A = 13 i_A = 13 o_A = 13 u_A = 13 precedence: top# = ok = active = __ = mark = and = isList = isNeList = isQid = isNePal = isPal = nil = tt = a = e = i = o = u partial status: pi(top#) = [] pi(ok) = [1] pi(active) = [] pi(__) = [2] pi(mark) = [] pi(and) = [2] pi(isList) = [1] pi(isNeList) = [] pi(isQid) = [1] pi(isNePal) = [1] pi(isPal) = [1] pi(nil) = [] pi(tt) = [] pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] The next rules are strictly ordered: p1 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: active#(and(X1,X2)) -> active#(X1) p2: active#(__(X1,X2)) -> active#(X2) p3: active#(__(X1,X2)) -> active#(X1) and R consists of: r1: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) r2: active(__(X,nil())) -> mark(X) r3: active(__(nil(),X)) -> mark(X) r4: active(and(tt(),X)) -> mark(X) r5: active(isList(V)) -> mark(isNeList(V)) r6: active(isList(nil())) -> mark(tt()) r7: active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) r8: active(isNeList(V)) -> mark(isQid(V)) r9: active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) r10: active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) r11: active(isNePal(V)) -> mark(isQid(V)) r12: active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) r13: active(isPal(V)) -> mark(isNePal(V)) r14: active(isPal(nil())) -> mark(tt()) r15: active(isQid(a())) -> mark(tt()) r16: active(isQid(e())) -> mark(tt()) r17: active(isQid(i())) -> mark(tt()) r18: active(isQid(o())) -> mark(tt()) r19: active(isQid(u())) -> mark(tt()) r20: active(__(X1,X2)) -> __(active(X1),X2) r21: active(__(X1,X2)) -> __(X1,active(X2)) r22: active(and(X1,X2)) -> and(active(X1),X2) r23: __(mark(X1),X2) -> mark(__(X1,X2)) r24: __(X1,mark(X2)) -> mark(__(X1,X2)) r25: and(mark(X1),X2) -> mark(and(X1,X2)) r26: proper(__(X1,X2)) -> __(proper(X1),proper(X2)) r27: proper(nil()) -> ok(nil()) r28: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) r29: proper(tt()) -> ok(tt()) r30: proper(isList(X)) -> isList(proper(X)) r31: proper(isNeList(X)) -> isNeList(proper(X)) r32: proper(isQid(X)) -> isQid(proper(X)) r33: proper(isNePal(X)) -> isNePal(proper(X)) r34: proper(isPal(X)) -> isPal(proper(X)) r35: proper(a()) -> ok(a()) r36: proper(e()) -> ok(e()) r37: proper(i()) -> ok(i()) r38: proper(o()) -> ok(o()) r39: proper(u()) -> ok(u()) r40: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) r41: and(ok(X1),ok(X2)) -> ok(and(X1,X2)) r42: isList(ok(X)) -> ok(isList(X)) r43: isNeList(ok(X)) -> ok(isNeList(X)) r44: isQid(ok(X)) -> ok(isQid(X)) r45: isNePal(ok(X)) -> ok(isNePal(X)) r46: isPal(ok(X)) -> ok(isPal(X)) r47: top(mark(X)) -> top(proper(X)) r48: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: max/plus interpretations on natural numbers: active#_A(x1) = max{2, x1 + 1} and_A(x1,x2) = max{x1, x2} ___A(x1,x2) = max{1, x1, x2} precedence: active# = and = __ partial status: pi(active#) = [1] pi(and) = [1, 2] pi(__) = [1, 2] 2. weighted path order base order: max/plus interpretations on natural numbers: active#_A(x1) = x1 + 3 and_A(x1,x2) = max{x1 + 1, x2} ___A(x1,x2) = max{x1 + 1, x2 + 1} precedence: active# = and = __ partial status: pi(active#) = [] pi(and) = [2] pi(__) = [2] The next rules are strictly ordered: p1, p2, p3 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: proper#(isPal(X)) -> proper#(X) p2: proper#(isNePal(X)) -> proper#(X) p3: proper#(isQid(X)) -> proper#(X) p4: proper#(isNeList(X)) -> proper#(X) p5: proper#(isList(X)) -> proper#(X) p6: proper#(and(X1,X2)) -> proper#(X2) p7: proper#(and(X1,X2)) -> proper#(X1) p8: proper#(__(X1,X2)) -> proper#(X2) p9: proper#(__(X1,X2)) -> proper#(X1) and R consists of: r1: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) r2: active(__(X,nil())) -> mark(X) r3: active(__(nil(),X)) -> mark(X) r4: active(and(tt(),X)) -> mark(X) r5: active(isList(V)) -> mark(isNeList(V)) r6: active(isList(nil())) -> mark(tt()) r7: active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) r8: active(isNeList(V)) -> mark(isQid(V)) r9: active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) r10: active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) r11: active(isNePal(V)) -> mark(isQid(V)) r12: active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) r13: active(isPal(V)) -> mark(isNePal(V)) r14: active(isPal(nil())) -> mark(tt()) r15: active(isQid(a())) -> mark(tt()) r16: active(isQid(e())) -> mark(tt()) r17: active(isQid(i())) -> mark(tt()) r18: active(isQid(o())) -> mark(tt()) r19: active(isQid(u())) -> mark(tt()) r20: active(__(X1,X2)) -> __(active(X1),X2) r21: active(__(X1,X2)) -> __(X1,active(X2)) r22: active(and(X1,X2)) -> and(active(X1),X2) r23: __(mark(X1),X2) -> mark(__(X1,X2)) r24: __(X1,mark(X2)) -> mark(__(X1,X2)) r25: and(mark(X1),X2) -> mark(and(X1,X2)) r26: proper(__(X1,X2)) -> __(proper(X1),proper(X2)) r27: proper(nil()) -> ok(nil()) r28: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) r29: proper(tt()) -> ok(tt()) r30: proper(isList(X)) -> isList(proper(X)) r31: proper(isNeList(X)) -> isNeList(proper(X)) r32: proper(isQid(X)) -> isQid(proper(X)) r33: proper(isNePal(X)) -> isNePal(proper(X)) r34: proper(isPal(X)) -> isPal(proper(X)) r35: proper(a()) -> ok(a()) r36: proper(e()) -> ok(e()) r37: proper(i()) -> ok(i()) r38: proper(o()) -> ok(o()) r39: proper(u()) -> ok(u()) r40: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) r41: and(ok(X1),ok(X2)) -> ok(and(X1,X2)) r42: isList(ok(X)) -> ok(isList(X)) r43: isNeList(ok(X)) -> ok(isNeList(X)) r44: isQid(ok(X)) -> ok(isQid(X)) r45: isNePal(ok(X)) -> ok(isNePal(X)) r46: isPal(ok(X)) -> ok(isPal(X)) r47: top(mark(X)) -> top(proper(X)) r48: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: max/plus interpretations on natural numbers: proper#_A(x1) = max{2, x1 + 1} isPal_A(x1) = max{1, x1} isNePal_A(x1) = max{1, x1} isQid_A(x1) = max{1, x1} isNeList_A(x1) = max{1, x1} isList_A(x1) = max{1, x1} and_A(x1,x2) = max{x1, x2} ___A(x1,x2) = max{x1, x2} precedence: proper# = isPal = isNePal = isQid = isNeList = isList = and = __ partial status: pi(proper#) = [1] pi(isPal) = [1] pi(isNePal) = [1] pi(isQid) = [1] pi(isNeList) = [1] pi(isList) = [1] pi(and) = [1, 2] pi(__) = [1, 2] 2. weighted path order base order: max/plus interpretations on natural numbers: proper#_A(x1) = x1 + 2 isPal_A(x1) = x1 + 1 isNePal_A(x1) = x1 + 1 isQid_A(x1) = x1 + 1 isNeList_A(x1) = x1 + 1 isList_A(x1) = x1 + 1 and_A(x1,x2) = max{x1 + 1, x2 + 1} ___A(x1,x2) = max{x1 + 1, x2 + 1} precedence: proper# = isPal = isNePal = isQid = isNeList = isList = and = __ partial status: pi(proper#) = [] pi(isPal) = [1] pi(isNePal) = [1] pi(isQid) = [1] pi(isNeList) = [1] pi(isList) = [1] pi(and) = [2] pi(__) = [2] The next rules are strictly ordered: p1, p2, p3, p4, p5, p6, p7, p8, p9 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: __#(mark(X1),X2) -> __#(X1,X2) p2: __#(ok(X1),ok(X2)) -> __#(X1,X2) p3: __#(X1,mark(X2)) -> __#(X1,X2) and R consists of: r1: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) r2: active(__(X,nil())) -> mark(X) r3: active(__(nil(),X)) -> mark(X) r4: active(and(tt(),X)) -> mark(X) r5: active(isList(V)) -> mark(isNeList(V)) r6: active(isList(nil())) -> mark(tt()) r7: active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) r8: active(isNeList(V)) -> mark(isQid(V)) r9: active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) r10: active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) r11: active(isNePal(V)) -> mark(isQid(V)) r12: active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) r13: active(isPal(V)) -> mark(isNePal(V)) r14: active(isPal(nil())) -> mark(tt()) r15: active(isQid(a())) -> mark(tt()) r16: active(isQid(e())) -> mark(tt()) r17: active(isQid(i())) -> mark(tt()) r18: active(isQid(o())) -> mark(tt()) r19: active(isQid(u())) -> mark(tt()) r20: active(__(X1,X2)) -> __(active(X1),X2) r21: active(__(X1,X2)) -> __(X1,active(X2)) r22: active(and(X1,X2)) -> and(active(X1),X2) r23: __(mark(X1),X2) -> mark(__(X1,X2)) r24: __(X1,mark(X2)) -> mark(__(X1,X2)) r25: and(mark(X1),X2) -> mark(and(X1,X2)) r26: proper(__(X1,X2)) -> __(proper(X1),proper(X2)) r27: proper(nil()) -> ok(nil()) r28: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) r29: proper(tt()) -> ok(tt()) r30: proper(isList(X)) -> isList(proper(X)) r31: proper(isNeList(X)) -> isNeList(proper(X)) r32: proper(isQid(X)) -> isQid(proper(X)) r33: proper(isNePal(X)) -> isNePal(proper(X)) r34: proper(isPal(X)) -> isPal(proper(X)) r35: proper(a()) -> ok(a()) r36: proper(e()) -> ok(e()) r37: proper(i()) -> ok(i()) r38: proper(o()) -> ok(o()) r39: proper(u()) -> ok(u()) r40: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) r41: and(ok(X1),ok(X2)) -> ok(and(X1,X2)) r42: isList(ok(X)) -> ok(isList(X)) r43: isNeList(ok(X)) -> ok(isNeList(X)) r44: isQid(ok(X)) -> ok(isQid(X)) r45: isNePal(ok(X)) -> ok(isNePal(X)) r46: isPal(ok(X)) -> ok(isPal(X)) r47: top(mark(X)) -> top(proper(X)) r48: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: max/plus interpretations on natural numbers: __#_A(x1,x2) = max{2, x1} mark_A(x1) = max{2, x1 + 1} ok_A(x1) = max{7, x1 + 3} precedence: __# = ok > mark partial status: pi(__#) = [1] pi(mark) = [1] pi(ok) = [1] 2. weighted path order base order: max/plus interpretations on natural numbers: __#_A(x1,x2) = 0 mark_A(x1) = x1 + 1 ok_A(x1) = max{3, x1 + 1} precedence: __# = ok > mark partial status: pi(__#) = [] pi(mark) = [1] pi(ok) = [] The next rules are strictly ordered: p1 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: __#(ok(X1),ok(X2)) -> __#(X1,X2) p2: __#(X1,mark(X2)) -> __#(X1,X2) and R consists of: r1: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) r2: active(__(X,nil())) -> mark(X) r3: active(__(nil(),X)) -> mark(X) r4: active(and(tt(),X)) -> mark(X) r5: active(isList(V)) -> mark(isNeList(V)) r6: active(isList(nil())) -> mark(tt()) r7: active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) r8: active(isNeList(V)) -> mark(isQid(V)) r9: active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) r10: active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) r11: active(isNePal(V)) -> mark(isQid(V)) r12: active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) r13: active(isPal(V)) -> mark(isNePal(V)) r14: active(isPal(nil())) -> mark(tt()) r15: active(isQid(a())) -> mark(tt()) r16: active(isQid(e())) -> mark(tt()) r17: active(isQid(i())) -> mark(tt()) r18: active(isQid(o())) -> mark(tt()) r19: active(isQid(u())) -> mark(tt()) r20: active(__(X1,X2)) -> __(active(X1),X2) r21: active(__(X1,X2)) -> __(X1,active(X2)) r22: active(and(X1,X2)) -> and(active(X1),X2) r23: __(mark(X1),X2) -> mark(__(X1,X2)) r24: __(X1,mark(X2)) -> mark(__(X1,X2)) r25: and(mark(X1),X2) -> mark(and(X1,X2)) r26: proper(__(X1,X2)) -> __(proper(X1),proper(X2)) r27: proper(nil()) -> ok(nil()) r28: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) r29: proper(tt()) -> ok(tt()) r30: proper(isList(X)) -> isList(proper(X)) r31: proper(isNeList(X)) -> isNeList(proper(X)) r32: proper(isQid(X)) -> isQid(proper(X)) r33: proper(isNePal(X)) -> isNePal(proper(X)) r34: proper(isPal(X)) -> isPal(proper(X)) r35: proper(a()) -> ok(a()) r36: proper(e()) -> ok(e()) r37: proper(i()) -> ok(i()) r38: proper(o()) -> ok(o()) r39: proper(u()) -> ok(u()) r40: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) r41: and(ok(X1),ok(X2)) -> ok(and(X1,X2)) r42: isList(ok(X)) -> ok(isList(X)) r43: isNeList(ok(X)) -> ok(isNeList(X)) r44: isQid(ok(X)) -> ok(isQid(X)) r45: isNePal(ok(X)) -> ok(isNePal(X)) r46: isPal(ok(X)) -> ok(isPal(X)) r47: top(mark(X)) -> top(proper(X)) r48: top(ok(X)) -> top(active(X)) The estimated dependency graph contains the following SCCs: {p1, p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: __#(ok(X1),ok(X2)) -> __#(X1,X2) p2: __#(X1,mark(X2)) -> __#(X1,X2) and R consists of: r1: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) r2: active(__(X,nil())) -> mark(X) r3: active(__(nil(),X)) -> mark(X) r4: active(and(tt(),X)) -> mark(X) r5: active(isList(V)) -> mark(isNeList(V)) r6: active(isList(nil())) -> mark(tt()) r7: active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) r8: active(isNeList(V)) -> mark(isQid(V)) r9: active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) r10: active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) r11: active(isNePal(V)) -> mark(isQid(V)) r12: active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) r13: active(isPal(V)) -> mark(isNePal(V)) r14: active(isPal(nil())) -> mark(tt()) r15: active(isQid(a())) -> mark(tt()) r16: active(isQid(e())) -> mark(tt()) r17: active(isQid(i())) -> mark(tt()) r18: active(isQid(o())) -> mark(tt()) r19: active(isQid(u())) -> mark(tt()) r20: active(__(X1,X2)) -> __(active(X1),X2) r21: active(__(X1,X2)) -> __(X1,active(X2)) r22: active(and(X1,X2)) -> and(active(X1),X2) r23: __(mark(X1),X2) -> mark(__(X1,X2)) r24: __(X1,mark(X2)) -> mark(__(X1,X2)) r25: and(mark(X1),X2) -> mark(and(X1,X2)) r26: proper(__(X1,X2)) -> __(proper(X1),proper(X2)) r27: proper(nil()) -> ok(nil()) r28: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) r29: proper(tt()) -> ok(tt()) r30: proper(isList(X)) -> isList(proper(X)) r31: proper(isNeList(X)) -> isNeList(proper(X)) r32: proper(isQid(X)) -> isQid(proper(X)) r33: proper(isNePal(X)) -> isNePal(proper(X)) r34: proper(isPal(X)) -> isPal(proper(X)) r35: proper(a()) -> ok(a()) r36: proper(e()) -> ok(e()) r37: proper(i()) -> ok(i()) r38: proper(o()) -> ok(o()) r39: proper(u()) -> ok(u()) r40: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) r41: and(ok(X1),ok(X2)) -> ok(and(X1,X2)) r42: isList(ok(X)) -> ok(isList(X)) r43: isNeList(ok(X)) -> ok(isNeList(X)) r44: isQid(ok(X)) -> ok(isQid(X)) r45: isNePal(ok(X)) -> ok(isNePal(X)) r46: isPal(ok(X)) -> ok(isPal(X)) r47: top(mark(X)) -> top(proper(X)) r48: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: max/plus interpretations on natural numbers: __#_A(x1,x2) = x2 + 1 ok_A(x1) = x1 + 2 mark_A(x1) = x1 + 1 precedence: __# = ok = mark partial status: pi(__#) = [] pi(ok) = [] pi(mark) = [] 2. weighted path order base order: max/plus interpretations on natural numbers: __#_A(x1,x2) = 0 ok_A(x1) = max{2, x1} mark_A(x1) = x1 precedence: __# = ok = mark partial status: pi(__#) = [] pi(ok) = [1] pi(mark) = [1] The next rules are strictly ordered: p1, p2 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: isNeList#(ok(X)) -> isNeList#(X) and R consists of: r1: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) r2: active(__(X,nil())) -> mark(X) r3: active(__(nil(),X)) -> mark(X) r4: active(and(tt(),X)) -> mark(X) r5: active(isList(V)) -> mark(isNeList(V)) r6: active(isList(nil())) -> mark(tt()) r7: active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) r8: active(isNeList(V)) -> mark(isQid(V)) r9: active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) r10: active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) r11: active(isNePal(V)) -> mark(isQid(V)) r12: active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) r13: active(isPal(V)) -> mark(isNePal(V)) r14: active(isPal(nil())) -> mark(tt()) r15: active(isQid(a())) -> mark(tt()) r16: active(isQid(e())) -> mark(tt()) r17: active(isQid(i())) -> mark(tt()) r18: active(isQid(o())) -> mark(tt()) r19: active(isQid(u())) -> mark(tt()) r20: active(__(X1,X2)) -> __(active(X1),X2) r21: active(__(X1,X2)) -> __(X1,active(X2)) r22: active(and(X1,X2)) -> and(active(X1),X2) r23: __(mark(X1),X2) -> mark(__(X1,X2)) r24: __(X1,mark(X2)) -> mark(__(X1,X2)) r25: and(mark(X1),X2) -> mark(and(X1,X2)) r26: proper(__(X1,X2)) -> __(proper(X1),proper(X2)) r27: proper(nil()) -> ok(nil()) r28: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) r29: proper(tt()) -> ok(tt()) r30: proper(isList(X)) -> isList(proper(X)) r31: proper(isNeList(X)) -> isNeList(proper(X)) r32: proper(isQid(X)) -> isQid(proper(X)) r33: proper(isNePal(X)) -> isNePal(proper(X)) r34: proper(isPal(X)) -> isPal(proper(X)) r35: proper(a()) -> ok(a()) r36: proper(e()) -> ok(e()) r37: proper(i()) -> ok(i()) r38: proper(o()) -> ok(o()) r39: proper(u()) -> ok(u()) r40: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) r41: and(ok(X1),ok(X2)) -> ok(and(X1,X2)) r42: isList(ok(X)) -> ok(isList(X)) r43: isNeList(ok(X)) -> ok(isNeList(X)) r44: isQid(ok(X)) -> ok(isQid(X)) r45: isNePal(ok(X)) -> ok(isNePal(X)) r46: isPal(ok(X)) -> ok(isPal(X)) r47: top(mark(X)) -> top(proper(X)) r48: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: max/plus interpretations on natural numbers: isNeList#_A(x1) = max{4, x1 + 3} ok_A(x1) = max{3, x1 + 2} precedence: isNeList# = ok partial status: pi(isNeList#) = [1] pi(ok) = [1] 2. weighted path order base order: max/plus interpretations on natural numbers: isNeList#_A(x1) = max{1, x1 - 1} ok_A(x1) = x1 precedence: isNeList# = ok partial status: pi(isNeList#) = [] pi(ok) = [1] The next rules are strictly ordered: p1 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: and#(mark(X1),X2) -> and#(X1,X2) p2: and#(ok(X1),ok(X2)) -> and#(X1,X2) and R consists of: r1: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) r2: active(__(X,nil())) -> mark(X) r3: active(__(nil(),X)) -> mark(X) r4: active(and(tt(),X)) -> mark(X) r5: active(isList(V)) -> mark(isNeList(V)) r6: active(isList(nil())) -> mark(tt()) r7: active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) r8: active(isNeList(V)) -> mark(isQid(V)) r9: active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) r10: active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) r11: active(isNePal(V)) -> mark(isQid(V)) r12: active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) r13: active(isPal(V)) -> mark(isNePal(V)) r14: active(isPal(nil())) -> mark(tt()) r15: active(isQid(a())) -> mark(tt()) r16: active(isQid(e())) -> mark(tt()) r17: active(isQid(i())) -> mark(tt()) r18: active(isQid(o())) -> mark(tt()) r19: active(isQid(u())) -> mark(tt()) r20: active(__(X1,X2)) -> __(active(X1),X2) r21: active(__(X1,X2)) -> __(X1,active(X2)) r22: active(and(X1,X2)) -> and(active(X1),X2) r23: __(mark(X1),X2) -> mark(__(X1,X2)) r24: __(X1,mark(X2)) -> mark(__(X1,X2)) r25: and(mark(X1),X2) -> mark(and(X1,X2)) r26: proper(__(X1,X2)) -> __(proper(X1),proper(X2)) r27: proper(nil()) -> ok(nil()) r28: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) r29: proper(tt()) -> ok(tt()) r30: proper(isList(X)) -> isList(proper(X)) r31: proper(isNeList(X)) -> isNeList(proper(X)) r32: proper(isQid(X)) -> isQid(proper(X)) r33: proper(isNePal(X)) -> isNePal(proper(X)) r34: proper(isPal(X)) -> isPal(proper(X)) r35: proper(a()) -> ok(a()) r36: proper(e()) -> ok(e()) r37: proper(i()) -> ok(i()) r38: proper(o()) -> ok(o()) r39: proper(u()) -> ok(u()) r40: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) r41: and(ok(X1),ok(X2)) -> ok(and(X1,X2)) r42: isList(ok(X)) -> ok(isList(X)) r43: isNeList(ok(X)) -> ok(isNeList(X)) r44: isQid(ok(X)) -> ok(isQid(X)) r45: isNePal(ok(X)) -> ok(isNePal(X)) r46: isPal(ok(X)) -> ok(isPal(X)) r47: top(mark(X)) -> top(proper(X)) r48: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: max/plus interpretations on natural numbers: and#_A(x1,x2) = max{2, x1 + 1, x2 + 1} mark_A(x1) = max{1, x1} ok_A(x1) = max{1, x1} precedence: and# = mark = ok partial status: pi(and#) = [1, 2] pi(mark) = [1] pi(ok) = [1] 2. weighted path order base order: max/plus interpretations on natural numbers: and#_A(x1,x2) = max{x1 + 1, x2 - 2} mark_A(x1) = x1 ok_A(x1) = max{4, x1 + 1} precedence: and# = mark = ok partial status: pi(and#) = [1] pi(mark) = [1] pi(ok) = [1] The next rules are strictly ordered: p1, p2 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: isList#(ok(X)) -> isList#(X) and R consists of: r1: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) r2: active(__(X,nil())) -> mark(X) r3: active(__(nil(),X)) -> mark(X) r4: active(and(tt(),X)) -> mark(X) r5: active(isList(V)) -> mark(isNeList(V)) r6: active(isList(nil())) -> mark(tt()) r7: active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) r8: active(isNeList(V)) -> mark(isQid(V)) r9: active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) r10: active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) r11: active(isNePal(V)) -> mark(isQid(V)) r12: active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) r13: active(isPal(V)) -> mark(isNePal(V)) r14: active(isPal(nil())) -> mark(tt()) r15: active(isQid(a())) -> mark(tt()) r16: active(isQid(e())) -> mark(tt()) r17: active(isQid(i())) -> mark(tt()) r18: active(isQid(o())) -> mark(tt()) r19: active(isQid(u())) -> mark(tt()) r20: active(__(X1,X2)) -> __(active(X1),X2) r21: active(__(X1,X2)) -> __(X1,active(X2)) r22: active(and(X1,X2)) -> and(active(X1),X2) r23: __(mark(X1),X2) -> mark(__(X1,X2)) r24: __(X1,mark(X2)) -> mark(__(X1,X2)) r25: and(mark(X1),X2) -> mark(and(X1,X2)) r26: proper(__(X1,X2)) -> __(proper(X1),proper(X2)) r27: proper(nil()) -> ok(nil()) r28: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) r29: proper(tt()) -> ok(tt()) r30: proper(isList(X)) -> isList(proper(X)) r31: proper(isNeList(X)) -> isNeList(proper(X)) r32: proper(isQid(X)) -> isQid(proper(X)) r33: proper(isNePal(X)) -> isNePal(proper(X)) r34: proper(isPal(X)) -> isPal(proper(X)) r35: proper(a()) -> ok(a()) r36: proper(e()) -> ok(e()) r37: proper(i()) -> ok(i()) r38: proper(o()) -> ok(o()) r39: proper(u()) -> ok(u()) r40: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) r41: and(ok(X1),ok(X2)) -> ok(and(X1,X2)) r42: isList(ok(X)) -> ok(isList(X)) r43: isNeList(ok(X)) -> ok(isNeList(X)) r44: isQid(ok(X)) -> ok(isQid(X)) r45: isNePal(ok(X)) -> ok(isNePal(X)) r46: isPal(ok(X)) -> ok(isPal(X)) r47: top(mark(X)) -> top(proper(X)) r48: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: max/plus interpretations on natural numbers: isList#_A(x1) = max{4, x1 + 3} ok_A(x1) = max{3, x1 + 2} precedence: isList# = ok partial status: pi(isList#) = [1] pi(ok) = [1] 2. weighted path order base order: max/plus interpretations on natural numbers: isList#_A(x1) = max{1, x1 - 1} ok_A(x1) = x1 precedence: isList# = ok partial status: pi(isList#) = [] pi(ok) = [1] The next rules are strictly ordered: p1 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: isQid#(ok(X)) -> isQid#(X) and R consists of: r1: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) r2: active(__(X,nil())) -> mark(X) r3: active(__(nil(),X)) -> mark(X) r4: active(and(tt(),X)) -> mark(X) r5: active(isList(V)) -> mark(isNeList(V)) r6: active(isList(nil())) -> mark(tt()) r7: active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) r8: active(isNeList(V)) -> mark(isQid(V)) r9: active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) r10: active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) r11: active(isNePal(V)) -> mark(isQid(V)) r12: active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) r13: active(isPal(V)) -> mark(isNePal(V)) r14: active(isPal(nil())) -> mark(tt()) r15: active(isQid(a())) -> mark(tt()) r16: active(isQid(e())) -> mark(tt()) r17: active(isQid(i())) -> mark(tt()) r18: active(isQid(o())) -> mark(tt()) r19: active(isQid(u())) -> mark(tt()) r20: active(__(X1,X2)) -> __(active(X1),X2) r21: active(__(X1,X2)) -> __(X1,active(X2)) r22: active(and(X1,X2)) -> and(active(X1),X2) r23: __(mark(X1),X2) -> mark(__(X1,X2)) r24: __(X1,mark(X2)) -> mark(__(X1,X2)) r25: and(mark(X1),X2) -> mark(and(X1,X2)) r26: proper(__(X1,X2)) -> __(proper(X1),proper(X2)) r27: proper(nil()) -> ok(nil()) r28: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) r29: proper(tt()) -> ok(tt()) r30: proper(isList(X)) -> isList(proper(X)) r31: proper(isNeList(X)) -> isNeList(proper(X)) r32: proper(isQid(X)) -> isQid(proper(X)) r33: proper(isNePal(X)) -> isNePal(proper(X)) r34: proper(isPal(X)) -> isPal(proper(X)) r35: proper(a()) -> ok(a()) r36: proper(e()) -> ok(e()) r37: proper(i()) -> ok(i()) r38: proper(o()) -> ok(o()) r39: proper(u()) -> ok(u()) r40: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) r41: and(ok(X1),ok(X2)) -> ok(and(X1,X2)) r42: isList(ok(X)) -> ok(isList(X)) r43: isNeList(ok(X)) -> ok(isNeList(X)) r44: isQid(ok(X)) -> ok(isQid(X)) r45: isNePal(ok(X)) -> ok(isNePal(X)) r46: isPal(ok(X)) -> ok(isPal(X)) r47: top(mark(X)) -> top(proper(X)) r48: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: max/plus interpretations on natural numbers: isQid#_A(x1) = max{4, x1 + 3} ok_A(x1) = max{3, x1 + 2} precedence: isQid# = ok partial status: pi(isQid#) = [1] pi(ok) = [1] 2. weighted path order base order: max/plus interpretations on natural numbers: isQid#_A(x1) = max{1, x1 - 1} ok_A(x1) = x1 precedence: isQid# = ok partial status: pi(isQid#) = [] pi(ok) = [1] The next rules are strictly ordered: p1 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: isPal#(ok(X)) -> isPal#(X) and R consists of: r1: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) r2: active(__(X,nil())) -> mark(X) r3: active(__(nil(),X)) -> mark(X) r4: active(and(tt(),X)) -> mark(X) r5: active(isList(V)) -> mark(isNeList(V)) r6: active(isList(nil())) -> mark(tt()) r7: active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) r8: active(isNeList(V)) -> mark(isQid(V)) r9: active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) r10: active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) r11: active(isNePal(V)) -> mark(isQid(V)) r12: active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) r13: active(isPal(V)) -> mark(isNePal(V)) r14: active(isPal(nil())) -> mark(tt()) r15: active(isQid(a())) -> mark(tt()) r16: active(isQid(e())) -> mark(tt()) r17: active(isQid(i())) -> mark(tt()) r18: active(isQid(o())) -> mark(tt()) r19: active(isQid(u())) -> mark(tt()) r20: active(__(X1,X2)) -> __(active(X1),X2) r21: active(__(X1,X2)) -> __(X1,active(X2)) r22: active(and(X1,X2)) -> and(active(X1),X2) r23: __(mark(X1),X2) -> mark(__(X1,X2)) r24: __(X1,mark(X2)) -> mark(__(X1,X2)) r25: and(mark(X1),X2) -> mark(and(X1,X2)) r26: proper(__(X1,X2)) -> __(proper(X1),proper(X2)) r27: proper(nil()) -> ok(nil()) r28: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) r29: proper(tt()) -> ok(tt()) r30: proper(isList(X)) -> isList(proper(X)) r31: proper(isNeList(X)) -> isNeList(proper(X)) r32: proper(isQid(X)) -> isQid(proper(X)) r33: proper(isNePal(X)) -> isNePal(proper(X)) r34: proper(isPal(X)) -> isPal(proper(X)) r35: proper(a()) -> ok(a()) r36: proper(e()) -> ok(e()) r37: proper(i()) -> ok(i()) r38: proper(o()) -> ok(o()) r39: proper(u()) -> ok(u()) r40: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) r41: and(ok(X1),ok(X2)) -> ok(and(X1,X2)) r42: isList(ok(X)) -> ok(isList(X)) r43: isNeList(ok(X)) -> ok(isNeList(X)) r44: isQid(ok(X)) -> ok(isQid(X)) r45: isNePal(ok(X)) -> ok(isNePal(X)) r46: isPal(ok(X)) -> ok(isPal(X)) r47: top(mark(X)) -> top(proper(X)) r48: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: max/plus interpretations on natural numbers: isPal#_A(x1) = max{4, x1 + 3} ok_A(x1) = max{3, x1 + 2} precedence: isPal# = ok partial status: pi(isPal#) = [1] pi(ok) = [1] 2. weighted path order base order: max/plus interpretations on natural numbers: isPal#_A(x1) = max{1, x1 - 1} ok_A(x1) = x1 precedence: isPal# = ok partial status: pi(isPal#) = [] pi(ok) = [1] The next rules are strictly ordered: p1 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: isNePal#(ok(X)) -> isNePal#(X) and R consists of: r1: active(__(__(X,Y),Z)) -> mark(__(X,__(Y,Z))) r2: active(__(X,nil())) -> mark(X) r3: active(__(nil(),X)) -> mark(X) r4: active(and(tt(),X)) -> mark(X) r5: active(isList(V)) -> mark(isNeList(V)) r6: active(isList(nil())) -> mark(tt()) r7: active(isList(__(V1,V2))) -> mark(and(isList(V1),isList(V2))) r8: active(isNeList(V)) -> mark(isQid(V)) r9: active(isNeList(__(V1,V2))) -> mark(and(isList(V1),isNeList(V2))) r10: active(isNeList(__(V1,V2))) -> mark(and(isNeList(V1),isList(V2))) r11: active(isNePal(V)) -> mark(isQid(V)) r12: active(isNePal(__(I,__(P,I)))) -> mark(and(isQid(I),isPal(P))) r13: active(isPal(V)) -> mark(isNePal(V)) r14: active(isPal(nil())) -> mark(tt()) r15: active(isQid(a())) -> mark(tt()) r16: active(isQid(e())) -> mark(tt()) r17: active(isQid(i())) -> mark(tt()) r18: active(isQid(o())) -> mark(tt()) r19: active(isQid(u())) -> mark(tt()) r20: active(__(X1,X2)) -> __(active(X1),X2) r21: active(__(X1,X2)) -> __(X1,active(X2)) r22: active(and(X1,X2)) -> and(active(X1),X2) r23: __(mark(X1),X2) -> mark(__(X1,X2)) r24: __(X1,mark(X2)) -> mark(__(X1,X2)) r25: and(mark(X1),X2) -> mark(and(X1,X2)) r26: proper(__(X1,X2)) -> __(proper(X1),proper(X2)) r27: proper(nil()) -> ok(nil()) r28: proper(and(X1,X2)) -> and(proper(X1),proper(X2)) r29: proper(tt()) -> ok(tt()) r30: proper(isList(X)) -> isList(proper(X)) r31: proper(isNeList(X)) -> isNeList(proper(X)) r32: proper(isQid(X)) -> isQid(proper(X)) r33: proper(isNePal(X)) -> isNePal(proper(X)) r34: proper(isPal(X)) -> isPal(proper(X)) r35: proper(a()) -> ok(a()) r36: proper(e()) -> ok(e()) r37: proper(i()) -> ok(i()) r38: proper(o()) -> ok(o()) r39: proper(u()) -> ok(u()) r40: __(ok(X1),ok(X2)) -> ok(__(X1,X2)) r41: and(ok(X1),ok(X2)) -> ok(and(X1,X2)) r42: isList(ok(X)) -> ok(isList(X)) r43: isNeList(ok(X)) -> ok(isNeList(X)) r44: isQid(ok(X)) -> ok(isQid(X)) r45: isNePal(ok(X)) -> ok(isNePal(X)) r46: isPal(ok(X)) -> ok(isPal(X)) r47: top(mark(X)) -> top(proper(X)) r48: top(ok(X)) -> top(active(X)) The set of usable rules consists of (no rules) Take the reduction pair: lexicographic combination of reduction pairs: 1. weighted path order base order: max/plus interpretations on natural numbers: isNePal#_A(x1) = max{4, x1 + 3} ok_A(x1) = max{3, x1 + 2} precedence: isNePal# = ok partial status: pi(isNePal#) = [1] pi(ok) = [1] 2. weighted path order base order: max/plus interpretations on natural numbers: isNePal#_A(x1) = max{1, x1 - 1} ok_A(x1) = x1 precedence: isNePal# = ok partial status: pi(isNePal#) = [] pi(ok) = [1] The next rules are strictly ordered: p1 We remove them from the problem. Then no dependency pair remains.