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()) mark(__(X1,X2)) -> active(__(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(tt()) -> active(tt()) mark(isList(X)) -> active(isList(X)) mark(isNeList(X)) -> active(isNeList(X)) mark(isQid(X)) -> active(isQid(X)) mark(isNePal(X)) -> active(isNePal(X)) mark(isPal(X)) -> active(isPal(X)) mark(a()) -> active(a()) mark(e()) -> active(e()) mark(i()) -> active(i()) mark(o()) -> active(o()) mark(u()) -> active(u()) __(mark(X1),X2) -> __(X1,X2) __(X1,mark(X2)) -> __(X1,X2) __(active(X1),X2) -> __(X1,X2) __(X1,active(X2)) -> __(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) isList(mark(X)) -> isList(X) isList(active(X)) -> isList(X) isNeList(mark(X)) -> isNeList(X) isNeList(active(X)) -> isNeList(X) isQid(mark(X)) -> isQid(X) isQid(active(X)) -> isQid(X) isNePal(mark(X)) -> isNePal(X) isNePal(active(X)) -> isNePal(X) isPal(mark(X)) -> isPal(X) isPal(active(X)) -> isPal(X) -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: active#(__(__(X,Y),Z)) -> mark#(__(X,__(Y,Z))) p2: active#(__(__(X,Y),Z)) -> __#(X,__(Y,Z)) p3: active#(__(__(X,Y),Z)) -> __#(Y,Z) p4: active#(__(X,nil())) -> mark#(X) p5: active#(__(nil(),X)) -> mark#(X) p6: active#(and(tt(),X)) -> mark#(X) p7: active#(isList(V)) -> mark#(isNeList(V)) p8: active#(isList(V)) -> isNeList#(V) p9: active#(isList(nil())) -> mark#(tt()) p10: active#(isList(__(V1,V2))) -> mark#(and(isList(V1),isList(V2))) p11: active#(isList(__(V1,V2))) -> and#(isList(V1),isList(V2)) p12: active#(isList(__(V1,V2))) -> isList#(V1) p13: active#(isList(__(V1,V2))) -> isList#(V2) p14: active#(isNeList(V)) -> mark#(isQid(V)) p15: active#(isNeList(V)) -> isQid#(V) p16: active#(isNeList(__(V1,V2))) -> mark#(and(isList(V1),isNeList(V2))) p17: active#(isNeList(__(V1,V2))) -> and#(isList(V1),isNeList(V2)) p18: active#(isNeList(__(V1,V2))) -> isList#(V1) p19: active#(isNeList(__(V1,V2))) -> isNeList#(V2) p20: active#(isNeList(__(V1,V2))) -> mark#(and(isNeList(V1),isList(V2))) p21: active#(isNeList(__(V1,V2))) -> and#(isNeList(V1),isList(V2)) p22: active#(isNeList(__(V1,V2))) -> isNeList#(V1) p23: active#(isNeList(__(V1,V2))) -> isList#(V2) p24: active#(isNePal(V)) -> mark#(isQid(V)) p25: active#(isNePal(V)) -> isQid#(V) p26: active#(isNePal(__(I,__(P,I)))) -> mark#(and(isQid(I),isPal(P))) p27: active#(isNePal(__(I,__(P,I)))) -> and#(isQid(I),isPal(P)) p28: active#(isNePal(__(I,__(P,I)))) -> isQid#(I) p29: active#(isNePal(__(I,__(P,I)))) -> isPal#(P) p30: active#(isPal(V)) -> mark#(isNePal(V)) p31: active#(isPal(V)) -> isNePal#(V) p32: active#(isPal(nil())) -> mark#(tt()) p33: active#(isQid(a())) -> mark#(tt()) p34: active#(isQid(e())) -> mark#(tt()) p35: active#(isQid(i())) -> mark#(tt()) p36: active#(isQid(o())) -> mark#(tt()) p37: active#(isQid(u())) -> mark#(tt()) p38: mark#(__(X1,X2)) -> active#(__(mark(X1),mark(X2))) p39: mark#(__(X1,X2)) -> __#(mark(X1),mark(X2)) p40: mark#(__(X1,X2)) -> mark#(X1) p41: mark#(__(X1,X2)) -> mark#(X2) p42: mark#(nil()) -> active#(nil()) p43: mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) p44: mark#(and(X1,X2)) -> and#(mark(X1),X2) p45: mark#(and(X1,X2)) -> mark#(X1) p46: mark#(tt()) -> active#(tt()) p47: mark#(isList(X)) -> active#(isList(X)) p48: mark#(isNeList(X)) -> active#(isNeList(X)) p49: mark#(isQid(X)) -> active#(isQid(X)) p50: mark#(isNePal(X)) -> active#(isNePal(X)) p51: mark#(isPal(X)) -> active#(isPal(X)) p52: mark#(a()) -> active#(a()) p53: mark#(e()) -> active#(e()) p54: mark#(i()) -> active#(i()) p55: mark#(o()) -> active#(o()) p56: mark#(u()) -> active#(u()) p57: __#(mark(X1),X2) -> __#(X1,X2) p58: __#(X1,mark(X2)) -> __#(X1,X2) p59: __#(active(X1),X2) -> __#(X1,X2) p60: __#(X1,active(X2)) -> __#(X1,X2) p61: and#(mark(X1),X2) -> and#(X1,X2) p62: and#(X1,mark(X2)) -> and#(X1,X2) p63: and#(active(X1),X2) -> and#(X1,X2) p64: and#(X1,active(X2)) -> and#(X1,X2) p65: isList#(mark(X)) -> isList#(X) p66: isList#(active(X)) -> isList#(X) p67: isNeList#(mark(X)) -> isNeList#(X) p68: isNeList#(active(X)) -> isNeList#(X) p69: isQid#(mark(X)) -> isQid#(X) p70: isQid#(active(X)) -> isQid#(X) p71: isNePal#(mark(X)) -> isNePal#(X) p72: isNePal#(active(X)) -> isNePal#(X) p73: isPal#(mark(X)) -> isPal#(X) p74: isPal#(active(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: mark(__(X1,X2)) -> active(__(mark(X1),mark(X2))) r21: mark(nil()) -> active(nil()) r22: mark(and(X1,X2)) -> active(and(mark(X1),X2)) r23: mark(tt()) -> active(tt()) r24: mark(isList(X)) -> active(isList(X)) r25: mark(isNeList(X)) -> active(isNeList(X)) r26: mark(isQid(X)) -> active(isQid(X)) r27: mark(isNePal(X)) -> active(isNePal(X)) r28: mark(isPal(X)) -> active(isPal(X)) r29: mark(a()) -> active(a()) r30: mark(e()) -> active(e()) r31: mark(i()) -> active(i()) r32: mark(o()) -> active(o()) r33: mark(u()) -> active(u()) r34: __(mark(X1),X2) -> __(X1,X2) r35: __(X1,mark(X2)) -> __(X1,X2) r36: __(active(X1),X2) -> __(X1,X2) r37: __(X1,active(X2)) -> __(X1,X2) r38: and(mark(X1),X2) -> and(X1,X2) r39: and(X1,mark(X2)) -> and(X1,X2) r40: and(active(X1),X2) -> and(X1,X2) r41: and(X1,active(X2)) -> and(X1,X2) r42: isList(mark(X)) -> isList(X) r43: isList(active(X)) -> isList(X) r44: isNeList(mark(X)) -> isNeList(X) r45: isNeList(active(X)) -> isNeList(X) r46: isQid(mark(X)) -> isQid(X) r47: isQid(active(X)) -> isQid(X) r48: isNePal(mark(X)) -> isNePal(X) r49: isNePal(active(X)) -> isNePal(X) r50: isPal(mark(X)) -> isPal(X) r51: isPal(active(X)) -> isPal(X) The estimated dependency graph contains the following SCCs: {p1, p4, p5, p6, p7, p10, p14, p16, p20, p24, p26, p30, p38, p40, p41, p43, p45, p47, p48, p49, p50, p51} {p57, p58, p59, p60} {p67, p68} {p61, p62, p63, p64} {p65, p66} {p69, p70} {p73, p74} {p71, p72} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: active#(__(__(X,Y),Z)) -> mark#(__(X,__(Y,Z))) p2: mark#(isPal(X)) -> active#(isPal(X)) p3: active#(isPal(V)) -> mark#(isNePal(V)) p4: mark#(isNePal(X)) -> active#(isNePal(X)) p5: active#(isNePal(__(I,__(P,I)))) -> mark#(and(isQid(I),isPal(P))) p6: mark#(isQid(X)) -> active#(isQid(X)) p7: active#(isNePal(V)) -> mark#(isQid(V)) p8: mark#(isNeList(X)) -> active#(isNeList(X)) p9: active#(isNeList(__(V1,V2))) -> mark#(and(isNeList(V1),isList(V2))) p10: mark#(isList(X)) -> active#(isList(X)) p11: active#(isNeList(__(V1,V2))) -> mark#(and(isList(V1),isNeList(V2))) p12: mark#(and(X1,X2)) -> mark#(X1) p13: mark#(and(X1,X2)) -> active#(and(mark(X1),X2)) p14: active#(isNeList(V)) -> mark#(isQid(V)) p15: mark#(__(X1,X2)) -> mark#(X2) p16: mark#(__(X1,X2)) -> mark#(X1) p17: mark#(__(X1,X2)) -> active#(__(mark(X1),mark(X2))) p18: active#(isList(__(V1,V2))) -> mark#(and(isList(V1),isList(V2))) p19: active#(isList(V)) -> mark#(isNeList(V)) p20: active#(and(tt(),X)) -> mark#(X) p21: active#(__(nil(),X)) -> mark#(X) p22: active#(__(X,nil())) -> mark#(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: mark(__(X1,X2)) -> active(__(mark(X1),mark(X2))) r21: mark(nil()) -> active(nil()) r22: mark(and(X1,X2)) -> active(and(mark(X1),X2)) r23: mark(tt()) -> active(tt()) r24: mark(isList(X)) -> active(isList(X)) r25: mark(isNeList(X)) -> active(isNeList(X)) r26: mark(isQid(X)) -> active(isQid(X)) r27: mark(isNePal(X)) -> active(isNePal(X)) r28: mark(isPal(X)) -> active(isPal(X)) r29: mark(a()) -> active(a()) r30: mark(e()) -> active(e()) r31: mark(i()) -> active(i()) r32: mark(o()) -> active(o()) r33: mark(u()) -> active(u()) r34: __(mark(X1),X2) -> __(X1,X2) r35: __(X1,mark(X2)) -> __(X1,X2) r36: __(active(X1),X2) -> __(X1,X2) r37: __(X1,active(X2)) -> __(X1,X2) r38: and(mark(X1),X2) -> and(X1,X2) r39: and(X1,mark(X2)) -> and(X1,X2) r40: and(active(X1),X2) -> and(X1,X2) r41: and(X1,active(X2)) -> and(X1,X2) r42: isList(mark(X)) -> isList(X) r43: isList(active(X)) -> isList(X) r44: isNeList(mark(X)) -> isNeList(X) r45: isNeList(active(X)) -> isNeList(X) r46: isQid(mark(X)) -> isQid(X) r47: isQid(active(X)) -> isQid(X) r48: isNePal(mark(X)) -> isNePal(X) r49: isNePal(active(X)) -> isNePal(X) r50: isPal(mark(X)) -> isPal(X) r51: isPal(active(X)) -> isPal(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, r47, r48, r49, r50, r51 Take the reduction pair: lexicographic combination of reduction pairs: 1. matrix interpretations: carrier: N^1 order: standard order interpretations: active#_A(x1) = x1 ___A(x1,x2) = x1 + x2 + 5 mark#_A(x1) = x1 isPal_A(x1) = x1 + 3 isNePal_A(x1) = x1 + 2 and_A(x1,x2) = x1 + x2 + 1 isQid_A(x1) = x1 + 1 isNeList_A(x1) = x1 + 2 isList_A(x1) = x1 + 3 mark_A(x1) = x1 tt_A() = 1 nil_A() = 1 active_A(x1) = x1 a_A() = 1 e_A() = 1 i_A() = 1 o_A() = 1 u_A() = 1 2. lexicographic path order with precedence: precedence: active > u > o > i > e > a > __ > mark > mark# > tt > nil > and > isList > active# > isNePal > isQid > isNeList > isPal argument filter: pi(active#) = [1] pi(__) = [1, 2] pi(mark#) = [1] pi(isPal) = [1] pi(isNePal) = [1] pi(and) = 2 pi(isQid) = [1] pi(isNeList) = [] pi(isList) = 1 pi(mark) = 1 pi(tt) = [] pi(nil) = [] pi(active) = 1 pi(a) = [] pi(e) = [] pi(i) = [] pi(o) = [] pi(u) = [] The next rules are strictly ordered: p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22 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: __#(X1,active(X2)) -> __#(X1,X2) p3: __#(active(X1),X2) -> __#(X1,X2) p4: __#(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: mark(__(X1,X2)) -> active(__(mark(X1),mark(X2))) r21: mark(nil()) -> active(nil()) r22: mark(and(X1,X2)) -> active(and(mark(X1),X2)) r23: mark(tt()) -> active(tt()) r24: mark(isList(X)) -> active(isList(X)) r25: mark(isNeList(X)) -> active(isNeList(X)) r26: mark(isQid(X)) -> active(isQid(X)) r27: mark(isNePal(X)) -> active(isNePal(X)) r28: mark(isPal(X)) -> active(isPal(X)) r29: mark(a()) -> active(a()) r30: mark(e()) -> active(e()) r31: mark(i()) -> active(i()) r32: mark(o()) -> active(o()) r33: mark(u()) -> active(u()) r34: __(mark(X1),X2) -> __(X1,X2) r35: __(X1,mark(X2)) -> __(X1,X2) r36: __(active(X1),X2) -> __(X1,X2) r37: __(X1,active(X2)) -> __(X1,X2) r38: and(mark(X1),X2) -> and(X1,X2) r39: and(X1,mark(X2)) -> and(X1,X2) r40: and(active(X1),X2) -> and(X1,X2) r41: and(X1,active(X2)) -> and(X1,X2) r42: isList(mark(X)) -> isList(X) r43: isList(active(X)) -> isList(X) r44: isNeList(mark(X)) -> isNeList(X) r45: isNeList(active(X)) -> isNeList(X) r46: isQid(mark(X)) -> isQid(X) r47: isQid(active(X)) -> isQid(X) r48: isNePal(mark(X)) -> isNePal(X) r49: isNePal(active(X)) -> isNePal(X) r50: isPal(mark(X)) -> isPal(X) r51: isPal(active(X)) -> isPal(X) The set of usable rules consists of (no rules) Take the monotone reduction pair: lexicographic combination of reduction pairs: 1. matrix interpretations: carrier: N^1 order: standard order interpretations: __#_A(x1,x2) = x1 + x2 mark_A(x1) = x1 + 1 active_A(x1) = x1 + 1 2. lexicographic path order with precedence: precedence: __# > active > mark argument filter: pi(__#) = [1, 2] pi(mark) = [1] pi(active) = [1] The next rules are strictly ordered: p1, p2, p3, p4 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, r47, r48, r49, r50, r51 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#(mark(X)) -> isNeList#(X) p2: isNeList#(active(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: mark(__(X1,X2)) -> active(__(mark(X1),mark(X2))) r21: mark(nil()) -> active(nil()) r22: mark(and(X1,X2)) -> active(and(mark(X1),X2)) r23: mark(tt()) -> active(tt()) r24: mark(isList(X)) -> active(isList(X)) r25: mark(isNeList(X)) -> active(isNeList(X)) r26: mark(isQid(X)) -> active(isQid(X)) r27: mark(isNePal(X)) -> active(isNePal(X)) r28: mark(isPal(X)) -> active(isPal(X)) r29: mark(a()) -> active(a()) r30: mark(e()) -> active(e()) r31: mark(i()) -> active(i()) r32: mark(o()) -> active(o()) r33: mark(u()) -> active(u()) r34: __(mark(X1),X2) -> __(X1,X2) r35: __(X1,mark(X2)) -> __(X1,X2) r36: __(active(X1),X2) -> __(X1,X2) r37: __(X1,active(X2)) -> __(X1,X2) r38: and(mark(X1),X2) -> and(X1,X2) r39: and(X1,mark(X2)) -> and(X1,X2) r40: and(active(X1),X2) -> and(X1,X2) r41: and(X1,active(X2)) -> and(X1,X2) r42: isList(mark(X)) -> isList(X) r43: isList(active(X)) -> isList(X) r44: isNeList(mark(X)) -> isNeList(X) r45: isNeList(active(X)) -> isNeList(X) r46: isQid(mark(X)) -> isQid(X) r47: isQid(active(X)) -> isQid(X) r48: isNePal(mark(X)) -> isNePal(X) r49: isNePal(active(X)) -> isNePal(X) r50: isPal(mark(X)) -> isPal(X) r51: isPal(active(X)) -> isPal(X) The set of usable rules consists of (no rules) Take the monotone reduction pair: lexicographic combination of reduction pairs: 1. matrix interpretations: carrier: N^1 order: standard order interpretations: isNeList#_A(x1) = x1 mark_A(x1) = x1 + 1 active_A(x1) = x1 + 1 2. lexicographic path order with precedence: precedence: isNeList# > active > mark argument filter: pi(isNeList#) = 1 pi(mark) = [1] pi(active) = [1] The next rules are strictly ordered: p1, p2 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, r47, r48, r49, r50, r51 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#(X1,active(X2)) -> and#(X1,X2) p3: and#(active(X1),X2) -> and#(X1,X2) p4: and#(X1,mark(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: mark(__(X1,X2)) -> active(__(mark(X1),mark(X2))) r21: mark(nil()) -> active(nil()) r22: mark(and(X1,X2)) -> active(and(mark(X1),X2)) r23: mark(tt()) -> active(tt()) r24: mark(isList(X)) -> active(isList(X)) r25: mark(isNeList(X)) -> active(isNeList(X)) r26: mark(isQid(X)) -> active(isQid(X)) r27: mark(isNePal(X)) -> active(isNePal(X)) r28: mark(isPal(X)) -> active(isPal(X)) r29: mark(a()) -> active(a()) r30: mark(e()) -> active(e()) r31: mark(i()) -> active(i()) r32: mark(o()) -> active(o()) r33: mark(u()) -> active(u()) r34: __(mark(X1),X2) -> __(X1,X2) r35: __(X1,mark(X2)) -> __(X1,X2) r36: __(active(X1),X2) -> __(X1,X2) r37: __(X1,active(X2)) -> __(X1,X2) r38: and(mark(X1),X2) -> and(X1,X2) r39: and(X1,mark(X2)) -> and(X1,X2) r40: and(active(X1),X2) -> and(X1,X2) r41: and(X1,active(X2)) -> and(X1,X2) r42: isList(mark(X)) -> isList(X) r43: isList(active(X)) -> isList(X) r44: isNeList(mark(X)) -> isNeList(X) r45: isNeList(active(X)) -> isNeList(X) r46: isQid(mark(X)) -> isQid(X) r47: isQid(active(X)) -> isQid(X) r48: isNePal(mark(X)) -> isNePal(X) r49: isNePal(active(X)) -> isNePal(X) r50: isPal(mark(X)) -> isPal(X) r51: isPal(active(X)) -> isPal(X) The set of usable rules consists of (no rules) Take the monotone reduction pair: lexicographic combination of reduction pairs: 1. matrix interpretations: carrier: N^1 order: standard order interpretations: and#_A(x1,x2) = x1 + x2 mark_A(x1) = x1 + 1 active_A(x1) = x1 + 1 2. lexicographic path order with precedence: precedence: and# > active > mark argument filter: pi(and#) = [1, 2] pi(mark) = [1] pi(active) = [1] The next rules are strictly ordered: p1, p2, p3, p4 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, r47, r48, r49, r50, r51 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#(mark(X)) -> isList#(X) p2: isList#(active(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: mark(__(X1,X2)) -> active(__(mark(X1),mark(X2))) r21: mark(nil()) -> active(nil()) r22: mark(and(X1,X2)) -> active(and(mark(X1),X2)) r23: mark(tt()) -> active(tt()) r24: mark(isList(X)) -> active(isList(X)) r25: mark(isNeList(X)) -> active(isNeList(X)) r26: mark(isQid(X)) -> active(isQid(X)) r27: mark(isNePal(X)) -> active(isNePal(X)) r28: mark(isPal(X)) -> active(isPal(X)) r29: mark(a()) -> active(a()) r30: mark(e()) -> active(e()) r31: mark(i()) -> active(i()) r32: mark(o()) -> active(o()) r33: mark(u()) -> active(u()) r34: __(mark(X1),X2) -> __(X1,X2) r35: __(X1,mark(X2)) -> __(X1,X2) r36: __(active(X1),X2) -> __(X1,X2) r37: __(X1,active(X2)) -> __(X1,X2) r38: and(mark(X1),X2) -> and(X1,X2) r39: and(X1,mark(X2)) -> and(X1,X2) r40: and(active(X1),X2) -> and(X1,X2) r41: and(X1,active(X2)) -> and(X1,X2) r42: isList(mark(X)) -> isList(X) r43: isList(active(X)) -> isList(X) r44: isNeList(mark(X)) -> isNeList(X) r45: isNeList(active(X)) -> isNeList(X) r46: isQid(mark(X)) -> isQid(X) r47: isQid(active(X)) -> isQid(X) r48: isNePal(mark(X)) -> isNePal(X) r49: isNePal(active(X)) -> isNePal(X) r50: isPal(mark(X)) -> isPal(X) r51: isPal(active(X)) -> isPal(X) The set of usable rules consists of (no rules) Take the monotone reduction pair: lexicographic combination of reduction pairs: 1. matrix interpretations: carrier: N^1 order: standard order interpretations: isList#_A(x1) = x1 mark_A(x1) = x1 + 1 active_A(x1) = x1 + 1 2. lexicographic path order with precedence: precedence: isList# > active > mark argument filter: pi(isList#) = 1 pi(mark) = [1] pi(active) = [1] The next rules are strictly ordered: p1, p2 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, r47, r48, r49, r50, r51 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#(mark(X)) -> isQid#(X) p2: isQid#(active(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: mark(__(X1,X2)) -> active(__(mark(X1),mark(X2))) r21: mark(nil()) -> active(nil()) r22: mark(and(X1,X2)) -> active(and(mark(X1),X2)) r23: mark(tt()) -> active(tt()) r24: mark(isList(X)) -> active(isList(X)) r25: mark(isNeList(X)) -> active(isNeList(X)) r26: mark(isQid(X)) -> active(isQid(X)) r27: mark(isNePal(X)) -> active(isNePal(X)) r28: mark(isPal(X)) -> active(isPal(X)) r29: mark(a()) -> active(a()) r30: mark(e()) -> active(e()) r31: mark(i()) -> active(i()) r32: mark(o()) -> active(o()) r33: mark(u()) -> active(u()) r34: __(mark(X1),X2) -> __(X1,X2) r35: __(X1,mark(X2)) -> __(X1,X2) r36: __(active(X1),X2) -> __(X1,X2) r37: __(X1,active(X2)) -> __(X1,X2) r38: and(mark(X1),X2) -> and(X1,X2) r39: and(X1,mark(X2)) -> and(X1,X2) r40: and(active(X1),X2) -> and(X1,X2) r41: and(X1,active(X2)) -> and(X1,X2) r42: isList(mark(X)) -> isList(X) r43: isList(active(X)) -> isList(X) r44: isNeList(mark(X)) -> isNeList(X) r45: isNeList(active(X)) -> isNeList(X) r46: isQid(mark(X)) -> isQid(X) r47: isQid(active(X)) -> isQid(X) r48: isNePal(mark(X)) -> isNePal(X) r49: isNePal(active(X)) -> isNePal(X) r50: isPal(mark(X)) -> isPal(X) r51: isPal(active(X)) -> isPal(X) The set of usable rules consists of (no rules) Take the monotone reduction pair: lexicographic combination of reduction pairs: 1. matrix interpretations: carrier: N^1 order: standard order interpretations: isQid#_A(x1) = x1 mark_A(x1) = x1 + 1 active_A(x1) = x1 + 1 2. lexicographic path order with precedence: precedence: isQid# > active > mark argument filter: pi(isQid#) = 1 pi(mark) = [1] pi(active) = [1] The next rules are strictly ordered: p1, p2 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, r47, r48, r49, r50, r51 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#(mark(X)) -> isPal#(X) p2: isPal#(active(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: mark(__(X1,X2)) -> active(__(mark(X1),mark(X2))) r21: mark(nil()) -> active(nil()) r22: mark(and(X1,X2)) -> active(and(mark(X1),X2)) r23: mark(tt()) -> active(tt()) r24: mark(isList(X)) -> active(isList(X)) r25: mark(isNeList(X)) -> active(isNeList(X)) r26: mark(isQid(X)) -> active(isQid(X)) r27: mark(isNePal(X)) -> active(isNePal(X)) r28: mark(isPal(X)) -> active(isPal(X)) r29: mark(a()) -> active(a()) r30: mark(e()) -> active(e()) r31: mark(i()) -> active(i()) r32: mark(o()) -> active(o()) r33: mark(u()) -> active(u()) r34: __(mark(X1),X2) -> __(X1,X2) r35: __(X1,mark(X2)) -> __(X1,X2) r36: __(active(X1),X2) -> __(X1,X2) r37: __(X1,active(X2)) -> __(X1,X2) r38: and(mark(X1),X2) -> and(X1,X2) r39: and(X1,mark(X2)) -> and(X1,X2) r40: and(active(X1),X2) -> and(X1,X2) r41: and(X1,active(X2)) -> and(X1,X2) r42: isList(mark(X)) -> isList(X) r43: isList(active(X)) -> isList(X) r44: isNeList(mark(X)) -> isNeList(X) r45: isNeList(active(X)) -> isNeList(X) r46: isQid(mark(X)) -> isQid(X) r47: isQid(active(X)) -> isQid(X) r48: isNePal(mark(X)) -> isNePal(X) r49: isNePal(active(X)) -> isNePal(X) r50: isPal(mark(X)) -> isPal(X) r51: isPal(active(X)) -> isPal(X) The set of usable rules consists of (no rules) Take the monotone reduction pair: lexicographic combination of reduction pairs: 1. matrix interpretations: carrier: N^1 order: standard order interpretations: isPal#_A(x1) = x1 mark_A(x1) = x1 + 1 active_A(x1) = x1 + 1 2. lexicographic path order with precedence: precedence: isPal# > active > mark argument filter: pi(isPal#) = 1 pi(mark) = [1] pi(active) = [1] The next rules are strictly ordered: p1, p2 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, r47, r48, r49, r50, r51 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#(mark(X)) -> isNePal#(X) p2: isNePal#(active(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: mark(__(X1,X2)) -> active(__(mark(X1),mark(X2))) r21: mark(nil()) -> active(nil()) r22: mark(and(X1,X2)) -> active(and(mark(X1),X2)) r23: mark(tt()) -> active(tt()) r24: mark(isList(X)) -> active(isList(X)) r25: mark(isNeList(X)) -> active(isNeList(X)) r26: mark(isQid(X)) -> active(isQid(X)) r27: mark(isNePal(X)) -> active(isNePal(X)) r28: mark(isPal(X)) -> active(isPal(X)) r29: mark(a()) -> active(a()) r30: mark(e()) -> active(e()) r31: mark(i()) -> active(i()) r32: mark(o()) -> active(o()) r33: mark(u()) -> active(u()) r34: __(mark(X1),X2) -> __(X1,X2) r35: __(X1,mark(X2)) -> __(X1,X2) r36: __(active(X1),X2) -> __(X1,X2) r37: __(X1,active(X2)) -> __(X1,X2) r38: and(mark(X1),X2) -> and(X1,X2) r39: and(X1,mark(X2)) -> and(X1,X2) r40: and(active(X1),X2) -> and(X1,X2) r41: and(X1,active(X2)) -> and(X1,X2) r42: isList(mark(X)) -> isList(X) r43: isList(active(X)) -> isList(X) r44: isNeList(mark(X)) -> isNeList(X) r45: isNeList(active(X)) -> isNeList(X) r46: isQid(mark(X)) -> isQid(X) r47: isQid(active(X)) -> isQid(X) r48: isNePal(mark(X)) -> isNePal(X) r49: isNePal(active(X)) -> isNePal(X) r50: isPal(mark(X)) -> isPal(X) r51: isPal(active(X)) -> isPal(X) The set of usable rules consists of (no rules) Take the monotone reduction pair: lexicographic combination of reduction pairs: 1. matrix interpretations: carrier: N^1 order: standard order interpretations: isNePal#_A(x1) = x1 mark_A(x1) = x1 + 1 active_A(x1) = x1 + 1 2. lexicographic path order with precedence: precedence: isNePal# > active > mark argument filter: pi(isNePal#) = 1 pi(mark) = [1] pi(active) = [1] The next rules are strictly ordered: p1, p2 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, r47, r48, r49, r50, r51 We remove them from the problem. Then no dependency pair remains.