YES

We show the termination of the TRS R:

  active(zeros()) -> mark(cons(|0|(),zeros()))
  active(U11(tt())) -> mark(tt())
  active(U21(tt())) -> mark(tt())
  active(U31(tt())) -> mark(tt())
  active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
  active(U42(tt())) -> mark(tt())
  active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
  active(U52(tt())) -> mark(tt())
  active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
  active(U62(tt())) -> mark(tt())
  active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
  active(U72(tt(),L)) -> mark(s(length(L)))
  active(U81(tt())) -> mark(nil())
  active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
  active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
  active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
  active(isNat(|0|())) -> mark(tt())
  active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
  active(isNat(s(V1))) -> mark(U21(isNat(V1)))
  active(isNatIList(V)) -> mark(U31(isNatList(V)))
  active(isNatIList(zeros())) -> mark(tt())
  active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
  active(isNatList(nil())) -> mark(tt())
  active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
  active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
  active(length(nil())) -> mark(|0|())
  active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
  active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
  active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
  mark(zeros()) -> active(zeros())
  mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
  mark(|0|()) -> active(|0|())
  mark(U11(X)) -> active(U11(mark(X)))
  mark(tt()) -> active(tt())
  mark(U21(X)) -> active(U21(mark(X)))
  mark(U31(X)) -> active(U31(mark(X)))
  mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
  mark(U42(X)) -> active(U42(mark(X)))
  mark(isNatIList(X)) -> active(isNatIList(X))
  mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
  mark(U52(X)) -> active(U52(mark(X)))
  mark(isNatList(X)) -> active(isNatList(X))
  mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
  mark(U62(X)) -> active(U62(mark(X)))
  mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
  mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
  mark(isNat(X)) -> active(isNat(X))
  mark(s(X)) -> active(s(mark(X)))
  mark(length(X)) -> active(length(mark(X)))
  mark(U81(X)) -> active(U81(mark(X)))
  mark(nil()) -> active(nil())
  mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
  mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
  mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
  mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
  cons(mark(X1),X2) -> cons(X1,X2)
  cons(X1,mark(X2)) -> cons(X1,X2)
  cons(active(X1),X2) -> cons(X1,X2)
  cons(X1,active(X2)) -> cons(X1,X2)
  U11(mark(X)) -> U11(X)
  U11(active(X)) -> U11(X)
  U21(mark(X)) -> U21(X)
  U21(active(X)) -> U21(X)
  U31(mark(X)) -> U31(X)
  U31(active(X)) -> U31(X)
  U41(mark(X1),X2) -> U41(X1,X2)
  U41(X1,mark(X2)) -> U41(X1,X2)
  U41(active(X1),X2) -> U41(X1,X2)
  U41(X1,active(X2)) -> U41(X1,X2)
  U42(mark(X)) -> U42(X)
  U42(active(X)) -> U42(X)
  isNatIList(mark(X)) -> isNatIList(X)
  isNatIList(active(X)) -> isNatIList(X)
  U51(mark(X1),X2) -> U51(X1,X2)
  U51(X1,mark(X2)) -> U51(X1,X2)
  U51(active(X1),X2) -> U51(X1,X2)
  U51(X1,active(X2)) -> U51(X1,X2)
  U52(mark(X)) -> U52(X)
  U52(active(X)) -> U52(X)
  isNatList(mark(X)) -> isNatList(X)
  isNatList(active(X)) -> isNatList(X)
  U61(mark(X1),X2) -> U61(X1,X2)
  U61(X1,mark(X2)) -> U61(X1,X2)
  U61(active(X1),X2) -> U61(X1,X2)
  U61(X1,active(X2)) -> U61(X1,X2)
  U62(mark(X)) -> U62(X)
  U62(active(X)) -> U62(X)
  U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
  U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
  U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
  U71(active(X1),X2,X3) -> U71(X1,X2,X3)
  U71(X1,active(X2),X3) -> U71(X1,X2,X3)
  U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
  U72(mark(X1),X2) -> U72(X1,X2)
  U72(X1,mark(X2)) -> U72(X1,X2)
  U72(active(X1),X2) -> U72(X1,X2)
  U72(X1,active(X2)) -> U72(X1,X2)
  isNat(mark(X)) -> isNat(X)
  isNat(active(X)) -> isNat(X)
  s(mark(X)) -> s(X)
  s(active(X)) -> s(X)
  length(mark(X)) -> length(X)
  length(active(X)) -> length(X)
  U81(mark(X)) -> U81(X)
  U81(active(X)) -> U81(X)
  U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
  U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
  U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
  U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
  U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
  U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
  U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
  U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
  U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
  U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
  U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
  U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
  U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
  U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
  U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
  U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
  U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
  U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
  U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
  U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
  U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
  U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
  U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
  U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
  take(mark(X1),X2) -> take(X1,X2)
  take(X1,mark(X2)) -> take(X1,X2)
  take(active(X1),X2) -> take(X1,X2)
  take(X1,active(X2)) -> take(X1,X2)

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: active#(zeros()) -> cons#(|0|(),zeros())
p3: active#(U11(tt())) -> mark#(tt())
p4: active#(U21(tt())) -> mark#(tt())
p5: active#(U31(tt())) -> mark#(tt())
p6: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p7: active#(U41(tt(),V2)) -> U42#(isNatIList(V2))
p8: active#(U41(tt(),V2)) -> isNatIList#(V2)
p9: active#(U42(tt())) -> mark#(tt())
p10: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p11: active#(U51(tt(),V2)) -> U52#(isNatList(V2))
p12: active#(U51(tt(),V2)) -> isNatList#(V2)
p13: active#(U52(tt())) -> mark#(tt())
p14: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p15: active#(U61(tt(),V2)) -> U62#(isNatIList(V2))
p16: active#(U61(tt(),V2)) -> isNatIList#(V2)
p17: active#(U62(tt())) -> mark#(tt())
p18: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p19: active#(U71(tt(),L,N)) -> U72#(isNat(N),L)
p20: active#(U71(tt(),L,N)) -> isNat#(N)
p21: active#(U72(tt(),L)) -> mark#(s(length(L)))
p22: active#(U72(tt(),L)) -> s#(length(L))
p23: active#(U72(tt(),L)) -> length#(L)
p24: active#(U81(tt())) -> mark#(nil())
p25: active#(U91(tt(),IL,M,N)) -> mark#(U92(isNat(M),IL,M,N))
p26: active#(U91(tt(),IL,M,N)) -> U92#(isNat(M),IL,M,N)
p27: active#(U91(tt(),IL,M,N)) -> isNat#(M)
p28: active#(U92(tt(),IL,M,N)) -> mark#(U93(isNat(N),IL,M,N))
p29: active#(U92(tt(),IL,M,N)) -> U93#(isNat(N),IL,M,N)
p30: active#(U92(tt(),IL,M,N)) -> isNat#(N)
p31: active#(U93(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p32: active#(U93(tt(),IL,M,N)) -> cons#(N,take(M,IL))
p33: active#(U93(tt(),IL,M,N)) -> take#(M,IL)
p34: active#(isNat(|0|())) -> mark#(tt())
p35: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p36: active#(isNat(length(V1))) -> U11#(isNatList(V1))
p37: active#(isNat(length(V1))) -> isNatList#(V1)
p38: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p39: active#(isNat(s(V1))) -> U21#(isNat(V1))
p40: active#(isNat(s(V1))) -> isNat#(V1)
p41: active#(isNatIList(V)) -> mark#(U31(isNatList(V)))
p42: active#(isNatIList(V)) -> U31#(isNatList(V))
p43: active#(isNatIList(V)) -> isNatList#(V)
p44: active#(isNatIList(zeros())) -> mark#(tt())
p45: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p46: active#(isNatIList(cons(V1,V2))) -> U41#(isNat(V1),V2)
p47: active#(isNatIList(cons(V1,V2))) -> isNat#(V1)
p48: active#(isNatList(nil())) -> mark#(tt())
p49: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p50: active#(isNatList(cons(V1,V2))) -> U51#(isNat(V1),V2)
p51: active#(isNatList(cons(V1,V2))) -> isNat#(V1)
p52: active#(isNatList(take(V1,V2))) -> mark#(U61(isNat(V1),V2))
p53: active#(isNatList(take(V1,V2))) -> U61#(isNat(V1),V2)
p54: active#(isNatList(take(V1,V2))) -> isNat#(V1)
p55: active#(length(nil())) -> mark#(|0|())
p56: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p57: active#(length(cons(N,L))) -> U71#(isNatList(L),L,N)
p58: active#(length(cons(N,L))) -> isNatList#(L)
p59: active#(take(|0|(),IL)) -> mark#(U81(isNatIList(IL)))
p60: active#(take(|0|(),IL)) -> U81#(isNatIList(IL))
p61: active#(take(|0|(),IL)) -> isNatIList#(IL)
p62: active#(take(s(M),cons(N,IL))) -> mark#(U91(isNatIList(IL),IL,M,N))
p63: active#(take(s(M),cons(N,IL))) -> U91#(isNatIList(IL),IL,M,N)
p64: active#(take(s(M),cons(N,IL))) -> isNatIList#(IL)
p65: mark#(zeros()) -> active#(zeros())
p66: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p67: mark#(cons(X1,X2)) -> cons#(mark(X1),X2)
p68: mark#(cons(X1,X2)) -> mark#(X1)
p69: mark#(|0|()) -> active#(|0|())
p70: mark#(U11(X)) -> active#(U11(mark(X)))
p71: mark#(U11(X)) -> U11#(mark(X))
p72: mark#(U11(X)) -> mark#(X)
p73: mark#(tt()) -> active#(tt())
p74: mark#(U21(X)) -> active#(U21(mark(X)))
p75: mark#(U21(X)) -> U21#(mark(X))
p76: mark#(U21(X)) -> mark#(X)
p77: mark#(U31(X)) -> active#(U31(mark(X)))
p78: mark#(U31(X)) -> U31#(mark(X))
p79: mark#(U31(X)) -> mark#(X)
p80: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p81: mark#(U41(X1,X2)) -> U41#(mark(X1),X2)
p82: mark#(U41(X1,X2)) -> mark#(X1)
p83: mark#(U42(X)) -> active#(U42(mark(X)))
p84: mark#(U42(X)) -> U42#(mark(X))
p85: mark#(U42(X)) -> mark#(X)
p86: mark#(isNatIList(X)) -> active#(isNatIList(X))
p87: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p88: mark#(U51(X1,X2)) -> U51#(mark(X1),X2)
p89: mark#(U51(X1,X2)) -> mark#(X1)
p90: mark#(U52(X)) -> active#(U52(mark(X)))
p91: mark#(U52(X)) -> U52#(mark(X))
p92: mark#(U52(X)) -> mark#(X)
p93: mark#(isNatList(X)) -> active#(isNatList(X))
p94: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p95: mark#(U61(X1,X2)) -> U61#(mark(X1),X2)
p96: mark#(U61(X1,X2)) -> mark#(X1)
p97: mark#(U62(X)) -> active#(U62(mark(X)))
p98: mark#(U62(X)) -> U62#(mark(X))
p99: mark#(U62(X)) -> mark#(X)
p100: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p101: mark#(U71(X1,X2,X3)) -> U71#(mark(X1),X2,X3)
p102: mark#(U71(X1,X2,X3)) -> mark#(X1)
p103: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p104: mark#(U72(X1,X2)) -> U72#(mark(X1),X2)
p105: mark#(U72(X1,X2)) -> mark#(X1)
p106: mark#(isNat(X)) -> active#(isNat(X))
p107: mark#(s(X)) -> active#(s(mark(X)))
p108: mark#(s(X)) -> s#(mark(X))
p109: mark#(s(X)) -> mark#(X)
p110: mark#(length(X)) -> active#(length(mark(X)))
p111: mark#(length(X)) -> length#(mark(X))
p112: mark#(length(X)) -> mark#(X)
p113: mark#(U81(X)) -> active#(U81(mark(X)))
p114: mark#(U81(X)) -> U81#(mark(X))
p115: mark#(U81(X)) -> mark#(X)
p116: mark#(nil()) -> active#(nil())
p117: mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4))
p118: mark#(U91(X1,X2,X3,X4)) -> U91#(mark(X1),X2,X3,X4)
p119: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p120: mark#(U92(X1,X2,X3,X4)) -> active#(U92(mark(X1),X2,X3,X4))
p121: mark#(U92(X1,X2,X3,X4)) -> U92#(mark(X1),X2,X3,X4)
p122: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p123: mark#(U93(X1,X2,X3,X4)) -> active#(U93(mark(X1),X2,X3,X4))
p124: mark#(U93(X1,X2,X3,X4)) -> U93#(mark(X1),X2,X3,X4)
p125: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p126: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p127: mark#(take(X1,X2)) -> take#(mark(X1),mark(X2))
p128: mark#(take(X1,X2)) -> mark#(X1)
p129: mark#(take(X1,X2)) -> mark#(X2)
p130: cons#(mark(X1),X2) -> cons#(X1,X2)
p131: cons#(X1,mark(X2)) -> cons#(X1,X2)
p132: cons#(active(X1),X2) -> cons#(X1,X2)
p133: cons#(X1,active(X2)) -> cons#(X1,X2)
p134: U11#(mark(X)) -> U11#(X)
p135: U11#(active(X)) -> U11#(X)
p136: U21#(mark(X)) -> U21#(X)
p137: U21#(active(X)) -> U21#(X)
p138: U31#(mark(X)) -> U31#(X)
p139: U31#(active(X)) -> U31#(X)
p140: U41#(mark(X1),X2) -> U41#(X1,X2)
p141: U41#(X1,mark(X2)) -> U41#(X1,X2)
p142: U41#(active(X1),X2) -> U41#(X1,X2)
p143: U41#(X1,active(X2)) -> U41#(X1,X2)
p144: U42#(mark(X)) -> U42#(X)
p145: U42#(active(X)) -> U42#(X)
p146: isNatIList#(mark(X)) -> isNatIList#(X)
p147: isNatIList#(active(X)) -> isNatIList#(X)
p148: U51#(mark(X1),X2) -> U51#(X1,X2)
p149: U51#(X1,mark(X2)) -> U51#(X1,X2)
p150: U51#(active(X1),X2) -> U51#(X1,X2)
p151: U51#(X1,active(X2)) -> U51#(X1,X2)
p152: U52#(mark(X)) -> U52#(X)
p153: U52#(active(X)) -> U52#(X)
p154: isNatList#(mark(X)) -> isNatList#(X)
p155: isNatList#(active(X)) -> isNatList#(X)
p156: U61#(mark(X1),X2) -> U61#(X1,X2)
p157: U61#(X1,mark(X2)) -> U61#(X1,X2)
p158: U61#(active(X1),X2) -> U61#(X1,X2)
p159: U61#(X1,active(X2)) -> U61#(X1,X2)
p160: U62#(mark(X)) -> U62#(X)
p161: U62#(active(X)) -> U62#(X)
p162: U71#(mark(X1),X2,X3) -> U71#(X1,X2,X3)
p163: U71#(X1,mark(X2),X3) -> U71#(X1,X2,X3)
p164: U71#(X1,X2,mark(X3)) -> U71#(X1,X2,X3)
p165: U71#(active(X1),X2,X3) -> U71#(X1,X2,X3)
p166: U71#(X1,active(X2),X3) -> U71#(X1,X2,X3)
p167: U71#(X1,X2,active(X3)) -> U71#(X1,X2,X3)
p168: U72#(mark(X1),X2) -> U72#(X1,X2)
p169: U72#(X1,mark(X2)) -> U72#(X1,X2)
p170: U72#(active(X1),X2) -> U72#(X1,X2)
p171: U72#(X1,active(X2)) -> U72#(X1,X2)
p172: isNat#(mark(X)) -> isNat#(X)
p173: isNat#(active(X)) -> isNat#(X)
p174: s#(mark(X)) -> s#(X)
p175: s#(active(X)) -> s#(X)
p176: length#(mark(X)) -> length#(X)
p177: length#(active(X)) -> length#(X)
p178: U81#(mark(X)) -> U81#(X)
p179: U81#(active(X)) -> U81#(X)
p180: U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4)
p181: U91#(X1,mark(X2),X3,X4) -> U91#(X1,X2,X3,X4)
p182: U91#(X1,X2,mark(X3),X4) -> U91#(X1,X2,X3,X4)
p183: U91#(X1,X2,X3,mark(X4)) -> U91#(X1,X2,X3,X4)
p184: U91#(active(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4)
p185: U91#(X1,active(X2),X3,X4) -> U91#(X1,X2,X3,X4)
p186: U91#(X1,X2,active(X3),X4) -> U91#(X1,X2,X3,X4)
p187: U91#(X1,X2,X3,active(X4)) -> U91#(X1,X2,X3,X4)
p188: U92#(mark(X1),X2,X3,X4) -> U92#(X1,X2,X3,X4)
p189: U92#(X1,mark(X2),X3,X4) -> U92#(X1,X2,X3,X4)
p190: U92#(X1,X2,mark(X3),X4) -> U92#(X1,X2,X3,X4)
p191: U92#(X1,X2,X3,mark(X4)) -> U92#(X1,X2,X3,X4)
p192: U92#(active(X1),X2,X3,X4) -> U92#(X1,X2,X3,X4)
p193: U92#(X1,active(X2),X3,X4) -> U92#(X1,X2,X3,X4)
p194: U92#(X1,X2,active(X3),X4) -> U92#(X1,X2,X3,X4)
p195: U92#(X1,X2,X3,active(X4)) -> U92#(X1,X2,X3,X4)
p196: U93#(mark(X1),X2,X3,X4) -> U93#(X1,X2,X3,X4)
p197: U93#(X1,mark(X2),X3,X4) -> U93#(X1,X2,X3,X4)
p198: U93#(X1,X2,mark(X3),X4) -> U93#(X1,X2,X3,X4)
p199: U93#(X1,X2,X3,mark(X4)) -> U93#(X1,X2,X3,X4)
p200: U93#(active(X1),X2,X3,X4) -> U93#(X1,X2,X3,X4)
p201: U93#(X1,active(X2),X3,X4) -> U93#(X1,X2,X3,X4)
p202: U93#(X1,X2,active(X3),X4) -> U93#(X1,X2,X3,X4)
p203: U93#(X1,X2,X3,active(X4)) -> U93#(X1,X2,X3,X4)
p204: take#(mark(X1),X2) -> take#(X1,X2)
p205: take#(X1,mark(X2)) -> take#(X1,X2)
p206: take#(active(X1),X2) -> take#(X1,X2)
p207: take#(X1,active(X2)) -> take#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p6, p10, p14, p18, p21, p25, p28, p31, p35, p38, p41, p45, p49, p52, p56, p59, p62, p65, p66, p68, p70, p72, p74, p76, p77, p79, p80, p82, p83, p85, p86, p87, p89, p90, p92, p93, p94, p96, p97, p99, p100, p102, p103, p105, p106, p107, p109, p110, p112, p113, p115, p117, p119, p120, p122, p123, p125, p126, p128, p129}
  {p144, p145}
  {p146, p147}
  {p152, p153}
  {p154, p155}
  {p160, p161}
  {p168, p169, p170, p171}
  {p172, p173}
  {p174, p175}
  {p176, p177}
  {p188, p189, p190, p191, p192, p193, p194, p195}
  {p196, p197, p198, p199, p200, p201, p202, p203}
  {p130, p131, p132, p133}
  {p204, p205, p206, p207}
  {p134, p135}
  {p136, p137}
  {p138, p139}
  {p140, p141, p142, p143}
  {p148, p149, p150, p151}
  {p156, p157, p158, p159}
  {p162, p163, p164, p165, p166, p167}
  {p178, p179}
  {p180, p181, p182, p183, p184, p185, p186, p187}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(take(X1,X2)) -> mark#(X2)
p4: mark#(take(X1,X2)) -> mark#(X1)
p5: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p6: active#(take(s(M),cons(N,IL))) -> mark#(U91(isNatIList(IL),IL,M,N))
p7: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(U93(X1,X2,X3,X4)) -> active#(U93(mark(X1),X2,X3,X4))
p9: active#(take(|0|(),IL)) -> mark#(U81(isNatIList(IL)))
p10: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p11: mark#(U92(X1,X2,X3,X4)) -> active#(U92(mark(X1),X2,X3,X4))
p12: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p13: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p14: mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4))
p15: active#(isNatList(take(V1,V2))) -> mark#(U61(isNat(V1),V2))
p16: mark#(U81(X)) -> mark#(X)
p17: mark#(U81(X)) -> active#(U81(mark(X)))
p18: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p19: mark#(length(X)) -> mark#(X)
p20: mark#(length(X)) -> active#(length(mark(X)))
p21: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p22: mark#(s(X)) -> mark#(X)
p23: mark#(s(X)) -> active#(s(mark(X)))
p24: active#(isNatIList(V)) -> mark#(U31(isNatList(V)))
p25: mark#(isNat(X)) -> active#(isNat(X))
p26: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p27: mark#(U72(X1,X2)) -> mark#(X1)
p28: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p29: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p30: mark#(U71(X1,X2,X3)) -> mark#(X1)
p31: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p32: active#(U93(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p33: mark#(U62(X)) -> mark#(X)
p34: mark#(U62(X)) -> active#(U62(mark(X)))
p35: active#(U92(tt(),IL,M,N)) -> mark#(U93(isNat(N),IL,M,N))
p36: mark#(U61(X1,X2)) -> mark#(X1)
p37: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p38: active#(U91(tt(),IL,M,N)) -> mark#(U92(isNat(M),IL,M,N))
p39: mark#(isNatList(X)) -> active#(isNatList(X))
p40: active#(U72(tt(),L)) -> mark#(s(length(L)))
p41: mark#(U52(X)) -> mark#(X)
p42: mark#(U52(X)) -> active#(U52(mark(X)))
p43: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p44: mark#(U51(X1,X2)) -> mark#(X1)
p45: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p46: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p47: mark#(isNatIList(X)) -> active#(isNatIList(X))
p48: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p49: mark#(U42(X)) -> mark#(X)
p50: mark#(U42(X)) -> active#(U42(mark(X)))
p51: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p52: mark#(U41(X1,X2)) -> mark#(X1)
p53: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p54: mark#(U31(X)) -> mark#(X)
p55: mark#(U31(X)) -> active#(U31(mark(X)))
p56: mark#(U21(X)) -> mark#(X)
p57: mark#(U21(X)) -> active#(U21(mark(X)))
p58: mark#(U11(X)) -> mark#(X)
p59: mark#(U11(X)) -> active#(U11(mark(X)))
p60: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p61: mark#(zeros()) -> active#(zeros())

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      active#_A(x1) = x1
      zeros_A() = (2,1)
      mark#_A(x1) = (2,0)
      cons_A(x1,x2) = (2,1)
      |0|_A() = (0,3)
      take_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (2,1)
      mark_A(x1) = ((1,1),(1,1)) x1 + (0,1)
      s_A(x1) = (1,2)
      U91_A(x1,x2,x3,x4) = (2,1)
      isNatIList_A(x1) = (2,1)
      U93_A(x1,x2,x3,x4) = (2,1)
      U81_A(x1) = (2,1)
      U92_A(x1,x2,x3,x4) = (2,1)
      length_A(x1) = ((0,0),(1,0)) x1 + (2,1)
      U71_A(x1,x2,x3) = (2,1)
      isNatList_A(x1) = ((0,0),(1,0)) x1 + (2,1)
      U61_A(x1,x2) = (2,1)
      isNat_A(x1) = (2,1)
      U51_A(x1,x2) = (2,1)
      U41_A(x1,x2) = (2,1)
      U31_A(x1) = (2,1)
      U21_A(x1) = (2,1)
      U72_A(x1,x2) = (2,1)
      U11_A(x1) = (2,1)
      tt_A() = (0,3)
      U62_A(x1) = (2,1)
      U52_A(x1) = (2,1)
      U42_A(x1) = (2,1)
      active_A(x1) = x1 + (1,4)
      nil_A() = (0,3)

The next rules are strictly ordered:

  p23

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(take(X1,X2)) -> mark#(X2)
p4: mark#(take(X1,X2)) -> mark#(X1)
p5: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p6: active#(take(s(M),cons(N,IL))) -> mark#(U91(isNatIList(IL),IL,M,N))
p7: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(U93(X1,X2,X3,X4)) -> active#(U93(mark(X1),X2,X3,X4))
p9: active#(take(|0|(),IL)) -> mark#(U81(isNatIList(IL)))
p10: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p11: mark#(U92(X1,X2,X3,X4)) -> active#(U92(mark(X1),X2,X3,X4))
p12: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p13: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p14: mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4))
p15: active#(isNatList(take(V1,V2))) -> mark#(U61(isNat(V1),V2))
p16: mark#(U81(X)) -> mark#(X)
p17: mark#(U81(X)) -> active#(U81(mark(X)))
p18: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p19: mark#(length(X)) -> mark#(X)
p20: mark#(length(X)) -> active#(length(mark(X)))
p21: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p22: mark#(s(X)) -> mark#(X)
p23: active#(isNatIList(V)) -> mark#(U31(isNatList(V)))
p24: mark#(isNat(X)) -> active#(isNat(X))
p25: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p26: mark#(U72(X1,X2)) -> mark#(X1)
p27: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p28: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p29: mark#(U71(X1,X2,X3)) -> mark#(X1)
p30: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p31: active#(U93(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p32: mark#(U62(X)) -> mark#(X)
p33: mark#(U62(X)) -> active#(U62(mark(X)))
p34: active#(U92(tt(),IL,M,N)) -> mark#(U93(isNat(N),IL,M,N))
p35: mark#(U61(X1,X2)) -> mark#(X1)
p36: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p37: active#(U91(tt(),IL,M,N)) -> mark#(U92(isNat(M),IL,M,N))
p38: mark#(isNatList(X)) -> active#(isNatList(X))
p39: active#(U72(tt(),L)) -> mark#(s(length(L)))
p40: mark#(U52(X)) -> mark#(X)
p41: mark#(U52(X)) -> active#(U52(mark(X)))
p42: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p43: mark#(U51(X1,X2)) -> mark#(X1)
p44: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p45: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p46: mark#(isNatIList(X)) -> active#(isNatIList(X))
p47: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p48: mark#(U42(X)) -> mark#(X)
p49: mark#(U42(X)) -> active#(U42(mark(X)))
p50: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p51: mark#(U41(X1,X2)) -> mark#(X1)
p52: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p53: mark#(U31(X)) -> mark#(X)
p54: mark#(U31(X)) -> active#(U31(mark(X)))
p55: mark#(U21(X)) -> mark#(X)
p56: mark#(U21(X)) -> active#(U21(mark(X)))
p57: mark#(U11(X)) -> mark#(X)
p58: mark#(U11(X)) -> active#(U11(mark(X)))
p59: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p60: mark#(zeros()) -> active#(zeros())

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p3: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p4: mark#(zeros()) -> active#(zeros())
p5: mark#(U11(X)) -> active#(U11(mark(X)))
p6: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p7: mark#(U11(X)) -> mark#(X)
p8: mark#(U21(X)) -> active#(U21(mark(X)))
p9: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p10: mark#(U21(X)) -> mark#(X)
p11: mark#(U31(X)) -> active#(U31(mark(X)))
p12: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p13: mark#(U31(X)) -> mark#(X)
p14: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p15: active#(U72(tt(),L)) -> mark#(s(length(L)))
p16: mark#(U41(X1,X2)) -> mark#(X1)
p17: mark#(U42(X)) -> active#(U42(mark(X)))
p18: active#(U91(tt(),IL,M,N)) -> mark#(U92(isNat(M),IL,M,N))
p19: mark#(U42(X)) -> mark#(X)
p20: mark#(isNatIList(X)) -> active#(isNatIList(X))
p21: active#(U92(tt(),IL,M,N)) -> mark#(U93(isNat(N),IL,M,N))
p22: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p23: active#(U93(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p24: mark#(U51(X1,X2)) -> mark#(X1)
p25: mark#(U52(X)) -> active#(U52(mark(X)))
p26: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p27: mark#(U52(X)) -> mark#(X)
p28: mark#(isNatList(X)) -> active#(isNatList(X))
p29: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p30: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p31: active#(isNatIList(V)) -> mark#(U31(isNatList(V)))
p32: mark#(U61(X1,X2)) -> mark#(X1)
p33: mark#(U62(X)) -> active#(U62(mark(X)))
p34: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p35: mark#(U62(X)) -> mark#(X)
p36: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p37: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p38: mark#(U71(X1,X2,X3)) -> mark#(X1)
p39: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p40: active#(isNatList(take(V1,V2))) -> mark#(U61(isNat(V1),V2))
p41: mark#(U72(X1,X2)) -> mark#(X1)
p42: mark#(isNat(X)) -> active#(isNat(X))
p43: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p44: mark#(s(X)) -> mark#(X)
p45: mark#(length(X)) -> active#(length(mark(X)))
p46: active#(take(|0|(),IL)) -> mark#(U81(isNatIList(IL)))
p47: mark#(length(X)) -> mark#(X)
p48: mark#(U81(X)) -> active#(U81(mark(X)))
p49: active#(take(s(M),cons(N,IL))) -> mark#(U91(isNatIList(IL),IL,M,N))
p50: mark#(U81(X)) -> mark#(X)
p51: mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4))
p52: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p53: mark#(U92(X1,X2,X3,X4)) -> active#(U92(mark(X1),X2,X3,X4))
p54: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p55: mark#(U93(X1,X2,X3,X4)) -> active#(U93(mark(X1),X2,X3,X4))
p56: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p57: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p58: mark#(take(X1,X2)) -> mark#(X1)
p59: mark#(take(X1,X2)) -> mark#(X2)
p60: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      active#_A(x1) = x1
      zeros_A() = (2,1)
      mark#_A(x1) = (2,0)
      cons_A(x1,x2) = (1,1)
      |0|_A() = (1,1)
      mark_A(x1) = (1,1)
      U41_A(x1,x2) = (2,1)
      tt_A() = (1,1)
      U42_A(x1) = (2,1)
      isNatIList_A(x1) = (2,1)
      U11_A(x1) = (1,1)
      U51_A(x1,x2) = (2,1)
      U52_A(x1) = (2,1)
      isNatList_A(x1) = (2,1)
      U21_A(x1) = (1,1)
      U61_A(x1,x2) = (2,1)
      U62_A(x1) = (1,1)
      U31_A(x1) = (1,1)
      U71_A(x1,x2,x3) = (2,1)
      U72_A(x1,x2) = (2,1)
      isNat_A(x1) = (2,1)
      s_A(x1) = (1,1)
      length_A(x1) = (2,1)
      U91_A(x1,x2,x3,x4) = (2,1)
      U92_A(x1,x2,x3,x4) = (2,1)
      U93_A(x1,x2,x3,x4) = (2,1)
      take_A(x1,x2) = (2,1)
      U81_A(x1) = (1,1)
      active_A(x1) = x1 + (1,0)
      nil_A() = (1,1)

The next rules are strictly ordered:

  p2, p5, p8, p11, p33, p48

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p3: mark#(zeros()) -> active#(zeros())
p4: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p5: mark#(U11(X)) -> mark#(X)
p6: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p7: mark#(U21(X)) -> mark#(X)
p8: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p9: mark#(U31(X)) -> mark#(X)
p10: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p11: active#(U72(tt(),L)) -> mark#(s(length(L)))
p12: mark#(U41(X1,X2)) -> mark#(X1)
p13: mark#(U42(X)) -> active#(U42(mark(X)))
p14: active#(U91(tt(),IL,M,N)) -> mark#(U92(isNat(M),IL,M,N))
p15: mark#(U42(X)) -> mark#(X)
p16: mark#(isNatIList(X)) -> active#(isNatIList(X))
p17: active#(U92(tt(),IL,M,N)) -> mark#(U93(isNat(N),IL,M,N))
p18: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p19: active#(U93(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p20: mark#(U51(X1,X2)) -> mark#(X1)
p21: mark#(U52(X)) -> active#(U52(mark(X)))
p22: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p23: mark#(U52(X)) -> mark#(X)
p24: mark#(isNatList(X)) -> active#(isNatList(X))
p25: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p26: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p27: active#(isNatIList(V)) -> mark#(U31(isNatList(V)))
p28: mark#(U61(X1,X2)) -> mark#(X1)
p29: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p30: mark#(U62(X)) -> mark#(X)
p31: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p32: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p33: mark#(U71(X1,X2,X3)) -> mark#(X1)
p34: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p35: active#(isNatList(take(V1,V2))) -> mark#(U61(isNat(V1),V2))
p36: mark#(U72(X1,X2)) -> mark#(X1)
p37: mark#(isNat(X)) -> active#(isNat(X))
p38: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p39: mark#(s(X)) -> mark#(X)
p40: mark#(length(X)) -> active#(length(mark(X)))
p41: active#(take(|0|(),IL)) -> mark#(U81(isNatIList(IL)))
p42: mark#(length(X)) -> mark#(X)
p43: active#(take(s(M),cons(N,IL))) -> mark#(U91(isNatIList(IL),IL,M,N))
p44: mark#(U81(X)) -> mark#(X)
p45: mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4))
p46: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p47: mark#(U92(X1,X2,X3,X4)) -> active#(U92(mark(X1),X2,X3,X4))
p48: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p49: mark#(U93(X1,X2,X3,X4)) -> active#(U93(mark(X1),X2,X3,X4))
p50: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p51: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p52: mark#(take(X1,X2)) -> mark#(X1)
p53: mark#(take(X1,X2)) -> mark#(X2)
p54: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(take(X1,X2)) -> mark#(X2)
p4: mark#(take(X1,X2)) -> mark#(X1)
p5: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p6: active#(take(s(M),cons(N,IL))) -> mark#(U91(isNatIList(IL),IL,M,N))
p7: mark#(U93(X1,X2,X3,X4)) -> mark#(X1)
p8: mark#(U93(X1,X2,X3,X4)) -> active#(U93(mark(X1),X2,X3,X4))
p9: active#(take(|0|(),IL)) -> mark#(U81(isNatIList(IL)))
p10: mark#(U92(X1,X2,X3,X4)) -> mark#(X1)
p11: mark#(U92(X1,X2,X3,X4)) -> active#(U92(mark(X1),X2,X3,X4))
p12: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p13: mark#(U91(X1,X2,X3,X4)) -> mark#(X1)
p14: mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4))
p15: active#(isNatList(take(V1,V2))) -> mark#(U61(isNat(V1),V2))
p16: mark#(U81(X)) -> mark#(X)
p17: mark#(length(X)) -> mark#(X)
p18: mark#(length(X)) -> active#(length(mark(X)))
p19: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p20: mark#(s(X)) -> mark#(X)
p21: mark#(isNat(X)) -> active#(isNat(X))
p22: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p23: mark#(U72(X1,X2)) -> mark#(X1)
p24: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p25: active#(isNatIList(V)) -> mark#(U31(isNatList(V)))
p26: mark#(U71(X1,X2,X3)) -> mark#(X1)
p27: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p28: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p29: mark#(U62(X)) -> mark#(X)
p30: mark#(U61(X1,X2)) -> mark#(X1)
p31: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p32: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p33: mark#(isNatList(X)) -> active#(isNatList(X))
p34: active#(U93(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p35: mark#(U52(X)) -> mark#(X)
p36: mark#(U52(X)) -> active#(U52(mark(X)))
p37: active#(U92(tt(),IL,M,N)) -> mark#(U93(isNat(N),IL,M,N))
p38: mark#(U51(X1,X2)) -> mark#(X1)
p39: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p40: active#(U91(tt(),IL,M,N)) -> mark#(U92(isNat(M),IL,M,N))
p41: mark#(isNatIList(X)) -> active#(isNatIList(X))
p42: active#(U72(tt(),L)) -> mark#(s(length(L)))
p43: mark#(U42(X)) -> mark#(X)
p44: mark#(U42(X)) -> active#(U42(mark(X)))
p45: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p46: mark#(U41(X1,X2)) -> mark#(X1)
p47: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p48: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p49: mark#(U31(X)) -> mark#(X)
p50: mark#(U21(X)) -> mark#(X)
p51: mark#(U11(X)) -> mark#(X)
p52: mark#(zeros()) -> active#(zeros())
p53: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p54: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      active#_A(x1) = ((0,1),(0,1)) x1
      zeros_A() = (1,3)
      mark#_A(x1) = ((1,0),(1,0)) x1 + (2,2)
      cons_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (0,1)
      |0|_A() = (0,1)
      take_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (2,4)
      mark_A(x1) = ((1,0),(1,0)) x1 + (0,1)
      s_A(x1) = ((1,0),(1,0)) x1 + (0,1)
      U91_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (1,3)
      isNatIList_A(x1) = (1,3)
      U93_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (1,3)
      U81_A(x1) = ((1,0),(1,0)) x1 + (1,1)
      U92_A(x1,x2,x3,x4) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (1,3)
      length_A(x1) = ((1,0),(1,0)) x1 + (2,4)
      U71_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (1,3)
      isNatList_A(x1) = (1,3)
      U61_A(x1,x2) = ((1,0),(1,0)) x1 + (0,2)
      isNat_A(x1) = (1,3)
      U51_A(x1,x2) = ((1,0),(1,0)) x1 + (0,2)
      U41_A(x1,x2) = ((1,0),(1,0)) x1 + (0,2)
      U72_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (1,3)
      U31_A(x1) = ((1,0),(1,0)) x1 + (0,1)
      U21_A(x1) = ((1,0),(1,0)) x1 + (0,1)
      U62_A(x1) = ((1,0),(1,0)) x1 + (0,1)
      U11_A(x1) = ((1,0),(1,0)) x1 + (0,1)
      tt_A() = (1,1)
      U52_A(x1) = ((1,0),(1,0)) x1 + (0,1)
      U42_A(x1) = x1
      active_A(x1) = ((1,0),(1,0)) x1 + (0,1)
      nil_A() = (2,1)

The next rules are strictly ordered:

  p3, p4, p7, p10, p13, p16, p17, p23, p26, p36, p44

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p4: active#(take(s(M),cons(N,IL))) -> mark#(U91(isNatIList(IL),IL,M,N))
p5: mark#(U93(X1,X2,X3,X4)) -> active#(U93(mark(X1),X2,X3,X4))
p6: active#(take(|0|(),IL)) -> mark#(U81(isNatIList(IL)))
p7: mark#(U92(X1,X2,X3,X4)) -> active#(U92(mark(X1),X2,X3,X4))
p8: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p9: mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4))
p10: active#(isNatList(take(V1,V2))) -> mark#(U61(isNat(V1),V2))
p11: mark#(length(X)) -> active#(length(mark(X)))
p12: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p13: mark#(s(X)) -> mark#(X)
p14: mark#(isNat(X)) -> active#(isNat(X))
p15: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p16: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p17: active#(isNatIList(V)) -> mark#(U31(isNatList(V)))
p18: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p19: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p20: mark#(U62(X)) -> mark#(X)
p21: mark#(U61(X1,X2)) -> mark#(X1)
p22: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p23: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p24: mark#(isNatList(X)) -> active#(isNatList(X))
p25: active#(U93(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p26: mark#(U52(X)) -> mark#(X)
p27: active#(U92(tt(),IL,M,N)) -> mark#(U93(isNat(N),IL,M,N))
p28: mark#(U51(X1,X2)) -> mark#(X1)
p29: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p30: active#(U91(tt(),IL,M,N)) -> mark#(U92(isNat(M),IL,M,N))
p31: mark#(isNatIList(X)) -> active#(isNatIList(X))
p32: active#(U72(tt(),L)) -> mark#(s(length(L)))
p33: mark#(U42(X)) -> mark#(X)
p34: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p35: mark#(U41(X1,X2)) -> mark#(X1)
p36: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p37: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p38: mark#(U31(X)) -> mark#(X)
p39: mark#(U21(X)) -> mark#(X)
p40: mark#(U11(X)) -> mark#(X)
p41: mark#(zeros()) -> active#(zeros())
p42: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p43: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(zeros()) -> active#(zeros())
p4: mark#(U11(X)) -> mark#(X)
p5: mark#(U21(X)) -> mark#(X)
p6: mark#(U31(X)) -> mark#(X)
p7: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p8: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p9: mark#(U41(X1,X2)) -> mark#(X1)
p10: mark#(U42(X)) -> mark#(X)
p11: mark#(isNatIList(X)) -> active#(isNatIList(X))
p12: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p13: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p14: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p15: mark#(U51(X1,X2)) -> mark#(X1)
p16: mark#(U52(X)) -> mark#(X)
p17: mark#(isNatList(X)) -> active#(isNatList(X))
p18: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p19: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p20: active#(U72(tt(),L)) -> mark#(s(length(L)))
p21: mark#(U61(X1,X2)) -> mark#(X1)
p22: mark#(U62(X)) -> mark#(X)
p23: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p24: active#(U91(tt(),IL,M,N)) -> mark#(U92(isNat(M),IL,M,N))
p25: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p26: active#(U92(tt(),IL,M,N)) -> mark#(U93(isNat(N),IL,M,N))
p27: mark#(isNat(X)) -> active#(isNat(X))
p28: active#(U93(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p29: mark#(s(X)) -> mark#(X)
p30: mark#(length(X)) -> active#(length(mark(X)))
p31: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p32: mark#(U91(X1,X2,X3,X4)) -> active#(U91(mark(X1),X2,X3,X4))
p33: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p34: mark#(U92(X1,X2,X3,X4)) -> active#(U92(mark(X1),X2,X3,X4))
p35: active#(isNatIList(V)) -> mark#(U31(isNatList(V)))
p36: mark#(U93(X1,X2,X3,X4)) -> active#(U93(mark(X1),X2,X3,X4))
p37: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p38: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p39: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p40: active#(isNatList(take(V1,V2))) -> mark#(U61(isNat(V1),V2))
p41: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p42: active#(take(|0|(),IL)) -> mark#(U81(isNatIList(IL)))
p43: active#(take(s(M),cons(N,IL))) -> mark#(U91(isNatIList(IL),IL,M,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      active#_A(x1) = ((0,1),(0,1)) x1
      zeros_A() = (1,2)
      mark#_A(x1) = ((1,1),(1,1)) x1
      cons_A(x1,x2) = ((0,0),(1,1)) x1 + (1,0)
      |0|_A() = (0,0)
      U11_A(x1) = ((0,0),(1,1)) x1
      U21_A(x1) = ((1,1),(0,0)) x1
      U31_A(x1) = ((0,0),(1,1)) x1
      U41_A(x1,x2) = ((1,1),(0,0)) x1
      mark_A(x1) = ((0,1),(1,0)) x1
      tt_A() = (0,0)
      U42_A(x1) = ((1,1),(0,0)) x1
      isNatIList_A(x1) = (0,0)
      U51_A(x1,x2) = ((1,1),(0,0)) x1
      U52_A(x1) = ((0,0),(1,1)) x1
      isNatList_A(x1) = (0,0)
      U61_A(x1,x2) = ((1,1),(0,0)) x1
      U62_A(x1) = ((0,0),(1,1)) x1
      U71_A(x1,x2,x3) = (0,0)
      U72_A(x1,x2) = (0,0)
      isNat_A(x1) = (0,0)
      s_A(x1) = ((1,1),(0,0)) x1
      length_A(x1) = (0,0)
      U91_A(x1,x2,x3,x4) = ((1,1),(0,0)) x1 + ((0,0),(1,1)) x4 + (1,6)
      U92_A(x1,x2,x3,x4) = ((0,0),(1,1)) x4 + (1,4)
      U93_A(x1,x2,x3,x4) = ((0,0),(1,1)) x4 + (1,2)
      take_A(x1,x2) = ((0,0),(1,1)) x2 + (1,7)
      U81_A(x1) = (1,1)
      active_A(x1) = ((0,1),(1,0)) x1
      nil_A() = (1,1)

The next rules are strictly ordered:

  p1, p2, p3, p24, p26, p28, p32, p34, p36, p38, p42, p43

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X)) -> mark#(X)
p2: mark#(U21(X)) -> mark#(X)
p3: mark#(U31(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p5: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p6: mark#(U41(X1,X2)) -> mark#(X1)
p7: mark#(U42(X)) -> mark#(X)
p8: mark#(isNatIList(X)) -> active#(isNatIList(X))
p9: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p10: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p11: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p12: mark#(U51(X1,X2)) -> mark#(X1)
p13: mark#(U52(X)) -> mark#(X)
p14: mark#(isNatList(X)) -> active#(isNatList(X))
p15: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p16: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p17: active#(U72(tt(),L)) -> mark#(s(length(L)))
p18: mark#(U61(X1,X2)) -> mark#(X1)
p19: mark#(U62(X)) -> mark#(X)
p20: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p21: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p22: mark#(isNat(X)) -> active#(isNat(X))
p23: mark#(s(X)) -> mark#(X)
p24: mark#(length(X)) -> active#(length(mark(X)))
p25: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p26: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p27: active#(isNatIList(V)) -> mark#(U31(isNatList(V)))
p28: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p29: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p30: active#(isNatList(take(V1,V2))) -> mark#(U61(isNat(V1),V2))
p31: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U11(X)) -> mark#(X)
p2: mark#(length(X)) -> active#(length(mark(X)))
p3: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p4: mark#(s(X)) -> mark#(X)
p5: mark#(isNat(X)) -> active#(isNat(X))
p6: active#(isNatList(take(V1,V2))) -> mark#(U61(isNat(V1),V2))
p7: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p8: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p9: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p10: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p11: mark#(U62(X)) -> mark#(X)
p12: mark#(U61(X1,X2)) -> mark#(X1)
p13: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p14: active#(isNatIList(V)) -> mark#(U31(isNatList(V)))
p15: mark#(isNatList(X)) -> active#(isNatList(X))
p16: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p17: mark#(U52(X)) -> mark#(X)
p18: mark#(U51(X1,X2)) -> mark#(X1)
p19: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p20: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p21: mark#(isNatIList(X)) -> active#(isNatIList(X))
p22: active#(U72(tt(),L)) -> mark#(s(length(L)))
p23: mark#(U42(X)) -> mark#(X)
p24: mark#(U41(X1,X2)) -> mark#(X1)
p25: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p26: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p27: mark#(U31(X)) -> mark#(X)
p28: mark#(U21(X)) -> mark#(X)
p29: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p30: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p31: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = ((0,1),(0,0)) x1
      U11_A(x1) = x1 + (1,1)
      length_A(x1) = ((0,1),(0,1)) x1 + (1,1)
      active#_A(x1) = ((0,1),(0,0)) x1
      mark_A(x1) = ((1,1),(0,1)) x1 + (1,0)
      cons_A(x1,x2) = x1 + x2 + (1,0)
      U71_A(x1,x2,x3) = ((0,1),(0,1)) x2 + (1,1)
      isNatList_A(x1) = x1 + (1,0)
      s_A(x1) = x1 + (1,0)
      isNat_A(x1) = x1 + (1,0)
      take_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (3,3)
      U61_A(x1,x2) = x1 + x2 + (1,3)
      U72_A(x1,x2) = ((0,1),(0,1)) x2 + (1,1)
      U51_A(x1,x2) = x1 + x2 + (1,0)
      isNatIList_A(x1) = ((0,1),(0,1)) x1 + (1,2)
      U41_A(x1,x2) = ((0,1),(0,1)) x1 + x2 + (1,2)
      U62_A(x1) = x1 + (1,1)
      U31_A(x1) = ((0,1),(0,1)) x1 + (1,1)
      U21_A(x1) = x1 + (1,0)
      U52_A(x1) = x1 + (1,0)
      tt_A() = (1,0)
      U42_A(x1) = x1 + (1,0)
      active_A(x1) = ((1,1),(0,1)) x1 + (1,0)
      zeros_A() = (1,1)
      |0|_A() = (1,0)
      U81_A(x1) = ((0,1),(0,1)) x1 + (1,1)
      nil_A() = (1,1)
      U91_A(x1,x2,x3,x4) = x2 + x3 + ((0,1),(0,1)) x4 + (1,3)
      U92_A(x1,x2,x3,x4) = x2 + x3 + ((0,1),(0,1)) x4 + (1,3)
      U93_A(x1,x2,x3,x4) = ((0,1),(0,0)) x1 + x2 + x3 + x4 + (1,3)

The next rules are strictly ordered:

  p1, p11, p12, p14, p24, p27

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p3: mark#(s(X)) -> mark#(X)
p4: mark#(isNat(X)) -> active#(isNat(X))
p5: active#(isNatList(take(V1,V2))) -> mark#(U61(isNat(V1),V2))
p6: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p7: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p8: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p9: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p10: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p11: mark#(isNatList(X)) -> active#(isNatList(X))
p12: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p13: mark#(U52(X)) -> mark#(X)
p14: mark#(U51(X1,X2)) -> mark#(X1)
p15: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p16: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p17: mark#(isNatIList(X)) -> active#(isNatIList(X))
p18: active#(U72(tt(),L)) -> mark#(s(length(L)))
p19: mark#(U42(X)) -> mark#(X)
p20: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p21: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p22: mark#(U21(X)) -> mark#(X)
p23: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p24: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p25: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p5: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p6: mark#(U42(X)) -> mark#(X)
p7: mark#(isNatIList(X)) -> active#(isNatIList(X))
p8: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p9: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p10: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p11: mark#(U51(X1,X2)) -> mark#(X1)
p12: mark#(U52(X)) -> mark#(X)
p13: mark#(isNatList(X)) -> active#(isNatList(X))
p14: active#(U72(tt(),L)) -> mark#(s(length(L)))
p15: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p16: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p17: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p18: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p19: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p20: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p21: mark#(isNat(X)) -> active#(isNat(X))
p22: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p23: mark#(s(X)) -> mark#(X)
p24: active#(isNatList(take(V1,V2))) -> mark#(U61(isNat(V1),V2))
p25: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = x1
      length_A(x1) = x1 + (2,0)
      active#_A(x1) = x1
      mark_A(x1) = ((1,0),(1,1)) x1
      U41_A(x1,x2) = x1 + x2 + (1,0)
      tt_A() = (1,0)
      U42_A(x1) = x1
      isNatIList_A(x1) = x1 + (2,1)
      U21_A(x1) = x1
      U51_A(x1,x2) = x1 + x2 + (1,0)
      U52_A(x1) = x1
      isNatList_A(x1) = ((1,0),(1,0)) x1 + (2,0)
      U61_A(x1,x2) = x1 + x2 + (4,0)
      U62_A(x1) = (5,0)
      U71_A(x1,x2,x3) = x2 + x3 + (2,0)
      U72_A(x1,x2) = x1 + x2 + (1,0)
      isNat_A(x1) = x1 + (1,1)
      s_A(x1) = x1
      U11_A(x1) = (3,0)
      cons_A(x1,x2) = x1 + x2
      take_A(x1,x2) = x1 + x2 + (3,1)
      active_A(x1) = ((1,0),(1,0)) x1
      zeros_A() = (1,1)
      |0|_A() = (0,1)
      U31_A(x1) = (1,1)
      U81_A(x1) = (2,1)
      nil_A() = (1,1)
      U91_A(x1,x2,x3,x4) = x2 + x3 + x4 + (3,0)
      U92_A(x1,x2,x3,x4) = x2 + x3 + x4 + (3,0)
      U93_A(x1,x2,x3,x4) = x2 + x3 + x4 + (3,0)

The next rules are strictly ordered:

  p11

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p5: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p6: mark#(U42(X)) -> mark#(X)
p7: mark#(isNatIList(X)) -> active#(isNatIList(X))
p8: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p9: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p10: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p11: mark#(U52(X)) -> mark#(X)
p12: mark#(isNatList(X)) -> active#(isNatList(X))
p13: active#(U72(tt(),L)) -> mark#(s(length(L)))
p14: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p15: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p16: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p17: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p18: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p19: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p20: mark#(isNat(X)) -> active#(isNat(X))
p21: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p22: mark#(s(X)) -> mark#(X)
p23: active#(isNatList(take(V1,V2))) -> mark#(U61(isNat(V1),V2))
p24: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p3: mark#(s(X)) -> mark#(X)
p4: mark#(isNat(X)) -> active#(isNat(X))
p5: active#(isNatList(take(V1,V2))) -> mark#(U61(isNat(V1),V2))
p6: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p7: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p8: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p9: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p10: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p11: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p12: mark#(isNatList(X)) -> active#(isNatList(X))
p13: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p14: mark#(U52(X)) -> mark#(X)
p15: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p16: active#(U72(tt(),L)) -> mark#(s(length(L)))
p17: mark#(isNatIList(X)) -> active#(isNatIList(X))
p18: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p19: mark#(U42(X)) -> mark#(X)
p20: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p21: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p22: mark#(U21(X)) -> mark#(X)
p23: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p24: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = ((0,1),(0,0)) x1
      length_A(x1) = ((0,1),(0,1)) x1
      active#_A(x1) = x1
      mark_A(x1) = x1
      cons_A(x1,x2) = x2 + (1,0)
      U71_A(x1,x2,x3) = ((0,1),(0,1)) x2
      isNatList_A(x1) = ((0,1),(0,1)) x1 + (1,1)
      s_A(x1) = x1
      isNat_A(x1) = (1,1)
      take_A(x1,x2) = ((0,1),(0,0)) x1 + x2 + (5,4)
      U61_A(x1,x2) = ((1,1),(1,1)) x1 + (1,1)
      U72_A(x1,x2) = ((0,1),(0,1)) x2
      U51_A(x1,x2) = ((0,1),(0,1)) x2 + (1,1)
      isNatIList_A(x1) = (3,3)
      U41_A(x1,x2) = ((1,1),(1,1)) x1 + (1,1)
      U21_A(x1) = x1
      U11_A(x1) = (1,1)
      U52_A(x1) = ((0,1),(0,1)) x1
      tt_A() = (1,1)
      U42_A(x1) = x1 + (3,0)
      U62_A(x1) = (1,3)
      active_A(x1) = x1
      zeros_A() = (1,0)
      |0|_A() = (0,0)
      U31_A(x1) = (1,1)
      U81_A(x1) = ((1,0),(1,0)) x1 + (1,1)
      nil_A() = (1,0)
      U91_A(x1,x2,x3,x4) = x1 + x2 + ((0,1),(0,0)) x3 + (2,4)
      U92_A(x1,x2,x3,x4) = ((1,1),(0,0)) x1 + x2 + ((0,1),(0,0)) x3 + (1,4)
      U93_A(x1,x2,x3,x4) = ((0,1),(1,0)) x1 + x2 + ((0,1),(0,0)) x3 + (1,3)

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p3: mark#(s(X)) -> mark#(X)
p4: mark#(isNat(X)) -> active#(isNat(X))
p5: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p6: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p7: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p8: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p9: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p10: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p11: mark#(isNatList(X)) -> active#(isNatList(X))
p12: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p13: mark#(U52(X)) -> mark#(X)
p14: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p15: active#(U72(tt(),L)) -> mark#(s(length(L)))
p16: mark#(isNatIList(X)) -> active#(isNatIList(X))
p17: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p18: mark#(U42(X)) -> mark#(X)
p19: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p20: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p21: mark#(U21(X)) -> mark#(X)
p22: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p23: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p5: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p6: mark#(U42(X)) -> mark#(X)
p7: mark#(isNatIList(X)) -> active#(isNatIList(X))
p8: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p9: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p10: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p11: mark#(U52(X)) -> mark#(X)
p12: mark#(isNatList(X)) -> active#(isNatList(X))
p13: active#(U72(tt(),L)) -> mark#(s(length(L)))
p14: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p15: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p16: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p17: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p18: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p19: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p20: mark#(isNat(X)) -> active#(isNat(X))
p21: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p22: mark#(s(X)) -> mark#(X)
p23: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      length_A(x1) = (1,0)
      active#_A(x1) = x1
      mark_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      U41_A(x1,x2) = x1 + ((0,1),(0,1)) x2 + (1,0)
      tt_A() = (0,0)
      U42_A(x1) = x1 + (1,0)
      isNatIList_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      U21_A(x1) = x1 + (1,0)
      U51_A(x1,x2) = x1 + ((0,1),(0,1)) x2 + (2,1)
      U52_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      isNatList_A(x1) = ((0,1),(0,1)) x1 + (2,1)
      U61_A(x1,x2) = (2,2)
      U62_A(x1) = (2,1)
      U71_A(x1,x2,x3) = (1,0)
      U72_A(x1,x2) = (1,0)
      isNat_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      s_A(x1) = x1 + (1,0)
      U11_A(x1) = (1,0)
      cons_A(x1,x2) = x1 + ((0,1),(0,1)) x2 + (1,0)
      active_A(x1) = x1
      zeros_A() = (1,0)
      |0|_A() = (0,0)
      U31_A(x1) = (1,0)
      U81_A(x1) = (2,1)
      nil_A() = (1,1)
      U91_A(x1,x2,x3,x4) = ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + ((0,1),(0,1)) x4 + (2,1)
      U92_A(x1,x2,x3,x4) = ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + ((0,1),(0,1)) x4 + (2,1)
      U93_A(x1,x2,x3,x4) = ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + ((0,1),(0,1)) x4 + (2,1)
      take_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (2,1)

The next rules are strictly ordered:

  p14

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p5: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p6: mark#(U42(X)) -> mark#(X)
p7: mark#(isNatIList(X)) -> active#(isNatIList(X))
p8: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p9: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p10: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p11: mark#(U52(X)) -> mark#(X)
p12: mark#(isNatList(X)) -> active#(isNatList(X))
p13: active#(U72(tt(),L)) -> mark#(s(length(L)))
p14: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p15: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p16: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p17: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p18: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p19: mark#(isNat(X)) -> active#(isNat(X))
p20: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p21: mark#(s(X)) -> mark#(X)
p22: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p3: mark#(s(X)) -> mark#(X)
p4: mark#(isNat(X)) -> active#(isNat(X))
p5: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p6: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p7: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p8: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p9: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p10: mark#(isNatList(X)) -> active#(isNatList(X))
p11: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p12: mark#(U52(X)) -> mark#(X)
p13: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p14: active#(U72(tt(),L)) -> mark#(s(length(L)))
p15: mark#(isNatIList(X)) -> active#(isNatIList(X))
p16: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p17: mark#(U42(X)) -> mark#(X)
p18: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p19: active#(U61(tt(),V2)) -> mark#(U62(isNatIList(V2)))
p20: mark#(U21(X)) -> mark#(X)
p21: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p22: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = (1,0)
      length_A(x1) = (1,1)
      active#_A(x1) = ((0,1),(0,0)) x1
      mark_A(x1) = (1,1)
      cons_A(x1,x2) = (1,1)
      U71_A(x1,x2,x3) = (1,1)
      isNatList_A(x1) = (1,1)
      s_A(x1) = (1,1)
      isNat_A(x1) = (1,1)
      U51_A(x1,x2) = (1,1)
      U72_A(x1,x2) = (1,1)
      isNatIList_A(x1) = (1,1)
      U41_A(x1,x2) = (1,1)
      U21_A(x1) = (1,1)
      U11_A(x1) = (1,1)
      U52_A(x1) = (1,1)
      tt_A() = (1,1)
      U42_A(x1) = (1,1)
      U61_A(x1,x2) = (1,2)
      U62_A(x1) = (1,1)
      active_A(x1) = (1,1)
      zeros_A() = (1,1)
      |0|_A() = (1,1)
      U31_A(x1) = (1,1)
      U81_A(x1) = (1,1)
      nil_A() = (1,1)
      U91_A(x1,x2,x3,x4) = (1,1)
      U92_A(x1,x2,x3,x4) = (1,1)
      U93_A(x1,x2,x3,x4) = (1,1)
      take_A(x1,x2) = (1,1)

The next rules are strictly ordered:

  p19

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p3: mark#(s(X)) -> mark#(X)
p4: mark#(isNat(X)) -> active#(isNat(X))
p5: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p6: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p7: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p8: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p9: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p10: mark#(isNatList(X)) -> active#(isNatList(X))
p11: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p12: mark#(U52(X)) -> mark#(X)
p13: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p14: active#(U72(tt(),L)) -> mark#(s(length(L)))
p15: mark#(isNatIList(X)) -> active#(isNatIList(X))
p16: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p17: mark#(U42(X)) -> mark#(X)
p18: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p19: mark#(U21(X)) -> mark#(X)
p20: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p21: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p5: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p6: mark#(U42(X)) -> mark#(X)
p7: mark#(isNatIList(X)) -> active#(isNatIList(X))
p8: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p9: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p10: active#(U72(tt(),L)) -> mark#(s(length(L)))
p11: mark#(U52(X)) -> mark#(X)
p12: mark#(isNatList(X)) -> active#(isNatList(X))
p13: active#(isNat(length(V1))) -> mark#(U11(isNatList(V1)))
p14: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p15: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p16: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p17: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p18: mark#(isNat(X)) -> active#(isNat(X))
p19: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p20: mark#(s(X)) -> mark#(X)
p21: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = ((0,1),(0,0)) x1 + (1,0)
      length_A(x1) = ((0,1),(0,1)) x1 + (3,2)
      active#_A(x1) = x1
      mark_A(x1) = x1
      U41_A(x1,x2) = ((0,1),(0,1)) x2 + (2,1)
      tt_A() = (2,1)
      U42_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      isNatIList_A(x1) = ((0,1),(0,1)) x1 + (2,1)
      U21_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      U51_A(x1,x2) = (2,1)
      U52_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      isNatList_A(x1) = (2,1)
      U71_A(x1,x2,x3) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (2,1)
      U72_A(x1,x2) = ((0,1),(0,1)) x2 + (3,2)
      isNat_A(x1) = (4,3)
      s_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      U11_A(x1) = ((0,1),(0,1)) x1 + (2,1)
      cons_A(x1,x2) = ((0,1),(0,1)) x2 + (1,0)
      active_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      zeros_A() = (2,1)
      |0|_A() = (2,1)
      U31_A(x1) = (2,1)
      U61_A(x1,x2) = (2,1)
      U62_A(x1) = (2,1)
      U81_A(x1) = (2,1)
      nil_A() = (2,1)
      U91_A(x1,x2,x3,x4) = ((0,1),(0,1)) x2 + (2,1)
      U92_A(x1,x2,x3,x4) = ((0,1),(0,1)) x2 + (2,1)
      U93_A(x1,x2,x3,x4) = ((0,1),(0,1)) x2 + (2,1)
      take_A(x1,x2) = ((0,1),(0,1)) x2 + (2,1)

The next rules are strictly ordered:

  p13

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p5: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p6: mark#(U42(X)) -> mark#(X)
p7: mark#(isNatIList(X)) -> active#(isNatIList(X))
p8: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p9: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p10: active#(U72(tt(),L)) -> mark#(s(length(L)))
p11: mark#(U52(X)) -> mark#(X)
p12: mark#(isNatList(X)) -> active#(isNatList(X))
p13: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p14: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p15: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p16: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p17: mark#(isNat(X)) -> active#(isNat(X))
p18: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p19: mark#(s(X)) -> mark#(X)
p20: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p3: mark#(s(X)) -> mark#(X)
p4: mark#(isNat(X)) -> active#(isNat(X))
p5: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p6: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p7: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNat(V1),V2))
p8: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p9: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p10: mark#(isNatList(X)) -> active#(isNatList(X))
p11: active#(U72(tt(),L)) -> mark#(s(length(L)))
p12: mark#(U52(X)) -> mark#(X)
p13: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p14: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p15: mark#(isNatIList(X)) -> active#(isNatIList(X))
p16: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p17: mark#(U42(X)) -> mark#(X)
p18: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p19: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p20: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = ((1,1),(0,0)) x1
      length_A(x1) = (0,0)
      active#_A(x1) = ((1,1),(0,0)) x1
      mark_A(x1) = ((0,0),(1,1)) x1
      cons_A(x1,x2) = x2 + (1,1)
      U71_A(x1,x2,x3) = (0,0)
      isNatList_A(x1) = (0,2)
      s_A(x1) = ((1,1),(0,0)) x1
      isNat_A(x1) = (0,2)
      U51_A(x1,x2) = ((1,1),(0,0)) x1
      U72_A(x1,x2) = (0,0)
      isNatIList_A(x1) = x1 + (1,3)
      U41_A(x1,x2) = x2 + (1,3)
      U21_A(x1) = ((1,1),(0,0)) x1
      tt_A() = (0,2)
      U52_A(x1) = ((0,0),(1,1)) x1
      U42_A(x1) = ((1,1),(0,0)) x1
      active_A(x1) = ((0,0),(1,1)) x1
      zeros_A() = (2,1)
      |0|_A() = (0,0)
      U11_A(x1) = (0,2)
      U31_A(x1) = ((1,1),(0,0)) x1 + (1,1)
      U61_A(x1,x2) = (0,2)
      U62_A(x1) = (1,1)
      U81_A(x1) = (0,1)
      nil_A() = (0,0)
      U91_A(x1,x2,x3,x4) = x2 + (0,3)
      U92_A(x1,x2,x3,x4) = x2 + (0,3)
      U93_A(x1,x2,x3,x4) = x2 + (1,2)
      take_A(x1,x2) = ((1,1),(0,0)) x2 + (0,1)

The next rules are strictly ordered:

  p7

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p3: mark#(s(X)) -> mark#(X)
p4: mark#(isNat(X)) -> active#(isNat(X))
p5: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p6: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p7: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p8: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p9: mark#(isNatList(X)) -> active#(isNatList(X))
p10: active#(U72(tt(),L)) -> mark#(s(length(L)))
p11: mark#(U52(X)) -> mark#(X)
p12: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p13: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p14: mark#(isNatIList(X)) -> active#(isNatIList(X))
p15: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p16: mark#(U42(X)) -> mark#(X)
p17: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p18: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p19: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p5: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p6: mark#(U42(X)) -> mark#(X)
p7: mark#(isNatIList(X)) -> active#(isNatIList(X))
p8: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p9: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p10: active#(U72(tt(),L)) -> mark#(s(length(L)))
p11: mark#(U52(X)) -> mark#(X)
p12: mark#(isNatList(X)) -> active#(isNatList(X))
p13: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p14: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p15: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p16: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p17: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p18: mark#(isNat(X)) -> active#(isNat(X))
p19: mark#(s(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = (2,0)
      length_A(x1) = (1,2)
      active#_A(x1) = ((0,1),(0,0)) x1
      mark_A(x1) = ((0,0),(1,0)) x1 + (1,1)
      U41_A(x1,x2) = (1,2)
      tt_A() = (1,1)
      U42_A(x1) = (1,0)
      isNatIList_A(x1) = (1,1)
      U21_A(x1) = (1,1)
      U51_A(x1,x2) = (1,2)
      U52_A(x1) = (1,1)
      isNatList_A(x1) = (1,2)
      U71_A(x1,x2,x3) = (1,2)
      U72_A(x1,x2) = (1,2)
      isNat_A(x1) = (1,2)
      s_A(x1) = (1,1)
      cons_A(x1,x2) = (1,1)
      active_A(x1) = (1,2)
      zeros_A() = (1,1)
      |0|_A() = (1,1)
      U11_A(x1) = (1,0)
      U31_A(x1) = (1,0)
      U61_A(x1,x2) = (1,0)
      U62_A(x1) = (1,0)
      U81_A(x1) = (1,1)
      nil_A() = (1,0)
      U91_A(x1,x2,x3,x4) = (1,0)
      U92_A(x1,x2,x3,x4) = (1,0)
      U93_A(x1,x2,x3,x4) = (1,0)
      take_A(x1,x2) = (1,0)

The next rules are strictly ordered:

  p7

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p5: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p6: mark#(U42(X)) -> mark#(X)
p7: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p8: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p9: active#(U72(tt(),L)) -> mark#(s(length(L)))
p10: mark#(U52(X)) -> mark#(X)
p11: mark#(isNatList(X)) -> active#(isNatList(X))
p12: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p13: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p14: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p15: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p16: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p17: mark#(isNat(X)) -> active#(isNat(X))
p18: mark#(s(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p3: mark#(s(X)) -> mark#(X)
p4: mark#(isNat(X)) -> active#(isNat(X))
p5: active#(isNatList(cons(V1,V2))) -> mark#(U51(isNat(V1),V2))
p6: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p7: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p8: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p9: active#(U72(tt(),L)) -> mark#(s(length(L)))
p10: mark#(isNatList(X)) -> active#(isNatList(X))
p11: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p12: mark#(U52(X)) -> mark#(X)
p13: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p14: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p15: mark#(U42(X)) -> mark#(X)
p16: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p17: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p18: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = ((1,0),(1,0)) x1 + (1,1)
      length_A(x1) = (0,1)
      active#_A(x1) = ((0,1),(0,1)) x1
      mark_A(x1) = ((1,0),(1,1)) x1
      cons_A(x1,x2) = x1 + x2 + (1,1)
      U71_A(x1,x2,x3) = (0,1)
      isNatList_A(x1) = ((0,1),(0,1)) x1 + (1,2)
      s_A(x1) = x1 + (0,1)
      isNat_A(x1) = ((1,0),(1,0)) x1 + (1,2)
      U51_A(x1,x2) = ((0,1),(0,1)) x2 + (1,2)
      U72_A(x1,x2) = (0,1)
      U21_A(x1) = x1 + (0,1)
      tt_A() = (0,1)
      U52_A(x1) = ((1,0),(1,0)) x1 + (0,1)
      U42_A(x1) = x1 + (0,1)
      U41_A(x1,x2) = (0,1)
      isNatIList_A(x1) = x1 + (0,1)
      active_A(x1) = ((1,0),(1,1)) x1
      zeros_A() = (2,1)
      |0|_A() = (0,1)
      U11_A(x1) = (1,1)
      U31_A(x1) = (0,1)
      U61_A(x1,x2) = (1,1)
      U62_A(x1) = (1,1)
      U81_A(x1) = ((0,0),(1,0)) x1 + (1,1)
      nil_A() = (1,1)
      U91_A(x1,x2,x3,x4) = x3 + x4 + (3,1)
      U92_A(x1,x2,x3,x4) = ((0,0),(1,0)) x1 + x4 + (1,2)
      U93_A(x1,x2,x3,x4) = x4 + (1,2)
      take_A(x1,x2) = x1 + x2 + (2,1)

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p3: mark#(s(X)) -> mark#(X)
p4: mark#(isNat(X)) -> active#(isNat(X))
p5: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p6: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p7: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p8: active#(U72(tt(),L)) -> mark#(s(length(L)))
p9: mark#(isNatList(X)) -> active#(isNatList(X))
p10: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p11: mark#(U52(X)) -> mark#(X)
p12: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p13: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p14: mark#(U42(X)) -> mark#(X)
p15: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p16: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p17: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p5: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p6: mark#(U42(X)) -> mark#(X)
p7: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p8: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p9: mark#(U52(X)) -> mark#(X)
p10: mark#(isNatList(X)) -> active#(isNatList(X))
p11: active#(U72(tt(),L)) -> mark#(s(length(L)))
p12: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p13: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p14: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p15: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p16: mark#(isNat(X)) -> active#(isNat(X))
p17: mark#(s(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = (2,0)
      length_A(x1) = (2,1)
      active#_A(x1) = x1
      mark_A(x1) = ((0,0),(1,1)) x1 + (0,1)
      U41_A(x1,x2) = (2,0)
      tt_A() = (1,1)
      U42_A(x1) = (0,2)
      isNatIList_A(x1) = (1,1)
      U21_A(x1) = (0,2)
      U51_A(x1,x2) = (2,0)
      U52_A(x1) = (0,2)
      isNatList_A(x1) = (1,1)
      U71_A(x1,x2,x3) = (2,0)
      U72_A(x1,x2) = (2,0)
      isNat_A(x1) = (2,1)
      s_A(x1) = (0,2)
      cons_A(x1,x2) = (1,1)
      active_A(x1) = (0,3)
      zeros_A() = (1,1)
      |0|_A() = (0,2)
      U11_A(x1) = (1,1)
      U31_A(x1) = (1,1)
      U61_A(x1,x2) = (0,2)
      U62_A(x1) = (0,2)
      U81_A(x1) = (1,1)
      nil_A() = (1,1)
      U91_A(x1,x2,x3,x4) = (1,1)
      U92_A(x1,x2,x3,x4) = (1,1)
      U93_A(x1,x2,x3,x4) = (1,1)
      take_A(x1,x2) = (1,1)

The next rules are strictly ordered:

  p10

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p5: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p6: mark#(U42(X)) -> mark#(X)
p7: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p8: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p9: mark#(U52(X)) -> mark#(X)
p10: active#(U72(tt(),L)) -> mark#(s(length(L)))
p11: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p12: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p13: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p14: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p15: mark#(isNat(X)) -> active#(isNat(X))
p16: mark#(s(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p3: mark#(s(X)) -> mark#(X)
p4: mark#(isNat(X)) -> active#(isNat(X))
p5: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p6: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p7: active#(U72(tt(),L)) -> mark#(s(length(L)))
p8: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p9: active#(U71(tt(),L,N)) -> mark#(U72(isNat(N),L))
p10: mark#(U52(X)) -> mark#(X)
p11: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p12: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p13: mark#(U42(X)) -> mark#(X)
p14: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p15: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p16: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = ((0,1),(0,0)) x1
      length_A(x1) = ((1,0),(1,1)) x1 + (0,2)
      active#_A(x1) = ((0,1),(0,0)) x1
      mark_A(x1) = x1
      cons_A(x1,x2) = x1 + ((1,1),(0,1)) x2
      U71_A(x1,x2,x3) = ((0,1),(1,1)) x1 + ((1,0),(1,1)) x2 + (0,1)
      isNatList_A(x1) = x1 + (1,0)
      s_A(x1) = x1 + (4,1)
      isNat_A(x1) = ((0,1),(0,0)) x1 + (4,4)
      U21_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      U72_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,0),(1,1)) x2 + (0,4)
      tt_A() = (0,4)
      U52_A(x1) = x1
      U51_A(x1,x2) = x2 + (1,0)
      U42_A(x1) = x1
      U41_A(x1,x2) = x2 + (1,4)
      isNatIList_A(x1) = x1 + (1,4)
      active_A(x1) = x1
      zeros_A() = (1,0)
      |0|_A() = (0,1)
      U11_A(x1) = ((1,1),(0,0)) x1 + (1,4)
      U31_A(x1) = (1,4)
      U61_A(x1,x2) = (1,4)
      U62_A(x1) = x1 + (1,0)
      U81_A(x1) = x1 + (1,4)
      nil_A() = (1,4)
      U91_A(x1,x2,x3,x4) = ((1,1),(0,0)) x2 + x3 + x4 + (6,4)
      U92_A(x1,x2,x3,x4) = ((1,1),(0,0)) x2 + x3 + x4 + (6,4)
      U93_A(x1,x2,x3,x4) = ((0,1),(0,1)) x1 + ((1,1),(0,0)) x2 + x3 + x4 + (2,0)
      take_A(x1,x2) = x1 + x2 + (2,4)

The next rules are strictly ordered:

  p3, p7, p9

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p3: mark#(isNat(X)) -> active#(isNat(X))
p4: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p5: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p6: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p7: mark#(U52(X)) -> mark#(X)
p8: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p9: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p10: mark#(U42(X)) -> mark#(X)
p11: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p12: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p13: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p5: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p6: mark#(U42(X)) -> mark#(X)
p7: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p8: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p9: mark#(U52(X)) -> mark#(X)
p10: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p11: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p12: mark#(U72(X1,X2)) -> active#(U72(mark(X1),X2))
p13: mark#(isNat(X)) -> active#(isNat(X))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = ((0,1),(0,0)) x1 + (2,0)
      length_A(x1) = x1 + (2,0)
      active#_A(x1) = ((1,1),(0,0)) x1
      mark_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      U41_A(x1,x2) = x2 + (2,0)
      tt_A() = (1,1)
      U42_A(x1) = ((0,1),(0,1)) x1
      isNatIList_A(x1) = x1
      U21_A(x1) = x1 + (1,0)
      U51_A(x1,x2) = x2 + (2,0)
      U52_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      isNatList_A(x1) = x1 + (1,0)
      isNat_A(x1) = x1 + (2,1)
      s_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      U71_A(x1,x2,x3) = x2 + (2,0)
      cons_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,1)) x2 + (1,0)
      U72_A(x1,x2) = x2 + (1,0)
      active_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      zeros_A() = (0,1)
      |0|_A() = (1,1)
      U11_A(x1) = x1 + (1,1)
      U31_A(x1) = x1 + (1,0)
      U61_A(x1,x2) = ((0,1),(0,1)) x2 + (1,0)
      U62_A(x1) = x1 + (1,0)
      U81_A(x1) = (1,1)
      nil_A() = (1,1)
      U91_A(x1,x2,x3,x4) = ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + ((0,1),(0,0)) x4 + (1,0)
      U92_A(x1,x2,x3,x4) = ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + ((0,1),(0,0)) x4 + (1,0)
      U93_A(x1,x2,x3,x4) = ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + ((0,1),(0,0)) x4 + (1,0)
      take_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (1,0)

The next rules are strictly ordered:

  p12

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p5: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p6: mark#(U42(X)) -> mark#(X)
p7: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p8: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p9: mark#(U52(X)) -> mark#(X)
p10: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p11: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p12: mark#(isNat(X)) -> active#(isNat(X))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(length(cons(N,L))) -> mark#(U71(isNatList(L),L,N))
p3: mark#(isNat(X)) -> active#(isNat(X))
p4: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p5: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p6: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p7: mark#(U52(X)) -> mark#(X)
p8: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p9: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p10: mark#(U42(X)) -> mark#(X)
p11: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p12: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = ((0,1),(0,0)) x1 + (1,0)
      length_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      active#_A(x1) = x1
      mark_A(x1) = x1 + (1,0)
      cons_A(x1,x2) = ((0,1),(0,0)) x1 + (1,2)
      U71_A(x1,x2,x3) = (2,1)
      isNatList_A(x1) = ((0,1),(0,0)) x1 + (1,3)
      isNat_A(x1) = (5,4)
      s_A(x1) = ((0,1),(0,0)) x1 + (1,0)
      U21_A(x1) = x1 + (1,0)
      U51_A(x1,x2) = (4,3)
      tt_A() = (1,2)
      U52_A(x1) = x1 + (1,0)
      U41_A(x1,x2) = (3,2)
      U42_A(x1) = x1 + (1,0)
      isNatIList_A(x1) = (1,2)
      active_A(x1) = x1 + (1,0)
      zeros_A() = (1,2)
      |0|_A() = (1,1)
      U11_A(x1) = ((0,1),(0,1)) x1 + (1,1)
      U31_A(x1) = ((0,1),(0,0)) x1 + (1,2)
      U61_A(x1,x2) = ((0,1),(0,0)) x1 + (1,3)
      U62_A(x1) = ((0,1),(0,1)) x1 + (1,1)
      U72_A(x1,x2) = ((0,1),(0,0)) x1 + (1,0)
      U81_A(x1) = ((0,1),(0,0)) x1 + (1,1)
      nil_A() = (1,1)
      U91_A(x1,x2,x3,x4) = ((0,1),(0,1)) x1 + (1,0)
      U92_A(x1,x2,x3,x4) = ((0,1),(0,0)) x1 + (1,2)
      U93_A(x1,x2,x3,x4) = ((0,1),(0,0)) x1 + (1,2)
      take_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (1,0)

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: mark#(length(X)) -> active#(length(mark(X)))
p2: mark#(isNat(X)) -> active#(isNat(X))
p3: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p4: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p5: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p6: mark#(U52(X)) -> mark#(X)
p7: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p8: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p9: mark#(U42(X)) -> mark#(X)
p10: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p11: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(length(X)) -> active#(length(mark(X)))
p2: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p5: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p6: mark#(U42(X)) -> mark#(X)
p7: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p8: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p9: mark#(U52(X)) -> mark#(X)
p10: mark#(U71(X1,X2,X3)) -> active#(U71(mark(X1),X2,X3))
p11: mark#(isNat(X)) -> active#(isNat(X))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = (3,2)
      length_A(x1) = (1,1)
      active#_A(x1) = ((1,1),(1,0)) x1
      mark_A(x1) = x1 + (1,0)
      U41_A(x1,x2) = (2,1)
      tt_A() = (0,1)
      U42_A(x1) = (1,1)
      isNatIList_A(x1) = (0,1)
      U21_A(x1) = (1,1)
      U51_A(x1,x2) = (2,1)
      U52_A(x1) = ((0,1),(0,0)) x1 + (1,1)
      isNatList_A(x1) = (0,1)
      isNat_A(x1) = (2,1)
      s_A(x1) = ((0,1),(0,0)) x1 + (1,0)
      U71_A(x1,x2,x3) = (1,1)
      active_A(x1) = ((0,1),(0,1)) x1
      zeros_A() = (0,1)
      cons_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (0,1)
      |0|_A() = (1,1)
      U11_A(x1) = (1,1)
      U31_A(x1) = ((0,1),(0,0)) x1 + (1,1)
      U61_A(x1,x2) = (1,1)
      U62_A(x1) = (0,1)
      U72_A(x1,x2) = ((0,1),(0,0)) x2 + (1,1)
      U81_A(x1) = ((0,1),(0,0)) x1 + (1,1)
      nil_A() = (1,1)
      U91_A(x1,x2,x3,x4) = ((0,1),(0,0)) x2 + ((0,1),(0,0)) x4 + (1,1)
      U92_A(x1,x2,x3,x4) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + ((0,1),(0,0)) x4 + (1,1)
      U93_A(x1,x2,x3,x4) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + ((0,1),(0,0)) x4 + (1,1)
      take_A(x1,x2) = ((0,1),(0,0)) x2 + (1,1)

The next rules are strictly ordered:

  p1, p10

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p2: mark#(U21(X)) -> mark#(X)
p3: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p4: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p5: mark#(U42(X)) -> mark#(X)
p6: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p7: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p8: mark#(U52(X)) -> mark#(X)
p9: mark#(isNat(X)) -> active#(isNat(X))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p2: mark#(isNat(X)) -> active#(isNat(X))
p3: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p4: mark#(U52(X)) -> mark#(X)
p5: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p6: active#(U51(tt(),V2)) -> mark#(U52(isNatList(V2)))
p7: mark#(U42(X)) -> mark#(X)
p8: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p9: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      active#_A(x1) = x1
      U41_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (1,2)
      tt_A() = (1,1)
      mark#_A(x1) = x1
      U42_A(x1) = x1 + (0,2)
      isNatIList_A(x1) = x1 + (1,3)
      isNat_A(x1) = ((0,0),(1,0)) x1 + (1,2)
      s_A(x1) = (1,1)
      U21_A(x1) = x1 + (0,2)
      U52_A(x1) = x1 + (0,2)
      U51_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,1),(0,0)) x2 + (2,1)
      mark_A(x1) = ((1,0),(1,1)) x1 + (0,1)
      isNatList_A(x1) = ((0,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,0),(1,1)) x1
      zeros_A() = (3,1)
      cons_A(x1,x2) = x1 + x2 + (1,1)
      |0|_A() = (0,0)
      U11_A(x1) = (1,2)
      U31_A(x1) = (1,2)
      U61_A(x1,x2) = x1 + (0,3)
      U62_A(x1) = (1,2)
      U71_A(x1,x2,x3) = (1,3)
      U72_A(x1,x2) = ((1,0),(1,0)) x1 + (0,1)
      length_A(x1) = ((0,0),(1,0)) x1 + (3,1)
      U81_A(x1) = (1,2)
      nil_A() = (1,1)
      U91_A(x1,x2,x3,x4) = x1 + x4 + (0,5)
      U92_A(x1,x2,x3,x4) = x4 + (1,4)
      U93_A(x1,x2,x3,x4) = x4 + (1,3)
      take_A(x1,x2) = x2 + (5,1)

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p2: mark#(isNat(X)) -> active#(isNat(X))
p3: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p4: mark#(U52(X)) -> mark#(X)
p5: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p6: mark#(U42(X)) -> mark#(X)
p7: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p8: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p2: mark#(U21(X)) -> mark#(X)
p3: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p4: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p5: mark#(U42(X)) -> mark#(X)
p6: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p7: mark#(U52(X)) -> mark#(X)
p8: mark#(isNat(X)) -> active#(isNat(X))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      active#_A(x1) = x1
      U41_A(x1,x2) = x1 + (2,0)
      tt_A() = (0,1)
      mark#_A(x1) = (2,0)
      U42_A(x1) = ((0,1),(0,0)) x1 + (1,1)
      isNatIList_A(x1) = ((0,1),(0,1)) x1 + (0,1)
      U21_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      mark_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      isNat_A(x1) = x1 + (2,0)
      s_A(x1) = x1 + (1,0)
      U51_A(x1,x2) = (1,1)
      U52_A(x1) = ((0,1),(0,0)) x1 + (1,1)
      active_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      zeros_A() = (0,2)
      cons_A(x1,x2) = ((0,1),(0,1)) x1 + (0,1)
      |0|_A() = (1,1)
      U11_A(x1) = (0,1)
      U31_A(x1) = (1,1)
      isNatList_A(x1) = ((0,1),(0,0)) x1 + (1,1)
      U61_A(x1,x2) = ((0,1),(0,0)) x1 + (1,1)
      U62_A(x1) = (1,1)
      U71_A(x1,x2,x3) = ((0,1),(0,0)) x1 + (1,1)
      U72_A(x1,x2) = ((0,1),(0,0)) x1 + (1,1)
      length_A(x1) = ((0,1),(0,0)) x1 + (0,1)
      U81_A(x1) = (1,1)
      nil_A() = (0,1)
      U91_A(x1,x2,x3,x4) = ((0,1),(0,1)) x4 + (1,1)
      U92_A(x1,x2,x3,x4) = ((0,1),(0,0)) x1 + x4 + (1,1)
      U93_A(x1,x2,x3,x4) = ((0,1),(0,0)) x1 + x4 + (0,1)
      take_A(x1,x2) = x2 + (0,1)

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p2: mark#(U21(X)) -> mark#(X)
p3: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p4: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p5: mark#(U42(X)) -> mark#(X)
p6: mark#(U52(X)) -> mark#(X)
p7: mark#(isNat(X)) -> active#(isNat(X))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(U41(tt(),V2)) -> mark#(U42(isNatIList(V2)))
p2: mark#(isNat(X)) -> active#(isNat(X))
p3: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p4: mark#(U52(X)) -> mark#(X)
p5: mark#(U42(X)) -> mark#(X)
p6: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p7: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      active#_A(x1) = x1
      U41_A(x1,x2) = ((0,1),(0,1)) x2 + (2,1)
      tt_A() = (1,1)
      mark#_A(x1) = x1
      U42_A(x1) = ((1,0),(1,0)) x1 + (0,1)
      isNatIList_A(x1) = ((0,1),(0,1)) x1 + (1,2)
      isNat_A(x1) = x1 + (1,1)
      s_A(x1) = ((1,0),(1,0)) x1 + (0,1)
      U21_A(x1) = ((1,0),(1,0)) x1
      U52_A(x1) = ((1,0),(1,0)) x1 + (0,1)
      mark_A(x1) = ((1,0),(1,1)) x1 + (0,1)
      active_A(x1) = ((1,0),(1,1)) x1 + (0,1)
      zeros_A() = (3,1)
      cons_A(x1,x2) = ((0,1),(0,1)) x2 + (1,1)
      |0|_A() = (1,1)
      U11_A(x1) = (1,1)
      U31_A(x1) = ((0,0),(1,0)) x1 + (1,1)
      U51_A(x1,x2) = ((0,1),(0,1)) x2 + (1,2)
      isNatList_A(x1) = ((0,1),(1,0)) x1 + (1,1)
      U61_A(x1,x2) = ((1,0),(1,0)) x1 + (1,3)
      U62_A(x1) = (1,1)
      U71_A(x1,x2,x3) = ((0,0),(1,0)) x1 + (1,1)
      U72_A(x1,x2) = (1,2)
      length_A(x1) = ((0,0),(1,0)) x1 + (1,1)
      U81_A(x1) = (1,1)
      nil_A() = (1,1)
      U91_A(x1,x2,x3,x4) = ((1,0),(1,0)) x3 + (3,1)
      U92_A(x1,x2,x3,x4) = ((1,0),(1,0)) x3 + (2,2)
      U93_A(x1,x2,x3,x4) = ((1,0),(1,0)) x3 + (2,2)
      take_A(x1,x2) = ((1,0),(1,0)) x1 + (3,1)

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: mark#(isNat(X)) -> active#(isNat(X))
p2: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p3: mark#(U52(X)) -> mark#(X)
p4: mark#(U42(X)) -> mark#(X)
p5: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p6: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(isNat(X)) -> active#(isNat(X))
p2: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U41(X1,X2)) -> active#(U41(mark(X1),X2))
p5: mark#(U42(X)) -> mark#(X)
p6: mark#(U52(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = (2,0)
      isNat_A(x1) = (2,1)
      active#_A(x1) = x1
      s_A(x1) = ((1,0),(1,0)) x1 + (0,1)
      U21_A(x1) = ((0,0),(1,0)) x1 + (1,1)
      U41_A(x1,x2) = ((0,0),(1,0)) x1 + (1,1)
      mark_A(x1) = ((1,0),(1,0)) x1 + (0,1)
      U42_A(x1) = ((0,0),(1,0)) x1 + (1,1)
      U52_A(x1) = (1,1)
      active_A(x1) = ((1,0),(1,0)) x1 + (0,1)
      zeros_A() = (0,0)
      cons_A(x1,x2) = ((1,0),(1,0)) x1 + (0,1)
      |0|_A() = (0,1)
      U11_A(x1) = ((0,0),(1,0)) x1 + (1,1)
      tt_A() = (1,1)
      U31_A(x1) = ((0,0),(1,0)) x1 + (1,1)
      isNatIList_A(x1) = ((1,0),(1,0)) x1 + (1,1)
      U51_A(x1,x2) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (1,1)
      isNatList_A(x1) = ((0,0),(1,0)) x1 + (3,1)
      U61_A(x1,x2) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (1,1)
      U62_A(x1) = ((0,0),(1,0)) x1 + (1,1)
      U71_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (1,1)
      U72_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (1,1)
      length_A(x1) = ((0,0),(1,0)) x1 + (1,1)
      U81_A(x1) = ((0,0),(1,0)) x1 + (1,1)
      nil_A() = (1,1)
      U91_A(x1,x2,x3,x4) = x3 + ((1,0),(1,0)) x4 + (1,1)
      U92_A(x1,x2,x3,x4) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x3 + ((1,0),(1,0)) x4 + (1,1)
      U93_A(x1,x2,x3,x4) = ((0,0),(1,0)) x1 + x3 + ((1,0),(1,0)) x4 + (0,1)
      take_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (1,1)

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(isNat(X)) -> active#(isNat(X))
p2: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U42(X)) -> mark#(X)
p5: mark#(U52(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(isNat(X)) -> active#(isNat(X))
p2: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p3: mark#(U52(X)) -> mark#(X)
p4: mark#(U42(X)) -> mark#(X)
p5: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  r62, r63, r98, r99

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = ((0,1),(1,1)) x1
      isNat_A(x1) = ((1,1),(0,0)) x1 + (1,1)
      active#_A(x1) = ((0,1),(1,0)) x1 + (0,1)
      s_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      U21_A(x1) = ((1,1),(0,1)) x1 + (1,0)
      U52_A(x1) = ((1,1),(1,1)) x1 + (1,0)
      U42_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(isNat(X)) -> active#(isNat(X))
p2: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p3: mark#(U52(X)) -> mark#(X)
p4: mark#(U21(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(isNat(X)) -> active#(isNat(X))
p2: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))
p3: mark#(U21(X)) -> mark#(X)
p4: mark#(U52(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  r62, r63, r98, r99

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = ((1,1),(1,0)) x1
      isNat_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active#_A(x1) = ((1,1),(1,0)) x1
      s_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      U21_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      U52_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

The next rules are strictly ordered:

  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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(isNat(X)) -> active#(isNat(X))
p2: active#(isNat(s(V1))) -> mark#(U21(isNat(V1)))

and R consists of:

  (no rules)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U42#(mark(X)) -> U42#(X)
p2: U42#(active(X)) -> U42#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      U42#_A(x1) = ((1,0),(1,1)) x1
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

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: isNatIList#(mark(X)) -> isNatIList#(X)
p2: isNatIList#(active(X)) -> isNatIList#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      isNatIList#_A(x1) = ((1,0),(1,1)) x1
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

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: U52#(mark(X)) -> U52#(X)
p2: U52#(active(X)) -> U52#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      U52#_A(x1) = ((1,0),(1,1)) x1
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

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: isNatList#(mark(X)) -> isNatList#(X)
p2: isNatList#(active(X)) -> isNatList#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      isNatList#_A(x1) = ((1,0),(1,1)) x1
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

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: U62#(mark(X)) -> U62#(X)
p2: U62#(active(X)) -> U62#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      U62#_A(x1) = ((1,0),(1,1)) x1
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

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: U72#(mark(X1),X2) -> U72#(X1,X2)
p2: U72#(X1,active(X2)) -> U72#(X1,X2)
p3: U72#(active(X1),X2) -> U72#(X1,X2)
p4: U72#(X1,mark(X2)) -> U72#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      U72#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2
      mark_A(x1) = ((0,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

The next rules are strictly ordered:

  p1, p2, p3, p4

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: isNat#(mark(X)) -> isNat#(X)
p2: isNat#(active(X)) -> isNat#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      isNat#_A(x1) = ((1,0),(1,1)) x1
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

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: s#(mark(X)) -> s#(X)
p2: s#(active(X)) -> s#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      s#_A(x1) = ((1,0),(1,1)) x1
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

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: length#(mark(X)) -> length#(X)
p2: length#(active(X)) -> length#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      length#_A(x1) = ((1,0),(1,1)) x1
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

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: U92#(mark(X1),X2,X3,X4) -> U92#(X1,X2,X3,X4)
p2: U92#(X1,X2,X3,active(X4)) -> U92#(X1,X2,X3,X4)
p3: U92#(X1,X2,active(X3),X4) -> U92#(X1,X2,X3,X4)
p4: U92#(X1,active(X2),X3,X4) -> U92#(X1,X2,X3,X4)
p5: U92#(active(X1),X2,X3,X4) -> U92#(X1,X2,X3,X4)
p6: U92#(X1,X2,X3,mark(X4)) -> U92#(X1,X2,X3,X4)
p7: U92#(X1,X2,mark(X3),X4) -> U92#(X1,X2,X3,X4)
p8: U92#(X1,mark(X2),X3,X4) -> U92#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      U92#_A(x1,x2,x3,x4) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + ((1,1),(1,1)) x4
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p8
  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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

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: U93#(mark(X1),X2,X3,X4) -> U93#(X1,X2,X3,X4)
p2: U93#(X1,X2,X3,active(X4)) -> U93#(X1,X2,X3,X4)
p3: U93#(X1,X2,active(X3),X4) -> U93#(X1,X2,X3,X4)
p4: U93#(X1,active(X2),X3,X4) -> U93#(X1,X2,X3,X4)
p5: U93#(active(X1),X2,X3,X4) -> U93#(X1,X2,X3,X4)
p6: U93#(X1,X2,X3,mark(X4)) -> U93#(X1,X2,X3,X4)
p7: U93#(X1,X2,mark(X3),X4) -> U93#(X1,X2,X3,X4)
p8: U93#(X1,mark(X2),X3,X4) -> U93#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      U93#_A(x1,x2,x3,x4) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + ((1,1),(1,1)) x4
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p8
  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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

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: cons#(mark(X1),X2) -> cons#(X1,X2)
p2: cons#(X1,active(X2)) -> cons#(X1,X2)
p3: cons#(active(X1),X2) -> cons#(X1,X2)
p4: cons#(X1,mark(X2)) -> cons#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      cons#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2
      mark_A(x1) = ((0,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

The next rules are strictly ordered:

  p1, p2, p3, p4

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: take#(mark(X1),X2) -> take#(X1,X2)
p2: take#(X1,active(X2)) -> take#(X1,X2)
p3: take#(active(X1),X2) -> take#(X1,X2)
p4: take#(X1,mark(X2)) -> take#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      take#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2
      mark_A(x1) = ((0,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

The next rules are strictly ordered:

  p1, p2, p3, p4

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: U11#(mark(X)) -> U11#(X)
p2: U11#(active(X)) -> U11#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      U11#_A(x1) = ((1,0),(1,1)) x1
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

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: U21#(mark(X)) -> U21#(X)
p2: U21#(active(X)) -> U21#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      U21#_A(x1) = ((1,0),(1,1)) x1
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

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: U31#(mark(X)) -> U31#(X)
p2: U31#(active(X)) -> U31#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      U31#_A(x1) = ((1,0),(1,1)) x1
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

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: U41#(mark(X1),X2) -> U41#(X1,X2)
p2: U41#(X1,active(X2)) -> U41#(X1,X2)
p3: U41#(active(X1),X2) -> U41#(X1,X2)
p4: U41#(X1,mark(X2)) -> U41#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      U41#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2
      mark_A(x1) = ((0,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

The next rules are strictly ordered:

  p1, p2, p3, p4

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: U51#(mark(X1),X2) -> U51#(X1,X2)
p2: U51#(X1,active(X2)) -> U51#(X1,X2)
p3: U51#(active(X1),X2) -> U51#(X1,X2)
p4: U51#(X1,mark(X2)) -> U51#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      U51#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2
      mark_A(x1) = ((0,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

The next rules are strictly ordered:

  p1, p2, p3, p4

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: U61#(mark(X1),X2) -> U61#(X1,X2)
p2: U61#(X1,active(X2)) -> U61#(X1,X2)
p3: U61#(active(X1),X2) -> U61#(X1,X2)
p4: U61#(X1,mark(X2)) -> U61#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      U61#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2
      mark_A(x1) = ((0,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

The next rules are strictly ordered:

  p1, p2, p3, p4

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: U71#(mark(X1),X2,X3) -> U71#(X1,X2,X3)
p2: U71#(X1,X2,active(X3)) -> U71#(X1,X2,X3)
p3: U71#(X1,active(X2),X3) -> U71#(X1,X2,X3)
p4: U71#(active(X1),X2,X3) -> U71#(X1,X2,X3)
p5: U71#(X1,X2,mark(X3)) -> U71#(X1,X2,X3)
p6: U71#(X1,mark(X2),X3) -> U71#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      U71#_A(x1,x2,x3) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6
  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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

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: U81#(mark(X)) -> U81#(X)
p2: U81#(active(X)) -> U81#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      U81#_A(x1) = ((1,0),(1,1)) x1
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

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: U91#(mark(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4)
p2: U91#(X1,X2,X3,active(X4)) -> U91#(X1,X2,X3,X4)
p3: U91#(X1,X2,active(X3),X4) -> U91#(X1,X2,X3,X4)
p4: U91#(X1,active(X2),X3,X4) -> U91#(X1,X2,X3,X4)
p5: U91#(active(X1),X2,X3,X4) -> U91#(X1,X2,X3,X4)
p6: U91#(X1,X2,X3,mark(X4)) -> U91#(X1,X2,X3,X4)
p7: U91#(X1,X2,mark(X3),X4) -> U91#(X1,X2,X3,X4)
p8: U91#(X1,mark(X2),X3,X4) -> U91#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U11(tt())) -> mark(tt())
r3: active(U21(tt())) -> mark(tt())
r4: active(U31(tt())) -> mark(tt())
r5: active(U41(tt(),V2)) -> mark(U42(isNatIList(V2)))
r6: active(U42(tt())) -> mark(tt())
r7: active(U51(tt(),V2)) -> mark(U52(isNatList(V2)))
r8: active(U52(tt())) -> mark(tt())
r9: active(U61(tt(),V2)) -> mark(U62(isNatIList(V2)))
r10: active(U62(tt())) -> mark(tt())
r11: active(U71(tt(),L,N)) -> mark(U72(isNat(N),L))
r12: active(U72(tt(),L)) -> mark(s(length(L)))
r13: active(U81(tt())) -> mark(nil())
r14: active(U91(tt(),IL,M,N)) -> mark(U92(isNat(M),IL,M,N))
r15: active(U92(tt(),IL,M,N)) -> mark(U93(isNat(N),IL,M,N))
r16: active(U93(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r17: active(isNat(|0|())) -> mark(tt())
r18: active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
r19: active(isNat(s(V1))) -> mark(U21(isNat(V1)))
r20: active(isNatIList(V)) -> mark(U31(isNatList(V)))
r21: active(isNatIList(zeros())) -> mark(tt())
r22: active(isNatIList(cons(V1,V2))) -> mark(U41(isNat(V1),V2))
r23: active(isNatList(nil())) -> mark(tt())
r24: active(isNatList(cons(V1,V2))) -> mark(U51(isNat(V1),V2))
r25: active(isNatList(take(V1,V2))) -> mark(U61(isNat(V1),V2))
r26: active(length(nil())) -> mark(|0|())
r27: active(length(cons(N,L))) -> mark(U71(isNatList(L),L,N))
r28: active(take(|0|(),IL)) -> mark(U81(isNatIList(IL)))
r29: active(take(s(M),cons(N,IL))) -> mark(U91(isNatIList(IL),IL,M,N))
r30: mark(zeros()) -> active(zeros())
r31: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r32: mark(|0|()) -> active(|0|())
r33: mark(U11(X)) -> active(U11(mark(X)))
r34: mark(tt()) -> active(tt())
r35: mark(U21(X)) -> active(U21(mark(X)))
r36: mark(U31(X)) -> active(U31(mark(X)))
r37: mark(U41(X1,X2)) -> active(U41(mark(X1),X2))
r38: mark(U42(X)) -> active(U42(mark(X)))
r39: mark(isNatIList(X)) -> active(isNatIList(X))
r40: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r41: mark(U52(X)) -> active(U52(mark(X)))
r42: mark(isNatList(X)) -> active(isNatList(X))
r43: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r44: mark(U62(X)) -> active(U62(mark(X)))
r45: mark(U71(X1,X2,X3)) -> active(U71(mark(X1),X2,X3))
r46: mark(U72(X1,X2)) -> active(U72(mark(X1),X2))
r47: mark(isNat(X)) -> active(isNat(X))
r48: mark(s(X)) -> active(s(mark(X)))
r49: mark(length(X)) -> active(length(mark(X)))
r50: mark(U81(X)) -> active(U81(mark(X)))
r51: mark(nil()) -> active(nil())
r52: mark(U91(X1,X2,X3,X4)) -> active(U91(mark(X1),X2,X3,X4))
r53: mark(U92(X1,X2,X3,X4)) -> active(U92(mark(X1),X2,X3,X4))
r54: mark(U93(X1,X2,X3,X4)) -> active(U93(mark(X1),X2,X3,X4))
r55: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r56: cons(mark(X1),X2) -> cons(X1,X2)
r57: cons(X1,mark(X2)) -> cons(X1,X2)
r58: cons(active(X1),X2) -> cons(X1,X2)
r59: cons(X1,active(X2)) -> cons(X1,X2)
r60: U11(mark(X)) -> U11(X)
r61: U11(active(X)) -> U11(X)
r62: U21(mark(X)) -> U21(X)
r63: U21(active(X)) -> U21(X)
r64: U31(mark(X)) -> U31(X)
r65: U31(active(X)) -> U31(X)
r66: U41(mark(X1),X2) -> U41(X1,X2)
r67: U41(X1,mark(X2)) -> U41(X1,X2)
r68: U41(active(X1),X2) -> U41(X1,X2)
r69: U41(X1,active(X2)) -> U41(X1,X2)
r70: U42(mark(X)) -> U42(X)
r71: U42(active(X)) -> U42(X)
r72: isNatIList(mark(X)) -> isNatIList(X)
r73: isNatIList(active(X)) -> isNatIList(X)
r74: U51(mark(X1),X2) -> U51(X1,X2)
r75: U51(X1,mark(X2)) -> U51(X1,X2)
r76: U51(active(X1),X2) -> U51(X1,X2)
r77: U51(X1,active(X2)) -> U51(X1,X2)
r78: U52(mark(X)) -> U52(X)
r79: U52(active(X)) -> U52(X)
r80: isNatList(mark(X)) -> isNatList(X)
r81: isNatList(active(X)) -> isNatList(X)
r82: U61(mark(X1),X2) -> U61(X1,X2)
r83: U61(X1,mark(X2)) -> U61(X1,X2)
r84: U61(active(X1),X2) -> U61(X1,X2)
r85: U61(X1,active(X2)) -> U61(X1,X2)
r86: U62(mark(X)) -> U62(X)
r87: U62(active(X)) -> U62(X)
r88: U71(mark(X1),X2,X3) -> U71(X1,X2,X3)
r89: U71(X1,mark(X2),X3) -> U71(X1,X2,X3)
r90: U71(X1,X2,mark(X3)) -> U71(X1,X2,X3)
r91: U71(active(X1),X2,X3) -> U71(X1,X2,X3)
r92: U71(X1,active(X2),X3) -> U71(X1,X2,X3)
r93: U71(X1,X2,active(X3)) -> U71(X1,X2,X3)
r94: U72(mark(X1),X2) -> U72(X1,X2)
r95: U72(X1,mark(X2)) -> U72(X1,X2)
r96: U72(active(X1),X2) -> U72(X1,X2)
r97: U72(X1,active(X2)) -> U72(X1,X2)
r98: isNat(mark(X)) -> isNat(X)
r99: isNat(active(X)) -> isNat(X)
r100: s(mark(X)) -> s(X)
r101: s(active(X)) -> s(X)
r102: length(mark(X)) -> length(X)
r103: length(active(X)) -> length(X)
r104: U81(mark(X)) -> U81(X)
r105: U81(active(X)) -> U81(X)
r106: U91(mark(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r107: U91(X1,mark(X2),X3,X4) -> U91(X1,X2,X3,X4)
r108: U91(X1,X2,mark(X3),X4) -> U91(X1,X2,X3,X4)
r109: U91(X1,X2,X3,mark(X4)) -> U91(X1,X2,X3,X4)
r110: U91(active(X1),X2,X3,X4) -> U91(X1,X2,X3,X4)
r111: U91(X1,active(X2),X3,X4) -> U91(X1,X2,X3,X4)
r112: U91(X1,X2,active(X3),X4) -> U91(X1,X2,X3,X4)
r113: U91(X1,X2,X3,active(X4)) -> U91(X1,X2,X3,X4)
r114: U92(mark(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r115: U92(X1,mark(X2),X3,X4) -> U92(X1,X2,X3,X4)
r116: U92(X1,X2,mark(X3),X4) -> U92(X1,X2,X3,X4)
r117: U92(X1,X2,X3,mark(X4)) -> U92(X1,X2,X3,X4)
r118: U92(active(X1),X2,X3,X4) -> U92(X1,X2,X3,X4)
r119: U92(X1,active(X2),X3,X4) -> U92(X1,X2,X3,X4)
r120: U92(X1,X2,active(X3),X4) -> U92(X1,X2,X3,X4)
r121: U92(X1,X2,X3,active(X4)) -> U92(X1,X2,X3,X4)
r122: U93(mark(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r123: U93(X1,mark(X2),X3,X4) -> U93(X1,X2,X3,X4)
r124: U93(X1,X2,mark(X3),X4) -> U93(X1,X2,X3,X4)
r125: U93(X1,X2,X3,mark(X4)) -> U93(X1,X2,X3,X4)
r126: U93(active(X1),X2,X3,X4) -> U93(X1,X2,X3,X4)
r127: U93(X1,active(X2),X3,X4) -> U93(X1,X2,X3,X4)
r128: U93(X1,X2,active(X3),X4) -> U93(X1,X2,X3,X4)
r129: U93(X1,X2,X3,active(X4)) -> U93(X1,X2,X3,X4)
r130: take(mark(X1),X2) -> take(X1,X2)
r131: take(X1,mark(X2)) -> take(X1,X2)
r132: take(active(X1),X2) -> take(X1,X2)
r133: take(X1,active(X2)) -> take(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      U91#_A(x1,x2,x3,x4) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + ((1,1),(1,1)) x4
      mark_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      active_A(x1) = ((1,1),(1,1)) x1 + (1,1)

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p8
  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, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133

We remove them from the problem.  Then no dependency pair remains.