YES

We show the termination of the TRS R:

  U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
  U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
  U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
  U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
  U15(tt(),V2) -> U16(isNat(activate(V2)))
  U16(tt()) -> tt()
  U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
  U22(tt(),V1) -> U23(isNat(activate(V1)))
  U23(tt()) -> tt()
  U31(tt(),V2) -> U32(isNatKind(activate(V2)))
  U32(tt()) -> tt()
  U41(tt()) -> tt()
  U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
  U52(tt(),N) -> activate(N)
  U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
  U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
  U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
  U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
  isNat(n__0()) -> tt()
  isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
  isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
  isNatKind(n__0()) -> tt()
  isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
  isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
  plus(N,|0|()) -> U51(isNat(N),N)
  plus(N,s(M)) -> U61(isNat(M),M,N)
  |0|() -> n__0()
  plus(X1,X2) -> n__plus(X1,X2)
  s(X) -> n__s(X)
  activate(n__0()) -> |0|()
  activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
  activate(n__s(X)) -> s(activate(X))
  activate(X) -> X

-- SCC decomposition.

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

p1: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
p2: U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
p3: U11#(tt(),V1,V2) -> activate#(V1)
p4: U11#(tt(),V1,V2) -> activate#(V2)
p5: U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
p6: U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
p7: U12#(tt(),V1,V2) -> activate#(V2)
p8: U12#(tt(),V1,V2) -> activate#(V1)
p9: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
p10: U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
p11: U13#(tt(),V1,V2) -> activate#(V2)
p12: U13#(tt(),V1,V2) -> activate#(V1)
p13: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
p14: U14#(tt(),V1,V2) -> isNat#(activate(V1))
p15: U14#(tt(),V1,V2) -> activate#(V1)
p16: U14#(tt(),V1,V2) -> activate#(V2)
p17: U15#(tt(),V2) -> U16#(isNat(activate(V2)))
p18: U15#(tt(),V2) -> isNat#(activate(V2))
p19: U15#(tt(),V2) -> activate#(V2)
p20: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
p21: U21#(tt(),V1) -> isNatKind#(activate(V1))
p22: U21#(tt(),V1) -> activate#(V1)
p23: U22#(tt(),V1) -> U23#(isNat(activate(V1)))
p24: U22#(tt(),V1) -> isNat#(activate(V1))
p25: U22#(tt(),V1) -> activate#(V1)
p26: U31#(tt(),V2) -> U32#(isNatKind(activate(V2)))
p27: U31#(tt(),V2) -> isNatKind#(activate(V2))
p28: U31#(tt(),V2) -> activate#(V2)
p29: U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
p30: U51#(tt(),N) -> isNatKind#(activate(N))
p31: U51#(tt(),N) -> activate#(N)
p32: U52#(tt(),N) -> activate#(N)
p33: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
p34: U61#(tt(),M,N) -> isNatKind#(activate(M))
p35: U61#(tt(),M,N) -> activate#(M)
p36: U61#(tt(),M,N) -> activate#(N)
p37: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
p38: U62#(tt(),M,N) -> isNat#(activate(N))
p39: U62#(tt(),M,N) -> activate#(N)
p40: U62#(tt(),M,N) -> activate#(M)
p41: U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
p42: U63#(tt(),M,N) -> isNatKind#(activate(N))
p43: U63#(tt(),M,N) -> activate#(N)
p44: U63#(tt(),M,N) -> activate#(M)
p45: U64#(tt(),M,N) -> s#(plus(activate(N),activate(M)))
p46: U64#(tt(),M,N) -> plus#(activate(N),activate(M))
p47: U64#(tt(),M,N) -> activate#(N)
p48: U64#(tt(),M,N) -> activate#(M)
p49: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
p50: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p51: isNat#(n__plus(V1,V2)) -> activate#(V1)
p52: isNat#(n__plus(V1,V2)) -> activate#(V2)
p53: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p54: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p55: isNat#(n__s(V1)) -> activate#(V1)
p56: isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
p57: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p58: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p59: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p60: isNatKind#(n__s(V1)) -> U41#(isNatKind(activate(V1)))
p61: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p62: isNatKind#(n__s(V1)) -> activate#(V1)
p63: plus#(N,|0|()) -> U51#(isNat(N),N)
p64: plus#(N,|0|()) -> isNat#(N)
p65: plus#(N,s(M)) -> U61#(isNat(M),M,N)
p66: plus#(N,s(M)) -> isNat#(M)
p67: activate#(n__0()) -> |0|#()
p68: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p69: activate#(n__plus(X1,X2)) -> activate#(X1)
p70: activate#(n__plus(X1,X2)) -> activate#(X2)
p71: activate#(n__s(X)) -> s#(activate(X))
p72: activate#(n__s(X)) -> activate#(X)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
r2: U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
r3: U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
r4: U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
r5: U15(tt(),V2) -> U16(isNat(activate(V2)))
r6: U16(tt()) -> tt()
r7: U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
r8: U22(tt(),V1) -> U23(isNat(activate(V1)))
r9: U23(tt()) -> tt()
r10: U31(tt(),V2) -> U32(isNatKind(activate(V2)))
r11: U32(tt()) -> tt()
r12: U41(tt()) -> tt()
r13: U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
r14: U52(tt(),N) -> activate(N)
r15: U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
r16: U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
r17: U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
r18: U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
r19: isNat(n__0()) -> tt()
r20: isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
r21: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r22: isNatKind(n__0()) -> tt()
r23: isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
r24: isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
r25: plus(N,|0|()) -> U51(isNat(N),N)
r26: plus(N,s(M)) -> U61(isNat(M),M,N)
r27: |0|() -> n__0()
r28: plus(X1,X2) -> n__plus(X1,X2)
r29: s(X) -> n__s(X)
r30: activate(n__0()) -> |0|()
r31: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r32: activate(n__s(X)) -> s(activate(X))
r33: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p18, p19, p20, p21, p22, p24, p25, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p61, p62, p63, p64, p65, p66, p68, p69, p70, p72}


-- Reduction pair.

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

p1: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
p2: U12#(tt(),V1,V2) -> activate#(V1)
p3: activate#(n__s(X)) -> activate#(X)
p4: activate#(n__plus(X1,X2)) -> activate#(X2)
p5: activate#(n__plus(X1,X2)) -> activate#(X1)
p6: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p7: plus#(N,s(M)) -> isNat#(M)
p8: isNat#(n__s(V1)) -> activate#(V1)
p9: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p10: isNatKind#(n__s(V1)) -> activate#(V1)
p11: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p12: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p13: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p14: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p15: isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
p16: U31#(tt(),V2) -> activate#(V2)
p17: U31#(tt(),V2) -> isNatKind#(activate(V2))
p18: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p19: U21#(tt(),V1) -> activate#(V1)
p20: U21#(tt(),V1) -> isNatKind#(activate(V1))
p21: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
p22: U22#(tt(),V1) -> activate#(V1)
p23: U22#(tt(),V1) -> isNat#(activate(V1))
p24: isNat#(n__plus(V1,V2)) -> activate#(V2)
p25: isNat#(n__plus(V1,V2)) -> activate#(V1)
p26: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p27: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
p28: U11#(tt(),V1,V2) -> activate#(V2)
p29: U11#(tt(),V1,V2) -> activate#(V1)
p30: U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
p31: plus#(N,s(M)) -> U61#(isNat(M),M,N)
p32: U61#(tt(),M,N) -> activate#(N)
p33: U61#(tt(),M,N) -> activate#(M)
p34: U61#(tt(),M,N) -> isNatKind#(activate(M))
p35: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
p36: U62#(tt(),M,N) -> activate#(M)
p37: U62#(tt(),M,N) -> activate#(N)
p38: U62#(tt(),M,N) -> isNat#(activate(N))
p39: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
p40: U63#(tt(),M,N) -> activate#(M)
p41: U63#(tt(),M,N) -> activate#(N)
p42: U63#(tt(),M,N) -> isNatKind#(activate(N))
p43: U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
p44: U64#(tt(),M,N) -> activate#(M)
p45: U64#(tt(),M,N) -> activate#(N)
p46: U64#(tt(),M,N) -> plus#(activate(N),activate(M))
p47: plus#(N,|0|()) -> isNat#(N)
p48: plus#(N,|0|()) -> U51#(isNat(N),N)
p49: U51#(tt(),N) -> activate#(N)
p50: U51#(tt(),N) -> isNatKind#(activate(N))
p51: U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
p52: U52#(tt(),N) -> activate#(N)
p53: U12#(tt(),V1,V2) -> activate#(V2)
p54: U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
p55: U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
p56: U13#(tt(),V1,V2) -> activate#(V1)
p57: U13#(tt(),V1,V2) -> activate#(V2)
p58: U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
p59: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
p60: U14#(tt(),V1,V2) -> activate#(V2)
p61: U14#(tt(),V1,V2) -> activate#(V1)
p62: U14#(tt(),V1,V2) -> isNat#(activate(V1))
p63: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
p64: U15#(tt(),V2) -> activate#(V2)
p65: U15#(tt(),V2) -> isNat#(activate(V2))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
r2: U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
r3: U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
r4: U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
r5: U15(tt(),V2) -> U16(isNat(activate(V2)))
r6: U16(tt()) -> tt()
r7: U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
r8: U22(tt(),V1) -> U23(isNat(activate(V1)))
r9: U23(tt()) -> tt()
r10: U31(tt(),V2) -> U32(isNatKind(activate(V2)))
r11: U32(tt()) -> tt()
r12: U41(tt()) -> tt()
r13: U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
r14: U52(tt(),N) -> activate(N)
r15: U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
r16: U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
r17: U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
r18: U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
r19: isNat(n__0()) -> tt()
r20: isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
r21: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r22: isNatKind(n__0()) -> tt()
r23: isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
r24: isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
r25: plus(N,|0|()) -> U51(isNat(N),N)
r26: plus(N,s(M)) -> U61(isNat(M),M,N)
r27: |0|() -> n__0()
r28: plus(X1,X2) -> n__plus(X1,X2)
r29: s(X) -> n__s(X)
r30: activate(n__0()) -> |0|()
r31: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r32: activate(n__s(X)) -> s(activate(X))
r33: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          U11#_A(x1,x2,x3) = max{x1 + 62, x2 + 54, x3 + 102}
          tt_A = 17
          U12#_A(x1,x2,x3) = max{x1 + 65, x2 + 53, x3 + 101}
          isNatKind_A(x1) = max{0, x1 - 87}
          activate_A(x1) = x1
          activate#_A(x1) = max{52, x1 - 60}
          n__s_A(x1) = max{122, x1}
          n__plus_A(x1,x2) = max{122, x1 + 59, x2 + 107}
          plus#_A(x1,x2) = max{60, x1 - 1, x2 - 4}
          s_A(x1) = max{122, x1}
          isNat#_A(x1) = max{61, x1 - 4}
          isNatKind#_A(x1) = max{1, x1 - 56}
          U31#_A(x1,x2) = max{65, x2 + 2}
          U21#_A(x1,x2) = max{63, x1 - 23, x2 - 4}
          U22#_A(x1,x2) = max{62, x2 - 4}
          U61#_A(x1,x2,x3) = max{96, x1 - 15, x2 - 4, x3 - 1}
          isNat_A(x1) = max{28, x1 - 71}
          U62#_A(x1,x2,x3) = max{96, x1 - 19, x2 - 4, x3 - 1}
          U63#_A(x1,x2,x3) = max{96, x1 + 68, x2 - 4, x3 - 1}
          U64#_A(x1,x2,x3) = max{x1 + 85, x2 - 4, x3 - 1}
          |0|_A = 105
          U51#_A(x1,x2) = max{53, x1 + 1, x2 - 2}
          U52#_A(x1,x2) = max{53, x1 - 3, x2 - 2}
          U13#_A(x1,x2,x3) = max{x1 + 81, x2 + 52, x3 + 100}
          U14#_A(x1,x2,x3) = max{x1 + 99, x2 + 52, x3 + 100}
          U15#_A(x1,x2) = max{x1 - 1, x2 + 62}
          U16_A(x1) = 18
          U15_A(x1,x2) = 18
          U64_A(x1,x2,x3) = max{x1 + 105, x2 + 107, x3 + 59}
          plus_A(x1,x2) = max{122, x1 + 59, x2 + 107}
          U14_A(x1,x2,x3) = 22
          U63_A(x1,x2,x3) = max{x2 + 107, x3 + 59}
          U13_A(x1,x2,x3) = 23
          U23_A(x1) = 28
          U52_A(x1,x2) = max{x1 - 89, x2 + 1}
          U62_A(x1,x2,x3) = max{109, x1 + 41, x2 + 107, x3 + 59}
          U12_A(x1,x2,x3) = max{26, x1 - 105}
          U22_A(x1,x2) = 28
          U32_A(x1) = max{18, x1 - 89}
          U51_A(x1,x2) = max{x1 - 105, x2 + 2}
          U61_A(x1,x2,x3) = max{108, x1 + 93, x2 + 107, x3 + 59}
          U11_A(x1,x2,x3) = max{x1 + 51, x2 - 12}
          U21_A(x1,x2) = max{29, x1 - 1}
          U31_A(x1,x2) = max{x1 - 105, x2 + 19}
          U41_A(x1) = max{35, x1 - 88}
          n__0_A = 105
    
      precedence:
      
        U11# = tt = U12# = isNatKind = activate = activate# = n__s = n__plus = plus# = s = isNat# = isNatKind# = U31# = U21# = U22# = U61# = isNat = U62# = U63# = U64# = |0| = U51# = U52# = U13# = U14# = U15# = U16 = U15 = U64 = plus = U14 = U63 = U13 = U23 = U52 = U62 = U12 = U22 = U32 = U51 = U61 = U11 = U21 = U31 = U41 = n__0
      
      partial status:
    
        pi(U11#) = []
        pi(tt) = []
        pi(U12#) = []
        pi(isNatKind) = []
        pi(activate) = []
        pi(activate#) = []
        pi(n__s) = []
        pi(n__plus) = []
        pi(plus#) = []
        pi(s) = []
        pi(isNat#) = []
        pi(isNatKind#) = []
        pi(U31#) = []
        pi(U21#) = []
        pi(U22#) = []
        pi(U61#) = []
        pi(isNat) = []
        pi(U62#) = []
        pi(U63#) = []
        pi(U64#) = []
        pi(|0|) = []
        pi(U51#) = []
        pi(U52#) = []
        pi(U13#) = []
        pi(U14#) = []
        pi(U15#) = []
        pi(U16) = []
        pi(U15) = []
        pi(U64) = []
        pi(plus) = []
        pi(U14) = []
        pi(U63) = []
        pi(U13) = []
        pi(U23) = []
        pi(U52) = []
        pi(U62) = []
        pi(U12) = []
        pi(U22) = []
        pi(U32) = []
        pi(U51) = []
        pi(U61) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(n__0) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          U11#_A(x1,x2,x3) = 541
          tt_A = 182
          U12#_A(x1,x2,x3) = 541
          isNatKind_A(x1) = 589
          activate_A(x1) = max{158, x1 + 42}
          activate#_A(x1) = 541
          n__s_A(x1) = 98
          n__plus_A(x1,x2) = 0
          plus#_A(x1,x2) = 540
          s_A(x1) = 141
          isNat#_A(x1) = 159
          isNatKind#_A(x1) = 541
          U31#_A(x1,x2) = 97
          U21#_A(x1,x2) = 159
          U22#_A(x1,x2) = 159
          U61#_A(x1,x2,x3) = 540
          isNat_A(x1) = 540
          U62#_A(x1,x2,x3) = 540
          U63#_A(x1,x2,x3) = 540
          U64#_A(x1,x2,x3) = 540
          |0|_A = 161
          U51#_A(x1,x2) = 541
          U52#_A(x1,x2) = 1
          U13#_A(x1,x2,x3) = 541
          U14#_A(x1,x2,x3) = 157
          U15#_A(x1,x2) = 134
          U16_A(x1) = 622
          U15_A(x1,x2) = 623
          U64_A(x1,x2,x3) = 157
          plus_A(x1,x2) = 157
          U14_A(x1,x2,x3) = 623
          U63_A(x1,x2,x3) = 157
          U13_A(x1,x2,x3) = 539
          U23_A(x1) = 542
          U52_A(x1,x2) = 985
          U62_A(x1,x2,x3) = 157
          U12_A(x1,x2,x3) = 539
          U22_A(x1,x2) = 543
          U32_A(x1) = 589
          U51_A(x1,x2) = 589
          U61_A(x1,x2,x3) = 157
          U11_A(x1,x2,x3) = 540
          U21_A(x1,x2) = 543
          U31_A(x1,x2) = 589
          U41_A(x1) = 97
          n__0_A = 120
    
      precedence:
      
        isNatKind > tt > activate > U31# > U52 > U51# = U52# = plus > U61 > U62 > U63 > U11# > U12# > U13# = U14# > n__plus = isNatKind# = U23 = U51 > activate# = plus# = s = U61# = U62# = U63# = U64# = U64 > n__0 > U16 > isNat# = U21# = U22# = |0| > isNat = U15 = U14 = U22 = U21 > U15# = U12 > n__s = U13 = U11 > U32 = U31 = U41
      
      partial status:
    
        pi(U11#) = []
        pi(tt) = []
        pi(U12#) = []
        pi(isNatKind) = []
        pi(activate) = [1]
        pi(activate#) = []
        pi(n__s) = []
        pi(n__plus) = []
        pi(plus#) = []
        pi(s) = []
        pi(isNat#) = []
        pi(isNatKind#) = []
        pi(U31#) = []
        pi(U21#) = []
        pi(U22#) = []
        pi(U61#) = []
        pi(isNat) = []
        pi(U62#) = []
        pi(U63#) = []
        pi(U64#) = []
        pi(|0|) = []
        pi(U51#) = []
        pi(U52#) = []
        pi(U13#) = []
        pi(U14#) = []
        pi(U15#) = []
        pi(U16) = []
        pi(U15) = []
        pi(U64) = []
        pi(plus) = []
        pi(U14) = []
        pi(U63) = []
        pi(U13) = []
        pi(U23) = []
        pi(U52) = []
        pi(U62) = []
        pi(U12) = []
        pi(U22) = []
        pi(U32) = []
        pi(U51) = []
        pi(U61) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(n__0) = []
    

The next rules are strictly ordered:

  p1, p2, p4, p5, p7, p8, p9, p12, p14, p15, p16, p17, p19, p20, p22, p24, p25, p26, p27, p28, p29, p32, p33, p34, p36, p37, p40, p41, p42, p44, p45, p48, p49, p50, p51, p52, p53, p55, p57, p58, p60, p61, p62, p63, p64, p65

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__s(X)) -> activate#(X)
p2: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p3: isNatKind#(n__s(V1)) -> activate#(V1)
p4: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p5: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p6: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p7: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
p8: U22#(tt(),V1) -> isNat#(activate(V1))
p9: U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
p10: plus#(N,s(M)) -> U61#(isNat(M),M,N)
p11: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
p12: U62#(tt(),M,N) -> isNat#(activate(N))
p13: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
p14: U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
p15: U64#(tt(),M,N) -> plus#(activate(N),activate(M))
p16: plus#(N,|0|()) -> isNat#(N)
p17: U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
p18: U13#(tt(),V1,V2) -> activate#(V1)
p19: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
r2: U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
r3: U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
r4: U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
r5: U15(tt(),V2) -> U16(isNat(activate(V2)))
r6: U16(tt()) -> tt()
r7: U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
r8: U22(tt(),V1) -> U23(isNat(activate(V1)))
r9: U23(tt()) -> tt()
r10: U31(tt(),V2) -> U32(isNatKind(activate(V2)))
r11: U32(tt()) -> tt()
r12: U41(tt()) -> tt()
r13: U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
r14: U52(tt(),N) -> activate(N)
r15: U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
r16: U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
r17: U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
r18: U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
r19: isNat(n__0()) -> tt()
r20: isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
r21: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r22: isNatKind(n__0()) -> tt()
r23: isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
r24: isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
r25: plus(N,|0|()) -> U51(isNat(N),N)
r26: plus(N,s(M)) -> U61(isNat(M),M,N)
r27: |0|() -> n__0()
r28: plus(X1,X2) -> n__plus(X1,X2)
r29: s(X) -> n__s(X)
r30: activate(n__0()) -> |0|()
r31: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r32: activate(n__s(X)) -> s(activate(X))
r33: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p4}
  {p1}
  {p10, p11, p13, p14, p15}
  {p6, p7, p8}


-- Reduction pair.

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

p1: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
r2: U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
r3: U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
r4: U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
r5: U15(tt(),V2) -> U16(isNat(activate(V2)))
r6: U16(tt()) -> tt()
r7: U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
r8: U22(tt(),V1) -> U23(isNat(activate(V1)))
r9: U23(tt()) -> tt()
r10: U31(tt(),V2) -> U32(isNatKind(activate(V2)))
r11: U32(tt()) -> tt()
r12: U41(tt()) -> tt()
r13: U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
r14: U52(tt(),N) -> activate(N)
r15: U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
r16: U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
r17: U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
r18: U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
r19: isNat(n__0()) -> tt()
r20: isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
r21: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r22: isNatKind(n__0()) -> tt()
r23: isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
r24: isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
r25: plus(N,|0|()) -> U51(isNat(N),N)
r26: plus(N,s(M)) -> U61(isNat(M),M,N)
r27: |0|() -> n__0()
r28: plus(X1,X2) -> n__plus(X1,X2)
r29: s(X) -> n__s(X)
r30: activate(n__0()) -> |0|()
r31: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r32: activate(n__s(X)) -> s(activate(X))
r33: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            isNatKind#_A(x1) = ((1,0),(1,0)) x1 + (1,1)
            n__s_A(x1) = ((1,0),(1,0)) x1 + (5,4)
            activate_A(x1) = x1
            U16_A(x1) = (1,8)
            tt_A() = (0,8)
            U15_A(x1,x2) = (8,8)
            isNat_A(x1) = ((0,1),(0,0)) x1 + (0,8)
            U14_A(x1,x2,x3) = ((0,1),(0,0)) x1 + (0,8)
            U13_A(x1,x2,x3) = ((1,1),(0,0)) x3 + (0,8)
            isNatKind_A(x1) = ((1,0),(1,1)) x1
            U23_A(x1) = (1,8)
            U32_A(x1) = ((1,0),(0,0)) x1 + (1,9)
            U64_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (8,7)
            s_A(x1) = ((1,0),(1,0)) x1 + (5,4)
            plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (3,0)
            U12_A(x1,x2,x3) = ((1,1),(0,0)) x3 + (0,8)
            U22_A(x1,x2) = (2,8)
            U31_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,0),(1,0)) x2 + (2,1)
            U41_A(x1) = (4,8)
            U63_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (0,7)
            U11_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (0,8)
            U21_A(x1,x2) = (3,8)
            U52_A(x1,x2) = x2 + (1,9)
            U62_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (8,7)
            n__0_A() = (1,9)
            n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (3,0)
            U51_A(x1,x2) = x2 + (2,9)
            U61_A(x1,x2,x3) = ((0,1),(0,1)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,1)) x3
            |0|_A() = (1,9)
    
      precedence:
      
        activate > |0| > plus = U51 > U52 > U62 = U61 > U63 > U21 > isNatKind > U31 > isNatKind# > U64 > n__0 > U41 > s > n__s > U22 > isNat > U11 > n__plus > U12 > U13 > U14 > U15 > U23 > U16 = tt = U32
      
      partial status:
    
        pi(isNatKind#) = []
        pi(n__s) = []
        pi(activate) = [1]
        pi(U16) = []
        pi(tt) = []
        pi(U15) = []
        pi(isNat) = []
        pi(U14) = []
        pi(U13) = []
        pi(isNatKind) = [1]
        pi(U23) = []
        pi(U32) = []
        pi(U64) = []
        pi(s) = []
        pi(plus) = [2]
        pi(U12) = []
        pi(U22) = []
        pi(U31) = []
        pi(U41) = []
        pi(U63) = []
        pi(U11) = []
        pi(U21) = []
        pi(U52) = []
        pi(U62) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(U51) = []
        pi(U61) = [3]
        pi(|0|) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            isNatKind#_A(x1) = (11,1)
            n__s_A(x1) = (16,2)
            activate_A(x1) = (10,1)
            U16_A(x1) = (12,2)
            tt_A() = (11,2)
            U15_A(x1,x2) = (13,2)
            isNat_A(x1) = (17,3)
            U14_A(x1,x2,x3) = (14,2)
            U13_A(x1,x2,x3) = (14,2)
            isNatKind_A(x1) = x1 + (13,3)
            U23_A(x1) = (0,0)
            U32_A(x1) = (12,2)
            U64_A(x1,x2,x3) = (19,2)
            s_A(x1) = (18,2)
            plus_A(x1,x2) = ((1,0),(0,0)) x2 + (15,5)
            U12_A(x1,x2,x3) = (15,2)
            U22_A(x1,x2) = (0,0)
            U31_A(x1,x2) = (25,2)
            U41_A(x1) = (16,5)
            U63_A(x1,x2,x3) = (20,2)
            U11_A(x1,x2,x3) = (16,3)
            U21_A(x1,x2) = (16,1)
            U52_A(x1,x2) = (11,1)
            U62_A(x1,x2,x3) = (21,2)
            n__0_A() = (1,1)
            n__plus_A(x1,x2) = (24,4)
            U51_A(x1,x2) = (16,3)
            U61_A(x1,x2,x3) = (22,2)
            |0|_A() = (1,1)
    
      precedence:
      
        |0| > plus > n__plus > U51 > n__0 > U31 > U32 > U52 > n__s > isNatKind# = activate > U64 > s > U21 = U61 > isNat > U11 > isNatKind > U12 > U14 = U13 > U15 > U16 = tt > U62 > U41 > U22 > U23 = U63
      
      partial status:
    
        pi(isNatKind#) = []
        pi(n__s) = []
        pi(activate) = []
        pi(U16) = []
        pi(tt) = []
        pi(U15) = []
        pi(isNat) = []
        pi(U14) = []
        pi(U13) = []
        pi(isNatKind) = [1]
        pi(U23) = []
        pi(U32) = []
        pi(U64) = []
        pi(s) = []
        pi(plus) = []
        pi(U12) = []
        pi(U22) = []
        pi(U31) = []
        pi(U41) = []
        pi(U63) = []
        pi(U11) = []
        pi(U21) = []
        pi(U52) = []
        pi(U62) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(U51) = []
        pi(U61) = []
        pi(|0|) = []
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: activate#(n__s(X)) -> activate#(X)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
r2: U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
r3: U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
r4: U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
r5: U15(tt(),V2) -> U16(isNat(activate(V2)))
r6: U16(tt()) -> tt()
r7: U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
r8: U22(tt(),V1) -> U23(isNat(activate(V1)))
r9: U23(tt()) -> tt()
r10: U31(tt(),V2) -> U32(isNatKind(activate(V2)))
r11: U32(tt()) -> tt()
r12: U41(tt()) -> tt()
r13: U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
r14: U52(tt(),N) -> activate(N)
r15: U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
r16: U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
r17: U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
r18: U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
r19: isNat(n__0()) -> tt()
r20: isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
r21: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r22: isNatKind(n__0()) -> tt()
r23: isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
r24: isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
r25: plus(N,|0|()) -> U51(isNat(N),N)
r26: plus(N,s(M)) -> U61(isNat(M),M,N)
r27: |0|() -> n__0()
r28: plus(X1,X2) -> n__plus(X1,X2)
r29: s(X) -> n__s(X)
r30: activate(n__0()) -> |0|()
r31: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r32: activate(n__s(X)) -> s(activate(X))
r33: activate(X) -> X

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          activate#_A(x1) = max{4, x1 + 3}
          n__s_A(x1) = max{3, x1 + 2}
    
      precedence:
      
        activate# = n__s
      
      partial status:
    
        pi(activate#) = [1]
        pi(n__s) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          activate#_A(x1) = max{1, x1 - 1}
          n__s_A(x1) = x1
    
      precedence:
      
        activate# = n__s
      
      partial status:
    
        pi(activate#) = []
        pi(n__s) = [1]
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: plus#(N,s(M)) -> U61#(isNat(M),M,N)
p2: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
p3: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
p4: U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
p5: U64#(tt(),M,N) -> plus#(activate(N),activate(M))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
r2: U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
r3: U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
r4: U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
r5: U15(tt(),V2) -> U16(isNat(activate(V2)))
r6: U16(tt()) -> tt()
r7: U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
r8: U22(tt(),V1) -> U23(isNat(activate(V1)))
r9: U23(tt()) -> tt()
r10: U31(tt(),V2) -> U32(isNatKind(activate(V2)))
r11: U32(tt()) -> tt()
r12: U41(tt()) -> tt()
r13: U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
r14: U52(tt(),N) -> activate(N)
r15: U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
r16: U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
r17: U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
r18: U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
r19: isNat(n__0()) -> tt()
r20: isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
r21: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r22: isNatKind(n__0()) -> tt()
r23: isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
r24: isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
r25: plus(N,|0|()) -> U51(isNat(N),N)
r26: plus(N,s(M)) -> U61(isNat(M),M,N)
r27: |0|() -> n__0()
r28: plus(X1,X2) -> n__plus(X1,X2)
r29: s(X) -> n__s(X)
r30: activate(n__0()) -> |0|()
r31: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r32: activate(n__s(X)) -> s(activate(X))
r33: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            plus#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,1),(1,1)) x2 + (3,0)
            s_A(x1) = x1 + (24,37)
            U61#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,0),(1,0)) x3 + (23,61)
            isNat_A(x1) = ((1,0),(1,1)) x1 + (3,38)
            tt_A() = (6,4)
            U62#_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + ((1,0),(1,0)) x3 + (18,36)
            isNatKind_A(x1) = ((0,1),(0,0)) x1 + (4,4)
            activate_A(x1) = x1 + (2,0)
            U63#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,0),(1,0)) x3 + (13,8)
            U64#_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((1,1),(1,1)) x2 + ((1,0),(1,0)) x3 + (4,4)
            U16_A(x1) = (6,4)
            U15_A(x1,x2) = x1 + (0,36)
            U64_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((0,1),(0,1)) x2 + ((1,1),(0,1)) x3 + (38,68)
            plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (15,31)
            U14_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((1,0),(1,1)) x2 + (1,76)
            U63_A(x1,x2,x3) = ((0,1),(0,1)) x2 + ((1,1),(0,1)) x3 + (45,68)
            U13_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((1,0),(1,1)) x2 + (3,78)
            U23_A(x1) = (7,5)
            U52_A(x1,x2) = x1 + x2 + (0,1)
            U62_A(x1,x2,x3) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + ((1,1),(0,1)) x3 + (44,64)
            U12_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((1,0),(1,1)) x2 + (5,80)
            U22_A(x1,x2) = ((0,0),(0,1)) x2 + (8,37)
            U32_A(x1) = (7,4)
            U51_A(x1,x2) = ((1,1),(0,1)) x2 + (6,5)
            U61_A(x1,x2,x3) = ((0,1),(0,1)) x2 + ((1,1),(0,1)) x3 + (51,68)
            U11_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((1,0),(1,1)) x2 + (8,82)
            U21_A(x1,x2) = x2 + (8,37)
            U31_A(x1,x2) = x1 + ((0,1),(0,0)) x2 + (10,0)
            U41_A(x1) = (23,4)
            |0|_A() = (8,7)
            n__0_A() = (7,7)
            n__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (15,31)
            n__s_A(x1) = x1 + (24,37)
    
      precedence:
      
        U12 > U64# > n__0 > tt = U16 = U32 = U41 > |0| > U61# = U62# > U11 > U51 > U52 > U14 = U13 > activate > plus > n__plus > U63 = U62 = U61 > s = U64 > plus# > isNat = n__s > U23 = U22 = U21 > isNatKind = U31 > U15 > U63#
      
      partial status:
    
        pi(plus#) = [2]
        pi(s) = []
        pi(U61#) = [2]
        pi(isNat) = [1]
        pi(tt) = []
        pi(U62#) = []
        pi(isNatKind) = []
        pi(activate) = [1]
        pi(U63#) = []
        pi(U64#) = []
        pi(U16) = []
        pi(U15) = [1]
        pi(U64) = [3]
        pi(plus) = []
        pi(U14) = []
        pi(U63) = []
        pi(U13) = []
        pi(U23) = []
        pi(U52) = [1, 2]
        pi(U62) = []
        pi(U12) = [2]
        pi(U22) = []
        pi(U32) = []
        pi(U51) = [2]
        pi(U61) = []
        pi(U11) = [2]
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(|0|) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(n__s) = [1]
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            plus#_A(x1,x2) = ((1,0),(1,1)) x2 + (1,0)
            s_A(x1) = (12,4)
            U61#_A(x1,x2,x3) = ((1,1),(0,1)) x2 + (6,17)
            isNat_A(x1) = (0,0)
            tt_A() = (3,3)
            U62#_A(x1,x2,x3) = (5,5)
            isNatKind_A(x1) = (1,2)
            activate_A(x1) = ((0,1),(0,1)) x1 + (9,5)
            U63#_A(x1,x2,x3) = (4,5)
            U64#_A(x1,x2,x3) = (1,1)
            U16_A(x1) = (4,4)
            U15_A(x1,x2) = (4,4)
            U64_A(x1,x2,x3) = x3 + (13,5)
            plus_A(x1,x2) = (7,3)
            U14_A(x1,x2,x3) = (10,4)
            U63_A(x1,x2,x3) = (4,2)
            U13_A(x1,x2,x3) = (11,4)
            U23_A(x1) = (0,0)
            U52_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,1),(0,0)) x2 + (10,2)
            U62_A(x1,x2,x3) = (5,2)
            U12_A(x1,x2,x3) = ((0,1),(0,1)) x2 + (12,5)
            U22_A(x1,x2) = (0,0)
            U32_A(x1) = (4,4)
            U51_A(x1,x2) = ((1,1),(1,0)) x2 + (24,3)
            U61_A(x1,x2,x3) = (6,2)
            U11_A(x1,x2,x3) = ((0,1),(1,0)) x2 + (8,2)
            U21_A(x1,x2) = (0,0)
            U31_A(x1,x2) = (1,1)
            U41_A(x1) = (1,1)
            |0|_A() = (9,5)
            n__0_A() = (4,0)
            n__plus_A(x1,x2) = (8,0)
            n__s_A(x1) = ((1,1),(1,1)) x1 + (13,2)
    
      precedence:
      
        isNat > isNatKind > U21 > U12 = U41 > U62# > s = activate = U64 > plus = U51 = U31 = n__plus > U61 > U62 > U63 > U61# > n__s > U32 > plus# = tt > U63# = U52 = U22 = |0| = n__0 > U11 > U13 > U14 > U23 > U15 > U16 > U64#
      
      partial status:
    
        pi(plus#) = []
        pi(s) = []
        pi(U61#) = []
        pi(isNat) = []
        pi(tt) = []
        pi(U62#) = []
        pi(isNatKind) = []
        pi(activate) = []
        pi(U63#) = []
        pi(U64#) = []
        pi(U16) = []
        pi(U15) = []
        pi(U64) = [3]
        pi(plus) = []
        pi(U14) = []
        pi(U63) = []
        pi(U13) = []
        pi(U23) = []
        pi(U52) = []
        pi(U62) = []
        pi(U12) = []
        pi(U22) = []
        pi(U32) = []
        pi(U51) = []
        pi(U61) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(|0|) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(n__s) = []
    

The next rules are strictly ordered:

  p1, p2, p5

We remove them from the problem.

-- SCC decomposition.

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

p1: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
p2: U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
r2: U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
r3: U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
r4: U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
r5: U15(tt(),V2) -> U16(isNat(activate(V2)))
r6: U16(tt()) -> tt()
r7: U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
r8: U22(tt(),V1) -> U23(isNat(activate(V1)))
r9: U23(tt()) -> tt()
r10: U31(tt(),V2) -> U32(isNatKind(activate(V2)))
r11: U32(tt()) -> tt()
r12: U41(tt()) -> tt()
r13: U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
r14: U52(tt(),N) -> activate(N)
r15: U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
r16: U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
r17: U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
r18: U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
r19: isNat(n__0()) -> tt()
r20: isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
r21: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r22: isNatKind(n__0()) -> tt()
r23: isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
r24: isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
r25: plus(N,|0|()) -> U51(isNat(N),N)
r26: plus(N,s(M)) -> U61(isNat(M),M,N)
r27: |0|() -> n__0()
r28: plus(X1,X2) -> n__plus(X1,X2)
r29: s(X) -> n__s(X)
r30: activate(n__0()) -> |0|()
r31: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r32: activate(n__s(X)) -> s(activate(X))
r33: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p2: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
p3: U22#(tt(),V1) -> isNat#(activate(V1))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
r2: U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
r3: U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
r4: U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
r5: U15(tt(),V2) -> U16(isNat(activate(V2)))
r6: U16(tt()) -> tt()
r7: U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
r8: U22(tt(),V1) -> U23(isNat(activate(V1)))
r9: U23(tt()) -> tt()
r10: U31(tt(),V2) -> U32(isNatKind(activate(V2)))
r11: U32(tt()) -> tt()
r12: U41(tt()) -> tt()
r13: U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
r14: U52(tt(),N) -> activate(N)
r15: U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
r16: U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
r17: U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
r18: U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
r19: isNat(n__0()) -> tt()
r20: isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
r21: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r22: isNatKind(n__0()) -> tt()
r23: isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
r24: isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
r25: plus(N,|0|()) -> U51(isNat(N),N)
r26: plus(N,s(M)) -> U61(isNat(M),M,N)
r27: |0|() -> n__0()
r28: plus(X1,X2) -> n__plus(X1,X2)
r29: s(X) -> n__s(X)
r30: activate(n__0()) -> |0|()
r31: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r32: activate(n__s(X)) -> s(activate(X))
r33: activate(X) -> X

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            isNat#_A(x1) = ((1,1),(1,1)) x1 + (7,7)
            n__s_A(x1) = ((0,1),(1,0)) x1 + (8,24)
            U21#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,1),(1,1)) x2 + (6,25)
            isNatKind_A(x1) = ((0,0),(1,1)) x1 + (10,11)
            activate_A(x1) = x1
            tt_A() = (6,6)
            U22#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,1),(1,1)) x2 + (1,7)
            U16_A(x1) = ((0,1),(0,0)) x1 + (1,6)
            U15_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(0,0)) x2 + (1,0)
            isNat_A(x1) = ((1,1),(0,0)) x1 + (9,11)
            U14_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (22,11)
            U13_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (17,11)
            U23_A(x1) = (7,7)
            U64_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (28,24)
            s_A(x1) = ((0,1),(1,0)) x1 + (8,24)
            plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (0,19)
            U12_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (27,11)
            U22_A(x1,x2) = (7,7)
            U63_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (29,24)
            U11_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (27,11)
            U21_A(x1,x2) = ((0,0),(1,0)) x1 + (7,1)
            U52_A(x1,x2) = x2 + (7,7)
            U62_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (30,24)
            U32_A(x1) = (7,6)
            U51_A(x1,x2) = x2 + (11,11)
            U61_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (31,51)
            n__0_A() = (7,11)
            n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (0,19)
            U31_A(x1,x2) = (8,20)
            U41_A(x1) = (9,6)
            |0|_A() = (7,11)
    
      precedence:
      
        U15 = U52 = U51 > U64 = U63 = U62 = U61 > n__s = activate = isNat = s = plus = U11 = n__plus = |0| > isNatKind = U31 > U41 > U16 = U13 = U12 > tt = U22# = U23 = U22 = U21 = U32 = n__0 > isNat# > U21# = U14
      
      partial status:
    
        pi(isNat#) = [1]
        pi(n__s) = []
        pi(U21#) = []
        pi(isNatKind) = []
        pi(activate) = [1]
        pi(tt) = []
        pi(U22#) = []
        pi(U16) = []
        pi(U15) = []
        pi(isNat) = []
        pi(U14) = []
        pi(U13) = []
        pi(U23) = []
        pi(U64) = []
        pi(s) = []
        pi(plus) = []
        pi(U12) = []
        pi(U22) = []
        pi(U63) = []
        pi(U11) = []
        pi(U21) = []
        pi(U52) = []
        pi(U62) = []
        pi(U32) = []
        pi(U51) = []
        pi(U61) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(U31) = []
        pi(U41) = []
        pi(|0|) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            isNat#_A(x1) = ((1,0),(0,0)) x1 + (7,1)
            n__s_A(x1) = (0,0)
            U21#_A(x1,x2) = (6,1)
            isNatKind_A(x1) = (6,2)
            activate_A(x1) = ((0,0),(1,1)) x1 + (13,2)
            tt_A() = (0,2)
            U22#_A(x1,x2) = (1,1)
            U16_A(x1) = (1,2)
            U15_A(x1,x2) = (2,2)
            isNat_A(x1) = (9,3)
            U14_A(x1,x2,x3) = (3,2)
            U13_A(x1,x2,x3) = (3,2)
            U23_A(x1) = (1,2)
            U64_A(x1,x2,x3) = (2,0)
            s_A(x1) = (1,0)
            plus_A(x1,x2) = (12,3)
            U12_A(x1,x2,x3) = (4,2)
            U22_A(x1,x2) = (7,3)
            U63_A(x1,x2,x3) = (7,0)
            U11_A(x1,x2,x3) = (5,2)
            U21_A(x1,x2) = (8,3)
            U52_A(x1,x2) = (1,3)
            U62_A(x1,x2,x3) = (8,0)
            U32_A(x1) = (1,2)
            U51_A(x1,x2) = (11,4)
            U61_A(x1,x2,x3) = (12,1)
            n__0_A() = (11,0)
            n__plus_A(x1,x2) = (7,2)
            U31_A(x1,x2) = (1,0)
            U41_A(x1) = (6,3)
            |0|_A() = (10,3)
    
      precedence:
      
        |0| > n__0 > U52 > U51 > U13 > U14 > isNat > U21 > U22 > U23 > s > isNatKind > U41 > tt > U11 > isNat# = activate = plus = U12 > U21# > n__plus > U61 > U62 > U22# > U31 > U63 > U64 > U15 > U16 > n__s = U32
      
      partial status:
    
        pi(isNat#) = []
        pi(n__s) = []
        pi(U21#) = []
        pi(isNatKind) = []
        pi(activate) = []
        pi(tt) = []
        pi(U22#) = []
        pi(U16) = []
        pi(U15) = []
        pi(isNat) = []
        pi(U14) = []
        pi(U13) = []
        pi(U23) = []
        pi(U64) = []
        pi(s) = []
        pi(plus) = []
        pi(U12) = []
        pi(U22) = []
        pi(U63) = []
        pi(U11) = []
        pi(U21) = []
        pi(U52) = []
        pi(U62) = []
        pi(U32) = []
        pi(U51) = []
        pi(U61) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(U31) = []
        pi(U41) = []
        pi(|0|) = []
    

The next rules are strictly ordered:

  p1, p3

We remove them from the problem.

-- SCC decomposition.

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

p1: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
r2: U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
r3: U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
r4: U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
r5: U15(tt(),V2) -> U16(isNat(activate(V2)))
r6: U16(tt()) -> tt()
r7: U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
r8: U22(tt(),V1) -> U23(isNat(activate(V1)))
r9: U23(tt()) -> tt()
r10: U31(tt(),V2) -> U32(isNatKind(activate(V2)))
r11: U32(tt()) -> tt()
r12: U41(tt()) -> tt()
r13: U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
r14: U52(tt(),N) -> activate(N)
r15: U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
r16: U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
r17: U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
r18: U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
r19: isNat(n__0()) -> tt()
r20: isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
r21: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r22: isNatKind(n__0()) -> tt()
r23: isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
r24: isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
r25: plus(N,|0|()) -> U51(isNat(N),N)
r26: plus(N,s(M)) -> U61(isNat(M),M,N)
r27: |0|() -> n__0()
r28: plus(X1,X2) -> n__plus(X1,X2)
r29: s(X) -> n__s(X)
r30: activate(n__0()) -> |0|()
r31: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r32: activate(n__s(X)) -> s(activate(X))
r33: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  (no SCCs)