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(X1,X2)
  activate(n__s(X)) -> s(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#(X1,X2)
p69: activate#(n__s(X)) -> s#(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(X1,X2)
r32: activate(n__s(X)) -> s(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}


-- 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__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,s(M)) -> isNat#(M)
p5: isNat#(n__s(V1)) -> activate#(V1)
p6: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p7: isNatKind#(n__s(V1)) -> activate#(V1)
p8: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p10: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p11: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p12: isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
p13: U31#(tt(),V2) -> activate#(V2)
p14: U31#(tt(),V2) -> isNatKind#(activate(V2))
p15: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p16: U21#(tt(),V1) -> activate#(V1)
p17: U21#(tt(),V1) -> isNatKind#(activate(V1))
p18: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
p19: U22#(tt(),V1) -> activate#(V1)
p20: U22#(tt(),V1) -> isNat#(activate(V1))
p21: isNat#(n__plus(V1,V2)) -> activate#(V2)
p22: isNat#(n__plus(V1,V2)) -> activate#(V1)
p23: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p24: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
p25: U11#(tt(),V1,V2) -> activate#(V2)
p26: U11#(tt(),V1,V2) -> activate#(V1)
p27: U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
p28: plus#(N,s(M)) -> U61#(isNat(M),M,N)
p29: U61#(tt(),M,N) -> activate#(N)
p30: U61#(tt(),M,N) -> activate#(M)
p31: U61#(tt(),M,N) -> isNatKind#(activate(M))
p32: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
p33: U62#(tt(),M,N) -> activate#(M)
p34: U62#(tt(),M,N) -> activate#(N)
p35: U62#(tt(),M,N) -> isNat#(activate(N))
p36: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
p37: U63#(tt(),M,N) -> activate#(M)
p38: U63#(tt(),M,N) -> activate#(N)
p39: U63#(tt(),M,N) -> isNatKind#(activate(N))
p40: U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
p41: U64#(tt(),M,N) -> activate#(M)
p42: U64#(tt(),M,N) -> activate#(N)
p43: U64#(tt(),M,N) -> plus#(activate(N),activate(M))
p44: plus#(N,|0|()) -> isNat#(N)
p45: plus#(N,|0|()) -> U51#(isNat(N),N)
p46: U51#(tt(),N) -> activate#(N)
p47: U51#(tt(),N) -> isNatKind#(activate(N))
p48: U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
p49: U52#(tt(),N) -> activate#(N)
p50: U12#(tt(),V1,V2) -> activate#(V2)
p51: U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
p52: U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
p53: U13#(tt(),V1,V2) -> activate#(V1)
p54: U13#(tt(),V1,V2) -> activate#(V2)
p55: U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
p56: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
p57: U14#(tt(),V1,V2) -> activate#(V2)
p58: U14#(tt(),V1,V2) -> activate#(V1)
p59: U14#(tt(),V1,V2) -> isNat#(activate(V1))
p60: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
p61: U15#(tt(),V2) -> activate#(V2)
p62: 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(X1,X2)
r32: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2,x3) = x1 + x2 + x3 + 9
          tt_A() = 7
          U12#_A(x1,x2,x3) = x1 + x2 + x3 + 5
          isNatKind_A(x1) = 10
          activate_A(x1) = x1
          activate#_A(x1) = x1 + 4
          n__plus_A(x1,x2) = x1 + x2 + 20
          plus#_A(x1,x2) = x1 + x2 + 20
          s_A(x1) = x1 + 11
          isNat#_A(x1) = x1 + 4
          n__s_A(x1) = x1 + 11
          isNatKind#_A(x1) = x1 + 6
          U31#_A(x1,x2) = x2 + 26
          U21#_A(x1,x2) = x2 + 14
          U22#_A(x1,x2) = x2 + 4
          U61#_A(x1,x2,x3) = x2 + x3 + 30
          isNat_A(x1) = x1 + 6
          U62#_A(x1,x2,x3) = x1 + x2 + x3 + 19
          U63#_A(x1,x2,x3) = x2 + x3 + 25
          U64#_A(x1,x2,x3) = x1 + x2 + x3 + 14
          |0|_A() = 13
          U51#_A(x1,x2) = x2 + 12
          U52#_A(x1,x2) = x1 + x2 + 1
          U13#_A(x1,x2,x3) = x1 + x2 + x3 + 1
          U14#_A(x1,x2,x3) = x2 + x3 + 6
          U15#_A(x1,x2) = x2 + 5
          U16_A(x1) = 8
          U15_A(x1,x2) = 9
          U64_A(x1,x2,x3) = x2 + x3 + 31
          plus_A(x1,x2) = x1 + x2 + 20
          U14_A(x1,x2,x3) = 10
          U63_A(x1,x2,x3) = x2 + x3 + 31
          U13_A(x1,x2,x3) = x1 + 4
          U23_A(x1) = 8
          U52_A(x1,x2) = x2 + 1
          U62_A(x1,x2,x3) = x2 + x3 + 31
          U12_A(x1,x2,x3) = 15
          U22_A(x1,x2) = x1 + 2
          U32_A(x1) = 8
          U51_A(x1,x2) = x2 + 2
          U61_A(x1,x2,x3) = x2 + x3 + 31
          U11_A(x1,x2,x3) = 16
          U21_A(x1,x2) = x2 + 13
          U31_A(x1,x2) = 9
          U41_A(x1) = 9
          n__0_A() = 13
  
    precedence:
    
      U52 = U51 = U41 > U32 > U21# > U22# > U14# > |0| = U13# = n__0 > U15 = U22 > activate > plus = U62 = U61 > U64 = U63 > U16 > U52# > isNatKind# = U31# = U51# > U11 > isNat > U64# > U63# > U12 > U14 = U13 > n__plus = plus# > isNat# > U12# > U61# > U62# > U11# > isNatKind > U31 > U21 > tt = activate# > U15# > U23 > s = n__s
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(U12#) = [3]
      pi(isNatKind) = []
      pi(activate) = [1]
      pi(activate#) = []
      pi(n__plus) = [2]
      pi(plus#) = []
      pi(s) = []
      pi(isNat#) = [1]
      pi(n__s) = []
      pi(isNatKind#) = []
      pi(U31#) = []
      pi(U21#) = []
      pi(U22#) = [2]
      pi(U61#) = []
      pi(isNat) = [1]
      pi(U62#) = [1]
      pi(U63#) = []
      pi(U64#) = []
      pi(|0|) = []
      pi(U51#) = []
      pi(U52#) = [2]
      pi(U13#) = []
      pi(U14#) = []
      pi(U15#) = [2]
      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) = [2]
      pi(U31) = []
      pi(U41) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p49

We remove them from the problem.

-- 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: U12#(tt(),V1,V2) -> activate#(V1)
p3: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,s(M)) -> isNat#(M)
p5: isNat#(n__s(V1)) -> activate#(V1)
p6: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p7: isNatKind#(n__s(V1)) -> activate#(V1)
p8: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p10: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p11: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p12: isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
p13: U31#(tt(),V2) -> activate#(V2)
p14: U31#(tt(),V2) -> isNatKind#(activate(V2))
p15: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p16: U21#(tt(),V1) -> activate#(V1)
p17: U21#(tt(),V1) -> isNatKind#(activate(V1))
p18: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
p19: U22#(tt(),V1) -> activate#(V1)
p20: U22#(tt(),V1) -> isNat#(activate(V1))
p21: isNat#(n__plus(V1,V2)) -> activate#(V2)
p22: isNat#(n__plus(V1,V2)) -> activate#(V1)
p23: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p24: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
p25: U11#(tt(),V1,V2) -> activate#(V2)
p26: U11#(tt(),V1,V2) -> activate#(V1)
p27: U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
p28: plus#(N,s(M)) -> U61#(isNat(M),M,N)
p29: U61#(tt(),M,N) -> activate#(N)
p30: U61#(tt(),M,N) -> activate#(M)
p31: U61#(tt(),M,N) -> isNatKind#(activate(M))
p32: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
p33: U62#(tt(),M,N) -> activate#(M)
p34: U62#(tt(),M,N) -> activate#(N)
p35: U62#(tt(),M,N) -> isNat#(activate(N))
p36: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
p37: U63#(tt(),M,N) -> activate#(M)
p38: U63#(tt(),M,N) -> activate#(N)
p39: U63#(tt(),M,N) -> isNatKind#(activate(N))
p40: U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
p41: U64#(tt(),M,N) -> activate#(M)
p42: U64#(tt(),M,N) -> activate#(N)
p43: U64#(tt(),M,N) -> plus#(activate(N),activate(M))
p44: plus#(N,|0|()) -> isNat#(N)
p45: plus#(N,|0|()) -> U51#(isNat(N),N)
p46: U51#(tt(),N) -> activate#(N)
p47: U51#(tt(),N) -> isNatKind#(activate(N))
p48: U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
p49: U12#(tt(),V1,V2) -> activate#(V2)
p50: U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
p51: U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
p52: U13#(tt(),V1,V2) -> activate#(V1)
p53: U13#(tt(),V1,V2) -> activate#(V2)
p54: U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
p55: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
p56: U14#(tt(),V1,V2) -> activate#(V2)
p57: U14#(tt(),V1,V2) -> activate#(V1)
p58: U14#(tt(),V1,V2) -> isNat#(activate(V1))
p59: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
p60: U15#(tt(),V2) -> activate#(V2)
p61: 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(X1,X2)
r32: activate(n__s(X)) -> s(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, 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, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61}


-- 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) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
p3: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
p4: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
p5: U15#(tt(),V2) -> isNat#(activate(V2))
p6: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
p7: U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
p8: isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
p9: U31#(tt(),V2) -> isNatKind#(activate(V2))
p10: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p11: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p12: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p13: plus#(N,|0|()) -> U51#(isNat(N),N)
p14: U51#(tt(),N) -> isNatKind#(activate(N))
p15: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p16: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p17: isNatKind#(n__s(V1)) -> activate#(V1)
p18: U51#(tt(),N) -> activate#(N)
p19: plus#(N,|0|()) -> isNat#(N)
p20: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p21: isNat#(n__plus(V1,V2)) -> activate#(V1)
p22: isNat#(n__plus(V1,V2)) -> activate#(V2)
p23: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p24: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
p25: U22#(tt(),V1) -> isNat#(activate(V1))
p26: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p27: isNat#(n__s(V1)) -> activate#(V1)
p28: U22#(tt(),V1) -> activate#(V1)
p29: U21#(tt(),V1) -> isNatKind#(activate(V1))
p30: U21#(tt(),V1) -> activate#(V1)
p31: plus#(N,s(M)) -> U61#(isNat(M),M,N)
p32: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
p33: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
p34: U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
p35: U64#(tt(),M,N) -> plus#(activate(N),activate(M))
p36: plus#(N,s(M)) -> isNat#(M)
p37: U64#(tt(),M,N) -> activate#(N)
p38: U64#(tt(),M,N) -> activate#(M)
p39: U63#(tt(),M,N) -> isNatKind#(activate(N))
p40: U63#(tt(),M,N) -> activate#(N)
p41: U63#(tt(),M,N) -> activate#(M)
p42: U62#(tt(),M,N) -> isNat#(activate(N))
p43: U62#(tt(),M,N) -> activate#(N)
p44: U62#(tt(),M,N) -> activate#(M)
p45: U61#(tt(),M,N) -> isNatKind#(activate(M))
p46: U61#(tt(),M,N) -> activate#(M)
p47: U61#(tt(),M,N) -> activate#(N)
p48: U31#(tt(),V2) -> activate#(V2)
p49: U11#(tt(),V1,V2) -> activate#(V1)
p50: U11#(tt(),V1,V2) -> activate#(V2)
p51: U15#(tt(),V2) -> activate#(V2)
p52: U14#(tt(),V1,V2) -> isNat#(activate(V1))
p53: U14#(tt(),V1,V2) -> activate#(V1)
p54: U14#(tt(),V1,V2) -> activate#(V2)
p55: U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
p56: U13#(tt(),V1,V2) -> activate#(V2)
p57: U13#(tt(),V1,V2) -> activate#(V1)
p58: U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
p59: U12#(tt(),V1,V2) -> activate#(V2)
p60: U12#(tt(),V1,V2) -> 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(X1,X2)
r32: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2,x3) = x2 + x3 + 11
          tt_A() = 6
          U12#_A(x1,x2,x3) = x2 + x3 + 10
          isNatKind_A(x1) = x1 + 2
          activate_A(x1) = x1
          U13#_A(x1,x2,x3) = x2 + x3 + 9
          U14#_A(x1,x2,x3) = x2 + x3 + 8
          U15#_A(x1,x2) = x1 + x2 + 1
          isNat_A(x1) = x1 + 6
          isNat#_A(x1) = x1
          n__plus_A(x1,x2) = x1 + x2 + 12
          isNatKind#_A(x1) = x1
          U31#_A(x1,x2) = x2 + 12
          activate#_A(x1) = x1
          plus#_A(x1,x2) = x1 + x2
          |0|_A() = 7
          U51#_A(x1,x2) = x2 + 1
          n__s_A(x1) = x1 + 9
          U21#_A(x1,x2) = x2 + 8
          U22#_A(x1,x2) = x2 + 7
          s_A(x1) = x1 + 9
          U61#_A(x1,x2,x3) = x2 + x3
          U62#_A(x1,x2,x3) = x2 + x3
          U63#_A(x1,x2,x3) = x2 + x3
          U64#_A(x1,x2,x3) = x2 + x3
          U16_A(x1) = 7
          U15_A(x1,x2) = 8
          U64_A(x1,x2,x3) = x2 + x3 + 21
          plus_A(x1,x2) = x1 + x2 + 12
          U14_A(x1,x2,x3) = x1 + 3
          U63_A(x1,x2,x3) = x2 + x3 + 21
          U13_A(x1,x2,x3) = x3 + 7
          U23_A(x1) = 7
          U52_A(x1,x2) = x2 + 7
          U62_A(x1,x2,x3) = x2 + x3 + 21
          U12_A(x1,x2,x3) = x3 + 8
          U22_A(x1,x2) = 8
          U32_A(x1) = 7
          U51_A(x1,x2) = x2 + 8
          U61_A(x1,x2,x3) = x2 + x3 + 21
          U11_A(x1,x2,x3) = x1 + x3 + 3
          U21_A(x1,x2) = x1 + 3
          U31_A(x1,x2) = x1 + 11
          U41_A(x1) = 7
          n__0_A() = 7
  
    precedence:
    
      activate = U61# = U62# = U63# = U32 = U31 > U51# > |0| > n__0 > U41 > tt = isNat = U16 = U22 = U21 > U13# = U64# > U14# > U11# > isNatKind = isNat# = U21# = U22# > U12# > U15# > activate# = plus# > U13 = U12 = U11 > U14 = U23 > n__plus = isNatKind# = U31# = plus = U51 > U52 > n__s = s = U15 = U64 = U63 = U62 = U61
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(U12#) = []
      pi(isNatKind) = []
      pi(activate) = []
      pi(U13#) = []
      pi(U14#) = []
      pi(U15#) = [1]
      pi(isNat) = [1]
      pi(isNat#) = []
      pi(n__plus) = []
      pi(isNatKind#) = []
      pi(U31#) = []
      pi(activate#) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(U51#) = []
      pi(n__s) = []
      pi(U21#) = [2]
      pi(U22#) = []
      pi(s) = []
      pi(U61#) = []
      pi(U62#) = []
      pi(U63#) = []
      pi(U64#) = []
      pi(U16) = []
      pi(U15) = []
      pi(U64) = []
      pi(plus) = [1, 2]
      pi(U14) = []
      pi(U63) = []
      pi(U13) = []
      pi(U23) = []
      pi(U52) = []
      pi(U62) = []
      pi(U12) = []
      pi(U22) = []
      pi(U32) = []
      pi(U51) = [2]
      pi(U61) = []
      pi(U11) = []
      pi(U21) = [1]
      pi(U31) = []
      pi(U41) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p51

We remove them from the problem.

-- 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: U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
p3: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
p4: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
p5: U15#(tt(),V2) -> isNat#(activate(V2))
p6: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
p7: U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
p8: isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
p9: U31#(tt(),V2) -> isNatKind#(activate(V2))
p10: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p11: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p12: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p13: plus#(N,|0|()) -> U51#(isNat(N),N)
p14: U51#(tt(),N) -> isNatKind#(activate(N))
p15: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p16: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p17: isNatKind#(n__s(V1)) -> activate#(V1)
p18: U51#(tt(),N) -> activate#(N)
p19: plus#(N,|0|()) -> isNat#(N)
p20: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p21: isNat#(n__plus(V1,V2)) -> activate#(V1)
p22: isNat#(n__plus(V1,V2)) -> activate#(V2)
p23: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p24: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
p25: U22#(tt(),V1) -> isNat#(activate(V1))
p26: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p27: isNat#(n__s(V1)) -> activate#(V1)
p28: U22#(tt(),V1) -> activate#(V1)
p29: U21#(tt(),V1) -> isNatKind#(activate(V1))
p30: U21#(tt(),V1) -> activate#(V1)
p31: plus#(N,s(M)) -> U61#(isNat(M),M,N)
p32: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
p33: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
p34: U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
p35: U64#(tt(),M,N) -> plus#(activate(N),activate(M))
p36: plus#(N,s(M)) -> isNat#(M)
p37: U64#(tt(),M,N) -> activate#(N)
p38: U64#(tt(),M,N) -> activate#(M)
p39: U63#(tt(),M,N) -> isNatKind#(activate(N))
p40: U63#(tt(),M,N) -> activate#(N)
p41: U63#(tt(),M,N) -> activate#(M)
p42: U62#(tt(),M,N) -> isNat#(activate(N))
p43: U62#(tt(),M,N) -> activate#(N)
p44: U62#(tt(),M,N) -> activate#(M)
p45: U61#(tt(),M,N) -> isNatKind#(activate(M))
p46: U61#(tt(),M,N) -> activate#(M)
p47: U61#(tt(),M,N) -> activate#(N)
p48: U31#(tt(),V2) -> activate#(V2)
p49: U11#(tt(),V1,V2) -> activate#(V1)
p50: U11#(tt(),V1,V2) -> activate#(V2)
p51: U14#(tt(),V1,V2) -> isNat#(activate(V1))
p52: U14#(tt(),V1,V2) -> activate#(V1)
p53: U14#(tt(),V1,V2) -> activate#(V2)
p54: U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
p55: U13#(tt(),V1,V2) -> activate#(V2)
p56: U13#(tt(),V1,V2) -> activate#(V1)
p57: U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
p58: U12#(tt(),V1,V2) -> activate#(V2)
p59: U12#(tt(),V1,V2) -> 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(X1,X2)
r32: activate(n__s(X)) -> s(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, 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}


-- 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__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,s(M)) -> isNat#(M)
p5: isNat#(n__s(V1)) -> activate#(V1)
p6: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p7: isNatKind#(n__s(V1)) -> activate#(V1)
p8: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p10: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p11: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p12: isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
p13: U31#(tt(),V2) -> activate#(V2)
p14: U31#(tt(),V2) -> isNatKind#(activate(V2))
p15: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p16: U21#(tt(),V1) -> activate#(V1)
p17: U21#(tt(),V1) -> isNatKind#(activate(V1))
p18: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
p19: U22#(tt(),V1) -> activate#(V1)
p20: U22#(tt(),V1) -> isNat#(activate(V1))
p21: isNat#(n__plus(V1,V2)) -> activate#(V2)
p22: isNat#(n__plus(V1,V2)) -> activate#(V1)
p23: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p24: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
p25: U11#(tt(),V1,V2) -> activate#(V2)
p26: U11#(tt(),V1,V2) -> activate#(V1)
p27: U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
p28: plus#(N,s(M)) -> U61#(isNat(M),M,N)
p29: U61#(tt(),M,N) -> activate#(N)
p30: U61#(tt(),M,N) -> activate#(M)
p31: U61#(tt(),M,N) -> isNatKind#(activate(M))
p32: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
p33: U62#(tt(),M,N) -> activate#(M)
p34: U62#(tt(),M,N) -> activate#(N)
p35: U62#(tt(),M,N) -> isNat#(activate(N))
p36: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
p37: U63#(tt(),M,N) -> activate#(M)
p38: U63#(tt(),M,N) -> activate#(N)
p39: U63#(tt(),M,N) -> isNatKind#(activate(N))
p40: U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
p41: U64#(tt(),M,N) -> activate#(M)
p42: U64#(tt(),M,N) -> activate#(N)
p43: U64#(tt(),M,N) -> plus#(activate(N),activate(M))
p44: plus#(N,|0|()) -> isNat#(N)
p45: plus#(N,|0|()) -> U51#(isNat(N),N)
p46: U51#(tt(),N) -> activate#(N)
p47: U51#(tt(),N) -> isNatKind#(activate(N))
p48: U12#(tt(),V1,V2) -> activate#(V2)
p49: U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
p50: U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
p51: U13#(tt(),V1,V2) -> activate#(V1)
p52: U13#(tt(),V1,V2) -> activate#(V2)
p53: U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
p54: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
p55: U14#(tt(),V1,V2) -> activate#(V2)
p56: U14#(tt(),V1,V2) -> activate#(V1)
p57: U14#(tt(),V1,V2) -> isNat#(activate(V1))
p58: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
p59: 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(X1,X2)
r32: activate(n__s(X)) -> s(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:

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

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- 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: U12#(tt(),V1,V2) -> activate#(V1)
p3: plus#(N,s(M)) -> isNat#(M)
p4: isNat#(n__s(V1)) -> activate#(V1)
p5: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p6: isNatKind#(n__s(V1)) -> activate#(V1)
p7: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p8: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p9: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p10: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p11: isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
p12: U31#(tt(),V2) -> activate#(V2)
p13: U31#(tt(),V2) -> isNatKind#(activate(V2))
p14: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p15: U21#(tt(),V1) -> activate#(V1)
p16: U21#(tt(),V1) -> isNatKind#(activate(V1))
p17: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
p18: U22#(tt(),V1) -> activate#(V1)
p19: U22#(tt(),V1) -> isNat#(activate(V1))
p20: isNat#(n__plus(V1,V2)) -> activate#(V2)
p21: isNat#(n__plus(V1,V2)) -> activate#(V1)
p22: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p23: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
p24: U11#(tt(),V1,V2) -> activate#(V2)
p25: U11#(tt(),V1,V2) -> activate#(V1)
p26: U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
p27: plus#(N,s(M)) -> U61#(isNat(M),M,N)
p28: U61#(tt(),M,N) -> activate#(N)
p29: U61#(tt(),M,N) -> activate#(M)
p30: U61#(tt(),M,N) -> isNatKind#(activate(M))
p31: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
p32: U62#(tt(),M,N) -> activate#(M)
p33: U62#(tt(),M,N) -> activate#(N)
p34: U62#(tt(),M,N) -> isNat#(activate(N))
p35: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
p36: U63#(tt(),M,N) -> activate#(M)
p37: U63#(tt(),M,N) -> activate#(N)
p38: U63#(tt(),M,N) -> isNatKind#(activate(N))
p39: U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
p40: U64#(tt(),M,N) -> activate#(M)
p41: U64#(tt(),M,N) -> activate#(N)
p42: U64#(tt(),M,N) -> plus#(activate(N),activate(M))
p43: plus#(N,|0|()) -> isNat#(N)
p44: plus#(N,|0|()) -> U51#(isNat(N),N)
p45: U51#(tt(),N) -> activate#(N)
p46: U51#(tt(),N) -> isNatKind#(activate(N))
p47: U12#(tt(),V1,V2) -> activate#(V2)
p48: U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
p49: U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
p50: U13#(tt(),V1,V2) -> activate#(V1)
p51: U13#(tt(),V1,V2) -> activate#(V2)
p52: U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
p53: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
p54: U14#(tt(),V1,V2) -> activate#(V2)
p55: U14#(tt(),V1,V2) -> activate#(V1)
p56: U14#(tt(),V1,V2) -> isNat#(activate(V1))
p57: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
p58: 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(X1,X2)
r32: activate(n__s(X)) -> s(X)
r33: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p27, p31, p35, p39, p42}
  {p1, p14, p17, p19, p23, p49, p53, p56, p57, p58}
  {p7, p10, p11, p13}


-- Reduction pair.

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

p1: U64#(tt(),M,N) -> plus#(activate(N),activate(M))
p2: plus#(N,s(M)) -> U61#(isNat(M),M,N)
p3: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
p4: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
p5: 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(X1,X2)
r32: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          U64#_A(x1,x2,x3) = x1 + x2 + 1
          tt_A() = 2
          plus#_A(x1,x2) = x2 + 1
          activate_A(x1) = x1
          s_A(x1) = x1 + 17
          U61#_A(x1,x2,x3) = x2 + 16
          isNat_A(x1) = x1 + 24
          U62#_A(x1,x2,x3) = x1 + x2 + 8
          isNatKind_A(x1) = 7
          U63#_A(x1,x2,x3) = x2 + 9
          U16_A(x1) = 3
          U15_A(x1,x2) = 25
          U64_A(x1,x2,x3) = x2 + x3 + 23
          plus_A(x1,x2) = x1 + x2 + 6
          U14_A(x1,x2,x3) = 26
          U63_A(x1,x2,x3) = x2 + x3 + 23
          U13_A(x1,x2,x3) = 27
          U23_A(x1) = 3
          U52_A(x1,x2) = x2 + 3
          U62_A(x1,x2,x3) = x2 + x3 + 23
          U12_A(x1,x2,x3) = 28
          U22_A(x1,x2) = 4
          U32_A(x1) = 3
          U51_A(x1,x2) = x2 + 4
          U61_A(x1,x2,x3) = x2 + x3 + 23
          U11_A(x1,x2,x3) = 29
          U21_A(x1,x2) = x2 + 5
          U31_A(x1,x2) = 4
          U41_A(x1) = x1
          |0|_A() = 17
          n__0_A() = 17
          n__plus_A(x1,x2) = x1 + x2 + 6
          n__s_A(x1) = x1 + 17
  
    precedence:
    
      U64# = tt = activate = isNatKind = U14 = U13 = U12 = U11 = U41 > plus = U61 > U16 = U15 > U22 = U21 > U23 > U32 = U31 > U62 > U63# = U63 > U64 > U62# > s = U61# = n__s > U52 > U51 > plus# = isNat = |0| > n__0 > n__plus
    
    partial status:
  
      pi(U64#) = []
      pi(tt) = []
      pi(plus#) = [2]
      pi(activate) = []
      pi(s) = [1]
      pi(U61#) = [2]
      pi(isNat) = [1]
      pi(U62#) = []
      pi(isNatKind) = []
      pi(U63#) = []
      pi(U16) = []
      pi(U15) = []
      pi(U64) = [3]
      pi(plus) = [2]
      pi(U14) = []
      pi(U63) = []
      pi(U13) = []
      pi(U23) = []
      pi(U52) = []
      pi(U62) = []
      pi(U12) = []
      pi(U22) = []
      pi(U32) = []
      pi(U51) = [2]
      pi(U61) = []
      pi(U11) = []
      pi(U21) = []
      pi(U31) = []
      pi(U41) = []
      pi(|0|) = []
      pi(n__0) = []
      pi(n__plus) = []
      pi(n__s) = [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: U64#(tt(),M,N) -> plus#(activate(N),activate(M))
p2: plus#(N,s(M)) -> U61#(isNat(M),M,N)
p3: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
p4: U62#(tt(),M,N) -> U63#(isNat(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(X1,X2)
r32: activate(n__s(X)) -> s(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: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
p2: U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
p3: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
p4: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
p5: U15#(tt(),V2) -> isNat#(activate(V2))
p6: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
p7: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p8: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
p9: U22#(tt(),V1) -> isNat#(activate(V1))
p10: U14#(tt(),V1,V2) -> 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(X1,X2)
r32: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2,x3) = x2 + x3 + 10
          tt_A() = 9
          U12#_A(x1,x2,x3) = x2 + x3 + 10
          isNatKind_A(x1) = x1 + 1
          activate_A(x1) = x1
          U13#_A(x1,x2,x3) = x2 + x3 + 10
          U14#_A(x1,x2,x3) = x2 + x3 + 10
          U15#_A(x1,x2) = x1 + x2 + 2
          isNat_A(x1) = x1 + 6
          isNat#_A(x1) = x1 + 10
          n__plus_A(x1,x2) = x1 + x2
          n__s_A(x1) = x1 + 8
          U21#_A(x1,x2) = x2 + 12
          U22#_A(x1,x2) = x2 + 11
          U16_A(x1) = 10
          U15_A(x1,x2) = x1 + 2
          U64_A(x1,x2,x3) = x2 + x3 + 8
          s_A(x1) = x1 + 8
          plus_A(x1,x2) = x1 + x2
          U14_A(x1,x2,x3) = x1 + x2 + 1
          U63_A(x1,x2,x3) = x2 + x3 + 8
          U13_A(x1,x2,x3) = x2 + x3 + 3
          U23_A(x1) = 10
          U52_A(x1,x2) = x2 + 1
          U62_A(x1,x2,x3) = x2 + x3 + 8
          U12_A(x1,x2,x3) = x2 + x3 + 4
          U22_A(x1,x2) = 11
          U32_A(x1) = x1
          U51_A(x1,x2) = x2 + 2
          U61_A(x1,x2,x3) = x2 + x3 + 8
          U11_A(x1,x2,x3) = x2 + x3 + 5
          U21_A(x1,x2) = 12
          U31_A(x1,x2) = x2 + 1
          U41_A(x1) = 9
          |0|_A() = 10
          n__0_A() = 10
  
    precedence:
    
      U11# = U12# = U13# = U14# = U15# = isNat# = U22# > U11 > U13 = U12 > U14 > tt = isNatKind = U16 = U15 = U32 = U31 = U41 > isNat > n__plus = plus = U52 = U51 = U61 > U62 > n__s = U64 = s = U63 > activate = |0| > n__0 > U21# > U22 = U21 > U23
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(U12#) = []
      pi(isNatKind) = []
      pi(activate) = [1]
      pi(U13#) = []
      pi(U14#) = []
      pi(U15#) = []
      pi(isNat) = [1]
      pi(isNat#) = []
      pi(n__plus) = []
      pi(n__s) = []
      pi(U21#) = []
      pi(U22#) = [2]
      pi(U16) = []
      pi(U15) = []
      pi(U64) = []
      pi(s) = []
      pi(plus) = []
      pi(U14) = [1]
      pi(U63) = []
      pi(U13) = []
      pi(U23) = []
      pi(U52) = []
      pi(U62) = [2, 3]
      pi(U12) = []
      pi(U22) = []
      pi(U32) = []
      pi(U51) = []
      pi(U61) = []
      pi(U11) = [3]
      pi(U21) = []
      pi(U31) = []
      pi(U41) = []
      pi(|0|) = []
      pi(n__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: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
p2: U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
p3: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
p4: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
p5: U15#(tt(),V2) -> isNat#(activate(V2))
p6: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
p7: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
p8: U22#(tt(),V1) -> isNat#(activate(V1))
p9: U14#(tt(),V1,V2) -> 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(X1,X2)
r32: activate(n__s(X)) -> s(X)
r33: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- 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) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
p3: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
p4: U14#(tt(),V1,V2) -> isNat#(activate(V1))
p5: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
p6: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
p7: 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(X1,X2)
r32: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2,x3) = x2 + x3 + 146
          tt_A() = 78
          U12#_A(x1,x2,x3) = x2 + x3 + 131
          isNatKind_A(x1) = x1 + 79
          activate_A(x1) = x1 + 7
          U13#_A(x1,x2,x3) = x2 + x3 + 116
          U14#_A(x1,x2,x3) = x2 + x3 + 101
          isNat#_A(x1) = x1 + 72
          n__plus_A(x1,x2) = x1 + x2 + 89
          U15#_A(x1,x2) = x1 + x2 + 2
          isNat_A(x1) = 91
          U16_A(x1) = 79
          U15_A(x1,x2) = 80
          U64_A(x1,x2,x3) = 3
          s_A(x1) = 2
          plus_A(x1,x2) = x1 + x2 + 92
          U14_A(x1,x2,x3) = 81
          U63_A(x1,x2,x3) = 4
          U13_A(x1,x2,x3) = 82
          U23_A(x1) = 79
          U52_A(x1,x2) = x2 + 72
          U62_A(x1,x2,x3) = 5
          U12_A(x1,x2,x3) = 83
          U22_A(x1,x2) = 80
          U32_A(x1) = 79
          U51_A(x1,x2) = x2 + 91
          U61_A(x1,x2,x3) = 6
          U11_A(x1,x2,x3) = 90
          U21_A(x1,x2) = 81
          U31_A(x1,x2) = x1 + 2
          U41_A(x1) = 79
          |0|_A() = 0
          n__0_A() = 0
          n__s_A(x1) = 1
  
    precedence:
    
      tt > U11# = U12# = U13# > U14# = isNat# = U15# > isNatKind = U32 = U31 > activate = n__plus = plus = U52 = U51 > s = U23 > U13 = U12 > U15 = U14 > U16 > U61 > isNat = U11 = U21 > U22 = U41 > |0| > n__0 = n__s > U64 = U63 = U62
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(U12#) = []
      pi(isNatKind) = [1]
      pi(activate) = []
      pi(U13#) = []
      pi(U14#) = []
      pi(isNat#) = []
      pi(n__plus) = []
      pi(U15#) = []
      pi(isNat) = []
      pi(U16) = []
      pi(U15) = []
      pi(U64) = []
      pi(s) = []
      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__s) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- 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: U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
p3: U14#(tt(),V1,V2) -> isNat#(activate(V1))
p4: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
p5: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
p6: 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(X1,X2)
r32: activate(n__s(X)) -> s(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: U31#(tt(),V2) -> isNatKind#(activate(V2))
p2: isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
p3: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p4: 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(X1,X2)
r32: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          U31#_A(x1,x2) = x2
          tt_A() = 0
          isNatKind#_A(x1) = x1
          activate_A(x1) = x1
          n__plus_A(x1,x2) = x1 + x2 + 2
          isNatKind_A(x1) = 1
          n__s_A(x1) = x1
          U16_A(x1) = 0
          U15_A(x1,x2) = 0
          isNat_A(x1) = 3
          U14_A(x1,x2,x3) = 0
          U13_A(x1,x2,x3) = 0
          U23_A(x1) = x1
          U64_A(x1,x2,x3) = x2 + x3 + 2
          s_A(x1) = x1
          plus_A(x1,x2) = x1 + x2 + 2
          U12_A(x1,x2,x3) = 0
          U22_A(x1,x2) = 3
          U63_A(x1,x2,x3) = x2 + x3 + 2
          U11_A(x1,x2,x3) = 0
          U21_A(x1,x2) = 3
          U52_A(x1,x2) = x1 + x2 + 5
          U62_A(x1,x2,x3) = x2 + x3 + 2
          U32_A(x1) = 0
          U51_A(x1,x2) = x2 + 6
          U61_A(x1,x2,x3) = x2 + x3 + 2
          n__0_A() = 4
          U31_A(x1,x2) = 0
          U41_A(x1) = 0
          |0|_A() = 4
  
    precedence:
    
      isNat = U11 = U21 > activate > U13 = U12 > n__plus = plus = U51 > U31# > U31 > U16 = U15 = U14 > U23 = U22 > n__0 = |0| > tt = isNatKind# = isNatKind = n__s = U64 = s = U63 = U52 = U62 = U32 = U61 = U41
    
    partial status:
  
      pi(U31#) = [2]
      pi(tt) = []
      pi(isNatKind#) = []
      pi(activate) = [1]
      pi(n__plus) = []
      pi(isNatKind) = []
      pi(n__s) = []
      pi(U16) = []
      pi(U15) = []
      pi(isNat) = []
      pi(U14) = []
      pi(U13) = []
      pi(U23) = []
      pi(U64) = []
      pi(s) = []
      pi(plus) = [1]
      pi(U12) = []
      pi(U22) = []
      pi(U63) = []
      pi(U11) = []
      pi(U21) = []
      pi(U52) = []
      pi(U62) = []
      pi(U32) = []
      pi(U51) = []
      pi(U61) = [3]
      pi(n__0) = []
      pi(U31) = []
      pi(U41) = []
      pi(|0|) = []

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: isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
p2: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p3: 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(X1,X2)
r32: activate(n__s(X)) -> s(X)
r33: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p2, p3}


-- Reduction pair.

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

p1: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p2: 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(X1,X2)
r32: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          isNatKind#_A(x1) = x1
          n__plus_A(x1,x2) = x1 + x2 + 4
          activate_A(x1) = x1
          n__s_A(x1) = x1 + 4
          U16_A(x1) = 2
          tt_A() = 1
          U15_A(x1,x2) = 3
          isNat_A(x1) = 7
          U14_A(x1,x2,x3) = 4
          U13_A(x1,x2,x3) = 5
          isNatKind_A(x1) = x1 + 9
          U23_A(x1) = 2
          U32_A(x1) = 2
          U64_A(x1,x2,x3) = x2 + x3 + 8
          s_A(x1) = x1 + 4
          plus_A(x1,x2) = x1 + x2 + 4
          U12_A(x1,x2,x3) = 5
          U22_A(x1,x2) = 5
          U31_A(x1,x2) = 3
          U41_A(x1) = x1 + 1
          U63_A(x1,x2,x3) = x2 + x3 + 8
          U11_A(x1,x2,x3) = 6
          U21_A(x1,x2) = 6
          U52_A(x1,x2) = x2 + 2
          U62_A(x1,x2,x3) = x2 + x3 + 8
          n__0_A() = 8
          U51_A(x1,x2) = x2 + 3
          U61_A(x1,x2,x3) = x2 + x3 + 8
          |0|_A() = 8
  
    precedence:
    
      isNat > U23 > U12 = U11 > U15 = U14 = U13 > U16 = tt > U22 = U21 = U52 > n__plus = plus = U41 = U51 > isNatKind# = activate = isNatKind > U32 = U31 > n__0 = U61 = |0| > n__s = U64 = s = U63 = U62
    
    partial status:
  
      pi(isNatKind#) = [1]
      pi(n__plus) = []
      pi(activate) = [1]
      pi(n__s) = []
      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) = [2]
      pi(U62) = [2, 3]
      pi(n__0) = []
      pi(U51) = []
      pi(U61) = [2]
      pi(|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: isNatKind#(n__plus(V1,V2)) -> 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(X1,X2)
r32: activate(n__s(X)) -> s(X)
r33: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: isNatKind#(n__plus(V1,V2)) -> 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(X1,X2)
r32: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          isNatKind#_A(x1) = x1 + 1
          n__plus_A(x1,x2) = x1 + 37
          activate_A(x1) = x1 + 17
          U16_A(x1) = 1
          tt_A() = 0
          U15_A(x1,x2) = 2
          isNat_A(x1) = 16
          U14_A(x1,x2,x3) = x1 + 3
          U13_A(x1,x2,x3) = 13
          isNatKind_A(x1) = 9
          U23_A(x1) = 0
          U32_A(x1) = 0
          U64_A(x1,x2,x3) = 5
          s_A(x1) = 4
          plus_A(x1,x2) = x1 + 53
          U12_A(x1,x2,x3) = 14
          U22_A(x1,x2) = 1
          U31_A(x1,x2) = 0
          U41_A(x1) = 1
          U63_A(x1,x2,x3) = 6
          U11_A(x1,x2,x3) = 15
          U21_A(x1,x2) = 2
          U52_A(x1,x2) = x2 + 18
          U62_A(x1,x2,x3) = 7
          n__0_A() = 1
          n__s_A(x1) = 3
          U51_A(x1,x2) = x1 + x2 + 36
          U61_A(x1,x2,x3) = 8
          |0|_A() = 17
  
    precedence:
    
      activate = isNat = U23 = U32 = U31 = U11 = U21 = U52 = n__0 = U51 > U15 = U14 > U13 = U12 = n__s > U16 = tt = isNatKind = U41 = |0| > plus = U61 > n__plus > isNatKind# > U22 = U62 > U64 = U63 > s
    
    partial status:
  
      pi(isNatKind#) = [1]
      pi(n__plus) = []
      pi(activate) = []
      pi(U16) = []
      pi(tt) = []
      pi(U15) = []
      pi(isNat) = []
      pi(U14) = []
      pi(U13) = []
      pi(isNatKind) = []
      pi(U23) = []
      pi(U32) = []
      pi(U64) = []
      pi(s) = []
      pi(plus) = [1]
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(U63) = []
      pi(U11) = []
      pi(U21) = []
      pi(U52) = []
      pi(U62) = []
      pi(n__0) = []
      pi(n__s) = []
      pi(U51) = []
      pi(U61) = []
      pi(|0|) = []

The next rules are strictly ordered:

  p1

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