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:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            U11#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (12,61)
            tt_A() = (7,58)
            U12#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + x3 + (11,140)
            isNatKind_A(x1) = x1 + (2,31)
            activate_A(x1) = ((1,0),(1,1)) x1 + (0,76)
            activate#_A(x1) = ((1,0),(1,0)) x1 + (8,62)
            n__s_A(x1) = ((1,0),(1,0)) x1 + (76,1)
            n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (13,46)
            plus#_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (0,57)
            s_A(x1) = ((1,0),(1,0)) x1 + (76,2)
            isNat#_A(x1) = x1 + (9,60)
            isNatKind#_A(x1) = ((1,0),(1,1)) x1 + (8,29)
            U31#_A(x1,x2) = ((1,0),(1,1)) x2 + (9,12)
            U21#_A(x1,x2) = ((1,0),(1,0)) x2 + (11,214)
            U22#_A(x1,x2) = ((1,0),(1,1)) x2 + (10,137)
            U61#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + x2 + ((1,0),(1,0)) x3 + (75,56)
            isNat_A(x1) = ((1,0),(0,0)) x1 + (2,58)
            U62#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (74,288)
            U63#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (10,211)
            U64#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (9,210)
            |0|_A() = (10,3)
            U51#_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (10,56)
            U52#_A(x1,x2) = ((1,0),(1,1)) x2 + (8,63)
            U13#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (10,63)
            U14#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(0,0)) x3 + (9,214)
            U15#_A(x1,x2) = x1 + x2 + (6,79)
            U16_A(x1) = (8,59)
            U15_A(x1,x2) = (9,60)
            U64_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (89,16)
            plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (13,47)
            U14_A(x1,x2,x3) = (10,61)
            U63_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (89,56)
            U13_A(x1,x2,x3) = (11,75)
            U23_A(x1) = (8,55)
            U52_A(x1,x2) = ((1,0),(0,0)) x2 + (8,1)
            U62_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (89,57)
            U12_A(x1,x2,x3) = (12,108)
            U22_A(x1,x2) = (9,56)
            U32_A(x1) = (7,59)
            U51_A(x1,x2) = ((1,0),(0,0)) x2 + (9,108)
            U61_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (89,57)
            U11_A(x1,x2,x3) = ((0,0),(1,0)) x2 + (14,109)
            U21_A(x1,x2) = ((0,0),(1,0)) x2 + (10,57)
            U31_A(x1,x2) = ((1,0),(1,0)) x1 + (12,70)
            U41_A(x1) = x1 + (75,1)
            n__0_A() = (10,3)
    
      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:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            U11#_A(x1,x2,x3) = ((0,0),(1,0)) x3 + (12,24)
            tt_A() = (5,0)
            U12#_A(x1,x2,x3) = ((0,0),(1,0)) x3 + (11,3)
            isNatKind_A(x1) = (10,0)
            activate_A(x1) = (2,4)
            activate#_A(x1) = (2,23)
            n__s_A(x1) = (4,24)
            n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + (2,6)
            plus#_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (0,1)
            s_A(x1) = (14,3)
            isNat#_A(x1) = ((1,0),(1,1)) x1 + (0,15)
            isNatKind#_A(x1) = (4,23)
            U31#_A(x1,x2) = (4,23)
            U21#_A(x1,x2) = (11,25)
            U22#_A(x1,x2) = (6,20)
            U61#_A(x1,x2,x3) = (0,11)
            isNat_A(x1) = (13,10)
            U62#_A(x1,x2,x3) = (0,10)
            U63#_A(x1,x2,x3) = (7,24)
            U64#_A(x1,x2,x3) = (6,3)
            |0|_A() = (14,25)
            U51#_A(x1,x2) = ((1,0),(0,0)) x2 + (2,24)
            U52#_A(x1,x2) = (6,25)
            U13#_A(x1,x2,x3) = (4,24)
            U14#_A(x1,x2,x3) = (3,22)
            U15#_A(x1,x2) = ((0,0),(1,0)) x1 + (6,19)
            U16_A(x1) = (6,2)
            U15_A(x1,x2) = (7,1)
            U64_A(x1,x2,x3) = (1,5)
            plus_A(x1,x2) = x1 + (16,9)
            U14_A(x1,x2,x3) = (8,2)
            U63_A(x1,x2,x3) = (6,6)
            U13_A(x1,x2,x3) = (9,3)
            U23_A(x1) = (6,4)
            U52_A(x1,x2) = (15,2)
            U62_A(x1,x2,x3) = (14,7)
            U12_A(x1,x2,x3) = (11,4)
            U22_A(x1,x2) = (7,3)
            U32_A(x1) = (6,1)
            U51_A(x1,x2) = (15,3)
            U61_A(x1,x2,x3) = (15,8)
            U11_A(x1,x2,x3) = (12,9)
            U21_A(x1,x2) = (9,25)
            U31_A(x1,x2) = (7,7)
            U41_A(x1) = (11,25)
            n__0_A() = (0,1)
    
      precedence:
      
        n__0 > plus > U12 > U61 > n__plus > U64# > U12# > s > n__s > U21# > U14# > activate > U62 > U63 > isNatKind = U64 = U31 = U41 > U21 > isNat > U14 > U13 > U11 > U15# > U15 > U13# > U52# > plus# = U51# > isNat# = isNatKind# = U31# > U32 > U22 > U51 > U52 > activate# = U22# > U61# > U16 > U62# > |0| > U63# > U11# > U23 > tt
      
      partial status:
    
        pi(U11#) = []
        pi(tt) = []
        pi(U12#) = []
        pi(isNatKind) = []
        pi(activate) = []
        pi(activate#) = []
        pi(n__s) = []
        pi(n__plus) = []
        pi(plus#) = [1]
        pi(s) = []
        pi(isNat#) = [1]
        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, p7, p8, p9, p10, p12, p13, p16, p18, p20, p21, p22, p23, p24, p26, p27, p28, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p42, p43, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p61, p62, p63, 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)) -> activate#(X2)
p3: activate#(n__plus(X1,X2)) -> activate#(X1)
p4: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p5: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p6: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p7: isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
p8: U31#(tt(),V2) -> isNatKind#(activate(V2))
p9: U21#(tt(),V1) -> activate#(V1)
p10: isNat#(n__plus(V1,V2)) -> activate#(V1)
p11: U11#(tt(),V1,V2) -> activate#(V1)
p12: U63#(tt(),M,N) -> activate#(N)
p13: U64#(tt(),M,N) -> activate#(M)
p14: U64#(tt(),M,N) -> activate#(N)
p15: U64#(tt(),M,N) -> plus#(activate(N),activate(M))
p16: plus#(N,|0|()) -> isNat#(N)
p17: U14#(tt(),V1,V2) -> activate#(V2)
p18: U15#(tt(),V2) -> 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:

  {p1, p2, p3, p4, p10, p16}
  {p5, p6, p7, p8}


-- Reduction pair.

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: plus#(N,|0|()) -> isNat#(N)
p4: isNat#(n__plus(V1,V2)) -> activate#(V1)
p5: activate#(n__plus(X1,X2)) -> activate#(X1)
p6: activate#(n__plus(X1,X2)) -> activate#(X2)

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: lexicographic order
          interpretations:
            activate#_A(x1) = x1 + (1,10)
            n__s_A(x1) = ((1,0),(1,0)) x1 + (4,23)
            n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (14,11)
            plus#_A(x1,x2) = x1 + x2 + (13,2)
            activate_A(x1) = ((1,0),(1,1)) x1 + (0,9)
            |0|_A() = (7,1)
            isNat#_A(x1) = x1 + (8,4)
            U16_A(x1) = (8,3)
            tt_A() = (7,2)
            U15_A(x1,x2) = (9,4)
            isNat_A(x1) = ((1,0),(0,0)) x1 + (0,31)
            U14_A(x1,x2,x3) = (10,5)
            U13_A(x1,x2,x3) = ((1,0),(0,0)) x3 + (11,6)
            isNatKind_A(x1) = ((1,0),(1,0)) x1 + (19,12)
            U23_A(x1) = ((1,0),(1,0)) x1 + (1,3)
            U32_A(x1) = (8,3)
            U64_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (18,47)
            s_A(x1) = ((1,0),(1,0)) x1 + (4,32)
            plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (14,15)
            U12_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (12,1)
            U22_A(x1,x2) = x2 + (2,0)
            U31_A(x1,x2) = ((0,0),(1,0)) x1 + (15,1)
            U41_A(x1) = (22,24)
            U63_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (18,48)
            U11_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (13,30)
            U21_A(x1,x2) = ((1,0),(1,1)) x2 + (3,13)
            U52_A(x1,x2) = ((1,0),(1,0)) x2 + (8,10)
            U62_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (18,49)
            n__0_A() = (7,1)
            U51_A(x1,x2) = x2 + (20,22)
            U61_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (18,50)
    
      precedence:
      
        isNatKind = U23 = U41 > activate = U64 > isNat = s = plus = U31 > U32 > U21 > U11 > U13 = U12 = U22 > activate# = n__s = n__plus = plus# = isNat# = U14 = U61 > U62 > U15 > U16 > |0| = tt = U63 = U52 = n__0 = U51
      
      partial status:
    
        pi(activate#) = [1]
        pi(n__s) = []
        pi(n__plus) = [1, 2]
        pi(plus#) = [1, 2]
        pi(activate) = [1]
        pi(|0|) = []
        pi(isNat#) = [1]
        pi(U16) = []
        pi(tt) = []
        pi(U15) = []
        pi(isNat) = []
        pi(U14) = []
        pi(U13) = []
        pi(isNatKind) = []
        pi(U23) = []
        pi(U32) = []
        pi(U64) = []
        pi(s) = []
        pi(plus) = [2]
        pi(U12) = [3]
        pi(U22) = [2]
        pi(U31) = []
        pi(U41) = []
        pi(U63) = []
        pi(U11) = []
        pi(U21) = [2]
        pi(U52) = []
        pi(U62) = []
        pi(n__0) = []
        pi(U51) = [2]
        pi(U61) = [3]
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            activate#_A(x1) = (8,18)
            n__s_A(x1) = (9,15)
            n__plus_A(x1,x2) = ((1,0),(0,0)) x2 + (17,17)
            plus#_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (1,1)
            activate_A(x1) = ((0,0),(1,0)) x1 + (3,2)
            |0|_A() = (3,2)
            isNat#_A(x1) = ((1,0),(1,1)) x1 + (5,3)
            U16_A(x1) = (0,0)
            tt_A() = (13,21)
            U15_A(x1,x2) = (0,0)
            isNat_A(x1) = (21,19)
            U14_A(x1,x2,x3) = (0,0)
            U13_A(x1,x2,x3) = (11,2)
            isNatKind_A(x1) = (16,16)
            U23_A(x1) = (1,22)
            U32_A(x1) = (14,22)
            U64_A(x1,x2,x3) = (11,22)
            s_A(x1) = (10,0)
            plus_A(x1,x2) = x2 + (19,18)
            U12_A(x1,x2,x3) = (12,3)
            U22_A(x1,x2) = ((1,0),(1,0)) x2 + (3,2)
            U31_A(x1,x2) = (15,23)
            U41_A(x1) = (14,22)
            U63_A(x1,x2,x3) = (12,23)
            U11_A(x1,x2,x3) = (18,18)
            U21_A(x1,x2) = ((1,0),(1,1)) x2 + (17,15)
            U52_A(x1,x2) = (14,22)
            U62_A(x1,x2,x3) = (22,24)
            n__0_A() = (0,22)
            U51_A(x1,x2) = (15,22)
            U61_A(x1,x2,x3) = (23,25)
    
      precedence:
      
        U31 > U51 > n__0 > U52 > U61 > isNat = isNatKind = U22 = U21 = U62 > U11 > activate = U13 > U32 > tt = s = U41 > U12 = U63 > U23 > U64 > plus > n__plus > activate# = plus# = |0| = isNat# > n__s > U14 > U15 > U16
      
      partial status:
    
        pi(activate#) = []
        pi(n__s) = []
        pi(n__plus) = []
        pi(plus#) = [1, 2]
        pi(activate) = []
        pi(|0|) = []
        pi(isNat#) = [1]
        pi(U16) = []
        pi(tt) = []
        pi(U15) = []
        pi(isNat) = []
        pi(U14) = []
        pi(U13) = []
        pi(isNatKind) = []
        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(U51) = []
        pi(U61) = []
    

The next rules are strictly ordered:

  p1, p3, p4

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: activate#(n__plus(X1,X2)) -> activate#(X1)
p3: activate#(n__plus(X1,X2)) -> activate#(X2)

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:

  {p2, p3}


-- Reduction pair.

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

p1: activate#(n__plus(X1,X2)) -> activate#(X1)
p2: activate#(n__plus(X1,X2)) -> activate#(X2)

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:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            activate#_A(x1) = ((1,0),(1,1)) x1
            n__plus_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (1,1)
    
      precedence:
      
        activate# > n__plus
      
      partial status:
    
        pi(activate#) = [1]
        pi(n__plus) = [1, 2]
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            activate#_A(x1) = x1
            n__plus_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (1,1)
    
      precedence:
      
        n__plus > activate#
      
      partial status:
    
        pi(activate#) = [1]
        pi(n__plus) = []
    

The next rules are strictly ordered:

  p1, p2

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

-- Reduction pair.

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

p1: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p2: isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
p3: U31#(tt(),V2) -> isNatKind#(activate(V2))
p4: 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(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: lexicographic order
          interpretations:
            isNatKind#_A(x1) = x1 + (7,9)
            n__s_A(x1) = ((1,0),(0,0)) x1 + (30,1)
            activate_A(x1) = x1 + (0,9)
            n__plus_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (6,43)
            U31#_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (1,1)
            isNatKind_A(x1) = x1 + (6,17)
            tt_A() = (26,5)
            U16_A(x1) = (27,4)
            U15_A(x1,x2) = (30,10)
            isNat_A(x1) = ((1,0),(1,0)) x1 + (29,2)
            U14_A(x1,x2,x3) = ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (31,11)
            U13_A(x1,x2,x3) = (32,4)
            U23_A(x1) = (27,6)
            U64_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (36,4)
            s_A(x1) = ((1,0),(0,0)) x1 + (30,3)
            plus_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (6,43)
            U12_A(x1,x2,x3) = (33,8)
            U22_A(x1,x2) = (28,7)
            U63_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (36,5)
            U11_A(x1,x2,x3) = ((0,0),(1,0)) x1 + (34,1)
            U21_A(x1,x2) = (29,27)
            U52_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (1,1)
            U62_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (36,15)
            U32_A(x1) = ((1,0),(0,0)) x1 + (26,6)
            U51_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (7,1)
            U61_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (36,25)
            n__0_A() = (21,23)
            U31_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (6,34)
            U41_A(x1) = x1 + (29,1)
            |0|_A() = (21,31)
    
      precedence:
      
        isNatKind# = n__s = activate = n__plus = U31# = isNatKind = tt = U16 = U15 = isNat = U14 = U13 = U23 = U64 = s = plus = U12 = U22 = U63 = U11 = U21 = U52 = U62 = U32 = U51 = U61 = n__0 = U31 = U41 = |0|
      
      partial status:
    
        pi(isNatKind#) = []
        pi(n__s) = []
        pi(activate) = []
        pi(n__plus) = []
        pi(U31#) = []
        pi(isNatKind) = []
        pi(tt) = []
        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(U31) = []
        pi(U41) = []
        pi(|0|) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            isNatKind#_A(x1) = x1 + (6,2)
            n__s_A(x1) = (23,23)
            activate_A(x1) = ((0,0),(1,0)) x1 + (12,16)
            n__plus_A(x1,x2) = (9,17)
            U31#_A(x1,x2) = x1 + (1,0)
            isNatKind_A(x1) = (13,15)
            tt_A() = (19,19)
            U16_A(x1) = (20,1)
            U15_A(x1,x2) = (21,2)
            isNat_A(x1) = (26,23)
            U14_A(x1,x2,x3) = (22,3)
            U13_A(x1,x2,x3) = (23,0)
            U23_A(x1) = (20,20)
            U64_A(x1,x2,x3) = (11,15)
            s_A(x1) = (1,1)
            plus_A(x1,x2) = (10,24)
            U12_A(x1,x2,x3) = (24,13)
            U22_A(x1,x2) = (21,21)
            U63_A(x1,x2,x3) = (18,17)
            U11_A(x1,x2,x3) = (25,14)
            U21_A(x1,x2) = (22,22)
            U52_A(x1,x2) = ((1,0),(1,0)) x2 + (13,17)
            U62_A(x1,x2,x3) = (27,24)
            U32_A(x1) = (20,20)
            U51_A(x1,x2) = (0,0)
            U61_A(x1,x2,x3) = (0,0)
            n__0_A() = (11,25)
            U31_A(x1,x2) = (10,18)
            U41_A(x1) = ((1,0),(1,0)) x1 + (1,1)
            |0|_A() = (11,24)
    
      precedence:
      
        activate = U62 > isNat > U31 > isNatKind# = U31# > U51 > U61 > plus > U16 = U64 > s > n__s > U22 = U21 > U11 > n__0 > isNatKind > U41 > U32 > n__plus > U63 > U23 > tt = U12 > U15 = U14 = U13 > U52 > |0|
      
      partial status:
    
        pi(isNatKind#) = [1]
        pi(n__s) = []
        pi(activate) = []
        pi(n__plus) = []
        pi(U31#) = []
        pi(isNatKind) = []
        pi(tt) = []
        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(U31) = []
        pi(U41) = []
        pi(|0|) = []
    

The next rules are strictly ordered:

  p1, p3, p4

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))

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)