YES

We show the termination of the TRS R:

  U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
  U12(tt(),V2) -> U13(isNat(activate(V2)))
  U13(tt()) -> tt()
  U21(tt(),V1) -> U22(isNat(activate(V1)))
  U22(tt()) -> tt()
  U31(tt(),N) -> activate(N)
  U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
  and(tt(),X) -> activate(X)
  isNat(n__0()) -> tt()
  isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
  isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
  isNatKind(n__0()) -> tt()
  isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
  isNatKind(n__s(V1)) -> isNatKind(activate(V1))
  plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
  plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
  |0|() -> n__0()
  plus(X1,X2) -> n__plus(X1,X2)
  isNatKind(X) -> n__isNatKind(X)
  s(X) -> n__s(X)
  and(X1,X2) -> n__and(X1,X2)
  activate(n__0()) -> |0|()
  activate(n__plus(X1,X2)) -> plus(X1,X2)
  activate(n__isNatKind(X)) -> isNatKind(X)
  activate(n__s(X)) -> s(X)
  activate(n__and(X1,X2)) -> and(X1,X2)
  activate(X) -> X

-- SCC decomposition.

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

p1: U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2))
p2: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p3: U11#(tt(),V1,V2) -> activate#(V1)
p4: U11#(tt(),V1,V2) -> activate#(V2)
p5: U12#(tt(),V2) -> U13#(isNat(activate(V2)))
p6: U12#(tt(),V2) -> isNat#(activate(V2))
p7: U12#(tt(),V2) -> activate#(V2)
p8: U21#(tt(),V1) -> U22#(isNat(activate(V1)))
p9: U21#(tt(),V1) -> isNat#(activate(V1))
p10: U21#(tt(),V1) -> activate#(V1)
p11: U31#(tt(),N) -> activate#(N)
p12: U41#(tt(),M,N) -> s#(plus(activate(N),activate(M)))
p13: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p14: U41#(tt(),M,N) -> activate#(N)
p15: U41#(tt(),M,N) -> activate#(M)
p16: and#(tt(),X) -> activate#(X)
p17: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p18: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p19: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p20: isNat#(n__plus(V1,V2)) -> activate#(V1)
p21: isNat#(n__plus(V1,V2)) -> activate#(V2)
p22: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p23: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p24: isNat#(n__s(V1)) -> activate#(V1)
p25: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p26: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p27: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p28: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p29: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p30: isNatKind#(n__s(V1)) -> activate#(V1)
p31: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p32: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p33: plus#(N,|0|()) -> isNat#(N)
p34: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p35: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p36: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p37: plus#(N,s(M)) -> isNat#(M)
p38: plus#(N,s(M)) -> isNat#(N)
p39: activate#(n__0()) -> |0|#()
p40: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p41: activate#(n__isNatKind(X)) -> isNatKind#(X)
p42: activate#(n__s(X)) -> s#(X)
p43: activate#(n__and(X1,X2)) -> and#(X1,X2)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p6, p7, p9, p10, p11, 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, p40, p41, p43}


-- Reduction pair.

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

p1: U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2))
p2: U12#(tt(),V2) -> activate#(V2)
p3: activate#(n__and(X1,X2)) -> and#(X1,X2)
p4: and#(tt(),X) -> activate#(X)
p5: activate#(n__isNatKind(X)) -> isNatKind#(X)
p6: isNatKind#(n__s(V1)) -> activate#(V1)
p7: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p8: plus#(N,s(M)) -> isNat#(N)
p9: isNat#(n__s(V1)) -> activate#(V1)
p10: isNat#(n__s(V1)) -> isNatKind#(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)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p16: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p17: U21#(tt(),V1) -> activate#(V1)
p18: U21#(tt(),V1) -> isNat#(activate(V1))
p19: isNat#(n__plus(V1,V2)) -> activate#(V2)
p20: isNat#(n__plus(V1,V2)) -> activate#(V1)
p21: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p22: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p23: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p24: U11#(tt(),V1,V2) -> activate#(V2)
p25: U11#(tt(),V1,V2) -> activate#(V1)
p26: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p27: plus#(N,s(M)) -> isNat#(M)
p28: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p29: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p30: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p31: U41#(tt(),M,N) -> activate#(M)
p32: U41#(tt(),M,N) -> activate#(N)
p33: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p34: plus#(N,|0|()) -> isNat#(N)
p35: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p36: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p37: U31#(tt(),N) -> activate#(N)
p38: U12#(tt(),V2) -> isNat#(activate(V2))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  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 + (20,76)
          tt_A() = (3,66)
          U12#_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (5,21)
          isNat_A(x1) = (21,70)
          activate_A(x1) = x1 + (0,25)
          activate#_A(x1) = ((1,0),(0,0)) x1 + (4,75)
          n__and_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (2,1)
          and#_A(x1,x2) = ((1,0),(0,0)) x2 + (5,86)
          n__isNatKind_A(x1) = ((1,0),(1,1)) x1 + (22,72)
          isNatKind#_A(x1) = ((1,0),(1,0)) x1 + (23,73)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (51,34)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (46,34)
          plus#_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (43,1)
          s_A(x1) = ((1,0),(0,0)) x1 + (51,35)
          isNat#_A(x1) = x1 + (2,40)
          isNatKind_A(x1) = ((1,0),(1,1)) x1 + (22,95)
          U21#_A(x1,x2) = ((1,0),(1,1)) x2 + (5,76)
          and_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (2,24)
          U41#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (52,76)
          |0|_A() = (1,37)
          U31#_A(x1,x2) = ((1,0),(1,0)) x2 + (5,76)
          U13_A(x1) = (4,67)
          U12_A(x1,x2) = (5,68)
          U22_A(x1) = (4,67)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (2,74)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (97,36)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (46,36)
          U11_A(x1,x2,x3) = (6,69)
          U21_A(x1,x2) = (21,70)
          n__0_A() = (1,36)
  
    precedence:
    
      U12# > U31# > activate > isNatKind# = and > isNatKind > |0| = U31 = n__0 > tt = isNat = U13 = U12 = U11 > plus# > and# > U11# = activate# = isNat# = U21# = U21 > U41# > n__plus = plus > U41 > s > n__s > n__isNatKind > n__and > U22
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(U12#) = []
      pi(isNat) = []
      pi(activate) = []
      pi(activate#) = []
      pi(n__and) = []
      pi(and#) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(s) = []
      pi(isNat#) = []
      pi(isNatKind) = [1]
      pi(U21#) = []
      pi(and) = []
      pi(U41#) = []
      pi(|0|) = []
      pi(U31#) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = [2]
      pi(U11) = []
      pi(U21) = []
      pi(n__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: U12#(tt(),V2) -> activate#(V2)
p2: activate#(n__and(X1,X2)) -> and#(X1,X2)
p3: and#(tt(),X) -> activate#(X)
p4: activate#(n__isNatKind(X)) -> isNatKind#(X)
p5: isNatKind#(n__s(V1)) -> activate#(V1)
p6: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p7: plus#(N,s(M)) -> isNat#(N)
p8: isNat#(n__s(V1)) -> activate#(V1)
p9: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p10: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p11: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p12: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p13: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p14: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p15: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p16: U21#(tt(),V1) -> activate#(V1)
p17: U21#(tt(),V1) -> isNat#(activate(V1))
p18: isNat#(n__plus(V1,V2)) -> activate#(V2)
p19: isNat#(n__plus(V1,V2)) -> activate#(V1)
p20: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p21: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p22: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p23: U11#(tt(),V1,V2) -> activate#(V2)
p24: U11#(tt(),V1,V2) -> activate#(V1)
p25: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p26: plus#(N,s(M)) -> isNat#(M)
p27: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p28: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p29: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p30: U41#(tt(),M,N) -> activate#(M)
p31: U41#(tt(),M,N) -> activate#(N)
p32: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p33: plus#(N,|0|()) -> isNat#(N)
p34: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p35: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p36: U31#(tt(),N) -> activate#(N)
p37: U12#(tt(),V2) -> isNat#(activate(V2))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {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}


-- Reduction pair.

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

p1: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
p3: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p5: U31#(tt(),N) -> activate#(N)
p6: activate#(n__isNatKind(X)) -> isNatKind#(X)
p7: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p8: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p10: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p11: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p12: isNatKind#(n__s(V1)) -> activate#(V1)
p13: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p14: plus#(N,|0|()) -> isNat#(N)
p15: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p16: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p17: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p18: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p19: isNat#(n__plus(V1,V2)) -> activate#(V1)
p20: isNat#(n__plus(V1,V2)) -> activate#(V2)
p21: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p22: U21#(tt(),V1) -> isNat#(activate(V1))
p23: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p24: isNat#(n__s(V1)) -> activate#(V1)
p25: U21#(tt(),V1) -> activate#(V1)
p26: U11#(tt(),V1,V2) -> activate#(V1)
p27: U11#(tt(),V1,V2) -> activate#(V2)
p28: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p29: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p30: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p31: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p32: plus#(N,s(M)) -> isNat#(M)
p33: plus#(N,s(M)) -> isNat#(N)
p34: U41#(tt(),M,N) -> activate#(N)
p35: U41#(tt(),M,N) -> activate#(M)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = ((1,0),(1,0)) x1 + (5,75)
          n__and_A(x1,x2) = x2 + (8,0)
          and#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (10,84)
          tt_A() = (2,19)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (13,19)
          plus#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (17,1)
          |0|_A() = (1,78)
          U31#_A(x1,x2) = x2 + (6,81)
          and_A(x1,x2) = ((1,0),(1,1)) x2 + (8,18)
          isNat_A(x1) = (5,22)
          n__isNatKind_A(x1) = x1 + (4,24)
          isNatKind#_A(x1) = x1 + (5,55)
          isNatKind_A(x1) = x1 + (4,50)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,23)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (6,24)
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (3,77)
          U11#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (6,78)
          U21#_A(x1,x2) = ((1,0),(0,0)) x2 + (7,76)
          s_A(x1) = ((1,0),(0,0)) x1 + (6,25)
          U41#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,1)) x3 + (18,76)
          U13_A(x1) = (3,20)
          U12_A(x1,x2) = ((0,0),(1,0)) x2 + (4,21)
          U22_A(x1) = (3,26)
          U31_A(x1,x2) = ((1,0),(0,0)) x2 + (2,1)
          U41_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (19,24)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (13,20)
          U11_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x3 + (4,22)
          U21_A(x1,x2) = (4,25)
          n__0_A() = (1,55)
  
    precedence:
    
      |0| = activate = U41# = U31 = U41 = plus > isNat = U11# > s > activate# = and# = n__plus = plus# = isNatKind# = isNat# = U21# > n__s = n__0 > tt = and = n__isNatKind = isNatKind = U13 = U12 = U22 = U11 = U21 > U31# > n__and
    
    partial status:
  
      pi(activate#) = []
      pi(n__and) = []
      pi(and#) = []
      pi(tt) = []
      pi(n__plus) = [2]
      pi(plus#) = []
      pi(|0|) = []
      pi(U31#) = [2]
      pi(and) = []
      pi(isNat) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(isNatKind) = []
      pi(activate) = []
      pi(n__s) = []
      pi(isNat#) = []
      pi(U11#) = []
      pi(U21#) = []
      pi(s) = []
      pi(U41#) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = []
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p15

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
p3: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p5: U31#(tt(),N) -> activate#(N)
p6: activate#(n__isNatKind(X)) -> isNatKind#(X)
p7: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p8: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p10: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p11: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p12: isNatKind#(n__s(V1)) -> activate#(V1)
p13: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p14: plus#(N,|0|()) -> isNat#(N)
p15: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p16: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p17: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p18: isNat#(n__plus(V1,V2)) -> activate#(V1)
p19: isNat#(n__plus(V1,V2)) -> activate#(V2)
p20: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p21: U21#(tt(),V1) -> isNat#(activate(V1))
p22: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p23: isNat#(n__s(V1)) -> activate#(V1)
p24: U21#(tt(),V1) -> activate#(V1)
p25: U11#(tt(),V1,V2) -> activate#(V1)
p26: U11#(tt(),V1,V2) -> activate#(V2)
p27: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p28: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p29: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p30: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p31: plus#(N,s(M)) -> isNat#(M)
p32: plus#(N,s(M)) -> isNat#(N)
p33: U41#(tt(),M,N) -> activate#(N)
p34: U41#(tt(),M,N) -> activate#(M)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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, p16, p17, p18, p19, p20, p21, p22, p23, p24, p27, p28, p29, p30, p31, p32, p33, p34}


-- Reduction pair.

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

p1: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
p3: activate#(n__isNatKind(X)) -> isNatKind#(X)
p4: isNatKind#(n__s(V1)) -> activate#(V1)
p5: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p6: plus#(N,s(M)) -> isNat#(N)
p7: isNat#(n__s(V1)) -> activate#(V1)
p8: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p10: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p11: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p12: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p13: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p14: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p15: U21#(tt(),V1) -> activate#(V1)
p16: U21#(tt(),V1) -> isNat#(activate(V1))
p17: isNat#(n__plus(V1,V2)) -> activate#(V2)
p18: isNat#(n__plus(V1,V2)) -> activate#(V1)
p19: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p20: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p21: plus#(N,s(M)) -> isNat#(M)
p22: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p23: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p24: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p25: U41#(tt(),M,N) -> activate#(M)
p26: U41#(tt(),M,N) -> activate#(N)
p27: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p28: plus#(N,|0|()) -> isNat#(N)
p29: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p30: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p31: U31#(tt(),N) -> activate#(N)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = ((1,0),(1,0)) x1 + (2,46)
          n__and_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (39,2)
          and#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (16,1)
          tt_A() = (1,57)
          n__isNatKind_A(x1) = ((1,0),(1,0)) x1 + (5,1)
          isNatKind#_A(x1) = ((1,0),(0,0)) x1 + (4,48)
          n__s_A(x1) = ((1,0),(1,0)) x1 + (90,47)
          n__plus_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (45,0)
          plus#_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (23,200)
          s_A(x1) = ((1,0),(1,0)) x1 + (90,48)
          isNat#_A(x1) = ((1,0),(1,1)) x1 + (46,1)
          activate_A(x1) = x1 + (0,95)
          isNatKind_A(x1) = ((1,0),(1,0)) x1 + (5,34)
          U21#_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (89,44)
          isNat_A(x1) = ((0,0),(1,0)) x1 + (4,95)
          and_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (39,38)
          U41#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(0,0)) x3 + (24,391)
          |0|_A() = (24,40)
          U31#_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (25,46)
          U13_A(x1) = (1,57)
          U12_A(x1,x2) = (2,96)
          U22_A(x1) = (2,58)
          U31_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (68,46)
          U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (135,94)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (45,55)
          U11_A(x1,x2,x3) = (3,97)
          U21_A(x1,x2) = ((0,0),(1,0)) x2 + (3,59)
          n__0_A() = (24,1)
  
    precedence:
    
      U31 > plus > U41 > activate > U21# > |0| = n__0 > U12 > U13 > isNatKind# > activate# = U31# > plus# = isNat# = isNatKind = and = U41# > n__and > and# > isNat = U11 > n__isNatKind > tt = n__s = n__plus = s = U22 = U21
    
    partial status:
  
      pi(activate#) = []
      pi(n__and) = []
      pi(and#) = []
      pi(tt) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(s) = []
      pi(isNat#) = []
      pi(activate) = [1]
      pi(isNatKind) = []
      pi(U21#) = [2]
      pi(isNat) = []
      pi(and) = []
      pi(U41#) = []
      pi(|0|) = []
      pi(U31#) = [2]
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = [2]
      pi(U11) = []
      pi(U21) = []
      pi(n__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: and#(tt(),X) -> activate#(X)
p2: activate#(n__isNatKind(X)) -> isNatKind#(X)
p3: isNatKind#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,s(M)) -> isNat#(N)
p6: isNat#(n__s(V1)) -> activate#(V1)
p7: isNat#(n__s(V1)) -> isNatKind#(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)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p13: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p14: U21#(tt(),V1) -> activate#(V1)
p15: U21#(tt(),V1) -> isNat#(activate(V1))
p16: isNat#(n__plus(V1,V2)) -> activate#(V2)
p17: isNat#(n__plus(V1,V2)) -> activate#(V1)
p18: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p19: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p20: plus#(N,s(M)) -> isNat#(M)
p21: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p22: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p23: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p24: U41#(tt(),M,N) -> activate#(M)
p25: U41#(tt(),M,N) -> activate#(N)
p26: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p27: plus#(N,|0|()) -> isNat#(N)
p28: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p29: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p30: U31#(tt(),N) -> activate#(N)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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}


-- Reduction pair.

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

p1: and#(tt(),X) -> activate#(X)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p4: U31#(tt(),N) -> activate#(N)
p5: activate#(n__isNatKind(X)) -> isNatKind#(X)
p6: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p7: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p8: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p9: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p10: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p11: isNatKind#(n__s(V1)) -> activate#(V1)
p12: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p13: plus#(N,|0|()) -> isNat#(N)
p14: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p15: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p16: isNat#(n__plus(V1,V2)) -> activate#(V1)
p17: isNat#(n__plus(V1,V2)) -> activate#(V2)
p18: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p19: U21#(tt(),V1) -> isNat#(activate(V1))
p20: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p21: isNat#(n__s(V1)) -> activate#(V1)
p22: U21#(tt(),V1) -> activate#(V1)
p23: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p24: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p25: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p26: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p27: plus#(N,s(M)) -> isNat#(M)
p28: plus#(N,s(M)) -> isNat#(N)
p29: U41#(tt(),M,N) -> activate#(N)
p30: U41#(tt(),M,N) -> activate#(M)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          and#_A(x1,x2) = ((1,0),(0,0)) x2 + (2,7)
          tt_A() = (0,3)
          activate#_A(x1) = ((1,0),(0,0)) x1 + (2,5)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (35,1)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (36,4)
          |0|_A() = (0,2)
          U31#_A(x1,x2) = ((1,0),(0,0)) x2 + (3,3)
          and_A(x1,x2) = x1 + x2 + (9,20)
          isNat_A(x1) = (5,15)
          n__isNatKind_A(x1) = ((1,0),(1,0)) x1 + (0,14)
          isNatKind#_A(x1) = ((1,0),(0,0)) x1 + (1,7)
          isNatKind_A(x1) = ((1,0),(1,0)) x1 + (0,16)
          activate_A(x1) = x1 + (0,20)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (3,6)
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (36,4)
          U21#_A(x1,x2) = ((1,0),(0,0)) x2 + (36,21)
          s_A(x1) = ((1,0),(0,0)) x1 + (3,8)
          U41#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (39,4)
          n__and_A(x1,x2) = x1 + x2 + (9,0)
          U13_A(x1) = (1,4)
          U12_A(x1,x2) = (5,5)
          U22_A(x1) = (1,15)
          U31_A(x1,x2) = ((1,0),(0,0)) x2 + (1,1)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (38,12)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (35,2)
          U11_A(x1,x2,x3) = (5,6)
          U21_A(x1,x2) = (4,14)
          n__0_A() = (0,1)
  
    precedence:
    
      U21# = U41 > n__plus = U31# = and = isNat = isNatKind = activate = s = U13 = U12 = plus = U11 > n__and > |0| = n__isNatKind = U22 = U31 > n__s = U21 > tt > and# = activate# = plus# = isNatKind# = isNat# = U41# = n__0
    
    partial status:
  
      pi(and#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(U31#) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(isNatKind) = []
      pi(activate) = []
      pi(n__s) = []
      pi(isNat#) = []
      pi(U21#) = []
      pi(s) = []
      pi(U41#) = []
      pi(n__and) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = []
      pi(U11) = []
      pi(U21) = []
      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: and#(tt(),X) -> activate#(X)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: U31#(tt(),N) -> activate#(N)
p4: activate#(n__isNatKind(X)) -> isNatKind#(X)
p5: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p6: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p7: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p8: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p9: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p10: isNatKind#(n__s(V1)) -> activate#(V1)
p11: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p12: plus#(N,|0|()) -> isNat#(N)
p13: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p14: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p15: isNat#(n__plus(V1,V2)) -> activate#(V1)
p16: isNat#(n__plus(V1,V2)) -> activate#(V2)
p17: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p18: U21#(tt(),V1) -> isNat#(activate(V1))
p19: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p20: isNat#(n__s(V1)) -> activate#(V1)
p21: U21#(tt(),V1) -> activate#(V1)
p22: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p23: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p24: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p25: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p26: plus#(N,s(M)) -> isNat#(M)
p27: plus#(N,s(M)) -> isNat#(N)
p28: U41#(tt(),M,N) -> activate#(N)
p29: U41#(tt(),M,N) -> activate#(M)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, 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}


-- Reduction pair.

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

p1: and#(tt(),X) -> activate#(X)
p2: activate#(n__isNatKind(X)) -> isNatKind#(X)
p3: isNatKind#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,s(M)) -> isNat#(N)
p6: isNat#(n__s(V1)) -> activate#(V1)
p7: isNat#(n__s(V1)) -> isNatKind#(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)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p13: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p14: U21#(tt(),V1) -> activate#(V1)
p15: U21#(tt(),V1) -> isNat#(activate(V1))
p16: isNat#(n__plus(V1,V2)) -> activate#(V2)
p17: isNat#(n__plus(V1,V2)) -> activate#(V1)
p18: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p19: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p20: plus#(N,s(M)) -> isNat#(M)
p21: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p22: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p23: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p24: U41#(tt(),M,N) -> activate#(M)
p25: U41#(tt(),M,N) -> activate#(N)
p26: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p27: plus#(N,|0|()) -> isNat#(N)
p28: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          and#_A(x1,x2) = ((1,0),(0,0)) x2 + (5,52)
          tt_A() = (12,58)
          activate#_A(x1) = ((1,0),(0,0)) x1 + (4,26)
          n__isNatKind_A(x1) = ((1,0),(1,1)) x1 + (9,35)
          isNatKind#_A(x1) = ((1,0),(1,1)) x1 + (8,36)
          n__s_A(x1) = x1 + (49,25)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (15,1)
          plus#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (2,57)
          s_A(x1) = x1 + (49,59)
          isNat#_A(x1) = ((1,0),(1,1)) x1 + (1,10)
          activate_A(x1) = x1 + (0,47)
          isNatKind_A(x1) = ((1,0),(1,1)) x1 + (9,38)
          U21#_A(x1,x2) = ((1,0),(0,0)) x2 + (48,84)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (15,54)
          and_A(x1,x2) = ((1,0),(1,0)) x2 + (1,46)
          n__and_A(x1,x2) = ((1,0),(1,0)) x2 + (1,45)
          U41#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (5,58)
          |0|_A() = (13,9)
          U13_A(x1) = (13,45)
          U12_A(x1,x2) = (16,46)
          U22_A(x1) = ((0,0),(1,0)) x1 + (13,47)
          U31_A(x1,x2) = x2 + (14,48)
          U41_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(0,0)) x3 + (64,119)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (15,12)
          U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (5,53)
          U21_A(x1,x2) = (14,1)
          n__0_A() = (13,8)
  
    precedence:
    
      activate = and = |0| = U31 = plus > isNatKind = U13 > isNat > U41 > n__and > U22 = U21 > n__0 > n__s = s = U21# > isNat# > plus# > tt > activate# = U41# > U12 > n__plus > and# > isNatKind# > n__isNatKind > U11
    
    partial status:
  
      pi(and#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__isNatKind) = [1]
      pi(isNatKind#) = [1]
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(s) = []
      pi(isNat#) = [1]
      pi(activate) = []
      pi(isNatKind) = [1]
      pi(U21#) = []
      pi(isNat) = []
      pi(and) = []
      pi(n__and) = []
      pi(U41#) = []
      pi(|0|) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = []
      pi(U11) = []
      pi(U21) = []
      pi(n__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: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: isNatKind#(n__s(V1)) -> activate#(V1)
p3: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,s(M)) -> isNat#(N)
p5: isNat#(n__s(V1)) -> activate#(V1)
p6: isNat#(n__s(V1)) -> isNatKind#(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)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p12: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p13: U21#(tt(),V1) -> activate#(V1)
p14: U21#(tt(),V1) -> isNat#(activate(V1))
p15: isNat#(n__plus(V1,V2)) -> activate#(V2)
p16: isNat#(n__plus(V1,V2)) -> activate#(V1)
p17: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p18: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p19: plus#(N,s(M)) -> isNat#(M)
p20: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p21: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p22: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p23: U41#(tt(),M,N) -> activate#(M)
p24: U41#(tt(),M,N) -> activate#(N)
p25: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p26: plus#(N,|0|()) -> isNat#(N)
p27: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p12, p13, p14, p15, p16, p17, p19, p22, p23, p24, p25, p26}


-- Reduction pair.

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

p1: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p3: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,|0|()) -> isNat#(N)
p6: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p7: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p8: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__s(V1)) -> activate#(V1)
p10: isNat#(n__plus(V1,V2)) -> activate#(V1)
p11: isNat#(n__plus(V1,V2)) -> activate#(V2)
p12: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p13: U21#(tt(),V1) -> isNat#(activate(V1))
p14: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p15: isNat#(n__s(V1)) -> activate#(V1)
p16: U21#(tt(),V1) -> activate#(V1)
p17: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p18: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p19: plus#(N,s(M)) -> isNat#(M)
p20: plus#(N,s(M)) -> isNat#(N)
p21: U41#(tt(),M,N) -> activate#(N)
p22: U41#(tt(),M,N) -> activate#(M)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = ((1,0),(1,1)) x1 + (5,31)
          n__isNatKind_A(x1) = x1 + (14,1)
          isNatKind#_A(x1) = ((1,0),(1,1)) x1 + (12,6)
          n__plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (158,30)
          activate_A(x1) = x1 + (0,28)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (5,31)
          |0|_A() = (3,42)
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (7,41)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (6,16)
          U21#_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (8,25)
          isNatKind_A(x1) = x1 + (14,12)
          tt_A() = (4,27)
          s_A(x1) = ((1,0),(0,0)) x1 + (6,28)
          U41#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (6,60)
          and_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (1,28)
          isNat_A(x1) = x1 + (66,66)
          n__and_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (1,0)
          U13_A(x1) = (5,28)
          U12_A(x1,x2) = (6,29)
          U22_A(x1) = ((1,0),(0,0)) x1 + (1,28)
          U31_A(x1,x2) = x2 + (4,27)
          U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (164,91)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (158,57)
          U11_A(x1,x2,x3) = x1 + (61,3)
          U21_A(x1,x2) = x2 + (68,29)
          n__0_A() = (3,15)
  
    precedence:
    
      isNat > U12 = U11 > U13 > n__isNatKind = activate = |0| = isNatKind = tt = and = n__and > U21 > n__plus = U41 = plus > activate# = isNatKind# = plus# = isNat# = n__s = U21# = s = U41# = U31 > n__0 > U22
    
    partial status:
  
      pi(activate#) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(n__plus) = []
      pi(activate) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(U21#) = []
      pi(isNatKind) = []
      pi(tt) = []
      pi(s) = []
      pi(U41#) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__and) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = []
      pi(U11) = [1]
      pi(U21) = [2]
      pi(n__0) = []

The next rules are strictly ordered:

  p21

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p3: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,|0|()) -> isNat#(N)
p6: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p7: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p8: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__s(V1)) -> activate#(V1)
p10: isNat#(n__plus(V1,V2)) -> activate#(V1)
p11: isNat#(n__plus(V1,V2)) -> activate#(V2)
p12: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p13: U21#(tt(),V1) -> isNat#(activate(V1))
p14: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p15: isNat#(n__s(V1)) -> activate#(V1)
p16: U21#(tt(),V1) -> activate#(V1)
p17: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p18: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p19: plus#(N,s(M)) -> isNat#(M)
p20: plus#(N,s(M)) -> isNat#(N)
p21: U41#(tt(),M,N) -> activate#(M)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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}


-- Reduction pair.

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

p1: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: isNatKind#(n__s(V1)) -> activate#(V1)
p3: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,s(M)) -> isNat#(N)
p5: isNat#(n__s(V1)) -> activate#(V1)
p6: isNat#(n__s(V1)) -> isNatKind#(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: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p12: U21#(tt(),V1) -> activate#(V1)
p13: U21#(tt(),V1) -> isNat#(activate(V1))
p14: isNat#(n__plus(V1,V2)) -> activate#(V2)
p15: isNat#(n__plus(V1,V2)) -> activate#(V1)
p16: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p17: plus#(N,s(M)) -> isNat#(M)
p18: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p19: U41#(tt(),M,N) -> activate#(M)
p20: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p21: plus#(N,|0|()) -> isNat#(N)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = ((1,0),(1,0)) x1 + (1,41)
          n__isNatKind_A(x1) = x1 + (41,0)
          isNatKind#_A(x1) = ((1,0),(0,0)) x1 + (1,81)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (28,2)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (30,0)
          plus#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (13,53)
          s_A(x1) = ((1,0),(0,0)) x1 + (28,3)
          isNat#_A(x1) = ((1,0),(1,0)) x1 + (26,52)
          activate_A(x1) = x1 + (0,39)
          U21#_A(x1,x2) = ((1,0),(1,1)) x2 + (27,40)
          isNatKind_A(x1) = x1 + (41,38)
          tt_A() = (12,42)
          U41#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,1)) x3 + (29,54)
          and_A(x1,x2) = x2 + (18,55)
          isNat_A(x1) = ((0,0),(1,0)) x1 + (42,46)
          n__and_A(x1,x2) = x2 + (18,54)
          |0|_A() = (13,5)
          U13_A(x1) = (13,0)
          U12_A(x1,x2) = (14,0)
          U22_A(x1) = (13,43)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (14,40)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (58,4)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (30,5)
          U11_A(x1,x2,x3) = (31,40)
          U21_A(x1,x2) = ((0,0),(1,0)) x1 + (14,32)
          n__0_A() = (13,5)
  
    precedence:
    
      activate = isNatKind = and = n__and = |0| = U11 = U21 > U13 = U12 > n__isNatKind = n__s = s = tt = isNat = U22 = U41 = plus = n__0 > activate# = isNatKind# = n__plus = plus# = isNat# = U21# = U41# > U31
    
    partial status:
  
      pi(activate#) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(s) = []
      pi(isNat#) = []
      pi(activate) = []
      pi(U21#) = [2]
      pi(isNatKind) = []
      pi(tt) = []
      pi(U41#) = [3]
      pi(and) = []
      pi(isNat) = []
      pi(n__and) = []
      pi(|0|) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = []
      pi(U11) = []
      pi(U21) = []
      pi(n__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: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,s(M)) -> isNat#(N)
p4: isNat#(n__s(V1)) -> activate#(V1)
p5: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p6: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p7: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p8: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p9: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p10: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p11: U21#(tt(),V1) -> activate#(V1)
p12: U21#(tt(),V1) -> isNat#(activate(V1))
p13: isNat#(n__plus(V1,V2)) -> activate#(V2)
p14: isNat#(n__plus(V1,V2)) -> activate#(V1)
p15: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p16: plus#(N,s(M)) -> isNat#(M)
p17: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p18: U41#(tt(),M,N) -> activate#(M)
p19: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p20: plus#(N,|0|()) -> isNat#(N)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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}


-- Reduction pair.

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

p1: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p3: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,|0|()) -> isNat#(N)
p6: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p7: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p8: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNat#(n__plus(V1,V2)) -> activate#(V1)
p10: isNat#(n__plus(V1,V2)) -> activate#(V2)
p11: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p12: U21#(tt(),V1) -> isNat#(activate(V1))
p13: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p14: isNat#(n__s(V1)) -> activate#(V1)
p15: U21#(tt(),V1) -> activate#(V1)
p16: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p17: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p18: plus#(N,s(M)) -> isNat#(M)
p19: plus#(N,s(M)) -> isNat#(N)
p20: U41#(tt(),M,N) -> activate#(M)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = x1 + (27,33)
          n__isNatKind_A(x1) = ((1,0),(1,0)) x1 + (2,4)
          isNatKind#_A(x1) = ((1,0),(0,0)) x1 + (29,33)
          n__plus_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (0,4)
          activate_A(x1) = x1 + (0,12)
          plus#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (26,36)
          |0|_A() = (9,4)
          isNat#_A(x1) = ((1,0),(1,0)) x1 + (29,34)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (31,11)
          U21#_A(x1,x2) = ((1,0),(1,0)) x2 + (30,35)
          isNatKind_A(x1) = ((1,0),(1,0)) x1 + (2,10)
          tt_A() = (10,1)
          s_A(x1) = ((1,0),(0,0)) x1 + (31,22)
          U41#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (32,35)
          and_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (0,3)
          isNat_A(x1) = (30,21)
          n__and_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (0,1)
          U13_A(x1) = (11,0)
          U12_A(x1,x2) = (12,2)
          U22_A(x1) = (11,2)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (8,17)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (31,23)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (0,12)
          U11_A(x1,x2,x3) = (13,3)
          U21_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (12,10)
          n__0_A() = (9,2)
  
    precedence:
    
      isNat = U11 = U21 > activate# = isNatKind# = plus# = isNat# = U21# = U41# > activate = isNatKind = tt = and = U13 = U31 = plus > |0| > n__and > n__isNatKind > n__plus > n__s = s = U22 = U41 > U12 = n__0
    
    partial status:
  
      pi(activate#) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(n__plus) = [2]
      pi(activate) = [1]
      pi(plus#) = []
      pi(|0|) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(U21#) = []
      pi(isNatKind) = []
      pi(tt) = []
      pi(s) = []
      pi(U41#) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__and) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = []
      pi(U11) = []
      pi(U21) = []
      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: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p3: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,|0|()) -> isNat#(N)
p5: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p6: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p7: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p8: isNat#(n__plus(V1,V2)) -> activate#(V1)
p9: isNat#(n__plus(V1,V2)) -> activate#(V2)
p10: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p11: U21#(tt(),V1) -> isNat#(activate(V1))
p12: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p13: isNat#(n__s(V1)) -> activate#(V1)
p14: U21#(tt(),V1) -> activate#(V1)
p15: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p16: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p17: plus#(N,s(M)) -> isNat#(M)
p18: plus#(N,s(M)) -> isNat#(N)
p19: U41#(tt(),M,N) -> activate#(M)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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}


-- Reduction pair.

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

p1: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p3: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,s(M)) -> isNat#(N)
p6: isNat#(n__s(V1)) -> activate#(V1)
p7: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p8: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p9: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p10: U21#(tt(),V1) -> activate#(V1)
p11: U21#(tt(),V1) -> isNat#(activate(V1))
p12: isNat#(n__plus(V1,V2)) -> activate#(V2)
p13: isNat#(n__plus(V1,V2)) -> activate#(V1)
p14: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p15: plus#(N,s(M)) -> isNat#(M)
p16: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p17: U41#(tt(),M,N) -> activate#(M)
p18: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p19: plus#(N,|0|()) -> isNat#(N)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          n__isNatKind_A(x1) = ((1,0),(1,0)) x1 + (2,3)
          isNatKind#_A(x1) = x1 + (1,2)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (8,24)
          activate_A(x1) = x1 + (0,23)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (46,24)
          plus#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (3,72)
          s_A(x1) = ((1,0),(0,0)) x1 + (8,25)
          isNat#_A(x1) = ((1,0),(1,1)) x1 + (2,106)
          U21#_A(x1,x2) = ((1,0),(0,0)) x2 + (3,130)
          isNatKind_A(x1) = ((1,0),(1,0)) x1 + (2,14)
          tt_A() = (1,13)
          U41#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (9,119)
          and_A(x1,x2) = ((1,0),(0,0)) x2 + (5,59)
          isNat_A(x1) = ((0,0),(1,0)) x1 + (10,59)
          n__and_A(x1,x2) = ((1,0),(0,0)) x2 + (5,57)
          |0|_A() = (0,35)
          U13_A(x1) = ((1,0),(0,0)) x1 + (0,14)
          U12_A(x1,x2) = (10,15)
          U22_A(x1) = (2,14)
          U31_A(x1,x2) = ((1,0),(0,0)) x2 + (1,36)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (54,26)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (46,25)
          U11_A(x1,x2,x3) = (10,16)
          U21_A(x1,x2) = (3,25)
          n__0_A() = (0,13)
  
    precedence:
    
      activate = isNatKind = tt = and = isNat = n__and = |0| = U13 = U12 = U11 = U21 > n__isNatKind > n__plus = U31 = U41 = plus > n__s = s > n__0 > isNatKind# > isNat# = U21# = U22 > U41# > activate# = plus#
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__isNatKind) = []
      pi(isNatKind#) = [1]
      pi(n__s) = []
      pi(activate) = []
      pi(n__plus) = []
      pi(plus#) = [1, 2]
      pi(s) = []
      pi(isNat#) = [1]
      pi(U21#) = []
      pi(isNatKind) = []
      pi(tt) = []
      pi(U41#) = [2, 3]
      pi(and) = []
      pi(isNat) = []
      pi(n__and) = []
      pi(|0|) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = []
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p10

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p3: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,s(M)) -> isNat#(N)
p6: isNat#(n__s(V1)) -> activate#(V1)
p7: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p8: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p9: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p10: U21#(tt(),V1) -> isNat#(activate(V1))
p11: isNat#(n__plus(V1,V2)) -> activate#(V2)
p12: isNat#(n__plus(V1,V2)) -> activate#(V1)
p13: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p14: plus#(N,s(M)) -> isNat#(M)
p15: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p16: U41#(tt(),M,N) -> activate#(M)
p17: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p18: plus#(N,|0|()) -> isNat#(N)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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}


-- Reduction pair.

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

p1: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p3: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,|0|()) -> isNat#(N)
p6: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p7: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p8: isNat#(n__plus(V1,V2)) -> activate#(V1)
p9: isNat#(n__plus(V1,V2)) -> activate#(V2)
p10: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p11: U21#(tt(),V1) -> isNat#(activate(V1))
p12: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p13: isNat#(n__s(V1)) -> activate#(V1)
p14: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p15: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p16: plus#(N,s(M)) -> isNat#(M)
p17: plus#(N,s(M)) -> isNat#(N)
p18: U41#(tt(),M,N) -> activate#(M)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = x1 + (1,166)
          n__isNatKind_A(x1) = ((1,0),(1,1)) x1 + (10,119)
          isNatKind#_A(x1) = ((1,0),(1,1)) x1 + (1,118)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (11,38)
          activate_A(x1) = x1 + (0,30)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (1,205)
          |0|_A() = (22,57)
          isNat#_A(x1) = ((1,0),(1,1)) x1 + (3,154)
          n__s_A(x1) = ((1,0),(1,0)) x1 + (4,147)
          U21#_A(x1,x2) = ((1,0),(0,0)) x2 + (5,185)
          isNatKind_A(x1) = ((1,0),(1,1)) x1 + (10,148)
          tt_A() = (7,165)
          s_A(x1) = ((1,0),(1,0)) x1 + (4,153)
          U41#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,236)
          and_A(x1,x2) = x2 + (11,31)
          isNat_A(x1) = x1 + (6,139)
          n__and_A(x1,x2) = x2 + (11,2)
          U13_A(x1) = (8,166)
          U12_A(x1,x2) = (9,167)
          U22_A(x1) = (8,165)
          U31_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (23,98)
          U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (15,165)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (11,39)
          U11_A(x1,x2,x3) = ((1,0),(1,0)) x2 + x3 + (9,168)
          U21_A(x1,x2) = (9,146)
          n__0_A() = (22,27)
  
    precedence:
    
      U13 > U12 > isNat = U21 > isNatKind# = U11 > n__isNatKind = isNatKind > U31 > n__and > activate = s > |0| = and > plus# = U41# > U41 > plus > tt > activate# > n__plus > isNat# > n__s = U21# = U22 = n__0
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(n__plus) = [1, 2]
      pi(activate) = [1]
      pi(plus#) = []
      pi(|0|) = []
      pi(isNat#) = [1]
      pi(n__s) = []
      pi(U21#) = []
      pi(isNatKind) = []
      pi(tt) = []
      pi(s) = []
      pi(U41#) = []
      pi(and) = []
      pi(isNat) = [1]
      pi(n__and) = [2]
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = [1, 2]
      pi(U11) = [3]
      pi(U21) = []
      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: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p3: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,|0|()) -> isNat#(N)
p5: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p6: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p7: isNat#(n__plus(V1,V2)) -> activate#(V1)
p8: isNat#(n__plus(V1,V2)) -> activate#(V2)
p9: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p10: U21#(tt(),V1) -> isNat#(activate(V1))
p11: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p12: isNat#(n__s(V1)) -> activate#(V1)
p13: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p14: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p15: plus#(N,s(M)) -> isNat#(M)
p16: plus#(N,s(M)) -> isNat#(N)
p17: U41#(tt(),M,N) -> activate#(M)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: U41#(tt(),M,N) -> activate#(M)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,s(M)) -> isNat#(N)
p4: isNat#(n__s(V1)) -> activate#(V1)
p5: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p6: U21#(tt(),V1) -> isNat#(activate(V1))
p7: isNat#(n__plus(V1,V2)) -> activate#(V2)
p8: isNat#(n__plus(V1,V2)) -> activate#(V1)
p9: plus#(N,s(M)) -> isNat#(M)
p10: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p11: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p12: plus#(N,|0|()) -> isNat#(N)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          U41#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (27,23)
          tt_A() = (19,4)
          activate#_A(x1) = ((1,0),(1,0)) x1 + (13,24)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (14,45)
          plus#_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (26,1)
          s_A(x1) = ((1,0),(0,0)) x1 + (23,19)
          isNat#_A(x1) = x1 + (0,23)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (23,5)
          U21#_A(x1,x2) = ((1,0),(0,0)) x2 + (23,1)
          isNatKind_A(x1) = ((1,0),(1,0)) x1 + (15,35)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,21)
          and_A(x1,x2) = ((1,0),(1,1)) x2 + (1,33)
          isNat_A(x1) = x1 + (22,2)
          n__isNatKind_A(x1) = ((1,0),(0,0)) x1 + (15,0)
          n__and_A(x1,x2) = ((1,0),(1,1)) x2 + (1,32)
          |0|_A() = (23,49)
          U13_A(x1) = (20,5)
          U12_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (2,1)
          U22_A(x1) = (20,5)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (24,47)
          U41_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (37,1)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (14,46)
          U11_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (6,28)
          U21_A(x1,x2) = (21,6)
          n__0_A() = (23,6)
  
    precedence:
    
      U41# = activate# = n__plus = plus# = isNatKind = activate = and = isNat = n__isNatKind = n__and = |0| = U31 = plus = U11 > U12 = U41 > U22 = U21 > tt = U13 > s = n__s = U21# > isNat# = n__0
    
    partial status:
  
      pi(U41#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(s) = []
      pi(isNat#) = [1]
      pi(n__s) = []
      pi(U21#) = []
      pi(isNatKind) = []
      pi(activate) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__isNatKind) = []
      pi(n__and) = []
      pi(|0|) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = []
      pi(U11) = []
      pi(U21) = []
      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: U41#(tt(),M,N) -> activate#(M)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p5: U21#(tt(),V1) -> isNat#(activate(V1))
p6: isNat#(n__plus(V1,V2)) -> activate#(V2)
p7: isNat#(n__plus(V1,V2)) -> activate#(V1)
p8: plus#(N,s(M)) -> isNat#(M)
p9: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p10: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p11: plus#(N,|0|()) -> isNat#(N)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: U41#(tt(),M,N) -> activate#(M)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,|0|()) -> isNat#(N)
p4: isNat#(n__plus(V1,V2)) -> activate#(V1)
p5: isNat#(n__plus(V1,V2)) -> activate#(V2)
p6: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p7: U21#(tt(),V1) -> isNat#(activate(V1))
p8: isNat#(n__s(V1)) -> activate#(V1)
p9: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p10: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p11: plus#(N,s(M)) -> isNat#(M)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          U41#_A(x1,x2,x3) = ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (12,18)
          tt_A() = (1,7)
          activate#_A(x1) = ((0,0),(1,0)) x1 + (12,1)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (15,2)
          plus#_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (12,6)
          |0|_A() = (2,2)
          isNat#_A(x1) = ((0,0),(1,0)) x1 + (12,3)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (13,10)
          U21#_A(x1,x2) = ((0,0),(1,0)) x2 + (12,4)
          isNatKind_A(x1) = ((1,0),(1,0)) x1 + (14,25)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,17)
          s_A(x1) = ((1,0),(0,0)) x1 + (13,11)
          and_A(x1,x2) = ((1,0),(0,0)) x2 + (15,20)
          isNat_A(x1) = x1 + (11,14)
          n__isNatKind_A(x1) = ((1,0),(0,0)) x1 + (14,20)
          n__and_A(x1,x2) = ((1,0),(0,0)) x2 + (15,0)
          U13_A(x1) = x1 + (1,1)
          U12_A(x1,x2) = ((1,0),(0,0)) x2 + (13,0)
          U22_A(x1) = (2,8)
          U31_A(x1,x2) = x2 + (2,14)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (28,12)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (15,13)
          U11_A(x1,x2,x3) = ((1,0),(0,0)) x3 + (14,1)
          U21_A(x1,x2) = (3,9)
          n__0_A() = (2,1)
  
    precedence:
    
      U41# = activate# = plus# = isNat# = U21# > |0| = activate = and = n__and > s = U41 = plus > U11 > tt = isNatKind = isNat = n__isNatKind = U22 = U21 > U12 > U13 > n__plus = n__s = U31 = n__0
    
    partial status:
  
      pi(U41#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(U21#) = []
      pi(isNatKind) = []
      pi(activate) = []
      pi(s) = []
      pi(and) = []
      pi(isNat) = [1]
      pi(n__isNatKind) = []
      pi(n__and) = []
      pi(U13) = [1]
      pi(U12) = []
      pi(U22) = []
      pi(U31) = [2]
      pi(U41) = []
      pi(plus) = []
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

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

p1: U41#(tt(),M,N) -> activate#(M)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,|0|()) -> isNat#(N)
p4: isNat#(n__plus(V1,V2)) -> activate#(V1)
p5: isNat#(n__plus(V1,V2)) -> activate#(V2)
p6: U21#(tt(),V1) -> isNat#(activate(V1))
p7: isNat#(n__s(V1)) -> activate#(V1)
p8: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p9: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p10: plus#(N,s(M)) -> isNat#(M)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: U41#(tt(),M,N) -> activate#(M)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,s(M)) -> isNat#(M)
p4: isNat#(n__s(V1)) -> activate#(V1)
p5: isNat#(n__plus(V1,V2)) -> activate#(V2)
p6: isNat#(n__plus(V1,V2)) -> activate#(V1)
p7: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p8: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p9: plus#(N,|0|()) -> isNat#(N)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          U41#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (47,75)
          tt_A() = (1,0)
          activate#_A(x1) = ((1,0),(1,1)) x1 + (3,13)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (3,13)
          plus#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,30)
          s_A(x1) = x1 + (46,32)
          isNat#_A(x1) = ((1,0),(1,1)) x1 + (1,107)
          n__s_A(x1) = x1 + (46,11)
          and_A(x1,x2) = ((1,0),(0,0)) x2 + (1,23)
          isNat_A(x1) = x1 + (0,109)
          n__isNatKind_A(x1) = x1 + (50,0)
          n__and_A(x1,x2) = ((1,0),(0,0)) x2 + (1,1)
          activate_A(x1) = x1 + (0,22)
          |0|_A() = (52,24)
          U13_A(x1) = (2,22)
          U12_A(x1,x2) = ((1,0),(1,0)) x1 + (2,22)
          U22_A(x1) = (2,0)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (1,109)
          U41_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (49,111)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (3,34)
          U11_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (2,22)
          U21_A(x1,x2) = x2 + (3,23)
          isNatKind_A(x1) = x1 + (50,10)
          n__0_A() = (52,24)
  
    precedence:
    
      n__plus = U31 = U41 = plus > and = activate = U13 = U12 = U11 = isNatKind > |0| > s = isNat = n__isNatKind = U22 = U21 > tt > U41# > activate# = plus# = isNat# = n__s > n__and = n__0
    
    partial status:
  
      pi(U41#) = []
      pi(tt) = []
      pi(activate#) = [1]
      pi(n__plus) = [1, 2]
      pi(plus#) = [2]
      pi(s) = [1]
      pi(isNat#) = [1]
      pi(n__s) = [1]
      pi(and) = []
      pi(isNat) = []
      pi(n__isNatKind) = [1]
      pi(n__and) = []
      pi(activate) = []
      pi(|0|) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = [1, 2]
      pi(U11) = []
      pi(U21) = []
      pi(isNatKind) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: U41#(tt(),M,N) -> activate#(M)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,s(M)) -> isNat#(M)
p4: isNat#(n__plus(V1,V2)) -> activate#(V2)
p5: isNat#(n__plus(V1,V2)) -> activate#(V1)
p6: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p7: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p8: plus#(N,|0|()) -> isNat#(N)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: U41#(tt(),M,N) -> activate#(M)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,|0|()) -> isNat#(N)
p4: isNat#(n__plus(V1,V2)) -> activate#(V1)
p5: isNat#(n__plus(V1,V2)) -> activate#(V2)
p6: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p7: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p8: plus#(N,s(M)) -> isNat#(M)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          U41#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(0,0)) x3 + (19,4)
          tt_A() = (6,11)
          activate#_A(x1) = ((1,0),(1,0)) x1 + (7,0)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (17,0)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (18,2)
          |0|_A() = (10,1)
          isNat#_A(x1) = x1 + (9,2)
          s_A(x1) = x1 + (8,1)
          and_A(x1,x2) = ((1,0),(1,1)) x2
          isNat_A(x1) = ((1,0),(1,0)) x1 + (2,10)
          n__isNatKind_A(x1) = (16,0)
          n__and_A(x1,x2) = x2
          activate_A(x1) = ((1,0),(1,1)) x1
          U13_A(x1) = (7,12)
          U12_A(x1,x2) = (8,1)
          U22_A(x1) = ((0,0),(1,0)) x1 + (6,6)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (11,2)
          U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (25,3)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (17,1)
          U11_A(x1,x2,x3) = ((0,0),(1,0)) x1 + (18,3)
          U21_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (7,3)
          isNatKind_A(x1) = (16,16)
          n__0_A() = (10,1)
          n__s_A(x1) = x1 + (8,0)
  
    precedence:
    
      U41# = tt = s = and = isNat = activate = U13 = U12 = U31 = U41 = U11 = U21 = isNatKind > |0| = U22 > n__plus = plus > activate# = plus# > n__isNatKind = n__and > isNat# = n__0 > n__s
    
    partial status:
  
      pi(U41#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(isNat#) = [1]
      pi(s) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__isNatKind) = []
      pi(n__and) = []
      pi(activate) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = []
      pi(U11) = []
      pi(U21) = []
      pi(isNatKind) = []
      pi(n__0) = []
      pi(n__s) = []

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: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p2: plus#(N,|0|()) -> isNat#(N)
p3: isNat#(n__plus(V1,V2)) -> activate#(V1)
p4: isNat#(n__plus(V1,V2)) -> activate#(V2)
p5: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p6: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p7: plus#(N,s(M)) -> isNat#(M)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p2: plus#(N,s(M)) -> isNat#(M)
p3: isNat#(n__plus(V1,V2)) -> activate#(V2)
p4: isNat#(n__plus(V1,V2)) -> activate#(V1)
p5: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p6: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p7: plus#(N,|0|()) -> isNat#(N)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = ((1,0),(0,0)) x1 + (7,2)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (0,81)
          plus#_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (2,1)
          s_A(x1) = ((1,0),(0,0)) x1 + (9,49)
          isNat#_A(x1) = x1 + (8,7)
          U41#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + x3 + (10,36)
          and_A(x1,x2) = x2 + (0,35)
          isNat_A(x1) = ((1,0),(1,0)) x1 + (21,24)
          n__isNatKind_A(x1) = x1 + (12,11)
          n__and_A(x1,x2) = x2 + (0,2)
          tt_A() = (18,2)
          activate_A(x1) = x1 + (0,34)
          |0|_A() = (7,6)
          U13_A(x1) = (19,1)
          U12_A(x1,x2) = ((1,0),(0,0)) x1 + (4,3)
          U22_A(x1) = (19,2)
          U31_A(x1,x2) = ((1,0),(0,0)) x2 + (6,35)
          U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (9,50)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (0,82)
          U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (8,4)
          U21_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (20,5)
          isNatKind_A(x1) = x1 + (12,12)
          n__0_A() = (7,5)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (9,16)
  
    precedence:
    
      U11 > U13 = U12 > n__plus = and = tt = activate = |0| = U22 = U31 = U41 = plus = isNatKind > activate# = plus# = U41# > isNat# > n__and > n__isNatKind > s = isNat = U21 = n__0 = n__s
    
    partial status:
  
      pi(activate#) = []
      pi(n__plus) = [2]
      pi(plus#) = []
      pi(s) = []
      pi(isNat#) = [1]
      pi(U41#) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__isNatKind) = [1]
      pi(n__and) = []
      pi(tt) = []
      pi(activate) = [1]
      pi(|0|) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = [2]
      pi(U11) = []
      pi(U21) = []
      pi(isNatKind) = []
      pi(n__0) = []
      pi(n__s) = []

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: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p2: isNat#(n__plus(V1,V2)) -> activate#(V2)
p3: isNat#(n__plus(V1,V2)) -> activate#(V1)
p4: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p5: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p6: plus#(N,|0|()) -> isNat#(N)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p2: plus#(N,|0|()) -> isNat#(N)
p3: isNat#(n__plus(V1,V2)) -> activate#(V1)
p4: isNat#(n__plus(V1,V2)) -> activate#(V2)
p5: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p6: U41#(tt(),M,N) -> plus#(activate(N),activate(M))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = ((1,0),(1,0)) x1 + (1,1)
          n__plus_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (9,3)
          plus#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (10,9)
          |0|_A() = (0,14)
          isNat#_A(x1) = ((1,0),(1,0)) x1
          s_A(x1) = ((1,0),(0,0)) x1 + (12,0)
          U41#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (11,9)
          and_A(x1,x2) = ((1,0),(0,0)) x2 + (9,12)
          isNat_A(x1) = x1 + (4,8)
          n__isNatKind_A(x1) = x1 + (1,0)
          n__and_A(x1,x2) = ((1,0),(0,0)) x2 + (9,0)
          tt_A() = (0,0)
          activate_A(x1) = x1 + (0,12)
          U13_A(x1) = (0,0)
          U12_A(x1,x2) = (1,0)
          U22_A(x1) = ((0,0),(1,0)) x1 + (1,1)
          U31_A(x1,x2) = x2 + (1,13)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (21,11)
          plus_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (9,14)
          U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (2,1)
          U21_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (13,6)
          isNatKind_A(x1) = x1 + (1,12)
          n__0_A() = (0,3)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (12,0)
  
    precedence:
    
      isNat = U13 > activate# = plus# = isNat# = U41# > n__plus = |0| = and = activate = U12 = U31 = U41 = plus = U11 = isNatKind > s = U22 = U21 = n__s > n__and = tt > n__0 > n__isNatKind
    
    partial status:
  
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(isNat#) = []
      pi(s) = []
      pi(U41#) = []
      pi(and) = []
      pi(isNat) = [1]
      pi(n__isNatKind) = []
      pi(n__and) = []
      pi(tt) = []
      pi(activate) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = []
      pi(U11) = []
      pi(U21) = []
      pi(isNatKind) = []
      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: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p2: plus#(N,|0|()) -> isNat#(N)
p3: isNat#(n__plus(V1,V2)) -> activate#(V2)
p4: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p5: U41#(tt(),M,N) -> plus#(activate(N),activate(M))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p2: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p3: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p4: plus#(N,|0|()) -> isNat#(N)
p5: isNat#(n__plus(V1,V2)) -> activate#(V2)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = ((1,0),(1,1)) x1 + (1,45)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (49,44)
          plus#_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (8,139)
          s_A(x1) = ((1,0),(0,0)) x1 + (5,82)
          U41#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,0)) x3 + (9,225)
          and_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (39,1)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (7,141)
          n__isNatKind_A(x1) = ((1,0),(1,1)) x1 + (9,58)
          n__and_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (39,1)
          tt_A() = (8,72)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,82)
          |0|_A() = (10,2)
          isNat#_A(x1) = x1 + (9,3)
          U13_A(x1) = ((0,0),(1,0)) x1 + (9,66)
          U12_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (10,132)
          U22_A(x1) = (9,73)
          U31_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (11,1)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (54,83)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (49,45)
          U11_A(x1,x2,x3) = ((0,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (50,142)
          U21_A(x1,x2) = ((1,0),(1,0)) x1 + (2,73)
          isNatKind_A(x1) = ((1,0),(1,1)) x1 + (9,60)
          n__0_A() = (10,1)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (5,81)
  
    precedence:
    
      U41# > activate# = plus# = and = n__and = isNat# > U31 > activate = |0| = plus > U41 = U11 > isNat = n__isNatKind = tt = U13 = U12 = U22 = U21 = isNatKind > n__plus = s = n__0 = n__s
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__plus) = [1, 2]
      pi(plus#) = [1, 2]
      pi(s) = []
      pi(U41#) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__isNatKind) = [1]
      pi(n__and) = []
      pi(tt) = []
      pi(activate) = [1]
      pi(|0|) = []
      pi(isNat#) = [1]
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = [1]
      pi(U11) = []
      pi(U21) = []
      pi(isNatKind) = [1]
      pi(n__0) = []
      pi(n__s) = []

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: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p2: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p3: plus#(N,|0|()) -> isNat#(N)
p4: isNat#(n__plus(V1,V2)) -> activate#(V2)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p2: U41#(tt(),M,N) -> plus#(activate(N),activate(M))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          plus#_A(x1,x2) = ((1,0),(1,0)) x2 + (2,19)
          s_A(x1) = ((1,0),(0,0)) x1 + (21,21)
          U41#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + (23,22)
          and_A(x1,x2) = ((1,0),(1,1)) x2 + (8,2)
          isNat_A(x1) = ((1,0),(1,0)) x1 + (12,39)
          n__isNatKind_A(x1) = ((1,0),(0,0)) x1 + (15,22)
          n__and_A(x1,x2) = ((1,0),(1,1)) x2 + (8,1)
          tt_A() = (1,20)
          activate_A(x1) = x1 + (0,19)
          U13_A(x1) = (1,20)
          U12_A(x1,x2) = ((0,0),(1,0)) x1 + (2,20)
          U22_A(x1) = (2,17)
          U31_A(x1,x2) = x2 + (1,20)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (35,22)
          plus_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (14,42)
          U11_A(x1,x2,x3) = x1 + ((0,0),(1,0)) x2 + (2,13)
          U21_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (22,17)
          isNatKind_A(x1) = ((1,0),(0,0)) x1 + (15,40)
          n__0_A() = (0,1)
          n__plus_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (14,41)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (21,3)
          |0|_A() = (0,2)
  
    precedence:
    
      and = activate = U31 = U41 = plus = isNatKind > U22 > U12 = U11 > tt = U13 > s > plus# = U41# = isNat > n__isNatKind = n__and = |0| > n__0 = n__plus > U21 = n__s
    
    partial status:
  
      pi(plus#) = []
      pi(s) = []
      pi(U41#) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__isNatKind) = []
      pi(n__and) = []
      pi(tt) = []
      pi(activate) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = []
      pi(U11) = []
      pi(U21) = []
      pi(isNatKind) = []
      pi(n__0) = []
      pi(n__plus) = [1]
      pi(n__s) = []
      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: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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: 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(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isNatKind#_A(x1) = x1 + (1,25)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (8,24)
          activate_A(x1) = ((1,0),(1,1)) x1
          n__s_A(x1) = ((1,0),(0,0)) x1 + (21,1)
          U13_A(x1) = (2,11)
          tt_A() = (1,10)
          U12_A(x1,x2) = (3,12)
          isNat_A(x1) = x1 + (5,1)
          U22_A(x1) = ((0,0),(1,0)) x1 + (2,10)
          U11_A(x1,x2,x3) = ((0,0),(1,0)) x3 + (4,13)
          U21_A(x1,x2) = (3,0)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (9,1)
          U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (29,30)
          s_A(x1) = ((1,0),(1,0)) x1 + (21,21)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (8,31)
          n__0_A() = (8,0)
          and_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (5,2)
          isNatKind_A(x1) = ((1,0),(1,0)) x1 + (4,3)
          n__isNatKind_A(x1) = ((1,0),(0,0)) x1 + (4,0)
          |0|_A() = (8,2)
          n__and_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (5,1)
  
    precedence:
    
      activate = tt = isNat = U22 = U21 = U31 = and = isNatKind = n__and > U11 > n__isNatKind > n__plus = U41 = s = plus > isNatKind# = n__s = U12 = n__0 = |0| > U13
    
    partial status:
  
      pi(isNatKind#) = [1]
      pi(n__plus) = []
      pi(activate) = []
      pi(n__s) = []
      pi(U13) = []
      pi(tt) = []
      pi(U12) = []
      pi(isNat) = []
      pi(U22) = []
      pi(U11) = []
      pi(U21) = []
      pi(U31) = []
      pi(U41) = []
      pi(s) = []
      pi(plus) = [2]
      pi(n__0) = []
      pi(and) = []
      pi(isNatKind) = []
      pi(n__isNatKind) = []
      pi(|0|) = []
      pi(n__and) = []

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__s(V1)) -> isNatKind#(activate(V1))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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__s(V1)) -> isNatKind#(activate(V1))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: activate(n__0()) -> |0|()
r23: activate(n__plus(X1,X2)) -> plus(X1,X2)
r24: activate(n__isNatKind(X)) -> isNatKind(X)
r25: activate(n__s(X)) -> s(X)
r26: activate(n__and(X1,X2)) -> and(X1,X2)
r27: 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isNatKind#_A(x1) = ((1,0),(1,1)) x1 + (1,2)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (24,9)
          activate_A(x1) = x1 + (0,8)
          U13_A(x1) = (6,7)
          tt_A() = (5,6)
          U12_A(x1,x2) = (7,1)
          isNat_A(x1) = (9,11)
          U22_A(x1) = ((0,0),(1,0)) x1 + (6,2)
          U11_A(x1,x2,x3) = ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (8,12)
          U21_A(x1,x2) = ((0,0),(1,0)) x2 + (7,1)
          U31_A(x1,x2) = x2 + (11,23)
          U41_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (30,6)
          s_A(x1) = ((1,0),(0,0)) x1 + (24,10)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (6,12)
          n__0_A() = (10,1)
          n__plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (6,5)
          and_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (1,4)
          isNatKind_A(x1) = ((1,0),(0,0)) x1 + (1,7)
          n__isNatKind_A(x1) = ((1,0),(0,0)) x1 + (1,1)
          |0|_A() = (10,8)
          n__and_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (1,3)
  
    precedence:
    
      n__s = activate = tt = isNat = U11 = U21 = U31 = U41 = s = plus = n__plus = and = isNatKind = n__isNatKind > isNatKind# = U22 = n__0 = |0| > U13 = U12 = n__and
    
    partial status:
  
      pi(isNatKind#) = [1]
      pi(n__s) = []
      pi(activate) = []
      pi(U13) = []
      pi(tt) = []
      pi(U12) = []
      pi(isNat) = []
      pi(U22) = []
      pi(U11) = []
      pi(U21) = []
      pi(U31) = []
      pi(U41) = []
      pi(s) = []
      pi(plus) = []
      pi(n__0) = []
      pi(n__plus) = []
      pi(and) = []
      pi(isNatKind) = []
      pi(n__isNatKind) = []
      pi(|0|) = []
      pi(n__and) = []

The next rules are strictly ordered:

  p1

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