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^1
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2,x3) = x2 + x3 + 10
          tt_A() = 0
          U12#_A(x1,x2) = x2 + 10
          isNat_A(x1) = x1
          activate_A(x1) = x1
          activate#_A(x1) = x1 + 9
          n__and_A(x1,x2) = x2
          and#_A(x1,x2) = x2 + 9
          n__isNatKind_A(x1) = x1
          isNatKind#_A(x1) = x1 + 4
          n__s_A(x1) = x1 + 6
          n__plus_A(x1,x2) = x1 + x2 + 11
          plus#_A(x1,x2) = x1 + x2 + 3
          s_A(x1) = x1 + 6
          isNat#_A(x1) = x1 + 5
          isNatKind_A(x1) = x1
          U21#_A(x1,x2) = x2 + 10
          and_A(x1,x2) = x2
          U41#_A(x1,x2,x3) = x2 + x3 + 9
          |0|_A() = 8
          U31#_A(x1,x2) = x2 + 10
          U13_A(x1) = 0
          U12_A(x1,x2) = 0
          U22_A(x1) = 0
          U31_A(x1,x2) = x2 + 19
          U41_A(x1,x2,x3) = x2 + x3 + 17
          plus_A(x1,x2) = x1 + x2 + 11
          U11_A(x1,x2,x3) = x3 + 1
          U21_A(x1,x2) = 6
          n__0_A() = 8
  
    precedence:
    
      activate = n__and = and = plus > s = U41 > n__s > U11# = n__plus = isNat# = U31# > isNat > |0| > tt = U12# = activate# = and# = n__isNatKind = isNatKind# = plus# = isNatKind = U21# = U41# = U13 = U12 = U22 = U31 = U11 = U21 = n__0
    
    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) = [2]
      pi(plus#) = [1, 2]
      pi(s) = []
      pi(isNat#) = [1]
      pi(isNatKind) = []
      pi(U21#) = []
      pi(and) = []
      pi(U41#) = []
      pi(|0|) = []
      pi(U31#) = [2]
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = []
      pi(U11) = [3]
      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: 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: 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:

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


-- 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) -> isNat#(activate(V2))
p3: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p4: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p5: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p6: and#(tt(),X) -> activate#(X)
p7: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p8: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p9: U31#(tt(),N) -> activate#(N)
p10: activate#(n__isNatKind(X)) -> isNatKind#(X)
p11: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p12: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p13: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p14: activate#(n__and(X1,X2)) -> and#(X1,X2)
p15: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p16: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p17: isNatKind#(n__s(V1)) -> activate#(V1)
p18: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p19: plus#(N,|0|()) -> isNat#(N)
p20: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p21: isNat#(n__plus(V1,V2)) -> activate#(V1)
p22: isNat#(n__plus(V1,V2)) -> activate#(V2)
p23: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p24: U21#(tt(),V1) -> isNat#(activate(V1))
p25: isNat#(n__s(V1)) -> activate#(V1)
p26: U21#(tt(),V1) -> activate#(V1)
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)
p35: U11#(tt(),V1,V2) -> activate#(V1)
p36: U11#(tt(),V1,V2) -> activate#(V2)
p37: U12#(tt(),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^1
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2,x3) = x2 + x3 + 14
          tt_A() = 1
          U12#_A(x1,x2) = x1 + x2 + 6
          isNat_A(x1) = x1 + 7
          activate_A(x1) = x1
          isNat#_A(x1) = x1 + 6
          n__plus_A(x1,x2) = x1 + x2 + 17
          and_A(x1,x2) = x2 + 4
          isNatKind_A(x1) = x1 + 18
          n__isNatKind_A(x1) = x1 + 18
          and#_A(x1,x2) = x2 + 3
          activate#_A(x1) = x1 + 2
          plus#_A(x1,x2) = x1 + x2 + 16
          |0|_A() = 19
          U31#_A(x1,x2) = x2 + 34
          isNatKind#_A(x1) = x1 + 5
          n__and_A(x1,x2) = x2 + 4
          n__s_A(x1) = x1 + 11
          U21#_A(x1,x2) = x2 + 7
          s_A(x1) = x1 + 11
          U41#_A(x1,x2,x3) = x2 + x3 + 26
          U13_A(x1) = 2
          U12_A(x1,x2) = 3
          U22_A(x1) = 2
          U31_A(x1,x2) = x2 + 20
          U41_A(x1,x2,x3) = x2 + x3 + 28
          plus_A(x1,x2) = x1 + x2 + 17
          U11_A(x1,x2,x3) = x2 + 8
          U21_A(x1,x2) = x2 + 8
          n__0_A() = 19
  
    precedence:
    
      U12# > U11# = tt = plus# = U41# > U11 > isNat > activate > isNatKind# > and > n__and > |0| > n__0 > plus > U41 > s > U31# > U12 > n__plus = U13 > isNat# = isNatKind = n__isNatKind > U31 > U21# > and# > activate# > n__s = U22 = U21
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(U12#) = []
      pi(isNat) = [1]
      pi(activate) = [1]
      pi(isNat#) = [1]
      pi(n__plus) = []
      pi(and) = [2]
      pi(isNatKind) = [1]
      pi(n__isNatKind) = [1]
      pi(and#) = []
      pi(activate#) = [1]
      pi(plus#) = [1]
      pi(|0|) = []
      pi(U31#) = []
      pi(isNatKind#) = [1]
      pi(n__and) = []
      pi(n__s) = [1]
      pi(U21#) = []
      pi(s) = []
      pi(U41#) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = [3]
      pi(plus) = []
      pi(U11) = []
      pi(U21) = [2]
      pi(n__0) = []

The next rules are strictly ordered:

  p34

We remove them from the problem.

-- SCC decomposition.

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

p1: U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2))
p2: U12#(tt(),V2) -> isNat#(activate(V2))
p3: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p4: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p5: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p6: and#(tt(),X) -> activate#(X)
p7: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p8: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p9: U31#(tt(),N) -> activate#(N)
p10: activate#(n__isNatKind(X)) -> isNatKind#(X)
p11: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p12: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p13: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p14: activate#(n__and(X1,X2)) -> and#(X1,X2)
p15: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p16: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p17: isNatKind#(n__s(V1)) -> activate#(V1)
p18: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p19: plus#(N,|0|()) -> isNat#(N)
p20: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p21: isNat#(n__plus(V1,V2)) -> activate#(V1)
p22: isNat#(n__plus(V1,V2)) -> activate#(V2)
p23: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p24: U21#(tt(),V1) -> isNat#(activate(V1))
p25: isNat#(n__s(V1)) -> activate#(V1)
p26: U21#(tt(),V1) -> activate#(V1)
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: U11#(tt(),V1,V2) -> activate#(V1)
p35: U11#(tt(),V1,V2) -> activate#(V2)
p36: U12#(tt(),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, 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: 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)) -> 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: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p17: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p18: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p19: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p20: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
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#(N)
p31: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p32: plus#(N,|0|()) -> isNat#(N)
p33: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p34: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p35: U31#(tt(),N) -> activate#(N)
p36: 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^1
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2,x3) = x2 + x3 + 21
          tt_A() = 8
          U12#_A(x1,x2) = x2 + 7
          isNat_A(x1) = 10
          activate_A(x1) = x1
          activate#_A(x1) = x1 + 7
          n__and_A(x1,x2) = x1 + x2
          and#_A(x1,x2) = x2 + 7
          n__isNatKind_A(x1) = x1 + 9
          isNatKind#_A(x1) = x1 + 15
          n__s_A(x1) = x1 + 27
          n__plus_A(x1,x2) = x1 + x2 + 14
          plus#_A(x1,x2) = x1 + x2 + 20
          s_A(x1) = x1 + 27
          isNat#_A(x1) = x1 + 7
          U21#_A(x1,x2) = x2 + 28
          isNatKind_A(x1) = x1 + 9
          and_A(x1,x2) = x1 + x2
          U41#_A(x1,x2,x3) = x2 + x3 + 47
          |0|_A() = 0
          U31#_A(x1,x2) = x2 + 8
          U13_A(x1) = 8
          U12_A(x1,x2) = x1
          U22_A(x1) = x1
          U31_A(x1,x2) = x2 + 1
          U41_A(x1,x2,x3) = x2 + x3 + 41
          plus_A(x1,x2) = x1 + x2 + 14
          U11_A(x1,x2,x3) = 10
          U21_A(x1,x2) = 10
          n__0_A() = 0
  
    precedence:
    
      activate > |0| = n__0 > U11# = U12# = activate# = and# = isNatKind# = plus# = isNat# = U21# = U41# = U31# > isNat = U22 = U21 > U13 = U12 = U11 > tt = n__and = n__isNatKind = isNatKind = and > n__s = n__plus = s = U31 = U41 = plus
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(U12#) = [2]
      pi(isNat) = []
      pi(activate) = [1]
      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(U21#) = []
      pi(isNatKind) = []
      pi(and) = []
      pi(U41#) = []
      pi(|0|) = []
      pi(U31#) = []
      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:

  p36

We remove them from the problem.

-- SCC decomposition.

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

p1: U11#(tt(),V1,V2) -> U12#(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)) -> 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: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p17: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p18: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p19: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p20: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
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#(N)
p31: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p32: plus#(N,|0|()) -> isNat#(N)
p33: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p34: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p35: 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, p31, p32, p33, p34, p35}


-- 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__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: and#(tt(),X) -> activate#(X)
p9: activate#(n__and(X1,X2)) -> and#(X1,X2)
p10: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p11: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p12: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p13: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p14: isNatKind#(n__s(V1)) -> activate#(V1)
p15: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p16: plus#(N,|0|()) -> isNat#(N)
p17: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p18: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p19: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p20: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p21: isNat#(n__plus(V1,V2)) -> activate#(V1)
p22: isNat#(n__plus(V1,V2)) -> activate#(V2)
p23: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p24: U21#(tt(),V1) -> isNat#(activate(V1))
p25: isNat#(n__s(V1)) -> activate#(V1)
p26: U21#(tt(),V1) -> activate#(V1)
p27: U11#(tt(),V1,V2) -> activate#(V1)
p28: U11#(tt(),V1,V2) -> activate#(V2)
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) -> plus#(activate(N),activate(M))
p31: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p32: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p33: plus#(N,s(M)) -> isNat#(M)
p34: plus#(N,s(M)) -> isNat#(N)
p35: U41#(tt(),M,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^1
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2,x3) = x2 + x3 + 29
          tt_A() = 1
          U12#_A(x1,x2) = x1 + x2 + 23
          isNat_A(x1) = 5
          activate_A(x1) = x1
          activate#_A(x1) = x1 + 23
          n__plus_A(x1,x2) = x1 + x2 + 22
          plus#_A(x1,x2) = x1 + x2 + 44
          |0|_A() = 0
          U31#_A(x1,x2) = x2 + 24
          and_A(x1,x2) = x1 + x2 + 1
          n__isNatKind_A(x1) = x1 + 4
          isNatKind#_A(x1) = x1 + 24
          and#_A(x1,x2) = x1 + x2 + 23
          isNatKind_A(x1) = x1 + 4
          n__and_A(x1,x2) = x1 + x2 + 1
          n__s_A(x1) = x1
          isNat#_A(x1) = x1 + 24
          U21#_A(x1,x2) = x2 + 24
          s_A(x1) = x1
          U41#_A(x1,x2,x3) = x2 + x3 + 44
          U13_A(x1) = 2
          U12_A(x1,x2) = 3
          U22_A(x1) = 2
          U31_A(x1,x2) = x2 + 1
          U41_A(x1,x2,x3) = x2 + x3 + 22
          plus_A(x1,x2) = x1 + x2 + 22
          U11_A(x1,x2,x3) = 4
          U21_A(x1,x2) = 3
          n__0_A() = 0
  
    precedence:
    
      n__isNatKind = isNatKind > U12# > isNat = activate = plus# = U41# > activate# = |0| = isNatKind# = and# = isNat# = U21# > U11# = U31# = and = plus > n__s = s = U31 = U41 = U11 > U13 = U12 > tt = U22 = U21 = n__0 > n__plus = n__and
    
    partial status:
  
      pi(U11#) = [2]
      pi(tt) = []
      pi(U12#) = []
      pi(isNat) = []
      pi(activate) = [1]
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(U31#) = []
      pi(and) = [2]
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(and#) = []
      pi(isNatKind) = []
      pi(n__and) = []
      pi(n__s) = []
      pi(isNat#) = []
      pi(U21#) = []
      pi(s) = []
      pi(U41#) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = [2]
      pi(U41) = [3]
      pi(plus) = [1, 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__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: and#(tt(),X) -> activate#(X)
p8: activate#(n__and(X1,X2)) -> and#(X1,X2)
p9: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p10: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p11: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p12: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p13: isNatKind#(n__s(V1)) -> activate#(V1)
p14: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p15: plus#(N,|0|()) -> isNat#(N)
p16: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p17: U11#(tt(),V1,V2) -> isNat#(activate(V1))
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: U21#(tt(),V1) -> isNat#(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)

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}


-- 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#(N)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__and(X1,X2)) -> and#(X1,X2)
p5: and#(tt(),X) -> activate#(X)
p6: activate#(n__isNatKind(X)) -> isNatKind#(X)
p7: isNatKind#(n__s(V1)) -> activate#(V1)
p8: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p10: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p11: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p12: isNatKind#(n__plus(V1,V2)) -> 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: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p21: U11#(tt(),V1,V2) -> activate#(V2)
p22: U11#(tt(),V1,V2) -> activate#(V1)
p23: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p24: plus#(N,s(M)) -> isNat#(M)
p25: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p26: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
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) -> activate#(N)
p29: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p30: plus#(N,|0|()) -> isNat#(N)
p31: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p32: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p33: 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^1
        order: lexicographic order
        interpretations:
          activate#_A(x1) = x1
          n__plus_A(x1,x2) = x1 + x2 + 4
          plus#_A(x1,x2) = x1 + x2 + 3
          s_A(x1) = x1 + 4
          isNat#_A(x1) = x1 + 2
          n__s_A(x1) = x1 + 4
          n__and_A(x1,x2) = x2 + 4
          and#_A(x1,x2) = x2 + 1
          tt_A() = 0
          n__isNatKind_A(x1) = x1 + 1
          isNatKind#_A(x1) = x1
          activate_A(x1) = x1
          isNatKind_A(x1) = x1 + 1
          U21#_A(x1,x2) = x2 + 6
          U11#_A(x1,x2,x3) = x2 + x3 + 6
          and_A(x1,x2) = x2 + 4
          isNat_A(x1) = 9
          U41#_A(x1,x2,x3) = x2 + x3 + 5
          |0|_A() = 0
          U31#_A(x1,x2) = x2 + 1
          U13_A(x1) = 1
          U12_A(x1,x2) = 2
          U22_A(x1) = 1
          U31_A(x1,x2) = x2 + 4
          U41_A(x1,x2,x3) = x2 + x3 + 8
          plus_A(x1,x2) = x1 + x2 + 4
          U11_A(x1,x2,x3) = 5
          U21_A(x1,x2) = 8
          n__0_A() = 0
  
    precedence:
    
      n__isNatKind = isNatKind > n__plus = n__and = and = plus > activate = U41 > U13 = U12 > activate# = plus# = s = isNat# = n__s = and# = tt = isNatKind# = U21# = U11# = isNat = U41# = |0| = U31# = U22 = U31 = U11 = U21 = n__0
    
    partial status:
  
      pi(activate#) = []
      pi(n__plus) = [1, 2]
      pi(plus#) = []
      pi(s) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(n__and) = []
      pi(and#) = []
      pi(tt) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(activate) = [1]
      pi(isNatKind) = []
      pi(U21#) = []
      pi(U11#) = []
      pi(and) = []
      pi(isNat) = []
      pi(U41#) = []
      pi(|0|) = []
      pi(U31#) = [2]
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = [2]
      pi(U41) = [3]
      pi(plus) = [1, 2]
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p33

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,s(M)) -> isNat#(N)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__and(X1,X2)) -> and#(X1,X2)
p5: and#(tt(),X) -> activate#(X)
p6: activate#(n__isNatKind(X)) -> isNatKind#(X)
p7: isNatKind#(n__s(V1)) -> activate#(V1)
p8: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p10: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p11: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p12: isNatKind#(n__plus(V1,V2)) -> 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: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p21: U11#(tt(),V1,V2) -> activate#(V2)
p22: U11#(tt(),V1,V2) -> activate#(V1)
p23: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p24: plus#(N,s(M)) -> isNat#(M)
p25: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p26: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
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) -> activate#(N)
p29: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p30: plus#(N,|0|()) -> isNat#(N)
p31: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p32: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),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, p31}


-- 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|()) -> and#(isNat(N),n__isNatKind(N))
p3: and#(tt(),X) -> activate#(X)
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: activate#(n__and(X1,X2)) -> and#(X1,X2)
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|()) -> isNat#(N)
p13: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p14: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p15: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p16: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p17: isNat#(n__plus(V1,V2)) -> activate#(V1)
p18: isNat#(n__plus(V1,V2)) -> activate#(V2)
p19: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p20: U21#(tt(),V1) -> isNat#(activate(V1))
p21: isNat#(n__s(V1)) -> activate#(V1)
p22: U21#(tt(),V1) -> activate#(V1)
p23: U11#(tt(),V1,V2) -> activate#(V1)
p24: U11#(tt(),V1,V2) -> activate#(V2)
p25: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p26: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p27: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p28: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p29: plus#(N,s(M)) -> isNat#(M)
p30: plus#(N,s(M)) -> isNat#(N)
p31: U41#(tt(),M,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^1
        order: lexicographic order
        interpretations:
          activate#_A(x1) = x1
          n__plus_A(x1,x2) = x1 + x2 + 13
          plus#_A(x1,x2) = x1 + x2 + 11
          |0|_A() = 0
          and#_A(x1,x2) = x2 + 1
          isNat_A(x1) = 5
          n__isNatKind_A(x1) = x1 + 6
          tt_A() = 1
          isNatKind#_A(x1) = x1 + 1
          isNatKind_A(x1) = x1 + 6
          activate_A(x1) = x1
          n__and_A(x1,x2) = x2 + 3
          n__s_A(x1) = x1
          isNat#_A(x1) = x1 + 5
          U11#_A(x1,x2,x3) = x2 + x3 + 14
          and_A(x1,x2) = x2 + 3
          U21#_A(x1,x2) = x2 + 5
          s_A(x1) = x1
          U41#_A(x1,x2,x3) = x2 + x3 + 11
          U13_A(x1) = 2
          U12_A(x1,x2) = 3
          U22_A(x1) = 2
          U31_A(x1,x2) = x2 + 1
          U41_A(x1,x2,x3) = x2 + x3 + 13
          plus_A(x1,x2) = x1 + x2 + 13
          U11_A(x1,x2,x3) = 4
          U21_A(x1,x2) = 4
          n__0_A() = 0
  
    precedence:
    
      n__plus = U22 = U31 = U41 = plus > n__s = s > |0| = isNat# = U11# = U21# = U12 = U11 = n__0 > U13 > tt > n__and = and > plus# = activate = U41# > isNat = isNatKind > n__isNatKind = isNatKind# > and# > activate# > U21
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__plus) = [1]
      pi(plus#) = []
      pi(|0|) = []
      pi(and#) = [2]
      pi(isNat) = []
      pi(n__isNatKind) = [1]
      pi(tt) = []
      pi(isNatKind#) = []
      pi(isNatKind) = []
      pi(activate) = [1]
      pi(n__and) = []
      pi(n__s) = []
      pi(isNat#) = []
      pi(U11#) = []
      pi(and) = []
      pi(U21#) = []
      pi(s) = []
      pi(U41#) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = [3]
      pi(plus) = [1]
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p22

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|()) -> and#(isNat(N),n__isNatKind(N))
p3: and#(tt(),X) -> activate#(X)
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: activate#(n__and(X1,X2)) -> and#(X1,X2)
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|()) -> isNat#(N)
p13: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p14: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p15: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p16: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p17: isNat#(n__plus(V1,V2)) -> activate#(V1)
p18: isNat#(n__plus(V1,V2)) -> activate#(V2)
p19: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p20: U21#(tt(),V1) -> isNat#(activate(V1))
p21: isNat#(n__s(V1)) -> activate#(V1)
p22: U11#(tt(),V1,V2) -> activate#(V1)
p23: U11#(tt(),V1,V2) -> activate#(V2)
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) -> plus#(activate(N),activate(M))
p26: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p27: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p28: plus#(N,s(M)) -> isNat#(M)
p29: plus#(N,s(M)) -> isNat#(N)
p30: U41#(tt(),M,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: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p2: plus#(N,s(M)) -> isNat#(N)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__and(X1,X2)) -> and#(X1,X2)
p5: and#(tt(),X) -> activate#(X)
p6: activate#(n__isNatKind(X)) -> isNatKind#(X)
p7: isNatKind#(n__s(V1)) -> activate#(V1)
p8: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p10: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p11: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p12: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p13: isNat#(n__s(V1)) -> U21#(isNatKind(activate(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: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p20: U11#(tt(),V1,V2) -> activate#(V2)
p21: U11#(tt(),V1,V2) -> activate#(V1)
p22: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p23: plus#(N,s(M)) -> isNat#(M)
p24: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(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)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
p27: U41#(tt(),M,N) -> activate#(N)
p28: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p29: plus#(N,|0|()) -> isNat#(N)
p30: 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^1
        order: lexicographic order
        interpretations:
          activate#_A(x1) = x1 + 7
          n__plus_A(x1,x2) = x1 + x2 + 11
          plus#_A(x1,x2) = x1 + x2 + 5
          s_A(x1) = x1 + 11
          isNat#_A(x1) = x1 + 7
          n__s_A(x1) = x1 + 11
          n__and_A(x1,x2) = x2 + 5
          and#_A(x1,x2) = x2 + 8
          tt_A() = 6
          n__isNatKind_A(x1) = x1 + 2
          isNatKind#_A(x1) = x1 + 1
          activate_A(x1) = x1
          isNatKind_A(x1) = x1 + 2
          U21#_A(x1,x2) = x2 + 8
          U11#_A(x1,x2,x3) = x2 + x3 + 12
          and_A(x1,x2) = x2 + 5
          isNat_A(x1) = 13
          U41#_A(x1,x2,x3) = x2 + x3 + 8
          |0|_A() = 9
          U13_A(x1) = 7
          U12_A(x1,x2) = 8
          U22_A(x1) = 7
          U31_A(x1,x2) = x2 + 1
          U41_A(x1,x2,x3) = x2 + x3 + 22
          plus_A(x1,x2) = x1 + x2 + 11
          U11_A(x1,x2,x3) = 12
          U21_A(x1,x2) = 10
          n__0_A() = 9
  
    precedence:
    
      n__isNatKind = isNatKind > U21# > plus# = isNat# = U11# = U41# > U22 > U31 > n__and = activate = and = plus > s = n__s = U41 > activate# = and# = isNatKind# > n__plus > tt = isNat = |0| = U13 = U12 = U11 = U21 = n__0
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__plus) = [2]
      pi(plus#) = []
      pi(s) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(n__and) = [2]
      pi(and#) = [2]
      pi(tt) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = [1]
      pi(activate) = [1]
      pi(isNatKind) = []
      pi(U21#) = [2]
      pi(U11#) = []
      pi(and) = [2]
      pi(isNat) = []
      pi(U41#) = []
      pi(|0|) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = [2]
      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: plus#(N,s(M)) -> isNat#(N)
p2: isNat#(n__s(V1)) -> activate#(V1)
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: 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) -> 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: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p18: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p19: U11#(tt(),V1,V2) -> activate#(V2)
p20: U11#(tt(),V1,V2) -> activate#(V1)
p21: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p22: plus#(N,s(M)) -> isNat#(M)
p23: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(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)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
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))

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:

  {p25, p27}
  {p12, p13, p18, p21}
  {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) -> plus#(activate(N),activate(M))
p2: 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 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^1
        order: lexicographic order
        interpretations:
          U41#_A(x1,x2,x3) = x2 + 5
          tt_A() = 9
          plus#_A(x1,x2) = x2
          activate_A(x1) = x1
          s_A(x1) = x1 + 6
          and_A(x1,x2) = x2 + 1
          isNat_A(x1) = x1 + 12
          n__isNatKind_A(x1) = x1 + 3
          n__and_A(x1,x2) = x2 + 1
          U13_A(x1) = 9
          U12_A(x1,x2) = 9
          U22_A(x1) = 10
          U31_A(x1,x2) = x2 + 12
          U41_A(x1,x2,x3) = x2 + x3 + 8
          plus_A(x1,x2) = x1 + x2 + 2
          U11_A(x1,x2,x3) = 10
          U21_A(x1,x2) = x1 + 2
          isNatKind_A(x1) = x1 + 3
          n__0_A() = 11
          n__plus_A(x1,x2) = x1 + x2 + 2
          n__s_A(x1) = x1 + 6
          |0|_A() = 11
  
    precedence:
    
      U22 = U31 = U41 = plus = n__plus > activate = s > isNatKind > U41# = and > U13 = U12 > isNat = U11 = U21 = n__s > |0| > tt = n__0 > plus# = n__isNatKind = n__and
    
    partial status:
  
      pi(U41#) = [2]
      pi(tt) = []
      pi(plus#) = []
      pi(activate) = [1]
      pi(s) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__isNatKind) = [1]
      pi(n__and) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = []
      pi(U11) = []
      pi(U21) = []
      pi(isNatKind) = [1]
      pi(n__0) = []
      pi(n__plus) = []
      pi(n__s) = [1]
      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: 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:

  (no SCCs)

-- Reduction pair.

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

p1: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p2: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p3: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p4: U21#(tt(),V1) -> isNat#(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^1
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2,x3) = x2
          tt_A() = 1
          isNat#_A(x1) = x1
          activate_A(x1) = x1
          n__plus_A(x1,x2) = x1 + x2
          and_A(x1,x2) = x2
          isNatKind_A(x1) = 1
          n__isNatKind_A(x1) = 1
          n__s_A(x1) = x1 + 5
          U21#_A(x1,x2) = x2 + 5
          U13_A(x1) = 2
          U12_A(x1,x2) = 3
          isNat_A(x1) = 4
          U22_A(x1) = 2
          U11_A(x1,x2,x3) = 4
          U21_A(x1,x2) = 3
          U31_A(x1,x2) = x2 + 1
          U41_A(x1,x2,x3) = x2 + x3 + 5
          s_A(x1) = x1 + 5
          plus_A(x1,x2) = x1 + x2
          n__0_A() = 5
          |0|_A() = 5
          n__and_A(x1,x2) = x2
  
    precedence:
    
      activate = and = isNatKind = isNat = U11 = U21 = plus > n__plus = U31 > tt = U13 = U12 = U22 > U41 > U11# = isNat# = n__isNatKind = n__s = U21# = s = n__0 = |0| = n__and
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(isNat#) = []
      pi(activate) = []
      pi(n__plus) = []
      pi(and) = []
      pi(isNatKind) = []
      pi(n__isNatKind) = []
      pi(n__s) = [1]
      pi(U21#) = []
      pi(U13) = []
      pi(U12) = []
      pi(isNat) = []
      pi(U22) = []
      pi(U11) = []
      pi(U21) = []
      pi(U31) = []
      pi(U41) = []
      pi(s) = [1]
      pi(plus) = []
      pi(n__0) = []
      pi(|0|) = []
      pi(n__and) = []

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: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p2: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p3: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),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, p2}


-- Reduction pair.

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

p1: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p2: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),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^1
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2,x3) = x2 + 4
          tt_A() = 2
          isNat#_A(x1) = x1 + 1
          activate_A(x1) = x1 + 2
          n__plus_A(x1,x2) = x1 + 10
          and_A(x1,x2) = x1 + x2
          isNatKind_A(x1) = 2
          n__isNatKind_A(x1) = 0
          U13_A(x1) = 3
          U12_A(x1,x2) = 6
          isNat_A(x1) = x1 + 5
          U22_A(x1) = 3
          U11_A(x1,x2,x3) = x2 + 7
          U21_A(x1,x2) = 4
          U31_A(x1,x2) = x2 + 3
          U41_A(x1,x2,x3) = 3
          s_A(x1) = 1
          plus_A(x1,x2) = x1 + 11
          n__0_A() = 0
          n__s_A(x1) = 0
          |0|_A() = 1
          n__and_A(x1,x2) = x1 + x2
  
    precedence:
    
      activate = and = isNatKind = n__isNatKind = isNat = U31 = U41 = s = plus = n__s > U13 = U12 > U11# > isNat# > U11 > n__0 > |0| = n__and > tt = n__plus = U22 = U21
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(isNat#) = [1]
      pi(activate) = []
      pi(n__plus) = [1]
      pi(and) = []
      pi(isNatKind) = []
      pi(n__isNatKind) = []
      pi(U13) = []
      pi(U12) = []
      pi(isNat) = []
      pi(U22) = []
      pi(U11) = []
      pi(U21) = []
      pi(U31) = []
      pi(U41) = []
      pi(s) = []
      pi(plus) = []
      pi(n__0) = []
      pi(n__s) = []
      pi(|0|) = []
      pi(n__and) = []

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: U11#(tt(),V1,V2) -> isNat#(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:

  (no SCCs)

-- 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__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p5: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p6: isNatKind#(n__plus(V1,V2)) -> 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)

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^1
        order: lexicographic order
        interpretations:
          activate#_A(x1) = x1 + 1
          n__and_A(x1,x2) = x1 + x2 + 2
          and#_A(x1,x2) = x1 + x2 + 1
          tt_A() = 2
          n__isNatKind_A(x1) = x1 + 7
          isNatKind#_A(x1) = x1 + 1
          n__plus_A(x1,x2) = x1 + x2 + 33
          isNatKind_A(x1) = x1 + 7
          activate_A(x1) = x1
          n__s_A(x1) = x1
          U13_A(x1) = 3
          U12_A(x1,x2) = 4
          isNat_A(x1) = 6
          U22_A(x1) = 2
          U11_A(x1,x2,x3) = 5
          U21_A(x1,x2) = 3
          U31_A(x1,x2) = x2 + 1
          U41_A(x1,x2,x3) = x2 + x3 + 33
          s_A(x1) = x1
          plus_A(x1,x2) = x1 + x2 + 33
          n__0_A() = 16
          and_A(x1,x2) = x1 + x2 + 2
          |0|_A() = 16
  
    precedence:
    
      activate > n__plus = plus > n__and = isNatKind = U21 = U41 = and > U31 > n__0 = |0| > tt = U22 > n__s = U12 = s > isNatKind# > activate# > and# > isNat > U11 > U13 > n__isNatKind
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__and) = []
      pi(and#) = [2]
      pi(tt) = []
      pi(n__isNatKind) = [1]
      pi(isNatKind#) = []
      pi(n__plus) = []
      pi(isNatKind) = []
      pi(activate) = []
      pi(n__s) = []
      pi(U13) = []
      pi(U12) = []
      pi(isNat) = []
      pi(U22) = []
      pi(U11) = []
      pi(U21) = []
      pi(U31) = []
      pi(U41) = []
      pi(s) = []
      pi(plus) = []
      pi(n__0) = []
      pi(and) = []
      pi(|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__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
p3: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p4: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p5: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p6: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p7: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p8: isNatKind#(n__s(V1)) -> 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:

  {p4, p7}
  {p1, p2}


-- Reduction pair.

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

p1: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p2: isNatKind#(n__plus(V1,V2)) -> 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^1
        order: lexicographic order
        interpretations:
          isNatKind#_A(x1) = x1
          n__s_A(x1) = x1
          activate_A(x1) = x1
          n__plus_A(x1,x2) = x1 + x2 + 7
          U13_A(x1) = 2
          tt_A() = 1
          U12_A(x1,x2) = 3
          isNat_A(x1) = 5
          U22_A(x1) = 2
          U11_A(x1,x2,x3) = 4
          U21_A(x1,x2) = 3
          U31_A(x1,x2) = x2 + 1
          U41_A(x1,x2,x3) = x2 + x3 + 7
          s_A(x1) = x1
          plus_A(x1,x2) = x1 + x2 + 7
          n__0_A() = 0
          and_A(x1,x2) = x2
          isNatKind_A(x1) = x1 + 6
          n__isNatKind_A(x1) = x1 + 6
          |0|_A() = 0
          n__and_A(x1,x2) = x2
  
    precedence:
    
      isNat > U11 > U22 = U21 > n__plus = plus > U31 = U41 = isNatKind = n__isNatKind > activate = U13 = tt = U12 = s = n__0 = and = |0| = n__and > isNatKind# = n__s
    
    partial status:
  
      pi(isNatKind#) = []
      pi(n__s) = []
      pi(activate) = [1]
      pi(n__plus) = [1]
      pi(U13) = []
      pi(tt) = []
      pi(U12) = []
      pi(isNat) = []
      pi(U22) = []
      pi(U11) = []
      pi(U21) = []
      pi(U31) = []
      pi(U41) = []
      pi(s) = []
      pi(plus) = [1]
      pi(n__0) = []
      pi(and) = [2]
      pi(isNatKind) = []
      pi(n__isNatKind) = []
      pi(|0|) = []
      pi(n__and) = [2]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: isNatKind#(n__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^1
        order: lexicographic order
        interpretations:
          isNatKind#_A(x1) = x1 + 1
          n__s_A(x1) = x1 + 9
          activate_A(x1) = x1
          U13_A(x1) = 2
          tt_A() = 1
          U12_A(x1,x2) = 3
          isNat_A(x1) = 5
          U22_A(x1) = 2
          U11_A(x1,x2,x3) = 4
          U21_A(x1,x2) = 3
          U31_A(x1,x2) = x2 + 1
          U41_A(x1,x2,x3) = x2 + x3 + 18
          s_A(x1) = x1 + 9
          plus_A(x1,x2) = x1 + x2 + 9
          n__0_A() = 9
          n__plus_A(x1,x2) = x1 + x2 + 9
          and_A(x1,x2) = x2 + 2
          isNatKind_A(x1) = x1 + 6
          n__isNatKind_A(x1) = x1 + 6
          |0|_A() = 9
          n__and_A(x1,x2) = x2 + 2
  
    precedence:
    
      isNat = U22 = U21 > isNatKind# > U31 = U41 = plus = n__plus > n__s = s > activate = n__0 = and = isNatKind = n__isNatKind = |0| = n__and > U13 = tt = U12 = U11
    
    partial status:
  
      pi(isNatKind#) = [1]
      pi(n__s) = [1]
      pi(activate) = [1]
      pi(U13) = []
      pi(tt) = []
      pi(U12) = []
      pi(isNat) = []
      pi(U22) = []
      pi(U11) = []
      pi(U21) = []
      pi(U31) = []
      pi(U41) = []
      pi(s) = [1]
      pi(plus) = [1]
      pi(n__0) = []
      pi(n__plus) = [1]
      pi(and) = [2]
      pi(isNatKind) = [1]
      pi(n__isNatKind) = [1]
      pi(|0|) = []
      pi(n__and) = [2]

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)

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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          activate#_A(x1) = x1 + 1
          n__and_A(x1,x2) = x1 + x2 + 1
          and#_A(x1,x2) = x1 + x2
          tt_A() = 2
  
    precedence:
    
      and# > activate# > n__and > tt
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__and) = []
      pi(and#) = []
      pi(tt) = []

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__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:

  (no SCCs)