YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

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

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p6, p7, p9, p10, p11, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p40, p41, p43}


-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          U11#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (19,0)
          tt_A() = (7,4)
          U12#_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,0),(0,0)) x2 + (13,0)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (6,5)
          activate_A(x1) = ((1,0),(1,1)) x1
          activate#_A(x1) = ((1,0),(0,0)) x1 + (11,0)
          n__and_A(x1,x2) = ((0,1),(1,0)) x1 + ((1,0),(1,1)) x2 + (0,4)
          and#_A(x1,x2) = ((1,0),(0,0)) x2 + (11,0)
          n__isNatKind_A(x1) = ((1,0),(0,0)) x1
          isNatKind#_A(x1) = ((1,0),(0,0)) x1 + (11,0)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (0,18)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (5,5)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (16,0)
          s_A(x1) = ((1,0),(1,0)) x1 + (0,18)
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (16,0)
          isNatKind_A(x1) = ((1,0),(1,0)) x1
          U21#_A(x1,x2) = ((1,0),(0,0)) x2 + (16,0)
          and_A(x1,x2) = ((0,1),(1,0)) x1 + ((1,0),(1,1)) x2 + (0,4)
          U41#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (16,0)
          |0|_A() = (12,4)
          U31#_A(x1,x2) = ((1,0),(0,0)) x2 + (11,0)
          U13_A(x1) = (8,4)
          U12_A(x1,x2) = ((0,0),(0,1)) x1 + (9,0)
          U22_A(x1) = (8,4)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (11,5)
          U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (5,23)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (5,5)
          U11_A(x1,x2,x3) = (10,5)
          U21_A(x1,x2) = ((0,1),(0,0)) x1 + (5,4)
          n__0_A() = (12,4)
  
    precedence:
    
      isNat = activate = U31 = U41 = plus > n__plus > U22 > n__and = and > tt = isNatKind = |0| = U13 = n__0 > n__isNatKind > s > U11# = U12# = activate# = and# = isNatKind# = plus# = isNat# = U21# = U41# = U31# > U12 = U11 > n__s = U21
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(U12#) = []
      pi(isNat) = []
      pi(activate) = []
      pi(activate#) = []
      pi(n__and) = []
      pi(and#) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(s) = []
      pi(isNat#) = []
      pi(isNatKind) = []
      pi(U21#) = []
      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:

  p15

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)) -> 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: 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)) -> isNatKind#(activate(V1))
p12: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p13: activate#(n__and(X1,X2)) -> and#(X1,X2)
p14: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p15: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p16: isNatKind#(n__s(V1)) -> activate#(V1)
p17: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p18: plus#(N,|0|()) -> isNat#(N)
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)) -> isNatKind#(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^2
        order: standard order
        interpretations:
          U11#_A(x1,x2,x3) = ((1,1),(0,0)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (23,98)
          tt_A() = (29,2)
          U12#_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + (29,55)
          isNat_A(x1) = ((1,1),(0,1)) x1 + (4,3)
          activate_A(x1) = x1 + (8,16)
          isNat#_A(x1) = ((1,1),(1,1)) x1 + (4,33)
          n__plus_A(x1,x2) = x1 + ((1,1),(1,1)) x2 + (167,0)
          and_A(x1,x2) = x2 + (8,41)
          isNatKind_A(x1) = ((0,0),(1,1)) x1 + (31,4)
          n__isNatKind_A(x1) = ((0,0),(1,1)) x1 + (23,3)
          and#_A(x1,x2) = ((1,1),(1,1)) x2 + (1,7)
          activate#_A(x1) = ((1,1),(1,1)) x1 + (1,0)
          plus#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (1,5)
          |0|_A() = (26,2)
          U31#_A(x1,x2) = ((1,1),(1,1)) x2 + (25,3)
          isNatKind#_A(x1) = ((1,1),(1,1)) x1 + (27,0)
          n__and_A(x1,x2) = x2 + (0,33)
          n__s_A(x1) = x1 + (32,16)
          U21#_A(x1,x2) = ((1,1),(1,1)) x2 + (28,57)
          s_A(x1) = x1 + (40,32)
          U41#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (50,53)
          U13_A(x1) = (30,3)
          U12_A(x1,x2) = (31,3)
          U22_A(x1) = (30,2)
          U31_A(x1,x2) = x2 + (27,16)
          U41_A(x1,x2,x3) = ((1,1),(1,1)) x2 + x3 + (240,88)
          plus_A(x1,x2) = x1 + ((1,1),(1,1)) x2 + (168,16)
          U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (123,3)
          U21_A(x1,x2) = (31,19)
          n__0_A() = (25,1)
  
    precedence:
    
      U12# = U11 > U41# > plus# = U31# > and# > U11# = isNat# = n__plus = U21# = U31 = U41 = plus > U12 > isNat = U21 > |0| = n__0 > n__s > activate = and = isNatKind = n__isNatKind = n__and > U13 > tt > U22 > activate# = isNatKind# = s
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(U12#) = []
      pi(isNat) = [1]
      pi(activate) = []
      pi(isNat#) = []
      pi(n__plus) = []
      pi(and) = []
      pi(isNatKind) = []
      pi(n__isNatKind) = []
      pi(and#) = []
      pi(activate#) = []
      pi(plus#) = [2]
      pi(|0|) = []
      pi(U31#) = []
      pi(isNatKind#) = []
      pi(n__and) = []
      pi(n__s) = []
      pi(U21#) = []
      pi(s) = []
      pi(U41#) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = [2]
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  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) -> U12#(isNat(activate(V1)),activate(V2))
p2: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p3: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p4: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p5: and#(tt(),X) -> activate#(X)
p6: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p7: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p8: U31#(tt(),N) -> activate#(N)
p9: activate#(n__isNatKind(X)) -> isNatKind#(X)
p10: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p11: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p12: activate#(n__and(X1,X2)) -> and#(X1,X2)
p13: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p14: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p15: isNatKind#(n__s(V1)) -> activate#(V1)
p16: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p17: plus#(N,|0|()) -> isNat#(N)
p18: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p19: isNat#(n__plus(V1,V2)) -> activate#(V1)
p20: isNat#(n__plus(V1,V2)) -> activate#(V2)
p21: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p22: U21#(tt(),V1) -> isNat#(activate(V1))
p23: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p24: isNat#(n__s(V1)) -> activate#(V1)
p25: U21#(tt(),V1) -> activate#(V1)
p26: 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) -> plus#(activate(N),activate(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)) -> and#(isNat(M),n__isNatKind(M))
p30: plus#(N,s(M)) -> isNat#(M)
p31: plus#(N,s(M)) -> isNat#(N)
p32: U41#(tt(),M,N) -> activate#(N)
p33: U41#(tt(),M,N) -> activate#(M)
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)) -> 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: 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)

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          U11#_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((1,1),(1,1)) x2 + ((0,1),(1,1)) x3 + (316,203)
          tt_A() = (149,202)
          U12#_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(1,1)) x2 + (1,37)
          isNat_A(x1) = x1 + (498,135)
          activate_A(x1) = x1 + (19,6)
          activate#_A(x1) = ((0,1),(0,1)) x1 + (202,1)
          n__and_A(x1,x2) = ((1,1),(0,1)) x2 + (109,2)
          and#_A(x1,x2) = ((0,1),(0,1)) x2 + (203,3)
          n__isNatKind_A(x1) = ((1,1),(1,1)) x1 + (3,127)
          isNatKind#_A(x1) = ((1,1),(1,1)) x1 + (307,128)
          n__s_A(x1) = ((0,1),(1,0)) x1 + (21,91)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (123,165)
          plus#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (366,1)
          s_A(x1) = ((0,1),(1,0)) x1 + (40,91)
          isNat#_A(x1) = ((1,1),(1,1)) x1 + (220,41)
          U21#_A(x1,x2) = ((1,1),(1,1)) x2 + (246,66)
          isNatKind_A(x1) = ((1,1),(1,1)) x1 + (21,128)
          and_A(x1,x2) = ((1,1),(0,1)) x2 + (128,8)
          U41#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (497,92)
          |0|_A() = (4,129)
          U31#_A(x1,x2) = ((0,1),(0,1)) x2 + (203,130)
          U13_A(x1) = (150,203)
          U12_A(x1,x2) = ((0,0),(0,1)) x2 + (518,203)
          U22_A(x1) = (150,203)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (20,130)
          U41_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (255,265)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (124,165)
          U11_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((0,1),(0,1)) x3 + (519,209)
          U21_A(x1,x2) = (518,203)
          n__0_A() = (0,129)
  
    precedence:
    
      U21# > U11# = isNat# > U12# > plus# = U31# > isNatKind# = U41# > activate = n__isNatKind = n__s = n__plus = s = isNatKind = |0| = U41 = plus > activate# > tt = and# = U13 > isNat > n__and = and = U12 = U31 = U11 = U21 = n__0 > U22
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(U12#) = []
      pi(isNat) = []
      pi(activate) = []
      pi(activate#) = []
      pi(n__and) = []
      pi(and#) = []
      pi(n__isNatKind) = [1]
      pi(isNatKind#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(s) = []
      pi(isNat#) = []
      pi(U21#) = []
      pi(isNatKind) = [1]
      pi(and) = [2]
      pi(U41#) = [3]
      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:

  p29

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)) -> 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: 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: U41#(tt(),M,N) -> activate#(M)
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, 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)) -> isNatKind#(activate(V1))
p8: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p9: activate#(n__and(X1,X2)) -> and#(X1,X2)
p10: and#(tt(),X) -> activate#(X)
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)) -> isNatKind#(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)) -> 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)

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

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

The next rules are strictly ordered:

  p15

We remove them from the problem.

-- SCC decomposition.

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

p1: 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)) -> isNatKind#(activate(V1))
p8: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p9: activate#(n__and(X1,X2)) -> and#(X1,X2)
p10: and#(tt(),X) -> activate#(X)
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: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p16: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p17: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p18: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p19: isNat#(n__plus(V1,V2)) -> activate#(V1)
p20: isNat#(n__plus(V1,V2)) -> activate#(V2)
p21: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p22: U21#(tt(),V1) -> isNat#(activate(V1))
p23: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p24: isNat#(n__s(V1)) -> activate#(V1)
p25: U21#(tt(),V1) -> activate#(V1)
p26: U11#(tt(),V1,V2) -> activate#(V1)
p27: U11#(tt(),V1,V2) -> activate#(V2)
p28: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p29: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p30: plus#(N,s(M)) -> isNat#(M)
p31: plus#(N,s(M)) -> isNat#(N)

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, 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: 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: 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,|0|()) -> and#(isNat(N),n__isNatKind(N))
p30: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p31: U31#(tt(),N) -> activate#(N)

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          U11#_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + ((1,0),(1,1)) x3 + (213,98)
          tt_A() = (61,8)
          U12#_A(x1,x2) = ((1,0),(1,0)) x2 + (121,19)
          isNat_A(x1) = ((0,1),(1,0)) x1 + (62,1)
          activate_A(x1) = x1 + (4,9)
          activate#_A(x1) = ((1,0),(1,0)) x1 + (120,19)
          n__and_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (290,15)
          and#_A(x1,x2) = ((1,0),(1,0)) x2 + (121,19)
          n__isNatKind_A(x1) = ((1,1),(1,0)) x1 + (1,225)
          isNatKind#_A(x1) = ((1,0),(1,0)) x1 + (2,20)
          n__s_A(x1) = x1 + (119,18)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (229,73)
          plus#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (348,160)
          s_A(x1) = x1 + (123,27)
          isNat#_A(x1) = ((1,1),(1,1)) x1 + (49,93)
          U21#_A(x1,x2) = ((1,1),(1,1)) x2 + (121,106)
          isNatKind_A(x1) = ((1,1),(1,0)) x1 + (5,226)
          and_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (293,24)
          |0|_A() = (63,1)
          U31#_A(x1,x2) = ((1,0),(1,1)) x2 + (121,224)
          U13_A(x1) = (61,9)
          U12_A(x1,x2) = ((0,1),(0,0)) x1 + (64,9)
          U22_A(x1) = (70,9)
          U31_A(x1,x2) = ((1,1),(0,1)) x2 + (64,9)
          U41_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (379,127)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (230,74)
          U11_A(x1,x2,x3) = ((1,0),(0,0)) x2 + (70,9)
          U21_A(x1,x2) = (71,19)
          n__0_A() = (60,0)
  
    precedence:
    
      isNat > activate = n__isNatKind = |0| = U31 = plus = n__0 > U11# = U12# > activate# = isNatKind# = n__plus = plus# = isNat# = U21# = U31# = U11 > U22 = U21 > U41 > n__and = and# = n__s = s = isNatKind = and > tt = U13 = U12
    
    partial status:
  
      pi(U11#) = [3]
      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#) = []
      pi(s) = [1]
      pi(isNat#) = [1]
      pi(U21#) = []
      pi(isNatKind) = []
      pi(and) = []
      pi(|0|) = []
      pi(U31#) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = [2]
      pi(plus) = []
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: U12#(tt(),V2) -> activate#(V2)
p2: activate#(n__and(X1,X2)) -> and#(X1,X2)
p3: and#(tt(),X) -> activate#(X)
p4: activate#(n__isNatKind(X)) -> isNatKind#(X)
p5: isNatKind#(n__s(V1)) -> activate#(V1)
p6: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p7: plus#(N,s(M)) -> isNat#(N)
p8: isNat#(n__s(V1)) -> activate#(V1)
p9: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p10: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p11: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p12: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p13: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p14: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p15: U21#(tt(),V1) -> activate#(V1)
p16: U21#(tt(),V1) -> isNat#(activate(V1))
p17: isNat#(n__plus(V1,V2)) -> activate#(V2)
p18: isNat#(n__plus(V1,V2)) -> activate#(V1)
p19: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p20: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p21: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p22: U11#(tt(),V1,V2) -> activate#(V2)
p23: U11#(tt(),V1,V2) -> activate#(V1)
p24: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p25: plus#(N,s(M)) -> isNat#(M)
p26: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p27: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p28: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p29: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p30: U31#(tt(),N) -> activate#(N)

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {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__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
p3: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p5: U31#(tt(),N) -> activate#(N)
p6: activate#(n__isNatKind(X)) -> isNatKind#(X)
p7: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p8: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p9: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p10: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p11: isNatKind#(n__s(V1)) -> activate#(V1)
p12: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p13: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p14: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p15: plus#(N,s(M)) -> isNat#(M)
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)) -> isNatKind#(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)) -> isNat#(N)

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          activate#_A(x1) = ((0,1),(0,1)) x1 + (49,15)
          n__and_A(x1,x2) = x1 + x2 + (0,2)
          and#_A(x1,x2) = ((0,1),(0,1)) x2 + (50,17)
          tt_A() = (1,157)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,1),(1,1)) x2 + (56,121)
          plus#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (169,135)
          |0|_A() = (3,152)
          U31#_A(x1,x2) = ((1,1),(1,1)) x2 + (50,153)
          and_A(x1,x2) = x1 + x2 + (3,8)
          isNat_A(x1) = ((0,1),(1,0)) x1 + (0,157)
          n__isNatKind_A(x1) = ((1,1),(0,1)) x1 + (46,0)
          isNatKind#_A(x1) = ((0,1),(0,1)) x1 + (49,0)
          activate_A(x1) = x1 + (4,6)
          n__s_A(x1) = x1 + (22,15)
          s_A(x1) = x1 + (25,16)
          isNat#_A(x1) = ((1,1),(0,1)) x1 + (26,6)
          U11#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(0,1)) x3 + (182,106)
          isNatKind_A(x1) = ((1,1),(0,1)) x1 + (47,6)
          U21#_A(x1,x2) = ((1,1),(0,1)) x2 + (53,15)
          U13_A(x1) = (2,158)
          U12_A(x1,x2) = (3,158)
          U22_A(x1) = (2,158)
          U31_A(x1,x2) = x2 + (5,153)
          U41_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,0),(1,1)) x3 + (96,157)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,1),(1,1)) x2 + (56,121)
          U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (4,161)
          U21_A(x1,x2) = ((0,1),(0,0)) x2 + (3,158)
          n__0_A() = (0,151)
  
    precedence:
    
      n__and = n__plus = plus > U11# > isNat# = U21# > activate# = and# = plus# = and = n__isNatKind = isNatKind# = activate = isNatKind = U31 > U31# > U41 > U13 > tt = isNat = U12 = U11 > s > |0| = n__0 > n__s = U22 = U21
    
    partial status:
  
      pi(activate#) = []
      pi(n__and) = []
      pi(and#) = []
      pi(tt) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(U31#) = []
      pi(and) = [2]
      pi(isNat) = []
      pi(n__isNatKind) = [1]
      pi(isNatKind#) = []
      pi(activate) = [1]
      pi(n__s) = [1]
      pi(s) = [1]
      pi(isNat#) = [1]
      pi(U11#) = [3]
      pi(isNatKind) = [1]
      pi(U21#) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = [2]
      pi(U41) = []
      pi(plus) = []
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p21

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
p3: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p5: U31#(tt(),N) -> activate#(N)
p6: activate#(n__isNatKind(X)) -> isNatKind#(X)
p7: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p8: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p9: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p10: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p11: isNatKind#(n__s(V1)) -> activate#(V1)
p12: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p13: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p14: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p15: plus#(N,s(M)) -> isNat#(M)
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__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p22: U21#(tt(),V1) -> isNat#(activate(V1))
p23: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p24: isNat#(n__s(V1)) -> activate#(V1)
p25: U21#(tt(),V1) -> activate#(V1)
p26: U11#(tt(),V1,V2) -> activate#(V1)
p27: U11#(tt(),V1,V2) -> activate#(V2)
p28: plus#(N,s(M)) -> isNat#(N)

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28}


-- Reduction pair.

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

p1: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
p3: activate#(n__isNatKind(X)) -> isNatKind#(X)
p4: isNatKind#(n__s(V1)) -> activate#(V1)
p5: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p6: plus#(N,s(M)) -> isNat#(N)
p7: isNat#(n__s(V1)) -> activate#(V1)
p8: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p10: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p11: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p12: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p13: 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#(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,|0|()) -> and#(isNat(N),n__isNatKind(N))
p27: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p28: U31#(tt(),N) -> activate#(N)

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          activate#_A(x1) = ((1,0),(1,0)) x1 + (37,0)
          n__and_A(x1,x2) = ((0,1),(0,1)) x1 + x2 + (0,1)
          and#_A(x1,x2) = ((1,0),(1,0)) x2 + (37,0)
          tt_A() = (14,0)
          n__isNatKind_A(x1) = ((1,0),(1,0)) x1 + (0,9)
          isNatKind#_A(x1) = ((1,0),(1,0)) x1 + (37,0)
          n__s_A(x1) = x1 + (0,62)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (18,19)
          plus#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (16,0)
          s_A(x1) = x1 + (0,62)
          isNat#_A(x1) = ((1,1),(1,1)) x1
          activate_A(x1) = x1
          U21#_A(x1,x2) = ((1,1),(1,1)) x2 + (62,1)
          isNatKind_A(x1) = ((1,0),(1,0)) x1 + (0,9)
          U11#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,0)) x3 + (37,0)
          and_A(x1,x2) = ((0,1),(0,1)) x1 + x2 + (0,1)
          isNat_A(x1) = ((0,0),(0,1)) x1 + (79,41)
          |0|_A() = (14,8)
          U31#_A(x1,x2) = ((1,0),(1,0)) x2 + (38,9)
          U13_A(x1) = (15,1)
          U12_A(x1,x2) = (16,1)
          U22_A(x1) = ((0,0),(0,1)) x1 + (15,0)
          U31_A(x1,x2) = ((1,1),(1,1)) x2 + (15,9)
          U41_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (19,81)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (18,19)
          U11_A(x1,x2,x3) = ((0,0),(1,0)) x2 + (17,1)
          U21_A(x1,x2) = ((0,0),(0,1)) x2 + (16,41)
          n__0_A() = (14,8)
  
    precedence:
    
      activate# = n__and = and# = tt = n__isNatKind = isNatKind# = n__s = n__plus = plus# = s = isNat# = activate = U21# = isNatKind = U11# = and = isNat = |0| = U31# = U13 = U12 = U22 = U31 = U41 = plus = U11 = U21 = n__0
    
    partial status:
  
      pi(activate#) = []
      pi(n__and) = []
      pi(and#) = []
      pi(tt) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(s) = []
      pi(isNat#) = []
      pi(activate) = []
      pi(U21#) = []
      pi(isNatKind) = []
      pi(U11#) = []
      pi(and) = []
      pi(isNat) = []
      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:

  p24

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
p3: activate#(n__isNatKind(X)) -> isNatKind#(X)
p4: isNatKind#(n__s(V1)) -> activate#(V1)
p5: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p6: plus#(N,s(M)) -> isNat#(N)
p7: isNat#(n__s(V1)) -> activate#(V1)
p8: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p10: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p11: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p12: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p13: 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#(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#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p25: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p26: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p27: 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}


-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          activate#_A(x1) = ((1,0),(1,0)) x1 + (155,0)
          n__and_A(x1,x2) = ((0,1),(0,0)) x1 + x2 + (0,14)
          and#_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,0),(1,0)) x2 + (23,0)
          tt_A() = (222,133)
          n__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(1,1)) x2 + (58,98)
          plus#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (212,0)
          |0|_A() = (224,223)
          U31#_A(x1,x2) = ((1,1),(1,1)) x2 + (225,0)
          and_A(x1,x2) = ((0,1),(0,0)) x1 + x2 + (0,14)
          isNat_A(x1) = ((0,1),(0,0)) x1 + (1,135)
          n__isNatKind_A(x1) = ((1,1),(0,0)) x1 + (0,122)
          isNatKind#_A(x1) = ((1,1),(1,0)) x1
          activate_A(x1) = x1 + (46,14)
          n__s_A(x1) = x1 + (47,108)
          s_A(x1) = x1 + (74,122)
          isNat#_A(x1) = ((1,1),(1,1)) x1 + (163,146)
          U11#_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,0)) x3 + (2,0)
          isNatKind_A(x1) = ((1,1),(0,0)) x1 + (40,136)
          U21#_A(x1,x2) = ((1,1),(1,1)) x2 + (224,206)
          U13_A(x1) = ((0,1),(0,0)) x1 + (90,134)
          U12_A(x1,x2) = ((1,0),(0,0)) x1 + (4,134)
          U22_A(x1) = ((1,0),(0,0)) x1 + (78,134)
          U31_A(x1,x2) = x2 + (225,224)
          U41_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(0,1)) x3 + (298,295)
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(1,1)) x2 + (103,99)
          U11_A(x1,x2,x3) = ((0,1),(0,0)) x2 + (45,134)
          U21_A(x1,x2) = ((0,1),(0,0)) x2 + (94,134)
          n__0_A() = (223,222)
  
    precedence:
    
      U31 > isNat > U13 = U12 = U11 > activate# = and# = plus# = U31# = isNatKind# = isNat# = U11# = U21# > n__and = and = n__isNatKind = activate = isNatKind > |0| > plus > U41 > n__s = s = U22 = U21 > tt > n__0 > n__plus
    
    partial status:
  
      pi(activate#) = []
      pi(n__and) = []
      pi(and#) = []
      pi(tt) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(U31#) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(activate) = []
      pi(n__s) = []
      pi(s) = []
      pi(isNat#) = []
      pi(U11#) = []
      pi(isNatKind) = []
      pi(U21#) = []
      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:

  p14

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26}


-- Reduction pair.

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

p1: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
p3: activate#(n__isNatKind(X)) -> isNatKind#(X)
p4: isNatKind#(n__s(V1)) -> activate#(V1)
p5: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p6: plus#(N,s(M)) -> isNat#(N)
p7: isNat#(n__s(V1)) -> activate#(V1)
p8: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p10: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p11: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p12: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p13: 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#(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)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p24: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p25: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p26: U31#(tt(),N) -> activate#(N)

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          activate#_A(x1) = ((1,0),(1,0)) x1 + (2,1)
          n__and_A(x1,x2) = ((0,1),(0,0)) x1 + x2 + (4,7)
          and#_A(x1,x2) = ((1,0),(1,0)) x2 + (5,1)
          tt_A() = (1,2)
          n__isNatKind_A(x1) = ((1,0),(1,0)) x1 + (1,3)
          isNatKind#_A(x1) = ((1,0),(1,0)) x1 + (1,1)
          n__s_A(x1) = x1 + (22,0)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,0),(1,0)) x2 + (7,3)
          plus#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (1,2)
          s_A(x1) = x1 + (22,0)
          isNat#_A(x1) = ((1,0),(1,1)) x1 + (0,1)
          activate_A(x1) = x1
          U21#_A(x1,x2) = ((1,0),(1,1)) x2 + (3,1)
          isNatKind_A(x1) = ((1,0),(1,0)) x1 + (1,3)
          U11#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,0)) x3 + (2,1)
          and_A(x1,x2) = ((0,1),(0,0)) x1 + x2 + (4,7)
          isNat_A(x1) = ((0,1),(0,0)) x1 + (4,11)
          |0|_A() = (8,1)
          U31#_A(x1,x2) = ((1,0),(1,0)) x2 + (8,1)
          U13_A(x1) = (2,3)
          U12_A(x1,x2) = (3,3)
          U22_A(x1) = (2,3)
          U31_A(x1,x2) = x2 + (1,1)
          U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,1),(1,1)) x3 + (29,3)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,0),(1,0)) x2 + (7,3)
          U11_A(x1,x2,x3) = (4,3)
          U21_A(x1,x2) = (3,11)
          n__0_A() = (8,1)
  
    precedence:
    
      n__and = n__isNatKind = activate = isNatKind = and = isNat = |0| = U31 = plus > U21 > U22 > tt > n__plus = U41 = U11 > isNatKind# > activate# = and# = plus# = isNat# = U21# = U11# > U31# > U12 > U13 > n__s = s = n__0
    
    partial status:
  
      pi(activate#) = []
      pi(n__and) = []
      pi(and#) = []
      pi(tt) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(n__s) = []
      pi(n__plus) = [1]
      pi(plus#) = []
      pi(s) = [1]
      pi(isNat#) = []
      pi(activate) = []
      pi(U21#) = []
      pi(isNatKind) = []
      pi(U11#) = []
      pi(and) = []
      pi(isNat) = []
      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:

  p11

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
p3: activate#(n__isNatKind(X)) -> isNatKind#(X)
p4: isNatKind#(n__s(V1)) -> activate#(V1)
p5: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p6: plus#(N,s(M)) -> isNat#(N)
p7: isNat#(n__s(V1)) -> activate#(V1)
p8: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p10: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p11: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p12: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p13: U21#(tt(),V1) -> activate#(V1)
p14: U21#(tt(),V1) -> isNat#(activate(V1))
p15: isNat#(n__plus(V1,V2)) -> activate#(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)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p23: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p24: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p25: 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}


-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

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

  p22

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24}


-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          activate#_A(x1) = ((0,1),(0,0)) x1 + (1,45)
          n__and_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (1,2)
          and#_A(x1,x2) = ((0,1),(0,0)) x2 + (2,45)
          tt_A() = (2,1)
          n__isNatKind_A(x1) = ((1,0),(1,1)) x1 + (11,7)
          isNatKind#_A(x1) = ((0,1),(0,0)) x1 + (8,45)
          n__s_A(x1) = x1 + (10,89)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (24,0)
          plus#_A(x1,x2) = ((1,1),(0,0)) x1 + ((1,1),(0,0)) x2 + (1,45)
          s_A(x1) = x1 + (10,89)
          isNat#_A(x1) = ((0,1),(0,0)) x1 + (15,45)
          activate_A(x1) = x1 + (5,0)
          U21#_A(x1,x2) = ((0,1),(0,0)) x2 + (16,45)
          isNatKind_A(x1) = ((1,0),(1,1)) x1 + (12,7)
          U11#_A(x1,x2,x3) = ((0,1),(0,0)) x2 + ((0,1),(0,0)) x3 + (15,45)
          and_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (3,2)
          isNat_A(x1) = (37,30)
          |0|_A() = (0,8)
          U31#_A(x1,x2) = ((0,1),(0,0)) x2 + (1,45)
          U13_A(x1) = ((1,0),(0,0)) x1 + (0,2)
          U12_A(x1,x2) = (37,2)
          U22_A(x1) = (3,2)
          U31_A(x1,x2) = ((1,1),(0,1)) x2 + (6,8)
          U41_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (44,99)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (24,0)
          U11_A(x1,x2,x3) = (37,30)
          U21_A(x1,x2) = (4,30)
          n__0_A() = (0,8)
  
    precedence:
    
      U21# > activate# = and# = isNatKind# = plus# = U31# > isNat# = U11# > n__and = activate = isNatKind = and > n__plus = U31 = plus > isNat = U41 = U21 > tt = n__isNatKind = U13 = U12 = U11 > s > n__s = U22 > |0| = n__0
    
    partial status:
  
      pi(activate#) = []
      pi(n__and) = []
      pi(and#) = []
      pi(tt) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(n__s) = []
      pi(n__plus) = [2]
      pi(plus#) = []
      pi(s) = []
      pi(isNat#) = []
      pi(activate) = []
      pi(U21#) = []
      pi(isNatKind) = []
      pi(U11#) = []
      pi(and) = []
      pi(isNat) = []
      pi(|0|) = []
      pi(U31#) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = [2]
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p18

We remove them from the problem.

-- SCC decomposition.

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

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


-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          activate#_A(x1) = ((0,1),(0,1)) x1 + (50,11)
          n__and_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (1,0)
          and#_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,1),(0,1)) x2 + (24,11)
          tt_A() = (27,1)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,1),(1,1)) x2 + (77,45)
          plus#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(0,1)) x2 + (94,56)
          |0|_A() = (104,23)
          U31#_A(x1,x2) = ((0,1),(1,1)) x2 + (105,24)
          and_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (2,0)
          isNat_A(x1) = (79,23)
          n__isNatKind_A(x1) = ((1,1),(0,1)) x1 + (13,0)
          isNatKind#_A(x1) = ((0,1),(0,1)) x1 + (27,11)
          activate_A(x1) = x1 + (2,23)
          n__s_A(x1) = x1 + (49,55)
          s_A(x1) = x1 + (50,78)
          isNat#_A(x1) = ((1,1),(1,1)) x1 + (1,79)
          U11#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((0,0),(0,1)) x3 + (51,104)
          isNatKind_A(x1) = ((1,1),(0,1)) x1 + (13,23)
          U21#_A(x1,x2) = ((1,1),(1,1)) x2 + (51,104)
          U13_A(x1) = (28,2)
          U12_A(x1,x2) = (29,2)
          U22_A(x1) = (78,23)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (3,24)
          U41_A(x1,x2,x3) = ((0,1),(1,1)) x2 + ((1,0),(1,1)) x3 + (154,173)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,1),(1,1)) x2 + (79,45)
          U11_A(x1,x2,x3) = (78,22)
          U21_A(x1,x2) = (79,23)
          n__0_A() = (103,0)
  
    precedence:
    
      |0| = isNatKind > tt = isNat = U11 > activate = U12 = U31 = plus > and > n__and > U41 > isNat# = U21# > n__plus = U11# > plus# > s > n__s > U21 > U13 > activate# = and# = U31# = n__isNatKind = isNatKind# = U22 = n__0
    
    partial status:
  
      pi(activate#) = []
      pi(n__and) = []
      pi(and#) = []
      pi(tt) = []
      pi(n__plus) = []
      pi(plus#) = [2]
      pi(|0|) = []
      pi(U31#) = []
      pi(and) = [2]
      pi(isNat) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(activate) = [1]
      pi(n__s) = []
      pi(s) = []
      pi(isNat#) = []
      pi(U11#) = []
      pi(isNatKind) = []
      pi(U21#) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = [2]
      pi(U41) = []
      pi(plus) = [1]
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p16

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
p3: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p5: U31#(tt(),N) -> activate#(N)
p6: activate#(n__isNatKind(X)) -> isNatKind#(X)
p7: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p8: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p9: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p10: isNatKind#(n__s(V1)) -> activate#(V1)
p11: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p12: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p13: plus#(N,s(M)) -> isNat#(N)
p14: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p15: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p16: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p17: isNat#(n__plus(V1,V2)) -> activate#(V1)
p18: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p19: U21#(tt(),V1) -> isNat#(activate(V1))
p20: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p21: U21#(tt(),V1) -> activate#(V1)
p22: U11#(tt(),V1,V2) -> 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, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22}


-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

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

  p7

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
p3: activate#(n__isNatKind(X)) -> isNatKind#(X)
p4: isNatKind#(n__s(V1)) -> activate#(V1)
p5: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p6: plus#(N,s(M)) -> isNat#(N)
p7: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p8: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p9: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p10: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p11: U21#(tt(),V1) -> activate#(V1)
p12: U21#(tt(),V1) -> isNat#(activate(V1))
p13: isNat#(n__plus(V1,V2)) -> activate#(V1)
p14: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p15: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p16: U11#(tt(),V1,V2) -> activate#(V1)
p17: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p18: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p19: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p20: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p21: 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}


-- Reduction pair.

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

p1: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
p3: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p5: U31#(tt(),N) -> activate#(N)
p6: activate#(n__isNatKind(X)) -> isNatKind#(X)
p7: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p8: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p9: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p10: isNatKind#(n__s(V1)) -> activate#(V1)
p11: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p12: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p13: plus#(N,s(M)) -> isNat#(N)
p14: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p15: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p16: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p17: isNat#(n__plus(V1,V2)) -> activate#(V1)
p18: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p19: U21#(tt(),V1) -> isNat#(activate(V1))
p20: U21#(tt(),V1) -> activate#(V1)
p21: U11#(tt(),V1,V2) -> activate#(V1)

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          activate#_A(x1) = ((1,0),(0,0)) x1 + (8,5)
          n__and_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (8,1)
          and#_A(x1,x2) = ((1,0),(0,0)) x2 + (8,5)
          tt_A() = (1,1)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,0),(1,1)) x2 + (8,6)
          plus#_A(x1,x2) = ((1,1),(0,0)) x1 + ((1,0),(0,0)) x2 + (16,5)
          |0|_A() = (0,0)
          U31#_A(x1,x2) = ((1,1),(0,0)) x2 + (9,5)
          and_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (8,1)
          isNat_A(x1) = ((0,1),(0,0)) x1 + (4,5)
          n__isNatKind_A(x1) = ((1,0),(1,0)) x1 + (3,1)
          isNatKind#_A(x1) = ((1,0),(0,0)) x1 + (11,5)
          activate_A(x1) = x1
          n__s_A(x1) = ((1,0),(0,0)) x1 + (20,5)
          s_A(x1) = ((1,0),(0,0)) x1 + (20,5)
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (3,5)
          U11#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + (9,5)
          isNatKind_A(x1) = ((1,0),(1,0)) x1 + (3,1)
          U21#_A(x1,x2) = ((1,0),(0,0)) x2 + (22,5)
          U13_A(x1) = (2,4)
          U12_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (2,4)
          U22_A(x1) = (2,2)
          U31_A(x1,x2) = ((1,1),(0,1)) x2 + (1,0)
          U41_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((1,0),(0,0)) x2 + ((1,1),(0,0)) x3 + (28,4)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,0),(1,1)) x2 + (8,6)
          U11_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((0,1),(0,0)) x3 + (9,4)
          U21_A(x1,x2) = (3,2)
          n__0_A() = (0,0)
  
    precedence:
    
      U22 > |0| = n__0 > isNat = U12 = U11 > n__isNatKind = isNatKind > n__plus = n__s = s = U31 = U41 = plus > isNat# = U21# > U11# > U31# > activate# = and# = isNatKind# > plus# > U13 > n__and = and > activate > tt > U21
    
    partial status:
  
      pi(activate#) = []
      pi(n__and) = []
      pi(and#) = []
      pi(tt) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(U31#) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(activate) = [1]
      pi(n__s) = []
      pi(s) = []
      pi(isNat#) = []
      pi(U11#) = []
      pi(isNatKind) = []
      pi(U21#) = []
      pi(U13) = []
      pi(U12) = []
      pi(U22) = []
      pi(U31) = []
      pi(U41) = []
      pi(plus) = []
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: and#(tt(),X) -> activate#(X)
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)) -> isNatKind#(activate(V1))
p7: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p8: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__s(V1)) -> activate#(V1)
p10: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p11: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
p12: plus#(N,s(M)) -> 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)) -> isNatKind#(activate(V1))
p16: isNat#(n__plus(V1,V2)) -> activate#(V1)
p17: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p18: U21#(tt(),V1) -> isNat#(activate(V1))
p19: U21#(tt(),V1) -> activate#(V1)
p20: U11#(tt(),V1,V2) -> 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:

  {p13, p14, p17, p18}
  {p1, p2, p5, p6, p7, p8, p9}


-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          isNat#_A(x1) = ((1,0),(1,0)) x1 + (1,0)
          n__s_A(x1) = x1 + (71,94)
          U21#_A(x1,x2) = ((1,0),(1,0)) x2 + (47,23)
          isNatKind_A(x1) = ((1,1),(0,1)) x1 + (3,24)
          activate_A(x1) = x1 + (23,24)
          tt_A() = (14,14)
          n__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(1,1)) x2 + (138,71)
          U11#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,0),(0,0)) x3 + (67,24)
          and_A(x1,x2) = ((0,1),(0,1)) x1 + ((1,1),(0,1)) x2 + (15,10)
          n__isNatKind_A(x1) = ((1,1),(0,1)) x1 + (3,0)
          U13_A(x1) = ((0,0),(1,0)) x1 + (17,0)
          U12_A(x1,x2) = (20,51)
          isNat_A(x1) = ((0,0),(1,1)) x1 + (51,5)
          U22_A(x1) = (15,15)
          U11_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (51,38)
          U21_A(x1,x2) = (50,15)
          U31_A(x1,x2) = x2 + (24,24)
          U41_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(0,1)) x3 + (303,236)
          s_A(x1) = x1 + (71,94)
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(1,1)) x2 + (138,71)
          n__0_A() = (29,14)
          |0|_A() = (52,15)
          n__and_A(x1,x2) = ((0,1),(0,1)) x1 + ((1,1),(0,1)) x2 + (15,10)
  
    precedence:
    
      isNatKind = n__plus = n__isNatKind = U41 = plus > U22 > and = n__and > isNat# = n__s = U21# = activate = tt = U13 = isNat = U11 = U21 = U31 = s = n__0 = |0| > U11# = U12
    
    partial status:
  
      pi(isNat#) = []
      pi(n__s) = []
      pi(U21#) = []
      pi(isNatKind) = []
      pi(activate) = [1]
      pi(tt) = []
      pi(n__plus) = []
      pi(U11#) = []
      pi(and) = []
      pi(n__isNatKind) = []
      pi(U13) = []
      pi(U12) = []
      pi(isNat) = []
      pi(U22) = []
      pi(U11) = []
      pi(U21) = []
      pi(U31) = [2]
      pi(U41) = []
      pi(s) = []
      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: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p2: U21#(tt(),V1) -> isNat#(activate(V1))
p3: 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 estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p2: U21#(tt(),V1) -> 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^2
        order: standard order
        interpretations:
          isNat#_A(x1) = x1 + (1,4)
          n__s_A(x1) = x1 + (5,8)
          U21#_A(x1,x2) = x2 + (4,6)
          isNatKind_A(x1) = ((0,1),(1,1)) x1 + (1,9)
          activate_A(x1) = x1 + (1,2)
          tt_A() = (1,6)
          U13_A(x1) = (1,7)
          U12_A(x1,x2) = (21,26)
          isNat_A(x1) = ((1,0),(1,1)) x1 + (19,23)
          U22_A(x1) = ((0,0),(1,0)) x1 + (2,5)
          U11_A(x1,x2,x3) = ((1,0),(0,0)) x3 + (22,99)
          U21_A(x1,x2) = ((1,0),(1,1)) x2 + (3,25)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (2,3)
          U41_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,0),(1,1)) x3 + (15,86)
          s_A(x1) = x1 + (6,8)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,1),(1,1)) x2 + (4,72)
          n__0_A() = (0,0)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,1),(1,1)) x2 + (4,72)
          and_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (0,1)
          n__isNatKind_A(x1) = ((0,1),(1,1)) x1 + (0,9)
          |0|_A() = (0,2)
          n__and_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (0,1)
  
    precedence:
    
      |0| > isNatKind = activate = U31 = n__0 = and = n__isNatKind > isNat = U21 > U41 = plus = n__plus > s = n__and > U22 > U13 = U12 = U11 > n__s = U21# = tt > isNat#
    
    partial status:
  
      pi(isNat#) = [1]
      pi(n__s) = []
      pi(U21#) = []
      pi(isNatKind) = []
      pi(activate) = []
      pi(tt) = []
      pi(U13) = []
      pi(U12) = []
      pi(isNat) = [1]
      pi(U22) = []
      pi(U11) = []
      pi(U21) = [2]
      pi(U31) = []
      pi(U41) = []
      pi(s) = []
      pi(plus) = [2]
      pi(n__0) = []
      pi(n__plus) = []
      pi(and) = []
      pi(n__isNatKind) = []
      pi(|0|) = []
      pi(n__and) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: 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 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__s(V1)) -> activate#(V1)
p5: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p6: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p7: 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^2
        order: standard order
        interpretations:
          activate#_A(x1) = ((0,1),(0,1)) x1 + (13,0)
          n__and_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (0,15)
          and#_A(x1,x2) = ((0,1),(0,1)) x2 + (14,0)
          tt_A() = (1,3)
          n__isNatKind_A(x1) = ((1,1),(1,1)) x1 + (7,0)
          isNatKind#_A(x1) = ((0,1),(0,1)) x1 + (12,0)
          n__s_A(x1) = x1 + (1,2)
          activate_A(x1) = x1
          n__plus_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (11,11)
          U13_A(x1) = (1,3)
          U12_A(x1,x2) = (2,3)
          isNat_A(x1) = ((0,1),(1,0)) x1 + (4,15)
          U22_A(x1) = (2,4)
          U11_A(x1,x2,x3) = ((0,1),(1,0)) x2 + ((1,1),(1,0)) x3 + (12,12)
          U21_A(x1,x2) = (3,4)
          U31_A(x1,x2) = x2 + (13,1)
          U41_A(x1,x2,x3) = ((1,0),(1,1)) x2 + x3 + (12,13)
          s_A(x1) = x1 + (1,2)
          plus_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (11,11)
          n__0_A() = (3,0)
          and_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (0,15)
          isNatKind_A(x1) = ((1,1),(1,1)) x1 + (7,0)
          |0|_A() = (3,0)
  
    precedence:
    
      n__and = n__s = activate = isNat = U21 = U31 = U41 = s = plus = and = isNatKind = |0| > n__plus > activate# = and# = isNatKind# = n__0 > n__isNatKind > tt = U13 = U12 = U22 = U11
    
    partial status:
  
      pi(activate#) = []
      pi(n__and) = []
      pi(and#) = []
      pi(tt) = []
      pi(n__isNatKind) = [1]
      pi(isNatKind#) = []
      pi(n__s) = []
      pi(activate) = []
      pi(n__plus) = []
      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(isNatKind) = []
      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: activate#(n__and(X1,X2)) -> and#(X1,X2)
p2: activate#(n__isNatKind(X)) -> isNatKind#(X)
p3: isNatKind#(n__s(V1)) -> activate#(V1)
p4: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p5: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p6: 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 estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p5, p6}


-- Reduction pair.

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

p1: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p3: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p4: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p5: 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^2
        order: standard order
        interpretations:
          activate#_A(x1) = ((0,1),(0,1)) x1 + (16,8)
          n__isNatKind_A(x1) = ((0,1),(1,1)) x1 + (33,0)
          isNatKind#_A(x1) = ((0,1),(0,1)) x1 + (15,4)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (15,25)
          activate_A(x1) = x1 + (0,1)
          n__s_A(x1) = x1 + (2,4)
          U13_A(x1) = ((0,1),(0,0)) x1 + (1,32)
          tt_A() = (16,16)
          U12_A(x1,x2) = x1 + ((1,1),(0,0)) x2 + (19,16)
          isNat_A(x1) = ((0,1),(1,1)) x1 + (31,32)
          U22_A(x1) = ((0,0),(1,0)) x1 + (16,0)
          U11_A(x1,x2,x3) = ((0,1),(1,1)) x2 + ((1,1),(0,0)) x3 + (53,71)
          U21_A(x1,x2) = ((0,0),(0,1)) x2 + (17,32)
          U31_A(x1,x2) = x2 + (1,2)
          U41_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (17,31)
          s_A(x1) = x1 + (2,4)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (15,25)
          n__0_A() = (17,0)
          and_A(x1,x2) = ((0,1),(0,1)) x1 + x2 + (1,24)
          isNatKind_A(x1) = ((0,1),(1,1)) x1 + (33,0)
          |0|_A() = (17,1)
          n__and_A(x1,x2) = ((0,1),(0,1)) x1 + x2 + (1,24)
  
    precedence:
    
      n__plus = activate = n__s = U31 = U41 = s = plus = |0| > n__0 > and = isNatKind > n__and > U12 = isNat = U11 > activate# = isNatKind# = U13 = U21 > n__isNatKind > tt = U22
    
    partial status:
  
      pi(activate#) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(n__plus) = []
      pi(activate) = []
      pi(n__s) = []
      pi(U13) = []
      pi(tt) = []
      pi(U12) = []
      pi(isNat) = []
      pi(U22) = []
      pi(U11) = []
      pi(U21) = []
      pi(U31) = []
      pi(U41) = []
      pi(s) = []
      pi(plus) = []
      pi(n__0) = []
      pi(and) = []
      pi(isNatKind) = []
      pi(|0|) = []
      pi(n__and) = []

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p3: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p4: 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, p2, p3, p4}


-- Reduction pair.

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

p1: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p3: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p4: 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^2
        order: standard order
        interpretations:
          activate#_A(x1) = ((1,0),(0,0)) x1 + (1,10)
          n__isNatKind_A(x1) = ((0,1),(0,0)) x1 + (40,21)
          isNatKind#_A(x1) = ((0,1),(0,0)) x1 + (6,10)
          n__s_A(x1) = x1 + (10,27)
          activate_A(x1) = x1 + (0,5)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,1),(1,1)) x2 + (12,10)
          U13_A(x1) = (2,1)
          tt_A() = (1,0)
          U12_A(x1,x2) = (3,1)
          isNat_A(x1) = ((0,1),(0,1)) x1 + (9,0)
          U22_A(x1) = (2,1)
          U11_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,1)) x3 + (13,5)
          U21_A(x1,x2) = (15,27)
          U31_A(x1,x2) = x2 + (11,5)
          U41_A(x1,x2,x3) = ((0,1),(1,1)) x2 + ((1,0),(1,1)) x3 + (28,47)
          s_A(x1) = x1 + (10,27)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,1),(1,1)) x2 + (12,10)
          n__0_A() = (10,0)
          and_A(x1,x2) = x2 + (5,5)
          isNatKind_A(x1) = ((0,1),(0,0)) x1 + (40,26)
          |0|_A() = (10,0)
          n__and_A(x1,x2) = x2 + (5,1)
  
    precedence:
    
      n__0 = |0| > n__s = n__plus = U41 = s = plus > n__isNatKind = activate = U31 = isNatKind > activate# = isNatKind# = and = n__and > isNat > U22 = U21 > U13 = tt = U12 > U11
    
    partial status:
  
      pi(activate#) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(n__s) = []
      pi(activate) = [1]
      pi(n__plus) = []
      pi(U13) = []
      pi(tt) = []
      pi(U12) = []
      pi(isNat) = []
      pi(U22) = []
      pi(U11) = [3]
      pi(U21) = []
      pi(U31) = [2]
      pi(U41) = []
      pi(s) = []
      pi(plus) = []
      pi(n__0) = []
      pi(and) = []
      pi(isNatKind) = []
      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: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p3: 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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: activate#(n__isNatKind(X)) -> isNatKind#(X)
p2: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p3: isNatKind#(n__plus(V1,V2)) -> activate#(V2)

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          activate#_A(x1) = ((0,1),(0,0)) x1 + (1,4)
          n__isNatKind_A(x1) = ((0,1),(0,1)) x1 + (21,26)
          isNatKind#_A(x1) = ((0,1),(0,0)) x1 + (21,4)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((0,1),(1,1)) x2 + (33,76)
          activate_A(x1) = x1 + (6,4)
          U13_A(x1) = ((0,0),(1,0)) x1 + (3,0)
          tt_A() = (2,0)
          U12_A(x1,x2) = (4,33)
          isNat_A(x1) = (33,33)
          U22_A(x1) = (3,0)
          U11_A(x1,x2,x3) = (5,33)
          U21_A(x1,x2) = (7,33)
          U31_A(x1,x2) = ((1,1),(0,1)) x2 + (7,33)
          U41_A(x1,x2,x3) = ((0,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (67,130)
          s_A(x1) = x1 + (20,34)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((0,1),(1,1)) x2 + (33,76)
          n__0_A() = (1,28)
          and_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (7,4)
          isNatKind_A(x1) = ((0,1),(0,1)) x1 + (27,30)
          n__s_A(x1) = x1 + (15,34)
          |0|_A() = (6,32)
          n__and_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (2,4)
  
    precedence:
    
      n__plus = plus > U31 > activate# = isNatKind# > isNat > U22 = U21 > U11 > U12 > U13 = tt > activate = U41 = s = n__0 = and = isNatKind = |0| = n__and > n__isNatKind > n__s
    
    partial status:
  
      pi(activate#) = []
      pi(n__isNatKind) = []
      pi(isNatKind#) = []
      pi(n__plus) = [1]
      pi(activate) = [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) = []
      pi(isNatKind) = []
      pi(n__s) = [1]
      pi(|0|) = []
      pi(n__and) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p2: isNatKind#(n__plus(V1,V2)) -> activate#(V2)

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          isNatKind#_A(x1) = ((1,1),(0,1)) x1 + (1,1)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((0,0),(1,0)) x2 + (9,6)
          activate_A(x1) = ((1,1),(0,1)) x1 + (7,6)
          U13_A(x1) = (20,4)
          tt_A() = (20,4)
          U12_A(x1,x2) = (20,4)
          isNat_A(x1) = ((1,0),(1,0)) x1 + (22,11)
          U22_A(x1) = (21,5)
          U11_A(x1,x2,x3) = (30,7)
          U21_A(x1,x2) = ((1,0),(0,0)) x1 + (2,5)
          U31_A(x1,x2) = ((1,1),(1,1)) x2 + (8,6)
          U41_A(x1,x2,x3) = ((1,1),(0,0)) x3 + (6,3)
          s_A(x1) = (5,2)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,0),(1,0)) x2 + (18,6)
          n__0_A() = (3,3)
          and_A(x1,x2) = ((1,1),(0,1)) x2 + (7,6)
          isNatKind_A(x1) = (20,9)
          n__isNatKind_A(x1) = (10,3)
          n__s_A(x1) = (5,1)
          |0|_A() = (3,3)
          n__and_A(x1,x2) = x2 + (0,4)
  
    precedence:
    
      n__plus = isNat > U12 = U11 > activate = U13 = tt = U21 = U31 = plus = and = isNatKind = |0| = n__and > U41 = s = n__0 = n__isNatKind = n__s > isNatKind# = U22
    
    partial status:
  
      pi(isNatKind#) = [1]
      pi(n__plus) = []
      pi(activate) = []
      pi(U13) = []
      pi(tt) = []
      pi(U12) = []
      pi(isNat) = []
      pi(U22) = []
      pi(U11) = []
      pi(U21) = []
      pi(U31) = []
      pi(U41) = []
      pi(s) = []
      pi(plus) = []
      pi(n__0) = []
      pi(and) = []
      pi(isNatKind) = []
      pi(n__isNatKind) = []
      pi(n__s) = []
      pi(|0|) = []
      pi(n__and) = []

The next rules are strictly ordered:

  p1

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