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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            U11#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(0,0)) x3 + (23,2)
            tt_A() = (2,9)
            U12#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (3,3)
            isNat_A(x1) = ((1,0),(0,0)) x1 + (0,34)
            activate_A(x1) = ((1,0),(1,1)) x1 + (0,12)
            activate#_A(x1) = ((1,0),(1,0)) x1 + (3,4)
            n__and_A(x1,x2) = ((1,0),(0,0)) x2 + (15,15)
            and#_A(x1,x2) = ((1,0),(0,0)) x2 + (4,20)
            n__isNatKind_A(x1) = ((1,0),(0,0)) x1 + (7,6)
            isNatKind#_A(x1) = ((1,0),(0,0)) x1 + (1,5)
            n__s_A(x1) = ((1,0),(0,0)) x1 + (6,13)
            n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (22,13)
            plus#_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (21,27)
            s_A(x1) = ((1,0),(0,0)) x1 + (6,14)
            isNat#_A(x1) = ((1,0),(1,0)) x1 + (3,22)
            isNatKind_A(x1) = ((1,0),(1,0)) x1 + (7,12)
            U21#_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (4,21)
            and_A(x1,x2) = ((1,0),(1,0)) x2 + (15,26)
            U41#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,1)) x3 + (26,3)
            |0|_A() = (2,21)
            U31#_A(x1,x2) = ((1,0),(1,1)) x2 + (4,22)
            U13_A(x1) = ((1,0),(1,1)) x1 + (2,2)
            U12_A(x1,x2) = ((1,0),(0,0)) x2 + (3,35)
            U22_A(x1) = ((1,0),(0,0)) x1 + (6,10)
            U31_A(x1,x2) = x2 + (3,12)
            U41_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (28,13)
            plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (22,46)
            U11_A(x1,x2,x3) = ((1,0),(1,0)) x3 + (4,36)
            U21_A(x1,x2) = ((1,0),(0,0)) x2 + (6,11)
            n__0_A() = (2,8)
    
      precedence:
      
        activate > U12 > isNatKind = U13 > n__plus = and = |0| = plus > isNat > n__and > U41# = U41 > n__isNatKind > n__s = s = isNat# = n__0 > U31# = U11 = U21 > U11# = tt = U12# = activate# = U21# = U22 > plus# > and# > U31 > isNatKind#
      
      partial status:
    
        pi(U11#) = [2]
        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#) = [1]
        pi(s) = []
        pi(isNat#) = []
        pi(isNatKind) = []
        pi(U21#) = [2]
        pi(and) = []
        pi(U41#) = []
        pi(|0|) = []
        pi(U31#) = [2]
        pi(U13) = [1]
        pi(U12) = []
        pi(U22) = []
        pi(U31) = []
        pi(U41) = []
        pi(plus) = []
        pi(U11) = []
        pi(U21) = []
        pi(n__0) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            U11#_A(x1,x2,x3) = (7,0)
            tt_A() = (11,17)
            U12#_A(x1,x2) = (6,1)
            isNat_A(x1) = (14,0)
            activate_A(x1) = (9,6)
            activate#_A(x1) = (4,9)
            n__and_A(x1,x2) = (1,3)
            and#_A(x1,x2) = (0,0)
            n__isNatKind_A(x1) = (15,18)
            isNatKind#_A(x1) = (3,4)
            n__s_A(x1) = (0,3)
            n__plus_A(x1,x2) = (0,1)
            plus#_A(x1,x2) = (3,8)
            s_A(x1) = (6,4)
            isNat#_A(x1) = (5,10)
            isNatKind_A(x1) = (15,18)
            U21#_A(x1,x2) = (5,10)
            and_A(x1,x2) = (2,5)
            U41#_A(x1,x2,x3) = (7,7)
            |0|_A() = (0,1)
            U31#_A(x1,x2) = (1,2)
            U13_A(x1) = ((0,0),(1,0)) x1 + (9,1)
            U12_A(x1,x2) = (10,16)
            U22_A(x1) = (12,18)
            U31_A(x1,x2) = (0,0)
            U41_A(x1,x2,x3) = (7,7)
            plus_A(x1,x2) = (8,2)
            U11_A(x1,x2,x3) = (11,0)
            U21_A(x1,x2) = (13,19)
            n__0_A() = (0,0)
    
      precedence:
      
        n__0 > U41 > U12 > activate = U41# > and > isNat > U11 > n__plus > isNatKind > isNat# = U21# = |0| > U11# > U12# > n__s > isNatKind# > U21 > U31 > U13 > s > n__isNatKind > and# > tt = activate# > plus# > U31# > plus > U22 > n__and
      
      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:

  p1, p2, p4, p5, p6, p7, p8, p9, p10, p12, p13, p19, p20, p21, p22, p23, p24, p25, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38

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: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p3: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p4: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p5: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p6: U21#(tt(),V1) -> activate#(V1)
p7: U21#(tt(),V1) -> isNat#(activate(V1))
p8: U11#(tt(),V1,V2) -> isNat#(activate(V1))

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p2, p3}
  {p5, p7}


-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            isNatKind#_A(x1) = ((1,0),(0,0)) x1 + (33,1)
            n__s_A(x1) = x1 + (33,20)
            activate_A(x1) = x1 + (0,6)
            n__plus_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (34,5)
            U13_A(x1) = (2,1)
            tt_A() = (1,2)
            U12_A(x1,x2) = (3,7)
            isNat_A(x1) = ((1,0),(0,0)) x1 + (68,24)
            U22_A(x1) = (33,6)
            U11_A(x1,x2,x3) = ((1,0),(1,1)) x1 + (68,5)
            U21_A(x1,x2) = ((1,0),(0,0)) x2 + (34,7)
            U31_A(x1,x2) = x2 + (1,7)
            U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + x3 + (67,37)
            s_A(x1) = x1 + (33,25)
            plus_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (34,5)
            n__0_A() = (0,0)
            and_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (1,2)
            isNatKind_A(x1) = ((1,0),(1,1)) x1 + (2,3)
            n__isNatKind_A(x1) = ((1,0),(1,1)) x1 + (2,1)
            |0|_A() = (0,5)
            n__and_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (1,1)
    
      precedence:
      
        U31 > n__plus = plus > isNatKind = n__isNatKind > U41 = and > activate > isNatKind# > |0| > s > n__s > isNat > U11 > U12 > U13 = U21 = n__0 > U22 = n__and > tt
      
      partial status:
    
        pi(isNatKind#) = []
        pi(n__s) = []
        pi(activate) = [1]
        pi(n__plus) = [1]
        pi(U13) = []
        pi(tt) = []
        pi(U12) = []
        pi(isNat) = []
        pi(U22) = []
        pi(U11) = [1]
        pi(U21) = []
        pi(U31) = [2]
        pi(U41) = []
        pi(s) = [1]
        pi(plus) = [1]
        pi(n__0) = []
        pi(and) = []
        pi(isNatKind) = [1]
        pi(n__isNatKind) = []
        pi(|0|) = []
        pi(n__and) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            isNatKind#_A(x1) = (0,0)
            n__s_A(x1) = (0,0)
            activate_A(x1) = (14,8)
            n__plus_A(x1,x2) = (12,4)
            U13_A(x1) = (11,3)
            tt_A() = (10,2)
            U12_A(x1,x2) = (15,4)
            isNat_A(x1) = (9,6)
            U22_A(x1) = (1,1)
            U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (6,5)
            U21_A(x1,x2) = (2,9)
            U31_A(x1,x2) = ((1,0),(0,0)) x2 + (15,9)
            U41_A(x1,x2,x3) = (12,10)
            s_A(x1) = (9,9)
            plus_A(x1,x2) = (13,5)
            n__0_A() = (7,1)
            and_A(x1,x2) = (2,2)
            isNatKind_A(x1) = (11,3)
            n__isNatKind_A(x1) = (1,2)
            |0|_A() = (8,7)
            n__and_A(x1,x2) = (1,1)
    
      precedence:
      
        isNatKind# = U31 > |0| > isNat > activate = U11 > U12 > plus > n__plus > isNatKind > and > n__isNatKind > n__0 > U41 > s > n__s > U13 > tt > n__and > U21 > U22
      
      partial status:
    
        pi(isNatKind#) = []
        pi(n__s) = []
        pi(activate) = []
        pi(n__plus) = []
        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(|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: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))

and R consists of:

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

The set of usable rules consists of

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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            isNatKind#_A(x1) = ((1,0),(1,1)) x1 + (1,2)
            n__s_A(x1) = ((1,0),(1,0)) x1 + (7,6)
            activate_A(x1) = ((1,0),(1,1)) x1 + (0,5)
            U13_A(x1) = (5,3)
            tt_A() = (4,2)
            U12_A(x1,x2) = ((1,0),(1,0)) x1 + (5,0)
            isNat_A(x1) = x1 + (0,28)
            U22_A(x1) = (5,1)
            U11_A(x1,x2,x3) = x2 + x3 + (6,1)
            U21_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (2,2)
            U31_A(x1,x2) = x2 + (6,1)
            U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (14,15)
            s_A(x1) = ((1,0),(1,0)) x1 + (7,7)
            plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (7,13)
            n__0_A() = (5,20)
            n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (7,12)
            and_A(x1,x2) = ((1,0),(0,0)) x2 + (5,11)
            isNatKind_A(x1) = x1 + (1,0)
            n__isNatKind_A(x1) = x1 + (1,0)
            |0|_A() = (5,29)
            n__and_A(x1,x2) = ((1,0),(0,0)) x2 + (5,8)
    
      precedence:
      
        isNatKind# > and > U31 > n__0 = |0| > activate = U41 = s = plus > tt > n__plus > n__s = U13 = U12 = isNat = U11 = U21 > n__and > U22 = isNatKind = n__isNatKind
      
      partial status:
    
        pi(isNatKind#) = [1]
        pi(n__s) = []
        pi(activate) = [1]
        pi(U13) = []
        pi(tt) = []
        pi(U12) = []
        pi(isNat) = [1]
        pi(U22) = []
        pi(U11) = [2, 3]
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(s) = []
        pi(plus) = [1, 2]
        pi(n__0) = []
        pi(n__plus) = [1, 2]
        pi(and) = []
        pi(isNatKind) = [1]
        pi(n__isNatKind) = []
        pi(|0|) = []
        pi(n__and) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            isNatKind#_A(x1) = (1,4)
            n__s_A(x1) = (4,5)
            activate_A(x1) = (3,3)
            U13_A(x1) = (2,2)
            tt_A() = (1,1)
            U12_A(x1,x2) = (4,4)
            isNat_A(x1) = ((1,0),(1,1)) x1 + (12,12)
            U22_A(x1) = (0,0)
            U11_A(x1,x2,x3) = ((0,0),(1,0)) x2 + x3 + (5,8)
            U21_A(x1,x2) = (0,1)
            U31_A(x1,x2) = (2,12)
            U41_A(x1,x2,x3) = (6,7)
            s_A(x1) = (5,6)
            plus_A(x1,x2) = ((1,0),(0,0)) x2 + (4,11)
            n__0_A() = (7,2)
            n__plus_A(x1,x2) = ((1,0),(0,0)) x2 + (0,1)
            and_A(x1,x2) = (6,8)
            isNatKind_A(x1) = (2,2)
            n__isNatKind_A(x1) = (0,10)
            |0|_A() = (7,9)
            n__and_A(x1,x2) = (6,7)
    
      precedence:
      
        U22 > plus > isNat = n__plus = n__isNatKind = |0| > n__and > n__s = U21 > U41 > and > activate = tt = U31 = s = n__0 > isNatKind# > U11 > U12 > U13 > isNatKind
      
      partial status:
    
        pi(isNatKind#) = []
        pi(n__s) = []
        pi(activate) = []
        pi(U13) = []
        pi(tt) = []
        pi(U12) = []
        pi(isNat) = [1]
        pi(U22) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(s) = []
        pi(plus) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(and) = []
        pi(isNatKind) = []
        pi(n__isNatKind) = []
        pi(|0|) = []
        pi(n__and) = []
    

The next rules are strictly ordered:

  p1

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

-- 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:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            isNat#_A(x1) = ((0,0),(1,0)) x1 + (4,1)
            n__s_A(x1) = ((1,0),(0,0)) x1 + (22,0)
            U21#_A(x1,x2) = ((0,0),(1,0)) x2 + (4,2)
            isNatKind_A(x1) = ((1,0),(0,0)) x1 + (5,15)
            activate_A(x1) = ((1,0),(1,1)) x1 + (0,3)
            tt_A() = (0,0)
            U13_A(x1) = (1,7)
            U12_A(x1,x2) = (2,6)
            isNat_A(x1) = (4,5)
            U22_A(x1) = (1,0)
            U11_A(x1,x2,x3) = (3,2)
            U21_A(x1,x2) = (2,4)
            U31_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (1,5)
            U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (30,7)
            s_A(x1) = ((1,0),(0,0)) x1 + (22,6)
            plus_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (8,10)
            n__0_A() = (0,7)
            n__plus_A(x1,x2) = x1 + x2 + (8,4)
            and_A(x1,x2) = ((1,0),(0,0)) x2 + (8,15)
            n__isNatKind_A(x1) = ((1,0),(0,0)) x1 + (5,7)
            |0|_A() = (0,8)
            n__and_A(x1,x2) = ((1,0),(0,0)) x2 + (8,5)
    
      precedence:
      
        U31 > isNat# = n__s = U21# = activate = U13 = s = n__0 = |0| > isNatKind = tt = isNat = U22 = U21 = n__isNatKind > and > U41 = plus = n__and > U12 = U11 = n__plus
      
      partial status:
    
        pi(isNat#) = []
        pi(n__s) = []
        pi(U21#) = []
        pi(isNatKind) = []
        pi(activate) = []
        pi(tt) = []
        pi(U13) = []
        pi(U12) = []
        pi(isNat) = []
        pi(U22) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = [2]
        pi(U41) = [3]
        pi(s) = []
        pi(plus) = [1, 2]
        pi(n__0) = []
        pi(n__plus) = []
        pi(and) = []
        pi(n__isNatKind) = []
        pi(|0|) = []
        pi(n__and) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            isNat#_A(x1) = (1,1)
            n__s_A(x1) = (5,0)
            U21#_A(x1,x2) = (0,0)
            isNatKind_A(x1) = (10,3)
            activate_A(x1) = (9,4)
            tt_A() = (2,2)
            U13_A(x1) = (0,0)
            U12_A(x1,x2) = (0,0)
            isNat_A(x1) = (11,0)
            U22_A(x1) = (3,0)
            U11_A(x1,x2,x3) = (0,0)
            U21_A(x1,x2) = (4,0)
            U31_A(x1,x2) = ((0,0),(1,0)) x2 + (10,5)
            U41_A(x1,x2,x3) = (7,1)
            s_A(x1) = (6,3)
            plus_A(x1,x2) = (8,3)
            n__0_A() = (3,0)
            n__plus_A(x1,x2) = (1,0)
            and_A(x1,x2) = (10,5)
            n__isNatKind_A(x1) = (0,0)
            |0|_A() = (3,0)
            n__and_A(x1,x2) = (0,0)
    
      precedence:
      
        n__and > plus = n__isNatKind > U31 > isNat# = activate = n__plus > U21# > U41 > isNat = U21 = s = n__0 = |0| > U22 > n__s > U11 > U12 > tt > isNatKind > and > U13
      
      partial status:
    
        pi(isNat#) = []
        pi(n__s) = []
        pi(U21#) = []
        pi(isNatKind) = []
        pi(activate) = []
        pi(tt) = []
        pi(U13) = []
        pi(U12) = []
        pi(isNat) = []
        pi(U22) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(s) = []
        pi(plus) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(and) = []
        pi(n__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: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  (no SCCs)