YES

We show the termination of the TRS R:

  U11(tt(),N) -> activate(N)
  U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
  and(tt(),X) -> activate(X)
  isNat(n__0()) -> tt()
  isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
  isNat(n__s(V1)) -> isNat(activate(V1))
  plus(N,|0|()) -> U11(isNat(N),N)
  plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
  |0|() -> n__0()
  plus(X1,X2) -> n__plus(X1,X2)
  isNat(X) -> n__isNat(X)
  s(X) -> n__s(X)
  activate(n__0()) -> |0|()
  activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
  activate(n__isNat(X)) -> isNat(X)
  activate(n__s(X)) -> s(activate(X))
  activate(X) -> X

-- SCC decomposition.

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

p1: U11#(tt(),N) -> activate#(N)
p2: U21#(tt(),M,N) -> s#(plus(activate(N),activate(M)))
p3: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p4: U21#(tt(),M,N) -> activate#(N)
p5: U21#(tt(),M,N) -> activate#(M)
p6: and#(tt(),X) -> activate#(X)
p7: isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p8: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p9: isNat#(n__plus(V1,V2)) -> activate#(V1)
p10: isNat#(n__plus(V1,V2)) -> activate#(V2)
p11: isNat#(n__s(V1)) -> isNat#(activate(V1))
p12: isNat#(n__s(V1)) -> activate#(V1)
p13: plus#(N,|0|()) -> U11#(isNat(N),N)
p14: plus#(N,|0|()) -> isNat#(N)
p15: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p16: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p17: plus#(N,s(M)) -> isNat#(M)
p18: activate#(n__0()) -> |0|#()
p19: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p20: activate#(n__plus(X1,X2)) -> activate#(X1)
p21: activate#(n__plus(X1,X2)) -> activate#(X2)
p22: activate#(n__isNat(X)) -> isNat#(X)
p23: activate#(n__s(X)) -> s#(activate(X))
p24: activate#(n__s(X)) -> activate#(X)

and R consists of:

r1: U11(tt(),N) -> activate(N)
r2: U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
r3: and(tt(),X) -> activate(X)
r4: isNat(n__0()) -> tt()
r5: isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
r6: isNat(n__s(V1)) -> isNat(activate(V1))
r7: plus(N,|0|()) -> U11(isNat(N),N)
r8: plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
r9: |0|() -> n__0()
r10: plus(X1,X2) -> n__plus(X1,X2)
r11: isNat(X) -> n__isNat(X)
r12: s(X) -> n__s(X)
r13: activate(n__0()) -> |0|()
r14: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(activate(X))
r17: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

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

and R consists of:

r1: U11(tt(),N) -> activate(N)
r2: U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
r3: and(tt(),X) -> activate(X)
r4: isNat(n__0()) -> tt()
r5: isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
r6: isNat(n__s(V1)) -> isNat(activate(V1))
r7: plus(N,|0|()) -> U11(isNat(N),N)
r8: plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
r9: |0|() -> n__0()
r10: plus(X1,X2) -> n__plus(X1,X2)
r11: isNat(X) -> n__isNat(X)
r12: s(X) -> n__s(X)
r13: activate(n__0()) -> |0|()
r14: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(activate(X))
r17: 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

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) = ((1,0),(1,0)) x2 + (14,41)
            tt_A() = (40,37)
            activate#_A(x1) = x1 + (13,40)
            n__s_A(x1) = ((1,0),(0,0)) x1 + (28,31)
            n__isNat_A(x1) = ((1,0),(0,0)) x1 + (15,7)
            isNat#_A(x1) = ((1,0),(0,0)) x1 + (27,39)
            n__plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (14,5)
            plus#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (26,14)
            activate_A(x1) = x1 + (0,32)
            s_A(x1) = ((1,0),(0,0)) x1 + (28,34)
            and#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (14,1)
            isNat_A(x1) = ((1,0),(0,0)) x1 + (15,38)
            U21#_A(x1,x2,x3) = x2 + x3 + (29,75)
            and_A(x1,x2) = ((1,0),(0,0)) x2 + (14,33)
            |0|_A() = (26,42)
            U11_A(x1,x2) = ((1,0),(1,0)) x2 + (27,33)
            U21_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (42,35)
            plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (14,36)
            n__0_A() = (26,11)
    
      precedence:
      
        tt = plus# = and > isNat# > U21 > activate = s = |0| > activate# = and# > n__s > U11# = U11 = plus > U21# > isNat > n__isNat = n__plus > n__0
      
      partial status:
    
        pi(U11#) = []
        pi(tt) = []
        pi(activate#) = []
        pi(n__s) = []
        pi(n__isNat) = []
        pi(isNat#) = []
        pi(n__plus) = []
        pi(plus#) = [2]
        pi(activate) = [1]
        pi(s) = []
        pi(and#) = [2]
        pi(isNat) = []
        pi(U21#) = [2, 3]
        pi(and) = []
        pi(|0|) = []
        pi(U11) = []
        pi(U21) = []
        pi(plus) = []
        pi(n__0) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            U11#_A(x1,x2) = (5,21)
            tt_A() = (3,6)
            activate#_A(x1) = (4,13)
            n__s_A(x1) = (10,2)
            n__isNat_A(x1) = (5,22)
            isNat#_A(x1) = (9,14)
            n__plus_A(x1,x2) = (6,12)
            plus#_A(x1,x2) = x2 + (2,5)
            activate_A(x1) = x1 + (2,2)
            s_A(x1) = (10,5)
            and#_A(x1,x2) = (5,4)
            isNat_A(x1) = (8,21)
            U21#_A(x1,x2,x3) = x2 + (11,13)
            and_A(x1,x2) = (5,11)
            |0|_A() = (10,15)
            U11_A(x1,x2) = (1,16)
            U21_A(x1,x2,x3) = (2,6)
            plus_A(x1,x2) = (7,13)
            n__0_A() = (9,14)
    
      precedence:
      
        U11# > U21# > plus# > n__0 > U11 > n__isNat > isNat# > |0| > plus > s = and > activate > n__s > isNat > tt > U21 > and# > activate# > n__plus
      
      partial status:
    
        pi(U11#) = []
        pi(tt) = []
        pi(activate#) = []
        pi(n__s) = []
        pi(n__isNat) = []
        pi(isNat#) = []
        pi(n__plus) = []
        pi(plus#) = [2]
        pi(activate) = []
        pi(s) = []
        pi(and#) = []
        pi(isNat) = []
        pi(U21#) = [2]
        pi(and) = []
        pi(|0|) = []
        pi(U11) = []
        pi(U21) = []
        pi(plus) = []
        pi(n__0) = []
    

The next rules are strictly ordered:

  p1, p3, p4, p7, p8, p11, p12, p13, p15, p16, p17, p18, p19, p20

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__s(X)) -> activate#(X)
p2: activate#(n__plus(X1,X2)) -> activate#(X2)
p3: activate#(n__plus(X1,X2)) -> activate#(X1)
p4: isNat#(n__s(V1)) -> isNat#(activate(V1))
p5: isNat#(n__plus(V1,V2)) -> activate#(V2)
p6: and#(tt(),X) -> activate#(X)
p7: plus#(N,|0|()) -> U11#(isNat(N),N)

and R consists of:

r1: U11(tt(),N) -> activate(N)
r2: U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
r3: and(tt(),X) -> activate(X)
r4: isNat(n__0()) -> tt()
r5: isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
r6: isNat(n__s(V1)) -> isNat(activate(V1))
r7: plus(N,|0|()) -> U11(isNat(N),N)
r8: plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
r9: |0|() -> n__0()
r10: plus(X1,X2) -> n__plus(X1,X2)
r11: isNat(X) -> n__isNat(X)
r12: s(X) -> n__s(X)
r13: activate(n__0()) -> |0|()
r14: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(activate(X))
r17: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p4}
  {p1, p2, p3}


-- Reduction pair.

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

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

and R consists of:

r1: U11(tt(),N) -> activate(N)
r2: U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
r3: and(tt(),X) -> activate(X)
r4: isNat(n__0()) -> tt()
r5: isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
r6: isNat(n__s(V1)) -> isNat(activate(V1))
r7: plus(N,|0|()) -> U11(isNat(N),N)
r8: plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
r9: |0|() -> n__0()
r10: plus(X1,X2) -> n__plus(X1,X2)
r11: isNat(X) -> n__isNat(X)
r12: s(X) -> n__s(X)
r13: activate(n__0()) -> |0|()
r14: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(activate(X))
r17: 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

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) = ((1,0),(0,0)) x1 + (12,1)
            n__s_A(x1) = ((1,0),(0,0)) x1 + (12,3)
            activate_A(x1) = ((1,0),(1,1)) x1 + (0,3)
            U11_A(x1,x2) = ((1,0),(1,1)) x2 + (1,4)
            tt_A() = (11,12)
            U21_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (24,10)
            s_A(x1) = ((1,0),(0,0)) x1 + (12,9)
            plus_A(x1,x2) = x1 + x2 + (12,5)
            and_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (1,1)
            isNat_A(x1) = ((1,0),(0,0)) x1 + (11,13)
            n__0_A() = (0,13)
            n__plus_A(x1,x2) = x1 + x2 + (12,2)
            n__isNat_A(x1) = ((1,0),(0,0)) x1 + (11,0)
            |0|_A() = (0,14)
    
      precedence:
      
        isNat# > |0| > isNat > n__s = s > and > n__0 > tt > activate = U11 = U21 = plus = n__plus = n__isNat
      
      partial status:
    
        pi(isNat#) = []
        pi(n__s) = []
        pi(activate) = []
        pi(U11) = [2]
        pi(tt) = []
        pi(U21) = []
        pi(s) = []
        pi(plus) = [1, 2]
        pi(and) = []
        pi(isNat) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(n__isNat) = []
        pi(|0|) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            isNat#_A(x1) = (0,0)
            n__s_A(x1) = (9,3)
            activate_A(x1) = (8,6)
            U11_A(x1,x2) = (9,7)
            tt_A() = (1,1)
            U21_A(x1,x2,x3) = (12,8)
            s_A(x1) = (10,7)
            plus_A(x1,x2) = ((1,0),(0,0)) x2 + (3,9)
            and_A(x1,x2) = (9,5)
            isNat_A(x1) = (7,2)
            n__0_A() = (2,0)
            n__plus_A(x1,x2) = (2,4)
            n__isNat_A(x1) = (1,3)
            |0|_A() = (3,8)
    
      precedence:
      
        plus = and = n__plus > U11 > U21 > tt > activate = n__0 > n__s = s = isNat = |0| > isNat# = n__isNat
      
      partial status:
    
        pi(isNat#) = []
        pi(n__s) = []
        pi(activate) = []
        pi(U11) = []
        pi(tt) = []
        pi(U21) = []
        pi(s) = []
        pi(plus) = []
        pi(and) = []
        pi(isNat) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(n__isNat) = []
        pi(|0|) = []
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

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

and R consists of:

r1: U11(tt(),N) -> activate(N)
r2: U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
r3: and(tt(),X) -> activate(X)
r4: isNat(n__0()) -> tt()
r5: isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
r6: isNat(n__s(V1)) -> isNat(activate(V1))
r7: plus(N,|0|()) -> U11(isNat(N),N)
r8: plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
r9: |0|() -> n__0()
r10: plus(X1,X2) -> n__plus(X1,X2)
r11: isNat(X) -> n__isNat(X)
r12: s(X) -> n__s(X)
r13: activate(n__0()) -> |0|()
r14: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(activate(X))
r17: activate(X) -> X

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            activate#_A(x1) = ((1,0),(1,1)) x1 + (1,2)
            n__s_A(x1) = ((1,0),(0,0)) x1 + (2,3)
            n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (2,1)
    
      precedence:
      
        n__plus > activate# > n__s
      
      partial status:
    
        pi(activate#) = [1]
        pi(n__s) = []
        pi(n__plus) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            activate#_A(x1) = (0,0)
            n__s_A(x1) = (1,1)
            n__plus_A(x1,x2) = (1,1)
    
      precedence:
      
        n__s > activate# = n__plus
      
      partial status:
    
        pi(activate#) = []
        pi(n__s) = []
        pi(n__plus) = []
    

The next rules are strictly ordered:

  p2, p3

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__s(X)) -> activate#(X)

and R consists of:

r1: U11(tt(),N) -> activate(N)
r2: U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
r3: and(tt(),X) -> activate(X)
r4: isNat(n__0()) -> tt()
r5: isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
r6: isNat(n__s(V1)) -> isNat(activate(V1))
r7: plus(N,|0|()) -> U11(isNat(N),N)
r8: plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
r9: |0|() -> n__0()
r10: plus(X1,X2) -> n__plus(X1,X2)
r11: isNat(X) -> n__isNat(X)
r12: s(X) -> n__s(X)
r13: activate(n__0()) -> |0|()
r14: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(activate(X))
r17: 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: activate#(n__s(X)) -> activate#(X)

and R consists of:

r1: U11(tt(),N) -> activate(N)
r2: U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
r3: and(tt(),X) -> activate(X)
r4: isNat(n__0()) -> tt()
r5: isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
r6: isNat(n__s(V1)) -> isNat(activate(V1))
r7: plus(N,|0|()) -> U11(isNat(N),N)
r8: plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
r9: |0|() -> n__0()
r10: plus(X1,X2) -> n__plus(X1,X2)
r11: isNat(X) -> n__isNat(X)
r12: s(X) -> n__s(X)
r13: activate(n__0()) -> |0|()
r14: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(activate(X))
r17: activate(X) -> X

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            activate#_A(x1) = ((1,0),(1,1)) x1 + (2,2)
            n__s_A(x1) = ((1,0),(0,0)) x1 + (1,1)
    
      precedence:
      
        activate# = n__s
      
      partial status:
    
        pi(activate#) = []
        pi(n__s) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            activate#_A(x1) = ((1,0),(0,0)) x1
            n__s_A(x1) = (1,1)
    
      precedence:
      
        n__s > activate#
      
      partial status:
    
        pi(activate#) = []
        pi(n__s) = []
    

The next rules are strictly ordered:

  p1

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