YES

We show the termination of the TRS R:

  U11(tt(),V2) -> U12(isNat(activate(V2)))
  U12(tt()) -> tt()
  U21(tt()) -> tt()
  U31(tt(),N) -> activate(N)
  U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
  U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
  isNat(n__0()) -> tt()
  isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
  isNat(n__s(V1)) -> U21(isNat(activate(V1)))
  plus(N,|0|()) -> U31(isNat(N),N)
  plus(N,s(M)) -> U41(isNat(M),M,N)
  |0|() -> n__0()
  plus(X1,X2) -> n__plus(X1,X2)
  s(X) -> n__s(X)
  activate(n__0()) -> |0|()
  activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
  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(),V2) -> U12#(isNat(activate(V2)))
p2: U11#(tt(),V2) -> isNat#(activate(V2))
p3: U11#(tt(),V2) -> activate#(V2)
p4: U31#(tt(),N) -> activate#(N)
p5: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p6: U41#(tt(),M,N) -> isNat#(activate(N))
p7: U41#(tt(),M,N) -> activate#(N)
p8: U41#(tt(),M,N) -> activate#(M)
p9: U42#(tt(),M,N) -> s#(plus(activate(N),activate(M)))
p10: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p11: U42#(tt(),M,N) -> activate#(N)
p12: U42#(tt(),M,N) -> activate#(M)
p13: isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
p14: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p15: isNat#(n__plus(V1,V2)) -> activate#(V1)
p16: isNat#(n__plus(V1,V2)) -> activate#(V2)
p17: isNat#(n__s(V1)) -> U21#(isNat(activate(V1)))
p18: isNat#(n__s(V1)) -> isNat#(activate(V1))
p19: isNat#(n__s(V1)) -> activate#(V1)
p20: plus#(N,|0|()) -> U31#(isNat(N),N)
p21: plus#(N,|0|()) -> isNat#(N)
p22: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p23: plus#(N,s(M)) -> isNat#(M)
p24: activate#(n__0()) -> |0|#()
p25: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p26: activate#(n__plus(X1,X2)) -> activate#(X1)
p27: activate#(n__plus(X1,X2)) -> activate#(X2)
p28: activate#(n__s(X)) -> s#(activate(X))
p29: activate#(n__s(X)) -> activate#(X)

and R consists of:

r1: U11(tt(),V2) -> U12(isNat(activate(V2)))
r2: U12(tt()) -> tt()
r3: U21(tt()) -> tt()
r4: U31(tt(),N) -> activate(N)
r5: U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
r6: U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
r7: isNat(n__0()) -> tt()
r8: isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
r9: isNat(n__s(V1)) -> U21(isNat(activate(V1)))
r10: plus(N,|0|()) -> U31(isNat(N),N)
r11: plus(N,s(M)) -> U41(isNat(M),M,N)
r12: |0|() -> n__0()
r13: plus(X1,X2) -> n__plus(X1,X2)
r14: s(X) -> n__s(X)
r15: activate(n__0()) -> |0|()
r16: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(X))
r18: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p5, p6, p7, p8, p10, p11, p12, p13, p14, p15, p16, p18, p19, p20, p21, p22, p23, p25, p26, p27, p29}


-- Reduction pair.

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

p1: U11#(tt(),V2) -> isNat#(activate(V2))
p2: isNat#(n__s(V1)) -> activate#(V1)
p3: activate#(n__s(X)) -> activate#(X)
p4: activate#(n__plus(X1,X2)) -> activate#(X2)
p5: activate#(n__plus(X1,X2)) -> activate#(X1)
p6: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p7: plus#(N,s(M)) -> isNat#(M)
p8: isNat#(n__s(V1)) -> isNat#(activate(V1))
p9: isNat#(n__plus(V1,V2)) -> activate#(V2)
p10: isNat#(n__plus(V1,V2)) -> activate#(V1)
p11: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p12: isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
p13: U11#(tt(),V2) -> activate#(V2)
p14: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p15: U41#(tt(),M,N) -> activate#(M)
p16: U41#(tt(),M,N) -> activate#(N)
p17: U41#(tt(),M,N) -> isNat#(activate(N))
p18: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p19: U42#(tt(),M,N) -> activate#(M)
p20: U42#(tt(),M,N) -> activate#(N)
p21: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p22: plus#(N,|0|()) -> isNat#(N)
p23: plus#(N,|0|()) -> U31#(isNat(N),N)
p24: U31#(tt(),N) -> activate#(N)

and R consists of:

r1: U11(tt(),V2) -> U12(isNat(activate(V2)))
r2: U12(tt()) -> tt()
r3: U21(tt()) -> tt()
r4: U31(tt(),N) -> activate(N)
r5: U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
r6: U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
r7: isNat(n__0()) -> tt()
r8: isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
r9: isNat(n__s(V1)) -> U21(isNat(activate(V1)))
r10: plus(N,|0|()) -> U31(isNat(N),N)
r11: plus(N,s(M)) -> U41(isNat(M),M,N)
r12: |0|() -> n__0()
r13: plus(X1,X2) -> n__plus(X1,X2)
r14: s(X) -> n__s(X)
r15: activate(n__0()) -> |0|()
r16: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(X))
r18: 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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        U11#_A(x1,x2) = x1 + ((1,0),(1,1)) x2
        tt_A() = (3,2)
        isNat#_A(x1) = ((1,0),(1,0)) x1 + (2,0)
        activate_A(x1) = x1
        n__s_A(x1) = x1 + (0,4)
        activate#_A(x1) = ((1,0),(1,0)) x1 + (1,0)
        n__plus_A(x1,x2) = x1 + ((1,1),(0,1)) x2 + (0,1)
        plus#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,1),(1,1)) x2
        s_A(x1) = x1 + (0,4)
        isNat_A(x1) = ((1,0),(1,0)) x1 + (1,0)
        U41#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,0),(1,0)) x3 + (3,0)
        U42#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,0),(1,0)) x3 + (2,0)
        |0|_A() = (2,1)
        U31#_A(x1,x2) = ((1,0),(1,0)) x2 + (2,0)
        U42_A(x1,x2,x3) = ((1,1),(0,1)) x2 + x3 + (1,5)
        plus_A(x1,x2) = x1 + ((1,1),(0,1)) x2 + (0,1)
        U12_A(x1) = x1 + (1,2)
        U31_A(x1,x2) = x2 + (1,0)
        U41_A(x1,x2,x3) = ((1,1),(0,1)) x2 + x3 + (2,5)
        U11_A(x1,x2) = ((0,1),(0,1)) x1 + x2 + (1,0)
        U21_A(x1) = x1
        n__0_A() = (2,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        U11#_A(x1,x2) = ((0,1),(0,0)) x2 + (18,0)
        tt_A() = (1,2)
        isNat#_A(x1) = x1 + (14,14)
        activate_A(x1) = ((1,1),(1,1)) x1 + (3,1)
        n__s_A(x1) = ((1,1),(1,1)) x1 + (4,3)
        activate#_A(x1) = ((1,1),(1,1)) x1 + (2,1)
        n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((0,1),(1,0)) x2 + (6,4)
        plus#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (9,5)
        s_A(x1) = ((1,1),(1,1)) x1 + (5,3)
        isNat_A(x1) = (3,2)
        U41#_A(x1,x2,x3) = ((0,0),(1,0)) x2
        U42#_A(x1,x2,x3) = x2 + (1,0)
        |0|_A() = (2,1)
        U31#_A(x1,x2) = (1,9)
        U42_A(x1,x2,x3) = (12,11)
        plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((0,1),(1,0)) x2 + (7,4)
        U12_A(x1) = x1 + (1,1)
        U31_A(x1,x2) = x2 + (9,7)
        U41_A(x1,x2,x3) = (11,10)
        U11_A(x1,x2) = (0,0)
        U21_A(x1) = (2,2)
        n__0_A() = (1,1)
    
    3. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        U11#_A(x1,x2) = (10,5)
        tt_A() = (2,1)
        isNat#_A(x1) = ((1,1),(0,0)) x1 + (7,6)
        activate_A(x1) = x1 + (2,9)
        n__s_A(x1) = x1 + (1,1)
        activate#_A(x1) = ((1,1),(0,0)) x1 + (9,6)
        n__plus_A(x1,x2) = ((0,0),(1,0)) x1 + (1,1)
        plus#_A(x1,x2) = ((1,1),(1,0)) x2
        s_A(x1) = x1 + (6,1)
        isNat_A(x1) = (1,1)
        U41#_A(x1,x2,x3) = (6,0)
        U42#_A(x1,x2,x3) = (7,1)
        |0|_A() = (4,3)
        U31#_A(x1,x2) = (8,5)
        U42_A(x1,x2,x3) = (1,3)
        plus_A(x1,x2) = ((0,0),(1,0)) x1 + (2,1)
        U12_A(x1) = x1 + (1,3)
        U31_A(x1,x2) = ((1,1),(0,1)) x2 + (3,9)
        U41_A(x1,x2,x3) = (0,2)
        U11_A(x1,x2) = (0,2)
        U21_A(x1) = (0,1)
        n__0_A() = (1,4)
    

The next rules are strictly ordered:

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

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