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. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          U11#_A(x1,x2) = max{x1 + 2, x2 + 6}
          tt_A = 6
          isNat#_A(x1) = max{5, x1 - 1}
          activate_A(x1) = x1
          n__s_A(x1) = max{7, x1}
          activate#_A(x1) = max{0, x1 - 2}
          n__plus_A(x1,x2) = max{x1 + 27, x2 + 25}
          plus#_A(x1,x2) = max{x1 + 21, x2 + 22}
          s_A(x1) = max{7, x1}
          isNat_A(x1) = x1 + 10
          U41#_A(x1,x2,x3) = max{x1 + 7, x2 + 22, x3 + 21}
          U42#_A(x1,x2,x3) = max{x1 + 10, x2 + 22, x3 + 21}
          |0|_A = 3
          U31#_A(x1,x2) = max{x1 + 1, x2 + 20}
          U42_A(x1,x2,x3) = max{x1 - 7, x2 + 25, x3 + 27}
          plus_A(x1,x2) = max{x1 + 27, x2 + 25}
          U12_A(x1) = max{19, x1 + 18}
          U31_A(x1,x2) = max{27, x1 + 17, x2 + 1}
          U41_A(x1,x2,x3) = max{x1 + 3, x2 + 25, x3 + 27}
          U11_A(x1,x2) = max{x1 + 18, x2 + 35}
          U21_A(x1) = 6
          n__0_A = 3
    
      precedence:
      
        n__plus = plus > isNat = U12 = U41 = U11 > U42 > isNat# > activate > s = U31 > |0| > U21 = n__0 > tt > U11# > plus# = U41# = U42# > n__s = activate# = U31#
      
      partial status:
    
        pi(U11#) = [1]
        pi(tt) = []
        pi(isNat#) = []
        pi(activate) = [1]
        pi(n__s) = []
        pi(activate#) = []
        pi(n__plus) = []
        pi(plus#) = []
        pi(s) = []
        pi(isNat) = [1]
        pi(U41#) = []
        pi(U42#) = []
        pi(|0|) = []
        pi(U31#) = [1, 2]
        pi(U42) = [2]
        pi(plus) = []
        pi(U12) = []
        pi(U31) = [2]
        pi(U41) = [3]
        pi(U11) = []
        pi(U21) = []
        pi(n__0) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          U11#_A(x1,x2) = max{52, x1}
          tt_A = 79
          isNat#_A(x1) = 52
          activate_A(x1) = 66
          n__s_A(x1) = 5
          activate#_A(x1) = 57
          n__plus_A(x1,x2) = 44
          plus#_A(x1,x2) = 55
          s_A(x1) = 6
          isNat_A(x1) = 52
          U41#_A(x1,x2,x3) = 55
          U42#_A(x1,x2,x3) = 55
          |0|_A = 65
          U31#_A(x1,x2) = x1 + 2
          U42_A(x1,x2,x3) = 146
          plus_A(x1,x2) = 147
          U12_A(x1) = 80
          U31_A(x1,x2) = x2 + 67
          U41_A(x1,x2,x3) = 147
          U11_A(x1,x2) = 51
          U21_A(x1) = 78
          n__0_A = 0
    
      precedence:
      
        |0| = U31 > activate = U42 = U12 > U11 = n__0 > s > plus > U41 > plus# = U41# = U42# > isNat# > U11# = U21 > activate# = U31# > n__plus > isNat > tt > n__s
      
      partial status:
    
        pi(U11#) = [1]
        pi(tt) = []
        pi(isNat#) = []
        pi(activate) = []
        pi(n__s) = []
        pi(activate#) = []
        pi(n__plus) = []
        pi(plus#) = []
        pi(s) = []
        pi(isNat) = []
        pi(U41#) = []
        pi(U42#) = []
        pi(|0|) = []
        pi(U31#) = [1]
        pi(U42) = []
        pi(plus) = []
        pi(U12) = []
        pi(U31) = [2]
        pi(U41) = []
        pi(U11) = []
        pi(U21) = []
        pi(n__0) = []
    

The next rules are strictly ordered:

  p1, p2, p6, p7, p9, p10, p12, p13, p15, p16, p17, p19, p20, p22

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)) -> isNat#(activate(V1))
p6: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p7: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p8: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p9: plus#(N,|0|()) -> U31#(isNat(N),N)
p10: 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 estimated dependency graph contains the following SCCs:

  {p6, p7, p8}
  {p1, p2, p3}
  {p4, p5}


-- Reduction pair.

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

p1: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p2: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p3: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),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. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            U42#_A(x1,x2,x3) = x1 + ((1,0),(1,1)) x2 + x3 + (1,14)
            tt_A() = (3,0)
            plus#_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (4,1)
            activate_A(x1) = x1 + (0,6)
            s_A(x1) = x1 + (30,0)
            U41#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + x3 + (34,30)
            isNat_A(x1) = (33,3)
            U42_A(x1,x2,x3) = ((1,0),(1,0)) x2 + x3 + (32,14)
            plus_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (2,7)
            U12_A(x1) = (4,1)
            U31_A(x1,x2) = ((1,0),(0,0)) x2 + (35,5)
            U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + x3 + (32,36)
            U11_A(x1,x2) = (5,2)
            U21_A(x1) = (31,1)
            |0|_A() = (34,4)
            n__0_A() = (34,3)
            n__plus_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (2,7)
            n__s_A(x1) = x1 + (30,0)
    
      precedence:
      
        isNat = U12 = U11 = U21 > tt > activate = |0| > U31 > plus = n__plus > U41 > U42 > s > n__s > n__0 > U42# > U41# > plus#
      
      partial status:
    
        pi(U42#) = [1, 2, 3]
        pi(tt) = []
        pi(plus#) = [1, 2]
        pi(activate) = [1]
        pi(s) = [1]
        pi(U41#) = [2]
        pi(isNat) = []
        pi(U42) = [3]
        pi(plus) = []
        pi(U12) = []
        pi(U31) = []
        pi(U41) = [3]
        pi(U11) = []
        pi(U21) = []
        pi(|0|) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(n__s) = [1]
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            U42#_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3
            tt_A() = (14,6)
            plus#_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (7,4)
            activate_A(x1) = x1 + (3,0)
            s_A(x1) = x1 + (1,2)
            U41#_A(x1,x2,x3) = (7,3)
            isNat_A(x1) = (0,0)
            U42_A(x1,x2,x3) = (7,5)
            plus_A(x1,x2) = (5,2)
            U12_A(x1) = (14,7)
            U31_A(x1,x2) = (0,0)
            U41_A(x1,x2,x3) = (2,1)
            U11_A(x1,x2) = (0,0)
            U21_A(x1) = (0,0)
            |0|_A() = (1,1)
            n__0_A() = (1,1)
            n__plus_A(x1,x2) = (4,3)
            n__s_A(x1) = ((1,0),(0,0)) x1 + (5,1)
    
      precedence:
      
        n__s > isNat = plus > U11 > U42 > n__plus > U31 > |0| = n__0 > U12 > U42# = plus# = activate = s = U41# = U41 > tt > U21
      
      partial status:
    
        pi(U42#) = []
        pi(tt) = []
        pi(plus#) = [1]
        pi(activate) = []
        pi(s) = []
        pi(U41#) = []
        pi(isNat) = []
        pi(U42) = []
        pi(plus) = []
        pi(U12) = []
        pi(U31) = []
        pi(U41) = []
        pi(U11) = []
        pi(U21) = []
        pi(|0|) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(n__s) = []
    

The next rules are strictly ordered:

  p1, p2, p3

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(),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

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          activate#_A(x1) = max{2, x1 + 1}
          n__s_A(x1) = max{1, x1}
          n__plus_A(x1,x2) = max{1, x1, x2}
    
      precedence:
      
        activate# = n__s = n__plus
      
      partial status:
    
        pi(activate#) = [1]
        pi(n__s) = [1]
        pi(n__plus) = [1, 2]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          activate#_A(x1) = x1 + 2
          n__s_A(x1) = x1 + 1
          n__plus_A(x1,x2) = max{x1 + 1, x2 + 1}
    
      precedence:
      
        activate# = n__s = n__plus
      
      partial status:
    
        pi(activate#) = []
        pi(n__s) = [1]
        pi(n__plus) = [2]
    

The next rules are strictly ordered:

  p1, p2, p3

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

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. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          isNat#_A(x1) = x1
          n__s_A(x1) = x1
          activate_A(x1) = x1
          n__plus_A(x1,x2) = max{x1 + 12, x2 + 4}
          U12_A(x1) = max{0, x1 - 1}
          tt_A = 0
          U11_A(x1,x2) = max{0, x1 - 2}
          isNat_A(x1) = 0
          U21_A(x1) = x1
          U42_A(x1,x2,x3) = max{x2 + 4, x3 + 12}
          s_A(x1) = x1
          plus_A(x1,x2) = max{x1 + 12, x2 + 4}
          U31_A(x1,x2) = max{x1 + 7, x2 + 1}
          U41_A(x1,x2,x3) = max{x1 - 1, x2 + 4, x3 + 12}
          n__0_A = 1
          |0|_A = 1
    
      precedence:
      
        isNat > n__plus = plus > U21 > n__0 = |0| > U31 > activate = U11 > U12 > tt = U41 > U42 > isNat# = n__s = s
      
      partial status:
    
        pi(isNat#) = []
        pi(n__s) = []
        pi(activate) = [1]
        pi(n__plus) = []
        pi(U12) = []
        pi(tt) = []
        pi(U11) = []
        pi(isNat) = []
        pi(U21) = []
        pi(U42) = [3]
        pi(s) = []
        pi(plus) = []
        pi(U31) = [1, 2]
        pi(U41) = []
        pi(n__0) = []
        pi(|0|) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          isNat#_A(x1) = 26
          n__s_A(x1) = 14
          activate_A(x1) = 25
          n__plus_A(x1,x2) = 25
          U12_A(x1) = 34
          tt_A = 37
          U11_A(x1,x2) = 35
          isNat_A(x1) = 36
          U21_A(x1) = 35
          U42_A(x1,x2,x3) = 25
          s_A(x1) = 14
          plus_A(x1,x2) = 25
          U31_A(x1,x2) = max{35, x1 - 1, x2 + 26}
          U41_A(x1,x2,x3) = 25
          n__0_A = 26
          |0|_A = 26
    
      precedence:
      
        isNat# = U21 = U31 > activate = s = plus = n__0 = |0| > U41 > U42 > n__s = n__plus = U12 = tt = U11 = isNat
      
      partial status:
    
        pi(isNat#) = []
        pi(n__s) = []
        pi(activate) = []
        pi(n__plus) = []
        pi(U12) = []
        pi(tt) = []
        pi(U11) = []
        pi(isNat) = []
        pi(U21) = []
        pi(U42) = []
        pi(s) = []
        pi(plus) = []
        pi(U31) = [2]
        pi(U41) = []
        pi(n__0) = []
        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: isNat#(n__s(V1)) -> isNat#(activate(V1))

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:

  {p1}


-- 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(),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. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            isNat#_A(x1) = ((1,0),(1,1)) x1 + (0,2)
            n__s_A(x1) = ((1,0),(0,0)) x1 + (1,5)
            activate_A(x1) = x1 + (0,4)
            U12_A(x1) = (4,2)
            tt_A() = (3,1)
            U11_A(x1,x2) = (5,3)
            isNat_A(x1) = x1 + (2,10)
            U21_A(x1) = ((1,0),(0,0)) x1 + (0,2)
            U42_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (7,10)
            s_A(x1) = ((1,0),(0,0)) x1 + (1,9)
            plus_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (6,11)
            U31_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (4,2)
            U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (7,11)
            n__0_A() = (3,0)
            n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (6,11)
            |0|_A() = (3,0)
    
      precedence:
      
        isNat# = n__s > activate = U12 = tt = U11 = isNat = U21 = U31 = n__0 = |0| > U42 = plus = U41 = n__plus > s
      
      partial status:
    
        pi(isNat#) = [1]
        pi(n__s) = []
        pi(activate) = [1]
        pi(U12) = []
        pi(tt) = []
        pi(U11) = []
        pi(isNat) = []
        pi(U21) = []
        pi(U42) = []
        pi(s) = []
        pi(plus) = [2]
        pi(U31) = [2]
        pi(U41) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(|0|) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            isNat#_A(x1) = (1,1)
            n__s_A(x1) = (3,2)
            activate_A(x1) = x1 + (2,3)
            U12_A(x1) = (7,6)
            tt_A() = (6,5)
            U11_A(x1,x2) = (8,7)
            isNat_A(x1) = (9,8)
            U21_A(x1) = (8,9)
            U42_A(x1,x2,x3) = (5,4)
            s_A(x1) = (4,3)
            plus_A(x1,x2) = (3,0)
            U31_A(x1,x2) = x2 + (4,4)
            U41_A(x1,x2,x3) = (9,9)
            n__0_A() = (0,0)
            n__plus_A(x1,x2) = (3,0)
            |0|_A() = (1,0)
    
      precedence:
      
        activate = U11 = U42 = |0| > tt = n__0 > U41 > s > isNat# = n__s = U12 = isNat = U21 = plus = U31 = n__plus
      
      partial status:
    
        pi(isNat#) = []
        pi(n__s) = []
        pi(activate) = []
        pi(U12) = []
        pi(tt) = []
        pi(U11) = []
        pi(isNat) = []
        pi(U21) = []
        pi(U42) = []
        pi(s) = []
        pi(plus) = []
        pi(U31) = []
        pi(U41) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(|0|) = []
    

The next rules are strictly ordered:

  p1

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