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(X1,X2)
  activate(n__isNat(X)) -> isNat(X)
  activate(n__s(X)) -> s(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#(X1,X2)
p20: activate#(n__isNat(X)) -> isNat#(X)
p21: activate#(n__s(X)) -> s#(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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(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}


-- Reduction pair.

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

p1: U11#(tt(),N) -> activate#(N)
p2: activate#(n__isNat(X)) -> isNat#(X)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,s(M)) -> isNat#(M)
p6: isNat#(n__s(V1)) -> isNat#(activate(V1))
p7: isNat#(n__plus(V1,V2)) -> activate#(V2)
p8: isNat#(n__plus(V1,V2)) -> activate#(V1)
p9: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p10: isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p11: and#(tt(),X) -> activate#(X)
p12: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p13: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p14: U21#(tt(),M,N) -> activate#(M)
p15: U21#(tt(),M,N) -> activate#(N)
p16: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p17: plus#(N,|0|()) -> isNat#(N)
p18: 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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          U11#_A(x1,x2) = ((1,1),(0,0)) x2 + (2,0)
          tt_A() = (3,1)
          activate#_A(x1) = ((1,1),(0,0)) x1 + (2,0)
          n__isNat_A(x1) = ((0,1),(1,0)) x1 + (0,1)
          isNat#_A(x1) = ((1,1),(0,0)) x1 + (2,0)
          n__s_A(x1) = x1
          n__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(0,1)) x2
          plus#_A(x1,x2) = ((1,1),(0,0)) x1 + ((1,1),(0,0)) x2 + (2,0)
          s_A(x1) = x1
          activate_A(x1) = x1
          and#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,1),(0,0)) x2 + (1,0)
          isNat_A(x1) = ((0,1),(1,0)) x1 + (0,1)
          U21#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (2,0)
          and_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2
          |0|_A() = (0,4)
          U11_A(x1,x2) = x2 + (4,4)
          U21_A(x1,x2,x3) = ((1,1),(0,1)) x2 + ((1,1),(0,1)) x3
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(0,1)) x2
          n__0_A() = (0,4)
  
    precedence:
    
      U11# = tt = activate# = n__isNat = isNat# = n__s = n__plus = plus# = s = activate = and# = isNat = U21# = and = |0| = U11 = U21 = plus = n__0
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__isNat) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(s) = []
      pi(activate) = []
      pi(and#) = []
      pi(isNat) = []
      pi(U21#) = []
      pi(and) = []
      pi(|0|) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p17

We remove them from the problem.

-- SCC decomposition.

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

p1: U11#(tt(),N) -> activate#(N)
p2: activate#(n__isNat(X)) -> isNat#(X)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,s(M)) -> isNat#(M)
p6: isNat#(n__s(V1)) -> isNat#(activate(V1))
p7: isNat#(n__plus(V1,V2)) -> activate#(V2)
p8: isNat#(n__plus(V1,V2)) -> activate#(V1)
p9: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p10: isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p11: and#(tt(),X) -> activate#(X)
p12: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p13: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p14: U21#(tt(),M,N) -> activate#(M)
p15: U21#(tt(),M,N) -> activate#(N)
p16: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p17: 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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(X)
r17: 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}


-- Reduction pair.

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

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

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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          U11#_A(x1,x2) = ((0,1),(0,0)) x2 + (21,10)
          tt_A() = (57,20)
          activate#_A(x1) = ((0,1),(0,0)) x1 + (21,10)
          n__plus_A(x1,x2) = x1 + x2 + (43,0)
          plus#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (3,10)
          |0|_A() = (1,18)
          isNat_A(x1) = ((0,1),(0,1)) x1 + (41,11)
          s_A(x1) = ((0,0),(0,1)) x1 + (25,20)
          U21#_A(x1,x2,x3) = ((0,1),(0,0)) x2 + ((0,1),(0,0)) x3 + (22,10)
          and_A(x1,x2) = ((1,1),(0,1)) x2 + (6,0)
          n__isNat_A(x1) = ((0,0),(0,1)) x1 + (24,11)
          activate_A(x1) = ((1,1),(0,1)) x1 + (6,0)
          and#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (1,10)
          isNat#_A(x1) = ((0,1),(0,0)) x1 + (23,10)
          n__s_A(x1) = ((0,0),(0,1)) x1 + (0,20)
          U11_A(x1,x2) = ((1,1),(0,1)) x2 + (7,18)
          U21_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(0,1)) x2 + ((0,0),(0,1)) x3 + (26,20)
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + x2 + (43,0)
          n__0_A() = (0,18)
  
    precedence:
    
      |0| = n__0 > U21# > U11# = activate# = plus# = and# = isNat# > tt = isNat = and = activate > U11 > U21 = plus > s = n__s > n__isNat > n__plus
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(isNat) = []
      pi(s) = []
      pi(U21#) = []
      pi(and) = []
      pi(n__isNat) = []
      pi(activate) = []
      pi(and#) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      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: U11#(tt(),N) -> activate#(N)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,|0|()) -> U11#(isNat(N),N)
p4: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p5: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p6: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p7: and#(tt(),X) -> activate#(X)
p8: activate#(n__isNat(X)) -> isNat#(X)
p9: isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p10: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p11: isNat#(n__plus(V1,V2)) -> activate#(V1)
p12: isNat#(n__plus(V1,V2)) -> activate#(V2)
p13: isNat#(n__s(V1)) -> isNat#(activate(V1))
p14: isNat#(n__s(V1)) -> activate#(V1)
p15: plus#(N,s(M)) -> isNat#(M)
p16: U21#(tt(),M,N) -> activate#(M)

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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(X)
r17: 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}


-- Reduction pair.

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

p1: U11#(tt(),N) -> activate#(N)
p2: activate#(n__isNat(X)) -> isNat#(X)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,s(M)) -> isNat#(M)
p6: isNat#(n__s(V1)) -> isNat#(activate(V1))
p7: isNat#(n__plus(V1,V2)) -> activate#(V2)
p8: isNat#(n__plus(V1,V2)) -> activate#(V1)
p9: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p10: isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p11: and#(tt(),X) -> activate#(X)
p12: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p13: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p14: U21#(tt(),M,N) -> activate#(M)
p15: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p16: 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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          U11#_A(x1,x2) = ((1,1),(0,0)) x2 + (4,2)
          tt_A() = (0,1)
          activate#_A(x1) = ((0,1),(0,0)) x1 + (3,2)
          n__isNat_A(x1) = ((1,1),(1,1)) x1 + (8,2)
          isNat#_A(x1) = ((0,1),(0,0)) x1 + (4,2)
          n__s_A(x1) = x1 + (2,3)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (0,4)
          plus#_A(x1,x2) = ((1,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (3,2)
          s_A(x1) = x1 + (2,4)
          activate_A(x1) = x1 + (0,1)
          and#_A(x1,x2) = ((0,1),(0,0)) x2 + (4,2)
          isNat_A(x1) = ((1,1),(1,1)) x1 + (8,2)
          U21#_A(x1,x2,x3) = ((0,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (6,2)
          and_A(x1,x2) = ((0,1),(0,0)) x1 + x2 + (0,2)
          |0|_A() = (1,2)
          U11_A(x1,x2) = ((1,0),(1,1)) x2 + (1,1)
          U21_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,10)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (0,4)
          n__0_A() = (1,1)
  
    precedence:
    
      activate = isNat = and = U11 = plus > tt > s = U21 > |0| > n__0 > n__s > U11# = activate# = n__isNat = isNat# = n__plus = plus# = and# = U21#
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__isNat) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(s) = []
      pi(activate) = []
      pi(and#) = []
      pi(isNat) = []
      pi(U21#) = []
      pi(and) = []
      pi(|0|) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      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: U11#(tt(),N) -> activate#(N)
p2: activate#(n__isNat(X)) -> isNat#(X)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,s(M)) -> isNat#(M)
p6: isNat#(n__s(V1)) -> isNat#(activate(V1))
p7: isNat#(n__plus(V1,V2)) -> activate#(V2)
p8: isNat#(n__plus(V1,V2)) -> activate#(V1)
p9: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p10: isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p11: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p12: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p13: U21#(tt(),M,N) -> activate#(M)
p14: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p15: 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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(X)
r17: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p12, p13, p14, p15}


-- Reduction pair.

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

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

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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          U11#_A(x1,x2) = ((1,0),(0,0)) x2
          tt_A() = (0,0)
          activate#_A(x1) = ((1,0),(0,0)) x1
          n__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(1,0)) x2 + (2,1)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2
          |0|_A() = (0,0)
          isNat_A(x1) = ((1,0),(0,0)) x1
          s_A(x1) = x1
          U21#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3
          and_A(x1,x2) = x2
          n__isNat_A(x1) = ((1,0),(0,0)) x1
          activate_A(x1) = x1
          isNat#_A(x1) = ((1,0),(0,0)) x1
          n__s_A(x1) = x1
          U11_A(x1,x2) = ((1,1),(0,1)) x2 + (1,1)
          U21_A(x1,x2,x3) = ((1,1),(1,0)) x2 + ((1,1),(0,1)) x3 + (2,1)
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(1,0)) x2 + (2,1)
          n__0_A() = (0,0)
  
    precedence:
    
      U11# = tt = activate# = n__plus = plus# = |0| = isNat = s = U21# = and = n__isNat = activate = isNat# = n__s = U11 = U21 = plus = n__0
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(isNat) = []
      pi(s) = []
      pi(U21#) = []
      pi(and) = []
      pi(n__isNat) = []
      pi(activate) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p10

We remove them from the problem.

-- SCC decomposition.

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

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

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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(X)
r17: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12}


-- Reduction pair.

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

p1: U11#(tt(),N) -> activate#(N)
p2: activate#(n__isNat(X)) -> isNat#(X)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,s(M)) -> isNat#(M)
p6: isNat#(n__s(V1)) -> isNat#(activate(V1))
p7: isNat#(n__plus(V1,V2)) -> activate#(V1)
p8: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p9: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p10: U21#(tt(),M,N) -> activate#(M)
p11: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p12: 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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          U11#_A(x1,x2) = ((0,1),(1,1)) x2 + (3,1)
          tt_A() = (1,0)
          activate#_A(x1) = ((0,1),(1,1)) x1 + (3,0)
          n__isNat_A(x1) = ((0,0),(1,1)) x1 + (2,0)
          isNat#_A(x1) = ((0,1),(1,1)) x1 + (3,0)
          n__s_A(x1) = x1
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,0),(1,1)) x2 + (4,2)
          plus#_A(x1,x2) = ((0,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (3,2)
          s_A(x1) = x1
          activate_A(x1) = x1
          U21#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((0,1),(1,1)) x3 + (3,2)
          and_A(x1,x2) = ((0,0),(1,1)) x1 + x2
          isNat_A(x1) = ((0,0),(1,1)) x1 + (2,0)
          |0|_A() = (0,0)
          U11_A(x1,x2) = ((1,1),(1,1)) x2 + (1,1)
          U21_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,1),(1,1)) x3 + (4,2)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,0),(1,1)) x2 + (4,2)
          n__0_A() = (0,0)
  
    precedence:
    
      U11# = tt = activate# = n__isNat = isNat# = n__s = n__plus = plus# = s = activate = U21# = and = isNat = |0| = U11 = U21 = plus = n__0
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__isNat) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(s) = []
      pi(activate) = []
      pi(U21#) = []
      pi(and) = []
      pi(isNat) = []
      pi(|0|) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p8

We remove them from the problem.

-- SCC decomposition.

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

p1: U11#(tt(),N) -> activate#(N)
p2: activate#(n__isNat(X)) -> isNat#(X)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,s(M)) -> isNat#(M)
p6: isNat#(n__s(V1)) -> isNat#(activate(V1))
p7: isNat#(n__plus(V1,V2)) -> activate#(V1)
p8: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p9: U21#(tt(),M,N) -> activate#(M)
p10: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p11: 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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(X)
r17: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11}


-- Reduction pair.

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

p1: U11#(tt(),N) -> activate#(N)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,|0|()) -> U11#(isNat(N),N)
p4: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p5: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p6: plus#(N,s(M)) -> isNat#(M)
p7: isNat#(n__plus(V1,V2)) -> activate#(V1)
p8: activate#(n__isNat(X)) -> isNat#(X)
p9: isNat#(n__s(V1)) -> isNat#(activate(V1))
p10: isNat#(n__s(V1)) -> activate#(V1)
p11: U21#(tt(),M,N) -> activate#(M)

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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          U11#_A(x1,x2) = ((1,1),(0,0)) x2 + (9,5)
          tt_A() = (7,0)
          activate#_A(x1) = ((1,0),(0,0)) x1 + (8,5)
          n__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,0),(1,1)) x2 + (6,1)
          plus#_A(x1,x2) = ((1,1),(0,0)) x1 + ((1,0),(0,0)) x2 + (10,5)
          |0|_A() = (1,1)
          isNat_A(x1) = ((1,0),(1,0)) x1 + (8,4)
          s_A(x1) = ((1,0),(0,0)) x1
          U21#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,1),(0,0)) x3 + (10,5)
          and_A(x1,x2) = ((0,1),(0,0)) x1 + x2 + (1,2)
          n__isNat_A(x1) = ((1,0),(1,0)) x1 + (8,4)
          activate_A(x1) = x1
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (9,5)
          n__s_A(x1) = ((1,0),(0,0)) x1
          U11_A(x1,x2) = x2
          U21_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,1),(0,1)) x3 + (6,1)
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,0),(1,1)) x2 + (6,1)
          n__0_A() = (1,1)
  
    precedence:
    
      U11# = tt = activate# = n__plus = plus# = |0| = isNat = s = U21# = and = n__isNat = activate = isNat# = n__s = U11 = U21 = plus = n__0
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(isNat) = []
      pi(s) = []
      pi(U21#) = []
      pi(and) = []
      pi(n__isNat) = []
      pi(activate) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      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: U11#(tt(),N) -> activate#(N)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,|0|()) -> U11#(isNat(N),N)
p4: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p5: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p6: plus#(N,s(M)) -> isNat#(M)
p7: isNat#(n__plus(V1,V2)) -> activate#(V1)
p8: activate#(n__isNat(X)) -> isNat#(X)
p9: isNat#(n__s(V1)) -> isNat#(activate(V1))
p10: isNat#(n__s(V1)) -> 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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(X)
r17: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10}


-- Reduction pair.

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

p1: U11#(tt(),N) -> activate#(N)
p2: activate#(n__isNat(X)) -> isNat#(X)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,s(M)) -> isNat#(M)
p6: isNat#(n__s(V1)) -> isNat#(activate(V1))
p7: isNat#(n__plus(V1,V2)) -> activate#(V1)
p8: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p9: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p10: 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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          U11#_A(x1,x2) = ((0,1),(0,1)) x2 + (5,6)
          tt_A() = (4,0)
          activate#_A(x1) = ((0,1),(0,1)) x1 + (5,6)
          n__isNat_A(x1) = ((0,1),(0,1)) x1 + (7,5)
          isNat#_A(x1) = ((0,1),(0,1)) x1 + (3,3)
          n__s_A(x1) = ((0,1),(0,1)) x1 + (3,3)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(0,1)) x2 + (7,5)
          plus#_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (1,1)
          s_A(x1) = ((0,1),(0,1)) x1 + (3,3)
          activate_A(x1) = x1
          U21#_A(x1,x2,x3) = ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + (2,1)
          and_A(x1,x2) = ((0,1),(0,1)) x1 + x2
          isNat_A(x1) = ((0,1),(0,1)) x1 + (7,5)
          |0|_A() = (1,5)
          U11_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,6)
          U21_A(x1,x2,x3) = ((0,1),(0,1)) x2 + ((1,1),(1,1)) x3 + (9,8)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(0,1)) x2 + (7,5)
          n__0_A() = (1,5)
  
    precedence:
    
      U11# = tt = activate# = n__isNat = isNat# = n__s = n__plus = plus# = s = activate = U21# = and = isNat = |0| = U11 = U21 = plus = n__0
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__isNat) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(s) = []
      pi(activate) = []
      pi(U21#) = []
      pi(and) = []
      pi(isNat) = []
      pi(|0|) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      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: U11#(tt(),N) -> activate#(N)
p2: activate#(n__isNat(X)) -> isNat#(X)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,s(M)) -> isNat#(M)
p6: isNat#(n__s(V1)) -> isNat#(activate(V1))
p7: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p8: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p9: 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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(X)
r17: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9}


-- Reduction pair.

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

p1: U11#(tt(),N) -> activate#(N)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,|0|()) -> U11#(isNat(N),N)
p4: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p5: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p6: plus#(N,s(M)) -> isNat#(M)
p7: isNat#(n__s(V1)) -> isNat#(activate(V1))
p8: isNat#(n__s(V1)) -> activate#(V1)
p9: activate#(n__isNat(X)) -> isNat#(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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          U11#_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,1),(1,0)) x2 + (2,2)
          tt_A() = (26,1)
          activate#_A(x1) = ((0,1),(1,0)) x1 + (27,2)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,1),(1,1)) x2 + (26,0)
          plus#_A(x1,x2) = ((1,1),(1,0)) x1 + ((0,1),(1,0)) x2 + (26,0)
          |0|_A() = (26,0)
          isNat_A(x1) = ((1,0),(1,1)) x1 + (23,2)
          s_A(x1) = x1 + (29,15)
          U21#_A(x1,x2,x3) = ((0,1),(1,0)) x2 + ((1,1),(1,0)) x3 + (41,16)
          and_A(x1,x2) = x2 + (23,1)
          n__isNat_A(x1) = ((1,0),(1,1)) x1 + (19,1)
          activate_A(x1) = x1 + (6,1)
          isNat#_A(x1) = ((0,1),(1,0)) x1 + (12,10)
          n__s_A(x1) = x1 + (28,15)
          U11_A(x1,x2) = x2 + (27,1)
          U21_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,0),(1,1)) x3 + (70,29)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,1),(1,1)) x2 + (27,0)
          n__0_A() = (21,0)
  
    precedence:
    
      U11# = activate# = plus# = U21# = isNat# = plus > isNat = and > tt = s = activate = n__s = U21 > U11 > n__plus = |0| = n__0 > n__isNat
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(isNat) = []
      pi(s) = []
      pi(U21#) = []
      pi(and) = []
      pi(n__isNat) = []
      pi(activate) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

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

p1: U11#(tt(),N) -> activate#(N)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,|0|()) -> U11#(isNat(N),N)
p4: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p5: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p6: isNat#(n__s(V1)) -> isNat#(activate(V1))
p7: isNat#(n__s(V1)) -> activate#(V1)
p8: activate#(n__isNat(X)) -> isNat#(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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(X)
r17: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: U11#(tt(),N) -> activate#(N)
p2: activate#(n__isNat(X)) -> isNat#(X)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p6: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p7: plus#(N,|0|()) -> U11#(isNat(N),N)
p8: 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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          U11#_A(x1,x2) = ((0,1),(0,0)) x2 + (4,12)
          tt_A() = (59,1)
          activate#_A(x1) = ((0,1),(0,0)) x1 + (3,12)
          n__isNat_A(x1) = ((1,1),(1,1)) x1 + (7,13)
          isNat#_A(x1) = ((1,1),(0,0)) x1 + (15,12)
          n__s_A(x1) = ((0,1),(1,0)) x1 + (3,12)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (41,44)
          plus#_A(x1,x2) = ((0,1),(0,0)) x1 + (5,12)
          s_A(x1) = ((0,1),(1,0)) x1 + (6,12)
          U21#_A(x1,x2,x3) = ((0,1),(0,0)) x3 + (5,12)
          and_A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (1,0)
          isNat_A(x1) = ((1,1),(1,1)) x1 + (7,13)
          activate_A(x1) = x1 + (4,0)
          |0|_A() = (60,1)
          U11_A(x1,x2) = ((1,1),(0,1)) x2 + (5,2)
          U21_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (59,62)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (42,44)
          n__0_A() = (60,1)
  
    precedence:
    
      isNat# = |0| = U11 = n__0 > U21 = plus > n__isNat = and = isNat > U11# = tt = activate# = n__s = n__plus = plus# = s = U21# = activate
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__isNat) = [1]
      pi(isNat#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(s) = []
      pi(U21#) = []
      pi(and) = []
      pi(isNat) = [1]
      pi(activate) = [1]
      pi(|0|) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      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: U11#(tt(),N) -> activate#(N)
p2: activate#(n__isNat(X)) -> isNat#(X)
p3: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p5: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p6: plus#(N,|0|()) -> U11#(isNat(N),N)
p7: 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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(X)
r17: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: U11#(tt(),N) -> activate#(N)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,|0|()) -> U11#(isNat(N),N)
p4: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p5: U21#(tt(),M,N) -> plus#(activate(N),activate(M))

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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(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:

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

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: U11#(tt(),N) -> activate#(N)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p4: U21#(tt(),M,N) -> plus#(activate(N),activate(M))

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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(X)
r17: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p3, p4}


-- Reduction pair.

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

p1: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p2: U21#(tt(),M,N) -> plus#(activate(N),activate(M))

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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          plus#_A(x1,x2) = ((0,1),(0,1)) x2 + (22,1)
          s_A(x1) = ((0,1),(0,1)) x1 + (24,6)
          U21#_A(x1,x2,x3) = ((0,1),(0,1)) x2 + (23,6)
          and_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,1),(0,1)) x2 + (1,0)
          isNat_A(x1) = ((0,1),(0,1)) x1 + (26,8)
          n__isNat_A(x1) = ((0,0),(0,1)) x1 + (1,8)
          tt_A() = (21,20)
          activate_A(x1) = ((1,1),(0,1)) x1 + (20,0)
          U11_A(x1,x2) = ((1,1),(0,1)) x2 + (21,1)
          U21_A(x1,x2,x3) = ((0,1),(0,1)) x2 + ((1,1),(0,1)) x3 + (24,6)
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (19,0)
          |0|_A() = (2,12)
          n__0_A() = (1,12)
          n__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,0),(0,1)) x2 + (19,0)
          n__s_A(x1) = ((0,0),(0,1)) x1 + (1,6)
  
    precedence:
    
      and = isNat = tt = activate = U11 = plus > s = U21 = |0| = n__0 > n__isNat = n__s > n__plus > plus# > U21#
    
    partial status:
  
      pi(plus#) = []
      pi(s) = []
      pi(U21#) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__isNat) = []
      pi(tt) = []
      pi(activate) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(|0|) = []
      pi(n__0) = []
      pi(n__plus) = []
      pi(n__s) = []

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(),M,N) -> plus#(activate(N),activate(M))

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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(X)
r17: 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: 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(X1,X2)
r15: activate(n__isNat(X)) -> isNat(X)
r16: activate(n__s(X)) -> s(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          isNat#_A(x1) = ((0,1),(0,0)) x1 + (1,2)
          n__s_A(x1) = ((0,0),(0,1)) x1 + (0,1)
          activate_A(x1) = ((1,1),(0,1)) x1 + (5,0)
          U11_A(x1,x2) = ((1,1),(0,1)) x2 + (7,4)
          tt_A() = (2,0)
          U21_A(x1,x2,x3) = ((0,0),(0,1)) x2 + ((0,0),(0,1)) x3 + (4,2)
          s_A(x1) = ((0,0),(0,1)) x1 + (3,1)
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + x2 + (2,1)
          and_A(x1,x2) = ((1,1),(0,1)) x2 + (5,0)
          isNat_A(x1) = ((0,1),(0,0)) x1 + (7,3)
          n__0_A() = (1,3)
          n__plus_A(x1,x2) = x1 + x2 + (1,1)
          n__isNat_A(x1) = ((0,1),(0,0)) x1 + (0,3)
          |0|_A() = (6,3)
  
    precedence:
    
      activate = U11 = tt = plus = and = isNat = n__0 = n__plus = n__isNat > isNat# > n__s = U21 = s = |0|
    
    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.