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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2) = ((1,0),(1,1)) x2 + (3,17)
          tt_A() = (1,27)
          activate#_A(x1) = ((1,0),(1,1)) x1 + (2,16)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (2,6)
          n__isNat_A(x1) = ((1,0),(0,0)) x1 + (5,20)
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (6,19)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (14,6)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (15,7)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,7)
          s_A(x1) = ((1,0),(0,0)) x1 + (2,14)
          and#_A(x1,x2) = ((1,0),(0,0)) x2 + (10,20)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (5,29)
          U21#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(0,0)) x3 + (16,15)
          and_A(x1,x2) = x2 + (13,8)
          |0|_A() = (0,11)
          U11_A(x1,x2) = ((1,0),(0,0)) x2 + (1,8)
          U21_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (16,15)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (14,19)
          n__0_A() = (0,11)
  
    precedence:
    
      |0| = n__0 > n__isNat = isNat > U21# > U11# = activate# = isNat# = plus# = and# > and > n__plus = activate = s = U21 = plus > tt = n__s = U11
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__s) = []
      pi(n__isNat) = []
      pi(isNat#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = [1]
      pi(s) = []
      pi(and#) = []
      pi(isNat) = []
      pi(U21#) = [2]
      pi(and) = [2]
      pi(|0|) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = [2]
      pi(n__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: 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)) -> 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)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p13: and#(tt(),X) -> activate#(X)
p14: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p15: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p16: U21#(tt(),M,N) -> activate#(M)
p17: U21#(tt(),M,N) -> activate#(N)
p18: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p19: plus#(N,|0|()) -> isNat#(N)
p20: 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:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20}


-- 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#(activate(X1),activate(X2))
p3: plus#(N,|0|()) -> U11#(isNat(N),N)
p4: plus#(N,|0|()) -> isNat#(N)
p5: isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p6: and#(tt(),X) -> activate#(X)
p7: activate#(n__plus(X1,X2)) -> activate#(X1)
p8: activate#(n__plus(X1,X2)) -> activate#(X2)
p9: activate#(n__isNat(X)) -> isNat#(X)
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)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p16: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p17: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p18: plus#(N,s(M)) -> isNat#(M)
p19: U21#(tt(),M,N) -> activate#(N)
p20: 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(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (7,75)
          tt_A() = (5,43)
          activate#_A(x1) = ((1,0),(0,0)) x1 + (6,51)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (25,0)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (4,34)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,15)
          |0|_A() = (4,45)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (5,44)
          isNat#_A(x1) = ((1,0),(1,1)) x1 + (7,28)
          and#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (2,47)
          n__isNat_A(x1) = ((1,0),(0,0)) x1 + (5,27)
          n__s_A(x1) = x1 + (13,1)
          s_A(x1) = x1 + (13,11)
          U21#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(0,0)) x3 + (12,50)
          and_A(x1,x2) = x2 + (7,16)
          U11_A(x1,x2) = ((1,0),(0,0)) x2 + (5,16)
          U21_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (38,25)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (25,13)
          n__0_A() = (4,44)
  
    precedence:
    
      n__plus > and > U11# = activate# = plus# = and# = U21# > activate = isNat = s = U11 = plus > tt > n__s > U21 > |0| = n__0 > isNat# > n__isNat
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(|0|) = []
      pi(isNat) = []
      pi(isNat#) = [1]
      pi(and#) = []
      pi(n__isNat) = []
      pi(n__s) = []
      pi(s) = []
      pi(U21#) = [2]
      pi(and) = [2]
      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#(activate(X1),activate(X2))
p3: plus#(N,|0|()) -> U11#(isNat(N),N)
p4: plus#(N,|0|()) -> isNat#(N)
p5: isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p6: and#(tt(),X) -> activate#(X)
p7: activate#(n__plus(X1,X2)) -> activate#(X1)
p8: activate#(n__plus(X1,X2)) -> activate#(X2)
p9: activate#(n__isNat(X)) -> isNat#(X)
p10: isNat#(n__plus(V1,V2)) -> activate#(V1)
p11: isNat#(n__plus(V1,V2)) -> activate#(V2)
p12: isNat#(n__s(V1)) -> isNat#(activate(V1))
p13: isNat#(n__s(V1)) -> activate#(V1)
p14: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p15: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p16: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p17: plus#(N,s(M)) -> isNat#(M)
p18: U21#(tt(),M,N) -> activate#(N)
p19: 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(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, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19}


-- 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)) -> 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)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p12: and#(tt(),X) -> activate#(X)
p13: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p14: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p15: U21#(tt(),M,N) -> activate#(M)
p16: U21#(tt(),M,N) -> activate#(N)
p17: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p18: plus#(N,|0|()) -> isNat#(N)
p19: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2) = ((1,0),(1,1)) x2 + (16,20)
          tt_A() = (14,1)
          activate#_A(x1) = x1 + (15,17)
          n__isNat_A(x1) = ((1,0),(1,0)) x1 + (7,4)
          isNat#_A(x1) = x1 + (16,20)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (6,1)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (16,0)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (27,1)
          activate_A(x1) = x1 + (0,15)
          s_A(x1) = ((1,0),(0,0)) x1 + (6,12)
          and#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (18,18)
          isNat_A(x1) = ((1,0),(1,0)) x1 + (7,11)
          U21#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + ((1,0),(0,0)) x3 + (28,12)
          and_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (1,2)
          |0|_A() = (15,3)
          U11_A(x1,x2) = ((1,0),(0,0)) x2 + (16,1)
          U21_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (22,13)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (16,14)
          n__0_A() = (15,2)
  
    precedence:
    
      n__plus = U21# = |0| = plus = n__0 > tt = activate# = n__isNat = isNat# = n__s = plus# = activate = s = and# = isNat = and = U11 = U21 > U11#
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__isNat) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(s) = []
      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:

  p1

We remove them from the problem.

-- SCC decomposition.

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

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

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = ((1,0),(1,1)) x1 + (12,2)
          n__isNat_A(x1) = ((1,0),(0,0)) x1
          isNat#_A(x1) = x1 + (1,4)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (18,0)
          and#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (13,3)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (0,5)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,6)
          tt_A() = (0,1)
          plus#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (2,15)
          |0|_A() = (0,6)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (14,11)
          s_A(x1) = ((1,0),(0,0)) x1 + (14,12)
          U21#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,1)) x3 + (15,13)
          and_A(x1,x2) = ((1,0),(1,1)) x2 + (17,7)
          U11_A(x1,x2) = ((1,0),(0,0)) x2 + (1,0)
          U21_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (32,13)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (18,0)
          n__0_A() = (0,2)
  
    precedence:
    
      n__isNat = n__plus = isNat = and = U11 = plus > activate > plus# = U21# > U21 > tt > and# = n__s = s > n__0 > isNat# > activate# = |0|
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__isNat) = []
      pi(isNat#) = []
      pi(n__plus) = []
      pi(and#) = [2]
      pi(isNat) = []
      pi(activate) = [1]
      pi(tt) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(n__s) = []
      pi(s) = []
      pi(U21#) = []
      pi(and) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(n__0) = []

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

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (16,2)
          n__plus_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (18,4)
          and#_A(x1,x2) = ((1,0),(0,0)) x2 + (6,2)
          isNat_A(x1) = ((0,0),(1,0)) x1 + (14,34)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,3)
          n__isNat_A(x1) = ((0,0),(1,0)) x1 + (14,34)
          tt_A() = (4,12)
          activate#_A(x1) = ((1,0),(0,0)) x1 + (5,8)
          plus#_A(x1,x2) = x1 + x2 + (2,1)
          s_A(x1) = ((1,0),(0,0)) x1 + (30,0)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (30,0)
          U21#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (31,5)
          and_A(x1,x2) = ((1,0),(1,1)) x2 + (0,3)
          |0|_A() = (15,13)
          U11_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2
          U21_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (48,34)
          plus_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (18,5)
          n__0_A() = (15,13)
  
    precedence:
    
      s = n__s = |0| = U21 = n__0 > isNat# = and# = isNat > activate = and > n__plus = plus# = U21# = U11 = plus > tt = activate# > n__isNat
    
    partial status:
  
      pi(isNat#) = []
      pi(n__plus) = []
      pi(and#) = []
      pi(isNat) = []
      pi(activate) = [1]
      pi(n__isNat) = []
      pi(tt) = []
      pi(activate#) = []
      pi(plus#) = [2]
      pi(s) = []
      pi(n__s) = []
      pi(U21#) = []
      pi(and) = [2]
      pi(|0|) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = [1, 2]
      pi(n__0) = []

The next rules are strictly ordered:

  p14

We remove them from the problem.

-- SCC decomposition.

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

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

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15}


-- Reduction pair.

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

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

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (1,15)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (16,10)
          and#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (8,1)
          isNat_A(x1) = x1 + (4,11)
          activate_A(x1) = ((1,0),(1,1)) x1
          n__isNat_A(x1) = x1 + (4,8)
          tt_A() = (14,12)
          activate#_A(x1) = ((1,0),(0,0)) x1 + (17,14)
          plus#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (15,13)
          |0|_A() = (11,14)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (18,0)
          s_A(x1) = ((1,0),(0,0)) x1 + (18,0)
          U21#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (19,1)
          and_A(x1,x2) = x1 + x2 + (1,1)
          U11_A(x1,x2) = ((1,0),(1,1)) x2 + (1,15)
          U21_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (34,12)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (16,11)
          n__0_A() = (11,13)
  
    precedence:
    
      n__plus = isNat = activate = n__isNat = |0| = s = and = U11 = U21 = plus = n__0 > n__s > tt > isNat# = and# = activate# = plus# = U21#
    
    partial status:
  
      pi(isNat#) = []
      pi(n__plus) = []
      pi(and#) = []
      pi(isNat) = []
      pi(activate) = []
      pi(n__isNat) = []
      pi(tt) = []
      pi(activate#) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(n__s) = []
      pi(s) = []
      pi(U21#) = []
      pi(and) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p9

We remove them from the problem.

-- SCC decomposition.

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

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


-- Reduction pair.

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

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

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isNat#_A(x1) = ((1,0),(1,0)) x1 + (23,2)
          n__plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (8,5)
          and#_A(x1,x2) = ((1,0),(1,0)) x2 + (15,3)
          isNat_A(x1) = ((1,0),(1,0)) x1 + (1,9)
          activate_A(x1) = x1 + (0,6)
          n__isNat_A(x1) = ((1,0),(1,0)) x1 + (1,6)
          tt_A() = (9,1)
          activate#_A(x1) = ((1,0),(1,0)) x1 + (14,2)
          plus#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (10,4)
          s_A(x1) = ((1,0),(0,0)) x1 + (13,3)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (13,1)
          U21#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (15,5)
          and_A(x1,x2) = x2 + (1,10)
          |0|_A() = (24,8)
          U11_A(x1,x2) = ((1,0),(1,0)) x2 + (1,7)
          U21_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (21,4)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (8,6)
          n__0_A() = (24,3)
  
    precedence:
    
      U11 > isNat# = isNat = activate = plus# = U21# > and > n__plus = n__isNat = s = n__s = U21 = plus > and# = tt = activate# > |0| = n__0
    
    partial status:
  
      pi(isNat#) = []
      pi(n__plus) = []
      pi(and#) = []
      pi(isNat) = []
      pi(activate) = [1]
      pi(n__isNat) = []
      pi(tt) = []
      pi(activate#) = []
      pi(plus#) = []
      pi(s) = []
      pi(n__s) = []
      pi(U21#) = []
      pi(and) = []
      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: isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p2: and#(tt(),X) -> activate#(X)
p3: activate#(n__plus(X1,X2)) -> activate#(X1)
p4: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p5: plus#(N,s(M)) -> isNat#(M)
p6: isNat#(n__s(V1)) -> activate#(V1)
p7: isNat#(n__plus(V1,V2)) -> activate#(V2)
p8: isNat#(n__plus(V1,V2)) -> activate#(V1)
p9: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p10: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p11: U21#(tt(),M,N) -> activate#(M)
p12: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p13: plus#(N,|0|()) -> isNat#(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:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13}


-- Reduction pair.

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

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

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isNat#_A(x1) = ((1,0),(1,0)) x1 + (10,35)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (8,18)
          and#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (1,1)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (8,21)
          activate_A(x1) = x1 + (0,19)
          n__isNat_A(x1) = ((1,0),(0,0)) x1 + (8,2)
          tt_A() = (14,18)
          activate#_A(x1) = ((1,0),(1,0)) x1 + (6,32)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (11,39)
          |0|_A() = (15,20)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (7,33)
          s_A(x1) = ((1,0),(0,0)) x1 + (7,34)
          U21#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (17,33)
          and_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (0,20)
          U11_A(x1,x2) = ((1,0),(0,0)) x2 + (1,21)
          U21_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (15,35)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (8,36)
          n__0_A() = (15,19)
  
    precedence:
    
      isNat# > and# = activate# = plus# = U21# > isNat = activate = |0| = and = U11 = U21 = plus > n__plus = n__isNat = n__s = s > tt = n__0
    
    partial status:
  
      pi(isNat#) = []
      pi(n__plus) = []
      pi(and#) = []
      pi(isNat) = []
      pi(activate) = []
      pi(n__isNat) = []
      pi(tt) = []
      pi(activate#) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(n__s) = []
      pi(s) = []
      pi(U21#) = []
      pi(and) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(n__0) = []

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

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          and#_A(x1,x2) = ((1,0),(0,0)) x2 + (7,1)
          tt_A() = (1,8)
          activate#_A(x1) = ((1,0),(1,0)) x1 + (6,9)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (6,0)
          plus#_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (5,4)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,3)
          s_A(x1) = ((1,0),(0,0)) x1 + (7,5)
          isNat#_A(x1) = ((1,0),(1,0)) x1 + (0,4)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (7,4)
          isNat_A(x1) = x1 + (3,2)
          n__isNat_A(x1) = x1 + (3,0)
          U21#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (8,8)
          and_A(x1,x2) = ((1,0),(1,1)) x2 + (5,5)
          |0|_A() = (0,9)
          U11_A(x1,x2) = ((1,0),(1,1)) x2 + (1,4)
          U21_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (13,6)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (6,7)
          n__0_A() = (0,9)
  
    precedence:
    
      n__plus > activate = isNat = U11 > and > |0| = n__0 > s = n__s = U21 = plus > tt > and# = activate# = plus# = isNat# = n__isNat > U21#
    
    partial status:
  
      pi(and#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = [1]
      pi(s) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(isNat) = []
      pi(n__isNat) = []
      pi(U21#) = []
      pi(and) = [2]
      pi(|0|) = []
      pi(U11) = [2]
      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: and#(tt(),X) -> activate#(X)
p2: activate#(n__plus(X1,X2)) -> activate#(X1)
p3: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p4: plus#(N,s(M)) -> isNat#(M)
p5: isNat#(n__s(V1)) -> activate#(V1)
p6: isNat#(n__plus(V1,V2)) -> activate#(V2)
p7: isNat#(n__plus(V1,V2)) -> activate#(V1)
p8: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p9: plus#(N,s(M)) -> U21#(and(isNat(M),n__isNat(N)),M,N)
p10: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p11: plus#(N,|0|()) -> isNat#(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:

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

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          and#_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,15)
          tt_A() = (0,3)
          activate#_A(x1) = x1 + (0,14)
          n__plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (11,0)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (10,13)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,2)
          |0|_A() = (3,15)
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (2,15)
          n__s_A(x1) = x1
          s_A(x1) = x1
          U21#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (10,13)
          and_A(x1,x2) = ((1,0),(1,0)) x2 + (1,3)
          isNat_A(x1) = ((1,0),(1,0)) x1 + (4,12)
          n__isNat_A(x1) = ((1,0),(1,0)) x1 + (4,7)
          U11_A(x1,x2) = ((1,0),(1,0)) x2 + (4,3)
          U21_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (11,8)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (11,8)
          n__0_A() = (3,11)
  
    precedence:
    
      n__plus = plus# = activate = n__s = s = U21# = and = isNat = n__isNat = U11 = U21 = plus > and# > tt = |0| = n__0 > isNat# > activate#
    
    partial status:
  
      pi(and#) = [2]
      pi(tt) = []
      pi(activate#) = [1]
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(|0|) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(s) = []
      pi(U21#) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__isNat) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(n__0) = []

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: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: plus#(N,|0|()) -> isNat#(N)
p3: isNat#(n__plus(V1,V2)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> activate#(X1)
p5: isNat#(n__plus(V1,V2)) -> activate#(V2)
p6: isNat#(n__s(V1)) -> 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,s(M)) -> and#(isNat(M),n__isNat(N))
p10: plus#(N,s(M)) -> isNat#(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(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, p2, p3, p4, p5, p6, p7, p8, p10}


-- Reduction pair.

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

p1: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: plus#(N,s(M)) -> isNat#(M)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> activate#(X1)
p5: isNat#(n__plus(V1,V2)) -> activate#(V2)
p6: isNat#(n__plus(V1,V2)) -> 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|()) -> isNat#(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = ((0,0),(1,0)) x1 + (6,16)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (7,15)
          plus#_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (6,15)
          activate_A(x1) = x1 + (0,6)
          s_A(x1) = ((1,0),(0,0)) x1 + (5,18)
          isNat#_A(x1) = ((0,0),(1,0)) x1 + (6,14)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (5,17)
          U21#_A(x1,x2,x3) = ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (6,19)
          and_A(x1,x2) = x2 + (0,7)
          isNat_A(x1) = x1 + (13,6)
          n__isNat_A(x1) = x1 + (13,1)
          tt_A() = (1,1)
          |0|_A() = (0,3)
          U11_A(x1,x2) = x2 + (1,21)
          U21_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (12,19)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (7,20)
          n__0_A() = (0,2)
  
    precedence:
    
      activate# = n__plus = plus# = activate = s = isNat# = and = isNat = tt = U11 = U21 = plus > n__s = |0| = n__0 > U21# > n__isNat
    
    partial status:
  
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(s) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(U21#) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__isNat) = []
      pi(tt) = []
      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: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: plus#(N,s(M)) -> isNat#(M)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> activate#(X1)
p5: isNat#(n__plus(V1,V2)) -> activate#(V2)
p6: isNat#(n__plus(V1,V2)) -> activate#(V1)
p7: U21#(tt(),M,N) -> plus#(activate(N),activate(M))
p8: plus#(N,|0|()) -> isNat#(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:

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


-- Reduction pair.

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

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

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = x1 + (1,14)
          n__plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (11,0)
          plus#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (10,2)
          activate_A(x1) = x1 + (0,11)
          |0|_A() = (0,16)
          isNat#_A(x1) = x1 + (9,2)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (8,13)
          s_A(x1) = ((1,0),(0,0)) x1 + (8,14)
          U11_A(x1,x2) = x2 + (1,12)
          tt_A() = (8,15)
          U21_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (19,7)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (11,10)
          and_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (1,1)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (9,20)
          n__0_A() = (0,16)
          n__isNat_A(x1) = ((1,0),(0,0)) x1 + (9,10)
  
    precedence:
    
      activate = |0| = U11 = U21 = plus > tt = and = isNat > n__plus = n__s = s = n__isNat > isNat# > activate# > plus# = n__0
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(|0|) = []
      pi(isNat#) = [1]
      pi(n__s) = []
      pi(s) = []
      pi(U11) = []
      pi(tt) = []
      pi(U21) = []
      pi(plus) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__0) = []
      pi(n__isNat) = []

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: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: plus#(N,|0|()) -> isNat#(N)
p3: isNat#(n__plus(V1,V2)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> activate#(X1)
p5: isNat#(n__plus(V1,V2)) -> activate#(V2)
p6: plus#(N,s(M)) -> isNat#(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(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, p2, p3, p4, p5, p6}


-- Reduction pair.

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

p1: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: plus#(N,s(M)) -> isNat#(M)
p3: isNat#(n__plus(V1,V2)) -> activate#(V2)
p4: activate#(n__plus(X1,X2)) -> activate#(X1)
p5: isNat#(n__plus(V1,V2)) -> activate#(V1)
p6: plus#(N,|0|()) -> isNat#(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = ((0,0),(1,0)) x1 + (0,11)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (10,10)
          plus#_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (0,3)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,6)
          s_A(x1) = ((1,0),(0,0)) x1 + (5,10)
          isNat#_A(x1) = ((0,0),(1,0)) x1 + (0,2)
          |0|_A() = (0,1)
          U11_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,0)
          tt_A() = (1,0)
          U21_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (15,11)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (10,11)
          and_A(x1,x2) = ((1,0),(1,1)) x2 + (0,7)
          isNat_A(x1) = ((0,0),(1,0)) x1 + (2,0)
          n__0_A() = (0,0)
          n__isNat_A(x1) = ((0,0),(1,0)) x1 + (2,0)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (5,0)
  
    precedence:
    
      s = tt = U21 = isNat = n__isNat > and > n__plus = activate = |0| = U11 = plus > plus# = isNat# = n__0 > activate# = n__s
    
    partial status:
  
      pi(activate#) = []
      pi(n__plus) = [1]
      pi(plus#) = []
      pi(activate) = [1]
      pi(s) = []
      pi(isNat#) = []
      pi(|0|) = []
      pi(U11) = [2]
      pi(tt) = []
      pi(U21) = []
      pi(plus) = [1]
      pi(and) = []
      pi(isNat) = []
      pi(n__0) = []
      pi(n__isNat) = []
      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: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: plus#(N,s(M)) -> isNat#(M)
p3: activate#(n__plus(X1,X2)) -> activate#(X1)
p4: isNat#(n__plus(V1,V2)) -> activate#(V1)
p5: plus#(N,|0|()) -> isNat#(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:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: plus#(N,|0|()) -> isNat#(N)
p3: isNat#(n__plus(V1,V2)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> activate#(X1)
p5: plus#(N,s(M)) -> isNat#(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(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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = x1 + (0,5)
          n__plus_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (12,6)
          plus#_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (2,1)
          activate_A(x1) = ((1,0),(1,1)) x1
          |0|_A() = (2,3)
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (3,4)
          s_A(x1) = ((1,0),(0,0)) x1 + (4,3)
          U11_A(x1,x2) = ((1,0),(0,0)) x2 + (3,4)
          tt_A() = (0,1)
          U21_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (16,4)
          plus_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (12,7)
          and_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (1,0)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (5,7)
          n__0_A() = (2,2)
          n__isNat_A(x1) = ((1,0),(0,0)) x1 + (5,2)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (4,0)
  
    precedence:
    
      n__plus = activate = tt = and = isNat > U11 > U21 = plus > plus# = isNat# > activate# > |0| = s = n__s > n__0 = n__isNat
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(|0|) = []
      pi(isNat#) = []
      pi(s) = []
      pi(U11) = []
      pi(tt) = []
      pi(U21) = []
      pi(plus) = []
      pi(and) = []
      pi(isNat) = []
      pi(n__0) = []
      pi(n__isNat) = []
      pi(n__s) = []

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: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: isNat#(n__plus(V1,V2)) -> activate#(V1)
p3: activate#(n__plus(X1,X2)) -> activate#(X1)
p4: plus#(N,s(M)) -> isNat#(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(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, p2, p3, p4}


-- Reduction pair.

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

p1: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: plus#(N,s(M)) -> isNat#(M)
p3: isNat#(n__plus(V1,V2)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> activate#(X1)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          activate#_A(x1) = x1 + (1,5)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (9,1)
          plus#_A(x1,x2) = ((1,0),(0,0)) x2 + (1,2)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,7)
          s_A(x1) = ((1,0),(0,0)) x1 + (8,12)
          isNat#_A(x1) = x1 + (1,3)
          U11_A(x1,x2) = ((1,0),(1,1)) x2 + (7,8)
          tt_A() = (12,10)
          U21_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (17,1)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (9,2)
          and_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (3,1)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (7,11)
          n__0_A() = (6,1)
          n__isNat_A(x1) = ((1,0),(0,0)) x1 + (7,2)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (8,1)
          |0|_A() = (6,12)
  
    precedence:
    
      activate = U21 = plus > plus# = isNat# = U11 > s = tt = and = isNat > activate# = n__plus = n__0 = n__isNat = n__s = |0|
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__plus) = [1]
      pi(plus#) = []
      pi(activate) = [1]
      pi(s) = []
      pi(isNat#) = []
      pi(U11) = [2]
      pi(tt) = []
      pi(U21) = []
      pi(plus) = [1, 2]
      pi(and) = []
      pi(isNat) = []
      pi(n__0) = []
      pi(n__isNat) = []
      pi(n__s) = []
      pi(|0|) = []

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: plus#(N,s(M)) -> isNat#(M)
p2: isNat#(n__plus(V1,V2)) -> activate#(V1)
p3: activate#(n__plus(X1,X2)) -> activate#(X1)

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:

  {p3}


-- Reduction pair.

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

p1: activate#(n__plus(X1,X2)) -> activate#(X1)

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 monotone reduction pair:

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

The next rules are strictly ordered:

  p1

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