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: standard order
        interpretations:
          U11#_A(x1,x2) = ((0,1),(0,1)) x2 + (22,18)
          tt_A() = (13,21)
          activate#_A(x1) = ((0,1),(0,1)) x1 + (22,13)
          n__s_A(x1) = ((0,0),(0,1)) x1 + (22,21)
          n__isNat_A(x1) = ((0,0),(0,1)) x1 + (25,4)
          isNat#_A(x1) = ((0,1),(0,1)) x1 + (12,17)
          n__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,0),(0,1)) x2 + (11,25)
          plus#_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (5,26)
          activate_A(x1) = ((1,1),(0,1)) x1 + (34,0)
          s_A(x1) = ((0,0),(0,1)) x1 + (24,21)
          and#_A(x1,x2) = ((0,1),(1,0)) x1 + ((0,1),(0,1)) x2 + (2,0)
          isNat_A(x1) = ((0,1),(0,1)) x1 + (38,4)
          U21#_A(x1,x2,x3) = ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + (23,26)
          and_A(x1,x2) = ((1,1),(1,1)) x2 + (34,0)
          |0|_A() = (2,17)
          U11_A(x1,x2) = ((1,1),(0,1)) x2 + (35,18)
          U21_A(x1,x2,x3) = ((0,0),(0,1)) x2 + ((0,0),(0,1)) x3 + (35,46)
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,0),(0,1)) x2 + (36,25)
          n__0_A() = (1,17)
  
    precedence:
    
      U11# = tt = activate# = n__s = n__isNat = isNat# = n__plus = plus# = activate = s = and# = isNat = U21# = and = |0| = U11 = U21 = plus = n__0
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__s) = []
      pi(n__isNat) = []
      pi(isNat#) = []
      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:

  p5

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__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#(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__isNat(X)) -> isNat#(X)
p9: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p10: isNat#(n__plus(V1,V2)) -> activate#(V1)
p11: activate#(n__s(X)) -> activate#(X)
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: standard order
        interpretations:
          U11#_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,0),(0,0)) x2 + (1,3)
          tt_A() = (1,2)
          activate#_A(x1) = ((1,0),(0,0)) x1 + (3,3)
          n__plus_A(x1,x2) = x1 + ((1,1),(0,1)) x2 + (0,4)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (3,3)
          activate_A(x1) = x1
          |0|_A() = (2,3)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (2,4)
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (5,3)
          and#_A(x1,x2) = ((1,0),(0,0)) x2 + (3,3)
          n__isNat_A(x1) = ((1,0),(0,0)) x1 + (2,4)
          n__s_A(x1) = x1 + (2,2)
          s_A(x1) = x1 + (2,2)
          U21#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (5,3)
          and_A(x1,x2) = x2
          U11_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (3,0)
          U21_A(x1,x2,x3) = ((1,1),(0,1)) x2 + x3 + (3,6)
          plus_A(x1,x2) = x1 + ((1,1),(0,1)) x2 + (0,4)
          n__0_A() = (2,3)
  
    precedence:
    
      tt = activate = |0| = isNat = n__isNat = and = U11 > n__s = s = U21 = plus > n__0 > U11# = activate# = n__plus = plus# = isNat# = and# = U21#
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(|0|) = []
      pi(isNat) = []
      pi(isNat#) = []
      pi(and#) = []
      pi(n__isNat) = []
      pi(n__s) = []
      pi(s) = []
      pi(U21#) = []
      pi(and) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p13

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__isNat(X)) -> isNat#(X)
p9: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p10: isNat#(n__plus(V1,V2)) -> activate#(V1)
p11: activate#(n__s(X)) -> activate#(X)
p12: isNat#(n__plus(V1,V2)) -> activate#(V2)
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__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#(X1)
p6: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p7: plus#(N,s(M)) -> isNat#(M)
p8: isNat#(n__plus(V1,V2)) -> activate#(V2)
p9: isNat#(n__plus(V1,V2)) -> activate#(V1)
p10: isNat#(n__plus(V1,V2)) -> isNat#(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: standard order
        interpretations:
          U11#_A(x1,x2) = ((0,1),(0,1)) x2 + (19,6)
          tt_A() = (15,6)
          activate#_A(x1) = ((0,1),(0,1)) x1 + (18,6)
          n__s_A(x1) = ((0,0),(0,1)) x1 + (8,0)
          n__isNat_A(x1) = ((0,0),(0,1)) x1 + (1,1)
          isNat#_A(x1) = ((0,1),(0,1)) x1 + (18,6)
          n__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,0),(0,1)) x2 + (22,0)
          plus#_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (18,6)
          activate_A(x1) = ((1,1),(0,1)) x1 + (9,0)
          s_A(x1) = ((0,0),(0,1)) x1 + (16,0)
          and#_A(x1,x2) = x1 + ((1,1),(0,1)) x2 + (4,4)
          isNat_A(x1) = ((0,1),(0,1)) x1 + (11,1)
          U21#_A(x1,x2,x3) = ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3 + (18,6)
          and_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,1),(0,1)) x2 + (8,0)
          |0|_A() = (20,5)
          U11_A(x1,x2) = ((1,1),(0,1)) x2 + (21,5)
          U21_A(x1,x2,x3) = ((0,0),(0,1)) x2 + ((0,1),(0,1)) x3 + (16,0)
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,0),(0,1)) x2 + (22,0)
          n__0_A() = (7,5)
  
    precedence:
    
      tt = activate# = n__isNat = isNat# = plus# = activate = and# = isNat = U21# = and = |0| = n__0 > U11# = plus > n__plus > U21 > n__s = 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) = []
      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:

  p19

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__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#(X1)
p6: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p7: plus#(N,s(M)) -> isNat#(M)
p8: isNat#(n__plus(V1,V2)) -> activate#(V2)
p9: isNat#(n__plus(V1,V2)) -> activate#(V1)
p10: isNat#(n__plus(V1,V2)) -> isNat#(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)

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:

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


-- Reduction pair.

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

p1: activate#(n__s(X)) -> activate#(X)
p2: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p3: plus#(N,|0|()) -> isNat#(N)
p4: isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p5: and#(tt(),X) -> activate#(X)
p6: activate#(n__plus(X1,X2)) -> activate#(X1)
p7: activate#(n__isNat(X)) -> isNat#(X)
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)) -> 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: standard order
        interpretations:
          activate#_A(x1) = ((0,1),(0,1)) x1 + (4,3)
          n__s_A(x1) = x1
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,1)) x2 + (5,0)
          plus#_A(x1,x2) = ((0,1),(0,1)) x1 + ((1,1),(1,1)) x2 + (4,3)
          activate_A(x1) = x1
          |0|_A() = (1,0)
          isNat#_A(x1) = ((0,1),(0,1)) x1 + (4,3)
          and#_A(x1,x2) = ((1,0),(1,0)) x1 + ((0,1),(0,1)) x2 + (1,0)
          isNat_A(x1) = ((1,1),(0,1)) x1 + (2,1)
          n__isNat_A(x1) = ((1,1),(0,1)) x1 + (2,1)
          tt_A() = (3,0)
          s_A(x1) = x1
          U21#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((0,1),(0,1)) x3 + (4,3)
          and_A(x1,x2) = x2 + (1,0)
          U11_A(x1,x2) = x2 + (2,1)
          U21_A(x1,x2,x3) = ((0,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (5,0)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,1)) x2 + (5,0)
          n__0_A() = (1,0)
  
    precedence:
    
      activate = isNat = and > tt > |0| > plus > n__plus = n__0 > U11 > activate# = plus# = isNat# = and# = U21# > n__isNat > n__s = s = U21
    
    partial status:
  
      pi(activate#) = []
      pi(n__s) = []
      pi(n__plus) = [1]
      pi(plus#) = []
      pi(activate) = []
      pi(|0|) = []
      pi(isNat#) = []
      pi(and#) = []
      pi(isNat) = []
      pi(n__isNat) = []
      pi(tt) = []
      pi(s) = []
      pi(U21#) = []
      pi(and) = []
      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__s(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)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p5: and#(tt(),X) -> activate#(X)
p6: activate#(n__plus(X1,X2)) -> activate#(X1)
p7: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p8: isNat#(n__plus(V1,V2)) -> activate#(V1)
p9: isNat#(n__plus(V1,V2)) -> activate#(V2)
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: activate#(n__s(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: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p9: isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p10: and#(tt(),X) -> activate#(X)
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: standard order
        interpretations:
          activate#_A(x1) = ((1,1),(1,1)) x1 + (1,10)
          n__s_A(x1) = ((0,1),(1,0)) x1 + (3,10)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((0,1),(1,0)) x2 + (15,22)
          plus#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (1,47)
          activate_A(x1) = x1
          s_A(x1) = ((0,1),(1,0)) x1 + (3,10)
          isNat#_A(x1) = ((1,1),(1,1)) x1 + (14,22)
          and#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,1),(1,1)) x2 + (1,10)
          isNat_A(x1) = ((0,0),(1,1)) x1 + (6,1)
          n__isNat_A(x1) = ((0,0),(1,1)) x1 + (6,1)
          tt_A() = (6,9)
          U21#_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (2,47)
          and_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (0,24)
          |0|_A() = (5,8)
          U11_A(x1,x2) = ((1,1),(1,1)) x2 + (1,9)
          U21_A(x1,x2,x3) = x2 + ((1,1),(1,1)) x3 + (25,25)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((0,1),(1,0)) x2 + (15,22)
          n__0_A() = (5,8)
  
    precedence:
    
      activate = isNat = and = U11 > n__plus = s = n__isNat = U21 = plus > |0| = n__0 > plus# = U21# > isNat# > activate# > and# > tt > n__s
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(s) = []
      pi(isNat#) = [1]
      pi(and#) = [2]
      pi(isNat) = []
      pi(n__isNat) = []
      pi(tt) = []
      pi(U21#) = [3]
      pi(and) = []
      pi(|0|) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = [1]
      pi(n__0) = []

The next rules are strictly ordered:

  p15

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__s(X)) -> activate#(X)
p2: activate#(n__plus(X1,X2)) -> activate#(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: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p9: isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p10: and#(tt(),X) -> activate#(X)
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: 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: activate#(n__s(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)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p5: and#(tt(),X) -> activate#(X)
p6: activate#(n__plus(X1,X2)) -> activate#(X1)
p7: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p8: isNat#(n__plus(V1,V2)) -> activate#(V1)
p9: isNat#(n__plus(V1,V2)) -> activate#(V2)
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) -> activate#(N)
p13: U21#(tt(),M,N) -> activate#(M)
p14: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p15: 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: standard order
        interpretations:
          activate#_A(x1) = ((0,1),(0,1)) x1 + (5,4)
          n__s_A(x1) = x1
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,1),(1,1)) x2 + (6,0)
          plus#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (5,4)
          activate_A(x1) = x1
          |0|_A() = (3,1)
          isNat#_A(x1) = ((1,1),(0,1)) x1 + (5,4)
          and#_A(x1,x2) = ((0,1),(1,0)) x1 + ((0,1),(0,1)) x2 + (5,0)
          isNat_A(x1) = ((1,0),(1,0)) x1 + (4,0)
          n__isNat_A(x1) = ((1,0),(1,0)) x1 + (4,0)
          tt_A() = (4,3)
          s_A(x1) = x1
          U21#_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((1,1),(1,1)) x2 + ((0,1),(0,1)) x3 + (3,4)
          and_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,1)
          U11_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (4,0)
          U21_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,0),(1,1)) x3 + (6,0)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,1),(1,1)) x2 + (6,0)
          n__0_A() = (3,1)
  
    precedence:
    
      activate# = plus# = isNat# = and# = U11 > activate = U21 = plus > s > n__s = n__plus = |0| = isNat = n__isNat = tt = U21# = and = n__0
    
    partial status:
  
      pi(activate#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = [1]
      pi(|0|) = []
      pi(isNat#) = []
      pi(and#) = []
      pi(isNat) = []
      pi(n__isNat) = []
      pi(tt) = []
      pi(s) = []
      pi(U21#) = []
      pi(and) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p12

We remove them from the problem.

-- SCC decomposition.

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

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


-- Reduction pair.

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

p1: activate#(n__s(X)) -> activate#(X)
p2: activate#(n__plus(X1,X2)) -> activate#(X1)
p3: activate#(n__plus(X1,X2)) -> 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: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p9: isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p10: and#(tt(),X) -> activate#(X)
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: 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: standard order
        interpretations:
          activate#_A(x1) = ((0,1),(0,0)) x1 + (15,14)
          n__s_A(x1) = ((0,0),(0,1)) x1 + (14,8)
          n__plus_A(x1,x2) = x1 + ((0,1),(0,1)) x2 + (26,13)
          plus#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (27,14)
          activate_A(x1) = x1 + (36,0)
          s_A(x1) = ((0,0),(0,1)) x1 + (32,8)
          isNat#_A(x1) = ((0,1),(0,0)) x1 + (32,14)
          and#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (1,14)
          isNat_A(x1) = (36,15)
          n__isNat_A(x1) = (0,15)
          tt_A() = (36,15)
          U21#_A(x1,x2,x3) = ((0,1),(0,0)) x2 + (31,14)
          and_A(x1,x2) = ((1,0),(0,0)) x1 + x2
          |0|_A() = (37,13)
          U11_A(x1,x2) = x2 + (38,14)
          U21_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,0),(0,1)) x2 + ((0,0),(0,1)) x3 + (33,6)
          plus_A(x1,x2) = x1 + ((0,1),(0,1)) x2 + (26,13)
          n__0_A() = (1,13)
  
    precedence:
    
      isNat# > activate# > U21# > plus# > n__s = n__plus = activate = s = and# = isNat = n__isNat = tt = and = |0| = U11 = U21 = plus = n__0
    
    partial status:
  
      pi(activate#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(s) = []
      pi(isNat#) = []
      pi(and#) = []
      pi(isNat) = []
      pi(n__isNat) = []
      pi(tt) = []
      pi(U21#) = []
      pi(and) = []
      pi(|0|) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p12

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__s(X)) -> activate#(X)
p2: activate#(n__plus(X1,X2)) -> activate#(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: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p9: isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p10: and#(tt(),X) -> activate#(X)
p11: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p12: U21#(tt(),M,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, p13}


-- Reduction pair.

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

p1: activate#(n__s(X)) -> activate#(X)
p2: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p3: plus#(N,|0|()) -> isNat#(N)
p4: isNat#(n__plus(V1,V2)) -> and#(isNat(activate(V1)),n__isNat(activate(V2)))
p5: and#(tt(),X) -> activate#(X)
p6: activate#(n__plus(X1,X2)) -> activate#(X1)
p7: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p8: isNat#(n__plus(V1,V2)) -> activate#(V1)
p9: isNat#(n__plus(V1,V2)) -> activate#(V2)
p10: isNat#(n__s(V1)) -> activate#(V1)
p11: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p12: 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: standard order
        interpretations:
          activate#_A(x1) = x1
          n__s_A(x1) = x1 + (8,0)
          n__plus_A(x1,x2) = x1 + ((1,1),(0,1)) x2 + (6,0)
          plus#_A(x1,x2) = x1 + x2 + (5,0)
          activate_A(x1) = x1
          |0|_A() = (4,1)
          isNat#_A(x1) = x1 + (8,0)
          and#_A(x1,x2) = x2 + (2,0)
          isNat_A(x1) = (5,0)
          n__isNat_A(x1) = (5,0)
          tt_A() = (5,0)
          s_A(x1) = x1 + (8,0)
          U11_A(x1,x2) = x2 + (5,1)
          U21_A(x1,x2,x3) = ((1,1),(0,1)) x2 + x3 + (14,0)
          plus_A(x1,x2) = x1 + ((1,1),(0,1)) x2 + (6,0)
          and_A(x1,x2) = x2
          n__0_A() = (4,1)
  
    precedence:
    
      activate = isNat = and > |0| = n__0 > U11 = U21 = plus > n__isNat = s > and# > activate# = n__s = n__plus = plus# = isNat# = tt
    
    partial status:
  
      pi(activate#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = [1, 2]
      pi(activate) = []
      pi(|0|) = []
      pi(isNat#) = [1]
      pi(and#) = [2]
      pi(isNat) = []
      pi(n__isNat) = []
      pi(tt) = []
      pi(s) = [1]
      pi(U11) = []
      pi(U21) = []
      pi(plus) = [1, 2]
      pi(and) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__s(X)) -> activate#(X)
p2: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p3: plus#(N,|0|()) -> isNat#(N)
p4: and#(tt(),X) -> activate#(X)
p5: activate#(n__plus(X1,X2)) -> activate#(X1)
p6: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p7: isNat#(n__plus(V1,V2)) -> activate#(V1)
p8: isNat#(n__plus(V1,V2)) -> activate#(V2)
p9: isNat#(n__s(V1)) -> activate#(V1)
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 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: activate#(n__s(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: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p9: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p10: and#(tt(),X) -> activate#(X)
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 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:
          activate#_A(x1) = ((0,1),(0,1)) x1 + (21,17)
          n__s_A(x1) = ((0,0),(0,1)) x1 + (14,16)
          n__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (11,0)
          plus#_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (20,10)
          activate_A(x1) = ((1,1),(0,1)) x1 + (16,0)
          s_A(x1) = ((0,0),(0,1)) x1 + (25,16)
          isNat#_A(x1) = ((0,1),(0,1)) x1 + (26,17)
          and#_A(x1,x2) = ((1,1),(1,1)) x2 + (22,17)
          isNat_A(x1) = ((0,1),(0,1)) x1 + (13,1)
          n__isNat_A(x1) = ((0,0),(0,1)) x1 + (1,1)
          tt_A() = (20,8)
          |0|_A() = (14,7)
          U11_A(x1,x2) = ((1,1),(0,1)) x2 + (17,1)
          U21_A(x1,x2,x3) = ((0,0),(0,1)) x2 + ((0,1),(0,1)) x3 + (26,16)
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (11,0)
          and_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,1),(0,1)) x2 + (9,0)
          n__0_A() = (1,7)
  
    precedence:
    
      activate# = n__s = n__plus = plus# = activate = s = isNat# = and# = isNat = n__isNat = tt = |0| = U11 = U21 = plus = and = n__0
    
    partial status:
  
      pi(activate#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(s) = []
      pi(isNat#) = []
      pi(and#) = []
      pi(isNat) = []
      pi(n__isNat) = []
      pi(tt) = []
      pi(|0|) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(and) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

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


-- Reduction pair.

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

p1: activate#(n__s(X)) -> activate#(X)
p2: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p3: plus#(N,|0|()) -> isNat#(N)
p4: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
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)) -> and#(isNat(M),n__isNat(N))
p10: and#(tt(),X) -> activate#(X)

and R consists of:

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

The set of usable rules consists of

  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:
          activate#_A(x1) = ((0,1),(0,0)) x1 + (2,5)
          n__s_A(x1) = x1 + (16,18)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,1),(1,1)) x2
          plus#_A(x1,x2) = ((1,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (1,5)
          activate_A(x1) = x1
          |0|_A() = (2,4)
          isNat#_A(x1) = ((1,1),(0,0)) x1 + (5,5)
          s_A(x1) = x1 + (16,18)
          and#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (1,5)
          isNat_A(x1) = (17,7)
          n__isNat_A(x1) = (17,7)
          tt_A() = (1,5)
          U11_A(x1,x2) = x2 + (3,5)
          U21_A(x1,x2,x3) = ((0,1),(1,1)) x2 + ((1,0),(1,1)) x3 + (17,18)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,1),(1,1)) x2
          and_A(x1,x2) = x2
          n__0_A() = (2,4)
  
    precedence:
    
      activate = isNat = n__isNat = U11 = plus = and > |0| = n__0 > tt > U21 > n__s = s > activate# = n__plus = plus# = isNat# = and#
    
    partial status:
  
      pi(activate#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(|0|) = []
      pi(isNat#) = []
      pi(s) = []
      pi(and#) = []
      pi(isNat) = []
      pi(n__isNat) = []
      pi(tt) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(and) = []
      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: activate#(n__s(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)) -> isNat#(activate(V1))
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: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p9: and#(tt(),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, p2, p3, p4, p5, p6, p7, p8, p9}


-- Reduction pair.

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

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

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: standard order
        interpretations:
          activate#_A(x1) = ((1,0),(0,0)) x1 + (12,3)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (17,3)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (3,10)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (4,3)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,4)
          s_A(x1) = ((1,0),(0,0)) x1 + (17,10)
          and#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (12,3)
          isNat_A(x1) = (2,10)
          n__isNat_A(x1) = (2,4)
          tt_A() = (1,2)
          |0|_A() = (5,10)
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (9,3)
          U11_A(x1,x2) = ((1,0),(1,1)) x2 + (7,11)
          U21_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (20,11)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (3,11)
          and_A(x1,x2) = ((1,0),(1,1)) x2 + (0,4)
          n__0_A() = (5,1)
  
    precedence:
    
      n__s = activate = s = isNat = n__isNat = |0| = U11 = U21 = plus = and > activate# = n__plus = plus# = and# = isNat# > n__0 > tt
    
    partial status:
  
      pi(activate#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(s) = []
      pi(and#) = []
      pi(isNat) = []
      pi(n__isNat) = []
      pi(tt) = []
      pi(|0|) = []
      pi(isNat#) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(and) = []
      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: activate#(n__s(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)) -> and#(isNat(M),n__isNat(N))
p5: and#(tt(),X) -> activate#(X)
p6: plus#(N,|0|()) -> isNat#(N)
p7: isNat#(n__plus(V1,V2)) -> activate#(V2)
p8: isNat#(n__plus(V1,V2)) -> activate#(V1)

and R consists of:

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

The 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: activate#(n__s(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: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p8: and#(tt(),X) -> activate#(X)

and R consists of:

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

The set of usable rules consists of

  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:
          activate#_A(x1) = ((0,1),(0,0)) x1 + (6,0)
          n__s_A(x1) = ((0,0),(0,1)) x1 + (0,1)
          n__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,0),(0,1)) x2 + (7,0)
          plus#_A(x1,x2) = ((0,1),(0,0)) x1 + (6,0)
          activate_A(x1) = ((1,1),(0,1)) x1 + (3,0)
          |0|_A() = (5,1)
          isNat#_A(x1) = ((0,1),(0,0)) x1 + (6,0)
          s_A(x1) = ((0,0),(0,1)) x1 + (1,1)
          and#_A(x1,x2) = ((0,1),(0,0)) x2 + (6,0)
          isNat_A(x1) = (11,0)
          n__isNat_A(x1) = (8,0)
          tt_A() = (2,0)
          U11_A(x1,x2) = ((1,1),(0,1)) x2 + (4,1)
          U21_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((0,0),(0,1)) x2 + ((1,1),(0,1)) x3 + (4,1)
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,0),(0,1)) x2 + (7,0)
          and_A(x1,x2) = ((1,1),(0,1)) x2 + (3,0)
          n__0_A() = (1,1)
  
    precedence:
    
      n__plus = activate = isNat = U11 = plus = and > U21 > |0| = n__0 > n__s = s = tt > activate# = plus# = isNat# = and# = n__isNat
    
    partial status:
  
      pi(activate#) = []
      pi(n__s) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(|0|) = []
      pi(isNat#) = []
      pi(s) = []
      pi(and#) = []
      pi(isNat) = []
      pi(n__isNat) = []
      pi(tt) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(and) = []
      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: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p7: and#(tt(),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, p2, p3, p4, p5, p6, p7}


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

and R consists of:

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

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17

Take the reduction pair:

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

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

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

and R consists of:

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

The set of usable rules consists of

  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:
          activate#_A(x1) = ((0,1),(0,1)) x1 + (9,8)
          n__plus_A(x1,x2) = x1 + ((0,0),(0,1)) x2 + (8,0)
          plus#_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (4,8)
          activate_A(x1) = x1 + (6,0)
          |0|_A() = (6,7)
          isNat#_A(x1) = ((0,1),(0,1)) x1 + (10,8)
          s_A(x1) = ((0,0),(0,1)) x1 + (2,0)
          and#_A(x1,x2) = ((0,1),(1,0)) x1 + ((0,1),(0,1)) x2 + (3,3)
          isNat_A(x1) = ((0,1),(0,1)) x1 + (5,0)
          n__isNat_A(x1) = ((0,1),(0,1)) x1 + (1,0)
          tt_A() = (5,7)
          U11_A(x1,x2) = x2 + (7,7)
          U21_A(x1,x2,x3) = ((0,0),(0,1)) x2 + ((0,0),(0,1)) x3 + (5,0)
          plus_A(x1,x2) = x1 + ((0,0),(0,1)) x2 + (8,0)
          and_A(x1,x2) = ((0,1),(0,1)) x1 + x2 + (2,0)
          n__0_A() = (1,7)
          n__s_A(x1) = ((0,0),(0,1)) x1 + (1,0)
  
    precedence:
    
      activate# = plus# = activate = |0| = and# = isNat = n__isNat = U11 = plus = and = n__0 > tt > s = U21 > n__plus > isNat# = n__s
    
    partial status:
  
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(|0|) = []
      pi(isNat#) = []
      pi(s) = []
      pi(and#) = []
      pi(isNat) = []
      pi(n__isNat) = []
      pi(tt) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(and) = []
      pi(n__0) = []
      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#(V2)
p3: activate#(n__plus(X1,X2)) -> activate#(X1)
p4: plus#(N,s(M)) -> and#(isNat(M),n__isNat(N))
p5: and#(tt(),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}


-- 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)) -> and#(isNat(M),n__isNat(N))
p3: and#(tt(),X) -> activate#(X)
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: standard order
        interpretations:
          activate#_A(x1) = x1 + (2,1)
          n__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(0,1)) x2 + (3,0)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (2,0)
          activate_A(x1) = ((1,1),(0,1)) x1
          s_A(x1) = (9,8)
          and#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (2,0)
          isNat_A(x1) = (4,0)
          n__isNat_A(x1) = (4,0)
          tt_A() = (1,0)
          U11_A(x1,x2) = ((1,1),(0,1)) x2 + (13,6)
          U21_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,1),(0,1)) x3 + (9,8)
          plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,1),(0,1)) x2 + (3,0)
          and_A(x1,x2) = ((1,1),(0,1)) x2
          |0|_A() = (5,6)
          n__0_A() = (0,6)
          n__s_A(x1) = (4,8)
  
    precedence:
    
      n__plus = activate = isNat = n__isNat = U11 = U21 = plus = and > |0| > tt > n__0 > s = n__s > plus# > activate# = and#
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__plus) = []
      pi(plus#) = [2]
      pi(activate) = []
      pi(s) = []
      pi(and#) = [2]
      pi(isNat) = []
      pi(n__isNat) = []
      pi(tt) = []
      pi(U11) = []
      pi(U21) = []
      pi(plus) = []
      pi(and) = []
      pi(|0|) = []
      pi(n__0) = []
      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: and#(tt(),X) -> activate#(X)
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: standard order
        interpretations:
          activate#_A(x1) = x1 + (1,1)
          n__plus_A(x1,x2) = x1 + ((1,1),(1,1)) x2 + (2,1)
  
    precedence:
    
      n__plus > activate#
    
    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.