YES

We show the termination of the TRS R:

  U11(tt(),V2) -> U12(isNat(activate(V2)))
  U12(tt()) -> tt()
  U21(tt()) -> tt()
  U31(tt(),N) -> activate(N)
  U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
  U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
  isNat(n__0()) -> tt()
  isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
  isNat(n__s(V1)) -> U21(isNat(activate(V1)))
  plus(N,|0|()) -> U31(isNat(N),N)
  plus(N,s(M)) -> U41(isNat(M),M,N)
  |0|() -> n__0()
  plus(X1,X2) -> n__plus(X1,X2)
  s(X) -> n__s(X)
  activate(n__0()) -> |0|()
  activate(n__plus(X1,X2)) -> plus(X1,X2)
  activate(n__s(X)) -> s(X)
  activate(X) -> X

-- SCC decomposition.

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

p1: U11#(tt(),V2) -> U12#(isNat(activate(V2)))
p2: U11#(tt(),V2) -> isNat#(activate(V2))
p3: U11#(tt(),V2) -> activate#(V2)
p4: U31#(tt(),N) -> activate#(N)
p5: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p6: U41#(tt(),M,N) -> isNat#(activate(N))
p7: U41#(tt(),M,N) -> activate#(N)
p8: U41#(tt(),M,N) -> activate#(M)
p9: U42#(tt(),M,N) -> s#(plus(activate(N),activate(M)))
p10: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p11: U42#(tt(),M,N) -> activate#(N)
p12: U42#(tt(),M,N) -> activate#(M)
p13: isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
p14: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p15: isNat#(n__plus(V1,V2)) -> activate#(V1)
p16: isNat#(n__plus(V1,V2)) -> activate#(V2)
p17: isNat#(n__s(V1)) -> U21#(isNat(activate(V1)))
p18: isNat#(n__s(V1)) -> isNat#(activate(V1))
p19: isNat#(n__s(V1)) -> activate#(V1)
p20: plus#(N,|0|()) -> U31#(isNat(N),N)
p21: plus#(N,|0|()) -> isNat#(N)
p22: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p23: plus#(N,s(M)) -> isNat#(M)
p24: activate#(n__0()) -> |0|#()
p25: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p26: activate#(n__s(X)) -> s#(X)

and R consists of:

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

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (3,6)
          tt_A() = (2,9)
          isNat#_A(x1) = x1 + (4,7)
          activate_A(x1) = x1
          n__s_A(x1) = ((1,0),(0,0)) x1 + (5,0)
          activate#_A(x1) = ((1,0),(1,1)) x1 + (4,1)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (0,12)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (4,8)
          s_A(x1) = ((1,0),(0,0)) x1 + (5,0)
          isNat_A(x1) = ((1,0),(1,1)) x1 + (0,18)
          U41#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (9,8)
          U42#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(0,0)) x3 + (5,2)
          |0|_A() = (3,1)
          U31#_A(x1,x2) = ((1,0),(0,0)) x2 + (7,9)
          U42_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (5,1)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (0,12)
          U12_A(x1) = ((1,0),(1,0)) x1 + (0,10)
          U31_A(x1,x2) = ((1,0),(0,0)) x2 + (2,0)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (5,2)
          U11_A(x1,x2) = ((1,0),(1,0)) x2 + (0,11)
          U21_A(x1) = (3,24)
          n__0_A() = (3,1)
  
    precedence:
    
      activate = |0| = plus = U31 > tt = n__plus = isNat = U12 = U11 = U21 = n__0 > U11# = isNat# = activate# = plus# = U41# = U42# = U31# > n__s = s = U42 = U41
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(isNat#) = []
      pi(activate) = []
      pi(n__s) = []
      pi(activate#) = []
      pi(n__plus) = [2]
      pi(plus#) = []
      pi(s) = []
      pi(isNat) = []
      pi(U41#) = []
      pi(U42#) = []
      pi(|0|) = []
      pi(U31#) = []
      pi(U42) = []
      pi(plus) = []
      pi(U12) = []
      pi(U31) = []
      pi(U41) = []
      pi(U11) = []
      pi(U21) = []
      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: U11#(tt(),V2) -> isNat#(activate(V2))
p2: isNat#(n__s(V1)) -> activate#(V1)
p3: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,s(M)) -> isNat#(M)
p5: isNat#(n__s(V1)) -> isNat#(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)) -> U11#(isNat(activate(V1)),activate(V2))
p10: U11#(tt(),V2) -> activate#(V2)
p11: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p12: U41#(tt(),M,N) -> activate#(M)
p13: U41#(tt(),M,N) -> activate#(N)
p14: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p15: U42#(tt(),M,N) -> activate#(M)
p16: U42#(tt(),M,N) -> activate#(N)
p17: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p18: plus#(N,|0|()) -> isNat#(N)
p19: plus#(N,|0|()) -> U31#(isNat(N),N)
p20: U31#(tt(),N) -> activate#(N)

and R consists of:

r1: U11(tt(),V2) -> U12(isNat(activate(V2)))
r2: U12(tt()) -> tt()
r3: U21(tt()) -> tt()
r4: U31(tt(),N) -> activate(N)
r5: U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
r6: U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
r7: isNat(n__0()) -> tt()
r8: isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
r9: isNat(n__s(V1)) -> U21(isNat(activate(V1)))
r10: plus(N,|0|()) -> U31(isNat(N),N)
r11: plus(N,s(M)) -> U41(isNat(M),M,N)
r12: |0|() -> n__0()
r13: plus(X1,X2) -> n__plus(X1,X2)
r14: s(X) -> n__s(X)
r15: activate(n__0()) -> |0|()
r16: activate(n__plus(X1,X2)) -> plus(X1,X2)
r17: activate(n__s(X)) -> s(X)
r18: 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(),V2) -> isNat#(activate(V2))
p2: isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
p3: U11#(tt(),V2) -> activate#(V2)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,|0|()) -> U31#(isNat(N),N)
p6: U31#(tt(),N) -> activate#(N)
p7: plus#(N,|0|()) -> isNat#(N)
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,s(M)) -> U41#(isNat(M),M,N)
p14: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p15: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p16: plus#(N,s(M)) -> isNat#(M)
p17: U42#(tt(),M,N) -> activate#(N)
p18: U42#(tt(),M,N) -> activate#(M)
p19: U41#(tt(),M,N) -> activate#(N)
p20: U41#(tt(),M,N) -> activate#(M)

and R consists of:

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

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2) = ((1,0),(1,1)) x2 + (3,18)
          tt_A() = (1,7)
          isNat#_A(x1) = ((1,0),(1,1)) x1 + (0,15)
          activate_A(x1) = x1 + (0,2)
          n__plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (4,21)
          isNat_A(x1) = (1,35)
          activate#_A(x1) = ((1,0),(0,0)) x1 + (2,2)
          plus#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,2)
          |0|_A() = (0,0)
          U31#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (2,1)
          n__s_A(x1) = ((1,0),(1,0)) x1 + (3,8)
          s_A(x1) = ((1,0),(1,0)) x1 + (3,9)
          U41#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (4,2)
          U42#_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (1,1)
          U42_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (7,13)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (4,22)
          U12_A(x1) = ((0,0),(1,0)) x1 + (1,7)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (1,3)
          U41_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (7,13)
          U11_A(x1,x2) = ((1,0),(1,0)) x1 + (0,21)
          U21_A(x1) = ((0,0),(1,0)) x1 + (1,7)
          n__0_A() = (0,0)
  
    precedence:
    
      U12 > U31 > plus > n__plus > tt > U11# = isNat# > isNat = plus# = U41# = U21 > U31# = U11 > activate# > U42# > U41 > activate = s = U42 > |0| = n__s = n__0
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(isNat#) = [1]
      pi(activate) = [1]
      pi(n__plus) = []
      pi(isNat) = []
      pi(activate#) = []
      pi(plus#) = [2]
      pi(|0|) = []
      pi(U31#) = []
      pi(n__s) = []
      pi(s) = []
      pi(U41#) = []
      pi(U42#) = []
      pi(U42) = []
      pi(plus) = []
      pi(U12) = []
      pi(U31) = [2]
      pi(U41) = []
      pi(U11) = []
      pi(U21) = []
      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(),V2) -> isNat#(activate(V2))
p2: isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
p3: U11#(tt(),V2) -> activate#(V2)
p4: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p5: plus#(N,|0|()) -> U31#(isNat(N),N)
p6: U31#(tt(),N) -> activate#(N)
p7: plus#(N,|0|()) -> isNat#(N)
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: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p14: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p15: plus#(N,s(M)) -> isNat#(M)
p16: U42#(tt(),M,N) -> activate#(N)
p17: U42#(tt(),M,N) -> activate#(M)
p18: U41#(tt(),M,N) -> activate#(N)
p19: U41#(tt(),M,N) -> activate#(M)

and R consists of:

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

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: U11#(tt(),V2) -> isNat#(activate(V2))
p2: isNat#(n__s(V1)) -> activate#(V1)
p3: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,s(M)) -> isNat#(M)
p5: isNat#(n__s(V1)) -> isNat#(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)) -> U11#(isNat(activate(V1)),activate(V2))
p10: U11#(tt(),V2) -> activate#(V2)
p11: plus#(N,|0|()) -> isNat#(N)
p12: plus#(N,|0|()) -> U31#(isNat(N),N)
p13: U31#(tt(),N) -> activate#(N)

and R consists of:

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

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          U11#_A(x1,x2) = ((1,0),(1,1)) x2 + (1,12)
          tt_A() = (0,9)
          isNat#_A(x1) = x1 + (1,11)
          activate_A(x1) = ((1,0),(1,1)) x1
          n__s_A(x1) = ((1,0),(0,0)) x1 + (5,0)
          activate#_A(x1) = ((1,0),(0,0)) x1 + (1,11)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (6,0)
          plus#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + (3,11)
          s_A(x1) = ((1,0),(0,0)) x1 + (5,5)
          isNat_A(x1) = (2,7)
          |0|_A() = (0,10)
          U31#_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (1,2)
          U42_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (11,6)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (6,5)
          U12_A(x1) = (0,10)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (1,1)
          U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (11,9)
          U11_A(x1,x2) = (0,11)
          U21_A(x1) = (1,10)
          n__0_A() = (0,10)
  
    precedence:
    
      isNat > tt > U11# > isNat# = n__s = activate# = plus# = U31# > activate = |0| > s = U12 = U11 = n__0 > n__plus = plus = U31 = U41 = U21 > U42
    
    partial status:
  
      pi(U11#) = [2]
      pi(tt) = []
      pi(isNat#) = [1]
      pi(activate) = [1]
      pi(n__s) = []
      pi(activate#) = []
      pi(n__plus) = [1]
      pi(plus#) = [1]
      pi(s) = []
      pi(isNat) = []
      pi(|0|) = []
      pi(U31#) = []
      pi(U42) = []
      pi(plus) = [1]
      pi(U12) = []
      pi(U31) = [2]
      pi(U41) = []
      pi(U11) = []
      pi(U21) = []
      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: U11#(tt(),V2) -> isNat#(activate(V2))
p2: isNat#(n__s(V1)) -> activate#(V1)
p3: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p4: plus#(N,s(M)) -> isNat#(M)
p5: isNat#(n__s(V1)) -> isNat#(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: U11#(tt(),V2) -> activate#(V2)
p10: plus#(N,|0|()) -> isNat#(N)
p11: plus#(N,|0|()) -> U31#(isNat(N),N)
p12: U31#(tt(),N) -> activate#(N)

and R consists of:

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

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: isNat#(n__s(V1)) -> activate#(V1)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,|0|()) -> U31#(isNat(N),N)
p4: U31#(tt(),N) -> activate#(N)
p5: plus#(N,|0|()) -> isNat#(N)
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)) -> isNat#(activate(V1))
p10: plus#(N,s(M)) -> isNat#(M)

and R consists of:

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

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isNat#_A(x1) = ((0,0),(1,0)) x1 + (1,4)
          n__s_A(x1) = ((1,0),(1,0)) x1 + (5,3)
          activate#_A(x1) = ((0,0),(1,0)) x1 + (1,6)
          n__plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (3,14)
          plus#_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (1,8)
          |0|_A() = (5,3)
          U31#_A(x1,x2) = ((0,0),(1,0)) x2 + (1,12)
          isNat_A(x1) = x1 + (4,0)
          tt_A() = (2,1)
          activate_A(x1) = x1 + (0,10)
          s_A(x1) = ((1,0),(1,0)) x1 + (5,5)
          U42_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (8,11)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (3,15)
          U12_A(x1) = ((0,0),(1,0)) x1 + (3,1)
          U31_A(x1,x2) = x2 + (1,11)
          U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (8,12)
          U11_A(x1,x2) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (3,9)
          U21_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          n__0_A() = (5,2)
  
    precedence:
    
      n__s = plus# = |0| = activate = s = U42 = plus = U31 = U41 > U31# > isNat# > isNat > activate# = n__plus = tt = U12 = U11 = U21 = n__0
    
    partial status:
  
      pi(isNat#) = []
      pi(n__s) = []
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(U31#) = []
      pi(isNat) = []
      pi(tt) = []
      pi(activate) = []
      pi(s) = []
      pi(U42) = []
      pi(plus) = []
      pi(U12) = []
      pi(U31) = []
      pi(U41) = []
      pi(U11) = []
      pi(U21) = []
      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: isNat#(n__s(V1)) -> activate#(V1)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,|0|()) -> U31#(isNat(N),N)
p4: U31#(tt(),N) -> activate#(N)
p5: plus#(N,|0|()) -> isNat#(N)
p6: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p7: isNat#(n__plus(V1,V2)) -> activate#(V1)
p8: isNat#(n__s(V1)) -> isNat#(activate(V1))
p9: plus#(N,s(M)) -> isNat#(M)

and R consists of:

r1: U11(tt(),V2) -> U12(isNat(activate(V2)))
r2: U12(tt()) -> tt()
r3: U21(tt()) -> tt()
r4: U31(tt(),N) -> activate(N)
r5: U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
r6: U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
r7: isNat(n__0()) -> tt()
r8: isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
r9: isNat(n__s(V1)) -> U21(isNat(activate(V1)))
r10: plus(N,|0|()) -> U31(isNat(N),N)
r11: plus(N,s(M)) -> U41(isNat(M),M,N)
r12: |0|() -> n__0()
r13: plus(X1,X2) -> n__plus(X1,X2)
r14: s(X) -> n__s(X)
r15: activate(n__0()) -> |0|()
r16: activate(n__plus(X1,X2)) -> plus(X1,X2)
r17: activate(n__s(X)) -> s(X)
r18: 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: isNat#(n__s(V1)) -> activate#(V1)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,s(M)) -> isNat#(M)
p4: isNat#(n__s(V1)) -> isNat#(activate(V1))
p5: isNat#(n__plus(V1,V2)) -> activate#(V1)
p6: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p7: plus#(N,|0|()) -> isNat#(N)
p8: plus#(N,|0|()) -> U31#(isNat(N),N)
p9: U31#(tt(),N) -> activate#(N)

and R consists of:

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

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isNat#_A(x1) = x1 + (0,1)
          n__s_A(x1) = ((1,0),(1,0)) x1 + (5,6)
          activate#_A(x1) = x1 + (0,1)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (0,7)
          plus#_A(x1,x2) = x1 + x2 + (0,1)
          s_A(x1) = ((1,0),(1,0)) x1 + (5,8)
          activate_A(x1) = x1 + (0,4)
          |0|_A() = (0,5)
          U31#_A(x1,x2) = x2 + (0,2)
          isNat_A(x1) = (4,7)
          tt_A() = (1,3)
          U42_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (5,17)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (0,8)
          U12_A(x1) = (2,4)
          U31_A(x1,x2) = x2 + (0,4)
          U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (5,18)
          U11_A(x1,x2) = (3,1)
          U21_A(x1) = ((0,0),(1,0)) x1 + (2,1)
          n__0_A() = (0,5)
  
    precedence:
    
      |0| = U21 = n__0 > n__plus = activate = isNat = tt = U42 = plus = U12 = U31 = U41 = U11 > plus# = U31# > s > isNat# > activate# > n__s
    
    partial status:
  
      pi(isNat#) = [1]
      pi(n__s) = []
      pi(activate#) = [1]
      pi(n__plus) = []
      pi(plus#) = [2]
      pi(s) = []
      pi(activate) = [1]
      pi(|0|) = []
      pi(U31#) = []
      pi(isNat) = []
      pi(tt) = []
      pi(U42) = []
      pi(plus) = [1, 2]
      pi(U12) = []
      pi(U31) = [2]
      pi(U41) = []
      pi(U11) = []
      pi(U21) = []
      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: isNat#(n__s(V1)) -> activate#(V1)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,s(M)) -> isNat#(M)
p4: isNat#(n__s(V1)) -> isNat#(activate(V1))
p5: isNat#(n__plus(V1,V2)) -> activate#(V1)
p6: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p7: plus#(N,|0|()) -> isNat#(N)
p8: U31#(tt(),N) -> activate#(N)

and R consists of:

r1: U11(tt(),V2) -> U12(isNat(activate(V2)))
r2: U12(tt()) -> tt()
r3: U21(tt()) -> tt()
r4: U31(tt(),N) -> activate(N)
r5: U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
r6: U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
r7: isNat(n__0()) -> tt()
r8: isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
r9: isNat(n__s(V1)) -> U21(isNat(activate(V1)))
r10: plus(N,|0|()) -> U31(isNat(N),N)
r11: plus(N,s(M)) -> U41(isNat(M),M,N)
r12: |0|() -> n__0()
r13: plus(X1,X2) -> n__plus(X1,X2)
r14: s(X) -> n__s(X)
r15: activate(n__0()) -> |0|()
r16: activate(n__plus(X1,X2)) -> plus(X1,X2)
r17: activate(n__s(X)) -> s(X)
r18: 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: isNat#(n__s(V1)) -> activate#(V1)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,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: isNat#(n__s(V1)) -> isNat#(activate(V1))
p7: plus#(N,s(M)) -> isNat#(M)

and R consists of:

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

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (6,0)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (7,2)
          activate#_A(x1) = ((1,0),(0,0)) x1 + (1,1)
          n__plus_A(x1,x2) = x1 + x2 + (2,1)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (2,1)
          |0|_A() = (4,11)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,6)
          s_A(x1) = ((1,0),(0,0)) x1 + (7,3)
          U12_A(x1) = ((1,0),(1,0)) x1 + (0,5)
          tt_A() = (3,7)
          U11_A(x1,x2) = (3,9)
          isNat_A(x1) = (3,10)
          U21_A(x1) = (3,8)
          U42_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (9,4)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (5,12)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (6,5)
          n__0_A() = (4,6)
  
    precedence:
    
      activate = isNat = plus = U31 = U41 > |0| = s = U42 > isNat# = n__s = activate# = plus# > n__plus = U12 = tt = U11 = U21 = n__0
    
    partial status:
  
      pi(isNat#) = []
      pi(n__s) = []
      pi(activate#) = []
      pi(n__plus) = [2]
      pi(plus#) = []
      pi(|0|) = []
      pi(activate) = []
      pi(s) = []
      pi(U12) = []
      pi(tt) = []
      pi(U11) = []
      pi(isNat) = []
      pi(U21) = []
      pi(U42) = []
      pi(plus) = []
      pi(U31) = []
      pi(U41) = []
      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: isNat#(n__s(V1)) -> activate#(V1)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,|0|()) -> isNat#(N)
p4: isNat#(n__plus(V1,V2)) -> activate#(V1)
p5: isNat#(n__s(V1)) -> isNat#(activate(V1))
p6: plus#(N,s(M)) -> isNat#(M)

and R consists of:

r1: U11(tt(),V2) -> U12(isNat(activate(V2)))
r2: U12(tt()) -> tt()
r3: U21(tt()) -> tt()
r4: U31(tt(),N) -> activate(N)
r5: U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
r6: U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
r7: isNat(n__0()) -> tt()
r8: isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
r9: isNat(n__s(V1)) -> U21(isNat(activate(V1)))
r10: plus(N,|0|()) -> U31(isNat(N),N)
r11: plus(N,s(M)) -> U41(isNat(M),M,N)
r12: |0|() -> n__0()
r13: plus(X1,X2) -> n__plus(X1,X2)
r14: s(X) -> n__s(X)
r15: activate(n__0()) -> |0|()
r16: activate(n__plus(X1,X2)) -> plus(X1,X2)
r17: activate(n__s(X)) -> s(X)
r18: 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: isNat#(n__s(V1)) -> activate#(V1)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,s(M)) -> isNat#(M)
p4: isNat#(n__s(V1)) -> isNat#(activate(V1))
p5: isNat#(n__plus(V1,V2)) -> activate#(V1)
p6: plus#(N,|0|()) -> isNat#(N)

and R consists of:

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

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (11,11)
          n__s_A(x1) = x1 + (12,2)
          activate#_A(x1) = ((1,0),(1,1)) x1 + (11,1)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (13,4)
          plus#_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (2,1)
          s_A(x1) = x1 + (12,3)
          activate_A(x1) = x1 + (0,3)
          |0|_A() = (10,15)
          U12_A(x1) = (1,17)
          tt_A() = (0,16)
          U11_A(x1,x2) = (12,19)
          isNat_A(x1) = x1 + (11,14)
          U21_A(x1) = (1,17)
          U42_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (25,12)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (13,5)
          U31_A(x1,x2) = x2 + (11,1)
          U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (25,16)
          n__0_A() = (10,13)
  
    precedence:
    
      activate > s = U42 = plus = U31 = U41 > U11 = isNat > |0| = n__0 > n__plus > isNat# > activate# = plus# = U12 = tt > n__s = U21
    
    partial status:
  
      pi(isNat#) = []
      pi(n__s) = []
      pi(activate#) = [1]
      pi(n__plus) = [1]
      pi(plus#) = [1]
      pi(s) = [1]
      pi(activate) = []
      pi(|0|) = []
      pi(U12) = []
      pi(tt) = []
      pi(U11) = []
      pi(isNat) = [1]
      pi(U21) = []
      pi(U42) = []
      pi(plus) = [1]
      pi(U31) = []
      pi(U41) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

r1: U11(tt(),V2) -> U12(isNat(activate(V2)))
r2: U12(tt()) -> tt()
r3: U21(tt()) -> tt()
r4: U31(tt(),N) -> activate(N)
r5: U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
r6: U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
r7: isNat(n__0()) -> tt()
r8: isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
r9: isNat(n__s(V1)) -> U21(isNat(activate(V1)))
r10: plus(N,|0|()) -> U31(isNat(N),N)
r11: plus(N,s(M)) -> U41(isNat(M),M,N)
r12: |0|() -> n__0()
r13: plus(X1,X2) -> n__plus(X1,X2)
r14: s(X) -> n__s(X)
r15: activate(n__0()) -> |0|()
r16: activate(n__plus(X1,X2)) -> plus(X1,X2)
r17: activate(n__s(X)) -> s(X)
r18: 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: isNat#(n__s(V1)) -> activate#(V1)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,s(M)) -> isNat#(M)
p4: isNat#(n__plus(V1,V2)) -> activate#(V1)
p5: isNat#(n__s(V1)) -> isNat#(activate(V1))

and R consists of:

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

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isNat#_A(x1) = ((1,0),(1,1)) x1 + (1,1)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (12,2)
          activate#_A(x1) = ((1,0),(1,1)) x1 + (1,3)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (9,4)
          plus#_A(x1,x2) = ((1,0),(1,1)) x2 + (1,3)
          s_A(x1) = ((1,0),(0,0)) x1 + (12,14)
          activate_A(x1) = x1 + (0,13)
          U12_A(x1) = ((1,0),(1,0)) x1 + (8,1)
          tt_A() = (8,0)
          U11_A(x1,x2) = ((1,0),(1,0)) x2 + (10,3)
          isNat_A(x1) = x1 + (1,8)
          U21_A(x1) = (9,1)
          U42_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (21,19)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,0)) x2 + (9,5)
          U31_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (1,9)
          U41_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + x3 + (21,12)
          n__0_A() = (9,0)
          |0|_A() = (9,9)
  
    precedence:
    
      isNat# = activate# > plus# = |0| > n__s = s = activate = U12 = U42 = plus = U31 = U41 > n__plus > n__0 > tt = U11 = isNat = U21
    
    partial status:
  
      pi(isNat#) = [1]
      pi(n__s) = []
      pi(activate#) = []
      pi(n__plus) = [1]
      pi(plus#) = []
      pi(s) = []
      pi(activate) = [1]
      pi(U12) = []
      pi(tt) = []
      pi(U11) = []
      pi(isNat) = []
      pi(U21) = []
      pi(U42) = []
      pi(plus) = [1]
      pi(U31) = [2]
      pi(U41) = []
      pi(n__0) = []
      pi(|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: isNat#(n__s(V1)) -> activate#(V1)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,s(M)) -> isNat#(M)
p4: isNat#(n__s(V1)) -> isNat#(activate(V1))

and R consists of:

r1: U11(tt(),V2) -> U12(isNat(activate(V2)))
r2: U12(tt()) -> tt()
r3: U21(tt()) -> tt()
r4: U31(tt(),N) -> activate(N)
r5: U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N))
r6: U42(tt(),M,N) -> s(plus(activate(N),activate(M)))
r7: isNat(n__0()) -> tt()
r8: isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2))
r9: isNat(n__s(V1)) -> U21(isNat(activate(V1)))
r10: plus(N,|0|()) -> U31(isNat(N),N)
r11: plus(N,s(M)) -> U41(isNat(M),M,N)
r12: |0|() -> n__0()
r13: plus(X1,X2) -> n__plus(X1,X2)
r14: s(X) -> n__s(X)
r15: activate(n__0()) -> |0|()
r16: activate(n__plus(X1,X2)) -> plus(X1,X2)
r17: activate(n__s(X)) -> s(X)
r18: 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: isNat#(n__s(V1)) -> activate#(V1)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: plus#(N,s(M)) -> isNat#(M)
p4: isNat#(n__s(V1)) -> isNat#(activate(V1))

and R consists of:

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

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (1,2)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (6,4)
          activate#_A(x1) = ((1,0),(1,0)) x1 + (1,1)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (7,1)
          plus#_A(x1,x2) = ((1,0),(1,0)) x2 + (1,2)
          s_A(x1) = ((1,0),(0,0)) x1 + (6,5)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,3)
          U12_A(x1) = ((1,0),(0,0)) x1 + (3,1)
          tt_A() = (4,9)
          U11_A(x1,x2) = ((1,0),(0,0)) x2 + (6,2)
          isNat_A(x1) = x1 + (2,12)
          U21_A(x1) = (5,8)
          U42_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 + (7,8)
          U31_A(x1,x2) = ((1,0),(0,0)) x2 + (4,4)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (13,7)
          n__0_A() = (3,10)
          |0|_A() = (3,11)
  
    precedence:
    
      plus = U31 > U42 = U41 = n__0 > n__s = s = activate = isNat = U21 > U12 = tt > n__plus = |0| > isNat# > activate# = plus# > U11
    
    partial status:
  
      pi(isNat#) = []
      pi(n__s) = []
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(s) = []
      pi(activate) = [1]
      pi(U12) = []
      pi(tt) = []
      pi(U11) = []
      pi(isNat) = []
      pi(U21) = []
      pi(U42) = []
      pi(plus) = []
      pi(U31) = []
      pi(U41) = []
      pi(n__0) = []
      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: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p2: plus#(N,s(M)) -> isNat#(M)
p3: isNat#(n__s(V1)) -> isNat#(activate(V1))

and R consists of:

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

and R consists of:

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

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          isNat#_A(x1) = ((1,0),(1,0)) x1 + (1,1)
          n__s_A(x1) = ((1,0),(1,0)) x1 + (5,2)
          activate_A(x1) = x1 + (0,9)
          U12_A(x1) = ((0,0),(1,0)) x1 + (9,8)
          tt_A() = (8,0)
          U11_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (2,8)
          isNat_A(x1) = ((1,0),(1,0)) x1 + (1,12)
          U21_A(x1) = ((1,0),(1,0)) x1 + (1,2)
          U42_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (9,10)
          s_A(x1) = ((1,0),(1,0)) x1 + (5,5)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (4,11)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (10,14)
          U41_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (9,3)
          n__0_A() = (9,5)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (4,10)
          |0|_A() = (9,13)
  
    precedence:
    
      n__s = activate = tt = U11 = isNat = U21 = U42 = s = plus = U31 = U41 > isNat# > U12 = n__0 = n__plus = |0|
    
    partial status:
  
      pi(isNat#) = []
      pi(n__s) = []
      pi(activate) = []
      pi(U12) = []
      pi(tt) = []
      pi(U11) = []
      pi(isNat) = []
      pi(U21) = []
      pi(U42) = []
      pi(s) = []
      pi(plus) = []
      pi(U31) = []
      pi(U41) = []
      pi(n__0) = []
      pi(n__plus) = [2]
      pi(|0|) = []

The next rules are strictly ordered:

  p1

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