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: standard order
        interpretations:
          U11#_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,1),(0,0)) x2 + (29,33)
          tt_A() = (19,1)
          isNat#_A(x1) = ((1,1),(0,0)) x1 + (25,34)
          activate_A(x1) = x1 + (1,2)
          n__s_A(x1) = ((0,1),(1,0)) x1 + (0,31)
          activate#_A(x1) = ((1,1),(0,0)) x1 + (20,34)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (0,12)
          plus#_A(x1,x2) = ((1,1),(0,0)) x1 + ((1,1),(0,0)) x2 + (14,34)
          s_A(x1) = ((0,1),(1,0)) x1 + (0,33)
          isNat_A(x1) = (36,1)
          U41#_A(x1,x2,x3) = ((1,1),(0,1)) x1 + ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (9,33)
          U42#_A(x1,x2,x3) = ((1,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (21,34)
          |0|_A() = (19,2)
          U31#_A(x1,x2) = ((1,1),(0,0)) x2 + (21,34)
          U42_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (19,39)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (0,12)
          U12_A(x1) = (19,1)
          U31_A(x1,x2) = x2 + (20,3)
          U41_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (26,45)
          U11_A(x1,x2) = (19,1)
          U21_A(x1) = ((0,0),(0,1)) x1 + (20,0)
          n__0_A() = (19,2)
  
    precedence:
    
      U11# > isNat# = activate# = plus# = U41# = U42# > isNat = U11 = U21 > U12 > n__plus = plus = U41 > U42 > tt = |0| = U31# = U31 = n__0 > activate = s > n__s
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(isNat#) = []
      pi(activate) = [1]
      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) = [2]
      pi(plus) = [2]
      pi(U12) = []
      pi(U31) = []
      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,s(M)) -> U41#(isNat(M),M,N)
p11: U41#(tt(),M,N) -> activate#(M)
p12: U41#(tt(),M,N) -> activate#(N)
p13: U41#(tt(),M,N) -> isNat#(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:

  {p2, p3, p4, p5, p6, p7, p8, 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: 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)) -> U41#(isNat(M),M,N)
p11: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p12: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p13: plus#(N,s(M)) -> isNat#(M)
p14: U42#(tt(),M,N) -> activate#(N)
p15: U42#(tt(),M,N) -> activate#(M)
p16: U41#(tt(),M,N) -> isNat#(activate(N))
p17: U41#(tt(),M,N) -> activate#(N)
p18: 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: standard order
        interpretations:
          isNat#_A(x1) = ((0,1),(0,0)) x1 + (20,0)
          n__s_A(x1) = x1 + (18,11)
          activate#_A(x1) = ((0,1),(0,0)) x1 + (20,0)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (19,0)
          plus#_A(x1,x2) = ((1,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (20,0)
          |0|_A() = (20,0)
          U31#_A(x1,x2) = ((1,1),(0,0)) x1 + ((0,1),(0,0)) x2
          isNat_A(x1) = ((1,0),(0,0)) x1 + (15,4)
          tt_A() = (27,3)
          activate_A(x1) = x1 + (2,0)
          s_A(x1) = x1 + (18,11)
          U41#_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (24,0)
          U42#_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + ((1,1),(0,0)) x3 + (20,0)
          U42_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (43,15)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (20,0)
          U12_A(x1) = ((0,1),(0,0)) x1 + (25,4)
          U31_A(x1,x2) = ((1,1),(0,1)) x2 + (21,1)
          U41_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (48,19)
          U11_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,0),(0,0)) x2 + (27,4)
          U21_A(x1) = (28,4)
          n__0_A() = (19,0)
  
    precedence:
    
      U41# > isNat# = activate# = n__plus = plus# = U31# = plus > isNat = U41 = U11 > U42# > U21 > |0| = tt = activate = U42 = U12 = n__0 > s > n__s > U31
    
    partial status:
  
      pi(isNat#) = []
      pi(n__s) = [1]
      pi(activate#) = []
      pi(n__plus) = [1, 2]
      pi(plus#) = []
      pi(|0|) = []
      pi(U31#) = []
      pi(isNat) = []
      pi(tt) = []
      pi(activate) = []
      pi(s) = [1]
      pi(U41#) = []
      pi(U42#) = []
      pi(U42) = []
      pi(plus) = [1, 2]
      pi(U12) = []
      pi(U31) = []
      pi(U41) = [2]
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p11

We remove them from the problem.

-- SCC decomposition.

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

p1: 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)) -> U41#(isNat(M),M,N)
p11: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p12: plus#(N,s(M)) -> isNat#(M)
p13: U42#(tt(),M,N) -> activate#(N)
p14: U42#(tt(),M,N) -> activate#(M)
p15: U41#(tt(),M,N) -> isNat#(activate(N))
p16: U41#(tt(),M,N) -> activate#(N)
p17: 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, p12, p15, p16, p17}


-- 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#(V2)
p6: isNat#(n__plus(V1,V2)) -> activate#(V1)
p7: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p8: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p9: U41#(tt(),M,N) -> activate#(M)
p10: U41#(tt(),M,N) -> activate#(N)
p11: U41#(tt(),M,N) -> isNat#(activate(N))
p12: plus#(N,|0|()) -> isNat#(N)
p13: plus#(N,|0|()) -> U31#(isNat(N),N)
p14: 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: standard order
        interpretations:
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (16,30)
          n__s_A(x1) = x1 + (36,29)
          activate#_A(x1) = ((1,0),(0,0)) x1 + (16,30)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (0,30)
          plus#_A(x1,x2) = ((1,1),(0,0)) x1 + ((1,0),(0,0)) x2 + (16,30)
          s_A(x1) = x1 + (36,38)
          activate_A(x1) = x1 + (0,9)
          U41#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,1),(0,0)) x3 + (37,30)
          isNat_A(x1) = ((0,0),(0,1)) x1 + (75,14)
          tt_A() = (17,22)
          |0|_A() = (0,30)
          U31#_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,0),(0,0)) x2 + (1,30)
          U42_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (55,87)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (0,31)
          U12_A(x1) = (17,22)
          U31_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,0),(1,1)) x2 + (1,31)
          U41_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (74,105)
          U11_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(0,1)) x2 + (0,22)
          U21_A(x1) = (37,22)
          n__0_A() = (0,21)
  
    precedence:
    
      n__plus = activate = plus = U31 > U42 = U41 > |0| > isNat = U11 = U21 = n__0 > isNat# = activate# = plus# = U41# = U31# > tt = U12 > n__s = s
    
    partial status:
  
      pi(isNat#) = []
      pi(n__s) = []
      pi(activate#) = []
      pi(n__plus) = [2]
      pi(plus#) = []
      pi(s) = [1]
      pi(activate) = [1]
      pi(U41#) = []
      pi(isNat) = []
      pi(tt) = []
      pi(|0|) = []
      pi(U31#) = []
      pi(U42) = []
      pi(plus) = [2]
      pi(U12) = []
      pi(U31) = []
      pi(U41) = []
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: isNat#(n__s(V1)) -> activate#(V1)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: isNat#(n__s(V1)) -> isNat#(activate(V1))
p4: isNat#(n__plus(V1,V2)) -> activate#(V2)
p5: isNat#(n__plus(V1,V2)) -> activate#(V1)
p6: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p7: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p8: U41#(tt(),M,N) -> activate#(M)
p9: U41#(tt(),M,N) -> activate#(N)
p10: U41#(tt(),M,N) -> isNat#(activate(N))
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 estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: isNat#(n__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)) -> U41#(isNat(M),M,N)
p11: U41#(tt(),M,N) -> isNat#(activate(N))
p12: U41#(tt(),M,N) -> activate#(N)
p13: 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: standard order
        interpretations:
          isNat#_A(x1) = ((1,1),(1,1)) x1 + (47,49)
          n__s_A(x1) = ((0,1),(1,0)) x1 + (1,48)
          activate#_A(x1) = ((1,0),(1,1)) x1 + (2,48)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (75,93)
          plus#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (1,1)
          |0|_A() = (55,0)
          U31#_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,1),(1,1)) x2 + (1,48)
          isNat_A(x1) = (57,53)
          tt_A() = (49,53)
          activate_A(x1) = x1 + (2,1)
          s_A(x1) = ((0,1),(1,0)) x1 + (2,49)
          U41#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,1),(1,1)) x3 + (51,52)
          U42_A(x1,x2,x3) = ((1,1),(0,0)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,131)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (76,94)
          U12_A(x1) = (50,53)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (56,1)
          U41_A(x1,x2,x3) = ((1,1),(1,0)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (16,88)
          U11_A(x1,x2) = (56,53)
          U21_A(x1) = (50,53)
          n__0_A() = (54,0)
  
    precedence:
    
      |0| > isNat# = n__0 > activate# = plus# = U31# = isNat = U41# = U21 > tt = U12 = U11 > activate = plus = U31 > n__s = n__plus = s = U42 = U41
    
    partial status:
  
      pi(isNat#) = []
      pi(n__s) = []
      pi(activate#) = []
      pi(n__plus) = [1]
      pi(plus#) = [1]
      pi(|0|) = []
      pi(U31#) = []
      pi(isNat) = []
      pi(tt) = []
      pi(activate) = [1]
      pi(s) = []
      pi(U41#) = []
      pi(U42) = []
      pi(plus) = [1]
      pi(U12) = []
      pi(U31) = [2]
      pi(U41) = [2]
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: isNat#(n__s(V1)) -> activate#(V1)
p2: activate#(n__plus(X1,X2)) -> plus#(X1,X2)
p3: U31#(tt(),N) -> activate#(N)
p4: plus#(N,|0|()) -> isNat#(N)
p5: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p6: isNat#(n__plus(V1,V2)) -> activate#(V1)
p7: isNat#(n__plus(V1,V2)) -> activate#(V2)
p8: isNat#(n__s(V1)) -> isNat#(activate(V1))
p9: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p10: U41#(tt(),M,N) -> isNat#(activate(N))
p11: U41#(tt(),M,N) -> activate#(N)
p12: 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, p4, p5, p6, p7, p8, p9, 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,s(M)) -> U41#(isNat(M),M,N)
p4: U41#(tt(),M,N) -> activate#(M)
p5: U41#(tt(),M,N) -> activate#(N)
p6: U41#(tt(),M,N) -> isNat#(activate(N))
p7: isNat#(n__s(V1)) -> isNat#(activate(V1))
p8: isNat#(n__plus(V1,V2)) -> activate#(V2)
p9: isNat#(n__plus(V1,V2)) -> activate#(V1)
p10: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p11: 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: standard order
        interpretations:
          isNat#_A(x1) = x1
          n__s_A(x1) = x1 + (2,0)
          activate#_A(x1) = x1
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (2,2)
          plus#_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,1),(0,1)) x2 + (0,1)
          s_A(x1) = x1 + (2,0)
          U41#_A(x1,x2,x3) = ((0,1),(0,0)) x1 + x2 + ((1,0),(1,1)) x3 + (0,1)
          isNat_A(x1) = ((0,0),(0,1)) x1 + (0,2)
          tt_A() = (0,0)
          activate_A(x1) = x1
          |0|_A() = (0,0)
          U42_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (4,2)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (2,2)
          U12_A(x1) = (0,1)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (1,1)
          U41_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (4,2)
          U11_A(x1,x2) = (0,1)
          U21_A(x1) = (0,1)
          n__0_A() = (0,0)
  
    precedence:
    
      s = activate = |0| = U42 = plus = U31 = U41 = n__0 > isNat = U12 = U11 = U21 > n__s = n__plus > isNat# = activate# = plus# = U41# = tt
    
    partial status:
  
      pi(isNat#) = []
      pi(n__s) = []
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = [1, 2]
      pi(s) = []
      pi(U41#) = [3]
      pi(isNat) = []
      pi(tt) = []
      pi(activate) = [1]
      pi(|0|) = []
      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:

  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)) -> U41#(isNat(M),M,N)
p4: U41#(tt(),M,N) -> activate#(N)
p5: U41#(tt(),M,N) -> isNat#(activate(N))
p6: isNat#(n__s(V1)) -> isNat#(activate(V1))
p7: isNat#(n__plus(V1,V2)) -> activate#(V2)
p8: isNat#(n__plus(V1,V2)) -> activate#(V1)
p9: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p10: 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 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: 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__plus(V1,V2)) -> activate#(V2)
p7: isNat#(n__s(V1)) -> isNat#(activate(V1))
p8: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p9: U41#(tt(),M,N) -> isNat#(activate(N))
p10: U41#(tt(),M,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: standard order
        interpretations:
          isNat#_A(x1) = ((1,1),(1,1)) x1 + (6,60)
          n__s_A(x1) = ((0,1),(1,0)) x1 + (0,67)
          activate#_A(x1) = ((1,1),(1,1)) x1 + (3,36)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (4,36)
          plus#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,0),(1,0)) x2 + (5,68)
          |0|_A() = (1,14)
          activate_A(x1) = x1 + (7,1)
          s_A(x1) = ((0,1),(1,0)) x1 + (0,67)
          U41#_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((1,1),(1,1)) x3 + (1,68)
          isNat_A(x1) = ((0,0),(0,1)) x1 + (6,1)
          tt_A() = (1,14)
          U42_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (53,87)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (4,36)
          U12_A(x1) = (2,15)
          U31_A(x1,x2) = ((1,1),(0,1)) x2 + (8,15)
          U41_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (70,103)
          U11_A(x1,x2) = ((0,0),(1,1)) x2 + (3,29)
          U21_A(x1) = (6,68)
          n__0_A() = (0,13)
  
    precedence:
    
      isNat# = activate# = plus# = |0| > U41# > n__plus = isNat = tt = plus = U41 = U21 > activate = U31 > s = U42 > n__s = n__0 > U12 = U11
    
    partial status:
  
      pi(isNat#) = []
      pi(n__s) = []
      pi(activate#) = []
      pi(n__plus) = [1]
      pi(plus#) = []
      pi(|0|) = []
      pi(activate) = [1]
      pi(s) = []
      pi(U41#) = []
      pi(isNat) = []
      pi(tt) = []
      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:

  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|()) -> isNat#(N)
p4: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p5: isNat#(n__plus(V1,V2)) -> activate#(V1)
p6: isNat#(n__plus(V1,V2)) -> activate#(V2)
p7: isNat#(n__s(V1)) -> isNat#(activate(V1))
p8: U41#(tt(),M,N) -> isNat#(activate(N))
p9: U41#(tt(),M,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__s(V1)) -> isNat#(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))

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: standard order
        interpretations:
          isNat#_A(x1) = ((1,1),(1,1)) x1 + (21,25)
          n__s_A(x1) = x1 + (27,31)
          activate#_A(x1) = ((1,1),(0,1)) x1 + (29,32)
          n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (30,32)
          plus#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(0,1)) x2 + (19,24)
          |0|_A() = (2,1)
          activate_A(x1) = x1 + (4,3)
          U12_A(x1) = (2,1)
          tt_A() = (0,0)
          U11_A(x1,x2) = (3,1)
          isNat_A(x1) = (29,2)
          U21_A(x1) = (28,2)
          U42_A(x1,x2,x3) = ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (75,77)
          s_A(x1) = x1 + (30,31)
          plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (30,32)
          U31_A(x1,x2) = ((1,1),(0,1)) x2 + (5,3)
          U41_A(x1,x2,x3) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (90,91)
          n__0_A() = (1,0)
  
    precedence:
    
      plus = U41 > U42 = s > n__plus = plus# > isNat# > n__s = activate# = activate = U11 = isNat = U21 = U31 > tt > |0| = U12 > n__0
    
    partial status:
  
      pi(isNat#) = [1]
      pi(n__s) = [1]
      pi(activate#) = [1]
      pi(n__plus) = [1, 2]
      pi(plus#) = [1, 2]
      pi(|0|) = []
      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) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

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

and R consists of:

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

The estimated dependency graph contains the following SCCs:

  {p3, p6}


-- Reduction pair.

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

p1: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p2: 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: standard order
        interpretations:
          isNat#_A(x1) = ((0,1),(1,1)) x1 + (1,49)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,1),(1,1)) x2 + (7,6)
          activate_A(x1) = x1 + (8,1)
          n__s_A(x1) = x1 + (28,57)
          U12_A(x1) = (85,71)
          tt_A() = (84,70)
          U11_A(x1,x2) = (86,71)
          isNat_A(x1) = ((1,1),(0,0)) x1 + (74,72)
          U21_A(x1) = (85,70)
          U42_A(x1,x2,x3) = ((0,1),(1,1)) x2 + ((1,0),(1,1)) x3 + (61,82)
          s_A(x1) = x1 + (36,58)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,1),(1,1)) x2 + (15,6)
          U31_A(x1,x2) = x2 + (15,2)
          U41_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((0,1),(1,1)) x2 + ((1,0),(1,1)) x3 + (0,100)
          n__0_A() = (14,1)
          |0|_A() = (14,1)
  
    precedence:
    
      isNat > tt = n__0 = |0| > isNat# = n__plus = activate = plus = U31 = U41 > U42 > s > U12 = U11 > U21 > n__s
    
    partial status:
  
      pi(isNat#) = []
      pi(n__plus) = []
      pi(activate) = []
      pi(n__s) = []
      pi(U12) = []
      pi(tt) = []
      pi(U11) = []
      pi(isNat) = []
      pi(U21) = []
      pi(U42) = []
      pi(s) = []
      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: 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}


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

The next rules are strictly ordered:

  p1

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