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(activate(X1),activate(X2))
  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(),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#(activate(X1),activate(X2))
p26: activate#(n__plus(X1,X2)) -> activate#(X1)
p27: activate#(n__plus(X1,X2)) -> activate#(X2)
p28: activate#(n__s(X)) -> s#(activate(X))
p29: activate#(n__s(X)) -> activate#(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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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, p26, p27, p29}


-- 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__s(X)) -> activate#(X)
p4: activate#(n__plus(X1,X2)) -> activate#(X2)
p5: activate#(n__plus(X1,X2)) -> activate#(X1)
p6: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p7: plus#(N,s(M)) -> isNat#(M)
p8: isNat#(n__s(V1)) -> isNat#(activate(V1))
p9: isNat#(n__plus(V1,V2)) -> activate#(V2)
p10: isNat#(n__plus(V1,V2)) -> activate#(V1)
p11: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p12: isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
p13: U11#(tt(),V2) -> activate#(V2)
p14: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p15: U41#(tt(),M,N) -> activate#(M)
p16: U41#(tt(),M,N) -> activate#(N)
p17: U41#(tt(),M,N) -> isNat#(activate(N))
p18: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p19: U42#(tt(),M,N) -> activate#(M)
p20: U42#(tt(),M,N) -> activate#(N)
p21: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p22: plus#(N,|0|()) -> isNat#(N)
p23: plus#(N,|0|()) -> U31#(isNat(N),N)
p24: 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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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) = ((0,0),(1,0)) x2 + (5,6)
          tt_A() = (6,8)
          isNat#_A(x1) = ((0,0),(1,0)) x1 + (5,2)
          activate_A(x1) = x1 + (0,12)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (4,5)
          activate#_A(x1) = ((0,0),(1,0)) x1 + (5,6)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (4,1)
          plus#_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (5,3)
          s_A(x1) = ((1,0),(0,0)) x1 + (4,7)
          isNat_A(x1) = ((1,0),(1,0)) x1 + (7,4)
          U41#_A(x1,x2,x3) = ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (5,7)
          U42#_A(x1,x2,x3) = ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (5,6)
          |0|_A() = (6,10)
          U31#_A(x1,x2) = ((0,0),(1,0)) x2 + (5,7)
          U42_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (8,9)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (4,12)
          U12_A(x1) = (7,1)
          U31_A(x1,x2) = ((1,0),(0,0)) x2 + (7,11)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (8,10)
          U11_A(x1,x2) = x1 + ((0,0),(1,0)) x2 + (2,5)
          U21_A(x1) = (7,8)
          n__0_A() = (6,9)
  
    precedence:
    
      activate = n__s = n__plus = s = |0| = U42 = plus = U31 = U41 > plus# > U41# > U42# > U31# > U11# = isNat# > U12 > tt = isNat = U11 = U21 = n__0 > activate#
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(isNat#) = []
      pi(activate) = []
      pi(n__s) = []
      pi(activate#) = []
      pi(n__plus) = []
      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:

  p24

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__s(X)) -> activate#(X)
p4: activate#(n__plus(X1,X2)) -> activate#(X2)
p5: activate#(n__plus(X1,X2)) -> activate#(X1)
p6: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p7: plus#(N,s(M)) -> isNat#(M)
p8: isNat#(n__s(V1)) -> isNat#(activate(V1))
p9: isNat#(n__plus(V1,V2)) -> activate#(V2)
p10: isNat#(n__plus(V1,V2)) -> activate#(V1)
p11: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p12: isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
p13: U11#(tt(),V2) -> activate#(V2)
p14: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p15: U41#(tt(),M,N) -> activate#(M)
p16: U41#(tt(),M,N) -> activate#(N)
p17: U41#(tt(),M,N) -> isNat#(activate(N))
p18: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p19: U42#(tt(),M,N) -> activate#(M)
p20: U42#(tt(),M,N) -> activate#(N)
p21: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p22: plus#(N,|0|()) -> isNat#(N)
p23: plus#(N,|0|()) -> U31#(isNat(N),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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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, p21, p22}


-- 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#(activate(X1),activate(X2))
p5: plus#(N,|0|()) -> isNat#(N)
p6: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p7: isNat#(n__plus(V1,V2)) -> activate#(V1)
p8: activate#(n__plus(X1,X2)) -> activate#(X1)
p9: activate#(n__plus(X1,X2)) -> activate#(X2)
p10: activate#(n__s(X)) -> activate#(X)
p11: isNat#(n__plus(V1,V2)) -> activate#(V2)
p12: isNat#(n__s(V1)) -> isNat#(activate(V1))
p13: isNat#(n__s(V1)) -> activate#(V1)
p14: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p15: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p16: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p17: plus#(N,s(M)) -> isNat#(M)
p18: U42#(tt(),M,N) -> activate#(N)
p19: U42#(tt(),M,N) -> activate#(M)
p20: U41#(tt(),M,N) -> isNat#(activate(N))
p21: U41#(tt(),M,N) -> activate#(N)
p22: 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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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)) x2 + (26,5)
          tt_A() = (36,29)
          isNat#_A(x1) = x1 + (7,0)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,4)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (27,0)
          isNat_A(x1) = (39,41)
          activate#_A(x1) = ((1,0),(1,1)) x1 + (7,28)
          plus#_A(x1,x2) = x1 + x2 + (26,46)
          |0|_A() = (40,0)
          n__s_A(x1) = ((1,0),(1,0)) x1 + (8,0)
          s_A(x1) = ((1,0),(1,0)) x1 + (8,0)
          U41#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (33,6)
          U42#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (33,30)
          U42_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (35,28)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (27,22)
          U12_A(x1) = (37,30)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (41,5)
          U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (35,29)
          U11_A(x1,x2) = ((0,0),(1,0)) x1 + (38,1)
          U21_A(x1) = (37,30)
          n__0_A() = (40,0)
  
    precedence:
    
      U11# = plus# = U41# = U42# > U31 > tt = activate = n__plus = isNat = activate# = |0| = n__s = s = U42 = plus = U12 = U41 = U11 = U21 = n__0 > isNat#
    
    partial status:
  
      pi(U11#) = []
      pi(tt) = []
      pi(isNat#) = [1]
      pi(activate) = [1]
      pi(n__plus) = []
      pi(isNat) = []
      pi(activate#) = [1]
      pi(plus#) = []
      pi(|0|) = []
      pi(n__s) = []
      pi(s) = []
      pi(U41#) = []
      pi(U42#) = []
      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:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
p2: U11#(tt(),V2) -> activate#(V2)
p3: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p4: plus#(N,|0|()) -> isNat#(N)
p5: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p6: isNat#(n__plus(V1,V2)) -> activate#(V1)
p7: activate#(n__plus(X1,X2)) -> activate#(X1)
p8: activate#(n__plus(X1,X2)) -> activate#(X2)
p9: activate#(n__s(X)) -> activate#(X)
p10: isNat#(n__plus(V1,V2)) -> activate#(V2)
p11: isNat#(n__s(V1)) -> isNat#(activate(V1))
p12: isNat#(n__s(V1)) -> activate#(V1)
p13: 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) -> isNat#(activate(N))
p20: U41#(tt(),M,N) -> activate#(N)
p21: 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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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, p21}


-- Reduction pair.

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

p1: isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
p2: U11#(tt(),V2) -> activate#(V2)
p3: activate#(n__s(X)) -> activate#(X)
p4: activate#(n__plus(X1,X2)) -> activate#(X2)
p5: activate#(n__plus(X1,X2)) -> activate#(X1)
p6: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p7: plus#(N,s(M)) -> isNat#(M)
p8: isNat#(n__s(V1)) -> activate#(V1)
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: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p14: U41#(tt(),M,N) -> activate#(M)
p15: U41#(tt(),M,N) -> activate#(N)
p16: U41#(tt(),M,N) -> isNat#(activate(N))
p17: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p18: U42#(tt(),M,N) -> activate#(M)
p19: U42#(tt(),M,N) -> activate#(N)
p20: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p21: 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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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 + (5,13)
          n__plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (6,0)
          U11#_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (2,2)
          isNat_A(x1) = (7,10)
          activate_A(x1) = x1 + (0,8)
          tt_A() = (2,12)
          activate#_A(x1) = ((1,0),(0,0)) x1 + (3,13)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (4,3)
          plus#_A(x1,x2) = x1 + x2 + (1,11)
          s_A(x1) = ((1,0),(0,0)) x1 + (4,4)
          U41#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (5,14)
          U42#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (4,5)
          |0|_A() = (8,2)
          U42_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (10,5)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (6,7)
          U12_A(x1) = (3,13)
          U31_A(x1,x2) = ((1,0),(0,0)) x2 + (9,1)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (10,6)
          U11_A(x1,x2) = (4,9)
          U21_A(x1) = (3,13)
          n__0_A() = (8,2)
  
    precedence:
    
      tt = U12 = U11 > isNat# = U11# = activate# = plus# = U41# = U42# > activate = n__s = s = |0| = U42 = n__0 > U21 > isNat > n__plus = plus = U31 = U41
    
    partial status:
  
      pi(isNat#) = []
      pi(n__plus) = []
      pi(U11#) = []
      pi(isNat) = []
      pi(activate) = []
      pi(tt) = []
      pi(activate#) = []
      pi(n__s) = []
      pi(plus#) = []
      pi(s) = []
      pi(U41#) = []
      pi(U42#) = []
      pi(|0|) = []
      pi(U42) = []
      pi(plus) = []
      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: isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
p2: U11#(tt(),V2) -> activate#(V2)
p3: activate#(n__s(X)) -> activate#(X)
p4: activate#(n__plus(X1,X2)) -> activate#(X2)
p5: activate#(n__plus(X1,X2)) -> activate#(X1)
p6: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p7: plus#(N,s(M)) -> isNat#(M)
p8: isNat#(n__s(V1)) -> 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: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p13: U41#(tt(),M,N) -> activate#(M)
p14: U41#(tt(),M,N) -> activate#(N)
p15: U41#(tt(),M,N) -> isNat#(activate(N))
p16: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p17: U42#(tt(),M,N) -> activate#(M)
p18: U42#(tt(),M,N) -> activate#(N)
p19: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p20: 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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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: isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
p2: U11#(tt(),V2) -> activate#(V2)
p3: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p4: plus#(N,|0|()) -> isNat#(N)
p5: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p6: isNat#(n__plus(V1,V2)) -> activate#(V1)
p7: activate#(n__plus(X1,X2)) -> activate#(X1)
p8: activate#(n__plus(X1,X2)) -> activate#(X2)
p9: activate#(n__s(X)) -> activate#(X)
p10: isNat#(n__plus(V1,V2)) -> activate#(V2)
p11: isNat#(n__s(V1)) -> activate#(V1)
p12: plus#(N,s(M)) -> U41#(isNat(M),M,N)
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) -> isNat#(activate(N))
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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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 + (6,6)
          n__plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (0,7)
          U11#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (0,1)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (4,21)
          activate_A(x1) = x1 + (0,8)
          tt_A() = (1,2)
          activate#_A(x1) = ((1,0),(1,0)) x1
          plus#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2
          |0|_A() = (7,2)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (8,9)
          s_A(x1) = ((1,0),(0,0)) x1 + (8,10)
          U41#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (8,7)
          U42#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (2,7)
          U42_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (8,11)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (0,12)
          U12_A(x1) = (2,3)
          U31_A(x1,x2) = ((1,0),(0,0)) x2 + (6,18)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (8,12)
          U11_A(x1,x2) = (3,20)
          U21_A(x1) = (2,1)
          n__0_A() = (7,1)
  
    precedence:
    
      isNat# > U11# > activate# = plus# = U41# = U42# > U31 > activate = U42 = plus = U41 > s > isNat > |0| = U21 = n__0 > n__plus = tt = n__s = U12 = U11
    
    partial status:
  
      pi(isNat#) = []
      pi(n__plus) = []
      pi(U11#) = []
      pi(isNat) = []
      pi(activate) = [1]
      pi(tt) = []
      pi(activate#) = []
      pi(plus#) = []
      pi(|0|) = []
      pi(n__s) = []
      pi(s) = []
      pi(U41#) = []
      pi(U42#) = []
      pi(U42) = []
      pi(plus) = []
      pi(U12) = []
      pi(U31) = []
      pi(U41) = []
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: isNat#(n__plus(V1,V2)) -> U11#(isNat(activate(V1)),activate(V2))
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: activate#(n__plus(X1,X2)) -> activate#(X2)
p8: activate#(n__s(X)) -> activate#(X)
p9: isNat#(n__plus(V1,V2)) -> activate#(V2)
p10: isNat#(n__s(V1)) -> activate#(V1)
p11: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p12: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p13: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p14: plus#(N,s(M)) -> isNat#(M)
p15: U42#(tt(),M,N) -> activate#(N)
p16: U42#(tt(),M,N) -> activate#(M)
p17: U41#(tt(),M,N) -> isNat#(activate(N))
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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(X))
r18: 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, p19}


-- Reduction pair.

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

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

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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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:
          activate#_A(x1) = ((1,0),(0,0)) x1 + (10,14)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (6,24)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (10,12)
          activate_A(x1) = x1 + (0,2)
          s_A(x1) = ((1,0),(0,0)) x1 + (6,16)
          isNat#_A(x1) = ((1,0),(0,0)) x1 + (4,28)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (6,15)
          U41#_A(x1,x2,x3) = x2 + ((1,0),(0,0)) x3 + (15,27)
          isNat_A(x1) = (6,23)
          tt_A() = (3,21)
          U42#_A(x1,x2,x3) = x1 + x2 + ((1,0),(0,0)) x3 + (8,1)
          |0|_A() = (1,16)
          U42_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (12,14)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (6,25)
          U12_A(x1) = (4,20)
          U31_A(x1,x2) = ((1,0),(1,1)) x2 + (2,26)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (12,21)
          U11_A(x1,x2) = ((0,0),(1,0)) x2 + (5,1)
          U21_A(x1) = (6,22)
          n__0_A() = (1,15)
  
    precedence:
    
      activate = plus = U12 = U31 > |0| > U42 = U41 > activate# = n__plus = plus# = s = isNat# = n__s = U41# = isNat = tt = U42# = U11 = U21 > n__0
    
    partial status:
  
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(s) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(U41#) = []
      pi(isNat) = []
      pi(tt) = []
      pi(U42#) = []
      pi(|0|) = []
      pi(U42) = []
      pi(plus) = []
      pi(U12) = []
      pi(U31) = []
      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: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: plus#(N,s(M)) -> isNat#(M)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> activate#(X2)
p5: activate#(n__plus(X1,X2)) -> activate#(X1)
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)) -> U41#(isNat(M),M,N)
p10: U41#(tt(),M,N) -> activate#(M)
p11: U41#(tt(),M,N) -> activate#(N)
p12: U41#(tt(),M,N) -> isNat#(activate(N))
p13: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p14: U42#(tt(),M,N) -> activate#(M)
p15: U42#(tt(),M,N) -> activate#(N)
p16: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p17: 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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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}


-- 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)) -> isNat#(activate(V1))
p4: isNat#(n__plus(V1,V2)) -> activate#(V1)
p5: activate#(n__plus(X1,X2)) -> activate#(X1)
p6: activate#(n__plus(X1,X2)) -> activate#(X2)
p7: isNat#(n__plus(V1,V2)) -> activate#(V2)
p8: isNat#(n__s(V1)) -> activate#(V1)
p9: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p10: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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:
          activate#_A(x1) = ((1,0),(1,1)) x1 + (10,2)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (10,16)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (0,20)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,9)
          |0|_A() = (13,11)
          isNat#_A(x1) = ((1,0),(1,1)) x1 + (1,32)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (13,11)
          s_A(x1) = ((1,0),(0,0)) x1 + (13,33)
          U41#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + x2 + x3 + (12,43)
          isNat_A(x1) = ((1,0),(1,1)) x1 + (11,1)
          tt_A() = (12,1)
          U42#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (11,30)
          U42_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (23,34)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (10,17)
          U12_A(x1) = (13,2)
          U31_A(x1,x2) = ((1,0),(1,0)) x2 + (1,10)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + x3 + (23,35)
          U11_A(x1,x2) = ((1,0),(0,0)) x1 + (10,11)
          U21_A(x1) = (14,2)
          n__0_A() = (13,2)
  
    precedence:
    
      n__s > activate# = n__plus = plus# = U41# = U42# > activate = s = U42 = plus = U31 = U41 > isNat > U11 > U12 > |0| > tt = U21 > n__0 > isNat#
    
    partial status:
  
      pi(activate#) = []
      pi(n__plus) = [1, 2]
      pi(plus#) = []
      pi(activate) = [1]
      pi(|0|) = []
      pi(isNat#) = [1]
      pi(n__s) = []
      pi(s) = []
      pi(U41#) = []
      pi(isNat) = []
      pi(tt) = []
      pi(U42#) = []
      pi(U42) = []
      pi(plus) = []
      pi(U12) = []
      pi(U31) = []
      pi(U41) = []
      pi(U11) = []
      pi(U21) = []
      pi(n__0) = []

The next rules are strictly ordered:

  p7

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: plus#(N,|0|()) -> isNat#(N)
p3: isNat#(n__plus(V1,V2)) -> isNat#(activate(V1))
p4: isNat#(n__plus(V1,V2)) -> activate#(V1)
p5: activate#(n__plus(X1,X2)) -> activate#(X1)
p6: activate#(n__plus(X1,X2)) -> activate#(X2)
p7: isNat#(n__s(V1)) -> activate#(V1)
p8: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p9: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p10: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p11: plus#(N,s(M)) -> isNat#(M)
p12: U42#(tt(),M,N) -> activate#(N)
p13: U42#(tt(),M,N) -> activate#(M)
p14: U41#(tt(),M,N) -> isNat#(activate(N))
p15: U41#(tt(),M,N) -> activate#(N)
p16: 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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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}


-- Reduction pair.

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

p1: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: plus#(N,s(M)) -> isNat#(M)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> activate#(X2)
p5: activate#(n__plus(X1,X2)) -> activate#(X1)
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: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p13: U42#(tt(),M,N) -> activate#(M)
p14: U42#(tt(),M,N) -> activate#(N)
p15: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p16: 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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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:
          activate#_A(x1) = ((0,0),(1,0)) x1 + (1,2)
          n__plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (3,1)
          plus#_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (1,5)
          activate_A(x1) = x1 + (0,11)
          s_A(x1) = ((1,0),(0,0)) x1 + (0,4)
          isNat#_A(x1) = ((0,0),(1,0)) x1 + (1,3)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (0,3)
          U41#_A(x1,x2,x3) = ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (1,5)
          isNat_A(x1) = (4,12)
          tt_A() = (0,0)
          U42#_A(x1,x2,x3) = ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (1,5)
          |0|_A() = (14,4)
          U42_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (3,5)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (3,11)
          U12_A(x1) = (1,0)
          U31_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (11,12)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (3,10)
          U11_A(x1,x2) = (2,10)
          U21_A(x1) = ((0,0),(1,0)) x1 + (1,9)
          n__0_A() = (14,0)
  
    precedence:
    
      n__plus = activate = isNat = |0| = U42 = plus = U31 = U41 = U21 > tt = U12 > s > n__s = U11 > n__0 > activate# = plus# = isNat# = U41# = U42#
    
    partial status:
  
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = []
      pi(s) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(U41#) = []
      pi(isNat) = []
      pi(tt) = []
      pi(U42#) = []
      pi(|0|) = []
      pi(U42) = []
      pi(plus) = []
      pi(U12) = []
      pi(U31) = []
      pi(U41) = []
      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: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: plus#(N,s(M)) -> isNat#(M)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> activate#(X2)
p5: activate#(n__plus(X1,X2)) -> activate#(X1)
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) -> U42#(isNat(activate(N)),activate(M),activate(N))
p12: U42#(tt(),M,N) -> activate#(M)
p13: U42#(tt(),M,N) -> activate#(N)
p14: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p15: 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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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}


-- 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)) -> isNat#(activate(V1))
p4: isNat#(n__plus(V1,V2)) -> activate#(V1)
p5: activate#(n__plus(X1,X2)) -> activate#(X1)
p6: activate#(n__plus(X1,X2)) -> activate#(X2)
p7: isNat#(n__s(V1)) -> activate#(V1)
p8: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p9: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p10: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p11: plus#(N,s(M)) -> isNat#(M)
p12: U42#(tt(),M,N) -> activate#(N)
p13: U42#(tt(),M,N) -> activate#(M)
p14: U41#(tt(),M,N) -> activate#(N)
p15: 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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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:
          activate#_A(x1) = ((1,0),(1,0)) x1 + (18,5)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (19,26)
          plus#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (20,25)
          activate_A(x1) = x1 + (0,56)
          |0|_A() = (25,54)
          isNat#_A(x1) = ((1,0),(1,1)) x1 + (26,49)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (23,1)
          s_A(x1) = ((1,0),(0,0)) x1 + (23,2)
          U41#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (22,47)
          isNat_A(x1) = ((1,0),(0,0)) x1 + (41,53)
          tt_A() = (57,4)
          U42#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,0)) x3 + (21,0)
          U42_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (42,3)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (19,27)
          U12_A(x1) = (58,3)
          U31_A(x1,x2) = x2 + (24,55)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (42,49)
          U11_A(x1,x2) = (59,53)
          U21_A(x1) = ((1,0),(0,0)) x1 + (0,52)
          n__0_A() = (25,3)
  
    precedence:
    
      activate = isNat = plus > U42 = U41 > n__s = s > U12 = U11 > activate# = n__plus = plus# = isNat# = U41# = U42# > |0| = tt > U31 = U21 = n__0
    
    partial status:
  
      pi(activate#) = []
      pi(n__plus) = []
      pi(plus#) = []
      pi(activate) = [1]
      pi(|0|) = []
      pi(isNat#) = []
      pi(n__s) = []
      pi(s) = []
      pi(U41#) = []
      pi(isNat) = []
      pi(tt) = []
      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:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: plus#(N,|0|()) -> isNat#(N)
p3: isNat#(n__plus(V1,V2)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> activate#(X1)
p5: activate#(n__plus(X1,X2)) -> activate#(X2)
p6: isNat#(n__s(V1)) -> activate#(V1)
p7: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p8: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p9: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p10: plus#(N,s(M)) -> isNat#(M)
p11: U42#(tt(),M,N) -> activate#(N)
p12: U42#(tt(),M,N) -> activate#(M)
p13: U41#(tt(),M,N) -> activate#(N)
p14: 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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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}


-- Reduction pair.

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

p1: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p2: plus#(N,s(M)) -> isNat#(M)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__plus(X1,X2)) -> activate#(X2)
p5: activate#(n__plus(X1,X2)) -> activate#(X1)
p6: isNat#(n__plus(V1,V2)) -> 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) -> U42#(isNat(activate(N)),activate(M),activate(N))
p11: U42#(tt(),M,N) -> activate#(M)
p12: U42#(tt(),M,N) -> activate#(N)
p13: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p14: 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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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:
          activate#_A(x1) = x1 + (13,15)
          n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (14,11)
          plus#_A(x1,x2) = x1 + x2 + (13,2)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,12)
          s_A(x1) = ((1,0),(0,0)) x1 + (27,14)
          isNat#_A(x1) = x1 + (2,13)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (27,13)
          U41#_A(x1,x2,x3) = x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (13,5)
          isNat_A(x1) = (16,10)
          tt_A() = (2,4)
          U42#_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (14,13)
          |0|_A() = (1,12)
          U42_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (41,37)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (14,12)
          U12_A(x1) = ((0,0),(1,0)) x1 + (3,1)
          U31_A(x1,x2) = ((1,0),(0,0)) x2 + (2,1)
          U41_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (41,36)
          U11_A(x1,x2) = (13,9)
          U21_A(x1) = (16,5)
          n__0_A() = (1,3)
  
    precedence:
    
      U41# = U42# > |0| = U12 = U31 > activate# = n__plus = plus# = isNat# > activate = plus > U42 = U41 > s > n__s > isNat = tt = U21 > n__0 > U11
    
    partial status:
  
      pi(activate#) = [1]
      pi(n__plus) = [1, 2]
      pi(plus#) = [1, 2]
      pi(activate) = [1]
      pi(s) = []
      pi(isNat#) = [1]
      pi(n__s) = []
      pi(U41#) = []
      pi(isNat) = []
      pi(tt) = []
      pi(U42#) = []
      pi(|0|) = []
      pi(U42) = []
      pi(plus) = [1]
      pi(U12) = []
      pi(U31) = []
      pi(U41) = []
      pi(U11) = []
      pi(U21) = []
      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: plus#(N,s(M)) -> isNat#(M)
p2: isNat#(n__s(V1)) -> activate#(V1)
p3: activate#(n__plus(X1,X2)) -> activate#(X2)
p4: activate#(n__plus(X1,X2)) -> activate#(X1)
p5: isNat#(n__plus(V1,V2)) -> activate#(V1)
p6: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p7: U41#(tt(),M,N) -> activate#(M)
p8: U41#(tt(),M,N) -> activate#(N)
p9: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),activate(N))
p10: U42#(tt(),M,N) -> activate#(M)
p11: U42#(tt(),M,N) -> activate#(N)
p12: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p13: 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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(X))
r18: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p6, p9, p12}
  {p3, p4}


-- Reduction pair.

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

p1: U42#(tt(),M,N) -> plus#(activate(N),activate(M))
p2: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p3: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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:
          U42#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (9,3)
          tt_A() = (7,3)
          plus#_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (0,1)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,1)
          s_A(x1) = ((1,0),(0,0)) x1 + (9,4)
          U41#_A(x1,x2,x3) = x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (2,0)
          isNat_A(x1) = ((0,0),(1,0)) x1 + (7,0)
          U42_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (15,5)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (6,7)
          U12_A(x1) = ((1,0),(0,0)) x1 + (0,4)
          U31_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (5,1)
          U41_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (8,6)
          U11_A(x1,x2) = (7,5)
          U21_A(x1) = ((1,0),(1,1)) x1 + (0,1)
          |0|_A() = (4,5)
          n__0_A() = (4,4)
          n__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (6,2)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (9,0)
  
    precedence:
    
      activate > plus = U31 = n__plus > U42 = U41 > isNat > plus# > s > |0| > n__s > tt = U12 = U11 = U21 > n__0 > U42# = U41#
    
    partial status:
  
      pi(U42#) = []
      pi(tt) = []
      pi(plus#) = [1]
      pi(activate) = []
      pi(s) = []
      pi(U41#) = []
      pi(isNat) = []
      pi(U42) = []
      pi(plus) = []
      pi(U12) = []
      pi(U31) = []
      pi(U41) = []
      pi(U11) = []
      pi(U21) = [1]
      pi(|0|) = []
      pi(n__0) = []
      pi(n__plus) = []
      pi(n__s) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: plus#(N,s(M)) -> U41#(isNat(M),M,N)
p2: U41#(tt(),M,N) -> U42#(isNat(activate(N)),activate(M),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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(X))
r18: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: activate#(n__plus(X1,X2)) -> activate#(X2)
p2: activate#(n__plus(X1,X2)) -> activate#(X1)

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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(X))
r18: activate(X) -> X

The set of usable rules consists of

  (no rules)

Take the reduction pair:

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

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)) -> activate#(X1)

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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(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: activate#(n__plus(X1,X2)) -> activate#(X1)

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(activate(X1),activate(X2))
r17: activate(n__s(X)) -> s(activate(X))
r18: activate(X) -> X

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

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

The next rules are strictly ordered:

  p1

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