YES

We show the termination of the TRS R:

  U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
  U12(tt(),V2) -> U13(isNat(activate(V2)))
  U13(tt()) -> tt()
  U21(tt(),V1) -> U22(isNat(activate(V1)))
  U22(tt()) -> tt()
  U31(tt(),N) -> activate(N)
  U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
  and(tt(),X) -> activate(X)
  isNat(n__0()) -> tt()
  isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
  isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
  isNatKind(n__0()) -> tt()
  isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
  isNatKind(n__s(V1)) -> isNatKind(activate(V1))
  plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
  plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
  |0|() -> n__0()
  plus(X1,X2) -> n__plus(X1,X2)
  isNatKind(X) -> n__isNatKind(X)
  s(X) -> n__s(X)
  and(X1,X2) -> n__and(X1,X2)
  isNat(X) -> n__isNat(X)
  activate(n__0()) -> |0|()
  activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
  activate(n__isNatKind(X)) -> isNatKind(X)
  activate(n__s(X)) -> s(activate(X))
  activate(n__and(X1,X2)) -> and(activate(X1),X2)
  activate(n__isNat(X)) -> isNat(X)
  activate(X) -> X

-- SCC decomposition.

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

p1: U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2))
p2: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p3: U11#(tt(),V1,V2) -> activate#(V1)
p4: U11#(tt(),V1,V2) -> activate#(V2)
p5: U12#(tt(),V2) -> U13#(isNat(activate(V2)))
p6: U12#(tt(),V2) -> isNat#(activate(V2))
p7: U12#(tt(),V2) -> activate#(V2)
p8: U21#(tt(),V1) -> U22#(isNat(activate(V1)))
p9: U21#(tt(),V1) -> isNat#(activate(V1))
p10: U21#(tt(),V1) -> activate#(V1)
p11: U31#(tt(),N) -> activate#(N)
p12: U41#(tt(),M,N) -> s#(plus(activate(N),activate(M)))
p13: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p14: U41#(tt(),M,N) -> activate#(N)
p15: U41#(tt(),M,N) -> activate#(M)
p16: and#(tt(),X) -> activate#(X)
p17: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p18: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p19: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p20: isNat#(n__plus(V1,V2)) -> activate#(V1)
p21: isNat#(n__plus(V1,V2)) -> activate#(V2)
p22: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p23: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p24: isNat#(n__s(V1)) -> activate#(V1)
p25: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p26: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p27: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p28: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p29: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p30: isNatKind#(n__s(V1)) -> activate#(V1)
p31: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p32: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p33: plus#(N,|0|()) -> isNat#(N)
p34: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
p35: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N)))
p36: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p37: plus#(N,s(M)) -> isNat#(M)
p38: activate#(n__0()) -> |0|#()
p39: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p40: activate#(n__plus(X1,X2)) -> activate#(X1)
p41: activate#(n__plus(X1,X2)) -> activate#(X2)
p42: activate#(n__isNatKind(X)) -> isNatKind#(X)
p43: activate#(n__s(X)) -> s#(activate(X))
p44: activate#(n__s(X)) -> activate#(X)
p45: activate#(n__and(X1,X2)) -> and#(activate(X1),X2)
p46: activate#(n__and(X1,X2)) -> activate#(X1)
p47: activate#(n__isNat(X)) -> isNat#(X)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: isNat(X) -> n__isNat(X)
r23: activate(n__0()) -> |0|()
r24: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r25: activate(n__isNatKind(X)) -> isNatKind(X)
r26: activate(n__s(X)) -> s(activate(X))
r27: activate(n__and(X1,X2)) -> and(activate(X1),X2)
r28: activate(n__isNat(X)) -> isNat(X)
r29: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p6, p7, p9, p10, p11, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p39, p40, p41, p42, p44, p45, p46, p47}


-- Reduction pair.

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

p1: U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2))
p2: U12#(tt(),V2) -> activate#(V2)
p3: activate#(n__isNat(X)) -> isNat#(X)
p4: isNat#(n__s(V1)) -> activate#(V1)
p5: activate#(n__and(X1,X2)) -> activate#(X1)
p6: activate#(n__and(X1,X2)) -> and#(activate(X1),X2)
p7: and#(tt(),X) -> activate#(X)
p8: activate#(n__s(X)) -> activate#(X)
p9: activate#(n__isNatKind(X)) -> isNatKind#(X)
p10: isNatKind#(n__s(V1)) -> activate#(V1)
p11: activate#(n__plus(X1,X2)) -> activate#(X2)
p12: activate#(n__plus(X1,X2)) -> activate#(X1)
p13: activate#(n__plus(X1,X2)) -> plus#(activate(X1),activate(X2))
p14: plus#(N,s(M)) -> isNat#(M)
p15: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p16: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p17: isNatKind#(n__plus(V1,V2)) -> activate#(V2)
p18: isNatKind#(n__plus(V1,V2)) -> activate#(V1)
p19: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p20: isNatKind#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p21: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p22: U21#(tt(),V1) -> activate#(V1)
p23: U21#(tt(),V1) -> isNat#(activate(V1))
p24: isNat#(n__plus(V1,V2)) -> activate#(V2)
p25: isNat#(n__plus(V1,V2)) -> activate#(V1)
p26: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p27: isNat#(n__plus(V1,V2)) -> and#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
p28: isNat#(n__plus(V1,V2)) -> U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
p29: U11#(tt(),V1,V2) -> activate#(V2)
p30: U11#(tt(),V1,V2) -> activate#(V1)
p31: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p32: plus#(N,s(M)) -> and#(isNat(M),n__isNatKind(M))
p33: plus#(N,s(M)) -> and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N)))
p34: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
p35: U41#(tt(),M,N) -> activate#(M)
p36: U41#(tt(),M,N) -> activate#(N)
p37: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p38: plus#(N,|0|()) -> isNat#(N)
p39: plus#(N,|0|()) -> and#(isNat(N),n__isNatKind(N))
p40: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)
p41: U31#(tt(),N) -> activate#(N)
p42: U12#(tt(),V2) -> isNat#(activate(V2))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: isNat(X) -> n__isNat(X)
r23: activate(n__0()) -> |0|()
r24: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r25: activate(n__isNatKind(X)) -> isNatKind(X)
r26: activate(n__s(X)) -> s(activate(X))
r27: activate(n__and(X1,X2)) -> and(activate(X1),X2)
r28: activate(n__isNat(X)) -> isNat(X)
r29: 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, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          U11#_A(x1,x2,x3) = max{x1 + 33, x2 + 102, x3 + 111}
          tt_A = 68
          U12#_A(x1,x2) = max{x1 - 64, x2 + 103}
          isNat_A(x1) = max{164, x1 + 42}
          activate_A(x1) = x1
          activate#_A(x1) = max{2, x1 - 1}
          n__isNat_A(x1) = max{164, x1 + 42}
          isNat#_A(x1) = max{3, x1}
          n__s_A(x1) = max{113, x1}
          n__and_A(x1,x2) = max{x1 + 31, x2 + 25}
          and#_A(x1,x2) = max{x1 + 1, x2 + 3}
          n__isNatKind_A(x1) = x1 + 47
          isNatKind#_A(x1) = max{3, x1}
          n__plus_A(x1,x2) = max{x1 + 120, x2 + 112}
          plus#_A(x1,x2) = max{x1 + 76, x2 + 105}
          s_A(x1) = max{113, x1}
          isNatKind_A(x1) = x1 + 47
          U21#_A(x1,x2) = max{x1 - 48, x2}
          and_A(x1,x2) = max{x1 + 31, x2 + 25}
          U41#_A(x1,x2,x3) = max{x1 - 22, x2 + 105, x3 + 76}
          |0|_A = 132
          U31#_A(x1,x2) = max{x1 + 3, x2 + 76}
          U13_A(x1) = max{69, x1 - 162}
          U12_A(x1,x2) = max{x1 - 65, x2 + 100}
          U22_A(x1) = max{1, x1}
          U31_A(x1,x2) = max{121, x2 + 120}
          U41_A(x1,x2,x3) = max{x2 + 112, x3 + 120}
          plus_A(x1,x2) = max{x1 + 120, x2 + 112}
          U11_A(x1,x2,x3) = max{163, x1 + 33, x2 + 162, x3 + 100}
          U21_A(x1,x2) = max{164, x1 - 5, x2 + 42}
          n__0_A = 132
    
      precedence:
      
        isNat = n__isNat = U11 > U12 > U13 > plus# = U41# > |0| = n__0 > U21 > U11# = n__plus = plus > U41 > U31# > n__s = n__isNatKind = s = isNatKind > n__and = and > activate > U12# > U31 > and# > activate# > tt = isNat# = isNatKind# = U21# = U22
      
      partial status:
    
        pi(U11#) = []
        pi(tt) = []
        pi(U12#) = []
        pi(isNat) = []
        pi(activate) = [1]
        pi(activate#) = []
        pi(n__isNat) = []
        pi(isNat#) = []
        pi(n__s) = []
        pi(n__and) = []
        pi(and#) = [1, 2]
        pi(n__isNatKind) = []
        pi(isNatKind#) = []
        pi(n__plus) = []
        pi(plus#) = []
        pi(s) = []
        pi(isNatKind) = []
        pi(U21#) = []
        pi(and) = []
        pi(U41#) = []
        pi(|0|) = []
        pi(U31#) = [1]
        pi(U13) = []
        pi(U12) = [2]
        pi(U22) = []
        pi(U31) = [2]
        pi(U41) = [3]
        pi(plus) = []
        pi(U11) = []
        pi(U21) = [2]
        pi(n__0) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          U11#_A(x1,x2,x3) = 1077
          tt_A = 92
          U12#_A(x1,x2) = 122
          isNat_A(x1) = 1076
          activate_A(x1) = x1 + 353
          activate#_A(x1) = 122
          n__isNat_A(x1) = 580
          isNat#_A(x1) = 148
          n__s_A(x1) = 75
          n__and_A(x1,x2) = 36
          and#_A(x1,x2) = max{120, x1 + 30}
          n__isNatKind_A(x1) = 115
          isNatKind#_A(x1) = 147
          n__plus_A(x1,x2) = 119
          plus#_A(x1,x2) = 121
          s_A(x1) = 75
          isNatKind_A(x1) = 116
          U21#_A(x1,x2) = 148
          and_A(x1,x2) = 75
          U41#_A(x1,x2,x3) = 121
          |0|_A = 1194
          U31#_A(x1,x2) = max{120, x1 + 46}
          U13_A(x1) = 1278
          U12_A(x1,x2) = x2 + 1279
          U22_A(x1) = 92
          U31_A(x1,x2) = max{1345, x2 + 354}
          U41_A(x1,x2,x3) = 76
          plus_A(x1,x2) = 472
          U11_A(x1,x2,x3) = 0
          U21_A(x1,x2) = 1076
          n__0_A = 0
    
      precedence:
      
        plus# = U41# > U11# = U12# > isNat# = isNatKind# = U21# > U13 > activate = n__isNatKind = U31 > isNat = plus > U31# > activate# = and# > U41 = n__0 > n__s = s > isNatKind = and > n__and = n__plus = |0| = U12 = U11 > U21 > U22 > tt > n__isNat
      
      partial status:
    
        pi(U11#) = []
        pi(tt) = []
        pi(U12#) = []
        pi(isNat) = []
        pi(activate) = []
        pi(activate#) = []
        pi(n__isNat) = []
        pi(isNat#) = []
        pi(n__s) = []
        pi(n__and) = []
        pi(and#) = [1]
        pi(n__isNatKind) = []
        pi(isNatKind#) = []
        pi(n__plus) = []
        pi(plus#) = []
        pi(s) = []
        pi(isNatKind) = []
        pi(U21#) = []
        pi(and) = []
        pi(U41#) = []
        pi(|0|) = []
        pi(U31#) = [1]
        pi(U13) = []
        pi(U12) = [2]
        pi(U22) = []
        pi(U31) = [2]
        pi(U41) = []
        pi(plus) = []
        pi(U11) = []
        pi(U21) = []
        pi(n__0) = []
    

The next rules are strictly ordered:

  p3, p6, p7, p9, p10, p12, p13, p14, p17, p18, p20, p27, p28, p29, p30, p32, p33, p35, p36, p38, p39, p41, p42

We remove them from the problem.

-- SCC decomposition.

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

p1: U11#(tt(),V1,V2) -> U12#(isNat(activate(V1)),activate(V2))
p2: U12#(tt(),V2) -> activate#(V2)
p3: isNat#(n__s(V1)) -> activate#(V1)
p4: activate#(n__and(X1,X2)) -> activate#(X1)
p5: activate#(n__s(X)) -> activate#(X)
p6: activate#(n__plus(X1,X2)) -> activate#(X2)
p7: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
p8: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p9: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p10: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
p11: U21#(tt(),V1) -> activate#(V1)
p12: U21#(tt(),V1) -> isNat#(activate(V1))
p13: isNat#(n__plus(V1,V2)) -> activate#(V2)
p14: isNat#(n__plus(V1,V2)) -> activate#(V1)
p15: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
p16: U11#(tt(),V1,V2) -> isNat#(activate(V1))
p17: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
p18: U41#(tt(),M,N) -> plus#(activate(N),activate(M))
p19: plus#(N,|0|()) -> U31#(and(isNat(N),n__isNatKind(N)),N)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: isNat(X) -> n__isNat(X)
r23: activate(n__0()) -> |0|()
r24: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r25: activate(n__isNatKind(X)) -> isNatKind(X)
r26: activate(n__s(X)) -> s(activate(X))
r27: activate(n__and(X1,X2)) -> and(activate(X1),X2)
r28: activate(n__isNat(X)) -> isNat(X)
r29: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p10, p12}
  {p4, p5, p6}
  {p8, p9}
  {p17, p18}


-- Reduction pair.

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

p1: U21#(tt(),V1) -> isNat#(activate(V1))
p2: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: isNat(X) -> n__isNat(X)
r23: activate(n__0()) -> |0|()
r24: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r25: activate(n__isNatKind(X)) -> isNatKind(X)
r26: activate(n__s(X)) -> s(activate(X))
r27: activate(n__and(X1,X2)) -> and(activate(X1),X2)
r28: activate(n__isNat(X)) -> isNat(X)
r29: 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, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            U21#_A(x1,x2) = ((1,0),(1,0)) x2 + (6,5)
            tt_A() = (0,0)
            isNat#_A(x1) = ((1,0),(1,0)) x1 + (2,5)
            activate_A(x1) = x1
            n__s_A(x1) = x1 + (5,4)
            isNatKind_A(x1) = ((0,1),(0,1)) x1 + (1,0)
            U13_A(x1) = ((0,0),(0,1)) x1 + (1,0)
            U12_A(x1,x2) = (3,3)
            isNat_A(x1) = ((1,0),(0,0)) x1 + (0,3)
            U22_A(x1) = (4,1)
            U11_A(x1,x2,x3) = (3,3)
            U21_A(x1,x2) = (5,2)
            U31_A(x1,x2) = ((1,0),(1,1)) x2 + (2,0)
            U41_A(x1,x2,x3) = ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (8,5)
            s_A(x1) = x1 + (5,4)
            plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (3,1)
            and_A(x1,x2) = ((0,1),(1,0)) x1 + x2
            n__0_A() = (1,0)
            n__plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (3,1)
            n__isNatKind_A(x1) = ((0,1),(0,1)) x1 + (1,0)
            |0|_A() = (1,0)
            n__and_A(x1,x2) = ((0,1),(1,0)) x1 + x2
            n__isNat_A(x1) = ((1,0),(0,0)) x1 + (0,3)
    
      precedence:
      
        U21# = tt = isNat# = activate = n__s = isNatKind = U13 = U12 = isNat = U22 = U11 = U21 = U31 = U41 = s = plus = and = n__0 = n__plus = n__isNatKind = |0| = n__and = n__isNat
      
      partial status:
    
        pi(U21#) = []
        pi(tt) = []
        pi(isNat#) = []
        pi(activate) = []
        pi(n__s) = []
        pi(isNatKind) = []
        pi(U13) = []
        pi(U12) = []
        pi(isNat) = []
        pi(U22) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(s) = []
        pi(plus) = []
        pi(and) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(n__isNatKind) = []
        pi(|0|) = []
        pi(n__and) = []
        pi(n__isNat) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            U21#_A(x1,x2) = ((1,1),(0,0)) x2 + (7,6)
            tt_A() = (0,1)
            isNat#_A(x1) = ((1,0),(0,0)) x1 + (4,5)
            activate_A(x1) = ((1,1),(0,1)) x1 + (2,5)
            n__s_A(x1) = (3,5)
            isNatKind_A(x1) = (6,9)
            U13_A(x1) = (0,1)
            U12_A(x1,x2) = (1,4)
            isNat_A(x1) = ((1,1),(1,1)) x1 + (1,5)
            U22_A(x1) = (0,0)
            U11_A(x1,x2,x3) = (1,12)
            U21_A(x1,x2) = (1,0)
            U31_A(x1,x2) = ((1,1),(0,1)) x2 + (2,5)
            U41_A(x1,x2,x3) = (5,6)
            s_A(x1) = (4,5)
            plus_A(x1,x2) = ((1,1),(0,1)) x1 + (8,7)
            and_A(x1,x2) = ((1,1),(1,1)) x2 + (2,5)
            n__0_A() = (0,0)
            n__plus_A(x1,x2) = ((1,1),(0,1)) x1 + (7,7)
            n__isNatKind_A(x1) = (0,4)
            |0|_A() = (0,4)
            n__and_A(x1,x2) = ((0,0),(1,1)) x2 + (1,4)
            n__isNat_A(x1) = ((1,1),(1,1)) x1
    
      precedence:
      
        |0| > U21# > U11 > U12 > U13 > U41 > n__s > U31 > activate = isNatKind = and > tt > plus = n__and > n__0 > s > isNat# > n__plus > isNat > n__isNatKind = n__isNat > U21 > U22
      
      partial status:
    
        pi(U21#) = []
        pi(tt) = []
        pi(isNat#) = []
        pi(activate) = []
        pi(n__s) = []
        pi(isNatKind) = []
        pi(U13) = []
        pi(U12) = []
        pi(isNat) = []
        pi(U22) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(s) = []
        pi(plus) = [1]
        pi(and) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(n__isNatKind) = []
        pi(|0|) = []
        pi(n__and) = []
        pi(n__isNat) = []
    

The next rules are strictly ordered:

  p1, p2

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

-- Reduction pair.

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

p1: activate#(n__and(X1,X2)) -> activate#(X1)
p2: activate#(n__plus(X1,X2)) -> activate#(X2)
p3: activate#(n__s(X)) -> activate#(X)

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: isNat(X) -> n__isNat(X)
r23: activate(n__0()) -> |0|()
r24: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r25: activate(n__isNatKind(X)) -> isNatKind(X)
r26: activate(n__s(X)) -> s(activate(X))
r27: activate(n__and(X1,X2)) -> and(activate(X1),X2)
r28: activate(n__isNat(X)) -> isNat(X)
r29: activate(X) -> X

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          activate#_A(x1) = x1 + 4
          n__and_A(x1,x2) = max{x1, x2}
          n__plus_A(x1,x2) = max{x1 - 1, x2}
          n__s_A(x1) = max{1, x1}
    
      precedence:
      
        activate# = n__and = n__plus = n__s
      
      partial status:
    
        pi(activate#) = [1]
        pi(n__and) = [1, 2]
        pi(n__plus) = [2]
        pi(n__s) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          activate#_A(x1) = x1 + 2
          n__and_A(x1,x2) = max{x1, x2}
          n__plus_A(x1,x2) = x2
          n__s_A(x1) = x1
    
      precedence:
      
        activate# = n__and = n__plus = n__s
      
      partial status:
    
        pi(activate#) = [1]
        pi(n__and) = [1, 2]
        pi(n__plus) = [2]
        pi(n__s) = [1]
    

The next rules are strictly ordered:

  p1, p2, p3

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

-- Reduction pair.

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

p1: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
p2: isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: isNat(X) -> n__isNat(X)
r23: activate(n__0()) -> |0|()
r24: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r25: activate(n__isNatKind(X)) -> isNatKind(X)
r26: activate(n__s(X)) -> s(activate(X))
r27: activate(n__and(X1,X2)) -> and(activate(X1),X2)
r28: activate(n__isNat(X)) -> isNat(X)
r29: 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, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          isNatKind#_A(x1) = x1 + 64
          n__s_A(x1) = max{3, x1}
          activate_A(x1) = x1
          n__plus_A(x1,x2) = max{63, x1 + 43, x2 + 15}
          U13_A(x1) = 16
          tt_A = 0
          U12_A(x1,x2) = 17
          isNat_A(x1) = x1 + 18
          U22_A(x1) = max{1, x1 - 20}
          U11_A(x1,x2,x3) = max{27, x2 + 16}
          U21_A(x1,x2) = max{2, x2 - 1}
          U31_A(x1,x2) = max{62, x1 - 18, x2 + 1}
          U41_A(x1,x2,x3) = max{63, x2 + 15, x3 + 43}
          s_A(x1) = max{3, x1}
          plus_A(x1,x2) = max{63, x1 + 43, x2 + 15}
          and_A(x1,x2) = max{x1 + 37, x2 + 15}
          n__0_A = 46
          isNatKind_A(x1) = x1 + 45
          n__isNatKind_A(x1) = x1 + 45
          |0|_A = 46
          n__and_A(x1,x2) = max{x1 + 37, x2 + 15}
          n__isNat_A(x1) = x1 + 18
    
      precedence:
      
        isNatKind# = n__s = activate = n__plus = U13 = tt = U12 = isNat = U22 = U11 = U21 = U31 = U41 = s = plus = and = n__0 = isNatKind = n__isNatKind = |0| = n__and = n__isNat
      
      partial status:
    
        pi(isNatKind#) = []
        pi(n__s) = []
        pi(activate) = []
        pi(n__plus) = []
        pi(U13) = []
        pi(tt) = []
        pi(U12) = []
        pi(isNat) = []
        pi(U22) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(s) = []
        pi(plus) = []
        pi(and) = []
        pi(n__0) = []
        pi(isNatKind) = []
        pi(n__isNatKind) = []
        pi(|0|) = []
        pi(n__and) = []
        pi(n__isNat) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          isNatKind#_A(x1) = 251
          n__s_A(x1) = 11
          activate_A(x1) = x1 + 184
          n__plus_A(x1,x2) = 166
          U13_A(x1) = 319
          tt_A = 320
          U12_A(x1,x2) = 319
          isNat_A(x1) = x1 + 235
          U22_A(x1) = 659
          U11_A(x1,x2,x3) = 400
          U21_A(x1,x2) = 318
          U31_A(x1,x2) = 1
          U41_A(x1,x2,x3) = 119
          s_A(x1) = 12
          plus_A(x1,x2) = 350
          and_A(x1,x2) = 82
          n__0_A = 43
          isNatKind_A(x1) = 82
          n__isNatKind_A(x1) = 82
          |0|_A = 44
          n__and_A(x1,x2) = 1
          n__isNat_A(x1) = x1 + 51
    
      precedence:
      
        activate = U41 = s = n__and > isNatKind# = n__s = n__plus = U13 = tt = U12 = isNat = U22 = U11 = U21 = U31 = plus = n__0 = isNatKind = n__isNat > and = n__isNatKind = |0|
      
      partial status:
    
        pi(isNatKind#) = []
        pi(n__s) = []
        pi(activate) = [1]
        pi(n__plus) = []
        pi(U13) = []
        pi(tt) = []
        pi(U12) = []
        pi(isNat) = [1]
        pi(U22) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(s) = []
        pi(plus) = []
        pi(and) = []
        pi(n__0) = []
        pi(isNatKind) = []
        pi(n__isNatKind) = []
        pi(|0|) = []
        pi(n__and) = []
        pi(n__isNat) = []
    

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: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: isNat(X) -> n__isNat(X)
r23: activate(n__0()) -> |0|()
r24: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r25: activate(n__isNatKind(X)) -> isNatKind(X)
r26: activate(n__s(X)) -> s(activate(X))
r27: activate(n__and(X1,X2)) -> and(activate(X1),X2)
r28: activate(n__isNat(X)) -> isNat(X)
r29: 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: isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: isNat(X) -> n__isNat(X)
r23: activate(n__0()) -> |0|()
r24: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r25: activate(n__isNatKind(X)) -> isNatKind(X)
r26: activate(n__s(X)) -> s(activate(X))
r27: activate(n__and(X1,X2)) -> and(activate(X1),X2)
r28: activate(n__isNat(X)) -> isNat(X)
r29: 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, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            isNatKind#_A(x1) = ((1,0),(1,0)) x1 + (0,1)
            n__s_A(x1) = ((1,0),(1,0)) x1 + (5,0)
            activate_A(x1) = x1
            U13_A(x1) = (3,2)
            tt_A() = (2,1)
            U12_A(x1,x2) = (4,2)
            isNat_A(x1) = ((1,1),(1,0)) x1 + (0,3)
            U22_A(x1) = (3,1)
            U11_A(x1,x2,x3) = ((0,0),(1,0)) x1 + (5,0)
            U21_A(x1,x2) = ((0,1),(0,0)) x1 + ((1,0),(1,0)) x2 + (3,1)
            U31_A(x1,x2) = ((1,1),(1,1)) x2 + (2,0)
            U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,1),(1,1)) x3 + (10,5)
            s_A(x1) = ((1,0),(1,0)) x1 + (5,0)
            plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,0),(1,1)) x2 + (5,0)
            and_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (4,1)
            n__0_A() = (1,1)
            n__plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,0),(1,1)) x2 + (5,0)
            isNatKind_A(x1) = ((1,0),(1,0)) x1 + (3,0)
            n__isNatKind_A(x1) = ((1,0),(1,0)) x1 + (3,0)
            |0|_A() = (1,1)
            n__and_A(x1,x2) = ((0,0),(1,0)) x1 + x2 + (4,1)
            n__isNat_A(x1) = ((1,1),(1,0)) x1 + (0,3)
    
      precedence:
      
        isNatKind = n__isNatKind > and = n__and > isNatKind# = activate = isNat = U41 = s = plus > U21 > U11 = n__plus > U12 = U22 > U13 = tt > |0| > U31 = n__isNat > n__0 > n__s
      
      partial status:
    
        pi(isNatKind#) = []
        pi(n__s) = []
        pi(activate) = [1]
        pi(U13) = []
        pi(tt) = []
        pi(U12) = []
        pi(isNat) = []
        pi(U22) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = [2]
        pi(U41) = []
        pi(s) = []
        pi(plus) = [2]
        pi(and) = [2]
        pi(n__0) = []
        pi(n__plus) = [2]
        pi(isNatKind) = []
        pi(n__isNatKind) = []
        pi(|0|) = []
        pi(n__and) = [2]
        pi(n__isNat) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            isNatKind#_A(x1) = (0,0)
            n__s_A(x1) = (0,1)
            activate_A(x1) = ((0,1),(0,1)) x1 + (5,0)
            U13_A(x1) = (2,1)
            tt_A() = (1,0)
            U12_A(x1,x2) = (6,1)
            isNat_A(x1) = (0,3)
            U22_A(x1) = (2,0)
            U11_A(x1,x2,x3) = (0,0)
            U21_A(x1,x2) = (0,2)
            U31_A(x1,x2) = (0,0)
            U41_A(x1,x2,x3) = (3,0)
            s_A(x1) = (2,0)
            plus_A(x1,x2) = (4,0)
            and_A(x1,x2) = ((0,1),(0,1)) x2 + (5,3)
            n__0_A() = (0,0)
            n__plus_A(x1,x2) = (1,0)
            isNatKind_A(x1) = (5,3)
            n__isNatKind_A(x1) = (0,0)
            |0|_A() = (5,0)
            n__and_A(x1,x2) = ((0,1),(0,1)) x2 + (5,3)
            n__isNat_A(x1) = (0,3)
    
      precedence:
      
        isNat = isNatKind = n__isNatKind > U11 > n__0 > U22 > plus > U31 > U13 = tt > U41 > and > isNatKind# = n__s = activate = U12 = U21 = s = n__isNat > n__and > |0| > n__plus
      
      partial status:
    
        pi(isNatKind#) = []
        pi(n__s) = []
        pi(activate) = []
        pi(U13) = []
        pi(tt) = []
        pi(U12) = []
        pi(isNat) = []
        pi(U22) = []
        pi(U11) = []
        pi(U21) = []
        pi(U31) = []
        pi(U41) = []
        pi(s) = []
        pi(plus) = []
        pi(and) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(isNatKind) = []
        pi(n__isNatKind) = []
        pi(|0|) = []
        pi(n__and) = []
        pi(n__isNat) = []
    

The next rules are strictly ordered:

  p1

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

-- Reduction pair.

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

p1: plus#(N,s(M)) -> U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
p2: U41#(tt(),M,N) -> plus#(activate(N),activate(M))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: isNat(X) -> n__isNat(X)
r23: activate(n__0()) -> |0|()
r24: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r25: activate(n__isNatKind(X)) -> isNatKind(X)
r26: activate(n__s(X)) -> s(activate(X))
r27: activate(n__and(X1,X2)) -> and(activate(X1),X2)
r28: activate(n__isNat(X)) -> isNat(X)
r29: 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, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            plus#_A(x1,x2) = ((0,1),(0,0)) x2 + (24,21)
            s_A(x1) = ((0,0),(0,1)) x1 + (21,20)
            U41#_A(x1,x2,x3) = ((0,1),(0,0)) x2 + (25,21)
            and_A(x1,x2) = ((1,1),(0,1)) x2 + (1,0)
            isNat_A(x1) = ((0,1),(0,1)) x1 + (35,15)
            n__isNatKind_A(x1) = ((0,0),(0,1)) x1 + (22,10)
            n__and_A(x1,x2) = ((1,1),(0,1)) x2
            n__isNat_A(x1) = ((0,0),(0,1)) x1 + (19,15)
            tt_A() = (34,19)
            activate_A(x1) = ((1,1),(0,1)) x1 + (1,0)
            U13_A(x1) = x1 + (1,0)
            U12_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,1)) x2 + (18,15)
            U22_A(x1) = ((0,0),(1,0)) x1 + (36,0)
            U31_A(x1,x2) = ((1,1),(0,1)) x2 + (8,9)
            U41_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((0,0),(0,1)) x2 + x3 + (3,20)
            plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,1),(0,1)) x2
            U11_A(x1,x2,x3) = ((0,1),(0,0)) x2 + ((0,1),(0,1)) x3 + (34,15)
            U21_A(x1,x2) = ((0,0),(0,1)) x2 + (36,35)
            isNatKind_A(x1) = ((0,1),(0,1)) x1 + (33,10)
            n__0_A() = (24,9)
            n__plus_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,1),(0,1)) x2
            n__s_A(x1) = ((0,0),(0,1)) x1 + (0,20)
            |0|_A() = (34,9)
    
      precedence:
      
        plus# > and = activate = U31 = isNatKind > isNat > n__and = n__isNat = U21 > plus > U22 = U41 > s = U41# > n__plus > U11 > U12 > tt = U13 > n__s > |0| > n__0 > n__isNatKind
      
      partial status:
    
        pi(plus#) = []
        pi(s) = []
        pi(U41#) = []
        pi(and) = []
        pi(isNat) = []
        pi(n__isNatKind) = []
        pi(n__and) = []
        pi(n__isNat) = []
        pi(tt) = []
        pi(activate) = []
        pi(U13) = [1]
        pi(U12) = []
        pi(U22) = []
        pi(U31) = [2]
        pi(U41) = []
        pi(plus) = []
        pi(U11) = []
        pi(U21) = []
        pi(isNatKind) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(n__s) = []
        pi(|0|) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: standard order
          interpretations:
            plus#_A(x1,x2) = (5,3)
            s_A(x1) = (7,3)
            U41#_A(x1,x2,x3) = (6,3)
            and_A(x1,x2) = (10,3)
            isNat_A(x1) = (4,2)
            n__isNatKind_A(x1) = (0,0)
            n__and_A(x1,x2) = (9,0)
            n__isNat_A(x1) = (1,2)
            tt_A() = (1,0)
            activate_A(x1) = (10,3)
            U13_A(x1) = (1,0)
            U12_A(x1,x2) = (2,2)
            U22_A(x1) = (2,1)
            U31_A(x1,x2) = (11,3)
            U41_A(x1,x2,x3) = (8,3)
            plus_A(x1,x2) = (8,3)
            U11_A(x1,x2,x3) = (3,2)
            U21_A(x1,x2) = (3,2)
            isNatKind_A(x1) = (10,3)
            n__0_A() = (0,1)
            n__plus_A(x1,x2) = (0,0)
            n__s_A(x1) = (1,2)
            |0|_A() = (0,1)
    
      precedence:
      
        |0| > n__0 > n__s > n__plus > n__isNat = plus > U41 > U31 > plus# = s = U41# > and = activate = isNatKind > isNat > U11 > U13 = U12 > n__isNatKind > U21 > U22 > tt > n__and
      
      partial status:
    
        pi(plus#) = []
        pi(s) = []
        pi(U41#) = []
        pi(and) = []
        pi(isNat) = []
        pi(n__isNatKind) = []
        pi(n__and) = []
        pi(n__isNat) = []
        pi(tt) = []
        pi(activate) = []
        pi(U13) = []
        pi(U12) = []
        pi(U22) = []
        pi(U31) = []
        pi(U41) = []
        pi(plus) = []
        pi(U11) = []
        pi(U21) = []
        pi(isNatKind) = []
        pi(n__0) = []
        pi(n__plus) = []
        pi(n__s) = []
        pi(|0|) = []
    

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: U41#(tt(),M,N) -> plus#(activate(N),activate(M))

and R consists of:

r1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
r2: U12(tt(),V2) -> U13(isNat(activate(V2)))
r3: U13(tt()) -> tt()
r4: U21(tt(),V1) -> U22(isNat(activate(V1)))
r5: U22(tt()) -> tt()
r6: U31(tt(),N) -> activate(N)
r7: U41(tt(),M,N) -> s(plus(activate(N),activate(M)))
r8: and(tt(),X) -> activate(X)
r9: isNat(n__0()) -> tt()
r10: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
r11: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
r12: isNatKind(n__0()) -> tt()
r13: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
r14: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
r15: plus(N,|0|()) -> U31(and(isNat(N),n__isNatKind(N)),N)
r16: plus(N,s(M)) -> U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
r17: |0|() -> n__0()
r18: plus(X1,X2) -> n__plus(X1,X2)
r19: isNatKind(X) -> n__isNatKind(X)
r20: s(X) -> n__s(X)
r21: and(X1,X2) -> n__and(X1,X2)
r22: isNat(X) -> n__isNat(X)
r23: activate(n__0()) -> |0|()
r24: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
r25: activate(n__isNatKind(X)) -> isNatKind(X)
r26: activate(n__s(X)) -> s(activate(X))
r27: activate(n__and(X1,X2)) -> and(activate(X1),X2)
r28: activate(n__isNat(X)) -> isNat(X)
r29: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  (no SCCs)