YES

We show the termination of the TRS R:

  a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2)
  a__U12(tt(),V2) -> a__U13(a__isNat(V2))
  a__U13(tt()) -> tt()
  a__U21(tt(),V1) -> a__U22(a__isNat(V1))
  a__U22(tt()) -> tt()
  a__U31(tt(),N) -> mark(N)
  a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
  a__and(tt(),X) -> mark(X)
  a__isNat(|0|()) -> tt()
  a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
  a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
  a__isNatKind(|0|()) -> tt()
  a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2))
  a__isNatKind(s(V1)) -> a__isNatKind(V1)
  a__plus(N,|0|()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N)
  a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
  mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
  mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
  mark(isNat(X)) -> a__isNat(X)
  mark(U13(X)) -> a__U13(mark(X))
  mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
  mark(U22(X)) -> a__U22(mark(X))
  mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
  mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
  mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
  mark(and(X1,X2)) -> a__and(mark(X1),X2)
  mark(isNatKind(X)) -> a__isNatKind(X)
  mark(tt()) -> tt()
  mark(s(X)) -> s(mark(X))
  mark(|0|()) -> |0|()
  a__U11(X1,X2,X3) -> U11(X1,X2,X3)
  a__U12(X1,X2) -> U12(X1,X2)
  a__isNat(X) -> isNat(X)
  a__U13(X) -> U13(X)
  a__U21(X1,X2) -> U21(X1,X2)
  a__U22(X) -> U22(X)
  a__U31(X1,X2) -> U31(X1,X2)
  a__U41(X1,X2,X3) -> U41(X1,X2,X3)
  a__plus(X1,X2) -> plus(X1,X2)
  a__and(X1,X2) -> and(X1,X2)
  a__isNatKind(X) -> isNatKind(X)

-- SCC decomposition.

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

p1: a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2)
p2: a__U11#(tt(),V1,V2) -> a__isNat#(V1)
p3: a__U12#(tt(),V2) -> a__U13#(a__isNat(V2))
p4: a__U12#(tt(),V2) -> a__isNat#(V2)
p5: a__U21#(tt(),V1) -> a__U22#(a__isNat(V1))
p6: a__U21#(tt(),V1) -> a__isNat#(V1)
p7: a__U31#(tt(),N) -> mark#(N)
p8: a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p9: a__U41#(tt(),M,N) -> mark#(N)
p10: a__U41#(tt(),M,N) -> mark#(M)
p11: a__and#(tt(),X) -> mark#(X)
p12: a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
p13: a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p14: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p16: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p17: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p18: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p19: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p20: a__plus#(N,|0|()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N)
p21: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p22: a__plus#(N,|0|()) -> a__isNat#(N)
p23: a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
p24: a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
p25: a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNatKind(M))
p26: a__plus#(N,s(M)) -> a__isNat#(M)
p27: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
p28: mark#(U11(X1,X2,X3)) -> mark#(X1)
p29: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p30: mark#(U12(X1,X2)) -> mark#(X1)
p31: mark#(isNat(X)) -> a__isNat#(X)
p32: mark#(U13(X)) -> a__U13#(mark(X))
p33: mark#(U13(X)) -> mark#(X)
p34: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p35: mark#(U21(X1,X2)) -> mark#(X1)
p36: mark#(U22(X)) -> a__U22#(mark(X))
p37: mark#(U22(X)) -> mark#(X)
p38: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p39: mark#(U31(X1,X2)) -> mark#(X1)
p40: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p41: mark#(U41(X1,X2,X3)) -> mark#(X1)
p42: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p43: mark#(plus(X1,X2)) -> mark#(X1)
p44: mark#(plus(X1,X2)) -> mark#(X2)
p45: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p46: mark#(and(X1,X2)) -> mark#(X1)
p47: mark#(isNatKind(X)) -> a__isNatKind#(X)
p48: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2)
r2: a__U12(tt(),V2) -> a__U13(a__isNat(V2))
r3: a__U13(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),N) -> mark(N)
r7: a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r8: a__and(tt(),X) -> mark(X)
r9: a__isNat(|0|()) -> tt()
r10: a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
r11: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r12: a__isNatKind(|0|()) -> tt()
r13: a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2))
r14: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r15: a__plus(N,|0|()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N)
r16: a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
r17: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r18: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r19: mark(isNat(X)) -> a__isNat(X)
r20: mark(U13(X)) -> a__U13(mark(X))
r21: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r22: mark(U22(X)) -> a__U22(mark(X))
r23: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r24: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r25: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r26: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r27: mark(isNatKind(X)) -> a__isNatKind(X)
r28: mark(tt()) -> tt()
r29: mark(s(X)) -> s(mark(X))
r30: mark(|0|()) -> |0|()
r31: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r32: a__U12(X1,X2) -> U12(X1,X2)
r33: a__isNat(X) -> isNat(X)
r34: a__U13(X) -> U13(X)
r35: a__U21(X1,X2) -> U21(X1,X2)
r36: a__U22(X) -> U22(X)
r37: a__U31(X1,X2) -> U31(X1,X2)
r38: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r39: a__plus(X1,X2) -> plus(X1,X2)
r40: a__and(X1,X2) -> and(X1,X2)
r41: a__isNatKind(X) -> isNatKind(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p4, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p33, p34, p35, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48}


-- Reduction pair.

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

p1: a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2)
p2: a__U12#(tt(),V2) -> a__isNat#(V2)
p3: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p4: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p5: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p6: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p7: a__and#(tt(),X) -> mark#(X)
p8: mark#(s(X)) -> mark#(X)
p9: mark#(isNatKind(X)) -> a__isNatKind#(X)
p10: mark#(and(X1,X2)) -> mark#(X1)
p11: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p12: mark#(plus(X1,X2)) -> mark#(X2)
p13: mark#(plus(X1,X2)) -> mark#(X1)
p14: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p15: a__plus#(N,s(M)) -> a__isNat#(M)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p19: a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p20: a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
p21: a__U11#(tt(),V1,V2) -> a__isNat#(V1)
p22: a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNatKind(M))
p23: a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
p24: a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
p25: a__U41#(tt(),M,N) -> mark#(M)
p26: mark#(U41(X1,X2,X3)) -> mark#(X1)
p27: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p28: a__U41#(tt(),M,N) -> mark#(N)
p29: mark#(U31(X1,X2)) -> mark#(X1)
p30: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p31: a__U31#(tt(),N) -> mark#(N)
p32: mark#(U22(X)) -> mark#(X)
p33: mark#(U21(X1,X2)) -> mark#(X1)
p34: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p35: mark#(U13(X)) -> mark#(X)
p36: mark#(isNat(X)) -> a__isNat#(X)
p37: mark#(U12(X1,X2)) -> mark#(X1)
p38: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p39: mark#(U11(X1,X2,X3)) -> mark#(X1)
p40: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
p41: a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p42: a__plus#(N,|0|()) -> a__isNat#(N)
p43: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p44: a__plus#(N,|0|()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2)
r2: a__U12(tt(),V2) -> a__U13(a__isNat(V2))
r3: a__U13(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),N) -> mark(N)
r7: a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r8: a__and(tt(),X) -> mark(X)
r9: a__isNat(|0|()) -> tt()
r10: a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
r11: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r12: a__isNatKind(|0|()) -> tt()
r13: a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2))
r14: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r15: a__plus(N,|0|()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N)
r16: a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
r17: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r18: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r19: mark(isNat(X)) -> a__isNat(X)
r20: mark(U13(X)) -> a__U13(mark(X))
r21: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r22: mark(U22(X)) -> a__U22(mark(X))
r23: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r24: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r25: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r26: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r27: mark(isNatKind(X)) -> a__isNatKind(X)
r28: mark(tt()) -> tt()
r29: mark(s(X)) -> s(mark(X))
r30: mark(|0|()) -> |0|()
r31: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r32: a__U12(X1,X2) -> U12(X1,X2)
r33: a__isNat(X) -> isNat(X)
r34: a__U13(X) -> U13(X)
r35: a__U21(X1,X2) -> U21(X1,X2)
r36: a__U22(X) -> U22(X)
r37: a__U31(X1,X2) -> U31(X1,X2)
r38: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r39: a__plus(X1,X2) -> plus(X1,X2)
r40: a__and(X1,X2) -> and(X1,X2)
r41: a__isNatKind(X) -> isNatKind(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, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__U11#_A(x1,x2,x3) = max{x1 + 1, x2 + 153, x3 + 56}
          tt_A = 10
          a__U12#_A(x1,x2) = max{153, x2 + 36}
          a__isNat_A(x1) = max{32, x1 - 82}
          a__isNat#_A(x1) = x1 + 35
          s_A(x1) = max{119, x1}
          a__isNatKind#_A(x1) = max{154, x1 - 11}
          plus_A(x1,x2) = max{187, x1 + 165, x2 + 21}
          a__and#_A(x1,x2) = max{154, x2 + 122}
          a__isNatKind_A(x1) = max{33, x1 - 132}
          isNatKind_A(x1) = max{33, x1 - 132}
          mark#_A(x1) = x1 + 121
          and_A(x1,x2) = max{33, x1 + 11, x2 + 21}
          mark_A(x1) = max{22, x1}
          a__plus#_A(x1,x2) = max{x1 + 165, x2 + 134}
          a__U21#_A(x1,x2) = max{153, x1 + 1, x2 + 35}
          a__and_A(x1,x2) = max{33, x1 + 11, x2 + 21}
          isNat_A(x1) = max{32, x1 - 82}
          a__U41#_A(x1,x2,x3) = max{x1 + 177, x2 + 134, x3 + 165}
          U41_A(x1,x2,x3) = max{187, x1 + 56, x2 + 21, x3 + 165}
          U31_A(x1,x2) = max{x1 + 4, x2 + 43}
          a__U31#_A(x1,x2) = max{x1 + 20, x2 + 164}
          U22_A(x1) = max{1, x1}
          U21_A(x1,x2) = max{32, x1 + 4, x2 - 82}
          U13_A(x1) = max{23, x1}
          U12_A(x1,x2) = max{32, x1, x2 - 66}
          U11_A(x1,x2,x3) = max{x1 + 1, x2 + 83, x3 - 65}
          |0|_A = 21
          a__U11_A(x1,x2,x3) = max{x1 + 1, x2 + 83, x3 - 65}
          a__U12_A(x1,x2) = max{32, x1, x2 - 66}
          a__U13_A(x1) = max{23, x1}
          a__U21_A(x1,x2) = max{32, x1 + 4, x2 - 82}
          a__U22_A(x1) = max{11, x1}
          a__U31_A(x1,x2) = max{x1 + 4, x2 + 43}
          a__U41_A(x1,x2,x3) = max{187, x1 + 56, x2 + 21, x3 + 165}
          a__plus_A(x1,x2) = max{187, x1 + 165, x2 + 21}
    
      precedence:
      
        a__U11# = tt = a__U12# = a__isNat = a__isNat# = s = a__isNatKind# = plus = a__and# = a__isNatKind = isNatKind = mark# = and = mark = a__plus# = a__U21# = a__and = isNat = a__U41# = U41 = U31 = a__U31# = U22 = U21 = U13 = U12 = U11 = |0| = a__U11 = a__U12 = a__U13 = a__U21 = a__U22 = a__U31 = a__U41 = a__plus
      
      partial status:
    
        pi(a__U11#) = []
        pi(tt) = []
        pi(a__U12#) = []
        pi(a__isNat) = []
        pi(a__isNat#) = []
        pi(s) = []
        pi(a__isNatKind#) = []
        pi(plus) = []
        pi(a__and#) = []
        pi(a__isNatKind) = []
        pi(isNatKind) = []
        pi(mark#) = []
        pi(and) = []
        pi(mark) = []
        pi(a__plus#) = []
        pi(a__U21#) = []
        pi(a__and) = []
        pi(isNat) = []
        pi(a__U41#) = []
        pi(U41) = []
        pi(U31) = []
        pi(a__U31#) = []
        pi(U22) = []
        pi(U21) = []
        pi(U13) = []
        pi(U12) = []
        pi(U11) = []
        pi(|0|) = []
        pi(a__U11) = []
        pi(a__U12) = []
        pi(a__U13) = []
        pi(a__U21) = []
        pi(a__U22) = []
        pi(a__U31) = []
        pi(a__U41) = []
        pi(a__plus) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__U11#_A(x1,x2,x3) = 31
          tt_A = 31
          a__U12#_A(x1,x2) = 31
          a__isNat_A(x1) = 34
          a__isNat#_A(x1) = 31
          s_A(x1) = 53
          a__isNatKind#_A(x1) = 31
          plus_A(x1,x2) = 0
          a__and#_A(x1,x2) = 31
          a__isNatKind_A(x1) = 101
          isNatKind_A(x1) = 50
          mark#_A(x1) = 31
          and_A(x1,x2) = 101
          mark_A(x1) = 101
          a__plus#_A(x1,x2) = 31
          a__U21#_A(x1,x2) = 31
          a__and_A(x1,x2) = 101
          isNat_A(x1) = 33
          a__U41#_A(x1,x2,x3) = 31
          U41_A(x1,x2,x3) = 54
          U31_A(x1,x2) = 32
          a__U31#_A(x1,x2) = 31
          U22_A(x1) = 32
          U21_A(x1,x2) = 32
          U13_A(x1) = 0
          U12_A(x1,x2) = 30
          U11_A(x1,x2,x3) = 34
          |0|_A = 101
          a__U11_A(x1,x2,x3) = 34
          a__U12_A(x1,x2) = 34
          a__U13_A(x1) = 34
          a__U21_A(x1,x2) = 33
          a__U22_A(x1) = 32
          a__U31_A(x1,x2) = 101
          a__U41_A(x1,x2,x3) = 54
          a__plus_A(x1,x2) = 101
    
      precedence:
      
        tt = a__U22 > isNat > s > a__isNat = a__isNatKind = mark = a__and = a__U31 = a__plus > a__U11 > a__U12 > a__U41 > U11 > a__U11# = a__U12# = a__isNat# = a__isNatKind# = a__and# = mark# = a__plus# = a__U21# = a__U41# = a__U31# > and = U21 > isNatKind = U41 > plus > a__U21 > U12 > |0| = a__U13 > U22 > U13 > U31
      
      partial status:
    
        pi(a__U11#) = []
        pi(tt) = []
        pi(a__U12#) = []
        pi(a__isNat) = []
        pi(a__isNat#) = []
        pi(s) = []
        pi(a__isNatKind#) = []
        pi(plus) = []
        pi(a__and#) = []
        pi(a__isNatKind) = []
        pi(isNatKind) = []
        pi(mark#) = []
        pi(and) = []
        pi(mark) = []
        pi(a__plus#) = []
        pi(a__U21#) = []
        pi(a__and) = []
        pi(isNat) = []
        pi(a__U41#) = []
        pi(U41) = []
        pi(U31) = []
        pi(a__U31#) = []
        pi(U22) = []
        pi(U21) = []
        pi(U13) = []
        pi(U12) = []
        pi(U11) = []
        pi(|0|) = []
        pi(a__U11) = []
        pi(a__U12) = []
        pi(a__U13) = []
        pi(a__U21) = []
        pi(a__U22) = []
        pi(a__U31) = []
        pi(a__U41) = []
        pi(a__plus) = []
    

The next rules are strictly ordered:

  p18

We remove them from the problem.

-- SCC decomposition.

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

p1: a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2)
p2: a__U12#(tt(),V2) -> a__isNat#(V2)
p3: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p4: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p5: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p6: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p7: a__and#(tt(),X) -> mark#(X)
p8: mark#(s(X)) -> mark#(X)
p9: mark#(isNatKind(X)) -> a__isNatKind#(X)
p10: mark#(and(X1,X2)) -> mark#(X1)
p11: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p12: mark#(plus(X1,X2)) -> mark#(X2)
p13: mark#(plus(X1,X2)) -> mark#(X1)
p14: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p15: a__plus#(N,s(M)) -> a__isNat#(M)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p19: a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
p20: a__U11#(tt(),V1,V2) -> a__isNat#(V1)
p21: a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNatKind(M))
p22: a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
p23: a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
p24: a__U41#(tt(),M,N) -> mark#(M)
p25: mark#(U41(X1,X2,X3)) -> mark#(X1)
p26: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p27: a__U41#(tt(),M,N) -> mark#(N)
p28: mark#(U31(X1,X2)) -> mark#(X1)
p29: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p30: a__U31#(tt(),N) -> mark#(N)
p31: mark#(U22(X)) -> mark#(X)
p32: mark#(U21(X1,X2)) -> mark#(X1)
p33: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p34: mark#(U13(X)) -> mark#(X)
p35: mark#(isNat(X)) -> a__isNat#(X)
p36: mark#(U12(X1,X2)) -> mark#(X1)
p37: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p38: mark#(U11(X1,X2,X3)) -> mark#(X1)
p39: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
p40: a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p41: a__plus#(N,|0|()) -> a__isNat#(N)
p42: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p43: a__plus#(N,|0|()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2)
r2: a__U12(tt(),V2) -> a__U13(a__isNat(V2))
r3: a__U13(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),N) -> mark(N)
r7: a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r8: a__and(tt(),X) -> mark(X)
r9: a__isNat(|0|()) -> tt()
r10: a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
r11: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r12: a__isNatKind(|0|()) -> tt()
r13: a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2))
r14: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r15: a__plus(N,|0|()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N)
r16: a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
r17: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r18: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r19: mark(isNat(X)) -> a__isNat(X)
r20: mark(U13(X)) -> a__U13(mark(X))
r21: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r22: mark(U22(X)) -> a__U22(mark(X))
r23: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r24: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r25: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r26: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r27: mark(isNatKind(X)) -> a__isNatKind(X)
r28: mark(tt()) -> tt()
r29: mark(s(X)) -> s(mark(X))
r30: mark(|0|()) -> |0|()
r31: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r32: a__U12(X1,X2) -> U12(X1,X2)
r33: a__isNat(X) -> isNat(X)
r34: a__U13(X) -> U13(X)
r35: a__U21(X1,X2) -> U21(X1,X2)
r36: a__U22(X) -> U22(X)
r37: a__U31(X1,X2) -> U31(X1,X2)
r38: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r39: a__plus(X1,X2) -> plus(X1,X2)
r40: a__and(X1,X2) -> and(X1,X2)
r41: a__isNatKind(X) -> isNatKind(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, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43}


-- Reduction pair.

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

p1: a__U11#(tt(),V1,V2) -> a__U12#(a__isNat(V1),V2)
p2: a__U12#(tt(),V2) -> a__isNat#(V2)
p3: a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
p4: a__U11#(tt(),V1,V2) -> a__isNat#(V1)
p5: a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p6: a__and#(tt(),X) -> mark#(X)
p7: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
p8: mark#(U11(X1,X2,X3)) -> mark#(X1)
p9: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p10: mark#(U12(X1,X2)) -> mark#(X1)
p11: mark#(isNat(X)) -> a__isNat#(X)
p12: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p13: a__U21#(tt(),V1) -> a__isNat#(V1)
p14: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p15: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p16: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p17: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p18: mark#(U13(X)) -> mark#(X)
p19: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p20: mark#(U21(X1,X2)) -> mark#(X1)
p21: mark#(U22(X)) -> mark#(X)
p22: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p23: a__U31#(tt(),N) -> mark#(N)
p24: mark#(U31(X1,X2)) -> mark#(X1)
p25: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p26: a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p27: a__plus#(N,|0|()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N)
p28: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p29: a__plus#(N,|0|()) -> a__isNat#(N)
p30: a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
p31: a__U41#(tt(),M,N) -> mark#(N)
p32: mark#(U41(X1,X2,X3)) -> mark#(X1)
p33: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p34: a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
p35: a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNatKind(M))
p36: a__plus#(N,s(M)) -> a__isNat#(M)
p37: mark#(plus(X1,X2)) -> mark#(X1)
p38: mark#(plus(X1,X2)) -> mark#(X2)
p39: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p40: mark#(and(X1,X2)) -> mark#(X1)
p41: mark#(isNatKind(X)) -> a__isNatKind#(X)
p42: mark#(s(X)) -> mark#(X)
p43: a__U41#(tt(),M,N) -> mark#(M)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2)
r2: a__U12(tt(),V2) -> a__U13(a__isNat(V2))
r3: a__U13(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),N) -> mark(N)
r7: a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r8: a__and(tt(),X) -> mark(X)
r9: a__isNat(|0|()) -> tt()
r10: a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
r11: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r12: a__isNatKind(|0|()) -> tt()
r13: a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2))
r14: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r15: a__plus(N,|0|()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N)
r16: a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
r17: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r18: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r19: mark(isNat(X)) -> a__isNat(X)
r20: mark(U13(X)) -> a__U13(mark(X))
r21: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r22: mark(U22(X)) -> a__U22(mark(X))
r23: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r24: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r25: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r26: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r27: mark(isNatKind(X)) -> a__isNatKind(X)
r28: mark(tt()) -> tt()
r29: mark(s(X)) -> s(mark(X))
r30: mark(|0|()) -> |0|()
r31: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r32: a__U12(X1,X2) -> U12(X1,X2)
r33: a__isNat(X) -> isNat(X)
r34: a__U13(X) -> U13(X)
r35: a__U21(X1,X2) -> U21(X1,X2)
r36: a__U22(X) -> U22(X)
r37: a__U31(X1,X2) -> U31(X1,X2)
r38: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r39: a__plus(X1,X2) -> plus(X1,X2)
r40: a__and(X1,X2) -> and(X1,X2)
r41: a__isNatKind(X) -> isNatKind(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, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__U11#_A(x1,x2,x3) = max{x1 + 17, x2 + 68, x3 + 67}
          tt_A = 0
          a__U12#_A(x1,x2) = max{x1 + 18, x2 + 65}
          a__isNat_A(x1) = max{49, x1 - 100}
          a__isNat#_A(x1) = max{16, x1 - 102}
          plus_A(x1,x2) = max{360, x1 + 230, x2 + 354}
          a__and_A(x1,x2) = max{x1 + 21, x2 + 73}
          a__isNatKind_A(x1) = max{3, x1 - 151}
          isNatKind_A(x1) = max{2, x1 - 151}
          a__and#_A(x1,x2) = max{x1 + 14, x2 + 59}
          mark#_A(x1) = max{15, x1 - 1}
          U11_A(x1,x2,x3) = max{x1 + 24, x2 + 97, x3 + 68}
          mark_A(x1) = max{5, x1}
          U12_A(x1,x2) = max{x1 + 48, x2 + 67}
          isNat_A(x1) = max{49, x1 - 100}
          s_A(x1) = max{204, x1}
          a__U21#_A(x1,x2) = max{x1 + 17, x2 - 102}
          a__isNatKind#_A(x1) = max{0, x1 - 153}
          U13_A(x1) = max{49, x1 + 17}
          U21_A(x1,x2) = max{55, x1 + 50, x2 - 100}
          U22_A(x1) = x1
          U31_A(x1,x2) = max{x1 + 145, x2 + 152}
          a__U31#_A(x1,x2) = max{x1 + 138, x2 + 16}
          U41_A(x1,x2,x3) = max{360, x1 + 205, x2 + 354, x3 + 230}
          a__U41#_A(x1,x2,x3) = max{x1 + 65, x2 + 9, x3 + 60}
          a__plus#_A(x1,x2) = max{x1 + 60, x2 + 9}
          |0|_A = 204
          and_A(x1,x2) = max{x1 + 21, x2 + 73}
          a__U11_A(x1,x2,x3) = max{x1 + 24, x2 + 97, x3 + 68}
          a__U12_A(x1,x2) = max{x1 + 48, x2 + 67}
          a__U13_A(x1) = max{49, x1 + 17}
          a__U21_A(x1,x2) = max{55, x1 + 50, x2 - 100}
          a__U22_A(x1) = max{1, x1}
          a__U31_A(x1,x2) = max{x1 + 145, x2 + 152}
          a__U41_A(x1,x2,x3) = max{360, x1 + 205, x2 + 354, x3 + 230}
          a__plus_A(x1,x2) = max{360, x1 + 230, x2 + 354}
    
      precedence:
      
        a__U41# = a__plus# > a__isNatKind = mark > a__isNat > plus = U31 = a__U13 = a__U21 = a__U31 = a__plus > tt = U41 = a__U41 > mark# = s > a__U12# > a__and# = isNat > U13 = a__U11 = a__U12 > isNatKind > U11 > a__isNat# = a__U21# > a__isNatKind# > a__and = and > a__U11# > |0| > U12 = a__U22 > U22 = a__U31# > U21
      
      partial status:
    
        pi(a__U11#) = [1, 2, 3]
        pi(tt) = []
        pi(a__U12#) = [1, 2]
        pi(a__isNat) = []
        pi(a__isNat#) = []
        pi(plus) = []
        pi(a__and) = [1]
        pi(a__isNatKind) = []
        pi(isNatKind) = []
        pi(a__and#) = [1, 2]
        pi(mark#) = []
        pi(U11) = [3]
        pi(mark) = []
        pi(U12) = []
        pi(isNat) = []
        pi(s) = []
        pi(a__U21#) = []
        pi(a__isNatKind#) = []
        pi(U13) = []
        pi(U21) = [1]
        pi(U22) = []
        pi(U31) = []
        pi(a__U31#) = []
        pi(U41) = []
        pi(a__U41#) = []
        pi(a__plus#) = []
        pi(|0|) = []
        pi(and) = []
        pi(a__U11) = []
        pi(a__U12) = []
        pi(a__U13) = []
        pi(a__U21) = []
        pi(a__U22) = []
        pi(a__U31) = []
        pi(a__U41) = [3]
        pi(a__plus) = []
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__U11#_A(x1,x2,x3) = max{x1 - 81, x2 + 499, x3 + 499}
          tt_A = 280
          a__U12#_A(x1,x2) = max{x1 + 289, x2 + 129}
          a__isNat_A(x1) = 210
          a__isNat#_A(x1) = 128
          plus_A(x1,x2) = 49
          a__and_A(x1,x2) = max{115, x1 + 67}
          a__isNatKind_A(x1) = 127
          isNatKind_A(x1) = 47
          a__and#_A(x1,x2) = max{45, x1 - 214}
          mark#_A(x1) = 65
          U11_A(x1,x2,x3) = 47
          mark_A(x1) = 127
          U12_A(x1,x2) = 0
          isNat_A(x1) = 210
          s_A(x1) = 127
          a__U21#_A(x1,x2) = 128
          a__isNatKind#_A(x1) = 66
          U13_A(x1) = 48
          U21_A(x1,x2) = 129
          U22_A(x1) = 125
          U31_A(x1,x2) = 0
          a__U31#_A(x1,x2) = 63
          U41_A(x1,x2,x3) = 64
          a__U41#_A(x1,x2,x3) = 63
          a__plus#_A(x1,x2) = 63
          |0|_A = 111
          and_A(x1,x2) = 115
          a__U11_A(x1,x2,x3) = 127
          a__U12_A(x1,x2) = 126
          a__U13_A(x1) = 127
          a__U21_A(x1,x2) = 210
          a__U22_A(x1) = 126
          a__U31_A(x1,x2) = 127
          a__U41_A(x1,x2,x3) = x3 + 128
          a__plus_A(x1,x2) = 127
    
      precedence:
      
        U12 > U31 = U41 > a__U41# = a__plus# = a__U12 > mark# > a__U31# > U21 > a__U11# = a__U12# = a__isNat# = a__U21# > a__isNatKind# > isNatKind = a__and# > U22 > a__and > a__U41 > mark = s = a__U31 = a__plus > a__isNatKind > tt > plus = U13 > a__U13 > a__isNat > isNat > a__U22 > a__U11 > U11 > a__U21 > |0| > and
      
      partial status:
    
        pi(a__U11#) = [3]
        pi(tt) = []
        pi(a__U12#) = []
        pi(a__isNat) = []
        pi(a__isNat#) = []
        pi(plus) = []
        pi(a__and) = [1]
        pi(a__isNatKind) = []
        pi(isNatKind) = []
        pi(a__and#) = []
        pi(mark#) = []
        pi(U11) = []
        pi(mark) = []
        pi(U12) = []
        pi(isNat) = []
        pi(s) = []
        pi(a__U21#) = []
        pi(a__isNatKind#) = []
        pi(U13) = []
        pi(U21) = []
        pi(U22) = []
        pi(U31) = []
        pi(a__U31#) = []
        pi(U41) = []
        pi(a__U41#) = []
        pi(a__plus#) = []
        pi(|0|) = []
        pi(and) = []
        pi(a__U11) = []
        pi(a__U12) = []
        pi(a__U13) = []
        pi(a__U21) = []
        pi(a__U22) = []
        pi(a__U31) = []
        pi(a__U41) = []
        pi(a__plus) = []
    

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p9, p11, p14, p15, p19, p22, p23, p25, p27, p28, p29, p31, p33, p34, p35, p36, p39, p41, p43

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U11(X1,X2,X3)) -> mark#(X1)
p2: mark#(U12(X1,X2)) -> mark#(X1)
p3: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p4: a__U21#(tt(),V1) -> a__isNat#(V1)
p5: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p6: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p7: mark#(U13(X)) -> mark#(X)
p8: mark#(U21(X1,X2)) -> mark#(X1)
p9: mark#(U22(X)) -> mark#(X)
p10: mark#(U31(X1,X2)) -> mark#(X1)
p11: a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p12: a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
p13: mark#(U41(X1,X2,X3)) -> mark#(X1)
p14: mark#(plus(X1,X2)) -> mark#(X1)
p15: mark#(plus(X1,X2)) -> mark#(X2)
p16: mark#(and(X1,X2)) -> mark#(X1)
p17: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2)
r2: a__U12(tt(),V2) -> a__U13(a__isNat(V2))
r3: a__U13(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),N) -> mark(N)
r7: a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r8: a__and(tt(),X) -> mark(X)
r9: a__isNat(|0|()) -> tt()
r10: a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
r11: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r12: a__isNatKind(|0|()) -> tt()
r13: a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2))
r14: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r15: a__plus(N,|0|()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N)
r16: a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
r17: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r18: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r19: mark(isNat(X)) -> a__isNat(X)
r20: mark(U13(X)) -> a__U13(mark(X))
r21: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r22: mark(U22(X)) -> a__U22(mark(X))
r23: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r24: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r25: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r26: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r27: mark(isNatKind(X)) -> a__isNatKind(X)
r28: mark(tt()) -> tt()
r29: mark(s(X)) -> s(mark(X))
r30: mark(|0|()) -> |0|()
r31: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r32: a__U12(X1,X2) -> U12(X1,X2)
r33: a__isNat(X) -> isNat(X)
r34: a__U13(X) -> U13(X)
r35: a__U21(X1,X2) -> U21(X1,X2)
r36: a__U22(X) -> U22(X)
r37: a__U31(X1,X2) -> U31(X1,X2)
r38: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r39: a__plus(X1,X2) -> plus(X1,X2)
r40: a__and(X1,X2) -> and(X1,X2)
r41: a__isNatKind(X) -> isNatKind(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p7, p8, p9, p10, p13, p14, p15, p16, p17}
  {p3, p4}
  {p5, p6}
  {p11, p12}


-- Reduction pair.

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

p1: mark#(U11(X1,X2,X3)) -> mark#(X1)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(and(X1,X2)) -> mark#(X1)
p4: mark#(plus(X1,X2)) -> mark#(X2)
p5: mark#(plus(X1,X2)) -> mark#(X1)
p6: mark#(U41(X1,X2,X3)) -> mark#(X1)
p7: mark#(U31(X1,X2)) -> mark#(X1)
p8: mark#(U22(X)) -> mark#(X)
p9: mark#(U21(X1,X2)) -> mark#(X1)
p10: mark#(U13(X)) -> mark#(X)
p11: mark#(U12(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2)
r2: a__U12(tt(),V2) -> a__U13(a__isNat(V2))
r3: a__U13(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),N) -> mark(N)
r7: a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r8: a__and(tt(),X) -> mark(X)
r9: a__isNat(|0|()) -> tt()
r10: a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
r11: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r12: a__isNatKind(|0|()) -> tt()
r13: a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2))
r14: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r15: a__plus(N,|0|()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N)
r16: a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
r17: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r18: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r19: mark(isNat(X)) -> a__isNat(X)
r20: mark(U13(X)) -> a__U13(mark(X))
r21: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r22: mark(U22(X)) -> a__U22(mark(X))
r23: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r24: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r25: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r26: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r27: mark(isNatKind(X)) -> a__isNatKind(X)
r28: mark(tt()) -> tt()
r29: mark(s(X)) -> s(mark(X))
r30: mark(|0|()) -> |0|()
r31: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r32: a__U12(X1,X2) -> U12(X1,X2)
r33: a__isNat(X) -> isNat(X)
r34: a__U13(X) -> U13(X)
r35: a__U21(X1,X2) -> U21(X1,X2)
r36: a__U22(X) -> U22(X)
r37: a__U31(X1,X2) -> U31(X1,X2)
r38: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r39: a__plus(X1,X2) -> plus(X1,X2)
r40: a__and(X1,X2) -> and(X1,X2)
r41: a__isNatKind(X) -> isNatKind(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          mark#_A(x1) = max{4, x1 + 3}
          U11_A(x1,x2,x3) = max{x1, x2 + 1, x3}
          s_A(x1) = max{1, x1}
          and_A(x1,x2) = max{x1, x2}
          plus_A(x1,x2) = max{x1 + 1, x2}
          U41_A(x1,x2,x3) = max{x1, x2, x3}
          U31_A(x1,x2) = max{x1, x2}
          U22_A(x1) = max{1, x1}
          U21_A(x1,x2) = max{x1, x2}
          U13_A(x1) = max{3, x1 + 2}
          U12_A(x1,x2) = max{x1, x2}
    
      precedence:
      
        mark# = U11 = s = and = plus = U41 = U31 = U22 = U21 = U13 = U12
      
      partial status:
    
        pi(mark#) = [1]
        pi(U11) = [1, 2, 3]
        pi(s) = [1]
        pi(and) = [1, 2]
        pi(plus) = [1, 2]
        pi(U41) = [1, 2, 3]
        pi(U31) = [1, 2]
        pi(U22) = [1]
        pi(U21) = [1, 2]
        pi(U13) = [1]
        pi(U12) = [1, 2]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          mark#_A(x1) = x1
          U11_A(x1,x2,x3) = max{x1, x2, x3}
          s_A(x1) = x1 + 1
          and_A(x1,x2) = max{x1, x2}
          plus_A(x1,x2) = max{x1, x2}
          U41_A(x1,x2,x3) = max{x1, x2, x3}
          U31_A(x1,x2) = max{x1, x2}
          U22_A(x1) = x1
          U21_A(x1,x2) = max{x1, x2}
          U13_A(x1) = x1
          U12_A(x1,x2) = max{x1, x2}
    
      precedence:
      
        mark# = U11 = s = and = plus = U41 = U31 = U22 = U21 = U13 = U12
      
      partial status:
    
        pi(mark#) = [1]
        pi(U11) = [1, 2, 3]
        pi(s) = [1]
        pi(and) = [1, 2]
        pi(plus) = [1, 2]
        pi(U41) = [1, 2, 3]
        pi(U31) = [1, 2]
        pi(U22) = [1]
        pi(U21) = [1, 2]
        pi(U13) = [1]
        pi(U12) = [1, 2]
    

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11

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: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p2: a__U21#(tt(),V1) -> a__isNat#(V1)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2)
r2: a__U12(tt(),V2) -> a__U13(a__isNat(V2))
r3: a__U13(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),N) -> mark(N)
r7: a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r8: a__and(tt(),X) -> mark(X)
r9: a__isNat(|0|()) -> tt()
r10: a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
r11: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r12: a__isNatKind(|0|()) -> tt()
r13: a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2))
r14: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r15: a__plus(N,|0|()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N)
r16: a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
r17: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r18: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r19: mark(isNat(X)) -> a__isNat(X)
r20: mark(U13(X)) -> a__U13(mark(X))
r21: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r22: mark(U22(X)) -> a__U22(mark(X))
r23: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r24: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r25: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r26: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r27: mark(isNatKind(X)) -> a__isNatKind(X)
r28: mark(tt()) -> tt()
r29: mark(s(X)) -> s(mark(X))
r30: mark(|0|()) -> |0|()
r31: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r32: a__U12(X1,X2) -> U12(X1,X2)
r33: a__isNat(X) -> isNat(X)
r34: a__U13(X) -> U13(X)
r35: a__U21(X1,X2) -> U21(X1,X2)
r36: a__U22(X) -> U22(X)
r37: a__U31(X1,X2) -> U31(X1,X2)
r38: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r39: a__plus(X1,X2) -> plus(X1,X2)
r40: a__and(X1,X2) -> and(X1,X2)
r41: a__isNatKind(X) -> isNatKind(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, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__isNat#_A(x1) = ((1,0),(1,0)) x1 + (3,1)
            s_A(x1) = ((1,0),(0,0)) x1 + (5,4)
            a__U21#_A(x1,x2) = ((1,0),(1,0)) x2 + (4,5)
            a__isNatKind_A(x1) = ((1,0),(1,0)) x1 + (7,7)
            tt_A() = (0,9)
            a__U11_A(x1,x2,x3) = (3,1)
            a__U12_A(x1,x2) = (2,11)
            a__isNat_A(x1) = (8,8)
            a__U13_A(x1) = (1,10)
            a__U21_A(x1,x2) = (0,11)
            a__U22_A(x1) = (0,10)
            a__U31_A(x1,x2) = ((1,0),(0,0)) x2 + (1,11)
            mark_A(x1) = ((1,0),(1,0)) x1 + (0,11)
            a__U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (17,15)
            a__plus_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (12,11)
            |0|_A() = (0,1)
            plus_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (12,10)
            a__and_A(x1,x2) = ((1,0),(0,0)) x2 + (11,22)
            isNatKind_A(x1) = ((1,0),(0,0)) x1 + (7,0)
            and_A(x1,x2) = ((1,0),(0,0)) x2 + (11,0)
            isNat_A(x1) = (8,5)
            U11_A(x1,x2,x3) = (3,0)
            U12_A(x1,x2) = (2,0)
            U13_A(x1) = (1,0)
            U21_A(x1,x2) = (0,11)
            U22_A(x1) = (0,0)
            U31_A(x1,x2) = ((1,0),(0,0)) x2 + (1,0)
            U41_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (17,14)
    
      precedence:
      
        a__isNat = a__U31 = mark = a__plus > a__isNatKind > a__U11 > a__U12 > a__and > a__U13 > tt > |0| > a__U41 = plus > s > a__U21# = U11 = U12 > a__isNat# > isNatKind = U13 > a__U21 > U31 > U21 > and = U41 > isNat > a__U22 > U22
      
      partial status:
    
        pi(a__isNat#) = []
        pi(s) = []
        pi(a__U21#) = []
        pi(a__isNatKind) = []
        pi(tt) = []
        pi(a__U11) = []
        pi(a__U12) = []
        pi(a__isNat) = []
        pi(a__U13) = []
        pi(a__U21) = []
        pi(a__U22) = []
        pi(a__U31) = []
        pi(mark) = []
        pi(a__U41) = []
        pi(a__plus) = []
        pi(|0|) = []
        pi(plus) = []
        pi(a__and) = []
        pi(isNatKind) = []
        pi(and) = []
        pi(isNat) = []
        pi(U11) = []
        pi(U12) = []
        pi(U13) = []
        pi(U21) = []
        pi(U22) = []
        pi(U31) = []
        pi(U41) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__isNat#_A(x1) = (3,5)
            s_A(x1) = (3,6)
            a__U21#_A(x1,x2) = (2,1)
            a__isNatKind_A(x1) = (2,4)
            tt_A() = (0,0)
            a__U11_A(x1,x2,x3) = (4,0)
            a__U12_A(x1,x2) = (3,2)
            a__isNat_A(x1) = (4,0)
            a__U13_A(x1) = (2,1)
            a__U21_A(x1,x2) = (4,0)
            a__U22_A(x1) = (3,5)
            a__U31_A(x1,x2) = (8,2)
            mark_A(x1) = (7,4)
            a__U41_A(x1,x2,x3) = (6,5)
            a__plus_A(x1,x2) = (5,3)
            |0|_A() = (1,1)
            plus_A(x1,x2) = (5,1)
            a__and_A(x1,x2) = (2,2)
            isNatKind_A(x1) = (0,2)
            and_A(x1,x2) = (1,1)
            isNat_A(x1) = (3,0)
            U11_A(x1,x2,x3) = (4,0)
            U12_A(x1,x2) = (1,0)
            U13_A(x1) = (0,0)
            U21_A(x1,x2) = (0,0)
            U22_A(x1) = (1,0)
            U31_A(x1,x2) = (0,0)
            U41_A(x1,x2,x3) = (1,1)
    
      precedence:
      
        a__U12 > U21 > U13 > a__U22 > U41 > s = isNat > a__U11 = a__isNat > U11 > isNatKind > a__isNat# > a__U21# = a__U41 > a__plus = U12 > a__U31 > U31 > U22 > a__isNatKind = mark = |0| = a__and = and > a__U13 > plus > a__U21 > tt
      
      partial status:
    
        pi(a__isNat#) = []
        pi(s) = []
        pi(a__U21#) = []
        pi(a__isNatKind) = []
        pi(tt) = []
        pi(a__U11) = []
        pi(a__U12) = []
        pi(a__isNat) = []
        pi(a__U13) = []
        pi(a__U21) = []
        pi(a__U22) = []
        pi(a__U31) = []
        pi(mark) = []
        pi(a__U41) = []
        pi(a__plus) = []
        pi(|0|) = []
        pi(plus) = []
        pi(a__and) = []
        pi(isNatKind) = []
        pi(and) = []
        pi(isNat) = []
        pi(U11) = []
        pi(U12) = []
        pi(U13) = []
        pi(U21) = []
        pi(U22) = []
        pi(U31) = []
        pi(U41) = []
    

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: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p2: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2)
r2: a__U12(tt(),V2) -> a__U13(a__isNat(V2))
r3: a__U13(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),N) -> mark(N)
r7: a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r8: a__and(tt(),X) -> mark(X)
r9: a__isNat(|0|()) -> tt()
r10: a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
r11: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r12: a__isNatKind(|0|()) -> tt()
r13: a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2))
r14: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r15: a__plus(N,|0|()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N)
r16: a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
r17: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r18: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r19: mark(isNat(X)) -> a__isNat(X)
r20: mark(U13(X)) -> a__U13(mark(X))
r21: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r22: mark(U22(X)) -> a__U22(mark(X))
r23: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r24: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r25: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r26: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r27: mark(isNatKind(X)) -> a__isNatKind(X)
r28: mark(tt()) -> tt()
r29: mark(s(X)) -> s(mark(X))
r30: mark(|0|()) -> |0|()
r31: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r32: a__U12(X1,X2) -> U12(X1,X2)
r33: a__isNat(X) -> isNat(X)
r34: a__U13(X) -> U13(X)
r35: a__U21(X1,X2) -> U21(X1,X2)
r36: a__U22(X) -> U22(X)
r37: a__U31(X1,X2) -> U31(X1,X2)
r38: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r39: a__plus(X1,X2) -> plus(X1,X2)
r40: a__and(X1,X2) -> and(X1,X2)
r41: a__isNatKind(X) -> isNatKind(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__isNatKind#_A(x1) = x1 + 3
          plus_A(x1,x2) = max{x1, x2}
          s_A(x1) = max{1, x1}
    
      precedence:
      
        a__isNatKind# = plus = s
      
      partial status:
    
        pi(a__isNatKind#) = [1]
        pi(plus) = [1, 2]
        pi(s) = [1]
    
    2. weighted path order
    
      base order:
    
        max/plus interpretations on natural numbers:
        
          a__isNatKind#_A(x1) = x1 + 1
          plus_A(x1,x2) = max{x1, x2}
          s_A(x1) = x1
    
      precedence:
      
        a__isNatKind# = plus = s
      
      partial status:
    
        pi(a__isNatKind#) = [1]
        pi(plus) = [1, 2]
        pi(s) = [1]
    

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: a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p2: a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2)
r2: a__U12(tt(),V2) -> a__U13(a__isNat(V2))
r3: a__U13(tt()) -> tt()
r4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
r5: a__U22(tt()) -> tt()
r6: a__U31(tt(),N) -> mark(N)
r7: a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r8: a__and(tt(),X) -> mark(X)
r9: a__isNat(|0|()) -> tt()
r10: a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
r11: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r12: a__isNatKind(|0|()) -> tt()
r13: a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2))
r14: a__isNatKind(s(V1)) -> a__isNatKind(V1)
r15: a__plus(N,|0|()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N)
r16: a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
r17: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r18: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
r19: mark(isNat(X)) -> a__isNat(X)
r20: mark(U13(X)) -> a__U13(mark(X))
r21: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r22: mark(U22(X)) -> a__U22(mark(X))
r23: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r24: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
r25: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r26: mark(and(X1,X2)) -> a__and(mark(X1),X2)
r27: mark(isNatKind(X)) -> a__isNatKind(X)
r28: mark(tt()) -> tt()
r29: mark(s(X)) -> s(mark(X))
r30: mark(|0|()) -> |0|()
r31: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r32: a__U12(X1,X2) -> U12(X1,X2)
r33: a__isNat(X) -> isNat(X)
r34: a__U13(X) -> U13(X)
r35: a__U21(X1,X2) -> U21(X1,X2)
r36: a__U22(X) -> U22(X)
r37: a__U31(X1,X2) -> U31(X1,X2)
r38: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
r39: a__plus(X1,X2) -> plus(X1,X2)
r40: a__and(X1,X2) -> and(X1,X2)
r41: a__isNatKind(X) -> isNatKind(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, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__U41#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (38,24)
            tt_A() = (11,21)
            a__plus#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (12,127)
            mark_A(x1) = ((1,0),(1,1)) x1 + (0,34)
            s_A(x1) = ((1,0),(1,0)) x1 + (39,1)
            a__and_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (1,1)
            a__isNat_A(x1) = (17,51)
            isNatKind_A(x1) = ((1,0),(1,1)) x1 + (1,19)
            and_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (1,1)
            isNat_A(x1) = (17,0)
            a__U11_A(x1,x2,x3) = (14,51)
            a__U12_A(x1,x2) = (13,50)
            a__U13_A(x1) = (12,49)
            a__U21_A(x1,x2) = ((0,0),(1,0)) x1 + (16,2)
            a__U22_A(x1) = (11,35)
            a__U31_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (1,36)
            a__U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (49,36)
            a__plus_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (10,36)
            a__isNatKind_A(x1) = ((1,0),(1,1)) x1 + (1,51)
            |0|_A() = (10,20)
            plus_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (10,35)
            U11_A(x1,x2,x3) = (14,34)
            U12_A(x1,x2) = (13,50)
            U13_A(x1) = (12,3)
            U21_A(x1,x2) = ((0,0),(1,0)) x1 + (16,1)
            U22_A(x1) = (11,35)
            U31_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (1,35)
            U41_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (49,35)
    
      precedence:
      
        a__and > mark = a__U21 = |0| > a__U11 = U11 > a__plus = plus > a__U12 = a__U41 = U12 = U41 > a__U13 = U13 > tt = a__plus# = s = a__isNat = a__U22 = a__isNatKind = U21 > a__U41# > isNatKind = isNat = U22 > and > U31 > a__U31
      
      partial status:
    
        pi(a__U41#) = []
        pi(tt) = []
        pi(a__plus#) = [2]
        pi(mark) = []
        pi(s) = []
        pi(a__and) = [1]
        pi(a__isNat) = []
        pi(isNatKind) = [1]
        pi(and) = [2]
        pi(isNat) = []
        pi(a__U11) = []
        pi(a__U12) = []
        pi(a__U13) = []
        pi(a__U21) = []
        pi(a__U22) = []
        pi(a__U31) = []
        pi(a__U41) = []
        pi(a__plus) = [2]
        pi(a__isNatKind) = []
        pi(|0|) = []
        pi(plus) = []
        pi(U11) = []
        pi(U12) = []
        pi(U13) = []
        pi(U21) = []
        pi(U22) = []
        pi(U31) = [2]
        pi(U41) = []
    
    2. weighted path order
    
      base order:
    
        matrix interpretations:
        
          carrier: N^2
          order: lexicographic order
          interpretations:
            a__U41#_A(x1,x2,x3) = (5,9)
            tt_A() = (3,1)
            a__plus#_A(x1,x2) = ((1,0),(1,1)) x2 + (1,1)
            mark_A(x1) = (3,4)
            s_A(x1) = (7,10)
            a__and_A(x1,x2) = ((1,0),(0,0)) x1
            a__isNat_A(x1) = (7,19)
            isNatKind_A(x1) = (1,1)
            and_A(x1,x2) = (3,0)
            isNat_A(x1) = (1,0)
            a__U11_A(x1,x2,x3) = (6,6)
            a__U12_A(x1,x2) = (5,5)
            a__U13_A(x1) = (4,4)
            a__U21_A(x1,x2) = (7,5)
            a__U22_A(x1) = (3,2)
            a__U31_A(x1,x2) = (3,12)
            a__U41_A(x1,x2,x3) = (8,11)
            a__plus_A(x1,x2) = ((0,0),(1,0)) x2 + (9,9)
            a__isNatKind_A(x1) = (3,2)
            |0|_A() = (2,5)
            plus_A(x1,x2) = (5,3)
            U11_A(x1,x2,x3) = (0,0)
            U12_A(x1,x2) = (0,0)
            U13_A(x1) = (0,0)
            U21_A(x1,x2) = (3,0)
            U22_A(x1) = (0,1)
            U31_A(x1,x2) = (0,13)
            U41_A(x1,x2,x3) = (1,12)
    
      precedence:
      
        a__U41# = a__plus# > isNat > and > U21 > a__U13 > mark = a__and = a__U31 = a__isNatKind = |0| = U31 > a__isNat = a__U21 > U22 > a__U22 > a__plus > a__U41 > s > tt > U13 > isNatKind > a__U11 > a__U12 > U12 > plus = U41 > U11
      
      partial status:
    
        pi(a__U41#) = []
        pi(tt) = []
        pi(a__plus#) = [2]
        pi(mark) = []
        pi(s) = []
        pi(a__and) = []
        pi(a__isNat) = []
        pi(isNatKind) = []
        pi(and) = []
        pi(isNat) = []
        pi(a__U11) = []
        pi(a__U12) = []
        pi(a__U13) = []
        pi(a__U21) = []
        pi(a__U22) = []
        pi(a__U31) = []
        pi(a__U41) = []
        pi(a__plus) = []
        pi(a__isNatKind) = []
        pi(|0|) = []
        pi(plus) = []
        pi(U11) = []
        pi(U12) = []
        pi(U13) = []
        pi(U21) = []
        pi(U22) = []
        pi(U31) = []
        pi(U41) = []
    

The next rules are strictly ordered:

  p1, p2

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