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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          a__U11#_A(x1,x2,x3) = 1
          tt_A() = 0
          a__U12#_A(x1,x2) = 1
          a__isNat_A(x1) = 0
          a__isNat#_A(x1) = 1
          s_A(x1) = x1 + 3
          a__isNatKind#_A(x1) = 1
          plus_A(x1,x2) = x1 + x2
          a__and#_A(x1,x2) = x2 + 1
          a__isNatKind_A(x1) = 0
          isNatKind_A(x1) = 0
          mark#_A(x1) = x1 + 1
          and_A(x1,x2) = x1 + x2
          mark_A(x1) = x1
          a__plus#_A(x1,x2) = x1 + x2
          a__U21#_A(x1,x2) = 1
          a__and_A(x1,x2) = x1 + x2
          isNat_A(x1) = 0
          a__U41#_A(x1,x2,x3) = x2 + x3 + 2
          U41_A(x1,x2,x3) = x1 + x2 + x3 + 3
          U31_A(x1,x2) = x1 + x2 + 3
          a__U31#_A(x1,x2) = x2 + 2
          U22_A(x1) = x1
          U21_A(x1,x2) = x1
          U13_A(x1) = x1
          U12_A(x1,x2) = x1
          U11_A(x1,x2,x3) = x1
          |0|_A() = 4
          a__U11_A(x1,x2,x3) = x1
          a__U12_A(x1,x2) = x1
          a__U13_A(x1) = x1
          a__U21_A(x1,x2) = x1
          a__U22_A(x1) = x1
          a__U31_A(x1,x2) = x1 + x2 + 3
          a__U41_A(x1,x2,x3) = x1 + x2 + x3 + 3
          a__plus_A(x1,x2) = x1 + x2
  
    precedence:
    
      a__plus# = a__U41# > a__U11# = a__U12# = a__isNat# = a__isNatKind# = a__and# = mark# = a__U21# > a__U31# > tt = a__isNat = s = plus = a__isNatKind = isNatKind = and = mark = a__and = isNat = U41 = 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) = []

The next rules are strictly ordered:

  p28

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__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: 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#(plus(V1,V2)) -> a__isNatKind#(V1)
p13: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p21: mark#(U21(X1,X2)) -> mark#(X1)
p22: mark#(U22(X)) -> mark#(X)
p23: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p24: a__U31#(tt(),N) -> mark#(N)
p25: mark#(U31(X1,X2)) -> mark#(X1)
p26: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p27: a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p28: a__plus#(N,|0|()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N)
p29: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p30: a__plus#(N,|0|()) -> a__isNat#(N)
p31: a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
p32: a__U41#(tt(),M,N) -> mark#(M)
p33: mark#(U41(X1,X2,X3)) -> mark#(X1)
p34: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p35: a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
p36: a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNatKind(M))
p37: a__plus#(N,s(M)) -> a__isNat#(M)
p38: mark#(plus(X1,X2)) -> mark#(X1)
p39: mark#(plus(X1,X2)) -> mark#(X2)
p40: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p41: mark#(and(X1,X2)) -> mark#(X1)
p42: mark#(isNatKind(X)) -> a__isNatKind#(X)
p43: 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 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:

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

The next rules are strictly ordered:

  p32

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#(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#(plus(V1,V2)) -> a__isNatKind#(V1)
p13: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p21: mark#(U21(X1,X2)) -> mark#(X1)
p22: mark#(U22(X)) -> mark#(X)
p23: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p24: a__U31#(tt(),N) -> mark#(N)
p25: mark#(U31(X1,X2)) -> mark#(X1)
p26: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p27: a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p28: a__plus#(N,|0|()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N)
p29: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p30: a__plus#(N,|0|()) -> a__isNat#(N)
p31: a__plus#(N,s(M)) -> a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,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)

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}


-- 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) -> a__plus#(mark(N),mark(M))
p26: a__plus#(N,|0|()) -> a__isNat#(N)
p27: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p28: a__plus#(N,|0|()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N)
p29: a__U31#(tt(),N) -> mark#(N)
p30: mark#(U41(X1,X2,X3)) -> mark#(X1)
p31: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p32: mark#(U31(X1,X2)) -> mark#(X1)
p33: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p34: mark#(U22(X)) -> mark#(X)
p35: mark#(U21(X1,X2)) -> mark#(X1)
p36: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p37: mark#(U13(X)) -> mark#(X)
p38: mark#(isNat(X)) -> a__isNat#(X)
p39: mark#(U12(X1,X2)) -> mark#(X1)
p40: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p41: mark#(U11(X1,X2,X3)) -> mark#(X1)
p42: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          a__U11#_A(x1,x2,x3) = 0
          tt_A() = 0
          a__U12#_A(x1,x2) = 0
          a__isNat_A(x1) = 0
          a__isNat#_A(x1) = 0
          s_A(x1) = x1 + 3
          a__isNatKind#_A(x1) = 0
          plus_A(x1,x2) = x1 + x2 + 2
          a__and#_A(x1,x2) = x2
          a__isNatKind_A(x1) = 0
          isNatKind_A(x1) = 0
          mark#_A(x1) = x1
          and_A(x1,x2) = x1 + x2
          mark_A(x1) = x1
          a__plus#_A(x1,x2) = x1 + x2 + 1
          a__U21#_A(x1,x2) = 0
          a__and_A(x1,x2) = x1 + x2
          isNat_A(x1) = 0
          a__U41#_A(x1,x2,x3) = x2 + x3 + 2
          |0|_A() = 3
          a__U31#_A(x1,x2) = x2 + 1
          U41_A(x1,x2,x3) = x1 + x2 + x3 + 5
          U31_A(x1,x2) = x1 + x2 + 2
          U22_A(x1) = x1
          U21_A(x1,x2) = x1
          U13_A(x1) = x1
          U12_A(x1,x2) = x1
          U11_A(x1,x2,x3) = x1
          a__U11_A(x1,x2,x3) = x1
          a__U12_A(x1,x2) = x1
          a__U13_A(x1) = x1
          a__U21_A(x1,x2) = x1
          a__U22_A(x1) = x1
          a__U31_A(x1,x2) = x1 + x2 + 2
          a__U41_A(x1,x2,x3) = x1 + x2 + x3 + 5
          a__plus_A(x1,x2) = x1 + x2 + 2
  
    precedence:
    
      a__U11# = a__U12# = a__isNat# = a__isNatKind# = a__and# = mark# = a__U21# > a__U41# > tt = a__isNat = s = plus = a__isNatKind = isNatKind = and = mark = a__plus# = a__and = isNat = |0| = a__U31# = U41 = U31 = U22 = U21 = U13 = U12 = U11 = 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#) = [2]
      pi(a__U21#) = []
      pi(a__and) = []
      pi(isNat) = []
      pi(a__U41#) = []
      pi(|0|) = []
      pi(a__U31#) = []
      pi(U41) = []
      pi(U31) = []
      pi(U22) = []
      pi(U21) = []
      pi(U13) = []
      pi(U12) = []
      pi(U11) = []
      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:

  p24

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__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__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p25: a__plus#(N,|0|()) -> a__isNat#(N)
p26: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p27: a__plus#(N,|0|()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N)
p28: a__U31#(tt(),N) -> mark#(N)
p29: mark#(U41(X1,X2,X3)) -> mark#(X1)
p30: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p31: mark#(U31(X1,X2)) -> mark#(X1)
p32: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p33: mark#(U22(X)) -> mark#(X)
p34: mark#(U21(X1,X2)) -> mark#(X1)
p35: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p36: mark#(U13(X)) -> mark#(X)
p37: mark#(isNat(X)) -> a__isNat#(X)
p38: mark#(U12(X1,X2)) -> mark#(X1)
p39: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p40: mark#(U11(X1,X2,X3)) -> mark#(X1)
p41: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)

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}


-- 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#(plus(V1,V2)) -> a__isNatKind#(V1)
p13: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p21: mark#(U21(X1,X2)) -> mark#(X1)
p22: mark#(U22(X)) -> mark#(X)
p23: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p24: a__U31#(tt(),N) -> mark#(N)
p25: mark#(U31(X1,X2)) -> mark#(X1)
p26: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3)
p27: a__U41#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p28: a__plus#(N,|0|()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N)
p29: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p30: a__plus#(N,|0|()) -> a__isNat#(N)
p31: a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
p32: a__plus#(N,s(M)) -> a__and#(a__isNat(M),isNatKind(M))
p33: a__plus#(N,s(M)) -> a__isNat#(M)
p34: mark#(U41(X1,X2,X3)) -> mark#(X1)
p35: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p36: mark#(plus(X1,X2)) -> mark#(X1)
p37: mark#(plus(X1,X2)) -> mark#(X2)
p38: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p39: mark#(and(X1,X2)) -> mark#(X1)
p40: mark#(isNatKind(X)) -> a__isNatKind#(X)
p41: 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 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:

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

The next rules are strictly ordered:

  p27

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


-- 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,|0|()) -> a__isNat#(N)
p25: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p26: a__plus#(N,|0|()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N)
p27: a__U31#(tt(),N) -> mark#(N)
p28: mark#(U41(X1,X2,X3)) -> mark#(X1)
p29: mark#(U31(X1,X2)) -> mark#(X1)
p30: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
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)

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:

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

  p22

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__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__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
p23: a__plus#(N,|0|()) -> a__isNat#(N)
p24: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p25: a__plus#(N,|0|()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N)
p26: a__U31#(tt(),N) -> mark#(N)
p27: mark#(U41(X1,X2,X3)) -> mark#(X1)
p28: mark#(U31(X1,X2)) -> mark#(X1)
p29: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p30: mark#(U22(X)) -> mark#(X)
p31: mark#(U21(X1,X2)) -> mark#(X1)
p32: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p33: mark#(U13(X)) -> mark#(X)
p34: mark#(isNat(X)) -> a__isNat#(X)
p35: mark#(U12(X1,X2)) -> mark#(X1)
p36: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p37: mark#(U11(X1,X2,X3)) -> mark#(X1)
p38: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)

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}


-- 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#(plus(V1,V2)) -> a__isNatKind#(V1)
p13: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p21: mark#(U21(X1,X2)) -> mark#(X1)
p22: mark#(U22(X)) -> mark#(X)
p23: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p24: a__U31#(tt(),N) -> mark#(N)
p25: mark#(U31(X1,X2)) -> mark#(X1)
p26: mark#(U41(X1,X2,X3)) -> mark#(X1)
p27: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p28: a__plus#(N,|0|()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N)
p29: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p30: a__plus#(N,|0|()) -> a__isNat#(N)
p31: a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
p32: a__plus#(N,s(M)) -> a__isNat#(M)
p33: mark#(plus(X1,X2)) -> mark#(X1)
p34: mark#(plus(X1,X2)) -> mark#(X2)
p35: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p36: mark#(and(X1,X2)) -> mark#(X1)
p37: mark#(isNatKind(X)) -> a__isNatKind#(X)
p38: 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 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:

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

The next rules are strictly ordered:

  p26

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#(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#(plus(V1,V2)) -> a__isNatKind#(V1)
p13: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p21: mark#(U21(X1,X2)) -> mark#(X1)
p22: mark#(U22(X)) -> mark#(X)
p23: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p24: a__U31#(tt(),N) -> mark#(N)
p25: mark#(U31(X1,X2)) -> mark#(X1)
p26: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
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__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
p31: a__plus#(N,s(M)) -> a__isNat#(M)
p32: mark#(plus(X1,X2)) -> mark#(X1)
p33: mark#(plus(X1,X2)) -> mark#(X2)
p34: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p35: mark#(and(X1,X2)) -> mark#(X1)
p36: mark#(isNatKind(X)) -> a__isNatKind#(X)
p37: 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, 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}


-- 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__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
p23: a__plus#(N,|0|()) -> a__isNat#(N)
p24: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p25: a__plus#(N,|0|()) -> a__U31#(a__and(a__isNat(N),isNatKind(N)),N)
p26: a__U31#(tt(),N) -> mark#(N)
p27: mark#(U31(X1,X2)) -> mark#(X1)
p28: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p29: mark#(U22(X)) -> mark#(X)
p30: mark#(U21(X1,X2)) -> mark#(X1)
p31: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p32: mark#(U13(X)) -> mark#(X)
p33: mark#(isNat(X)) -> a__isNat#(X)
p34: mark#(U12(X1,X2)) -> mark#(X1)
p35: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p36: mark#(U11(X1,X2,X3)) -> mark#(X1)
p37: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          a__U11#_A(x1,x2,x3) = 2
          tt_A() = 0
          a__U12#_A(x1,x2) = 2
          a__isNat_A(x1) = 0
          a__isNat#_A(x1) = 2
          s_A(x1) = x1
          a__isNatKind#_A(x1) = 2
          plus_A(x1,x2) = x1 + x2
          a__and#_A(x1,x2) = x1 + x2 + 2
          a__isNatKind_A(x1) = 0
          isNatKind_A(x1) = 0
          mark#_A(x1) = x1 + 2
          and_A(x1,x2) = x1 + x2
          mark_A(x1) = x1
          a__plus#_A(x1,x2) = x1 + x2 + 2
          a__U21#_A(x1,x2) = 2
          a__and_A(x1,x2) = x1 + x2
          isNat_A(x1) = 0
          |0|_A() = 1
          a__U31#_A(x1,x2) = x2 + 2
          U31_A(x1,x2) = x1 + x2 + 1
          U22_A(x1) = x1
          U21_A(x1,x2) = x1
          U13_A(x1) = x1
          U12_A(x1,x2) = x1
          U11_A(x1,x2,x3) = x1
          a__U11_A(x1,x2,x3) = x1
          a__U12_A(x1,x2) = x1
          a__U13_A(x1) = x1
          a__U21_A(x1,x2) = x1
          a__U22_A(x1) = x1
          a__U31_A(x1,x2) = x1 + x2 + 1
          a__U41_A(x1,x2,x3) = x2 + x3
          a__plus_A(x1,x2) = x1 + x2
          U41_A(x1,x2,x3) = x2 + x3
  
    precedence:
    
      a__U31# > a__U11# = a__U12# = a__isNat# = a__isNatKind# = a__and# = mark# = a__plus# = a__U21# > tt = a__isNat = a__isNatKind = and = mark = a__and = isNat = U22 = U21 = U13 = U12 = U11 = a__U11 = a__U12 = a__U13 = a__U21 = a__U22 = a__U31 = a__plus > U31 > isNatKind = a__U41 = U41 > s = plus = |0|
    
    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(|0|) = []
      pi(a__U31#) = []
      pi(U31) = [2]
      pi(U22) = []
      pi(U21) = []
      pi(U13) = []
      pi(U12) = []
      pi(U11) = []
      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(U41) = []

The next rules are strictly ordered:

  p25

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__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__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
p23: a__plus#(N,|0|()) -> a__isNat#(N)
p24: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p25: a__U31#(tt(),N) -> mark#(N)
p26: mark#(U31(X1,X2)) -> mark#(X1)
p27: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p28: mark#(U22(X)) -> mark#(X)
p29: mark#(U21(X1,X2)) -> mark#(X1)
p30: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p31: mark#(U13(X)) -> mark#(X)
p32: mark#(isNat(X)) -> a__isNat#(X)
p33: mark#(U12(X1,X2)) -> mark#(X1)
p34: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p35: mark#(U11(X1,X2,X3)) -> mark#(X1)
p36: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)

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}


-- 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#(plus(V1,V2)) -> a__isNatKind#(V1)
p13: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p21: mark#(U21(X1,X2)) -> mark#(X1)
p22: mark#(U22(X)) -> mark#(X)
p23: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p24: a__U31#(tt(),N) -> mark#(N)
p25: mark#(U31(X1,X2)) -> mark#(X1)
p26: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p27: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p28: a__plus#(N,|0|()) -> a__isNat#(N)
p29: a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
p30: a__plus#(N,s(M)) -> a__isNat#(M)
p31: mark#(plus(X1,X2)) -> mark#(X1)
p32: mark#(plus(X1,X2)) -> mark#(X2)
p33: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p34: mark#(and(X1,X2)) -> mark#(X1)
p35: mark#(isNatKind(X)) -> a__isNatKind#(X)
p36: 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 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:

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

The next rules are strictly ordered:

  p30

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#(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#(plus(V1,V2)) -> a__isNatKind#(V1)
p13: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p21: mark#(U21(X1,X2)) -> mark#(X1)
p22: mark#(U22(X)) -> mark#(X)
p23: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p24: a__U31#(tt(),N) -> mark#(N)
p25: mark#(U31(X1,X2)) -> mark#(X1)
p26: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p27: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p28: a__plus#(N,|0|()) -> a__isNat#(N)
p29: a__plus#(N,s(M)) -> a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
p30: mark#(plus(X1,X2)) -> mark#(X1)
p31: mark#(plus(X1,X2)) -> mark#(X2)
p32: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p33: mark#(and(X1,X2)) -> mark#(X1)
p34: mark#(isNatKind(X)) -> a__isNatKind#(X)
p35: 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, 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}


-- 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__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
p16: a__plus#(N,|0|()) -> a__isNat#(N)
p17: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p18: a__U21#(tt(),V1) -> a__isNat#(V1)
p19: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p20: a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p21: a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
p22: a__U11#(tt(),V1,V2) -> a__isNat#(V1)
p23: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p24: mark#(U31(X1,X2)) -> mark#(X1)
p25: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p26: a__U31#(tt(),N) -> mark#(N)
p27: mark#(U22(X)) -> mark#(X)
p28: mark#(U21(X1,X2)) -> mark#(X1)
p29: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p30: mark#(U13(X)) -> mark#(X)
p31: mark#(isNat(X)) -> a__isNat#(X)
p32: mark#(U12(X1,X2)) -> mark#(X1)
p33: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p34: mark#(U11(X1,X2,X3)) -> mark#(X1)
p35: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          a__U11#_A(x1,x2,x3) = 2
          tt_A() = 0
          a__U12#_A(x1,x2) = 2
          a__isNat_A(x1) = 0
          a__isNat#_A(x1) = 2
          s_A(x1) = x1 + 2
          a__isNatKind#_A(x1) = 2
          plus_A(x1,x2) = x1 + x2
          a__and#_A(x1,x2) = x2 + 2
          a__isNatKind_A(x1) = 0
          isNatKind_A(x1) = 0
          mark#_A(x1) = x1 + 2
          and_A(x1,x2) = x1 + x2
          mark_A(x1) = x1
          a__plus#_A(x1,x2) = x2 + 1
          a__and_A(x1,x2) = x1 + x2
          isNat_A(x1) = 0
          |0|_A() = 3
          a__U21#_A(x1,x2) = 2
          U31_A(x1,x2) = x1 + x2
          a__U31#_A(x1,x2) = x2 + 2
          U22_A(x1) = x1
          U21_A(x1,x2) = x1
          U13_A(x1) = x1
          U12_A(x1,x2) = x1
          U11_A(x1,x2,x3) = x1
          a__U11_A(x1,x2,x3) = x1
          a__U12_A(x1,x2) = x1
          a__U13_A(x1) = x1
          a__U21_A(x1,x2) = x1
          a__U22_A(x1) = x1
          a__U31_A(x1,x2) = x1 + x2
          a__U41_A(x1,x2,x3) = x2 + x3 + 2
          a__plus_A(x1,x2) = x1 + x2
          U41_A(x1,x2,x3) = x2 + x3 + 2
  
    precedence:
    
      a__U11# = a__U12# = a__isNat# = a__isNatKind# = a__and# = mark# = a__plus# = a__U21# = a__U31# > a__isNatKind = isNatKind = and = mark = a__and = |0| = U31 = a__U31 > plus = a__plus > tt = a__isNat = s = isNat = U22 = U21 = U13 = U12 = U11 = a__U11 = a__U12 = a__U13 = a__U21 = a__U22 = a__U41 = U41
    
    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__and) = []
      pi(isNat) = []
      pi(|0|) = []
      pi(a__U21#) = []
      pi(U31) = []
      pi(a__U31#) = []
      pi(U22) = []
      pi(U21) = []
      pi(U13) = []
      pi(U12) = []
      pi(U11) = []
      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(U41) = []

The next rules are strictly ordered:

  p15

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,|0|()) -> a__isNat#(N)
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,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p23: mark#(U31(X1,X2)) -> mark#(X1)
p24: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p25: a__U31#(tt(),N) -> mark#(N)
p26: mark#(U22(X)) -> mark#(X)
p27: mark#(U21(X1,X2)) -> mark#(X1)
p28: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p29: mark#(U13(X)) -> mark#(X)
p30: mark#(isNat(X)) -> a__isNat#(X)
p31: mark#(U12(X1,X2)) -> mark#(X1)
p32: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p33: mark#(U11(X1,X2,X3)) -> mark#(X1)
p34: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)

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}


-- 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#(plus(V1,V2)) -> a__isNatKind#(V1)
p13: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p21: mark#(U21(X1,X2)) -> mark#(X1)
p22: mark#(U22(X)) -> mark#(X)
p23: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p24: a__U31#(tt(),N) -> mark#(N)
p25: mark#(U31(X1,X2)) -> mark#(X1)
p26: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p27: a__plus#(N,|0|()) -> a__and#(a__isNat(N),isNatKind(N))
p28: a__plus#(N,|0|()) -> a__isNat#(N)
p29: mark#(plus(X1,X2)) -> mark#(X1)
p30: mark#(plus(X1,X2)) -> mark#(X2)
p31: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p32: mark#(and(X1,X2)) -> mark#(X1)
p33: mark#(isNatKind(X)) -> a__isNatKind#(X)
p34: 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 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:

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

The next rules are strictly ordered:

  p27

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#(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#(plus(V1,V2)) -> a__isNatKind#(V1)
p13: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p21: mark#(U21(X1,X2)) -> mark#(X1)
p22: mark#(U22(X)) -> mark#(X)
p23: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p24: a__U31#(tt(),N) -> mark#(N)
p25: mark#(U31(X1,X2)) -> mark#(X1)
p26: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p27: a__plus#(N,|0|()) -> a__isNat#(N)
p28: mark#(plus(X1,X2)) -> mark#(X1)
p29: mark#(plus(X1,X2)) -> mark#(X2)
p30: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p31: mark#(and(X1,X2)) -> mark#(X1)
p32: mark#(isNatKind(X)) -> a__isNatKind#(X)
p33: 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, 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}


-- 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,|0|()) -> a__isNat#(N)
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: mark#(U31(X1,X2)) -> mark#(X1)
p23: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p24: a__U31#(tt(),N) -> mark#(N)
p25: mark#(U22(X)) -> mark#(X)
p26: mark#(U21(X1,X2)) -> mark#(X1)
p27: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p28: mark#(U13(X)) -> mark#(X)
p29: mark#(isNat(X)) -> a__isNat#(X)
p30: mark#(U12(X1,X2)) -> mark#(X1)
p31: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p32: mark#(U11(X1,X2,X3)) -> mark#(X1)
p33: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          a__U11#_A(x1,x2,x3) = 0
          tt_A() = 0
          a__U12#_A(x1,x2) = 0
          a__isNat_A(x1) = 0
          a__isNat#_A(x1) = 0
          s_A(x1) = x1
          a__isNatKind#_A(x1) = 0
          plus_A(x1,x2) = x1 + x2 + 2
          a__and#_A(x1,x2) = x2
          a__isNatKind_A(x1) = 0
          isNatKind_A(x1) = 0
          mark#_A(x1) = x1
          and_A(x1,x2) = x1 + x2
          mark_A(x1) = x1
          a__plus#_A(x1,x2) = x1 + 1
          |0|_A() = 1
          a__U21#_A(x1,x2) = 0
          a__and_A(x1,x2) = x1 + x2
          U31_A(x1,x2) = x1 + x2 + 2
          a__U31#_A(x1,x2) = x1 + x2 + 1
          U22_A(x1) = x1
          U21_A(x1,x2) = x1
          U13_A(x1) = x1
          isNat_A(x1) = 0
          U12_A(x1,x2) = x1
          U11_A(x1,x2,x3) = x1
          a__U11_A(x1,x2,x3) = x1
          a__U12_A(x1,x2) = x1
          a__U13_A(x1) = x1
          a__U21_A(x1,x2) = x1
          a__U22_A(x1) = x1
          a__U31_A(x1,x2) = x1 + x2 + 2
          a__U41_A(x1,x2,x3) = x2 + x3 + 2
          a__plus_A(x1,x2) = x1 + x2 + 2
          U41_A(x1,x2,x3) = x2 + x3 + 2
  
    precedence:
    
      a__U31# > a__isNat = a__isNatKind = mark = a__and = a__U11 > a__U21 = a__U22 > |0| > a__plus > a__U41 > s > plus > U41 > U21 > isNatKind > and = U22 = U13 = U11 = a__U12 = a__U13 > tt > U31 = isNat = a__U31 > U12 > a__U11# = a__U12# = a__isNat# = a__isNatKind# = a__and# = mark# = a__plus# = a__U21#
    
    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#) = [1]
      pi(|0|) = []
      pi(a__U21#) = []
      pi(a__and) = []
      pi(U31) = []
      pi(a__U31#) = [1]
      pi(U22) = []
      pi(U21) = []
      pi(U13) = []
      pi(isNat) = []
      pi(U12) = []
      pi(U11) = []
      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(U41) = []

The next rules are strictly ordered:

  p23

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,|0|()) -> a__isNat#(N)
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: mark#(U31(X1,X2)) -> mark#(X1)
p23: a__U31#(tt(),N) -> mark#(N)
p24: mark#(U22(X)) -> mark#(X)
p25: mark#(U21(X1,X2)) -> mark#(X1)
p26: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p27: mark#(U13(X)) -> mark#(X)
p28: mark#(isNat(X)) -> a__isNat#(X)
p29: mark#(U12(X1,X2)) -> mark#(X1)
p30: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p31: mark#(U11(X1,X2,X3)) -> mark#(X1)
p32: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)

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, p24, p25, p26, p27, p28, p29, p30, p31, p32}


-- 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#(plus(V1,V2)) -> a__isNatKind#(V1)
p13: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p21: mark#(U21(X1,X2)) -> mark#(X1)
p22: mark#(U22(X)) -> mark#(X)
p23: mark#(U31(X1,X2)) -> mark#(X1)
p24: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p25: a__plus#(N,|0|()) -> a__isNat#(N)
p26: mark#(plus(X1,X2)) -> mark#(X1)
p27: mark#(plus(X1,X2)) -> mark#(X2)
p28: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p29: mark#(and(X1,X2)) -> mark#(X1)
p30: mark#(isNatKind(X)) -> a__isNatKind#(X)
p31: 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 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:

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

The next rules are strictly ordered:

  p26

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#(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#(plus(V1,V2)) -> a__isNatKind#(V1)
p13: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p21: mark#(U21(X1,X2)) -> mark#(X1)
p22: mark#(U22(X)) -> mark#(X)
p23: mark#(U31(X1,X2)) -> mark#(X1)
p24: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p25: a__plus#(N,|0|()) -> a__isNat#(N)
p26: mark#(plus(X1,X2)) -> mark#(X2)
p27: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p28: mark#(and(X1,X2)) -> mark#(X1)
p29: mark#(isNatKind(X)) -> a__isNatKind#(X)
p30: 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, 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}


-- 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)) -> a__plus#(mark(X1),mark(X2))
p14: a__plus#(N,|0|()) -> a__isNat#(N)
p15: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p16: a__U21#(tt(),V1) -> a__isNat#(V1)
p17: a__isNat#(plus(V1,V2)) -> a__isNatKind#(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: mark#(U31(X1,X2)) -> mark#(X1)
p22: mark#(U22(X)) -> mark#(X)
p23: mark#(U21(X1,X2)) -> mark#(X1)
p24: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p25: mark#(U13(X)) -> mark#(X)
p26: mark#(isNat(X)) -> a__isNat#(X)
p27: mark#(U12(X1,X2)) -> mark#(X1)
p28: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p29: mark#(U11(X1,X2,X3)) -> mark#(X1)
p30: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)

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:

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

The next rules are strictly ordered:

  p14

We remove them from the problem.

-- SCC decomposition.

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

p1: 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)) -> a__plus#(mark(X1),mark(X2))
p14: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p15: a__U21#(tt(),V1) -> a__isNat#(V1)
p16: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p17: a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p18: a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
p19: a__U11#(tt(),V1,V2) -> a__isNat#(V1)
p20: mark#(U31(X1,X2)) -> mark#(X1)
p21: mark#(U22(X)) -> mark#(X)
p22: mark#(U21(X1,X2)) -> mark#(X1)
p23: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p24: mark#(U13(X)) -> mark#(X)
p25: mark#(isNat(X)) -> a__isNat#(X)
p26: mark#(U12(X1,X2)) -> mark#(X1)
p27: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p28: mark#(U11(X1,X2,X3)) -> mark#(X1)
p29: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)

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, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29}


-- 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#(plus(V1,V2)) -> a__isNatKind#(V1)
p13: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p21: mark#(U21(X1,X2)) -> mark#(X1)
p22: mark#(U22(X)) -> mark#(X)
p23: mark#(U31(X1,X2)) -> mark#(X1)
p24: mark#(plus(X1,X2)) -> mark#(X2)
p25: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p26: mark#(and(X1,X2)) -> mark#(X1)
p27: mark#(isNatKind(X)) -> a__isNatKind#(X)
p28: 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 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:

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

The next rules are strictly ordered:

  p28

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#(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#(plus(V1,V2)) -> a__isNatKind#(V1)
p13: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p21: mark#(U21(X1,X2)) -> mark#(X1)
p22: mark#(U22(X)) -> mark#(X)
p23: mark#(U31(X1,X2)) -> mark#(X1)
p24: mark#(plus(X1,X2)) -> mark#(X2)
p25: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p26: mark#(and(X1,X2)) -> mark#(X1)
p27: mark#(isNatKind(X)) -> a__isNatKind#(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, 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}


-- 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#(isNatKind(X)) -> a__isNatKind#(X)
p9: mark#(and(X1,X2)) -> mark#(X1)
p10: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p11: mark#(plus(X1,X2)) -> mark#(X2)
p12: mark#(U31(X1,X2)) -> mark#(X1)
p13: mark#(U22(X)) -> mark#(X)
p14: mark#(U21(X1,X2)) -> mark#(X1)
p15: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p16: a__U21#(tt(),V1) -> a__isNat#(V1)
p17: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),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: mark#(U13(X)) -> mark#(X)
p23: mark#(isNat(X)) -> a__isNat#(X)
p24: mark#(U12(X1,X2)) -> mark#(X1)
p25: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p26: mark#(U11(X1,X2,X3)) -> mark#(X1)
p27: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          a__U11#_A(x1,x2,x3) = 2
          tt_A() = 0
          a__U12#_A(x1,x2) = 2
          a__isNat_A(x1) = 0
          a__isNat#_A(x1) = 2
          s_A(x1) = x1 + 1
          a__isNatKind#_A(x1) = 2
          plus_A(x1,x2) = x1 + x2 + 1
          a__and#_A(x1,x2) = x2 + 2
          a__isNatKind_A(x1) = 0
          isNatKind_A(x1) = 0
          mark#_A(x1) = x1 + 2
          and_A(x1,x2) = x1 + x2
          mark_A(x1) = x1
          U31_A(x1,x2) = x1 + x2 + 1
          U22_A(x1) = x1
          U21_A(x1,x2) = x1
          a__U21#_A(x1,x2) = 2
          a__and_A(x1,x2) = x1 + x2
          U13_A(x1) = x1
          isNat_A(x1) = 0
          U12_A(x1,x2) = x1
          U11_A(x1,x2,x3) = x1
          a__U11_A(x1,x2,x3) = x1
          a__U12_A(x1,x2) = x1
          a__U13_A(x1) = x1
          a__U21_A(x1,x2) = x1
          a__U22_A(x1) = x1
          a__U31_A(x1,x2) = x1 + x2 + 1
          a__U41_A(x1,x2,x3) = x2 + x3 + 2
          a__plus_A(x1,x2) = x1 + x2 + 1
          |0|_A() = 2
          U41_A(x1,x2,x3) = x2 + x3 + 2
  
    precedence:
    
      a__U11# = tt = a__U12# = a__isNat = a__isNat# = s = a__isNatKind# = plus = a__and# = a__isNatKind = isNatKind = mark# = and = mark = U31 = U22 = U21 = a__U21# = a__and = U13 = isNat = U12 = U11 = a__U11 = a__U12 = a__U13 = a__U21 = a__U22 = a__U31 = a__U41 = a__plus = |0| = U41
    
    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(U31) = []
      pi(U22) = []
      pi(U21) = []
      pi(a__U21#) = []
      pi(a__and) = []
      pi(U13) = []
      pi(isNat) = []
      pi(U12) = []
      pi(U11) = []
      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(|0|) = []
      pi(U41) = []

The next rules are strictly ordered:

  p11

We remove them from the problem.

-- SCC decomposition.

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

p1: 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#(isNatKind(X)) -> a__isNatKind#(X)
p9: mark#(and(X1,X2)) -> mark#(X1)
p10: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p11: mark#(U31(X1,X2)) -> mark#(X1)
p12: mark#(U22(X)) -> mark#(X)
p13: mark#(U21(X1,X2)) -> mark#(X1)
p14: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p15: a__U21#(tt(),V1) -> a__isNat#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__isNat#(plus(V1,V2)) -> a__isNatKind#(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: mark#(U13(X)) -> mark#(X)
p22: mark#(isNat(X)) -> a__isNat#(X)
p23: mark#(U12(X1,X2)) -> mark#(X1)
p24: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p25: mark#(U11(X1,X2,X3)) -> mark#(X1)
p26: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)

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}


-- 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#(plus(V1,V2)) -> a__isNatKind#(V1)
p13: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p21: mark#(U21(X1,X2)) -> mark#(X1)
p22: mark#(U22(X)) -> mark#(X)
p23: mark#(U31(X1,X2)) -> mark#(X1)
p24: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p25: mark#(and(X1,X2)) -> mark#(X1)
p26: mark#(isNatKind(X)) -> a__isNatKind#(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 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:

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

The next rules are strictly ordered:

  p11

We remove them from the problem.

-- SCC decomposition.

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

p1: 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: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p12: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p13: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p14: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p15: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p16: a__U21#(tt(),V1) -> a__isNat#(V1)
p17: a__isNat#(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)) -> mark#(X1)
p23: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p24: mark#(and(X1,X2)) -> mark#(X1)
p25: mark#(isNatKind(X)) -> a__isNatKind#(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, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25}


-- 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#(isNatKind(X)) -> a__isNatKind#(X)
p9: mark#(and(X1,X2)) -> mark#(X1)
p10: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p11: mark#(U31(X1,X2)) -> mark#(X1)
p12: mark#(U22(X)) -> mark#(X)
p13: mark#(U21(X1,X2)) -> mark#(X1)
p14: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p15: a__U21#(tt(),V1) -> a__isNat#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__isNat#(plus(V1,V2)) -> a__isNatKind#(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: mark#(U13(X)) -> mark#(X)
p22: mark#(U12(X1,X2)) -> mark#(X1)
p23: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p24: mark#(U11(X1,X2,X3)) -> mark#(X1)
p25: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)

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:

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

The next rules are strictly ordered:

  p11

We remove them from the problem.

-- SCC decomposition.

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

p1: 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#(isNatKind(X)) -> a__isNatKind#(X)
p9: mark#(and(X1,X2)) -> mark#(X1)
p10: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p11: mark#(U22(X)) -> mark#(X)
p12: mark#(U21(X1,X2)) -> mark#(X1)
p13: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p14: a__U21#(tt(),V1) -> a__isNat#(V1)
p15: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p16: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p17: a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p18: a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
p19: a__U11#(tt(),V1,V2) -> a__isNat#(V1)
p20: mark#(U13(X)) -> mark#(X)
p21: mark#(U12(X1,X2)) -> mark#(X1)
p22: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p23: mark#(U11(X1,X2,X3)) -> mark#(X1)
p24: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)

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}


-- 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#(U13(X)) -> mark#(X)
p12: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p13: a__U21#(tt(),V1) -> a__isNat#(V1)
p14: a__isNat#(plus(V1,V2)) -> 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: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p19: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p20: mark#(U21(X1,X2)) -> mark#(X1)
p21: mark#(U22(X)) -> mark#(X)
p22: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p23: mark#(and(X1,X2)) -> mark#(X1)
p24: mark#(isNatKind(X)) -> a__isNatKind#(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 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:

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

The next rules are strictly ordered:

  p7

We remove them from the problem.

-- SCC decomposition.

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

p1: 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)) -> mark#(X1)
p8: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p9: mark#(U12(X1,X2)) -> mark#(X1)
p10: mark#(U13(X)) -> mark#(X)
p11: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p12: a__U21#(tt(),V1) -> a__isNat#(V1)
p13: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p14: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p15: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p16: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p17: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p18: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p19: mark#(U21(X1,X2)) -> mark#(X1)
p20: mark#(U22(X)) -> mark#(X)
p21: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p22: mark#(and(X1,X2)) -> mark#(X1)
p23: mark#(isNatKind(X)) -> a__isNatKind#(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, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23}


-- 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#(isNatKind(X)) -> a__isNatKind#(X)
p9: mark#(and(X1,X2)) -> mark#(X1)
p10: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p11: mark#(U22(X)) -> mark#(X)
p12: mark#(U21(X1,X2)) -> mark#(X1)
p13: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p14: a__U21#(tt(),V1) -> a__isNat#(V1)
p15: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p16: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p17: a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p18: a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
p19: a__U11#(tt(),V1,V2) -> a__isNat#(V1)
p20: mark#(U13(X)) -> mark#(X)
p21: mark#(U12(X1,X2)) -> mark#(X1)
p22: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p23: mark#(U11(X1,X2,X3)) -> 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

  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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          a__U11#_A(x1,x2,x3) = x2 + x3 + 3
          tt_A() = 0
          a__U12#_A(x1,x2) = x2 + 2
          a__isNat_A(x1) = x1 + 8
          a__isNat#_A(x1) = x1 + 1
          s_A(x1) = x1 + 4
          a__isNatKind#_A(x1) = 1
          plus_A(x1,x2) = x1 + x2 + 11
          a__and#_A(x1,x2) = x2 + 1
          a__isNatKind_A(x1) = 0
          isNatKind_A(x1) = 0
          mark#_A(x1) = x1 + 1
          and_A(x1,x2) = x1 + x2
          mark_A(x1) = x1
          U22_A(x1) = x1 + 2
          U21_A(x1,x2) = x1 + x2 + 11
          a__U21#_A(x1,x2) = x1 + x2 + 2
          a__and_A(x1,x2) = x1 + x2
          U13_A(x1) = x1
          U12_A(x1,x2) = x1 + x2 + 9
          U11_A(x1,x2,x3) = x1 + x2 + x3 + 18
          a__U11_A(x1,x2,x3) = x1 + x2 + x3 + 18
          a__U12_A(x1,x2) = x1 + x2 + 9
          a__U13_A(x1) = x1
          a__U21_A(x1,x2) = x1 + x2 + 11
          a__U22_A(x1) = x1 + 2
          a__U31_A(x1,x2) = x2 + 1
          a__U41_A(x1,x2,x3) = x2 + x3 + 15
          a__plus_A(x1,x2) = x1 + x2 + 11
          |0|_A() = 2
          isNat_A(x1) = x1 + 8
          U31_A(x1,x2) = x2 + 1
          U41_A(x1,x2,x3) = x2 + x3 + 15
  
    precedence:
    
      a__U11# = a__isNat = a__isNat# = a__isNatKind# = a__and# = a__isNatKind = mark# = mark = a__U21# = a__and = a__U22 > isNatKind > a__U31 > a__U41 = a__plus > s > and > a__U11 = |0| > a__U12# > plus = U22 > U11 = a__U21 > U21 > U41 > U31 > a__U12 > tt = a__U13 > U12 > U13 = isNat
    
    partial status:
  
      pi(a__U11#) = []
      pi(tt) = []
      pi(a__U12#) = []
      pi(a__isNat) = []
      pi(a__isNat#) = [1]
      pi(s) = []
      pi(a__isNatKind#) = []
      pi(plus) = [2]
      pi(a__and#) = []
      pi(a__isNatKind) = []
      pi(isNatKind) = []
      pi(mark#) = []
      pi(and) = []
      pi(mark) = []
      pi(U22) = []
      pi(U21) = [2]
      pi(a__U21#) = [1]
      pi(a__and) = []
      pi(U13) = []
      pi(U12) = [2]
      pi(U11) = [2]
      pi(a__U11) = [3]
      pi(a__U12) = []
      pi(a__U13) = []
      pi(a__U21) = []
      pi(a__U22) = []
      pi(a__U31) = []
      pi(a__U41) = []
      pi(a__plus) = [2]
      pi(|0|) = []
      pi(isNat) = []
      pi(U31) = []
      pi(U41) = [2]

The next rules are strictly ordered:

  p13

We remove them from the problem.

-- SCC decomposition.

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

p1: 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#(isNatKind(X)) -> a__isNatKind#(X)
p9: mark#(and(X1,X2)) -> mark#(X1)
p10: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p11: mark#(U22(X)) -> mark#(X)
p12: mark#(U21(X1,X2)) -> mark#(X1)
p13: a__U21#(tt(),V1) -> a__isNat#(V1)
p14: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p15: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p16: a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p17: a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
p18: a__U11#(tt(),V1,V2) -> a__isNat#(V1)
p19: mark#(U13(X)) -> mark#(X)
p20: mark#(U12(X1,X2)) -> mark#(X1)
p21: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p22: mark#(U11(X1,X2,X3)) -> 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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22}


-- Reduction pair.

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

p1: 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)) -> mark#(X1)
p8: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p9: mark#(U12(X1,X2)) -> mark#(X1)
p10: mark#(U13(X)) -> mark#(X)
p11: mark#(U21(X1,X2)) -> mark#(X1)
p12: mark#(U22(X)) -> mark#(X)
p13: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p14: mark#(and(X1,X2)) -> mark#(X1)
p15: mark#(isNatKind(X)) -> a__isNatKind#(X)
p16: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p17: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p18: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p19: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p20: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p21: a__U21#(tt(),V1) -> a__isNat#(V1)
p22: a__isNat#(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

  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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          a__U11#_A(x1,x2,x3) = x2 + x3 + 15
          tt_A() = 0
          a__U12#_A(x1,x2) = x2 + 15
          a__isNat_A(x1) = x1 + 3
          a__isNat#_A(x1) = x1 + 8
          plus_A(x1,x2) = x1 + x2 + 7
          a__and_A(x1,x2) = x1 + x2
          a__isNatKind_A(x1) = 0
          isNatKind_A(x1) = 0
          a__and#_A(x1,x2) = x2 + 10
          mark#_A(x1) = x1 + 10
          U11_A(x1,x2,x3) = x1 + x2 + x3 + 9
          U12_A(x1,x2) = x1 + x2 + 5
          mark_A(x1) = x1
          U13_A(x1) = x1 + 1
          U21_A(x1,x2) = x1 + x2 + 12
          U22_A(x1) = x1 + 1
          and_A(x1,x2) = x1 + x2
          a__isNatKind#_A(x1) = 10
          s_A(x1) = x1 + 11
          a__U21#_A(x1,x2) = x2 + 10
          a__U11_A(x1,x2,x3) = x1 + x2 + x3 + 9
          a__U12_A(x1,x2) = x1 + x2 + 5
          a__U13_A(x1) = x1 + 1
          a__U21_A(x1,x2) = x1 + x2 + 12
          a__U22_A(x1) = x1 + 1
          a__U31_A(x1,x2) = x2 + 1
          a__U41_A(x1,x2,x3) = x2 + x3 + 18
          a__plus_A(x1,x2) = x1 + x2 + 7
          |0|_A() = 2
          isNat_A(x1) = x1 + 3
          U31_A(x1,x2) = x2 + 1
          U41_A(x1,x2,x3) = x2 + x3 + 18
  
    precedence:
    
      a__and = a__isNatKind = isNatKind = mark = and = a__U31 = |0| > plus = s = a__U41 = a__plus > a__isNat = isNat > a__U11# = a__isNat# = a__U21# > a__and# = mark# = a__isNatKind# > a__U12# > U41 > U31 > U21 = U22 = a__U21 = a__U22 > tt = U11 = U12 = U13 = a__U11 = a__U12 = a__U13
    
    partial status:
  
      pi(a__U11#) = []
      pi(tt) = []
      pi(a__U12#) = [2]
      pi(a__isNat) = []
      pi(a__isNat#) = []
      pi(plus) = []
      pi(a__and) = []
      pi(a__isNatKind) = []
      pi(isNatKind) = []
      pi(a__and#) = []
      pi(mark#) = []
      pi(U11) = []
      pi(U12) = []
      pi(mark) = []
      pi(U13) = []
      pi(U21) = []
      pi(U22) = []
      pi(and) = []
      pi(a__isNatKind#) = []
      pi(s) = []
      pi(a__U21#) = []
      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(|0|) = []
      pi(isNat) = []
      pi(U31) = [2]
      pi(U41) = [2, 3]

The next rules are strictly ordered:

  p5

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#(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__and#(tt(),X) -> mark#(X)
p6: mark#(U11(X1,X2,X3)) -> mark#(X1)
p7: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p8: mark#(U12(X1,X2)) -> mark#(X1)
p9: mark#(U13(X)) -> mark#(X)
p10: mark#(U21(X1,X2)) -> mark#(X1)
p11: mark#(U22(X)) -> mark#(X)
p12: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p13: mark#(and(X1,X2)) -> mark#(X1)
p14: mark#(isNatKind(X)) -> a__isNatKind#(X)
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: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p19: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p20: a__U21#(tt(),V1) -> a__isNat#(V1)
p21: a__isNat#(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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21}


-- Reduction pair.

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

p1: 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#(isNatKind(X)) -> a__isNatKind#(X)
p9: mark#(and(X1,X2)) -> mark#(X1)
p10: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p11: mark#(U22(X)) -> mark#(X)
p12: mark#(U21(X1,X2)) -> mark#(X1)
p13: mark#(U13(X)) -> mark#(X)
p14: mark#(U12(X1,X2)) -> mark#(X1)
p15: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p16: mark#(U11(X1,X2,X3)) -> mark#(X1)
p17: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p18: a__U21#(tt(),V1) -> a__isNat#(V1)
p19: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
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)

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:

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

The next rules are strictly ordered:

  p20

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#(isNatKind(X)) -> a__isNatKind#(X)
p9: mark#(and(X1,X2)) -> mark#(X1)
p10: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p11: mark#(U22(X)) -> mark#(X)
p12: mark#(U21(X1,X2)) -> mark#(X1)
p13: mark#(U13(X)) -> mark#(X)
p14: mark#(U12(X1,X2)) -> mark#(X1)
p15: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p16: mark#(U11(X1,X2,X3)) -> mark#(X1)
p17: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p18: a__U21#(tt(),V1) -> a__isNat#(V1)
p19: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p20: a__U11#(tt(),V1,V2) -> 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 estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19}


-- Reduction pair.

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

p1: a__U12#(tt(),V2) -> a__isNat#(V2)
p2: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p3: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p4: a__and#(tt(),X) -> mark#(X)
p5: mark#(U11(X1,X2,X3)) -> mark#(X1)
p6: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p7: mark#(U12(X1,X2)) -> mark#(X1)
p8: mark#(U13(X)) -> mark#(X)
p9: mark#(U21(X1,X2)) -> mark#(X1)
p10: mark#(U22(X)) -> mark#(X)
p11: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p12: mark#(and(X1,X2)) -> mark#(X1)
p13: mark#(isNatKind(X)) -> a__isNatKind#(X)
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p17: a__U21#(tt(),V1) -> a__isNat#(V1)
p18: a__isNat#(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

  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:

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

The next rules are strictly ordered:

  p16

We remove them from the problem.

-- SCC decomposition.

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

p1: a__U12#(tt(),V2) -> a__isNat#(V2)
p2: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p3: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p4: a__and#(tt(),X) -> mark#(X)
p5: mark#(U11(X1,X2,X3)) -> mark#(X1)
p6: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p7: mark#(U12(X1,X2)) -> mark#(X1)
p8: mark#(U13(X)) -> mark#(X)
p9: mark#(U21(X1,X2)) -> mark#(X1)
p10: mark#(U22(X)) -> mark#(X)
p11: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p12: mark#(and(X1,X2)) -> mark#(X1)
p13: mark#(isNatKind(X)) -> a__isNatKind#(X)
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p16: a__U21#(tt(),V1) -> a__isNat#(V1)
p17: a__isNat#(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 estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: a__U12#(tt(),V2) -> a__isNat#(V2)
p2: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p3: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p4: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p5: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p6: a__and#(tt(),X) -> mark#(X)
p7: mark#(isNatKind(X)) -> a__isNatKind#(X)
p8: mark#(and(X1,X2)) -> mark#(X1)
p9: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p10: mark#(U22(X)) -> mark#(X)
p11: mark#(U21(X1,X2)) -> mark#(X1)
p12: mark#(U13(X)) -> mark#(X)
p13: mark#(U12(X1,X2)) -> mark#(X1)
p14: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p15: mark#(U11(X1,X2,X3)) -> mark#(X1)
p16: a__isNat#(plus(V1,V2)) -> 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

  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:

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

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: a__U12#(tt(),V2) -> a__isNat#(V2)
p2: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p3: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p4: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p5: a__and#(tt(),X) -> mark#(X)
p6: mark#(isNatKind(X)) -> a__isNatKind#(X)
p7: mark#(and(X1,X2)) -> mark#(X1)
p8: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p9: mark#(U22(X)) -> mark#(X)
p10: mark#(U21(X1,X2)) -> mark#(X1)
p11: mark#(U13(X)) -> mark#(X)
p12: mark#(U12(X1,X2)) -> mark#(X1)
p13: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p14: mark#(U11(X1,X2,X3)) -> mark#(X1)
p15: a__isNat#(plus(V1,V2)) -> 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 estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: a__U12#(tt(),V2) -> a__isNat#(V2)
p2: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p3: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p4: a__and#(tt(),X) -> mark#(X)
p5: mark#(U11(X1,X2,X3)) -> mark#(X1)
p6: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p7: mark#(U12(X1,X2)) -> mark#(X1)
p8: mark#(U13(X)) -> mark#(X)
p9: mark#(U21(X1,X2)) -> mark#(X1)
p10: mark#(U22(X)) -> mark#(X)
p11: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p12: mark#(and(X1,X2)) -> mark#(X1)
p13: mark#(isNatKind(X)) -> a__isNatKind#(X)
p14: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: 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

  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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          a__U12#_A(x1,x2) = x2 + 11
          tt_A() = 0
          a__isNat#_A(x1) = 10
          plus_A(x1,x2) = x1 + x2 + 11
          a__isNatKind#_A(x1) = 9
          a__and#_A(x1,x2) = x2 + 9
          a__isNatKind_A(x1) = 0
          isNatKind_A(x1) = 0
          mark#_A(x1) = x1 + 9
          U11_A(x1,x2,x3) = x1 + x2 + x3 + 10
          U12_A(x1,x2) = x1 + x2 + 5
          mark_A(x1) = x1
          U13_A(x1) = x1 + 1
          U21_A(x1,x2) = x1 + x2 + 6
          U22_A(x1) = x1 + 1
          and_A(x1,x2) = x1 + x2
          s_A(x1) = x1 + 7
          a__U11_A(x1,x2,x3) = x1 + x2 + x3 + 10
          a__U12_A(x1,x2) = x1 + x2 + 5
          a__isNat_A(x1) = x1 + 4
          a__U13_A(x1) = x1 + 1
          a__U21_A(x1,x2) = x1 + x2 + 6
          a__U22_A(x1) = x1 + 1
          a__U31_A(x1,x2) = x2 + 12
          a__U41_A(x1,x2,x3) = x2 + x3 + 18
          a__plus_A(x1,x2) = x1 + x2 + 11
          a__and_A(x1,x2) = x1 + x2
          |0|_A() = 1
          isNat_A(x1) = x1 + 4
          U31_A(x1,x2) = x2 + 12
          U41_A(x1,x2,x3) = x2 + x3 + 18
  
    precedence:
    
      a__U12# > a__isNat# > a__isNatKind# = a__and# = mark# > tt = a__isNatKind = U11 = U12 = mark = and = a__U11 = a__U12 = a__U21 = a__and = |0| > a__isNat = isNat > a__U22 = a__U31 = a__plus > a__U41 > U41 > U22 = U31 > plus = isNatKind > s > U13 = a__U13 > U21
    
    partial status:
  
      pi(a__U12#) = [2]
      pi(tt) = []
      pi(a__isNat#) = []
      pi(plus) = [2]
      pi(a__isNatKind#) = []
      pi(a__and#) = []
      pi(a__isNatKind) = []
      pi(isNatKind) = []
      pi(mark#) = []
      pi(U11) = []
      pi(U12) = []
      pi(mark) = []
      pi(U13) = [1]
      pi(U21) = []
      pi(U22) = []
      pi(and) = []
      pi(s) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__isNat) = [1]
      pi(a__U13) = [1]
      pi(a__U21) = []
      pi(a__U22) = [1]
      pi(a__U31) = []
      pi(a__U41) = []
      pi(a__plus) = []
      pi(a__and) = []
      pi(|0|) = []
      pi(isNat) = [1]
      pi(U31) = [2]
      pi(U41) = [3]

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: a__U12#(tt(),V2) -> a__isNat#(V2)
p2: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p3: a__and#(tt(),X) -> mark#(X)
p4: mark#(U11(X1,X2,X3)) -> mark#(X1)
p5: mark#(U12(X1,X2)) -> a__U12#(mark(X1),X2)
p6: mark#(U12(X1,X2)) -> mark#(X1)
p7: mark#(U13(X)) -> mark#(X)
p8: mark#(U21(X1,X2)) -> mark#(X1)
p9: mark#(U22(X)) -> mark#(X)
p10: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p11: mark#(and(X1,X2)) -> mark#(X1)
p12: mark#(isNatKind(X)) -> a__isNatKind#(X)
p13: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p14: 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 estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p6, p7, p8, p9, p10, p11, p12, p13, p14}


-- Reduction pair.

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

p1: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p2: a__and#(tt(),X) -> mark#(X)
p3: mark#(isNatKind(X)) -> a__isNatKind#(X)
p4: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p5: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p6: mark#(and(X1,X2)) -> mark#(X1)
p7: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
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)
p12: mark#(U11(X1,X2,X3)) -> 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

  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:

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

The next rules are strictly ordered:

  p12

We remove them from the problem.

-- SCC decomposition.

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

p1: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p2: a__and#(tt(),X) -> mark#(X)
p3: mark#(isNatKind(X)) -> a__isNatKind#(X)
p4: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p5: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p6: mark#(and(X1,X2)) -> mark#(X1)
p7: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
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 estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p2: a__and#(tt(),X) -> mark#(X)
p3: mark#(U12(X1,X2)) -> mark#(X1)
p4: mark#(U13(X)) -> mark#(X)
p5: mark#(U21(X1,X2)) -> mark#(X1)
p6: mark#(U22(X)) -> mark#(X)
p7: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p8: mark#(and(X1,X2)) -> mark#(X1)
p9: mark#(isNatKind(X)) -> a__isNatKind#(X)
p10: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p11: 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

  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:

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

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p2: a__and#(tt(),X) -> mark#(X)
p3: mark#(U13(X)) -> mark#(X)
p4: mark#(U21(X1,X2)) -> mark#(X1)
p5: mark#(U22(X)) -> mark#(X)
p6: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p7: mark#(and(X1,X2)) -> mark#(X1)
p8: mark#(isNatKind(X)) -> a__isNatKind#(X)
p9: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p10: 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 estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p2: a__and#(tt(),X) -> mark#(X)
p3: mark#(isNatKind(X)) -> a__isNatKind#(X)
p4: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p5: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p6: mark#(and(X1,X2)) -> mark#(X1)
p7: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p8: mark#(U22(X)) -> mark#(X)
p9: mark#(U21(X1,X2)) -> mark#(X1)
p10: mark#(U13(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 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          a__isNatKind#_A(x1) = 10
          plus_A(x1,x2) = x1 + 13
          a__and#_A(x1,x2) = x1 + x2 + 3
          a__isNatKind_A(x1) = 7
          isNatKind_A(x1) = 0
          tt_A() = 7
          mark#_A(x1) = x1 + 10
          s_A(x1) = 11
          and_A(x1,x2) = x1 + x2
          mark_A(x1) = x1 + 7
          U22_A(x1) = x1
          U21_A(x1,x2) = x1 + 1
          U13_A(x1) = x1
          a__U11_A(x1,x2,x3) = 8
          a__U12_A(x1,x2) = 8
          a__isNat_A(x1) = 8
          a__U13_A(x1) = x1
          a__U21_A(x1,x2) = x1 + 1
          a__U22_A(x1) = x1
          a__U31_A(x1,x2) = x2 + 7
          a__U41_A(x1,x2,x3) = 12
          a__plus_A(x1,x2) = x1 + 13
          a__and_A(x1,x2) = x1 + x2
          |0|_A() = 9
          isNat_A(x1) = 2
          U11_A(x1,x2,x3) = 2
          U12_A(x1,x2) = 6
          U31_A(x1,x2) = x2 + 1
          U41_A(x1,x2,x3) = 6
  
    precedence:
    
      a__isNat > s = a__U31 = |0| = U31 > a__isNatKind# = a__and# = mark# = U11 > plus = a__isNatKind = isNatKind = tt = and = mark = U22 = U21 = U13 = a__U11 = a__U12 = a__U13 = a__U21 = a__U22 = a__U41 = a__plus = a__and = isNat = U12 = U41
    
    partial status:
  
      pi(a__isNatKind#) = []
      pi(plus) = []
      pi(a__and#) = []
      pi(a__isNatKind) = []
      pi(isNatKind) = []
      pi(tt) = []
      pi(mark#) = []
      pi(s) = []
      pi(and) = []
      pi(mark) = []
      pi(U22) = []
      pi(U21) = []
      pi(U13) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__isNat) = []
      pi(a__U13) = []
      pi(a__U21) = []
      pi(a__U22) = []
      pi(a__U31) = []
      pi(a__U41) = []
      pi(a__plus) = []
      pi(a__and) = []
      pi(|0|) = []
      pi(isNat) = []
      pi(U11) = []
      pi(U12) = []
      pi(U31) = []
      pi(U41) = []

The next rules are strictly ordered:

  p9

We remove them from the problem.

-- SCC decomposition.

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

p1: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p2: a__and#(tt(),X) -> mark#(X)
p3: mark#(isNatKind(X)) -> a__isNatKind#(X)
p4: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p5: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p6: mark#(and(X1,X2)) -> mark#(X1)
p7: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p8: mark#(U22(X)) -> mark#(X)
p9: mark#(U13(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, p3, p4, p5, p6, p7, p8, p9}


-- Reduction pair.

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

p1: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p2: a__and#(tt(),X) -> mark#(X)
p3: mark#(U13(X)) -> mark#(X)
p4: mark#(U22(X)) -> mark#(X)
p5: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p6: mark#(and(X1,X2)) -> mark#(X1)
p7: mark#(isNatKind(X)) -> a__isNatKind#(X)
p8: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p9: 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

  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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          a__isNatKind#_A(x1) = x1 + 1
          plus_A(x1,x2) = x1 + x2 + 12
          a__and#_A(x1,x2) = x2 + 1
          a__isNatKind_A(x1) = x1 + 2
          isNatKind_A(x1) = x1 + 2
          tt_A() = 0
          mark#_A(x1) = x1
          U13_A(x1) = x1 + 1
          U22_A(x1) = x1 + 1
          and_A(x1,x2) = x1 + x2 + 1
          mark_A(x1) = x1
          s_A(x1) = x1 + 8
          a__U11_A(x1,x2,x3) = x2 + x3 + 11
          a__U12_A(x1,x2) = x1 + x2 + 6
          a__isNat_A(x1) = x1 + 4
          a__U13_A(x1) = x1 + 1
          a__U21_A(x1,x2) = x2 + 6
          a__U22_A(x1) = x1 + 1
          a__U31_A(x1,x2) = x2 + 1
          a__U41_A(x1,x2,x3) = x2 + x3 + 20
          a__plus_A(x1,x2) = x1 + x2 + 12
          a__and_A(x1,x2) = x1 + x2 + 1
          |0|_A() = 3
          isNat_A(x1) = x1 + 4
          U11_A(x1,x2,x3) = x2 + x3 + 11
          U12_A(x1,x2) = x1 + x2 + 6
          U21_A(x1,x2) = x2 + 6
          U31_A(x1,x2) = x2 + 1
          U41_A(x1,x2,x3) = x2 + x3 + 20
  
    precedence:
    
      a__U11 = U11 > |0| > plus = a__U31 = a__U41 = a__plus = U31 = U41 > mark = s = a__U22 > a__and > a__and# = mark# = U22 = and > tt > a__U12 = a__isNat = a__U13 = U12 > a__U21 = isNat = U21 > a__isNatKind# = a__isNatKind = isNatKind > U13
    
    partial status:
  
      pi(a__isNatKind#) = [1]
      pi(plus) = []
      pi(a__and#) = []
      pi(a__isNatKind) = [1]
      pi(isNatKind) = []
      pi(tt) = []
      pi(mark#) = [1]
      pi(U13) = []
      pi(U22) = [1]
      pi(and) = []
      pi(mark) = [1]
      pi(s) = []
      pi(a__U11) = [3]
      pi(a__U12) = [2]
      pi(a__isNat) = [1]
      pi(a__U13) = []
      pi(a__U21) = []
      pi(a__U22) = []
      pi(a__U31) = []
      pi(a__U41) = []
      pi(a__plus) = []
      pi(a__and) = []
      pi(|0|) = []
      pi(isNat) = [1]
      pi(U11) = [3]
      pi(U12) = [2]
      pi(U21) = []
      pi(U31) = []
      pi(U41) = []

The next rules are strictly ordered:

  p7

We remove them from the problem.

-- SCC decomposition.

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

p1: a__isNatKind#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p2: a__and#(tt(),X) -> mark#(X)
p3: mark#(U13(X)) -> mark#(X)
p4: mark#(U22(X)) -> mark#(X)
p5: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p6: mark#(and(X1,X2)) -> mark#(X1)
p7: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p8: 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 estimated dependency graph contains the following SCCs:

  {p7, p8}
  {p2, p3, p4, p5, p6}


-- Reduction pair.

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

p1: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p2: a__isNatKind#(plus(V1,V2)) -> 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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          a__isNatKind#_A(x1) = x1 + 1
          s_A(x1) = x1 + 2
          plus_A(x1,x2) = x1 + x2 + 2
  
    precedence:
    
      a__isNatKind# > s = plus
    
    partial status:
  
      pi(a__isNatKind#) = [1]
      pi(s) = [1]
      pi(plus) = []

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: 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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          a__isNatKind#_A(x1) = x1
          s_A(x1) = x1 + 1
  
    precedence:
    
      a__isNatKind# = s
    
    partial status:
  
      pi(a__isNatKind#) = [1]
      pi(s) = [1]

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: a__and#(tt(),X) -> mark#(X)
p2: mark#(and(X1,X2)) -> mark#(X1)
p3: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p4: mark#(U22(X)) -> mark#(X)
p5: mark#(U13(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 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          a__and#_A(x1,x2) = x2 + 4
          tt_A() = 0
          mark#_A(x1) = x1 + 1
          and_A(x1,x2) = x1 + x2 + 5
          mark_A(x1) = x1
          U22_A(x1) = x1 + 2
          U13_A(x1) = x1 + 2
          a__U11_A(x1,x2,x3) = x3 + 14
          a__U12_A(x1,x2) = x2 + 13
          a__isNat_A(x1) = x1 + 10
          a__U13_A(x1) = x1 + 2
          a__U21_A(x1,x2) = x2 + 13
          a__U22_A(x1) = x1 + 2
          a__U31_A(x1,x2) = x2 + 1
          a__U41_A(x1,x2,x3) = x2 + x3 + 9
          s_A(x1) = x1 + 4
          a__plus_A(x1,x2) = x1 + x2 + 5
          a__and_A(x1,x2) = x1 + x2 + 5
          |0|_A() = 0
          plus_A(x1,x2) = x1 + x2 + 5
          a__isNatKind_A(x1) = x1
          isNatKind_A(x1) = x1
          isNat_A(x1) = x1 + 10
          U11_A(x1,x2,x3) = x3 + 14
          U12_A(x1,x2) = x2 + 13
          U21_A(x1,x2) = x2 + 13
          U31_A(x1,x2) = x2 + 1
          U41_A(x1,x2,x3) = x2 + x3 + 9
  
    precedence:
    
      mark# > tt = mark = U22 = U13 = a__U12 = a__isNat = a__U13 = a__U21 = a__U22 = a__U31 = a__U41 = s = a__plus = |0| = plus = a__isNatKind = isNatKind = isNat = U12 = U21 = U31 = U41 > a__U11 > a__and > a__and# = and = U11
    
    partial status:
  
      pi(a__and#) = []
      pi(tt) = []
      pi(mark#) = [1]
      pi(and) = []
      pi(mark) = [1]
      pi(U22) = []
      pi(U13) = []
      pi(a__U11) = []
      pi(a__U12) = [2]
      pi(a__isNat) = [1]
      pi(a__U13) = []
      pi(a__U21) = [2]
      pi(a__U22) = [1]
      pi(a__U31) = []
      pi(a__U41) = []
      pi(s) = []
      pi(a__plus) = [1, 2]
      pi(a__and) = [2]
      pi(|0|) = []
      pi(plus) = [1, 2]
      pi(a__isNatKind) = []
      pi(isNatKind) = []
      pi(isNat) = []
      pi(U11) = []
      pi(U12) = []
      pi(U21) = []
      pi(U31) = []
      pi(U41) = []

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: mark#(and(X1,X2)) -> mark#(X1)
p2: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)
p3: mark#(U22(X)) -> mark#(X)
p4: mark#(U13(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, p3, p4}


-- Reduction pair.

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

p1: mark#(and(X1,X2)) -> mark#(X1)
p2: mark#(U13(X)) -> mark#(X)
p3: mark#(U22(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 set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1
          and_A(x1,x2) = x1 + x2 + 1
          U13_A(x1) = x1 + 1
          U22_A(x1) = x1 + 1
  
    precedence:
    
      mark# = and = U13 = U22
    
    partial status:
  
      pi(mark#) = [1]
      pi(and) = [1, 2]
      pi(U13) = [1]
      pi(U22) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U13(X)) -> mark#(X)
p2: mark#(U22(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}


-- Reduction pair.

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

p1: mark#(U13(X)) -> mark#(X)
p2: mark#(U22(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 set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1
          U13_A(x1) = x1 + 1
          U22_A(x1) = x1 + 1
  
    precedence:
    
      mark# = U13 = U22
    
    partial status:
  
      pi(mark#) = [1]
      pi(U13) = [1]
      pi(U22) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U22(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}


-- Reduction pair.

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

p1: mark#(U22(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 set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^1
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1
          U22_A(x1) = x1 + 1
  
    precedence:
    
      mark# = U22
    
    partial status:
  
      pi(mark#) = [1]
      pi(U22) = [1]

The next rules are strictly ordered:

  p1

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