YES

We show the termination of the TRS R:

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

-- SCC decomposition.

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

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

and R consists of:

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

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

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

and R consists of:

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

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^1
      order: standard 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 + 1
        a__isNatKind#_A(x1) = 0
        plus_A(x1,x2) = x1 + x2 + 1
        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
        U41_A(x1,x2,x3) = x1 + x2 + x3 + 2
        U31_A(x1,x2) = x1 + x2 + 2
        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() = 1
        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 + 2
        a__plus_A(x1,x2) = x1 + x2 + 1
    
    2. matrix interpretations:
    
      carrier: N^1
      order: standard order
      interpretations:
        a__U11#_A(x1,x2,x3) = 1
        tt_A() = 2
        a__U12#_A(x1,x2) = 1
        a__isNat_A(x1) = 5
        a__isNat#_A(x1) = 1
        s_A(x1) = 1
        a__isNatKind#_A(x1) = 1
        plus_A(x1,x2) = 3
        a__and#_A(x1,x2) = 1
        a__isNatKind_A(x1) = 3
        isNatKind_A(x1) = 1
        mark#_A(x1) = 1
        and_A(x1,x2) = x2 + 1
        mark_A(x1) = x1 + 2
        a__plus#_A(x1,x2) = 1
        a__U21#_A(x1,x2) = 1
        a__and_A(x1,x2) = x2 + 2
        isNat_A(x1) = 4
        a__U41#_A(x1,x2,x3) = 0
        U41_A(x1,x2,x3) = 1
        U31_A(x1,x2) = 2
        a__U31#_A(x1,x2) = 1
        U22_A(x1) = 2
        U21_A(x1,x2) = 3
        U13_A(x1) = 1
        U12_A(x1,x2) = 2
        U11_A(x1,x2,x3) = 3
        |0|_A() = 1
        a__U11_A(x1,x2,x3) = 4
        a__U12_A(x1,x2) = 3
        a__U13_A(x1) = 2
        a__U21_A(x1,x2) = 4
        a__U22_A(x1) = 3
        a__U31_A(x1,x2) = 3
        a__U41_A(x1,x2,x3) = 2
        a__plus_A(x1,x2) = 4
    

The next rules are strictly ordered:

  p8, p12, p13, p15, p22, p23, p24, p25, p26, p27, p28, p29, p31, p41, p42, p43

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#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p12: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p13: a__U21#(tt(),V1) -> a__isNat#(V1)
p14: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p15: a__isNat#(plus(V1,V2)) -> a__and#(a__isNatKind(V1),isNatKind(V2))
p16: a__isNat#(plus(V1,V2)) -> a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
p17: a__U11#(tt(),V1,V2) -> a__isNat#(V1)
p18: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p19: mark#(U22(X)) -> mark#(X)
p20: mark#(U21(X1,X2)) -> mark#(X1)
p21: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
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)
p28: 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, p12, p13, p14, p15, p16, p17, 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#(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#(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 set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^1
      order: standard order
      interpretations:
        a__U11#_A(x1,x2,x3) = x2 + x3 + 2
        tt_A() = 0
        a__U12#_A(x1,x2) = x2 + 1
        a__isNat_A(x1) = x1 + 1
        a__isNat#_A(x1) = x1
        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 + 2
        mark#_A(x1) = x1 + 2
        U11_A(x1,x2,x3) = x1 + x2 + x3 + 5
        mark_A(x1) = x1
        U12_A(x1,x2) = x1 + x2 + 3
        isNat_A(x1) = x1 + 1
        a__isNatKind#_A(x1) = 2
        s_A(x1) = x1 + 3
        a__U21#_A(x1,x2) = x2 + 2
        U13_A(x1) = x1 + 1
        U21_A(x1,x2) = x1 + x2 + 3
        U22_A(x1) = x1 + 1
        and_A(x1,x2) = x1 + x2
        a__U11_A(x1,x2,x3) = x1 + x2 + x3 + 5
        a__U12_A(x1,x2) = x1 + x2 + 3
        a__U13_A(x1) = x1 + 1
        a__U21_A(x1,x2) = x1 + x2 + 3
        a__U22_A(x1) = x1 + 1
        a__U31_A(x1,x2) = x2 + 1
        a__U41_A(x1,x2,x3) = x2 + x3 + 7
        a__plus_A(x1,x2) = x1 + x2 + 4
        |0|_A() = 1
        U31_A(x1,x2) = x2 + 1
        U41_A(x1,x2,x3) = x2 + x3 + 7
    
    2. matrix interpretations:
    
      carrier: N^1
      order: standard order
      interpretations:
        a__U11#_A(x1,x2,x3) = x2 + 2
        tt_A() = 2
        a__U12#_A(x1,x2) = 0
        a__isNat_A(x1) = 3
        a__isNat#_A(x1) = x1 + 1
        plus_A(x1,x2) = x1 + 4
        a__and_A(x1,x2) = x1 + x2
        a__isNatKind_A(x1) = 2
        isNatKind_A(x1) = 0
        a__and#_A(x1,x2) = x2 + 6
        mark#_A(x1) = x1 + 6
        U11_A(x1,x2,x3) = 1
        mark_A(x1) = x1 + 2
        U12_A(x1,x2) = x1 + 1
        isNat_A(x1) = 2
        a__isNatKind#_A(x1) = 6
        s_A(x1) = 2
        a__U21#_A(x1,x2) = 2
        U13_A(x1) = x1 + 1
        U21_A(x1,x2) = x1 + 2
        U22_A(x1) = x1 + 2
        and_A(x1,x2) = x1 + x2
        a__U11_A(x1,x2,x3) = 2
        a__U12_A(x1,x2) = x1 + 1
        a__U13_A(x1) = x1 + 1
        a__U21_A(x1,x2) = x1 + 2
        a__U22_A(x1) = x1 + 2
        a__U31_A(x1,x2) = x2 + 5
        a__U41_A(x1,x2,x3) = x3 + 3
        a__plus_A(x1,x2) = x1 + 4
        |0|_A() = 1
        U31_A(x1,x2) = x2 + 4
        U41_A(x1,x2,x3) = x3 + 2
    

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p7, p8, p9, p10, p11, p12, p16, p17, p18, p19, p20, p21, p22

We remove them from the problem.

-- SCC decomposition.

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

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


-- Reduction pair.

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

p1: a__and#(tt(),X) -> mark#(X)
p2: mark#(isNatKind(X)) -> a__isNatKind#(X)
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: mark#(and(X1,X2)) -> mark#(X1)
p7: mark#(and(X1,X2)) -> a__and#(mark(X1),X2)

and R consists of:

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

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^1
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = x2 + 1
        tt_A() = 1
        mark#_A(x1) = x1
        isNatKind_A(x1) = x1 + 1
        a__isNatKind#_A(x1) = x1 + 1
        s_A(x1) = x1 + 2
        plus_A(x1,x2) = x1 + x2 + 4
        a__isNatKind_A(x1) = x1 + 1
        and_A(x1,x2) = x1 + x2 + 2
        mark_A(x1) = x1
        a__U11_A(x1,x2,x3) = x2 + 5
        a__U12_A(x1,x2) = 5
        a__isNat_A(x1) = x1 + 1
        a__U13_A(x1) = 2
        a__U21_A(x1,x2) = x2 + 2
        a__U22_A(x1) = 1
        a__U31_A(x1,x2) = x2 + 1
        a__U41_A(x1,x2,x3) = x2 + x3 + 6
        a__plus_A(x1,x2) = x1 + x2 + 4
        a__and_A(x1,x2) = x1 + x2 + 2
        |0|_A() = 1
        isNat_A(x1) = x1 + 1
        U11_A(x1,x2,x3) = x2 + 5
        U12_A(x1,x2) = 5
        U13_A(x1) = 2
        U21_A(x1,x2) = x2 + 2
        U22_A(x1) = 1
        U31_A(x1,x2) = x2 + 1
        U41_A(x1,x2,x3) = x2 + x3 + 6
    
    2. matrix interpretations:
    
      carrier: N^1
      order: standard order
      interpretations:
        a__and#_A(x1,x2) = x2 + 2
        tt_A() = 4
        mark#_A(x1) = x1 + 1
        isNatKind_A(x1) = 0
        a__isNatKind#_A(x1) = 0
        s_A(x1) = 1
        plus_A(x1,x2) = 2
        a__isNatKind_A(x1) = 1
        and_A(x1,x2) = x2 + 2
        mark_A(x1) = x1 + 4
        a__U11_A(x1,x2,x3) = 3
        a__U12_A(x1,x2) = 3
        a__isNat_A(x1) = 3
        a__U13_A(x1) = 2
        a__U21_A(x1,x2) = x2 + 4
        a__U22_A(x1) = 5
        a__U31_A(x1,x2) = 2
        a__U41_A(x1,x2,x3) = 2
        a__plus_A(x1,x2) = 3
        a__and_A(x1,x2) = x2 + 3
        |0|_A() = 1
        isNat_A(x1) = 1
        U11_A(x1,x2,x3) = 1
        U12_A(x1,x2) = 1
        U13_A(x1) = 1
        U21_A(x1,x2) = x2 + 1
        U22_A(x1) = 2
        U31_A(x1,x2) = 1
        U41_A(x1,x2,x3) = 1
    

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7

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