YES

We show the termination of the TRS R:

  a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
  a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
  a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
  a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
  a__U15(tt(),V2) -> a__U16(a__isNat(V2))
  a__U16(tt()) -> tt()
  a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
  a__U22(tt(),V1) -> a__U23(a__isNat(V1))
  a__U23(tt()) -> tt()
  a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
  a__U32(tt()) -> tt()
  a__U41(tt()) -> tt()
  a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
  a__U52(tt(),N) -> mark(N)
  a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
  a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
  a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
  a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
  a__isNat(|0|()) -> tt()
  a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
  a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
  a__isNatKind(|0|()) -> tt()
  a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
  a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
  a__plus(N,|0|()) -> a__U51(a__isNat(N),N)
  a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
  mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
  mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
  mark(isNatKind(X)) -> a__isNatKind(X)
  mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
  mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
  mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
  mark(isNat(X)) -> a__isNat(X)
  mark(U16(X)) -> a__U16(mark(X))
  mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
  mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
  mark(U23(X)) -> a__U23(mark(X))
  mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
  mark(U32(X)) -> a__U32(mark(X))
  mark(U41(X)) -> a__U41(mark(X))
  mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
  mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
  mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
  mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
  mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
  mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
  mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
  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,X3) -> U12(X1,X2,X3)
  a__isNatKind(X) -> isNatKind(X)
  a__U13(X1,X2,X3) -> U13(X1,X2,X3)
  a__U14(X1,X2,X3) -> U14(X1,X2,X3)
  a__U15(X1,X2) -> U15(X1,X2)
  a__isNat(X) -> isNat(X)
  a__U16(X) -> U16(X)
  a__U21(X1,X2) -> U21(X1,X2)
  a__U22(X1,X2) -> U22(X1,X2)
  a__U23(X) -> U23(X)
  a__U31(X1,X2) -> U31(X1,X2)
  a__U32(X) -> U32(X)
  a__U41(X) -> U41(X)
  a__U51(X1,X2) -> U51(X1,X2)
  a__U52(X1,X2) -> U52(X1,X2)
  a__U61(X1,X2,X3) -> U61(X1,X2,X3)
  a__U62(X1,X2,X3) -> U62(X1,X2,X3)
  a__U63(X1,X2,X3) -> U63(X1,X2,X3)
  a__U64(X1,X2,X3) -> U64(X1,X2,X3)
  a__plus(X1,X2) -> plus(X1,X2)

-- SCC decomposition.

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

p1: a__U11#(tt(),V1,V2) -> a__U12#(a__isNatKind(V1),V1,V2)
p2: a__U11#(tt(),V1,V2) -> a__isNatKind#(V1)
p3: a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2)
p4: a__U12#(tt(),V1,V2) -> a__isNatKind#(V2)
p5: a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2)
p6: a__U13#(tt(),V1,V2) -> a__isNatKind#(V2)
p7: a__U14#(tt(),V1,V2) -> a__U15#(a__isNat(V1),V2)
p8: a__U14#(tt(),V1,V2) -> a__isNat#(V1)
p9: a__U15#(tt(),V2) -> a__U16#(a__isNat(V2))
p10: a__U15#(tt(),V2) -> a__isNat#(V2)
p11: a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),V1)
p12: a__U21#(tt(),V1) -> a__isNatKind#(V1)
p13: a__U22#(tt(),V1) -> a__U23#(a__isNat(V1))
p14: a__U22#(tt(),V1) -> a__isNat#(V1)
p15: a__U31#(tt(),V2) -> a__U32#(a__isNatKind(V2))
p16: a__U31#(tt(),V2) -> a__isNatKind#(V2)
p17: a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
p18: a__U51#(tt(),N) -> a__isNatKind#(N)
p19: a__U52#(tt(),N) -> mark#(N)
p20: a__U61#(tt(),M,N) -> a__U62#(a__isNatKind(M),M,N)
p21: a__U61#(tt(),M,N) -> a__isNatKind#(M)
p22: a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N)
p23: a__U62#(tt(),M,N) -> a__isNat#(N)
p24: a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
p25: a__U63#(tt(),M,N) -> a__isNatKind#(N)
p26: a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p27: a__U64#(tt(),M,N) -> mark#(N)
p28: a__U64#(tt(),M,N) -> mark#(M)
p29: a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2)
p30: a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
p31: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p32: a__isNat#(s(V1)) -> a__isNatKind#(V1)
p33: a__isNatKind#(plus(V1,V2)) -> a__U31#(a__isNatKind(V1),V2)
p34: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
p35: a__isNatKind#(s(V1)) -> a__U41#(a__isNatKind(V1))
p36: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p37: a__plus#(N,|0|()) -> a__U51#(a__isNat(N),N)
p38: a__plus#(N,|0|()) -> a__isNat#(N)
p39: a__plus#(N,s(M)) -> a__U61#(a__isNat(M),M,N)
p40: a__plus#(N,s(M)) -> a__isNat#(M)
p41: mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
p42: mark#(U11(X1,X2,X3)) -> mark#(X1)
p43: mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
p44: mark#(U12(X1,X2,X3)) -> mark#(X1)
p45: mark#(isNatKind(X)) -> a__isNatKind#(X)
p46: mark#(U13(X1,X2,X3)) -> a__U13#(mark(X1),X2,X3)
p47: mark#(U13(X1,X2,X3)) -> mark#(X1)
p48: mark#(U14(X1,X2,X3)) -> a__U14#(mark(X1),X2,X3)
p49: mark#(U14(X1,X2,X3)) -> mark#(X1)
p50: mark#(U15(X1,X2)) -> a__U15#(mark(X1),X2)
p51: mark#(U15(X1,X2)) -> mark#(X1)
p52: mark#(isNat(X)) -> a__isNat#(X)
p53: mark#(U16(X)) -> a__U16#(mark(X))
p54: mark#(U16(X)) -> mark#(X)
p55: mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
p56: mark#(U21(X1,X2)) -> mark#(X1)
p57: mark#(U22(X1,X2)) -> a__U22#(mark(X1),X2)
p58: mark#(U22(X1,X2)) -> mark#(X1)
p59: mark#(U23(X)) -> a__U23#(mark(X))
p60: mark#(U23(X)) -> mark#(X)
p61: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
p62: mark#(U31(X1,X2)) -> mark#(X1)
p63: mark#(U32(X)) -> a__U32#(mark(X))
p64: mark#(U32(X)) -> mark#(X)
p65: mark#(U41(X)) -> a__U41#(mark(X))
p66: mark#(U41(X)) -> mark#(X)
p67: mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2)
p68: mark#(U51(X1,X2)) -> mark#(X1)
p69: mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
p70: mark#(U52(X1,X2)) -> mark#(X1)
p71: mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3)
p72: mark#(U61(X1,X2,X3)) -> mark#(X1)
p73: mark#(U62(X1,X2,X3)) -> a__U62#(mark(X1),X2,X3)
p74: mark#(U62(X1,X2,X3)) -> mark#(X1)
p75: mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
p76: mark#(U63(X1,X2,X3)) -> mark#(X1)
p77: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p78: mark#(U64(X1,X2,X3)) -> mark#(X1)
p79: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p80: mark#(plus(X1,X2)) -> mark#(X1)
p81: mark#(plus(X1,X2)) -> mark#(X2)
p82: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
r2: a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
r3: a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
r4: a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
r5: a__U15(tt(),V2) -> a__U16(a__isNat(V2))
r6: a__U16(tt()) -> tt()
r7: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r8: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r9: a__U23(tt()) -> tt()
r10: a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
r11: a__U32(tt()) -> tt()
r12: a__U41(tt()) -> tt()
r13: a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
r14: a__U52(tt(),N) -> mark(N)
r15: a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
r16: a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
r17: a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
r18: a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r19: a__isNat(|0|()) -> tt()
r20: a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
r21: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r22: a__isNatKind(|0|()) -> tt()
r23: a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
r24: a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
r25: a__plus(N,|0|()) -> a__U51(a__isNat(N),N)
r26: a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
r27: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r28: mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
r29: mark(isNatKind(X)) -> a__isNatKind(X)
r30: mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
r31: mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
r32: mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
r33: mark(isNat(X)) -> a__isNat(X)
r34: mark(U16(X)) -> a__U16(mark(X))
r35: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r36: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r37: mark(U23(X)) -> a__U23(mark(X))
r38: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r39: mark(U32(X)) -> a__U32(mark(X))
r40: mark(U41(X)) -> a__U41(mark(X))
r41: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r42: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r43: mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
r44: mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
r45: mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
r46: mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
r47: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r48: mark(tt()) -> tt()
r49: mark(s(X)) -> s(mark(X))
r50: mark(|0|()) -> |0|()
r51: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r52: a__U12(X1,X2,X3) -> U12(X1,X2,X3)
r53: a__isNatKind(X) -> isNatKind(X)
r54: a__U13(X1,X2,X3) -> U13(X1,X2,X3)
r55: a__U14(X1,X2,X3) -> U14(X1,X2,X3)
r56: a__U15(X1,X2) -> U15(X1,X2)
r57: a__isNat(X) -> isNat(X)
r58: a__U16(X) -> U16(X)
r59: a__U21(X1,X2) -> U21(X1,X2)
r60: a__U22(X1,X2) -> U22(X1,X2)
r61: a__U23(X) -> U23(X)
r62: a__U31(X1,X2) -> U31(X1,X2)
r63: a__U32(X) -> U32(X)
r64: a__U41(X) -> U41(X)
r65: a__U51(X1,X2) -> U51(X1,X2)
r66: a__U52(X1,X2) -> U52(X1,X2)
r67: a__U61(X1,X2,X3) -> U61(X1,X2,X3)
r68: a__U62(X1,X2,X3) -> U62(X1,X2,X3)
r69: a__U63(X1,X2,X3) -> U63(X1,X2,X3)
r70: a__U64(X1,X2,X3) -> U64(X1,X2,X3)
r71: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p17, p19, p20, p22, p24, p26, p27, p28, p37, p39, p42, p44, p47, p49, p51, p54, p56, p58, p60, p62, p64, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82}
  {p1, p3, p5, p7, p8, p10, p11, p14, p29, p31}
  {p16, p33, p34, p36}


-- Reduction pair.

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

p1: a__U64#(tt(),M,N) -> mark#(M)
p2: mark#(s(X)) -> mark#(X)
p3: mark#(plus(X1,X2)) -> mark#(X2)
p4: mark#(plus(X1,X2)) -> mark#(X1)
p5: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p6: a__plus#(N,s(M)) -> a__U61#(a__isNat(M),M,N)
p7: a__U61#(tt(),M,N) -> a__U62#(a__isNatKind(M),M,N)
p8: a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N)
p9: a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
p10: a__U64#(tt(),M,N) -> mark#(N)
p11: mark#(U64(X1,X2,X3)) -> mark#(X1)
p12: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p13: a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p14: a__plus#(N,|0|()) -> a__U51#(a__isNat(N),N)
p15: a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
p16: a__U52#(tt(),N) -> mark#(N)
p17: mark#(U63(X1,X2,X3)) -> mark#(X1)
p18: mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
p19: mark#(U62(X1,X2,X3)) -> mark#(X1)
p20: mark#(U62(X1,X2,X3)) -> a__U62#(mark(X1),X2,X3)
p21: mark#(U61(X1,X2,X3)) -> mark#(X1)
p22: mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3)
p23: mark#(U52(X1,X2)) -> mark#(X1)
p24: mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
p25: mark#(U51(X1,X2)) -> mark#(X1)
p26: mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2)
p27: mark#(U41(X)) -> mark#(X)
p28: mark#(U32(X)) -> mark#(X)
p29: mark#(U31(X1,X2)) -> mark#(X1)
p30: mark#(U23(X)) -> mark#(X)
p31: mark#(U22(X1,X2)) -> mark#(X1)
p32: mark#(U21(X1,X2)) -> mark#(X1)
p33: mark#(U16(X)) -> mark#(X)
p34: mark#(U15(X1,X2)) -> mark#(X1)
p35: mark#(U14(X1,X2,X3)) -> mark#(X1)
p36: mark#(U13(X1,X2,X3)) -> mark#(X1)
p37: mark#(U12(X1,X2,X3)) -> mark#(X1)
p38: mark#(U11(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
r2: a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
r3: a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
r4: a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
r5: a__U15(tt(),V2) -> a__U16(a__isNat(V2))
r6: a__U16(tt()) -> tt()
r7: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r8: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r9: a__U23(tt()) -> tt()
r10: a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
r11: a__U32(tt()) -> tt()
r12: a__U41(tt()) -> tt()
r13: a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
r14: a__U52(tt(),N) -> mark(N)
r15: a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
r16: a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
r17: a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
r18: a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r19: a__isNat(|0|()) -> tt()
r20: a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
r21: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r22: a__isNatKind(|0|()) -> tt()
r23: a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
r24: a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
r25: a__plus(N,|0|()) -> a__U51(a__isNat(N),N)
r26: a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
r27: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r28: mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
r29: mark(isNatKind(X)) -> a__isNatKind(X)
r30: mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
r31: mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
r32: mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
r33: mark(isNat(X)) -> a__isNat(X)
r34: mark(U16(X)) -> a__U16(mark(X))
r35: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r36: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r37: mark(U23(X)) -> a__U23(mark(X))
r38: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r39: mark(U32(X)) -> a__U32(mark(X))
r40: mark(U41(X)) -> a__U41(mark(X))
r41: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r42: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r43: mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
r44: mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
r45: mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
r46: mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
r47: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r48: mark(tt()) -> tt()
r49: mark(s(X)) -> s(mark(X))
r50: mark(|0|()) -> |0|()
r51: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r52: a__U12(X1,X2,X3) -> U12(X1,X2,X3)
r53: a__isNatKind(X) -> isNatKind(X)
r54: a__U13(X1,X2,X3) -> U13(X1,X2,X3)
r55: a__U14(X1,X2,X3) -> U14(X1,X2,X3)
r56: a__U15(X1,X2) -> U15(X1,X2)
r57: a__isNat(X) -> isNat(X)
r58: a__U16(X) -> U16(X)
r59: a__U21(X1,X2) -> U21(X1,X2)
r60: a__U22(X1,X2) -> U22(X1,X2)
r61: a__U23(X) -> U23(X)
r62: a__U31(X1,X2) -> U31(X1,X2)
r63: a__U32(X) -> U32(X)
r64: a__U41(X) -> U41(X)
r65: a__U51(X1,X2) -> U51(X1,X2)
r66: a__U52(X1,X2) -> U52(X1,X2)
r67: a__U61(X1,X2,X3) -> U61(X1,X2,X3)
r68: a__U62(X1,X2,X3) -> U62(X1,X2,X3)
r69: a__U63(X1,X2,X3) -> U63(X1,X2,X3)
r70: a__U64(X1,X2,X3) -> U64(X1,X2,X3)
r71: a__plus(X1,X2) -> plus(X1,X2)

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, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      a__U64#_A(x1,x2,x3) = x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3
      tt_A() = (1,1)
      mark#_A(x1) = ((1,0),(1,0)) x1 + (1,0)
      s_A(x1) = x1 + (3,1)
      plus_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (0,1)
      a__plus#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2
      mark_A(x1) = ((1,0),(1,0)) x1 + (0,2)
      a__U61#_A(x1,x2,x3) = x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (1,0)
      a__isNat_A(x1) = (1,2)
      a__U62#_A(x1,x2,x3) = ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (2,0)
      a__isNatKind_A(x1) = (1,1)
      a__U63#_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3
      U64_A(x1,x2,x3) = x1 + x2 + x3 + (2,1)
      |0|_A() = (1,1)
      a__U51#_A(x1,x2) = ((1,0),(1,0)) x2 + (1,0)
      a__U52#_A(x1,x2) = ((1,0),(1,0)) x2 + (1,0)
      U63_A(x1,x2,x3) = ((1,0),(1,0)) x1 + x2 + ((1,0),(1,0)) x3 + (2,1)
      U62_A(x1,x2,x3) = x1 + x2 + x3 + (2,1)
      U61_A(x1,x2,x3) = x1 + x2 + ((1,0),(1,0)) x3 + (2,1)
      U52_A(x1,x2) = x1 + x2 + (0,1)
      U51_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (0,1)
      U41_A(x1) = x1 + (0,1)
      U32_A(x1) = x1 + (0,1)
      U31_A(x1,x2) = x1 + (0,1)
      U23_A(x1) = x1
      U22_A(x1,x2) = x1 + (0,2)
      U21_A(x1,x2) = x1 + (0,2)
      U16_A(x1) = x1 + (0,1)
      U15_A(x1,x2) = x1 + (0,1)
      U14_A(x1,x2,x3) = x1 + (0,1)
      U13_A(x1,x2,x3) = x1 + (0,1)
      U12_A(x1,x2,x3) = x1 + (0,1)
      U11_A(x1,x2,x3) = x1 + (0,1)
      a__U11_A(x1,x2,x3) = ((1,0),(1,0)) x1 + (0,1)
      a__U12_A(x1,x2,x3) = ((1,0),(1,0)) x1 + (0,1)
      a__U13_A(x1,x2,x3) = x1 + (0,1)
      a__U14_A(x1,x2,x3) = x1 + (0,1)
      a__U15_A(x1,x2) = x1 + (0,1)
      a__U16_A(x1) = x1 + (0,1)
      a__U21_A(x1,x2) = x1 + (0,2)
      a__U22_A(x1,x2) = x1 + (0,2)
      a__U23_A(x1) = x1
      a__U31_A(x1,x2) = x1 + (0,1)
      a__U32_A(x1) = x1 + (0,1)
      a__U41_A(x1) = x1 + (0,1)
      a__U51_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (0,2)
      a__U52_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (0,2)
      a__U61_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (2,1)
      a__U62_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (2,1)
      a__U63_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (2,1)
      a__U64_A(x1,x2,x3) = x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (2,2)
      a__plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (0,1)
      isNatKind_A(x1) = (1,1)
      isNat_A(x1) = (1,1)

The next rules are strictly ordered:

  p2, p5, p6, p11, p12, p13, 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__U64#(tt(),M,N) -> mark#(M)
p2: mark#(plus(X1,X2)) -> mark#(X2)
p3: mark#(plus(X1,X2)) -> mark#(X1)
p4: a__U61#(tt(),M,N) -> a__U62#(a__isNatKind(M),M,N)
p5: a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N)
p6: a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
p7: a__U64#(tt(),M,N) -> mark#(N)
p8: a__plus#(N,|0|()) -> a__U51#(a__isNat(N),N)
p9: a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
p10: a__U52#(tt(),N) -> mark#(N)
p11: mark#(U52(X1,X2)) -> mark#(X1)
p12: mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
p13: mark#(U51(X1,X2)) -> mark#(X1)
p14: mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2)
p15: mark#(U41(X)) -> mark#(X)
p16: mark#(U32(X)) -> mark#(X)
p17: mark#(U31(X1,X2)) -> mark#(X1)
p18: mark#(U23(X)) -> mark#(X)
p19: mark#(U22(X1,X2)) -> mark#(X1)
p20: mark#(U21(X1,X2)) -> mark#(X1)
p21: mark#(U16(X)) -> mark#(X)
p22: mark#(U15(X1,X2)) -> mark#(X1)
p23: mark#(U14(X1,X2,X3)) -> mark#(X1)
p24: mark#(U13(X1,X2,X3)) -> mark#(X1)
p25: mark#(U12(X1,X2,X3)) -> mark#(X1)
p26: mark#(U11(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
r2: a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
r3: a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
r4: a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
r5: a__U15(tt(),V2) -> a__U16(a__isNat(V2))
r6: a__U16(tt()) -> tt()
r7: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r8: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r9: a__U23(tt()) -> tt()
r10: a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
r11: a__U32(tt()) -> tt()
r12: a__U41(tt()) -> tt()
r13: a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
r14: a__U52(tt(),N) -> mark(N)
r15: a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
r16: a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
r17: a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
r18: a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r19: a__isNat(|0|()) -> tt()
r20: a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
r21: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r22: a__isNatKind(|0|()) -> tt()
r23: a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
r24: a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
r25: a__plus(N,|0|()) -> a__U51(a__isNat(N),N)
r26: a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
r27: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r28: mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
r29: mark(isNatKind(X)) -> a__isNatKind(X)
r30: mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
r31: mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
r32: mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
r33: mark(isNat(X)) -> a__isNat(X)
r34: mark(U16(X)) -> a__U16(mark(X))
r35: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r36: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r37: mark(U23(X)) -> a__U23(mark(X))
r38: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r39: mark(U32(X)) -> a__U32(mark(X))
r40: mark(U41(X)) -> a__U41(mark(X))
r41: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r42: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r43: mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
r44: mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
r45: mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
r46: mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
r47: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r48: mark(tt()) -> tt()
r49: mark(s(X)) -> s(mark(X))
r50: mark(|0|()) -> |0|()
r51: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r52: a__U12(X1,X2,X3) -> U12(X1,X2,X3)
r53: a__isNatKind(X) -> isNatKind(X)
r54: a__U13(X1,X2,X3) -> U13(X1,X2,X3)
r55: a__U14(X1,X2,X3) -> U14(X1,X2,X3)
r56: a__U15(X1,X2) -> U15(X1,X2)
r57: a__isNat(X) -> isNat(X)
r58: a__U16(X) -> U16(X)
r59: a__U21(X1,X2) -> U21(X1,X2)
r60: a__U22(X1,X2) -> U22(X1,X2)
r61: a__U23(X) -> U23(X)
r62: a__U31(X1,X2) -> U31(X1,X2)
r63: a__U32(X) -> U32(X)
r64: a__U41(X) -> U41(X)
r65: a__U51(X1,X2) -> U51(X1,X2)
r66: a__U52(X1,X2) -> U52(X1,X2)
r67: a__U61(X1,X2,X3) -> U61(X1,X2,X3)
r68: a__U62(X1,X2,X3) -> U62(X1,X2,X3)
r69: a__U63(X1,X2,X3) -> U63(X1,X2,X3)
r70: a__U64(X1,X2,X3) -> U64(X1,X2,X3)
r71: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p2, p3, 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: mark#(plus(X1,X2)) -> mark#(X2)
p2: mark#(U11(X1,X2,X3)) -> mark#(X1)
p3: mark#(U12(X1,X2,X3)) -> mark#(X1)
p4: mark#(U13(X1,X2,X3)) -> mark#(X1)
p5: mark#(U14(X1,X2,X3)) -> mark#(X1)
p6: mark#(U15(X1,X2)) -> mark#(X1)
p7: mark#(U16(X)) -> mark#(X)
p8: mark#(U21(X1,X2)) -> mark#(X1)
p9: mark#(U22(X1,X2)) -> mark#(X1)
p10: mark#(U23(X)) -> mark#(X)
p11: mark#(U31(X1,X2)) -> mark#(X1)
p12: mark#(U32(X)) -> mark#(X)
p13: mark#(U41(X)) -> mark#(X)
p14: mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2)
p15: a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
p16: a__U52#(tt(),N) -> mark#(N)
p17: mark#(U51(X1,X2)) -> mark#(X1)
p18: mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
p19: mark#(U52(X1,X2)) -> mark#(X1)
p20: mark#(plus(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
r2: a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
r3: a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
r4: a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
r5: a__U15(tt(),V2) -> a__U16(a__isNat(V2))
r6: a__U16(tt()) -> tt()
r7: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r8: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r9: a__U23(tt()) -> tt()
r10: a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
r11: a__U32(tt()) -> tt()
r12: a__U41(tt()) -> tt()
r13: a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
r14: a__U52(tt(),N) -> mark(N)
r15: a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
r16: a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
r17: a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
r18: a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r19: a__isNat(|0|()) -> tt()
r20: a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
r21: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r22: a__isNatKind(|0|()) -> tt()
r23: a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
r24: a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
r25: a__plus(N,|0|()) -> a__U51(a__isNat(N),N)
r26: a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
r27: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r28: mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
r29: mark(isNatKind(X)) -> a__isNatKind(X)
r30: mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
r31: mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
r32: mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
r33: mark(isNat(X)) -> a__isNat(X)
r34: mark(U16(X)) -> a__U16(mark(X))
r35: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r36: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r37: mark(U23(X)) -> a__U23(mark(X))
r38: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r39: mark(U32(X)) -> a__U32(mark(X))
r40: mark(U41(X)) -> a__U41(mark(X))
r41: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r42: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r43: mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
r44: mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
r45: mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
r46: mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
r47: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r48: mark(tt()) -> tt()
r49: mark(s(X)) -> s(mark(X))
r50: mark(|0|()) -> |0|()
r51: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r52: a__U12(X1,X2,X3) -> U12(X1,X2,X3)
r53: a__isNatKind(X) -> isNatKind(X)
r54: a__U13(X1,X2,X3) -> U13(X1,X2,X3)
r55: a__U14(X1,X2,X3) -> U14(X1,X2,X3)
r56: a__U15(X1,X2) -> U15(X1,X2)
r57: a__isNat(X) -> isNat(X)
r58: a__U16(X) -> U16(X)
r59: a__U21(X1,X2) -> U21(X1,X2)
r60: a__U22(X1,X2) -> U22(X1,X2)
r61: a__U23(X) -> U23(X)
r62: a__U31(X1,X2) -> U31(X1,X2)
r63: a__U32(X) -> U32(X)
r64: a__U41(X) -> U41(X)
r65: a__U51(X1,X2) -> U51(X1,X2)
r66: a__U52(X1,X2) -> U52(X1,X2)
r67: a__U61(X1,X2,X3) -> U61(X1,X2,X3)
r68: a__U62(X1,X2,X3) -> U62(X1,X2,X3)
r69: a__U63(X1,X2,X3) -> U63(X1,X2,X3)
r70: a__U64(X1,X2,X3) -> U64(X1,X2,X3)
r71: a__plus(X1,X2) -> plus(X1,X2)

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, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = ((1,1),(0,1)) x1
      plus_A(x1,x2) = ((0,1),(1,1)) x1 + x2 + (2,3)
      U11_A(x1,x2,x3) = x1
      U12_A(x1,x2,x3) = x1
      U13_A(x1,x2,x3) = x1
      U14_A(x1,x2,x3) = x1
      U15_A(x1,x2) = x1
      U16_A(x1) = x1
      U21_A(x1,x2) = x1
      U22_A(x1,x2) = x1
      U23_A(x1) = x1
      U31_A(x1,x2) = x1
      U32_A(x1) = x1
      U41_A(x1) = x1
      U51_A(x1,x2) = ((0,1),(1,1)) x1 + ((0,1),(1,1)) x2 + (1,1)
      a__U51#_A(x1,x2) = x1 + ((1,1),(1,1)) x2
      mark_A(x1) = ((0,1),(1,1)) x1 + (1,1)
      tt_A() = (1,1)
      a__U52#_A(x1,x2) = ((1,1),(1,1)) x2
      a__isNatKind_A(x1) = (1,1)
      U52_A(x1,x2) = ((0,1),(1,1)) x1 + ((0,1),(1,1)) x2 + (1,1)
      a__U11_A(x1,x2,x3) = x1
      a__U12_A(x1,x2,x3) = x1
      a__U13_A(x1,x2,x3) = x1
      a__U14_A(x1,x2,x3) = x1
      a__U15_A(x1,x2) = x1
      a__isNat_A(x1) = (1,1)
      a__U16_A(x1) = x1
      a__U21_A(x1,x2) = x1
      a__U22_A(x1,x2) = x1
      a__U23_A(x1) = x1
      a__U31_A(x1,x2) = x1
      a__U32_A(x1) = x1
      a__U41_A(x1) = x1
      a__U51_A(x1,x2) = ((0,1),(1,1)) x1 + ((0,1),(1,1)) x2 + (1,1)
      a__U52_A(x1,x2) = ((0,1),(1,1)) x1 + ((0,1),(1,1)) x2 + (1,1)
      a__U61_A(x1,x2,x3) = x3 + (1,1)
      a__U62_A(x1,x2,x3) = x3 + (1,1)
      a__U63_A(x1,x2,x3) = (1,1)
      a__U64_A(x1,x2,x3) = (1,1)
      s_A(x1) = (0,1)
      a__plus_A(x1,x2) = ((0,1),(1,1)) x1 + x2 + (2,3)
      |0|_A() = (1,0)
      isNat_A(x1) = (1,1)
      U61_A(x1,x2,x3) = x3 + (1,1)
      U62_A(x1,x2,x3) = x3 + (1,0)
      U63_A(x1,x2,x3) = (1,1)
      U64_A(x1,x2,x3) = (1,0)
      isNatKind_A(x1) = (1,1)

The next rules are strictly ordered:

  p1, p14, p15, p17, p18, p19, p20

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U11(X1,X2,X3)) -> mark#(X1)
p2: mark#(U12(X1,X2,X3)) -> mark#(X1)
p3: mark#(U13(X1,X2,X3)) -> mark#(X1)
p4: mark#(U14(X1,X2,X3)) -> mark#(X1)
p5: mark#(U15(X1,X2)) -> mark#(X1)
p6: mark#(U16(X)) -> mark#(X)
p7: mark#(U21(X1,X2)) -> mark#(X1)
p8: mark#(U22(X1,X2)) -> mark#(X1)
p9: mark#(U23(X)) -> mark#(X)
p10: mark#(U31(X1,X2)) -> mark#(X1)
p11: mark#(U32(X)) -> mark#(X)
p12: mark#(U41(X)) -> mark#(X)
p13: a__U52#(tt(),N) -> mark#(N)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
r2: a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
r3: a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
r4: a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
r5: a__U15(tt(),V2) -> a__U16(a__isNat(V2))
r6: a__U16(tt()) -> tt()
r7: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r8: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r9: a__U23(tt()) -> tt()
r10: a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
r11: a__U32(tt()) -> tt()
r12: a__U41(tt()) -> tt()
r13: a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
r14: a__U52(tt(),N) -> mark(N)
r15: a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
r16: a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
r17: a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
r18: a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r19: a__isNat(|0|()) -> tt()
r20: a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
r21: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r22: a__isNatKind(|0|()) -> tt()
r23: a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
r24: a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
r25: a__plus(N,|0|()) -> a__U51(a__isNat(N),N)
r26: a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
r27: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r28: mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
r29: mark(isNatKind(X)) -> a__isNatKind(X)
r30: mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
r31: mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
r32: mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
r33: mark(isNat(X)) -> a__isNat(X)
r34: mark(U16(X)) -> a__U16(mark(X))
r35: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r36: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r37: mark(U23(X)) -> a__U23(mark(X))
r38: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r39: mark(U32(X)) -> a__U32(mark(X))
r40: mark(U41(X)) -> a__U41(mark(X))
r41: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r42: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r43: mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
r44: mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
r45: mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
r46: mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
r47: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r48: mark(tt()) -> tt()
r49: mark(s(X)) -> s(mark(X))
r50: mark(|0|()) -> |0|()
r51: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r52: a__U12(X1,X2,X3) -> U12(X1,X2,X3)
r53: a__isNatKind(X) -> isNatKind(X)
r54: a__U13(X1,X2,X3) -> U13(X1,X2,X3)
r55: a__U14(X1,X2,X3) -> U14(X1,X2,X3)
r56: a__U15(X1,X2) -> U15(X1,X2)
r57: a__isNat(X) -> isNat(X)
r58: a__U16(X) -> U16(X)
r59: a__U21(X1,X2) -> U21(X1,X2)
r60: a__U22(X1,X2) -> U22(X1,X2)
r61: a__U23(X) -> U23(X)
r62: a__U31(X1,X2) -> U31(X1,X2)
r63: a__U32(X) -> U32(X)
r64: a__U41(X) -> U41(X)
r65: a__U51(X1,X2) -> U51(X1,X2)
r66: a__U52(X1,X2) -> U52(X1,X2)
r67: a__U61(X1,X2,X3) -> U61(X1,X2,X3)
r68: a__U62(X1,X2,X3) -> U62(X1,X2,X3)
r69: a__U63(X1,X2,X3) -> U63(X1,X2,X3)
r70: a__U64(X1,X2,X3) -> U64(X1,X2,X3)
r71: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: mark#(U11(X1,X2,X3)) -> mark#(X1)
p2: mark#(U41(X)) -> mark#(X)
p3: mark#(U32(X)) -> mark#(X)
p4: mark#(U31(X1,X2)) -> mark#(X1)
p5: mark#(U23(X)) -> mark#(X)
p6: mark#(U22(X1,X2)) -> mark#(X1)
p7: mark#(U21(X1,X2)) -> mark#(X1)
p8: mark#(U16(X)) -> mark#(X)
p9: mark#(U15(X1,X2)) -> mark#(X1)
p10: mark#(U14(X1,X2,X3)) -> mark#(X1)
p11: mark#(U13(X1,X2,X3)) -> mark#(X1)
p12: mark#(U12(X1,X2,X3)) -> mark#(X1)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
r2: a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
r3: a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
r4: a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
r5: a__U15(tt(),V2) -> a__U16(a__isNat(V2))
r6: a__U16(tt()) -> tt()
r7: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r8: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r9: a__U23(tt()) -> tt()
r10: a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
r11: a__U32(tt()) -> tt()
r12: a__U41(tt()) -> tt()
r13: a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
r14: a__U52(tt(),N) -> mark(N)
r15: a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
r16: a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
r17: a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
r18: a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r19: a__isNat(|0|()) -> tt()
r20: a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
r21: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r22: a__isNatKind(|0|()) -> tt()
r23: a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
r24: a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
r25: a__plus(N,|0|()) -> a__U51(a__isNat(N),N)
r26: a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
r27: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r28: mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
r29: mark(isNatKind(X)) -> a__isNatKind(X)
r30: mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
r31: mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
r32: mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
r33: mark(isNat(X)) -> a__isNat(X)
r34: mark(U16(X)) -> a__U16(mark(X))
r35: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r36: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r37: mark(U23(X)) -> a__U23(mark(X))
r38: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r39: mark(U32(X)) -> a__U32(mark(X))
r40: mark(U41(X)) -> a__U41(mark(X))
r41: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r42: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r43: mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
r44: mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
r45: mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
r46: mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
r47: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r48: mark(tt()) -> tt()
r49: mark(s(X)) -> s(mark(X))
r50: mark(|0|()) -> |0|()
r51: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r52: a__U12(X1,X2,X3) -> U12(X1,X2,X3)
r53: a__isNatKind(X) -> isNatKind(X)
r54: a__U13(X1,X2,X3) -> U13(X1,X2,X3)
r55: a__U14(X1,X2,X3) -> U14(X1,X2,X3)
r56: a__U15(X1,X2) -> U15(X1,X2)
r57: a__isNat(X) -> isNat(X)
r58: a__U16(X) -> U16(X)
r59: a__U21(X1,X2) -> U21(X1,X2)
r60: a__U22(X1,X2) -> U22(X1,X2)
r61: a__U23(X) -> U23(X)
r62: a__U31(X1,X2) -> U31(X1,X2)
r63: a__U32(X) -> U32(X)
r64: a__U41(X) -> U41(X)
r65: a__U51(X1,X2) -> U51(X1,X2)
r66: a__U52(X1,X2) -> U52(X1,X2)
r67: a__U61(X1,X2,X3) -> U61(X1,X2,X3)
r68: a__U62(X1,X2,X3) -> U62(X1,X2,X3)
r69: a__U63(X1,X2,X3) -> U63(X1,X2,X3)
r70: a__U64(X1,X2,X3) -> U64(X1,X2,X3)
r71: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      mark#_A(x1) = ((1,1),(1,1)) x1
      U11_A(x1,x2,x3) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,1)
      U41_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      U32_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      U31_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (1,1)
      U23_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      U22_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,0)) x2 + (1,1)
      U21_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (1,1)
      U16_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      U15_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,0),(1,1)) x2 + (1,1)
      U14_A(x1,x2,x3) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,1)
      U13_A(x1,x2,x3) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,1)
      U12_A(x1,x2,x3) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,1)

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12
  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, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71

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__U11#(tt(),V1,V2) -> a__U12#(a__isNatKind(V1),V1,V2)
p2: a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2)
p3: a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2)
p4: a__U14#(tt(),V1,V2) -> a__isNat#(V1)
p5: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p6: a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),V1)
p7: a__U22#(tt(),V1) -> a__isNat#(V1)
p8: a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2)
p9: a__U14#(tt(),V1,V2) -> a__U15#(a__isNat(V1),V2)
p10: a__U15#(tt(),V2) -> a__isNat#(V2)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
r2: a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
r3: a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
r4: a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
r5: a__U15(tt(),V2) -> a__U16(a__isNat(V2))
r6: a__U16(tt()) -> tt()
r7: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r8: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r9: a__U23(tt()) -> tt()
r10: a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
r11: a__U32(tt()) -> tt()
r12: a__U41(tt()) -> tt()
r13: a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
r14: a__U52(tt(),N) -> mark(N)
r15: a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
r16: a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
r17: a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
r18: a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r19: a__isNat(|0|()) -> tt()
r20: a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
r21: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r22: a__isNatKind(|0|()) -> tt()
r23: a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
r24: a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
r25: a__plus(N,|0|()) -> a__U51(a__isNat(N),N)
r26: a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
r27: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r28: mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
r29: mark(isNatKind(X)) -> a__isNatKind(X)
r30: mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
r31: mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
r32: mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
r33: mark(isNat(X)) -> a__isNat(X)
r34: mark(U16(X)) -> a__U16(mark(X))
r35: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r36: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r37: mark(U23(X)) -> a__U23(mark(X))
r38: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r39: mark(U32(X)) -> a__U32(mark(X))
r40: mark(U41(X)) -> a__U41(mark(X))
r41: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r42: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r43: mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
r44: mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
r45: mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
r46: mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
r47: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r48: mark(tt()) -> tt()
r49: mark(s(X)) -> s(mark(X))
r50: mark(|0|()) -> |0|()
r51: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r52: a__U12(X1,X2,X3) -> U12(X1,X2,X3)
r53: a__isNatKind(X) -> isNatKind(X)
r54: a__U13(X1,X2,X3) -> U13(X1,X2,X3)
r55: a__U14(X1,X2,X3) -> U14(X1,X2,X3)
r56: a__U15(X1,X2) -> U15(X1,X2)
r57: a__isNat(X) -> isNat(X)
r58: a__U16(X) -> U16(X)
r59: a__U21(X1,X2) -> U21(X1,X2)
r60: a__U22(X1,X2) -> U22(X1,X2)
r61: a__U23(X) -> U23(X)
r62: a__U31(X1,X2) -> U31(X1,X2)
r63: a__U32(X) -> U32(X)
r64: a__U41(X) -> U41(X)
r65: a__U51(X1,X2) -> U51(X1,X2)
r66: a__U52(X1,X2) -> U52(X1,X2)
r67: a__U61(X1,X2,X3) -> U61(X1,X2,X3)
r68: a__U62(X1,X2,X3) -> U62(X1,X2,X3)
r69: a__U63(X1,X2,X3) -> U63(X1,X2,X3)
r70: a__U64(X1,X2,X3) -> U64(X1,X2,X3)
r71: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r19, r20, r21, r22, r23, r24, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      a__U11#_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((0,1),(1,1)) x2 + ((1,1),(1,1)) x3 + (2,1)
      tt_A() = (1,1)
      a__U12#_A(x1,x2,x3) = x1 + ((0,1),(0,1)) x2 + ((1,1),(1,1)) x3 + (2,0)
      a__isNatKind_A(x1) = ((0,0),(1,0)) x1 + (1,1)
      a__U13#_A(x1,x2,x3) = x1 + ((0,1),(0,1)) x2 + ((1,1),(0,1)) x3 + (2,0)
      a__U14#_A(x1,x2,x3) = ((0,1),(0,0)) x1 + ((0,1),(0,1)) x2 + ((0,1),(0,1)) x3
      a__isNat#_A(x1) = ((0,1),(0,1)) x1 + (1,0)
      s_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      a__U21#_A(x1,x2) = ((1,1),(1,1)) x2 + (1,0)
      a__U22#_A(x1,x2) = ((1,1),(1,1)) x2 + (1,0)
      plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (1,2)
      a__U15#_A(x1,x2) = ((0,1),(0,1)) x2 + (1,0)
      a__isNat_A(x1) = ((1,0),(1,1)) x1 + (1,1)
      a__U16_A(x1) = (1,1)
      U16_A(x1) = (0,0)
      a__U15_A(x1,x2) = ((0,0),(1,0)) x1 + ((0,1),(0,1)) x2 + (1,0)
      U15_A(x1,x2) = x2
      a__U14_A(x1,x2,x3) = x1 + ((0,0),(1,1)) x2 + ((0,1),(0,1)) x3 + (1,0)
      U14_A(x1,x2,x3) = x1 + ((0,0),(1,0)) x2
      a__U13_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((0,0),(1,1)) x2 + ((0,1),(1,1)) x3
      a__U23_A(x1) = ((0,0),(1,0)) x1 + (1,0)
      U13_A(x1,x2,x3) = ((0,0),(1,0)) x1
      U23_A(x1) = (0,0)
      a__U12_A(x1,x2,x3) = ((0,0),(1,1)) x2 + ((1,1),(1,1)) x3 + (1,1)
      a__U22_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(1,1)) x2 + (0,1)
      a__U32_A(x1) = x1 + (1,0)
      U12_A(x1,x2,x3) = (0,0)
      U22_A(x1,x2) = ((0,1),(0,0)) x1
      U32_A(x1) = (0,0)
      a__U11_A(x1,x2,x3) = ((0,1),(1,1)) x1 + ((0,1),(1,1)) x2 + ((1,1),(1,1)) x3
      a__U21_A(x1,x2) = ((0,0),(1,1)) x1 + ((1,1),(1,1)) x2 + (1,0)
      a__U31_A(x1,x2) = x1 + ((0,0),(1,1)) x2 + (1,0)
      a__U41_A(x1) = x1 + (1,0)
      U11_A(x1,x2,x3) = (0,0)
      U21_A(x1,x2) = (0,0)
      U31_A(x1,x2) = (0,0)
      U41_A(x1) = (0,0)
      |0|_A() = (1,1)
      isNatKind_A(x1) = (0,0)
      isNat_A(x1) = (0,0)

The next rules are strictly ordered:

  p1, p3, p5

We remove them from the problem.

-- SCC decomposition.

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

p1: a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2)
p2: a__U14#(tt(),V1,V2) -> a__isNat#(V1)
p3: a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),V1)
p4: a__U22#(tt(),V1) -> a__isNat#(V1)
p5: a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2)
p6: a__U14#(tt(),V1,V2) -> a__U15#(a__isNat(V1),V2)
p7: a__U15#(tt(),V2) -> a__isNat#(V2)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
r2: a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
r3: a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
r4: a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
r5: a__U15(tt(),V2) -> a__U16(a__isNat(V2))
r6: a__U16(tt()) -> tt()
r7: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r8: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r9: a__U23(tt()) -> tt()
r10: a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
r11: a__U32(tt()) -> tt()
r12: a__U41(tt()) -> tt()
r13: a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
r14: a__U52(tt(),N) -> mark(N)
r15: a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
r16: a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
r17: a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
r18: a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r19: a__isNat(|0|()) -> tt()
r20: a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
r21: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r22: a__isNatKind(|0|()) -> tt()
r23: a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
r24: a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
r25: a__plus(N,|0|()) -> a__U51(a__isNat(N),N)
r26: a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
r27: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r28: mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
r29: mark(isNatKind(X)) -> a__isNatKind(X)
r30: mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
r31: mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
r32: mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
r33: mark(isNat(X)) -> a__isNat(X)
r34: mark(U16(X)) -> a__U16(mark(X))
r35: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r36: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r37: mark(U23(X)) -> a__U23(mark(X))
r38: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r39: mark(U32(X)) -> a__U32(mark(X))
r40: mark(U41(X)) -> a__U41(mark(X))
r41: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r42: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r43: mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
r44: mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
r45: mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
r46: mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
r47: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r48: mark(tt()) -> tt()
r49: mark(s(X)) -> s(mark(X))
r50: mark(|0|()) -> |0|()
r51: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r52: a__U12(X1,X2,X3) -> U12(X1,X2,X3)
r53: a__isNatKind(X) -> isNatKind(X)
r54: a__U13(X1,X2,X3) -> U13(X1,X2,X3)
r55: a__U14(X1,X2,X3) -> U14(X1,X2,X3)
r56: a__U15(X1,X2) -> U15(X1,X2)
r57: a__isNat(X) -> isNat(X)
r58: a__U16(X) -> U16(X)
r59: a__U21(X1,X2) -> U21(X1,X2)
r60: a__U22(X1,X2) -> U22(X1,X2)
r61: a__U23(X) -> U23(X)
r62: a__U31(X1,X2) -> U31(X1,X2)
r63: a__U32(X) -> U32(X)
r64: a__U41(X) -> U41(X)
r65: a__U51(X1,X2) -> U51(X1,X2)
r66: a__U52(X1,X2) -> U52(X1,X2)
r67: a__U61(X1,X2,X3) -> U61(X1,X2,X3)
r68: a__U62(X1,X2,X3) -> U62(X1,X2,X3)
r69: a__U63(X1,X2,X3) -> U63(X1,X2,X3)
r70: a__U64(X1,X2,X3) -> U64(X1,X2,X3)
r71: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

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

p1: a__isNatKind#(plus(V1,V2)) -> a__U31#(a__isNatKind(V1),V2)
p2: a__U31#(tt(),V2) -> a__isNatKind#(V2)
p3: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p4: a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
r2: a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
r3: a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
r4: a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
r5: a__U15(tt(),V2) -> a__U16(a__isNat(V2))
r6: a__U16(tt()) -> tt()
r7: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r8: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r9: a__U23(tt()) -> tt()
r10: a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
r11: a__U32(tt()) -> tt()
r12: a__U41(tt()) -> tt()
r13: a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
r14: a__U52(tt(),N) -> mark(N)
r15: a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
r16: a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
r17: a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
r18: a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r19: a__isNat(|0|()) -> tt()
r20: a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
r21: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r22: a__isNatKind(|0|()) -> tt()
r23: a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
r24: a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
r25: a__plus(N,|0|()) -> a__U51(a__isNat(N),N)
r26: a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
r27: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r28: mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
r29: mark(isNatKind(X)) -> a__isNatKind(X)
r30: mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
r31: mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
r32: mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
r33: mark(isNat(X)) -> a__isNat(X)
r34: mark(U16(X)) -> a__U16(mark(X))
r35: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r36: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r37: mark(U23(X)) -> a__U23(mark(X))
r38: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r39: mark(U32(X)) -> a__U32(mark(X))
r40: mark(U41(X)) -> a__U41(mark(X))
r41: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r42: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r43: mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
r44: mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
r45: mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
r46: mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
r47: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r48: mark(tt()) -> tt()
r49: mark(s(X)) -> s(mark(X))
r50: mark(|0|()) -> |0|()
r51: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r52: a__U12(X1,X2,X3) -> U12(X1,X2,X3)
r53: a__isNatKind(X) -> isNatKind(X)
r54: a__U13(X1,X2,X3) -> U13(X1,X2,X3)
r55: a__U14(X1,X2,X3) -> U14(X1,X2,X3)
r56: a__U15(X1,X2) -> U15(X1,X2)
r57: a__isNat(X) -> isNat(X)
r58: a__U16(X) -> U16(X)
r59: a__U21(X1,X2) -> U21(X1,X2)
r60: a__U22(X1,X2) -> U22(X1,X2)
r61: a__U23(X) -> U23(X)
r62: a__U31(X1,X2) -> U31(X1,X2)
r63: a__U32(X) -> U32(X)
r64: a__U41(X) -> U41(X)
r65: a__U51(X1,X2) -> U51(X1,X2)
r66: a__U52(X1,X2) -> U52(X1,X2)
r67: a__U61(X1,X2,X3) -> U61(X1,X2,X3)
r68: a__U62(X1,X2,X3) -> U62(X1,X2,X3)
r69: a__U63(X1,X2,X3) -> U63(X1,X2,X3)
r70: a__U64(X1,X2,X3) -> U64(X1,X2,X3)
r71: a__plus(X1,X2) -> plus(X1,X2)

The set of usable rules consists of

  r10, r11, r12, r22, r23, r24, r53, r62, r63, r64

Take the reduction pair:

  matrix interpretations:
  
    carrier: N^2
    order: standard order
    interpretations:
      a__isNatKind#_A(x1) = ((1,0),(1,0)) x1
      plus_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,0),(1,1)) x2 + (1,1)
      a__U31#_A(x1,x2) = ((0,1),(0,1)) x1 + ((1,0),(1,0)) x2
      a__isNatKind_A(x1) = ((0,1),(0,1)) x1 + (1,1)
      tt_A() = (1,1)
      s_A(x1) = ((1,1),(1,1)) x1 + (1,1)
      a__U32_A(x1) = ((0,1),(0,1)) x1
      U32_A(x1) = ((0,1),(0,1)) x1
      a__U31_A(x1,x2) = x1 + ((0,1),(1,1)) x2
      a__U41_A(x1) = ((0,1),(1,0)) x1
      U31_A(x1,x2) = (0,0)
      U41_A(x1) = (0,0)
      |0|_A() = (1,1)
      isNatKind_A(x1) = (0,0)

The next rules are strictly ordered:

  p2, p3, p4

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__U31#(a__isNatKind(V1),V2)

and R consists of:

r1: a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
r2: a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
r3: a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
r4: a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
r5: a__U15(tt(),V2) -> a__U16(a__isNat(V2))
r6: a__U16(tt()) -> tt()
r7: a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
r8: a__U22(tt(),V1) -> a__U23(a__isNat(V1))
r9: a__U23(tt()) -> tt()
r10: a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
r11: a__U32(tt()) -> tt()
r12: a__U41(tt()) -> tt()
r13: a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
r14: a__U52(tt(),N) -> mark(N)
r15: a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
r16: a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
r17: a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
r18: a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
r19: a__isNat(|0|()) -> tt()
r20: a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
r21: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
r22: a__isNatKind(|0|()) -> tt()
r23: a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
r24: a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
r25: a__plus(N,|0|()) -> a__U51(a__isNat(N),N)
r26: a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
r27: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
r28: mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
r29: mark(isNatKind(X)) -> a__isNatKind(X)
r30: mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
r31: mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
r32: mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
r33: mark(isNat(X)) -> a__isNat(X)
r34: mark(U16(X)) -> a__U16(mark(X))
r35: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
r36: mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
r37: mark(U23(X)) -> a__U23(mark(X))
r38: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
r39: mark(U32(X)) -> a__U32(mark(X))
r40: mark(U41(X)) -> a__U41(mark(X))
r41: mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
r42: mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
r43: mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
r44: mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
r45: mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
r46: mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
r47: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
r48: mark(tt()) -> tt()
r49: mark(s(X)) -> s(mark(X))
r50: mark(|0|()) -> |0|()
r51: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
r52: a__U12(X1,X2,X3) -> U12(X1,X2,X3)
r53: a__isNatKind(X) -> isNatKind(X)
r54: a__U13(X1,X2,X3) -> U13(X1,X2,X3)
r55: a__U14(X1,X2,X3) -> U14(X1,X2,X3)
r56: a__U15(X1,X2) -> U15(X1,X2)
r57: a__isNat(X) -> isNat(X)
r58: a__U16(X) -> U16(X)
r59: a__U21(X1,X2) -> U21(X1,X2)
r60: a__U22(X1,X2) -> U22(X1,X2)
r61: a__U23(X) -> U23(X)
r62: a__U31(X1,X2) -> U31(X1,X2)
r63: a__U32(X) -> U32(X)
r64: a__U41(X) -> U41(X)
r65: a__U51(X1,X2) -> U51(X1,X2)
r66: a__U52(X1,X2) -> U52(X1,X2)
r67: a__U61(X1,X2,X3) -> U61(X1,X2,X3)
r68: a__U62(X1,X2,X3) -> U62(X1,X2,X3)
r69: a__U63(X1,X2,X3) -> U63(X1,X2,X3)
r70: a__U64(X1,X2,X3) -> U64(X1,X2,X3)
r71: a__plus(X1,X2) -> plus(X1,X2)

The estimated dependency graph contains the following SCCs:

  (no SCCs)