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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U64#_A(x1,x2,x3) = x1 + x2 + x3 + (43,11)
          tt_A() = (0,18)
          mark#_A(x1) = ((1,0),(0,0)) x1 + (22,29)
          s_A(x1) = ((1,0),(0,0)) x1 + (47,23)
          plus_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (21,18)
          a__plus#_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (43,28)
          mark_A(x1) = x1 + (0,18)
          a__U61#_A(x1,x2,x3) = x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (46,5)
          a__isNat_A(x1) = ((0,0),(1,0)) x1 + (0,22)
          a__U62#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (45,23)
          a__isNatKind_A(x1) = ((0,0),(1,0)) x1
          a__U63#_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (44,19)
          U64_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (68,30)
          |0|_A() = (19,1)
          a__U51#_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (23,11)
          a__U52#_A(x1,x2) = x1 + x2 + (23,29)
          U63_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (68,18)
          U62_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (68,15)
          U61_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (68,17)
          U52_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (20,19)
          U51_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (20,21)
          U41_A(x1) = ((1,0),(0,0)) x1 + (0,18)
          U32_A(x1) = x1 + (0,18)
          U31_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,19)
          U23_A(x1) = ((1,0),(0,0)) x1 + (0,18)
          U22_A(x1,x2) = ((1,0),(0,0)) x1 + (0,18)
          U21_A(x1,x2) = ((1,0),(0,0)) x1 + (0,18)
          U16_A(x1) = ((1,0),(0,0)) x1 + (0,19)
          U15_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,20)
          U14_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x3 + (0,20)
          U13_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x3 + (0,18)
          U12_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x3 + (0,18)
          U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x3 + (0,18)
          a__U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x3 + (0,24)
          a__U12_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x3 + (0,23)
          a__U13_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x3 + (0,22)
          a__U14_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x3 + (0,21)
          a__U15_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,20)
          a__U16_A(x1) = ((1,0),(0,0)) x1 + (0,19)
          a__U21_A(x1,x2) = ((1,0),(0,0)) x1 + (0,24)
          a__U22_A(x1,x2) = ((1,0),(0,0)) x1 + (0,19)
          a__U23_A(x1) = ((1,0),(0,0)) x1 + (0,19)
          a__U31_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,20)
          a__U32_A(x1) = x1 + (0,18)
          a__U41_A(x1) = ((1,0),(0,0)) x1 + (0,24)
          a__U51_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,1)) x2 + (20,38)
          a__U52_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (20,19)
          a__U61_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (68,34)
          a__U62_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (68,33)
          a__U63_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (68,32)
          a__U64_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(0,0)) x3 + (68,31)
          a__plus_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (21,18)
          isNatKind_A(x1) = ((0,0),(1,0)) x1
          isNat_A(x1) = ((0,0),(1,0)) x1 + (0,5)
  
    precedence:
    
      a__U51 > U51 > a__U14 > U16 = U15 = a__U15 = a__U16 > U12 = a__U22 > tt = U23 = a__U23 > plus = mark = a__isNatKind = U63 = U62 = U52 = U31 = a__U31 = a__U52 = a__U61 = a__U62 = a__U63 = a__U64 = a__plus > U32 = U11 = a__U11 = a__U12 = a__U32 > a__isNat > U64 = U61 > U13 = a__U13 > isNat > U41 = a__U41 > U21 = a__U21 > a__U64# = mark# = s = a__plus# = a__U61# = a__U62# = a__U51# = a__U52# > a__U63# > U22 > isNatKind > |0| = U14
    
    partial status:
  
      pi(a__U64#) = [3]
      pi(tt) = []
      pi(mark#) = []
      pi(s) = []
      pi(plus) = []
      pi(a__plus#) = []
      pi(mark) = [1]
      pi(a__U61#) = []
      pi(a__isNat) = []
      pi(a__U62#) = []
      pi(a__isNatKind) = []
      pi(a__U63#) = []
      pi(U64) = []
      pi(|0|) = []
      pi(a__U51#) = []
      pi(a__U52#) = []
      pi(U63) = []
      pi(U62) = []
      pi(U61) = []
      pi(U52) = []
      pi(U51) = []
      pi(U41) = []
      pi(U32) = [1]
      pi(U31) = []
      pi(U23) = []
      pi(U22) = []
      pi(U21) = []
      pi(U16) = []
      pi(U15) = []
      pi(U14) = []
      pi(U13) = []
      pi(U12) = []
      pi(U11) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__U13) = []
      pi(a__U14) = []
      pi(a__U15) = []
      pi(a__U16) = []
      pi(a__U21) = []
      pi(a__U22) = []
      pi(a__U23) = []
      pi(a__U31) = []
      pi(a__U32) = [1]
      pi(a__U41) = []
      pi(a__U51) = []
      pi(a__U52) = []
      pi(a__U61) = []
      pi(a__U62) = []
      pi(a__U63) = []
      pi(a__U64) = []
      pi(a__plus) = [1]
      pi(isNatKind) = []
      pi(isNat) = []

The next rules are strictly ordered:

  p8

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#(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__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
p9: a__U64#(tt(),M,N) -> mark#(N)
p10: mark#(U64(X1,X2,X3)) -> mark#(X1)
p11: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p12: a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p13: a__plus#(N,|0|()) -> a__U51#(a__isNat(N),N)
p14: a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
p15: a__U52#(tt(),N) -> mark#(N)
p16: mark#(U63(X1,X2,X3)) -> mark#(X1)
p17: mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
p18: mark#(U62(X1,X2,X3)) -> mark#(X1)
p19: mark#(U62(X1,X2,X3)) -> a__U62#(mark(X1),X2,X3)
p20: mark#(U61(X1,X2,X3)) -> mark#(X1)
p21: mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3)
p22: mark#(U52(X1,X2)) -> mark#(X1)
p23: mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
p24: mark#(U51(X1,X2)) -> mark#(X1)
p25: mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2)
p26: mark#(U41(X)) -> mark#(X)
p27: mark#(U32(X)) -> mark#(X)
p28: mark#(U31(X1,X2)) -> mark#(X1)
p29: mark#(U23(X)) -> mark#(X)
p30: mark#(U22(X1,X2)) -> mark#(X1)
p31: mark#(U21(X1,X2)) -> mark#(X1)
p32: mark#(U16(X)) -> mark#(X)
p33: mark#(U15(X1,X2)) -> mark#(X1)
p34: mark#(U14(X1,X2,X3)) -> mark#(X1)
p35: mark#(U13(X1,X2,X3)) -> mark#(X1)
p36: mark#(U12(X1,X2,X3)) -> mark#(X1)
p37: 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:

  {p1, p2, p3, p4, p5, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p20, 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__U64#(tt(),M,N) -> mark#(M)
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#(U61(X1,X2,X3)) -> mark#(X1)
p21: mark#(U62(X1,X2,X3)) -> mark#(X1)
p22: mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
p23: a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
p24: a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M))
p25: a__plus#(N,|0|()) -> a__U51#(a__isNat(N),N)
p26: a__U64#(tt(),M,N) -> mark#(N)
p27: mark#(U63(X1,X2,X3)) -> mark#(X1)
p28: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p29: mark#(U64(X1,X2,X3)) -> mark#(X1)
p30: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p31: mark#(plus(X1,X2)) -> mark#(X1)
p32: mark#(plus(X1,X2)) -> mark#(X2)
p33: 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 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U64#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (7,19)
          tt_A() = (0,2)
          mark#_A(x1) = ((1,0),(0,0)) x1 + (1,20)
          U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,1)
          U12_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,6)
          U13_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,5)
          U14_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,4)
          U15_A(x1,x2) = ((1,0),(0,0)) x1 + (0,3)
          U16_A(x1) = ((1,0),(0,0)) x1 + (0,1)
          U21_A(x1,x2) = ((1,0),(0,0)) x1 + (0,3)
          U22_A(x1,x2) = ((1,0),(0,0)) x1 + (0,2)
          U23_A(x1) = ((1,0),(0,0)) x1 + (0,1)
          U31_A(x1,x2) = ((1,0),(0,0)) x1 + (0,15)
          U32_A(x1) = ((1,0),(0,0)) x1
          U41_A(x1) = ((1,0),(0,0)) x1
          U51_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (4,1)
          a__U51#_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (3,5)
          mark_A(x1) = ((1,0),(0,0)) x1 + (0,16)
          a__U52#_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,19)
          a__isNatKind_A(x1) = (0,16)
          U52_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (3,14)
          U61_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (9,1)
          U62_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (9,3)
          U63_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (9,2)
          a__U63#_A(x1,x2,x3) = x1 + ((1,0),(1,1)) x2 + ((1,0),(1,0)) x3 + (8,18)
          a__plus#_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (6,2)
          |0|_A() = (0,13)
          a__isNat_A(x1) = (0,9)
          U64_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (9,1)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (5,16)
          s_A(x1) = ((1,0),(0,0)) x1 + (4,0)
          a__U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,8)
          a__U12_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,7)
          a__U13_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,6)
          a__U14_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,5)
          a__U15_A(x1,x2) = ((1,0),(0,0)) x1 + (0,4)
          a__U16_A(x1) = ((1,0),(0,0)) x1 + (0,3)
          a__U21_A(x1,x2) = ((1,0),(0,0)) x1 + (0,4)
          a__U22_A(x1,x2) = ((1,0),(0,0)) x1 + (0,3)
          a__U23_A(x1) = ((1,0),(0,0)) x1 + (0,2)
          a__U31_A(x1,x2) = ((1,0),(0,0)) x1 + (0,15)
          a__U32_A(x1) = ((1,0),(0,0)) x1 + (0,15)
          a__U41_A(x1) = ((1,0),(0,0)) x1 + (0,16)
          a__U51_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (4,14)
          a__U52_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (3,15)
          a__U61_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (9,16)
          a__U62_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (9,3)
          a__U63_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (9,3)
          a__U64_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (9,3)
          a__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (5,16)
          isNatKind_A(x1) = (0,16)
          isNat_A(x1) = (0,1)
  
    precedence:
    
      a__U52# > a__U31 > U31 > a__U64# = a__U63# > a__U23 > U23 > mark# > a__U51# = a__plus# > |0| > mark = plus = a__U51 = a__plus > U61 = U63 = U64 = s = a__U61 = a__U62 = a__U63 = a__U64 > U21 = a__isNat = a__U21 > U11 = a__U11 > U13 = a__isNatKind = a__U12 = a__U13 = a__U14 = a__U15 = a__U52 = isNatKind > U51 > isNat > U14 > U15 > tt = U41 = a__U41 > U32 = a__U32 > U22 = a__U22 > U16 = a__U16 > U12 = U52 = U62
    
    partial status:
  
      pi(a__U64#) = []
      pi(tt) = []
      pi(mark#) = []
      pi(U11) = []
      pi(U12) = []
      pi(U13) = []
      pi(U14) = []
      pi(U15) = []
      pi(U16) = []
      pi(U21) = []
      pi(U22) = []
      pi(U23) = []
      pi(U31) = []
      pi(U32) = []
      pi(U41) = []
      pi(U51) = []
      pi(a__U51#) = []
      pi(mark) = []
      pi(a__U52#) = [2]
      pi(a__isNatKind) = []
      pi(U52) = []
      pi(U61) = []
      pi(U62) = []
      pi(U63) = []
      pi(a__U63#) = [1]
      pi(a__plus#) = [2]
      pi(|0|) = []
      pi(a__isNat) = []
      pi(U64) = []
      pi(plus) = []
      pi(s) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__U13) = []
      pi(a__U14) = []
      pi(a__U15) = []
      pi(a__U16) = []
      pi(a__U21) = []
      pi(a__U22) = []
      pi(a__U23) = []
      pi(a__U31) = []
      pi(a__U32) = []
      pi(a__U41) = []
      pi(a__U51) = []
      pi(a__U52) = []
      pi(a__U61) = []
      pi(a__U62) = []
      pi(a__U63) = []
      pi(a__U64) = []
      pi(a__plus) = []
      pi(isNatKind) = []
      pi(isNat) = []

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__U64#(tt(),M,N) -> mark#(M)
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#(U61(X1,X2,X3)) -> mark#(X1)
p21: mark#(U62(X1,X2,X3)) -> mark#(X1)
p22: mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
p23: a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
p24: a__plus#(N,|0|()) -> a__U51#(a__isNat(N),N)
p25: a__U64#(tt(),M,N) -> mark#(N)
p26: mark#(U63(X1,X2,X3)) -> mark#(X1)
p27: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p28: mark#(U64(X1,X2,X3)) -> mark#(X1)
p29: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
p30: mark#(plus(X1,X2)) -> mark#(X1)
p31: mark#(plus(X1,X2)) -> mark#(X2)
p32: 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:

  {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}


-- 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,|0|()) -> a__U51#(a__isNat(N),N)
p7: a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
p8: a__U52#(tt(),N) -> mark#(N)
p9: mark#(U64(X1,X2,X3)) -> mark#(X1)
p10: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p11: a__U64#(tt(),M,N) -> mark#(N)
p12: mark#(U63(X1,X2,X3)) -> mark#(X1)
p13: mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
p14: a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
p15: mark#(U62(X1,X2,X3)) -> mark#(X1)
p16: mark#(U61(X1,X2,X3)) -> mark#(X1)
p17: mark#(U52(X1,X2)) -> mark#(X1)
p18: mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
p19: mark#(U51(X1,X2)) -> mark#(X1)
p20: mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2)
p21: mark#(U41(X)) -> mark#(X)
p22: mark#(U32(X)) -> mark#(X)
p23: mark#(U31(X1,X2)) -> mark#(X1)
p24: mark#(U23(X)) -> mark#(X)
p25: mark#(U22(X1,X2)) -> mark#(X1)
p26: mark#(U21(X1,X2)) -> mark#(X1)
p27: mark#(U16(X)) -> mark#(X)
p28: mark#(U15(X1,X2)) -> mark#(X1)
p29: mark#(U14(X1,X2,X3)) -> mark#(X1)
p30: mark#(U13(X1,X2,X3)) -> mark#(X1)
p31: mark#(U12(X1,X2,X3)) -> mark#(X1)
p32: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U64#_A(x1,x2,x3) = x1 + ((1,0),(0,0)) x2 + x3 + (1,2)
          tt_A() = (6,39)
          mark#_A(x1) = ((1,0),(1,1)) x1 + (5,40)
          s_A(x1) = ((1,0),(0,0)) x1 + (12,112)
          plus_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (54,13)
          a__plus#_A(x1,x2) = x1 + ((0,0),(1,0)) x2 + (9,96)
          mark_A(x1) = ((1,0),(1,1)) x1 + (0,12)
          |0|_A() = (9,36)
          a__U51#_A(x1,x2) = x2 + (8,37)
          a__isNat_A(x1) = ((0,0),(1,0)) x1 + (6,35)
          a__U52#_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (1,32)
          a__isNatKind_A(x1) = (6,39)
          U64_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + x3 + (60,7)
          U63_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + x3 + (60,35)
          a__U63#_A(x1,x2,x3) = x1 + x2 + x3 + (2,124)
          U62_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + x3 + (60,9)
          U61_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + x3 + (60,11)
          U52_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (3,14)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (4,13)
          U41_A(x1) = x1
          U32_A(x1) = x1
          U31_A(x1,x2) = x1
          U23_A(x1) = x1 + (0,11)
          U22_A(x1,x2) = x1 + ((0,0),(1,0)) x2 + (0,7)
          U21_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (0,1)
          U16_A(x1) = x1 + (0,1)
          U15_A(x1,x2) = x1 + ((0,0),(1,0)) x2 + (0,1)
          U14_A(x1,x2,x3) = x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (0,41)
          U13_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (0,41)
          U12_A(x1,x2,x3) = x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (0,48)
          U11_A(x1,x2,x3) = x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (0,49)
          a__U11_A(x1,x2,x3) = x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (0,49)
          a__U12_A(x1,x2,x3) = x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (0,48)
          a__U13_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (0,41)
          a__U14_A(x1,x2,x3) = x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (0,41)
          a__U15_A(x1,x2) = x1 + ((0,0),(1,0)) x2 + (0,1)
          a__U16_A(x1) = x1 + (0,1)
          a__U21_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (0,1)
          a__U22_A(x1,x2) = x1 + ((0,0),(1,0)) x2 + (0,7)
          a__U23_A(x1) = x1 + (0,11)
          a__U31_A(x1,x2) = x1
          a__U32_A(x1) = x1
          a__U41_A(x1) = x1
          a__U51_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (4,17)
          a__U52_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (3,17)
          a__U61_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + x3 + (60,71)
          a__U62_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + x3 + (60,68)
          a__U63_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + x3 + (60,107)
          a__U64_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + x3 + (60,67)
          a__plus_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (54,14)
          isNatKind_A(x1) = (6,39)
          isNat_A(x1) = ((0,0),(1,0)) x1 + (6,18)
  
    precedence:
    
      |0| > mark > a__U62 = a__U63 > tt = a__isNatKind = a__U31 = a__U32 = a__U41 > U63 > U62 > a__U64 > U15 = a__U15 > U31 > U64 > mark# = plus = a__U63# = U61 = a__U61 = a__plus > U21 = a__U21 > a__plus# > a__isNat = isNat > U41 > a__U22 > U23 = a__U23 > U14 = U13 = U12 = U11 = a__U11 = a__U12 = a__U13 = a__U14 = a__U51 = a__U52 > U51 > a__U64# = U16 = a__U16 > s > U52 = U22 = isNatKind > a__U51# = a__U52# = U32
    
    partial status:
  
      pi(a__U64#) = [1]
      pi(tt) = []
      pi(mark#) = []
      pi(s) = []
      pi(plus) = []
      pi(a__plus#) = [1]
      pi(mark) = [1]
      pi(|0|) = []
      pi(a__U51#) = [2]
      pi(a__isNat) = []
      pi(a__U52#) = []
      pi(a__isNatKind) = []
      pi(U64) = []
      pi(U63) = []
      pi(a__U63#) = [1, 2]
      pi(U62) = []
      pi(U61) = []
      pi(U52) = [2]
      pi(U51) = [1, 2]
      pi(U41) = []
      pi(U32) = []
      pi(U31) = []
      pi(U23) = []
      pi(U22) = [1]
      pi(U21) = []
      pi(U16) = [1]
      pi(U15) = [1]
      pi(U14) = []
      pi(U13) = []
      pi(U12) = []
      pi(U11) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__U13) = []
      pi(a__U14) = []
      pi(a__U15) = [1]
      pi(a__U16) = [1]
      pi(a__U21) = []
      pi(a__U22) = []
      pi(a__U23) = []
      pi(a__U31) = []
      pi(a__U32) = []
      pi(a__U41) = []
      pi(a__U51) = []
      pi(a__U52) = []
      pi(a__U61) = []
      pi(a__U62) = [3]
      pi(a__U63) = [3]
      pi(a__U64) = [1]
      pi(a__plus) = []
      pi(isNatKind) = []
      pi(isNat) = []

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__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: a__plus#(N,|0|()) -> a__U51#(a__isNat(N),N)
p6: a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
p7: a__U52#(tt(),N) -> mark#(N)
p8: mark#(U64(X1,X2,X3)) -> mark#(X1)
p9: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p10: a__U64#(tt(),M,N) -> mark#(N)
p11: mark#(U63(X1,X2,X3)) -> mark#(X1)
p12: mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
p13: a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
p14: mark#(U62(X1,X2,X3)) -> mark#(X1)
p15: mark#(U61(X1,X2,X3)) -> mark#(X1)
p16: mark#(U52(X1,X2)) -> mark#(X1)
p17: mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
p18: mark#(U51(X1,X2)) -> mark#(X1)
p19: mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2)
p20: mark#(U41(X)) -> mark#(X)
p21: mark#(U32(X)) -> mark#(X)
p22: mark#(U31(X1,X2)) -> mark#(X1)
p23: mark#(U23(X)) -> mark#(X)
p24: mark#(U22(X1,X2)) -> mark#(X1)
p25: mark#(U21(X1,X2)) -> mark#(X1)
p26: mark#(U16(X)) -> mark#(X)
p27: mark#(U15(X1,X2)) -> mark#(X1)
p28: mark#(U14(X1,X2,X3)) -> mark#(X1)
p29: mark#(U13(X1,X2,X3)) -> mark#(X1)
p30: mark#(U12(X1,X2,X3)) -> mark#(X1)
p31: 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:

  {p1, p2, p3, 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}


-- Reduction pair.

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

p1: a__U64#(tt(),M,N) -> mark#(M)
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#(U61(X1,X2,X3)) -> mark#(X1)
p21: mark#(U62(X1,X2,X3)) -> mark#(X1)
p22: mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
p23: a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
p24: a__U64#(tt(),M,N) -> mark#(N)
p25: mark#(U63(X1,X2,X3)) -> mark#(X1)
p26: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p27: mark#(U64(X1,X2,X3)) -> mark#(X1)
p28: mark#(plus(X1,X2)) -> mark#(X1)
p29: mark#(plus(X1,X2)) -> mark#(X2)
p30: 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 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U64#_A(x1,x2,x3) = ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (37,20)
          tt_A() = (23,27)
          mark#_A(x1) = ((1,0),(0,0)) x1 + (37,20)
          U11_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3
          U12_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (0,21)
          U13_A(x1,x2,x3) = x1 + ((0,0),(1,0)) x2 + (0,2)
          U14_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,26)
          U15_A(x1,x2) = x1 + (0,19)
          U16_A(x1) = ((1,0),(0,0)) x1 + (0,26)
          U21_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,42)
          U22_A(x1,x2) = ((1,0),(1,1)) x1
          U23_A(x1) = ((1,0),(0,0)) x1 + (0,26)
          U31_A(x1,x2) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (0,1)
          U32_A(x1) = ((1,0),(0,0)) x1 + (0,24)
          U41_A(x1) = ((1,0),(0,0)) x1 + (0,19)
          U51_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (3,1)
          a__U51#_A(x1,x2) = ((1,0),(0,0)) x2 + (39,19)
          mark_A(x1) = ((1,0),(1,1)) x1 + (0,3)
          a__U52#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + (38,20)
          a__isNatKind_A(x1) = ((0,0),(1,0)) x1 + (23,6)
          U52_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (3,1)
          U61_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (35,2)
          U62_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (35,2)
          U63_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (35,4)
          a__U63#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (38,5)
          U64_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (35,21)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (22,0)
          s_A(x1) = ((1,0),(0,0)) x1 + (36,43)
          a__U11_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3
          a__U12_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (0,22)
          a__U13_A(x1,x2,x3) = x1 + ((0,0),(1,0)) x2 + (0,2)
          a__U14_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,28)
          a__U15_A(x1,x2) = x1 + (0,19)
          a__isNat_A(x1) = ((0,0),(1,0)) x1 + (23,9)
          a__U16_A(x1) = ((1,0),(0,0)) x1 + (0,28)
          a__U21_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,44)
          a__U22_A(x1,x2) = ((1,0),(1,1)) x1
          a__U23_A(x1) = ((1,0),(0,0)) x1 + (0,28)
          a__U31_A(x1,x2) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (0,4)
          a__U32_A(x1) = ((1,0),(0,0)) x1 + (0,27)
          a__U41_A(x1) = ((1,0),(1,0)) x1 + (0,19)
          a__U51_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (3,2)
          a__U52_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (3,2)
          a__U61_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (35,39)
          a__U62_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (35,38)
          a__U63_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (35,5)
          a__U64_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (35,44)
          a__plus_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (22,0)
          |0|_A() = (27,24)
          isNat_A(x1) = ((0,0),(1,0)) x1 + (23,1)
          isNatKind_A(x1) = ((0,0),(1,0)) x1 + (23,1)
  
    precedence:
    
      a__U64# > isNat > U52 = a__U52 > U41 = a__isNatKind = a__U41 > mark# = a__U52# = a__U63# > U32 = mark = a__U31 = a__U32 > plus = a__plus > a__U61 = a__U62 > tt = U11 = U21 = a__U11 = a__isNat = a__U21 = a__U22 = a__U23 = a__U51 > U51 > U22 > |0| > U63 = a__U63 > s = a__U64 > U64 > U12 = U23 = a__U51# = U62 = a__U12 = a__U13 > U14 = a__U14 = a__U15 > U15 > U61 > U13 > U31 > U16 = a__U16 > isNatKind
    
    partial status:
  
      pi(a__U64#) = []
      pi(tt) = []
      pi(mark#) = []
      pi(U11) = []
      pi(U12) = []
      pi(U13) = []
      pi(U14) = []
      pi(U15) = [1]
      pi(U16) = []
      pi(U21) = []
      pi(U22) = []
      pi(U23) = []
      pi(U31) = []
      pi(U32) = []
      pi(U41) = []
      pi(U51) = []
      pi(a__U51#) = []
      pi(mark) = [1]
      pi(a__U52#) = []
      pi(a__isNatKind) = []
      pi(U52) = []
      pi(U61) = []
      pi(U62) = []
      pi(U63) = []
      pi(a__U63#) = []
      pi(U64) = []
      pi(plus) = []
      pi(s) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__U13) = []
      pi(a__U14) = []
      pi(a__U15) = []
      pi(a__isNat) = []
      pi(a__U16) = []
      pi(a__U21) = []
      pi(a__U22) = []
      pi(a__U23) = []
      pi(a__U31) = []
      pi(a__U32) = []
      pi(a__U41) = []
      pi(a__U51) = []
      pi(a__U52) = []
      pi(a__U61) = []
      pi(a__U62) = []
      pi(a__U63) = [1]
      pi(a__U64) = []
      pi(a__plus) = [2]
      pi(|0|) = []
      pi(isNat) = []
      pi(isNatKind) = []

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__U64#(tt(),M,N) -> mark#(M)
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: a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
p15: a__U52#(tt(),N) -> mark#(N)
p16: mark#(U51(X1,X2)) -> mark#(X1)
p17: mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
p18: mark#(U52(X1,X2)) -> mark#(X1)
p19: mark#(U61(X1,X2,X3)) -> mark#(X1)
p20: mark#(U62(X1,X2,X3)) -> mark#(X1)
p21: mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
p22: a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
p23: a__U64#(tt(),M,N) -> mark#(N)
p24: mark#(U63(X1,X2,X3)) -> mark#(X1)
p25: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p26: mark#(U64(X1,X2,X3)) -> mark#(X1)
p27: mark#(plus(X1,X2)) -> mark#(X1)
p28: mark#(plus(X1,X2)) -> mark#(X2)
p29: 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:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, 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__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#(U64(X1,X2,X3)) -> mark#(X1)
p6: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p7: a__U64#(tt(),M,N) -> mark#(N)
p8: mark#(U63(X1,X2,X3)) -> mark#(X1)
p9: mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
p10: a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
p11: mark#(U62(X1,X2,X3)) -> mark#(X1)
p12: mark#(U61(X1,X2,X3)) -> mark#(X1)
p13: mark#(U52(X1,X2)) -> mark#(X1)
p14: mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
p15: a__U52#(tt(),N) -> mark#(N)
p16: mark#(U51(X1,X2)) -> mark#(X1)
p17: mark#(U41(X)) -> mark#(X)
p18: mark#(U32(X)) -> mark#(X)
p19: mark#(U31(X1,X2)) -> mark#(X1)
p20: mark#(U23(X)) -> mark#(X)
p21: mark#(U22(X1,X2)) -> mark#(X1)
p22: mark#(U21(X1,X2)) -> mark#(X1)
p23: mark#(U16(X)) -> mark#(X)
p24: mark#(U15(X1,X2)) -> mark#(X1)
p25: mark#(U14(X1,X2,X3)) -> mark#(X1)
p26: mark#(U13(X1,X2,X3)) -> mark#(X1)
p27: mark#(U12(X1,X2,X3)) -> mark#(X1)
p28: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U64#_A(x1,x2,x3) = ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (1,19)
          tt_A() = (2,20)
          mark#_A(x1) = ((0,0),(1,0)) x1 + (1,17)
          s_A(x1) = ((1,0),(0,0)) x1 + (15,0)
          plus_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (8,0)
          U64_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + x3 + (21,16)
          mark_A(x1) = x1 + (0,24)
          U63_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + x3 + (21,25)
          a__U63#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (1,26)
          a__isNatKind_A(x1) = ((0,0),(1,0)) x1 + (2,6)
          U62_A(x1,x2,x3) = x1 + ((1,0),(0,0)) x2 + x3 + (21,7)
          U61_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + x3 + (21,1)
          U52_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (8,1)
          a__U52#_A(x1,x2) = ((0,0),(1,0)) x2 + (1,18)
          U51_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (21,0)
          U41_A(x1) = ((1,0),(1,0)) x1 + (0,18)
          U32_A(x1) = ((1,0),(1,1)) x1 + (0,1)
          U31_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2
          U23_A(x1) = ((1,0),(1,1)) x1
          U22_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2
          U21_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2
          U16_A(x1) = ((1,0),(1,1)) x1
          U15_A(x1,x2) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (0,16)
          U14_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x3
          U13_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x3
          U12_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x3
          U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x3
          a__U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x3 + (0,23)
          a__U12_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x3 + (0,21)
          a__U13_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x3 + (0,22)
          a__U14_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x3 + (0,21)
          a__U15_A(x1,x2) = ((1,0),(1,0)) x1 + ((0,0),(1,0)) x2 + (0,16)
          a__isNat_A(x1) = ((0,0),(1,0)) x1 + (2,16)
          a__U16_A(x1) = ((1,0),(1,1)) x1
          a__U21_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,22)
          a__U22_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,21)
          a__U23_A(x1) = ((1,0),(1,1)) x1
          a__U31_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,10)
          a__U32_A(x1) = ((1,0),(1,1)) x1 + (0,1)
          a__U41_A(x1) = ((1,0),(1,0)) x1 + (0,18)
          a__U51_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (21,14)
          a__U52_A(x1,x2) = ((1,0),(1,0)) x1 + x2 + (8,23)
          a__U61_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + x3 + (21,12)
          a__U62_A(x1,x2,x3) = x1 + ((1,0),(0,0)) x2 + x3 + (21,7)
          a__U63_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + x3 + (21,26)
          a__U64_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + x3 + (21,23)
          a__plus_A(x1,x2) = x1 + ((1,0),(1,0)) x2 + (8,0)
          |0|_A() = (15,21)
          isNat_A(x1) = ((0,0),(1,0)) x1 + (2,0)
          isNatKind_A(x1) = ((0,0),(1,0)) x1 + (2,0)
  
    precedence:
    
      a__U11 > a__isNat = isNat > mark# = a__U52# > mark = U52 = U12 = a__U12 = a__U13 = a__U21 = a__U52 > U31 = a__U31 > a__U32 > U41 = U32 = a__U41 > U61 > a__U61 > U21 > U62 = a__U62 > a__U63 > U14 = U13 = a__U14 > a__U15 > a__U16 > U15 > U22 = a__U22 > U63 > tt = a__U23 > a__isNatKind = isNatKind > a__U51 > U51 > a__plus > U23 > a__U63# > a__U64# > plus = a__U64 = |0| > U64 > U16 > s = U11
    
    partial status:
  
      pi(a__U64#) = []
      pi(tt) = []
      pi(mark#) = []
      pi(s) = []
      pi(plus) = []
      pi(U64) = [3]
      pi(mark) = []
      pi(U63) = []
      pi(a__U63#) = []
      pi(a__isNatKind) = []
      pi(U62) = []
      pi(U61) = []
      pi(U52) = []
      pi(a__U52#) = []
      pi(U51) = []
      pi(U41) = []
      pi(U32) = []
      pi(U31) = []
      pi(U23) = []
      pi(U22) = []
      pi(U21) = []
      pi(U16) = []
      pi(U15) = []
      pi(U14) = []
      pi(U13) = []
      pi(U12) = []
      pi(U11) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__U13) = []
      pi(a__U14) = []
      pi(a__U15) = []
      pi(a__isNat) = []
      pi(a__U16) = []
      pi(a__U21) = []
      pi(a__U22) = []
      pi(a__U23) = []
      pi(a__U31) = []
      pi(a__U32) = [1]
      pi(a__U41) = []
      pi(a__U51) = []
      pi(a__U52) = []
      pi(a__U61) = []
      pi(a__U62) = []
      pi(a__U63) = []
      pi(a__U64) = []
      pi(a__plus) = []
      pi(|0|) = []
      pi(isNat) = []
      pi(isNatKind) = []

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__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#(U64(X1,X2,X3)) -> mark#(X1)
p6: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p7: a__U64#(tt(),M,N) -> mark#(N)
p8: mark#(U63(X1,X2,X3)) -> mark#(X1)
p9: mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
p10: a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
p11: mark#(U62(X1,X2,X3)) -> mark#(X1)
p12: mark#(U61(X1,X2,X3)) -> mark#(X1)
p13: mark#(U52(X1,X2)) -> mark#(X1)
p14: mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
p15: mark#(U51(X1,X2)) -> mark#(X1)
p16: mark#(U41(X)) -> mark#(X)
p17: mark#(U32(X)) -> mark#(X)
p18: mark#(U31(X1,X2)) -> mark#(X1)
p19: mark#(U23(X)) -> mark#(X)
p20: mark#(U22(X1,X2)) -> mark#(X1)
p21: mark#(U21(X1,X2)) -> mark#(X1)
p22: mark#(U16(X)) -> mark#(X)
p23: mark#(U15(X1,X2)) -> mark#(X1)
p24: mark#(U14(X1,X2,X3)) -> mark#(X1)
p25: mark#(U13(X1,X2,X3)) -> mark#(X1)
p26: mark#(U12(X1,X2,X3)) -> mark#(X1)
p27: 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:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, 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__U64#(tt(),M,N) -> mark#(M)
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)) -> mark#(X1)
p15: mark#(U52(X1,X2)) -> mark#(X1)
p16: mark#(U61(X1,X2,X3)) -> mark#(X1)
p17: mark#(U62(X1,X2,X3)) -> mark#(X1)
p18: mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
p19: a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
p20: a__U64#(tt(),M,N) -> mark#(N)
p21: mark#(U63(X1,X2,X3)) -> mark#(X1)
p22: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p23: mark#(U64(X1,X2,X3)) -> mark#(X1)
p24: mark#(plus(X1,X2)) -> mark#(X1)
p25: mark#(plus(X1,X2)) -> mark#(X2)
p26: 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 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U64#_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + ((1,0),(1,1)) x3 + (0,1)
          tt_A() = (4,17)
          mark#_A(x1) = ((1,0),(0,0)) x1 + (0,28)
          U11_A(x1,x2,x3) = ((1,0),(1,0)) x1 + (0,28)
          U12_A(x1,x2,x3) = ((1,0),(1,0)) x1 + (0,29)
          U13_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,25)
          U14_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,19)
          U15_A(x1,x2) = ((1,0),(0,0)) x1 + (0,17)
          U16_A(x1) = ((1,0),(0,0)) x1 + (0,29)
          U21_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,18)
          U22_A(x1,x2) = ((1,0),(0,0)) x1 + (0,17)
          U23_A(x1) = ((1,0),(0,0)) x1 + (0,29)
          U31_A(x1,x2) = ((1,0),(1,0)) x1 + (0,29)
          U32_A(x1) = ((1,0),(0,0)) x1 + (0,15)
          U41_A(x1) = ((1,0),(1,0)) x1 + (0,15)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,16)
          U52_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,1)
          U61_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (11,27)
          U62_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (11,27)
          U63_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (11,27)
          a__U63#_A(x1,x2,x3) = ((1,0),(0,0)) x1 + x2 + ((1,0),(0,0)) x3 + (1,39)
          mark_A(x1) = x1 + (0,15)
          a__isNatKind_A(x1) = ((0,0),(1,0)) x1 + (4,38)
          U64_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (11,14)
          plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (10,42)
          s_A(x1) = ((1,0),(0,0)) x1 + (5,16)
          a__U11_A(x1,x2,x3) = ((1,0),(1,0)) x1 + (0,37)
          a__U12_A(x1,x2,x3) = ((1,0),(1,0)) x1 + (0,36)
          a__U13_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,39)
          a__U14_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,33)
          a__U15_A(x1,x2) = ((1,0),(0,0)) x1 + (0,31)
          a__isNat_A(x1) = ((0,0),(1,0)) x1 + (4,32)
          a__U16_A(x1) = ((1,0),(0,0)) x1 + (0,30)
          a__U21_A(x1,x2) = ((1,0),(0,0)) x1 + ((0,0),(1,0)) x2 + (0,32)
          a__U22_A(x1,x2) = ((1,0),(0,0)) x1 + (0,31)
          a__U23_A(x1) = ((1,0),(0,0)) x1 + (0,30)
          a__U31_A(x1,x2) = ((1,0),(1,0)) x1 + (0,30)
          a__U32_A(x1) = ((1,0),(0,0)) x1 + (0,18)
          a__U41_A(x1) = ((1,0),(1,0)) x1 + (0,15)
          a__U51_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,16)
          a__U52_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,2)
          a__U61_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (11,41)
          a__U62_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (11,36)
          a__U63_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (11,39)
          a__U64_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (11,14)
          a__plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (10,42)
          |0|_A() = (7,1)
          isNat_A(x1) = ((0,0),(1,0)) x1 + (4,31)
          isNatKind_A(x1) = ((0,0),(1,0)) x1 + (4,24)
  
    precedence:
    
      U31 = U41 = a__U41 > U52 > plus = a__plus > mark > U51 = a__U51 > a__U61 > a__U63 > U22 > a__U62 > U63 > U12 = U61 > a__U11 = a__U12 > a__U13 = a__U14 > U11 = U13 > isNatKind > a__U64 > U64 > U14 > a__U64# = mark# > |0| > a__U52 > U32 > a__isNat > a__U21 > s > a__U22 > U23 = a__U23 > tt = a__U31 = a__U32 > U21 > a__U63# = a__isNatKind > U62 > U16 = a__U15 = a__U16 = isNat > U15
    
    partial status:
  
      pi(a__U64#) = [1]
      pi(tt) = []
      pi(mark#) = []
      pi(U11) = []
      pi(U12) = []
      pi(U13) = []
      pi(U14) = []
      pi(U15) = []
      pi(U16) = []
      pi(U21) = []
      pi(U22) = []
      pi(U23) = []
      pi(U31) = []
      pi(U32) = []
      pi(U41) = []
      pi(U51) = [1]
      pi(U52) = []
      pi(U61) = []
      pi(U62) = []
      pi(U63) = []
      pi(a__U63#) = []
      pi(mark) = [1]
      pi(a__isNatKind) = []
      pi(U64) = []
      pi(plus) = []
      pi(s) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__U13) = []
      pi(a__U14) = []
      pi(a__U15) = []
      pi(a__isNat) = []
      pi(a__U16) = []
      pi(a__U21) = []
      pi(a__U22) = []
      pi(a__U23) = []
      pi(a__U31) = []
      pi(a__U32) = []
      pi(a__U41) = []
      pi(a__U51) = [1]
      pi(a__U52) = []
      pi(a__U61) = []
      pi(a__U62) = []
      pi(a__U63) = []
      pi(a__U64) = []
      pi(a__plus) = []
      pi(|0|) = []
      pi(isNat) = []
      pi(isNatKind) = []

The next rules are strictly ordered:

  p19

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#(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)) -> mark#(X1)
p15: mark#(U52(X1,X2)) -> mark#(X1)
p16: mark#(U61(X1,X2,X3)) -> mark#(X1)
p17: mark#(U62(X1,X2,X3)) -> mark#(X1)
p18: mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
p19: a__U64#(tt(),M,N) -> mark#(N)
p20: mark#(U63(X1,X2,X3)) -> mark#(X1)
p21: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p22: mark#(U64(X1,X2,X3)) -> mark#(X1)
p23: mark#(plus(X1,X2)) -> mark#(X1)
p24: mark#(plus(X1,X2)) -> mark#(X2)
p25: 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:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p19, p20, p21, p22, p23, p24, p25}


-- 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#(U64(X1,X2,X3)) -> mark#(X1)
p6: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p7: a__U64#(tt(),M,N) -> mark#(N)
p8: mark#(U63(X1,X2,X3)) -> mark#(X1)
p9: mark#(U62(X1,X2,X3)) -> mark#(X1)
p10: mark#(U61(X1,X2,X3)) -> mark#(X1)
p11: mark#(U52(X1,X2)) -> mark#(X1)
p12: mark#(U51(X1,X2)) -> mark#(X1)
p13: mark#(U41(X)) -> mark#(X)
p14: mark#(U32(X)) -> mark#(X)
p15: mark#(U31(X1,X2)) -> mark#(X1)
p16: mark#(U23(X)) -> mark#(X)
p17: mark#(U22(X1,X2)) -> mark#(X1)
p18: mark#(U21(X1,X2)) -> mark#(X1)
p19: mark#(U16(X)) -> mark#(X)
p20: mark#(U15(X1,X2)) -> mark#(X1)
p21: mark#(U14(X1,X2,X3)) -> mark#(X1)
p22: mark#(U13(X1,X2,X3)) -> mark#(X1)
p23: mark#(U12(X1,X2,X3)) -> mark#(X1)
p24: 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U64#_A(x1,x2,x3) = ((0,0),(1,0)) x2 + ((0,0),(1,0)) x3 + (1,5)
          tt_A() = (17,27)
          mark#_A(x1) = ((0,0),(1,0)) x1 + (1,4)
          s_A(x1) = ((1,0),(1,0)) x1 + (36,3)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (2,0)
          U64_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (21,6)
          mark_A(x1) = ((1,0),(1,1)) x1 + (0,26)
          U63_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (21,32)
          U62_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (21,0)
          U61_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(1,0)) x3 + (21,0)
          U52_A(x1,x2) = x1 + x2 + (2,1)
          U51_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (20,3)
          U41_A(x1) = ((1,0),(0,0)) x1 + (0,3)
          U32_A(x1) = ((1,0),(1,0)) x1 + (0,3)
          U31_A(x1,x2) = ((1,0),(0,0)) x1 + (0,3)
          U23_A(x1) = ((1,0),(0,0)) x1 + (0,3)
          U22_A(x1,x2) = ((1,0),(1,1)) x1
          U21_A(x1,x2) = ((1,0),(0,0)) x1 + (0,25)
          U16_A(x1) = ((1,0),(0,0)) x1 + (0,3)
          U15_A(x1,x2) = ((1,0),(0,0)) x1 + (0,5)
          U14_A(x1,x2,x3) = ((1,0),(1,0)) x1 + (0,5)
          U13_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,6)
          U12_A(x1,x2,x3) = x1 + (0,6)
          U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,14)
          a__U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,39)
          a__U12_A(x1,x2,x3) = x1 + (0,6)
          a__isNatKind_A(x1) = (17,32)
          a__U13_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,31)
          a__U14_A(x1,x2,x3) = ((1,0),(1,0)) x1 + (0,13)
          a__U15_A(x1,x2) = ((1,0),(0,0)) x1 + (0,29)
          a__isNat_A(x1) = (17,51)
          a__U16_A(x1) = ((1,0),(0,0)) x1 + (0,28)
          a__U21_A(x1,x2) = ((1,0),(0,0)) x1 + (0,50)
          a__U22_A(x1,x2) = ((1,0),(1,1)) x1
          a__U23_A(x1) = ((1,0),(0,0)) x1 + (0,28)
          a__U31_A(x1,x2) = ((1,0),(1,0)) x1 + (0,12)
          a__U32_A(x1) = ((1,0),(1,0)) x1 + (0,11)
          a__U41_A(x1) = ((1,0),(0,0)) x1 + (0,28)
          a__U51_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + (20,35)
          a__U52_A(x1,x2) = x1 + x2 + (2,2)
          a__U61_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (21,35)
          a__U62_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (21,34)
          a__U63_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (21,33)
          a__U64_A(x1,x2,x3) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + ((1,0),(1,0)) x3 + (21,11)
          a__plus_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (2,0)
          |0|_A() = (36,0)
          isNatKind_A(x1) = (17,1)
          isNat_A(x1) = (17,9)
  
    precedence:
    
      a__plus > a__U15 > U64 = U15 = a__U32 > plus = U32 > U41 > mark = U63 = U62 = U13 = a__U13 = a__U14 = a__U51 = a__U52 = a__U62 = a__U63 > a__isNat = a__U21 > a__isNatKind > U22 = a__U22 > U11 = a__U11 > U21 > U14 = a__U61 > a__U64# = mark# > a__U41 > tt = a__U16 = isNatKind > isNat > a__U12 > U23 > U61 = U52 > U31 = a__U23 = a__U31 > U51 > s = a__U64 > U16 = U12 = |0|
    
    partial status:
  
      pi(a__U64#) = []
      pi(tt) = []
      pi(mark#) = []
      pi(s) = []
      pi(plus) = []
      pi(U64) = []
      pi(mark) = []
      pi(U63) = []
      pi(U62) = []
      pi(U61) = []
      pi(U52) = [1]
      pi(U51) = []
      pi(U41) = []
      pi(U32) = []
      pi(U31) = []
      pi(U23) = []
      pi(U22) = []
      pi(U21) = []
      pi(U16) = []
      pi(U15) = []
      pi(U14) = []
      pi(U13) = []
      pi(U12) = []
      pi(U11) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__isNatKind) = []
      pi(a__U13) = []
      pi(a__U14) = []
      pi(a__U15) = []
      pi(a__isNat) = []
      pi(a__U16) = []
      pi(a__U21) = []
      pi(a__U22) = []
      pi(a__U23) = []
      pi(a__U31) = []
      pi(a__U32) = []
      pi(a__U41) = []
      pi(a__U51) = []
      pi(a__U52) = []
      pi(a__U61) = []
      pi(a__U62) = []
      pi(a__U63) = []
      pi(a__U64) = []
      pi(a__plus) = []
      pi(|0|) = []
      pi(isNatKind) = []
      pi(isNat) = []

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#(s(X)) -> mark#(X)
p2: mark#(plus(X1,X2)) -> mark#(X2)
p3: mark#(plus(X1,X2)) -> mark#(X1)
p4: mark#(U64(X1,X2,X3)) -> mark#(X1)
p5: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p6: a__U64#(tt(),M,N) -> mark#(N)
p7: mark#(U63(X1,X2,X3)) -> mark#(X1)
p8: mark#(U62(X1,X2,X3)) -> mark#(X1)
p9: mark#(U61(X1,X2,X3)) -> mark#(X1)
p10: mark#(U52(X1,X2)) -> mark#(X1)
p11: mark#(U51(X1,X2)) -> mark#(X1)
p12: mark#(U41(X)) -> mark#(X)
p13: mark#(U32(X)) -> mark#(X)
p14: mark#(U31(X1,X2)) -> mark#(X1)
p15: mark#(U23(X)) -> mark#(X)
p16: mark#(U22(X1,X2)) -> mark#(X1)
p17: mark#(U21(X1,X2)) -> mark#(X1)
p18: mark#(U16(X)) -> mark#(X)
p19: mark#(U15(X1,X2)) -> mark#(X1)
p20: mark#(U14(X1,X2,X3)) -> mark#(X1)
p21: mark#(U13(X1,X2,X3)) -> mark#(X1)
p22: mark#(U12(X1,X2,X3)) -> mark#(X1)
p23: 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:

  {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: mark#(s(X)) -> mark#(X)
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)) -> mark#(X1)
p15: mark#(U52(X1,X2)) -> mark#(X1)
p16: mark#(U61(X1,X2,X3)) -> mark#(X1)
p17: mark#(U62(X1,X2,X3)) -> mark#(X1)
p18: mark#(U63(X1,X2,X3)) -> mark#(X1)
p19: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p20: a__U64#(tt(),M,N) -> mark#(N)
p21: mark#(U64(X1,X2,X3)) -> mark#(X1)
p22: mark#(plus(X1,X2)) -> mark#(X1)
p23: mark#(plus(X1,X2)) -> mark#(X2)

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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((0,0),(1,0)) x1 + (7,7)
          s_A(x1) = ((1,0),(0,0)) x1 + (4,2)
          U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,7)
          U12_A(x1,x2,x3) = ((1,0),(0,0)) x1
          U13_A(x1,x2,x3) = ((1,0),(0,0)) x1
          U14_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,4)
          U15_A(x1,x2) = ((1,0),(1,0)) x1 + (0,1)
          U16_A(x1) = ((1,0),(0,0)) x1
          U21_A(x1,x2) = ((1,0),(0,0)) x1
          U22_A(x1,x2) = ((1,0),(0,0)) x1
          U23_A(x1) = ((1,0),(0,0)) x1
          U31_A(x1,x2) = ((1,0),(0,0)) x1
          U32_A(x1) = ((1,0),(0,0)) x1
          U41_A(x1) = ((1,0),(0,0)) x1
          U51_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (2,0)
          U52_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,0)
          U61_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (6,8)
          U62_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (6,0)
          U63_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (6,0)
          U64_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (6,3)
          a__U64#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + ((0,0),(1,0)) x3 + (7,12)
          mark_A(x1) = x1 + (0,12)
          tt_A() = (2,0)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (4,0)
          a__U11_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,7)
          a__U12_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,6)
          a__isNatKind_A(x1) = (2,4)
          a__U13_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,5)
          a__U14_A(x1,x2,x3) = ((1,0),(0,0)) x1 + (0,4)
          a__U15_A(x1,x2) = ((1,0),(1,0)) x1 + (0,1)
          a__isNat_A(x1) = (2,11)
          a__U16_A(x1) = ((1,0),(0,0)) x1 + (0,1)
          a__U21_A(x1,x2) = ((1,0),(0,0)) x1 + (0,5)
          a__U22_A(x1,x2) = ((1,0),(0,0)) x1 + (0,1)
          a__U23_A(x1) = ((1,0),(0,0)) x1 + (0,1)
          a__U31_A(x1,x2) = ((1,0),(0,0)) x1 + (0,3)
          a__U32_A(x1) = ((1,0),(0,0)) x1 + (0,3)
          a__U41_A(x1) = ((1,0),(0,0)) x1 + (0,3)
          a__U51_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (2,0)
          a__U52_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,1)
          a__U61_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (6,9)
          a__U62_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (6,6)
          a__U63_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (6,5)
          a__U64_A(x1,x2,x3) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + ((1,0),(0,0)) x3 + (6,4)
          a__plus_A(x1,x2) = ((1,0),(0,0)) x1 + ((1,0),(0,0)) x2 + (4,10)
          |0|_A() = (1,0)
          isNatKind_A(x1) = (2,0)
          isNat_A(x1) = (2,0)
  
    precedence:
    
      tt = a__U41 > mark# > a__isNatKind > U51 = U62 = mark = plus = a__U31 = a__U32 = a__U51 = a__U52 = a__U62 = a__plus > U31 > U11 = a__U11 > s = U63 = a__U63 = a__U64 > U12 = U13 = U64 = a__U12 = a__U13 > a__isNat = isNat > a__U64# > U14 = U15 = a__U14 = a__U15 = a__U16 > a__U21 = a__U22 > a__U61 > U61 > isNatKind > U41 > U22 > U52 > U16 > U21 > a__U23 > U23 = |0| > U32
    
    partial status:
  
      pi(mark#) = []
      pi(s) = []
      pi(U11) = []
      pi(U12) = []
      pi(U13) = []
      pi(U14) = []
      pi(U15) = []
      pi(U16) = []
      pi(U21) = []
      pi(U22) = []
      pi(U23) = []
      pi(U31) = []
      pi(U32) = []
      pi(U41) = []
      pi(U51) = []
      pi(U52) = []
      pi(U61) = []
      pi(U62) = []
      pi(U63) = []
      pi(U64) = []
      pi(a__U64#) = []
      pi(mark) = []
      pi(tt) = []
      pi(plus) = []
      pi(a__U11) = []
      pi(a__U12) = []
      pi(a__isNatKind) = []
      pi(a__U13) = []
      pi(a__U14) = []
      pi(a__U15) = []
      pi(a__isNat) = []
      pi(a__U16) = []
      pi(a__U21) = []
      pi(a__U22) = []
      pi(a__U23) = []
      pi(a__U31) = []
      pi(a__U32) = []
      pi(a__U41) = []
      pi(a__U51) = []
      pi(a__U52) = []
      pi(a__U61) = []
      pi(a__U62) = []
      pi(a__U63) = []
      pi(a__U64) = []
      pi(a__plus) = []
      pi(|0|) = []
      pi(isNatKind) = []
      pi(isNat) = []

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: mark#(s(X)) -> mark#(X)
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)) -> mark#(X1)
p15: mark#(U52(X1,X2)) -> mark#(X1)
p16: mark#(U61(X1,X2,X3)) -> mark#(X1)
p17: mark#(U62(X1,X2,X3)) -> mark#(X1)
p18: mark#(U63(X1,X2,X3)) -> mark#(X1)
p19: mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
p20: mark#(U64(X1,X2,X3)) -> mark#(X1)
p21: mark#(plus(X1,X2)) -> mark#(X1)
p22: mark#(plus(X1,X2)) -> mark#(X2)

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, p13, p14, p15, p16, p17, p18, p20, p21, p22}


-- Reduction pair.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(plus(X1,X2)) -> mark#(X2)
p3: mark#(plus(X1,X2)) -> mark#(X1)
p4: mark#(U64(X1,X2,X3)) -> mark#(X1)
p5: mark#(U63(X1,X2,X3)) -> mark#(X1)
p6: mark#(U62(X1,X2,X3)) -> mark#(X1)
p7: mark#(U61(X1,X2,X3)) -> mark#(X1)
p8: mark#(U52(X1,X2)) -> mark#(X1)
p9: mark#(U51(X1,X2)) -> mark#(X1)
p10: mark#(U41(X)) -> mark#(X)
p11: mark#(U32(X)) -> mark#(X)
p12: mark#(U31(X1,X2)) -> mark#(X1)
p13: mark#(U23(X)) -> mark#(X)
p14: mark#(U22(X1,X2)) -> mark#(X1)
p15: mark#(U21(X1,X2)) -> mark#(X1)
p16: mark#(U16(X)) -> mark#(X)
p17: mark#(U15(X1,X2)) -> mark#(X1)
p18: mark#(U14(X1,X2,X3)) -> mark#(X1)
p19: mark#(U13(X1,X2,X3)) -> mark#(X1)
p20: mark#(U12(X1,X2,X3)) -> mark#(X1)
p21: 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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          s_A(x1) = x1 + (2,1)
          plus_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (2,1)
          U64_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (2,3)
          U63_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,1)
          U62_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,1)
          U61_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,3)
          U52_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (2,3)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,3)
          U41_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          U32_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U31_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (2,1)
          U23_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U22_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
          U21_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,3)
          U16_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U15_A(x1,x2) = x1 + (2,1)
          U14_A(x1,x2,x3) = x1 + (2,1)
          U13_A(x1,x2,x3) = x1 + ((1,0),(1,1)) x3 + (2,1)
          U12_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (2,3)
          U11_A(x1,x2,x3) = x1 + x2 + x3 + (2,3)
  
    precedence:
    
      U22 > U64 = U52 = U21 > U23 > s = plus = U63 = U62 = U61 = U16 > mark# = U51 = U41 = U32 = U31 = U15 = U14 = U13 = U12 = U11
    
    partial status:
  
      pi(mark#) = [1]
      pi(s) = [1]
      pi(plus) = [1, 2]
      pi(U64) = [1]
      pi(U63) = [1, 3]
      pi(U62) = [1, 3]
      pi(U61) = [1, 2]
      pi(U52) = [1]
      pi(U51) = [1, 2]
      pi(U41) = [1]
      pi(U32) = [1]
      pi(U31) = []
      pi(U23) = []
      pi(U22) = [1]
      pi(U21) = [1, 2]
      pi(U16) = [1]
      pi(U15) = []
      pi(U14) = [1]
      pi(U13) = [1, 3]
      pi(U12) = [1, 3]
      pi(U11) = [1, 3]

The next rules are strictly ordered:

  p21

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(plus(X1,X2)) -> mark#(X2)
p3: mark#(plus(X1,X2)) -> mark#(X1)
p4: mark#(U64(X1,X2,X3)) -> mark#(X1)
p5: mark#(U63(X1,X2,X3)) -> mark#(X1)
p6: mark#(U62(X1,X2,X3)) -> mark#(X1)
p7: mark#(U61(X1,X2,X3)) -> mark#(X1)
p8: mark#(U52(X1,X2)) -> mark#(X1)
p9: mark#(U51(X1,X2)) -> mark#(X1)
p10: mark#(U41(X)) -> mark#(X)
p11: mark#(U32(X)) -> mark#(X)
p12: mark#(U31(X1,X2)) -> mark#(X1)
p13: mark#(U23(X)) -> mark#(X)
p14: mark#(U22(X1,X2)) -> mark#(X1)
p15: mark#(U21(X1,X2)) -> mark#(X1)
p16: mark#(U16(X)) -> mark#(X)
p17: mark#(U15(X1,X2)) -> mark#(X1)
p18: mark#(U14(X1,X2,X3)) -> mark#(X1)
p19: mark#(U13(X1,X2,X3)) -> mark#(X1)
p20: 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 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}


-- Reduction pair.

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

p1: mark#(s(X)) -> mark#(X)
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: mark#(U51(X1,X2)) -> mark#(X1)
p14: mark#(U52(X1,X2)) -> mark#(X1)
p15: mark#(U61(X1,X2,X3)) -> mark#(X1)
p16: mark#(U62(X1,X2,X3)) -> mark#(X1)
p17: mark#(U63(X1,X2,X3)) -> mark#(X1)
p18: mark#(U64(X1,X2,X3)) -> mark#(X1)
p19: mark#(plus(X1,X2)) -> mark#(X1)
p20: mark#(plus(X1,X2)) -> mark#(X2)

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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          s_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          U12_A(x1,x2,x3) = x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,3)
          U13_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((0,0),(1,0)) x3 + (2,3)
          U14_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,1)
          U15_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,3)
          U16_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U21_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,3)
          U22_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (2,1)
          U23_A(x1) = x1 + (2,1)
          U31_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,1)
          U32_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U41_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,3)
          U52_A(x1,x2) = x1 + x2 + (2,3)
          U61_A(x1,x2,x3) = x1 + x3 + (2,3)
          U62_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,1)
          U63_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x3 + (2,3)
          U64_A(x1,x2,x3) = x1 + x2 + x3 + (2,1)
          plus_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (2,3)
  
    precedence:
    
      mark# = U51 > U32 = U41 = U52 > s = U12 = U21 = U31 = U61 > U13 = U14 > U15 = U16 = U22 = U62 = U63 = U64 > U23 = plus
    
    partial status:
  
      pi(mark#) = [1]
      pi(s) = [1]
      pi(U12) = [3]
      pi(U13) = [1, 2]
      pi(U14) = [1, 2]
      pi(U15) = [1]
      pi(U16) = [1]
      pi(U21) = []
      pi(U22) = [1]
      pi(U23) = [1]
      pi(U31) = [1]
      pi(U32) = [1]
      pi(U41) = [1]
      pi(U51) = [1, 2]
      pi(U52) = [2]
      pi(U61) = [1, 3]
      pi(U62) = [1]
      pi(U63) = [1, 3]
      pi(U64) = []
      pi(plus) = [1]

The next rules are strictly ordered:

  p10

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(s(X)) -> mark#(X)
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#(U32(X)) -> mark#(X)
p11: mark#(U41(X)) -> mark#(X)
p12: mark#(U51(X1,X2)) -> mark#(X1)
p13: mark#(U52(X1,X2)) -> mark#(X1)
p14: mark#(U61(X1,X2,X3)) -> mark#(X1)
p15: mark#(U62(X1,X2,X3)) -> mark#(X1)
p16: mark#(U63(X1,X2,X3)) -> mark#(X1)
p17: mark#(U64(X1,X2,X3)) -> mark#(X1)
p18: mark#(plus(X1,X2)) -> mark#(X1)
p19: mark#(plus(X1,X2)) -> mark#(X2)

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, p13, p14, p15, p16, p17, p18, p19}


-- Reduction pair.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(plus(X1,X2)) -> mark#(X2)
p3: mark#(plus(X1,X2)) -> mark#(X1)
p4: mark#(U64(X1,X2,X3)) -> mark#(X1)
p5: mark#(U63(X1,X2,X3)) -> mark#(X1)
p6: mark#(U62(X1,X2,X3)) -> mark#(X1)
p7: mark#(U61(X1,X2,X3)) -> mark#(X1)
p8: mark#(U52(X1,X2)) -> mark#(X1)
p9: mark#(U51(X1,X2)) -> mark#(X1)
p10: mark#(U41(X)) -> mark#(X)
p11: mark#(U32(X)) -> mark#(X)
p12: mark#(U23(X)) -> mark#(X)
p13: mark#(U22(X1,X2)) -> mark#(X1)
p14: mark#(U21(X1,X2)) -> mark#(X1)
p15: mark#(U16(X)) -> mark#(X)
p16: mark#(U15(X1,X2)) -> mark#(X1)
p17: mark#(U14(X1,X2,X3)) -> mark#(X1)
p18: mark#(U13(X1,X2,X3)) -> mark#(X1)
p19: 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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,0)) x1 + (2,2)
          s_A(x1) = ((1,0),(1,1)) x1 + (3,3)
          plus_A(x1,x2) = x1 + x2 + (3,1)
          U64_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + (3,3)
          U63_A(x1,x2,x3) = x1 + x3 + (1,1)
          U62_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + (3,1)
          U61_A(x1,x2,x3) = x1 + x2 + (1,1)
          U52_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (3,3)
          U51_A(x1,x2) = x1 + x2 + (1,1)
          U41_A(x1) = ((1,0),(1,1)) x1 + (3,3)
          U32_A(x1) = ((1,0),(1,0)) x1 + (3,3)
          U23_A(x1) = ((1,0),(1,1)) x1 + (3,1)
          U22_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (3,1)
          U21_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (3,1)
          U16_A(x1) = ((1,0),(0,0)) x1 + (1,1)
          U15_A(x1,x2) = x1 + x2 + (3,1)
          U14_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + (3,3)
          U13_A(x1,x2,x3) = x1 + x3 + (1,1)
          U12_A(x1,x2,x3) = ((1,0),(0,0)) x1 + x3 + (3,1)
  
    precedence:
    
      mark# = U32 > U22 > U16 = U14 > U23 > U21 > U62 = U61 = U52 = U51 = U41 = U15 > s = U64 = U63 = U13 = U12 > plus
    
    partial status:
  
      pi(mark#) = []
      pi(s) = []
      pi(plus) = []
      pi(U64) = [2]
      pi(U63) = [1]
      pi(U62) = []
      pi(U61) = [2]
      pi(U52) = [2]
      pi(U51) = [2]
      pi(U41) = [1]
      pi(U32) = []
      pi(U23) = [1]
      pi(U22) = []
      pi(U21) = []
      pi(U16) = []
      pi(U15) = [2]
      pi(U14) = []
      pi(U13) = [3]
      pi(U12) = [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: mark#(s(X)) -> mark#(X)
p2: mark#(plus(X1,X2)) -> mark#(X2)
p3: mark#(plus(X1,X2)) -> mark#(X1)
p4: mark#(U64(X1,X2,X3)) -> mark#(X1)
p5: mark#(U63(X1,X2,X3)) -> mark#(X1)
p6: mark#(U62(X1,X2,X3)) -> mark#(X1)
p7: mark#(U61(X1,X2,X3)) -> mark#(X1)
p8: mark#(U52(X1,X2)) -> mark#(X1)
p9: mark#(U51(X1,X2)) -> mark#(X1)
p10: mark#(U41(X)) -> mark#(X)
p11: mark#(U23(X)) -> mark#(X)
p12: mark#(U22(X1,X2)) -> mark#(X1)
p13: mark#(U21(X1,X2)) -> mark#(X1)
p14: mark#(U16(X)) -> mark#(X)
p15: mark#(U15(X1,X2)) -> mark#(X1)
p16: mark#(U14(X1,X2,X3)) -> mark#(X1)
p17: mark#(U13(X1,X2,X3)) -> mark#(X1)
p18: 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 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}


-- Reduction pair.

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

p1: mark#(s(X)) -> mark#(X)
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#(U41(X)) -> mark#(X)
p11: mark#(U51(X1,X2)) -> mark#(X1)
p12: mark#(U52(X1,X2)) -> mark#(X1)
p13: mark#(U61(X1,X2,X3)) -> mark#(X1)
p14: mark#(U62(X1,X2,X3)) -> mark#(X1)
p15: mark#(U63(X1,X2,X3)) -> mark#(X1)
p16: mark#(U64(X1,X2,X3)) -> mark#(X1)
p17: mark#(plus(X1,X2)) -> mark#(X1)
p18: mark#(plus(X1,X2)) -> mark#(X2)

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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          s_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          U12_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + x3 + (2,3)
          U13_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x3 + (2,3)
          U14_A(x1,x2,x3) = x1 + x3 + (2,1)
          U15_A(x1,x2) = x1 + (2,1)
          U16_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          U21_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,1)
          U22_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
          U23_A(x1) = x1 + (2,1)
          U41_A(x1) = x1 + (2,1)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
          U52_A(x1,x2) = x1 + x2 + (2,1)
          U61_A(x1,x2,x3) = x1 + x2 + x3 + (2,1)
          U62_A(x1,x2,x3) = x1 + x2 + x3 + (2,1)
          U63_A(x1,x2,x3) = x1 + x2 + x3 + (2,1)
          U64_A(x1,x2,x3) = x1 + x2 + x3 + (2,1)
          plus_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (2,1)
  
    precedence:
    
      mark# = U52 = U61 = U62 = plus > U51 > U41 > s = U22 = U23 > U14 = U15 = U16 = U21 > U12 = U63 = U64 > U13
    
    partial status:
  
      pi(mark#) = [1]
      pi(s) = []
      pi(U12) = [3]
      pi(U13) = []
      pi(U14) = [1, 3]
      pi(U15) = [1]
      pi(U16) = [1]
      pi(U21) = [1, 2]
      pi(U22) = []
      pi(U23) = [1]
      pi(U41) = [1]
      pi(U51) = [1, 2]
      pi(U52) = [1]
      pi(U61) = [1, 3]
      pi(U62) = [1]
      pi(U63) = [1, 3]
      pi(U64) = [1, 3]
      pi(plus) = [2]

The next rules are strictly ordered:

  p18

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(s(X)) -> mark#(X)
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#(U41(X)) -> mark#(X)
p11: mark#(U51(X1,X2)) -> mark#(X1)
p12: mark#(U52(X1,X2)) -> mark#(X1)
p13: mark#(U61(X1,X2,X3)) -> mark#(X1)
p14: mark#(U62(X1,X2,X3)) -> mark#(X1)
p15: mark#(U63(X1,X2,X3)) -> mark#(X1)
p16: mark#(U64(X1,X2,X3)) -> mark#(X1)
p17: 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 estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: mark#(s(X)) -> mark#(X)
p2: mark#(plus(X1,X2)) -> mark#(X1)
p3: mark#(U64(X1,X2,X3)) -> mark#(X1)
p4: mark#(U63(X1,X2,X3)) -> mark#(X1)
p5: mark#(U62(X1,X2,X3)) -> mark#(X1)
p6: mark#(U61(X1,X2,X3)) -> mark#(X1)
p7: mark#(U52(X1,X2)) -> mark#(X1)
p8: mark#(U51(X1,X2)) -> mark#(X1)
p9: mark#(U41(X)) -> mark#(X)
p10: mark#(U23(X)) -> mark#(X)
p11: mark#(U22(X1,X2)) -> mark#(X1)
p12: mark#(U21(X1,X2)) -> mark#(X1)
p13: mark#(U16(X)) -> mark#(X)
p14: mark#(U15(X1,X2)) -> mark#(X1)
p15: mark#(U14(X1,X2,X3)) -> mark#(X1)
p16: mark#(U13(X1,X2,X3)) -> mark#(X1)
p17: 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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,1)) x1 + (1,2)
          s_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
          U64_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + ((0,0),(1,0)) x3 + (2,3)
          U63_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + x3 + (2,3)
          U62_A(x1,x2,x3) = x1 + x2 + x3 + (2,1)
          U61_A(x1,x2,x3) = x1 + x2 + x3 + (2,1)
          U52_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (2,3)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
          U41_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          U23_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U22_A(x1,x2) = x1 + x2 + (2,1)
          U21_A(x1,x2) = x1 + x2 + (2,1)
          U16_A(x1) = x1 + (2,1)
          U15_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (2,1)
          U14_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + x3 + (2,3)
          U13_A(x1,x2,x3) = ((1,0),(1,1)) x1 + (2,3)
          U12_A(x1,x2,x3) = x1 + x2 + x3 + (2,3)
  
    precedence:
    
      mark# = U41 = U22 > U16 > U62 = U23 > U21 = U14 > s = plus = U64 = U61 > U63 = U52 > U51 = U15 = U13 = U12
    
    partial status:
  
      pi(mark#) = [1]
      pi(s) = []
      pi(plus) = []
      pi(U64) = [1]
      pi(U63) = [1, 3]
      pi(U62) = [1, 3]
      pi(U61) = [1, 3]
      pi(U52) = [1]
      pi(U51) = [1, 2]
      pi(U41) = [1]
      pi(U23) = [1]
      pi(U22) = [1, 2]
      pi(U21) = [2]
      pi(U16) = [1]
      pi(U15) = [1]
      pi(U14) = []
      pi(U13) = [1]
      pi(U12) = [1, 3]

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#(plus(X1,X2)) -> mark#(X1)
p2: mark#(U64(X1,X2,X3)) -> mark#(X1)
p3: mark#(U63(X1,X2,X3)) -> mark#(X1)
p4: mark#(U62(X1,X2,X3)) -> mark#(X1)
p5: mark#(U61(X1,X2,X3)) -> mark#(X1)
p6: mark#(U52(X1,X2)) -> mark#(X1)
p7: mark#(U51(X1,X2)) -> mark#(X1)
p8: mark#(U41(X)) -> mark#(X)
p9: mark#(U23(X)) -> mark#(X)
p10: mark#(U22(X1,X2)) -> mark#(X1)
p11: mark#(U21(X1,X2)) -> mark#(X1)
p12: mark#(U16(X)) -> mark#(X)
p13: mark#(U15(X1,X2)) -> mark#(X1)
p14: mark#(U14(X1,X2,X3)) -> mark#(X1)
p15: mark#(U13(X1,X2,X3)) -> mark#(X1)
p16: 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 estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: mark#(plus(X1,X2)) -> 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#(U41(X)) -> mark#(X)
p11: mark#(U51(X1,X2)) -> mark#(X1)
p12: mark#(U52(X1,X2)) -> mark#(X1)
p13: mark#(U61(X1,X2,X3)) -> mark#(X1)
p14: mark#(U62(X1,X2,X3)) -> mark#(X1)
p15: mark#(U63(X1,X2,X3)) -> mark#(X1)
p16: mark#(U64(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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,1)
          U12_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((0,0),(1,0)) x3 + (2,3)
          U13_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + x3 + (2,3)
          U14_A(x1,x2,x3) = x1 + x2 + x3 + (2,1)
          U15_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,1)
          U16_A(x1) = x1 + (2,1)
          U21_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
          U22_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,3)
          U23_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U41_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,1)
          U52_A(x1,x2) = x1 + x2 + (2,1)
          U61_A(x1,x2,x3) = x1 + x3 + (2,1)
          U62_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,3)
          U63_A(x1,x2,x3) = x1 + x2 + x3 + (2,3)
          U64_A(x1,x2,x3) = x1 + x2 + x3 + (2,3)
  
    precedence:
    
      mark# = U41 = U51 = U63 > U23 > U22 = U52 > U61 > plus = U12 = U13 = U14 = U15 = U16 = U21 = U62 = U64
    
    partial status:
  
      pi(mark#) = [1]
      pi(plus) = [1, 2]
      pi(U12) = [1, 2]
      pi(U13) = [1, 3]
      pi(U14) = [1, 3]
      pi(U15) = [1, 2]
      pi(U16) = [1]
      pi(U21) = [1, 2]
      pi(U22) = [1]
      pi(U23) = [1]
      pi(U41) = [1]
      pi(U51) = [1]
      pi(U52) = [1, 2]
      pi(U61) = [1, 3]
      pi(U62) = [1]
      pi(U63) = [1]
      pi(U64) = [3]

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#(U12(X1,X2,X3)) -> mark#(X1)
p2: mark#(U13(X1,X2,X3)) -> mark#(X1)
p3: mark#(U14(X1,X2,X3)) -> mark#(X1)
p4: mark#(U15(X1,X2)) -> mark#(X1)
p5: mark#(U16(X)) -> mark#(X)
p6: mark#(U21(X1,X2)) -> mark#(X1)
p7: mark#(U22(X1,X2)) -> mark#(X1)
p8: mark#(U23(X)) -> mark#(X)
p9: mark#(U41(X)) -> mark#(X)
p10: mark#(U51(X1,X2)) -> mark#(X1)
p11: mark#(U52(X1,X2)) -> mark#(X1)
p12: mark#(U61(X1,X2,X3)) -> mark#(X1)
p13: mark#(U62(X1,X2,X3)) -> mark#(X1)
p14: mark#(U63(X1,X2,X3)) -> mark#(X1)
p15: mark#(U64(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:

  {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: mark#(U12(X1,X2,X3)) -> mark#(X1)
p2: mark#(U64(X1,X2,X3)) -> mark#(X1)
p3: mark#(U63(X1,X2,X3)) -> mark#(X1)
p4: mark#(U62(X1,X2,X3)) -> mark#(X1)
p5: mark#(U61(X1,X2,X3)) -> mark#(X1)
p6: mark#(U52(X1,X2)) -> mark#(X1)
p7: mark#(U51(X1,X2)) -> mark#(X1)
p8: mark#(U41(X)) -> mark#(X)
p9: mark#(U23(X)) -> mark#(X)
p10: mark#(U22(X1,X2)) -> mark#(X1)
p11: mark#(U21(X1,X2)) -> mark#(X1)
p12: mark#(U16(X)) -> mark#(X)
p13: mark#(U15(X1,X2)) -> mark#(X1)
p14: mark#(U14(X1,X2,X3)) -> mark#(X1)
p15: mark#(U13(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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,0)) x1 + (2,2)
          U12_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + x3 + (3,3)
          U64_A(x1,x2,x3) = x1 + x3 + (1,1)
          U63_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + x3 + (3,3)
          U62_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + x3 + (3,1)
          U61_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + x3 + (3,3)
          U52_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (3,3)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (3,1)
          U41_A(x1) = x1 + (3,1)
          U23_A(x1) = x1 + (3,1)
          U22_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (3,1)
          U21_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (3,1)
          U16_A(x1) = x1 + (3,1)
          U15_A(x1,x2) = x1 + x2 + (1,1)
          U14_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + x3 + (3,3)
          U13_A(x1,x2,x3) = ((1,0),(0,0)) x1 + x3 + (1,3)
  
    precedence:
    
      mark# > U16 > U22 > U21 > U23 > U12 = U62 = U61 = U52 = U41 > U15 > U64 = U63 = U51 = U14 = U13
    
    partial status:
  
      pi(mark#) = []
      pi(U12) = [1]
      pi(U64) = []
      pi(U63) = [3]
      pi(U62) = [3]
      pi(U61) = [3]
      pi(U52) = [1]
      pi(U51) = []
      pi(U41) = [1]
      pi(U23) = [1]
      pi(U22) = []
      pi(U21) = [2]
      pi(U16) = []
      pi(U15) = []
      pi(U14) = [3]
      pi(U13) = [3]

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: mark#(U12(X1,X2,X3)) -> mark#(X1)
p2: mark#(U64(X1,X2,X3)) -> mark#(X1)
p3: mark#(U63(X1,X2,X3)) -> mark#(X1)
p4: mark#(U62(X1,X2,X3)) -> mark#(X1)
p5: mark#(U61(X1,X2,X3)) -> mark#(X1)
p6: mark#(U52(X1,X2)) -> mark#(X1)
p7: mark#(U51(X1,X2)) -> mark#(X1)
p8: mark#(U41(X)) -> mark#(X)
p9: mark#(U23(X)) -> mark#(X)
p10: mark#(U22(X1,X2)) -> mark#(X1)
p11: mark#(U21(X1,X2)) -> mark#(X1)
p12: mark#(U16(X)) -> mark#(X)
p13: mark#(U15(X1,X2)) -> mark#(X1)
p14: mark#(U13(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:

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


-- Reduction pair.

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

p1: mark#(U12(X1,X2,X3)) -> mark#(X1)
p2: mark#(U13(X1,X2,X3)) -> mark#(X1)
p3: mark#(U15(X1,X2)) -> mark#(X1)
p4: mark#(U16(X)) -> mark#(X)
p5: mark#(U21(X1,X2)) -> mark#(X1)
p6: mark#(U22(X1,X2)) -> mark#(X1)
p7: mark#(U23(X)) -> mark#(X)
p8: mark#(U41(X)) -> mark#(X)
p9: mark#(U51(X1,X2)) -> mark#(X1)
p10: mark#(U52(X1,X2)) -> mark#(X1)
p11: mark#(U61(X1,X2,X3)) -> mark#(X1)
p12: mark#(U62(X1,X2,X3)) -> mark#(X1)
p13: mark#(U63(X1,X2,X3)) -> mark#(X1)
p14: mark#(U64(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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          U12_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,3)
          U13_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (2,1)
          U15_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,3)
          U16_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          U21_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,3)
          U22_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,1)
          U23_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U41_A(x1) = x1 + (2,1)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
          U52_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,1)
          U61_A(x1,x2,x3) = x1 + x2 + x3 + (2,1)
          U62_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,1)
          U63_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + (2,1)
          U64_A(x1,x2,x3) = x1 + x3 + (2,3)
  
    precedence:
    
      mark# = U51 = U52 = U62 > U12 = U13 = U15 = U16 > U21 = U22 = U41 > U23 = U61 > U63 = U64
    
    partial status:
  
      pi(mark#) = [1]
      pi(U12) = [1, 3]
      pi(U13) = [1, 3]
      pi(U15) = [1, 2]
      pi(U16) = [1]
      pi(U21) = [1, 2]
      pi(U22) = [1, 2]
      pi(U23) = []
      pi(U41) = [1]
      pi(U51) = [1, 2]
      pi(U52) = [1, 2]
      pi(U61) = [1, 2]
      pi(U62) = [1, 3]
      pi(U63) = [2]
      pi(U64) = [1, 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: mark#(U12(X1,X2,X3)) -> mark#(X1)
p2: mark#(U13(X1,X2,X3)) -> mark#(X1)
p3: mark#(U15(X1,X2)) -> mark#(X1)
p4: mark#(U16(X)) -> mark#(X)
p5: mark#(U22(X1,X2)) -> mark#(X1)
p6: mark#(U23(X)) -> mark#(X)
p7: mark#(U41(X)) -> mark#(X)
p8: mark#(U51(X1,X2)) -> mark#(X1)
p9: mark#(U52(X1,X2)) -> mark#(X1)
p10: mark#(U61(X1,X2,X3)) -> mark#(X1)
p11: mark#(U62(X1,X2,X3)) -> mark#(X1)
p12: mark#(U63(X1,X2,X3)) -> mark#(X1)
p13: mark#(U64(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:

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


-- Reduction pair.

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

p1: mark#(U12(X1,X2,X3)) -> mark#(X1)
p2: mark#(U64(X1,X2,X3)) -> mark#(X1)
p3: mark#(U63(X1,X2,X3)) -> mark#(X1)
p4: mark#(U62(X1,X2,X3)) -> mark#(X1)
p5: mark#(U61(X1,X2,X3)) -> mark#(X1)
p6: mark#(U52(X1,X2)) -> mark#(X1)
p7: mark#(U51(X1,X2)) -> mark#(X1)
p8: mark#(U41(X)) -> mark#(X)
p9: mark#(U23(X)) -> mark#(X)
p10: mark#(U22(X1,X2)) -> mark#(X1)
p11: mark#(U16(X)) -> mark#(X)
p12: mark#(U15(X1,X2)) -> mark#(X1)
p13: mark#(U13(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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          U12_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + x3 + (2,3)
          U64_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((0,0),(1,0)) x3 + (2,3)
          U63_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((0,0),(1,0)) x3 + (2,1)
          U62_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,1)
          U61_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((0,0),(1,0)) x3 + (2,3)
          U52_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,1)
          U51_A(x1,x2) = x1 + x2 + (2,1)
          U41_A(x1) = x1 + (2,1)
          U23_A(x1) = x1 + (2,1)
          U22_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (2,3)
          U16_A(x1) = x1 + (2,1)
          U15_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,3)
          U13_A(x1,x2,x3) = ((1,0),(1,0)) x1 + x2 + x3 + (2,1)
  
    precedence:
    
      U23 > U62 = U15 > U64 > U41 > U52 > U16 > U61 = U22 > U12 = U63 = U13 > mark# = U51
    
    partial status:
  
      pi(mark#) = [1]
      pi(U12) = [1, 3]
      pi(U64) = [1, 2]
      pi(U63) = [1, 2]
      pi(U62) = [1, 3]
      pi(U61) = [1, 2]
      pi(U52) = [1, 2]
      pi(U51) = [1, 2]
      pi(U41) = [1]
      pi(U23) = []
      pi(U22) = [1]
      pi(U16) = [1]
      pi(U15) = [1]
      pi(U13) = [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: mark#(U12(X1,X2,X3)) -> mark#(X1)
p2: mark#(U64(X1,X2,X3)) -> mark#(X1)
p3: mark#(U63(X1,X2,X3)) -> mark#(X1)
p4: mark#(U62(X1,X2,X3)) -> mark#(X1)
p5: mark#(U61(X1,X2,X3)) -> mark#(X1)
p6: mark#(U52(X1,X2)) -> mark#(X1)
p7: mark#(U51(X1,X2)) -> mark#(X1)
p8: mark#(U41(X)) -> mark#(X)
p9: mark#(U23(X)) -> mark#(X)
p10: mark#(U22(X1,X2)) -> mark#(X1)
p11: mark#(U15(X1,X2)) -> mark#(X1)
p12: mark#(U13(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:

  {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#(U12(X1,X2,X3)) -> mark#(X1)
p2: mark#(U13(X1,X2,X3)) -> mark#(X1)
p3: mark#(U15(X1,X2)) -> mark#(X1)
p4: mark#(U22(X1,X2)) -> mark#(X1)
p5: mark#(U23(X)) -> mark#(X)
p6: mark#(U41(X)) -> mark#(X)
p7: mark#(U51(X1,X2)) -> mark#(X1)
p8: mark#(U52(X1,X2)) -> mark#(X1)
p9: mark#(U61(X1,X2,X3)) -> mark#(X1)
p10: mark#(U62(X1,X2,X3)) -> mark#(X1)
p11: mark#(U63(X1,X2,X3)) -> mark#(X1)
p12: mark#(U64(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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          U12_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (2,3)
          U13_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (2,1)
          U15_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,1)
          U22_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,1)
          U23_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U41_A(x1) = x1 + (2,3)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
          U52_A(x1,x2) = x1 + x2 + (2,1)
          U61_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + (2,1)
          U62_A(x1,x2,x3) = x1 + ((0,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (2,3)
          U63_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + ((1,0),(1,1)) x3 + (2,1)
          U64_A(x1,x2,x3) = ((1,0),(1,0)) x1 + x3 + (2,3)
  
    precedence:
    
      U61 > U12 = U15 = U52 > mark# = U22 = U23 = U41 = U62 = U63 > U13 = U51 = U64
    
    partial status:
  
      pi(mark#) = [1]
      pi(U12) = [1, 3]
      pi(U13) = [1, 3]
      pi(U15) = [1, 2]
      pi(U22) = [1, 2]
      pi(U23) = []
      pi(U41) = [1]
      pi(U51) = [1, 2]
      pi(U52) = [1, 2]
      pi(U61) = [1, 2]
      pi(U62) = [1, 3]
      pi(U63) = [1, 3]
      pi(U64) = [3]

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: mark#(U12(X1,X2,X3)) -> mark#(X1)
p2: mark#(U13(X1,X2,X3)) -> mark#(X1)
p3: mark#(U15(X1,X2)) -> mark#(X1)
p4: mark#(U22(X1,X2)) -> mark#(X1)
p5: mark#(U23(X)) -> mark#(X)
p6: mark#(U41(X)) -> mark#(X)
p7: mark#(U51(X1,X2)) -> mark#(X1)
p8: mark#(U52(X1,X2)) -> mark#(X1)
p9: mark#(U61(X1,X2,X3)) -> mark#(X1)
p10: mark#(U62(X1,X2,X3)) -> mark#(X1)
p11: mark#(U63(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:

  {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: mark#(U12(X1,X2,X3)) -> mark#(X1)
p2: mark#(U63(X1,X2,X3)) -> mark#(X1)
p3: mark#(U62(X1,X2,X3)) -> mark#(X1)
p4: mark#(U61(X1,X2,X3)) -> mark#(X1)
p5: mark#(U52(X1,X2)) -> mark#(X1)
p6: mark#(U51(X1,X2)) -> mark#(X1)
p7: mark#(U41(X)) -> mark#(X)
p8: mark#(U23(X)) -> mark#(X)
p9: mark#(U22(X1,X2)) -> mark#(X1)
p10: mark#(U15(X1,X2)) -> mark#(X1)
p11: mark#(U13(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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          U12_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + x3 + (2,3)
          U63_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,3)
          U62_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,1)
          U61_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,1)
          U52_A(x1,x2) = x1 + x2 + (2,1)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,1)
          U41_A(x1) = x1 + (2,1)
          U23_A(x1) = x1 + (2,1)
          U22_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (2,3)
          U15_A(x1,x2) = x1 + x2 + (2,3)
          U13_A(x1,x2,x3) = ((1,0),(1,0)) x1 + x2 + x3 + (2,3)
  
    precedence:
    
      U52 > mark# = U41 > U23 > U51 = U22 > U12 = U62 = U15 > U61 = U13 > U63
    
    partial status:
  
      pi(mark#) = [1]
      pi(U12) = [1, 3]
      pi(U63) = []
      pi(U62) = [1, 3]
      pi(U61) = [1, 3]
      pi(U52) = [1]
      pi(U51) = [1, 2]
      pi(U41) = [1]
      pi(U23) = [1]
      pi(U22) = [1, 2]
      pi(U15) = [2]
      pi(U13) = [3]

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(U12(X1,X2,X3)) -> mark#(X1)
p2: mark#(U63(X1,X2,X3)) -> mark#(X1)
p3: mark#(U62(X1,X2,X3)) -> mark#(X1)
p4: mark#(U52(X1,X2)) -> mark#(X1)
p5: mark#(U51(X1,X2)) -> mark#(X1)
p6: mark#(U41(X)) -> mark#(X)
p7: mark#(U23(X)) -> mark#(X)
p8: mark#(U22(X1,X2)) -> mark#(X1)
p9: mark#(U15(X1,X2)) -> mark#(X1)
p10: mark#(U13(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:

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


-- Reduction pair.

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

p1: mark#(U12(X1,X2,X3)) -> mark#(X1)
p2: mark#(U13(X1,X2,X3)) -> mark#(X1)
p3: mark#(U15(X1,X2)) -> mark#(X1)
p4: mark#(U22(X1,X2)) -> mark#(X1)
p5: mark#(U23(X)) -> mark#(X)
p6: mark#(U41(X)) -> mark#(X)
p7: mark#(U51(X1,X2)) -> mark#(X1)
p8: mark#(U52(X1,X2)) -> mark#(X1)
p9: mark#(U62(X1,X2,X3)) -> mark#(X1)
p10: mark#(U63(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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,0)) x1 + (1,2)
          U12_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + x3 + (2,3)
          U13_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + x3 + (2,3)
          U15_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
          U22_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,1)
          U23_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          U41_A(x1) = x1 + (2,1)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
          U52_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
          U62_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + x3 + (2,3)
          U63_A(x1,x2,x3) = ((1,0),(0,0)) x1 + x2 + x3 + (2,1)
  
    precedence:
    
      mark# > U62 > U13 > U63 > U12 = U52 > U15 = U22 = U23 = U41 = U51
    
    partial status:
  
      pi(mark#) = []
      pi(U12) = [3]
      pi(U13) = []
      pi(U15) = [2]
      pi(U22) = [2]
      pi(U23) = [1]
      pi(U41) = [1]
      pi(U51) = [1, 2]
      pi(U52) = [1]
      pi(U62) = [3]
      pi(U63) = [3]

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(X1,X2,X3)) -> mark#(X1)
p2: mark#(U15(X1,X2)) -> mark#(X1)
p3: mark#(U22(X1,X2)) -> mark#(X1)
p4: mark#(U23(X)) -> mark#(X)
p5: mark#(U41(X)) -> mark#(X)
p6: mark#(U51(X1,X2)) -> mark#(X1)
p7: mark#(U52(X1,X2)) -> mark#(X1)
p8: mark#(U62(X1,X2,X3)) -> mark#(X1)
p9: mark#(U63(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:

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


-- Reduction pair.

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

p1: mark#(U13(X1,X2,X3)) -> mark#(X1)
p2: mark#(U63(X1,X2,X3)) -> mark#(X1)
p3: mark#(U62(X1,X2,X3)) -> mark#(X1)
p4: mark#(U52(X1,X2)) -> mark#(X1)
p5: mark#(U51(X1,X2)) -> mark#(X1)
p6: mark#(U41(X)) -> mark#(X)
p7: mark#(U23(X)) -> mark#(X)
p8: mark#(U22(X1,X2)) -> mark#(X1)
p9: mark#(U15(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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          U13_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,1)
          U63_A(x1,x2,x3) = x1 + x3 + (2,1)
          U62_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x2 + ((1,0),(1,1)) x3 + (2,1)
          U52_A(x1,x2) = x1 + x2 + (2,1)
          U51_A(x1,x2) = x1 + x2 + (2,1)
          U41_A(x1) = x1 + (2,1)
          U23_A(x1) = ((1,0),(1,1)) x1 + (2,3)
          U22_A(x1,x2) = x1 + x2 + (2,3)
          U15_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (2,3)
  
    precedence:
    
      U22 > mark# = U41 = U23 > U63 > U52 = U15 > U13 > U62 > U51
    
    partial status:
  
      pi(mark#) = [1]
      pi(U13) = [1, 3]
      pi(U63) = []
      pi(U62) = []
      pi(U52) = [1, 2]
      pi(U51) = [1]
      pi(U41) = [1]
      pi(U23) = [1]
      pi(U22) = [2]
      pi(U15) = [1, 2]

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#(U63(X1,X2,X3)) -> mark#(X1)
p2: mark#(U62(X1,X2,X3)) -> mark#(X1)
p3: mark#(U52(X1,X2)) -> mark#(X1)
p4: mark#(U51(X1,X2)) -> mark#(X1)
p5: mark#(U41(X)) -> mark#(X)
p6: mark#(U23(X)) -> mark#(X)
p7: mark#(U22(X1,X2)) -> mark#(X1)
p8: mark#(U15(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 estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: mark#(U63(X1,X2,X3)) -> mark#(X1)
p2: mark#(U15(X1,X2)) -> mark#(X1)
p3: mark#(U22(X1,X2)) -> mark#(X1)
p4: mark#(U23(X)) -> mark#(X)
p5: mark#(U41(X)) -> mark#(X)
p6: mark#(U51(X1,X2)) -> mark#(X1)
p7: mark#(U52(X1,X2)) -> mark#(X1)
p8: mark#(U62(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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          U63_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((0,0),(1,0)) x3 + (2,1)
          U15_A(x1,x2) = x1 + (2,3)
          U22_A(x1,x2) = x1 + x2 + (2,1)
          U23_A(x1) = x1 + (2,3)
          U41_A(x1) = x1 + (2,3)
          U51_A(x1,x2) = x1 + x2 + (2,3)
          U52_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (2,3)
          U62_A(x1,x2,x3) = ((1,0),(1,1)) x1 + x3 + (2,3)
  
    precedence:
    
      U51 = U52 > U23 = U62 > U63 > U15 > U22 > U41 > mark#
    
    partial status:
  
      pi(mark#) = [1]
      pi(U63) = []
      pi(U15) = []
      pi(U22) = []
      pi(U23) = [1]
      pi(U41) = []
      pi(U51) = []
      pi(U52) = []
      pi(U62) = [1, 3]

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#(U15(X1,X2)) -> mark#(X1)
p2: mark#(U22(X1,X2)) -> mark#(X1)
p3: mark#(U23(X)) -> mark#(X)
p4: mark#(U41(X)) -> mark#(X)
p5: mark#(U51(X1,X2)) -> mark#(X1)
p6: mark#(U52(X1,X2)) -> mark#(X1)
p7: mark#(U62(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:

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


-- Reduction pair.

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

p1: mark#(U15(X1,X2)) -> mark#(X1)
p2: mark#(U62(X1,X2,X3)) -> mark#(X1)
p3: mark#(U52(X1,X2)) -> mark#(X1)
p4: mark#(U51(X1,X2)) -> mark#(X1)
p5: mark#(U41(X)) -> mark#(X)
p6: mark#(U23(X)) -> mark#(X)
p7: mark#(U22(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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          U15_A(x1,x2) = x1 + x2 + (2,1)
          U62_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((1,0),(1,1)) x3 + (2,1)
          U52_A(x1,x2) = ((1,0),(1,1)) x1 + ((0,0),(1,0)) x2 + (2,3)
          U51_A(x1,x2) = x1 + x2 + (2,3)
          U41_A(x1) = ((1,0),(0,0)) x1 + (2,1)
          U23_A(x1) = ((1,0),(0,0)) x1 + (2,1)
          U22_A(x1,x2) = x1 + x2 + (2,3)
  
    precedence:
    
      U62 = U52 = U51 = U41 > mark# = U23 = U22 > U15
    
    partial status:
  
      pi(mark#) = [1]
      pi(U15) = []
      pi(U62) = []
      pi(U52) = []
      pi(U51) = []
      pi(U41) = []
      pi(U23) = []
      pi(U22) = [2]

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#(U62(X1,X2,X3)) -> mark#(X1)
p2: mark#(U52(X1,X2)) -> mark#(X1)
p3: mark#(U51(X1,X2)) -> mark#(X1)
p4: mark#(U41(X)) -> mark#(X)
p5: mark#(U23(X)) -> mark#(X)
p6: mark#(U22(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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

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

p1: mark#(U62(X1,X2,X3)) -> mark#(X1)
p2: mark#(U22(X1,X2)) -> mark#(X1)
p3: mark#(U23(X)) -> mark#(X)
p4: mark#(U41(X)) -> mark#(X)
p5: mark#(U51(X1,X2)) -> mark#(X1)
p6: mark#(U52(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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (2,2)
          U62_A(x1,x2,x3) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + ((0,0),(1,0)) x3 + (3,1)
          U22_A(x1,x2) = x1 + x2 + (1,1)
          U23_A(x1) = ((1,0),(0,0)) x1 + (3,3)
          U41_A(x1) = ((1,0),(0,0)) x1 + (3,1)
          U51_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (3,1)
          U52_A(x1,x2) = x1 + x2 + (3,3)
  
    precedence:
    
      U62 > U22 = U23 = U51 = U52 > U41 > mark#
    
    partial status:
  
      pi(mark#) = [1]
      pi(U62) = []
      pi(U22) = [1, 2]
      pi(U23) = []
      pi(U41) = []
      pi(U51) = [2]
      pi(U52) = [1, 2]

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(X1,X2)) -> mark#(X1)
p2: mark#(U23(X)) -> mark#(X)
p3: mark#(U41(X)) -> mark#(X)
p4: mark#(U51(X1,X2)) -> mark#(X1)
p5: mark#(U52(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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: mark#(U22(X1,X2)) -> mark#(X1)
p2: mark#(U52(X1,X2)) -> mark#(X1)
p3: mark#(U51(X1,X2)) -> mark#(X1)
p4: mark#(U41(X)) -> mark#(X)
p5: mark#(U23(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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(0,0)) x1
          U22_A(x1,x2) = ((1,0),(0,0)) x1 + (1,1)
          U52_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,1)
          U51_A(x1,x2) = x1 + x2 + (1,1)
          U41_A(x1) = ((1,0),(0,0)) x1 + (1,1)
          U23_A(x1) = ((1,0),(0,0)) x1 + (1,1)
  
    precedence:
    
      U23 > U52 = U41 > U22 > mark# = U51
    
    partial status:
  
      pi(mark#) = []
      pi(U22) = []
      pi(U52) = [2]
      pi(U51) = [1, 2]
      pi(U41) = []
      pi(U23) = []

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#(U52(X1,X2)) -> mark#(X1)
p2: mark#(U51(X1,X2)) -> mark#(X1)
p3: mark#(U41(X)) -> mark#(X)
p4: mark#(U23(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:

  {p1, p2, p3, p4}


-- Reduction pair.

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

p1: mark#(U52(X1,X2)) -> mark#(X1)
p2: mark#(U23(X)) -> mark#(X)
p3: mark#(U41(X)) -> mark#(X)
p4: mark#(U51(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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1
          U52_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (1,1)
          U23_A(x1) = ((1,0),(1,1)) x1 + (1,1)
          U41_A(x1) = x1 + (1,1)
          U51_A(x1,x2) = x1 + x2 + (1,1)
  
    precedence:
    
      U52 > U23 > U41 > mark# = U51
    
    partial status:
  
      pi(mark#) = [1]
      pi(U52) = [1, 2]
      pi(U23) = []
      pi(U41) = [1]
      pi(U51) = [1, 2]

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#(U23(X)) -> mark#(X)
p2: mark#(U41(X)) -> mark#(X)
p3: mark#(U51(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 estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: mark#(U23(X)) -> mark#(X)
p2: mark#(U51(X1,X2)) -> mark#(X1)
p3: mark#(U41(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 set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = x1 + (1,2)
          U23_A(x1) = x1 + (2,3)
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (2,1)
          U41_A(x1) = x1 + (2,3)
  
    precedence:
    
      mark# = U51 > U23 > U41
    
    partial status:
  
      pi(mark#) = [1]
      pi(U23) = [1]
      pi(U51) = [1, 2]
      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: mark#(U23(X)) -> mark#(X)
p2: mark#(U51(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 estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: mark#(U23(X)) -> mark#(X)
p2: mark#(U51(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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,0)) x1 + (1,2)
          U23_A(x1) = ((1,0),(0,0)) x1 + (2,1)
          U51_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (2,1)
  
    precedence:
    
      mark# = U23 = U51
    
    partial status:
  
      pi(mark#) = []
      pi(U23) = []
      pi(U51) = [2]

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#(U51(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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: mark#(U51(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

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          mark#_A(x1) = ((1,0),(1,1)) x1
          U51_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (1,1)
  
    precedence:
    
      mark# = U51
    
    partial status:
  
      pi(mark#) = [1]
      pi(U51) = [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__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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U11#_A(x1,x2,x3) = ((0,0),(1,0)) x1 + x2 + ((1,0),(1,0)) x3 + (14,3)
          tt_A() = (11,0)
          a__U12#_A(x1,x2,x3) = x2 + ((1,0),(0,0)) x3 + (14,13)
          a__isNatKind_A(x1) = ((1,0),(1,0)) x1 + (7,15)
          a__U13#_A(x1,x2,x3) = x2 + ((1,0),(0,0)) x3 + (13,12)
          a__U14#_A(x1,x2,x3) = x2 + ((1,0),(0,0)) x3 + (12,11)
          a__isNat#_A(x1) = x1 + (9,9)
          s_A(x1) = ((1,0),(1,0)) x1 + (15,0)
          a__U21#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (13,1)
          a__U22#_A(x1,x2) = x2 + (12,10)
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + ((1,0),(1,1)) x2 + (6,2)
          a__U15#_A(x1,x2) = ((1,0),(0,0)) x2 + (10,10)
          a__isNat_A(x1) = ((0,0),(1,0)) x1 + (17,12)
          a__U16_A(x1) = (12,1)
          U16_A(x1) = (1,2)
          a__U15_A(x1,x2) = (13,2)
          U15_A(x1,x2) = (0,0)
          a__U14_A(x1,x2,x3) = (14,3)
          U14_A(x1,x2,x3) = (0,0)
          a__U13_A(x1,x2,x3) = (15,16)
          a__U23_A(x1) = (12,1)
          U13_A(x1,x2,x3) = ((0,0),(1,0)) x3 + (1,17)
          U23_A(x1) = (0,0)
          a__U12_A(x1,x2,x3) = (16,17)
          a__U22_A(x1,x2) = (13,2)
          a__U32_A(x1) = (12,1)
          U12_A(x1,x2,x3) = (0,0)
          U22_A(x1,x2) = (0,0)
          U32_A(x1) = (1,2)
          a__U11_A(x1,x2,x3) = ((0,0),(1,0)) x1 + (16,7)
          a__U21_A(x1,x2) = ((0,0),(1,0)) x1 + (14,5)
          a__U31_A(x1,x2) = ((0,0),(1,0)) x1 + (12,2)
          a__U41_A(x1) = ((0,0),(1,0)) x1 + (16,1)
          U11_A(x1,x2,x3) = (0,0)
          U21_A(x1,x2) = (0,0)
          U31_A(x1,x2) = (11,1)
          U41_A(x1) = (0,0)
          |0|_A() = (5,0)
          isNatKind_A(x1) = (1,16)
          isNat_A(x1) = (1,1)
  
    precedence:
    
      U32 > U14 = U12 > U23 > a__isNatKind = a__U32 = a__U31 > |0| > U13 = isNatKind > isNat > a__isNat = a__U12 = a__U11 > a__U14# = a__isNat# = a__U21# > a__U11# = a__U12# = a__U13# = plus > s = U11 > a__U15# = U16 = a__U23 > a__U13 > a__U15 = a__U14 > tt = a__U16 = a__U22 = U22 = a__U21 > a__U22# = U15 = a__U41 = U21 = U31 > U41
    
    partial status:
  
      pi(a__U11#) = []
      pi(tt) = []
      pi(a__U12#) = []
      pi(a__isNatKind) = []
      pi(a__U13#) = []
      pi(a__U14#) = []
      pi(a__isNat#) = []
      pi(s) = []
      pi(a__U21#) = []
      pi(a__U22#) = []
      pi(plus) = [2]
      pi(a__U15#) = []
      pi(a__isNat) = []
      pi(a__U16) = []
      pi(U16) = []
      pi(a__U15) = []
      pi(U15) = []
      pi(a__U14) = []
      pi(U14) = []
      pi(a__U13) = []
      pi(a__U23) = []
      pi(U13) = []
      pi(U23) = []
      pi(a__U12) = []
      pi(a__U22) = []
      pi(a__U32) = []
      pi(U12) = []
      pi(U22) = []
      pi(U32) = []
      pi(a__U11) = []
      pi(a__U21) = []
      pi(a__U31) = []
      pi(a__U41) = []
      pi(U11) = []
      pi(U21) = []
      pi(U31) = []
      pi(U41) = []
      pi(|0|) = []
      pi(isNatKind) = []
      pi(isNat) = []

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__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__U14#(tt(),V1,V2) -> a__isNat#(V1)
p4: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p5: a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),V1)
p6: a__U22#(tt(),V1) -> a__isNat#(V1)
p7: a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2)
p8: a__U14#(tt(),V1,V2) -> a__U15#(a__isNat(V1),V2)
p9: 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:

  {p4, p5, p6}


-- Reduction pair.

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

p1: a__U22#(tt(),V1) -> a__isNat#(V1)
p2: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p3: a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__U22#_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(1,1)) x2 + (3,4)
          tt_A() = (1,2)
          a__isNat#_A(x1) = ((1,0),(0,0)) x1 + (2,4)
          s_A(x1) = ((1,0),(1,1)) x1 + (6,10)
          a__U21#_A(x1,x2) = x2 + (4,0)
          a__isNatKind_A(x1) = ((1,0),(1,1)) x1 + (5,3)
          a__U32_A(x1) = (2,1)
          U32_A(x1) = (0,0)
          a__U31_A(x1,x2) = (3,2)
          a__U41_A(x1) = ((1,0),(1,1)) x1 + (2,1)
          U31_A(x1,x2) = (0,0)
          U41_A(x1) = (0,0)
          |0|_A() = (2,3)
          plus_A(x1,x2) = (4,4)
          isNatKind_A(x1) = (0,0)
  
    precedence:
    
      tt = a__isNatKind = |0| > a__isNat# = a__U32 = U32 = a__U31 = plus > s > a__U21# > a__U22# = a__U41 = U41 = isNatKind > U31
    
    partial status:
  
      pi(a__U22#) = []
      pi(tt) = []
      pi(a__isNat#) = []
      pi(s) = [1]
      pi(a__U21#) = [2]
      pi(a__isNatKind) = [1]
      pi(a__U32) = []
      pi(U32) = []
      pi(a__U31) = []
      pi(a__U41) = [1]
      pi(U31) = []
      pi(U41) = []
      pi(|0|) = []
      pi(plus) = []
      pi(isNatKind) = []

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: a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
p2: a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),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 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:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__isNatKind#_A(x1) = x1 + (2,2)
          plus_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (7,3)
          a__U31#_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (4,2)
          a__isNatKind_A(x1) = ((1,0),(1,1)) x1 + (4,6)
          tt_A() = (1,1)
          s_A(x1) = ((1,0),(1,1)) x1 + (7,7)
          a__U32_A(x1) = (2,2)
          U32_A(x1) = (0,0)
          a__U31_A(x1,x2) = ((1,0),(1,1)) x1 + (2,1)
          a__U41_A(x1) = ((1,0),(1,1)) x1 + (2,2)
          U31_A(x1,x2) = (0,0)
          U41_A(x1) = (1,1)
          |0|_A() = (2,2)
          isNatKind_A(x1) = (0,0)
  
    precedence:
    
      plus > a__U31# > a__isNatKind# > a__isNatKind > tt = s = U32 = a__U31 = U31 = |0| = isNatKind > U41 > a__U41 > a__U32
    
    partial status:
  
      pi(a__isNatKind#) = []
      pi(plus) = []
      pi(a__U31#) = []
      pi(a__isNatKind) = [1]
      pi(tt) = []
      pi(s) = [1]
      pi(a__U32) = []
      pi(U32) = []
      pi(a__U31) = []
      pi(a__U41) = []
      pi(U31) = []
      pi(U41) = []
      pi(|0|) = []
      pi(isNatKind) = []

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: a__U31#(tt(),V2) -> a__isNatKind#(V2)
p2: a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
p3: 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 estimated dependency graph contains the following SCCs:

  {p2, p3}


-- 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__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 reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__isNatKind#_A(x1) = ((1,0),(0,0)) x1 + (2,2)
          s_A(x1) = ((1,0),(0,0)) x1 + (3,3)
          plus_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (1,1)
  
    precedence:
    
      a__isNatKind# = s > plus
    
    partial status:
  
      pi(a__isNatKind#) = []
      pi(s) = []
      pi(plus) = []

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: 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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: 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

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: lexicographic order
        interpretations:
          a__isNatKind#_A(x1) = ((1,0),(1,1)) x1
          plus_A(x1,x2) = ((1,0),(1,1)) x1 + x2 + (1,1)
  
    precedence:
    
      a__isNatKind# > plus
    
    partial status:
  
      pi(a__isNatKind#) = [1]
      pi(plus) = [1, 2]

The next rules are strictly ordered:

  p1

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