/export/starexec/sandbox/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES
We consider the system theBenchmark.

We are asked to determine termination of the following first-order TRS.

  0 : [] --> o 
  U11 : [o * o * o] --> o 
  U12 : [o * o * o] --> o 
  U13 : [o * o * o] --> o 
  U14 : [o * o * o] --> o 
  U15 : [o * o] --> o 
  U16 : [o] --> o 
  U21 : [o * o] --> o 
  U22 : [o * o] --> o 
  U23 : [o] --> o 
  U31 : [o * o] --> o 
  U32 : [o] --> o 
  U41 : [o] --> o 
  U51 : [o * o] --> o 
  U52 : [o * o] --> o 
  U61 : [o * o * o] --> o 
  U62 : [o * o * o] --> o 
  U63 : [o * o * o] --> o 
  U64 : [o * o * o] --> o 
  a!6220!6220U11 : [o * o * o] --> o 
  a!6220!6220U12 : [o * o * o] --> o 
  a!6220!6220U13 : [o * o * o] --> o 
  a!6220!6220U14 : [o * o * o] --> o 
  a!6220!6220U15 : [o * o] --> o 
  a!6220!6220U16 : [o] --> o 
  a!6220!6220U21 : [o * o] --> o 
  a!6220!6220U22 : [o * o] --> o 
  a!6220!6220U23 : [o] --> o 
  a!6220!6220U31 : [o * o] --> o 
  a!6220!6220U32 : [o] --> o 
  a!6220!6220U41 : [o] --> o 
  a!6220!6220U51 : [o * o] --> o 
  a!6220!6220U52 : [o * o] --> o 
  a!6220!6220U61 : [o * o * o] --> o 
  a!6220!6220U62 : [o * o * o] --> o 
  a!6220!6220U63 : [o * o * o] --> o 
  a!6220!6220U64 : [o * o * o] --> o 
  a!6220!6220isNat : [o] --> o 
  a!6220!6220isNatKind : [o] --> o 
  a!6220!6220plus : [o * o] --> o 
  isNat : [o] --> o 
  isNatKind : [o] --> o 
  mark : [o] --> o 
  plus : [o * o] --> o 
  s : [o] --> o 
  tt : [] --> o 

  a!6220!6220U11(tt, X, Y) => a!6220!6220U12(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220U12(tt, X, Y) => a!6220!6220U13(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U13(tt, X, Y) => a!6220!6220U14(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U14(tt, X, Y) => a!6220!6220U15(a!6220!6220isNat(X), Y) 
  a!6220!6220U15(tt, X) => a!6220!6220U16(a!6220!6220isNat(X)) 
  a!6220!6220U16(tt) => tt 
  a!6220!6220U21(tt, X) => a!6220!6220U22(a!6220!6220isNatKind(X), X) 
  a!6220!6220U22(tt, X) => a!6220!6220U23(a!6220!6220isNat(X)) 
  a!6220!6220U23(tt) => tt 
  a!6220!6220U31(tt, X) => a!6220!6220U32(a!6220!6220isNatKind(X)) 
  a!6220!6220U32(tt) => tt 
  a!6220!6220U41(tt) => tt 
  a!6220!6220U51(tt, X) => a!6220!6220U52(a!6220!6220isNatKind(X), X) 
  a!6220!6220U52(tt, X) => mark(X) 
  a!6220!6220U61(tt, X, Y) => a!6220!6220U62(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220U62(tt, X, Y) => a!6220!6220U63(a!6220!6220isNat(Y), X, Y) 
  a!6220!6220U63(tt, X, Y) => a!6220!6220U64(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U64(tt, X, Y) => s(a!6220!6220plus(mark(Y), mark(X))) 
  a!6220!6220isNat(0) => tt 
  a!6220!6220isNat(plus(X, Y)) => a!6220!6220U11(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220isNat(s(X)) => a!6220!6220U21(a!6220!6220isNatKind(X), X) 
  a!6220!6220isNatKind(0) => tt 
  a!6220!6220isNatKind(plus(X, Y)) => a!6220!6220U31(a!6220!6220isNatKind(X), Y) 
  a!6220!6220isNatKind(s(X)) => a!6220!6220U41(a!6220!6220isNatKind(X)) 
  a!6220!6220plus(X, 0) => a!6220!6220U51(a!6220!6220isNat(X), X) 
  a!6220!6220plus(X, s(Y)) => a!6220!6220U61(a!6220!6220isNat(Y), Y, X) 
  mark(U11(X, Y, Z)) => a!6220!6220U11(mark(X), Y, Z) 
  mark(U12(X, Y, Z)) => a!6220!6220U12(mark(X), Y, Z) 
  mark(isNatKind(X)) => a!6220!6220isNatKind(X) 
  mark(U13(X, Y, Z)) => a!6220!6220U13(mark(X), Y, Z) 
  mark(U14(X, Y, Z)) => a!6220!6220U14(mark(X), Y, Z) 
  mark(U15(X, Y)) => a!6220!6220U15(mark(X), Y) 
  mark(isNat(X)) => a!6220!6220isNat(X) 
  mark(U16(X)) => a!6220!6220U16(mark(X)) 
  mark(U21(X, Y)) => a!6220!6220U21(mark(X), Y) 
  mark(U22(X, Y)) => a!6220!6220U22(mark(X), Y) 
  mark(U23(X)) => a!6220!6220U23(mark(X)) 
  mark(U31(X, Y)) => a!6220!6220U31(mark(X), Y) 
  mark(U32(X)) => a!6220!6220U32(mark(X)) 
  mark(U41(X)) => a!6220!6220U41(mark(X)) 
  mark(U51(X, Y)) => a!6220!6220U51(mark(X), Y) 
  mark(U52(X, Y)) => a!6220!6220U52(mark(X), Y) 
  mark(U61(X, Y, Z)) => a!6220!6220U61(mark(X), Y, Z) 
  mark(U62(X, Y, Z)) => a!6220!6220U62(mark(X), Y, Z) 
  mark(U63(X, Y, Z)) => a!6220!6220U63(mark(X), Y, Z) 
  mark(U64(X, Y, Z)) => a!6220!6220U64(mark(X), Y, Z) 
  mark(plus(X, Y)) => a!6220!6220plus(mark(X), mark(Y)) 
  mark(tt) => tt 
  mark(s(X)) => s(mark(X)) 
  mark(0) => 0 
  a!6220!6220U11(X, Y, Z) => U11(X, Y, Z) 
  a!6220!6220U12(X, Y, Z) => U12(X, Y, Z) 
  a!6220!6220isNatKind(X) => isNatKind(X) 
  a!6220!6220U13(X, Y, Z) => U13(X, Y, Z) 
  a!6220!6220U14(X, Y, Z) => U14(X, Y, Z) 
  a!6220!6220U15(X, Y) => U15(X, Y) 
  a!6220!6220isNat(X) => isNat(X) 
  a!6220!6220U16(X) => U16(X) 
  a!6220!6220U21(X, Y) => U21(X, Y) 
  a!6220!6220U22(X, Y) => U22(X, Y) 
  a!6220!6220U23(X) => U23(X) 
  a!6220!6220U31(X, Y) => U31(X, Y) 
  a!6220!6220U32(X) => U32(X) 
  a!6220!6220U41(X) => U41(X) 
  a!6220!6220U51(X, Y) => U51(X, Y) 
  a!6220!6220U52(X, Y) => U52(X, Y) 
  a!6220!6220U61(X, Y, Z) => U61(X, Y, Z) 
  a!6220!6220U62(X, Y, Z) => U62(X, Y, Z) 
  a!6220!6220U63(X, Y, Z) => U63(X, Y, Z) 
  a!6220!6220U64(X, Y, Z) => U64(X, Y, Z) 
  a!6220!6220plus(X, Y) => plus(X, Y) 

We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs in [Kop12, Ch. 7.8]).

We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] a!6220!6220U11#(tt, X, Y) =#> a!6220!6220U12#(a!6220!6220isNatKind(X), X, Y)   
    1] a!6220!6220U11#(tt, X, Y) =#> a!6220!6220isNatKind#(X)   
    2] a!6220!6220U12#(tt, X, Y) =#> a!6220!6220U13#(a!6220!6220isNatKind(Y), X, Y)   
    3] a!6220!6220U12#(tt, X, Y) =#> a!6220!6220isNatKind#(Y)   
    4] a!6220!6220U13#(tt, X, Y) =#> a!6220!6220U14#(a!6220!6220isNatKind(Y), X, Y)   
    5] a!6220!6220U13#(tt, X, Y) =#> a!6220!6220isNatKind#(Y)   
    6] a!6220!6220U14#(tt, X, Y) =#> a!6220!6220U15#(a!6220!6220isNat(X), Y)   
    7] a!6220!6220U14#(tt, X, Y) =#> a!6220!6220isNat#(X)   
    8] a!6220!6220U15#(tt, X) =#> a!6220!6220U16#(a!6220!6220isNat(X))   
    9] a!6220!6220U15#(tt, X) =#> a!6220!6220isNat#(X)   
    10] a!6220!6220U21#(tt, X) =#> a!6220!6220U22#(a!6220!6220isNatKind(X), X)   
    11] a!6220!6220U21#(tt, X) =#> a!6220!6220isNatKind#(X)   
    12] a!6220!6220U22#(tt, X) =#> a!6220!6220U23#(a!6220!6220isNat(X))   
    13] a!6220!6220U22#(tt, X) =#> a!6220!6220isNat#(X)   
    14] a!6220!6220U31#(tt, X) =#> a!6220!6220U32#(a!6220!6220isNatKind(X))   
    15] a!6220!6220U31#(tt, X) =#> a!6220!6220isNatKind#(X)   
    16] a!6220!6220U51#(tt, X) =#> a!6220!6220U52#(a!6220!6220isNatKind(X), X)   
    17] a!6220!6220U51#(tt, X) =#> a!6220!6220isNatKind#(X)   
    18] a!6220!6220U52#(tt, X) =#> mark#(X)   
    19] a!6220!6220U61#(tt, X, Y) =#> a!6220!6220U62#(a!6220!6220isNatKind(X), X, Y)   
    20] a!6220!6220U61#(tt, X, Y) =#> a!6220!6220isNatKind#(X)   
    21] a!6220!6220U62#(tt, X, Y) =#> a!6220!6220U63#(a!6220!6220isNat(Y), X, Y)   
    22] a!6220!6220U62#(tt, X, Y) =#> a!6220!6220isNat#(Y)   
    23] a!6220!6220U63#(tt, X, Y) =#> a!6220!6220U64#(a!6220!6220isNatKind(Y), X, Y)   
    24] a!6220!6220U63#(tt, X, Y) =#> a!6220!6220isNatKind#(Y)   
    25] a!6220!6220U64#(tt, X, Y) =#> a!6220!6220plus#(mark(Y), mark(X))   
    26] a!6220!6220U64#(tt, X, Y) =#> mark#(Y)   
    27] a!6220!6220U64#(tt, X, Y) =#> mark#(X)   
    28] a!6220!6220isNat#(plus(X, Y)) =#> a!6220!6220U11#(a!6220!6220isNatKind(X), X, Y)   
    29] a!6220!6220isNat#(plus(X, Y)) =#> a!6220!6220isNatKind#(X)   
    30] a!6220!6220isNat#(s(X)) =#> a!6220!6220U21#(a!6220!6220isNatKind(X), X)   
    31] a!6220!6220isNat#(s(X)) =#> a!6220!6220isNatKind#(X)   
    32] a!6220!6220isNatKind#(plus(X, Y)) =#> a!6220!6220U31#(a!6220!6220isNatKind(X), Y)   
    33] a!6220!6220isNatKind#(plus(X, Y)) =#> a!6220!6220isNatKind#(X)   
    34] a!6220!6220isNatKind#(s(X)) =#> a!6220!6220U41#(a!6220!6220isNatKind(X))   
    35] a!6220!6220isNatKind#(s(X)) =#> a!6220!6220isNatKind#(X)   
    36] a!6220!6220plus#(X, 0) =#> a!6220!6220U51#(a!6220!6220isNat(X), X)   
    37] a!6220!6220plus#(X, 0) =#> a!6220!6220isNat#(X)   
    38] a!6220!6220plus#(X, s(Y)) =#> a!6220!6220U61#(a!6220!6220isNat(Y), Y, X)   
    39] a!6220!6220plus#(X, s(Y)) =#> a!6220!6220isNat#(Y)   
    40] mark#(U11(X, Y, Z)) =#> a!6220!6220U11#(mark(X), Y, Z)   
    41] mark#(U11(X, Y, Z)) =#> mark#(X)   
    42] mark#(U12(X, Y, Z)) =#> a!6220!6220U12#(mark(X), Y, Z)   
    43] mark#(U12(X, Y, Z)) =#> mark#(X)   
    44] mark#(isNatKind(X)) =#> a!6220!6220isNatKind#(X)   
    45] mark#(U13(X, Y, Z)) =#> a!6220!6220U13#(mark(X), Y, Z)   
    46] mark#(U13(X, Y, Z)) =#> mark#(X)   
    47] mark#(U14(X, Y, Z)) =#> a!6220!6220U14#(mark(X), Y, Z)   
    48] mark#(U14(X, Y, Z)) =#> mark#(X)   
    49] mark#(U15(X, Y)) =#> a!6220!6220U15#(mark(X), Y)   
    50] mark#(U15(X, Y)) =#> mark#(X)   
    51] mark#(isNat(X)) =#> a!6220!6220isNat#(X)   
    52] mark#(U16(X)) =#> a!6220!6220U16#(mark(X))   
    53] mark#(U16(X)) =#> mark#(X)   
    54] mark#(U21(X, Y)) =#> a!6220!6220U21#(mark(X), Y)   
    55] mark#(U21(X, Y)) =#> mark#(X)   
    56] mark#(U22(X, Y)) =#> a!6220!6220U22#(mark(X), Y)   
    57] mark#(U22(X, Y)) =#> mark#(X)   
    58] mark#(U23(X)) =#> a!6220!6220U23#(mark(X))   
    59] mark#(U23(X)) =#> mark#(X)   
    60] mark#(U31(X, Y)) =#> a!6220!6220U31#(mark(X), Y)   
    61] mark#(U31(X, Y)) =#> mark#(X)   
    62] mark#(U32(X)) =#> a!6220!6220U32#(mark(X))   
    63] mark#(U32(X)) =#> mark#(X)   
    64] mark#(U41(X)) =#> a!6220!6220U41#(mark(X))   
    65] mark#(U41(X)) =#> mark#(X)   
    66] mark#(U51(X, Y)) =#> a!6220!6220U51#(mark(X), Y)   
    67] mark#(U51(X, Y)) =#> mark#(X)   
    68] mark#(U52(X, Y)) =#> a!6220!6220U52#(mark(X), Y)   
    69] mark#(U52(X, Y)) =#> mark#(X)   
    70] mark#(U61(X, Y, Z)) =#> a!6220!6220U61#(mark(X), Y, Z)   
    71] mark#(U61(X, Y, Z)) =#> mark#(X)   
    72] mark#(U62(X, Y, Z)) =#> a!6220!6220U62#(mark(X), Y, Z)   
    73] mark#(U62(X, Y, Z)) =#> mark#(X)   
    74] mark#(U63(X, Y, Z)) =#> a!6220!6220U63#(mark(X), Y, Z)   
    75] mark#(U63(X, Y, Z)) =#> mark#(X)   
    76] mark#(U64(X, Y, Z)) =#> a!6220!6220U64#(mark(X), Y, Z)   
    77] mark#(U64(X, Y, Z)) =#> mark#(X)   
    78] mark#(plus(X, Y)) =#> a!6220!6220plus#(mark(X), mark(Y))   
    79] mark#(plus(X, Y)) =#> mark#(X)   
    80] mark#(plus(X, Y)) =#> mark#(Y)   
    81] mark#(s(X)) =#> mark#(X)   

  Rules R_0:

    a!6220!6220U11(tt, X, Y) => a!6220!6220U12(a!6220!6220isNatKind(X), X, Y) 
    a!6220!6220U12(tt, X, Y) => a!6220!6220U13(a!6220!6220isNatKind(Y), X, Y) 
    a!6220!6220U13(tt, X, Y) => a!6220!6220U14(a!6220!6220isNatKind(Y), X, Y) 
    a!6220!6220U14(tt, X, Y) => a!6220!6220U15(a!6220!6220isNat(X), Y) 
    a!6220!6220U15(tt, X) => a!6220!6220U16(a!6220!6220isNat(X)) 
    a!6220!6220U16(tt) => tt 
    a!6220!6220U21(tt, X) => a!6220!6220U22(a!6220!6220isNatKind(X), X) 
    a!6220!6220U22(tt, X) => a!6220!6220U23(a!6220!6220isNat(X)) 
    a!6220!6220U23(tt) => tt 
    a!6220!6220U31(tt, X) => a!6220!6220U32(a!6220!6220isNatKind(X)) 
    a!6220!6220U32(tt) => tt 
    a!6220!6220U41(tt) => tt 
    a!6220!6220U51(tt, X) => a!6220!6220U52(a!6220!6220isNatKind(X), X) 
    a!6220!6220U52(tt, X) => mark(X) 
    a!6220!6220U61(tt, X, Y) => a!6220!6220U62(a!6220!6220isNatKind(X), X, Y) 
    a!6220!6220U62(tt, X, Y) => a!6220!6220U63(a!6220!6220isNat(Y), X, Y) 
    a!6220!6220U63(tt, X, Y) => a!6220!6220U64(a!6220!6220isNatKind(Y), X, Y) 
    a!6220!6220U64(tt, X, Y) => s(a!6220!6220plus(mark(Y), mark(X))) 
    a!6220!6220isNat(0) => tt 
    a!6220!6220isNat(plus(X, Y)) => a!6220!6220U11(a!6220!6220isNatKind(X), X, Y) 
    a!6220!6220isNat(s(X)) => a!6220!6220U21(a!6220!6220isNatKind(X), X) 
    a!6220!6220isNatKind(0) => tt 
    a!6220!6220isNatKind(plus(X, Y)) => a!6220!6220U31(a!6220!6220isNatKind(X), Y) 
    a!6220!6220isNatKind(s(X)) => a!6220!6220U41(a!6220!6220isNatKind(X)) 
    a!6220!6220plus(X, 0) => a!6220!6220U51(a!6220!6220isNat(X), X) 
    a!6220!6220plus(X, s(Y)) => a!6220!6220U61(a!6220!6220isNat(Y), Y, X) 
    mark(U11(X, Y, Z)) => a!6220!6220U11(mark(X), Y, Z) 
    mark(U12(X, Y, Z)) => a!6220!6220U12(mark(X), Y, Z) 
    mark(isNatKind(X)) => a!6220!6220isNatKind(X) 
    mark(U13(X, Y, Z)) => a!6220!6220U13(mark(X), Y, Z) 
    mark(U14(X, Y, Z)) => a!6220!6220U14(mark(X), Y, Z) 
    mark(U15(X, Y)) => a!6220!6220U15(mark(X), Y) 
    mark(isNat(X)) => a!6220!6220isNat(X) 
    mark(U16(X)) => a!6220!6220U16(mark(X)) 
    mark(U21(X, Y)) => a!6220!6220U21(mark(X), Y) 
    mark(U22(X, Y)) => a!6220!6220U22(mark(X), Y) 
    mark(U23(X)) => a!6220!6220U23(mark(X)) 
    mark(U31(X, Y)) => a!6220!6220U31(mark(X), Y) 
    mark(U32(X)) => a!6220!6220U32(mark(X)) 
    mark(U41(X)) => a!6220!6220U41(mark(X)) 
    mark(U51(X, Y)) => a!6220!6220U51(mark(X), Y) 
    mark(U52(X, Y)) => a!6220!6220U52(mark(X), Y) 
    mark(U61(X, Y, Z)) => a!6220!6220U61(mark(X), Y, Z) 
    mark(U62(X, Y, Z)) => a!6220!6220U62(mark(X), Y, Z) 
    mark(U63(X, Y, Z)) => a!6220!6220U63(mark(X), Y, Z) 
    mark(U64(X, Y, Z)) => a!6220!6220U64(mark(X), Y, Z) 
    mark(plus(X, Y)) => a!6220!6220plus(mark(X), mark(Y)) 
    mark(tt) => tt 
    mark(s(X)) => s(mark(X)) 
    mark(0) => 0 
    a!6220!6220U11(X, Y, Z) => U11(X, Y, Z) 
    a!6220!6220U12(X, Y, Z) => U12(X, Y, Z) 
    a!6220!6220isNatKind(X) => isNatKind(X) 
    a!6220!6220U13(X, Y, Z) => U13(X, Y, Z) 
    a!6220!6220U14(X, Y, Z) => U14(X, Y, Z) 
    a!6220!6220U15(X, Y) => U15(X, Y) 
    a!6220!6220isNat(X) => isNat(X) 
    a!6220!6220U16(X) => U16(X) 
    a!6220!6220U21(X, Y) => U21(X, Y) 
    a!6220!6220U22(X, Y) => U22(X, Y) 
    a!6220!6220U23(X) => U23(X) 
    a!6220!6220U31(X, Y) => U31(X, Y) 
    a!6220!6220U32(X) => U32(X) 
    a!6220!6220U41(X) => U41(X) 
    a!6220!6220U51(X, Y) => U51(X, Y) 
    a!6220!6220U52(X, Y) => U52(X, Y) 
    a!6220!6220U61(X, Y, Z) => U61(X, Y, Z) 
    a!6220!6220U62(X, Y, Z) => U62(X, Y, Z) 
    a!6220!6220U63(X, Y, Z) => U63(X, Y, Z) 
    a!6220!6220U64(X, Y, Z) => U64(X, Y, Z) 
    a!6220!6220plus(X, Y) => plus(X, Y) 

Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_0, minimal, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 2, 3 
    * 1 : 32, 33, 34, 35 
    * 2 : 4, 5 
    * 3 : 32, 33, 34, 35 
    * 4 : 6, 7 
    * 5 : 32, 33, 34, 35 
    * 6 : 8, 9 
    * 7 : 28, 29, 30, 31 
    * 8 :  
    * 9 : 28, 29, 30, 31 
    * 10 : 12, 13 
    * 11 : 32, 33, 34, 35 
    * 12 :  
    * 13 : 28, 29, 30, 31 
    * 14 :  
    * 15 : 32, 33, 34, 35 
    * 16 : 18 
    * 17 : 32, 33, 34, 35 
    * 18 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 19 : 21, 22 
    * 20 : 32, 33, 34, 35 
    * 21 : 23, 24 
    * 22 : 28, 29, 30, 31 
    * 23 : 25, 26, 27 
    * 24 : 32, 33, 34, 35 
    * 25 : 36, 37, 38, 39 
    * 26 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 27 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 28 : 0, 1 
    * 29 : 32, 33, 34, 35 
    * 30 : 10, 11 
    * 31 : 32, 33, 34, 35 
    * 32 : 14, 15 
    * 33 : 32, 33, 34, 35 
    * 34 :  
    * 35 : 32, 33, 34, 35 
    * 36 : 16, 17 
    * 37 : 28, 29, 30, 31 
    * 38 : 19, 20 
    * 39 : 28, 29, 30, 31 
    * 40 : 0, 1 
    * 41 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 42 : 2, 3 
    * 43 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 44 : 32, 33, 34, 35 
    * 45 : 4, 5 
    * 46 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 47 : 6, 7 
    * 48 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 49 : 8, 9 
    * 50 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 51 : 28, 29, 30, 31 
    * 52 :  
    * 53 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 54 : 10, 11 
    * 55 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 56 : 12, 13 
    * 57 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 58 :  
    * 59 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 60 : 14, 15 
    * 61 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 62 :  
    * 63 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 64 :  
    * 65 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 66 : 16, 17 
    * 67 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 68 : 18 
    * 69 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 70 : 19, 20 
    * 71 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 72 : 21, 22 
    * 73 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 74 : 23, 24 
    * 75 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 76 : 25, 26, 27 
    * 77 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 78 : 36, 37, 38, 39 
    * 79 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 80 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 
    * 81 : 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 

This graph has the following strongly connected components:

  P_1:

    a!6220!6220U11#(tt, X, Y) =#> a!6220!6220U12#(a!6220!6220isNatKind(X), X, Y)   
    a!6220!6220U12#(tt, X, Y) =#> a!6220!6220U13#(a!6220!6220isNatKind(Y), X, Y)   
    a!6220!6220U13#(tt, X, Y) =#> a!6220!6220U14#(a!6220!6220isNatKind(Y), X, Y)   
    a!6220!6220U14#(tt, X, Y) =#> a!6220!6220U15#(a!6220!6220isNat(X), Y)   
    a!6220!6220U14#(tt, X, Y) =#> a!6220!6220isNat#(X)   
    a!6220!6220U15#(tt, X) =#> a!6220!6220isNat#(X)   
    a!6220!6220U21#(tt, X) =#> a!6220!6220U22#(a!6220!6220isNatKind(X), X)   
    a!6220!6220U22#(tt, X) =#> a!6220!6220isNat#(X)   
    a!6220!6220isNat#(plus(X, Y)) =#> a!6220!6220U11#(a!6220!6220isNatKind(X), X, Y)   
    a!6220!6220isNat#(s(X)) =#> a!6220!6220U21#(a!6220!6220isNatKind(X), X)   

  P_2:

    a!6220!6220U31#(tt, X) =#> a!6220!6220isNatKind#(X)   
    a!6220!6220isNatKind#(plus(X, Y)) =#> a!6220!6220U31#(a!6220!6220isNatKind(X), Y)   
    a!6220!6220isNatKind#(plus(X, Y)) =#> a!6220!6220isNatKind#(X)   
    a!6220!6220isNatKind#(s(X)) =#> a!6220!6220isNatKind#(X)   

  P_3:

    a!6220!6220U51#(tt, X) =#> a!6220!6220U52#(a!6220!6220isNatKind(X), X)   
    a!6220!6220U52#(tt, X) =#> mark#(X)   
    a!6220!6220U61#(tt, X, Y) =#> a!6220!6220U62#(a!6220!6220isNatKind(X), X, Y)   
    a!6220!6220U62#(tt, X, Y) =#> a!6220!6220U63#(a!6220!6220isNat(Y), X, Y)   
    a!6220!6220U63#(tt, X, Y) =#> a!6220!6220U64#(a!6220!6220isNatKind(Y), X, Y)   
    a!6220!6220U64#(tt, X, Y) =#> a!6220!6220plus#(mark(Y), mark(X))   
    a!6220!6220U64#(tt, X, Y) =#> mark#(Y)   
    a!6220!6220U64#(tt, X, Y) =#> mark#(X)   
    a!6220!6220plus#(X, 0) =#> a!6220!6220U51#(a!6220!6220isNat(X), X)   
    a!6220!6220plus#(X, s(Y)) =#> a!6220!6220U61#(a!6220!6220isNat(Y), Y, X)   
    mark#(U11(X, Y, Z)) =#> mark#(X)   
    mark#(U12(X, Y, Z)) =#> mark#(X)   
    mark#(U13(X, Y, Z)) =#> mark#(X)   
    mark#(U14(X, Y, Z)) =#> mark#(X)   
    mark#(U15(X, Y)) =#> mark#(X)   
    mark#(U16(X)) =#> mark#(X)   
    mark#(U21(X, Y)) =#> mark#(X)   
    mark#(U22(X, Y)) =#> mark#(X)   
    mark#(U23(X)) =#> mark#(X)   
    mark#(U31(X, Y)) =#> mark#(X)   
    mark#(U32(X)) =#> mark#(X)   
    mark#(U41(X)) =#> mark#(X)   
    mark#(U51(X, Y)) =#> a!6220!6220U51#(mark(X), Y)   
    mark#(U51(X, Y)) =#> mark#(X)   
    mark#(U52(X, Y)) =#> a!6220!6220U52#(mark(X), Y)   
    mark#(U52(X, Y)) =#> mark#(X)   
    mark#(U61(X, Y, Z)) =#> a!6220!6220U61#(mark(X), Y, Z)   
    mark#(U61(X, Y, Z)) =#> mark#(X)   
    mark#(U62(X, Y, Z)) =#> a!6220!6220U62#(mark(X), Y, Z)   
    mark#(U62(X, Y, Z)) =#> mark#(X)   
    mark#(U63(X, Y, Z)) =#> a!6220!6220U63#(mark(X), Y, Z)   
    mark#(U63(X, Y, Z)) =#> mark#(X)   
    mark#(U64(X, Y, Z)) =#> a!6220!6220U64#(mark(X), Y, Z)   
    mark#(U64(X, Y, Z)) =#> mark#(X)   
    mark#(plus(X, Y)) =#> a!6220!6220plus#(mark(X), mark(Y))   
    mark#(plus(X, Y)) =#> mark#(X)   
    mark#(plus(X, Y)) =#> mark#(Y)   
    mark#(s(X)) =#> mark#(X)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f), (P_2, R_0, m, f) and (P_3, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_3, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_3, R_0, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  a!6220!6220U51#(tt, X) >? a!6220!6220U52#(a!6220!6220isNatKind(X), X) 
  a!6220!6220U52#(tt, X) >? mark#(X) 
  a!6220!6220U61#(tt, X, Y) >? a!6220!6220U62#(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220U62#(tt, X, Y) >? a!6220!6220U63#(a!6220!6220isNat(Y), X, Y) 
  a!6220!6220U63#(tt, X, Y) >? a!6220!6220U64#(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U64#(tt, X, Y) >? a!6220!6220plus#(mark(Y), mark(X)) 
  a!6220!6220U64#(tt, X, Y) >? mark#(Y) 
  a!6220!6220U64#(tt, X, Y) >? mark#(X) 
  a!6220!6220plus#(X, 0) >? a!6220!6220U51#(a!6220!6220isNat(X), X) 
  a!6220!6220plus#(X, s(Y)) >? a!6220!6220U61#(a!6220!6220isNat(Y), Y, X) 
  mark#(U11(X, Y, Z)) >? mark#(X) 
  mark#(U12(X, Y, Z)) >? mark#(X) 
  mark#(U13(X, Y, Z)) >? mark#(X) 
  mark#(U14(X, Y, Z)) >? mark#(X) 
  mark#(U15(X, Y)) >? mark#(X) 
  mark#(U16(X)) >? mark#(X) 
  mark#(U21(X, Y)) >? mark#(X) 
  mark#(U22(X, Y)) >? mark#(X) 
  mark#(U23(X)) >? mark#(X) 
  mark#(U31(X, Y)) >? mark#(X) 
  mark#(U32(X)) >? mark#(X) 
  mark#(U41(X)) >? mark#(X) 
  mark#(U51(X, Y)) >? a!6220!6220U51#(mark(X), Y) 
  mark#(U51(X, Y)) >? mark#(X) 
  mark#(U52(X, Y)) >? a!6220!6220U52#(mark(X), Y) 
  mark#(U52(X, Y)) >? mark#(X) 
  mark#(U61(X, Y, Z)) >? a!6220!6220U61#(mark(X), Y, Z) 
  mark#(U61(X, Y, Z)) >? mark#(X) 
  mark#(U62(X, Y, Z)) >? a!6220!6220U62#(mark(X), Y, Z) 
  mark#(U62(X, Y, Z)) >? mark#(X) 
  mark#(U63(X, Y, Z)) >? a!6220!6220U63#(mark(X), Y, Z) 
  mark#(U63(X, Y, Z)) >? mark#(X) 
  mark#(U64(X, Y, Z)) >? a!6220!6220U64#(mark(X), Y, Z) 
  mark#(U64(X, Y, Z)) >? mark#(X) 
  mark#(plus(X, Y)) >? a!6220!6220plus#(mark(X), mark(Y)) 
  mark#(plus(X, Y)) >? mark#(X) 
  mark#(plus(X, Y)) >? mark#(Y) 
  mark#(s(X)) >? mark#(X) 
  a!6220!6220U11(tt, X, Y) >= a!6220!6220U12(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220U12(tt, X, Y) >= a!6220!6220U13(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U13(tt, X, Y) >= a!6220!6220U14(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U14(tt, X, Y) >= a!6220!6220U15(a!6220!6220isNat(X), Y) 
  a!6220!6220U15(tt, X) >= a!6220!6220U16(a!6220!6220isNat(X)) 
  a!6220!6220U16(tt) >= tt 
  a!6220!6220U21(tt, X) >= a!6220!6220U22(a!6220!6220isNatKind(X), X) 
  a!6220!6220U22(tt, X) >= a!6220!6220U23(a!6220!6220isNat(X)) 
  a!6220!6220U23(tt) >= tt 
  a!6220!6220U31(tt, X) >= a!6220!6220U32(a!6220!6220isNatKind(X)) 
  a!6220!6220U32(tt) >= tt 
  a!6220!6220U41(tt) >= tt 
  a!6220!6220U51(tt, X) >= a!6220!6220U52(a!6220!6220isNatKind(X), X) 
  a!6220!6220U52(tt, X) >= mark(X) 
  a!6220!6220U61(tt, X, Y) >= a!6220!6220U62(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220U62(tt, X, Y) >= a!6220!6220U63(a!6220!6220isNat(Y), X, Y) 
  a!6220!6220U63(tt, X, Y) >= a!6220!6220U64(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U64(tt, X, Y) >= s(a!6220!6220plus(mark(Y), mark(X))) 
  a!6220!6220isNat(0) >= tt 
  a!6220!6220isNat(plus(X, Y)) >= a!6220!6220U11(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220isNat(s(X)) >= a!6220!6220U21(a!6220!6220isNatKind(X), X) 
  a!6220!6220isNatKind(0) >= tt 
  a!6220!6220isNatKind(plus(X, Y)) >= a!6220!6220U31(a!6220!6220isNatKind(X), Y) 
  a!6220!6220isNatKind(s(X)) >= a!6220!6220U41(a!6220!6220isNatKind(X)) 
  a!6220!6220plus(X, 0) >= a!6220!6220U51(a!6220!6220isNat(X), X) 
  a!6220!6220plus(X, s(Y)) >= a!6220!6220U61(a!6220!6220isNat(Y), Y, X) 
  mark(U11(X, Y, Z)) >= a!6220!6220U11(mark(X), Y, Z) 
  mark(U12(X, Y, Z)) >= a!6220!6220U12(mark(X), Y, Z) 
  mark(isNatKind(X)) >= a!6220!6220isNatKind(X) 
  mark(U13(X, Y, Z)) >= a!6220!6220U13(mark(X), Y, Z) 
  mark(U14(X, Y, Z)) >= a!6220!6220U14(mark(X), Y, Z) 
  mark(U15(X, Y)) >= a!6220!6220U15(mark(X), Y) 
  mark(isNat(X)) >= a!6220!6220isNat(X) 
  mark(U16(X)) >= a!6220!6220U16(mark(X)) 
  mark(U21(X, Y)) >= a!6220!6220U21(mark(X), Y) 
  mark(U22(X, Y)) >= a!6220!6220U22(mark(X), Y) 
  mark(U23(X)) >= a!6220!6220U23(mark(X)) 
  mark(U31(X, Y)) >= a!6220!6220U31(mark(X), Y) 
  mark(U32(X)) >= a!6220!6220U32(mark(X)) 
  mark(U41(X)) >= a!6220!6220U41(mark(X)) 
  mark(U51(X, Y)) >= a!6220!6220U51(mark(X), Y) 
  mark(U52(X, Y)) >= a!6220!6220U52(mark(X), Y) 
  mark(U61(X, Y, Z)) >= a!6220!6220U61(mark(X), Y, Z) 
  mark(U62(X, Y, Z)) >= a!6220!6220U62(mark(X), Y, Z) 
  mark(U63(X, Y, Z)) >= a!6220!6220U63(mark(X), Y, Z) 
  mark(U64(X, Y, Z)) >= a!6220!6220U64(mark(X), Y, Z) 
  mark(plus(X, Y)) >= a!6220!6220plus(mark(X), mark(Y)) 
  mark(tt) >= tt 
  mark(s(X)) >= s(mark(X)) 
  mark(0) >= 0 
  a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) 
  a!6220!6220U12(X, Y, Z) >= U12(X, Y, Z) 
  a!6220!6220isNatKind(X) >= isNatKind(X) 
  a!6220!6220U13(X, Y, Z) >= U13(X, Y, Z) 
  a!6220!6220U14(X, Y, Z) >= U14(X, Y, Z) 
  a!6220!6220U15(X, Y) >= U15(X, Y) 
  a!6220!6220isNat(X) >= isNat(X) 
  a!6220!6220U16(X) >= U16(X) 
  a!6220!6220U21(X, Y) >= U21(X, Y) 
  a!6220!6220U22(X, Y) >= U22(X, Y) 
  a!6220!6220U23(X) >= U23(X) 
  a!6220!6220U31(X, Y) >= U31(X, Y) 
  a!6220!6220U32(X) >= U32(X) 
  a!6220!6220U41(X) >= U41(X) 
  a!6220!6220U51(X, Y) >= U51(X, Y) 
  a!6220!6220U52(X, Y) >= U52(X, Y) 
  a!6220!6220U61(X, Y, Z) >= U61(X, Y, Z) 
  a!6220!6220U62(X, Y, Z) >= U62(X, Y, Z) 
  a!6220!6220U63(X, Y, Z) >= U63(X, Y, Z) 
  a!6220!6220U64(X, Y, Z) >= U64(X, Y, Z) 
  a!6220!6220plus(X, Y) >= plus(X, Y) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 1 
  U11 = \y0y1y2.y0 
  U12 = \y0y1y2.y0 
  U13 = \y0y1y2.y0 
  U14 = \y0y1y2.y0 
  U15 = \y0y1.y0 
  U16 = \y0.y0 
  U21 = \y0y1.y0 
  U22 = \y0y1.y0 
  U23 = \y0.y0 
  U31 = \y0y1.y0 
  U32 = \y0.y0 
  U41 = \y0.2y0 
  U51 = \y0y1.1 + y1 + 2y0 
  U52 = \y0y1.y1 + 2y0 
  U61 = \y0y1y2.y2 + 2y0 + 2y1 
  U62 = \y0y1y2.y2 + 2y0 + 2y1 
  U63 = \y0y1y2.y2 + 2y0 + 2y1 
  U64 = \y0y1y2.y0 + y2 + 2y1 
  a!6220!6220U11 = \y0y1y2.y0 
  a!6220!6220U12 = \y0y1y2.y0 
  a!6220!6220U13 = \y0y1y2.y0 
  a!6220!6220U14 = \y0y1y2.y0 
  a!6220!6220U15 = \y0y1.y0 
  a!6220!6220U16 = \y0.y0 
  a!6220!6220U21 = \y0y1.y0 
  a!6220!6220U22 = \y0y1.y0 
  a!6220!6220U23 = \y0.y0 
  a!6220!6220U31 = \y0y1.y0 
  a!6220!6220U32 = \y0.y0 
  a!6220!6220U41 = \y0.2y0 
  a!6220!6220U51 = \y0y1.1 + y1 + 2y0 
  a!6220!6220U51# = \y0y1.1 + 2y1 
  a!6220!6220U52 = \y0y1.y1 + 2y0 
  a!6220!6220U52# = \y0y1.2y1 
  a!6220!6220U61 = \y0y1y2.y2 + 2y0 + 2y1 
  a!6220!6220U61# = \y0y1y2.2y1 + 2y2 
  a!6220!6220U62 = \y0y1y2.y2 + 2y0 + 2y1 
  a!6220!6220U62# = \y0y1y2.2y1 + 2y2 
  a!6220!6220U63 = \y0y1y2.y2 + 2y0 + 2y1 
  a!6220!6220U63# = \y0y1y2.2y1 + 2y2 
  a!6220!6220U64 = \y0y1y2.y0 + y2 + 2y1 
  a!6220!6220U64# = \y0y1y2.2y1 + 2y2 
  a!6220!6220isNat = \y0.0 
  a!6220!6220isNatKind = \y0.0 
  a!6220!6220plus = \y0y1.y0 + 2y1 
  a!6220!6220plus# = \y0y1.2y0 + 2y1 
  isNat = \y0.0 
  isNatKind = \y0.0 
  mark = \y0.y0 
  mark# = \y0.2y0 
  plus = \y0y1.y0 + 2y1 
  s = \y0.y0 
  tt = 0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220U51#(tt, _x0)]] = 1 + 2x0 > 2x0 = [[a!6220!6220U52#(a!6220!6220isNatKind(_x0), _x0)]] 
  [[a!6220!6220U52#(tt, _x0)]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220U61#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U62#(a!6220!6220isNatKind(_x0), _x0, _x1)]] 
  [[a!6220!6220U62#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U63#(a!6220!6220isNat(_x1), _x0, _x1)]] 
  [[a!6220!6220U63#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U64#(a!6220!6220isNatKind(_x1), _x0, _x1)]] 
  [[a!6220!6220U64#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220plus#(mark(_x1), mark(_x0))]] 
  [[a!6220!6220U64#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x1 = [[mark#(_x1)]] 
  [[a!6220!6220U64#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220plus#(_x0, 0)]] = 2 + 2x0 > 1 + 2x0 = [[a!6220!6220U51#(a!6220!6220isNat(_x0), _x0)]] 
  [[a!6220!6220plus#(_x0, s(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U61#(a!6220!6220isNat(_x1), _x1, _x0)]] 
  [[mark#(U11(_x0, _x1, _x2))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U12(_x0, _x1, _x2))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U13(_x0, _x1, _x2))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U14(_x0, _x1, _x2))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U15(_x0, _x1))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U16(_x0))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U21(_x0, _x1))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U22(_x0, _x1))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U23(_x0))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U31(_x0, _x1))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U32(_x0))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U41(_x0))]] = 4x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U51(_x0, _x1))]] = 2 + 2x1 + 4x0 > 1 + 2x1 = [[a!6220!6220U51#(mark(_x0), _x1)]] 
  [[mark#(U51(_x0, _x1))]] = 2 + 2x1 + 4x0 > 2x0 = [[mark#(_x0)]] 
  [[mark#(U52(_x0, _x1))]] = 2x1 + 4x0 >= 2x1 = [[a!6220!6220U52#(mark(_x0), _x1)]] 
  [[mark#(U52(_x0, _x1))]] = 2x1 + 4x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U61(_x0, _x1, _x2))]] = 2x2 + 4x0 + 4x1 >= 2x1 + 2x2 = [[a!6220!6220U61#(mark(_x0), _x1, _x2)]] 
  [[mark#(U61(_x0, _x1, _x2))]] = 2x2 + 4x0 + 4x1 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U62(_x0, _x1, _x2))]] = 2x2 + 4x0 + 4x1 >= 2x1 + 2x2 = [[a!6220!6220U62#(mark(_x0), _x1, _x2)]] 
  [[mark#(U62(_x0, _x1, _x2))]] = 2x2 + 4x0 + 4x1 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U63(_x0, _x1, _x2))]] = 2x2 + 4x0 + 4x1 >= 2x1 + 2x2 = [[a!6220!6220U63#(mark(_x0), _x1, _x2)]] 
  [[mark#(U63(_x0, _x1, _x2))]] = 2x2 + 4x0 + 4x1 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U64(_x0, _x1, _x2))]] = 2x0 + 2x2 + 4x1 >= 2x1 + 2x2 = [[a!6220!6220U64#(mark(_x0), _x1, _x2)]] 
  [[mark#(U64(_x0, _x1, _x2))]] = 2x0 + 2x2 + 4x1 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(plus(_x0, _x1))]] = 2x0 + 4x1 >= 2x0 + 2x1 = [[a!6220!6220plus#(mark(_x0), mark(_x1))]] 
  [[mark#(plus(_x0, _x1))]] = 2x0 + 4x1 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(plus(_x0, _x1))]] = 2x0 + 4x1 >= 2x1 = [[mark#(_x1)]] 
  [[mark#(s(_x0))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220U11(tt, _x0, _x1)]] = 0 >= 0 = [[a!6220!6220U12(a!6220!6220isNatKind(_x0), _x0, _x1)]] 
  [[a!6220!6220U12(tt, _x0, _x1)]] = 0 >= 0 = [[a!6220!6220U13(a!6220!6220isNatKind(_x1), _x0, _x1)]] 
  [[a!6220!6220U13(tt, _x0, _x1)]] = 0 >= 0 = [[a!6220!6220U14(a!6220!6220isNatKind(_x1), _x0, _x1)]] 
  [[a!6220!6220U14(tt, _x0, _x1)]] = 0 >= 0 = [[a!6220!6220U15(a!6220!6220isNat(_x0), _x1)]] 
  [[a!6220!6220U15(tt, _x0)]] = 0 >= 0 = [[a!6220!6220U16(a!6220!6220isNat(_x0))]] 
  [[a!6220!6220U16(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U21(tt, _x0)]] = 0 >= 0 = [[a!6220!6220U22(a!6220!6220isNatKind(_x0), _x0)]] 
  [[a!6220!6220U22(tt, _x0)]] = 0 >= 0 = [[a!6220!6220U23(a!6220!6220isNat(_x0))]] 
  [[a!6220!6220U23(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U31(tt, _x0)]] = 0 >= 0 = [[a!6220!6220U32(a!6220!6220isNatKind(_x0))]] 
  [[a!6220!6220U32(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U41(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U51(tt, _x0)]] = 1 + x0 >= x0 = [[a!6220!6220U52(a!6220!6220isNatKind(_x0), _x0)]] 
  [[a!6220!6220U52(tt, _x0)]] = x0 >= x0 = [[mark(_x0)]] 
  [[a!6220!6220U61(tt, _x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220U62(a!6220!6220isNatKind(_x0), _x0, _x1)]] 
  [[a!6220!6220U62(tt, _x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220U63(a!6220!6220isNat(_x1), _x0, _x1)]] 
  [[a!6220!6220U63(tt, _x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220U64(a!6220!6220isNatKind(_x1), _x0, _x1)]] 
  [[a!6220!6220U64(tt, _x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[s(a!6220!6220plus(mark(_x1), mark(_x0)))]] 
  [[a!6220!6220isNat(0)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220isNat(plus(_x0, _x1))]] = 0 >= 0 = [[a!6220!6220U11(a!6220!6220isNatKind(_x0), _x0, _x1)]] 
  [[a!6220!6220isNat(s(_x0))]] = 0 >= 0 = [[a!6220!6220U21(a!6220!6220isNatKind(_x0), _x0)]] 
  [[a!6220!6220isNatKind(0)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220isNatKind(plus(_x0, _x1))]] = 0 >= 0 = [[a!6220!6220U31(a!6220!6220isNatKind(_x0), _x1)]] 
  [[a!6220!6220isNatKind(s(_x0))]] = 0 >= 0 = [[a!6220!6220U41(a!6220!6220isNatKind(_x0))]] 
  [[a!6220!6220plus(_x0, 0)]] = 2 + x0 >= 1 + x0 = [[a!6220!6220U51(a!6220!6220isNat(_x0), _x0)]] 
  [[a!6220!6220plus(_x0, s(_x1))]] = x0 + 2x1 >= x0 + 2x1 = [[a!6220!6220U61(a!6220!6220isNat(_x1), _x1, _x0)]] 
  [[mark(U11(_x0, _x1, _x2))]] = x0 >= x0 = [[a!6220!6220U11(mark(_x0), _x1, _x2)]] 
  [[mark(U12(_x0, _x1, _x2))]] = x0 >= x0 = [[a!6220!6220U12(mark(_x0), _x1, _x2)]] 
  [[mark(isNatKind(_x0))]] = 0 >= 0 = [[a!6220!6220isNatKind(_x0)]] 
  [[mark(U13(_x0, _x1, _x2))]] = x0 >= x0 = [[a!6220!6220U13(mark(_x0), _x1, _x2)]] 
  [[mark(U14(_x0, _x1, _x2))]] = x0 >= x0 = [[a!6220!6220U14(mark(_x0), _x1, _x2)]] 
  [[mark(U15(_x0, _x1))]] = x0 >= x0 = [[a!6220!6220U15(mark(_x0), _x1)]] 
  [[mark(isNat(_x0))]] = 0 >= 0 = [[a!6220!6220isNat(_x0)]] 
  [[mark(U16(_x0))]] = x0 >= x0 = [[a!6220!6220U16(mark(_x0))]] 
  [[mark(U21(_x0, _x1))]] = x0 >= x0 = [[a!6220!6220U21(mark(_x0), _x1)]] 
  [[mark(U22(_x0, _x1))]] = x0 >= x0 = [[a!6220!6220U22(mark(_x0), _x1)]] 
  [[mark(U23(_x0))]] = x0 >= x0 = [[a!6220!6220U23(mark(_x0))]] 
  [[mark(U31(_x0, _x1))]] = x0 >= x0 = [[a!6220!6220U31(mark(_x0), _x1)]] 
  [[mark(U32(_x0))]] = x0 >= x0 = [[a!6220!6220U32(mark(_x0))]] 
  [[mark(U41(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U41(mark(_x0))]] 
  [[mark(U51(_x0, _x1))]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[a!6220!6220U51(mark(_x0), _x1)]] 
  [[mark(U52(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220U52(mark(_x0), _x1)]] 
  [[mark(U61(_x0, _x1, _x2))]] = x2 + 2x0 + 2x1 >= x2 + 2x0 + 2x1 = [[a!6220!6220U61(mark(_x0), _x1, _x2)]] 
  [[mark(U62(_x0, _x1, _x2))]] = x2 + 2x0 + 2x1 >= x2 + 2x0 + 2x1 = [[a!6220!6220U62(mark(_x0), _x1, _x2)]] 
  [[mark(U63(_x0, _x1, _x2))]] = x2 + 2x0 + 2x1 >= x2 + 2x0 + 2x1 = [[a!6220!6220U63(mark(_x0), _x1, _x2)]] 
  [[mark(U64(_x0, _x1, _x2))]] = x0 + x2 + 2x1 >= x0 + x2 + 2x1 = [[a!6220!6220U64(mark(_x0), _x1, _x2)]] 
  [[mark(plus(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[a!6220!6220plus(mark(_x0), mark(_x1))]] 
  [[mark(tt)]] = 0 >= 0 = [[tt]] 
  [[mark(s(_x0))]] = x0 >= x0 = [[s(mark(_x0))]] 
  [[mark(0)]] = 1 >= 1 = [[0]] 
  [[a!6220!6220U11(_x0, _x1, _x2)]] = x0 >= x0 = [[U11(_x0, _x1, _x2)]] 
  [[a!6220!6220U12(_x0, _x1, _x2)]] = x0 >= x0 = [[U12(_x0, _x1, _x2)]] 
  [[a!6220!6220isNatKind(_x0)]] = 0 >= 0 = [[isNatKind(_x0)]] 
  [[a!6220!6220U13(_x0, _x1, _x2)]] = x0 >= x0 = [[U13(_x0, _x1, _x2)]] 
  [[a!6220!6220U14(_x0, _x1, _x2)]] = x0 >= x0 = [[U14(_x0, _x1, _x2)]] 
  [[a!6220!6220U15(_x0, _x1)]] = x0 >= x0 = [[U15(_x0, _x1)]] 
  [[a!6220!6220isNat(_x0)]] = 0 >= 0 = [[isNat(_x0)]] 
  [[a!6220!6220U16(_x0)]] = x0 >= x0 = [[U16(_x0)]] 
  [[a!6220!6220U21(_x0, _x1)]] = x0 >= x0 = [[U21(_x0, _x1)]] 
  [[a!6220!6220U22(_x0, _x1)]] = x0 >= x0 = [[U22(_x0, _x1)]] 
  [[a!6220!6220U23(_x0)]] = x0 >= x0 = [[U23(_x0)]] 
  [[a!6220!6220U31(_x0, _x1)]] = x0 >= x0 = [[U31(_x0, _x1)]] 
  [[a!6220!6220U32(_x0)]] = x0 >= x0 = [[U32(_x0)]] 
  [[a!6220!6220U41(_x0)]] = 2x0 >= 2x0 = [[U41(_x0)]] 
  [[a!6220!6220U51(_x0, _x1)]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[U51(_x0, _x1)]] 
  [[a!6220!6220U52(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[U52(_x0, _x1)]] 
  [[a!6220!6220U61(_x0, _x1, _x2)]] = x2 + 2x0 + 2x1 >= x2 + 2x0 + 2x1 = [[U61(_x0, _x1, _x2)]] 
  [[a!6220!6220U62(_x0, _x1, _x2)]] = x2 + 2x0 + 2x1 >= x2 + 2x0 + 2x1 = [[U62(_x0, _x1, _x2)]] 
  [[a!6220!6220U63(_x0, _x1, _x2)]] = x2 + 2x0 + 2x1 >= x2 + 2x0 + 2x1 = [[U63(_x0, _x1, _x2)]] 
  [[a!6220!6220U64(_x0, _x1, _x2)]] = x0 + x2 + 2x1 >= x0 + x2 + 2x1 = [[U64(_x0, _x1, _x2)]] 
  [[a!6220!6220plus(_x0, _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[plus(_x0, _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_3, R_0, minimal, formative) by (P_4, R_0, minimal, formative), where P_4 consists of:

  a!6220!6220U52#(tt, X) =#> mark#(X)   
  a!6220!6220U61#(tt, X, Y) =#> a!6220!6220U62#(a!6220!6220isNatKind(X), X, Y)   
  a!6220!6220U62#(tt, X, Y) =#> a!6220!6220U63#(a!6220!6220isNat(Y), X, Y)   
  a!6220!6220U63#(tt, X, Y) =#> a!6220!6220U64#(a!6220!6220isNatKind(Y), X, Y)   
  a!6220!6220U64#(tt, X, Y) =#> a!6220!6220plus#(mark(Y), mark(X))   
  a!6220!6220U64#(tt, X, Y) =#> mark#(Y)   
  a!6220!6220U64#(tt, X, Y) =#> mark#(X)   
  a!6220!6220plus#(X, s(Y)) =#> a!6220!6220U61#(a!6220!6220isNat(Y), Y, X)   
  mark#(U11(X, Y, Z)) =#> mark#(X)   
  mark#(U12(X, Y, Z)) =#> mark#(X)   
  mark#(U13(X, Y, Z)) =#> mark#(X)   
  mark#(U14(X, Y, Z)) =#> mark#(X)   
  mark#(U15(X, Y)) =#> mark#(X)   
  mark#(U16(X)) =#> mark#(X)   
  mark#(U21(X, Y)) =#> mark#(X)   
  mark#(U22(X, Y)) =#> mark#(X)   
  mark#(U23(X)) =#> mark#(X)   
  mark#(U31(X, Y)) =#> mark#(X)   
  mark#(U32(X)) =#> mark#(X)   
  mark#(U41(X)) =#> mark#(X)   
  mark#(U52(X, Y)) =#> a!6220!6220U52#(mark(X), Y)   
  mark#(U52(X, Y)) =#> mark#(X)   
  mark#(U61(X, Y, Z)) =#> a!6220!6220U61#(mark(X), Y, Z)   
  mark#(U61(X, Y, Z)) =#> mark#(X)   
  mark#(U62(X, Y, Z)) =#> a!6220!6220U62#(mark(X), Y, Z)   
  mark#(U62(X, Y, Z)) =#> mark#(X)   
  mark#(U63(X, Y, Z)) =#> a!6220!6220U63#(mark(X), Y, Z)   
  mark#(U63(X, Y, Z)) =#> mark#(X)   
  mark#(U64(X, Y, Z)) =#> a!6220!6220U64#(mark(X), Y, Z)   
  mark#(U64(X, Y, Z)) =#> mark#(X)   
  mark#(plus(X, Y)) =#> a!6220!6220plus#(mark(X), mark(Y))   
  mark#(plus(X, Y)) =#> mark#(X)   
  mark#(plus(X, Y)) =#> mark#(Y)   
  mark#(s(X)) =#> mark#(X)   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_4, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_4, R_0, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  a!6220!6220U52#(tt, X) >? mark#(X) 
  a!6220!6220U61#(tt, X, Y) >? a!6220!6220U62#(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220U62#(tt, X, Y) >? a!6220!6220U63#(a!6220!6220isNat(Y), X, Y) 
  a!6220!6220U63#(tt, X, Y) >? a!6220!6220U64#(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U64#(tt, X, Y) >? a!6220!6220plus#(mark(Y), mark(X)) 
  a!6220!6220U64#(tt, X, Y) >? mark#(Y) 
  a!6220!6220U64#(tt, X, Y) >? mark#(X) 
  a!6220!6220plus#(X, s(Y)) >? a!6220!6220U61#(a!6220!6220isNat(Y), Y, X) 
  mark#(U11(X, Y, Z)) >? mark#(X) 
  mark#(U12(X, Y, Z)) >? mark#(X) 
  mark#(U13(X, Y, Z)) >? mark#(X) 
  mark#(U14(X, Y, Z)) >? mark#(X) 
  mark#(U15(X, Y)) >? mark#(X) 
  mark#(U16(X)) >? mark#(X) 
  mark#(U21(X, Y)) >? mark#(X) 
  mark#(U22(X, Y)) >? mark#(X) 
  mark#(U23(X)) >? mark#(X) 
  mark#(U31(X, Y)) >? mark#(X) 
  mark#(U32(X)) >? mark#(X) 
  mark#(U41(X)) >? mark#(X) 
  mark#(U52(X, Y)) >? a!6220!6220U52#(mark(X), Y) 
  mark#(U52(X, Y)) >? mark#(X) 
  mark#(U61(X, Y, Z)) >? a!6220!6220U61#(mark(X), Y, Z) 
  mark#(U61(X, Y, Z)) >? mark#(X) 
  mark#(U62(X, Y, Z)) >? a!6220!6220U62#(mark(X), Y, Z) 
  mark#(U62(X, Y, Z)) >? mark#(X) 
  mark#(U63(X, Y, Z)) >? a!6220!6220U63#(mark(X), Y, Z) 
  mark#(U63(X, Y, Z)) >? mark#(X) 
  mark#(U64(X, Y, Z)) >? a!6220!6220U64#(mark(X), Y, Z) 
  mark#(U64(X, Y, Z)) >? mark#(X) 
  mark#(plus(X, Y)) >? a!6220!6220plus#(mark(X), mark(Y)) 
  mark#(plus(X, Y)) >? mark#(X) 
  mark#(plus(X, Y)) >? mark#(Y) 
  mark#(s(X)) >? mark#(X) 
  a!6220!6220U11(tt, X, Y) >= a!6220!6220U12(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220U12(tt, X, Y) >= a!6220!6220U13(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U13(tt, X, Y) >= a!6220!6220U14(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U14(tt, X, Y) >= a!6220!6220U15(a!6220!6220isNat(X), Y) 
  a!6220!6220U15(tt, X) >= a!6220!6220U16(a!6220!6220isNat(X)) 
  a!6220!6220U16(tt) >= tt 
  a!6220!6220U21(tt, X) >= a!6220!6220U22(a!6220!6220isNatKind(X), X) 
  a!6220!6220U22(tt, X) >= a!6220!6220U23(a!6220!6220isNat(X)) 
  a!6220!6220U23(tt) >= tt 
  a!6220!6220U31(tt, X) >= a!6220!6220U32(a!6220!6220isNatKind(X)) 
  a!6220!6220U32(tt) >= tt 
  a!6220!6220U41(tt) >= tt 
  a!6220!6220U51(tt, X) >= a!6220!6220U52(a!6220!6220isNatKind(X), X) 
  a!6220!6220U52(tt, X) >= mark(X) 
  a!6220!6220U61(tt, X, Y) >= a!6220!6220U62(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220U62(tt, X, Y) >= a!6220!6220U63(a!6220!6220isNat(Y), X, Y) 
  a!6220!6220U63(tt, X, Y) >= a!6220!6220U64(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U64(tt, X, Y) >= s(a!6220!6220plus(mark(Y), mark(X))) 
  a!6220!6220isNat(0) >= tt 
  a!6220!6220isNat(plus(X, Y)) >= a!6220!6220U11(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220isNat(s(X)) >= a!6220!6220U21(a!6220!6220isNatKind(X), X) 
  a!6220!6220isNatKind(0) >= tt 
  a!6220!6220isNatKind(plus(X, Y)) >= a!6220!6220U31(a!6220!6220isNatKind(X), Y) 
  a!6220!6220isNatKind(s(X)) >= a!6220!6220U41(a!6220!6220isNatKind(X)) 
  a!6220!6220plus(X, 0) >= a!6220!6220U51(a!6220!6220isNat(X), X) 
  a!6220!6220plus(X, s(Y)) >= a!6220!6220U61(a!6220!6220isNat(Y), Y, X) 
  mark(U11(X, Y, Z)) >= a!6220!6220U11(mark(X), Y, Z) 
  mark(U12(X, Y, Z)) >= a!6220!6220U12(mark(X), Y, Z) 
  mark(isNatKind(X)) >= a!6220!6220isNatKind(X) 
  mark(U13(X, Y, Z)) >= a!6220!6220U13(mark(X), Y, Z) 
  mark(U14(X, Y, Z)) >= a!6220!6220U14(mark(X), Y, Z) 
  mark(U15(X, Y)) >= a!6220!6220U15(mark(X), Y) 
  mark(isNat(X)) >= a!6220!6220isNat(X) 
  mark(U16(X)) >= a!6220!6220U16(mark(X)) 
  mark(U21(X, Y)) >= a!6220!6220U21(mark(X), Y) 
  mark(U22(X, Y)) >= a!6220!6220U22(mark(X), Y) 
  mark(U23(X)) >= a!6220!6220U23(mark(X)) 
  mark(U31(X, Y)) >= a!6220!6220U31(mark(X), Y) 
  mark(U32(X)) >= a!6220!6220U32(mark(X)) 
  mark(U41(X)) >= a!6220!6220U41(mark(X)) 
  mark(U51(X, Y)) >= a!6220!6220U51(mark(X), Y) 
  mark(U52(X, Y)) >= a!6220!6220U52(mark(X), Y) 
  mark(U61(X, Y, Z)) >= a!6220!6220U61(mark(X), Y, Z) 
  mark(U62(X, Y, Z)) >= a!6220!6220U62(mark(X), Y, Z) 
  mark(U63(X, Y, Z)) >= a!6220!6220U63(mark(X), Y, Z) 
  mark(U64(X, Y, Z)) >= a!6220!6220U64(mark(X), Y, Z) 
  mark(plus(X, Y)) >= a!6220!6220plus(mark(X), mark(Y)) 
  mark(tt) >= tt 
  mark(s(X)) >= s(mark(X)) 
  mark(0) >= 0 
  a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) 
  a!6220!6220U12(X, Y, Z) >= U12(X, Y, Z) 
  a!6220!6220isNatKind(X) >= isNatKind(X) 
  a!6220!6220U13(X, Y, Z) >= U13(X, Y, Z) 
  a!6220!6220U14(X, Y, Z) >= U14(X, Y, Z) 
  a!6220!6220U15(X, Y) >= U15(X, Y) 
  a!6220!6220isNat(X) >= isNat(X) 
  a!6220!6220U16(X) >= U16(X) 
  a!6220!6220U21(X, Y) >= U21(X, Y) 
  a!6220!6220U22(X, Y) >= U22(X, Y) 
  a!6220!6220U23(X) >= U23(X) 
  a!6220!6220U31(X, Y) >= U31(X, Y) 
  a!6220!6220U32(X) >= U32(X) 
  a!6220!6220U41(X) >= U41(X) 
  a!6220!6220U51(X, Y) >= U51(X, Y) 
  a!6220!6220U52(X, Y) >= U52(X, Y) 
  a!6220!6220U61(X, Y, Z) >= U61(X, Y, Z) 
  a!6220!6220U62(X, Y, Z) >= U62(X, Y, Z) 
  a!6220!6220U63(X, Y, Z) >= U63(X, Y, Z) 
  a!6220!6220U64(X, Y, Z) >= U64(X, Y, Z) 
  a!6220!6220plus(X, Y) >= plus(X, Y) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 2 
  U11 = \y0y1y2.y0 
  U12 = \y0y1y2.y0 
  U13 = \y0y1y2.y0 
  U14 = \y0y1y2.2y0 
  U15 = \y0y1.y0 
  U16 = \y0.y0 
  U21 = \y0y1.y0 
  U22 = \y0y1.2y0 
  U23 = \y0.y0 
  U31 = \y0y1.2y0 
  U32 = \y0.y0 
  U41 = \y0.y0 
  U51 = \y0y1.2 + y1 
  U52 = \y0y1.2 + y1 + 2y0 
  U61 = \y0y1y2.y1 + y2 + 2y0 
  U62 = \y0y1y2.y0 + y1 + y2 
  U63 = \y0y1y2.y1 + y2 + 2y0 
  U64 = \y0y1y2.y1 + y2 + 2y0 
  a!6220!6220U11 = \y0y1y2.y0 
  a!6220!6220U12 = \y0y1y2.y0 
  a!6220!6220U13 = \y0y1y2.y0 
  a!6220!6220U14 = \y0y1y2.2y0 
  a!6220!6220U15 = \y0y1.y0 
  a!6220!6220U16 = \y0.y0 
  a!6220!6220U21 = \y0y1.y0 
  a!6220!6220U22 = \y0y1.2y0 
  a!6220!6220U23 = \y0.y0 
  a!6220!6220U31 = \y0y1.2y0 
  a!6220!6220U32 = \y0.y0 
  a!6220!6220U41 = \y0.y0 
  a!6220!6220U51 = \y0y1.2 + y1 
  a!6220!6220U52 = \y0y1.2 + y1 + 2y0 
  a!6220!6220U52# = \y0y1.1 + 2y1 
  a!6220!6220U61 = \y0y1y2.y1 + y2 + 2y0 
  a!6220!6220U61# = \y0y1y2.2y1 + 2y2 
  a!6220!6220U62 = \y0y1y2.y0 + y1 + y2 
  a!6220!6220U62# = \y0y1y2.2y1 + 2y2 
  a!6220!6220U63 = \y0y1y2.y1 + y2 + 2y0 
  a!6220!6220U63# = \y0y1y2.2y1 + 2y2 
  a!6220!6220U64 = \y0y1y2.y1 + y2 + 2y0 
  a!6220!6220U64# = \y0y1y2.2y1 + 2y2 
  a!6220!6220isNat = \y0.0 
  a!6220!6220isNatKind = \y0.0 
  a!6220!6220plus = \y0y1.y0 + y1 
  a!6220!6220plus# = \y0y1.2y0 + 2y1 
  isNat = \y0.0 
  isNatKind = \y0.0 
  mark = \y0.y0 
  mark# = \y0.2y0 
  plus = \y0y1.y0 + y1 
  s = \y0.y0 
  tt = 0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220U52#(tt, _x0)]] = 1 + 2x0 > 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220U61#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U62#(a!6220!6220isNatKind(_x0), _x0, _x1)]] 
  [[a!6220!6220U62#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U63#(a!6220!6220isNat(_x1), _x0, _x1)]] 
  [[a!6220!6220U63#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U64#(a!6220!6220isNatKind(_x1), _x0, _x1)]] 
  [[a!6220!6220U64#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220plus#(mark(_x1), mark(_x0))]] 
  [[a!6220!6220U64#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x1 = [[mark#(_x1)]] 
  [[a!6220!6220U64#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220plus#(_x0, s(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U61#(a!6220!6220isNat(_x1), _x1, _x0)]] 
  [[mark#(U11(_x0, _x1, _x2))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U12(_x0, _x1, _x2))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U13(_x0, _x1, _x2))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U14(_x0, _x1, _x2))]] = 4x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U15(_x0, _x1))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U16(_x0))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U21(_x0, _x1))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U22(_x0, _x1))]] = 4x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U23(_x0))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U31(_x0, _x1))]] = 4x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U32(_x0))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U41(_x0))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U52(_x0, _x1))]] = 4 + 2x1 + 4x0 > 1 + 2x1 = [[a!6220!6220U52#(mark(_x0), _x1)]] 
  [[mark#(U52(_x0, _x1))]] = 4 + 2x1 + 4x0 > 2x0 = [[mark#(_x0)]] 
  [[mark#(U61(_x0, _x1, _x2))]] = 2x1 + 2x2 + 4x0 >= 2x1 + 2x2 = [[a!6220!6220U61#(mark(_x0), _x1, _x2)]] 
  [[mark#(U61(_x0, _x1, _x2))]] = 2x1 + 2x2 + 4x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U62(_x0, _x1, _x2))]] = 2x0 + 2x1 + 2x2 >= 2x1 + 2x2 = [[a!6220!6220U62#(mark(_x0), _x1, _x2)]] 
  [[mark#(U62(_x0, _x1, _x2))]] = 2x0 + 2x1 + 2x2 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U63(_x0, _x1, _x2))]] = 2x1 + 2x2 + 4x0 >= 2x1 + 2x2 = [[a!6220!6220U63#(mark(_x0), _x1, _x2)]] 
  [[mark#(U63(_x0, _x1, _x2))]] = 2x1 + 2x2 + 4x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U64(_x0, _x1, _x2))]] = 2x1 + 2x2 + 4x0 >= 2x1 + 2x2 = [[a!6220!6220U64#(mark(_x0), _x1, _x2)]] 
  [[mark#(U64(_x0, _x1, _x2))]] = 2x1 + 2x2 + 4x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(plus(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220plus#(mark(_x0), mark(_x1))]] 
  [[mark#(plus(_x0, _x1))]] = 2x0 + 2x1 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(plus(_x0, _x1))]] = 2x0 + 2x1 >= 2x1 = [[mark#(_x1)]] 
  [[mark#(s(_x0))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220U11(tt, _x0, _x1)]] = 0 >= 0 = [[a!6220!6220U12(a!6220!6220isNatKind(_x0), _x0, _x1)]] 
  [[a!6220!6220U12(tt, _x0, _x1)]] = 0 >= 0 = [[a!6220!6220U13(a!6220!6220isNatKind(_x1), _x0, _x1)]] 
  [[a!6220!6220U13(tt, _x0, _x1)]] = 0 >= 0 = [[a!6220!6220U14(a!6220!6220isNatKind(_x1), _x0, _x1)]] 
  [[a!6220!6220U14(tt, _x0, _x1)]] = 0 >= 0 = [[a!6220!6220U15(a!6220!6220isNat(_x0), _x1)]] 
  [[a!6220!6220U15(tt, _x0)]] = 0 >= 0 = [[a!6220!6220U16(a!6220!6220isNat(_x0))]] 
  [[a!6220!6220U16(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U21(tt, _x0)]] = 0 >= 0 = [[a!6220!6220U22(a!6220!6220isNatKind(_x0), _x0)]] 
  [[a!6220!6220U22(tt, _x0)]] = 0 >= 0 = [[a!6220!6220U23(a!6220!6220isNat(_x0))]] 
  [[a!6220!6220U23(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U31(tt, _x0)]] = 0 >= 0 = [[a!6220!6220U32(a!6220!6220isNatKind(_x0))]] 
  [[a!6220!6220U32(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U41(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U51(tt, _x0)]] = 2 + x0 >= 2 + x0 = [[a!6220!6220U52(a!6220!6220isNatKind(_x0), _x0)]] 
  [[a!6220!6220U52(tt, _x0)]] = 2 + x0 >= x0 = [[mark(_x0)]] 
  [[a!6220!6220U61(tt, _x0, _x1)]] = x0 + x1 >= x0 + x1 = [[a!6220!6220U62(a!6220!6220isNatKind(_x0), _x0, _x1)]] 
  [[a!6220!6220U62(tt, _x0, _x1)]] = x0 + x1 >= x0 + x1 = [[a!6220!6220U63(a!6220!6220isNat(_x1), _x0, _x1)]] 
  [[a!6220!6220U63(tt, _x0, _x1)]] = x0 + x1 >= x0 + x1 = [[a!6220!6220U64(a!6220!6220isNatKind(_x1), _x0, _x1)]] 
  [[a!6220!6220U64(tt, _x0, _x1)]] = x0 + x1 >= x0 + x1 = [[s(a!6220!6220plus(mark(_x1), mark(_x0)))]] 
  [[a!6220!6220isNat(0)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220isNat(plus(_x0, _x1))]] = 0 >= 0 = [[a!6220!6220U11(a!6220!6220isNatKind(_x0), _x0, _x1)]] 
  [[a!6220!6220isNat(s(_x0))]] = 0 >= 0 = [[a!6220!6220U21(a!6220!6220isNatKind(_x0), _x0)]] 
  [[a!6220!6220isNatKind(0)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220isNatKind(plus(_x0, _x1))]] = 0 >= 0 = [[a!6220!6220U31(a!6220!6220isNatKind(_x0), _x1)]] 
  [[a!6220!6220isNatKind(s(_x0))]] = 0 >= 0 = [[a!6220!6220U41(a!6220!6220isNatKind(_x0))]] 
  [[a!6220!6220plus(_x0, 0)]] = 2 + x0 >= 2 + x0 = [[a!6220!6220U51(a!6220!6220isNat(_x0), _x0)]] 
  [[a!6220!6220plus(_x0, s(_x1))]] = x0 + x1 >= x0 + x1 = [[a!6220!6220U61(a!6220!6220isNat(_x1), _x1, _x0)]] 
  [[mark(U11(_x0, _x1, _x2))]] = x0 >= x0 = [[a!6220!6220U11(mark(_x0), _x1, _x2)]] 
  [[mark(U12(_x0, _x1, _x2))]] = x0 >= x0 = [[a!6220!6220U12(mark(_x0), _x1, _x2)]] 
  [[mark(isNatKind(_x0))]] = 0 >= 0 = [[a!6220!6220isNatKind(_x0)]] 
  [[mark(U13(_x0, _x1, _x2))]] = x0 >= x0 = [[a!6220!6220U13(mark(_x0), _x1, _x2)]] 
  [[mark(U14(_x0, _x1, _x2))]] = 2x0 >= 2x0 = [[a!6220!6220U14(mark(_x0), _x1, _x2)]] 
  [[mark(U15(_x0, _x1))]] = x0 >= x0 = [[a!6220!6220U15(mark(_x0), _x1)]] 
  [[mark(isNat(_x0))]] = 0 >= 0 = [[a!6220!6220isNat(_x0)]] 
  [[mark(U16(_x0))]] = x0 >= x0 = [[a!6220!6220U16(mark(_x0))]] 
  [[mark(U21(_x0, _x1))]] = x0 >= x0 = [[a!6220!6220U21(mark(_x0), _x1)]] 
  [[mark(U22(_x0, _x1))]] = 2x0 >= 2x0 = [[a!6220!6220U22(mark(_x0), _x1)]] 
  [[mark(U23(_x0))]] = x0 >= x0 = [[a!6220!6220U23(mark(_x0))]] 
  [[mark(U31(_x0, _x1))]] = 2x0 >= 2x0 = [[a!6220!6220U31(mark(_x0), _x1)]] 
  [[mark(U32(_x0))]] = x0 >= x0 = [[a!6220!6220U32(mark(_x0))]] 
  [[mark(U41(_x0))]] = x0 >= x0 = [[a!6220!6220U41(mark(_x0))]] 
  [[mark(U51(_x0, _x1))]] = 2 + x1 >= 2 + x1 = [[a!6220!6220U51(mark(_x0), _x1)]] 
  [[mark(U52(_x0, _x1))]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[a!6220!6220U52(mark(_x0), _x1)]] 
  [[mark(U61(_x0, _x1, _x2))]] = x1 + x2 + 2x0 >= x1 + x2 + 2x0 = [[a!6220!6220U61(mark(_x0), _x1, _x2)]] 
  [[mark(U62(_x0, _x1, _x2))]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[a!6220!6220U62(mark(_x0), _x1, _x2)]] 
  [[mark(U63(_x0, _x1, _x2))]] = x1 + x2 + 2x0 >= x1 + x2 + 2x0 = [[a!6220!6220U63(mark(_x0), _x1, _x2)]] 
  [[mark(U64(_x0, _x1, _x2))]] = x1 + x2 + 2x0 >= x1 + x2 + 2x0 = [[a!6220!6220U64(mark(_x0), _x1, _x2)]] 
  [[mark(plus(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[a!6220!6220plus(mark(_x0), mark(_x1))]] 
  [[mark(tt)]] = 0 >= 0 = [[tt]] 
  [[mark(s(_x0))]] = x0 >= x0 = [[s(mark(_x0))]] 
  [[mark(0)]] = 2 >= 2 = [[0]] 
  [[a!6220!6220U11(_x0, _x1, _x2)]] = x0 >= x0 = [[U11(_x0, _x1, _x2)]] 
  [[a!6220!6220U12(_x0, _x1, _x2)]] = x0 >= x0 = [[U12(_x0, _x1, _x2)]] 
  [[a!6220!6220isNatKind(_x0)]] = 0 >= 0 = [[isNatKind(_x0)]] 
  [[a!6220!6220U13(_x0, _x1, _x2)]] = x0 >= x0 = [[U13(_x0, _x1, _x2)]] 
  [[a!6220!6220U14(_x0, _x1, _x2)]] = 2x0 >= 2x0 = [[U14(_x0, _x1, _x2)]] 
  [[a!6220!6220U15(_x0, _x1)]] = x0 >= x0 = [[U15(_x0, _x1)]] 
  [[a!6220!6220isNat(_x0)]] = 0 >= 0 = [[isNat(_x0)]] 
  [[a!6220!6220U16(_x0)]] = x0 >= x0 = [[U16(_x0)]] 
  [[a!6220!6220U21(_x0, _x1)]] = x0 >= x0 = [[U21(_x0, _x1)]] 
  [[a!6220!6220U22(_x0, _x1)]] = 2x0 >= 2x0 = [[U22(_x0, _x1)]] 
  [[a!6220!6220U23(_x0)]] = x0 >= x0 = [[U23(_x0)]] 
  [[a!6220!6220U31(_x0, _x1)]] = 2x0 >= 2x0 = [[U31(_x0, _x1)]] 
  [[a!6220!6220U32(_x0)]] = x0 >= x0 = [[U32(_x0)]] 
  [[a!6220!6220U41(_x0)]] = x0 >= x0 = [[U41(_x0)]] 
  [[a!6220!6220U51(_x0, _x1)]] = 2 + x1 >= 2 + x1 = [[U51(_x0, _x1)]] 
  [[a!6220!6220U52(_x0, _x1)]] = 2 + x1 + 2x0 >= 2 + x1 + 2x0 = [[U52(_x0, _x1)]] 
  [[a!6220!6220U61(_x0, _x1, _x2)]] = x1 + x2 + 2x0 >= x1 + x2 + 2x0 = [[U61(_x0, _x1, _x2)]] 
  [[a!6220!6220U62(_x0, _x1, _x2)]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[U62(_x0, _x1, _x2)]] 
  [[a!6220!6220U63(_x0, _x1, _x2)]] = x1 + x2 + 2x0 >= x1 + x2 + 2x0 = [[U63(_x0, _x1, _x2)]] 
  [[a!6220!6220U64(_x0, _x1, _x2)]] = x1 + x2 + 2x0 >= x1 + x2 + 2x0 = [[U64(_x0, _x1, _x2)]] 
  [[a!6220!6220plus(_x0, _x1)]] = x0 + x1 >= x0 + x1 = [[plus(_x0, _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_4, R_0, minimal, formative) by (P_5, R_0, minimal, formative), where P_5 consists of:

  a!6220!6220U61#(tt, X, Y) =#> a!6220!6220U62#(a!6220!6220isNatKind(X), X, Y)   
  a!6220!6220U62#(tt, X, Y) =#> a!6220!6220U63#(a!6220!6220isNat(Y), X, Y)   
  a!6220!6220U63#(tt, X, Y) =#> a!6220!6220U64#(a!6220!6220isNatKind(Y), X, Y)   
  a!6220!6220U64#(tt, X, Y) =#> a!6220!6220plus#(mark(Y), mark(X))   
  a!6220!6220U64#(tt, X, Y) =#> mark#(Y)   
  a!6220!6220U64#(tt, X, Y) =#> mark#(X)   
  a!6220!6220plus#(X, s(Y)) =#> a!6220!6220U61#(a!6220!6220isNat(Y), Y, X)   
  mark#(U11(X, Y, Z)) =#> mark#(X)   
  mark#(U12(X, Y, Z)) =#> mark#(X)   
  mark#(U13(X, Y, Z)) =#> mark#(X)   
  mark#(U14(X, Y, Z)) =#> mark#(X)   
  mark#(U15(X, Y)) =#> mark#(X)   
  mark#(U16(X)) =#> mark#(X)   
  mark#(U21(X, Y)) =#> mark#(X)   
  mark#(U22(X, Y)) =#> mark#(X)   
  mark#(U23(X)) =#> mark#(X)   
  mark#(U31(X, Y)) =#> mark#(X)   
  mark#(U32(X)) =#> mark#(X)   
  mark#(U41(X)) =#> mark#(X)   
  mark#(U61(X, Y, Z)) =#> a!6220!6220U61#(mark(X), Y, Z)   
  mark#(U61(X, Y, Z)) =#> mark#(X)   
  mark#(U62(X, Y, Z)) =#> a!6220!6220U62#(mark(X), Y, Z)   
  mark#(U62(X, Y, Z)) =#> mark#(X)   
  mark#(U63(X, Y, Z)) =#> a!6220!6220U63#(mark(X), Y, Z)   
  mark#(U63(X, Y, Z)) =#> mark#(X)   
  mark#(U64(X, Y, Z)) =#> a!6220!6220U64#(mark(X), Y, Z)   
  mark#(U64(X, Y, Z)) =#> mark#(X)   
  mark#(plus(X, Y)) =#> a!6220!6220plus#(mark(X), mark(Y))   
  mark#(plus(X, Y)) =#> mark#(X)   
  mark#(plus(X, Y)) =#> mark#(Y)   
  mark#(s(X)) =#> mark#(X)   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_5, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_5, R_0, minimal, formative).

We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  a!6220!6220U61#(tt, X, Y) >? a!6220!6220U62#(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220U62#(tt, X, Y) >? a!6220!6220U63#(a!6220!6220isNat(Y), X, Y) 
  a!6220!6220U63#(tt, X, Y) >? a!6220!6220U64#(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U64#(tt, X, Y) >? a!6220!6220plus#(mark(Y), mark(X)) 
  a!6220!6220U64#(tt, X, Y) >? mark#(Y) 
  a!6220!6220U64#(tt, X, Y) >? mark#(X) 
  a!6220!6220plus#(X, s(Y)) >? a!6220!6220U61#(a!6220!6220isNat(Y), Y, X) 
  mark#(U11(X, Y, Z)) >? mark#(X) 
  mark#(U12(X, Y, Z)) >? mark#(X) 
  mark#(U13(X, Y, Z)) >? mark#(X) 
  mark#(U14(X, Y, Z)) >? mark#(X) 
  mark#(U15(X, Y)) >? mark#(X) 
  mark#(U16(X)) >? mark#(X) 
  mark#(U21(X, Y)) >? mark#(X) 
  mark#(U22(X, Y)) >? mark#(X) 
  mark#(U23(X)) >? mark#(X) 
  mark#(U31(X, Y)) >? mark#(X) 
  mark#(U32(X)) >? mark#(X) 
  mark#(U41(X)) >? mark#(X) 
  mark#(U61(X, Y, Z)) >? a!6220!6220U61#(mark(X), Y, Z) 
  mark#(U61(X, Y, Z)) >? mark#(X) 
  mark#(U62(X, Y, Z)) >? a!6220!6220U62#(mark(X), Y, Z) 
  mark#(U62(X, Y, Z)) >? mark#(X) 
  mark#(U63(X, Y, Z)) >? a!6220!6220U63#(mark(X), Y, Z) 
  mark#(U63(X, Y, Z)) >? mark#(X) 
  mark#(U64(X, Y, Z)) >? a!6220!6220U64#(mark(X), Y, Z) 
  mark#(U64(X, Y, Z)) >? mark#(X) 
  mark#(plus(X, Y)) >? a!6220!6220plus#(mark(X), mark(Y)) 
  mark#(plus(X, Y)) >? mark#(X) 
  mark#(plus(X, Y)) >? mark#(Y) 
  mark#(s(X)) >? mark#(X) 
  a!6220!6220U11(tt, X, Y) >= a!6220!6220U12(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220U12(tt, X, Y) >= a!6220!6220U13(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U13(tt, X, Y) >= a!6220!6220U14(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U14(tt, X, Y) >= a!6220!6220U15(a!6220!6220isNat(X), Y) 
  a!6220!6220U15(tt, X) >= a!6220!6220U16(a!6220!6220isNat(X)) 
  a!6220!6220U16(tt) >= tt 
  a!6220!6220U21(tt, X) >= a!6220!6220U22(a!6220!6220isNatKind(X), X) 
  a!6220!6220U22(tt, X) >= a!6220!6220U23(a!6220!6220isNat(X)) 
  a!6220!6220U23(tt) >= tt 
  a!6220!6220U31(tt, X) >= a!6220!6220U32(a!6220!6220isNatKind(X)) 
  a!6220!6220U32(tt) >= tt 
  a!6220!6220U41(tt) >= tt 
  a!6220!6220U51(tt, X) >= a!6220!6220U52(a!6220!6220isNatKind(X), X) 
  a!6220!6220U52(tt, X) >= mark(X) 
  a!6220!6220U61(tt, X, Y) >= a!6220!6220U62(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220U62(tt, X, Y) >= a!6220!6220U63(a!6220!6220isNat(Y), X, Y) 
  a!6220!6220U63(tt, X, Y) >= a!6220!6220U64(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U64(tt, X, Y) >= s(a!6220!6220plus(mark(Y), mark(X))) 
  a!6220!6220isNat(0) >= tt 
  a!6220!6220isNat(plus(X, Y)) >= a!6220!6220U11(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220isNat(s(X)) >= a!6220!6220U21(a!6220!6220isNatKind(X), X) 
  a!6220!6220isNatKind(0) >= tt 
  a!6220!6220isNatKind(plus(X, Y)) >= a!6220!6220U31(a!6220!6220isNatKind(X), Y) 
  a!6220!6220isNatKind(s(X)) >= a!6220!6220U41(a!6220!6220isNatKind(X)) 
  a!6220!6220plus(X, 0) >= a!6220!6220U51(a!6220!6220isNat(X), X) 
  a!6220!6220plus(X, s(Y)) >= a!6220!6220U61(a!6220!6220isNat(Y), Y, X) 
  mark(U11(X, Y, Z)) >= a!6220!6220U11(mark(X), Y, Z) 
  mark(U12(X, Y, Z)) >= a!6220!6220U12(mark(X), Y, Z) 
  mark(isNatKind(X)) >= a!6220!6220isNatKind(X) 
  mark(U13(X, Y, Z)) >= a!6220!6220U13(mark(X), Y, Z) 
  mark(U14(X, Y, Z)) >= a!6220!6220U14(mark(X), Y, Z) 
  mark(U15(X, Y)) >= a!6220!6220U15(mark(X), Y) 
  mark(isNat(X)) >= a!6220!6220isNat(X) 
  mark(U16(X)) >= a!6220!6220U16(mark(X)) 
  mark(U21(X, Y)) >= a!6220!6220U21(mark(X), Y) 
  mark(U22(X, Y)) >= a!6220!6220U22(mark(X), Y) 
  mark(U23(X)) >= a!6220!6220U23(mark(X)) 
  mark(U31(X, Y)) >= a!6220!6220U31(mark(X), Y) 
  mark(U32(X)) >= a!6220!6220U32(mark(X)) 
  mark(U41(X)) >= a!6220!6220U41(mark(X)) 
  mark(U51(X, Y)) >= a!6220!6220U51(mark(X), Y) 
  mark(U52(X, Y)) >= a!6220!6220U52(mark(X), Y) 
  mark(U61(X, Y, Z)) >= a!6220!6220U61(mark(X), Y, Z) 
  mark(U62(X, Y, Z)) >= a!6220!6220U62(mark(X), Y, Z) 
  mark(U63(X, Y, Z)) >= a!6220!6220U63(mark(X), Y, Z) 
  mark(U64(X, Y, Z)) >= a!6220!6220U64(mark(X), Y, Z) 
  mark(plus(X, Y)) >= a!6220!6220plus(mark(X), mark(Y)) 
  mark(tt) >= tt 
  mark(s(X)) >= s(mark(X)) 
  mark(0) >= 0 
  a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) 
  a!6220!6220U12(X, Y, Z) >= U12(X, Y, Z) 
  a!6220!6220isNatKind(X) >= isNatKind(X) 
  a!6220!6220U13(X, Y, Z) >= U13(X, Y, Z) 
  a!6220!6220U14(X, Y, Z) >= U14(X, Y, Z) 
  a!6220!6220U15(X, Y) >= U15(X, Y) 
  a!6220!6220isNat(X) >= isNat(X) 
  a!6220!6220U16(X) >= U16(X) 
  a!6220!6220U21(X, Y) >= U21(X, Y) 
  a!6220!6220U22(X, Y) >= U22(X, Y) 
  a!6220!6220U23(X) >= U23(X) 
  a!6220!6220U31(X, Y) >= U31(X, Y) 
  a!6220!6220U32(X) >= U32(X) 
  a!6220!6220U41(X) >= U41(X) 
  a!6220!6220U51(X, Y) >= U51(X, Y) 
  a!6220!6220U52(X, Y) >= U52(X, Y) 
  a!6220!6220U61(X, Y, Z) >= U61(X, Y, Z) 
  a!6220!6220U62(X, Y, Z) >= U62(X, Y, Z) 
  a!6220!6220U63(X, Y, Z) >= U63(X, Y, Z) 
  a!6220!6220U64(X, Y, Z) >= U64(X, Y, Z) 
  a!6220!6220plus(X, Y) >= plus(X, Y) 

We orient these requirements with a polynomial interpretation in the natural numbers.

The following interpretation satisfies the requirements:

  0 = 0 
  U11 = \y0y1y2.y0 
  U12 = \y0y1y2.y0 
  U13 = \y0y1y2.y0 
  U14 = \y0y1y2.y0 
  U15 = \y0y1.y0 
  U16 = \y0.y0 
  U21 = \y0y1.y0 
  U22 = \y0y1.2y0 
  U23 = \y0.2y0 
  U31 = \y0y1.y0 
  U32 = \y0.y0 
  U41 = \y0.2y0 
  U51 = \y0y1.y1 
  U52 = \y0y1.y1 
  U61 = \y0y1y2.1 + y2 + 2y0 + 2y1 
  U62 = \y0y1y2.1 + y0 + y2 + 2y1 
  U63 = \y0y1y2.1 + y0 + y2 + 2y1 
  U64 = \y0y1y2.1 + y0 + y2 + 2y1 
  a!6220!6220U11 = \y0y1y2.y0 
  a!6220!6220U12 = \y0y1y2.y0 
  a!6220!6220U13 = \y0y1y2.y0 
  a!6220!6220U14 = \y0y1y2.y0 
  a!6220!6220U15 = \y0y1.y0 
  a!6220!6220U16 = \y0.y0 
  a!6220!6220U21 = \y0y1.y0 
  a!6220!6220U22 = \y0y1.2y0 
  a!6220!6220U23 = \y0.2y0 
  a!6220!6220U31 = \y0y1.y0 
  a!6220!6220U32 = \y0.y0 
  a!6220!6220U41 = \y0.2y0 
  a!6220!6220U51 = \y0y1.y1 
  a!6220!6220U52 = \y0y1.y1 
  a!6220!6220U61 = \y0y1y2.1 + y2 + 2y0 + 2y1 
  a!6220!6220U61# = \y0y1y2.y2 + 2y1 
  a!6220!6220U62 = \y0y1y2.1 + y0 + y2 + 2y1 
  a!6220!6220U62# = \y0y1y2.y2 + 2y1 
  a!6220!6220U63 = \y0y1y2.1 + y0 + y2 + 2y1 
  a!6220!6220U63# = \y0y1y2.y2 + 2y1 
  a!6220!6220U64 = \y0y1y2.1 + y0 + y2 + 2y1 
  a!6220!6220U64# = \y0y1y2.y2 + 2y1 
  a!6220!6220isNat = \y0.0 
  a!6220!6220isNatKind = \y0.0 
  a!6220!6220plus = \y0y1.y0 + 2y1 
  a!6220!6220plus# = \y0y1.y0 + 2y1 
  isNat = \y0.0 
  isNatKind = \y0.0 
  mark = \y0.y0 
  mark# = \y0.y0 
  plus = \y0y1.y0 + 2y1 
  s = \y0.1 + y0 
  tt = 0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220U61#(tt, _x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220U62#(a!6220!6220isNatKind(_x0), _x0, _x1)]] 
  [[a!6220!6220U62#(tt, _x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220U63#(a!6220!6220isNat(_x1), _x0, _x1)]] 
  [[a!6220!6220U63#(tt, _x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220U64#(a!6220!6220isNatKind(_x1), _x0, _x1)]] 
  [[a!6220!6220U64#(tt, _x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220plus#(mark(_x1), mark(_x0))]] 
  [[a!6220!6220U64#(tt, _x0, _x1)]] = x1 + 2x0 >= x1 = [[mark#(_x1)]] 
  [[a!6220!6220U64#(tt, _x0, _x1)]] = x1 + 2x0 >= x0 = [[mark#(_x0)]] 
  [[a!6220!6220plus#(_x0, s(_x1))]] = 2 + x0 + 2x1 > x0 + 2x1 = [[a!6220!6220U61#(a!6220!6220isNat(_x1), _x1, _x0)]] 
  [[mark#(U11(_x0, _x1, _x2))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(U12(_x0, _x1, _x2))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(U13(_x0, _x1, _x2))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(U14(_x0, _x1, _x2))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(U15(_x0, _x1))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(U16(_x0))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(U21(_x0, _x1))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(U22(_x0, _x1))]] = 2x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(U23(_x0))]] = 2x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(U31(_x0, _x1))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(U32(_x0))]] = x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(U41(_x0))]] = 2x0 >= x0 = [[mark#(_x0)]] 
  [[mark#(U61(_x0, _x1, _x2))]] = 1 + x2 + 2x0 + 2x1 > x2 + 2x1 = [[a!6220!6220U61#(mark(_x0), _x1, _x2)]] 
  [[mark#(U61(_x0, _x1, _x2))]] = 1 + x2 + 2x0 + 2x1 > x0 = [[mark#(_x0)]] 
  [[mark#(U62(_x0, _x1, _x2))]] = 1 + x0 + x2 + 2x1 > x2 + 2x1 = [[a!6220!6220U62#(mark(_x0), _x1, _x2)]] 
  [[mark#(U62(_x0, _x1, _x2))]] = 1 + x0 + x2 + 2x1 > x0 = [[mark#(_x0)]] 
  [[mark#(U63(_x0, _x1, _x2))]] = 1 + x0 + x2 + 2x1 > x2 + 2x1 = [[a!6220!6220U63#(mark(_x0), _x1, _x2)]] 
  [[mark#(U63(_x0, _x1, _x2))]] = 1 + x0 + x2 + 2x1 > x0 = [[mark#(_x0)]] 
  [[mark#(U64(_x0, _x1, _x2))]] = 1 + x0 + x2 + 2x1 > x2 + 2x1 = [[a!6220!6220U64#(mark(_x0), _x1, _x2)]] 
  [[mark#(U64(_x0, _x1, _x2))]] = 1 + x0 + x2 + 2x1 > x0 = [[mark#(_x0)]] 
  [[mark#(plus(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[a!6220!6220plus#(mark(_x0), mark(_x1))]] 
  [[mark#(plus(_x0, _x1))]] = x0 + 2x1 >= x0 = [[mark#(_x0)]] 
  [[mark#(plus(_x0, _x1))]] = x0 + 2x1 >= x1 = [[mark#(_x1)]] 
  [[mark#(s(_x0))]] = 1 + x0 > x0 = [[mark#(_x0)]] 
  [[a!6220!6220U11(tt, _x0, _x1)]] = 0 >= 0 = [[a!6220!6220U12(a!6220!6220isNatKind(_x0), _x0, _x1)]] 
  [[a!6220!6220U12(tt, _x0, _x1)]] = 0 >= 0 = [[a!6220!6220U13(a!6220!6220isNatKind(_x1), _x0, _x1)]] 
  [[a!6220!6220U13(tt, _x0, _x1)]] = 0 >= 0 = [[a!6220!6220U14(a!6220!6220isNatKind(_x1), _x0, _x1)]] 
  [[a!6220!6220U14(tt, _x0, _x1)]] = 0 >= 0 = [[a!6220!6220U15(a!6220!6220isNat(_x0), _x1)]] 
  [[a!6220!6220U15(tt, _x0)]] = 0 >= 0 = [[a!6220!6220U16(a!6220!6220isNat(_x0))]] 
  [[a!6220!6220U16(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U21(tt, _x0)]] = 0 >= 0 = [[a!6220!6220U22(a!6220!6220isNatKind(_x0), _x0)]] 
  [[a!6220!6220U22(tt, _x0)]] = 0 >= 0 = [[a!6220!6220U23(a!6220!6220isNat(_x0))]] 
  [[a!6220!6220U23(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U31(tt, _x0)]] = 0 >= 0 = [[a!6220!6220U32(a!6220!6220isNatKind(_x0))]] 
  [[a!6220!6220U32(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U41(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U51(tt, _x0)]] = x0 >= x0 = [[a!6220!6220U52(a!6220!6220isNatKind(_x0), _x0)]] 
  [[a!6220!6220U52(tt, _x0)]] = x0 >= x0 = [[mark(_x0)]] 
  [[a!6220!6220U61(tt, _x0, _x1)]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[a!6220!6220U62(a!6220!6220isNatKind(_x0), _x0, _x1)]] 
  [[a!6220!6220U62(tt, _x0, _x1)]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[a!6220!6220U63(a!6220!6220isNat(_x1), _x0, _x1)]] 
  [[a!6220!6220U63(tt, _x0, _x1)]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[a!6220!6220U64(a!6220!6220isNatKind(_x1), _x0, _x1)]] 
  [[a!6220!6220U64(tt, _x0, _x1)]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[s(a!6220!6220plus(mark(_x1), mark(_x0)))]] 
  [[a!6220!6220isNat(0)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220isNat(plus(_x0, _x1))]] = 0 >= 0 = [[a!6220!6220U11(a!6220!6220isNatKind(_x0), _x0, _x1)]] 
  [[a!6220!6220isNat(s(_x0))]] = 0 >= 0 = [[a!6220!6220U21(a!6220!6220isNatKind(_x0), _x0)]] 
  [[a!6220!6220isNatKind(0)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220isNatKind(plus(_x0, _x1))]] = 0 >= 0 = [[a!6220!6220U31(a!6220!6220isNatKind(_x0), _x1)]] 
  [[a!6220!6220isNatKind(s(_x0))]] = 0 >= 0 = [[a!6220!6220U41(a!6220!6220isNatKind(_x0))]] 
  [[a!6220!6220plus(_x0, 0)]] = x0 >= x0 = [[a!6220!6220U51(a!6220!6220isNat(_x0), _x0)]] 
  [[a!6220!6220plus(_x0, s(_x1))]] = 2 + x0 + 2x1 >= 1 + x0 + 2x1 = [[a!6220!6220U61(a!6220!6220isNat(_x1), _x1, _x0)]] 
  [[mark(U11(_x0, _x1, _x2))]] = x0 >= x0 = [[a!6220!6220U11(mark(_x0), _x1, _x2)]] 
  [[mark(U12(_x0, _x1, _x2))]] = x0 >= x0 = [[a!6220!6220U12(mark(_x0), _x1, _x2)]] 
  [[mark(isNatKind(_x0))]] = 0 >= 0 = [[a!6220!6220isNatKind(_x0)]] 
  [[mark(U13(_x0, _x1, _x2))]] = x0 >= x0 = [[a!6220!6220U13(mark(_x0), _x1, _x2)]] 
  [[mark(U14(_x0, _x1, _x2))]] = x0 >= x0 = [[a!6220!6220U14(mark(_x0), _x1, _x2)]] 
  [[mark(U15(_x0, _x1))]] = x0 >= x0 = [[a!6220!6220U15(mark(_x0), _x1)]] 
  [[mark(isNat(_x0))]] = 0 >= 0 = [[a!6220!6220isNat(_x0)]] 
  [[mark(U16(_x0))]] = x0 >= x0 = [[a!6220!6220U16(mark(_x0))]] 
  [[mark(U21(_x0, _x1))]] = x0 >= x0 = [[a!6220!6220U21(mark(_x0), _x1)]] 
  [[mark(U22(_x0, _x1))]] = 2x0 >= 2x0 = [[a!6220!6220U22(mark(_x0), _x1)]] 
  [[mark(U23(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U23(mark(_x0))]] 
  [[mark(U31(_x0, _x1))]] = x0 >= x0 = [[a!6220!6220U31(mark(_x0), _x1)]] 
  [[mark(U32(_x0))]] = x0 >= x0 = [[a!6220!6220U32(mark(_x0))]] 
  [[mark(U41(_x0))]] = 2x0 >= 2x0 = [[a!6220!6220U41(mark(_x0))]] 
  [[mark(U51(_x0, _x1))]] = x1 >= x1 = [[a!6220!6220U51(mark(_x0), _x1)]] 
  [[mark(U52(_x0, _x1))]] = x1 >= x1 = [[a!6220!6220U52(mark(_x0), _x1)]] 
  [[mark(U61(_x0, _x1, _x2))]] = 1 + x2 + 2x0 + 2x1 >= 1 + x2 + 2x0 + 2x1 = [[a!6220!6220U61(mark(_x0), _x1, _x2)]] 
  [[mark(U62(_x0, _x1, _x2))]] = 1 + x0 + x2 + 2x1 >= 1 + x0 + x2 + 2x1 = [[a!6220!6220U62(mark(_x0), _x1, _x2)]] 
  [[mark(U63(_x0, _x1, _x2))]] = 1 + x0 + x2 + 2x1 >= 1 + x0 + x2 + 2x1 = [[a!6220!6220U63(mark(_x0), _x1, _x2)]] 
  [[mark(U64(_x0, _x1, _x2))]] = 1 + x0 + x2 + 2x1 >= 1 + x0 + x2 + 2x1 = [[a!6220!6220U64(mark(_x0), _x1, _x2)]] 
  [[mark(plus(_x0, _x1))]] = x0 + 2x1 >= x0 + 2x1 = [[a!6220!6220plus(mark(_x0), mark(_x1))]] 
  [[mark(tt)]] = 0 >= 0 = [[tt]] 
  [[mark(s(_x0))]] = 1 + x0 >= 1 + x0 = [[s(mark(_x0))]] 
  [[mark(0)]] = 0 >= 0 = [[0]] 
  [[a!6220!6220U11(_x0, _x1, _x2)]] = x0 >= x0 = [[U11(_x0, _x1, _x2)]] 
  [[a!6220!6220U12(_x0, _x1, _x2)]] = x0 >= x0 = [[U12(_x0, _x1, _x2)]] 
  [[a!6220!6220isNatKind(_x0)]] = 0 >= 0 = [[isNatKind(_x0)]] 
  [[a!6220!6220U13(_x0, _x1, _x2)]] = x0 >= x0 = [[U13(_x0, _x1, _x2)]] 
  [[a!6220!6220U14(_x0, _x1, _x2)]] = x0 >= x0 = [[U14(_x0, _x1, _x2)]] 
  [[a!6220!6220U15(_x0, _x1)]] = x0 >= x0 = [[U15(_x0, _x1)]] 
  [[a!6220!6220isNat(_x0)]] = 0 >= 0 = [[isNat(_x0)]] 
  [[a!6220!6220U16(_x0)]] = x0 >= x0 = [[U16(_x0)]] 
  [[a!6220!6220U21(_x0, _x1)]] = x0 >= x0 = [[U21(_x0, _x1)]] 
  [[a!6220!6220U22(_x0, _x1)]] = 2x0 >= 2x0 = [[U22(_x0, _x1)]] 
  [[a!6220!6220U23(_x0)]] = 2x0 >= 2x0 = [[U23(_x0)]] 
  [[a!6220!6220U31(_x0, _x1)]] = x0 >= x0 = [[U31(_x0, _x1)]] 
  [[a!6220!6220U32(_x0)]] = x0 >= x0 = [[U32(_x0)]] 
  [[a!6220!6220U41(_x0)]] = 2x0 >= 2x0 = [[U41(_x0)]] 
  [[a!6220!6220U51(_x0, _x1)]] = x1 >= x1 = [[U51(_x0, _x1)]] 
  [[a!6220!6220U52(_x0, _x1)]] = x1 >= x1 = [[U52(_x0, _x1)]] 
  [[a!6220!6220U61(_x0, _x1, _x2)]] = 1 + x2 + 2x0 + 2x1 >= 1 + x2 + 2x0 + 2x1 = [[U61(_x0, _x1, _x2)]] 
  [[a!6220!6220U62(_x0, _x1, _x2)]] = 1 + x0 + x2 + 2x1 >= 1 + x0 + x2 + 2x1 = [[U62(_x0, _x1, _x2)]] 
  [[a!6220!6220U63(_x0, _x1, _x2)]] = 1 + x0 + x2 + 2x1 >= 1 + x0 + x2 + 2x1 = [[U63(_x0, _x1, _x2)]] 
  [[a!6220!6220U64(_x0, _x1, _x2)]] = 1 + x0 + x2 + 2x1 >= 1 + x0 + x2 + 2x1 = [[U64(_x0, _x1, _x2)]] 
  [[a!6220!6220plus(_x0, _x1)]] = x0 + 2x1 >= x0 + 2x1 = [[plus(_x0, _x1)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_5, R_0, minimal, formative) by (P_6, R_0, minimal, formative), where P_6 consists of:

  a!6220!6220U61#(tt, X, Y) =#> a!6220!6220U62#(a!6220!6220isNatKind(X), X, Y)   
  a!6220!6220U62#(tt, X, Y) =#> a!6220!6220U63#(a!6220!6220isNat(Y), X, Y)   
  a!6220!6220U63#(tt, X, Y) =#> a!6220!6220U64#(a!6220!6220isNatKind(Y), X, Y)   
  a!6220!6220U64#(tt, X, Y) =#> a!6220!6220plus#(mark(Y), mark(X))   
  a!6220!6220U64#(tt, X, Y) =#> mark#(Y)   
  a!6220!6220U64#(tt, X, Y) =#> mark#(X)   
  mark#(U11(X, Y, Z)) =#> mark#(X)   
  mark#(U12(X, Y, Z)) =#> mark#(X)   
  mark#(U13(X, Y, Z)) =#> mark#(X)   
  mark#(U14(X, Y, Z)) =#> mark#(X)   
  mark#(U15(X, Y)) =#> mark#(X)   
  mark#(U16(X)) =#> mark#(X)   
  mark#(U21(X, Y)) =#> mark#(X)   
  mark#(U22(X, Y)) =#> mark#(X)   
  mark#(U23(X)) =#> mark#(X)   
  mark#(U31(X, Y)) =#> mark#(X)   
  mark#(U32(X)) =#> mark#(X)   
  mark#(U41(X)) =#> mark#(X)   
  mark#(plus(X, Y)) =#> a!6220!6220plus#(mark(X), mark(Y))   
  mark#(plus(X, Y)) =#> mark#(X)   
  mark#(plus(X, Y)) =#> mark#(Y)   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_6, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_6, R_0, minimal, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 1 
    * 1 : 2 
    * 2 : 3, 4, 5 
    * 3 :  
    * 4 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 
    * 5 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 
    * 6 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 
    * 7 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 
    * 8 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 
    * 9 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 
    * 10 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 
    * 11 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 
    * 12 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 
    * 13 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 
    * 14 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 
    * 15 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 
    * 16 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 
    * 17 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 
    * 18 :  
    * 19 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 
    * 20 : 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 

This graph has the following strongly connected components:

  P_7:

    mark#(U11(X, Y, Z)) =#> mark#(X)   
    mark#(U12(X, Y, Z)) =#> mark#(X)   
    mark#(U13(X, Y, Z)) =#> mark#(X)   
    mark#(U14(X, Y, Z)) =#> mark#(X)   
    mark#(U15(X, Y)) =#> mark#(X)   
    mark#(U16(X)) =#> mark#(X)   
    mark#(U21(X, Y)) =#> mark#(X)   
    mark#(U22(X, Y)) =#> mark#(X)   
    mark#(U23(X)) =#> mark#(X)   
    mark#(U31(X, Y)) =#> mark#(X)   
    mark#(U32(X)) =#> mark#(X)   
    mark#(U41(X)) =#> mark#(X)   
    mark#(plus(X, Y)) =#> mark#(X)   
    mark#(plus(X, Y)) =#> mark#(Y)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_6, R_0, m, f) by (P_7, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_2, R_0, minimal, formative) and (P_7, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_7, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(mark#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(mark#(U11(X, Y, Z))) = U11(X, Y, Z) |> X = nu(mark#(X)) 
  nu(mark#(U12(X, Y, Z))) = U12(X, Y, Z) |> X = nu(mark#(X)) 
  nu(mark#(U13(X, Y, Z))) = U13(X, Y, Z) |> X = nu(mark#(X)) 
  nu(mark#(U14(X, Y, Z))) = U14(X, Y, Z) |> X = nu(mark#(X)) 
  nu(mark#(U15(X, Y))) = U15(X, Y) |> X = nu(mark#(X)) 
  nu(mark#(U16(X))) = U16(X) |> X = nu(mark#(X)) 
  nu(mark#(U21(X, Y))) = U21(X, Y) |> X = nu(mark#(X)) 
  nu(mark#(U22(X, Y))) = U22(X, Y) |> X = nu(mark#(X)) 
  nu(mark#(U23(X))) = U23(X) |> X = nu(mark#(X)) 
  nu(mark#(U31(X, Y))) = U31(X, Y) |> X = nu(mark#(X)) 
  nu(mark#(U32(X))) = U32(X) |> X = nu(mark#(X)) 
  nu(mark#(U41(X))) = U41(X) |> X = nu(mark#(X)) 
  nu(mark#(plus(X, Y))) = plus(X, Y) |> X = nu(mark#(X)) 
  nu(mark#(plus(X, Y))) = plus(X, Y) |> Y = nu(mark#(Y)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_7, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_2, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_2, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(a!6220!6220U31#) = 2 
  nu(a!6220!6220isNatKind#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(a!6220!6220U31#(tt, X)) = X = X = nu(a!6220!6220isNatKind#(X)) 
  nu(a!6220!6220isNatKind#(plus(X, Y))) = plus(X, Y) |> Y = nu(a!6220!6220U31#(a!6220!6220isNatKind(X), Y)) 
  nu(a!6220!6220isNatKind#(plus(X, Y))) = plus(X, Y) |> X = nu(a!6220!6220isNatKind#(X)) 
  nu(a!6220!6220isNatKind#(s(X))) = s(X) |> X = nu(a!6220!6220isNatKind#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_2, R_0, minimal, f) by (P_8, R_0, minimal, f), where P_8 contains:

  a!6220!6220U31#(tt, X) =#> a!6220!6220isNatKind#(X)   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative) and (P_8, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_8, R_0, minimal, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 :  

This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.

Thus, the original system is terminating if (P_1, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_1, R_0, minimal, formative).

We will use the reduction pair processor with usable rules [Kop12, Thm. 7.44].  The usable rules of (P_1, R_0) are:

  a!6220!6220U11(tt, X, Y) => a!6220!6220U12(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220U12(tt, X, Y) => a!6220!6220U13(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U13(tt, X, Y) => a!6220!6220U14(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U14(tt, X, Y) => a!6220!6220U15(a!6220!6220isNat(X), Y) 
  a!6220!6220U15(tt, X) => a!6220!6220U16(a!6220!6220isNat(X)) 
  a!6220!6220U16(tt) => tt 
  a!6220!6220U21(tt, X) => a!6220!6220U22(a!6220!6220isNatKind(X), X) 
  a!6220!6220U22(tt, X) => a!6220!6220U23(a!6220!6220isNat(X)) 
  a!6220!6220U23(tt) => tt 
  a!6220!6220U31(tt, X) => a!6220!6220U32(a!6220!6220isNatKind(X)) 
  a!6220!6220U32(tt) => tt 
  a!6220!6220U41(tt) => tt 
  a!6220!6220isNat(0) => tt 
  a!6220!6220isNat(plus(X, Y)) => a!6220!6220U11(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220isNat(s(X)) => a!6220!6220U21(a!6220!6220isNatKind(X), X) 
  a!6220!6220isNatKind(0) => tt 
  a!6220!6220isNatKind(plus(X, Y)) => a!6220!6220U31(a!6220!6220isNatKind(X), Y) 
  a!6220!6220isNatKind(s(X)) => a!6220!6220U41(a!6220!6220isNatKind(X)) 
  a!6220!6220U11(X, Y, Z) => U11(X, Y, Z) 
  a!6220!6220U12(X, Y, Z) => U12(X, Y, Z) 
  a!6220!6220isNatKind(X) => isNatKind(X) 
  a!6220!6220U13(X, Y, Z) => U13(X, Y, Z) 
  a!6220!6220U14(X, Y, Z) => U14(X, Y, Z) 
  a!6220!6220U15(X, Y) => U15(X, Y) 
  a!6220!6220isNat(X) => isNat(X) 
  a!6220!6220U16(X) => U16(X) 
  a!6220!6220U21(X, Y) => U21(X, Y) 
  a!6220!6220U22(X, Y) => U22(X, Y) 
  a!6220!6220U23(X) => U23(X) 
  a!6220!6220U31(X, Y) => U31(X, Y) 
  a!6220!6220U32(X) => U32(X) 
  a!6220!6220U41(X) => U41(X) 

It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:

  a!6220!6220U11#(tt, X, Y) >? a!6220!6220U12#(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220U12#(tt, X, Y) >? a!6220!6220U13#(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U13#(tt, X, Y) >? a!6220!6220U14#(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U14#(tt, X, Y) >? a!6220!6220U15#(a!6220!6220isNat(X), Y) 
  a!6220!6220U14#(tt, X, Y) >? a!6220!6220isNat#(X) 
  a!6220!6220U15#(tt, X) >? a!6220!6220isNat#(X) 
  a!6220!6220U21#(tt, X) >? a!6220!6220U22#(a!6220!6220isNatKind(X), X) 
  a!6220!6220U22#(tt, X) >? a!6220!6220isNat#(X) 
  a!6220!6220isNat#(plus(X, Y)) >? a!6220!6220U11#(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220isNat#(s(X)) >? a!6220!6220U21#(a!6220!6220isNatKind(X), X) 
  a!6220!6220U11(tt, X, Y) >= a!6220!6220U12(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220U12(tt, X, Y) >= a!6220!6220U13(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U13(tt, X, Y) >= a!6220!6220U14(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U14(tt, X, Y) >= a!6220!6220U15(a!6220!6220isNat(X), Y) 
  a!6220!6220U15(tt, X) >= a!6220!6220U16(a!6220!6220isNat(X)) 
  a!6220!6220U16(tt) >= tt 
  a!6220!6220U21(tt, X) >= a!6220!6220U22(a!6220!6220isNatKind(X), X) 
  a!6220!6220U22(tt, X) >= a!6220!6220U23(a!6220!6220isNat(X)) 
  a!6220!6220U23(tt) >= tt 
  a!6220!6220U31(tt, X) >= a!6220!6220U32(a!6220!6220isNatKind(X)) 
  a!6220!6220U32(tt) >= tt 
  a!6220!6220U41(tt) >= tt 
  a!6220!6220isNat(0) >= tt 
  a!6220!6220isNat(plus(X, Y)) >= a!6220!6220U11(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220isNat(s(X)) >= a!6220!6220U21(a!6220!6220isNatKind(X), X) 
  a!6220!6220isNatKind(0) >= tt 
  a!6220!6220isNatKind(plus(X, Y)) >= a!6220!6220U31(a!6220!6220isNatKind(X), Y) 
  a!6220!6220isNatKind(s(X)) >= a!6220!6220U41(a!6220!6220isNatKind(X)) 
  a!6220!6220U11(X, Y, Z) >= U11(X, Y, Z) 
  a!6220!6220U12(X, Y, Z) >= U12(X, Y, Z) 
  a!6220!6220isNatKind(X) >= isNatKind(X) 
  a!6220!6220U13(X, Y, Z) >= U13(X, Y, Z) 
  a!6220!6220U14(X, Y, Z) >= U14(X, Y, Z) 
  a!6220!6220U15(X, Y) >= U15(X, Y) 
  a!6220!6220isNat(X) >= isNat(X) 
  a!6220!6220U16(X) >= U16(X) 
  a!6220!6220U21(X, Y) >= U21(X, Y) 
  a!6220!6220U22(X, Y) >= U22(X, Y) 
  a!6220!6220U23(X) >= U23(X) 
  a!6220!6220U31(X, Y) >= U31(X, Y) 
  a!6220!6220U32(X) >= U32(X) 
  a!6220!6220U41(X) >= U41(X) 

We orient these requirements with a polynomial interpretation in the natural numbers.

We consider usable_rules with respect to the following argument filtering:

  a!6220!6220U11#(x_1,x_2,x_3) = a!6220!6220U11#(x_2x_3) 
  a!6220!6220U12#(x_1,x_2,x_3) = a!6220!6220U12#(x_2x_3) 
  a!6220!6220U13#(x_1,x_2,x_3) = a!6220!6220U13#(x_2x_3) 
  a!6220!6220U14#(x_1,x_2,x_3) = a!6220!6220U14#(x_2x_3) 
  a!6220!6220U15#(x_1,x_2) = a!6220!6220U15#(x_2) 
  a!6220!6220U21#(x_1,x_2) = a!6220!6220U21#(x_2) 
  a!6220!6220U22#(x_1,x_2) = a!6220!6220U22#(x_2) 

This leaves the following ordering requirements:

  a!6220!6220U11#(tt, X, Y) >= a!6220!6220U12#(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220U12#(tt, X, Y) >= a!6220!6220U13#(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U13#(tt, X, Y) >= a!6220!6220U14#(a!6220!6220isNatKind(Y), X, Y) 
  a!6220!6220U14#(tt, X, Y) >= a!6220!6220U15#(a!6220!6220isNat(X), Y) 
  a!6220!6220U14#(tt, X, Y) >= a!6220!6220isNat#(X) 
  a!6220!6220U15#(tt, X) >= a!6220!6220isNat#(X) 
  a!6220!6220U21#(tt, X) >= a!6220!6220U22#(a!6220!6220isNatKind(X), X) 
  a!6220!6220U22#(tt, X) >= a!6220!6220isNat#(X) 
  a!6220!6220isNat#(plus(X, Y)) >= a!6220!6220U11#(a!6220!6220isNatKind(X), X, Y) 
  a!6220!6220isNat#(s(X)) > a!6220!6220U21#(a!6220!6220isNatKind(X), X) 

The following interpretation satisfies the requirements:

  0 = 3 
  U11 = \y0y1y2.0 
  U12 = \y0y1y2.0 
  U13 = \y0y1y2.0 
  U14 = \y0y1y2.0 
  U15 = \y0y1.0 
  U16 = \y0.0 
  U21 = \y0y1.0 
  U22 = \y0y1.0 
  U23 = \y0.0 
  U31 = \y0y1.0 
  U32 = \y0.0 
  U41 = \y0.0 
  a!6220!6220U11 = \y0y1y2.0 
  a!6220!6220U11# = \y0y1y2.3y1 + 3y2 
  a!6220!6220U12 = \y0y1y2.0 
  a!6220!6220U12# = \y0y1y2.3y1 + 3y2 
  a!6220!6220U13 = \y0y1y2.0 
  a!6220!6220U13# = \y0y1y2.3y1 + 3y2 
  a!6220!6220U14 = \y0y1y2.0 
  a!6220!6220U14# = \y0y1y2.3y1 + 3y2 
  a!6220!6220U15 = \y0y1.0 
  a!6220!6220U15# = \y0y1.3y1 
  a!6220!6220U16 = \y0.0 
  a!6220!6220U21 = \y0y1.0 
  a!6220!6220U21# = \y0y1.3y1 
  a!6220!6220U22 = \y0y1.0 
  a!6220!6220U22# = \y0y1.3y1 
  a!6220!6220U23 = \y0.0 
  a!6220!6220U31 = \y0y1.0 
  a!6220!6220U32 = \y0.0 
  a!6220!6220U41 = \y0.0 
  a!6220!6220isNat = \y0.0 
  a!6220!6220isNatKind = \y0.0 
  a!6220!6220isNat# = \y0.3y0 
  isNat = \y0.0 
  isNatKind = \y0.0 
  plus = \y0y1.3 + y1 + 2y0 
  s = \y0.3 + y0 
  tt = 0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220U11#(tt, _x0, _x1)]] = 3x0 + 3x1 >= 3x0 + 3x1 = [[a!6220!6220U12#(a!6220!6220isNatKind(_x0), _x0, _x1)]] 
  [[a!6220!6220U12#(tt, _x0, _x1)]] = 3x0 + 3x1 >= 3x0 + 3x1 = [[a!6220!6220U13#(a!6220!6220isNatKind(_x1), _x0, _x1)]] 
  [[a!6220!6220U13#(tt, _x0, _x1)]] = 3x0 + 3x1 >= 3x0 + 3x1 = [[a!6220!6220U14#(a!6220!6220isNatKind(_x1), _x0, _x1)]] 
  [[a!6220!6220U14#(tt, _x0, _x1)]] = 3x0 + 3x1 >= 3x1 = [[a!6220!6220U15#(a!6220!6220isNat(_x0), _x1)]] 
  [[a!6220!6220U14#(tt, _x0, _x1)]] = 3x0 + 3x1 >= 3x0 = [[a!6220!6220isNat#(_x0)]] 
  [[a!6220!6220U15#(tt, _x0)]] = 3x0 >= 3x0 = [[a!6220!6220isNat#(_x0)]] 
  [[a!6220!6220U21#(tt, _x0)]] = 3x0 >= 3x0 = [[a!6220!6220U22#(a!6220!6220isNatKind(_x0), _x0)]] 
  [[a!6220!6220U22#(tt, _x0)]] = 3x0 >= 3x0 = [[a!6220!6220isNat#(_x0)]] 
  [[a!6220!6220isNat#(plus(_x0, _x1))]] = 9 + 3x1 + 6x0 > 3x0 + 3x1 = [[a!6220!6220U11#(a!6220!6220isNatKind(_x0), _x0, _x1)]] 
  [[a!6220!6220isNat#(s(_x0))]] = 9 + 3x0 > 3x0 = [[a!6220!6220U21#(a!6220!6220isNatKind(_x0), _x0)]] 

By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace the dependency pair problem (P_1, R_0, minimal, formative) by (P_9, R_0, minimal, formative), where P_9 consists of:

  a!6220!6220U11#(tt, X, Y) =#> a!6220!6220U12#(a!6220!6220isNatKind(X), X, Y)   
  a!6220!6220U12#(tt, X, Y) =#> a!6220!6220U13#(a!6220!6220isNatKind(Y), X, Y)   
  a!6220!6220U13#(tt, X, Y) =#> a!6220!6220U14#(a!6220!6220isNatKind(Y), X, Y)   
  a!6220!6220U14#(tt, X, Y) =#> a!6220!6220U15#(a!6220!6220isNat(X), Y)   
  a!6220!6220U14#(tt, X, Y) =#> a!6220!6220isNat#(X)   
  a!6220!6220U15#(tt, X) =#> a!6220!6220isNat#(X)   
  a!6220!6220U21#(tt, X) =#> a!6220!6220U22#(a!6220!6220isNatKind(X), X)   
  a!6220!6220U22#(tt, X) =#> a!6220!6220isNat#(X)   

Thus, the original system is terminating if (P_9, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_9, R_0, minimal, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 1 
    * 1 : 2 
    * 2 : 3, 4 
    * 3 : 5 
    * 4 :  
    * 5 :  
    * 6 : 7 
    * 7 :  

This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.