/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 
  U12 : [o] --> o 
  U21 : [o] --> o 
  U31 : [o * o] --> o 
  U41 : [o * o * o] --> o 
  U42 : [o * o * o] --> o 
  a!6220!6220U11 : [o * o] --> o 
  a!6220!6220U12 : [o] --> o 
  a!6220!6220U21 : [o] --> o 
  a!6220!6220U31 : [o * o] --> o 
  a!6220!6220U41 : [o * o * o] --> o 
  a!6220!6220U42 : [o * o * o] --> o 
  a!6220!6220isNat : [o] --> o 
  a!6220!6220plus : [o * o] --> o 
  isNat : [o] --> o 
  mark : [o] --> o 
  plus : [o * o] --> o 
  s : [o] --> o 
  tt : [] --> o 

  a!6220!6220U11(tt, X) => a!6220!6220U12(a!6220!6220isNat(X)) 
  a!6220!6220U12(tt) => tt 
  a!6220!6220U21(tt) => tt 
  a!6220!6220U31(tt, X) => mark(X) 
  a!6220!6220U41(tt, X, Y) => a!6220!6220U42(a!6220!6220isNat(Y), X, Y) 
  a!6220!6220U42(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!6220isNat(X), Y) 
  a!6220!6220isNat(s(X)) => a!6220!6220U21(a!6220!6220isNat(X)) 
  a!6220!6220plus(X, 0) => a!6220!6220U31(a!6220!6220isNat(X), X) 
  a!6220!6220plus(X, s(Y)) => a!6220!6220U41(a!6220!6220isNat(Y), Y, X) 
  mark(U11(X, Y)) => a!6220!6220U11(mark(X), Y) 
  mark(U12(X)) => a!6220!6220U12(mark(X)) 
  mark(isNat(X)) => a!6220!6220isNat(X) 
  mark(U21(X)) => a!6220!6220U21(mark(X)) 
  mark(U31(X, Y)) => a!6220!6220U31(mark(X), Y) 
  mark(U41(X, Y, Z)) => a!6220!6220U41(mark(X), Y, Z) 
  mark(U42(X, Y, Z)) => a!6220!6220U42(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) => U11(X, Y) 
  a!6220!6220U12(X) => U12(X) 
  a!6220!6220isNat(X) => isNat(X) 
  a!6220!6220U21(X) => U21(X) 
  a!6220!6220U31(X, Y) => U31(X, Y) 
  a!6220!6220U41(X, Y, Z) => U41(X, Y, Z) 
  a!6220!6220U42(X, Y, Z) => U42(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) =#> a!6220!6220U12#(a!6220!6220isNat(X))   
    1] a!6220!6220U11#(tt, X) =#> a!6220!6220isNat#(X)   
    2] a!6220!6220U31#(tt, X) =#> mark#(X)   
    3] a!6220!6220U41#(tt, X, Y) =#> a!6220!6220U42#(a!6220!6220isNat(Y), X, Y)   
    4] a!6220!6220U41#(tt, X, Y) =#> a!6220!6220isNat#(Y)   
    5] a!6220!6220U42#(tt, X, Y) =#> a!6220!6220plus#(mark(Y), mark(X))   
    6] a!6220!6220U42#(tt, X, Y) =#> mark#(Y)   
    7] a!6220!6220U42#(tt, X, Y) =#> mark#(X)   
    8] a!6220!6220isNat#(plus(X, Y)) =#> a!6220!6220U11#(a!6220!6220isNat(X), Y)   
    9] a!6220!6220isNat#(plus(X, Y)) =#> a!6220!6220isNat#(X)   
    10] a!6220!6220isNat#(s(X)) =#> a!6220!6220U21#(a!6220!6220isNat(X))   
    11] a!6220!6220isNat#(s(X)) =#> a!6220!6220isNat#(X)   
    12] a!6220!6220plus#(X, 0) =#> a!6220!6220U31#(a!6220!6220isNat(X), X)   
    13] a!6220!6220plus#(X, 0) =#> a!6220!6220isNat#(X)   
    14] a!6220!6220plus#(X, s(Y)) =#> a!6220!6220U41#(a!6220!6220isNat(Y), Y, X)   
    15] a!6220!6220plus#(X, s(Y)) =#> a!6220!6220isNat#(Y)   
    16] mark#(U11(X, Y)) =#> a!6220!6220U11#(mark(X), Y)   
    17] mark#(U11(X, Y)) =#> mark#(X)   
    18] mark#(U12(X)) =#> a!6220!6220U12#(mark(X))   
    19] mark#(U12(X)) =#> mark#(X)   
    20] mark#(isNat(X)) =#> a!6220!6220isNat#(X)   
    21] mark#(U21(X)) =#> a!6220!6220U21#(mark(X))   
    22] mark#(U21(X)) =#> mark#(X)   
    23] mark#(U31(X, Y)) =#> a!6220!6220U31#(mark(X), Y)   
    24] mark#(U31(X, Y)) =#> mark#(X)   
    25] mark#(U41(X, Y, Z)) =#> a!6220!6220U41#(mark(X), Y, Z)   
    26] mark#(U41(X, Y, Z)) =#> mark#(X)   
    27] mark#(U42(X, Y, Z)) =#> a!6220!6220U42#(mark(X), Y, Z)   
    28] mark#(U42(X, Y, Z)) =#> mark#(X)   
    29] mark#(plus(X, Y)) =#> a!6220!6220plus#(mark(X), mark(Y))   
    30] mark#(plus(X, Y)) =#> mark#(X)   
    31] mark#(plus(X, Y)) =#> mark#(Y)   
    32] mark#(s(X)) =#> mark#(X)   

  Rules R_0:

    a!6220!6220U11(tt, X) => a!6220!6220U12(a!6220!6220isNat(X)) 
    a!6220!6220U12(tt) => tt 
    a!6220!6220U21(tt) => tt 
    a!6220!6220U31(tt, X) => mark(X) 
    a!6220!6220U41(tt, X, Y) => a!6220!6220U42(a!6220!6220isNat(Y), X, Y) 
    a!6220!6220U42(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!6220isNat(X), Y) 
    a!6220!6220isNat(s(X)) => a!6220!6220U21(a!6220!6220isNat(X)) 
    a!6220!6220plus(X, 0) => a!6220!6220U31(a!6220!6220isNat(X), X) 
    a!6220!6220plus(X, s(Y)) => a!6220!6220U41(a!6220!6220isNat(Y), Y, X) 
    mark(U11(X, Y)) => a!6220!6220U11(mark(X), Y) 
    mark(U12(X)) => a!6220!6220U12(mark(X)) 
    mark(isNat(X)) => a!6220!6220isNat(X) 
    mark(U21(X)) => a!6220!6220U21(mark(X)) 
    mark(U31(X, Y)) => a!6220!6220U31(mark(X), Y) 
    mark(U41(X, Y, Z)) => a!6220!6220U41(mark(X), Y, Z) 
    mark(U42(X, Y, Z)) => a!6220!6220U42(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) => U11(X, Y) 
    a!6220!6220U12(X) => U12(X) 
    a!6220!6220isNat(X) => isNat(X) 
    a!6220!6220U21(X) => U21(X) 
    a!6220!6220U31(X, Y) => U31(X, Y) 
    a!6220!6220U41(X, Y, Z) => U41(X, Y, Z) 
    a!6220!6220U42(X, Y, Z) => U42(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 :  
    * 1 : 8, 9, 10, 11 
    * 2 : 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 
    * 3 : 5, 6, 7 
    * 4 : 8, 9, 10, 11 
    * 5 : 12, 13, 14, 15 
    * 6 : 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 
    * 7 : 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 
    * 8 : 0, 1 
    * 9 : 8, 9, 10, 11 
    * 10 :  
    * 11 : 8, 9, 10, 11 
    * 12 : 2 
    * 13 : 8, 9, 10, 11 
    * 14 : 3, 4 
    * 15 : 8, 9, 10, 11 
    * 16 : 0, 1 
    * 17 : 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 
    * 18 :  
    * 19 : 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 
    * 20 : 8, 9, 10, 11 
    * 21 :  
    * 22 : 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 
    * 23 : 2 
    * 24 : 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 
    * 25 : 3, 4 
    * 26 : 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 
    * 27 : 5, 6, 7 
    * 28 : 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 
    * 29 : 12, 13, 14, 15 
    * 30 : 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 
    * 31 : 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 
    * 32 : 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 

This graph has the following strongly connected components:

  P_1:

    a!6220!6220U11#(tt, X) =#> a!6220!6220isNat#(X)   
    a!6220!6220isNat#(plus(X, Y)) =#> a!6220!6220U11#(a!6220!6220isNat(X), Y)   
    a!6220!6220isNat#(plus(X, Y)) =#> a!6220!6220isNat#(X)   
    a!6220!6220isNat#(s(X)) =#> a!6220!6220isNat#(X)   

  P_2:

    a!6220!6220U31#(tt, X) =#> mark#(X)   
    a!6220!6220U41#(tt, X, Y) =#> a!6220!6220U42#(a!6220!6220isNat(Y), X, Y)   
    a!6220!6220U42#(tt, X, Y) =#> a!6220!6220plus#(mark(Y), mark(X))   
    a!6220!6220U42#(tt, X, Y) =#> mark#(Y)   
    a!6220!6220U42#(tt, X, Y) =#> mark#(X)   
    a!6220!6220plus#(X, 0) =#> a!6220!6220U31#(a!6220!6220isNat(X), X)   
    a!6220!6220plus#(X, s(Y)) =#> a!6220!6220U41#(a!6220!6220isNat(Y), Y, X)   
    mark#(U11(X, Y)) =#> mark#(X)   
    mark#(U12(X)) =#> mark#(X)   
    mark#(U21(X)) =#> mark#(X)   
    mark#(U31(X, Y)) =#> a!6220!6220U31#(mark(X), Y)   
    mark#(U31(X, Y)) =#> mark#(X)   
    mark#(U41(X, Y, Z)) =#> a!6220!6220U41#(mark(X), Y, Z)   
    mark#(U41(X, Y, Z)) =#> mark#(X)   
    mark#(U42(X, Y, Z)) =#> a!6220!6220U42#(mark(X), Y, Z)   
    mark#(U42(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) and (P_2, R_0, m, f).

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 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!6220U31#(tt, X) >? mark#(X) 
  a!6220!6220U41#(tt, X, Y) >? a!6220!6220U42#(a!6220!6220isNat(Y), X, Y) 
  a!6220!6220U42#(tt, X, Y) >? a!6220!6220plus#(mark(Y), mark(X)) 
  a!6220!6220U42#(tt, X, Y) >? mark#(Y) 
  a!6220!6220U42#(tt, X, Y) >? mark#(X) 
  a!6220!6220plus#(X, 0) >? a!6220!6220U31#(a!6220!6220isNat(X), X) 
  a!6220!6220plus#(X, s(Y)) >? a!6220!6220U41#(a!6220!6220isNat(Y), Y, X) 
  mark#(U11(X, Y)) >? mark#(X) 
  mark#(U12(X)) >? mark#(X) 
  mark#(U21(X)) >? mark#(X) 
  mark#(U31(X, Y)) >? a!6220!6220U31#(mark(X), Y) 
  mark#(U31(X, Y)) >? mark#(X) 
  mark#(U41(X, Y, Z)) >? a!6220!6220U41#(mark(X), Y, Z) 
  mark#(U41(X, Y, Z)) >? mark#(X) 
  mark#(U42(X, Y, Z)) >? a!6220!6220U42#(mark(X), Y, Z) 
  mark#(U42(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) >= a!6220!6220U12(a!6220!6220isNat(X)) 
  a!6220!6220U12(tt) >= tt 
  a!6220!6220U21(tt) >= tt 
  a!6220!6220U31(tt, X) >= mark(X) 
  a!6220!6220U41(tt, X, Y) >= a!6220!6220U42(a!6220!6220isNat(Y), X, Y) 
  a!6220!6220U42(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!6220isNat(X), Y) 
  a!6220!6220isNat(s(X)) >= a!6220!6220U21(a!6220!6220isNat(X)) 
  a!6220!6220plus(X, 0) >= a!6220!6220U31(a!6220!6220isNat(X), X) 
  a!6220!6220plus(X, s(Y)) >= a!6220!6220U41(a!6220!6220isNat(Y), Y, X) 
  mark(U11(X, Y)) >= a!6220!6220U11(mark(X), Y) 
  mark(U12(X)) >= a!6220!6220U12(mark(X)) 
  mark(isNat(X)) >= a!6220!6220isNat(X) 
  mark(U21(X)) >= a!6220!6220U21(mark(X)) 
  mark(U31(X, Y)) >= a!6220!6220U31(mark(X), Y) 
  mark(U41(X, Y, Z)) >= a!6220!6220U41(mark(X), Y, Z) 
  mark(U42(X, Y, Z)) >= a!6220!6220U42(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) >= U11(X, Y) 
  a!6220!6220U12(X) >= U12(X) 
  a!6220!6220isNat(X) >= isNat(X) 
  a!6220!6220U21(X) >= U21(X) 
  a!6220!6220U31(X, Y) >= U31(X, Y) 
  a!6220!6220U41(X, Y, Z) >= U41(X, Y, Z) 
  a!6220!6220U42(X, Y, Z) >= U42(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 = \y0y1.y0 
  U12 = \y0.y0 
  U21 = \y0.y0 
  U31 = \y0y1.y1 + 2y0 
  U41 = \y0y1y2.1 + y0 + y2 + 2y1 
  U42 = \y0y1y2.1 + y0 + y2 + 2y1 
  a!6220!6220U11 = \y0y1.y0 
  a!6220!6220U12 = \y0.y0 
  a!6220!6220U21 = \y0.y0 
  a!6220!6220U31 = \y0y1.y1 + 2y0 
  a!6220!6220U31# = \y0y1.2y1 
  a!6220!6220U41 = \y0y1y2.1 + y0 + y2 + 2y1 
  a!6220!6220U41# = \y0y1y2.2y1 + 2y2 
  a!6220!6220U42 = \y0y1y2.1 + y0 + y2 + 2y1 
  a!6220!6220U42# = \y0y1y2.2y1 + 2y2 
  a!6220!6220isNat = \y0.0 
  a!6220!6220plus = \y0y1.1 + y0 + 2y1 
  a!6220!6220plus# = \y0y1.2y0 + 2y1 
  isNat = \y0.0 
  mark = \y0.y0 
  mark# = \y0.2y0 
  plus = \y0y1.1 + y0 + 2y1 
  s = \y0.y0 
  tt = 0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220U31#(tt, _x0)]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220U41#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U42#(a!6220!6220isNat(_x1), _x0, _x1)]] 
  [[a!6220!6220U42#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220plus#(mark(_x1), mark(_x0))]] 
  [[a!6220!6220U42#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x1 = [[mark#(_x1)]] 
  [[a!6220!6220U42#(tt, _x0, _x1)]] = 2x0 + 2x1 >= 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220plus#(_x0, 0)]] = 2x0 >= 2x0 = [[a!6220!6220U31#(a!6220!6220isNat(_x0), _x0)]] 
  [[a!6220!6220plus#(_x0, s(_x1))]] = 2x0 + 2x1 >= 2x0 + 2x1 = [[a!6220!6220U41#(a!6220!6220isNat(_x1), _x1, _x0)]] 
  [[mark#(U11(_x0, _x1))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U12(_x0))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U21(_x0))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U31(_x0, _x1))]] = 2x1 + 4x0 >= 2x1 = [[a!6220!6220U31#(mark(_x0), _x1)]] 
  [[mark#(U31(_x0, _x1))]] = 2x1 + 4x0 >= 2x0 = [[mark#(_x0)]] 
  [[mark#(U41(_x0, _x1, _x2))]] = 2 + 2x0 + 2x2 + 4x1 > 2x1 + 2x2 = [[a!6220!6220U41#(mark(_x0), _x1, _x2)]] 
  [[mark#(U41(_x0, _x1, _x2))]] = 2 + 2x0 + 2x2 + 4x1 > 2x0 = [[mark#(_x0)]] 
  [[mark#(U42(_x0, _x1, _x2))]] = 2 + 2x0 + 2x2 + 4x1 > 2x1 + 2x2 = [[a!6220!6220U42#(mark(_x0), _x1, _x2)]] 
  [[mark#(U42(_x0, _x1, _x2))]] = 2 + 2x0 + 2x2 + 4x1 > 2x0 = [[mark#(_x0)]] 
  [[mark#(plus(_x0, _x1))]] = 2 + 2x0 + 4x1 > 2x0 + 2x1 = [[a!6220!6220plus#(mark(_x0), mark(_x1))]] 
  [[mark#(plus(_x0, _x1))]] = 2 + 2x0 + 4x1 > 2x0 = [[mark#(_x0)]] 
  [[mark#(plus(_x0, _x1))]] = 2 + 2x0 + 4x1 > 2x1 = [[mark#(_x1)]] 
  [[mark#(s(_x0))]] = 2x0 >= 2x0 = [[mark#(_x0)]] 
  [[a!6220!6220U11(tt, _x0)]] = 0 >= 0 = [[a!6220!6220U12(a!6220!6220isNat(_x0))]] 
  [[a!6220!6220U12(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U21(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U31(tt, _x0)]] = x0 >= x0 = [[mark(_x0)]] 
  [[a!6220!6220U41(tt, _x0, _x1)]] = 1 + x1 + 2x0 >= 1 + x1 + 2x0 = [[a!6220!6220U42(a!6220!6220isNat(_x1), _x0, _x1)]] 
  [[a!6220!6220U42(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!6220isNat(_x0), _x1)]] 
  [[a!6220!6220isNat(s(_x0))]] = 0 >= 0 = [[a!6220!6220U21(a!6220!6220isNat(_x0))]] 
  [[a!6220!6220plus(_x0, 0)]] = 1 + x0 >= x0 = [[a!6220!6220U31(a!6220!6220isNat(_x0), _x0)]] 
  [[a!6220!6220plus(_x0, s(_x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[a!6220!6220U41(a!6220!6220isNat(_x1), _x1, _x0)]] 
  [[mark(U11(_x0, _x1))]] = x0 >= x0 = [[a!6220!6220U11(mark(_x0), _x1)]] 
  [[mark(U12(_x0))]] = x0 >= x0 = [[a!6220!6220U12(mark(_x0))]] 
  [[mark(isNat(_x0))]] = 0 >= 0 = [[a!6220!6220isNat(_x0)]] 
  [[mark(U21(_x0))]] = x0 >= x0 = [[a!6220!6220U21(mark(_x0))]] 
  [[mark(U31(_x0, _x1))]] = x1 + 2x0 >= x1 + 2x0 = [[a!6220!6220U31(mark(_x0), _x1)]] 
  [[mark(U41(_x0, _x1, _x2))]] = 1 + x0 + x2 + 2x1 >= 1 + x0 + x2 + 2x1 = [[a!6220!6220U41(mark(_x0), _x1, _x2)]] 
  [[mark(U42(_x0, _x1, _x2))]] = 1 + x0 + x2 + 2x1 >= 1 + x0 + x2 + 2x1 = [[a!6220!6220U42(mark(_x0), _x1, _x2)]] 
  [[mark(plus(_x0, _x1))]] = 1 + x0 + 2x1 >= 1 + x0 + 2x1 = [[a!6220!6220plus(mark(_x0), mark(_x1))]] 
  [[mark(tt)]] = 0 >= 0 = [[tt]] 
  [[mark(s(_x0))]] = x0 >= x0 = [[s(mark(_x0))]] 
  [[mark(0)]] = 0 >= 0 = [[0]] 
  [[a!6220!6220U11(_x0, _x1)]] = x0 >= x0 = [[U11(_x0, _x1)]] 
  [[a!6220!6220U12(_x0)]] = x0 >= x0 = [[U12(_x0)]] 
  [[a!6220!6220isNat(_x0)]] = 0 >= 0 = [[isNat(_x0)]] 
  [[a!6220!6220U21(_x0)]] = x0 >= x0 = [[U21(_x0)]] 
  [[a!6220!6220U31(_x0, _x1)]] = x1 + 2x0 >= x1 + 2x0 = [[U31(_x0, _x1)]] 
  [[a!6220!6220U41(_x0, _x1, _x2)]] = 1 + x0 + x2 + 2x1 >= 1 + x0 + x2 + 2x1 = [[U41(_x0, _x1, _x2)]] 
  [[a!6220!6220U42(_x0, _x1, _x2)]] = 1 + x0 + x2 + 2x1 >= 1 + x0 + x2 + 2x1 = [[U42(_x0, _x1, _x2)]] 
  [[a!6220!6220plus(_x0, _x1)]] = 1 + x0 + 2x1 >= 1 + 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_2, R_0, minimal, formative) by (P_3, R_0, minimal, formative), where P_3 consists of:

  a!6220!6220U31#(tt, X) =#> mark#(X)   
  a!6220!6220U41#(tt, X, Y) =#> a!6220!6220U42#(a!6220!6220isNat(Y), X, Y)   
  a!6220!6220U42#(tt, X, Y) =#> a!6220!6220plus#(mark(Y), mark(X))   
  a!6220!6220U42#(tt, X, Y) =#> mark#(Y)   
  a!6220!6220U42#(tt, X, Y) =#> mark#(X)   
  a!6220!6220plus#(X, 0) =#> a!6220!6220U31#(a!6220!6220isNat(X), X)   
  a!6220!6220plus#(X, s(Y)) =#> a!6220!6220U41#(a!6220!6220isNat(Y), Y, X)   
  mark#(U11(X, Y)) =#> mark#(X)   
  mark#(U12(X)) =#> mark#(X)   
  mark#(U21(X)) =#> mark#(X)   
  mark#(U31(X, Y)) =#> a!6220!6220U31#(mark(X), Y)   
  mark#(U31(X, Y)) =#> mark#(X)   
  mark#(s(X)) =#> mark#(X)   

Thus, the original system is terminating if each of (P_1, 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 place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 7, 8, 9, 10, 11, 12 
    * 1 : 2, 3, 4 
    * 2 : 5, 6 
    * 3 : 7, 8, 9, 10, 11, 12 
    * 4 : 7, 8, 9, 10, 11, 12 
    * 5 : 0 
    * 6 : 1 
    * 7 : 7, 8, 9, 10, 11, 12 
    * 8 : 7, 8, 9, 10, 11, 12 
    * 9 : 7, 8, 9, 10, 11, 12 
    * 10 : 0 
    * 11 : 7, 8, 9, 10, 11, 12 
    * 12 : 7, 8, 9, 10, 11, 12 

This graph has the following strongly connected components:

  P_4:

    a!6220!6220U31#(tt, X) =#> mark#(X)   
    mark#(U11(X, Y)) =#> mark#(X)   
    mark#(U12(X)) =#> mark#(X)   
    mark#(U21(X)) =#> mark#(X)   
    mark#(U31(X, Y)) =#> a!6220!6220U31#(mark(X), Y)   
    mark#(U31(X, Y)) =#> mark#(X)   
    mark#(s(X)) =#> mark#(X)   

  P_5:

    a!6220!6220U41#(tt, X, Y) =#> a!6220!6220U42#(a!6220!6220isNat(Y), X, Y)   
    a!6220!6220U42#(tt, X, Y) =#> a!6220!6220plus#(mark(Y), mark(X))   
    a!6220!6220plus#(X, s(Y)) =#> a!6220!6220U41#(a!6220!6220isNat(Y), Y, X)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_3, R_0, m, f) by (P_4, R_0, m, f) and (P_5, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_4, 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!6220U41#(tt, X, Y) >? a!6220!6220U42#(a!6220!6220isNat(Y), X, Y) 
  a!6220!6220U42#(tt, X, Y) >? a!6220!6220plus#(mark(Y), mark(X)) 
  a!6220!6220plus#(X, s(Y)) >? a!6220!6220U41#(a!6220!6220isNat(Y), Y, X) 
  a!6220!6220U11(tt, X) >= a!6220!6220U12(a!6220!6220isNat(X)) 
  a!6220!6220U12(tt) >= tt 
  a!6220!6220U21(tt) >= tt 
  a!6220!6220U31(tt, X) >= mark(X) 
  a!6220!6220U41(tt, X, Y) >= a!6220!6220U42(a!6220!6220isNat(Y), X, Y) 
  a!6220!6220U42(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!6220isNat(X), Y) 
  a!6220!6220isNat(s(X)) >= a!6220!6220U21(a!6220!6220isNat(X)) 
  a!6220!6220plus(X, 0) >= a!6220!6220U31(a!6220!6220isNat(X), X) 
  a!6220!6220plus(X, s(Y)) >= a!6220!6220U41(a!6220!6220isNat(Y), Y, X) 
  mark(U11(X, Y)) >= a!6220!6220U11(mark(X), Y) 
  mark(U12(X)) >= a!6220!6220U12(mark(X)) 
  mark(isNat(X)) >= a!6220!6220isNat(X) 
  mark(U21(X)) >= a!6220!6220U21(mark(X)) 
  mark(U31(X, Y)) >= a!6220!6220U31(mark(X), Y) 
  mark(U41(X, Y, Z)) >= a!6220!6220U41(mark(X), Y, Z) 
  mark(U42(X, Y, Z)) >= a!6220!6220U42(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) >= U11(X, Y) 
  a!6220!6220U12(X) >= U12(X) 
  a!6220!6220isNat(X) >= isNat(X) 
  a!6220!6220U21(X) >= U21(X) 
  a!6220!6220U31(X, Y) >= U31(X, Y) 
  a!6220!6220U41(X, Y, Z) >= U41(X, Y, Z) 
  a!6220!6220U42(X, Y, Z) >= U42(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 = \y0y1.0 
  U12 = \y0.0 
  U21 = \y0.0 
  U31 = \y0y1.y1 
  U41 = \y0y1y2.1 + y1 + y2 
  U42 = \y0y1y2.1 + y1 + y2 
  a!6220!6220U11 = \y0y1.0 
  a!6220!6220U12 = \y0.0 
  a!6220!6220U21 = \y0.0 
  a!6220!6220U31 = \y0y1.y1 
  a!6220!6220U41 = \y0y1y2.1 + y1 + y2 
  a!6220!6220U41# = \y0y1y2.2y1 
  a!6220!6220U42 = \y0y1y2.1 + y1 + y2 
  a!6220!6220U42# = \y0y1y2.2y1 
  a!6220!6220isNat = \y0.0 
  a!6220!6220plus = \y0y1.y0 + y1 
  a!6220!6220plus# = \y0y1.2y1 
  isNat = \y0.0 
  mark = \y0.y0 
  plus = \y0y1.y0 + y1 
  s = \y0.1 + y0 
  tt = 0 

Using this interpretation, the requirements translate to:

  [[a!6220!6220U41#(tt, _x0, _x1)]] = 2x0 >= 2x0 = [[a!6220!6220U42#(a!6220!6220isNat(_x1), _x0, _x1)]] 
  [[a!6220!6220U42#(tt, _x0, _x1)]] = 2x0 >= 2x0 = [[a!6220!6220plus#(mark(_x1), mark(_x0))]] 
  [[a!6220!6220plus#(_x0, s(_x1))]] = 2 + 2x1 > 2x1 = [[a!6220!6220U41#(a!6220!6220isNat(_x1), _x1, _x0)]] 
  [[a!6220!6220U11(tt, _x0)]] = 0 >= 0 = [[a!6220!6220U12(a!6220!6220isNat(_x0))]] 
  [[a!6220!6220U12(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U21(tt)]] = 0 >= 0 = [[tt]] 
  [[a!6220!6220U31(tt, _x0)]] = x0 >= x0 = [[mark(_x0)]] 
  [[a!6220!6220U41(tt, _x0, _x1)]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[a!6220!6220U42(a!6220!6220isNat(_x1), _x0, _x1)]] 
  [[a!6220!6220U42(tt, _x0, _x1)]] = 1 + x0 + x1 >= 1 + 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!6220isNat(_x0), _x1)]] 
  [[a!6220!6220isNat(s(_x0))]] = 0 >= 0 = [[a!6220!6220U21(a!6220!6220isNat(_x0))]] 
  [[a!6220!6220plus(_x0, 0)]] = x0 >= x0 = [[a!6220!6220U31(a!6220!6220isNat(_x0), _x0)]] 
  [[a!6220!6220plus(_x0, s(_x1))]] = 1 + x0 + x1 >= 1 + x0 + x1 = [[a!6220!6220U41(a!6220!6220isNat(_x1), _x1, _x0)]] 
  [[mark(U11(_x0, _x1))]] = 0 >= 0 = [[a!6220!6220U11(mark(_x0), _x1)]] 
  [[mark(U12(_x0))]] = 0 >= 0 = [[a!6220!6220U12(mark(_x0))]] 
  [[mark(isNat(_x0))]] = 0 >= 0 = [[a!6220!6220isNat(_x0)]] 
  [[mark(U21(_x0))]] = 0 >= 0 = [[a!6220!6220U21(mark(_x0))]] 
  [[mark(U31(_x0, _x1))]] = x1 >= x1 = [[a!6220!6220U31(mark(_x0), _x1)]] 
  [[mark(U41(_x0, _x1, _x2))]] = 1 + x1 + x2 >= 1 + x1 + x2 = [[a!6220!6220U41(mark(_x0), _x1, _x2)]] 
  [[mark(U42(_x0, _x1, _x2))]] = 1 + x1 + x2 >= 1 + x1 + x2 = [[a!6220!6220U42(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))]] = 1 + x0 >= 1 + x0 = [[s(mark(_x0))]] 
  [[mark(0)]] = 0 >= 0 = [[0]] 
  [[a!6220!6220U11(_x0, _x1)]] = 0 >= 0 = [[U11(_x0, _x1)]] 
  [[a!6220!6220U12(_x0)]] = 0 >= 0 = [[U12(_x0)]] 
  [[a!6220!6220isNat(_x0)]] = 0 >= 0 = [[isNat(_x0)]] 
  [[a!6220!6220U21(_x0)]] = 0 >= 0 = [[U21(_x0)]] 
  [[a!6220!6220U31(_x0, _x1)]] = x1 >= x1 = [[U31(_x0, _x1)]] 
  [[a!6220!6220U41(_x0, _x1, _x2)]] = 1 + x1 + x2 >= 1 + x1 + x2 = [[U41(_x0, _x1, _x2)]] 
  [[a!6220!6220U42(_x0, _x1, _x2)]] = 1 + x1 + x2 >= 1 + x1 + x2 = [[U42(_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_5, R_0, minimal, formative) by (P_6, R_0, minimal, formative), where P_6 consists of:

  a!6220!6220U41#(tt, X, Y) =#> a!6220!6220U42#(a!6220!6220isNat(Y), X, Y)   
  a!6220!6220U42#(tt, X, Y) =#> a!6220!6220plus#(mark(Y), mark(X))   

Thus, the original system is terminating if each of (P_1, R_0, minimal, formative), (P_4, 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 :  

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 each of (P_1, 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 apply the subterm criterion with the following projection function:

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

Thus, we can orient the dependency pairs as follows:

  nu(a!6220!6220U31#(tt, X)) = X = X = nu(mark#(X)) 
  nu(mark#(U11(X, Y))) = U11(X, Y) |> X = nu(mark#(X)) 
  nu(mark#(U12(X))) = U12(X) |> X = nu(mark#(X)) 
  nu(mark#(U21(X))) = U21(X) |> X = nu(mark#(X)) 
  nu(mark#(U31(X, Y))) = U31(X, Y) |> Y = nu(a!6220!6220U31#(mark(X), Y)) 
  nu(mark#(U31(X, Y))) = U31(X, Y) |> X = nu(mark#(X)) 
  nu(mark#(s(X))) = s(X) |> X = nu(mark#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_4, R_0, minimal, f) by (P_7, R_0, minimal, f), where P_7 contains:

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

Thus, the original system is terminating if each of (P_1, 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 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 apply the subterm criterion with the following projection function:

  nu(a!6220!6220U11#) = 2 
  nu(a!6220!6220isNat#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(a!6220!6220U11#(tt, X)) = X = X = nu(a!6220!6220isNat#(X)) 
  nu(a!6220!6220isNat#(plus(X, Y))) = plus(X, Y) |> Y = nu(a!6220!6220U11#(a!6220!6220isNat(X), Y)) 
  nu(a!6220!6220isNat#(plus(X, Y))) = plus(X, Y) |> X = nu(a!6220!6220isNat#(X)) 
  nu(a!6220!6220isNat#(s(X))) = s(X) |> X = nu(a!6220!6220isNat#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_1, R_0, minimal, f) by (P_8, R_0, minimal, f), where P_8 contains:

  a!6220!6220U11#(tt, X) =#> a!6220!6220isNat#(X)   

Thus, the original system is terminating if (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.

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.