/export/starexec/sandbox2/solver/bin/starexec_run_FirstOrder /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/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 
  2ndsneg : [o * o] --> o 
  2ndspos : [o * o] --> o 
  activate : [o] --> o 
  cons : [o * o] --> o 
  from : [o] --> o 
  n!6220!6220cons : [o * o] --> o 
  n!6220!6220from : [o] --> o 
  n!6220!6220s : [o] --> o 
  negrecip : [o] --> o 
  pi : [o] --> o 
  plus : [o * o] --> o 
  posrecip : [o] --> o 
  rcons : [o * o] --> o 
  rnil : [] --> o 
  s : [o] --> o 
  square : [o] --> o 
  times : [o * o] --> o 

  from(X) => cons(X, n!6220!6220from(n!6220!6220s(X))) 
  2ndspos(0, X) => rnil 
  2ndspos(s(X), cons(Y, n!6220!6220cons(Z, U))) => rcons(posrecip(activate(Z)), 2ndsneg(X, activate(U))) 
  2ndsneg(0, X) => rnil 
  2ndsneg(s(X), cons(Y, n!6220!6220cons(Z, U))) => rcons(negrecip(activate(Z)), 2ndspos(X, activate(U))) 
  pi(X) => 2ndspos(X, from(0)) 
  plus(0, X) => X 
  plus(s(X), Y) => s(plus(X, Y)) 
  times(0, X) => 0 
  times(s(X), Y) => plus(Y, times(X, Y)) 
  square(X) => times(X, X) 
  from(X) => n!6220!6220from(X) 
  s(X) => n!6220!6220s(X) 
  cons(X, Y) => n!6220!6220cons(X, Y) 
  activate(n!6220!6220from(X)) => from(activate(X)) 
  activate(n!6220!6220s(X)) => s(activate(X)) 
  activate(n!6220!6220cons(X, Y)) => cons(activate(X), Y) 
  activate(X) => X 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  from(X) >? cons(X, n!6220!6220from(n!6220!6220s(X))) 
  2ndspos(0, X) >? rnil 
  2ndspos(s(X), cons(Y, n!6220!6220cons(Z, U))) >? rcons(posrecip(activate(Z)), 2ndsneg(X, activate(U))) 
  2ndsneg(0, X) >? rnil 
  2ndsneg(s(X), cons(Y, n!6220!6220cons(Z, U))) >? rcons(negrecip(activate(Z)), 2ndspos(X, activate(U))) 
  pi(X) >? 2ndspos(X, from(0)) 
  plus(0, X) >? X 
  plus(s(X), Y) >? s(plus(X, Y)) 
  times(0, X) >? 0 
  times(s(X), Y) >? plus(Y, times(X, Y)) 
  square(X) >? times(X, X) 
  from(X) >? n!6220!6220from(X) 
  s(X) >? n!6220!6220s(X) 
  cons(X, Y) >? n!6220!6220cons(X, Y) 
  activate(n!6220!6220from(X)) >? from(activate(X)) 
  activate(n!6220!6220s(X)) >? s(activate(X)) 
  activate(n!6220!6220cons(X, Y)) >? cons(activate(X), Y) 
  activate(X) >? X 

about to try horpo

We use a recursive path ordering as defined in [Kop12, Chapter 5].

Argument functions:

  [[0]] = _|_ 
  [[posrecip(x_1)]] = x_1 
  [[rnil]] = _|_ 

We choose Lex = {2ndsneg, 2ndspos} and Mul = {activate, cons, from, n!6220!6220cons, n!6220!6220from, n!6220!6220s, negrecip, pi, plus, rcons, s, square, times}, and the following precedence: pi > square > times > plus > 2ndsneg = 2ndspos > negrecip > rcons > activate > from > n!6220!6220s = s > n!6220!6220from > cons > n!6220!6220cons

Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:

  from(X) > cons(X, n!6220!6220from(n!6220!6220s(X))) 
  2ndspos(_|_, X) >= _|_ 
  2ndspos(s(X), cons(Y, n!6220!6220cons(Z, U))) >= rcons(activate(Z), 2ndsneg(X, activate(U))) 
  2ndsneg(_|_, X) >= _|_ 
  2ndsneg(s(X), cons(Y, n!6220!6220cons(Z, U))) >= rcons(negrecip(activate(Z)), 2ndspos(X, activate(U))) 
  pi(X) > 2ndspos(X, from(_|_)) 
  plus(_|_, X) >= X 
  plus(s(X), Y) >= s(plus(X, Y)) 
  times(_|_, X) >= _|_ 
  times(s(X), Y) > plus(Y, times(X, Y)) 
  square(X) >= times(X, X) 
  from(X) >= n!6220!6220from(X) 
  s(X) >= n!6220!6220s(X) 
  cons(X, Y) > n!6220!6220cons(X, Y) 
  activate(n!6220!6220from(X)) > from(activate(X)) 
  activate(n!6220!6220s(X)) >= s(activate(X)) 
  activate(n!6220!6220cons(X, Y)) > cons(activate(X), Y) 
  activate(X) >= X 

With these choices, we have:

  1] from(X) > cons(X, n!6220!6220from(n!6220!6220s(X)))  because [2], by definition 
  2] from*(X) >= cons(X, n!6220!6220from(n!6220!6220s(X)))  because from > cons, [3] and [5], by (Copy) 
  3] from*(X) >= X  because [4], by (Select) 
  4] X >= X  by (Meta) 
  5] from*(X) >= n!6220!6220from(n!6220!6220s(X))  because from > n!6220!6220from and [6], by (Copy) 
  6] from*(X) >= n!6220!6220s(X)  because from > n!6220!6220s and [3], by (Copy) 

  7] 2ndspos(_|_, X) >= _|_  by (Bot) 

  8] 2ndspos(s(X), cons(Y, n!6220!6220cons(Z, U))) >= rcons(activate(Z), 2ndsneg(X, activate(U)))  because [9], by (Star) 
  9] 2ndspos*(s(X), cons(Y, n!6220!6220cons(Z, U))) >= rcons(activate(Z), 2ndsneg(X, activate(U)))  because 2ndspos > rcons, [10] and [17], by (Copy) 
  10] 2ndspos*(s(X), cons(Y, n!6220!6220cons(Z, U))) >= activate(Z)  because 2ndspos > activate and [11], by (Copy) 
  11] 2ndspos*(s(X), cons(Y, n!6220!6220cons(Z, U))) >= Z  because [12], by (Select) 
  12] cons(Y, n!6220!6220cons(Z, U)) >= Z  because [13], by (Star) 
  13] cons*(Y, n!6220!6220cons(Z, U)) >= Z  because [14], by (Select) 
  14] n!6220!6220cons(Z, U) >= Z  because [15], by (Star) 
  15] n!6220!6220cons*(Z, U) >= Z  because [16], by (Select) 
  16] Z >= Z  by (Meta) 
  17] 2ndspos*(s(X), cons(Y, n!6220!6220cons(Z, U))) >= 2ndsneg(X, activate(U))  because 2ndspos = 2ndsneg, [18], [21] and [23], by (Stat) 
  18] s(X) > X  because [19], by definition 
  19] s*(X) >= X  because [20], by (Select) 
  20] X >= X  by (Meta) 
  21] 2ndspos*(s(X), cons(Y, n!6220!6220cons(Z, U))) >= X  because [22], by (Select) 
  22] s(X) >= X  because [19], by (Star) 
  23] 2ndspos*(s(X), cons(Y, n!6220!6220cons(Z, U))) >= activate(U)  because 2ndspos > activate and [24], by (Copy) 
  24] 2ndspos*(s(X), cons(Y, n!6220!6220cons(Z, U))) >= U  because [25], by (Select) 
  25] cons(Y, n!6220!6220cons(Z, U)) >= U  because [26], by (Star) 
  26] cons*(Y, n!6220!6220cons(Z, U)) >= U  because [27], by (Select) 
  27] n!6220!6220cons(Z, U) >= U  because [28], by (Star) 
  28] n!6220!6220cons*(Z, U) >= U  because [29], by (Select) 
  29] U >= U  by (Meta) 

  30] 2ndsneg(_|_, X) >= _|_  by (Bot) 

  31] 2ndsneg(s(X), cons(Y, n!6220!6220cons(Z, U))) >= rcons(negrecip(activate(Z)), 2ndspos(X, activate(U)))  because [32], by (Star) 
  32] 2ndsneg*(s(X), cons(Y, n!6220!6220cons(Z, U))) >= rcons(negrecip(activate(Z)), 2ndspos(X, activate(U)))  because 2ndsneg > rcons, [33] and [36], by (Copy) 
  33] 2ndsneg*(s(X), cons(Y, n!6220!6220cons(Z, U))) >= negrecip(activate(Z))  because 2ndsneg > negrecip and [34], by (Copy) 
  34] 2ndsneg*(s(X), cons(Y, n!6220!6220cons(Z, U))) >= activate(Z)  because 2ndsneg > activate and [35], by (Copy) 
  35] 2ndsneg*(s(X), cons(Y, n!6220!6220cons(Z, U))) >= Z  because [12], by (Select) 
  36] 2ndsneg*(s(X), cons(Y, n!6220!6220cons(Z, U))) >= 2ndspos(X, activate(U))  because 2ndsneg = 2ndspos, [18], [37] and [38], by (Stat) 
  37] 2ndsneg*(s(X), cons(Y, n!6220!6220cons(Z, U))) >= X  because [22], by (Select) 
  38] 2ndsneg*(s(X), cons(Y, n!6220!6220cons(Z, U))) >= activate(U)  because 2ndsneg > activate and [39], by (Copy) 
  39] 2ndsneg*(s(X), cons(Y, n!6220!6220cons(Z, U))) >= U  because [25], by (Select) 

  40] pi(X) > 2ndspos(X, from(_|_))  because [41], by definition 
  41] pi*(X) >= 2ndspos(X, from(_|_))  because pi > 2ndspos, [42] and [43], by (Copy) 
  42] pi*(X) >= X  because [4], by (Select) 
  43] pi*(X) >= from(_|_)  because pi > from and [44], by (Copy) 
  44] pi*(X) >= _|_  by (Bot) 

  45] plus(_|_, X) >= X  because [46], by (Star) 
  46] plus*(_|_, X) >= X  because [16], by (Select) 

  47] plus(s(X), Y) >= s(plus(X, Y))  because [48], by (Star) 
  48] plus*(s(X), Y) >= s(plus(X, Y))  because plus > s and [49], by (Copy) 
  49] plus*(s(X), Y) >= plus(X, Y)  because plus in Mul, [50] and [52], by (Stat) 
  50] s(X) > X  because [51], by definition 
  51] s*(X) >= X  because [4], by (Select) 
  52] Y >= Y  by (Meta) 

  53] times(_|_, X) >= _|_  by (Bot) 

  54] times(s(X), Y) > plus(Y, times(X, Y))  because [55], by definition 
  55] times*(s(X), Y) >= plus(Y, times(X, Y))  because times > plus, [56] and [57], by (Copy) 
  56] times*(s(X), Y) >= Y  because [52], by (Select) 
  57] times*(s(X), Y) >= times(X, Y)  because times in Mul, [50] and [52], by (Stat) 

  58] square(X) >= times(X, X)  because [59], by (Star) 
  59] square*(X) >= times(X, X)  because square > times, [60] and [60], by (Copy) 
  60] square*(X) >= X  because [4], by (Select) 

  61] from(X) >= n!6220!6220from(X)  because [62], by (Star) 
  62] from*(X) >= n!6220!6220from(X)  because from > n!6220!6220from and [3], by (Copy) 

  63] s(X) >= n!6220!6220s(X)  because s = n!6220!6220s, s in Mul and [64], by (Fun) 
  64] X >= X  by (Meta) 

  65] cons(X, Y) > n!6220!6220cons(X, Y)  because [66], by definition 
  66] cons*(X, Y) >= n!6220!6220cons(X, Y)  because cons > n!6220!6220cons, [67] and [69], by (Copy) 
  67] cons*(X, Y) >= X  because [68], by (Select) 
  68] X >= X  by (Meta) 
  69] cons*(X, Y) >= Y  because [70], by (Select) 
  70] Y >= Y  by (Meta) 

  71] activate(n!6220!6220from(X)) > from(activate(X))  because [72], by definition 
  72] activate*(n!6220!6220from(X)) >= from(activate(X))  because activate > from and [73], by (Copy) 
  73] activate*(n!6220!6220from(X)) >= activate(X)  because activate in Mul and [74], by (Stat) 
  74] n!6220!6220from(X) > X  because [75], by definition 
  75] n!6220!6220from*(X) >= X  because [64], by (Select) 

  76] activate(n!6220!6220s(X)) >= s(activate(X))  because [77], by (Star) 
  77] activate*(n!6220!6220s(X)) >= s(activate(X))  because activate > s and [78], by (Copy) 
  78] activate*(n!6220!6220s(X)) >= activate(X)  because activate in Mul and [79], by (Stat) 
  79] n!6220!6220s(X) > X  because [80], by definition 
  80] n!6220!6220s*(X) >= X  because [64], by (Select) 

  81] activate(n!6220!6220cons(X, Y)) > cons(activate(X), Y)  because [82], by definition 
  82] activate*(n!6220!6220cons(X, Y)) >= cons(activate(X), Y)  because activate > cons, [83] and [86], by (Copy) 
  83] activate*(n!6220!6220cons(X, Y)) >= activate(X)  because activate in Mul and [84], by (Stat) 
  84] n!6220!6220cons(X, Y) > X  because [85], by definition 
  85] n!6220!6220cons*(X, Y) >= X  because [68], by (Select) 
  86] activate*(n!6220!6220cons(X, Y)) >= Y  because [87], by (Select) 
  87] n!6220!6220cons(X, Y) >= Y  because [88], by (Star) 
  88] n!6220!6220cons*(X, Y) >= Y  because [70], by (Select) 

  89] activate(X) >= X  because [90], by (Star) 
  90] activate*(X) >= X  because [64], by (Select) 

We can thus remove the following rules:

  from(X) => cons(X, n!6220!6220from(n!6220!6220s(X))) 
  pi(X) => 2ndspos(X, from(0)) 
  times(s(X), Y) => plus(Y, times(X, Y)) 
  cons(X, Y) => n!6220!6220cons(X, Y) 
  activate(n!6220!6220from(X)) => from(activate(X)) 
  activate(n!6220!6220cons(X, Y)) => cons(activate(X), Y) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  2ndspos(0, X) >? rnil 
  2ndspos(s(X), cons(Y, n!6220!6220cons(Z, U))) >? rcons(posrecip(activate(Z)), 2ndsneg(X, activate(U))) 
  2ndsneg(0, X) >? rnil 
  2ndsneg(s(X), cons(Y, n!6220!6220cons(Z, U))) >? rcons(negrecip(activate(Z)), 2ndspos(X, activate(U))) 
  plus(0, X) >? X 
  plus(s(X), Y) >? s(plus(X, Y)) 
  times(0, X) >? 0 
  square(X) >? times(X, X) 
  from(X) >? n!6220!6220from(X) 
  s(X) >? n!6220!6220s(X) 
  activate(n!6220!6220s(X)) >? s(activate(X)) 
  activate(X) >? X 

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

The following interpretation satisfies the requirements:

  0 = 0 
  2ndsneg = \y0y1.2 + y0 + 2y1 
  2ndspos = \y0y1.1 + y0 + 2y1 
  activate = \y0.2y0 
  cons = \y0y1.3 + y0 + 3y1 
  from = \y0.3 + 3y0 
  n!6220!6220cons = \y0y1.3 + 3y0 + 3y1 
  n!6220!6220from = \y0.y0 
  n!6220!6220s = \y0.y0 
  negrecip = \y0.y0 
  plus = \y0y1.y1 + 3y0 
  posrecip = \y0.y0 
  rcons = \y0y1.y0 + y1 
  rnil = 0 
  s = \y0.y0 
  square = \y0.3 + 3y0 
  times = \y0y1.2 + y0 + y1 

Using this interpretation, the requirements translate to:

  [[2ndspos(0, _x0)]] = 1 + 2x0 > 0 = [[rnil]] 
  [[2ndspos(s(_x0), cons(_x1, n!6220!6220cons(_x2, _x3)))]] = 25 + x0 + 2x1 + 18x2 + 18x3 > 2 + x0 + 2x2 + 4x3 = [[rcons(posrecip(activate(_x2)), 2ndsneg(_x0, activate(_x3)))]] 
  [[2ndsneg(0, _x0)]] = 2 + 2x0 > 0 = [[rnil]] 
  [[2ndsneg(s(_x0), cons(_x1, n!6220!6220cons(_x2, _x3)))]] = 26 + x0 + 2x1 + 18x2 + 18x3 > 1 + x0 + 2x2 + 4x3 = [[rcons(negrecip(activate(_x2)), 2ndspos(_x0, activate(_x3)))]] 
  [[plus(0, _x0)]] = x0 >= x0 = [[_x0]] 
  [[plus(s(_x0), _x1)]] = x1 + 3x0 >= x1 + 3x0 = [[s(plus(_x0, _x1))]] 
  [[times(0, _x0)]] = 2 + x0 > 0 = [[0]] 
  [[square(_x0)]] = 3 + 3x0 > 2 + 2x0 = [[times(_x0, _x0)]] 
  [[from(_x0)]] = 3 + 3x0 > x0 = [[n!6220!6220from(_x0)]] 
  [[s(_x0)]] = x0 >= x0 = [[n!6220!6220s(_x0)]] 
  [[activate(n!6220!6220s(_x0))]] = 2x0 >= 2x0 = [[s(activate(_x0))]] 
  [[activate(_x0)]] = 2x0 >= x0 = [[_x0]] 

We can thus remove the following rules:

  2ndspos(0, X) => rnil 
  2ndspos(s(X), cons(Y, n!6220!6220cons(Z, U))) => rcons(posrecip(activate(Z)), 2ndsneg(X, activate(U))) 
  2ndsneg(0, X) => rnil 
  2ndsneg(s(X), cons(Y, n!6220!6220cons(Z, U))) => rcons(negrecip(activate(Z)), 2ndspos(X, activate(U))) 
  times(0, X) => 0 
  square(X) => times(X, X) 
  from(X) => n!6220!6220from(X) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  plus(0, X) >? X 
  plus(s(X), Y) >? s(plus(X, Y)) 
  s(X) >? n!6220!6220s(X) 
  activate(n!6220!6220s(X)) >? s(activate(X)) 
  activate(X) >? X 

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

The following interpretation satisfies the requirements:

  0 = 3 
  activate = \y0.2y0 
  n!6220!6220s = \y0.1 + y0 
  plus = \y0y1.2y1 + 3y0 
  s = \y0.1 + y0 

Using this interpretation, the requirements translate to:

  [[plus(0, _x0)]] = 9 + 2x0 > x0 = [[_x0]] 
  [[plus(s(_x0), _x1)]] = 3 + 2x1 + 3x0 > 1 + 2x1 + 3x0 = [[s(plus(_x0, _x1))]] 
  [[s(_x0)]] = 1 + x0 >= 1 + x0 = [[n!6220!6220s(_x0)]] 
  [[activate(n!6220!6220s(_x0))]] = 2 + 2x0 > 1 + 2x0 = [[s(activate(_x0))]] 
  [[activate(_x0)]] = 2x0 >= x0 = [[_x0]] 

We can thus remove the following rules:

  plus(0, X) => X 
  plus(s(X), Y) => s(plus(X, Y)) 
  activate(n!6220!6220s(X)) => s(activate(X)) 

We use rule removal, following [Kop12, Theorem 2.23].

This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):

  s(X) >? n!6220!6220s(X) 
  activate(X) >? X 

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

The following interpretation satisfies the requirements:

  activate = \y0.3 + y0 
  n!6220!6220s = \y0.y0 
  s = \y0.3 + 3y0 

Using this interpretation, the requirements translate to:

  [[s(_x0)]] = 3 + 3x0 > x0 = [[n!6220!6220s(_x0)]] 
  [[activate(_x0)]] = 3 + x0 > x0 = [[_x0]] 

We can thus remove the following rules:

  s(X) => n!6220!6220s(X) 
  activate(X) => X 

All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.


+++ Citations +++

[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.