/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.

  a : [] --> o 
  b : [o * o] --> o 
  f!62200 : [o] --> o 
  f!62201 : [o] --> o 
  f!622010 : [o] --> o 
  f!62202 : [o] --> o 
  f!62203 : [o] --> o 
  f!62204 : [o] --> o 
  f!62205 : [o] --> o 
  f!62206 : [o] --> o 
  f!62207 : [o] --> o 
  f!62208 : [o] --> o 
  f!62209 : [o] --> o 
  g!62201 : [o * o] --> o 
  g!622010 : [o * o] --> o 
  g!62202 : [o * o] --> o 
  g!62203 : [o * o] --> o 
  g!62204 : [o * o] --> o 
  g!62205 : [o * o] --> o 
  g!62206 : [o * o] --> o 
  g!62207 : [o * o] --> o 
  g!62208 : [o * o] --> o 
  g!62209 : [o * o] --> o 
  s : [o] --> o 

  f!62200(X) => a 
  f!62201(X) => g!62201(X, X) 
  g!62201(s(X), Y) => b(f!62200(Y), g!62201(X, Y)) 
  f!62202(X) => g!62202(X, X) 
  g!62202(s(X), Y) => b(f!62201(Y), g!62202(X, Y)) 
  f!62203(X) => g!62203(X, X) 
  g!62203(s(X), Y) => b(f!62202(Y), g!62203(X, Y)) 
  f!62204(X) => g!62204(X, X) 
  g!62204(s(X), Y) => b(f!62203(Y), g!62204(X, Y)) 
  f!62205(X) => g!62205(X, X) 
  g!62205(s(X), Y) => b(f!62204(Y), g!62205(X, Y)) 
  f!62206(X) => g!62206(X, X) 
  g!62206(s(X), Y) => b(f!62205(Y), g!62206(X, Y)) 
  f!62207(X) => g!62207(X, X) 
  g!62207(s(X), Y) => b(f!62206(Y), g!62207(X, Y)) 
  f!62208(X) => g!62208(X, X) 
  g!62208(s(X), Y) => b(f!62207(Y), g!62208(X, Y)) 
  f!62209(X) => g!62209(X, X) 
  g!62209(s(X), Y) => b(f!62208(Y), g!62209(X, Y)) 
  f!622010(X) => g!622010(X, X) 
  g!622010(s(X), Y) => b(f!62209(Y), g!622010(X, Y)) 

As the system is orthogonal, it is terminating if it is innermost terminating by [Gra95].  Then, by [FuhGieParSchSwi11], it suffices to prove (innermost) termination of the typed system, with sort annotations chosen to respect the rules, as follows:

  a : [] --> lh 
  b : [lh * lh] --> lh 
  f!62200 : [dh] --> lh 
  f!62201 : [dh] --> lh 
  f!622010 : [dh] --> lh 
  f!62202 : [dh] --> lh 
  f!62203 : [dh] --> lh 
  f!62204 : [dh] --> lh 
  f!62205 : [dh] --> lh 
  f!62206 : [dh] --> lh 
  f!62207 : [dh] --> lh 
  f!62208 : [dh] --> lh 
  f!62209 : [dh] --> lh 
  g!62201 : [dh * dh] --> lh 
  g!622010 : [dh * dh] --> lh 
  g!62202 : [dh * dh] --> lh 
  g!62203 : [dh * dh] --> lh 
  g!62204 : [dh * dh] --> lh 
  g!62205 : [dh * dh] --> lh 
  g!62206 : [dh * dh] --> lh 
  g!62207 : [dh * dh] --> lh 
  g!62208 : [dh * dh] --> lh 
  g!62209 : [dh * dh] --> lh 
  s : [dh] --> dh 

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

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

  f!62200(X) >? a 
  f!62201(X) >? g!62201(X, X) 
  g!62201(s(X), Y) >? b(f!62200(Y), g!62201(X, Y)) 
  f!62202(X) >? g!62202(X, X) 
  g!62202(s(X), Y) >? b(f!62201(Y), g!62202(X, Y)) 
  f!62203(X) >? g!62203(X, X) 
  g!62203(s(X), Y) >? b(f!62202(Y), g!62203(X, Y)) 
  f!62204(X) >? g!62204(X, X) 
  g!62204(s(X), Y) >? b(f!62203(Y), g!62204(X, Y)) 
  f!62205(X) >? g!62205(X, X) 
  g!62205(s(X), Y) >? b(f!62204(Y), g!62205(X, Y)) 
  f!62206(X) >? g!62206(X, X) 
  g!62206(s(X), Y) >? b(f!62205(Y), g!62206(X, Y)) 
  f!62207(X) >? g!62207(X, X) 
  g!62207(s(X), Y) >? b(f!62206(Y), g!62207(X, Y)) 
  f!62208(X) >? g!62208(X, X) 
  g!62208(s(X), Y) >? b(f!62207(Y), g!62208(X, Y)) 
  f!62209(X) >? g!62209(X, X) 
  g!62209(s(X), Y) >? b(f!62208(Y), g!62209(X, Y)) 
  f!622010(X) >? g!622010(X, X) 
  g!622010(s(X), Y) >? b(f!62209(Y), g!622010(X, Y)) 

about to try horpo

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

Argument functions:

  [[a]] = _|_ 
  [[f!62200(x_1)]] = x_1 
  [[g!62201(x_1, x_2)]] = g!62201(x_2, x_1) 

We choose Lex = {g!62201} and Mul = {b, f!62201, f!622010, f!62202, f!62203, f!62204, f!62205, f!62206, f!62207, f!62208, f!62209, g!622010, g!62202, g!62203, g!62204, g!62205, g!62206, g!62207, g!62208, g!62209, s}, and the following precedence: f!622010 > g!622010 > f!62209 > g!62209 > f!62208 > f!62207 = g!62208 > g!62207 > f!62206 > g!62206 > f!62205 > g!62205 > f!62204 > g!62204 > f!62203 > f!62202 = g!62203 > s > f!62201 = g!62202 > g!62201 > b

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

  X >= _|_ 
  f!62201(X) > g!62201(X, X) 
  g!62201(s(X), Y) >= b(Y, g!62201(X, Y)) 
  f!62202(X) > g!62202(X, X) 
  g!62202(s(X), Y) >= b(f!62201(Y), g!62202(X, Y)) 
  f!62203(X) > g!62203(X, X) 
  g!62203(s(X), Y) >= b(f!62202(Y), g!62203(X, Y)) 
  f!62204(X) > g!62204(X, X) 
  g!62204(s(X), Y) >= b(f!62203(Y), g!62204(X, Y)) 
  f!62205(X) >= g!62205(X, X) 
  g!62205(s(X), Y) >= b(f!62204(Y), g!62205(X, Y)) 
  f!62206(X) >= g!62206(X, X) 
  g!62206(s(X), Y) > b(f!62205(Y), g!62206(X, Y)) 
  f!62207(X) > g!62207(X, X) 
  g!62207(s(X), Y) >= b(f!62206(Y), g!62207(X, Y)) 
  f!62208(X) >= g!62208(X, X) 
  g!62208(s(X), Y) >= b(f!62207(Y), g!62208(X, Y)) 
  f!62209(X) > g!62209(X, X) 
  g!62209(s(X), Y) > b(f!62208(Y), g!62209(X, Y)) 
  f!622010(X) >= g!622010(X, X) 
  g!622010(s(X), Y) > b(f!62209(Y), g!622010(X, Y)) 

With these choices, we have:

  1] X >= _|_  by (Bot) 

  2] f!62201(X) > g!62201(X, X)  because [3], by definition 
  3] f!62201*(X) >= g!62201(X, X)  because f!62201 > g!62201, [4] and [4], by (Copy) 
  4] f!62201*(X) >= X  because [5], by (Select) 
  5] X >= X  by (Meta) 

  6] g!62201(s(X), Y) >= b(Y, g!62201(X, Y))  because [7], by (Star) 
  7] g!62201*(s(X), Y) >= b(Y, g!62201(X, Y))  because g!62201 > b, [8] and [10], by (Copy) 
  8] g!62201*(s(X), Y) >= Y  because [9], by (Select) 
  9] Y >= Y  by (Meta) 
  10] g!62201*(s(X), Y) >= g!62201(X, Y)  because [11], [13], [14] and [8], by (Stat) 
  11] s(X) > X  because [12], by definition 
  12] s*(X) >= X  because [5], by (Select) 
  13] Y >= Y  by (Meta) 
  14] g!62201*(s(X), Y) >= X  because [15], by (Select) 
  15] s(X) >= X  because [12], by (Star) 

  16] f!62202(X) > g!62202(X, X)  because [17], by definition 
  17] f!62202*(X) >= g!62202(X, X)  because f!62202 > g!62202, [18] and [18], by (Copy) 
  18] f!62202*(X) >= X  because [5], by (Select) 

  19] g!62202(s(X), Y) >= b(f!62201(Y), g!62202(X, Y))  because [20], by (Star) 
  20] g!62202*(s(X), Y) >= b(f!62201(Y), g!62202(X, Y))  because g!62202 > b, [21] and [22], by (Copy) 
  21] g!62202*(s(X), Y) >= f!62201(Y)  because g!62202 = f!62201, g!62202 in Mul and [13], by (Stat) 
  22] g!62202*(s(X), Y) >= g!62202(X, Y)  because g!62202 in Mul, [11] and [13], by (Stat) 

  23] f!62203(X) > g!62203(X, X)  because [24], by definition 
  24] f!62203*(X) >= g!62203(X, X)  because f!62203 > g!62203, [25] and [25], by (Copy) 
  25] f!62203*(X) >= X  because [5], by (Select) 

  26] g!62203(s(X), Y) >= b(f!62202(Y), g!62203(X, Y))  because [27], by (Star) 
  27] g!62203*(s(X), Y) >= b(f!62202(Y), g!62203(X, Y))  because g!62203 > b, [28] and [29], by (Copy) 
  28] g!62203*(s(X), Y) >= f!62202(Y)  because g!62203 = f!62202, g!62203 in Mul and [13], by (Stat) 
  29] g!62203*(s(X), Y) >= g!62203(X, Y)  because g!62203 in Mul, [11] and [13], by (Stat) 

  30] f!62204(X) > g!62204(X, X)  because [31], by definition 
  31] f!62204*(X) >= g!62204(X, X)  because f!62204 > g!62204, [32] and [32], by (Copy) 
  32] f!62204*(X) >= X  because [5], by (Select) 

  33] g!62204(s(X), Y) >= b(f!62203(Y), g!62204(X, Y))  because [34], by (Star) 
  34] g!62204*(s(X), Y) >= b(f!62203(Y), g!62204(X, Y))  because g!62204 > b, [35] and [37], by (Copy) 
  35] g!62204*(s(X), Y) >= f!62203(Y)  because g!62204 > f!62203 and [36], by (Copy) 
  36] g!62204*(s(X), Y) >= Y  because [13], by (Select) 
  37] g!62204*(s(X), Y) >= g!62204(X, Y)  because g!62204 in Mul, [11] and [13], by (Stat) 

  38] f!62205(X) >= g!62205(X, X)  because [39], by (Star) 
  39] f!62205*(X) >= g!62205(X, X)  because f!62205 > g!62205, [40] and [40], by (Copy) 
  40] f!62205*(X) >= X  because [5], by (Select) 

  41] g!62205(s(X), Y) >= b(f!62204(Y), g!62205(X, Y))  because [42], by (Star) 
  42] g!62205*(s(X), Y) >= b(f!62204(Y), g!62205(X, Y))  because g!62205 > b, [43] and [45], by (Copy) 
  43] g!62205*(s(X), Y) >= f!62204(Y)  because g!62205 > f!62204 and [44], by (Copy) 
  44] g!62205*(s(X), Y) >= Y  because [13], by (Select) 
  45] g!62205*(s(X), Y) >= g!62205(X, Y)  because g!62205 in Mul, [11] and [13], by (Stat) 

  46] f!62206(X) >= g!62206(X, X)  because [47], by (Star) 
  47] f!62206*(X) >= g!62206(X, X)  because f!62206 > g!62206, [48] and [48], by (Copy) 
  48] f!62206*(X) >= X  because [5], by (Select) 

  49] g!62206(s(X), Y) > b(f!62205(Y), g!62206(X, Y))  because [50], by definition 
  50] g!62206*(s(X), Y) >= b(f!62205(Y), g!62206(X, Y))  because g!62206 > b, [51] and [53], by (Copy) 
  51] g!62206*(s(X), Y) >= f!62205(Y)  because g!62206 > f!62205 and [52], by (Copy) 
  52] g!62206*(s(X), Y) >= Y  because [13], by (Select) 
  53] g!62206*(s(X), Y) >= g!62206(X, Y)  because g!62206 in Mul, [11] and [13], by (Stat) 

  54] f!62207(X) > g!62207(X, X)  because [55], by definition 
  55] f!62207*(X) >= g!62207(X, X)  because f!62207 > g!62207, [56] and [56], by (Copy) 
  56] f!62207*(X) >= X  because [5], by (Select) 

  57] g!62207(s(X), Y) >= b(f!62206(Y), g!62207(X, Y))  because [58], by (Star) 
  58] g!62207*(s(X), Y) >= b(f!62206(Y), g!62207(X, Y))  because g!62207 > b, [59] and [61], by (Copy) 
  59] g!62207*(s(X), Y) >= f!62206(Y)  because g!62207 > f!62206 and [60], by (Copy) 
  60] g!62207*(s(X), Y) >= Y  because [13], by (Select) 
  61] g!62207*(s(X), Y) >= g!62207(X, Y)  because g!62207 in Mul, [11] and [13], by (Stat) 

  62] f!62208(X) >= g!62208(X, X)  because [63], by (Star) 
  63] f!62208*(X) >= g!62208(X, X)  because f!62208 > g!62208, [64] and [64], by (Copy) 
  64] f!62208*(X) >= X  because [5], by (Select) 

  65] g!62208(s(X), Y) >= b(f!62207(Y), g!62208(X, Y))  because [66], by (Star) 
  66] g!62208*(s(X), Y) >= b(f!62207(Y), g!62208(X, Y))  because g!62208 > b, [67] and [68], by (Copy) 
  67] g!62208*(s(X), Y) >= f!62207(Y)  because g!62208 = f!62207, g!62208 in Mul and [13], by (Stat) 
  68] g!62208*(s(X), Y) >= g!62208(X, Y)  because g!62208 in Mul, [11] and [13], by (Stat) 

  69] f!62209(X) > g!62209(X, X)  because [70], by definition 
  70] f!62209*(X) >= g!62209(X, X)  because f!62209 > g!62209, [71] and [71], by (Copy) 
  71] f!62209*(X) >= X  because [5], by (Select) 

  72] g!62209(s(X), Y) > b(f!62208(Y), g!62209(X, Y))  because [73], by definition 
  73] g!62209*(s(X), Y) >= b(f!62208(Y), g!62209(X, Y))  because g!62209 > b, [74] and [76], by (Copy) 
  74] g!62209*(s(X), Y) >= f!62208(Y)  because g!62209 > f!62208 and [75], by (Copy) 
  75] g!62209*(s(X), Y) >= Y  because [13], by (Select) 
  76] g!62209*(s(X), Y) >= g!62209(X, Y)  because g!62209 in Mul, [11] and [13], by (Stat) 

  77] f!622010(X) >= g!622010(X, X)  because [78], by (Star) 
  78] f!622010*(X) >= g!622010(X, X)  because f!622010 > g!622010, [79] and [79], by (Copy) 
  79] f!622010*(X) >= X  because [5], by (Select) 

  80] g!622010(s(X), Y) > b(f!62209(Y), g!622010(X, Y))  because [81], by definition 
  81] g!622010*(s(X), Y) >= b(f!62209(Y), g!622010(X, Y))  because g!622010 > b, [82] and [84], by (Copy) 
  82] g!622010*(s(X), Y) >= f!62209(Y)  because g!622010 > f!62209 and [83], by (Copy) 
  83] g!622010*(s(X), Y) >= Y  because [13], by (Select) 
  84] g!622010*(s(X), Y) >= g!622010(X, Y)  because g!622010 in Mul, [11] and [13], by (Stat) 

We can thus remove the following rules:

  f!62201(X) => g!62201(X, X) 
  f!62202(X) => g!62202(X, X) 
  f!62203(X) => g!62203(X, X) 
  f!62204(X) => g!62204(X, X) 
  g!62206(s(X), Y) => b(f!62205(Y), g!62206(X, Y)) 
  f!62207(X) => g!62207(X, X) 
  f!62209(X) => g!62209(X, X) 
  g!62209(s(X), Y) => b(f!62208(Y), g!62209(X, Y)) 
  g!622010(s(X), Y) => b(f!62209(Y), g!622010(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]):

  f!62200(X) >? a 
  g!62201(s(X), Y) >? b(f!62200(Y), g!62201(X, Y)) 
  g!62202(s(X), Y) >? b(f!62201(Y), g!62202(X, Y)) 
  g!62203(s(X), Y) >? b(f!62202(Y), g!62203(X, Y)) 
  g!62204(s(X), Y) >? b(f!62203(Y), g!62204(X, Y)) 
  f!62205(X) >? g!62205(X, X) 
  g!62205(s(X), Y) >? b(f!62204(Y), g!62205(X, Y)) 
  f!62206(X) >? g!62206(X, X) 
  g!62207(s(X), Y) >? b(f!62206(Y), g!62207(X, Y)) 
  f!62208(X) >? g!62208(X, X) 
  g!62208(s(X), Y) >? b(f!62207(Y), g!62208(X, Y)) 
  f!622010(X) >? g!622010(X, X) 

about to try horpo

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

Argument functions:

  [[a]] = _|_ 
  [[f!62200(x_1)]] = x_1 
  [[f!62202(x_1)]] = x_1 

We choose Lex = {} and Mul = {b, f!62201, f!622010, f!62203, f!62204, f!62205, f!62206, f!62207, f!62208, g!62201, g!622010, g!62202, g!62203, g!62204, g!62205, g!62206, g!62207, g!62208, s}, and the following precedence: g!62204 > g!62202 > f!622010 > g!62207 > f!62205 > f!62203 > g!62205 > f!62201 > f!62204 > f!62206 > f!62208 > g!62206 > f!62207 = g!62208 > g!622010 > g!62201 > g!62203 > s > b

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

  X >= _|_ 
  g!62201(s(X), Y) > b(Y, g!62201(X, Y)) 
  g!62202(s(X), Y) >= b(f!62201(Y), g!62202(X, Y)) 
  g!62203(s(X), Y) >= b(Y, g!62203(X, Y)) 
  g!62204(s(X), Y) >= b(f!62203(Y), g!62204(X, Y)) 
  f!62205(X) >= g!62205(X, X) 
  g!62205(s(X), Y) >= b(f!62204(Y), g!62205(X, Y)) 
  f!62206(X) > g!62206(X, X) 
  g!62207(s(X), Y) >= b(f!62206(Y), g!62207(X, Y)) 
  f!62208(X) >= g!62208(X, X) 
  g!62208(s(X), Y) >= b(f!62207(Y), g!62208(X, Y)) 
  f!622010(X) >= g!622010(X, X) 

With these choices, we have:

  1] X >= _|_  by (Bot) 

  2] g!62201(s(X), Y) > b(Y, g!62201(X, Y))  because [3], by definition 
  3] g!62201*(s(X), Y) >= b(Y, g!62201(X, Y))  because g!62201 > b, [4] and [6], by (Copy) 
  4] g!62201*(s(X), Y) >= Y  because [5], by (Select) 
  5] Y >= Y  by (Meta) 
  6] g!62201*(s(X), Y) >= g!62201(X, Y)  because g!62201 in Mul, [7] and [10], by (Stat) 
  7] s(X) > X  because [8], by definition 
  8] s*(X) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 
  10] Y >= Y  by (Meta) 

  11] g!62202(s(X), Y) >= b(f!62201(Y), g!62202(X, Y))  because [12], by (Star) 
  12] g!62202*(s(X), Y) >= b(f!62201(Y), g!62202(X, Y))  because g!62202 > b, [13] and [15], by (Copy) 
  13] g!62202*(s(X), Y) >= f!62201(Y)  because g!62202 > f!62201 and [14], by (Copy) 
  14] g!62202*(s(X), Y) >= Y  because [10], by (Select) 
  15] g!62202*(s(X), Y) >= g!62202(X, Y)  because g!62202 in Mul, [7] and [10], by (Stat) 

  16] g!62203(s(X), Y) >= b(Y, g!62203(X, Y))  because [17], by (Star) 
  17] g!62203*(s(X), Y) >= b(Y, g!62203(X, Y))  because g!62203 > b, [18] and [19], by (Copy) 
  18] g!62203*(s(X), Y) >= Y  because [10], by (Select) 
  19] g!62203*(s(X), Y) >= g!62203(X, Y)  because g!62203 in Mul, [7] and [10], by (Stat) 

  20] g!62204(s(X), Y) >= b(f!62203(Y), g!62204(X, Y))  because [21], by (Star) 
  21] g!62204*(s(X), Y) >= b(f!62203(Y), g!62204(X, Y))  because g!62204 > b, [22] and [24], by (Copy) 
  22] g!62204*(s(X), Y) >= f!62203(Y)  because g!62204 > f!62203 and [23], by (Copy) 
  23] g!62204*(s(X), Y) >= Y  because [10], by (Select) 
  24] g!62204*(s(X), Y) >= g!62204(X, Y)  because g!62204 in Mul, [7] and [10], by (Stat) 

  25] f!62205(X) >= g!62205(X, X)  because [26], by (Star) 
  26] f!62205*(X) >= g!62205(X, X)  because f!62205 > g!62205, [27] and [27], by (Copy) 
  27] f!62205*(X) >= X  because [9], by (Select) 

  28] g!62205(s(X), Y) >= b(f!62204(Y), g!62205(X, Y))  because [29], by (Star) 
  29] g!62205*(s(X), Y) >= b(f!62204(Y), g!62205(X, Y))  because g!62205 > b, [30] and [32], by (Copy) 
  30] g!62205*(s(X), Y) >= f!62204(Y)  because g!62205 > f!62204 and [31], by (Copy) 
  31] g!62205*(s(X), Y) >= Y  because [10], by (Select) 
  32] g!62205*(s(X), Y) >= g!62205(X, Y)  because g!62205 in Mul, [7] and [10], by (Stat) 

  33] f!62206(X) > g!62206(X, X)  because [34], by definition 
  34] f!62206*(X) >= g!62206(X, X)  because f!62206 > g!62206, [35] and [35], by (Copy) 
  35] f!62206*(X) >= X  because [9], by (Select) 

  36] g!62207(s(X), Y) >= b(f!62206(Y), g!62207(X, Y))  because [37], by (Star) 
  37] g!62207*(s(X), Y) >= b(f!62206(Y), g!62207(X, Y))  because g!62207 > b, [38] and [40], by (Copy) 
  38] g!62207*(s(X), Y) >= f!62206(Y)  because g!62207 > f!62206 and [39], by (Copy) 
  39] g!62207*(s(X), Y) >= Y  because [10], by (Select) 
  40] g!62207*(s(X), Y) >= g!62207(X, Y)  because g!62207 in Mul, [7] and [10], by (Stat) 

  41] f!62208(X) >= g!62208(X, X)  because [42], by (Star) 
  42] f!62208*(X) >= g!62208(X, X)  because f!62208 > g!62208, [43] and [43], by (Copy) 
  43] f!62208*(X) >= X  because [9], by (Select) 

  44] g!62208(s(X), Y) >= b(f!62207(Y), g!62208(X, Y))  because [45], by (Star) 
  45] g!62208*(s(X), Y) >= b(f!62207(Y), g!62208(X, Y))  because g!62208 > b, [46] and [47], by (Copy) 
  46] g!62208*(s(X), Y) >= f!62207(Y)  because g!62208 = f!62207, g!62208 in Mul and [10], by (Stat) 
  47] g!62208*(s(X), Y) >= g!62208(X, Y)  because g!62208 in Mul, [7] and [10], by (Stat) 

  48] f!622010(X) >= g!622010(X, X)  because [49], by (Star) 
  49] f!622010*(X) >= g!622010(X, X)  because f!622010 > g!622010, [50] and [50], by (Copy) 
  50] f!622010*(X) >= X  because [9], by (Select) 

We can thus remove the following rules:

  g!62201(s(X), Y) => b(f!62200(Y), g!62201(X, Y)) 
  f!62206(X) => g!62206(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]):

  f!62200(X) >? a 
  g!62202(s(X), Y) >? b(f!62201(Y), g!62202(X, Y)) 
  g!62203(s(X), Y) >? b(f!62202(Y), g!62203(X, Y)) 
  g!62204(s(X), Y) >? b(f!62203(Y), g!62204(X, Y)) 
  f!62205(X) >? g!62205(X, X) 
  g!62205(s(X), Y) >? b(f!62204(Y), g!62205(X, Y)) 
  g!62207(s(X), Y) >? b(f!62206(Y), g!62207(X, Y)) 
  f!62208(X) >? g!62208(X, X) 
  g!62208(s(X), Y) >? b(f!62207(Y), g!62208(X, Y)) 
  f!622010(X) >? g!622010(X, X) 

about to try horpo

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

Argument functions:

  [[a]] = _|_ 
  [[f!62200(x_1)]] = x_1 
  [[f!62202(x_1)]] = x_1 
  [[f!62203(x_1)]] = x_1 
  [[f!62204(x_1)]] = x_1 
  [[f!62206(x_1)]] = x_1 

We choose Lex = {} and Mul = {b, f!62201, f!622010, f!62205, f!62207, f!62208, g!622010, g!62202, g!62203, g!62204, g!62205, g!62207, g!62208, s}, and the following precedence: f!622010 > s > g!62207 > f!62208 > g!62208 > f!62201 = g!62202 > g!62204 > g!62203 > g!622010 > f!62205 > g!62205 > f!62207 > b

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

  X >= _|_ 
  g!62202(s(X), Y) >= b(f!62201(Y), g!62202(X, Y)) 
  g!62203(s(X), Y) >= b(Y, g!62203(X, Y)) 
  g!62204(s(X), Y) > b(Y, g!62204(X, Y)) 
  f!62205(X) >= g!62205(X, X) 
  g!62205(s(X), Y) >= b(Y, g!62205(X, Y)) 
  g!62207(s(X), Y) > b(Y, g!62207(X, Y)) 
  f!62208(X) >= g!62208(X, X) 
  g!62208(s(X), Y) >= b(f!62207(Y), g!62208(X, Y)) 
  f!622010(X) >= g!622010(X, X) 

With these choices, we have:

  1] X >= _|_  by (Bot) 

  2] g!62202(s(X), Y) >= b(f!62201(Y), g!62202(X, Y))  because [3], by (Star) 
  3] g!62202*(s(X), Y) >= b(f!62201(Y), g!62202(X, Y))  because g!62202 > b, [4] and [6], by (Copy) 
  4] g!62202*(s(X), Y) >= f!62201(Y)  because g!62202 = f!62201, g!62202 in Mul and [5], by (Stat) 
  5] Y >= Y  by (Meta) 
  6] g!62202*(s(X), Y) >= g!62202(X, Y)  because g!62202 in Mul, [7] and [5], by (Stat) 
  7] s(X) > X  because [8], by definition 
  8] s*(X) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 

  10] g!62203(s(X), Y) >= b(Y, g!62203(X, Y))  because [11], by (Star) 
  11] g!62203*(s(X), Y) >= b(Y, g!62203(X, Y))  because g!62203 > b, [12] and [13], by (Copy) 
  12] g!62203*(s(X), Y) >= Y  because [5], by (Select) 
  13] g!62203*(s(X), Y) >= g!62203(X, Y)  because g!62203 in Mul, [7] and [5], by (Stat) 

  14] g!62204(s(X), Y) > b(Y, g!62204(X, Y))  because [15], by definition 
  15] g!62204*(s(X), Y) >= b(Y, g!62204(X, Y))  because g!62204 > b, [16] and [17], by (Copy) 
  16] g!62204*(s(X), Y) >= Y  because [5], by (Select) 
  17] g!62204*(s(X), Y) >= g!62204(X, Y)  because g!62204 in Mul, [7] and [5], by (Stat) 

  18] f!62205(X) >= g!62205(X, X)  because [19], by (Star) 
  19] f!62205*(X) >= g!62205(X, X)  because f!62205 > g!62205, [20] and [20], by (Copy) 
  20] f!62205*(X) >= X  because [9], by (Select) 

  21] g!62205(s(X), Y) >= b(Y, g!62205(X, Y))  because [22], by (Star) 
  22] g!62205*(s(X), Y) >= b(Y, g!62205(X, Y))  because g!62205 > b, [23] and [24], by (Copy) 
  23] g!62205*(s(X), Y) >= Y  because [5], by (Select) 
  24] g!62205*(s(X), Y) >= g!62205(X, Y)  because g!62205 in Mul, [7] and [5], by (Stat) 

  25] g!62207(s(X), Y) > b(Y, g!62207(X, Y))  because [26], by definition 
  26] g!62207*(s(X), Y) >= b(Y, g!62207(X, Y))  because g!62207 > b, [27] and [28], by (Copy) 
  27] g!62207*(s(X), Y) >= Y  because [5], by (Select) 
  28] g!62207*(s(X), Y) >= g!62207(X, Y)  because g!62207 in Mul, [7] and [5], by (Stat) 

  29] f!62208(X) >= g!62208(X, X)  because [30], by (Star) 
  30] f!62208*(X) >= g!62208(X, X)  because f!62208 > g!62208, [31] and [31], by (Copy) 
  31] f!62208*(X) >= X  because [9], by (Select) 

  32] g!62208(s(X), Y) >= b(f!62207(Y), g!62208(X, Y))  because [33], by (Star) 
  33] g!62208*(s(X), Y) >= b(f!62207(Y), g!62208(X, Y))  because g!62208 > b, [34] and [36], by (Copy) 
  34] g!62208*(s(X), Y) >= f!62207(Y)  because g!62208 > f!62207 and [35], by (Copy) 
  35] g!62208*(s(X), Y) >= Y  because [5], by (Select) 
  36] g!62208*(s(X), Y) >= g!62208(X, Y)  because g!62208 in Mul, [7] and [5], by (Stat) 

  37] f!622010(X) >= g!622010(X, X)  because [38], by (Star) 
  38] f!622010*(X) >= g!622010(X, X)  because f!622010 > g!622010, [39] and [39], by (Copy) 
  39] f!622010*(X) >= X  because [9], by (Select) 

We can thus remove the following rules:

  g!62204(s(X), Y) => b(f!62203(Y), g!62204(X, Y)) 
  g!62207(s(X), Y) => b(f!62206(Y), g!62207(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]):

  f!62200(X) >? a 
  g!62202(s(X), Y) >? b(f!62201(Y), g!62202(X, Y)) 
  g!62203(s(X), Y) >? b(f!62202(Y), g!62203(X, Y)) 
  f!62205(X) >? g!62205(X, X) 
  g!62205(s(X), Y) >? b(f!62204(Y), g!62205(X, Y)) 
  f!62208(X) >? g!62208(X, X) 
  g!62208(s(X), Y) >? b(f!62207(Y), g!62208(X, Y)) 
  f!622010(X) >? g!622010(X, X) 

about to try horpo

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

Argument functions:

  [[a]] = _|_ 
  [[f!62200(x_1)]] = x_1 
  [[f!62202(x_1)]] = x_1 
  [[f!62204(x_1)]] = x_1 
  [[f!62207(x_1)]] = x_1 

We choose Lex = {} and Mul = {b, f!62201, f!622010, f!62205, f!62208, g!622010, g!62202, g!62203, g!62205, g!62208, s}, and the following precedence: f!62208 > s > f!62201 = g!62202 > f!622010 > g!622010 > f!62205 > g!62205 > g!62208 > g!62203 > b

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

  X >= _|_ 
  g!62202(s(X), Y) > b(f!62201(Y), g!62202(X, Y)) 
  g!62203(s(X), Y) >= b(Y, g!62203(X, Y)) 
  f!62205(X) >= g!62205(X, X) 
  g!62205(s(X), Y) >= b(Y, g!62205(X, Y)) 
  f!62208(X) >= g!62208(X, X) 
  g!62208(s(X), Y) >= b(Y, g!62208(X, Y)) 
  f!622010(X) >= g!622010(X, X) 

With these choices, we have:

  1] X >= _|_  by (Bot) 

  2] g!62202(s(X), Y) > b(f!62201(Y), g!62202(X, Y))  because [3], by definition 
  3] g!62202*(s(X), Y) >= b(f!62201(Y), g!62202(X, Y))  because g!62202 > b, [4] and [6], by (Copy) 
  4] g!62202*(s(X), Y) >= f!62201(Y)  because g!62202 = f!62201, g!62202 in Mul and [5], by (Stat) 
  5] Y >= Y  by (Meta) 
  6] g!62202*(s(X), Y) >= g!62202(X, Y)  because g!62202 in Mul, [7] and [5], by (Stat) 
  7] s(X) > X  because [8], by definition 
  8] s*(X) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 

  10] g!62203(s(X), Y) >= b(Y, g!62203(X, Y))  because [11], by (Star) 
  11] g!62203*(s(X), Y) >= b(Y, g!62203(X, Y))  because g!62203 > b, [12] and [13], by (Copy) 
  12] g!62203*(s(X), Y) >= Y  because [5], by (Select) 
  13] g!62203*(s(X), Y) >= g!62203(X, Y)  because g!62203 in Mul, [7] and [5], by (Stat) 

  14] f!62205(X) >= g!62205(X, X)  because [15], by (Star) 
  15] f!62205*(X) >= g!62205(X, X)  because f!62205 > g!62205, [16] and [16], by (Copy) 
  16] f!62205*(X) >= X  because [9], by (Select) 

  17] g!62205(s(X), Y) >= b(Y, g!62205(X, Y))  because [18], by (Star) 
  18] g!62205*(s(X), Y) >= b(Y, g!62205(X, Y))  because g!62205 > b, [19] and [20], by (Copy) 
  19] g!62205*(s(X), Y) >= Y  because [5], by (Select) 
  20] g!62205*(s(X), Y) >= g!62205(X, Y)  because g!62205 in Mul, [7] and [5], by (Stat) 

  21] f!62208(X) >= g!62208(X, X)  because [22], by (Star) 
  22] f!62208*(X) >= g!62208(X, X)  because f!62208 > g!62208, [23] and [23], by (Copy) 
  23] f!62208*(X) >= X  because [9], by (Select) 

  24] g!62208(s(X), Y) >= b(Y, g!62208(X, Y))  because [25], by (Star) 
  25] g!62208*(s(X), Y) >= b(Y, g!62208(X, Y))  because g!62208 > b, [26] and [27], by (Copy) 
  26] g!62208*(s(X), Y) >= Y  because [5], by (Select) 
  27] g!62208*(s(X), Y) >= g!62208(X, Y)  because g!62208 in Mul, [7] and [5], by (Stat) 

  28] f!622010(X) >= g!622010(X, X)  because [29], by (Star) 
  29] f!622010*(X) >= g!622010(X, X)  because f!622010 > g!622010, [30] and [30], by (Copy) 
  30] f!622010*(X) >= X  because [9], by (Select) 

We can thus remove the following rules:

  g!62202(s(X), Y) => b(f!62201(Y), g!62202(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]):

  f!62200(X) >? a 
  g!62203(s(X), Y) >? b(f!62202(Y), g!62203(X, Y)) 
  f!62205(X) >? g!62205(X, X) 
  g!62205(s(X), Y) >? b(f!62204(Y), g!62205(X, Y)) 
  f!62208(X) >? g!62208(X, X) 
  g!62208(s(X), Y) >? b(f!62207(Y), g!62208(X, Y)) 
  f!622010(X) >? g!622010(X, X) 

about to try horpo

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

Argument functions:

  [[a]] = _|_ 
  [[f!62200(x_1)]] = x_1 
  [[f!62202(x_1)]] = x_1 
  [[f!62204(x_1)]] = x_1 
  [[f!62207(x_1)]] = x_1 

We choose Lex = {} and Mul = {b, f!622010, f!62205, f!62208, g!622010, g!62203, g!62205, g!62208, s}, and the following precedence: f!62208 > g!62208 > s > g!62203 > f!622010 > g!622010 > f!62205 > g!62205 > b

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

  X >= _|_ 
  g!62203(s(X), Y) >= b(Y, g!62203(X, Y)) 
  f!62205(X) >= g!62205(X, X) 
  g!62205(s(X), Y) >= b(Y, g!62205(X, Y)) 
  f!62208(X) >= g!62208(X, X) 
  g!62208(s(X), Y) > b(Y, g!62208(X, Y)) 
  f!622010(X) >= g!622010(X, X) 

With these choices, we have:

  1] X >= _|_  by (Bot) 

  2] g!62203(s(X), Y) >= b(Y, g!62203(X, Y))  because [3], by (Star) 
  3] g!62203*(s(X), Y) >= b(Y, g!62203(X, Y))  because g!62203 > b, [4] and [6], by (Copy) 
  4] g!62203*(s(X), Y) >= Y  because [5], by (Select) 
  5] Y >= Y  by (Meta) 
  6] g!62203*(s(X), Y) >= g!62203(X, Y)  because g!62203 in Mul, [7] and [10], by (Stat) 
  7] s(X) > X  because [8], by definition 
  8] s*(X) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 
  10] Y >= Y  by (Meta) 

  11] f!62205(X) >= g!62205(X, X)  because [12], by (Star) 
  12] f!62205*(X) >= g!62205(X, X)  because f!62205 > g!62205, [13] and [13], by (Copy) 
  13] f!62205*(X) >= X  because [9], by (Select) 

  14] g!62205(s(X), Y) >= b(Y, g!62205(X, Y))  because [15], by (Star) 
  15] g!62205*(s(X), Y) >= b(Y, g!62205(X, Y))  because g!62205 > b, [16] and [17], by (Copy) 
  16] g!62205*(s(X), Y) >= Y  because [10], by (Select) 
  17] g!62205*(s(X), Y) >= g!62205(X, Y)  because g!62205 in Mul, [7] and [10], by (Stat) 

  18] f!62208(X) >= g!62208(X, X)  because [19], by (Star) 
  19] f!62208*(X) >= g!62208(X, X)  because f!62208 > g!62208, [20] and [20], by (Copy) 
  20] f!62208*(X) >= X  because [9], by (Select) 

  21] g!62208(s(X), Y) > b(Y, g!62208(X, Y))  because [22], by definition 
  22] g!62208*(s(X), Y) >= b(Y, g!62208(X, Y))  because g!62208 > b, [23] and [24], by (Copy) 
  23] g!62208*(s(X), Y) >= Y  because [10], by (Select) 
  24] g!62208*(s(X), Y) >= g!62208(X, Y)  because g!62208 in Mul, [7] and [10], by (Stat) 

  25] f!622010(X) >= g!622010(X, X)  because [26], by (Star) 
  26] f!622010*(X) >= g!622010(X, X)  because f!622010 > g!622010, [27] and [27], by (Copy) 
  27] f!622010*(X) >= X  because [9], by (Select) 

We can thus remove the following rules:

  g!62208(s(X), Y) => b(f!62207(Y), g!62208(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]):

  f!62200(X) >? a 
  g!62203(s(X), Y) >? b(f!62202(Y), g!62203(X, Y)) 
  f!62205(X) >? g!62205(X, X) 
  g!62205(s(X), Y) >? b(f!62204(Y), g!62205(X, Y)) 
  f!62208(X) >? g!62208(X, X) 
  f!622010(X) >? g!622010(X, X) 

about to try horpo

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

Argument functions:

  [[a]] = _|_ 
  [[f!62200(x_1)]] = x_1 
  [[f!62202(x_1)]] = x_1 
  [[f!62204(x_1)]] = x_1 

We choose Lex = {} and Mul = {b, f!622010, f!62205, f!62208, g!622010, g!62203, g!62205, g!62208, s}, and the following precedence: g!62203 > f!622010 > g!622010 > s > f!62208 > g!62208 > f!62205 > g!62205 > b

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

  X >= _|_ 
  g!62203(s(X), Y) > b(Y, g!62203(X, Y)) 
  f!62205(X) >= g!62205(X, X) 
  g!62205(s(X), Y) >= b(Y, g!62205(X, Y)) 
  f!62208(X) >= g!62208(X, X) 
  f!622010(X) >= g!622010(X, X) 

With these choices, we have:

  1] X >= _|_  by (Bot) 

  2] g!62203(s(X), Y) > b(Y, g!62203(X, Y))  because [3], by definition 
  3] g!62203*(s(X), Y) >= b(Y, g!62203(X, Y))  because g!62203 > b, [4] and [6], by (Copy) 
  4] g!62203*(s(X), Y) >= Y  because [5], by (Select) 
  5] Y >= Y  by (Meta) 
  6] g!62203*(s(X), Y) >= g!62203(X, Y)  because g!62203 in Mul, [7] and [10], by (Stat) 
  7] s(X) > X  because [8], by definition 
  8] s*(X) >= X  because [9], by (Select) 
  9] X >= X  by (Meta) 
  10] Y >= Y  by (Meta) 

  11] f!62205(X) >= g!62205(X, X)  because [12], by (Star) 
  12] f!62205*(X) >= g!62205(X, X)  because f!62205 > g!62205, [13] and [13], by (Copy) 
  13] f!62205*(X) >= X  because [9], by (Select) 

  14] g!62205(s(X), Y) >= b(Y, g!62205(X, Y))  because [15], by (Star) 
  15] g!62205*(s(X), Y) >= b(Y, g!62205(X, Y))  because g!62205 > b, [16] and [17], by (Copy) 
  16] g!62205*(s(X), Y) >= Y  because [10], by (Select) 
  17] g!62205*(s(X), Y) >= g!62205(X, Y)  because g!62205 in Mul, [7] and [10], by (Stat) 

  18] f!62208(X) >= g!62208(X, X)  because [19], by (Star) 
  19] f!62208*(X) >= g!62208(X, X)  because f!62208 > g!62208, [20] and [20], by (Copy) 
  20] f!62208*(X) >= X  because [9], by (Select) 

  21] f!622010(X) >= g!622010(X, X)  because [22], by (Star) 
  22] f!622010*(X) >= g!622010(X, X)  because f!622010 > g!622010, [23] and [23], by (Copy) 
  23] f!622010*(X) >= X  because [9], by (Select) 

We can thus remove the following rules:

  g!62203(s(X), Y) => b(f!62202(Y), g!62203(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]):

  f!62200(X) >? a 
  f!62205(X) >? g!62205(X, X) 
  g!62205(s(X), Y) >? b(f!62204(Y), g!62205(X, Y)) 
  f!62208(X) >? g!62208(X, X) 
  f!622010(X) >? g!622010(X, X) 

about to try horpo

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

Argument functions:

  [[a]] = _|_ 
  [[f!62200(x_1)]] = x_1 

We choose Lex = {} and Mul = {b, f!622010, f!62204, f!62205, f!62208, g!622010, g!62205, g!62208, s}, and the following precedence: f!622010 > f!62208 > g!622010 > g!62208 > s > f!62205 > g!62205 > f!62204 > b

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

  X >= _|_ 
  f!62205(X) > g!62205(X, X) 
  g!62205(s(X), Y) >= b(f!62204(Y), g!62205(X, Y)) 
  f!62208(X) >= g!62208(X, X) 
  f!622010(X) >= g!622010(X, X) 

With these choices, we have:

  1] X >= _|_  by (Bot) 

  2] f!62205(X) > g!62205(X, X)  because [3], by definition 
  3] f!62205*(X) >= g!62205(X, X)  because f!62205 > g!62205, [4] and [4], by (Copy) 
  4] f!62205*(X) >= X  because [5], by (Select) 
  5] X >= X  by (Meta) 

  6] g!62205(s(X), Y) >= b(f!62204(Y), g!62205(X, Y))  because [7], by (Star) 
  7] g!62205*(s(X), Y) >= b(f!62204(Y), g!62205(X, Y))  because g!62205 > b, [8] and [11], by (Copy) 
  8] g!62205*(s(X), Y) >= f!62204(Y)  because g!62205 > f!62204 and [9], by (Copy) 
  9] g!62205*(s(X), Y) >= Y  because [10], by (Select) 
  10] Y >= Y  by (Meta) 
  11] g!62205*(s(X), Y) >= g!62205(X, Y)  because g!62205 in Mul, [12] and [14], by (Stat) 
  12] s(X) > X  because [13], by definition 
  13] s*(X) >= X  because [5], by (Select) 
  14] Y >= Y  by (Meta) 

  15] f!62208(X) >= g!62208(X, X)  because [16], by (Star) 
  16] f!62208*(X) >= g!62208(X, X)  because f!62208 > g!62208, [17] and [17], by (Copy) 
  17] f!62208*(X) >= X  because [5], by (Select) 

  18] f!622010(X) >= g!622010(X, X)  because [19], by (Star) 
  19] f!622010*(X) >= g!622010(X, X)  because f!622010 > g!622010, [20] and [20], by (Copy) 
  20] f!622010*(X) >= X  because [5], by (Select) 

We can thus remove the following rules:

  f!62205(X) => g!62205(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]):

  f!62200(X) >? a 
  g!62205(s(X), Y) >? b(f!62204(Y), g!62205(X, Y)) 
  f!62208(X) >? g!62208(X, X) 
  f!622010(X) >? g!622010(X, X) 

about to try horpo

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

Argument functions:

  [[a]] = _|_ 
  [[f!62200(x_1)]] = x_1 

We choose Lex = {} and Mul = {b, f!622010, f!62204, f!62208, g!622010, g!62205, g!62208, s}, and the following precedence: g!62205 > s > f!62208 > f!62204 > g!62208 > b > f!622010 > g!622010

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

  X >= _|_ 
  g!62205(s(X), Y) > b(f!62204(Y), g!62205(X, Y)) 
  f!62208(X) >= g!62208(X, X) 
  f!622010(X) >= g!622010(X, X) 

With these choices, we have:

  1] X >= _|_  by (Bot) 

  2] g!62205(s(X), Y) > b(f!62204(Y), g!62205(X, Y))  because [3], by definition 
  3] g!62205*(s(X), Y) >= b(f!62204(Y), g!62205(X, Y))  because g!62205 > b, [4] and [7], by (Copy) 
  4] g!62205*(s(X), Y) >= f!62204(Y)  because g!62205 > f!62204 and [5], by (Copy) 
  5] g!62205*(s(X), Y) >= Y  because [6], by (Select) 
  6] Y >= Y  by (Meta) 
  7] g!62205*(s(X), Y) >= g!62205(X, Y)  because g!62205 in Mul, [8] and [11], by (Stat) 
  8] s(X) > X  because [9], by definition 
  9] s*(X) >= X  because [10], by (Select) 
  10] X >= X  by (Meta) 
  11] Y >= Y  by (Meta) 

  12] f!62208(X) >= g!62208(X, X)  because [13], by (Star) 
  13] f!62208*(X) >= g!62208(X, X)  because f!62208 > g!62208, [14] and [14], by (Copy) 
  14] f!62208*(X) >= X  because [10], by (Select) 

  15] f!622010(X) >= g!622010(X, X)  because [16], by (Star) 
  16] f!622010*(X) >= g!622010(X, X)  because f!622010 > g!622010, [17] and [17], by (Copy) 
  17] f!622010*(X) >= X  because [10], by (Select) 

We can thus remove the following rules:

  g!62205(s(X), Y) => b(f!62204(Y), g!62205(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]):

  f!62200(X) >? a 
  f!62208(X) >? g!62208(X, X) 
  f!622010(X) >? g!622010(X, X) 

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

The following interpretation satisfies the requirements:

  a = 0 
  f!62200 = \y0.3 + y0 
  f!622010 = \y0.3 + 3y0 
  f!62208 = \y0.3 + 3y0 
  g!622010 = \y0y1.y0 + y1 
  g!62208 = \y0y1.y0 + y1 

Using this interpretation, the requirements translate to:

  [[f!62200(_x0)]] = 3 + x0 > 0 = [[a]] 
  [[f!62208(_x0)]] = 3 + 3x0 > 2x0 = [[g!62208(_x0, _x0)]] 
  [[f!622010(_x0)]] = 3 + 3x0 > 2x0 = [[g!622010(_x0, _x0)]] 

We can thus remove the following rules:

  f!62200(X) => a 
  f!62208(X) => g!62208(X, X) 
  f!622010(X) => g!622010(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 +++

[FuhGieParSchSwi11]  C. Fuhs, J. Giesl, M. Parting, P. Schneider-Kamp, and S. Swiderski.  Proving Termination by Dependency Pairs and Inductive Theorem Proving.  In volume 47(2) of Journal of Automated Reasoning.  133--160, 2011.
[Gra95]  B. Gramlich.  Abstract Relations Between Restricted Termination and Confluence Properties of Rewrite Systems.  In volume 24(1-2) of Fundamentae Informaticae.  3--23, 1995.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.