/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.

  a : [] --> o 
  b : [o * o] --> o 
  f!62200 : [o] --> o 
  f!62201 : [o] --> o 
  f!62202 : [o] --> o 
  f!62203 : [o] --> o 
  f!62204 : [o] --> o 
  f!62205 : [o] --> o 
  g!62201 : [o * o] --> o 
  g!62202 : [o * o] --> o 
  g!62203 : [o * o] --> o 
  g!62204 : [o * o] --> o 
  g!62205 : [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)) 

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 : [] --> kd 
  b : [kd * kd] --> kd 
  f!62200 : [cd] --> kd 
  f!62201 : [cd] --> kd 
  f!62202 : [cd] --> kd 
  f!62203 : [cd] --> kd 
  f!62204 : [cd] --> kd 
  f!62205 : [cd] --> kd 
  g!62201 : [cd * cd] --> kd 
  g!62202 : [cd * cd] --> kd 
  g!62203 : [cd * cd] --> kd 
  g!62204 : [cd * cd] --> kd 
  g!62205 : [cd * cd] --> kd 
  s : [cd] --> cd 

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)) 

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!62201, f!62202, f!62203, f!62204, f!62205, g!62201, g!62202, g!62203, g!62204, g!62205, s}, and the following precedence: f!62205 > g!62205 > f!62204 > g!62204 > f!62203 > g!62203 > s > f!62202 > 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)) 

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 g!62201 in Mul, [11] and [13], by (Stat) 
  11] s(X) > X  because [12], by definition 
  12] s*(X) >= X  because [5], by (Select) 
  13] Y >= Y  by (Meta) 

  14] f!62202(X) >= g!62202(X, X)  because [15], by (Star) 
  15] f!62202*(X) >= g!62202(X, X)  because f!62202 > g!62202, [16] and [16], by (Copy) 
  16] f!62202*(X) >= X  because [5], by (Select) 

  17] g!62202(s(X), Y) >= b(f!62201(Y), g!62202(X, Y))  because [18], by (Star) 
  18] g!62202*(s(X), Y) >= b(f!62201(Y), g!62202(X, Y))  because g!62202 > b, [19] and [20], by (Copy) 
  19] g!62202*(s(X), Y) >= f!62201(Y)  because g!62202 = f!62201, g!62202 in Mul and [13], by (Stat) 
  20] g!62202*(s(X), Y) >= g!62202(X, Y)  because g!62202 in Mul, [11] and [13], by (Stat) 

  21] f!62203(X) > g!62203(X, X)  because [22], by definition 
  22] f!62203*(X) >= g!62203(X, X)  because f!62203 > g!62203, [23] and [23], by (Copy) 
  23] f!62203*(X) >= X  because [5], by (Select) 

  24] g!62203(s(X), Y) >= b(f!62202(Y), g!62203(X, Y))  because [25], by (Star) 
  25] g!62203*(s(X), Y) >= b(f!62202(Y), g!62203(X, Y))  because g!62203 > b, [26] and [28], by (Copy) 
  26] g!62203*(s(X), Y) >= f!62202(Y)  because g!62203 > f!62202 and [27], by (Copy) 
  27] g!62203*(s(X), Y) >= Y  because [13], by (Select) 
  28] g!62203*(s(X), Y) >= g!62203(X, Y)  because g!62203 in Mul, [11] and [13], by (Stat) 

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

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

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

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

We can thus remove the following rules:

  f!62201(X) => g!62201(X, X) 
  f!62203(X) => g!62203(X, X) 
  f!62204(X) => g!62204(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!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)) 
  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)) 

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!62201(x_1)]] = x_1 
  [[f!62203(x_1)]] = x_1 
  [[f!62204(x_1)]] = x_1 

We choose Lex = {} and Mul = {b, f!62202, f!62205, g!62201, g!62202, g!62203, g!62204, g!62205, s}, and the following precedence: g!62203 > f!62202 > s > g!62204 > f!62205 > g!62202 > g!62205 > g!62201 > 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)) 
  f!62202(X) >= g!62202(X, X) 
  g!62202(s(X), Y) > b(Y, g!62202(X, Y)) 
  g!62203(s(X), Y) >= b(f!62202(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)) 

With these choices, we have:

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

  2] g!62201(s(X), Y) >= b(Y, g!62201(X, Y))  because [3], by (Star) 
  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] f!62202(X) >= g!62202(X, X)  because [12], by (Star) 
  12] f!62202*(X) >= g!62202(X, X)  because f!62202 > g!62202, [13] and [13], by (Copy) 
  13] f!62202*(X) >= X  because [9], by (Select) 

  14] g!62202(s(X), Y) > b(Y, g!62202(X, Y))  because [15], by definition 
  15] g!62202*(s(X), Y) >= b(Y, g!62202(X, Y))  because g!62202 > b, [16] and [17], by (Copy) 
  16] g!62202*(s(X), Y) >= Y  because [10], by (Select) 
  17] g!62202*(s(X), Y) >= g!62202(X, Y)  because g!62202 in Mul, [7] and [10], by (Stat) 

  18] g!62203(s(X), Y) >= b(f!62202(Y), g!62203(X, Y))  because [19], by (Star) 
  19] g!62203*(s(X), Y) >= b(f!62202(Y), g!62203(X, Y))  because g!62203 > b, [20] and [22], by (Copy) 
  20] g!62203*(s(X), Y) >= f!62202(Y)  because g!62203 > f!62202 and [21], by (Copy) 
  21] g!62203*(s(X), Y) >= Y  because [10], by (Select) 
  22] g!62203*(s(X), Y) >= g!62203(X, Y)  because g!62203 in Mul, [7] and [10], by (Stat) 

  23] g!62204(s(X), Y) >= b(Y, g!62204(X, Y))  because [24], by (Star) 
  24] g!62204*(s(X), Y) >= b(Y, g!62204(X, Y))  because g!62204 > b, [25] and [26], by (Copy) 
  25] g!62204*(s(X), Y) >= Y  because [10], by (Select) 
  26] g!62204*(s(X), Y) >= g!62204(X, Y)  because g!62204 in Mul, [7] and [10], by (Stat) 

  27] f!62205(X) >= g!62205(X, X)  because [28], by (Star) 
  28] f!62205*(X) >= g!62205(X, X)  because f!62205 > g!62205, [29] and [29], by (Copy) 
  29] f!62205*(X) >= X  because [9], by (Select) 

  30] g!62205(s(X), Y) > b(Y, g!62205(X, Y))  because [31], by definition 
  31] g!62205*(s(X), Y) >= b(Y, g!62205(X, Y))  because g!62205 > b, [32] and [33], by (Copy) 
  32] g!62205*(s(X), Y) >= Y  because [10], by (Select) 
  33] g!62205*(s(X), Y) >= g!62205(X, Y)  because g!62205 in Mul, [7] and [10], by (Stat) 

We can thus remove the following rules:

  g!62202(s(X), Y) => b(f!62201(Y), g!62202(X, Y)) 
  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 
  g!62201(s(X), Y) >? b(f!62200(Y), g!62201(X, Y)) 
  f!62202(X) >? g!62202(X, X) 
  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) 

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!62203(x_1)]] = x_1 

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

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)) 
  f!62202(X) >= g!62202(X, X) 
  g!62203(s(X), Y) > b(f!62202(Y), g!62203(X, Y)) 
  g!62204(s(X), Y) >= b(Y, g!62204(X, Y)) 
  f!62205(X) >= g!62205(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] f!62202(X) >= g!62202(X, X)  because [12], by (Star) 
  12] f!62202*(X) >= g!62202(X, X)  because f!62202 > g!62202, [13] and [13], by (Copy) 
  13] f!62202*(X) >= X  because [9], by (Select) 

  14] g!62203(s(X), Y) > b(f!62202(Y), g!62203(X, Y))  because [15], by definition 
  15] g!62203*(s(X), Y) >= b(f!62202(Y), g!62203(X, Y))  because g!62203 > b, [16] and [18], by (Copy) 
  16] g!62203*(s(X), Y) >= f!62202(Y)  because g!62203 > f!62202 and [17], by (Copy) 
  17] g!62203*(s(X), Y) >= Y  because [10], by (Select) 
  18] g!62203*(s(X), Y) >= g!62203(X, Y)  because g!62203 in Mul, [7] and [10], by (Stat) 

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

  23] f!62205(X) >= g!62205(X, X)  because [24], by (Star) 
  24] f!62205*(X) >= g!62205(X, X)  because f!62205 > g!62205, [25] and [25], by (Copy) 
  25] f!62205*(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)) 
  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!62202(X) >? g!62202(X, X) 
  g!62204(s(X), Y) >? b(f!62203(Y), g!62204(X, Y)) 
  f!62205(X) >? g!62205(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!62202, f!62203, f!62205, g!62202, g!62204, g!62205, s}, and the following precedence: g!62204 > b > s > f!62202 > f!62205 > f!62203 > g!62202 > g!62205

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

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

With these choices, we have:

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

  2] f!62202(X) >= g!62202(X, X)  because [3], by (Star) 
  3] f!62202*(X) >= g!62202(X, X)  because f!62202 > g!62202, [4] and [4], by (Copy) 
  4] f!62202*(X) >= X  because [5], by (Select) 
  5] X >= X  by (Meta) 

  6] g!62204(s(X), Y) > b(f!62203(Y), g!62204(X, Y))  because [7], by definition 
  7] g!62204*(s(X), Y) >= b(f!62203(Y), g!62204(X, Y))  because g!62204 > b, [8] and [11], by (Copy) 
  8] g!62204*(s(X), Y) >= f!62203(Y)  because g!62204 > f!62203 and [9], by (Copy) 
  9] g!62204*(s(X), Y) >= Y  because [10], by (Select) 
  10] Y >= Y  by (Meta) 
  11] g!62204*(s(X), Y) >= g!62204(X, Y)  because g!62204 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!62205(X) >= g!62205(X, X)  because [16], by (Star) 
  16] f!62205*(X) >= g!62205(X, X)  because f!62205 > g!62205, [17] and [17], by (Copy) 
  17] f!62205*(X) >= X  because [5], by (Select) 

We can thus remove the following rules:

  g!62204(s(X), Y) => b(f!62203(Y), g!62204(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!62202(X) >? g!62202(X, X) 
  f!62205(X) >? g!62205(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!62202 = \y0.3 + 3y0 
  f!62205 = \y0.3 + 3y0 
  g!62202 = \y0y1.y0 + y1 
  g!62205 = \y0y1.y0 + y1 

Using this interpretation, the requirements translate to:

  [[f!62200(_x0)]] = 3 + x0 > 0 = [[a]] 
  [[f!62202(_x0)]] = 3 + 3x0 > 2x0 = [[g!62202(_x0, _x0)]] 
  [[f!62205(_x0)]] = 3 + 3x0 > 2x0 = [[g!62205(_x0, _x0)]] 

We can thus remove the following rules:

  f!62200(X) => a 
  f!62202(X) => g!62202(X, X) 
  f!62205(X) => g!62205(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.