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


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YES
We consider the system theBenchmark.

  Alphabet:

    0 : [] --> a 
    rec : [a -> b -> b * b * a] --> b 
    s : [a] --> a 

  Rules:

    rec(f, x, 0) => x 
    rec(f, x, s(y)) => f s(y) rec(f, x, y) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

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

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

  rec(F, X, 0) >? X 
  rec(F, X, s(Y)) >? F s(Y) rec(F, X, Y) 

about to try horpo

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

We choose Lex = {} and Mul = {0, @_{o -> o -> o}, @_{o -> o}, rec, s}, and the following precedence: 0 > rec > @_{o -> o -> o} > @_{o -> o} > s

With these choices, we have:

  1] rec(F, X, 0) > X  because [2], by definition 
  2] rec*(F, X, 0) >= X  because [3], by (Select) 
  3] X >= X  by (Meta) 

  4] rec(F, X, s(Y)) > @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y))  because [5], by definition 
  5] rec*(F, X, s(Y)) >= @_{o -> o}(@_{o -> o -> o}(F, s(Y)), rec(F, X, Y))  because rec > @_{o -> o}, [6] and [12], by (Copy) 
  6] rec*(F, X, s(Y)) >= @_{o -> o -> o}(F, s(Y))  because rec > @_{o -> o -> o}, [7] and [9], by (Copy) 
  7] rec*(F, X, s(Y)) >= F  because [8], by (Select) 
  8] F >= F  by (Meta) 
  9] rec*(F, X, s(Y)) >= s(Y)  because [10], by (Select) 
  10] s(Y) >= s(Y)  because s in Mul and [11], by (Fun) 
  11] Y >= Y  by (Meta) 
  12] rec*(F, X, s(Y)) >= rec(F, X, Y)  because rec in Mul, [13], [14] and [15], by (Stat) 
  13] F >= F  by (Meta) 
  14] X >= X  by (Meta) 
  15] s(Y) > Y  because [16], by definition 
  16] s*(Y) >= Y  because [11], by (Select) 

We can thus remove the following rules:

  rec(F, X, 0) => X 
  rec(F, X, s(Y)) => F s(Y) rec(F, X, Y) 

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.