/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:

    cons : [t * f] --> f 
    heightf : [f] --> N 
    heightt : [t] --> N 
    leaf : [] --> t 
    max : [N * N] --> N 
    nil : [] --> f 
    node : [f] --> t 
    s : [N] --> N 
    z : [] --> N 

  Rules:

    heightf(nil) => z 
    heightf(cons(x, y)) => max(heightt(x), heightf(y)) 
    heightt(leaf) => z 
    heightt(node(x)) => s(heightf(x)) 

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]):

  heightf(nil) >? z 
  heightf(cons(X, Y)) >? max(heightt(X), heightf(Y)) 
  heightt(leaf) >? z 
  heightt(node(X)) >? s(heightf(X)) 

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

The following interpretation satisfies the requirements:

  cons = \y0y1.3 + 3y0 + 3y1 
  heightf = \y0.y0 
  heightt = \y0.y0 
  leaf = 3 
  max = \y0y1.y0 + y1 
  nil = 3 
  node = \y0.3 + 3y0 
  s = \y0.y0 
  z = 0 

Using this interpretation, the requirements translate to:

  [[heightf(nil)]] = 3 > 0 = [[z]] 
  [[heightf(cons(_x0, _x1))]] = 3 + 3x0 + 3x1 > x0 + x1 = [[max(heightt(_x0), heightf(_x1))]] 
  [[heightt(leaf)]] = 3 > 0 = [[z]] 
  [[heightt(node(_x0))]] = 3 + 3x0 > x0 = [[s(heightf(_x0))]] 

We can thus remove the following rules:

  heightf(nil) => z 
  heightf(cons(X, Y)) => max(heightt(X), heightf(Y)) 
  heightt(leaf) => z 
  heightt(node(X)) => s(heightf(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.