We consider the system 05height.

  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 the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs.

We thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):

  Dependency Pairs P_0:

    0] heightf#(cons(X, Y)) =#> heightt#(X)   
    1] heightf#(cons(X, Y)) =#> heightf#(Y)   
    2] heightt#(node(X)) =#> heightf#(X)   

  Rules R_0:

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

Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite.

We consider the dependency pair problem (P_0, R_0, minimal, formative).

We apply the subterm criterion with the following projection function:

  nu(heightf#) = 1 
  nu(heightt#) = 1 

Thus, we can orient the dependency pairs as follows:

  nu(heightf#(cons(X, Y))) = cons(X, Y) |> X = nu(heightt#(X)) 
  nu(heightf#(cons(X, Y))) = cons(X, Y) |> Y = nu(heightf#(Y)) 
  nu(heightt#(node(X))) = node(X) |> X = nu(heightf#(X)) 

By [Kop12, Thm. 7.35], we may replace a dependency pair problem (P_0, R_0, minimal, f) by ({}, R_0, minimal, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ Citations +++

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