Linear Depth Increase of Lambda Terms in Leftmost-Outermost Beta-Reduction Rewrite Sequences

Accattoli and Dal Lago have recently proved that the number of steps in a leftmost-outermost β-reduction rewrite sequence to normal form provides an invariant cost model for the Lambda Calculus. They sketch how to implement leftmost-outermost rewrite sequences on a reasonable machine, with a polynomial overhead, by using simulating rewrite sequences in the linear explicit substitution calculus. I am interested in an implementation that demonstrates this result, but uses graph reduction techniques similar to those that are employed by runtime evaluators of functional programs. As a crucial stepping stone I prove here the following property of leftmost-outermost β-reduction rewrite sequences in the Lambda Calculus: For every λ-term M with depth d it holds that in every step of a leftmost-outermost β-reduction rewrite sequence starting on M the term depth increases by at most d, and hence that the depth of the n-th reduct of M in such a rewrite sequence is bounded by d · (n+1). Dedicated to Albert Visser on the occasion of his retirement, with much gratitude for my time in his group in Utrecht, and with my very best wishes for the future!