Proving Divide and Conquer Complexities in Isabelle/HOL - Eberl, Manuel (Journal of automated reasoning, 2017)
The Akra–Bazzi method (Akra and Bazzi in Comput Optim Appl 10(2):195–210, 1998 . doi: 10.1023/A:1018373005182 ), a generalisation of the well-known Master Theorem, is a useful tool for analysing the complexity of Divide and Conquer algorithms. This work describes a formalisation of the Akra–Bazzi method (as generalised by Leighton in Notes on better Master theorems for divide-and-conquer recurrences, 1996 . http://courses.csail.mit.edu/6.046/spring04/handouts/akrabazzi.pdf ) in the interactive theorem prover Isabelle/HOL and the derivation of a generalised version of the Master Theorem from it. We also provide some automated proof methods that facilitate the application of this Master Theorem and allow mostly automatic verification of Θ -bounds for these Divide and Conquer recurrences. To our knowledge, this is the first formalisation of theorems for the analysis of such recurrences.