Deep learning has had a profound impact on computer science in recent years, with applications to image recognition, language processing, bioinformatics, and more. Recently, Cohen et al. provided theoretical evidence for the superiority of deep learning over shallow learning. We formalized their mathematical proof using Isabelle/HOL. The Isabelle development simplifies and generalizes the original proof, while working around the limitations of the HOL type system. To support the formalization, we developed reusable libraries of formalized mathematics, including results about the matrix rank, the Borel measure, and multivariate polynomials as well as a library for tensor analysis.

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A series of rigid rod polyimides 1 af derived from pyromellitic anhydride (PMDA) and 2,5di-n-alkoxy-l,4phenylenediamine containing flexible alkoxy side chains with 4 to 16 carbon atoms was prepared by solution polycondensation. As demonstrated by model experiments full conversion to the imide structure is achieved by reaction of PMDA with 2,5-dialkoxyphenylene diisocyanate, the latter compound being blocked by imidazole. The best solvent for the polycondensation, found in the course of the model reaction, is 1,3-dimethyl-2-oxo-perhydropyrimidine together with 4dimethylaminopyridine as an accelerator. Wide-angle X-ray diffractograms showed that all polyimides under consideration here are highly crystalline at room temperature. The occurrence of a sharp reflection at the region of low angles together with its higher orders demonstrates the presence of a layered structure. The layer spacing was found to increase linearly with the length of the alkyl side chains. Polyimides bearing long side chains (n 2 10) exhibit two broad endotherms in the DSC analysis. As is revealed by wideangle X-ray analysis the fiist transition is related to a disordering process of the side chains. The second transition leads to a layered high-temperature phase akin to the modification (B) found in a recent study of a similar rigid polyester bearing flexible side chains. a) Part 3: Cf.3). b, Part of the thesis of M. Wenzel. 0025-1 16X/87/%03.00

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