If a circuit \(c\) has depth \(0\), then \(c\) is a single literal: there exists a literal \(l\) with \(c = \mathrm{Circuit.lit}\, l\).

Base case (\(d_{\mathrm{inner}} = 2\)) of the composed layer-2 bound. Here the top circuit \(c_{\mathrm{top}}\) has depth \(0\), hence is a single literal, so the composed function is one DNF gate or its negation. Under a Bernoulli random restriction with parameter \(\mathrm{composedDelta}\, l\, l\, 2\), the probability that the restricted function has decision-tree depth exceeding \(t\) is at most

under the hypotheses \(0 \lt l\), \(0 \lt n\), \(c_{\mathrm{top}}.\mathrm{depth}+2\le 2\), \(c_{\mathrm{top}}.\mathrm{size}\le s_{\mathrm{rem}}\), \(\mathrm{data.width}\le l\), and \(0 \lt s_{\mathrm{rem}}\).

Suppose for each gate \(i\) the restricted function \(\mathrm{restrictFn}\, (\mathrm{gates}\, i).\mathrm{eval}\, \rho _1\) has decision-tree depth at most \(l\). Then there exist DNFs \(\mathrm{gates}'\) such that each \((\mathrm{gates}'\, i)\) has width at most \(l\), evaluates to the corresponding restricted gate function on every input, and is clean: within each term every literal is determined by its variable (injectivity on variables) and each term has no duplicate literals.