Boolean circuits: basis, fan-in, size, and depth

A Boolean circuit on $n$ inputs is a finite directed acyclic graph with inputs $x_1,\dots ,x_n$, constants $0$ and $1$, and NOT, AND, and OR gates; AND and OR have fan-in two, while NOT has fan-in one. One node is designated as the output. The circuit's value is the value at that node after evaluating the graph in topological order. Its size is its number of non-input gates, and its depth is the largest number of non-input gates on a directed path ending at the output (a path may begin at an input or a constant).

If two fixed finite bounded-arity bases can simulate every gate (and every constant) of one another by fixed finite circuits, then changing between them costs only constant factors in size and depth. In particular this holds for any two such functionally complete bases with simulable constants. No such claim is made for an arbitrary finite basis that may fail to express one of the Boolean operations.