Area of a circle trigonometry, uses the coarea formula. Define a function formula_10 by formula_11. Note ρ is a Lipschitz function whose gradient is a unit vector formula_12 (almost everywhere). Let "D" be the disc formula_13 in formula_14. We will show that formula_15, where formula_16 is the two-dimensional Lebesgue measure in formula_14. We shall assume that the one-dimensional Hausdorff measure of the circle formula_18 is formula_19, the circumference of the circle of radius "r". (This can be taken as the definition of circumference.) Then, by the coarea formula, Similar to the onion proof outlined above, we could exploit calculus in a different way in
Straight-line grammar definition of the more general formalism of straight-line context-free tree grammars can be found in Lohrey et al. An SLG in Chomsky normal form is equivalent to a straight-line program. Straight-line grammar A straight-line grammar (sometimes abbreviated as SLG) is a formal grammar that generates exactly one string. Consequently, it does not branch (every non-terminal has only one associated production rule) nor loop (if non-terminal "A" appears in a derivation of "B", then "B" does not appear in a derivation of "A"). Straight-line grammars are widely used in the development of algorithms that execute directly on compressed structures (without prior