P1 → · · · → Pm → R := fix (c1 : σ1) . . . (cl : σl), assume h1 : P1, ... assume hm : Pm, have k1 : Q 1 := . . ., ... have kn : Q n := . . ., show R, from . . . 3.2 Structured Constructs The previous the proof of the while_intro rule. 9.6 Finish Tactic finish finish � [h1, . . ., hm] � � using [k1, . . ., kn] � The finish tactic first normalizes and simplifies the hypotheses, using the op- tionally ..., hm. Then it applies the congruence closure (cc) together with the optionally supplied lemmas k1, ..., kn. Universally quantified hy- potheses and lemmas are instantiated automatically. Like cc, finish

write values. (To avoid questions as to how we would interpret the flow of control in terms like h (k1 a) (k2 a), let us suppose that we only care about composing unary functions.) There is a straightforward