Section LINEARIZATION.
Variable prog: LTL.program.
Variable tprog: Linear.program.
Hypothesis TRANSF: match_prog prog tprog.
Let ge := Genv.globalenv prog.
Let tge := Genv.globalenv tprog.
Proof (Genv.find_funct_transf_partial TRANSF).
Proof (Genv.find_funct_ptr_transf_partial TRANSF).
Proof (Genv.find_symbol_transf_partial TRANSF).
Proof (Genv.senv_transf_partial TRANSF).
(** * Correctness of reachability analysis *)
(** The entry point of the function is reachable. *)
(** The successors of a reachable instruction are reachable. *)
(** * Properties of node enumeration *)
(** An enumeration of CFG nodes is correct if the following conditions hold: - All nodes for reachable basic blocks must be in the list. - The list is without repetition (so that no code duplication occurs).  We prove that the result of the [enumerate] function satisfies both conditions. *)
(** * Properties related to labels *)
(** If labels are globally unique and the Linear code [c] contains a subsequence [Llabel lbl :: c1], then [find_label lbl c] returns [c1]. *)
Fixpoint unique_labels (c: code) : Prop :=
match c with
| nil => True
| Llabel lbl :: c => ~(In (Llabel lbl) c) /\ unique_labels c
| i :: c => unique_labels c
end.
(** Correctness of the [starts_with] test. *)

Lemma starts_with_correct:
forall lbl c1 c2 c3 s f sp ls m,
is_tail c1 c2 ->
unique_labels c2 ->
starts_with lbl c1 = true ->
find_label lbl c2 = Some c3 ->
plus step tge (State s f sp c1 ls m)
E0 (State s f sp c3 ls m).