Section OVERLAP.
(** We consider a parallel move problem [mu] that satisfies the following conditions, which are stronger, overlap-aware variants of the [move_no_temp mu] and [is_mill mu] conditions used previously. *)
Variable mu: moves.
(** Sources and destinations are disjoint from all temporary registers. *)
Hypothesis mu_no_temporaries_src:
forall s, In s (srcs mu) -> disjoint_temps s.
Hypothesis mu_no_temporaries_dst:
forall d, In d (dests mu) -> disjoint_temps d.
(** Destinations are pairwise disjoint. *)
Hypothesis mu_dest_pairwise_disjoint:
pairwise_disjoint (dests mu).
(** Sources and destinations do not partially overlap. *)
Hypothesis mu_src_dst_no_overlap:
forall s d, In s (srcs mu) -> In d (dests mu) -> no_overlap s d.
(** Distinct temporaries do not partially overlap. *)
Hypothesis temps_no_overlap:
forall t1 t2, is_temp t1 -> is_temp t2 -> no_overlap t1 t2.
(** The following lemmas show that [mu] is a windmill and does not contain temporary registers. *)

Lemma disjoint_temps_not_temp:
forall r, disjoint_temps r -> is_not_temp r.