(** We define the ``adherence'' of the problem [mu] as the set of registers that partially overlap with one of the registers possibly assigned by the parallel move: destinations and temporaries. Again, we define the complement of the ``adherence'' set, which is more convenient for Coq reasoning. *)
Definition no_adherence (r: reg) : Prop :=
forall x, In x (dests mu) \/ is_temp x -> no_overlap x r.
(** As a consequence of the hypotheses on [mu], none of the destination registers, source registers, and temporaries belong to the adherence. *)

(* Specifications: *)
Lemma no_overlap_pairwise:
forall r1 r2 m, pairwise_disjoint m -> In r1 m -> In r2 m -> no_overlap r1 r2.
Lemma no_adherence_dst:
forall d, In d (dests mu) -> no_adherence d.
Lemma no_adherence_src:
forall s, In s (srcs mu) -> no_adherence s.
Lemma no_adherence_tmp:
forall t, is_temp t -> no_adherence t.
Lemma no_overlap_pairwise:
forall r1 r2 m, pairwise_disjoint m -> In r1 m -> In r2 m -> no_overlap r1 r2.
Lemma no_adherence_dst:
forall d, In d (dests mu) -> no_adherence d.
Lemma no_adherence_src:
forall s, In s (srcs mu) -> no_adherence s.
Lemma no_adherence_tmp:
forall t, is_temp t -> no_adherence t.