Section SMX_facts.
Context {State Symbol : Set}.
Local Definition Config := @Config State Symbol.
(* reachable in at most n steps *)
Inductive reachable_n (M: SMX) : nat -> Config -> Config -> Prop :=
| reach_refl (n: nat) (X: Config) : reachable_n M n X X
| reach_step (n: nat) (X Y Z: Config) : step M X Y -> reachable_n M n Y Z -> reachable_n M (1+n) X Z.
Definition terminal (M: SMX) (X: Config) : Prop :=
forall (Y: Config), not (step M X Y).
(* terminating or reachable in at most n steps *)
Definition maybe_reachable (M: SMX) (n: nat) (X Y: Config) : Prop :=
reachable_n M n X Y \/ (exists Z, reachable_n M n X Z /\ terminal M Z).

Lemma first_step {M} (op : Instruction) (v w: list Symbol) {n X Z} :
1 <= n -> In op M ->
let '((r, s, x), (r', s', y), b) := op in
X = (r ++ v, s ++ w, x) ->
reachable_n M (n-1) (if b is true then (s' ++ w, r' ++ v, y) else (r' ++ v, s' ++ w, y)) Z ->
reachable_n M n X Z.