(* Autor(s): Andrej Dudenhefner (1) Affiliation(s): (1) Saarland University, Saarbrücken, Germany *)
(* Facts on: Confluent Simple Stack Machines *)
(* uniform boundedness of deterministic simple stack machines *)
From Undecidability.StackMachines.Util Require Import Facts.
(* width of a configuration *)
Definition width : config -> nat := fun '(A, B, _) => length A + length B.
(* joinability of configurations *)
Definition equiv (X Y: config) := exists (Z: config), reachable M X Z /\ reachable M Y Z.
(* list all states occurring in M' *)
Fixpoint enum_states (M': ssm) : list state :=
match M' with
| [] => []
| (x, y, _, _, _) :: M' => x :: y :: enum_states M'
end.
Definition get_state : config -> state :=
fun '(_,  _, x) => x.
(* list all configurations with stack lengths lA and lB *)
Definition enum_configs (lA lB: nat) : list config :=
list_prod (list_prod (enum_stacks lA) (enum_stacks lB)) (enum_states M).
(* space X is an overapproximation of reachable states from X *)
Definition space (X: config) : list config :=
X :: flat_map (fun i => enum_configs i (width X - i)) (seq 0 (width X + 1)).

(* reachability preserves configuration width *)
Lemma reachable_width {X Y: config} : reachable M X Y -> width X = width Y.