(** Correctness of clean-up *)

Inductive match_stackframes: stackframe -> stackframe -> Prop :=
  | match_stackframe_intro:
      forall f sp ls c,
      incl c f.(fn_code) ->
      match_stackframes
        (Stackframe f sp ls c)
        (Stackframe (transf_function f) sp ls
          (remove_unused_labels (labels_branched_to f.(fn_code)) c)).(** Correctness of clean-up *)

Inductive match_stackframes: stackframe -> stackframe -> Prop :=
  | match_stackframe_intro:
      forall f sp ls c,
      incl c f.(fn_code) ->
      match_stackframes
        (Stackframe f sp ls c)
        (Stackframe (transf_function f) sp ls
          (remove_unused_labels (labels_branched_to f.(fn_code)) c)).Inductive match_states: state -> state -> Prop :=
  | match_states_intro:
      forall s f sp c ls m ts
        (STACKS: list_forall2 match_stackframes s ts)
        (INCL: incl c f.(fn_code)),
      match_states (State s f sp c ls m)
                   (State ts (transf_function f) sp (remove_unused_labels (labels_branched_to f.(fn_code)) c) ls m)
  | match_states_call:
      forall s f ls m ts,
      list_forall2 match_stackframes s ts ->
      match_states (Callstate s f ls m)
                   (Callstate ts (transf_fundef f) ls m)
  | match_states_return:
      forall s ls m ts,
      list_forall2 match_stackframes s ts ->
      match_states (Returnstate s ls m)
                   (Returnstate ts ls m).Inductive match_states: state -> state -> Prop :=
  | match_states_intro:
      forall s f sp c ls m ts
        (STACKS: list_forall2 match_stackframes s ts)
        (INCL: incl c f.(fn_code)),
      match_states (State s f sp c ls m)
                   (State ts (transf_function f) sp (remove_unused_labels (labels_branched_to f.(fn_code)) c) ls m)
  | match_states_call:
      forall s f ls m ts,
      list_forall2 match_stackframes s ts ->
      match_states (Callstate s f ls m)
                   (Callstate ts (transf_fundef f) ls m)
  | match_states_return:
      forall s ls m ts,
      list_forall2 match_stackframes s ts ->
      match_states (Returnstate s ls m)
                   (Returnstate ts ls m).Inductive match_states: state -> state -> Prop :=
  | match_states_intro:
      forall s f sp c ls m ts
        (STACKS: list_forall2 match_stackframes s ts)
        (INCL: incl c f.(fn_code)),
      match_states (State s f sp c ls m)
                   (State ts (transf_function f) sp (remove_unused_labels (labels_branched_to f.(fn_code)) c) ls m)
  | match_states_call:
      forall s f ls m ts,
      list_forall2 match_stackframes s ts ->
      match_states (Callstate s f ls m)
                   (Callstate ts (transf_fundef f) ls m)
  | match_states_return:
      forall s ls m ts,
      list_forall2 match_stackframes s ts ->
      match_states (Returnstate s ls m)
                   (Returnstate ts ls m).Inductive match_states: state -> state -> Prop :=
  | match_states_intro:
      forall s f sp c ls m ts
        (STACKS: list_forall2 match_stackframes s ts)
        (INCL: incl c f.(fn_code)),
      match_states (State s f sp c ls m)
                   (State ts (transf_function f) sp (remove_unused_labels (labels_branched_to f.(fn_code)) c) ls m)
  | match_states_call:
      forall s f ls m ts,
      list_forall2 match_stackframes s ts ->
      match_states (Callstate s f ls m)
                   (Callstate ts (transf_fundef f) ls m)
  | match_states_return:
      forall s ls m ts,
      list_forall2 match_stackframes s ts ->
      match_states (Returnstate s ls m)
                   (Returnstate ts ls m).Definition measure (st: state) : nat :=
  match st with
  | State s f sp c ls m => List.length c
  | _ => O
  end.

Lemma transf_initial_states: forall st1, initial_state prog st1 -> exists st2, initial_state tprog st2 /\ match_states st1 st2.