(** Consider an execution of a call/move/nop/return sequence in the
  original code and the corresponding tailcall in the transformed code.
  The transition sequences are of the following form
  (left: original code, right: transformed code).
  [f] is the calling function and [fd] the called function.
<<
     State stk f (Icall instruction)       State stk' f' (Itailcall)

     Callstate (frame::stk) fd args        Callstate stk' fd' args'
            .                                       .
            .                                       .
            .                                       .
     Returnstate (frame::stk) res          Returnstate stk' res'

     State stk f (move/nop/return seq)
            .
            .
            .
     State stk f (return instr)

     Returnstate stk res
>>
The simulation invariant must therefore account for two kinds of
mismatches between the transition sequences:
- The call stack of the original program can contain more frames
  than that of the transformed program (e.g. [frame] in the example above).
- The regular states corresponding to executing the move/nop/return
  sequence must all correspond to the single [Returnstate stk' res']
  state of the transformed program.

We first define the simulation invariant between call stacks.
The first two cases are standard, but the third case corresponds
to a frame that was eliminated by the transformation. *)

Inductive match_stackframes: list stackframe -> list stackframe -> Prop :=
  | match_stackframes_nil:
      match_stackframes nil nil
  | match_stackframes_normal: forall stk stk' res sp pc rs rs' f,
      match_stackframes stk stk' ->
      regs_lessdef rs rs' ->
      match_stackframes
        (Stackframe res f (Vptr sp Ptrofs.zero) pc rs :: stk)
        (Stackframe res (transf_function f) (Vptr sp Ptrofs.zero) pc rs' :: stk')
  | match_stackframes_tail: forall stk stk' res sp pc rs f,
      match_stackframes stk stk' ->
      is_return_spec f pc res ->
      f.(fn_stacksize) = 0 ->
      match_stackframes
        (Stackframe res f (Vptr sp Ptrofs.zero) pc rs :: stk)
        stk'.(** Consider an execution of a call/move/nop/return sequence in the
  original code and the corresponding tailcall in the transformed code.
  The transition sequences are of the following form
  (left: original code, right: transformed code).
  [f] is the calling function and [fd] the called function.
<<
     State stk f (Icall instruction)       State stk' f' (Itailcall)

     Callstate (frame::stk) fd args        Callstate stk' fd' args'
            .                                       .
            .                                       .
            .                                       .
     Returnstate (frame::stk) res          Returnstate stk' res'

     State stk f (move/nop/return seq)
            .
            .
            .
     State stk f (return instr)

     Returnstate stk res
>>
The simulation invariant must therefore account for two kinds of
mismatches between the transition sequences:
- The call stack of the original program can contain more frames
  than that of the transformed program (e.g. [frame] in the example above).
- The regular states corresponding to executing the move/nop/return
  sequence must all correspond to the single [Returnstate stk' res']
  state of the transformed program.

We first define the simulation invariant between call stacks.
The first two cases are standard, but the third case corresponds
to a frame that was eliminated by the transformation. *)

Inductive match_stackframes: list stackframe -> list stackframe -> Prop :=
  | match_stackframes_nil:
      match_stackframes nil nil
  | match_stackframes_normal: forall stk stk' res sp pc rs rs' f,
      match_stackframes stk stk' ->
      regs_lessdef rs rs' ->
      match_stackframes
        (Stackframe res f (Vptr sp Ptrofs.zero) pc rs :: stk)
        (Stackframe res (transf_function f) (Vptr sp Ptrofs.zero) pc rs' :: stk')
  | match_stackframes_tail: forall stk stk' res sp pc rs f,
      match_stackframes stk stk' ->
      is_return_spec f pc res ->
      f.(fn_stacksize) = 0 ->
      match_stackframes
        (Stackframe res f (Vptr sp Ptrofs.zero) pc rs :: stk)
        stk'.(** Consider an execution of a call/move/nop/return sequence in the
  original code and the corresponding tailcall in the transformed code.
  The transition sequences are of the following form
  (left: original code, right: transformed code).
  [f] is the calling function and [fd] the called function.
<<
     State stk f (Icall instruction)       State stk' f' (Itailcall)

     Callstate (frame::stk) fd args        Callstate stk' fd' args'
            .                                       .
            .                                       .
            .                                       .
     Returnstate (frame::stk) res          Returnstate stk' res'

     State stk f (move/nop/return seq)
            .
            .
            .
     State stk f (return instr)

     Returnstate stk res
>>
The simulation invariant must therefore account for two kinds of
mismatches between the transition sequences:
- The call stack of the original program can contain more frames
  than that of the transformed program (e.g. [frame] in the example above).
- The regular states corresponding to executing the move/nop/return
  sequence must all correspond to the single [Returnstate stk' res']
  state of the transformed program.

We first define the simulation invariant between call stacks.
The first two cases are standard, but the third case corresponds
to a frame that was eliminated by the transformation. *)

Inductive match_stackframes: list stackframe -> list stackframe -> Prop :=
  | match_stackframes_nil:
      match_stackframes nil nil
  | match_stackframes_normal: forall stk stk' res sp pc rs rs' f,
      match_stackframes stk stk' ->
      regs_lessdef rs rs' ->
      match_stackframes
        (Stackframe res f (Vptr sp Ptrofs.zero) pc rs :: stk)
        (Stackframe res (transf_function f) (Vptr sp Ptrofs.zero) pc rs' :: stk')
  | match_stackframes_tail: forall stk stk' res sp pc rs f,
      match_stackframes stk stk' ->
      is_return_spec f pc res ->
      f.(fn_stacksize) = 0 ->
      match_stackframes
        (Stackframe res f (Vptr sp Ptrofs.zero) pc rs :: stk)
        stk'.(** Consider an execution of a call/move/nop/return sequence in the
  original code and the corresponding tailcall in the transformed code.
  The transition sequences are of the following form
  (left: original code, right: transformed code).
  [f] is the calling function and [fd] the called function.
<<
     State stk f (Icall instruction)       State stk' f' (Itailcall)

     Callstate (frame::stk) fd args        Callstate stk' fd' args'
            .                                       .
            .                                       .
            .                                       .
     Returnstate (frame::stk) res          Returnstate stk' res'

     State stk f (move/nop/return seq)
            .
            .
            .
     State stk f (return instr)

     Returnstate stk res
>>
The simulation invariant must therefore account for two kinds of
mismatches between the transition sequences:
- The call stack of the original program can contain more frames
  than that of the transformed program (e.g. [frame] in the example above).
- The regular states corresponding to executing the move/nop/return
  sequence must all correspond to the single [Returnstate stk' res']
  state of the transformed program.

We first define the simulation invariant between call stacks.
The first two cases are standard, but the third case corresponds
to a frame that was eliminated by the transformation. *)

Inductive match_stackframes: list stackframe -> list stackframe -> Prop :=
  | match_stackframes_nil:
      match_stackframes nil nil
  | match_stackframes_normal: forall stk stk' res sp pc rs rs' f,
      match_stackframes stk stk' ->
      regs_lessdef rs rs' ->
      match_stackframes
        (Stackframe res f (Vptr sp Ptrofs.zero) pc rs :: stk)
        (Stackframe res (transf_function f) (Vptr sp Ptrofs.zero) pc rs' :: stk')
  | match_stackframes_tail: forall stk stk' res sp pc rs f,
      match_stackframes stk stk' ->
      is_return_spec f pc res ->
      f.(fn_stacksize) = 0 ->
      match_stackframes
        (Stackframe res f (Vptr sp Ptrofs.zero) pc rs :: stk)
        stk'.(** The last case of [match_states] corresponds to the execution
  of a move/nop/return sequence in the original code that was
  eliminated by the transformation:
<<
     State stk f (move/nop/return seq)  ~~  Returnstate stk' res'
            .
            .
            .
     State stk f (return instr)         ~~  Returnstate stk' res'
>>
  To preserve non-terminating behaviors, we need to make sure
  that the execution of this sequence in the original code cannot
  diverge.  For this, we introduce the following complicated
  measure over states, which will decrease strictly whenever
  the original code makes a transition but the transformed code
  does not. *)

Definition measure (st: state) : nat :=
  match st with
  | State s f sp pc rs m => (List.length s * (niter + 2) + return_measure f.(fn_code) pc + 1)%nat
  | Callstate s f args m => 0%nat
  | Returnstate s v m => (List.length s * (niter + 2))%nat
  end.Ltac TransfInstr :=
  match goal with
  | H: (PTree.get _ (fn_code _) = _) |- _ =>
      destruct (transf_instr_lookup _ _ _ H) as [i' [TINSTR TSPEC]]; inv TSPEC
  end.Ltac EliminatedInstr :=
  match goal with
  | H: (is_return_spec _ _ _) |- _ => inv H; try congruence
  | _ => idtac
  end.

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