Inductive match_states: state -> state -> Prop :=
  | match_states_regular: forall s f sp pc rs m ts tsp trs tm j
         (STACKS: match_stacks j s ts sp tsp)
         (KEPT: forall id, ref_function f id -> kept id)
         (SPINJ: j sp = Some(tsp, 0))
         (REGINJ: regset_inject j rs trs)
         (MEMINJ: Mem.inject j m tm),
      match_states (State s f (Vptr sp Ptrofs.zero) pc rs m)
                   (State ts f (Vptr tsp Ptrofs.zero) pc trs tm)
  | match_states_call: forall s fd args m ts targs tm j
         (STACKS: match_stacks j s ts (Mem.nextblock m) (Mem.nextblock tm))
         (KEPT: forall id, ref_fundef fd id -> kept id)
         (ARGINJ: Val.inject_list j args targs)
         (MEMINJ: Mem.inject j m tm),
      match_states (Callstate s fd args m)
                   (Callstate ts fd targs tm)
  | match_states_return: forall s res m ts tres tm j
         (STACKS: match_stacks j s ts (Mem.nextblock m) (Mem.nextblock tm))
         (RESINJ: Val.inject j res tres)
         (MEMINJ: Mem.inject j m tm),
      match_states (Returnstate s res m)
                   (Returnstate ts tres tm).Inductive match_states: state -> state -> Prop :=
  | match_states_regular: forall s f sp pc rs m ts tsp trs tm j
         (STACKS: match_stacks j s ts sp tsp)
         (KEPT: forall id, ref_function f id -> kept id)
         (SPINJ: j sp = Some(tsp, 0))
         (REGINJ: regset_inject j rs trs)
         (MEMINJ: Mem.inject j m tm),
      match_states (State s f (Vptr sp Ptrofs.zero) pc rs m)
                   (State ts f (Vptr tsp Ptrofs.zero) pc trs tm)
  | match_states_call: forall s fd args m ts targs tm j
         (STACKS: match_stacks j s ts (Mem.nextblock m) (Mem.nextblock tm))
         (KEPT: forall id, ref_fundef fd id -> kept id)
         (ARGINJ: Val.inject_list j args targs)
         (MEMINJ: Mem.inject j m tm),
      match_states (Callstate s fd args m)
                   (Callstate ts fd targs tm)
  | match_states_return: forall s res m ts tres tm j
         (STACKS: match_stacks j s ts (Mem.nextblock m) (Mem.nextblock tm))
         (RESINJ: Val.inject j res tres)
         (MEMINJ: Mem.inject j m tm),
      match_states (Returnstate s res m)
                   (Returnstate ts tres tm).Inductive match_states: state -> state -> Prop :=
  | match_states_regular: forall s f sp pc rs m ts tsp trs tm j
         (STACKS: match_stacks j s ts sp tsp)
         (KEPT: forall id, ref_function f id -> kept id)
         (SPINJ: j sp = Some(tsp, 0))
         (REGINJ: regset_inject j rs trs)
         (MEMINJ: Mem.inject j m tm),
      match_states (State s f (Vptr sp Ptrofs.zero) pc rs m)
                   (State ts f (Vptr tsp Ptrofs.zero) pc trs tm)
  | match_states_call: forall s fd args m ts targs tm j
         (STACKS: match_stacks j s ts (Mem.nextblock m) (Mem.nextblock tm))
         (KEPT: forall id, ref_fundef fd id -> kept id)
         (ARGINJ: Val.inject_list j args targs)
         (MEMINJ: Mem.inject j m tm),
      match_states (Callstate s fd args m)
                   (Callstate ts fd targs tm)
  | match_states_return: forall s res m ts tres tm j
         (STACKS: match_stacks j s ts (Mem.nextblock m) (Mem.nextblock tm))
         (RESINJ: Val.inject j res tres)
         (MEMINJ: Mem.inject j m tm),
      match_states (Returnstate s res m)
                   (Returnstate ts tres tm).Inductive match_states: state -> state -> Prop :=
  | match_states_regular: forall s f sp pc rs m ts tsp trs tm j
         (STACKS: match_stacks j s ts sp tsp)
         (KEPT: forall id, ref_function f id -> kept id)
         (SPINJ: j sp = Some(tsp, 0))
         (REGINJ: regset_inject j rs trs)
         (MEMINJ: Mem.inject j m tm),
      match_states (State s f (Vptr sp Ptrofs.zero) pc rs m)
                   (State ts f (Vptr tsp Ptrofs.zero) pc trs tm)
  | match_states_call: forall s fd args m ts targs tm j
         (STACKS: match_stacks j s ts (Mem.nextblock m) (Mem.nextblock tm))
         (KEPT: forall id, ref_fundef fd id -> kept id)
         (ARGINJ: Val.inject_list j args targs)
         (MEMINJ: Mem.inject j m tm),
      match_states (Callstate s fd args m)
                   (Callstate ts fd targs tm)
  | match_states_return: forall s res m ts tres tm j
         (STACKS: match_stacks j s ts (Mem.nextblock m) (Mem.nextblock tm))
         (RESINJ: Val.inject j res tres)
         (MEMINJ: Mem.inject j m tm),
      match_states (Returnstate s res m)
                   (Returnstate ts tres tm).

Lemma transf_initial_states: forall S1, initial_state p S1 -> exists S2, initial_state tp S2 /\ match_states S1 S2.