Section TRANSLATION.
Variable prog: Csharpminor.program.
Variable tprog: program.
Hypothesis TRANSL: match_prog prog tprog.
Let ge : Csharpminor.genv := Genv.globalenv prog.
Let tge: genv := Genv.globalenv tprog.
Proof (Genv.find_symbol_transf_partial TRANSL).
Proof (Genv.senv_transf_partial TRANSL).
Proof (Genv.find_funct_ptr_transf_partial TRANSL).
Proof (Genv.find_funct_transf_partial TRANSL).
(** * Derived properties of memory operations *)
(** * Correspondence between C#minor's and Cminor's environments and memory states *)
(** In C#minor, every variable is stored in a separate memory block. In the corresponding Cminor code, these variables become sub-blocks of the stack data block.  We capture these changes in memory via a memory injection [f]: [f b = Some(b', ofs)] means that C#minor block [b] corresponds to a sub-block of Cminor block [b] at offset [ofs].  A memory injection [f] defines a relation [Val.inject f] between values and a relation [Mem.inject f] between memory states.  These relations will be used intensively in our proof of simulation between C#minor and Cminor executions. *)
(** ** Matching between Cshaprminor's temporaries and Cminor's variables *)
Definition match_temps (f: meminj) (le: Csharpminor.temp_env) (te: env) : Prop :=
forall id v, le!id = Some v -> exists v', te!(id) = Some v' /\ Val.inject f v v'.
(** ** Matching between C#minor's variable environment and Cminor's stack pointer *)
Inductive match_var (f: meminj) (sp: block): option (block * Z) -> option Z -> Prop :=
| match_var_local: forall b sz ofs,
Val.inject f (Vptr b Ptrofs.zero) (Vptr sp (Ptrofs.repr ofs)) ->
match_var f sp (Some(b, sz)) (Some ofs)
| match_var_global:
match_var f sp None None.
(** Matching between a C#minor environment [e] and a Cminor stack pointer [sp]. The [lo] and [hi] parameters delimit the range of addresses for the blocks referenced from [te]. *)
Record match_env (f: meminj) (cenv: compilenv)
(e: Csharpminor.env) (sp: block)
(lo hi: block) : Prop :=
mk_match_env {
(** C#minor local variables match sub-blocks of the Cminor stack data block. *)
me_vars:
forall id, match_var f sp (e!id) (cenv!id);
(** [lo, hi] is a proper interval. *)
me_low_high:
Ple lo hi;
(** Every block appearing in the C#minor environment [e] must be in the range [lo, hi]. *)
me_bounded:
forall id b sz, PTree.get id e = Some(b, sz) -> Ple lo b /\ Plt b hi;
(** All blocks mapped to sub-blocks of the Cminor stack data must be images of variables from the C#minor environment [e] *)
me_inv:
forall b delta,
f b = Some(sp, delta) ->
exists id, exists sz, PTree.get id e = Some(b, sz);
(** All C#minor blocks below [lo] (i.e. allocated before the blocks referenced from [e]) must map to blocks that are below [sp] (i.e. allocated before the stack data for the current Cminor function). *)
me_incr:
forall b tb delta,
f b = Some(tb, delta) -> Plt b lo -> Plt tb sp
}.
Ltac geninv x :=
let H := fresh in (generalize x; intro H; inv H).
(** [match_env] and external calls *)
Remark inject_incr_separated_same:
forall f1 f2 m1 m1',
inject_incr f1 f2 -> inject_separated f1 f2 m1 m1' ->
forall b, Mem.valid_block m1 b -> f2 b = f1 b.
Remark inject_incr_separated_same':
forall f1 f2 m1 m1',
inject_incr f1 f2 -> inject_separated f1 f2 m1 m1' ->
forall b b' delta,
f2 b = Some(b', delta) -> Mem.valid_block m1' b' -> f1 b = Some(b', delta).
(** [match_env] and allocations *)

Lemma match_env_alloc:
forall f1 id cenv e sp lo m1 sz m2 b ofs f2,
match_env f1 (PTree.remove id cenv) e sp lo (Mem.nextblock m1) ->
Mem.alloc m1 0 sz = (m2, b) ->
cenv!id = Some ofs ->
inject_incr f1 f2 ->
f2 b = Some(sp, ofs) ->
(forall b', b' <> b -> f2 b' = f1 b') ->
e!id = None ->
match_env f2 cenv (PTree.set id (b, sz) e) sp lo (Mem.nextblock m2).