Section TRANSLATION.
Variable prog: Csharpminor.program.
Variable tprog: program.
Hypothesis TRANSL: transl_program prog = OK tprog.
Let ge : Csharpminor.genv := Genv.globalenv prog.
Let tge: genv := Genv.globalenv tprog.
Proof (Genv.find_symbol_transf_partial transl_fundef _ TRANSL).
Proof (Genv.find_funct_ptr_transf_partial transl_fundef _ TRANSL).
Proof (Genv.find_funct_transf_partial transl_fundef _ TRANSL).
Proof (Genv.find_var_info_transf_partial transl_fundef _ 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 Int.zero) (Vptr sp (Int.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 *)
(** The sizes of blocks appearing in [e] are respected. *)
Definition match_bounds (e: Csharpminor.env) (m: mem) : Prop :=
forall id b sz ofs p,
PTree.get id e = Some(b, sz) -> Mem.perm m b ofs Max p -> 0 <= ofs < sz.
(** ** Permissions on the Cminor stack block *)
(** The parts of the Cminor stack data block that are not images of C#minor local variable blocks remain freeable at all times. *)
Inductive is_reachable_from_env (f: meminj) (e: Csharpminor.env) (sp: block) (ofs: Z) : Prop :=
| is_reachable_intro: forall id b sz delta,
e!id = Some(b, sz) ->
f b = Some(sp, delta) ->
delta <= ofs < delta + sz ->
is_reachable_from_env f e sp ofs.
Definition padding_freeable (f: meminj) (e: Csharpminor.env) (tm: mem) (sp: block) (sz: Z) : Prop :=
forall ofs,
0 <= ofs < sz -> Mem.perm tm sp ofs Cur Freeable \/ is_reachable_from_env f e sp ofs.
(** Decidability of the [is_reachable_from_env] predicate. *)
(** * Correspondence between global environments *)
(** Global environments match if the memory injection [f] leaves unchanged the references to global symbols and functions. *)
Inductive match_globalenvs (f: meminj) (bound: block): Prop :=
| mk_match_globalenvs
(DOMAIN: forall b,
Plt b bound ->
((exists id, Genv.find_symbol ge id = Some b)
\/ (exists gv, Genv.find_var_info (Genv.globalenv prog) b = Some gv)
\/ (exists fptr, Genv.find_funct_ptr (Genv.globalenv prog) b = Some fptr)) ->
f b = Some(b, 0))
(IMAGE: forall b1 b2 delta gv,
Genv.find_var_info ge b2 = Some gv ->
f b1 = Some(b2, delta) -> Plt b2 bound -> b1 = b2)
(SYMBOLS: forall id b, Genv.find_symbol ge id = Some b -> Plt b bound)
(FUNCTIONS: forall b fd, Genv.find_funct_ptr ge b = Some fd -> Plt b bound)
(VARINFOS: forall b gv, Genv.find_var_info ge b = Some gv -> Plt b bound).
Remark inj_preserves_globals:
forall f hi,
match_globalenvs f hi ->
meminj_preserves_globals ge f.
(** * Invariant on abstract call stacks  *)
(** Call stacks represent abstractly the execution state of the current C#minor and Cminor functions, as well as the states of the calling functions.  A call stack is a list of frames, each frame collecting information on the current execution state of a C#minor function and its Cminor translation. *)
Inductive frame : Type :=
Frame(cenv: compilenv)
(tf: Cminor.function)
(e: Csharpminor.env)
(le: Csharpminor.temp_env)
(te: Cminor.env)
(sp: block)
(lo hi: block).
Definition callstack : Type := list frame.
(** Matching of call stacks imply: - matching of environments for each of the frames - matching of the global environments - separation conditions over the memory blocks allocated for C#minor local variables; - separation conditions over the memory blocks allocated for Cminor stack data; - freeable permissions on the parts of the Cminor stack data blocks that are not images of C#minor local variable blocks. *)
Inductive match_callstack (f: meminj) (m: mem) (tm: mem):
callstack -> block -> block -> Prop :=
| mcs_nil:
forall hi bound tbound,
match_globalenvs f hi ->
Ple hi bound -> Ple hi tbound ->
match_callstack f m tm nil bound tbound
| mcs_cons:
forall cenv tf e le te sp lo hi cs bound tbound
(BOUND: Ple hi bound)
(TBOUND: Plt sp tbound)
(MTMP: match_temps f le te)
(MENV: match_env f cenv e sp lo hi)
(BOUND: match_bounds e m)
(PERM: padding_freeable f e tm sp tf.(fn_stackspace))
(MCS: match_callstack f m tm cs lo sp),
match_callstack f m tm (Frame cenv tf e le te sp lo hi :: cs) bound tbound.
(** [match_callstack] implies [match_globalenvs]. *)
(** Invariance properties for [match_callstack]. *)
(** Assigning a temporary variable. *)
(** Preservation of [match_callstack] by freeing all blocks allocated for local variables at function entry (on the C#minor side) and simultaneously freeing the Cminor stack data block. *)

Lemma match_callstack_freelist:
forall f cenv tf e le te sp lo hi cs m m' tm,
Mem.inject f m tm ->
Mem.free_list m (blocks_of_env e) = Some m' ->
match_callstack f m tm (Frame cenv tf e le te sp lo hi :: cs) (Mem.nextblock m) (Mem.nextblock tm) ->
exists tm',
Mem.free tm sp 0 tf.(fn_stackspace) = Some tm'
/\ match_callstack f m' tm' cs (Mem.nextblock m') (Mem.nextblock tm')
/\ Mem.inject f m' tm'.