(*Require Import ExtAxioms.*)
(*Will need : *)
Require Export Coq.Logic.FunctionalExtensionality.
Notation val_inject:= Val.inject.
(** * Structured Injections *)
Record SM_Injection :=
{ (** The blocks allocated by *this* module in the source language. *)
locBlocksSrc : block -> bool
(** The blocks allocated by *this* module in the target language. *)
; locBlocksTgt : block -> bool
(** The subset of local source blocks that have been made public. Must be mapped. *)
; pubBlocksSrc : block -> bool
(** The subset of local target blocks that have been made public. Contains the image of the public source blocks. *)
; pubBlocksTgt : block -> bool
(** Memory injection on blocks allocated by *this* module. *)
; local_of: meminj
(** The blocks allocated by *other* modules at the source level. *)
; extBlocksSrc: block -> bool
(** The blocks allocated by *other* modules at the target level. *)
; extBlocksTgt: block -> bool
(** The subset of extern source blocks that have been made public. Must be mapped. *)
; frgnBlocksSrc : block -> bool
(** The subset of extern source blocks that have been made public. Must be mapped. *)
; frgnBlocksTgt : block -> bool
(** Memory injection on blocks allocated by OTHER modules; the injection is not modified by coresteps, and is partitioned by frgnBlocksSrc/Tgt into foreign (leaked to this module) and unknown (non-leaked) components, where the former blocks are accessible by *this* module. This module behaves uniformly over block mentioned by frgnInj, and assumes that blocks in frgnInj remain mapped. Compilation of *this* module neither merges nor unmaps blocks from here, nor does it spill into them. *)
; extern_of: meminj }.
(** The four projections: *)
Definition unknown_of (mu: SM_Injection) : meminj :=
match mu with
Build_SM_Injection locBSrc locBTgt pSrc pTgt local extBSrc extBTgt fSrc fTgt extern =>
fun b => if locBSrc b then None else if fSrc b then None else extern b
end.
Definition foreign_of (mu: SM_Injection) : meminj :=
match mu with
Build_SM_Injection locBSrc locBTgt pSrc pTgt local extBSrc extBTgt fSrc fTgt extern =>
fun b => if fSrc b then extern b else None
end.
Definition pub_of (mu: SM_Injection) : meminj :=
match mu with
Build_SM_Injection locBSrc locBTgt pSrc pTgt local extBSrc extBTgt fSrc fTgt extern =>
fun b => if pSrc b then local b else None
end.
Definition priv_of (mu: SM_Injection) : meminj :=
match mu with
Build_SM_Injection locBSrc locBTgt pSrc pTgt local extBSrc extBTgt fSrc fTgt extern =>
fun b => if pSrc b then None else local b
end.
(** * Lemmas *)

(* Specifications: *)
Lemma compose_meminjI_Some: forall j k b1 b2 d1 b3 d2
(J:j b1 = Some(b2,d1)) (K:k b2 = Some(b3,d2)),
compose_meminj j k b1 = Some(b3, d1+d2).
Lemma compose_meminjI_None: forall j k b1
(H: (j b1 = None) \/
(exists b2 d1, j b1 =Some(b2,d1) /\ k b2=None)),
compose_meminj j k b1 = None.
Lemma inject_incr_coincide: forall f g (INC: inject_incr f g) b p
(G:g b = Some p) q (F:f b = Some q),p=q.
Lemma inject_incr_inv: forall f g (INC: inject_incr f g) b
(G: g b =None), f b = None.
Lemma forall2_val_inject_D: forall vals1 vals2 j (ValInjMu : Forall2 (val_inject j) vals1 vals2)
v1 (IN: In v1 vals1), exists v2, val_inject j v1 v2 /\ In v2 vals2.