Definition globvar_eq {V1 V2: Type} (v1:globvar V1) (v2:globvar V2) :=
  match v1, v2 with
  | mkglobvar _ init1 readonly1 volatile1,
    mkglobvar _ init2 readonly2 volatile2 =>
    init1 = init2 /\ readonly1 =  readonly2 /\ volatile1 = volatile2
  end.Inductive external_description :=
| extern_func: signature -> external_description
| extern_globvar: external_description.Inductive external_description :=
| extern_func: signature -> external_description
| extern_globvar: external_description.Inductive external_description :=
| extern_func: signature -> external_description
| extern_globvar: external_description.Definition entryPts_ok  {F1 V1 F2 V2:Type}
  (P1 : AST.program F1 V1)    (P2 : AST.program F2 V2)
  (ExternIdents: list (ident * external_description))
  (entryPts: list (val * val * signature)): Prop :=
  forall e d, In (e,d) ExternIdents ->
    exists b, Genv.find_symbol  (Genv.globalenv P1) e = Some b /\
      Genv.find_symbol (Genv.globalenv P2) e = Some b /\
      match d with
        extern_func sig => In (Vptr b Int.zero,Vptr b Int.zero, sig) entryPts /\
        exists f1, exists f2, Genv.find_funct_ptr (Genv.globalenv P1) b = Some f1 /\
          Genv.find_funct_ptr (Genv.globalenv P2) b = Some f2
        | extern_globvar  => exists v1, exists v2,
          Genv.find_var_info (Genv.globalenv P1) b = Some v1 /\
          Genv.find_var_info (Genv.globalenv P2) b = Some v2 /\
          globvar_eq v1 v2
      end.Definition entryPts_inject_ok {F1 V1 F2 V2:Type}
  (P1 : AST.program F1 V1) (P2 : AST.program F2 V2) (j: meminj)
  (ExternIdents : list (ident * external_description))
  (entryPts: list (val * val * signature)): Prop :=
  forall e d, In (e,d) ExternIdents ->
    exists b1, exists b2, Genv.find_symbol (Genv.globalenv P1) e = Some b1 /\
      Genv.find_symbol (Genv.globalenv P2) e = Some b2 /\
      j b1 = Some(b2,0) /\
      match d with
      | extern_func sig =>
        In (Vptr b1 Int.zero,Vptr b2 Int.zero, sig) entryPts /\
        exists f1, exists f2,
          Genv.find_funct_ptr (Genv.globalenv P1) b1 = Some f1 /\
          Genv.find_funct_ptr (Genv.globalenv P2) b2 = Some f2
      | extern_globvar =>
        exists v1, exists v2,
          Genv.find_var_info  (Genv.globalenv P1) b1 = Some v1 /\
          Genv.find_var_info  (Genv.globalenv P2) b2 = Some v2 /\
          globvar_eq v1 v2
      end.Definition externvars_ok  {F1 V1:Type}  (P1 : AST.program F1 V1)
  (ExternIdents : list (ident * external_description)) : Prop :=
  forall b v, Genv.find_var_info  (Genv.globalenv P1) b = Some v ->
    exists e, Genv.find_symbol (Genv.globalenv P1) e = Some b /\
      In (e,extern_globvar) ExternIdents.Definition GenvHyp {F1 V1 F2 V2}
  (P1 : AST.program F1 V1) (P2 : AST.program F2 V2): Prop :=
  (forall id : ident,
    Genv.find_symbol (Genv.globalenv P2) id =
    Genv.find_symbol (Genv.globalenv P1) id) /\
  (forall b : block,
    block_is_volatile (Genv.globalenv P2) b =
    block_is_volatile (Genv.globalenv P1) b).Inductive core_correctness
  (I: forall F C V
      (Sem : CoreSemantics (Genv.t F V) C mem)
      (P : AST.program F V),Prop)
  (ExternIdents: list (ident * external_description))
  entrypoints :
  forall (F1 C1 V1 F2 C2 V2:Type)
    (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
    (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
    (P1 : AST.program F1 V1)
    (P2 : AST.program F2 V2), Type :=
    corec_eq : forall  (F1 C1 V1 F2 C2 V2:Type)
      (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
      (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
      (P1 : AST.program F1 V1)
      (P2 : AST.program F2 V2)
      (Eq_init : forall m1 : mem,
            Genv.init_mem P1 = Some m1 -> Genv.init_mem P2 = Some m1)
      (*(Eq_init: forall m1, initial_mem Sem1  (Genv.globalenv P1)  m1 P1.(prog_defs)->
        (exists m2, initial_mem Sem2  (Genv.globalenv P2)  m2 P2.(prog_defs)
          /\ m1 = m2))*)
      (ePts_ok: entryPts_ok P1 P2 ExternIdents entrypoints)
      (R:Forward_simulation_eq.Forward_simulation_equals Sem1 Sem2
        (Genv.globalenv P1) (Genv.globalenv P2) entrypoints),
      prog_main P1 = prog_main P2 ->
      (*HERE IS THE INJECTION OF THE GENV-ASSUMPTIONS INTO THE PROOF:*)
      GenvHyp P1 P2 ->
      I _ _ _  Sem1 P1 -> I _ _ _  Sem2 P2 ->
      core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2
| corec_ext: forall (F1 C1 V1 F2 C2 V2:Type)
  (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
  (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
  (P1 : AST.program F1 V1)
  (P2 : AST.program F2 V2)
  (Extends_init : forall m1 : mem, Genv.init_mem P1 = Some m1 ->
    exists m2, Genv.init_mem P2 = Some m2 /\ Mem.extends m1 m2)
  (*(Extends_init: forall m1, initial_mem Sem1  (Genv.globalenv P1)  m1 P1.(prog_defs)->
    (exists m2, initial_mem Sem2  (Genv.globalenv P2)  m2 P2.(prog_defs)
      /\ Mem.extends m1 m2))*)
  (ePts_ok: entryPts_ok P1 P2 ExternIdents entrypoints)
  (R:Forward_simulation_ext.Forward_simulation_extends Sem1 Sem2
    (Genv.globalenv P1) (Genv.globalenv P2) entrypoints),
  prog_main P1 = prog_main P2 ->
  (*HERE IS THE INJECTION OF THE GENV-ASSUMPTIONS INTO THE PROOF:*)
  GenvHyp P1 P2 ->
  I _ _ _ Sem1 P1 -> I _ _ _ Sem2 P2 ->
  core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2

| corec_inj : forall (F1 C1 V1 F2 C2 V2:Type)
  (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
  (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
  (P1 : AST.program F1 V1)
  (P2 : AST.program F2 V2)
  jInit
  (Inj_init : forall m1 : mem, Genv.init_mem P1 = Some m1 ->
      exists m2, Genv.init_mem P2 = Some m2 /\ Mem.inject jInit m1 m2)
  (*(Inj_init: forall m1, initial_mem Sem1  (Genv.globalenv P1)  m1 P1.(prog_defs)->
    (exists m2, initial_mem Sem2  (Genv.globalenv P2)  m2 P2.(prog_defs)
      /\ Mem.inject jInit m1 m2))*)
  (ePts_ok: entryPts_inject_ok P1 P2 jInit ExternIdents entrypoints)
  (preserves_globals: meminj_preserves_globals (Genv.globalenv P1) jInit)
  (R:Forward_simulation_inj.Forward_simulation_inject Sem1 Sem2
    (Genv.globalenv P1) (Genv.globalenv P2) entrypoints),
  prog_main P1 = prog_main P2 ->
 (*HERE IS THE INJECTION OF THE GENV-ASSUMPTIONS INTO THE PROOF:*)
  GenvHyp P1 P2 ->
  externvars_ok P1 ExternIdents ->
  I _ _ _ Sem1 P1 -> I _ _ _ Sem2 P2 ->
  core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2

| corec_trans: forall  (F1 C1 V1 F2 C2 V2 F3 C3 V3:Type)
  (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
  (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
  (Sem3 : CoreSemantics (Genv.t F3 V3) C3 mem)
  (P1 : AST.program F1 V1)
  (P2 : AST.program F2 V2)
  (P3 : AST.program F3 V3),
  core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2 ->
  core_correctness I ExternIdents entrypoints F2 C2 V2 F3 C3 V3 Sem2 Sem3 P2 P3 ->
  core_correctness I ExternIdents entrypoints F1 C1 V1 F3 C3 V3 Sem1 Sem3 P1 P3.Inductive core_correctness
  (I: forall F C V
      (Sem : CoreSemantics (Genv.t F V) C mem)
      (P : AST.program F V),Prop)
  (ExternIdents: list (ident * external_description))
  entrypoints :
  forall (F1 C1 V1 F2 C2 V2:Type)
    (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
    (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
    (P1 : AST.program F1 V1)
    (P2 : AST.program F2 V2), Type :=
    corec_eq : forall  (F1 C1 V1 F2 C2 V2:Type)
      (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
      (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
      (P1 : AST.program F1 V1)
      (P2 : AST.program F2 V2)
      (Eq_init : forall m1 : mem,
            Genv.init_mem P1 = Some m1 -> Genv.init_mem P2 = Some m1)
      (*(Eq_init: forall m1, initial_mem Sem1  (Genv.globalenv P1)  m1 P1.(prog_defs)->
        (exists m2, initial_mem Sem2  (Genv.globalenv P2)  m2 P2.(prog_defs)
          /\ m1 = m2))*)
      (ePts_ok: entryPts_ok P1 P2 ExternIdents entrypoints)
      (R:Forward_simulation_eq.Forward_simulation_equals Sem1 Sem2
        (Genv.globalenv P1) (Genv.globalenv P2) entrypoints),
      prog_main P1 = prog_main P2 ->
      (*HERE IS THE INJECTION OF THE GENV-ASSUMPTIONS INTO THE PROOF:*)
      GenvHyp P1 P2 ->
      I _ _ _  Sem1 P1 -> I _ _ _  Sem2 P2 ->
      core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2
| corec_ext: forall (F1 C1 V1 F2 C2 V2:Type)
  (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
  (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
  (P1 : AST.program F1 V1)
  (P2 : AST.program F2 V2)
  (Extends_init : forall m1 : mem, Genv.init_mem P1 = Some m1 ->
    exists m2, Genv.init_mem P2 = Some m2 /\ Mem.extends m1 m2)
  (*(Extends_init: forall m1, initial_mem Sem1  (Genv.globalenv P1)  m1 P1.(prog_defs)->
    (exists m2, initial_mem Sem2  (Genv.globalenv P2)  m2 P2.(prog_defs)
      /\ Mem.extends m1 m2))*)
  (ePts_ok: entryPts_ok P1 P2 ExternIdents entrypoints)
  (R:Forward_simulation_ext.Forward_simulation_extends Sem1 Sem2
    (Genv.globalenv P1) (Genv.globalenv P2) entrypoints),
  prog_main P1 = prog_main P2 ->
  (*HERE IS THE INJECTION OF THE GENV-ASSUMPTIONS INTO THE PROOF:*)
  GenvHyp P1 P2 ->
  I _ _ _ Sem1 P1 -> I _ _ _ Sem2 P2 ->
  core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2

| corec_inj : forall (F1 C1 V1 F2 C2 V2:Type)
  (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
  (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
  (P1 : AST.program F1 V1)
  (P2 : AST.program F2 V2)
  jInit
  (Inj_init : forall m1 : mem, Genv.init_mem P1 = Some m1 ->
      exists m2, Genv.init_mem P2 = Some m2 /\ Mem.inject jInit m1 m2)
  (*(Inj_init: forall m1, initial_mem Sem1  (Genv.globalenv P1)  m1 P1.(prog_defs)->
    (exists m2, initial_mem Sem2  (Genv.globalenv P2)  m2 P2.(prog_defs)
      /\ Mem.inject jInit m1 m2))*)
  (ePts_ok: entryPts_inject_ok P1 P2 jInit ExternIdents entrypoints)
  (preserves_globals: meminj_preserves_globals (Genv.globalenv P1) jInit)
  (R:Forward_simulation_inj.Forward_simulation_inject Sem1 Sem2
    (Genv.globalenv P1) (Genv.globalenv P2) entrypoints),
  prog_main P1 = prog_main P2 ->
 (*HERE IS THE INJECTION OF THE GENV-ASSUMPTIONS INTO THE PROOF:*)
  GenvHyp P1 P2 ->
  externvars_ok P1 ExternIdents ->
  I _ _ _ Sem1 P1 -> I _ _ _ Sem2 P2 ->
  core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2

| corec_trans: forall  (F1 C1 V1 F2 C2 V2 F3 C3 V3:Type)
  (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
  (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
  (Sem3 : CoreSemantics (Genv.t F3 V3) C3 mem)
  (P1 : AST.program F1 V1)
  (P2 : AST.program F2 V2)
  (P3 : AST.program F3 V3),
  core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2 ->
  core_correctness I ExternIdents entrypoints F2 C2 V2 F3 C3 V3 Sem2 Sem3 P2 P3 ->
  core_correctness I ExternIdents entrypoints F1 C1 V1 F3 C3 V3 Sem1 Sem3 P1 P3.Inductive core_correctness
  (I: forall F C V
      (Sem : CoreSemantics (Genv.t F V) C mem)
      (P : AST.program F V),Prop)
  (ExternIdents: list (ident * external_description))
  entrypoints :
  forall (F1 C1 V1 F2 C2 V2:Type)
    (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
    (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
    (P1 : AST.program F1 V1)
    (P2 : AST.program F2 V2), Type :=
    corec_eq : forall  (F1 C1 V1 F2 C2 V2:Type)
      (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
      (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
      (P1 : AST.program F1 V1)
      (P2 : AST.program F2 V2)
      (Eq_init : forall m1 : mem,
            Genv.init_mem P1 = Some m1 -> Genv.init_mem P2 = Some m1)
      (*(Eq_init: forall m1, initial_mem Sem1  (Genv.globalenv P1)  m1 P1.(prog_defs)->
        (exists m2, initial_mem Sem2  (Genv.globalenv P2)  m2 P2.(prog_defs)
          /\ m1 = m2))*)
      (ePts_ok: entryPts_ok P1 P2 ExternIdents entrypoints)
      (R:Forward_simulation_eq.Forward_simulation_equals Sem1 Sem2
        (Genv.globalenv P1) (Genv.globalenv P2) entrypoints),
      prog_main P1 = prog_main P2 ->
      (*HERE IS THE INJECTION OF THE GENV-ASSUMPTIONS INTO THE PROOF:*)
      GenvHyp P1 P2 ->
      I _ _ _  Sem1 P1 -> I _ _ _  Sem2 P2 ->
      core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2
| corec_ext: forall (F1 C1 V1 F2 C2 V2:Type)
  (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
  (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
  (P1 : AST.program F1 V1)
  (P2 : AST.program F2 V2)
  (Extends_init : forall m1 : mem, Genv.init_mem P1 = Some m1 ->
    exists m2, Genv.init_mem P2 = Some m2 /\ Mem.extends m1 m2)
  (*(Extends_init: forall m1, initial_mem Sem1  (Genv.globalenv P1)  m1 P1.(prog_defs)->
    (exists m2, initial_mem Sem2  (Genv.globalenv P2)  m2 P2.(prog_defs)
      /\ Mem.extends m1 m2))*)
  (ePts_ok: entryPts_ok P1 P2 ExternIdents entrypoints)
  (R:Forward_simulation_ext.Forward_simulation_extends Sem1 Sem2
    (Genv.globalenv P1) (Genv.globalenv P2) entrypoints),
  prog_main P1 = prog_main P2 ->
  (*HERE IS THE INJECTION OF THE GENV-ASSUMPTIONS INTO THE PROOF:*)
  GenvHyp P1 P2 ->
  I _ _ _ Sem1 P1 -> I _ _ _ Sem2 P2 ->
  core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2

| corec_inj : forall (F1 C1 V1 F2 C2 V2:Type)
  (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
  (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
  (P1 : AST.program F1 V1)
  (P2 : AST.program F2 V2)
  jInit
  (Inj_init : forall m1 : mem, Genv.init_mem P1 = Some m1 ->
      exists m2, Genv.init_mem P2 = Some m2 /\ Mem.inject jInit m1 m2)
  (*(Inj_init: forall m1, initial_mem Sem1  (Genv.globalenv P1)  m1 P1.(prog_defs)->
    (exists m2, initial_mem Sem2  (Genv.globalenv P2)  m2 P2.(prog_defs)
      /\ Mem.inject jInit m1 m2))*)
  (ePts_ok: entryPts_inject_ok P1 P2 jInit ExternIdents entrypoints)
  (preserves_globals: meminj_preserves_globals (Genv.globalenv P1) jInit)
  (R:Forward_simulation_inj.Forward_simulation_inject Sem1 Sem2
    (Genv.globalenv P1) (Genv.globalenv P2) entrypoints),
  prog_main P1 = prog_main P2 ->
 (*HERE IS THE INJECTION OF THE GENV-ASSUMPTIONS INTO THE PROOF:*)
  GenvHyp P1 P2 ->
  externvars_ok P1 ExternIdents ->
  I _ _ _ Sem1 P1 -> I _ _ _ Sem2 P2 ->
  core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2

| corec_trans: forall  (F1 C1 V1 F2 C2 V2 F3 C3 V3:Type)
  (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
  (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
  (Sem3 : CoreSemantics (Genv.t F3 V3) C3 mem)
  (P1 : AST.program F1 V1)
  (P2 : AST.program F2 V2)
  (P3 : AST.program F3 V3),
  core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2 ->
  core_correctness I ExternIdents entrypoints F2 C2 V2 F3 C3 V3 Sem2 Sem3 P2 P3 ->
  core_correctness I ExternIdents entrypoints F1 C1 V1 F3 C3 V3 Sem1 Sem3 P1 P3.Inductive core_correctness
  (I: forall F C V
      (Sem : CoreSemantics (Genv.t F V) C mem)
      (P : AST.program F V),Prop)
  (ExternIdents: list (ident * external_description))
  entrypoints :
  forall (F1 C1 V1 F2 C2 V2:Type)
    (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
    (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
    (P1 : AST.program F1 V1)
    (P2 : AST.program F2 V2), Type :=
    corec_eq : forall  (F1 C1 V1 F2 C2 V2:Type)
      (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
      (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
      (P1 : AST.program F1 V1)
      (P2 : AST.program F2 V2)
      (Eq_init : forall m1 : mem,
            Genv.init_mem P1 = Some m1 -> Genv.init_mem P2 = Some m1)
      (*(Eq_init: forall m1, initial_mem Sem1  (Genv.globalenv P1)  m1 P1.(prog_defs)->
        (exists m2, initial_mem Sem2  (Genv.globalenv P2)  m2 P2.(prog_defs)
          /\ m1 = m2))*)
      (ePts_ok: entryPts_ok P1 P2 ExternIdents entrypoints)
      (R:Forward_simulation_eq.Forward_simulation_equals Sem1 Sem2
        (Genv.globalenv P1) (Genv.globalenv P2) entrypoints),
      prog_main P1 = prog_main P2 ->
      (*HERE IS THE INJECTION OF THE GENV-ASSUMPTIONS INTO THE PROOF:*)
      GenvHyp P1 P2 ->
      I _ _ _  Sem1 P1 -> I _ _ _  Sem2 P2 ->
      core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2
| corec_ext: forall (F1 C1 V1 F2 C2 V2:Type)
  (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
  (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
  (P1 : AST.program F1 V1)
  (P2 : AST.program F2 V2)
  (Extends_init : forall m1 : mem, Genv.init_mem P1 = Some m1 ->
    exists m2, Genv.init_mem P2 = Some m2 /\ Mem.extends m1 m2)
  (*(Extends_init: forall m1, initial_mem Sem1  (Genv.globalenv P1)  m1 P1.(prog_defs)->
    (exists m2, initial_mem Sem2  (Genv.globalenv P2)  m2 P2.(prog_defs)
      /\ Mem.extends m1 m2))*)
  (ePts_ok: entryPts_ok P1 P2 ExternIdents entrypoints)
  (R:Forward_simulation_ext.Forward_simulation_extends Sem1 Sem2
    (Genv.globalenv P1) (Genv.globalenv P2) entrypoints),
  prog_main P1 = prog_main P2 ->
  (*HERE IS THE INJECTION OF THE GENV-ASSUMPTIONS INTO THE PROOF:*)
  GenvHyp P1 P2 ->
  I _ _ _ Sem1 P1 -> I _ _ _ Sem2 P2 ->
  core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2

| corec_inj : forall (F1 C1 V1 F2 C2 V2:Type)
  (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
  (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
  (P1 : AST.program F1 V1)
  (P2 : AST.program F2 V2)
  jInit
  (Inj_init : forall m1 : mem, Genv.init_mem P1 = Some m1 ->
      exists m2, Genv.init_mem P2 = Some m2 /\ Mem.inject jInit m1 m2)
  (*(Inj_init: forall m1, initial_mem Sem1  (Genv.globalenv P1)  m1 P1.(prog_defs)->
    (exists m2, initial_mem Sem2  (Genv.globalenv P2)  m2 P2.(prog_defs)
      /\ Mem.inject jInit m1 m2))*)
  (ePts_ok: entryPts_inject_ok P1 P2 jInit ExternIdents entrypoints)
  (preserves_globals: meminj_preserves_globals (Genv.globalenv P1) jInit)
  (R:Forward_simulation_inj.Forward_simulation_inject Sem1 Sem2
    (Genv.globalenv P1) (Genv.globalenv P2) entrypoints),
  prog_main P1 = prog_main P2 ->
 (*HERE IS THE INJECTION OF THE GENV-ASSUMPTIONS INTO THE PROOF:*)
  GenvHyp P1 P2 ->
  externvars_ok P1 ExternIdents ->
  I _ _ _ Sem1 P1 -> I _ _ _ Sem2 P2 ->
  core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2

| corec_trans: forall  (F1 C1 V1 F2 C2 V2 F3 C3 V3:Type)
  (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
  (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
  (Sem3 : CoreSemantics (Genv.t F3 V3) C3 mem)
  (P1 : AST.program F1 V1)
  (P2 : AST.program F2 V2)
  (P3 : AST.program F3 V3),
  core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2 ->
  core_correctness I ExternIdents entrypoints F2 C2 V2 F3 C3 V3 Sem2 Sem3 P2 P3 ->
  core_correctness I ExternIdents entrypoints F1 C1 V1 F3 C3 V3 Sem1 Sem3 P1 P3.Inductive core_correctness
  (I: forall F C V
      (Sem : CoreSemantics (Genv.t F V) C mem)
      (P : AST.program F V),Prop)
  (ExternIdents: list (ident * external_description))
  entrypoints :
  forall (F1 C1 V1 F2 C2 V2:Type)
    (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
    (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
    (P1 : AST.program F1 V1)
    (P2 : AST.program F2 V2), Type :=
    corec_eq : forall  (F1 C1 V1 F2 C2 V2:Type)
      (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
      (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
      (P1 : AST.program F1 V1)
      (P2 : AST.program F2 V2)
      (Eq_init : forall m1 : mem,
            Genv.init_mem P1 = Some m1 -> Genv.init_mem P2 = Some m1)
      (*(Eq_init: forall m1, initial_mem Sem1  (Genv.globalenv P1)  m1 P1.(prog_defs)->
        (exists m2, initial_mem Sem2  (Genv.globalenv P2)  m2 P2.(prog_defs)
          /\ m1 = m2))*)
      (ePts_ok: entryPts_ok P1 P2 ExternIdents entrypoints)
      (R:Forward_simulation_eq.Forward_simulation_equals Sem1 Sem2
        (Genv.globalenv P1) (Genv.globalenv P2) entrypoints),
      prog_main P1 = prog_main P2 ->
      (*HERE IS THE INJECTION OF THE GENV-ASSUMPTIONS INTO THE PROOF:*)
      GenvHyp P1 P2 ->
      I _ _ _  Sem1 P1 -> I _ _ _  Sem2 P2 ->
      core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2
| corec_ext: forall (F1 C1 V1 F2 C2 V2:Type)
  (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
  (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
  (P1 : AST.program F1 V1)
  (P2 : AST.program F2 V2)
  (Extends_init : forall m1 : mem, Genv.init_mem P1 = Some m1 ->
    exists m2, Genv.init_mem P2 = Some m2 /\ Mem.extends m1 m2)
  (*(Extends_init: forall m1, initial_mem Sem1  (Genv.globalenv P1)  m1 P1.(prog_defs)->
    (exists m2, initial_mem Sem2  (Genv.globalenv P2)  m2 P2.(prog_defs)
      /\ Mem.extends m1 m2))*)
  (ePts_ok: entryPts_ok P1 P2 ExternIdents entrypoints)
  (R:Forward_simulation_ext.Forward_simulation_extends Sem1 Sem2
    (Genv.globalenv P1) (Genv.globalenv P2) entrypoints),
  prog_main P1 = prog_main P2 ->
  (*HERE IS THE INJECTION OF THE GENV-ASSUMPTIONS INTO THE PROOF:*)
  GenvHyp P1 P2 ->
  I _ _ _ Sem1 P1 -> I _ _ _ Sem2 P2 ->
  core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2

| corec_inj : forall (F1 C1 V1 F2 C2 V2:Type)
  (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
  (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
  (P1 : AST.program F1 V1)
  (P2 : AST.program F2 V2)
  jInit
  (Inj_init : forall m1 : mem, Genv.init_mem P1 = Some m1 ->
      exists m2, Genv.init_mem P2 = Some m2 /\ Mem.inject jInit m1 m2)
  (*(Inj_init: forall m1, initial_mem Sem1  (Genv.globalenv P1)  m1 P1.(prog_defs)->
    (exists m2, initial_mem Sem2  (Genv.globalenv P2)  m2 P2.(prog_defs)
      /\ Mem.inject jInit m1 m2))*)
  (ePts_ok: entryPts_inject_ok P1 P2 jInit ExternIdents entrypoints)
  (preserves_globals: meminj_preserves_globals (Genv.globalenv P1) jInit)
  (R:Forward_simulation_inj.Forward_simulation_inject Sem1 Sem2
    (Genv.globalenv P1) (Genv.globalenv P2) entrypoints),
  prog_main P1 = prog_main P2 ->
 (*HERE IS THE INJECTION OF THE GENV-ASSUMPTIONS INTO THE PROOF:*)
  GenvHyp P1 P2 ->
  externvars_ok P1 ExternIdents ->
  I _ _ _ Sem1 P1 -> I _ _ _ Sem2 P2 ->
  core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2

| corec_trans: forall  (F1 C1 V1 F2 C2 V2 F3 C3 V3:Type)
  (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem)
  (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem)
  (Sem3 : CoreSemantics (Genv.t F3 V3) C3 mem)
  (P1 : AST.program F1 V1)
  (P2 : AST.program F2 V2)
  (P3 : AST.program F3 V3),
  core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2 ->
  core_correctness I ExternIdents entrypoints F2 C2 V2 F3 C3 V3 Sem2 Sem3 P2 P3 ->
  core_correctness I ExternIdents entrypoints F1 C1 V1 F3 C3 V3 Sem1 Sem3 P1 P3.

Lemma corec_I: forall {F1 C1 V1 F2 C2 V2} (Sem1 : CoreSemantics (Genv.t F1 V1) C1 mem) (Sem2 : CoreSemantics (Genv.t F2 V2) C2 mem) (P1 : AST.program F1 V1) (P2 : AST.program F2 V2) ExternIdents I entrypoints, core_correctness I ExternIdents entrypoints F1 C1 V1 F2 C2 V2 Sem1 Sem2 P1 P2 -> I _ _ _ Sem1 P1 /\ I _ _ _ Sem2 P2.