Require Import Coqlib.
Require Import Errors.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Op.
Require Import Asm.
Require Import Events.
Require Import Globalenvs.
Require Import Smallstep.
Require Import Values.
Require Import Memory.
Require Import Maps.
Require Import CommonTactic.
Require Import AuxLemma.
Require Import FlatMemory.
Require Import AuxStateDataType.
Require Import Constant.
Require Import GlobIdent.
Require Import RealParams.
Require Import LoadStoreSem1.
Require Import AsmImplLemma.
Require Import GenSem.
Require Import RefinementTactic.
Require Import PrimSemantics.
Require Import XOmega.

Require Import liblayers.logic.PTreeModules.
Require Import liblayers.logic.LayerLogicImpl.
Require Import liblayers.compcertx.Stencil.
Require Import liblayers.compcertx.MakeProgram.
Require Import liblayers.compat.CompatLayers.
Require Import liblayers.compat.CompatGenSem.
Require Import compcert.cfrontend.Ctypes.
Require Import LayerCalculusLemma.
Require Import AbstractDataType.
Require Import DeviceStateDataType.

Require Import MCurID.
Require Import MBoot.
Require Import CurIDGenSpec.
Require Import LAsmModuleSemAux.

Section Refinement.

  Local Open Scope string_scope.
  Local Open Scope error_monad_scope.
  Local Open Scope Z_scope.

  Context `{real_params: RealParams}.
  Context `{multi_oracle_prop: MultiOracleProp}.

  Notation HDATA := RData.
  Notation LDATA := RData.

  Notation HDATAOps := (cdata (cdata_ops := mcurid_data_ops) HDATA).
  Notation LDATAOps := (cdata (cdata_ops := mboot_data_ops) LDATA).

  Section WITHMEM.

    Context `{Hstencil: Stencil}.
    Context `{Hmem: Mem.MemoryModelX}.
    Context `{Hmwd: UseMemWithData mem}.
    Context `{wait_time: WaitTime}.

    Section REFINEMENT_REL.

        Inductive match_CurID: stencil → (ZMap.t Z) → mem → meminj → Prop :=
        | MATCH_CURID:
            ∀ m b f cid s,
              (∀ i, 0 ≤ i < TOTAL_CPU →
                         ∃ v,
                           Mem.load Mint32 m b (i × 4) = Some (Vint v)
                           ∧ Mem.valid_access m Mint32 b (i × 4) Writable
                           ∧ ZMap.get i cid = (Int.unsigned v))
              → find_symbol s CURID_LOC = Some b
              → match_CurID s cid m f.

        Inductive match_RData: stencil → HDATA → mem → meminj → Prop :=
        | MATCH_RDATA:
            ∀ hadt m f s,
              match_CurID s (cid hadt) m f
              → match_RData s hadt m f.

        Record relate_RData (f: meminj) (hadt: HDATA) (ladt: LDATA) :=
          mkrelate_RData {
              flatmem_re: FlatMem.flatmem_inj (HP hadt) (HP ladt);
              MM_re: MM hadt = MM ladt;
              MMSize_re: MMSize hadt = MMSize ladt;
              vmxinfo_re: vmxinfo hadt = vmxinfo ladt;
              CR3_re: CR3 hadt = CR3 ladt;
              ikern_re: ikern hadt = ikern ladt;
              pg_re: pg hadt = pg ladt;
              ihost_re: ihost hadt = ihost ladt;
              ti_fst_re: (fst (ti hadt)) = (fst (ti ladt));
              ti_snd_re: val_inject f (snd (ti hadt)) (snd (ti ladt));
              init_re: init hadt = init ladt;

              buffer_re: buffer hadt = buffer ladt;

              CPU_ID_re: CPU_ID hadt = CPU_ID ladt;
              multi_oracle_re: multi_oracle hadt = multi_oracle ladt;
              multi_log_re: multi_log hadt = multi_log ladt;
              com1_re: com1 ladt = com1 hadt;
              ioapic_re: ioapic ladt = ioapic hadt;
              lapic_re: lapic ladt = lapic hadt;
              intr_flag_re: intr_flag ladt = intr_flag hadt;
              saved_intr_flags_re: saved_intr_flags ladt = saved_intr_flags hadt;
              curr_intr_num_re: curr_intr_num ladt = curr_intr_num hadt;
              in_intr_re: in_intr hadt = in_intr ladt;
              tf_re: tfs_inj f (tf hadt) (tf ladt)
            }.

        Global Instance rel_ops: CompatRelOps HDATAOps LDATAOps :=
          {
            relate_AbData s f d1 d2 := relate_RData f d1 d2;
            match_AbData s d1 m f := match_RData s d1 m f;
            new_glbl := CURID_LOC :: nil
          }.

    End REFINEMENT_REL.

    Section Rel_Property.

      Lemma inject_match_correct:
        ∀ s d1 m2 f m2´ j,
          match_RData s d1 m2 f →
          Mem.inject j m2 m2´ →
          inject_incr (Mem.flat_inj (genv_next s)) j →
          match_RData s d1 m2´ (compose_meminj f j).
      Proof.
        inversion 1; subst; intros.
        inv H0.
        assert (HFB0: j b = Some (b, 0)).
        {
          eapply stencil_find_symbol_inject´; eauto.
        }
        econstructor; eauto; intros.
        econstructor; eauto; intros.
        specialize (H3 _ H0).
        destruct H3 as (v & HLD & HV & HM).
        specialize (Mem.load_inject _ _ _ _ _ _ _ _ _ H1 HLD HFB0).
        repeat rewrite Z.add_0_r.
        intros [v1´[HLD1´ HV1´]].
        inv HV1´.
        refine_split´; eauto.
        specialize(Mem.valid_access_inject _ _ _ _ _ _ _ _ _ HFB0 H1 HV).
        rewrite Z.add_0_r; trivial.
      Qed.

      Lemma store_match_correct:
        ∀ s abd m0 m0´ f b2 v v´ chunk,
          match_RData s abd m0 f →
          (∀ i b,
             In i new_glbl →
             find_symbol s i = Some b → b ≠ b2) →
          Mem.store chunk m0 b2 v v´ = Some m0´ →
          match_RData s abd m0´ f.
      Proof.
        intros. inv H. inv H2.
        econstructor; eauto.
        econstructor; eauto.
        eapply H0 in H3; simpl; eauto.
        intros. specialize (H _ H2).
        destruct H as (v0 & HLD & HV & HM).
        repeat rewrite (Mem.load_store_other _ _ _ _ _ _ H1); auto.
        refine_split´; eauto.
        eapply Mem.store_valid_access_1; eauto.
      Qed.

      Lemma storebytes_match_correct:
        ∀ s abd m0 m0´ f b2 v v´,
          match_RData s abd m0 f →
          (∀ i b,
             In i new_glbl →
             find_symbol s i = Some b → b ≠ b2) →
          Mem.storebytes m0 b2 v v´ = Some m0´ →
          match_RData s abd m0´ f.
      Proof.
        intros. inv H. inv H2.
        econstructor; eauto.
        econstructor; eauto.
        eapply H0 in H3; simpl; eauto.
        intros. specialize (H _ H2).
        destruct H as (v0 & HLD & HV & HM).
        repeat rewrite (Mem.load_storebytes_other _ _ _ _ _ H1); eauto.
        refine_split´; eauto.
        eapply Mem.storebytes_valid_access_1; eauto.
      Qed.

      Lemma free_match_correct:
        ∀ s abd m0 m0´ f ofs sz b2,
          match_RData s abd m0 f→
          (∀ i b,
             In i new_glbl →
             find_symbol s i = Some b → b ≠ b2) →
          Mem.free m0 b2 ofs sz = Some m0´ →
          match_RData s abd m0´ f.
      Proof.
        intros; inv H; inv H2.
        econstructor; eauto.
        econstructor; eauto.
        eapply H0 in H3; simpl; eauto.
        intros. specialize (H _ H2).
        destruct H as (v0 & HLD & HV & HM).
        repeat rewrite (Mem.load_free _ _ _ _ _ H1); auto.
        refine_split´; eauto.
        eapply Mem.valid_access_free_1; eauto.
      Qed.

      Lemma alloc_match_correct:
        ∀ s abd m´0 m´1 f f´ ofs sz b0 b´1,
          match_RData s abd m´0 f→
          Mem.alloc m´0 ofs sz = (m´1, b´1) →
          f´ b0 = Some (b´1, 0%Z) →
          (∀ b : block, b ≠ b0 → f´ b = f b) →
          inject_incr f f´ →
          (∀ i b,
             In i new_glbl →
             find_symbol s i = Some b → b ≠ b0) →
          match_RData s abd m´1 f´.
      Proof.
        intros. rename H1 into HF1, H2 into HB. inv H; inv H1.
        econstructor; eauto.
        econstructor; eauto.
        intros. specialize (H _ H1).
        destruct H as (v0 & HLD & HV & HM).
        refine_split´; eauto.
        - apply (Mem.load_alloc_other _ _ _ _ _ H0); trivial.
        - eapply Mem.valid_access_alloc_other; eauto.
      Qed.

      Lemma relate_incr:
        ∀ abd abd´ f f´,
          relate_RData f abd abd´
          → inject_incr f f´
          → relate_RData f´ abd abd´.
      Proof.
        inversion 1; subst; intros; inv H; constructor; eauto.
        eapply tfs_inj_incr; eauto.
      Qed.

      Global Instance rel_prf: CompatRel HDATAOps LDATAOps.
      Proof.
        constructor.
        - apply inject_match_correct.
        - apply store_match_correct.
        - apply alloc_match_correct.
        - apply free_match_correct.
        - apply storebytes_match_correct.
        - intros. eapply relate_incr; eauto.
      Qed.

    End Rel_Property.

    Section OneStep_Forward_Relation.

      Ltac pattern2_refinement_simpl:=
        pattern2_refinement_simpl´ (@relate_AbData).

      Section FRESH_PRIM.

        Lemma get_curid_spec_ref:
          compatsim (crel HDATA LDATA) (gensem get_curid_spec) get_curid_spec_low.
        Proof.
          compatsim_simpl (@match_AbData). inv H.
          assert(HOS: kernel_mode d2).
          {
            simpl; inv match_related.
            functional inversion H2; repeat split; trivial; congruence.
          }
          exploit CPU_ID_range; eauto.
          intros Hrange.
          specialize (H0 _ Hrange).
          destruct H0 as (v & HLD & _ & HM).
          assert (HP: v = z).
          {
            functional inversion H2; subst.
            erewrite <- CPU_ID_re in HM; eauto.
            rewrite HM in H.
            apply Int_unsigned_eq in H. trivial.
          }
          refine_split; eauto; econstructor; eauto.
          lift_unfold. trivial.
        Qed.

        Lemma set_curid_spec_ref:
          compatsim (crel HDATA LDATA) (gensem set_curid_spec) set_curid_spec_low.
        Proof.
          compatsim_simpl (@match_AbData).
          assert (Hkern: kernel_mode d2 ∧ CPU_ID d2 = CPU_ID d1).
          {
            inv match_related. functional inversion H1; subst.
            repeat split; try congruence; eauto.
          }
          destruct Hkern as (Hkern & HCPU_ID).
          inv H. pose proof H0 as HLV.
          exploit CPU_ID_range; eauto.
          intros Hrange.
          specialize (H0 _ Hrange).
          destruct H0 as (_ & _ & HV & _).
          exploit CPU_ID_re; eauto.
          intros Heq.
          specialize (Mem.valid_access_store _ _ _ _ (Vint i) HV); intros [m´ HST].
          refine_split.
          - econstructor; eauto.
            instantiate (2:= m´).
            instantiate (1:= d2).
            simpl; lift_trivial. subrewrite´.
          - constructor.
          - pose proof H1 as Hspec.
            functional inversion Hspec; subst.
            split; eauto; pattern2_refinement_simpl.
            econstructor; simpl; eauto.
            econstructor; eauto; intros.
            destruct (zeq i0 (CPU_ID d1)); subst.
            {
              rewrite HCPU_ID in ×.
              refine_split´; eauto.
              - eapply Mem.load_store_same in HST; eauto.
              - eapply Mem.store_valid_access_1; eauto.
              - rewrite ZMap.gss. trivial.
            }
            {
              specialize (HLV _ H).
              destruct HLV as (v & HLD & HV´ & HM).
              refine_split´; trivial.
              - erewrite Mem.load_store_other; eauto.
                right.
                simpl; destruct (zlt i0 (CPU_ID d1));
                [left; omega|right; omega].
              - eapply Mem.store_valid_access_1; eauto.
              - rewrite ZMap.gso; auto.
            }
          - apply inject_incr_refl.
        Qed.

        Lemma set_curid_init_spec_ref:
          compatsim (crel HDATA LDATA) (gensem set_curid_init_spec) set_curid_init_spec_low.
        Proof.
          compatsim_simpl (@match_AbData).
          assert (Hkern: kernel_mode d2).
          {
            inv match_related. functional inversion H1; subst.
            repeat split; try congruence; eauto.
          }
          assert (index_range: 0 ≤ (Int.unsigned i) < 8).
          { functional inversion H1; simpl; auto. }
          inv H. pose proof H0 as HLV.
          generalize index_range.
          intros Hrange.
          specialize (H0 _ Hrange).
          destruct H0 as (_ & _ & HV & _).
          specialize (Mem.valid_access_store _ _ _ _ (Vint (Int.repr (Int.unsigned i + 1))) HV); intros [m´ HST].
          refine_split.
          - econstructor; eauto.
            instantiate (2:= m´).
            instantiate (1:= d2).
            simpl; lift_trivial.
            subrewrite´.
          - constructor.
          - pose proof H1 as Hspec.
            functional inversion Hspec; subst.
            split; eauto; pattern2_refinement_simpl.
            econstructor; simpl; eauto.
            econstructor; eauto; intros.
            destruct (zeq i0 (Int.unsigned i)); subst.
            {
              refine_split´; eauto.
              - eapply Mem.load_store_same in HST; eauto.
              - eapply Mem.store_valid_access_1; eauto.
              - rewrite ZMap.gss.
                rewrite Int.unsigned_repr.
                reflexivity.
                generalize max_unsigned_val; intro.
                omega.
            }
            {
              specialize (HLV _ H).
              destruct HLV as (v & HLD & HV´ & HM).
              refine_split´; trivial.
              - erewrite Mem.load_store_other; eauto.
                right.
                simpl; destruct (zlt i0 (Int.unsigned i));
                [left; omega|right; omega].
              - eapply Mem.store_valid_access_1; eauto.
              - rewrite ZMap.gso; auto.
            }
          - apply inject_incr_refl.
        Qed.

      End FRESH_PRIM.

      Section PASSTHROUGH_PRIM.

        Section Exists.


          Lemma log_incr_exist:
            ∀ habd habd´ labd i ofs f,
              log_incr_spec i ofs habd = Some habd´
              → relate_RData f habd labd
              → ∃ labd´, log_incr_spec i ofs labd = Some labd´ ∧ relate_RData f habd´ labd´
                               ∧ cid habd´ = cid habd.
          Proof.
            unfold log_incr_spec; intros until f; exist_simpl.
          Qed.

          Lemma log_init_exist:
            ∀ habd habd´ labd n ofs f,
              log_init_spec n ofs habd = Some habd´
              → relate_RData f habd labd
              → ∃ labd´, log_init_spec n ofs labd = Some labd´ ∧ relate_RData f habd´ labd´
                               ∧ cid habd´ = cid habd.
          Proof.
            unfold log_init_spec; intros until f; exist_simpl.
          Qed.

          Lemma atomic_FAI_exist:
            ∀ habd habd´ labd n t bound i ofs f,
              atomic_FAI_spec bound i ofs habd = Some (habd´, (t, n))
              → relate_RData f habd labd
              → ∃ labd´, atomic_FAI_spec bound i ofs labd = Some (labd´, (t, n)) ∧ relate_RData f habd´ labd´
                               ∧ cid habd´ = cid habd.
          Proof.
            unfold atomic_FAI_spec; intros until f; exist_simpl.
          Qed.

          Lemma atomic_FAI_match:
            ∀ s bound i ofs d d´ m f t n,
              atomic_FAI_spec bound i ofs d = Some (d´, (t, n))
              → match_AbData s d m f
              → match_AbData s d´ m f.
          Proof.
            unfold atomic_FAI_spec; intros. subdestruct.
            inv H. inv H0. econstructor; eauto.
          Qed.

          Lemma atomic_FAI_sim:
            ∀ id,
              sim (crel RData RData)
                  (id ↦ primcall_atomic_FAI_compatsem atomic_FAI_spec)
                  (id ↦ primcall_atomic_FAI_compatsem atomic_FAI_spec).
          Proof.
            intros. layer_sim_simpl. compatsim_simpl (@match_AbData).
            inv match_extcall_states.
            exploit atomic_FAI_exist; eauto 1; intros (labd´ & HP & HM & Hcid).
            assert (Hf: ι b = Some (b, 0)).
            { eapply stencil_find_symbol_inject´; eauto. }
            exploit Mem.store_mapped_inject; eauto.
            rewrite Z.add_0_r.
            intros (m2´ & Hst & Hinj).
            revert Hstore1. intros.
            exploit Mem.store_mapped_inject; eauto.
            rewrite Z.add_0_r.
            intros (m3´ & Hst´ & Hinj´).
            eapply (extcall_args_with_data (D:= HDATAOps) d1) in Harg.
            exploit (extcall_args_inject (D1:= HDATAOps) (D2:= HDATAOps) d1 d2); eauto.
            intros (varg´ & Hargs & Hlist).
            eapply extcall_args_without_data in Hargs.
            refine_split;
              match goal with
                | |- inject_incr ?f ?f ⇒ apply inject_incr_refl
                | _ ⇒ (econstructor; eauto ; try congruence)
              end;
              match goal with
                | |- _ PC = Vptr _ _ ⇒ eapply reg_symbol_inject; eassumption
                | |- _ → val_inject _ _ _ ⇒ val_inject_simpl
                | _ ⇒ idtac
              end.
            econstructor; eauto ; try congruence.
            -
              assert (Hid: ∀ i0 b0,
                             In i0 new_glbl → find_symbol s i0 = Some b0 → b0 ≠ b).
              {
                intros.
                destruct (peq i0 TICKET_LOCK_LOC).
                + subst. inv H; inv H1.
                + red; intros; subst. elim n0.
                  eapply (genv_vars_inj s i0 TICKET_LOCK_LOC); eauto.
              }
              eapply store_match_correct; eauto.
              eapply store_match_correct; eauto.
              eapply atomic_FAI_match; eauto.
            -
              erewrite Mem.nextblock_store; eauto.
              erewrite Mem.nextblock_store; eauto.
              eapply Mem.nextblock_store in Hst; eauto.
              eapply Mem.nextblock_store in Hst´; eauto.
              rewrite Hst´, Hst. assumption.
            -
              intros. red; intros.
              eapply match_newglob_noperm; eauto.
              eapply Mem.perm_store_2; eauto.
              eapply Mem.perm_store_2; eauto.
            -
              eapply Mem.nextblock_store in Hstore1; eauto.
              rewrite Hstore1.
              erewrite Mem.nextblock_store; eauto.
          Qed.

          Lemma log_get_exist:
            ∀ habd habd´ labd n t i ofs f,
              log_get_spec i ofs habd = Some (habd´, (t, n))
              → relate_RData f habd labd
              → ∃ labd´, log_get_spec i ofs labd = Some (labd´, (t, n)) ∧ relate_RData f habd´ labd´
                               ∧ cid habd´ = cid habd.
          Proof.
            unfold log_get_spec; intros until f; exist_simpl.
          Qed.

          Lemma log_get_match:
            ∀ s i ofs d d´ m f t n,
              log_get_spec i ofs d = Some (d´, (t, n))
              → match_AbData s d m f
              → match_AbData s d´ m f.
          Proof.
            unfold log_get_spec; intros. subdestruct.
            inv H. inv H0. econstructor; eauto.
          Qed.

          Lemma log_get_sim:
            ∀ id,
              sim (crel RData RData)
                  (id ↦ primcall_atomic_GET_compatsem log_get_spec)
                  (id ↦ primcall_atomic_GET_compatsem log_get_spec).
          Proof.
            intros. layer_sim_simpl. compatsim_simpl (@match_AbData).
            inv match_extcall_states.
            exploit log_get_exist; eauto 1; intros (labd´ & HP & HM & Hcid).
            assert (Hf: ι b = Some (b, 0)).
            { eapply stencil_find_symbol_inject´; eauto. }
            exploit Mem.store_mapped_inject; eauto.
            rewrite Z.add_0_r.
            intros (m2´ & Hst & Hinj).
            revert Hstore1. intros.
            exploit Mem.store_mapped_inject; eauto.
            rewrite Z.add_0_r.
            intros (m3´ & Hst´ & Hinj´).
            eapply (extcall_args_with_data (D:= HDATAOps) d1) in Harg.
            exploit (extcall_args_inject (D1:= HDATAOps) (D2:= HDATAOps) d1 d2); eauto.
            intros (varg´ & Hargs & Hlist).
            eapply extcall_args_without_data in Hargs.
            refine_split;
              match goal with
                | |- inject_incr ?f ?f ⇒ apply inject_incr_refl
                | _ ⇒ (econstructor; eauto ; try congruence)
              end;
              match goal with
                | |- _ PC = Vptr _ _ ⇒ eapply reg_symbol_inject; eassumption
                | |- _ → val_inject _ _ _ ⇒ val_inject_simpl
                | _ ⇒ idtac
              end.
            econstructor; eauto ; try congruence.
            -
              assert (Hid: ∀ i0 b0,
                             In i0 new_glbl → find_symbol s i0 = Some b0 → b0 ≠ b).
              {
                intros.
                destruct (peq i0 TICKET_LOCK_LOC).
                + subst. inv H; inv H1.
                + red; intros; subst. elim n0.
                  eapply (genv_vars_inj s i0 TICKET_LOCK_LOC); eauto.
              }
              eapply store_match_correct; eauto.
              eapply store_match_correct; eauto.
              eapply log_get_match; eauto.
            -
              erewrite Mem.nextblock_store; eauto.
              erewrite Mem.nextblock_store; eauto.
              eapply Mem.nextblock_store in Hst; eauto.
              eapply Mem.nextblock_store in Hst´; eauto.
              rewrite Hst´, Hst. assumption.
            -
              intros. red; intros.
              eapply match_newglob_noperm; eauto.
              eapply Mem.perm_store_2; eauto.
              eapply Mem.perm_store_2; eauto.
            -
              eapply Mem.nextblock_store in Hstore1; eauto.
              rewrite Hstore1.
              erewrite Mem.nextblock_store; eauto.
          Qed.


        End Exists.

        Global Instance: (LoadStoreProp (hflatmem_store:= flatmem_store´) (lflatmem_store:= flatmem_store´)).
        Proof.
          accessor_prop_tac.
          - eapply flatmem_store´_exists; eauto.
        Qed.

        Lemma passthrough_correct:
          sim (crel HDATA LDATA) mcurid_passthrough mboot.
        Proof.
          sim_oplus.
          - apply fload´_sim.
          - apply fstore´_sim.
          - apply page_copy´´´_sim.
          - apply page_copy_back´_sim.
          - apply vmxinfo_get_sim.
          - apply setPG_sim.
          - apply setCR3_sim.
          - apply get_size_sim.
          - apply is_mm_usable_sim.
          - apply get_mm_s_sim.
          - apply get_mm_l_sim.
          - apply bootloader0_sim.
          - apply get_CPU_ID_sim.
          - apply release_shared0_sim.
          - apply (acquire_shared0_sim (valid_id_args:= Shared2ID_valid0)).
            intros. inv H. trivial. inv H0.
          - apply atomic_FAI_sim.
          - apply log_get_sim.
          - layer_sim_simpl; compatsim_simpl (@match_AbData); intros.
            exploit log_incr_exist; eauto 1; intros (labd´ & HP & HM & CID).
            match_external_states_simpl.
          - apply log_hold_sim.
          - layer_sim_simpl; compatsim_simpl (@match_AbData); intros.
            exploit log_init_exist; eauto 1; intros (labd´ & HP & HM & CID).
            match_external_states_simpl.
          - apply trapin_sim.
          - apply trapout´_sim.
          - apply hostin_sim.
          - apply hostout´_sim.
          - apply proc_create_postinit_sim.
          - apply trap_info_get_sim.
          - apply trap_info_ret_sim.
          - apply serial_irq_check_sim.
          - apply iret_sim.
          - apply cli_sim.
          - apply sti_sim.
          - apply serial_irq_current_sim.
          - apply ic_intr_sim.
          - apply save_context_sim.
          - apply restore_context_sim.
          - apply local_irq_save_sim.
          - apply local_irq_restore_sim.
          - apply serial_in_sim.
          - apply serial_out_sim.
          - apply serial_hw_intr_sim.
          - apply ioapic_read_sim.
          - apply ioapic_write_sim.
          - apply lapic_read_sim.
          - apply lapic_write_sim.
          - layer_sim_simpl.
            + eapply load_correct1.
            + eapply store_correct1.
        Qed.

      End PASSTHROUGH_PRIM.

    End OneStep_Forward_Relation.

  End WITHMEM.

End Refinement.