Require Import Coqlib.
Require Import Maps.
Require Import ASTExtra.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import Memory.
Require Import Events.
Require Import Stacklayout.
Require Import Globalenvs.
Require Import AsmX.
Require Import Smallstep.
Require Import AuxStateDataType.
Require Import Constant.
Require Import GlobIdent.
Require Import FlatMemory.
Require Import CommonTactic.
Require Import AuxLemma.
Require Import RealParams.
Require Import PrimSemantics.
Require Import LAsm.
Require Import LoadStoreSem1.
Require Import XOmega.

Require Import liblayers.logic.PTreeModules.
Require Import liblayers.logic.LayerLogicImpl.
Require Import liblayers.compat.CompatLayers.
Require Import liblayers.compat.CompatGenSem.

Require Import INVLemmaDriver.

Require Import AbstractDataType.
Require Import FutureTactic.

Require Export DHandlerOp.
Require Export ObjInterruptManagement.
Require Import INVLemmaInterrupt.
Require Import DeviceStateDataType.

Section WITHMEM.

  Local Open Scope Z_scope.

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

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

  Section INV.

    Global Instance get_curr_intr_num_inv: PreservesInvariants get_curr_intr_num_spec.
    Proof.
      preserves_invariants_simpl low_level_invariant high_level_invariant; auto.
    Qed.

    Global Instance intr_default_handler_inv: PreservesInvariants intr_default_handler_spec.
    Proof.
      preserves_invariants_simpl low_level_invariant high_level_invariant; auto.
    Qed.

    Global Instance serial_intr_disable_concrete_inv: PreservesInvariants serial_intr_disable_concrete_spec.
    Proof.
      constructor; simpl; intros; inv_generic_sem H.
      Opaque serial_intr_disable_concrete_aux.
      - functional inversion H2; subst; simpl in ×.
        assert (low_level_invariant m d0).
        {
          remember 100%nat as n.
          clear Heqn.
          Transparent serial_intr_disable_concrete_aux.
          clear H1 H3 H4 H5 H2.
          generalize dependent d.
          generalize dependent d0.
          induction n.
          {
            simpl in ×.
            intros.
            inv H7; eauto.
          }
          {
            simpl in ×.
            intros.
            subdestruct.
            eapply IHn in H7; eauto.
            functional inversion Hdestruct; subst.
            functional inversion H5; simpl in *; functional inversion H4; simpl in *; functional inversion H3; simpl in *; functional inversion H1; simpl in *; functional inversion H26; simpl in *; subst; inv H0; econstructor; simpl in *; eauto.
            inv H0; econstructor; simpl in *; eauto.
            functional inversion Hdestruct; subst.
            functional inversion H5; simpl in *; functional inversion H4; simpl in *; functional inversion H3; simpl in *; functional inversion H1; simpl in *; functional inversion H26; simpl in *; subst; inv H0; simpl; assumption.
            simpl; assumption.
            inv H7; inv H0; econstructor; simpl in *; eauto.
          }
        }
        inv H; econstructor; eauto 2 with serial_intr_disable_invariantdb.
      Opaque serial_intr_disable_concrete_aux.
      - functional inversion H2; subst; simpl in ×.
        remember 100%nat as n.
        clear Heqn.
        Transparent serial_intr_disable_concrete_aux.
        clear H4 H5 H2.
        generalize dependent d.
        generalize dependent d0.
        induction n.
        {
          simpl in ×.
          intros.
          inv H7; eauto.
          inv H0; econstructor; simpl in *; eauto.
          rewrite replace_preserves_Zlength.
          assumption.
          intros.
          generalize (valid_ioapic_state H _ _ H0); intros (vval & (b & nev)).
          assert (ncase: n = 4%nat ∨ n ≠ 4%nat) by omega.
          case_eq ncase; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ nev); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.
        }
        {
          simpl in ×.
          intros.
          subdestruct.
          eapply IHn in H7; eauto.
          functional inversion Hdestruct; subst.
          functional inversion H9; simpl in *; functional inversion H8; simpl in *; functional inversion H5; simpl in *; functional inversion H2; simpl in *; functional inversion H28; simpl in *; subst; unfold s_ioapic´ in *; inv H0; econstructor; simpl in *; eauto.

          unfold ioapic_trans_intr.
          destruct (s (ioapic d)).
          destruct (nth_error IoApicIrqs (Z.to_nat 4)).
          destruct (nth_error IoApicEnables (Z.to_nat 4)).
          destruct b.
          simpl; assumption.
          simpl; assumption.
          simpl; assumption.
          simpl; assumption.

          rewrite replace_replace.
          rewrite replace_preserves_Zlength.
          unfold ioapic_trans_intr.
          destruct (s (ioapic d)).
          destruct (nth_error IoApicIrqs (Z.to_nat 4)).
          destruct (nth_error IoApicEnables (Z.to_nat 4)).
          destruct b.
          simpl; assumption.
          simpl; assumption.
          simpl; assumption.
          simpl; assumption.

          unfold ioapic_trans_intr.
          destruct (s (ioapic d)).
          destruct (nth_error IoApicIrqs (Z.to_nat 4)).
          destruct (nth_error IoApicEnables (Z.to_nat 4)).
          destruct b.
          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H20.
          assert (n0case: n0 = 4%nat ∨ n0 ≠ 4%nat) by omega.
          case_eq n0case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H20); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H20.
          assert (n0case: n0 = 4%nat ∨ n0 ≠ 4%nat) by omega.
          case_eq n0case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H20); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H20.
          assert (n0case: n0 = 4%nat ∨ n0 ≠ 4%nat) by omega.
          case_eq n0case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H20); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H20.
          assert (n0case: n0 = 4%nat ∨ n0 ≠ 4%nat) by omega.
          case_eq n0case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H20); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          unfold ioapic_trans_intr.
          destruct (s (ioapic d)).
          destruct (nth_error IoApicIrqs (Z.to_nat 4)).
          destruct (nth_error IoApicEnables (Z.to_nat 4)).
          destruct b.
          simpl; assumption.
          simpl; assumption.
          simpl; assumption.
          simpl; assumption.

          rewrite replace_replace.
          rewrite replace_preserves_Zlength.
          unfold ioapic_trans_intr.
          destruct (s (ioapic d)).
          destruct (nth_error IoApicIrqs (Z.to_nat 4)).
          destruct (nth_error IoApicEnables (Z.to_nat 4)).
          destruct b.
          simpl; assumption.
          simpl; assumption.
          simpl; assumption.
          simpl; assumption.

          unfold ioapic_trans_intr.
          destruct (s (ioapic d)).
          destruct (nth_error IoApicIrqs (Z.to_nat 4)).
          destruct (nth_error IoApicEnables (Z.to_nat 4)).
          destruct b.
          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H20.
          assert (n0case: n0 = 4%nat ∨ n0 ≠ 4%nat) by omega.
          case_eq n0case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H20); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H20.
          assert (n0case: n0 = 4%nat ∨ n0 ≠ 4%nat) by omega.
          case_eq n0case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H20); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H20.
          assert (n0case: n0 = 4%nat ∨ n0 ≠ 4%nat) by omega.
          case_eq n0case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H20); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H20.
          assert (n0case: n0 = 4%nat ∨ n0 ≠ 4%nat) by omega.
          case_eq n0case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H20); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          inv H0; econstructor; simpl in *; eauto.

          functional inversion Hdestruct; subst.
          functional inversion H9; simpl in *; auto.
          simpl; auto.
          functional inversion Hdestruct; subst.
          functional inversion H9; simpl in *; auto.
          simpl; assumption.

          subst.
          functional inversion Hdestruct; subst.
          functional inversion H9; simpl in *; functional inversion H8; simpl in *; functional inversion H5; simpl in *; functional inversion H2; simpl in *; subst; assumption.
          simpl; assumption.
          inv H7; inv H0; econstructor; simpl in *; eauto.

          rewrite replace_preserves_Zlength.
          assumption.

          intros;
          simpl in ×.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H4.
          assert (n0case: n0 = 4%nat ∨ n0 ≠ 4%nat) by omega.
          case_eq n0case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H4); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.
        }
      Opaque serial_intr_disable_concrete_aux.
      - functional inversion H2; subst; simpl in ×.
        remember 100%nat as n.
        clear Heqn.
        Transparent serial_intr_disable_concrete_aux.
        clear H1 H3 H2.
        generalize dependent d.
        generalize dependent d0.
        induction n.
        {
          simpl in ×.
          intros.
          inv H7; eauto.
        }
        {
          simpl in ×.
          intros.
          subdestruct.
          assert(ikern r = true ∧ ihost r = true ∧ init r = true).
          {
            functional inversion Hdestruct; subst.
            functional inversion H9; simpl in *; functional inversion H8; simpl in *; functional inversion H3; simpl in *; functional inversion H1; simpl in *; functional inversion H28; simpl in *; subst; auto.
            simpl; auto.
          }
          destruct H.
          destruct H1.
          eapply IHn in H7; eauto.
          inv H7; auto.
        }
    Qed.


    Global Instance serial_intr_enable_concrete_inv: PreservesInvariants serial_intr_enable_concrete_spec.
    Proof.
      constructor; simpl; intros; inv_generic_sem H.
      Opaque serial_intr_enable_concrete_aux.
      - functional inversion H2; subst; simpl in ×.
        assert (low_level_invariant m d0).
        {
          remember 100%nat as n.
          clear Heqn.
          Transparent serial_intr_enable_concrete_aux.
          clear H1 H3 H4 H5 H2.
          generalize dependent d.
          generalize dependent d0.
          induction n.
          {
            simpl in ×.
            intros.
            inv H7; eauto.
          }
          {
            simpl in ×.
            intros.
            subdestruct.
            eapply IHn in H7; eauto.
            functional inversion Hdestruct; subst.
            functional inversion H5; simpl in *; functional inversion H4; simpl in *; simpl in *; subst; functional inversion H8; functional inversion H2; functional inversion H24; simpl in *; subst; inv H0; econstructor; simpl in *; eauto.
            functional inversion Hdestruct; subst.
            functional inversion H5; simpl in *; functional inversion H4; simpl in *; simpl in *; subst; functional inversion H8; functional inversion H2; functional inversion H24; simpl in *; subst; simpl; eassumption.
            inv H7; inv H0; econstructor; simpl in *; eauto.
          }
        }
        inv H; econstructor; eauto.
      Opaque serial_intr_enable_concrete_aux.
      - functional inversion H2; subst; simpl in ×.
        remember 100%nat as n.
        clear Heqn.
        Transparent serial_intr_enable_concrete_aux.
        clear H4 H5 H2.
        generalize dependent d.
        generalize dependent d0.
        induction n.
        {
          simpl in ×.
          intros.
          inv H7; eauto.
          inv H0; econstructor; simpl in *; eauto.
          rewrite replace_preserves_Zlength.
          assumption.
          intros.
          generalize (valid_ioapic_state H _ _ H0); intros (vval & (b & nev)).
          assert (ncase: n = 4%nat ∨ n ≠ 4%nat) by omega.
          case_eq ncase; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ nev); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.
        }
        {
          simpl in ×.
          intros.
          subdestruct.
          eapply IHn in H7; eauto.
          functional inversion Hdestruct; subst.
          functional inversion H10; simpl in *; functional inversion H9; simpl in *; subst; functional inversion H8; functional inversion H4; simpl in *; subst; functional inversion H26; simpl in *; subst; unfold s_ioapic´ in *; inv H0; econstructor; simpl in *; eauto.

          unfold ioapic_trans_intr.
          destruct (s (ioapic d)).
          destruct (nth_error IoApicIrqs (Z.to_nat 4)).
          destruct (nth_error IoApicEnables (Z.to_nat 4)).
          destruct b.
          intros;
          simpl in ×.
          assumption.
          assumption.
          assumption.
          assumption.

          unfold ioapic_trans_intr.
          destruct (s (ioapic d)).
          destruct (nth_error IoApicIrqs (Z.to_nat 4)).
          destruct (nth_error IoApicEnables (Z.to_nat 4)).
          destruct b.
          intros;
          simpl in ×.
          rewrite replace_replace.
          rewrite replace_preserves_Zlength.
          assumption.
          rewrite replace_replace.
          rewrite replace_preserves_Zlength.
          assumption.
          rewrite replace_replace.
          rewrite replace_preserves_Zlength.
          assumption.
          rewrite replace_replace.
          rewrite replace_preserves_Zlength.
          assumption.

          unfold ioapic_trans_intr.
          destruct (s (ioapic d)).
          destruct (nth_error IoApicIrqs (Z.to_nat 4)).
          destruct (nth_error IoApicEnables (Z.to_nat 4)).
          destruct b.
          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H30.
          assert (n1case: n1 = 4%nat ∨ n1 ≠ 4%nat) by omega.
          case_eq n1case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H30); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H30.
          assert (n1case: n1 = 4%nat ∨ n1 ≠ 4%nat) by omega.
          case_eq n1case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H30); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H30.
          assert (n1case: n1 = 4%nat ∨ n1 ≠ 4%nat) by omega.
          case_eq n1case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H30); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H30.
          assert (n1case: n1 = 4%nat ∨ n1 ≠ 4%nat) by omega.
          case_eq n1case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H30); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          unfold ioapic_trans_intr.
          destruct (s (ioapic d)).
          destruct (nth_error IoApicIrqs (Z.to_nat 4)).
          destruct (nth_error IoApicEnables (Z.to_nat 4)).
          destruct b.
          simpl in *; assumption.
          simpl in *; assumption.
          simpl in *; assumption.
          simpl in *; assumption.

          rewrite replace_replace.
          rewrite replace_preserves_Zlength.
          unfold ioapic_trans_intr.
          destruct (s (ioapic d)).
          destruct (nth_error IoApicIrqs (Z.to_nat 4)).
          destruct (nth_error IoApicEnables (Z.to_nat 4)).
          destruct b.
          simpl in *; assumption.
          simpl in *; assumption.
          simpl in *; assumption.
          simpl in *; assumption.

          unfold ioapic_trans_intr.
          destruct (s (ioapic d)).
          destruct (nth_error IoApicIrqs (Z.to_nat 4)).
          destruct (nth_error IoApicEnables (Z.to_nat 4)).
          destruct b.
          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H30.
          assert (n1case: n1 = 4%nat ∨ n1 ≠ 4%nat) by omega.
          case_eq n1case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H30); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H30.
          assert (n1case: n1 = 4%nat ∨ n1 ≠ 4%nat) by omega.
          case_eq n1case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H30); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H30.
          assert (n1case: n1 = 4%nat ∨ n1 ≠ 4%nat) by omega.
          case_eq n1case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H30); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          intros;
          simpl in ×.
          rewrite replace_replace.
          generalize (valid_ioapic_state H _ _ H0); intro tmp.
          destruct tmp.
          destruct H30.
          assert (n1case: n1 = 4%nat ∨ n1 ≠ 4%nat) by omega.
          case_eq n1case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ H30); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.

          functional inversion Hdestruct; subst.
          functional inversion H10; simpl in *; auto.
          functional inversion Hdestruct; subst.
          functional inversion H10; simpl in *; auto.
          functional inversion Hdestruct; subst.
          functional inversion H10; simpl in *; functional inversion H9; simpl in *; functional inversion H8; simpl in *; functional inversion H4; simpl in *; subst; assumption.
          inv H7; inv H0; econstructor; simpl in *; eauto.
          rewrite replace_preserves_Zlength.
          assumption.
          intros.
          generalize (valid_ioapic_state H _ _ H0); intros (vval & (b & nev)).
          assert (n0case: n0 = 4%nat ∨ n0 ≠ 4%nat) by omega.
          case_eq n0case; intros.
          subst.
          rewrite nth_error_replace_gss.
          split.
          reflexivity.
          esplit; reflexivity.
          generalize (nth_error_range _ _ _ _ nev); intro.
          xomega.
          rewrite nth_error_replace_gso.
          eapply valid_ioapic_state; eauto.
          change (Pos.to_nat 4%positive) with 4%nat.
          omega.
        }
      Opaque serial_intr_enable_concrete_aux.
      - functional inversion H2; subst; simpl in ×.
        remember 100%nat as n.
        clear Heqn.
        Transparent serial_intr_enable_concrete_aux.
        clear H1 H3 H2.
        generalize dependent d.
        generalize dependent d0.
        induction n.
        {
          simpl in ×.
          intros.
          inv H7; eauto.
        }
        {
          simpl in ×.
          intros.
          subdestruct.
          assert(ikern r = true ∧ ihost r = true ∧ init r = true).
          {
            functional inversion Hdestruct; subst.
            functional inversion H10; simpl in *; functional inversion H9; simpl in *; subst; functional inversion H8; functional inversion H2; simpl in *; subst; functional inversion H26; simpl in *; subst; eauto.
          }
          destruct H.
          destruct H1.
          eapply IHn in H7; eauto.
          inv H7; auto.
        }
    Qed.

  End INV.

  Definition dhandler_fresh : compatlayer (cdata RData) :=
    serial_intr_disable ↦ gensem serial_intr_disable_concrete_spec
                        ⊕ serial_intr_enable ↦ gensem serial_intr_enable_concrete_spec.

  Definition dhandler_passthrough : compatlayer (cdata RData) :=
    fload ↦ gensem fload´_spec
          ⊕ fstore ↦ gensem fstore´_spec
          
          ⊕ page_copy ↦ gensem page_copy´_spec
          ⊕ page_copy_back ↦ gensem page_copy_back´_spec

          ⊕ vmxinfo_get ↦ gensem vmxinfo_get_spec
          ⊕ set_pg ↦ gensem setPG_spec
          ⊕ set_cr3 ↦ setCR3_compatsem setCR3_spec
          ⊕ get_size ↦ gensem MMSize
          ⊕ is_usable ↦ gensem is_mm_usable_spec
          ⊕ get_mms ↦ gensem get_mm_s_spec
          ⊕ get_mml ↦ gensem get_mm_l_spec
          ⊕ get_CPU_ID ↦ gensem get_CPU_ID_spec
          ⊕ get_curid ↦ gensem get_curid_spec
          ⊕ set_curid ↦ gensem set_curid_spec
          ⊕ set_curid_init ↦ gensem set_curid_init_spec

          ⊕ release_lock ↦ primcall_release_lock_compatsem release_lock release_lock_spec0
          ⊕ acquire_lock ↦ primcall_acquire_lock_compatsem acquire_lock_spec0
          ⊕ ticket_lock_init ↦ gensem ticket_lock_init0_spec
          ⊕ cli ↦ gensem cli_spec
          ⊕ sti ↦ gensem sti_spec
          ⊕ serial_putc ↦ gensem serial_putc_spec
          ⊕ cons_buf_read ↦ gensem cons_buf_read_spec
          ⊕ trap_in ↦ primcall_general_compatsem trapin_spec
          ⊕ trap_out ↦ primcall_general_compatsem trapout´_spec
          ⊕ host_in ↦ primcall_general_compatsem hostin_spec
          ⊕ host_out ↦ primcall_general_compatsem hostout´_spec
          ⊕ proc_create_postinit ↦ gensem proc_create_postinit_spec
          ⊕ trap_get ↦ primcall_trap_info_get_compatsem trap_info_get_spec
          ⊕ trap_set ↦ primcall_trap_info_ret_compatsem trap_info_ret_spec
          ⊕ accessors ↦ {| exec_load := (@exec_loadex _ _ _ _ _ Hmwd);
                           exec_store := (@exec_storeex _ _ _ _ _ Hmwd) |}.

  Definition dhandler : compatlayer (cdata RData) :=
    dhandler_fresh ⊕ dhandler_passthrough.

End WITHMEM.