Library mcertikos.proc.PQThread


This file defines the abstract data and the primitives for the PThread layer, which will introduce the primtives of thread
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 LoadStoreSem2.
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 CalRealPTPool.
Require Import CalRealPT.
Require Import CalRealIDPDE.
Require Import CalRealInitPTE.
Require Import CalRealSMSPool.
Require Import CalRealProcModule.
Require Import CalRealIntelModule.

Require Import INVLemmaContainer.
Require Import INVLemmaMemory.
Require Import INVLemmaThread.
Require Import INVLemmaIntel.

Require Import AbstractDataType.

Require Export ObjCPU.
Require Export ObjFlatMem.
Require Export ObjContainer.
Require Export ObjVMM.
Require Export ObjLMM.
Require Export ObjShareMem.
Require Export ObjThread.
Require Export ObjMultiprocessor.

Require Import CalTicketLock.
Require Import INVLemmaQLock.
Require Import INVLemmaInterrupt.
Require Import INVLemmaDriver.
Require Import DeviceStateDataType.
Require Import FutureTactic.
Require Export ObjQLock.
Require Export ObjInterruptManagement.
Require Export ObjInterruptController.
Require Export ObjConsole.
Require Export ObjSerialDriver.
Require Export ObjInterruptDriver.
Require Export ObjQueue.
Require Export ObjCV.
Require Export ObjBigThread.

Require Import IntelPrimSemantics.

Abstract Data and Primitives at this layer

Section WITHMEM.

  Local Open Scope Z_scope.

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

**Definition of the invariants at MPTNew layer 0th page map is reserved for the kernel thread
  Record high_level_invariant (abd: RData) :=
    mkInvariant {
        valid_nps: pg abd = true → kern_low ≤ nps abd ≤ maxpage;
        
        valid_kern: ipt abd = false → pg abd = true;
        valid_iptt: ipt abd = true → ikern abd = true;
        valid_iptf: ikern abd = false → ipt abd = false;
        valid_ihost: ihost abd = false → pg abd = true ∧ ikern abd = true;
        valid_container: Container_valid (AC abd);
        
        init_pperm: pg abd = false → (pperm abd) = ZMap.init PGUndef;
        valid_PMap: pg abd = true →
                    (∀ i, 0≤ i < num_proc →
                               PMap_valid (ZMap.get i (ptpool abd)));
        
        valid_PT_kern: pg abd = true → ipt abd = true → (PT abd) = 0;
        valid_PMap_kern: pg abd = true → PMap_kern (ZMap.get 0 (ptpool abd));
        valid_PT: pg abd = true → 0≤ PT abd < num_proc;
        valid_dirty: dirty_ppage (pperm abd) (HP abd);

        valid_idpde: pg abd = true → IDPDE_init (idpde abd);
        
        valid_root: pg abd = true → cused (ZMap.get 0 (AC abd)) = true;

        
        
        

        
        
        valid_cons_buf_rpos: 0 ≤ rpos (console abd) < CONSOLE_BUFFER_SIZE;
        valid_cons_buf_length: 0 ≤ Zlength (cons_buf (console abd)) < CONSOLE_BUFFER_SIZE;

        CPU_ID_range: 0 ≤ (CPU_ID abd) < TOTAL_CPU;
        valid_curid: 0 ≤ ZMap.get (CPU_ID abd) (cid abd) < num_proc;
        valid_current_CPU_ID: CPU_ID abd = current_CPU_ID;
        valid_big_oracle: ∀ l e cpu,
               In (TEVENT cpu e) (ZMap.get lock_range (multi_oracle abd) current_CPU_ID l) →
               cpu ≠ current_CPU_ID
      }.

Definition of the abstract state ops

  Global Instance pqthread_data_ops : CompatDataOps RData :=
    {
      empty_data := init_adt multi_oracle_init7;
      high_level_invariant := high_level_invariant;
      low_level_invariant := low_level_invariant;
      kernel_mode adt := ikern adt = true ∧ ihost adt = true
    }.

Proofs that the initial abstract_data should satisfy the invariants

  Section Property_Abstract_Data.

    Lemma empty_data_high_level_invariant:
      high_level_invariant (init_adt multi_oracle_init7).
    Proof.
      constructor; simpl; intros; auto; try inv H.
      - apply empty_container_valid.
      - eapply dirty_ppage_init.
      - apply current_CPU_ID_range.
      - rewrite ZMap.gi; intuition.
      - eapply valid_global_oracle7; eauto.
    Qed.

Definition of the abstract state

    Global Instance pqthread_data_prf : CompatData RData.
    Proof.
      constructor.
      - apply low_level_invariant_incr.
      - apply empty_data_low_level_invariant.
      - apply empty_data_high_level_invariant.
    Qed.

  End Property_Abstract_Data.

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

Proofs that the primitives satisfies the invariants at this layer

  Section INV.


    Section RECEIVE.

      Lemma page_copy_back_high_level_inv:
        ∀ d d´ chid cound addr,
          page_copy_back_spec chid cound addr d = Some d´
          → high_level_invariant d
          → high_level_invariant d´.
      Proof.
        intros. functional inversion H; subst; eauto.
        inv H0.
        constructor; simpl; intros;
        try eapply dirty_ppage_gss_page_copy_back; eauto.
      Qed.

      Lemma page_copy_back_low_level_inv:
        ∀ d d´ chid cound addr n,
          page_copy_back_spec chid cound addr d = Some d´
          → low_level_invariant n d
          → low_level_invariant n d´.
      Proof.
        intros. functional inversion H; subst; eauto.
        inv H0. constructor; eauto 2.
      Qed.

      Lemma page_copy_back_kernel_mode:
        ∀ d d´ chid cound addr,
          page_copy_back_spec chid cound addr d = Some d´
          → kernel_mode d
          → kernel_mode d´.
      Proof.
        intros. functional inversion H; subst; eauto.
      Qed.

      Lemma ipc_receive_body_high_level_inv:
        ∀ fromid vaddr count d d´ n,
          ipc_receive_body_spec fromid vaddr count d = Some (d´, n) →
          high_level_invariant d →
          high_level_invariant d´.
      Proof.
        intros. functional inversion H; subst; eauto.
        exploit page_copy_back_high_level_inv; eauto.
        intros Hh. inv Hh. constructor; eauto 2; simpl; intros.
      Qed.

      Lemma ipc_receive_body_low_level_inv:
        ∀ fromid vaddr count d d´ n n´,
          ipc_receive_body_spec fromid vaddr count d = Some (d´, n) →
          low_level_invariant n´ d →
          low_level_invariant n´ d´.
      Proof.
        intros. functional inversion H; subst; eauto.
        exploit page_copy_back_low_level_inv; eauto.
        intros Hh. inv Hh. constructor; eauto 2.
      Qed.

      Lemma ipc_receive_body_kernel_mode:
        ∀ fromid vaddr count d d´ n,
          ipc_receive_body_spec fromid vaddr count d = Some (d´, n) →
          kernel_mode d →
          kernel_mode d´.
      Proof.
        intros. functional inversion H; subst; eauto.
        exploit page_copy_back_kernel_mode; eauto.
      Qed.

      Global Instance ipc_receive_body_inv: PreservesInvariants ipc_receive_body_spec.
      Proof.
        preserves_invariants_simpl´.
        - eapply ipc_receive_body_low_level_inv; eassumption.
        - eapply ipc_receive_body_high_level_inv; eassumption.
        - eapply ipc_receive_body_kernel_mode; eassumption.
      Qed.

    End RECEIVE.

    Section SEND.

      Lemma page_copy_high_level_inv:
        ∀ d d´ chid cound addr,
          page_copy_spec chid cound addr d = Some d´
          → high_level_invariant d
          → high_level_invariant d´.
      Proof.
        intros. functional inversion H; subst; eauto.
        inv H0.
        constructor; simpl; intros; eauto.
      Qed.

      Lemma page_copy_low_level_inv:
        ∀ d d´ chid cound addr n,
          page_copy_spec chid cound addr d = Some d´
          → low_level_invariant n d
          → low_level_invariant n d´.
      Proof.
        intros. functional inversion H; subst; eauto.
        inv H0. constructor; eauto 2.
      Qed.

      Lemma page_copy_kernel_mode:
        ∀ d d´ chid cound addr,
          page_copy_spec chid cound addr d = Some d´
          → kernel_mode d
          → kernel_mode d´.
      Proof.
        intros. functional inversion H; subst; eauto.
      Qed.

      Lemma ipc_send_body_high_level_inv:
        ∀ fromid vaddr count d d´ n,
          ipc_send_body_spec fromid vaddr count d = Some (d´, n) →
          high_level_invariant d →
          high_level_invariant d´.
      Proof.
        intros. functional inversion H; subst; eauto.
        exploit page_copy_high_level_inv; eauto.
        intros Hh. inv Hh. constructor; eauto 2; simpl; intros.
      Qed.

      Lemma ipc_send_body_low_level_inv:
        ∀ fromid vaddr count d d´ n n´,
          ipc_send_body_spec fromid vaddr count d = Some (d´, n) →
          low_level_invariant n´ d →
          low_level_invariant n´ d´.
      Proof.
        intros. functional inversion H; subst; eauto.
        exploit page_copy_low_level_inv; eauto.
        intros Hh. inv Hh. constructor; eauto 2.
      Qed.

      Lemma ipc_send_body_kernel_mode:
        ∀ fromid vaddr count d d´ n,
          ipc_send_body_spec fromid vaddr count d = Some (d´, n) →
          kernel_mode d →
          kernel_mode d´.
      Proof.
        intros. functional inversion H; subst; eauto.
        exploit page_copy_kernel_mode; eauto.
      Qed.

      Global Instance ipc_send_body_inv: PreservesInvariants ipc_send_body_spec.
      Proof.
        preserves_invariants_simpl´.
        - eapply ipc_send_body_low_level_inv; eassumption.
        - eapply ipc_send_body_high_level_inv; eassumption.
        - eapply ipc_send_body_kernel_mode; eassumption.
      Qed.

    End SEND.

    Global Instance set_sync_chan_busy_inv: PreservesInvariants set_sync_chan_busy_spec.
    Proof.
      preserves_invariants_simpl_auto.
    Qed.



    Global Instance cli_inv: PreservesInvariants cli_spec.
    Proof.
      preserves_invariants_direct low_level_invariant high_level_invariant; eauto 2.
    Qed.

    Global Instance sti_inv: PreservesInvariants sti_spec.
    Proof.
      preserves_invariants_direct low_level_invariant high_level_invariant; eauto 2.
    Qed.

    Global Instance cons_buf_read_inv:
      PreservesInvariants cons_buf_read_spec.
    Proof.
      preserves_invariants_nested low_level_invariant high_level_invariant; eauto 2.
    Qed.

    Global Instance serial_putc_inv:
      PreservesInvariants serial_putc_spec.
    Proof.
      preserves_invariants_simpl_auto.
    Qed.

    Global Instance serial_intr_disable_inv: PreservesInvariants serial_intr_disable_spec.
    Proof.
      constructor; simpl; intros; inv_generic_sem H.
      - inversion H0; econstructor; eauto 2 with serial_intr_disable_invariantdb.
        generalize (serial_intr_disable_preserves_tf _ _ H2); intro tmprw; rewrite <- tmprw; assumption.
      - inversion H0; econstructor; eauto 2 with serial_intr_disable_invariantdb; rest.
      - eauto 2 with serial_intr_disable_invariantdb.
   Qed.

    Global Instance serial_intr_enable_inv:
      PreservesInvariants serial_intr_enable_spec.
    Proof.
      constructor; simpl; intros; inv_generic_sem H.
      - inversion H0; econstructor; eauto 2 with serial_intr_enable_invariantdb.
        generalize (serial_intr_enable_preserves_tf _ _ H2); intro tmprw; rewrite <- tmprw; assumption.
      - inversion H0; econstructor; eauto 2 with serial_intr_enable_invariantdb; rest.
      - eauto 2 with serial_intr_enable_invariantdb.
    Qed.


    Section SC_lock.

      Lemma acquire_lock_SC_high_level_inv:
        ∀ d d´ i,
          acquire_lock_SC_spec i d = Some d´ →
          high_level_invariant d →
          high_level_invariant d´.
      Proof.
        unfold acquire_lock_SC_spec; intros.
        subdestruct; inv H; eauto.
        - inv H0. constructor; simpl; eauto; intros.
        - inv H0. constructor; simpl; eauto; intros.
      Qed.

      Lemma acquire_lock_SC_low_level_inv:
        ∀ d d´ i n,
          acquire_lock_SC_spec i d = Some d´ →
          low_level_invariant n d →
          low_level_invariant n d´.
      Proof.
        unfold acquire_lock_SC_spec; intros.
        subdestruct; inv H; eauto.
        - inv H0. constructor; simpl; eauto.
        - inv H0. constructor; simpl; eauto.
      Qed.

      Lemma acquire_lock_SC_kernel_mode:
        ∀ d d´ i,
          acquire_lock_SC_spec i d = Some d´ →
          kernel_mode d →
          kernel_mode d´.
      Proof.
        unfold acquire_lock_SC_spec; intros.
        subdestruct; inv H; eauto.
      Qed.

      Global Instance acquire_lock_SC_inv: PreservesInvariants acquire_lock_SC_spec.
      Proof.
        preserves_invariants_simpl´.
        - eapply acquire_lock_SC_low_level_inv; eassumption.
        - eapply acquire_lock_SC_high_level_inv; eassumption.
        - eapply acquire_lock_SC_kernel_mode; eassumption.
      Qed.

      Lemma release_lock_SC_high_level_inv:
        ∀ d d´ i,
          release_lock_SC_spec i d = Some d´ →
          high_level_invariant d →
          high_level_invariant d´.
      Proof.
        intros. unfold release_lock_SC_spec in *; subdestruct; inv H.
        - inv H0. constructor; simpl; eauto; intros.
      Qed.

      Lemma release_lock_SC_low_level_inv:
        ∀ d d´ i n,
          release_lock_SC_spec i d = Some d´ →
          low_level_invariant n d →
          low_level_invariant n d´.
      Proof.
        intros. unfold release_lock_SC_spec in *; subdestruct; inv H.
        - inv H0. constructor; simpl; eauto.
      Qed.

      Lemma release_lock_SC_kernel_mode:
        ∀ d d´ i,
          release_lock_SC_spec i d = Some d´ →
          kernel_mode d →
          kernel_mode d´.
      Proof.
        intros. unfold release_lock_SC_spec in *; subdestruct; inv H.
        - inv H0; eauto.
      Qed.

      Global Instance release_lock_SC_inv: PreservesInvariants release_lock_SC_spec.
      Proof.
        preserves_invariants_simpl´.
        - eapply release_lock_SC_low_level_inv; eassumption.
        - eapply release_lock_SC_high_level_inv; eassumption.
        - eapply release_lock_SC_kernel_mode; eassumption.
      Qed.

    End SC_lock.

    Section PALLOC.

      Lemma palloc_high_level_inv:
        ∀ d d´ i n,
          palloc_spec i d = Some (d´, n) →
          high_level_invariant d →
          high_level_invariant d´.
      Proof.
        unfold palloc_spec; intros.
        subdestruct; inv H; subst; eauto;
        inv H0; constructor; simpl; eauto; intros.
        + rewrite <- Hdestruct3.
          eapply alloc_container_valid´; eauto.
        + subst; simpl;
          intros; congruence.
        + eapply dirty_ppage_gso_alloc; eauto.
        + intros. destruct (zeq i 0); subst.
          × rewrite ZMap.gss; trivial.
          × rewrite ZMap.gso; auto.
      Qed.

      Lemma palloc_low_level_inv:
        ∀ d d´ i n n´,
          palloc_spec i d = Some (d´, n) →
          low_level_invariant n´ d →
          low_level_invariant n´ d´.
      Proof.
        unfold palloc_spec; intros.
        subdestruct; inv H; subst; eauto;
        inv H0; constructor; eauto.
      Qed.

      Lemma palloc_kernel_mode:
        ∀ d d´ i n,
          palloc_spec i d = Some (d´, n) →
          kernel_mode d →
          kernel_mode d´.
      Proof.
        unfold palloc_spec; intros.
        subdestruct; inv H; simpl; eauto.
      Qed.

      Global Instance palloc_inv: PreservesInvariants palloc_spec.
      Proof.
        preserves_invariants_simpl´.
        - eapply palloc_low_level_inv; eassumption.
        - eapply palloc_high_level_inv; eassumption.
        - eapply palloc_kernel_mode; eassumption.
      Qed.

    End PALLOC.


    Global Instance trapin_inv: PrimInvariants trapin_spec.
    Proof.
      PrimInvariants_simpl_auto.
    Qed.

    Global Instance trapout_inv: PrimInvariants trapout_spec.
    Proof.
      PrimInvariants_simpl_auto.
    Qed.

    Global Instance hostin_inv: PrimInvariants hostin_spec.
    Proof.
      PrimInvariants_simpl_auto.
    Qed.

    Global Instance hostout_inv: PrimInvariants hostout_spec.
    Proof.
      PrimInvariants_simpl_auto.
    Qed.

    Global Instance ptin_inv: PrimInvariants ptin_spec.
    Proof.
      PrimInvariants_simpl_auto.
    Qed.

    Global Instance ptout_inv: PrimInvariants ptout_spec.
    Proof.
      PrimInvariants_simpl_auto.
    Qed.

    Global Instance fstore_inv: PreservesInvariants fstore_spec.
    Proof.
      split; intros; inv_generic_sem H; inv H0; functional inversion H2.
      - functional inversion H. split; trivial.
      - functional inversion H.
        split; subst; simpl;
        try (eapply dirty_ppage_store_unmaped; try reflexivity; try eassumption); trivial.
      - functional inversion H0.
        split; simpl; try assumption.
    Qed.

    Global Instance setPT_inv: PreservesInvariants setPT_spec.
    Proof.
      preserves_invariants_simpl_auto.
    Qed.

    Section PTINSERT.

      Section PTINSERT_PTE.

        Lemma ptInsertPTE_high_level_inv:
          ∀ d d´ n vadr padr p,
            ptInsertPTE0_spec n vadr padr p d = Some d´ →
            high_level_invariant d →
            high_level_invariant d´.
        Proof.
          intros. functional inversion H; subst; eauto.
          inv H0; constructor_gso_simpl_tac; intros.
          - eapply PMap_valid_gso_valid; eauto.
          - functional inversion H2. functional inversion H1.
            eapply PMap_kern_gso; eauto.
        Qed.

        Lemma ptInsertPTE_low_level_inv:
          ∀ d d´ n vadr padr p n´,
            ptInsertPTE0_spec n vadr padr p d = Some d´ →
            low_level_invariant n´ d →
            low_level_invariant n´ d´.
        Proof.
          intros. functional inversion H; subst; eauto.
          inv H0. constructor; eauto.
        Qed.

        Lemma ptInsertPTE_kernel_mode:
          ∀ d d´ n vadr padr p,
            ptInsertPTE0_spec n vadr padr p d = Some d´ →
            kernel_mode d →
            kernel_mode d´.
        Proof.
          intros. functional inversion H; subst; eauto.
        Qed.

      End PTINSERT_PTE.

      Section PTPALLOCPDE.

        Lemma palloc_inv_prop:
          ∀ n d d´ v,
            palloc_spec n d = Some (d´, v) →
            ptpool d´ = ptpool d
            ∧ (v ≠ 0 →
                ZMap.get v (pperm d´) = PGAlloc
                ∧ ZMap.get v (LAT d´) = LATCValid nil
                ∧ 0 < v < nps d´)
            ∧ pg d´ = true.
        Proof.
          unfold palloc_spec. intros.
          subdestruct; inv H; simpl; subst.
          - repeat rewrite ZMap.gss. refine_split´; trivial.
            destruct a0 as (? & ? & ?). intros.
            refine_split´; trivial.
          - refine_split´; trivial.
            intros HF; elim HF. trivial.
          - refine_split´; trivial.
            intros HF; elim HF. trivial.
        Qed.

        Lemma ptAllocPDE_high_level_inv:
          ∀ d d´ n vadr v,
            ptAllocPDE0_spec n vadr d = Some (d´, v) →
            high_level_invariant d →
            high_level_invariant d´.
        Proof.
          intros. functional inversion H; subst; eauto.
          - eapply palloc_high_level_inv; eauto.
          - exploit palloc_high_level_inv; eauto.
            intros.
            exploit palloc_inv_prop; eauto. intros (HPT & Halloc & Hpg).
            clear H11.
            rewrite <- HPT in ×.
            inv H1; constructor_gso_simpl_tac; try (intros; congruence); intros.
            + eapply PMap_valid_gso_pde_unp; eauto.
              eapply real_init_PTE_defined.
            + functional inversion H3.
              eapply PMap_kern_gso; eauto.
            + eapply dirty_ppage_gss; eauto.
        Qed.

        Lemma ptAllocPDE_low_level_inv:
          ∀ d d´ n vadr v n´,
            ptAllocPDE0_spec n vadr d = Some (d´, v) →
            low_level_invariant n´ d →
            low_level_invariant n´ d´.
        Proof.
          intros. functional inversion H; subst; eauto.
          - eapply palloc_low_level_inv; eauto.
          - exploit palloc_low_level_inv; eauto.
            intros. inv H1. constructor; eauto.
        Qed.

        Lemma ptAllocPDE_kernel_mode:
          ∀ d d´ n vadr v,
            ptAllocPDE0_spec n vadr d = Some (d´, v) →
            kernel_mode d →
            kernel_mode d´.
        Proof.
          intros. functional inversion H; subst; eauto.
          - eapply palloc_kernel_mode; eauto.
          - exploit palloc_kernel_mode; eauto.
        Qed.

      End PTPALLOCPDE.

      Lemma ptInsert_high_level_inv:
        ∀ d d´ n vadr padr p v,
          ptInsert0_spec n vadr padr p d = Some (d´, v) →
          high_level_invariant d →
          high_level_invariant d´.
      Proof.
        intros. functional inversion H; subst; eauto.
        - eapply ptInsertPTE_high_level_inv; eassumption.
        - eapply ptAllocPDE_high_level_inv; eassumption.
        - eapply ptInsertPTE_high_level_inv; try eassumption.
          eapply ptAllocPDE_high_level_inv; eassumption.
      Qed.

      Lemma ptInsert_low_level_inv:
        ∀ d d´ n vadr padr p n´ v,
          ptInsert0_spec n vadr padr p d = Some (d´, v) →
          low_level_invariant n´ d →
          low_level_invariant n´ d´.
      Proof.
        intros. functional inversion H; subst; eauto.
        - eapply ptInsertPTE_low_level_inv; eassumption.
        - eapply ptAllocPDE_low_level_inv; eassumption.
        - eapply ptInsertPTE_low_level_inv; try eassumption.
          eapply ptAllocPDE_low_level_inv; eassumption.
      Qed.

      Lemma ptInsert_kernel_mode:
        ∀ d d´ n vadr padr p v,
          ptInsert0_spec n vadr padr p d = Some (d´, v) →
          kernel_mode d →
          kernel_mode d´.
      Proof.
        intros. functional inversion H; subst; eauto.
        - eapply ptInsertPTE_kernel_mode; eassumption.
        - eapply ptAllocPDE_kernel_mode; eassumption.
        - eapply ptInsertPTE_kernel_mode; try eassumption.
          eapply ptAllocPDE_kernel_mode; eassumption.
      Qed.

    End PTINSERT.

    Section PTRESV.

      Lemma ptResv_high_level_inv:
        ∀ d d´ n vadr p v,
          ptResv_spec n vadr p d = Some (d´, v) →
          high_level_invariant d →
          high_level_invariant d´.
      Proof.
        intros. functional inversion H; subst; eauto; clear H.
        - eapply palloc_high_level_inv; eassumption.
        - eapply ptInsert_high_level_inv; try eassumption.
          eapply palloc_high_level_inv; eassumption.
      Qed.

      Lemma ptResv_low_level_inv:
        ∀ d d´ n vadr p n´ v,
          ptResv_spec n vadr p d = Some (d´, v) →
          low_level_invariant n´ d →
          low_level_invariant n´ d´.
      Proof.
        intros. functional inversion H; subst; eauto.
        eapply palloc_low_level_inv; eassumption.
        eapply ptInsert_low_level_inv; try eassumption.
        eapply palloc_low_level_inv; eassumption.
      Qed.

      Lemma ptResv_kernel_mode:
        ∀ d d´ n vadr p v,
          ptResv_spec n vadr p d = Some (d´, v) →
          kernel_mode d →
          kernel_mode d´.
      Proof.
        intros. functional inversion H; subst; eauto.
        eapply palloc_kernel_mode; eassumption.
        eapply ptInsert_kernel_mode; try eassumption.
        eapply palloc_kernel_mode; eassumption.
      Qed.

      Global Instance ptResv_inv: PreservesInvariants ptResv_spec.
      Proof.
        preserves_invariants_simpl´.
        - eapply ptResv_low_level_inv; eassumption.
        - eapply ptResv_high_level_inv; eassumption.
        - eapply ptResv_kernel_mode; eassumption.
      Qed.

    End PTRESV.

    Section PTRESV2.

      Lemma ptResv2_high_level_inv:
        ∀ d d´ n vadr p n´ vadr´ p´ v,
          ptResv2_spec n vadr p n´ vadr´ p´ d = Some (d´, v) →
          high_level_invariant d →
          high_level_invariant d´.
      Proof.
        intros; functional inversion H; subst; eauto; clear H.
        - eapply palloc_high_level_inv; eassumption.
        - eapply ptInsert_high_level_inv; try eassumption.
          eapply palloc_high_level_inv; eassumption.
        - eapply ptInsert_high_level_inv; try eassumption.
          eapply ptInsert_high_level_inv; try eassumption.
          eapply palloc_high_level_inv; eassumption.
      Qed.

      Lemma ptResv2_low_level_inv:
        ∀ d d´ n vadr p n´ vadr´ p´ l v,
          ptResv2_spec n vadr p n´ vadr´ p´ d = Some (d´, v) →
          low_level_invariant l d →
          low_level_invariant l d´.
      Proof.
        intros; functional inversion H; subst; eauto.
        - eapply palloc_low_level_inv; eassumption.
        - eapply ptInsert_low_level_inv; try eassumption.
          eapply palloc_low_level_inv; eassumption.
        - eapply ptInsert_low_level_inv; try eassumption.
          eapply ptInsert_low_level_inv; try eassumption.
          eapply palloc_low_level_inv; eassumption.
      Qed.

      Lemma ptResv2_kernel_mode:
        ∀ d d´ n vadr p n´ vadr´ p´ v,
          ptResv2_spec n vadr p n´ vadr´ p´ d = Some (d´, v) →
          kernel_mode d →
          kernel_mode d´.
      Proof.
        intros; functional inversion H; subst; eauto.
        - eapply palloc_kernel_mode; eassumption.
        - eapply ptInsert_kernel_mode; try eassumption.
          eapply palloc_kernel_mode; eassumption.
        - eapply ptInsert_kernel_mode; try eassumption.
          eapply ptInsert_kernel_mode; try eassumption.
          eapply palloc_kernel_mode; eassumption.
      Qed.

    End PTRESV2.

    Section OFFER_SHARE.

      Global Instance offer_shared_mem_inv:
        PreservesInvariants offer_shared_mem_spec.
      Proof.
        preserves_invariants_simpl´;
        functional inversion H2; subst; eauto 2; try (inv H0; constructor; trivial; fail).
        - exploit ptResv2_low_level_inv; eauto.
          intros HP; inv HP. constructor; trivial.
        - exploit ptResv2_low_level_inv; eauto.
          intros HP; inv HP. constructor; trivial.
        - exploit ptResv2_high_level_inv; eauto.
          intros HP; inv HP. constructor; trivial.
        - exploit ptResv2_high_level_inv; eauto.
          intros HP; inv HP. constructor; trivial.
        - exploit ptResv2_kernel_mode; eauto.
        - exploit ptResv2_kernel_mode; eauto.
      Qed.

    End OFFER_SHARE.

    Global Instance shared_mem_status_inv:
      PreservesInvariants shared_mem_status_spec.
    Proof.
      preserves_invariants_simpl low_level_invariant high_level_invariant; eauto 2.
    Qed.

    Global Instance thread_wakeup_inv: PreservesInvariants biglow_thread_wakeup_spec.
    Proof.
      preserves_invariants_simpl´.
      - intros. functional inversion H2; try subst; eauto.
        + inv H0. econstructor; eauto.
        + subst me l´ l1 to. inv H0. econstructor; eauto.
        + inv H0. econstructor; eauto.
      - intros. functional inversion H2; try subst; eauto.
        + inv H0. econstructor; eauto.
        + subst me l´ l1 to. inv H0. econstructor; eauto.
        + inv H0. econstructor; eauto.

      - intros. functional inversion H2; try subst; eauto; simpl.
      Qed.

    Global Instance thread_spawn_inv: DNewInvariants biglow_thread_spawn_spec.
    Proof.
      constructor; intros; inv H0;
      unfold biglow_thread_spawn_spec in *;
      subdestruct; inv H; simpl; auto.

      -
        constructor; trivial; intros; simpl in ×.
        eapply kctxt_inject_neutral_gss_flatinj´; eauto.
        eapply kctxt_inject_neutral_gss_flatinj; eauto.

      -
        assert (Hrdyq: 0 ≤ rdy_q_id (CPU_ID d) < num_chan).
        {
          unfold rdy_q_id. omega.
        }
        constructor; simpl; eauto 2; try congruence; intros.
        + exploit split_container_valid; eauto.
          simpl. omega.
          rewrite Hdestruct3.
          auto.
        + destruct (zeq 0 (id × 8 + 1 + Z.of_nat (length (cchildren (ZMap.get id (AC d)))))); subst.
          rewrite e.
          rewrite ZMap.gss; simpl; split; auto.
          rewrite ZMap.gso; auto.
          destruct (zeq 0 id); subst.
          rewrite ZMap.gss; simpl; auto.
          rewrite ZMap.gso; auto.
    Qed.

    Global Instance sched_init_inv: PreservesInvariants biglow_sched_init_spec.
    Proof.
      preserves_invariants_simpl low_level_invariant high_level_invariant.
      - apply real_nps_range.
      - apply AC_init_container_valid.
      - eapply real_pt_PMap_valid; eauto.
      - apply real_pt_PMap_kern.
      - omega.
      - assumption.
      - apply real_idpde_init.
      - assumption.
      - erewrite real_valid_cidpool; eauto.
        omega.
      - eapply valid_big_oracle0; eauto.
    Qed.

    Local Opaque remove.

    Section YIELD.

      Global Instance thread_yield_inv: ThreadScheduleInvariants biglow_thread_yield_spec.
      Proof.
        constructor; intros; functional inversion H.
        - inv H1.
          constructor; trivial; simpl.
          eapply kctxt_inject_neutral_gss_mem; eauto.
        - inv H0.
          constructor; auto; simpl in *; intros; try congruence.
          + rewrite ZMap.gss. omega.
      Qed.

    End YIELD.

    Global Instance thread_sleep_inv: ThreadTransferInvariants biglow_thread_sleep_spec.
    Proof.
      constructor; intros; functional inversion H.
      - inv H1.
        constructor; trivial.
        eapply kctxt_inject_neutral_gss_mem; eauto.
      - inv H0.
        constructor; auto; simpl in *; intros; try congruence.
        + rewrite ZMap.gss. omega.
    Qed.

    Global Instance proc_create_postinit_inv:
      PreservesInvariants proc_create_postinit_spec.
    Proof.
      preserves_invariants_simpl low_level_invariant high_level_invariant; eauto 2.
    Qed.



  End INV.

  Definition exec_loadex {F V} := exec_loadex2 (F := F) (V := V).

  Definition exec_storeex {F V} := exec_storeex2 (flatmem_store:= flatmem_store) (F := F) (V := V).

  Global Instance flatmem_store_inv: FlatmemStoreInvariant (flatmem_store:= flatmem_store).
  Proof.
    split; inversion 1; intros.
    - functional inversion H0. split; trivial.
    - functional inversion H1.
      split; simpl; try (eapply dirty_ppage_store_unmaped´; try reflexivity; try eassumption); trivial.
  Qed.

  Global Instance trapinfo_set_inv: TrapinfoSetInvariant.
  Proof.
    split; inversion 1; intros; constructor; auto.
  Qed.

  Definition pqthread_passthrough : compatlayer (cdata RData) :=
    fload ↦ gensem fload_spec
          ⊕ fstore ↦ gensem fstore_spec
          ⊕ vmxinfo_get ↦ gensem vmxinfo_get_spec
          ⊕ palloc ↦ gensem palloc_spec
          
          ⊕ set_pt ↦ gensem setPT_spec
          ⊕ pt_read ↦ gensem ptRead_spec
          ⊕ pt_resv ↦ gensem ptResv_spec
          ⊕ shared_mem_status ↦ gensem shared_mem_status_spec
          ⊕ offer_shared_mem ↦ gensem offer_shared_mem_spec
          
          ⊕ thread_spawn ↦ dnew_compatsem biglow_thread_spawn_spec
          ⊕ thread_wakeup ↦ gensem biglow_thread_wakeup_spec
          ⊕ thread_yield ↦ primcall_thread_schedule_compatsem biglow_thread_yield_spec (prim_ident:= thread_yield)
          ⊕ thread_sleep ↦ primcall_thread_transfer_compatsem biglow_thread_sleep_spec
          ⊕ sched_init ↦ gensem biglow_sched_init_spec

          ⊕ pt_in ↦ primcall_general_compatsem´ ptin_spec (prim_ident:= pt_in)
          ⊕ pt_out ↦ primcall_general_compatsem´ ptout_spec (prim_ident:= pt_out)
          ⊕ container_get_nchildren ↦ gensem container_get_nchildren_spec
          ⊕ container_get_quota ↦ gensem container_get_quota_spec
          ⊕ container_get_usage ↦ gensem container_get_usage_spec
          ⊕ container_can_consume ↦ gensem container_can_consume_spec

          ⊕ get_CPU_ID ↦ gensem get_CPU_ID_spec
          ⊕ get_curid ↦ gensem get_curid_spec
          

          ⊕ acquire_lock_CHAN ↦ gensem acquire_lock_SC_spec
          ⊕ release_lock_CHAN ↦ gensem release_lock_SC_spec
          ⊕ get_sync_chan_busy ↦ gensem get_sync_chan_busy_spec
          ⊕ set_sync_chan_busy ↦ gensem set_sync_chan_busy_spec

          
          
          
          
          
          ⊕ ipc_send_body ↦ gensem ipc_send_body_spec
          ⊕ ipc_receive_body ↦ gensem ipc_receive_body_spec
          
          
          

          ⊕ cli ↦ gensem cli_spec
          ⊕ sti ↦ gensem sti_spec
          ⊕ serial_intr_disable ↦ gensem serial_intr_disable_spec
          ⊕ serial_intr_enable ↦ gensem serial_intr_enable_spec
          ⊕ serial_putc ↦ gensem serial_putc_spec
          ⊕ cons_buf_read ↦ gensem cons_buf_read_spec

          ⊕ proc_create_postinit ↦ gensem proc_create_postinit_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
          ⊕ 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; exec_store := @exec_storeex |}.

  Definition pqthread_fresh : compatlayer (cdata RData) :=
    (∅ ⊕ ∅).
  Definition pqthread :=
    pqthread_fresh ⊕ pqthread_passthrough.

End WITHMEM.