Library mcertikos.mm.ALOpGenLink

Require Import Coqlib.
Require Import Maps.
Require Import AST.
Require Import Integers.
Require Import Floats.
Require Import Values.
Require Import MemoryX.
Require Import MemWithData.
Require Import EventsX.
Require Import Globalenvs.
Require Import LAsm.
Require Import Smallstep.
Require Import ClightBigstep.
Require Import Cop.
Require Import ZArith.Zwf.
Require Import VCGen.
Require Import RealParams.
Require Import liblayers.compcertx.Stencil.
Require Import liblayers.compcertx.MakeProgram.
Require Import liblayers.compat.CompatLayers.
Require Import liblayers.compat.CompatGenSem.
Require Import CompatClightSem.
Require Import PrimSemantics.
Require Import Clight.
Require Import CDataTypes.
Require Import Ctypes.
Require Import I64Layer.
Require Import MALInitCode.
Require Import ALOpGen.
Require Import ALOpGenSpec.
Require Import LinkTactic.
Require Import CompCertiKOSproof.
Require Import MALInitCSource.
Require Import LayerCalculusLemma.
Require Import ALOpGenLinkSource.

Require Import FlatMemory.
Require Import Decision.
Require Import LAsmModuleSem.
Require Import Soundness.
Require Import CompatExternalCalls.

Require Import MALOp.
Require Import MALInit.

Section WITHCOMPCERTIKOS.

  Context `{compcertikos_prf: CompCertiKOS}.
  Context `{real_params_prf : RealParams}.
  Context `{multi_oracle_prop: MultiOracleProp}.

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

  Notation HDATAOps := (cdata (cdata_ops := malop_data_ops) HDATA).
  Notation LDATAOps := (cdata (cdata_ops := malinit_data_ops) LDATA).

  Lemma mem_init_correct:
    ∀ COMPILABLE: LayerCompilable (malinit ⊕ L64),
    ∀ MOK: ModuleOK (mem_init ↦ f_mem_init),
    ∀ M2: LAsm.module,
      CompCertiKOS.transf_module (mem_init ↦ f_mem_init) = OK M2 →
      cl_sim _ _
             (crel HDATA LDATA)
             (mem_init ↦ gensem mem_init_spec)
             (〚 M2 〛(malinit ⊕ L64)).
  Proof.
    intros.
    eapply link_compiler; eauto.
    - eapply mem_init_spec_ref.
    - link_nextblock.
    - link_kernel_mode.
    - eapply MALINITCODE.mem_init_code_correct.
    - le_oplus.
  Qed.

  Record MALOp_impl_inverted (M: module) : Prop:=
    {
      MALOP_mem_init: module;
      MALOP_mem_init_transf: CompCertiKOS.transf_module (mem_init ↦ f_mem_init) = ret MALOP_mem_init;
      MALOP_M: M = (MALOP_mem_init ⊕ ∅);
      MALOP_Mok: LayerOK (〚M 〛 (malinit ⊕ L64) ⊕ malinit ⊕ L64)
    }.

  Lemma module_impl_imply:
    ∀ M, MALOp_impl = OK M → MALOp_impl_inverted M.
  Proof.
    unfold MALOp_impl. intros M HM.
    inv_monad´ HM.
    econstructor.
    eassumption.
    inv HM1. reflexivity.
    apply x1.
  Qed.

  Lemma link_correct_aux:
    ∀ M, MALOp_impl = OK M →
              malinit ⊕ L64
                    ⊢ (path_inj (crel HDATA LDATA), M)
              : malop ⊕ L64.
  Proof.

    intros M HM.
    eapply module_impl_imply in HM; destruct HM; subst.

    eapply conseq_le_assoc_comm.
    hcomp_tac.
    {
      layer_link_split_tac.
      apply mem_init_correct; code_correct_tac.
    }
    {
      eapply layer_link_new_glbl_both.
      apply oplus_sim_monotonic.
      apply passthrough_correct.
      apply L64_auto_sim.
    }
  Qed.

  Lemma init_correct:
    ∀ M, MALOp_impl = OK M →
              ModuleOK M →
              cl_init_sim HDATAOps LDATAOps (crel HDATA LDATA) (malop ⊕ L64) M (malinit ⊕ L64).
  Proof.
    intros.
    pose proof (fun i ⇒ module_ok_variable M i (H0 i)) as MOK; clear H0.
    apply cl_init_sim_intro.
    - constructor; simpl; trivial.
      constructor; simpl; trivial. apply FlatMem.flatmem_empty_inj.
      + econstructor.
      + constructor.
    - intros; inv H0.
    - intros; inv H0.
    - intros; inv H0.
    - intros.

      eapply module_impl_imply in H; destruct H; subst.

      transf_none i. specialize (MOK i).
      assert (get_module_variable i (MALOP_mem_init ⊕ ∅) = OK None).
      {
        get_module_variable_relfexivity.
      }
      congruence.
    - decision.
  Qed.

  Theorem cl_backward_simulation:
    ∀ (s: stencil) (CTXT M: module) pl ph
           (builtin_idents_norepet_prf: CompCertBuiltins.BuiltinIdentsNorepet),
      MALOp_impl = OK M →
      make_program s CTXT (malop ⊕ L64) = OK ph →
      make_program s (CTXT ⊕ M) (malinit ⊕ L64) = OK pl →
      backward_simulation
        (LAsm.semantics (lcfg_ops := LC (malop ⊕ L64)) ph)
        (LAsm.semantics (lcfg_ops := LC (malinit ⊕ L64)) pl).
  Proof.
    intros.
    eapply (soundness (crel HDATA LDATA)); eauto; try decision.

    eapply link_correct_aux; eauto.
    eapply init_correct; eauto.
    eapply make_program_oplus_right; eassumption.

    eapply module_impl_imply in H.
    destruct H. assumption.
  Qed.

  Require Import LAsmModuleSemMakeProgram.

  Theorem make_program_exists:
    ∀ (s: stencil) (CTXT M: module) pl,
      MALOp_impl = OK M →
      make_program s (CTXT ⊕ M) (malinit ⊕ L64) = OK pl →
      ∃ ph, make_program s CTXT (malop ⊕ L64) = OK ph.
  Proof.
    intros.
    exploit link_correct_aux; eauto.
    eapply make_program_vertical´ in H0.
    destruct H0 as [p´ Hmake].
    intros Hle.
    eapply make_program_sim_monotonic_exists.
    2: eapply Hle.
    reflexivity.
    eassumption.

    eapply module_impl_imply in H.
    destruct H. assumption.
  Qed.

End WITHCOMPCERTIKOS.