Module Qcert.NRAEnv.Optim.NRAEnvRewriteContext



Section NRAEnvRewriteContext.
  Require Import Equivalence.
  Require Import Morphisms.
  Require Import Setoid.
  Require Import EquivDec.
  Require Import Program.
  Require Import Arith.
  Require Import List.
  Require Import Utils.
  Require Import CommonRuntime.
  Require Import NRARuntime.
  Require Import cNRAEnvRuntime.
  Require Import NRAContext.
  Require Import NRARewriteContext.
  Require Import cNRAEnvContext.
  Require Import cNRAEnvContextLift.
  Require Import NRAEnvRewrite.
  
  Local Open Scope nraenv_core_ctxt.

  Context {fruntime:foreign_runtime}.

  Ltac simpl_domain_eq_const
    := repeat
         match goal with
         | [H: domain ?x = nil |- _ ] =>
           apply domain_nil in H; subst x
         | [H: domain ?x = _::_ |- _ ] =>
           let n := fresh "n" in
           let a := fresh "a" in
           destruct x as [|[n a]]; simpl in H; inversion H; clear H;
           try subst n
         end.

  Ltac simpl_plugins
    := match goal with
         [H1:is_list_sorted lt_dec ?x = true,
             H2:equivlist ?x ?y |- _ ] =>
         apply insertion_sort_equivlist_nat in H2;
           rewrite (insertion_sort_sorted_is_id _ _ H1) in H2;
           simpl in H2; simpl_domain_eq_const; clear H1
       end.

  Ltac simpl_ctxt_equiv :=
    try apply <- nraenv_core_ctxt_equiv_strict_equiv;
    red; simpl; intros; simpl_plugins; simpl.

  Local Open Scope nraenv_core_ctxt_scope.
  Lemma envctxt_and_comm_ctxt :
    $2 ∧ $1 ≡ₑ $1 ∧ $2.
Proof.
    generalize ctxt_and_comm_ctxt; intros pf.
    apply lift_nra_context_proper in pf.
    simpl in pf; trivial.
  Qed.
  
  Lemma ctxt_envunion_assoc :
    ($1 ⋃ $2) ⋃ $3 ≡ₑ $1 ⋃ ($2 ⋃ $3).
Proof.
    simpl_ctxt_equiv.
    apply envunion_assoc.
  Qed.

  Lemma ctxt_app_over_merge :
    (ENV ⊗ $1) ◯ $2 ≡ₑ ENV ⊗ ($1 ◯ $2).
Proof.
    simpl_ctxt_equiv.
    apply app_over_merge.
  Qed.

  Lemma envctxt_select_union_distr :
    σ⟨ $0 ⟩($1 ⋃ $2) ≡ₑ σ⟨ $0 ⟩($1) ⋃ σ⟨ $0 ⟩($2).
Proof.
    generalize ctxt_select_union_distr; intros pf.
    apply lift_nra_context_proper in pf.
    simpl in pf; trivial.
  Qed.


End NRAEnvRewriteContext.