Require Import PArith.
Require Import List.
Import List.ListNotations.
Open Scope list_scope.

Require Import Rustre.Common.
Require Import Rustre.Dataflow.Syntax.
Require Import Rustre.Dataflow.IsVariable.
Require Import Rustre.Dataflow.IsDefined.
Require Import Rustre.Dataflow.Memories.

Inductive NoDup_defs : list equation → Prop :=
| NDDNil: NoDup_defs nil
| NDDEqDef:
    ∀ x ck e eqs,
      NoDup_defs eqs →
      ¬Is_defined_in_eqs x eqs →
      NoDup_defs (EqDef x ck e :: eqs)
| NDDEqApp:
    ∀ x ck f e eqs,
      NoDup_defs eqs →
      ¬Is_defined_in_eqs x eqs →
      NoDup_defs (EqApp x ck f e :: eqs)
| NDDEqFby:
    ∀ x ck v e eqs,
      NoDup_defs eqs →
      ¬Is_defined_in_eqs x eqs →
      NoDup_defs (EqFby x ck v e :: eqs).

Lemma NoDup_defs_cons:
  ∀ eq eqs,
    NoDup_defs (eq :: eqs) → NoDup_defs eqs.
Proof.
  intros eq eqs Hndd.
  destruct eq; inversion_clear Hndd; assumption.
Qed.

Lemma not_Is_variable_in_memories:
  ∀ x eqs,
    PS.In x (memories eqs)
    → NoDup_defs eqs
    → ¬Is_variable_in_eqs x eqs.
Proof.
  intros x eqs Hinm Hndd.
  pose proof (Is_defined_in_memories _ _ Hinm) as Hdin.
  unfold memories, Is_variable_in_eqs in ×.
  induction eqs as [eq|eq eqs IH].
  - simpl in *; intro; not_In_empty.
  - apply Is_defined_in_cons in Hdin.
    inversion Hndd
      as [|y ck e ? Hndds Hndi|y ck f e ? Hndds Hndi|y ck v0 e ? Hndds Hndi];
      subst; clear Hndd.
    + destruct Hdin as [Hdin|[Hndin Hdins]].
      simpl in Hinm.
      intro He; apply List.Exists_cons in He; destruct He as [He|He].
      apply Is_defined_in_memories in Hinm.
      inversion He; subst; clear He.
      contradiction.

      inversion Hdin; subst; clear Hdin.
      apply Is_variable_in_eqs_Is_defined_in_eqs in He.
      contradiction.

      simpl in Hinm.
      apply IH with (2:=Hndds) (3:=Hdins) in Hinm.
      intro He; apply List.Exists_cons in He; destruct He as [He|He].
      inversion He; subst; clear He.
      apply Hndin; now constructor.
      contradiction.
    + destruct Hdin as [Hdin|[Hndin Hdins]].
      simpl in Hinm.
      intro He; apply List.Exists_cons in He; destruct He as [He|He].
      apply Is_defined_in_memories in Hinm.
      inversion He; subst; clear He.
      contradiction.

      inversion Hdin; subst; clear Hdin.
      apply Is_variable_in_eqs_Is_defined_in_eqs in He.
      contradiction.

      simpl in Hinm.
      apply IH with (2:=Hndds) (3:=Hdins) in Hinm.
      intro He; apply List.Exists_cons in He; destruct He as [He|He].
      inversion He; subst; clear He.
      apply Hndin; now constructor.
      contradiction.
    + destruct Hdin as [Hdin|[Hndin Hdins]].
      simpl in Hinm.
      intro He; apply List.Exists_cons in He; destruct He as [He|He].
      inversion He; subst; clear He.
      inversion Hdin; subst; clear Hdin.

      apply Is_variable_in_eqs_Is_defined_in_eqs in He.
      contradiction.

      simpl in Hinm.
      apply In_fold_left_memory_eq in Hinm.
      destruct Hinm as [Hinm|Hinm].
      apply IH with (2:=Hndds) (3:=Hdins) in Hinm.
      intro He; apply List.Exists_cons in He; destruct He as [He|He].

      inversion He; subst; clear He.
      contradiction.

      apply PS.add_spec in Hinm.
      destruct Hinm as [Hinm|Hinm]; [|now not_In_empty].
      rewrite Hinm in ×.
      exfalso; apply Hndin; now constructor.
Qed.