tactics.v

From stdpp Require Import gmap.
From iris.bi Require Export bi.
From iris Require Import options.
Import bi.

Module bi_reflection. Section bi_reflection.
  Context {PROP : bi}.

  Inductive expr :=
    | EEmp : expr
    | EVar : nat → expr
    | ESep : expr → expr → expr.
  Fixpoint eval (Σ : list PROP) (e : expr) : PROP :=
    match e with
    | EEmp => emp
    | EVar n => default emp (Σ !! n)
    | ESep e1 e2 => eval Σ e1 ∗ eval Σ e2
    end%I.
  Fixpoint flatten (e : expr) : list nat :=
    match e with
    | EEmp => []
    | EVar n => [n]
    | ESep e1 e2 => flatten e1 ++ flatten e2
    end.

  Notation eval_list Σ l := ([∗ list] n ∈ l, default emp (Σ !! n))%I.

  Lemma eval_flatten Σ e : eval Σ e ⊣⊢ eval_list Σ (flatten e).
  Proof.
    induction e as [| |e1 IH1 e2 IH2];
      rewrite /= ?right_id ?big_opL_app ?IH1 ?IH2 //.
  Qed.

  (* Can be related to the RHS being affine *)
  Lemma flatten_entails `{BiAffine PROP} Σ e1 e2 :
    flatten e2 ⊆+ flatten e1 → eval Σ e1 ⊢ eval Σ e2.
  Proof. intros. rewrite !eval_flatten. by apply big_sepL_submseteq. Qed.
  Lemma flatten_equiv Σ e1 e2 :
    flatten e2 ≡ₚ flatten e1 → eval Σ e1 ⊣⊢ eval Σ e2.
  Proof. intros He. by rewrite !eval_flatten He. Qed.

  Fixpoint prune (e : expr) : expr :=
    match e with
    | EEmp => EEmp
    | EVar n => EVar n
    | ESep e1 e2 =>
       match prune e1, prune e2 with
       | EEmp, e2' => e2'
       | e1', EEmp => e1'
       | e1', e2' => ESep e1' e2'
       end
    end.
  Lemma flatten_prune e : flatten (prune e) = flatten e.
  Proof.
    induction e as [| |e1 IH1 e2 IH2]; simplify_eq/=; auto.
    rewrite -IH1 -IH2. by repeat case_match; rewrite ?right_id_L.
  Qed.
  Lemma prune_correct Σ e : eval Σ (prune e) ⊣⊢ eval Σ e.
  Proof. by rewrite !eval_flatten flatten_prune. Qed.

  Fixpoint cancel_go (n : nat) (e : expr) : option expr :=
    match e with
    | EEmp => None
    | EVar n' => if decide (n = n') then Some EEmp else None
    | ESep e1 e2 =>
       match cancel_go n e1 with
       | Some e1' => Some (ESep e1' e2)
       | None => ESep e1 <$> cancel_go n e2
       end
    end.
  Definition cancel (ns : list nat) (e: expr) : option expr :=
    prune <$> fold_right (mbind ∘ cancel_go) (Some e) ns.
  Lemma flatten_cancel_go e e' n :
    cancel_go n e = Some e' → flatten e ≡ₚ n :: flatten e'.
  Proof.
    revert e'; induction e as [| |e1 IH1 e2 IH2]; intros;
      repeat (simplify_option_eq || case_match); auto.
    - by rewrite IH1 //.
    - by rewrite IH2 // Permutation_middle.
  Qed.
  Lemma flatten_cancel e e' ns :
    cancel ns e = Some e' → flatten e ≡ₚ ns ++ flatten e'.
  Proof.
    rewrite /cancel fmap_Some=> -[{e'}-e' [He ->]]; rewrite flatten_prune.
    revert e' He; induction ns as [|n ns IH]=> e' He; simplify_option_eq; auto.
    rewrite Permutation_middle -flatten_cancel_go //; eauto.
  Qed.
  Lemma cancel_entails Σ e1 e2 e1' e2' ns :
    cancel ns e1 = Some e1' → cancel ns e2 = Some e2' →
    (eval Σ e1' ⊢ eval Σ e2') → eval Σ e1 ⊢ eval Σ e2.
  Proof.
    intros ??. rewrite !eval_flatten.
    rewrite (flatten_cancel e1 e1' ns) // (flatten_cancel e2 e2' ns) //; csimpl.
    rewrite !big_opL_app. apply sep_mono_r.
  Qed.

  Fixpoint to_expr (l : list nat) : expr :=
    match l with
    | [] => EEmp
    | [n] => EVar n
    | n :: l => ESep (EVar n) (to_expr l)
    end.
  Arguments to_expr !_ / : simpl nomatch.
  Lemma eval_to_expr Σ l : eval Σ (to_expr l) ⊣⊢ eval_list Σ l.
  Proof.
    induction l as [|n1 [|n2 l] IH]; csimpl; rewrite ?right_id //.
    by rewrite IH.
  Qed.

  Lemma split_l Σ e ns e' :
    cancel ns e = Some e' → eval Σ e ⊣⊢ (eval Σ (to_expr ns) ∗ eval Σ e').
  Proof.
    intros He%flatten_cancel.
    by rewrite eval_flatten He big_opL_app eval_to_expr eval_flatten.
  Qed.
  Lemma split_r Σ e ns e' :
    cancel ns e = Some e' → eval Σ e ⊣⊢ (eval Σ e' ∗ eval Σ (to_expr ns)).
  Proof. intros. rewrite /= comm. by apply split_l. Qed.

  Class Quote (Σ1 Σ2 : list PROP) (P : PROP) (e : expr) := {}.
  Global Instance quote_True Σ : Quote Σ Σ emp%I EEmp := {}.
  Global Instance quote_var Σ1 Σ2 P i:
    rlist.QuoteLookup Σ1 Σ2 P i → Quote Σ1 Σ2 P (EVar i) | 1000 := {}.
  Global Instance quote_sep Σ1 Σ2 Σ3 P1 P2 e1 e2 :
    Quote Σ1 Σ2 P1 e1 → Quote Σ2 Σ3 P2 e2 → Quote Σ1 Σ3 (P1 ∗ P2)%I (ESep e1 e2) := {}.

  Class QuoteArgs (Σ : list PROP) (Ps : list PROP) (ns : list nat) := {}.
  Global Instance quote_args_nil Σ : QuoteArgs Σ nil nil := {}.
  Global Instance quote_args_cons Σ Ps P ns n :
    rlist.QuoteLookup Σ Σ P n →
    QuoteArgs Σ Ps ns → QuoteArgs Σ (P :: Ps) (n :: ns) := {}.
  End bi_reflection.

  Ltac quote :=
    match goal with
    | |- ?P1 ⊢ ?P2 =>
      lazymatch type of (_ : Quote [] _ P1 _) with Quote _ ?Σ2 _ ?e1 =>
      lazymatch type of (_ : Quote Σ2 _ P2 _) with Quote _ ?Σ3 _ ?e2 =>
        change (eval Σ3 e1 ⊢ eval Σ3 e2) end end
    end.
  Ltac quote_l :=
    match goal with
    | |- ?P1 ⊢ ?P2 =>
      lazymatch type of (_ : Quote [] _ P1 _) with Quote _ ?Σ2 _ ?e1 =>
        change (eval Σ2 e1 ⊢ P2) end
    end.
End bi_reflection.

Tactic Notation "solve_sep_entails" :=
  bi_reflection.quote;
  first
    [apply bi_reflection.flatten_entails (* for affine BIs *)
    |apply equiv_entails, bi_reflection.flatten_equiv (* for other BIs *) ];
  apply (bool_decide_unpack _); vm_compute; exact Logic.I.

Tactic Notation "solve_sep_equiv" :=
  bi_reflection.quote; apply bi_reflection.flatten_equiv;
  apply (bool_decide_unpack _); vm_compute; exact Logic.I.

Ltac close_uPreds Ps tac :=
  let PROP := match goal with |- @bi_entails ?PROP _ _ => PROP end in
  let rec go Ps Qs :=
    lazymatch Ps with
    | [] => let Qs' := eval cbv [reverse rev_append] in (reverse Qs) in tac Qs'
    | ?P :: ?Ps => find_pat P ltac:(fun Q => go Ps (Q :: Qs))
    end in
  (* avoid evars in case Ps = @nil ?A *)
  try match Ps with [] => unify Ps (@nil PROP) end;
  go Ps (@nil PROP).

Tactic Notation "cancel" constr(Ps) :=
  bi_reflection.quote;
  let Σ := match goal with |- bi_reflection.eval ?Σ _ ⊢ _ => Σ end in
  let ns' := lazymatch type of (_ : bi_reflection.QuoteArgs Σ Ps _) with
             | bi_reflection.QuoteArgs _ _ ?ns' => ns'
             end in
  eapply bi_reflection.cancel_entails with (ns:=ns');
    [cbv; reflexivity|cbv; reflexivity|simpl].

Tactic Notation "ecancel" open_constr(Ps) :=
  close_uPreds Ps ltac:(fun Qs => cancel Qs).

(** [to_front [P1, P2, ..]] rewrites in the premise of ⊢ such that
    the assumptions P1, P2, ... appear at the front, in that order. *)
Tactic Notation "to_front" open_constr(Ps) :=
  close_uPreds Ps ltac:(fun Ps =>
    bi_reflection.quote_l;
    let Σ := match goal with |- bi_reflection.eval ?Σ _ ⊢ _ => Σ end in
    let ns' := lazymatch type of (_ : bi_reflection.QuoteArgs Σ Ps _) with
               | bi_reflection.QuoteArgs _ _ ?ns' => ns'
               end in
    eapply entails_equiv_l;
      first (apply bi_reflection.split_l with (ns:=ns'); cbv; reflexivity);
      simpl).

Tactic Notation "to_back" open_constr(Ps) :=
  close_uPreds Ps ltac:(fun Ps =>
    bi_reflection.quote_l;
    let Σ := match goal with |- bi_reflection.eval ?Σ _ ⊢ _ => Σ end in
    let ns' := lazymatch type of (_ : bi_reflection.QuoteArgs Σ Ps _) with
               | bi_reflection.QuoteArgs _ _ ?ns' => ns'
               end in
    eapply entails_equiv_l;
      first (apply bi_reflection.split_r with (ns:=ns'); cbv; reflexivity);
      simpl).

(** [sep_split] is used to introduce a (∗).
    Use [sep_split left: [P1, P2, ...]] to define which assertions will be
    taken to the left; the rest will be available on the right.
    [sep_split right: [P1, P2, ...]] works the other way around. *)
Tactic Notation "sep_split" "right:" open_constr(Ps) :=
  to_back Ps; apply sep_mono.
Tactic Notation "sep_split" "left:" open_constr(Ps) :=
  to_front Ps; apply sep_mono.