ltac_tactics.v 108 KB
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From iris.proofmode Require Import coq_tactics reduction.
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From iris.proofmode Require Import base intro_patterns spec_patterns sel_patterns.
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From iris.bi Require Export bi telescopes.
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From stdpp Require Import namespaces.
From iris.proofmode Require Export classes notation.
From stdpp Require Import hlist pretty.
Set Default Proof Using "Type".
Export ident.

(** For most of the tactics, we want to have tight control over the order and
way in which type class inference is performed. To that end, many tactics make
use of [notypeclasses refine] and the [iSolveTC] tactic to manually invoke type
class inference.

The tactic [iSolveTC] does not use [apply _], as that often leads to issues
because it will try to solve all evars whose type is a typeclass, in
dependency order (according to Matthieu). If one fails, it aborts. However, we
generally rely on progress on the main goal to be solved to make progress
elsewhere. With [typeclasses eauto], that seems to work better.

A drawback of [typeclasses eauto] is that it is multi-success, i.e. whenever
subsequent tactics fail, it will backtrack to [typeclasses eauto] to try the
next type class instance. This is almost always undesired and leads to poor
performance and horrible error messages, so we wrap it in a [once]. *)
Ltac iSolveTC :=
  solve [once (typeclasses eauto)].

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(** Tactic used for solving side-conditions arising from TC resolution in iMod
and iInv. *)
Ltac iSolveSideCondition :=
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  split_and?; try solve [ fast_done | solve_ndisj ].
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(** Used for printing [string]s and [ident]s. *)
Ltac pretty_ident H :=
  lazymatch H with
  | INamed ?H => H
  | ?H => H
  end.

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(** * Misc *)

Ltac iMissingHyps Hs :=
  let Δ :=
    lazymatch goal with
    | |- envs_entails ?Δ _ => Δ
    | |- context[ envs_split _ _ ?Δ ] => Δ
    end in
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  let Hhyps := pm_eval (envs_dom Δ) in
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  eval vm_compute in (list_difference Hs Hhyps).

Ltac iTypeOf H :=
  let Δ := match goal with |- envs_entails ?Δ _ => Δ end in
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  pm_eval (envs_lookup H Δ).
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Tactic Notation "iMatchHyp" tactic1(tac) :=
  match goal with
  | |- context[ environments.Esnoc _ ?x ?P ] => tac x P
  end.

(** * Start a proof *)
Tactic Notation "iStartProof" :=
  lazymatch goal with
  | |- envs_entails _ _ => idtac
  | |- ?φ => notypeclasses refine (as_emp_valid_2 φ _ _);
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               [iSolveTC || fail "iStartProof: not a BI assertion"
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               |apply tac_adequate]
  end.

(* Same as above, with 2 differences :
   - We can specify a BI in which we want the proof to be done
   - If the goal starts with a let or a ∀, they are automatically
     introduced. *)
Tactic Notation "iStartProof" uconstr(PROP) :=
  lazymatch goal with
  | |- @envs_entails ?PROP' _ _ =>
    (* This cannot be shared with the other [iStartProof], because
    type_term has a non-negligeable performance impact. *)
    let x := type_term (eq_refl : @eq Type PROP PROP') in idtac

  (* We eta-expand [as_emp_valid_2], in order to make sure that
     [iStartProof PROP] works even if [PROP] is the carrier type. In
     this case, typing this expression will end up unifying PROP with
     [bi_car _], and hence trigger the canonical structures mechanism
     to find the corresponding bi. *)
  | |- ?φ => notypeclasses refine ((λ P : PROP, @as_emp_valid_2 φ _ P) _ _ _);
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               [iSolveTC || fail "iStartProof: not a BI assertion"
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               |apply tac_adequate]
  end.

(** * Generate a fresh identifier *)
(* Tactic Notation tactics cannot return terms *)
Ltac iFresh :=
  (* We need to increment the environment counter using [tac_fresh].
     But because [iFresh] returns a value, we have to let bind
     [tac_fresh] wrapped under a match to force evaluation of this
     side-effect. See https://stackoverflow.com/a/46178884 *)
  let do_incr :=
      lazymatch goal with
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      | _ => iStartProof; eapply tac_fresh; first by (pm_reflexivity)
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      end in
  lazymatch goal with
  |- envs_entails ?Δ _ =>
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    let n := pm_eval (env_counter Δ) in
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    constr:(IAnon n)
  end.

(** * Simplification *)
Tactic Notation "iEval" tactic(t) :=
  iStartProof;
  eapply tac_eval;
    [let x := fresh in intros x; t; unfold x; reflexivity
    |].

Tactic Notation "iEval" tactic(t) "in" constr(H) :=
  iStartProof;
  eapply tac_eval_in with _ H _ _ _;
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    [pm_reflexivity || fail "iEval:" H "not found"
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    |let x := fresh in intros x; t; unfold x; reflexivity
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    |pm_reflexivity
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    |].

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Tactic Notation "iSimpl" := iEval (simpl).
Tactic Notation "iSimpl" "in" constr(H) := iEval (simpl) in H.
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(* It would be nice to also have an `iSsrRewrite`, however, for this we need to
pass arguments to Ssreflect's `rewrite` like `/= foo /bar` in Ltac, see:

  https://sympa.inria.fr/sympa/arc/coq-club/2018-01/msg00000.html

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PMP told me (= Robbert) in person that this is not possible with the current
Ltac, but it may be possible in Ltac2. *)
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(** * Context manipulation *)
Tactic Notation "iRename" constr(H1) "into" constr(H2) :=
  eapply tac_rename with _ H1 H2 _ _; (* (i:=H1) (j:=H2) *)
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    [pm_reflexivity ||
     let H1 := pretty_ident H1 in
     fail "iRename:" H1 "not found"
    |pm_reflexivity ||
     let H2 := pretty_ident H2 in
     fail "iRename:" H2 "not fresh"|].
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Local Inductive esel_pat :=
  | ESelPure
  | ESelIdent : bool  ident  esel_pat.

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Local Ltac iElaborateSelPat_go pat Δ Hs :=
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  lazymatch pat with
  | [] => eval cbv in Hs
  | SelPure :: ?pat =>  iElaborateSelPat_go pat Δ (ESelPure :: Hs)
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  | SelIntuitionistic :: ?pat =>
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    let Hs' := pm_eval (env_dom (env_intuitionistic Δ)) in
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    let Δ' := pm_eval (envs_clear_intuitionistic Δ) in
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    iElaborateSelPat_go pat Δ' ((ESelIdent true <$> Hs') ++ Hs)
  | SelSpatial :: ?pat =>
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    let Hs' := pm_eval (env_dom (env_spatial Δ)) in
    let Δ' := pm_eval (envs_clear_spatial Δ) in
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    iElaborateSelPat_go pat Δ' ((ESelIdent false <$> Hs') ++ Hs)
  | SelIdent ?H :: ?pat =>
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    lazymatch pm_eval (envs_lookup_delete false H Δ) with
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    | Some (?p,_,?Δ') =>  iElaborateSelPat_go pat Δ' (ESelIdent p H :: Hs)
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    | None =>
      let H := pretty_ident H in
      fail "iElaborateSelPat:" H "not found"
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    end
  end.
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Ltac iElaborateSelPat pat :=
  lazymatch goal with
  | |- envs_entails ?Δ _ =>
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    let pat := sel_pat.parse pat in iElaborateSelPat_go pat Δ (@nil esel_pat)
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  end.

Local Ltac iClearHyp H :=
  eapply tac_clear with _ H _ _; (* (i:=H) *)
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    [pm_reflexivity ||
     let H := pretty_ident H in
     fail "iClear:" H "not found"
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    |pm_reduce; iSolveTC ||
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     let H := pretty_ident H in
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     let P := match goal with |- TCOr (Affine ?P) _ => P end in
     fail "iClear:" H ":" P "not affine and the goal not absorbing"
    |].

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Local Ltac iClear_go Hs :=
  lazymatch Hs with
  | [] => idtac
  | ESelPure :: ?Hs => clear; iClear_go Hs
  | ESelIdent _ ?H :: ?Hs => iClearHyp H; iClear_go Hs
  end.
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Tactic Notation "iClear" constr(Hs) :=
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  iStartProof; let Hs := iElaborateSelPat Hs in iClear_go Hs.
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Tactic Notation "iClear" "(" ident_list(xs) ")" constr(Hs) :=
  iClear Hs; clear xs.

(** * Assumptions *)
Tactic Notation "iExact" constr(H) :=
  eapply tac_assumption with _ H _ _; (* (i:=H) *)
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    [pm_reflexivity ||
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     let H := pretty_ident H in
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     fail "iExact:" H "not found"
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    |iSolveTC ||
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     let H := pretty_ident H in
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     let P := match goal with |- FromAssumption _ ?P _ => P end in
     fail "iExact:" H ":" P "does not match goal"
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    |pm_reduce; iSolveTC ||
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     let H := pretty_ident H in
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     fail "iExact:" H "not absorbing and the remaining hypotheses not affine"].

Tactic Notation "iAssumptionCore" :=
  let rec find Γ i P :=
    lazymatch Γ with
    | Esnoc ?Γ ?j ?Q => first [unify P Q; unify i j|find Γ i P]
    end in
  match goal with
  | |- envs_lookup ?i (Envs ?Γp ?Γs _) = Some (_, ?P) =>
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     first [is_evar i; fail 1 | pm_reflexivity]
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  | |- envs_lookup ?i (Envs ?Γp ?Γs _) = Some (_, ?P) =>
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     is_evar i; first [find Γp i P | find Γs i P]; pm_reflexivity
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  | |- envs_lookup_delete _ ?i (Envs ?Γp ?Γs _) = Some (_, ?P, _) =>
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     first [is_evar i; fail 1 | pm_reflexivity]
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  | |- envs_lookup_delete _ ?i (Envs ?Γp ?Γs _) = Some (_, ?P, _) =>
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     is_evar i; first [find Γp i P | find Γs i P]; pm_reflexivity
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  end.

Tactic Notation "iAssumption" :=
  let Hass := fresh in
  let rec find p Γ Q :=
    lazymatch Γ with
    | Esnoc ?Γ ?j ?P => first
       [pose proof (_ : FromAssumption p P Q) as Hass;
        eapply (tac_assumption _ _ j p P);
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          [pm_reflexivity
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          |apply Hass
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          |pm_reduce; iSolveTC ||
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           fail 1 "iAssumption:" j "not absorbing and the remaining hypotheses not affine"]
       |assert (P = False%I) as Hass by reflexivity;
        apply (tac_false_destruct _ j p P);
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          [pm_reflexivity
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          |exact Hass]
       |find p Γ Q]
    end in
  lazymatch goal with
  | |- envs_entails (Envs ?Γp ?Γs _) ?Q =>
     first [find true Γp Q | find false Γs Q
           |fail "iAssumption:" Q "not found"]
  end.

(** * False *)
Tactic Notation "iExFalso" := apply tac_ex_falso.

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(** * Making hypotheses intuitionistic or pure *)
Local Tactic Notation "iIntuitionistic" constr(H) :=
  eapply tac_intuitionistic with _ H _ _ _; (* (i:=H) *)
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    [pm_reflexivity ||
     let H := pretty_ident H in
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     fail "iIntuitionistic:" H "not found"
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    |iSolveTC ||
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     let P := match goal with |- IntoPersistent _ ?P _ => P end in
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     fail "iIntuitionistic:" P "not persistent"
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    |pm_reduce; iSolveTC ||
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     let P := match goal with |- TCOr (Affine ?P) _ => P end in
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     fail "iIntuitionistic:" P "not affine and the goal not absorbing"
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    |pm_reflexivity|].
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Local Tactic Notation "iPure" constr(H) "as" simple_intropattern(pat) :=
  eapply tac_pure with _ H _ _ _; (* (i:=H1) *)
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    [pm_reflexivity ||
     let H := pretty_ident H in
     fail "iPure:" H "not found"
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    |iSolveTC ||
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     let P := match goal with |- IntoPure ?P _ => P end in
     fail "iPure:" P "not pure"
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    |pm_reduce; iSolveTC ||
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     let P := match goal with |- TCOr (Affine ?P) _ => P end in
     fail "iPure:" P "not affine and the goal not absorbing"
    |intros pat].

Tactic Notation "iEmpIntro" :=
  iStartProof;
  eapply tac_emp_intro;
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    [pm_reduce; iSolveTC ||
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     fail "iEmpIntro: spatial context contains non-affine hypotheses"].

Tactic Notation "iPureIntro" :=
  iStartProof;
  eapply tac_pure_intro;
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    [pm_reflexivity
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    |iSolveTC ||
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     let P := match goal with |- FromPure _ ?P _ => P end in
     fail "iPureIntro:" P "not pure"
    |].

(** Framing *)
Local Ltac iFrameFinish :=
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  pm_prettify;
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  try match goal with
  | |- envs_entails _ True => by iPureIntro
  | |- envs_entails _ emp => iEmpIntro
  end.

Local Ltac iFramePure t :=
  iStartProof;
  let φ := type of t in
  eapply (tac_frame_pure _ _ _ _ t);
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    [iSolveTC || fail "iFrame: cannot frame" φ
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    |iFrameFinish].

Local Ltac iFrameHyp H :=
  iStartProof;
  eapply tac_frame with _ H _ _ _;
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    [pm_reflexivity ||
     let H := pretty_ident H in
     fail "iFrame:" H "not found"
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    |iSolveTC ||
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     let R := match goal with |- Frame _ ?R _ _ => R end in
     fail "iFrame: cannot frame" R
    |iFrameFinish].

Local Ltac iFrameAnyPure :=
  repeat match goal with H : _ |- _ => iFramePure H end.

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Local Ltac iFrameAnyIntuitionistic :=
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  iStartProof;
  let rec go Hs :=
    match Hs with [] => idtac | ?H :: ?Hs => repeat iFrameHyp H; go Hs end in
  match goal with
  | |- envs_entails ?Δ _ =>
     let Hs := eval cbv in (env_dom (env_intuitionistic Δ)) in go Hs
  end.

Local Ltac iFrameAnySpatial :=
  iStartProof;
  let rec go Hs :=
    match Hs with [] => idtac | ?H :: ?Hs => try iFrameHyp H; go Hs end in
  match goal with
  | |- envs_entails ?Δ _ =>
     let Hs := eval cbv in (env_dom (env_spatial Δ)) in go Hs
  end.

Tactic Notation "iFrame" := iFrameAnySpatial.

Tactic Notation "iFrame" "(" constr(t1) ")" :=
  iFramePure t1.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) ")" :=
  iFramePure t1; iFrame ( t2 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) ")" :=
  iFramePure t1; iFrame ( t2 t3 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4) ")" :=
  iFramePure t1; iFrame ( t2 t3 t4 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) ")" :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) ")" :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) constr(t7) ")" :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 t7 ).
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) constr(t7) constr(t8)")" :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 t7 t8 ).

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Local Ltac iFrame_go Hs :=
  lazymatch Hs with
  | [] => idtac
  | SelPure :: ?Hs => iFrameAnyPure; iFrame_go Hs
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  | SelIntuitionistic :: ?Hs => iFrameAnyIntuitionistic; iFrame_go Hs
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  | SelSpatial :: ?Hs => iFrameAnySpatial; iFrame_go Hs
  | SelIdent ?H :: ?Hs => iFrameHyp H; iFrame_go Hs
  end.

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Tactic Notation "iFrame" constr(Hs) :=
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  let Hs := sel_pat.parse Hs in iFrame_go Hs.
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Tactic Notation "iFrame" "(" constr(t1) ")" constr(Hs) :=
  iFramePure t1; iFrame Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) ")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) ")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4) ")"
    constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 t4 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) ")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) ")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) constr(t7) ")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 t7 ) Hs.
Tactic Notation "iFrame" "(" constr(t1) constr(t2) constr(t3) constr(t4)
    constr(t5) constr(t6) constr(t7) constr(t8)")" constr(Hs) :=
  iFramePure t1; iFrame ( t2 t3 t4 t5 t6 t7 t8 ) Hs.

(** * Basic introduction tactics *)
Local Tactic Notation "iIntro" "(" simple_intropattern(x) ")" :=
  (* In the case the goal starts with an [let x := _ in _], we do not
     want to unfold x and start the proof mode. Instead, we want to
     use intros. So [iStartProof] has to be called only if [intros]
     fails *)
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  (* We use [_ || _] instead of [first [..|..]] so that the error in the second
  branch propagates upwards. *)
  (
    (* introduction at the meta level *)
    intros x
  ) || (
    (* introduction in the logic *)
    iStartProof;
    lazymatch goal with
    | |- envs_entails _ _ =>
      eapply tac_forall_intro;
        [iSolveTC ||
         let P := match goal with |- FromForall ?P _ => P end in
         fail "iIntro: cannot turn" P "into a universal quantifier"
        |pm_prettify; intros x
         (* subgoal *)]
    end).
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Local Tactic Notation "iIntro" constr(H) :=
  iStartProof;
  first
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  [(* (?Q → _) *)
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    eapply tac_impl_intro with _ H _ _ _; (* (i:=H) *)
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      [iSolveTC
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      |pm_reduce; iSolveTC ||
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       let P := lazymatch goal with |- Persistent ?P => P end in
       fail 1 "iIntro: introducing non-persistent" H ":" P
              "into non-empty spatial context"
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      |pm_reflexivity ||
       let H := pretty_ident H in
       fail 1 "iIntro:" H "not fresh"
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      |iSolveTC
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      |(* subgoal *)]
  |(* (_ -∗ _) *)
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    eapply tac_wand_intro with _ H _ _; (* (i:=H) *)
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      [iSolveTC
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      | pm_reflexivity ||
        let H := pretty_ident H in
        fail 1 "iIntro:" H "not fresh"
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      |(* subgoal *)]
  | fail 1 "iIntro: nothing to introduce" ].
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Local Tactic Notation "iIntro" "#" constr(H) :=
  iStartProof;
  first
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  [(* (?P → _) *)
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   eapply tac_impl_intro_intuitionistic with _ H _ _ _; (* (i:=H) *)
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     [iSolveTC
     |iSolveTC ||
      let P := match goal with |- IntoPersistent _ ?P _ => P end in
      fail 1 "iIntro:" P "not persistent"
     |pm_reflexivity ||
      let H := pretty_ident H in
      fail 1 "iIntro:" H "not fresh"
     |(* subgoal *)]
  |(* (?P -∗ _) *)
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   eapply tac_wand_intro_intuitionistic with _ H _ _ _; (* (i:=H) *)
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     [iSolveTC
     |iSolveTC ||
      let P := match goal with |- IntoPersistent _ ?P _ => P end in
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      fail 1 "iIntro:" P "not intuitionistic"
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     |iSolveTC ||
      let P := match goal with |- TCOr (Affine ?P) _ => P end in
      fail 1 "iIntro:" P "not affine and the goal not absorbing"
     |pm_reflexivity ||
      let H := pretty_ident H in
      fail 1 "iIntro:" H "not fresh"
     |(* subgoal *)]
  |fail 1 "iIntro: nothing to introduce"].
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Local Tactic Notation "iIntro" "_" :=
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  iStartProof;
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  first
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  [(* (?Q → _) *)
   eapply tac_impl_intro_drop;
     [iSolveTC
     |(* subgoal *)]
  |(* (_ -∗ _) *)
   eapply tac_wand_intro_drop;
     [iSolveTC
     |iSolveTC ||
      let P := match goal with |- TCOr (Affine ?P) _ => P end in
      fail 1 "iIntro:" P "not affine and the goal not absorbing"
     |(* subgoal *)]
  |(* (∀ _, _) *)
   iIntro (_)
   (* subgoal *)
  |fail 1 "iIntro: nothing to introduce"].
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Local Tactic Notation "iIntroForall" :=
  lazymatch goal with
  | |-  _, ?P => fail (* actually an →, this is handled by iIntro below *)
  | |-  _, _ => intro
  | |- let _ := _ in _ => intro
  | |- _ =>
    iStartProof;
    lazymatch goal with
    | |- envs_entails _ ( x : _, _) => let x' := fresh x in iIntro (x')
    end
  end.
Local Tactic Notation "iIntro" :=
  lazymatch goal with
  | |- _  ?P => intro
  | |- _ =>
    iStartProof;
    lazymatch goal with
    | |- envs_entails _ (_ - _) => iIntro (?) || let H := iFresh in iIntro #H || iIntro H
    | |- envs_entails _ (_  _) => iIntro (?) || let H := iFresh in iIntro #H || iIntro H
    end
  end.

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(** * Revert *)
Local Tactic Notation "iForallRevert" ident(x) :=
  let err x :=
    intros x;
    iMatchHyp (fun H P =>
      lazymatch P with
      | context [x] => fail 2 "iRevert:" x "is used in hypothesis" H
      end) in
  iStartProof;
  let A := type of x in
  lazymatch type of A with
  | Prop => revert x; first [apply tac_pure_revert|err x]
  | _ => revert x; first [apply tac_forall_revert|err x]
  end.

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(** The tactic [iRevertHyp H with tac] reverts the hypothesis [H] and calls
[tac] with a Boolean that is [true] iff [H] was in the intuitionistic context. *)
Tactic Notation "iRevertHyp" constr(H) "with" tactic1(tac) :=
  (* Create a Boolean evar [p] to keep track of whether the hypothesis [H] was
  in the intuitionistic context. *)
  let p := fresh in evar (p : bool);
  let p' := eval unfold p in p in clear p;
  eapply tac_revert with _ H p' _;
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    [pm_reflexivity ||
     let H := pretty_ident H in
     fail "iRevert:" H "not found"
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    |pm_reduce; tac p'].

Tactic Notation "iRevertHyp" constr(H) := iRevertHyp H with (fun _ => idtac).
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Tactic Notation "iRevert" constr(Hs) :=
  let rec go Hs :=
    lazymatch Hs with
    | [] => idtac
    | ESelPure :: ?Hs =>
       repeat match goal with x : _ |- _ => revert x end;
       go Hs
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    | ESelIdent _ ?H :: ?Hs => iRevertHyp H; go Hs
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    end in
  iStartProof; let Hs := iElaborateSelPat Hs in go Hs.

Tactic Notation "iRevert" "(" ident(x1) ")" :=
  iForallRevert x1.
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ")" :=
  iForallRevert x2; iRevert ( x1 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ")" :=
  iForallRevert x3; iRevert ( x1 x2 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4) ")" :=
  iForallRevert x4; iRevert ( x1 x2 x3 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ")" :=
  iForallRevert x5; iRevert ( x1 x2 x3 x4 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ")" :=
  iForallRevert x6; iRevert ( x1 x2 x3 x4 x5 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ident(x7) ")" :=
  iForallRevert x7; iRevert ( x1 x2 x3 x4 x5 x6 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ident(x7) ident(x8) ")" :=
  iForallRevert x8; iRevert ( x1 x2 x3 x4 x5 x6 x7 ).

Tactic Notation "iRevert" "(" ident(x1) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4) ")"
    constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 x4 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 x4 x5 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 x4 x5 x6 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ident(x7) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 x4 x5 x6 x7 ).
Tactic Notation "iRevert" "(" ident(x1) ident(x2) ident(x3) ident(x4)
    ident(x5) ident(x6) ident(x7) ident(x8) ")" constr(Hs) :=
  iRevert Hs; iRevert ( x1 x2 x3 x4 x5 x6 x7 x8 ).

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(** * The specialize and pose proof tactics *)
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Record iTrm {X As S} :=
  ITrm { itrm : X ; itrm_vars : hlist As ; itrm_hyps : S }.
Arguments ITrm {_ _ _} _ _ _.

Notation "( H $! x1 .. xn )" :=
  (ITrm H (hcons x1 .. (hcons xn hnil) ..) "") (at level 0, x1, xn at level 9).
Notation "( H $! x1 .. xn 'with' pat )" :=
  (ITrm H (hcons x1 .. (hcons xn hnil) ..) pat) (at level 0, x1, xn at level 9).
Notation "( H 'with' pat )" := (ITrm H hnil pat) (at level 0).

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(* The tactic [iIntoEmpValid] tactic solves a goal [bi_emp_valid Q]. The
argument [t] must be a Coq term whose type is of the following shape:

[∀ (x_1 : A_1) .. (x_n : A_n), φ]

and so that we have an instance `AsValid φ Q`.

Examples of such [φ]s are

- [bi_emp_valid P], in which case [Q] should be [P]
- [P1 ⊢ P2], in which case [Q] should be [P1 -∗ P2]
- [P1 ⊣⊢ P2], in which case [Q] should be [P1 ↔ P2]

The tactic instantiates each dependent argument [x_i] with an evar and generates
a goal [R] for each non-dependent argument [x_i : R].  For example, if the
original goal was [Q] and [t] has type [∀ x, P x → Q], then it generates an evar
[?x] for [x] and a subgoal [P ?x]. *)
Local Ltac iIntoEmpValid t :=
  let go_specialize t tT :=
    lazymatch tT with                (* We do not use hnf of tT, because, if
                                        entailment is not opaque, then it would
                                        unfold it. *)
    | ?P  ?Q => let H := fresh in assert P as H; [|iIntoEmpValid uconstr:(t H); clear H]
    |  _ : ?T, _ =>
      (* Put [T] inside an [id] to avoid TC inference from being invoked. *)
      (* This is a workarround for Coq bug #6583. *)
      let e := fresh in evar (e:id T);
      let e' := eval unfold e in e in clear e; iIntoEmpValid (t e')
    end
  in
    (* We try two reduction tactics for the type of t before trying to
       specialize it. We first try the head normal form in order to
       unfold all the definition that could hide an entailment.  Then,
       we try the much weaker [eval cbv zeta], because entailment is
       not necessarilly opaque, and could be unfolded by [hnf].

       However, for calling type class search, we only use [cbv zeta]
       in order to make sure we do not unfold [bi_emp_valid]. *)
    let tT := type of t in
    first
      [ let tT' := eval hnf in tT in go_specialize t tT'
      | let tT' := eval cbv zeta in tT in go_specialize t tT'
      | let tT' := eval cbv zeta in tT in
        notypeclasses refine (as_emp_valid_1 tT _ _);
          [iSolveTC || fail 1 "iPoseProof: not a BI assertion"
          |exact t]].

Tactic Notation "iPoseProofCoreHyp" constr(H) "as" constr(Hnew) :=
  eapply tac_pose_proof_hyp with _ _ H _ Hnew _;
    [pm_reflexivity ||
     let H := pretty_ident H in
     fail "iPoseProof:" H "not found"
    |pm_reflexivity ||
     let Htmp := pretty_ident Hnew in
     fail "iPoseProof:" Hnew "not fresh"
    |].

Tactic Notation "iPoseProofCoreLem"
    constr(lem) "as" constr(Hnew) "before_tc" tactic(tac) :=
  eapply tac_pose_proof with _ Hnew _; (* (j:=H) *)
    [iIntoEmpValid lem
    |pm_reflexivity ||
     let Htmp := pretty_ident Hnew in
     fail "iPoseProof:" Hnew "not fresh"
    |tac];
  (* Solve all remaining TC premises generated by [iIntoEmpValid] *)
  try iSolveTC.

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(** There is some hacky stuff going on here: because of Coq bug #6583, unresolved
type classes in the arguments `xs` are resolved at arbitrary moments. Tactics
like `apply`, `split` and `eexists` wrongly trigger type class search to resolve
these holes. To avoid TC being triggered too eagerly, this tactic uses `refine`
at most places instead of `apply`. *)
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Local Ltac iSpecializeArgs_go H xs :=
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  lazymatch xs with
  | hnil => idtac
  | hcons ?x ?xs =>
     notypeclasses refine (tac_forall_specialize _ _ H _ _ _ _ _ _ _);
       [pm_reflexivity ||
        let H := pretty_ident H in
        fail "iSpecialize:" H "not found"
       |iSolveTC ||
        let P := match goal with |- IntoForall ?P _ => P end in
        fail "iSpecialize: cannot instantiate" P "with" x
       |lazymatch goal with (* Force [A] in [ex_intro] to deal with coercions. *)
        | |-  _ : ?A, _ =>
          notypeclasses refine (@ex_intro A _ x (conj _ _))
        end; [shelve..|pm_reflexivity|iSpecializeArgs_go H xs]]
  end.
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Local Tactic Notation "iSpecializeArgs" constr(H) open_constr(xs) :=
  iSpecializeArgs_go H xs.
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Ltac iSpecializePat_go H1 pats :=
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  let solve_to_wand H1 :=
    iSolveTC ||
    let P := match goal with |- IntoWand _ _ ?P _ _ => P end in
    fail "iSpecialize:" P "not an implication/wand" in
  let solve_done d :=
    lazymatch d with
    | true =>
       done ||
       let Q := match goal with |- envs_entails _ ?Q => Q end in
       fail "iSpecialize: cannot solve" Q "using done"
    | false => idtac
    end in
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  lazymatch pats with
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    | [] => idtac
    | SForall :: ?pats =>
       idtac "[IPM] The * specialization pattern is deprecated because it is applied implicitly.";
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       iSpecializePat_go H1 pats
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    | SIdent ?H2 [] :: ?pats =>
       (* If we not need to specialize [H2] we can avoid a lot of unncessary
       context manipulation. *)
       notypeclasses refine (tac_specialize false _ _ _ H2 _ H1 _ _ _ _ _ _ _ _ _ _);
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         [pm_reflexivity ||
          let H2 := pretty_ident H2 in
          fail "iSpecialize:" H2 "not found"
         |pm_reflexivity ||
          let H1 := pretty_ident H1 in
          fail "iSpecialize:" H1 "not found"
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         |iSolveTC ||
          let P := match goal with |- IntoWand _ _ ?P ?Q _ => P end in
          let Q := match goal with |- IntoWand _ _ ?P ?Q _ => Q end in
          fail "iSpecialize: cannot instantiate" P "with" Q
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         |pm_reflexivity|iSpecializePat_go H1 pats]
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    | SIdent ?H2 ?pats1 :: ?pats =>
       (* If [H2] is in the intuitionistic context, we copy it into a new
       hypothesis [Htmp], so that it can be used multiple times. *)
       let H2tmp := iFresh in
       iPoseProofCoreHyp H2 as H2tmp;
       (* Revert [H1] and re-introduce it later so that it will not be consumsed
       by [pats1]. *)
       iRevertHyp H1 with (fun p =>
         iSpecializePat_go H2tmp pats1;
           [.. (* side-conditions of [iSpecialize] *)
           |lazymatch p with
            | true => iIntro #H1
            | _ => iIntro H1
            end]);
         (* We put the stuff below outside of the closure to get less verbose
         Ltac backtraces (which would otherwise include the whole closure). *)
         [.. (* side-conditions of [iSpecialize] *)
         |(* Use [remove_intuitionistic = true] to remove the copy [Htmp]. *)
          notypeclasses refine (tac_specialize true _ _ _ H2tmp _ H1 _ _ _ _ _ _ _ _ _ _);
            [pm_reflexivity ||
             let H2tmp := pretty_ident H2tmp in
             fail "iSpecialize:" H2tmp "not found"
            |pm_reflexivity ||
             let H1 := pretty_ident H1 in
             fail "iSpecialize:" H1 "not found"
            |iSolveTC ||
             let P := match goal with |- IntoWand _ _ ?P ?Q _ => P end in
             let Q := match goal with |- IntoWand _ _ ?P ?Q _ => Q end in
             fail "iSpecialize: cannot instantiate" P "with" Q
            |pm_reflexivity|iSpecializePat_go H1 pats]]
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    | SPureGoal ?d :: ?pats =>
       notypeclasses refine (tac_specialize_assert_pure _ _ H1 _ _ _ _ _ _ _ _ _ _ _ _);
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         [pm_reflexivity ||
          let H1 := pretty_ident H1 in
          fail "iSpecialize:" H1 "not found"
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         |solve_to_wand H1
         |iSolveTC ||
          let Q := match goal with |- FromPure _ ?Q _ => Q end in
          fail "iSpecialize:" Q "not pure"
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         |pm_reflexivity
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         |solve_done d (*goal*)
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         |iSpecializePat_go H1 pats]
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    | SGoal (SpecGoal GIntuitionistic false ?Hs_frame [] ?d) :: ?pats =>
       notypeclasses refine (tac_specialize_assert_intuitionistic _ _ _ H1 _ _ _ _ _ _ _ _ _ _ _ _ _);
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         [pm_reflexivity ||
          let H1 := pretty_ident H1 in
          fail "iSpecialize:" H1 "not found"
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         |solve_to_wand H1
         |iSolveTC ||
          let Q := match goal with |- Persistent ?Q => Q end in
          fail "iSpecialize:" Q "not persistent"
         |iSolveTC
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         |pm_reflexivity
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         |iFrame Hs_frame; solve_done d (*goal*)
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         |iSpecializePat_go H1 pats]
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    | SGoal (SpecGoal GIntuitionistic _ _ _ _) :: ?pats =>
       fail "iSpecialize: cannot select hypotheses for intuitionistic premise"
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    | SGoal (SpecGoal ?m ?lr ?Hs_frame ?Hs ?d) :: ?pats =>
       let Hs' := eval cbv in (if lr then Hs else Hs_frame ++ Hs) in
       notypeclasses refine (tac_specialize_assert _ _ _ _ H1 _ lr Hs' _ _ _ _ _ _ _ _ _ _ _);
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         [pm_reflexivity ||
          let H1 := pretty_ident H1 in
          fail "iSpecialize:" H1 "not found"
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         |solve_to_wand H1
         |lazymatch m with
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          | GSpatial => class_apply add_modal_id
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          | GModal => iSolveTC || fail "iSpecialize: goal not a modality"
          end
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         |pm_reflexivity ||
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          let Hs' := iMissingHyps Hs' in
          fail "iSpecialize: hypotheses" Hs' "not found"
         |iFrame Hs_frame; solve_done d (*goal*)
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         |iSpecializePat_go H1 pats]
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    | SAutoFrame GIntuitionistic :: ?pats =>
       notypeclasses refine (tac_specialize_assert_intuitionistic _ _ _ H1 _ _ _ _ _ _ _ _ _ _ _ _ _);
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         [pm_reflexivity ||
          let H1 := pretty_ident H1 in
          fail "iSpecialize:" H1 "not found"
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         |solve_to_wand H1
         |iSolveTC ||
          let Q := match goal with |- Persistent ?Q => Q end in
          fail "iSpecialize:" Q "not persistent"
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         |pm_reflexivity
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         |solve [iFrame "∗ #"]
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         |iSpecializePat_go H1 pats]
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    | SAutoFrame ?m :: ?pats =>
       notypeclasses refine (tac_specialize_frame _ _ H1 _ _ _ _ _ _ _ _ _ _ _ _);
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         [pm_reflexivity ||
          let H1 := pretty_ident H1 in
          fail "iSpecialize:" H1 "not found"
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         |solve_to_wand H1
         |lazymatch m with
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          | GSpatial => class_apply add_modal_id
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          | GModal => iSolveTC || fail "iSpecialize: goal not a modality"
          end
         |first
            [notypeclasses refine (tac_unlock_emp _ _ _)
            |notypeclasses refine (tac_unlock_True _ _ _)
            |iFrame "∗ #"; notypeclasses refine (tac_unlock _ _ _)
            |fail "iSpecialize: premise cannot be solved by framing"]
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         |exact eq_refl]; iIntro H1; iSpecializePat_go H1 pats
    end.

Local Tactic Notation "iSpecializePat" open_constr(H) constr(pat) :=
  let pats := spec_pat.parse pat in iSpecializePat_go H pats.
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(* The argument [p] denotes whether the conclusion of the specialized term is
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intuitionistic. If so, one can use all spatial hypotheses for both proving the
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premises and the remaning goal. The argument [p] can either be a Boolean or an
introduction pattern, which will be coerced into [true] when it solely contains
`#` or `%` patterns at the top-level.

In case the specialization pattern in [t] states that the modality of the goal
should be kept for one of the premises (i.e. [>[H1 .. Hn]] is used) then [p]
defaults to [false] (i.e. spatial hypotheses are not preserved). *)
Tactic Notation "iSpecializeCore" open_constr(H)
    "with" open_constr(xs) open_constr(pat) "as" constr(p) :=
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  let p := intro_pat_intuitionistic p in
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  let pat := spec_pat.parse pat in
  let H :=
    lazymatch type of H with
    | string => constr:(INamed H)
    | _ => H
    end in
  iSpecializeArgs H xs; [..|
  lazymatch type of H with
  | ident =>
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    (* The lemma [tac_specialize_intuitionistic_helper] allows one to use all
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    spatial hypotheses for both proving the premises of the lemma we
    specialize as well as those of the remaining goal. We can only use it when
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    the result of the specialization is intuitionistic, and no modality is
    eliminated. We do not use [tac_specialize_intuitionistic_helper] in the case
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    only universal quantifiers and no implications or wands are instantiated
    (i.e [pat = []]) because it is a.) not needed, and b.) more efficient. *)
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    let pat := spec_pat.parse pat in
    lazymatch eval compute in
      (p && bool_decide (pat  []) && negb (existsb spec_pat_modal pat)) with
    | true =>
       (* FIXME: do something reasonable when the BI is not affine *)
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       notypeclasses refine (tac_specialize_intuitionistic_helper _ _ H _ _ _ _ _ _ _ _ _ _ _);
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         [pm_reflexivity ||
          let H := pretty_ident H in
          fail "iSpecialize:" H "not found"
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         |iSpecializePat H pat;
           [..
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           |notypeclasses refine (tac_specialize_intuitionistic_helper_done _ H _ _ _);
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            pm_reflexivity]
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         |iSolveTC ||
          let Q := match goal with |- IntoPersistent _ ?Q _ => Q end in
          fail "iSpecialize:" Q "not persistent"
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         |pm_reduce; iSolveTC ||
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          let Q := match goal with |- TCAnd _ (Affine ?Q) => Q end in
          fail "iSpecialize:" Q "not affine"
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         |pm_reflexivity
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         |(* goal *)]
    | false => iSpecializePat H pat
    end
  | _ => fail "iSpecialize:" H "should be a hypothesis, use iPoseProof instead"
  end].

Tactic Notation "iSpecializeCore" open_constr(t) "as" constr(p) :=
  lazymatch type of t with
  | string => iSpecializeCore t with hnil "" as p
  | ident => iSpecializeCore t with hnil "" as p
  | _ =>
    lazymatch t with
    | ITrm ?H ?xs ?pat => iSpecializeCore H with xs pat as p
    | _ => fail "iSpecialize:" t "should be a proof mode term"
    end
  end.

Tactic Notation "iSpecialize" open_constr(t) :=
  iSpecializeCore t as false.
Tactic Notation "iSpecialize" open_constr(t) "as" "#" :=
  iSpecializeCore t as true.

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(** The tactic [iPoseProofCore lem as p lazy_tc tac] inserts the resource
described by [lem] into the context. The tactic takes a continuation [tac] as
its argument, which is called with a temporary fresh name [H] that refers to
the hypothesis containing [lem].

There are a couple of additional arguments:

- The argument [p] is like that of [iSpecialize]. It is a Boolean that denotes
  whether the conclusion of the specialized term [lem] is persistent.
- The argument [lazy_tc] denotes whether type class inference on the premises
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  of [lem] should be performed before (if [lazy_tc = false]) or after (if
  [lazy_tc = true]) [tac H] is called.
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Both variants of [lazy_tc] are used in other tactics that build on top of
[iPoseProofCore]:

- The tactic [iApply] uses lazy type class inference (i.e. [lazy_tc = true]),
  so that evars can first be matched against the goal before being solved by
  type class inference.
- The tactic [iDestruct] uses eager type class inference (i.e. [lazy_tc = false])
  because it may be not possible to eliminate logical connectives before all
  type class constraints have been resolved. *)
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Tactic Notation "iPoseProofCore" open_constr(lem)
    "as" constr(p) constr(lazy_tc) tactic(tac) :=
  iStartProof;
  let Htmp := iFresh in
  let t := lazymatch lem with ITrm ?t ?xs ?pat => t | _ => lem end in
  let t := lazymatch type of t with string => constr:(INamed t) | _ => t end in
  let spec_tac _ :=
    lazymatch lem with
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    | ITrm _ ?xs ?pat => iSpecializeCore (ITrm Htmp xs pat) as p
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    | _ => idtac
    end in
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  lazymatch type of t with
  | ident => iPoseProofCoreHyp t as Htmp; spec_tac (); [..|tac Htmp]
  | _ =>
     lazymatch eval compute in lazy_tc with
     | true =>
        iPoseProofCoreLem t as Htmp before_tc (spec_tac (); [..|tac Htmp])
     | false =>
        iPoseProofCoreLem t as Htmp before_tc (spec_tac ()); [..|tac Htmp]
     end
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  end.

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(** * The apply tactic *)
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(** [iApply lem] takes an argument [lem : P₁ -∗ .. -∗ Pₙ -∗ Q] (after the
specialization patterns in [lem] have been executed), where [Q] should match
the goal, and generates new goals [P1] ... [Pₙ]. Depending on the number of
premises [n], the tactic will have the following behavior:

- If [n = 0], it will immediately solve the goal (i.e. it will not generate any
  subgoals). When working in a general BI, this means that the tactic can fail
  in case there are non-affine spatial hypotheses in the context prior to using
  the [iApply] tactic. Note that if [n = 0], the tactic behaves exactly like
  [iExact lem].
- If [n > 0] it will generate a goals [P₁] ... [Pₙ]. All spatial hypotheses
  will be transferred to the last goal, i.e. [Pₙ]; the other goals will receive
  no spatial hypotheses. If you want to control more precisely how the spatial
  hypotheses are subdivided, you should add additional introduction patterns to
  [lem]. *)

(* The helper [iApplyHypExact] takes care of the [n=0] case. It fails with level
0 if we should proceed to the [n > 0] case, and with level 1 if there is an
actual error. *)
Local Ltac iApplyHypExact H :=
  first
    [eapply tac_assumption with _ H _ _; (* (i:=H) *)
       [pm_reflexivity || fail 1
       |iSolveTC || fail 1
       |pm_reduce; iSolveTC]
    |lazymatch iTypeOf H with
     | Some (_,?Q) =>
        fail 2 "iApply:" Q "not absorbing and the remaining hypotheses not affine"
     end].
Local Ltac iApplyHypLoop H :=
  first
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    [eapply tac_apply with _ H _ _ _;
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      [pm_reflexivity
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      |iSolveTC
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      |pm_prettify (* reduce redexes created by instantiation *)]
    |iSpecializePat H "[]"; last iApplyHypLoop H].

Tactic Notation "iApplyHyp" constr(H) :=
  first
    [iApplyHypExact H
    |iApplyHypLoop H
    |lazymatch iTypeOf H with
     | Some (_,?Q) => fail 1 "iApply: cannot apply" Q
     end].
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Tactic Notation "iApply" open_constr(lem) :=
  iPoseProofCore lem as false true (fun H => iApplyHyp H).

(** * Disjunction *)
Tactic Notation "iLeft" :=
  iStartProof;
  eapply tac_or_l;
    [iSolveTC ||
     let P := match goal with |- FromOr ?P _ _ => P end in
     fail "iLeft:" P "not a disjunction"
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    |(* subgoal *)].
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Tactic Notation "iRight" :=
  iStartProof;
  eapply tac_or_r;
    [iSolveTC ||
     let P := match goal with |- FromOr ?P _ _ => P end in
     fail "iRight:" P "not a disjunction"
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    |(* subgoal *)].
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Local Tactic Notation "iOrDestruct" constr(H) "as" constr(H1) constr(H2) :=
  eapply tac_or_destruct with _ _ H _ H1 H2 _ _ _; (* (i:=H) (j1:=H1) (j2:=H2) *)
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