tactics.v 77.9 KB
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From iris.proofmode Require Import coq_tactics.
From iris.proofmode Require Import intro_patterns spec_patterns sel_patterns.
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From iris.base_logic Require Export base_logic.
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From iris.proofmode Require Export classes notation.
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From iris.proofmode Require Import class_instances.
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From stdpp Require Import stringmap hlist.
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From iris.proofmode Require Import strings.
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Set Default Proof Using "Type".
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Declare Reduction env_cbv := cbv [
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  beq ascii_beq string_beq
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  env_lookup env_lookup_delete env_delete env_app env_replace env_dom
  env_persistent env_spatial env_spatial_is_nil envs_dom
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  envs_lookup envs_lookup_delete envs_delete envs_snoc envs_app
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    envs_simple_replace envs_replace envs_split
    envs_clear_spatial envs_clear_persistent
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    envs_split_go envs_split].
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Ltac env_cbv :=
  match goal with |- ?u => let v := eval env_cbv in u in change v end.
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Ltac env_reflexivity := env_cbv; exact eq_refl.
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(** * Misc *)
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(* Tactic Notation tactics cannot return terms *)
Ltac iFresh' H :=
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  lazymatch goal with
  |- of_envs ?Δ  _ =>
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     (* [vm_compute fails] if any of the hypotheses in [Δ] contain evars, so
     first use [cbv] to compute the domain of [Δ] *)
     let Hs := eval cbv in (envs_dom Δ) in
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     eval vm_compute in (fresh_string_of_set H (of_list Hs))
  | _ => H
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  end.
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Ltac iFresh := iFresh' "~".
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Ltac iMissingHyps Hs :=
  let Δ :=
    lazymatch goal with
    | |- of_envs ?Δ  _ => Δ
    | |- context[ envs_split _ _ ?Δ ] => Δ
    end in
  let Hhyps := eval env_cbv in (envs_dom Δ) in
  eval vm_compute in (list_difference Hs Hhyps).

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

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Class AsValid {M} (φ : Prop) (P : uPred M) := as_entails : φ  P.
Arguments AsValid {_} _%type _%I.

Instance as_valid_valid {M} (P : uPred M) : AsValid (uPred_valid P) P | 0.
Proof. by rewrite /AsValid. Qed.

Instance as_valid_entails {M} (P Q : uPred M) : AsValid (P  Q) (P - Q) | 1.
Proof. split. apply uPred.entails_wand. apply uPred.wand_entails. Qed.

Instance as_valid_equiv {M} (P Q : uPred M) : AsValid (P  Q) (P  Q).
Proof. split. apply uPred.equiv_iff. apply uPred.iff_equiv. Qed.

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(** * Start a proof *)
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Ltac iStartProof :=
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  lazymatch goal with
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  | |- of_envs _  _ => idtac
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  | |- ?P =>
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    apply (proj2 (_ : AsValid P _)), tac_adequate
    || fail "iStartProof: not a uPred"
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  end.

(** * Context manipulation *)
Tactic Notation "iRename" constr(H1) "into" constr(H2) :=
  eapply tac_rename with _ H1 H2 _ _; (* (i:=H1) (j:=H2) *)
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    [env_reflexivity || fail "iRename:" H1 "not found"
    |env_reflexivity || fail "iRename:" H2 "not fresh"|].
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Local Inductive esel_pat :=
  | ESelPure
  | ESelName : bool  string  esel_pat.

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Ltac iElaborateSelPat pat :=
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  let rec go pat Δ Hs :=
    lazymatch pat with
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    | [] => eval cbv in Hs
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    | SelPure :: ?pat => go pat Δ (ESelPure :: Hs)
    | SelPersistent :: ?pat =>
       let Hs' := eval env_cbv in (env_dom (env_persistent Δ)) in
       let Δ' := eval env_cbv in (envs_clear_persistent Δ) in
       go pat Δ' ((ESelName true <$> Hs') ++ Hs)
    | SelSpatial :: ?pat =>
       let Hs' := eval env_cbv in (env_dom (env_spatial Δ)) in
       let Δ' := eval env_cbv in (envs_clear_spatial Δ) in
       go pat Δ' ((ESelName false <$> Hs') ++ Hs)
    | SelName ?H :: ?pat =>
       lazymatch eval env_cbv in (envs_lookup_delete H Δ) with
       | Some (?p,_,?Δ') => go pat Δ' (ESelName p H :: Hs)
       | None => fail "iElaborateSelPat:" H "not found"
       end
    end in
  lazymatch goal with
  | |- of_envs ?Δ  _ =>
    let pat := sel_pat.parse pat in go pat Δ (@nil esel_pat)
  end.

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Tactic Notation "iClear" constr(Hs) :=
  let rec go Hs :=
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    lazymatch Hs with
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    | [] => idtac
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    | ESelPure :: ?Hs => clear; go Hs
    | ESelName _ ?H :: ?Hs =>
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       eapply tac_clear with _ H _ _; (* (i:=H) *)
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         [env_reflexivity || fail "iClear:" H "not found"|go Hs]
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    end in
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  let Hs := iElaborateSelPat Hs in go Hs.
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Tactic Notation "iClear" "(" ident_list(xs) ")" constr(Hs) :=
  iClear Hs; clear xs.
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(** * Assumptions *)
Tactic Notation "iExact" constr(H) :=
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  eapply tac_assumption with H _ _; (* (i:=H) *)
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    [env_reflexivity || fail "iExact:" H "not found"
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    |apply _ ||
     let P := match goal with |- FromAssumption _ ?P _ => P end in
     fail "iExact:" H ":" P "does not match goal"].
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Tactic Notation "iAssumptionCore" :=
  let rec find Γ i P :=
    match 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 | env_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]; env_reflexivity
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  | |- envs_lookup_delete ?i (Envs ?Γp ?Γs) = Some (_, ?P, _) =>
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     first [is_evar i; fail 1 | env_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]; env_reflexivity
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  end.
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Tactic Notation "iAssumption" :=
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  let Hass := fresh in
  let rec find p Γ Q :=
    match Γ with
    | Esnoc ?Γ ?j ?P => first
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       [pose proof (_ : FromAssumption p P Q) as Hass;
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        apply (tac_assumption _ j p P); [env_reflexivity|apply Hass]
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       |find p Γ Q]
    end in
  match goal with
  | |- of_envs (Envs ?Γp ?Γs)  ?Q =>
     first [find true Γp Q | find false Γs Q
           |fail "iAssumption:" Q "not found"]
  end.
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(** * False *)
Tactic Notation "iExFalso" := apply tac_ex_falso.

(** * Making hypotheses persistent or pure *)
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Local Tactic Notation "iPersistent" constr(H) :=
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  eapply tac_persistent with _ H _ _ _; (* (i:=H) *)
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    [env_reflexivity || fail "iPersistent:" H "not found"
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    |apply _ ||
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     let Q := match goal with |- IntoPersistentP _ ?Q _ => Q end in
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     fail "iPersistent:" Q "not persistent"
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    |env_reflexivity|].
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Local Tactic Notation "iPure" constr(H) "as" simple_intropattern(pat) :=
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  eapply tac_pure with _ H _ _ _; (* (i:=H1) *)
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    [env_reflexivity || fail "iPure:" H "not found"
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    |apply _ ||
     let P := match goal with |- IntoPure ?P _ => P end in
     fail "iPure:" P "not pure"
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    |intros pat].

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Tactic Notation "iPureIntro" :=
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  iStartProof;
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  eapply tac_pure_intro;
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    [apply _ ||
     let P := match goal with |- FromPure ?P _ => P end in
     fail "iPureIntro:" P "not pure"
    |].
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(** Framing *)
Local Ltac iFrameFinish :=
  lazy iota beta;
  try match goal with
  | |- _  True => exact (uPred.pure_intro _ _ I)
  end.

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

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

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

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

Local Ltac iFrameAnySpatial :=
  let rec go Hs :=
    match Hs with [] => idtac | ?H :: ?Hs => try iFrameHyp H; go Hs end in
  match goal with
  | |- of_envs ?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 ).

Tactic Notation "iFrame" constr(Hs) :=
  let rec go Hs :=
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    lazymatch Hs with
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    | [] => idtac
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    | SelPure :: ?Hs => iFrameAnyPure; go Hs
    | SelPersistent :: ?Hs => iFrameAnyPersistent; go Hs
    | SelSpatial :: ?Hs => iFrameAnySpatial; go Hs
    | SelName ?H :: ?Hs => iFrameHyp H; go Hs
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    end
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  in let Hs := sel_pat.parse Hs in 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.

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(** * Basic introduction tactics *)
Local Tactic Notation "iIntro" "(" simple_intropattern(x) ")" :=
  try iStartProof;
  try first
    [(* (∀ _, _) *) apply tac_forall_intro
    |(* (?P → _) *) eapply tac_impl_intro_pure;
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      [apply _ ||
       let P := match goal with |- IntoPure ?P _ => P end in
       fail "iIntro:" P "not pure"
      |]
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    |(* (?P -∗ _) *) eapply tac_wand_intro_pure;
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      [apply _ ||
       let P := match goal with |- IntoPure ?P _ => P end in
       fail "iIntro:" P "not pure"
      |]
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    |(* ⌜∀ _, _⌝ *) apply tac_pure_forall_intro
    |(* ⌜_ → _⌝ *) apply tac_pure_impl_intro];
  intros x.

Local Tactic Notation "iIntro" constr(H) :=
  iStartProof;
  first
  [ (* (?Q → _) *)
    eapply tac_impl_intro with _ H; (* (i:=H) *)
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      [env_cbv; apply _ ||
       let P := lazymatch goal with |- PersistentP ?P => P end in
       fail 1 "iIntro: introducing non-persistent" H ":" P
              "into non-empty spatial context"
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      |env_reflexivity || fail "iIntro:" H "not fresh"
      |]
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  | (* (_ -∗ _) *)
    eapply tac_wand_intro with _ H; (* (i:=H) *)
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      [env_reflexivity || fail 1 "iIntro:" H "not fresh"
      |]
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  | fail 1 "iIntro: nothing to introduce" ].

Local Tactic Notation "iIntro" "#" constr(H) :=
  iStartProof;
  first
  [ (* (?P → _) *)
    eapply tac_impl_intro_persistent with _ H _; (* (i:=H) *)
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      [apply _ || 
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       let P := match goal with |- IntoPersistentP _ ?P _ => P end in
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       fail 1 "iIntro: " P " not persistent"
      |env_reflexivity || fail 1 "iIntro:" H "not fresh"
      |]
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  | (* (?P -∗ _) *)
    eapply tac_wand_intro_persistent with _ H _; (* (i:=H) *)
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      [apply _ || 
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       let P := match goal with |- IntoPersistentP _ ?P _ => P end in
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       fail 1 "iIntro: " P " not persistent"
      |env_reflexivity || fail 1 "iIntro:" H "not fresh"
      |]
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  | fail 1 "iIntro: nothing to introduce" ].

Local Tactic Notation "iIntro" "_" :=
  try iStartProof;
  first
  [ (* (?Q → _) *) apply tac_impl_intro_drop
  | (* (_ -∗ _) *) apply tac_wand_intro_drop
  | (* (∀ _, _) *) iIntro (_)
  | fail 1 "iIntro: nothing to introduce" ].

Local Tactic Notation "iIntroForall" :=
  try iStartProof;
  lazymatch goal with
  | |-  _, ?P => fail
  | |-  _, _ => intro
  | |- _  ( x : _, _) => let x' := fresh x in iIntro (x')
  end.
Local Tactic Notation "iIntro" :=
  try iStartProof;
  lazymatch goal with
  | |- _  ?P => intro
  | |- _  (_ - _) => iIntro (?) || let H := iFresh in iIntro #H || iIntro H
  | |- _  (_  _) => iIntro (?) || let H := iFresh in iIntro #H || iIntro H
  end.

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(** * Specialize *)
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Record iTrm {X As} :=
  ITrm { itrm : X ; itrm_vars : hlist As ; itrm_hyps : string }.
Arguments ITrm {_ _} _ _ _.

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

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(*
There is some hacky stuff going on here (most probably there is a Coq bug).
Holes -- like unresolved type class instances -- in the argument `xs` are
resolved at arbitrary moments. It seems that tactics like `apply`, `split` and
`eexists` trigger type class search to resolve these holes. To avoid TC being
triggered too eagerly, this tactic uses `refine` at various places instead of
`apply`.

TODO: Investigate what really is going on. Is there a related to Cog bug #5752?
When should holes in an `open_constr` be resolved?
*)
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Local Tactic Notation "iSpecializeArgs" constr(H) open_constr(xs) :=
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  let rec go xs :=
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    lazymatch xs with
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    | hnil => apply id (* Finally, trigger TC *)
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    | hcons ?x ?xs =>
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       eapply tac_forall_specialize with _ H _ _ _; (* (i:=H) (a:=x) *)
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         [env_reflexivity || fail "iSpecialize:" H "not found"
         |typeclasses eauto ||
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          let P := match goal with |- IntoForall ?P _ => P end in
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          fail "iSpecialize: cannot instantiate" P "with" x
         |lazymatch goal with (* Force [A] in [ex_intro] to deal with coercions. *)
          | |-  _ : ?A, _ => refine (@ex_intro A _ x (conj _ _))
          end; [env_reflexivity|go xs]]
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    end in
  go xs.
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Local Tactic Notation "iSpecializePat" open_constr(H) constr(pat) :=
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  let solve_to_wand H1 :=
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    apply _ ||
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    let P := match goal with |- IntoWand _ ?P _ _ => P end in
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    fail "iSpecialize:" P "not an implication/wand" in
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  let rec go H1 pats :=
    lazymatch pats with
    | [] => idtac
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    | SForall :: ?pats =>
       idtac "the * specialization pattern is deprecated because it is applied implicitly";
       go H1 pats
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    | SName ?H2 :: ?pats =>
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       eapply tac_specialize with _ _ H2 _ H1 _ _ _ _; (* (j:=H1) (i:=H2) *)
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         [env_reflexivity || fail "iSpecialize:" H2 "not found"
         |env_reflexivity || fail "iSpecialize:" H1 "not found"
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         |apply _ ||
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          let P := match goal with |- IntoWand _ ?P ?Q _ => P end in
          let Q := match goal with |- IntoWand _ ?P ?Q _ => Q end in
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          fail "iSpecialize: cannot instantiate" P "with" Q
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         |env_reflexivity|go H1 pats]
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    | SPureGoal ?d :: ?pats =>
       eapply tac_specialize_assert_pure with _ H1 _ _ _ _ _;
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         [env_reflexivity || fail "iSpecialize:" H1 "not found"
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         |solve_to_wand H1
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         |apply _ ||
          let Q := match goal with |- FromPure ?Q _ => Q end in
          fail "iSpecialize:" Q "not pure"
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         |env_reflexivity
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         |done_if d (*goal*)
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         |go H1 pats]
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    | SGoal (SpecGoal GPersistent false ?Hs_frame [] ?d) :: ?pats =>
       eapply tac_specialize_assert_persistent with _ _ H1 _ _ _ _;
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         [env_reflexivity || fail "iSpecialize:" H1 "not found"
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         |solve_to_wand H1
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         |apply _ ||
          let Q := match goal with |- PersistentP ?Q => Q end in
          fail "iSpecialize:" Q "not persistent"
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         |env_reflexivity
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         |iFrame Hs_frame; done_if d (*goal*)
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         |go H1 pats]
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    | SGoal (SpecGoal GPersistent _ _ _ _) :: ?pats =>
       fail "iSpecialize: cannot select hypotheses for persistent premise"
    | SGoal (SpecGoal ?m ?lr ?Hs_frame ?Hs ?d) :: ?pats =>
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       let Hs' := eval cbv in (if lr then Hs else Hs_frame ++ Hs) in
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       eapply tac_specialize_assert with _ _ _ H1 _ lr Hs' _ _ _ _;
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         [env_reflexivity || fail "iSpecialize:" H1 "not found"
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         |solve_to_wand H1
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         |lazymatch m with
          | GSpatial => apply elim_modal_dummy
          | GModal => apply _ || fail "iSpecialize: goal not a modality"
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          end
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         |env_reflexivity ||
          let Hs' := iMissingHyps Hs' in
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          fail "iSpecialize: hypotheses" Hs' "not found"
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         |iFrame Hs_frame; done_if d (*goal*)
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         |go H1 pats]
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    | SAutoFrame GPersistent :: ?pats =>
       eapply tac_specialize_assert_persistent with _ _ H1 _ _ _ _;
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         [env_reflexivity || fail "iSpecialize:" H1 "not found"
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         |solve_to_wand H1
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         |apply _ ||
          let Q := match goal with |- PersistentP ?Q => Q end in
          fail "iSpecialize:" Q "not persistent"
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         |env_reflexivity
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         |solve [iFrame "∗ #"]
         |go H1 pats]
    | SAutoFrame ?m :: ?pats =>
       eapply tac_specialize_frame with _ H1 _ _ _ _ _ _;
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         [env_reflexivity || fail "iSpecialize:" H1 "not found"
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         |solve_to_wand H1
         |lazymatch m with
          | GSpatial => apply elim_modal_dummy
          | GModal => apply _ || fail "iSpecialize: goal not a modality"
          end
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         |iFrame "∗ #"; apply tac_unlock ||
          fail "iSpecialize: premise cannot be solved by framing"
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         |reflexivity]; iIntro H1; go H1 pats
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    end in let pats := spec_pat.parse pat in go H pats.

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(* The argument [p] denotes whether the conclusion of the specialized term is
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persistent. If so, one can use all spatial hypotheses for both proving the
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
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`#` 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). *)
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Tactic Notation "iSpecializeCore" open_constr(t) "as" constr(p) :=
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  let p := intro_pat_persistent p in
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  let t :=
    match type of t with string => constr:(ITrm t hnil "") | _ => t end in
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  lazymatch t with
  | ITrm ?H ?xs ?pat =>
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    let pat := spec_pat.parse pat in
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    lazymatch type of H with
    | string =>
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      (* The lemma [tac_specialize_persistent_helper] allows one to use all
      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
      the result of the specialization is persistent, and no modality is
      eliminated. As an optimization, we do not use this when only universal
      quantifiers are instantiated. *)
      lazymatch eval compute in
        (bool_decide (pat  []) && p && negb (existsb spec_pat_modal pat)) with
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      | true =>
         eapply tac_specialize_persistent_helper with _ H _ _ _;
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           [env_reflexivity || fail "iSpecialize:" H "not found"
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           |iSpecializeArgs H xs; iSpecializePat H pat; last (iExact H)
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           |apply _ ||
            let Q := match goal with |- PersistentP ?Q => Q end in
            fail "iSpecialize:" Q "not persistent"
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           |env_reflexivity|(* goal *)]
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      | false => iSpecializeArgs H xs; iSpecializePat H pat
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      end
    | _ => fail "iSpecialize:" H "should be a hypothesis, use iPoseProof instead"
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    end
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  | _ => fail "iSpecialize:" t "should be a proof mode term"
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  end.
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Tactic Notation "iSpecialize" open_constr(t) :=
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  iSpecializeCore t as false.
Tactic Notation "iSpecialize" open_constr(t) "as" "#" :=
  iSpecializeCore t as true.
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(** * Pose proof *)
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(* The tactic [iIntoValid] tactic solves a goal [uPred_valid Q]. The
arguments [t] is a Coq term whose type is of the following shape:
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- [∀ (x_1 : A_1) .. (x_n : A_n), uPred_valid Q]
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- [∀ (x_1 : A_1) .. (x_n : A_n), P1 ⊢ P2], in which case [Q] becomes [P1 -∗ P2]
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- [∀ (x_1 : A_1) .. (x_n : A_n), P1 ⊣⊢ P2], in which case [Q] becomes [P1 ↔ P2]

The tactic instantiates each dependent argument [x_i] with an evar and generates
a goal [P] for non-dependent arguments [x_i : P]. *)
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Tactic Notation "iIntoValid" open_constr(t) :=
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  let rec go t :=
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    let tT := type of t in
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    lazymatch eval hnf in tT with
    | ?P  ?Q => let H := fresh in assert P as H; [|go 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 #4969. *)
      let e := fresh in evar (e:id T);
      let e' := eval unfold e in e in clear e; go (t e')
    | _ =>
      let tT' := eval cbv zeta in tT in apply (proj1 (_ : AsValid tT' _) t)
      || fail "iPoseProof: not a uPred"
    end in
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  go t.
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(* The tactic [tac] is called with a temporary fresh name [H]. The argument
[lazy_tc] denotes whether type class inference on the premises of [lem] should
be performed before (if false) or after (if true) [tac H] is called. *)
Tactic Notation "iPoseProofCore" open_constr(lem)
    "as" constr(p) constr(lazy_tc) tactic(tac) :=
  try iStartProof;
  let Htmp := iFresh in
  let t :=
    lazymatch lem with ITrm ?t ?xs ?pat => t | _ => lem end in
  let spec_tac _ :=
    lazymatch lem with
    | ITrm ?t ?xs ?pat => iSpecializeCore (ITrm Htmp xs pat) as p
    | _ => idtac
    end in
  let go goal_tac :=
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    lazymatch type of t with
    | string =>
       eapply tac_pose_proof_hyp with _ _ t _ Htmp _;
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         [env_reflexivity || fail "iPoseProof:" t "not found"
         |env_reflexivity || fail "iPoseProof:" Htmp "not fresh"
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         |goal_tac ()]
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    | _ =>
       eapply tac_pose_proof with _ Htmp _; (* (j:=H) *)
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         [iIntoValid t
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         |env_reflexivity || fail "iPoseProof:" Htmp "not fresh"
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         |goal_tac ()]
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    end;
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    try (apply _) in
  lazymatch eval compute in lazy_tc with
  | true => go ltac:(fun _ => spec_tac (); last (tac Htmp))
  | false => go spec_tac; last (tac Htmp)
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  end.

(** * Apply *)
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Tactic Notation "iApplyHyp" constr(H) :=
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  let rec go H := first
    [eapply tac_apply with _ H _ _ _;
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      [env_reflexivity
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      |apply _
      |lazy beta (* reduce betas created by instantiation *)]
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    |iSpecializePat H "[]"; last go H] in
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  iExact H ||
  go H ||
  lazymatch iTypeOf H with
  | Some (_,?Q) => fail "iApply: cannot apply" Q
  end.

Tactic Notation "iApply" open_constr(lem) :=
  iPoseProofCore lem as false true (fun H => iApplyHyp H).
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(** * Revert *)
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Local Tactic Notation "iForallRevert" ident(x) :=
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  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
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  iStartProof;
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  let A := type of x in
  lazymatch type of A with
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  | Prop => revert x; first [apply tac_pure_revert|err x]
  | _ => revert x; first [apply tac_forall_revert|err x]
  end.
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Tactic Notation "iRevert" constr(Hs) :=
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  let rec go Hs :=
    lazymatch Hs with
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    | [] => idtac
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    | ESelPure :: ?Hs =>
       repeat match goal with x : _ |- _ => revert x end;
       go Hs
    | ESelName _ ?H :: ?Hs =>
       eapply tac_revert with _ H _ _; (* (i:=H2) *)
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         [env_reflexivity || fail "iRevert:" H "not found"
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         |env_cbv; go Hs]
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    end in
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  let Hs := iElaborateSelPat Hs in iStartProof; go Hs.
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Tactic Notation "iRevert" "(" ident(x1) ")" :=
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  iForallRevert x1.
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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) ")"
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    constr(Hs) :=
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  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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(** * Disjunction *)
Tactic Notation "iLeft" :=
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  iStartProof;
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  eapply tac_or_l;
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    [apply _ ||
     let P := match goal with |- FromOr ?P _ _ => P end in
     fail "iLeft:" P "not a disjunction"
    |].
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Tactic Notation "iRight" :=
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  iStartProof;
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  eapply tac_or_r;
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    [apply _ ||
     let P := match goal with |- FromOr ?P _ _ => P end in
     fail "iRight:" P "not a disjunction"
    |].
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Local Tactic Notation "iOrDestruct" constr(H) "as" constr(H1) constr(H2) :=
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  eapply tac_or_destruct with _ _ H _ H1 H2 _ _ _; (* (i:=H) (j1:=H1) (j2:=H2) *)
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    [env_reflexivity || fail "iOrDestruct:" H "not found"
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    |apply _ ||
     let P := match goal with |- IntoOr ?P _ _ => P end in
     fail "iOrDestruct: cannot destruct" P
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    |env_reflexivity || fail "iOrDestruct:" H1 "not fresh"
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    |env_reflexivity || fail "iOrDestruct:" H2 "not fresh"
    | |].
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(** * Conjunction and separating conjunction *)
Tactic Notation "iSplit" :=
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  iStartProof;
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  lazymatch goal with
  | |- _  _ =>
    eapply tac_and_split;
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      [apply _ ||
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       let P := match goal with |- FromAnd _ ?P _ _ => P end in
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       fail "iSplit:" P "not a conjunction"| |]
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  end.
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Tactic Notation "iSplitL" constr(Hs) :=
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  iStartProof;
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  let Hs := words Hs in
  eapply tac_sep_split with _ _ false Hs _ _; (* (js:=Hs) *)
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    [apply _ ||
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     let P := match goal with |- FromAnd _ ?P _ _ => P end in
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     fail "iSplitL:" P "not a separating conjunction"
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    |env_reflexivity ||
     let Hs := iMissingHyps Hs in
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     fail "iSplitL: hypotheses" Hs "not found"
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    | |].

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Tactic Notation "iSplitR" constr(Hs) :=
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  iStartProof;
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  let Hs := words Hs in
  eapply tac_sep_split with _ _ true Hs _ _; (* (js:=Hs) *)
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    [apply _ ||
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     let P := match goal with |- FromAnd _ ?P _ _ => P end in
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     fail "iSplitR:" P "not a separating conjunction"
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    |env_reflexivity ||
     let Hs := iMissingHyps Hs in
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     fail "iSplitR: hypotheses" Hs "not found"
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    | |].
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Tactic Notation "iSplitL" := iSplitR "".
Tactic Notation "iSplitR" := iSplitL "".

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Local Tactic Notation "iAndDestruct" constr(H) "as" constr(H1) constr(H2) :=
  eapply tac_and_destruct with _ H _ H1 H2 _ _ _; (* (i:=H) (j1:=H1) (j2:=H2) *)
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    [env_reflexivity || fail "iAndDestruct:" H "not found"
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    |apply _ ||
     let P := match goal with |- IntoAnd _ ?P _ _ => P end in
     fail "iAndDestruct: cannot destruct" P
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    |env_reflexivity || fail "iAndDestruct:" H1 "or" H2 " not fresh"|].
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Local Tactic Notation "iAndDestructChoice" constr(H) "as" constr(lr) constr(H') :=
  eapply tac_and_destruct_choice with _ H _ lr H' _ _ _;
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    [env_reflexivity || fail "iAndDestructChoice:" H "not found"
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    |apply _ ||
     let P := match goal with |- IntoAnd _ ?P _ _ => P end in
     fail "iAndDestructChoice: cannot destruct" P
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    |env_reflexivity || fail "iAndDestructChoice:" H' " not fresh"|].
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(** * Combinining hypotheses *)
Tactic Notation "iCombine" constr(Hs) "as" constr(H) :=
  let Hs := words Hs in
  eapply tac_combine with _ _ Hs _ _ H _;
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    [env_reflexivity ||
     let Hs := iMissingHyps Hs in
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     fail "iCombine: hypotheses" Hs "not found"
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    |apply _
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    |env_reflexivity || fail "iCombine:" H "not fresh"|].
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Tactic Notation "iCombine" constr(H1) constr(H2) "as" constr(H) :=
  iCombine [H1;H2] as H.

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(** * Existential *)
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Tactic Notation "iExists" uconstr(x1) :=
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  iStartProof;
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  eapply tac_exist;
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    [apply _ ||
     let P := match goal with |- FromExist ?P _ => P end in
     fail "iExists:" P "not an existential"
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    |cbv beta; eexists x1].
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Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) :=
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  iExists x1; iExists x2.
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Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) :=
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  iExists x1; iExists x2, x3.
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Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) :=
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  iExists x1; iExists x2, x3, x4.
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Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) :=
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  iExists x1; iExists x2, x3, x4, x5.
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Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) :=
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  iExists x1; iExists x2, x3, x4, x5, x6.
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Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) "," uconstr(x7) :=
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  iExists x1; iExists x2, x3, x4, x5, x6, x7.
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Tactic Notation "iExists" uconstr(x1) "," uconstr(x2) "," uconstr(x3) ","
    uconstr(x4) "," uconstr(x5) "," uconstr(x6) "," uconstr(x7) ","
    uconstr(x8) :=
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  iExists x1; iExists x2, x3, x4, x5, x6, x7, x8.

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Local Tactic Notation "iExistDestruct" constr(H)
    "as" simple_intropattern(x) constr(Hx) :=
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  eapply tac_exist_destruct with H _ Hx _ _; (* (i:=H) (j:=Hx) *)
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    [env_reflexivity || fail "iExistDestruct:" H "not found"
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    |apply _ ||
     let P := match goal with |- IntoExist ?P</