CFTactics.v 104 KB
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Set Implicit Arguments.
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Require Import LibCore Shared CFHeaps.
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Require Export (* LibInt *) CFPrint.
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(*--------------------------------------------------------*)
(* ** TODO: move to LibTactics.v *)

Inductive ltac_goal_to_discard := ltac_goal_to_discard_intro.

Ltac forwards_nounfold_skip_sides_then S cont :=
  let MARK := fresh in 
  generalize ltac_goal_to_discard_intro; 
  intro MARK;
  forwards_nounfold_then S ltac:(fun K =>
    clear MARK;
    cont K);
  match goal with
  | MARK: ltac_goal_to_discard |- _ => skip 
  | _ => idtac
  end.


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(* [xinduction IH: E X] now accepts for [E] either
   - a proof of [wf R] for [R] of type [A->A->Prop]
   - a binary relation of type [A->A->Prop]
   - a measure of type [A->nat]
 *)
 
Ltac induction_wf_core_then IH E X cont :=
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  let T := type of E in
  let T := eval hnf in T in
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  let clearX tt := 
    first [ clear X | fail 3 "the variable on which the induction is done appears in the hypotheses" ] in
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  match T with
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  | ?A -> nat =>
     induction_wf_core_then IH (measure_wf E) X cont
     (* TODO: error message might not show up in this case *)
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  | ?A -> ?A -> Prop =>
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     pattern X; 
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     first [
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       applys well_founded_ind E; 
       clearX tt;
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       [ change well_founded with wf; auto with wf
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       | intros X IH; cont tt ]
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     | fail 2 ]
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  | _ => 
    pattern X; 
    applys well_founded_ind E; 
    clearX tt; 
    intros X IH; 
    cont tt
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  end.

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Ltac induction_wf_core IH E X :=
  induction_wf_core_then IH E X ltac:(fun _ => idtac).

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Tactic Notation "induction_wf" ident(IH) ":" constr(E) ident(X) :=
  induction_wf_core IH E X.
Tactic Notation "induction_wf" ":" constr(E) ident(X) :=
  let IH := fresh "IH" in induction_wf IH: E X.
Tactic Notation "induction_wf" ":" constr(E) ident(X) :=
  induction_wf: E X.


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(* FOR LibList. v *)

Lemma app_eq_prefix_inv_l : forall A (l1 l2 l2' : list A),
  l1 ++ l2 = l1 ++ l2' -> l2 = l2'.
Proof using.
  introv E. induction l1; rew_list in *. auto. inverts* E.
Qed.

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(*--------------------------------------------------------*)
(* ** [=>] and [=>>] tactics for introduction *)

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(* [=> I1 .. IN] is the same as [intros I1 .. IN] *)

Ltac ltac_intros_post := idtac.
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Tactic Notation "=>" :=
  intros.
Tactic Notation "=>" simple_intropattern(I1) :=
  intros I1.
Tactic Notation "=>" simple_intropattern(I1) simple_intropattern(I2) :=
  intros I1 I2.
Tactic Notation "=>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) :=
  intros I1 I2 I3.
Tactic Notation "=>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) :=
  intros I1 I2 I3 I4.
Tactic Notation "=>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5) :=
  intros I1 I2 I3 I4 I5.
Tactic Notation "=>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
 simple_intropattern(I6) :=
  intros I1 I2 I3 I4 I5 I6.
Tactic Notation "=>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
 simple_intropattern(I6) simple_intropattern(I7) :=
  intros I1 I2 I3 I4 I5 I6 I7.
Tactic Notation "=>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
 simple_intropattern(I6) simple_intropattern(I7) simple_intropattern(I8) :=
  intros I1 I2 I3 I4 I5 I6 I7 I8.
Tactic Notation "=>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
 simple_intropattern(I6) simple_intropattern(I7) simple_intropattern(I8)
 simple_intropattern(I9) :=
  intros I1 I2 I3 I4 I5 I6 I7 I8 I9.
Tactic Notation "=>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
 simple_intropattern(I6) simple_intropattern(I7) simple_intropattern(I8)
 simple_intropattern(I9) simple_intropattern(I10) :=
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  intros I1 I2 I3 I4 I5 I6 I7 I8 I9 I10.
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(* [=>>] first introduces all non-dependent variables,
   then behaves as [intros]. It unfolds the head of the goal using [hnf] 
   if there are not head visible quantifiers. 
   
   Remark: instances of [Inhab] are treated as non-dependent and
   are introduced automatically. *)
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Ltac intro_nondeps_aux is_already_hnf := 
  match goal with
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  | |- (?P -> ?Q) => idtac
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  | |- (Inhab _) -> _ => intro; intro_nondeps_aux true 
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  | |- (forall _,_) => intros ?; intro_nondeps_aux true
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  | |- _ => 
     match is_already_hnf with
     | true => idtac
     | false => hnf; intro_nondeps_aux true  
     end
  end.
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Ltac intro_nondeps tt := intro_nondeps_aux false.

Tactic Notation "=>>" :=
  intro_nondeps tt.
Tactic Notation "=>>" simple_intropattern(I1) :=
  =>>; intros I1.
Tactic Notation "=>>" simple_intropattern(I1) simple_intropattern(I2) :=
  =>>; intros I1 I2.
Tactic Notation "=>>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) :=
  =>>; intros I1 I2 I3.
Tactic Notation "=>>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) :=
  =>>; intros I1 I2 I3 I4.
Tactic Notation "=>>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5) :=
  =>>; intros I1 I2 I3 I4 I5.
Tactic Notation "=>>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
 simple_intropattern(I6) :=
  =>>; intros I1 I2 I3 I4 I5 I6.
Tactic Notation "=>>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
 simple_intropattern(I6) simple_intropattern(I7) :=
  =>>; intros I1 I2 I3 I4 I5 I6 I7.
Tactic Notation "=>>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
 simple_intropattern(I6) simple_intropattern(I7) simple_intropattern(I8) :=
  =>>; intros I1 I2 I3 I4 I5 I6 I7 I8.
Tactic Notation "=>>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
 simple_intropattern(I6) simple_intropattern(I7) simple_intropattern(I8)
 simple_intropattern(I9) :=
  =>>; intros I1 I2 I3 I4 I5 I6 I7 I8 I9.
Tactic Notation "=>>" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
 simple_intropattern(I6) simple_intropattern(I7) simple_intropattern(I8)
 simple_intropattern(I9) simple_intropattern(I10) :=
  =>>; intros I1 I2 I3 I4 I5 I6 I7 I8 I9 I10.
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(* ********************************************************************** *)
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(** * Introduction 
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Section IntrovTest.
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Variables P1 P2 P3 : nat -> Prop.
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Lemma demo_introv_1 :
  forall a b, P1 a -> P2 b -> forall c d, P3 c -> P1 d -> c = b.
Proof using.
  dup 4.
  (* [introv] introduces all the variables which are not hypotheses,
     more precisely all the variables used dependently. *) 
  introv.
  (* if there is no more head variables, and no definition can 
     be unfolded at head of the goal, it does not do anything *)
  introv. skip.
  (* [introv A] introduces all variables, then does [intros A] *)
  introv A. introv B. introv. intros C D. skip.
  (* [introv] may take several arguments, as illustrated below *)
  introv A B. introv. skip.
  introv A B C D. skip.
Qed.
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Definition Same (x y : nat) := True -> x = y.
Definition Sym (x:nat) := forall y, x = y -> Same y x.
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Lemma demo_introv_2 :
  forall a, Sym a.
Proof using.
  dup 4.
  (* [introv] introduces a variable but no subsequent definition *)
  introv. 
  (* [introv] unfolds definition if no variable is visible *)
  introv. skip.
  (* [introv E] unfolds definitions until finding an hypothesis *)
  introv E. introv F. skip.
  (* [introv E F] unfolds several definitions if needed *)
  introv E F. skip.
  (* [introv] may unfold definition without any introduction *)
  introv E. introv. skip.
Qed.
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Lemma demo_introv_3 :
  forall a, a = 0 -> Sym a.
Proof using.
  dup 5. (* more examples *)
  (* introduces [a] only *)
  introv. skip.
  (* introduces [a = 0] *)
  introv E. skip.
  (* introduces [a = 0] and [a = y] *)
  introv E F. skip.
  (* introduces [a = 0] and [a = y] and [True] *)
  introv E F G. skip.
  (* introduction of more names fails *)
  try (introv E F G H). skip.
Qed.
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Definition TestSym := (forall a, a = 0 -> Sym a).
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Lemma demo_introv_4 :
  TestSym.
Proof using.
  dup 2. (* same as before, except the goal itself is a definition *)
  (* introduces [a] only *)
  introv. skip.
  (* introduces [a = 0] *)
  introv E. skip.
Qed.
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Lemma demo_introv_5 :
  forall a, a = 0 -> ~ Sym a.
Proof using.
  dup 2. (* playing with negation *)
  (* introduces [a = 0] *)
  introv E. skip.
  (* introduces [a = 0] and [Sym a] *)
  introv E F. skip.
Qed.
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(* Iterated unfolding to get hypotheses *)
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Definition AllSameAs (x:nat) := forall y, Same y x.
Definition AllSame := forall x, AllSameAs x.
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Lemma demo_introv_6 :
  AllSame.
Proof using.
  dup 2. 
  (* introduces only [x], then only [y] *)
  introv. introv. skip.
  (* introduces [x] and [y] and [True] *)
  introv E. skip.
Qed.
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Definition AllSameAgain := AllSame.
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Lemma demo_introv_7 :
  AllSameAgain.
Proof using.
  dup 2.  
  (* introduces only [x], then only [y] *)
  introv. introv. skip.
  (* introduces [x] and [y] and [True] *)
  introv E. skip.
Qed.
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Lemma demo_introv_8 :
  forall a (c:nat) b, P1 a -> P2 b -> True.
Proof using.
  (* notice that variables which are not used dependently
     are treated as hypotheses, e.g. variable [c] below.
     This might not be the desired behaviour, but that's
     all I'm able to implement in Ltac. *)
  introv c E F. skip.
Qed.
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Definition IMP P A (x y : A) := P -> x = y.
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Lemma demo_intros_all :
     (forall a b, P1 a -> P2 b -> IMP H1 a b)
  /\ (forall a b, a = 0 -> b = 1 -> ~ (a = b)).
Proof using.
  split.
  (* [intros_all] introduces as many arguments as possible *)
  intros_all. skip.
  intros_all. skip.
Qed.

(* An example showing that [intro] is not very-well
   specified with respect to beta-reduction, explaining
   why [introv] isn't doing better. *)

Definition testing f :=
  forall (x:nat), f x -> True.

Lemma demo_introv_what_to_do : testing (fun a => a = 0).
Proof using.   
  dup.
    intro. skip. (* does beta-reduce f *)
    hnf. intro. skip. (* does not beta-reduce f *)
Qed.

End IntrovTest.
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*)
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(********************************************************************)
(* ** Shared auxiliary functions *)
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(*--------------------------------------------------------*)
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(* ** Tools for manipulating post-conditions *)
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Ltac is_evar_as_bool E :=
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  constr:(ltac:(first 
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    [ is_evar E; exact true 
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    | exact false ])).
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Ltac cfml_get_postcondition tt :=
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  match goal with |- ?E => 
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  match get_fun_arg E with (_,?Q) => constr:(Q) 
  end end.

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(** [cfml_postcondition_is_evar tt] returns a boolean indicating
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    whether the post-condition of the current goal is an evar. *)
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Ltac cfml_postcondition_is_evar tt := 
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  let Q := cfml_get_postcondition tt in
  is_evar_as_bool Q.
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(*--------------------------------------------------------*)
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(* ** Auxiliary functions for reasoning on applications/specs *)
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(* [cfml_get_goal_fun] returns the function associated with the 
   [app] or [spec] at the head of the goal. *)
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Ltac cfml_get_goal_fun tt :=
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  match goal with 
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  | |- spec ?f ?n ?P => constr:(f)
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  | |- app ?f ?xs ?H ?Q => constr:(f)
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  | |- tag tag_apply (app ?f ?xs) ?H ?Q => constr:(f)
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  end.
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(* [cfml_get_goal_arity] returns the arity associated with the 
   [app] or [spec] at the head of the goal. *)
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Ltac cfml_get_goal_arity tt :=
  let aux xs :=
    let n := eval compute in (List.length xs) in constr:(n) in
  match goal with 
  | |- spec ?f ?n ?P => constr:(n)
  | |- app ?f ?xs ?H ?Q => aux xs
  | |- tag tag_apply (app ?f ?xs ?H ?Q) => aux xs
  end .
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(* [cfml_show_types_dyn_list L] displays the types of the 
   arguments in the list [L] *)

Ltac cfml_show_types_dyn_list L :=
  match L with
  | nil => idtac
  | (@dyn ?T ?x) :: ?R => idtac T "->"; cfml_show_types_dyn_list R
  end.

(* [cfml_get_app_args E] returns the arguments associated with the 
   [app] in the term [E] provided *)

Ltac cfml_get_app_args E :=
  match E with 
  | app ?f ?xs ?H ?Q => constr:(xs)
  | tag tag_apply (app ?f ?xs) ?H ?Q => constr:(xs)
  end.

(* [cfml_get_goal_app_args tt] returns the arguments associated with the 
   [app] at the head of the goal. *)

Ltac cfml_get_goal_app_args tt :=
  match goal with |- ?G => cfml_get_app_args G end.

(* [cfml_get_goal_app_ret E] returns the arguments associated with the 
   [app] in the term [E] provided. *)

Ltac cfml_get_app_ret E :=
  match E with 
  | @app_def ?f ?xs ?B ?H ?Q => constr:(B)
  | tag tag_apply (@app_def ?f ?xs ?B) ?H ?Q => constr:(B)
  end.

(* [cfml_get_goal_app_ret tt] returns the arguments associated with the 
   [app] at the head of the goal. *)

Ltac cfml_get_goal_app_ret tt :=
  match goal with |- ?G => cfml_get_app_ret G end.

(* [cfml_show_app_type E] displays the type of the
   function application in [E] *)

Ltac cfml_show_app_type E :=
  let L := cfml_get_app_args E in
  cfml_show_types_dyn_list L;
  let B := cfml_get_app_ret E in
  idtac B.

(* [cfml_show_app_type_goal tt] displays the type of 
   the function application in the goal.
   (calls [intros] if needed. *)

Ltac cfml_show_app_type_goal tt :=
  match goal with |- ?G => cfml_show_app_type G end.

(* [cfml_show_app_type_concl S] displays the type of 
   the function application in the conclusion of [S] *)

Ltac cfml_show_app_type_concl S :=
  forwards_nounfold_then S ltac:(fun K =>
    cfml_show_app_type K).


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(********************************************************************)
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(* ** Simplification, Automation, and Cleaning  *)


(*--------------------------------------------------------*)
(* ** [xclean] *)

(** [xclean] is a tactic that performs some clean up throughout
    the goal by taking care of:
    - rewriting facts such as [true = isTrue P] into [P]
    - simplifying partially applied equality, e.g. [(= 3) n] to [n = 3].

    See [reflect_clean tt] from the Shared.v file.

    Remark: this tactic is automatically called by [xextract]. 
*)

Ltac xclean_core :=
  reflect_clean tt.

Tactic Notation "xclean" :=
  reflect_clean tt.

(* If [xclean] is too long to type, then use:
  Tactic Notation "x" := xclean.
*)

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(*--------------------------------------------------------*)
(* ** [xauto] *)

(* [xauto] is a specialized version of [auto] that works
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   well in program verification. 
   
   - it will not attempt any work if the head of the goal
     has a tag (i.e. if it is a characteristif formula),
   - it is able to conclude a goal using [xok]
   - it calls [substs] to substitute all equalities before trying
     to call automation. 
     
   Tactics [xauto], [xauto~] and [xauto*] can be configured 
   independently. 
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   [xsimpl~] is equivalent to [xsimpl; xauto~].
   [xsimpl*] is equivalent to [xsimpl; xauto*].
*)
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Ltac xok_core cont :=  (* see [xok] spec further *)
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  solve [ hnf; apply rel_le_refl
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        | apply pred_le_refl
        | apply hsimpl_to_qunit; reflexivity
        | hsimpl; cont tt ].

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Ltac math_0 ::= xclean. (* TODO: why needed? *)
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Ltac xauto_common cont :=
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  cfml_check_not_tagged tt;  
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  try solve [ cont tt 
            | solve [ apply refl_equal ]
            | xok_core ltac:(fun _ => solve [ cont tt | substs; cont tt ] ) 
            | substs; if_eq; solve [ cont tt | apply refl_equal ]  ].

Ltac xauto_tilde_default cont := xauto_common cont.
Ltac xauto_star_default cont := xauto_common cont.

Ltac xauto_tilde := xauto_tilde_default ltac:(fun _ => auto_tilde).
Ltac xauto_star := xauto_star_default ltac:(fun _ => auto_star). 

Tactic Notation "xauto" "~" := xauto_tilde.
Tactic Notation "xauto" "*" := xauto_star.
Tactic Notation "xauto" := xauto~.

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(*--------------------------------------------------------*)
(* ** [xok] *)

(** [xok] proves a goal of the form [H ==> ?H'] or [Q ===> ?Q']
    by unifying the right-hand-side with the left-hand-side. 

    It also tries to call [hsimpl] to see if it solves the goal. 
    TODO: is this last feature of [xok] useful? *)

Tactic Notation "xok" := 
  xok_core ltac:(idcont).
(* [xok~] and [xok*] defined further *)
Tactic Notation "xok" "~" := 
  xok_core ltac:(fun _ => xauto~).
Tactic Notation "xok" "*" := 
  xok_core ltac:(fun _ => xauto*).



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(*--------------------------------------------------------*)
(* ** [xsimpl] *)

(** [xsimpl] performs a heap entailement simplification using 
    [hsimpl], then calls the tactic [xclean] for clean up. *)

Ltac xsimpl_core := hsimpl; xclean.

Tactic Notation "xsimpl" := xsimpl_core. 
Tactic Notation "xsimpl" "~" := xsimpl; xauto~.
Tactic Notation "xsimpl" "*" := xsimpl; xauto*.

(* TODO: factorize below with above *)

Ltac xsimpl_core_with E := hsimpl E; xclean.
Tactic Notation "xsimpl" constr(E) := xsimpl_core_with E. 
Tactic Notation "xsimpl" "~" constr(E) := xsimpl E; xauto~.
Tactic Notation "xsimpl" "*" constr(E) := xsimpl E; xauto*.


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Tactic Notation "hsimpl" "~" constr(L) :=
  hsimpl L; xauto~.
Tactic Notation "hsimpl" "~" constr(X1) constr(X2) :=
  hsimpl X1 X2; xauto~.
Tactic Notation "hsimpl" "~" constr(X1) constr(X2) constr(X3) :=
  hsimpl X1 X2 X3; xauto~.


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(*--------------------------------------------------------*)
(* ** [xlocal] *)

(** [xlocal] proves a goal of the form [is_local F]
    or [is_local_pred F], by exploiting the fact that
    - [F] may be of the form [local F']
    - [is_local F] may be an assumption
    - [is_local_pred S] may be an assumption, with [F = S x]
*)

Ltac xlocal_core tt ::=
  first [ assumption
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        | apply app_local; (* then prove [xs <> nil] *)
          let E := fresh in intro E; discriminate E
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        | apply local_is_local 
        (*| apply app_local_pred  --TODO fix *)
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        | match goal with H: is_local_pred ?S |- is_local (?S _) => apply H end ].



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(********************************************************************)
(* ** Tactics for Structural Rules *)


(*--------------------------------------------------------*)
(* ** [xpre] *)

(** [xpre] applies to a goal of the form [F H Q] and 
    allows to strengthen the pre-condition [H].
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    It produces [F ?H' Q] and [H ==> ?H' \* \GC],
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    where [Hexists HG, HG] can be instantiated with garbage
    to collect. *)

Tactic Notation "xpre" constr(H) :=
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  eapply (@local_gc_pre H); [ try xlocal | xtag_goal | ].
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(*--------------------------------------------------------*)
(* ** [xpost] *)

(** [xpost] applies to a goal of the form [F H Q] and 
    allows to weaken the pre-condition [Q].
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    It produces [F H ?Q'] and [?Q' ==> ?Q \* \GC].
    
    [xpost Q'] applies to a goal of the form [F H Q]
    and leaves the goals [F H Q'] and [Q' ===> Q]. 
    
    [xpost H'] is a shorthand for [xpost (#H')].

*)
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(* Lemma used by [xpost] *)

Lemma xpost_lemma : forall B Q' Q (F:~~B) H,
  is_local F -> 
  F H Q' -> 
  Q' ===> Q ->
  F H Q.
Proof using. intros. applys* local_weaken. Qed.

Tactic Notation "xpost" :=
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  eapply xpost_lemma; [ try xlocal | xtag_goal | ].
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Lemma fix_post : forall (B:Type) (Q':B->hprop) (F:~~B) (H:hprop) Q,
  is_local F ->
  F H Q' -> 
  Q' ===> Q ->
  F H Q.
Proof. intros. apply* local_weaken. Qed.

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Ltac xpost_core Q :=
  let Q' := match type of Q with
    | hprop => constr:(fun (_:unit) => Q)
    | _ => constr:(Q)
    end in
  match cfml_postcondition_is_evar tt with
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  | true => apply (@fix_post _ Q'); [ try xlocal | xtag_goal | apply rel_le_refl ]
  | false => apply (@fix_post _ Q'); [ try xlocal | xtag_goal | ]
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  end.

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Tactic Notation "xpost" constr(Q) := 
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  xpost_core Q.
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(*--------------------------------------------------------*)
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(* ** [xextract] *)

(** [xextract_check_not_needed tt] applies to a goal of the form [F H Q]
    and raises an error if [H] contains existentials or pure propositions
    that could have been extracted using [xextract] *)

Ltac xextract_check_not_needed tt :=
  match goal with |- ?R ?H ?Q => hextractible_rec H end.

(** Auxiliary definitions for [xextract]. *)

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Ltac xextract_core tt :=
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  match goal with
  | |- _ ==> _ => hextract; xclean
  | |- _ ===> _ => let r := fresh "r" in intros r; hextract; xclean
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  | |- _ => simpl; hclean; instantiate
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  end.
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(** [xextract] extracts existentials and pure propositions from
    the precondition [H] of a goal of the form [F H Q],
    or from the left-hand-side [H] of a goal of the form [H ==> H']
    or [H ===> H']. 
    
    It calls [xclean] which is useful for cleaning up *)
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Tactic Notation "xextract" := 
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  xextract_core tt; xtag_goal.
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(** [xextracts] calls [xextract], assumes that this call produces
   an equality at the head of the goal, and it then substitutes 
   this equality away. *)

Tactic Notation "xextracts" :=
  let E := fresh "TEMP" in xextract; intros E; subst_hyp E.

(** DEPRECATED
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   [xextract as I1 .. IN] should now be written [xextract. => I1 .. IN]
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Tactic Notation "xextract" "as" simple_intropattern(I1) := 
  xextract; intros I1; xclean.
Tactic Notation "xextract" "as" simple_intropattern(I1) simple_intropattern(I2) := 
  xextract; intros I1 I2; xclean. 
Tactic Notation "xextract" "as" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) := 
  xextract; intros I1 I2 I3; xclean.
Tactic Notation "xextract" "as" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) := 
  xextract; intros I1 I2 I3 I4; xclean.
Tactic Notation "xextract" "as" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5) := 
  xextract; intros I1 I2 I3 I4 I5; xclean.
Tactic Notation "xextract" "as" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
 simple_intropattern(I6) := 
  xextract; intros I1 I2 I3 I4 I5 I6; xclean.
Tactic Notation "xextract" "as" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
 simple_intropattern(I6) simple_intropattern(I7) := 
  xextract; intros I1 I2 I3 I4 I5 I6 I7; xclean.
Tactic Notation "xextract" "as" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
 simple_intropattern(I6) simple_intropattern(I7) simple_intropattern(I8) := 
  xextract; intros I1 I2 I3 I4 I5 I6 I7 I8; xclean.
Tactic Notation "xextract" "as" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
 simple_intropattern(I6) simple_intropattern(I7) simple_intropattern(I8) 
 simple_intropattern(I9) := 
  xextract; intros I1 I2 I3 I4 I5 I6 I7 I8 I9; xclean.
Tactic Notation "xextract" "as" simple_intropattern(I1) simple_intropattern(I2) 
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
 simple_intropattern(I6) simple_intropattern(I7) simple_intropattern(I8) 
 simple_intropattern(I9) simple_intropattern(I10) := 
  xextract; intros I1 I2 I3 I4 I5 I6 I7 I8 I9 I10; xclean.
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*)
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(*--------------------------------------------------------*)
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(* ** [xgc], [xgc_but], [xgc_all], [xgc_post] *)
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(** [xgc H'] applies to a goal of the form [F H Q]
    and substracts [H'] from [H]. 
    
    [xgc_but H'] substracts everything but [H'] from [H].  
    
    [xgc_all] substracts everything from [H], leaving [F \[] Q].
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    It is equivalent to [xgc_but \[]].
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    [xgc_post] applied to the goal [F H Q] replaces the goal 
    with [F H ?Q'] and [Q' ===> Q \*+ \GC], which allows to 
    perform garbage collection.
*)
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Ltac xgc_remove_core H :=
  xextract_check_not_needed tt;
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  eapply (@local_gc_pre_on H);
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    [ try xlocal
    | hsimpl
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    | xtag_goal ].
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Ltac xgc_keep_core H :=
  xextract_check_not_needed tt;
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  eapply (@local_gc_pre H);
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    [ try xlocal
    | hsimpl
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    | xtag_goal ].
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Ltac xgc_post_core :=
  xextract_check_not_needed tt;
  eapply local_gc_post; 
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  [ try xlocal | | xtag_goal ].
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Tactic Notation "xgc" constr(H) := 
  xgc_remove_core H.
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Tactic Notation "xgc_but" constr(H) := 
  xgc_keep_core H.
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Tactic Notation "xgc_post" := 
  xgc_post_core.

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(* DEPRECATED
Ltac xgc_post_if_not_evar_then cont :=
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  match cfml_postcondition_is_evar tt with
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  | true => cont tt
  | false => xgc_post; [ cont tt | ]
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  end.
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*)
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(* Lemma used by [xgc_all], to remove everything from the 
   pre-condition *)

Lemma local_gc_pre_all : forall B Q (F:~~B) H,
  is_local F -> 
  F \[] Q ->
  F H Q.
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Proof using. intros. apply* (@local_gc_pre_on H). hsimpl. Qed.
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Tactic Notation "xgc_all" := 
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  eapply local_gc_pre_all; [ try xlocal | xtag_goal ].
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(*--------------------------------------------------------*)
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(* ** [xframe] and [xframe_but] and [xframe_on] *)
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(** [xframe H'] applies to a goal of the form [F H Q]
    and substracts [H'] from both [H] and [Q].
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    [xframe_but H'] substracts everything but [H']. *)
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(* Lemma used by [xframe] *)
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Lemma xframe_lemma : forall H1 H2 B Q1 (F:~~B) H Q,
  is_local F -> 
  H ==> H1 \* H2 -> 
  F H1 Q1 -> 
  Q1 \*+ H2 ===> Q ->
  F H Q.
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Proof using. intros. apply* local_frame. Qed.
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Ltac xframe_remove_core H :=
  xextract_check_not_needed tt;
  eapply xframe_lemma with (H2 := H);
    [ try xlocal
    | hsimpl
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    | xtag_goal
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    | ].
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Tactic Notation "xframe" constr(H) := 
  xframe_remove_core H.
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Ltac xframe_keep_core H :=
  xextract_check_not_needed tt;
  eapply xframe_lemma with (H1 := H);
    [ try xlocal
    | hsimpl
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    | xtag_goal
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    | ].
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Tactic Notation "xframe_but" constr(H) := 
  xframe_keep_core H.
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(** [xframe_on p] applies to a goal of the form [F H Q]
    and calls [xframe (p ~> M)], where [M] is guessed 
    from the occurence of [p ~> M] that can be found in
    the pre-condition [H].
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    Variants: [xframe_on p1 p2] and [xframe_on p1 p2 p3]. *)
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Ltac xframes_core_1 s := 
  match goal with |- ?R ?H ?Q =>
    match H with context [ s ~> ?M ] =>
      xframe (s ~> M) end end.
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Tactic Notation "xframes" constr(s1) := 
  xframes_core_1 s1.
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Ltac xframes_core_2 s1 s2 := 
  match goal with |- ?R ?H ?Q =>
    match H with context [ s1 ~> ?M1 ] =>
      match H with context [ s2 ~> ?M2 ] =>
        xframe (s1 ~> M1 \* s2 ~> M2) end end end.
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Tactic Notation "xframes" constr(s1) constr(s2) := 
  xframes_core_2 s1 s2.
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Ltac xframes_core_3 s1 s2 s3 := 
  match goal with |- ?R ?H ?Q =>
    match H with context [ s1 ~> ?M1 ] =>
      match H with context [ s2 ~> ?M2 ] =>
        match H with context [ s3 ~> ?M3 ] =>
          xframe (s1 ~> M1 \* s2 ~> M2 \* s3 ~> M3) 
  end end end end.
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Tactic Notation "xframes" constr(s1) constr(s2) constr(s3) := 
  xframes_core_3 s1 s2 s3.
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(*--------------------------------------------------------*)
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(* ** [xapply] and [xapplys] *)
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(** [xapply E] applies a lemma [E] modulo frame/weakening. 
    It applies to a goal of the form [F H Q], and replaces it
    with [F ?H' ?Q'], applies [E] to the goal, then produces
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    the side condition [H ==> ?H'] and,
    - if [Q] is instantiated, then leaves [?Q' ===> Q \* \GC]
    - otherwise instantiates [Q] with [Q'].
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    [xapplys E] is like [xapply E] but also attemps to simplify
    the subgoals.
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*)
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Ltac xapply_core H cont1 cont2 :=
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  forwards_nounfold_then H ltac:(fun K =>
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    match cfml_postcondition_is_evar tt with
    | true => eapply local_frame; [ xlocal | sapply K | cont1 tt | try xok ]
    | false => eapply local_frame_gc; [ xlocal | sapply K | cont1 tt | cont2 tt ]
    end).
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Ltac xapply_base H :=
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  xextract_check_not_needed tt;
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  xapply_core H ltac:(fun tt => idtac) ltac:(fun tt => idtac). 
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Ltac xapplys_base H :=
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  xextract_check_not_needed tt;
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  xapply_core H ltac:(fun tt => hcancel) ltac:(fun tt => hsimpl). 
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Tactic Notation "xapply" constr(H) :=
  xapply_base H.
Tactic Notation "xapply" "~" constr(H) :=
  xapply H; xauto~.
Tactic Notation "xapply" "*" constr(H) :=
  xapply H; xauto*.
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Tactic Notation "xapplys" constr(H) :=
  xapplys_base H.
Tactic Notation "xapplys" "~" constr(H) :=
  xapplys H; xauto~.
Tactic Notation "xapplys" "*" constr(H) :=
  xapplys H; xauto*.
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(* -- commented out for faster compilation
Lemma xapply_demo_1 : forall B R H H' (Q:B->hprop),
  is_local R ->
  R H Q ->
  R (H \* H') (Q \*+ H').
Proof using.
  introv HR M. dup 2.
  { xapply M. xsimpl. xsimpl. }
  { xapplys M. }
Qed.
*)
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(*--------------------------------------------------------*)
(* ** [xchange] *)
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(** [xchange E] applies to a goal of the form [F H Q]
    and to a lemma [E] of type [H1 ==> H2] or [H1 = H2]. 
    It replaces the goal with [F H' Q], where [H']
    is computed by replacing [H1] with [H2] in [H].
    
    The substraction is computed by solving [H ==> H1 \* ?H']
    with [hsimpl]. If you need to solve this implication by hand,
    use [xchange_no_simpl E] instead.
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    [xchange <- E] is useful when [E] has type [H2 = H1]
      instead of [H1 = H2].
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    [xchange_show E] is useful to visualize the instantiation
    of the lemma used to implement [xchange].
    *)
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(* Lemma used by [xchange] *)
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Lemma xchange_lemma : forall H1 H1' H2 B H Q (F:~~B),
  is_local F -> 
  (H1 ==> H1') -> 
  (H ==> H1 \* H2) -> 
  F (H1' \* H2) Q -> 
  F H Q.
Proof using.
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  introv W1 L W2 M. applys local_frame __ \[]; eauto.
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  hsimpl. hchange~ W2. hsimpl~. rew_heap~. 
Qed.
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Ltac xchange_apply L cont :=
   eapply xchange_lemma; 
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     [ try xlocal | applys L | cont tt | xtag_goal ].
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  (* note: modif should be himpl_proj1 or himpl_proj2 
     ---what does this mean? *)
Ltac xchange_forwards L modif cont :=
  forwards_nounfold_then L ltac:(fun K =>
  match modif with
  | __ => 
     match type of K with
     | _ = _ => xchange_apply (@pred_le_proj1 _ _ _ K) cont
     | _ => xchange_apply K cont
     end
  | _ => xchange_apply (@modif _ _ _ K) cont
  end).
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Ltac xchange_with_core cont H H' :=
  eapply xchange_lemma with (H1:=H) (H1':=H'); 
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    [ try xlocal | | cont tt | xtag_goal ].
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Ltac xchange_core cont E modif :=
  match E with
  | ?H ==> ?H' => xchange_with_core cont H H'
  | _ => xchange_forwards E modif ltac:(fun _ => instantiate; cont tt)
  end.
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Ltac xchange_base cont E modif :=
  xextract_check_not_needed tt;
  match goal with
  | |- _ ==> _ => hchange_base E modif
  | |- _ ===> _ => hchange_base E modif
  | _ => xchange_core cont E modif
  end.
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Tactic Notation "xchange" constr(E) :=
  xchange_base ltac:(fun tt => hsimpl) E __.
Tactic Notation "xchange" "~" constr(E) :=
  xchange E; xauto~.
Tactic Notation "xchange" "*" constr(E) :=
  xchange E; xauto*.
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Tactic Notation "xchange" "<-" constr(E) :=
  xchange_base ltac:(fun tt => hsimpl) E pred_le_proj2.
Tactic Notation "xchange" "~" "<-" constr(E) :=
  xchange <- E; xauto~.
Tactic Notation "xchange" "*" "<-" constr(E) :=
  xchange <- E; xauto*.

Tactic Notation "xchange_no_simpl" constr(E) :=
  xchange_base ltac:(fun tt => idtac) E __.
Tactic Notation "xchange_no_simpl" "<-" constr(E) :=
  xchange_base ltac:(fun tt => idtac) E pred_le_proj2.

Tactic Notation "xchange_show" constr(E) :=
  xchange_forwards E __ ltac:(idcont).
Tactic Notation "xchange_show" "<-" constr(E) :=
  xchange_forwards pred_le_proj2 ltac:(idcont).
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(* DEPRECATED: 
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Tactic Notation "xchange" "->" constr(E) :=
  xchange_base ltac:(fun tt => hsimpl) E pred_le_proj1.
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Tactic Notation "xchange" constr(E) "as" := 
  xchange E; try xextract.
Tactic Notation "xchange" constr(E) "as" simple_intropattern(I1) := 
  xchange E; try xextract as I1.
Tactic Notation "xchange" constr(E) "as" simple_intropattern(I1) simple_intropattern(I2) 
  := 
  xchange E; try xextract as I1 I2.
Tactic Notation "xchange" constr(E) "as" simple_intropattern(I1) simple_intropattern(I2)
 simple_intropattern(I3) := 
  xchange E; try xextract as I1 I2 I3.
Tactic Notation "xchange" constr(E) "as" simple_intropattern(I1) simple_intropattern(I2)
 simple_intropattern(I3) simple_intropattern(I4) := 
  xchange E; try xextract as I1 I2 I3 I4. 
Tactic Notation "xchange" constr(E) "as" simple_intropattern(I1) simple_intropattern(I2)
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5) := 
  xchange E; try xextract as I1 I2 I3 I4 I5. 
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Tactic Notation "xchange" "~" constr(E) "as" := 
  xchange~ E; try xextract.
Tactic Notation "xchange" "~" constr(E) "as" simple_intropattern(I1) := 
  xchange~ E; try xextract as I1.
Tactic Notation "xchange" "~" constr(E) "as" simple_intropattern(I1) simple_intropattern(I2) := 
  xchange~ E; try xextract as I1 I2.
Tactic Notation "xchange" "~" constr(E) "as" simple_intropattern(I1) simple_intropattern(I2)
 simple_intropattern(I3) := 
  xchange~ E; try xextract as I1 I2 I3.
Tactic Notation "xchange" "~" constr(E) "as" simple_intropattern(I1) simple_intropattern(I2)
 simple_intropattern(I3) simple_intropattern(I4) := 
  xchange~ E; try xextract as I1 I2 I3 I4. 
Tactic Notation "xchange" "~" constr(E) "as" simple_intropattern(I1) simple_intropattern(I2)
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5) := 
  xchange~ E; try xextract as I1 I2 I3 I4 I5. 
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Tactic Notation "xchange" "*" constr(E) "as" := 
  xchange* E; try xextract.
Tactic Notation "xchange" "*" constr(E) "as" simple_intropattern(I1) := 
  xchange* E; try xextract as I1.
Tactic Notation "xchange" "*" constr(E) "as" simple_intropattern(I1) simple_intropattern(I2) := 
  xchange* E; try xextract as I1 I2.
Tactic Notation "xchange" "*" constr(E) "as" simple_intropattern(I1) simple_intropattern(I2)
 simple_intropattern(I3) := 
  xchange* E; try xextract as I1 I2 I3.
Tactic Notation "xchange" "*" constr(E) "as" simple_intropattern(I1) simple_intropattern(I2)
 simple_intropattern(I3) simple_intropattern(I4) := 
  xchange* E; try xextract as I1 I2 I3 I4. 
Tactic Notation "xchange" "*" constr(E) "as" simple_intropattern(I1) simple_intropattern(I2)
 simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5) := 
  xchange* E; try xextract as I1 I2 I3 I4 I5.
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*)
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(* DEPRECATED: chaining xchange operations.
Tactic Notation "xchange" constr(E1) constr(E2) :=
  xchange E1; xchange E2.
Tactic Notation "xchange" constr(E1) constr(E2) constr(E3) :=
  xchange E1; xchange E2 E3.
*)
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(************************************************************)
(* ** [xfocus] *)
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(** [xfocus] is a tactic for applying [xchange] without having
    to explicitly specify the name of a focusing lemma.
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    [xfocus p] applies to a goal of the form 
    [F H Q] or [H ==> H'] or [Q ===> Q'].
    It first searches for the pattern [p ~> T] in the pre-condition,
    then looks up in a database for the focus lemma [E] associated with
    the form [T], and then calls [xchange E].
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    [xfocus_show p] shows the lemma found, it is useful for debugging.
   
    Example for registering a focusing lemma:
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     Hint Extern 1 (Register focus (Tree _)) => 
       Provide tree_focus_contents.
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    Then, use: [xfocus p].  *)
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Ltac get_refocus_args tt :=
  match goal with 
  | |- ?Q1 ===> ?Q2 => constr:(Q1,Q2)
  | |- ?H1 ==> ?H2 => constr:(H1,H2)
  | |- ?R ?H1 ?Q2 => constr:(H1,Q2)
  (* TODO: gérer le cas de fonctions appliquées à R *)
  end.
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Ltac get_refocus_constr_in H t :=
  match H with context [ t ~> ?T ] => constr:(T) end.
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Ltac xfocus_constr t :=
  match get_refocus_args tt with (?H1,?H2) =>
  get_refocus_constr_in H1 t end.
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Ltac xfocus_core t :=
  let C1 := xfocus_constr t in
  ltac_database_get database_spec_focus C1;
  let K := fresh "TEMP" in
  intros K; xchange (K t); clear K.
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Ltac xfocus_show t := 
  let C1 := xfocus_constr t in
  pose C1; try ltac_database_get database_spec_focus C1; intros.
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Tactic Notation "xfocus" constr(t) :=
  xfocus_core t.
Tactic Notation "xfocus" "~" constr(t) :=
  xfocus t; xauto~.
Tactic Notation "xfocus" "*" constr(t) :=
  xfocus t; xauto*.
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(* DEPRECATED
Tactic Notation "xfocus" constr(t) "as" simple_intropattern(I1) :=
  xfocus t; xextract as I1.
Tactic Notation "xfocus" constr(t) "as" simple_intropattern(I1) simple_intropattern(I2) :=
  xfocus t; xextract as I1 I2.
Tactic Notation "xfocus" constr(t) "as" simple_intropattern(I1) simple_intropattern(I2)
  simple_intropattern(I3) :=
  xfocus t; xextract as I1 I2 I3.
Tactic Notation "xfocus" constr(t) "as" simple_intropattern(I1) simple_intropattern(I2)
  simple_intropattern(I3) simple_intropattern(I4) :=
  xfocus t; xextract as I1 I2 I3 I4.
Tactic Notation "xfocus" constr(t) "as" simple_intropattern(I1) simple_intropattern(I2)
  simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5) :=
  xfocus t; xextract as I1 I2 I3 I4 I5.
*)
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(************************************************************)
(* ** [xunfocus] *)
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(** [xunfocus] is the symmetricall of [xfocus]. It works in the 
    same way, except that it looks for an unfocusing lemma.
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    [xfocus p] applies to a goal of the form 
    [F H Q] or [H ==> H'] or [Q ===> Q'].
    It first searches for the pattern [p ~> T] in the pre-condition,
    then looks up in a database for the unfocus lemma [E] associated with
    the form [T], and then calls [xchange E].
   
    [xunfocus_show p] shows the lemma found, it is useful for debugging.
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    Example for registering a focusing lemma:
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     Hint Extern 1 (Register unfocus (Ref Id (MNode _ _ _))) => 
        Provide tree_node_unfocus.
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    Then, use: [xunfocus p]. *)
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Ltac xunfocus_constr t :=
  match get_refocus_args tt with (?H1,?H2) =>
  get_refocus_constr_in H1 t end.
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Ltac xunfocus_core t :=
  let C1 := xunfocus_constr t in
  ltac_database_get database_spec_unfocus C1;
  let K := fresh "TEMP" in
  intros K; xchange (K t); clear K.
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Ltac xunfocus_show t := 
  let C1 := xunfocus_constr t in
  pose C1; try ltac_database_get database_spec_unfocus C1; intros.
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Tactic Notation "xunfocus" constr(t) :=
  xunfocus_core t.
Tactic Notation "xunfocus" "~" constr(t) :=
  xunfocus t; xauto~.
Tactic Notation "xunfocus" "*" constr(t) :=
  xunfocus t; xauto*.
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(** It is often useful to chain unfocus operations.
    The syntax [xunfocus p1 .. pn] is provided for that purpose. *)
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Tactic Notation "xunfocus" constr(t1) constr(t2) :=
  xunfocus t1; xunfocus t2.
Tactic Notation "xunfocus" constr(t1) constr(t2) constr(t3) :=
  xunfocus t1; xunfocus t2 t3