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(* Copyright (c) 2012-2015, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file collects general purpose tactics that are used throughout
the development. *)
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From Coq Require Import Omega.
From Coq Require Export Psatz.
From prelude Require Export base.
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Lemma f_equal_dep {A B} (f g :  x : A, B x) x : f = g  f x = g x.
Proof. intros ->; reflexivity. Qed.
Lemma f_equal_help {A B} (f g : A  B) x y : f = g  x = y  f x = g y.
Proof. intros -> ->; reflexivity. Qed.
Ltac f_equal :=
  let rec go :=
    match goal with
    | _ => reflexivity
    | _ => apply f_equal_help; [go|try reflexivity]
    | |- ?f ?x = ?g ?x => apply (f_equal_dep f g); go
    end in
  try go.

(** We declare hint databases [f_equal], [congruence] and [lia] and containing
solely the tactic corresponding to its name. These hint database are useful in
to be combined in combination with other hint database. *)
Hint Extern 998 (_ = _) => f_equal : f_equal.
Hint Extern 999 => congruence : congruence.
Hint Extern 1000 => lia : lia.
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Hint Extern 1000 => omega : omega.
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Hint Extern 1001 => progress subst : subst. (** backtracking on this one will
be very bad, so use with care! *)
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(** The tactic [intuition] expands to [intuition auto with *] by default. This
is rather efficient when having big hint databases, or expensive [Hint Extern]
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declarations as the ones above. *)
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Tactic Notation "intuition" := intuition auto.

(** A slightly modified version of Ssreflect's finishing tactic [done]. It
also performs [reflexivity] and uses symmetry of negated equalities. Compared
to Ssreflect's [done], it does not compute the goal's [hnf] so as to avoid
unfolding setoid equalities. Note that this tactic performs much better than
Coq's [easy] tactic as it does not perform [inversion]. *)
Ltac done :=
  trivial; intros; solve
  [ repeat first
    [ solve [trivial]
    | solve [symmetry; trivial]
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    | eassumption
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    | reflexivity
    | discriminate
    | contradiction
    | solve [apply not_symmetry; trivial]
    | split ]
  | match goal with H : ¬_ |- _ => solve [destruct H; trivial] end ].
Tactic Notation "by" tactic(tac) :=
  tac; done.

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(** Aliases for trans and etrans that are easier to type *)
Tactic Notation "trans" constr(A) := transitivity A.
Tactic Notation "etrans" := etransitivity.

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(** Tactics for splitting conjunctions:

- [split_and] : split the goal if is syntactically of the shape [_  _]
- [split_ands?] : split the goal repeatedly (perhaps zero times) while it is
  of the shape [_  _].
- [split_ands!] : works similarly, but at least one split should succeed. In
  order to do so, it will head normalize the goal first to possibly expose a
  conjunction.

Note that [split_and] differs from [split] by only splitting conjunctions. The
[split] tactic splits any inductive with one constructor. *)
Tactic Notation "split_and" := match goal with |- _  _ => split end.
Tactic Notation "split_and" "?" := repeat split_and.
Tactic Notation "split_and" "!" := hnf; split_and; split_and?.
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(** The tactic [case_match] destructs an arbitrary match in the conclusion or
assumptions, and generates a corresponding equality. This tactic is best used
together with the [repeat] tactical. *)
Ltac case_match :=
  match goal with
  | H : context [ match ?x with _ => _ end ] |- _ => destruct x eqn:?
  | |- context [ match ?x with _ => _ end ] => destruct x eqn:?
  end.

(** The tactic [unless T by tac_fail] succeeds if [T] is not provable by
the tactic [tac_fail]. *)
Tactic Notation "unless" constr(T) "by" tactic3(tac_fail) :=
  first [assert T by tac_fail; fail 1 | idtac].

(** The tactic [repeat_on_hyps tac] repeatedly applies [tac] in unspecified
order on all hypotheses until it cannot be applied to any hypothesis anymore. *)
Tactic Notation "repeat_on_hyps" tactic3(tac) :=
  repeat match goal with H : _ |- _ => progress tac H end.

(** The tactic [clear dependent H1 ... Hn] clears the hypotheses [Hi] and
their dependencies. *)
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) :=
  clear dependent H1; clear dependent H2.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) :=
  clear dependent H1 H2; clear dependent H3.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) hyp(H4) :=
  clear dependent H1 H2 H3; clear dependent H4.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) hyp(H4)
  hyp(H5) := clear dependent H1 H2 H3 H4; clear dependent H5.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) hyp(H4) hyp(H5)
  hyp (H6) := clear dependent H1 H2 H3 H4 H5; clear dependent H6.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) hyp(H4) hyp(H5)
  hyp (H6) hyp(H7) := clear dependent H1 H2 H3 H4 H5 H6; clear dependent H7.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) hyp(H4) hyp(H5)
  hyp (H6) hyp(H7) hyp(H8) :=
  clear dependent H1 H2 H3 H4 H5 H6 H7; clear dependent H8.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) hyp(H4) hyp(H5)
  hyp (H6) hyp(H7) hyp(H8) hyp(H9) :=
  clear dependent H1 H2 H3 H4 H5 H6 H7 H8; clear dependent H9.
Tactic Notation "clear" "dependent" hyp(H1) hyp(H2) hyp(H3) hyp(H4) hyp(H5)
  hyp (H6) hyp(H7) hyp(H8) hyp(H9) hyp(H10) :=
  clear dependent H1 H2 H3 H4 H5 H6 H7 H8 H9; clear dependent H10.

(** The tactic [is_non_dependent H] determines whether the goal's conclusion or
hypotheses depend on [H]. *)
Tactic Notation "is_non_dependent" constr(H) :=
  match goal with
  | _ : context [ H ] |- _ => fail 1
  | |- context [ H ] => fail 1
  | _ => idtac
  end.

(** The tactic [var_eq x y] fails if [x] and [y] are unequal, and [var_neq]
does the converse. *)
Ltac var_eq x1 x2 := match x1 with x2 => idtac | _ => fail 1 end.
Ltac var_neq x1 x2 := match x1 with x2 => fail 1 | _ => idtac end.

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(** Operational type class projections in recursive calls are not folded back
appropriately by [simpl]. The tactic [csimpl] uses the [fold_classes] tactics
to refold recursive calls of [fmap], [mbind], [omap] and [alter]. A
self-contained example explaining the problem can be found in the following
Coq-club message:

https://sympa.inria.fr/sympa/arc/coq-club/2012-10/msg00147.html *)
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Ltac fold_classes :=
  repeat match goal with
  | |- appcontext [ ?F ] =>
    progress match type of F with
    | FMap _ =>
       change F with (@fmap _ F);
       repeat change (@fmap _ (@fmap _ F)) with (@fmap _ F)
    | MBind _ =>
       change F with (@mbind _ F);
       repeat change (@mbind _ (@mbind _ F)) with (@mbind _ F)
    | OMap _ =>
       change F with (@omap _ F);
       repeat change (@omap _ (@omap _ F)) with (@omap _ F)
    | Alter _ _ _ =>
       change F with (@alter _ _ _ F);
       repeat change (@alter _ _ _ (@alter _ _ _ F)) with (@alter _ _ _ F)
    end
  end.
Ltac fold_classes_hyps H :=
  repeat match type of H with
  | appcontext [ ?F ] =>
    progress match type of F with
    | FMap _ =>
       change F with (@fmap _ F) in H;
       repeat change (@fmap _ (@fmap _ F)) with (@fmap _ F) in H
    | MBind _ =>
       change F with (@mbind _ F) in H;
       repeat change (@mbind _ (@mbind _ F)) with (@mbind _ F) in H
    | OMap _ =>
       change F with (@omap _ F) in H;
       repeat change (@omap _ (@omap _ F)) with (@omap _ F) in H
    | Alter _ _ _ =>
       change F with (@alter _ _ _ F) in H;
       repeat change (@alter _ _ _ (@alter _ _ _ F)) with (@alter _ _ _ F) in H
    end
  end.
Tactic Notation "csimpl" "in" hyp(H) :=
  try (progress simpl in H; fold_classes_hyps H).
Tactic Notation "csimpl" := try (progress simpl; fold_classes).
Tactic Notation "csimpl" "in" "*" :=
  repeat_on_hyps (fun H => csimpl in H); csimpl.

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(** The tactic [simplify_eq] repeatedly substitutes, discriminates,
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and injects equalities, and tries to contradict impossible inequalities. *)
Tactic Notation "simplify_eq" := repeat
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  match goal with
  | H : _  _ |- _ => by destruct H
  | H : _ = _  False |- _ => by destruct H
  | H : ?x = _ |- _ => subst x
  | H : _ = ?x |- _ => subst x
  | H : _ = _ |- _ => discriminate H
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  | H : ?f _ = ?f _ |- _ => apply (inj f) in H
  | H : ?f _ _ = ?f _ _ |- _ => apply (inj2 f) in H; destruct H
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    (* before [injection] to circumvent bug #2939 in some situations *)
  | H : ?f _ = ?f _ |- _ => injection H as H
    (* first hyp will be named [H], subsequent hyps will be given fresh names *)
  | H : ?f _ _ = ?f _ _ |- _ => injection H as H
  | H : ?f _ _ _ = ?f _ _ _ |- _ => injection H as H
  | H : ?f _ _ _ _ = ?f _ _ _ _ |- _ => injection H as H
  | H : ?f _ _ _ _ _ = ?f _ _ _ _ _ |- _ => injection H as H
  | H : ?f _ _ _ _ _ _ = ?f _ _ _ _ _ _ |- _ => injection H as H
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  | H : ?x = ?x |- _ => clear H
    (* unclear how to generalize the below *)
  | H1 : ?o = Some ?x, H2 : ?o = Some ?y |- _ =>
    assert (y = x) by congruence; clear H2
  | H1 : ?o = Some ?x, H2 : ?o = None |- _ => congruence
  end.
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Tactic Notation "simplify_eq" "/=" :=
  repeat (progress csimpl in * || simplify_eq).
Tactic Notation "f_equal" "/=" := csimpl in *; f_equal.
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Ltac setoid_subst_aux R x :=
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  match goal with
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  | H : R x ?y |- _ =>
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     is_var x;
     try match y with x _ => fail 2 end;
     repeat match goal with
     | |- context [ x ] => setoid_rewrite H
     | H' : context [ x ] |- _ =>
        try match H' with H => fail 2 end;
        setoid_rewrite H in H'
     end;
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     clear x H
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  end.
Ltac setoid_subst :=
  repeat match goal with
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  | _ => progress simplify_eq/=
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  | H : @equiv ?A ?e ?x _ |- _ => setoid_subst_aux (@equiv A e) x
  | H : @equiv ?A ?e _ ?x |- _ => symmetry in H; setoid_subst_aux (@equiv A e) x
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  end.

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(** f_equiv solves goals of the form "f _ = f _", for any relation and any
    number of arguments, by looking for appropriate "Proper" instances.
    If it cannot solve an equality, it will leave that to the user. *)
Ltac f_equiv :=
  (* Deal with "pointwise_relation" *)
  try lazymatch goal with
    | |- pointwise_relation _ _ _ _ => intros ?
    end;
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  (* Normalize away equalities. *)
  subst;
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  (* repeatedly apply congruence lemmas and use the equalities in the hypotheses. *)
  first [ reflexivity | assumption | symmetry; assumption |
          match goal with
          (* We support matches on both sides, *if* they concern the same
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             variable.
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             TODO: We should support different variables, provided that we can
             derive contradictions for the off-diagonal cases. *)
          | |- ?R (match ?x with _ => _ end) (match ?x with _ => _ end) =>
            destruct x; f_equiv
          (* First assume that the arguments need the same relation as the result *)
          | |- ?R (?f ?x) (?f _) =>
            let H := fresh "Proper" in
            assert (Proper (R ==> R) f) as H by (eapply _);
            apply H; clear H; f_equiv
          | |- ?R (?f ?x ?y) (?f _ _) =>
            let H := fresh "Proper" in
            assert (Proper (R ==> R ==> R) f) as H by (eapply _);
            apply H; clear H; f_equiv
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          (* Next, try to infer the relation. Unfortunately, there is an instance
             of Proper for (eq ==> _), which will always be matched. *)
          (* TODO: Can we exclude that instance? *)
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          (* TODO: If some of the arguments are the same, we could also
             query for "pointwise_relation"'s. But that leads to a combinatorial
             explosion about which arguments are and which are not the same. *)
          | |- ?R (?f ?x) (?f _) =>
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            let R1 := fresh "R" in let H := fresh "HProp" in
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            let T := type of x in evar (R1: relation T);
            assert (Proper (R1 ==> R) f) as H by (subst R1; eapply _);
            subst R1; apply H; clear H; f_equiv
          | |- ?R (?f ?x ?y) (?f _ _) =>
            let R1 := fresh "R" in let R2 := fresh "R" in
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            let H := fresh "HProp" in
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            let T1 := type of x in evar (R1: relation T1);
            let T2 := type of y in evar (R2: relation T2);
            assert (Proper (R1 ==> R2 ==> R) f) as H by (subst R1 R2; eapply _);
            subst R1 R2; apply H; clear H; f_equiv
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           (* In case the function symbol differs, but the arguments are the same,
              maybe we have a pointwise_relation in our context. *)
           | H : pointwise_relation _ ?R ?f ?g |- ?R (?f ?x) (?g ?x) =>
             apply H; f_equiv
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         end | idtac (* Let the user solve this goal *)
        ].

(** solve_proper solves goals of the form "Proper (R1 ==> R2)", for any
    number of relations. All the actual work is done by f_equiv;
    solve_proper just introduces the assumptions and unfolds the first
    head symbol. *)
Ltac solve_proper :=
  (* Introduce everything *)
  intros;
  repeat lazymatch goal with
         | |- Proper _ _ => intros ???
         | |- (_ ==> _)%signature _ _ => intros ???
         end;
  (* Unfold the head symbol, which is the one we are proving a new property about *)
  lazymatch goal with
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  | |- ?R (?f _ _ _ _ _ _ _ _) (?f _ _ _ _ _ _ _ _) => unfold f
  | |- ?R (?f _ _ _ _ _ _ _) (?f _ _ _ _ _ _ _) => unfold f
  | |- ?R (?f _ _ _ _ _ _) (?f _ _ _ _ _ _) => unfold f
  | |- ?R (?f _ _ _ _ _) (?f _ _ _ _ _) => unfold f
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  | |- ?R (?f _ _ _ _) (?f _ _ _ _) => unfold f
  | |- ?R (?f _ _ _) (?f _ _ _) => unfold f
  | |- ?R (?f _ _) (?f _ _) => unfold f
  | |- ?R (?f _) (?f _) => unfold f
  end;
  solve [ f_equiv ].

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(** The tactic [intros_revert tac] introduces all foralls/arrows, performs tac,
and then reverts them. *)
Ltac intros_revert tac :=
  lazymatch goal with
  | |-  _, _ => let H := fresh in intro H; intros_revert tac; revert H
  | |- _ => tac
  end.
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(** Given a tactic [tac2] generating a list of terms, [iter tac1 tac2]
runs [tac x] for each element [x] until [tac x] succeeds. If it does not
suceed for any element of the generated list, the whole tactic wil fail. *)
Tactic Notation "iter" tactic(tac) tactic(l) :=
  let rec go l :=
  match l with ?x :: ?l => tac x || go l end in go l.

(** Given H : [A_1  ...  A_n  B] (where each [A_i] is non-dependent), the
tactic [feed tac H tac_by] creates a subgoal for each [A_i] and calls [tac p]
with the generated proof [p] of [B]. *)
Tactic Notation "feed" tactic(tac) constr(H) :=
  let rec go H :=
  let T := type of H in
  lazymatch eval hnf in T with
  | ?T1  ?T2 =>
    (* Use a separate counter for fresh names to make it more likely that
    the generated name is "fresh" with respect to those generated before
    calling the [feed] tactic. In particular, this hack makes sure that
    tactics like [let H' := fresh in feed (fun p => pose proof p as H') H] do
    not break. *)
    let HT1 := fresh "feed" in assert T1 as HT1;
      [| go (H HT1); clear HT1 ]
  | ?T1 => tac H
  end in go H.

(** The tactic [efeed tac H] is similar to [feed], but it also instantiates
dependent premises of [H] with evars. *)
Tactic Notation "efeed" constr(H) "using" tactic3(tac) "by" tactic3 (bytac) :=
  let rec go H :=
  let T := type of H in
  lazymatch eval hnf in T with
  | ?T1  ?T2 =>
    let HT1 := fresh "feed" in assert T1 as HT1;
      [bytac | go (H HT1); clear HT1 ]
  | ?T1  _ =>
    let e := fresh "feed" in evar (e:T1);
    let e' := eval unfold e in e in
    clear e; go (H e')
  | ?T1 => tac H
  end in go H.
Tactic Notation "efeed" constr(H) "using" tactic3(tac) :=
  efeed H using tac by idtac.

(** The following variants of [pose proof], [specialize], [inversion], and
[destruct], use the [feed] tactic before invoking the actual tactic. *)
Tactic Notation "feed" "pose" "proof" constr(H) "as" ident(H') :=
  feed (fun p => pose proof p as H') H.
Tactic Notation "feed" "pose" "proof" constr(H) :=
  feed (fun p => pose proof p) H.

Tactic Notation "efeed" "pose" "proof" constr(H) "as" ident(H') :=
  efeed H using (fun p => pose proof p as H').
Tactic Notation "efeed" "pose" "proof" constr(H) :=
  efeed H using (fun p => pose proof p).

Tactic Notation "feed" "specialize" hyp(H) :=
  feed (fun p => specialize p) H.
Tactic Notation "efeed" "specialize" hyp(H) :=
  efeed H using (fun p => specialize p).

Tactic Notation "feed" "inversion" constr(H) :=
  feed (fun p => let H':=fresh in pose proof p as H'; inversion H') H.
Tactic Notation "feed" "inversion" constr(H) "as" simple_intropattern(IP) :=
  feed (fun p => let H':=fresh in pose proof p as H'; inversion H' as IP) H.

Tactic Notation "feed" "destruct" constr(H) :=
  feed (fun p => let H':=fresh in pose proof p as H'; destruct H') H.
Tactic Notation "feed" "destruct" constr(H) "as" simple_intropattern(IP) :=
  feed (fun p => let H':=fresh in pose proof p as H'; destruct H' as IP) H.

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(** The following tactic can be used to add support for patterns to tactic notation:
It will search for the first subterm of the goal matching [pat], and then call [tac]
with that subterm. *)
Ltac find_pat pat tac :=
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  match goal with |- context [?x] =>
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                  unify pat x with typeclass_instances;
                  tryif tac x then idtac else fail 2
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end.

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(** Coq's [firstorder] tactic fails or loops on rather small goals already. In 
particular, on those generated by the tactic [unfold_elem_ofs] which is used
to solve propositions on collections. The [naive_solver] tactic implements an
ad-hoc and incomplete [firstorder]-like solver using Ltac's backtracking
mechanism. The tactic suffers from the following limitations:
- It might leave unresolved evars as Ltac provides no way to detect that.
- To avoid the tactic becoming too slow, we allow a universally quantified
  hypothesis to be instantiated only once during each search path.
- It does not perform backtracking on instantiation of universally quantified
  assumptions.

We use a counter to make the search breath first. Breath first search ensures
that a minimal number of hypotheses is instantiated, and thus reduced the
posibility that an evar remains unresolved.

Despite these limitations, it works much better than Coq's [firstorder] tactic
for the purposes of this development. This tactic either fails or proves the
goal. *)
Lemma forall_and_distr (A : Type) (P Q : A  Prop) :
  ( x, P x  Q x)  ( x, P x)  ( x, Q x).
Proof. firstorder. Qed.

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(** The tactic [no_new_unsolved_evars tac] executes [tac] and fails if it
creates any new evars. This trick is by Jonathan Leivent, see:
https://coq.inria.fr/bugs/show_bug.cgi?id=3872 *)

Ltac no_new_unsolved_evars tac := exact ltac:(tac).

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Tactic Notation "naive_solver" tactic(tac) :=
  unfold iff, not in *;
  repeat match goal with
  | H : context [ _, _  _ ] |- _ =>
    repeat setoid_rewrite forall_and_distr in H; revert H
  end;
  let rec go n :=
  repeat match goal with
  (**i intros *)
  | |-  _, _ => intro
  (**i simplification of assumptions *)
  | H : False |- _ => destruct H
  | H : _  _ |- _ => destruct H
  | H :  _, _  |- _ => destruct H
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  | H : ?P  ?Q, H2 : ?P |- _ => specialize (H H2)
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  (**i simplify and solve equalities *)
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  | |- _ => progress simplify_eq/=
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  (**i solve the goal *)
  | |- _ =>
    solve
    [ eassumption
    | symmetry; eassumption
    | apply not_symmetry; eassumption
    | reflexivity ]
  (**i operations that generate more subgoals *)
  | |- _  _ => split
  | H : _  _ |- _ => destruct H
  (**i solve the goal using the user supplied tactic *)
  | |- _ => solve [tac]
  end;
  (**i use recursion to enable backtracking on the following clauses. *)
  match goal with
  (**i instantiation of the conclusion *)
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  | |-  x, _ => no_new_unsolved_evars ltac:(eexists; go n)
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  | |- _  _ => first [left; go n | right; go n]
  | _ =>
    (**i instantiations of assumptions. *)
    lazymatch n with
    | S ?n' =>
      (**i we give priority to assumptions that fit on the conclusion. *)
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      match goal with
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      | H : _  _ |- _ =>
        is_non_dependent H;
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        no_new_unsolved_evars
          ltac:(first [eapply H | efeed pose proof H]; clear H; go n')
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      end
    end
  end
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  in iter (fun n' => go n') (eval compute in (seq 1 6)).
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Tactic Notation "naive_solver" := naive_solver eauto.