Library stdpp.tactics

This file collects general purpose tactics that are used throughout the development.

From Coq Require Export Lia.
From stdpp Require Export decidable.
From stdpp Require Import options.

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.

Global Hint Extern 998 (_ = _) ⇒ f_equal : f_equal.
Global Hint Extern 999 ⇒ congruence : congruence.
Global Hint Extern 1000 ⇒ lia : lia.
Global Hint Extern 1001 ⇒ progress subst : subst.

backtracking on this one will be very bad, so use with care!

The tactic intuition expands to intuition auto with × by default. This is rather inefficient when having big hint databases, or expensive Hint Extern declarations as the ones above.

Ltac intuition_solver ::= auto.

The fast_reflexivity tactic only works on syntactically equal terms. It can be used to avoid expensive failing unification.

Ltac fast_reflexivity :=
  match goal with
  | |- _ ?x ?x ⇒ solve [simple apply reflexivity]
  end.

done can get slow as it calls "trivial". fast_done can solve way less goals, but it will also always finish quickly. We do 'reflexivity' last because for goals of the form ?x = y, if we have x = y in the context, we will typically want to use the assumption and not reflexivity

Ltac fast_done :=
  solve
    [ eassumption
    | symmetry; eassumption
    | apply not_symmetry; eassumption
    | reflexivity ].
Tactic Notation "fast_by" tactic(tac) :=
  tac; fast_done.

Class TCFastDone (P : Prop) : Prop := tc_fast_done : P.
Global Hint Extern 1 (TCFastDone ?P) ⇒ (change P; fast_done) : typeclass_instances.

We avoid calling done recursively as that can lead to an unresolved evar.

Global Hint Extern 0 (is_Some _) ⇒ eexists; fast_done : core.

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.

Aliases for transitivity and etransitivity that are easier to type

Tactic Notation "trans" constr(A) := transitivity A.
Tactic Notation "etrans" := etransitivity.

Tactics for splitting conjunctions:

split_and : split the goal if is syntactically of the shape _ ∧ _
split_and? : split the goal repeatedly (perhaps zero times) while it is of the shape _ ∧ _.
split_and! : 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.

destruct_and? H : destruct assumption H repeatedly (perhaps zero times) while it is of the shape _ ∧ _.
destruct_and! H : works similarly, but at least one destruct should succeed. In order to do so, it will head normalize the goal first to possibly expose a conjunction.
destruct_and? iterates destruct_or? H on every matching assumption H.
destruct_and! works similarly, but at least one destruct should succeed.

Tactic Notation "split_and" :=
  match goal with
  | |- _ ∧ _ ⇒ split
  | |- Is_true (_ && _) ⇒ apply andb_True; split
  end.
Tactic Notation "split_and" "?" := repeat split_and.
Tactic Notation "split_and" "!" := hnf; split_and; split_and?.

Ltac destruct_and_go H :=
  try lazymatch type of H with
  | True ⇒ clear H
  | _ ∧ _ ⇒
    let H1 := fresh in
    let H2 := fresh in
    destruct H as [ H1 H2 ];
    destruct_and_go H1; destruct_and_go H2
  | Is_true (bool_decide _) ⇒
    apply (bool_decide_unpack _) in H;
    destruct_and_go H
  | Is_true (_ && _) ⇒
    apply andb_True in H;
    destruct_and_go H
  end.

Tactic Notation "destruct_and" "?" ident(H) :=
  destruct_and_go H.
Tactic Notation "destruct_and" "!" ident(H) :=
  hnf in H; progress (destruct_and? H).

Tactic Notation "destruct_and" "?" :=
  repeat match goal with H : _ |- _ ⇒ progress (destruct_and? H) end.
Tactic Notation "destruct_and" "!" :=
  progress destruct_and?.

Tactics for splitting disjunctions in an assumption:

destruct_or? H : destruct the assumption H repeatedly (perhaps zero times) while it is of the shape _ ∨ _.
destruct_or! H : works similarly, but at least one destruct should succeed. In order to do so, it will head normalize the goal first to possibly expose a disjunction.
destruct_or? iterates destruct_or? H on every matching assumption H.
destruct_or! works similarly, but at least one destruct should succeed.

Tactic Notation "destruct_or" "?" ident(H) :=
  repeat match type of H with
  | False ⇒ destruct H
  | _ ∨ _ ⇒ destruct H as [H|H]
  | Is_true (bool_decide _) ⇒ apply (bool_decide_unpack _) in H
  | Is_true (_ || _) ⇒ apply orb_True in H; destruct H as [H|H]
  end.
Tactic Notation "destruct_or" "!" ident(H) := hnf in H; progress (destruct_or? H).

Tactic Notation "destruct_or" "?" :=
  repeat match goal with H : _ |- _ ⇒ progress (destruct_or? H) end.
Tactic Notation "destruct_or" "!" :=
  progress destruct_or?.

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.

Tactic Notation "case_match" "eqn" ":" ident(Hd) :=
  match goal with
  | H : context [ match ?x with _ ⇒ _ end ] |- _ ⇒ destruct x eqn:Hd
  | |- context [ match ?x with _ ⇒ _ end ] ⇒ destruct x eqn:Hd
  end.
Ltac case_match :=
  let H := fresh in case_match eqn:H.

Tactic Notation "case_guard" "as" ident(Hx) :=
  match goal with
  | H : context C [@guard_or ?E ?e ?M ?T ?R ?P ?dec] |- _ ⇒
      change (@guard_or E e M T R P dec) with (
        match @decide P dec with left H' ⇒ @mret M R P H' | _ ⇒ @mthrow E M T P e end) in *;
      destruct_decide (@decide P dec) as Hx
  | |- context C [@guard_or ?E ?e ?M ?T ?R ?P ?dec] ⇒
      change (@guard_or E e M T R P dec) with (
        match @decide P dec with left H' ⇒ @mret M R P H' | _ ⇒ @mthrow E M T P e end) in *;
      destruct_decide (@decide P dec) as Hx
  end.
Tactic Notation "case_guard" :=
  let H := fresh in case_guard as H.

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. This provides an n-ary variant of Coq's standard clear dependent.

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.

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.

The tactic mk_evar T returns a new evar of type T, without affecting the current context.

This is usually a more useful behavior than Coq's evar, which is a side-effecting tactic (not returning anything) that introduces a local definition into the context that holds the evar. Note that the obvious alternative open_constr (_:T) has subtly different behavior, see std++ issue 115.

Usually, Ltacs cannot return a value and have a side-effect, but we use the trick described at <https://stackoverflow.com/questions/45949064/check-for-evars-in-a-tactic-that-returns-a-value/4617888446178884> to work around that: wrap the side-effect in a [match goal].

Ltac mk_evar T :=
  let T := constr:(T : Type) in
  let e := fresh in
  let _ := match goal with _ ⇒ evar (e:T) end in
  let e' := eval unfold e in e in
  let _ := match goal with _ ⇒ clear e end in
  e'.

The tactic get_head t returns the head function f when t is of the shape f a1 ... aN. This is purely syntactic, no unification is performed.

Ltac get_head e :=
  lazymatch e with
  | ?h _ ⇒ get_head h
  | _ ⇒ e
  end.

The tactic eunify x y succeeds if x and y can be unified, and fails otherwise. If it succeeds, it will instantiate necessary evars in x and y.

Contrary to Coq's standard unify tactic, which uses constr for the arguments x and y, eunify uses open_constr so that one can use holes (i.e., _s). For example, it allows one to write eunify x (S _), which will test if x unifies a successor.

Tactic Notation "eunify" open_constr(x) open_constr(y) :=
unify x y.

The tactic no_new_unsolved_evars tac executes tac and fails if it creates any new evars or leaves behind any subgoals.

Ltac no_new_unsolved_evars tac := solve [unshelve tac].

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

Ltac fold_classes :=
  repeat match goal with
  | |- context [ ?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
  | context [ ?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.

The tactic simplify_eq repeatedly substitutes, discriminates, and injects equalities, and tries to contradict impossible inequalities.

Tactic Notation "simplify_eq" := repeat
  match goal with
  | H : _ ≠ _ |- _ ⇒ by case H; try clear H
  | H : _ = _ → False |- _ ⇒ by case H; try clear H
  | H : ?x = _ |- _ ⇒ subst x
  | H : _ = ?x |- _ ⇒ subst x
  | H : _ = _ |- _ ⇒ discriminate H
  | H : _ ≡ _ |- _ ⇒ apply leibniz_equiv in H
  | H : ?f _ = ?f _ |- _ ⇒ apply (inj f) in H
  | H : ?f _ _ = ?f _ _ |- _ ⇒ apply (inj2 f) in H; destruct H

  | H : ?f _ = ?f _ |- _ ⇒ progress injection H as H

  | H : ?f _ _ = ?f _ _ |- _ ⇒ progress injection H as H
  | H : ?f _ _ _ = ?f _ _ _ |- _ ⇒ progress injection H as H
  | H : ?f _ _ _ _ = ?f _ _ _ _ |- _ ⇒ progress injection H as H
  | H : ?f _ _ _ _ _ = ?f _ _ _ _ _ |- _ ⇒ progress injection H as H
  | H : ?f _ _ _ _ _ _ = ?f _ _ _ _ _ _ |- _ ⇒ progress injection H as H
  | H : ?x = ?x |- _ ⇒ clear H

  | H1 : ?o = Some ?x, H2 : ?o = Some ?y |- _ ⇒
    assert (y = x) by congruence; clear H2
  | H1 : ?o = Some ?x, H2 : ?o = None |- _ ⇒ congruence
  | H : @existT ?A _ _ _ = existT _ _ |- _ ⇒
     apply (Eqdep_dec.inj_pair2_eq_dec _ (decide_rel (=@{A}))) in H
  end.
Tactic Notation "simplify_eq" "/=" :=
  repeat (progress csimpl in × || simplify_eq).
Tactic Notation "f_equal" "/=" := csimpl in *; f_equal.

Ltac setoid_subst_aux R x :=
  match goal with
  | H : R x ?y |- _ ⇒
     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;
     clear x H
  end.
Ltac setoid_subst :=
  repeat match goal with
  | _ ⇒ progress simplify_eq/=
  | 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
  end.

A little helper for f_equiv and solve_proper that simplifies away flip relations.

Ltac clean_flip :=
  repeat match goal with
  | |- (flip ?R) ?x ?y ⇒ change (R y x)
  | H : (flip ?R) ?x ?y |- _ ⇒ change (R y x) in H
  end.

f_equiv works on goals of the form f _ = f _, for any relation and any number of arguments. It looks for an appropriate Proper instance, and applies it. The tactic is somewhat limited, since it cannot be used to backtrack on the Proper instances that has been found. To that end, we try to avoid the trivial instance in which the resulting goals have an eq. More generally, we try to "maintain" the relation of the current goal. For example, when having Proper (equiv ==> dist) f and Proper (dist ==> dist) f, it will favor the second because the relation (dist) stays the same.

Ltac f_equiv :=

  clean_flip;

  match goal with

  | |- _ ?x ?x ⇒ fast_reflexivity

  | |- pointwise_relation _ _ _ _ ⇒ intros ?

  | |- ?R (match ?x with _ ⇒ _ end) (match ?x with _ ⇒ _ end) ⇒
    destruct x
  | H : ?R ?x ?y |- ?R2 (match ?x with _ ⇒ _ end) (match ?y with _ ⇒ _ end) ⇒
     destruct H

  | |- ?R (?f _) (?f _) ⇒ simple apply (_ : Proper (R ==> R) f)
  | |- ?R (?f _ _) (?f _ _) ⇒ simple apply (_ : Proper (R ==> R ==> R) f)
  | |- ?R (?f _ _ _) (?f _ _ _) ⇒ simple apply (_ : Proper (R ==> R ==> R ==> R) f)
  | |- ?R (?f _ _ _ _) (?f _ _ _ _) ⇒ simple apply (_ : Proper (R ==> R ==> R ==> R ==> R) f)
  | |- ?R (?f _ _ _ _ _) (?f _ _ _ _ _) ⇒ simple apply (_ : Proper (R ==> R ==> R ==> R ==> R ==> R) f)

  | |- (?R _) (?f _) (?f _) ⇒ simple apply (_ : Proper (R _ ==> R _) f)
  | |- (?R _ _) (?f _) (?f _) ⇒ simple apply (_ : Proper (R _ _ ==> R _ _) f)
  | |- (?R _ _ _) (?f _) (?f _) ⇒ simple apply (_ : Proper (R _ _ _ ==> R _ _ _) f)

  | |- (?R _) (?f _ _) (?f _ _) ⇒ simple apply (_ : Proper (R _ ==> R _ ==> R _) f)
  | |- (?R _ _) (?f _ _) (?f _ _) ⇒ simple apply (_ : Proper (R _ _ ==> R _ _ ==> R _ _) f)
  | |- (?R _ _ _) (?f _ _) (?f _ _) ⇒ simple apply (_ : Proper (R _ _ _ ==> R _ _ _ ==> R _ _ _) f)

  | |- (?R _) (?f _ _ _) (?f _ _ _) ⇒ simple apply (_ : Proper (R _ ==> R _ ==> R _ ==> R _) f)
  | |- (?R _ _) (?f _ _ _) (?f _ _ _) ⇒ simple apply (_ : Proper (R _ _ ==> R _ _ ==> R _ _ ==> R _ _) f)
  | |- (?R _ _ _) (?f _ _ _) (?f _ _ _) ⇒ simple apply (_ : Proper (R _ _ _ ==> R _ _ _ ==> R _ _ _ ==> R _ _ _) f)

  | |- (?R _) (?f _ _ _ _) (?f _ _ _ _) ⇒ simple apply (_ : Proper (R _ ==> R _ ==> R _ ==> R _ ==> R _) f)
  | |- (?R _ _) (?f _ _ _ _) (?f _ _ _ _) ⇒ simple apply (_ : Proper (R _ _ ==> R _ _ ==> R _ _ ==> R _ _ ==> R _ _) f)
  | |- (?R _ _ _) (?f _ _ _ _) (?f _ _ _ _) ⇒ simple apply (_ : Proper (R _ _ _ ==> R _ _ _ ==> R _ _ _ ==> R _ _ _ ==> R _ _ _) f)

  | |- (?R _) (?f _ _ _ _ _) (?f _ _ _ _ _) ⇒ simple apply (_ : Proper (R _ ==> R _ ==> R _ ==> R _ ==> R _ ==> R _) f)
  | |- (?R _ _) (?f _ _ _ _ _) (?f _ _ _ _ _) ⇒ simple apply (_ : Proper (R _ _ ==> R _ _ ==> R _ _ ==> R _ _ ==> R _ _ ==> R _ _) f)
  | |- (?R _ _ _) (?f _ _ _ _ _) (?f _ _ _ _ _) ⇒ simple apply (_ : Proper (R _ _ _ ==> R _ _ _ ==> R _ _ _ ==> R _ _ _ ==> R _ _ _ ==> R _ _ _) f)

  | H : _ ?f ?g |- ?R (?f ?x) (?g ?x) ⇒ solve [simple apply H]
  | H : _ ?f ?g |- ?R (?f ?x ?y) (?g ?x ?y) ⇒ solve [simple apply H]



  | |- ?R (?f _) _ ⇒ simple apply (_ : Proper (_ ==> R) f)
  | |- ?R (?f _ _) _ ⇒ simple apply (_ : Proper (_ ==> _ ==> R) f)
  | |- ?R (?f _ _ _) _ ⇒ simple apply (_ : Proper (_ ==> _ ==> _ ==> R) f)
  | |- ?R (?f _ _ _ _) _ ⇒ simple apply (_ : Proper (_ ==> _ ==> _ ==> _ ==> R) f)
  | |- ?R (?f _ _ _ _ _) _ ⇒ simple apply (_ : Proper (_ ==> _ ==> _ ==> _ ==> _ ==> R) f)
  end;

  try fast_reflexivity.
Tactic Notation "f_equiv" "/=" := csimpl in *; f_equiv.

The typeclass SolveProperSubrelation is used by the solve_proper tactic when the goal is of the form R1 x y and there are assumptions of the form R2 x y. We cannot use Coq's subrelation class here as adding the subrelation instances causes lots of backtracking in the Proper hint search, resulting in very slow/diverging rewrites due to exponential instance search.

Class SolveProperSubrelation {A} (R R' : relation A) :=
is_solve_proper_subrelation x y : R x y → R' x y.

We use ! to handle indexed relations such as dist, where we can have an R n assumption and a R ?m goal.

Global Hint Mode SolveProperSubrelation + ! ! : typeclass_instances.
Global Arguments is_solve_proper_subrelation {A R R' _ x y}.

Global Instance subrelation_solve_proper_subrelation {A} (R R' : relation A) :
subrelation R R' →
SolveProperSubrelation R R'.
Proof. intros ???. apply is_subrelation. Qed.

The tactic solve_proper_unfold unfolds the first head symbol, so that we proceed by repeatedly using f_equiv.

Ltac solve_proper_unfold :=

  try lazymatch goal with
  | |- ?R ?t1 ?t2 ⇒
    let h1 := get_head t1 in
    let h2 := get_head t2 in
    unify h1 h2;
    unfold h1
  end.

solve_proper_prepare does some preparation work before the main solve_proper loop. Having this as a separate tactic is useful for debugging solve_proper failure.

Ltac solve_proper_prepare :=

  intros;
  repeat lazymatch goal with
  | |- Proper _ _ ⇒ intros ???
  | |- (_ ==> _)%signature _ _ ⇒ intros ???
  | |- pointwise_relation _ _ _ _ ⇒ intros ?
  | |- ?R ?f _ ⇒

     let f' := constr:(λ x, f x) in

     intros ?; intros
  end;

  simplify_eq;

  (solve_proper_unfold + idtac); simpl.

solve_proper_finish is basically a version of eassumption that can also take into account subrelation.

Ltac solve_proper_finish :=

  eassumption ||
  match goal with
  | H : ?R1 ?x ?y |- ?R2 ?x ?y ⇒
    no_new_unsolved_evars ltac:(eapply (is_solve_proper_subrelation H))
  end.

The tactic solve_proper_core tac solves goals of the form "Proper (R1 ==> R2)", for any number of relations. The actual work is done by repeatedly applying tac.

Ltac solve_proper_core tac :=
  solve_proper_prepare;

  solve [repeat (clean_flip; first [solve_proper_finish | tac ()]) ].

Finally, solve_proper tries to apply f_equiv in a loop.

Ltac solve_proper := solve_proper_core ltac:(fun _ ⇒ f_equiv).

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.

The tactic iter tac l runs tac x for each element x ∈ l until tac x succeeds. If it does not succeed 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.

Runs tac on the n-th hypothesis that can be introduced from the goal.

Ltac num_tac n tac :=
  intros until n;
  lazymatch goal with

  | H : _ |- _ ⇒ tac H
  end.

The tactic inv is a fixed version of inversion_clear from the standard library that works around <https://github.com/coq/coq/issues/2465>. It also has a shorter name since clearing is the default for destruct, why wouldn't it also be the default for inversion? This is inspired by CompCert's inv tactic <https://github.com/AbsInt/CompCert/blob/5f761eb8456609d102acd8bc780b6fd3481131ef/lib/Coqlib.vL30>.

Tactic Notation "inv" ident(H) "as" simple_intropattern(ipat) :=
  inversion H as ipat; clear H; simplify_eq.
Tactic Notation "inv" ident(H) :=
  inversion H; clear H; simplify_eq.

Tactic Notation "inv" integer(n) "as" simple_intropattern(ipat) :=
  num_tac n ltac:(fun H ⇒ inv H as ipat).
Tactic Notation "inv" integer(n) :=
  num_tac n ltac:(fun H ⇒ inv H).

The "o" family of tactics equips pose proof, destruct, inversion,

generalize and specialize with support for "o"pen terms. You can leave underscores that become evars or subgoals, similar to refine. You can suffix the tactic with × (e.g., opose proof×) to eliminate all remaining ∀ and → (i.e., add underscores for the remaining arguments). For odestruct and oinversion, eliminating all remaining ∀ and → is the default (hence there is no × version).

The helper opose_core p tac takes a uconstr p and turns it into a constr that is passed to tac. All underscores inside p become evars, and the ones that are unifiable (i.e, appear in the type of other evars) are shelved.

This is similar to creating a open_constr, except that we have control over what does and does not get shelved. Creating a open_constr would shelve every created evar, which is not what we want, and it is hard to avoid since it happens very early (before we can easily wrap things in unshelve).

Ltac opose_core p tac :=

  let i := fresh "opose_internal" in
  unshelve (epose _ as i);
    [shelve
    |refine p
    |
     let t := eval unfold i in i in

     clear i;
     tac t];

  shelve_unifiable.

Turn all leading ∀ and → of p into evars (∀-evars will be shelved), and call tac with the term applied with those evars. This fill unfold definitions to find leading ∀/→. The type of p will be normalized by calling the normalizer function.

_name_guard is an unused argument where you can pass anything you want. If the argument is an intro pattern, those will be taken into account by the fresh that is inside this tactic, avoiding name collisions that can otherwise arise. This is a work-around for https://github.com/coq/coq/issues/18109.

Ltac evar_foralls p _name_guard normalizer tac :=
  let T := type of p in
  lazymatch normalizer T with
  | ?T1 → ?T2 ⇒

    let pT1 := fresh "__evar_foralls_internal" in
    assert T1 as pT1; [| evar_foralls (p pT1) _name_guard normalizer tac; clear pT1]
  | ∀ x : ?T1, _ ⇒
    let e := mk_evar T1 in
    evar_foralls (p e) _name_guard normalizer tac
  | ?T1 ⇒ tac p
  end.

Ltac opose_specialize_foralls_core p _name_guard tac :=
  opose_core p ltac:(fun p ⇒
    evar_foralls p _name_guard ltac:(fun t ⇒ eval hnf in t) tac).

Tactic Notation "opose" "proof" uconstr(p) "as" simple_intropattern(pat) :=
  opose_core p ltac:(fun p ⇒ pose proof p as pat).
Tactic Notation "opose" "proof" "*" uconstr(p) "as" simple_intropattern(pat) :=
  opose_specialize_foralls_core p pat ltac:(fun p ⇒ pose proof p as pat).

Tactic Notation "opose" "proof" uconstr(p) := opose proof p as ?.
Tactic Notation "opose" "proof" "*" uconstr(p) := opose proof× p as ?.

Tactic Notation "ogeneralize" uconstr(p) :=
  opose_core p ltac:(fun p ⇒ generalize p).
Tactic Notation "ogeneralize" "*" uconstr(p) :=
  opose_specialize_foralls_core p () ltac:(fun p ⇒ generalize p).

Similar to edestruct, odestruct will never clear the destructed variable. No × versions for odestruct and oinversion: we always specialize all foralls and implications; otherwise it does not make sense to destruct/invert. We also do not support eqn:EQ; this would not make sense for most users of this tactic since the term being destructed is some_lemma ?evar ?proofterm.

Tactic Notation "odestruct" uconstr(p) :=
  opose_specialize_foralls_core p () ltac:(fun p ⇒ destruct p).
Tactic Notation "odestruct" uconstr(p) "as" simple_intropattern(pat) :=
  opose_specialize_foralls_core p pat ltac:(fun p ⇒ destruct p as pat).

Tactic Notation "oinversion" uconstr(p) "as" simple_intropattern(pat) :=
  opose_specialize_foralls_core p pat ltac:(fun p ⇒

    let Hp := fresh in pose proof p as Hp; inversion Hp as pat; clear Hp).
Tactic Notation "oinversion" uconstr(p) :=
  opose_specialize_foralls_core p () ltac:(fun p ⇒
    let Hp := fresh in pose proof p as Hp; inversion Hp; clear Hp).

Tactic Notation "oinv" uconstr(p) "as" simple_intropattern(pat) :=
  opose_specialize_foralls_core p pat ltac:(fun p ⇒

    tryif is_var p then
      inv p as pat
    else

      let Hp := fresh in pose proof p as Hp; inv Hp as pat).
Tactic Notation "oinv" uconstr(p) :=
  opose_specialize_foralls_core p () ltac:(fun p ⇒
    tryif is_var p then
      inv p
    else
      let Hp := fresh in pose proof p as Hp; inv Hp).

Tactic Notation "oinv" integer(n) "as" simple_intropattern(ipat) :=
  num_tac n ltac:(fun H ⇒ oinv H as ipat).
Tactic Notation "oinv" integer(n) :=
  num_tac n ltac:(fun H ⇒ oinv H).

Helper for ospecialize: call tac with the name of the head term *if* that term is a variable.

Written in CPS to get around weird thunking limitations.

Ltac ospecialize_ident_head_of t tac :=
  let h := get_head t in
  tryif is_var h then tac h else
    fail "ospecialize can only specialize a local hypothesis;"
         "use opose proof instead".

Tactic Notation "ospecialize" uconstr(p) :=

  opose_core p ltac:(fun p ⇒
    ospecialize_ident_head_of p ltac:(fun H ⇒

      let H' := fresh in
      pose proof p as H'; clear H; rename H' into H
  )).
Tactic Notation "ospecialize" "*" uconstr(p) :=
  opose_specialize_foralls_core p () ltac:(fun p ⇒
    ospecialize_ident_head_of p ltac:(fun H ⇒

      let H' := fresh in
      pose proof p as H'; clear H; rename H' into H
  )).

The block definitions are taken from Coq.Program.Equality and can be used by tactics to separate their goal from hypotheses they generalize over.

Definition block {A : Type} (a : A) := a.

Ltac block_goal := match goal with [ |- ?T ] ⇒ change (block T) end.
Ltac unblock_goal := unfold block in ×.

learn_hyp p as H and learn_hyp p, where p is a proof of P, add P to the context and fail if P already exists in the context. This is a simple form of the learning pattern. These tactics are inspired by Program.Tactics.add_hypothesis.

Tactic Notation "learn_hyp" constr(p) "as" ident(H') :=
  let P := type of p in
  match goal with
  | H : P |- _ ⇒ fail 1
  | _ ⇒ pose proof p as H'
  end.
Tactic Notation "learn_hyp" constr(p) :=
  let H := fresh in learn_hyp p as H.

The tactic select pat tac finds the last (i.e., bottommost) hypothesis matching pat and passes it to the continuation tac. Its main advantage over using match goal with directly is that it is shorter. If pat matches multiple hypotheses and tac fails, then select tac will not backtrack on subsequent matching hypotheses.

The tactic select is written in CPS and does not return the name of the hypothesis due to limitations in the Ltac1 tactic runtime (see https://gitter.im/coq/coq?at=5e96c82f85b01628f04bbb89).

Tactic Notation "select" open_constr(pat) tactic3(tac) :=
lazymatch goal with

Before running tac on the hypothesis H we must first unify the pattern pat with the term it matched against. This forces every evar coming from pat (and in particular from the holes _ it contains and from the implicit arguments it uses) to be instantiated. If we do not do so then shelved goals are produced for every such evar.

| H : pat |- _ ⇒ let T := (type of H) in unify T pat; tac H
end.

We provide select variants of some widely used tactics.

select_revert reverts the first hypothesis matching pat.

Tactic Notation "revert" "select" open_constr(pat) := select pat (fun H ⇒ revert H).

Tactic Notation "rename" "select" open_constr(pat) "into" ident(name) :=
  select pat (fun H ⇒ rename H into name).

Tactic Notation "destruct" "select" open_constr(pat) :=
  select pat (fun H ⇒ destruct H).
Tactic Notation "destruct" "select" open_constr(pat) "as" simple_intropattern(ipat) :=
  select pat (fun H ⇒ destruct H as ipat).

Tactic Notation "inversion" "select" open_constr(pat) :=
  select pat (fun H ⇒ inversion H).
Tactic Notation "inversion" "select" open_constr(pat) "as" simple_intropattern(ipat) :=
  select pat (fun H ⇒ inversion H as ipat).
Tactic Notation "inv" "select" open_constr(pat) :=
  select pat (fun H ⇒ inv H).
Tactic Notation "inv" "select" open_constr(pat) "as" simple_intropattern(ipat) :=
  select pat (fun H ⇒ inv H as ipat).

The tactic is_closed_term t succeeds if t is a closed term and fails otherwise. By closed we mean that t does not depend on any variable bound in the context. axioms are considered closed terms by this tactic (but Section variables are not). A function application is considered closed if the function and the argument are closed, without considering the body of the function (or whether it is opaque or not). This tactic is useful for example to decide whether to call vm_compute on t.

This trick was originally suggested by Jason Gross: https://coq.zulipchat.com/narrow/stream/237977-Coq-users/topic/Check.20that.20a.20term.20is.20closed.20in.20Ltac/near/240885618

Ltac is_closed_term t :=
first [

We use the assert_succeeds sandbox to be able to freely change the context.

assert_succeeds (

Make sure that the goal only contains t. (We use const False t instead of let x := t in False as the let-binding in the latter would be unfolded by the unfold later.)

exfalso; change_no_check (const False t);

Clear all hypotheses.

repeat match goal with H : _ |- _ ⇒ try unfold H in *; clear H end;

If there are still hypotheses left, t is not closed.

          lazymatch goal with H : _ |- _ ⇒ fail | _ ⇒ idtac end
        ) |
      fail 1 "The term" t "is not closed"
    ].

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 sets. 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.

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

  | |- _ ⇒ fast_done

  | |- ∀ _, _ ⇒ intro

  | H : False |- _ ⇒ destruct H
  | H : is_Some None |- _ ⇒ destruct H as [? [=]]
| H : _ ∧ _ |- _ ⇒

     let H1 := fresh in let H2 := fresh in
     destruct H as [H1 H2]; try clear H
  | H : ∃ _, _ |- _ ⇒
     let x := fresh in let Hx := fresh in
     destruct H as [x Hx]; try clear H
  | H : ?P → ?Q, H2 : ?P |- _ ⇒ specialize (H H2)
  | H : Is_true (bool_decide _) |- _ ⇒ apply (bool_decide_unpack _) in H
  | H : Is_true (_ && _) |- _ ⇒ apply andb_True in H; destruct H

  | |- _ ⇒ progress simplify_eq/=

  | |- _ ∧ _ ⇒ split
  | |- Is_true (bool_decide _) ⇒ apply (bool_decide_pack _)
  | |- Is_true (_ && _) ⇒ apply andb_True; split
  | H : _ ∨ _ |- _ ⇒
     let H1 := fresh in destruct H as [H1|H1]; try clear H
  | H : Is_true (_ || _) |- _ ⇒
     apply orb_True in H; let H1 := fresh in destruct H as [H1|H1]; try clear H

  | |- _ ⇒ no_new_unsolved_evars (tac)
  end;

  match goal with

  | |- ∃ x, _ ⇒ no_new_unsolved_evars ltac:(eexists; go n)
  | |- _ ∨ _ ⇒ first [left; go n | right; go n]
  | |- Is_true (_ || _) ⇒ apply orb_True; first [left; go n | right; go n]
  | _ ⇒

    lazymatch n with
    | S ?n' ⇒

      match goal with
      | H : _ → _ |- _ ⇒
        is_non_dependent H;
        no_new_unsolved_evars
          ltac:(first [eapply H | opose proof× H]; clear H; go n')
      end
    end
  end
  in iter (fun n' ⇒ go n') (eval compute in (seq 1 6)).
Tactic Notation "naive_solver" := naive_solver eauto.