(* 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. *)
From Coq Require Import Omega.
From Coq Require Export Lia.
From stdpp Require Export decidable.
Set Default Proof Using "Type*".
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.
Hint Extern 1000 => omega : omega.
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 efficient when having big hint databases, or expensive [Hint Extern]
declarations as the ones above. *)
Tactic Notation "intuition" := intuition auto.
(* [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.
(** 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 :=
solve
[ repeat first
[ fast_done
| solve [trivial]
(* All the tactics below will introduce themselves anyway, or make no sense
for goals of product type. So this is a good place for us to do it. *)
| progress intros
| solve [symmetry; trivial]
| solve [apply not_symmetry; trivial]
| discriminate
| contradiction
| split
| match goal with H : ¬_ |- _ => case H; clear H; fast_done end ]
].
Tactic Notation "by" tactic(tac) :=
tac; done.
(** Aliases for trans and etrans 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_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
| |- Is_true (_ && _) => apply andb_True; split
end.
Tactic Notation "split_and" "?" := repeat split_and.
Tactic Notation "split_and" "!" := hnf; split_and; split_and?.
Tactic Notation "destruct_and" "?" :=
repeat match goal with
| H : False |- _ => destruct H
| H : _ ∧ _ |- _ => destruct H
| H : Is_true (bool_decide _) |- _ => apply (bool_decide_unpack _) in H
| H : Is_true (_ && _) |- _ => apply andb_True in H; destruct H
end.
Tactic Notation "destruct_and" "!" := progress (destruct_and?).
(** 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.
(** 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
(* before [injection] to circumvent bug #2939 in some situations *)
| H : ?f _ = ?f _ |- _ => progress injection H as H
(* first hyp will be named [H], subsequent hyps will be given fresh names *)
| 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
(* 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
| H : @existT ?A _ _ _ = existT _ _ |- _ =>
apply (Eqdep_dec.inj_pair2_eq_dec _ (decide_rel (@eq 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.
(** 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 :=
match goal with
| _ => reflexivity
| |- pointwise_relation _ _ _ _ => intros ?
(* We support matches on both sides, *if* they concern the same variable, or
variables in some relation. *)
| |- ?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
(* First assume that the arguments need the same relation as the result *)
| |- ?R (?f ?x) (?f _) => apply (_ : Proper (R ==> R) f)
(* For the case in which R is polymorphic, or an operational type class,
like equiv. *)
| |- (?R _) (?f ?x) (?f _) => apply (_ : Proper (R _ ==> _) f)
| |- (?R _ _) (?f ?x) (?f _) => apply (_ : Proper (R _ _ ==> _) f)
| |- (?R _ _ _) (?f ?x) (?f _) => apply (_ : Proper (R _ _ _ ==> _) f)
| |- (?R _) (?f ?x ?y) (?f _ _) => apply (_ : Proper (R _ ==> R _ ==> _) f)
| |- (?R _ _) (?f ?x ?y) (?f _ _) => apply (_ : Proper (R _ _ ==> R _ _ ==> _) f)
| |- (?R _ _ _) (?f ?x ?y) (?f _ _) => apply (_ : Proper (R _ _ _ ==> R _ _ _ ==> _) f)
| |- (?R _ _ _ _) (?f ?x ?y) (?f _ _) => apply (_ : Proper (R _ _ _ _ ==> R _ _ _ _ ==> _) f)
(* 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? *)
(* 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 _) => apply (_ : Proper (_ ==> R) f)
| |- ?R (?f ?x ?y) (?f _ _) => apply (_ : Proper (_ ==> _ ==> R) f)
(* 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
end;
try reflexivity.
(* The tactic [preprocess_solve_proper] unfolds the first head symbol, so that
we proceed by repeatedly using [f_equiv]. *)
Ltac preprocess_solve_proper :=
(* Introduce everything *)
intros;
repeat lazymatch goal with
| |- Proper _ _ => intros ???
| |- (_ ==> _)%signature _ _ => intros ???
| |- pointwise_relation _ _ _ _ => intros ?
| |- ?R ?f _ => try let f' := constr:(λ x, f x) in intros ?
end; simpl;
(* Unfold the head symbol, which is the one we are proving a new property about *)
lazymatch goal with
| |- ?R (?f _ _ _ _ _ _ _ _) (?f _ _ _ _ _ _ _ _) => unfold f
| |- ?R (?f _ _ _ _ _ _ _) (?f _ _ _ _ _ _ _) => unfold f
| |- ?R (?f _ _ _ _ _ _) (?f _ _ _ _ _ _) => unfold f
| |- ?R (?f _ _ _ _ _) (?f _ _ _ _ _) => unfold f
| |- ?R (?f _ _ _ _) (?f _ _ _ _) => unfold f
| |- ?R (?f _ _ _) (?f _ _ _) => unfold f
| |- ?R (?f _ _) (?f _ _) => unfold f
| |- ?R (?f _) (?f _) => unfold f
end;
simplify_eq.
(** The tactic [solve_proper] solves goals of the form "Proper (R1 ==> R2)", for
any number of relations. The actual work is done by repeatedly applying
[f_equiv]. *)
Ltac solve_proper :=
preprocess_solve_proper;
solve [repeat (f_equiv; try eassumption)].
(** 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.
(** 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.
(** 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 *.
(** 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 :=
match goal with
|- context [?x] =>
unify pat x with typeclass_instances;
tryif tac x then idtac else fail 2
end.
(** 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.
(** 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).
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 : _ ∧ _ |- _ =>
(* Work around bug https://coq.inria.fr/bugs/show_bug.cgi?id=2901 *)
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
(**i simplify and solve equalities *)
| |- _ => progress simplify_eq/=
(**i solve the goal *)
| |- _ => fast_done
(**i operations that generate more subgoals *)
| |- _ ∧ _ => 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
(**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 *)
| |- ∃ x, _ => no_new_unsolved_evars ltac:(eexists; go n)
| |- _ ∨ _ => 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. *)
match goal with
| H : _ → _ |- _ =>
is_non_dependent H;
no_new_unsolved_evars
ltac:(first [eapply H | efeed pose 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.