Library Coq.Init.Tactics

Require Import Notations.
Require Import Logic.
Require Import Specif.

Useful tactics

Ex falso quodlibet : a tactic for proving False instead of the current goal. This is just a nicer name for tactics such as elimtype False and other cut False.

Ltac exfalso := elimtype False.

A tactic for proof by contradiction. With contradict H,
• H:~A |- B gives |- A
• H:~A |- ~B gives H: B |- A
• H: A |- B gives |- ~A
• H: A |- ~B gives H: B |- ~A
• H:False leads to a resolved subgoal.
Moreover, negations may be in unfolded forms, and A or B may live in Type

Ltac contradict H :=
let save tac H := let x:=fresh in intro x; tac H; rename x into H
in
let negpos H := case H; clear H
in
let negneg H := save negpos H
in
let pospos H :=
let A := type of H in (exfalso; revert H; try fold (~A))
in
let posneg H := save pospos H
in
let neg H := match goal with
| |- (~_) => negneg H
| |- (_->False) => negneg H
| |- _ => negpos H
end in
let pos H := match goal with
| |- (~_) => posneg H
| |- (_->False) => posneg H
| |- _ => pospos H
end in
match type of H with
| (~_) => neg H
| (_->False) => neg H
| _ => (elim H;fail) || pos H
end.

Ltac swap H :=
idtac "swap is OBSOLETE: use contradict instead.";
intro; apply H; clear H.

Ltac absurd_hyp H :=
idtac "absurd_hyp is OBSOLETE: use contradict instead.";
let T := type of H in
absurd T.

Ltac false_hyp H G :=
let T := type of H in absurd T; [ apply G | assumption ].

Ltac case_eq x := generalize (refl_equal x); pattern x at -1; case x.

Tactic Notation "destruct_with_eqn" constr(x) :=
destruct x as []_eqn.
Tactic Notation "destruct_with_eqn" ident(n) :=
try intros until n; destruct n as []_eqn.
Tactic Notation "destruct_with_eqn" ":" ident(H) constr(x) :=
destruct x as []_eqn:H.
Tactic Notation "destruct_with_eqn" ":" ident(H) ident(n) :=
try intros until n; destruct n as []_eqn:H.

Break every hypothesis of a certain type

Ltac destruct_all t :=
match goal with
| x : t |- _ => destruct x; destruct_all t
| _ => idtac
end.

Tactic Notation "rewrite_all" constr(eq) := repeat rewrite eq in *.
Tactic Notation "rewrite_all" "<-" constr(eq) := repeat rewrite <- eq in *.

Tactics for applying equivalences.

The following code provides tactics "apply -> t", "apply <- t", "apply -> t in H" and "apply <- t in H". Here t is a term whose type consists of nested dependent and nondependent products with an equivalence A <-> B as the conclusion. The tactics with "->" in their names apply A -> B while those with "<-" in the name apply B -> A.

Ltac find_equiv H :=
let T := type of H in
lazymatch T with
| ?A -> ?B =>
let H1 := fresh in
let H2 := fresh in
cut A;
[intro H1; pose proof (H H1) as H2; clear H H1;
rename H2 into H; find_equiv H |
clear H]
| forall x : ?t, _ =>
let a := fresh "a" with
H1 := fresh "H" in
evar (a : t); pose proof (H a) as H1; unfold a in H1;
clear a; clear H; rename H1 into H; find_equiv H
| ?A <-> ?B => idtac
| _ => fail "The given statement does not seem to end with an equivalence."
end.

Ltac bapply lemma todo :=
let H := fresh in
pose proof lemma as H;
find_equiv H; [todo H; clear H | .. ].

Tactic Notation "apply" "->" constr(lemma) :=
bapply lemma ltac:(fun H => destruct H as [H _]; apply H).

Tactic Notation "apply" "<-" constr(lemma) :=
bapply lemma ltac:(fun H => destruct H as [_ H]; apply H).

Tactic Notation "apply" "->" constr(lemma) "in" hyp(J) :=
bapply lemma ltac:(fun H => destruct H as [H _]; apply H in J).

Tactic Notation "apply" "<-" constr(lemma) "in" hyp(J) :=
bapply lemma ltac:(fun H => destruct H as [_ H]; apply H in J).

An experimental tactic simpler than auto that is useful for ending proofs "in one step"

Ltac easy :=
let rec use_hyp H :=
match type of H with
| _ /\ _ => exact H || destruct_hyp H
| _ => try solve [inversion H]
end
with do_intro := let H := fresh in intro H; use_hyp H
with destruct_hyp H := case H; clear H; do_intro; do_intro in
let rec use_hyps :=
match goal with
| H : _ /\ _ |- _ => exact H || (destruct_hyp H; use_hyps)
| H : _ |- _ => solve [inversion H]
| _ => idtac
end in
let rec do_atom :=
solve [reflexivity | symmetry; trivial] ||
(split; do_atom)
with do_ccl := trivial with eq_true; repeat do_intro; do_atom in
(use_hyps; do_ccl) || fail "Cannot solve this goal".

Tactic Notation "now" tactic(t) := t; easy.

A tactic to document or check what is proved at some point of a script

Ltac now_show c := change c.

Support for rewriting decidability statements

Lemma decide_left : forall (C:Prop) (decide:{C}+{~C}),
C -> forall P:{C}+{~C}->Prop, (forall H:C, P (left _ H)) -> P decide.

Lemma decide_right : forall (C:Prop) (decide:{C}+{~C}),
~C -> forall P:{C}+{~C}->Prop, (forall H:~C, P (right _ H)) -> P decide.

Tactic Notation "decide" constr(lemma) "with" constr(H) :=
let try_to_merge_hyps H :=
try (clear H; intro H) ||
(let H' := fresh H "bis" in intro H'; try clear H') ||
(let H' := fresh in intro H'; try clear H') in
match type of H with
| ~ ?C => apply (decide_right lemma H); try_to_merge_hyps H
| ?C -> False => apply (decide_right lemma H); try_to_merge_hyps H
| _ => apply (decide_left lemma H); try_to_merge_hyps H
end.

Clear an hypothesis and its dependencies

Tactic Notation "clear" "dependent" hyp(h) :=
let rec depclear h :=
clear h ||
match goal with
| H : context [ h ] |- _ => depclear H; depclear h
end ||
fail "hypothesis to clear is used in the conclusion (maybe indirectly)"
in depclear h.

Revert an hypothesis and its dependencies : this is actually generalize dependent...

Tactic Notation "revert" "dependent" hyp(h) :=
generalize dependent h.