(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
(** * 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.
(* To contradict an hypothesis without copying its type. *)
Ltac absurd_hyp H :=
idtac "absurd_hyp is OBSOLETE: use contradict instead.";
let T := type of H in
absurd T.
(* A useful complement to contradict. Here H:A while G allows concluding ~A *)
Ltac false_hyp H G :=
let T := type of H in absurd T; [ apply G | assumption ].
(* A case with no loss of information. *)
Ltac case_eq x := generalize (eq_refl x); pattern x at -1; case x.
(* use either discriminate or injection on a hypothesis *)
Ltac destr_eq H := discriminate H || (try (injection H as H)).
(* Similar variants of destruct *)
Tactic Notation "destruct_with_eqn" constr(x) :=
destruct x eqn:?.
Tactic Notation "destruct_with_eqn" ident(n) :=
try intros until n; destruct n eqn:?.
Tactic Notation "destruct_with_eqn" ":" ident(H) constr(x) :=
destruct x eqn:H.
Tactic Notation "destruct_with_eqn" ":" ident(H) ident(n) :=
try intros until n; destruct n 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.
(* Rewriting in all hypothesis several times everywhere *)
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. *)
(* The idea of the tactics is to first provide a term in the context whose type is the implication (in one of the directions), and then apply it. The first idea is to produce a statement "forall ..., A -> B" (call this type T) and then do "assert (H : T)" for a fresh H. Thus, T can be proved from the original equivalence and then used to perform the application. However, currently in Ltac it is difficult to produce such T from the original formula.  Therefore, we first pose the original equivalence as H. If the type of H is a dependent product, we create an existential variable and apply H to this variable. If the type of H has the form C -> D, then we do a cut on C. Once we eliminate all products, we split (i.e., destruct) the conjunction into two parts and apply the relevant one. *)
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 do_atom :=
solve [ trivial with eq_true | reflexivity | symmetry; trivial | contradiction ] in
let rec do_ccl :=
try do_atom;
repeat (do_intro; try do_atom);
solve [ split; do_ccl ] in
solve [ do_atom | use_hyps; do_ccl ] ||
fail "Cannot solve this goal".
Tactic Notation "now" tactic(t) := t; easy.
(** Slightly more than [easy]*)
Ltac easy' := repeat split; simpl; easy || now destruct 1.
(** 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 *)
Set Implicit Arguments.

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