# Library Coq.Init.Tactics

# Useful tactics

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.

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 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idtac "absurd_hyp is OBSOLETE: use contradict instead.";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 (eq_refl x); pattern x at -1; case x.

Ltac destr_eq H := discriminate H || (try (injection H as H)).

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.

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 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.

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.