Library Coq.Init.Tactics
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idtac "swap is OBSOLETE: use contradict instead.";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idtac "absurd_hyp is OBSOLETE: use contradict instead.";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 (
eq_refl x);
pattern x at -1;
case x.
Ltac destr_eq H :=
discriminate H || (
try (
injection H;
clear H;
intro 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 rec do_atom :=
solve [
reflexivity |
symmetry;
trivial] ||
contradiction ||
(
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.
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.