Library Coq.Program.Tactics
This module implements various tactics used to simplify the goals produced by Program,
which are also generally useful.
The do tactic but using a Coq-side nat.
Ltac do_nat n tac :=
match n with
| 0 =>
idtac
|
S ?n' =>
tac ;
do_nat n' tac
end.
Do something on the last hypothesis, or fail
Ltac on_last_hyp tac :=
match goal with [ H : _ |- _ ] => tac H || fail 1 end.
Destructs one pair, without care regarding naming.
Ltac destruct_one_pair :=
match goal with
| [
H : (
_ /\
_) |-
_] =>
destruct H
| [
H :
prod _ _ |-
_] =>
destruct H
end.
Repeateadly destruct pairs.
Ltac destruct_pairs := repeat (destruct_one_pair).
Destruct one existential package, keeping the name of the hypothesis for the first component.
Ltac destruct_one_ex :=
let tac H :=
let ph :=
fresh "H"
in (
destruct H as [
H ph])
in
let tacT H :=
let ph :=
fresh "X"
in (
destruct H as [
H ph])
in
match goal with
| [
H : (
ex _) |-
_] =>
tac H
| [
H : (
sig ?P) |-
_ ] =>
tac H
| [
H : (
sigT ?P) |-
_ ] =>
tacT H
| [
H : (
ex2 _) |-
_] =>
tac H
end.
Repeateadly destruct existentials.
Ltac destruct_exists := repeat (destruct_one_ex).
Repeateadly destruct conjunctions and existentials.
Ltac destruct_conjs := repeat (destruct_one_pair || destruct_one_ex).
Destruct an existential hypothesis t keeping its name for the first component
and using Ht for the second
Tactic Notation "destruct" "exist" ident(t) ident(Ht) := destruct t as [t Ht].
Destruct a disjunction keeping its name in both subgoals.
Tactic Notation "destruct" "or" ident(H) := destruct H as [H|H].
Discriminate that also work on a x <> x hypothesis.
Ltac discriminates :=
match goal with
| [ H : ?x <> ?x |- _ ] => elim H ; reflexivity
| _ => discriminate
end.
Revert the last hypothesis.
Ltac revert_last :=
match goal with
[ H : _ |- _ ] => revert H
end.
Reapeateadly reverse the last hypothesis, putting everything in the goal.
Ltac reverse := repeat revert_last.
Clear duplicated hypotheses
Ltac clear_dup :=
match goal with
| [ H : ?X |- _ ] =>
match goal with
| [ H' : ?Y |- _ ] =>
match H with
| H' => fail 2
| _ => unify X Y ; (clear H' || clear H)
end
end
end.
Ltac clear_dups := repeat clear_dup.
A non-failing subst that substitutes as much as possible.
Ltac subst_no_fail :=
repeat (match goal with
[ H : ?X = ?Y |- _ ] => subst X || subst Y
end).
Tactic Notation "subst" "*" := subst_no_fail.
Ltac on_application f tac T :=
match T with
| context [f ?x ?y ?z ?w ?v ?u ?a ?b ?c] => tac (f x y z w v u a b c)
| context [f ?x ?y ?z ?w ?v ?u ?a ?b] => tac (f x y z w v u a b)
| context [f ?x ?y ?z ?w ?v ?u ?a] => tac (f x y z w v u a)
| context [f ?x ?y ?z ?w ?v ?u] => tac (f x y z w v u)
| context [f ?x ?y ?z ?w ?v] => tac (f x y z w v)
| context [f ?x ?y ?z ?w] => tac (f x y z w)
| context [f ?x ?y ?z] => tac (f x y z)
| context [f ?x ?y] => tac (f x y)
| context [f ?x] => tac (f x)
end.
A variant of apply using refine, doing as much conversion as necessary.
Ltac rapply p :=
refine (p _ _ _ _ _ _ _ _ _ _ _ _ _ _ _) ||
refine (p _ _ _ _ _ _ _ _ _ _ _ _ _ _) ||
refine (p _ _ _ _ _ _ _ _ _ _ _ _ _) ||
refine (p _ _ _ _ _ _ _ _ _ _ _ _) ||
refine (p _ _ _ _ _ _ _ _ _ _ _) ||
refine (p _ _ _ _ _ _ _ _ _ _) ||
refine (p _ _ _ _ _ _ _ _ _) ||
refine (p _ _ _ _ _ _ _ _) ||
refine (p _ _ _ _ _ _ _) ||
refine (p _ _ _ _ _ _) ||
refine (p _ _ _ _ _) ||
refine (p _ _ _ _) ||
refine (p _ _ _) ||
refine (p _ _) ||
refine (p _) ||
refine p.
Tactical on_call f tac applies tac on any application of f in the hypothesis or goal.
Ltac on_call f tac :=
match goal with
| |- ?T => on_application f tac T
| H : ?T |- _ => on_application f tac T
end.
Ltac destruct_call f :=
let tac t := (destruct t) in on_call f tac.
Ltac destruct_calls f := repeat destruct_call f.
Ltac destruct_call_in f H :=
let tac t := (destruct t) in
let T := type of H in
on_application f tac T.
Ltac destruct_call_as f l :=
let tac t := (destruct t as l) in on_call f tac.
Ltac destruct_call_as_in f l H :=
let tac t := (destruct t as l) in
let T := type of H in
on_application f tac T.
Tactic Notation "destruct_call" constr(f) := destruct_call f.
Permit to name the results of destructing the call to f.
Tactic Notation "destruct_call" constr(f) "as" simple_intropattern(l) :=
destruct_call_as f l.
Specify the hypothesis in which the call occurs as well.
Tactic Notation "destruct_call" constr(f) "in" hyp(id) :=
destruct_call_in f id.
Tactic Notation "destruct_call" constr(f) "as" simple_intropattern(l) "in" hyp(id) :=
destruct_call_as_in f l id.
Try to inject any potential constructor equality hypothesis.
Ltac autoinjection tac :=
match goal with
| [ H : ?f ?a = ?f' ?a' |- _ ] => tac H
end.
Ltac inject H := progress (inversion H ; subst*; clear_dups) ; clear H.
Ltac autoinjections := repeat (clear_dups ; autoinjection ltac:inject).
Destruct an hypothesis by first copying it to avoid dependencies.
Ltac destruct_nondep H := let H0 := fresh "H" in assert(H0 := H); destruct H0.
If bang appears in the goal, it means that we have a proof of False and the goal is solved.
Ltac bang :=
match goal with
| |- ?x =>
match x with
|
context [
False_rect _ ?p] =>
elim p
end
end.
A tactic to show contradiction by first asserting an automatically provable hypothesis.
Tactic Notation "contradiction" "by" constr(t) :=
let H := fresh in assert t as H by auto with * ; contradiction.
A tactic that adds H:=p:typeof(p) to the context if no hypothesis of the same type appears in the goal.
Useful to do saturation using tactics.
Ltac add_hypothesis H' p :=
match type of p with
?X =>
match goal with
| [ H : X |- _ ] => fail 1
| _ => set (H':=p) ; try (change p with H') ; clearbody H'
end
end.
A tactic to replace an hypothesis by another term.
Ltac replace_hyp H c :=
let H' := fresh "H" in
assert(H' := c) ; clear H ; rename H' into H.
A tactic to refine an hypothesis by supplying some of its arguments.
Ltac refine_hyp c :=
let tac H := replace_hyp H c in
match c with
| ?H _ => tac H
| ?H _ _ => tac H
| ?H _ _ _ => tac H
| ?H _ _ _ _ => tac H
| ?H _ _ _ _ _ => tac H
| ?H _ _ _ _ _ _ => tac H
| ?H _ _ _ _ _ _ _ => tac H
| ?H _ _ _ _ _ _ _ _ => tac H
end.
The default simplification tactic used by Program is defined by program_simpl, sometimes auto
is not enough, better rebind using Obligation Tactic := tac in this case,
possibly using program_simplify to use standard goal-cleaning tactics.
Ltac program_simplify :=
simpl ;
intros ;
destruct_conjs ;
simpl proj1_sig in * ;
subst* ;
autoinjections ;
try discriminates ;
try (
solve [
red ;
intros ;
destruct_conjs ;
autoinjections ;
discriminates ]).
Ltac program_simpl :=
program_simplify ;
auto.
Ltac obligation_tactic :=
program_simpl.