183 lines
6 KiB
Coq
183 lines
6 KiB
Coq
Require Export Omega.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* [false] replaces the current goal with [False]. *)
|
|
|
|
Ltac false :=
|
|
elimtype False.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* [tc] is a shortcut for [eauto with typeclass_instances]. For some reason,
|
|
it is often necessary to use [rewrite ... by tc]. *)
|
|
|
|
Ltac tc :=
|
|
eauto with typeclass_instances.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* The tactic [obvious] does nothing by default, but can be extended with hints
|
|
so as to solve relatively easy goals -- e.g., proving that a term is a value,
|
|
proving a size inequality, proving that a substitution is a renaming, etc.
|
|
These predicates are sometimes interrelated (e.g., size is preserved by
|
|
renamings; the property of being a value is preserved by renamings) so it
|
|
would be counterproductive to distinguish several hint databases. *)
|
|
|
|
Create HintDb obvious.
|
|
|
|
Ltac obvious :=
|
|
eauto with obvious typeclass_instances.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* The tactic [pick R k] picks a hypothesis [h] whose statement is an
|
|
application of the inductive predicate [R], and passes [h] to the (Ltac)
|
|
continuation [k]. *)
|
|
|
|
Ltac pick R k :=
|
|
match goal with
|
|
| h: R _ |- _ => k h
|
|
| h: R _ _ |- _ => k h
|
|
| h: R _ _ _ |- _ => k h
|
|
| h: R _ _ _ _ |- _ => k h
|
|
| h: R _ _ _ _ _ |- _ => k h
|
|
end.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* The tactic [invert h] case-analyzes the hypothesis [h], whose statement
|
|
should be an application of an inductive predicate. *)
|
|
|
|
Ltac invert h :=
|
|
inversion h; clear h; try subst.
|
|
|
|
(* The tactic [inv R] looks for a hypothesis [h] whose statement is an
|
|
application of the inductive predicate [R], and case-analyzes this
|
|
hypothesis. *)
|
|
|
|
Ltac inv R :=
|
|
pick R invert.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* The tactic [unpack] decomposes conjunctions and existential quantifiers
|
|
in the hypotheses. Then, it attempts to perform substitutions, if possible. *)
|
|
|
|
Ltac unpack :=
|
|
repeat match goal with
|
|
| h: _ /\ _ |- _ =>
|
|
destruct h
|
|
| h: exists _, _ |- _ =>
|
|
destruct h
|
|
end;
|
|
try subst.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* The tactic [forward H] introduces the term [H] as a new hypothesis, and
|
|
unpacks it (if necessary). *)
|
|
|
|
Ltac forward H :=
|
|
generalize H; intro; unpack.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* [push h] moves the hypothesis [h] into the goal. *)
|
|
|
|
Ltac push h :=
|
|
generalize h; clear h.
|
|
|
|
(* [ltac_Mark] and [ltac_mark] are dummies. They are used as sentinels by
|
|
certain tactics, to mark a position in the context or in the goal. *)
|
|
|
|
Inductive ltac_Mark : Type :=
|
|
| ltac_mark : ltac_Mark.
|
|
|
|
(* [push_until_mark] moves all hypotheses from the context into the goal,
|
|
starting from the bottom and stopping as soon as a mark (that is, a
|
|
hypothesis of type [ltac_Mark]) is reached. The mark is thrown away. The
|
|
tactic fails if no mark appears in the context. *)
|
|
|
|
Ltac push_until_mark :=
|
|
match goal with h: ?T |- _ =>
|
|
match T with
|
|
| ltac_Mark => clear h
|
|
| _ => push h; push_until_mark
|
|
end end.
|
|
|
|
(** [pop_until_mark] moves all hypotheses from the goal into the context,
|
|
until a hypothesis of type [ltac_Mark] is reached. The mark is thrown
|
|
away. The tactic fails if no mark appears in the goal. *)
|
|
|
|
Ltac pop_until_mark :=
|
|
match goal with
|
|
| |- (ltac_Mark -> _) => intros _
|
|
| _ => intro; pop_until_mark
|
|
end.
|
|
|
|
Ltac injections :=
|
|
(* Place an initial mark, so as to not disturb the goal. *)
|
|
generalize ltac_mark;
|
|
(* Look at every equality hypothesis. *)
|
|
repeat match goal with
|
|
| h: _ = _ |- _ =>
|
|
(* Try to apply the primitive tactic [injection] to this hypothesis.
|
|
If this succeeds, clear [h] and replace it with the results of
|
|
[injection]. Another mark is used for this purpose. If this fails,
|
|
just push [h] into the goal, as we do not wish to see it any more. *)
|
|
first [
|
|
generalize ltac_mark; injection h; clear h; pop_until_mark
|
|
| push h ]
|
|
end;
|
|
(* Pop all of the hypotheses that have been set aside above. *)
|
|
pop_until_mark.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* The following incantation means that [eauto with omega] can solve a goal
|
|
of the form [_ < _]. The tactic [zify] is a preprocessor which increases
|
|
the number of goals that [omega] can accept; e.g., it expands away [min]
|
|
and [max]. *)
|
|
|
|
Hint Extern 1 (le _ _) => (zify; omega) : omega.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* A little extra help for [eauto with omega]. *)
|
|
|
|
Lemma arith_le_SS: forall x y, x < y -> S x < S y.
|
|
Proof. intros. omega. Qed.
|
|
Lemma arith_SS_le: forall x y, S x < S y -> x < y.
|
|
Proof. intros. omega. Qed.
|
|
|
|
Hint Resolve arith_le_SS arith_SS_le : omega.
|
|
|
|
(* -------------------------------------------------------------------------- *)
|
|
|
|
(* [dblib_by_cases] simplifies goals in which a decidable integer comparison
|
|
appears. *)
|
|
|
|
Ltac dblib_intro_case_clear :=
|
|
let h := fresh in
|
|
intro h; case h; clear h.
|
|
|
|
Ltac dblib_inspect_cases :=
|
|
match goal with
|
|
| |- context [le_gt_dec ?n ?n'] =>
|
|
case (le_gt_dec n n')
|
|
| h: context [le_gt_dec ?n ?n'] |- _ =>
|
|
revert h; case (le_gt_dec n n'); intro h
|
|
| |- context [eq_nat_dec ?n ?n'] =>
|
|
case (eq_nat_dec n n')
|
|
| h: context [eq_nat_dec ?n ?n'] |- _ =>
|
|
revert h; case (eq_nat_dec n n'); intro h
|
|
| |- context [(lt_eq_lt_dec ?n ?n')] =>
|
|
case (lt_eq_lt_dec n n'); [ dblib_intro_case_clear | idtac ]
|
|
| h: context [(lt_eq_lt_dec ?n ?n')] |- _ =>
|
|
revert h; case (lt_eq_lt_dec n n'); [ dblib_intro_case_clear | idtac ]; intro h
|
|
end.
|
|
|
|
Ltac dblib_by_cases :=
|
|
repeat dblib_inspect_cases; try solve [ intros; elimtype False; omega ]; intros.
|