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LibTactics.v
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LibTactics.v
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(**************************************************************************
* Useful General-Purpose Tactics for Coq *
* Arthur Chargueraud *
* Distributed under the terms of the LGPL-v3 license *
* Intended for use with Coq version 8.1pl4 and 8.2 *
* Compatible with Coq version 8.3
* Release of 2009-02-20 *
***************************************************************************)
(** This file contains a set of tactics that extends the set of builtin
tactics provided with the standard distribution of Coq. It intends
to overcome a number of limitations of the standard set of tactics,
and thereby to help user to write shorter and more robust scripts.
Hopefully, Coq tactics will be improved as time goes by, and this
file should ultimately be useless. In the meanwhile, you will
probably find it very useful.
*)
(** The main features offered are:
- More convenient syntax for naming hypotheses, with tactics for
introduction and inversion that take as input only the name of
hypotheses of type [Prop], rather than the name of all variables.
- Tactics providing true support for manipulating N-ary conjunctions,
disjunctions and existentials, hidding the fact that the underlying
implementation is based on binary predicates.
- Convenient support for automation: tactic followed with the symbol
"~" or "*" will call automation on the generated subgoals.
Symbol "~" stands for [auto] and "*" for [intuition eauto].
These bindings can be customized.
- Forward-chaining tactics are provided to instantiate lemmas
either with variable or hypotheses or a mix of both.
- A more powerful implementation of [apply] is provided (it is based
on [refine] and thus behaves better with respect to conversion).
- An improved inversion tactic which substitutes equalities on variables
generated by the standard inversion mecanism. Moreover, it supports
the elimination of dependently-typed equalities (requires axiom [K],
which is a weak form of Proof Irrelevance).
- An improved induction tactic that saves the relevant information
by introducing equalities before doing the induction and
substitutes these equality in each subgoal generated by [induction].
- Tactics for saving time when writing proofs, with tactics to
asserts hypotheses or sub-goals, and improved tactics for
clearing, renaming, and sorting hypotheses.
*)
(** External credits:
- thanks to Xavier Leroy for providing with the idea of tactic [forward],
- thanks to Georges Gonthier for the implementation trick in [applys],
- thanks to Hugo Herbelin for useful feedback on several tactics.
*)
Set Implicit Arguments.
(* ********************************************************************** *)
(** * Additional notations for Coq *)
(* ---------------------------------------------------------------------- *)
(** ** N-ary Existentials *)
(** [exist T1 ... TN, P] is a shorthand for
[exists T1, ..., exists TN, P]. *)
Notation "'exist' x1 ',' P" :=
(exists x1, P)
(at level 200, x1 ident,
right associativity) : type_scope.
Notation "'exist' x1 x2 ',' P" :=
(exists x1, exists x2, P)
(at level 200, x1 ident, x2 ident,
right associativity) : type_scope.
Notation "'exist' x1 x2 x3 ',' P" :=
(exists x1, exists x2, exists x3, P)
(at level 200, x1 ident, x2 ident, x3 ident,
right associativity) : type_scope.
Notation "'exist' x1 x2 x3 x4 ',' P" :=
(exists x1, exists x2, exists x3, exists x4, P)
(at level 200, x1 ident, x2 ident, x3 ident, x4 ident,
right associativity) : type_scope.
Notation "'exist' x1 x2 x3 x4 x5 ',' P" :=
(exists x1, exists x2, exists x3, exists x4, exists x5, P)
(at level 200, x1 ident, x2 ident, x3 ident, x4 ident, x5 ident,
right associativity) : type_scope.
Notation "'exist' x1 x2 x3 x4 x5 x6 ',' P" :=
(exists x1, exists x2, exists x3, exists x4, exists x5, exists x6, P)
(at level 200, x1 ident, x2 ident, x3 ident, x4 ident, x5 ident,
x6 ident,
right associativity) : type_scope.
Notation "'exist' x1 x2 x3 x4 x5 x6 x7 ',' P" :=
(exists x1, exists x2, exists x3, exists x4, exists x5, exists x6,
exists x7, P)
(at level 200, x1 ident, x2 ident, x3 ident, x4 ident, x5 ident,
x6 ident, x7 ident,
right associativity) : type_scope.
Notation "'exist' x1 x2 x3 x4 x5 x6 x7 x8 ',' P" :=
(exists x1, exists x2, exists x3, exists x4, exists x5, exists x6,
exists x7, exists x8, P)
(at level 200, x1 ident, x2 ident, x3 ident, x4 ident, x5 ident,
x6 ident, x7 ident, x8 ident,
right associativity) : type_scope.
(* ---------------------------------------------------------------------- *)
(** ** Partial application of equality *)
(** [= x] is a unary predicate which holds of values equal to [x].
It simply denotes the partial application of equality. *)
Notation "'=' x" := (fun y => y = x) (at level 71).
(* ---------------------------------------------------------------------- *)
(** ** Notation for projections *)
Notation "'proj21' P" := (proj1 P) (at level 69, only parsing).
Notation "'proj22' P" := (proj2 P) (at level 69, only parsing).
Notation "'proj31' P" := (proj1 P) (at level 69).
Notation "'proj32' P" := (proj1 (proj2 P)) (at level 69).
Notation "'proj33' P" := (proj2 (proj2 P)) (at level 69).
Notation "'proj41' P" := (proj1 P) (at level 69).
Notation "'proj42' P" := (proj1 (proj2 P)) (at level 69).
Notation "'proj43' P" := (proj1 (proj2 (proj2 P))) (at level 69).
Notation "'proj44' P" := (proj2 (proj2 (proj2 P))) (at level 69).
Notation "'proj51' P" := (proj1 P) (at level 69).
Notation "'proj52' P" := (proj1 (proj2 P)) (at level 69).
Notation "'proj53' P" := (proj1 (proj2 (proj2 P))) (at level 69).
Notation "'proj54' P" := (proj1 (proj2 (proj2 (proj2 P)))) (at level 69).
Notation "'proj55' P" := (proj2 (proj2 (proj2 (proj2 P)))) (at level 69).
(* ********************************************************************** *)
(** * Tools for programming with Ltac *)
(* ---------------------------------------------------------------------- *)
(** ** Programming tactics *)
(** [ltac_no_arg] is a constant that can be used to simulate
optional arguments in tactic definitions.
Use [mytactic ltac_no_arg] on the tactic invokation,
and use [match arg with ltac_no_arg => ..] or
[match type of arg with ltac_No_arg => ..] to
test whether an argument was provided. *)
Inductive ltac_No_arg : Set :=
| ltac_no_arg : ltac_No_arg.
(* ---------------------------------------------------------------------- *)
(** ** Returning values *)
(** Ltac tactics are not allowed to both perform side-effect on the goal
and return a value in the same time. To work around this limitation,
we can use either the current goal or the proof context as a stack
to place return values. To avoid interferences, we box the return
values.
When the goal is used as stack, we use [ltac_tag_result] to box values.
When the context is used, we use the type [Carrier] to box values. *)
Definition ltac_tag_result (A:Type) (x:A) := x.
(** [build_result E] changes the goal from [G] to
[ltac_tag_result T -> G] where [T] is the type of [E]. *)
Ltac build_result t :=
match type of t with ?T =>
let H := fresh "TEMP" in
assert (H : ltac_tag_result T);
[ unfold ltac_tag_result; exact t | generalize H; clear H ]
end.
(** [if_is_result] is the identity on a goal of the form [ltac_tag_result T -> G]
and fails otherwise. *)
Ltac if_is_result :=
match goal with |- ltac_tag_result _ -> _ => idtac end.
(** [name_result H] expects a of the form [ltac_tag_result T -> G]
and changes the goal to [G] by introducing an hypothesis [H:T]. *)
Tactic Notation "name_result" simple_intropattern(H) :=
match goal with |- ltac_tag_result _ -> _ =>
unfold ltac_tag_result at 1;
first [ intros H
| let H' := fresh "NameAlreadyUsed" in intros H'] end.
(** With the type [Carrier], we implement the three following tactics:
- [_put x] is used to return a value (leaving it on the stack)
- [_get] is used to obtain the last returned value
- [_rem] is used to remove the last returned value from the
context.
The typicall usage is: [mytactic args; let result := _get in _rem; ...].
*)
Inductive Carrier : forall A, A -> Type :=
| carrier : forall A x, @Carrier A x.
(** [_put x] adds on hypothesis of type [Carrier x].
[_put2 x y] and [_put3 x y z] can be used for functions
that return pairs or triples of values. *)
Ltac _put x :=
generalize (carrier x); intro.
Ltac _put2 x y :=
_put y; _put x.
Ltac _put3 x y z :=
_put z; _put y; _put x.
(** [_get] returns the value [x] contained in the last hypothesis of
type [Carrier x] available in the context. If fails if there
is no such hypothesis. *)
Ltac _get :=
match goal with H: Carrier ?t |- _ => t end.
(** [_rem] clears the last hypothesis of type [Carrier _].
If fails if there is no such hypothesis. *)
Ltac _rem :=
match goal with H: Carrier ?t |- _ => clear H end.
(* ---------------------------------------------------------------------- *)
(** ** List of arguments for tactics *)
Require Import List.
(** [ltac_wild] is a constant that can be used to simulate
wildcard arguments in tactic definitions. Notation is [__]. *)
Inductive ltac_Wild : Set :=
| ltac_wild : ltac_Wild.
Notation "'__'" := ltac_wild : ltac_scope.
(** [ltac_wilds] is another constant that can be used to simulate
a sequence of [N] wildcards, with [N] chosen appropriately
depending on the context. Notation is [___]. *)
Inductive ltac_Wilds : Set :=
| ltac_wilds : ltac_Wilds.
Notation "'___'" := ltac_wilds : ltac_scope.
(** [Boxer] is a datatype such that the type [list Boxer] can be used
to manipulate list of values in ltac. *)
Inductive Boxer : Type :=
| boxer : forall (A:Type), A -> Boxer.
Notation "'>>>'" :=
(@nil Boxer)
(at level 0)
: ltac_scope.
Notation "'>>>' v1" :=
((boxer v1)::nil)
(at level 0, v1 at level 0)
: ltac_scope.
Notation "'>>>' v1 v2" :=
((boxer v1)::(boxer v2)::nil)
(at level 0, v1 at level 0, v2 at level 0)
: ltac_scope.
Notation "'>>>' v1 v2 v3" :=
((boxer v1)::(boxer v2)::(boxer v3)::nil)
(at level 0, v1 at level 0, v2 at level 0, v3 at level 0)
: ltac_scope.
Notation "'>>>' v1 v2 v3 v4" :=
((boxer v1)::(boxer v2)::(boxer v3)::(boxer v4)::nil)
(at level 0, v1 at level 0, v2 at level 0, v3 at level 0,
v4 at level 0)
: ltac_scope.
Notation "'>>>' v1 v2 v3 v4 v5" :=
((boxer v1)::(boxer v2)::(boxer v3)::(boxer v4)::(boxer v5)::nil)
(at level 0, v1 at level 0, v2 at level 0, v3 at level 0,
v4 at level 0, v5 at level 0)
: ltac_scope.
Notation "'>>>' v1 v2 v3 v4 v5 v6" :=
((boxer v1)::(boxer v2)::(boxer v3)::(boxer v4)::(boxer v5)
::(boxer v6)::nil)
(at level 0, v1 at level 0, v2 at level 0, v3 at level 0,
v4 at level 0, v5 at level 0, v6 at level 0)
: ltac_scope.
Notation "'>>>' v1 v2 v3 v4 v5 v6 v7" :=
((boxer v1)::(boxer v2)::(boxer v3)::(boxer v4)::(boxer v5)
::(boxer v6)::(boxer v7)::nil)
(at level 0, v1 at level 0, v2 at level 0, v3 at level 0,
v4 at level 0, v5 at level 0, v6 at level 0, v7 at level 0)
: ltac_scope.
Open Scope ltac_scope.
(** [ltac_inst] is a datatype that describes the four instantiation
modes that can be used in tactics [specializes], [lets],
[applys] and [forwards].
- [Args]: all arguments are to be provided,
- [Hyps]: only hypotheses are to be provided
(hypotheses = arguments not used dependently)
- [Vars]: only variables are to be provided
(variables = arguments used dependently)
- [Hnts]: the arguments provided are used as hints and are
affected to the first argument of matching type.
*)
Inductive ltac_inst : Set :=
| Args : ltac_inst
| Hyps : ltac_inst
| Vars : ltac_inst
| Hnts : ltac_inst.
(** [ltac_args] inputs a term [E] and returns a term of type "list boxer":
- if [E] is already of type "list Boxer" that starts with the
value of an instantiation mode, it returns [E],
- otherwise if [E] is already of type "list Boxer", it returns
[(boxer Hnts)::E], in other words, mode [Hnts] is the default,
- otherwise, it returns the list containing
[(boxer Hnts)::(boxer E)::nil], describing the fact that there
is only one argument provided, in the mode [Hnts]. *)
Ltac ltac_args E :=
match type of E with
| List.list Boxer =>
match E with
| (@boxer ltac_inst _)::_ => constr:(E)
| _ => constr:((boxer Hnts)::E)
end
| _ => constr:((boxer Hnts)::(boxer E)::nil)
end.
(* ---------------------------------------------------------------------- *)
(** ** Testing tactics *)
(** [show tac] executes a tactic [tac] that produces a result (not
performing any side-effect on the goal) and then display its result. *)
Tactic Notation "show" tactic(tac) :=
let R := tac in pose R.
(** [dup N] produces [N] copies of the current goal. It is useful
for building examples on which to illustrate behaviour of tactics.
[dup] is short for [dup 2]. *)
Lemma dup_lemma : forall P, P -> P -> P.
Proof. auto. Qed.
Ltac dup_tactic N :=
match N with
| 0 => idtac
| S 0 => idtac
| S ?N' => apply dup_lemma; [ | dup_tactic N' ]
end.
Tactic Notation "dup" constr(N) :=
dup_tactic N.
Tactic Notation "dup" :=
dup 2.
(* ---------------------------------------------------------------------- *)
(** ** Deconstructing terms *)
(** [get_head E] is a tactic that returns the head constant of the
term [E], ie, when applied to a term of the form [P x1 ... xN]
it returns [P]. If [E] is not an application, it returns [E]. *)
Ltac get_head E :=
match E with
| ?P _ _ _ _ _ _ _ _ _ => constr:(P)
| ?P _ _ _ _ _ _ _ _ => constr:(P)
| ?P _ _ _ _ _ _ _ => constr:(P)
| ?P _ _ _ _ _ _ => constr:(P)
| ?P _ _ _ _ _ => constr:(P)
| ?P _ _ _ _ => constr:(P)
| ?P _ _ _ => constr:(P)
| ?P _ _ => constr:(P)
| ?P _ => constr:(P)
| ?P => constr:(P)
end.
(* [is_metavar E] returns whether E is a meta-variable.
However, its implementation is approximative, since it
returns [true] also for constants (i.e. not-applicative terms). *)
Ltac is_metavar E :=
match E with
| ?P _ _ _ _ _ _ _ _ _ => constr:(false)
| ?P _ _ _ _ _ _ _ _ => constr:(false)
| ?P _ _ _ _ _ _ _ => constr:(false)
| ?P _ _ _ _ _ _ => constr:(false)
| ?P _ _ _ _ _ => constr:(false)
| ?P _ _ _ _ => constr:(false)
| ?P _ _ _ => constr:(false)
| ?P _ _ => constr:(false)
| ?P _ => constr:(false)
| ?P => constr:(true)
end.
(* ********************************************************************** *)
(** * Backward and forward chaining *)
(* ---------------------------------------------------------------------- *)
(** ** Adding assumptions *)
(** [lets H: E] adds an hypothesis [H : T] to the context, where [T] is
the type of term [E]. If [H] is an introduction pattern, it will
destruct [H] according to the pattern. *)
Tactic Notation "lets" simple_intropattern(I) ":" constr(E) :=
generalize E; intros I.
(** [lets H1 .. HN : E] is the same as
[lets \[H1 \[H2 \[.. HN\]\]\]\]:E], and thus equivalent to
[destruct E as \[H1 \[H2 \[.. HN\]\]\]\]]. *)
Tactic Notation "lets" simple_intropattern(I1)
simple_intropattern(I2) ":" constr(E) :=
lets [I1 I2]: E.
Tactic Notation "lets" simple_intropattern(I1)
simple_intropattern(I2) simple_intropattern(I3) ":" constr(E) :=
lets [I1 [I2 I3]]: E.
Tactic Notation "lets" simple_intropattern(I1)
simple_intropattern(I2) simple_intropattern(I3)
simple_intropattern(I4) ":" constr(E) :=
lets [I1 [I2 [I3 I4]]]: E.
Tactic Notation "lets" simple_intropattern(I1)
simple_intropattern(I2) simple_intropattern(I3)
simple_intropattern(I4) simple_intropattern(I5) ":" constr(E) :=
lets [I1 [I2 [I3 [I4 I5]]]]: E.
Tactic Notation "lets" simple_intropattern(I1)
simple_intropattern(I2) simple_intropattern(I3)
simple_intropattern(I4) simple_intropattern(I5)
simple_intropattern(I6) ":" constr(E) :=
lets [I1 [I2 [I3 [I4 [I5 I6]]]]]: E.
(** [lets_simpl H: E] is the same as [lets H: E] excepts that it
calls [simpl] on the hypothesis H. *)
Tactic Notation "lets_simpl" ident(H) ":" constr(E) :=
lets H: E; simpl in H.
(** [lets_hnf H: E] is the same as [lets H: E] excepts that it
calls [hnf] to set the definition in head normal form. *)
Tactic Notation "lets_hnf" ident(H) ":" constr(E) :=
lets H: E; hnf in H.
(** [lets: E] is equivalent to [lets H: E], only the name [H] is
automatically chosen by Coq. It is useful to type-check a
term (like the top-level command [Check]), but also to add
facts that are going to be used by automation.
Syntax [lets: E1 .. EN] is short for [lets: E1; ..; lets: EN]. *)
Tactic Notation "lets" ":" constr(E1) :=
generalize E1; intro.
Tactic Notation "lets" ":" constr(E1) constr(E2) :=
lets: E1; lets: E2.
Tactic Notation "lets" ":" constr(E1) constr(E2) constr(E3) :=
lets: E1; lets: E2; lets: E3.
(** [lets_simpl: E] is the same as [lets_simpl H: E] with
the name [H] being choosed automatically. *)
Tactic Notation "lets_simpl" ":" constr(T) :=
let H := fresh in lets_simpl H: T.
(** [lets_hnf: E] is the same as [lets_hnf H: E] with
the name [H] being choosed automatically. *)
Tactic Notation "lets_hnf" ":" constr(T) :=
let H := fresh in lets_hnf H: T.
(** [put X: E] is a synonymous for [pose (X := E)].
Other syntaxes are [put: E]. *)
Tactic Notation "put" ident(X) ":" constr(E) :=
pose (X := E).
Tactic Notation "put" ":" constr(E) :=
let X := fresh "X" in pose (X := E).
(* ---------------------------------------------------------------------- *)
(** ** Application *)
(** [applys] is a tactic similar to [eapply] except that it is
based on the [refine] tactics, and thus is strictly more
powerful (at least in theory :). In short, it is able to perform
on-the-fly conversions when required for arguments to match,
and it is able to instantiate existentials when required. *)
Tactic Notation "applys" constr(t) :=
first
[ refine (@t)
| refine (@t _)
| refine (@t _ _)
| refine (@t _ _ _)
| refine (@t _ _ _ _)
| refine (@t _ _ _ _ _)
| refine (@t _ _ _ _ _ _)
| refine (@t _ _ _ _ _ _ _)
| refine (@t _ _ _ _ _ _ _ _)
| refine (@t _ _ _ _ _ _ _ _ _)
| refine (@t _ _ _ _ _ _ _ _ _ _)
| refine (@t _ _ _ _ _ _ _ _ _ _ _)
| refine (@t _ _ _ _ _ _ _ _ _ _ _ _)
| refine (@t _ _ _ _ _ _ _ _ _ _ _ _ _)
].
(** The tactics [applys_N T], where [N] is a natural number,
provides a more efficient way of using [applys T]. It avoids
trying out all possible arities, by specifying explicitely
the arity of function [T]. This version is to be preferred
for programming intensively-used tactics. *)
Tactic Notation "applys_0" constr(t) :=
refine (@t).
Tactic Notation "applys_1" constr(t) :=
refine (@t _).
Tactic Notation "applys_2" constr(t) :=
refine (@t _ _).
Tactic Notation "applys_3" constr(t) :=
refine (@t _ _ _).
Tactic Notation "applys_4" constr(t) :=
refine (@t _ _ _ _).
Tactic Notation "applys_5" constr(t) :=
refine (@t _ _ _ _ _).
Tactic Notation "applys_6" constr(t) :=
refine (@t _ _ _ _ _ _).
Tactic Notation "applys_7" constr(t) :=
refine (@t _ _ _ _ _ _ _).
Tactic Notation "applys_8" constr(t) :=
refine (@t _ _ _ _ _ _ _ _).
Tactic Notation "applys_9" constr(t) :=
refine (@t _ _ _ _ _ _ _ _ _).
Tactic Notation "applys_10" constr(t) :=
refine (@t _ _ _ _ _ _ _ _ _ _).
(** [applys_to H E] transform the type of hypothesis [H] by
replacing it by the result of the application of the term
[E] to [H]. Intuitively, it is equivalent to [lets H: (E H)]. *)
Tactic Notation "applys_to" hyp(H) constr(E) :=
let H' := fresh in rename H into H';
(first [ lets H: (E H')
| lets H: (E _ H')
| lets H: (E _ _ H')
| lets H: (E _ _ _ H')
| lets H: (E _ _ _ _ H')
| lets H: (E _ _ _ _ _ H')
| lets H: (E _ _ _ _ _ _ H')
| lets H: (E _ _ _ _ _ _ _ H')
| lets H: (E _ _ _ _ _ _ _ _ H')
| lets H: (E _ _ _ _ _ _ _ _ _ H') ]
); clear H'.
(** [applys_in H E] transform the hypothesis [H] by replacing it
by the result of the application of [H] to the term [E]
Intuitively, it is equivalent to [lets H: (H E)].
DEPRECATED: use [specializes H E] instead. *)
Tactic Notation "applys_in" hyp(H) constr(E) :=
let H' := fresh in rename H into H';
(first [ lets H: (H' E)
| lets H: (H' _ E)
| lets H: (H' _ _ E)
| lets H: (H' _ _ _ E)
| lets H: (H' _ _ _ _ E)
| lets H: (H' _ _ _ _ _ E)
| lets H: (H' _ _ _ _ _ _ E)
| lets H: (H' _ _ _ _ _ _ _ E)
| lets H: (H' _ _ _ _ _ _ _ _ E)
| lets H: (H' _ _ _ _ _ _ _ _ _ E) ]
); clear H'.
(** [applys_clear E] performs [applys E] and then calls [clear] on
the head term of [E]. It fails if this head term is not an hypothesis
or if it used dependently. *)
Tactic Notation "applys_clear" constr(E) :=
applys E; let H := get_head E in clear E.
Tactic Notation "apply_clear" constr(E) :=
applys_clear E.
(* ---------------------------------------------------------------------- *)
(** ** Assertions *)
(** [false_goal] replaces any goal by the goal [False].
Contrary to the tactic [false] (below), it does not try to do
anything else *)
Tactic Notation "false_goal" :=
assert False; [ | contradiction ].
(** [false] replaces any goal by the goal [False].
Furthermore, it discharges the obligation if the context contains
[False] or an hypothesis of the form [C x1 .. xN = D y1 .. yM]. *)
Tactic Notation "false" :=
false_goal; try assumption; try discriminate.
(** [tryfalse] tries to solve a goal by contradiction, and leaves
the goal unchanged if it cannot solve it.
It is equivalent to [try solve \[ false \]]. *)
Tactic Notation "tryfalse" :=
try solve [ false ].
(** [tryfalse by tac /] is that same as [tryfalse] except that
it tries to solve the goal using tactic [tac] if [assumption]
and [discriminate] do not apply.
It is equivalent to [try solve \[ false; tac \]]. *)
Tactic Notation "tryfalse" "by" tactic(tac) "/" :=
try solve [ false; tac ].
(** [false T] is equivalent to [false; apply T]. It also
solves the goal if [T] has type [C x1 .. xN = D y1 .. yM]. *)
Ltac false_with_plus T Tac :=
false_goal; first
[ first [ apply T | eapply T | applys T]; Tac
| let H := fresh in lets H: T; discriminate H ].
Tactic Notation "false" constr(T) :=
false_with_plus T ltac:(idtac).
(** [asserts H: T] is another syntax for [assert (H : T)], which
also works with introduction patterns. For instance, we can write:
[asserts \[x P\] (exists n, n = 3)], or
[asserts \[H|H\] (n = 0 \/ n = 1). *)
Tactic Notation "asserts" simple_intropattern(I) ":" constr(T) :=
let H := fresh in assert (H : T);
[ | generalize H; clear H; intros I ].
(** [asserts H1 .. HN: T] is the same as
[asserts \[H1 \[H2 \[.. HN\]\]\]\]: T]. *)
Tactic Notation "asserts" simple_intropattern(I1)
simple_intropattern(I2) ":" constr(T) :=
asserts [I1 I2]: T.
Tactic Notation "asserts" simple_intropattern(I1)
simple_intropattern(I2) simple_intropattern(I3) ":" constr(T) :=
asserts [I1 [I2 I3]]: T.
Tactic Notation "asserts" simple_intropattern(I1)
simple_intropattern(I2) simple_intropattern(I3)
simple_intropattern(I4) ":" constr(T) :=
asserts [I1 [I2 [I3 I4]]]: T.
Tactic Notation "asserts" simple_intropattern(I1)
simple_intropattern(I2) simple_intropattern(I3)
simple_intropattern(I4) simple_intropattern(I5) ":" constr(T) :=
asserts [I1 [I2 [I3 [I4 I5]]]]: T.
Tactic Notation "asserts" simple_intropattern(I1)
simple_intropattern(I2) simple_intropattern(I3)
simple_intropattern(I4) simple_intropattern(I5)
simple_intropattern(I6) ":" constr(T) :=
asserts [I1 [I2 [I3 [I4 [I5 I6]]]]]: T.
(** [asserts: T as H] is another syntax for [asserts H: T] *)
Tactic Notation "asserts" ":" constr(T) "as" simple_intropattern(I) :=
asserts I : T.
(** [asserts: T] is [asserts H: T] with [H] being chosen automatically. *)
Tactic Notation "asserts" ":" constr(T) :=
let H := fresh in asserts H : T.
(** [cuts H: T] is the same as [asserts H: T] except that the two subgoals
generated are swapped: the subgoal [T] comes second. Note that contrary
to [cut], it introduces the hypothesis. *)
Tactic Notation "cuts" simple_intropattern(I) ":" constr(T) :=
cut (T); [ intros I | idtac ].
(** [cuts: T] is [cuts H: T] with [H] being chosen automatically. *)
Tactic Notation "cuts" ":" constr(T) :=
let H := fresh in cuts H: T.
(** [cuts H1 .. HN: T] is the same as
[cuts \[H1 \[H2 \[.. HN\]\]\]\]: T]. *)
Tactic Notation "cuts" simple_intropattern(I1)
simple_intropattern(I2) ":" constr(T) :=
cuts [I1 I2]: T.
Tactic Notation "cuts" simple_intropattern(I1)
simple_intropattern(I2) simple_intropattern(I3) ":" constr(T) :=
cuts [I1 [I2 I3]]: T.
Tactic Notation "cuts" simple_intropattern(I1)
simple_intropattern(I2) simple_intropattern(I3)
simple_intropattern(I4) ":" constr(T) :=
cuts [I1 [I2 [I3 I4]]]: T.
Tactic Notation "cuts" simple_intropattern(I1)
simple_intropattern(I2) simple_intropattern(I3)
simple_intropattern(I4) simple_intropattern(I5) ":" constr(T) :=
cuts [I1 [I2 [I3 [I4 I5]]]]: T.
Tactic Notation "cuts" simple_intropattern(I1)
simple_intropattern(I2) simple_intropattern(I3)
simple_intropattern(I4) simple_intropattern(I5)
simple_intropattern(I6) ":" constr(T) :=
cuts [I1 [I2 [I3 [I4 [I5 I6]]]]]: T.
(* ---------------------------------------------------------------------- *)
(** ** Instantiation and forward-chaining *)
(** The instantiation tactics are used to instantiate a lemma [E]
(whose type is a product) on some arguments. The type of [E] is
made of implications and universal quantifications, e.g.
[forall x, P x -> forall y z, Q x y z -> R z].
The first possibility is to provide arguments in order: first [x],
then a proof of [P x], then [y] etc... In this mode, called "Args",
all the arguments are to be provided. If a wildcard is provided
(written [__]), then an existential variable will be introduced in
place of the argument.
It often saves a lot of time to give only the dependent variables,
(here [x], [y] and [z]), and have the hypotheses generated as
subgoals. In this "Vars" mode, only variables are to be provided.
For instance, lemma [E] applied to [3] and [4] is a term
of type [forall z, Q 3 4 z -> R z], and [P 3] is a new subgoal.
It is possible to use wildcards to introduce existential variables.
However, there are situations where some of the hypotheses already
exists, and it saves time to instantiate the lemma [E] using the
hypotheses. For instance, suppose [F] is a term of type [P 2].
Then the application of [E] to [F] in this "Hyps" mode is a term of type
[forall y z, Q 2 y z -> R z]. Each wildcard use
will generate an assertion instead, for instance if [G] has type
[Q 2 3 4], then the application of [E] to a wildcard and to [G]
in mode-h is a term of type [R 4], and [P 2] is a new subgoal.
It is very convenient to give some arguments the lemma should be
instantiated on, and let the tactic find out automatically where
underscores should be insterted. The technique is simple: try to
place an argument, and if it does not work insert an underscore.
In this "Hints" mode ([Hnts] for short), underscore [__] would be useless,
since they can be omitted. So, we interpret underscore as follows:
an underscore means that we want to skip the argument that has the
same type as the next real argument provided (real means not an
underscore). If there is no real argument after underscore, then the
the underscore is used for the first possible argument.
There are four modes of instantiation:
- "Args": give all arguments,
- "Vars": give only variables,
- "Hyps": give only hypotheses,
- "Hnts": give some arguments.
The general syntax is [tactic (>>>Mode E1 .. EN)] where [tactic] is
the name of the tactic (possibly with some arguments) and [Mode]
is the name of the mode, and [Ei] are the arguments.
If [Mode] is omitted, [Hnts] will be inserted.
If [>>>Mode] is omitted, [>>>Hnts] will be inserted.
Moreover, some tactics accept the syntax [tactic E1 .. EN]
as short for [tactic (>>>Hnts E1 .. EN)].
Finally, if the argument [EN] given is a triple-underscore [___],
then it is equivalent to providing a list of wildcards, with
the appropriate number of wildcards. This means that all
the remaining arguments of the lemma will be instantiated.
*)
(* Underlying implementation *)
(* -- to be used once v8.1pl4 is no longer supported
Ltac app_darg t A v cont :=
let x := fresh "TEMP" in
evar (x:A);
instantiate (1:=v) in (Value of x);
let t' := constr:(t x) in
let t'' := (eval unfold x in t') in
subst x; cont t''.
*)
Ltac app_arg t P v cont :=
let H := fresh "TEMP" in
assert (H : P); [ apply v | cont(t H); try clear H ].
Ltac app_assert t P cont :=
let H := fresh "TEMP" in
assert (H : P); [ | cont(t H); clear H ].
Ltac app_evar t A cont :=
let x := fresh "TEMP" in
evar (x:A);
let t' := constr:(t x) in
let t'' := (eval unfold x in t') in
subst x; cont t''.
Ltac build_app_alls t final := (* vs is dummy *)
let rec go t :=
match type of t with
| ?P -> ?Q => app_assert t P go
| forall _:?A, _ => app_evar t A go
| _ => final t
end in
go t.
Ltac build_app_args t vs final :=
let rec go t vs :=
match vs with
| nil => first [ final t | fail 1 ]
| (boxer ltac_wilds)::_ => first [ build_app_alls t final | fail 1 ]
| (boxer ?v)::?vs' =>
let cont t' := go t' vs' in
match v with
| ltac_wild =>
match type of t with
| ?P -> ?Q => first [ app_assert t P cont | fail 3 ]
| forall _:?A, _ => first [ app_evar t A cont | fail 20 ]
end
| _ =>
match type of t with
| ?P -> ?Q => first [ app_arg t P v cont | fail 3 ]
| forall _:?A, _ => first [ cont (t v) | fail 3 ]
(*todo: v8.2, use: app_darg t A v cont *)
end
end
end in
go t vs.
Ltac build_app_vars t vs final :=
let rec go t vs :=
match vs with
| nil => first [ final t | fail 1 ]
| (boxer ltac_wilds)::_ => first [ build_app_alls t final | fail 1 ]
| (boxer ?v)::?vs' =>
match type of t with
| ?P -> ?Q =>
let cont t' := go t' vs in
first [ app_assert t P cont | fail 3 ]
| forall _:?A, _ =>
let cont t' := go t' vs' in
match v with
| ltac_wild => first [ app_evar t A cont | fail 3 ]
| _ => first [ cont(t v) | fail 3 ]
end
end
end in
go t vs.
Ltac build_app_hyps t vs final :=
let rec go t vs :=
match vs with
| nil => first [ final t | fail 1 ]
| (boxer ltac_wilds)::_ => first [ build_app_alls t final | fail 1 ]
| (boxer ?v)::?vs' =>
match type of t with
| ?P -> ?Q =>
let cont t' := go t' vs' in
match v with
| ltac_wild => first [ app_assert t P cont | fail 3 ]
| _ => first [ app_arg t P v cont | fail 3 ]
(* if mismatch is authorized
first [ app_arg t P v cont
| let cont' t' := go t' vs in
app_assert t P cont' ] *)
end
| forall _:?A, _ =>
let cont t' := go t' vs in
first [ app_evar t A cont | fail 3 ]
end
end in
go t vs.
Ltac boxerlist_next_type vs :=
match vs with
| nil => constr:(ltac_wild)
| (boxer ltac_wild)::?vs' => boxerlist_next_type vs'
| (boxer ltac_wilds)::_ => constr:(ltac_wild)
| (@boxer ?T _)::_ => constr:(T)
end.
Ltac build_app_hnts t vs final :=
let rec go t vs :=
match vs with
| nil => first [ final t | fail 1 ]
| (boxer ltac_wilds)::_ => first [ build_app_alls t final | fail 1 ]
| (boxer ?v)::?vs' =>
let cont t' := go t' vs in
let cont' t' := go t' vs' in
match v with
| ltac_wild =>
first [ let T := boxerlist_next_type vs' in
match T with
| ltac_wild =>
match type of t with
| ?P -> ?Q => first [ app_assert t P cont' | fail 3 ]
| forall _:?A, _ => first [ app_evar t A cont' | fail 3 ]
end
| _ =>
match type of t with (* should test T for unifiability *)
| T -> ?Q => first [ app_assert t T cont' | fail 3 ]
| forall _:T, _ => first [ app_evar t T cont' | fail 3 ]
| ?P -> ?Q => first [ app_assert t P cont | fail 3 ]
| forall _:?A, _ => first [ app_evar t A cont | fail 3 ]
end
end
| fail 2 ]
| _ =>
match type of t with
| ?P -> ?Q => first [ app_arg t P v cont'
| app_assert t P cont
| fail 3 ]
| forall _:?A, _ => first [ cont' (t v)
| app_evar t A cont
| fail 3 ]
end
end
end in
go t vs.
Ltac build_app t boxlist final :=
let args := ltac_args boxlist in
first [
match args with (boxer ?mode)::?vs =>
match mode with
| Args => build_app_args t vs final
| Vars => build_app_vars t vs final
| Hyps => build_app_hyps t vs final
| Hnts => build_app_hnts t vs final
end end
| fail 1 "Instantiation fails for:" t args].
(** [lets H: E of (>>> E1 E2 .. EN)] will instantiate lemma [E]
on the arguments [Ei] (which may be wildcards [__]),
and name [H] the resulting term ([H] may be an introduction
pattern or a sequence of introduction patterns [I1 I2 IN]).
The keyword "ok" may be replaced with "of_vars" or "of_hyps"
for providing only variables or only hypotheses. If the last
argument [EN] is [___] (triple-underscore), then all
arguments of [H] will be instantiated. *)
Ltac lets_build I E boxlist :=
build_app E boxlist ltac:(fun R => lets I: R).
Tactic Notation "lets" simple_intropattern(I) ":" constr(E) constr(A) :=
lets_build I E A.
Tactic Notation "lets" ":" constr(E) constr(A) :=
let H := fresh in lets H: E A.
Tactic Notation "lets" simple_intropattern(I1) simple_intropattern(I2)
":" constr(E) constr(A) :=
lets [I1 I2]: E A.
Tactic Notation "lets" simple_intropattern(I1) simple_intropattern(I2)
simple_intropattern(I3) ":" constr(E) constr(A) :=
lets [I1 [I2 I3]]: E A.
Tactic Notation "lets" simple_intropattern(I1) simple_intropattern(I2)
simple_intropattern(I3) simple_intropattern(I4) ":" constr(E) constr(A) :=
lets [I1 [I2 [I3 I4]]]: E A.
Tactic Notation "lets" simple_intropattern(I1) simple_intropattern(I2)
simple_intropattern(I3) simple_intropattern(I4) simple_intropattern(I5)
":" constr(E) constr(A) :=
lets [I1 [I2 [I3 [I4 I5]]]]: E A.
Tactic Notation "lets" simple_intropattern(I) ":" constr(E)
constr(A1) constr(A2) :=
lets I: E (>>> A1 A2).
Tactic Notation "lets" simple_intropattern(I) ":" constr(E)
constr(A1) constr(A2) constr(A3) :=
lets I: E (>>> A1 A2 A3).
Tactic Notation "lets" simple_intropattern(I) ":" constr(E)
constr(A1) constr(A2) constr(A3) constr(A4) :=
lets I: E (>>> A1 A2 A3 A4).
Tactic Notation "lets" simple_intropattern(I) ":" constr(E)
constr(A1) constr(A2) constr(A3) constr(A4) constr(A5) :=
lets I: E (>>> A1 A2 A3 A4 A5).
(** [forwards I: E A] is short for [lets I: E A], which means
that it will instantiate E on all arguments, and name [I] the
result. Similarly, [fowards_in H: E] is short for
[applys_in H ___] and will instantiate all arguments of [H]. *)
Tactic Notation "forwards" simple_intropattern(I) ":" constr(E) :=
lets I: E ___.
Tactic Notation "forwards" simple_intropattern(I) ":" constr(E) constr(A) :=
let A' := ltac_args A in
let A'' := (eval simpl in (A' ++ ((boxer ___)::nil))) in
lets I: E A''.
Tactic Notation "forwards" ":" constr(E) :=
let H := fresh in forwards H: E.
Tactic Notation "forwards" ":" constr(E) constr(A) :=
let H := fresh in forwards H: E A.
Tactic Notation "forwards" simple_intropattern(I1) simple_intropattern(I2)
":" constr(E) :=
forwards [I1 I2]: E.