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LibTactics.v
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LibTactics.v
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(** * LibTactics: A Collection of Handy General-Purpose Tactics *)
(* Chapter maintained by Arthur Chargueraud *)
(** 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, serious Coq
users will probably find it very useful.
The present file contains the implementation and the detailed
documentation of those tactics. The SF reader need not read this
file; instead, he/she is encouraged to read the chapter named
UseTactics.v, which is gentle introduction to the most useful
tactics from the LibTactic library. *)
(** 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).
- 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 the idea of tactic [forward],
- thanks to Georges Gonthier for the implementation trick in [rapply],
*)
Set Implicit Arguments.
(* ********************************************************************** *)
(** * Additional notations for Coq *)
(* ---------------------------------------------------------------------- *)
(** ** N-ary Existentials *)
(** [exists T1 ... TN, P] is a shorthand for
[exists T1, ..., exists TN, P]. Note that
[Coq.Program.Syntax] already defines exists
for arity up to 4. *)
Notation "'exists' x1 ',' P" :=
(exists x1, P)
(at level 200, x1 ident,
right associativity) : type_scope.
Notation "'exists' x1 x2 ',' P" :=
(exists x1, exists x2, P)
(at level 200, x1 ident, x2 ident,
right associativity) : type_scope.
Notation "'exists' 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 "'exists' 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 "'exists' 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 "'exists' 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 "'exists' 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 "'exists' 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.
Notation "'exists' x1 x2 x3 x4 x5 x6 x7 x8 x9 ',' P" :=
(exists x1, exists x2, exists x3, exists x4, exists x5, exists x6,
exists x7, exists x8, exists x9, P)
(at level 200, x1 ident, x2 ident, x3 ident, x4 ident, x5 ident,
x6 ident, x7 ident, x8 ident, x9 ident,
right associativity) : type_scope.
Notation "'exists' x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 ',' P" :=
(exists x1, exists x2, exists x3, exists x4, exists x5, exists x6,
exists x7, exists x8, exists x9, exists x10, P)
(at level 200, x1 ident, x2 ident, x3 ident, x4 ident, x5 ident,
x6 ident, x7 ident, x8 ident, x9 ident, x10 ident,
right associativity) : type_scope.
(* ********************************************************************** *)
(** * Tools for programming with Ltac *)
(* ---------------------------------------------------------------------- *)
(** ** Identity continuation *)
Ltac idcont tt :=
idtac.
(* ---------------------------------------------------------------------- *)
(** ** Untyped arguments for tactics *)
(** Any Coq value can be boxed into the type [Boxer]. This is
useful to use Coq computations for implementing tactics. *)
Inductive Boxer : Type :=
| boxer : forall (A:Type), A -> Boxer.
(* ---------------------------------------------------------------------- *)
(** ** Optional arguments for 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.
(* ---------------------------------------------------------------------- *)
(** ** Wildcard arguments for tactics *)
(** [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 is typically 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.
Open Scope ltac_scope.
(* ---------------------------------------------------------------------- *)
(** ** Position markers *)
(** [ltac_Mark] and [ltac_mark] are dummy definitions used as sentinel
by tactics, to mark a certain position in the context or in the goal. *)
Inductive ltac_Mark : Type :=
| ltac_mark : ltac_Mark.
(** [gen_until_mark] repeats [generalize] on hypotheses from the
context, starting from the bottom and stopping as soon as reaching
an hypothesis of type [Mark]. If fails if [Mark] does not
appear in the context. *)
Ltac gen_until_mark :=
match goal with H: ?T |- _ =>
match T with
| ltac_Mark => clear H
| _ => generalize H; clear H; gen_until_mark
end end.
(** [intro_until_mark] repeats [intro] until reaching an hypothesis of
type [Mark]. It throws away the hypothesis [Mark].
It fails if [Mark] does not appear as an hypothesis in the
goal. *)
Ltac intro_until_mark :=
match goal with
| |- (ltac_Mark -> _) => intros _
| _ => intro; intro_until_mark
end.
(* ---------------------------------------------------------------------- *)
(** ** List of arguments for tactics *)
(** A datatype of type [list Boxer] is used to manipulate list of
Coq values in ltac. Notation is [>> v1 v2 ... vN] for building
a list containing the values [v1] through [vN]. *)
Require Import List.
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.
Notation "'>>' v1 v2 v3 v4 v5 v6 v7 v8" :=
((boxer v1)::(boxer v2)::(boxer v3)::(boxer v4)::(boxer v5)
::(boxer v6)::(boxer v7)::(boxer v8)::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,
v8 at level 0)
: ltac_scope.
Notation "'>>' v1 v2 v3 v4 v5 v6 v7 v8 v9" :=
((boxer v1)::(boxer v2)::(boxer v3)::(boxer v4)::(boxer v5)
::(boxer v6)::(boxer v7)::(boxer v8)::(boxer v9)::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,
v8 at level 0, v9 at level 0)
: ltac_scope.
Notation "'>>' v1 v2 v3 v4 v5 v6 v7 v8 v9 v10" :=
((boxer v1)::(boxer v2)::(boxer v3)::(boxer v4)::(boxer v5)
::(boxer v6)::(boxer v7)::(boxer v8)::(boxer v9)::(boxer v10)::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,
v8 at level 0, v9 at level 0, v10 at level 0)
: ltac_scope.
Notation "'>>' v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11" :=
((boxer v1)::(boxer v2)::(boxer v3)::(boxer v4)::(boxer v5)
::(boxer v6)::(boxer v7)::(boxer v8)::(boxer v9)::(boxer v10)
::(boxer v11)::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,
v8 at level 0, v9 at level 0, v10 at level 0, v11 at level 0)
: ltac_scope.
Notation "'>>' v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12" :=
((boxer v1)::(boxer v2)::(boxer v3)::(boxer v4)::(boxer v5)
::(boxer v6)::(boxer v7)::(boxer v8)::(boxer v9)::(boxer v10)
::(boxer v11)::(boxer v12)::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,
v8 at level 0, v9 at level 0, v10 at level 0, v11 at level 0,
v12 at level 0)
: ltac_scope.
Notation "'>>' v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13" :=
((boxer v1)::(boxer v2)::(boxer v3)::(boxer v4)::(boxer v5)
::(boxer v6)::(boxer v7)::(boxer v8)::(boxer v9)::(boxer v10)
::(boxer v11)::(boxer v12)::(boxer v13)::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,
v8 at level 0, v9 at level 0, v10 at level 0, v11 at level 0,
v12 at level 0, v13 at level 0)
: ltac_scope.
(** The tactic [list_boxer_of] inputs a term [E] and returns a term
of type "list boxer", according to the following rules:
- if [E] is already of type "list Boxer", then it returns [E];
- otherwise, it returns the list [(boxer E)::nil]. *)
Ltac list_boxer_of E :=
match type of E with
| List.list Boxer => constr:(E)
| _ => constr:((boxer E)::nil)
end.
(* ---------------------------------------------------------------------- *)
(** ** Databases of lemmas *)
(** Use the hint facility to implement a database mapping
terms to terms. To declare a new database, use a definition:
[Definition mydatabase := True.]
Then, to map [mykey] to [myvalue], write the hint:
[Hint Extern 1 (Register mydatabase mykey) => Provide myvalue.]
Finally, to query the value associated with a key, run the
tactic [ltac_database_get mydatabase mykey]. This will leave
at the head of the goal the term [myvalue]. It can then be
named and exploited using [intro]. *)
Definition ltac_database (D:Boxer) (T:Boxer) (A:Boxer) := True.
Notation "'Register' D T" := (ltac_database (boxer D) (boxer T) _)
(at level 69, D at level 0, T at level 0).
Lemma ltac_database_provide : forall (A:Boxer) (D:Boxer) (T:Boxer),
ltac_database D T A.
Proof. split. Qed.
Ltac Provide T := apply (@ltac_database_provide (boxer T)).
Ltac ltac_database_get D T :=
let A := fresh "TEMP" in evar (A:Boxer);
let H := fresh "TEMP" in
assert (H : ltac_database (boxer D) (boxer T) A);
[ subst A; auto
| subst A; match type of H with ltac_database _ _ (boxer ?L) =>
generalize L end; clear H ].
(* ---------------------------------------------------------------------- *)
(** ** On-the-fly removal of hypotheses *)
(** In a list of arguments [>> H1 H2 .. HN] passed to a tactic
such as [lets] or [applys] or [forwards] or [specializes],
the term [rm], an identity function, can be placed in front
of the name of an hypothesis to be deleted. *)
Definition rm (A:Type) (X:A) := X.
(** [rm_term E] removes one hypothesis that admits the same
type as [E]. *)
Ltac rm_term E :=
let T := type of E in
match goal with H: T |- _ => try clear H end.
(** [rm_inside E] calls [rm_term Ei] for any subterm
of the form [rm Ei] found in E *)
Ltac rm_inside E :=
let go E := rm_inside E in
match E with
| rm ?X => rm_term X
| ?X1 ?X2 =>
go X1; go X2
| ?X1 ?X2 ?X3 =>
go X1; go X2; go X3
| ?X1 ?X2 ?X3 ?X4 =>
go X1; go X2; go X3; go X4
| ?X1 ?X2 ?X3 ?X4 ?X5 =>
go X1; go X2; go X3; go X4; go X5
| ?X1 ?X2 ?X3 ?X4 ?X5 ?X6 =>
go X1; go X2; go X3; go X4; go X5; go X6
| ?X1 ?X2 ?X3 ?X4 ?X5 ?X6 ?X7 =>
go X1; go X2; go X3; go X4; go X5; go X6; go X7
| ?X1 ?X2 ?X3 ?X4 ?X5 ?X6 ?X7 ?X8 =>
go X1; go X2; go X3; go X4; go X5; go X6; go X7; go X8
| ?X1 ?X2 ?X3 ?X4 ?X5 ?X6 ?X7 ?X8 ?X9 =>
go X1; go X2; go X3; go X4; go X5; go X6; go X7; go X8; go X9
| ?X1 ?X2 ?X3 ?X4 ?X5 ?X6 ?X7 ?X8 ?X9 ?X10 =>
go X1; go X2; go X3; go X4; go X5; go X6; go X7; go X8; go X9; go X10
| _ => idtac
end.
(** For faster performance, one may deactivate [rm_inside] by
replacing the body of this definition with [idtac]. *)
Ltac fast_rm_inside E :=
rm_inside E.
(* ---------------------------------------------------------------------- *)
(** ** Numbers as arguments *)
(** When tactic takes a natural number as argument, it may be
parsed either as a natural number or as a relative number.
In order for tactics to convert their arguments into natural numbers,
we provide a conversion tactic. *)
Require BinPos Coq.ZArith.BinInt.
Definition ltac_nat_from_int (x:BinInt.Z) : nat :=
match x with
| BinInt.Z0 => 0%nat
| BinInt.Zpos p => BinPos.nat_of_P p
| BinInt.Zneg p => 0%nat
end.
Ltac nat_from_number N :=
match type of N with
| nat => constr:(N)
| BinInt.Z => let N' := constr:(ltac_nat_from_int N) in eval compute in N'
end.
(** [ltac_pattern E at K] is the same as [pattern E at K] except that
[K] is a Coq natural rather than a Ltac integer. Syntax
[ltac_pattern E as K in H] is also available. *)
Tactic Notation "ltac_pattern" constr(E) "at" constr(K) :=
match nat_from_number K with
| 1 => pattern E at 1
| 2 => pattern E at 2
| 3 => pattern E at 3
| 4 => pattern E at 4
| 5 => pattern E at 5
| 6 => pattern E at 6
| 7 => pattern E at 7
| 8 => pattern E at 8
end.
Tactic Notation "ltac_pattern" constr(E) "at" constr(K) "in" hyp(H) :=
match nat_from_number K with
| 1 => pattern E at 1 in H
| 2 => pattern E at 2 in H
| 3 => pattern E at 3 in H
| 4 => pattern E at 4 in H
| 5 => pattern E at 5 in H
| 6 => pattern E at 6 in H
| 7 => pattern E at 7 in H
| 8 => pattern E at 8 in H
end.
(* ---------------------------------------------------------------------- *)
(** ** Testing tactics *)
(** [show tac] executes a tactic [tac] that produces a result,
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 nat_from_number 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.
(* ---------------------------------------------------------------------- *)
(** ** Check no evar in goal *)
Ltac check_noevar M :=
match M with M => idtac end.
Ltac check_noevar_hyp H := (* todo: imlement using check_noevar *)
let T := type of H in
match type of H with T => idtac end.
Ltac check_noevar_goal := (* todo: imlement using check_noevar *)
match goal with |- ?G => match G with G => idtac end end.
(* ---------------------------------------------------------------------- *)
(** ** Tagging of hypotheses *)
(** [get_last_hyp tt] is a function that returns the last hypothesis
at the bottom of the context. It is useful to obtain the default
name associated with the hypothesis, e.g.
[intro; let H := get_last_hyp tt in let H' := fresh "P" H in ...] *)
Ltac get_last_hyp tt :=
match goal with H: _ |- _ => constr:(H) end.
(* ---------------------------------------------------------------------- *)
(** ** Tagging of hypotheses *)
(** [ltac_tag_subst] is a specific marker for hypotheses
which is used to tag hypotheses that are equalities to
be substituted. *)
Definition ltac_tag_subst (A:Type) (x:A) := x.
(** [ltac_to_generalize] is a specific marker for hypotheses
to be generalized. *)
Definition ltac_to_generalize (A:Type) (x:A) := x.
Ltac gen_to_generalize :=
repeat match goal with
H: ltac_to_generalize _ |- _ => generalize H; clear H end.
Ltac mark_to_generalize H :=
let T := type of H in
change T with (ltac_to_generalize T) in H.
(* ---------------------------------------------------------------------- *)
(** ** 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].
Warning: the tactic seems to loop in some cases when the goal is
a product and one uses the result of this function. *)
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)
| ?P _ _ => constr:(P)
| ?P _ => constr:(P)
| ?P => constr:(P)
end.
(** [get_fun_arg E] is a tactic that decomposes an application
term [E], ie, when applied to a term of the form [X1 ... XN]
it returns a pair made of [X1 .. X(N-1)] and [XN]. *)
Ltac get_fun_arg E :=
match E with
| ?X1 ?X2 ?X3 ?X4 ?X5 ?X6 ?X7 ?X => constr:((X1 X2 X3 X4 X5 X6,X))
| ?X1 ?X2 ?X3 ?X4 ?X5 ?X6 ?X => constr:((X1 X2 X3 X4 X5,X))
| ?X1 ?X2 ?X3 ?X4 ?X5 ?X => constr:((X1 X2 X3 X4,X))
| ?X1 ?X2 ?X3 ?X4 ?X => constr:((X1 X2 X3,X))
| ?X1 ?X2 ?X3 ?X => constr:((X1 X2,X))
| ?X1 ?X2 ?X => constr:((X1,X))
| ?X1 ?X => constr:((X1,X))
end.
(* ---------------------------------------------------------------------- *)
(** ** Action at occurence and action not at occurence *)
(** [ltac_action_at K of E do Tac] isolates the [K]-th occurence of [E] in the
goal, setting it in the form [P E] for some named pattern [P],
then calls tactic [Tac], and finally unfolds [P]. Syntax
[ltac_action_at K of E in H do Tac] is also available. *)
Tactic Notation "ltac_action_at" constr(K) "of" constr(E) "do" tactic(Tac) :=
let p := fresh in ltac_pattern E at K;
match goal with |- ?P _ => set (p:=P) end;
Tac; unfold p; clear p.
Tactic Notation "ltac_action_at" constr(K) "of" constr(E) "in" hyp(H) "do" tactic(Tac) :=
let p := fresh in ltac_pattern E at K in H;
match type of H with ?P _ => set (p:=P) in H end;
Tac; unfold p in H; clear p.
(** [protects E do Tac] temporarily assigns a name to the expression [E]
so that the execution of tactic [Tac] will not modify [E]. This is
useful for instance to restrict the action of [simpl]. *)
Tactic Notation "protects" constr(E) "do" tactic(Tac) :=
(* let x := fresh "TEMP" in sets_eq x: E; T; subst x. *)
let x := fresh "TEMP" in let H := fresh "TEMP" in
set (X := E) in *; assert (H : X = E) by reflexivity;
clearbody X; Tac; subst x.
Tactic Notation "protects" constr(E) "do" tactic(Tac) "/" :=
protects E do Tac.
(* ---------------------------------------------------------------------- *)
(** ** An alias for [eq] *)
(** [eq'] is an alias for [eq] to be used for equalities in
inductive definitions, so that they don't get mixed with
equalities generated by [inversion]. *)
Definition eq' := @eq.
Hint Unfold eq'.
Notation "x '='' y" := (@eq' _ x y)
(at level 70, arguments at next level).
(* ********************************************************************** *)
(** * Backward and forward chaining *)
(* ---------------------------------------------------------------------- *)
(** ** Application *)
(** [rapply] 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 "rapply" constr(t) :=
first (* todo: les @ sont inutiles *)
[ eexact (@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 _ _ _ _ _ _ _ _ _ _ _ _)
| 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]. *)
Tactic Notation "rapply_0" constr(t) :=
refine (@t).
Tactic Notation "rapply_1" constr(t) :=
refine (@t _).
Tactic Notation "rapply_2" constr(t) :=
refine (@t _ _).
Tactic Notation "rapply_3" constr(t) :=
refine (@t _ _ _).
Tactic Notation "rapply_4" constr(t) :=
refine (@t _ _ _ _).
Tactic Notation "rapply_5" constr(t) :=
refine (@t _ _ _ _ _).
Tactic Notation "rapply_6" constr(t) :=
refine (@t _ _ _ _ _ _).
Tactic Notation "rapply_7" constr(t) :=
refine (@t _ _ _ _ _ _ _).
Tactic Notation "rapply_8" constr(t) :=
refine (@t _ _ _ _ _ _ _ _).
Tactic Notation "rapply_9" constr(t) :=
refine (@t _ _ _ _ _ _ _ _ _).
Tactic Notation "rapply_10" constr(t) :=
refine (@t _ _ _ _ _ _ _ _ _ _).
(** [lets_base 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. *)
Ltac lets_base I E := generalize E; intros I.
(** [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_base H (E H')
| lets_base H (E _ H')
| lets_base H (E _ _ H')
| lets_base H (E _ _ _ H')
| lets_base H (E _ _ _ _ H')
| lets_base H (E _ _ _ _ _ H')
| lets_base H (E _ _ _ _ _ _ H')
| lets_base H (E _ _ _ _ _ _ _ H')
| lets_base H (E _ _ _ _ _ _ _ _ H')
| lets_base H (E _ _ _ _ _ _ _ _ _ H') ]
); clear H'.
(** [constructors] calls [constructor] or [econstructor]. *)
Tactic Notation "constructors" :=
first [ constructor | econstructor ]; unfold eq'.
(* ---------------------------------------------------------------------- *)
(** ** 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" :=
elimtype False.
(** [false_post] is the underlying tactic used to prove goals
of the form [False]. In the default implementation, it proves
the goal if the context contains [False] or an hypothesis of the
form [C x1 .. xN = D y1 .. yM], or if the [congruence] tactic
finds a proof of [x <> x] for some [x]. *)
Ltac false_post :=
solve [ assumption | discriminate | congruence ].
(** [false] replaces any goal by the goal [False], and calls [false_post] *)
Tactic Notation "false" :=
false_goal; try false_post.
(** [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 \[ false; tac \]].
Example: [tryfalse by congruence/] *)
Tactic Notation "tryfalse" "by" tactic(tac) "/" :=
try solve [ false; instantiate; tac ].
(** [false T] tries [false; apply T], or otherwise adds [T] as
an assumption and calls [false]. *)
Tactic Notation "false" constr(T) "by" tactic(tac) "/" :=
false_goal; first
[ first [ apply T | eapply T | rapply T]; instantiate; tac (* todo: sapply?*)
| let H := fresh in lets_base H T;
first [ discriminate H (* optimization *)
| false; instantiate; tac ] ].
(* todo: false (>> H X1 X2)... *)
Tactic Notation "false" constr(T) :=
false T by idtac/.
(** [false_invert] proves any goal provided there is at least
one hypothesis [H] in the context that can be proved absurd
by calling [inversion H]. *)
Ltac false_invert_tactic :=
match goal with H:_ |- _ =>
solve [ inversion H
| clear H; false_invert_tactic
| fail 2 ] end.
Tactic Notation "false_invert" :=
false_invert_tactic.
(** [tryfalse_invert] tries to prove the goal using
[false] or [false_invert], and leaves the goal
unchanged if it does not succeed. *)
Tactic Notation "tryfalse_invert" :=
try solve [ false | false_invert ].
(** [asserts H: T] is another syntax for [assert (H : T)], which
also works with introduction patterns. For instance, one 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 a shorthand for
[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] 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 a shorthand for
[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. Underscore arguments [__] are
interpret 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 underscore is used for the first possible argument.
The general syntax is [tactic (>> E1 .. EN)] where [tactic] is
the name of the tactic (possibly with some arguments) and [Ei]
are the arguments. Moreover, some tactics accept the syntax
[tactic E1 .. EN] as short for [tactic (>>Hnts E1 .. EN)] for
values of [N] up to 5.
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 *)
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 app_arg t P v cont :=
let H := fresh "TEMP" in
assert (H : P); [ apply v | cont(t H); try clear H ].
Ltac build_app_alls t final :=
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 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
let T := type of t in
let T := eval hnf in T in
match v with
| ltac_wild =>
first [ let U := boxerlist_next_type vs' in
match U with
| ltac_wild =>
match T with
| ?P -> ?Q => first [ app_assert t P cont' | fail 3 ]
| forall _:?A, _ => first [ app_evar t A cont' | fail 3 ]
end
| _ =>
match T with (* should test T for unifiability *)
| U -> ?Q => first [ app_assert t U cont' | fail 3 ]
| forall _:U, _ => first [ app_evar t U 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 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 args final :=
first [
match args with (@boxer ?T ?t)::?vs =>
let t := constr:(t:T) in
build_app_hnts t vs final
end
| fail 1 "Instantiation fails for:" args].
Ltac unfold_head_until_product T :=
eval hnf in T.
Ltac args_unfold_head_if_not_product args :=
match args with (@boxer ?T ?t)::?vs =>
let T' := unfold_head_until_product T in
constr:((@boxer T' t)::vs)
end.
Ltac args_unfold_head_if_not_product_but_params args :=
match args with
| (boxer ?t)::(boxer ?v)::?vs =>
args_unfold_head_if_not_product args