LibTacticsA 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 propositions.
Convenient support for automation: tactics followed with the symbol "~" or "*" will call automation on the generated subgoals. The 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.

Require Import List.

(* Very important to remove hint trans_eq_bool from LibBool,
   otherwise eauto slows down dramatically:
  Lemma test : forall b, b = false.
  time eauto 7. (* takes over 4 seconds  to fail! *) *)

Remove Hints Bool.trans_eq_bool.

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 : ∀ (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.

gen_until_mark_with_processing F is similar to gen_until_mark except that it calls F on each hypothesis immediately before generalizing it. This is useful for processing the hypotheses.

Ltac gen_until_mark_with_processing cont :=
  match goal with H: ?T ⊢ _ ⇒
  match T with
  | ltac_Mark ⇒ clear H
  | _ ⇒ cont H; generalize H; clear H;
         gen_until_mark_with_processing cont
  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.

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.

Inductive Ltac_database_token : Prop := ltac_database_token.

Definition ltac_database (D:Boxer) (T:Boxer) (A:Boxer) := Ltac_database_token.

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 : ∀ (A:Boxer) (D:Boxer) (T:Boxer),
ltac_database D T A.
Proof using. 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 ].

(* Note for a possible alternative implementation of the ltac_database_token:
   Inductive Ltac_database : Type :=
     | ltac_database : forall A, A -> Ltac_database.
   Implicit Arguments ltac_database A.
*)

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.

Require Coq.Numbers.BinNums Coq.ZArith.BinInt.

Definition ltac_int_to_nat (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 number_to_nat N :=
  match type of N with
  | nat ⇒ constr:(N)
  | BinInt.Z ⇒ let N' := constr:(ltac_int_to_nat 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 number (nat or Z) rather than a Ltac integer. Syntax ltac_pattern E as K in H is also available.

ltac_set (x := E) at K is the same as set (x := E) at K except that K is a Coq number (nat or Z) rather than a Ltac integer.

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 : ∀ P, P → P → P.
Proof using. auto. Qed.

Ltac dup_tactic N :=
  match number_to_nat 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.

Testing evars and non-evars

is_not_evar E succeeds only if E is not an evar; it fails otherwise. It thus implements the negation of is_evar

Ltac is_not_evar E :=
first [ is_evar E; fail 1
| idtac ].

is_evar_as_bool E evaluates to true if E is an evar and to false otherwise.

Ltac is_evar_as_bool E :=
  constr:(ltac:(first
    [ is_evar E; exact true
    | exact false ])).

Check No Evar in Goal

Ltac check_noevar M :=
first [ has_evar M; fail 2 | idtac ].

Ltac check_noevar_hyp H :=
let T := type of H in check_noevar T.

Ltac check_noevar_goal :=
match goal with ⊢ ?G ⇒ check_noevar G end.

Helper Function for Introducing Evars

with_evar T (fun M ⇒ tac) creates a new evar that can be used in the tactic tac under the name M.

Ltac with_evar_base T cont :=
let x := fresh "TEMP" in evar (x:T); cont x; subst x.

Tactic Notation "with_evar" constr(T) tactic(cont) :=
with_evar_base T cont.

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.

More 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.

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 X₇,X))
  | ?X₁ ?X₂ ?X₃ ?X₄ ?X₅ ?X₆ ?X ⇒ constr:((X₁ X₂ X₃ X₄ X₅ X₆,X))
  | ?X₁ ?X₂ ?X₃ ?X₄ ?X₅ ?X ⇒ constr:((X₁ X₂ X₃ X₄ X₅,X))
  | ?X₁ ?X₂ ?X₃ ?X₄ ?X ⇒ constr:((X₁ X₂ X₃ X₄,X))
  | ?X₁ ?X₂ ?X₃ ?X ⇒ constr:((X₁ X₂ X₃,X))
  | ?X₁ ?X₂ ?X ⇒ constr:((X₁ X₂,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 "TEMP" 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 "TEMP" 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, y at next level).

Common Tactics for Simplifying Goals Like intuition

Ltac jauto_set_hyps :=
  repeat match goal with H: ?T ⊢ _ ⇒
    match T with
    | _ ∧ _ ⇒ destruct H
    | ∃ a, _ ⇒ destruct H
    | _ ⇒ generalize H; clear H
    end
  end.

Ltac jauto_set_goal :=
  repeat match goal with
  | ⊢ ∃ a, _ ⇒ esplit
  | ⊢ _ ∧ _ ⇒ split
  end.

Ltac jauto_set :=
  intros; jauto_set_hyps;
  intros; jauto_set_goal;
  unfold not in ×.

Backward and Forward Chaining

Application

Ltac old_refine f :=
refine f. (* ; shelve_unifiable. *)

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.

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) :=
  old_refine (@t).
Tactic Notation "rapply_1" constr(t) :=
  old_refine (@t _).
Tactic Notation "rapply_2" constr(t) :=
  old_refine (@t _ _).
Tactic Notation "rapply_3" constr(t) :=
  old_refine (@t _ _ _).
Tactic Notation "rapply_4" constr(t) :=
  old_refine (@t _ _ _ _).
Tactic Notation "rapply_5" constr(t) :=
  old_refine (@t _ _ _ _ _).
Tactic Notation "rapply_6" constr(t) :=
  old_refine (@t _ _ _ _ _ _).
Tactic Notation "rapply_7" constr(t) :=
  old_refine (@t _ _ _ _ _ _ _).
Tactic Notation "rapply_8" constr(t) :=
  old_refine (@t _ _ _ _ _ _ _ _).
Tactic Notation "rapply_9" constr(t) :=
  old_refine (@t _ _ _ _ _ _ _ _ _).
Tactic Notation "rapply_10" constr(t) :=
  old_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).

applys_to H₁,...,HN E applys E to several hypotheses

Tactic Notation "applys_to" hyp(H₁) "," hyp(H₂) constr(E) :=
  applys_to H₁ E; applys_to H₂ E.
Tactic Notation "applys_to" hyp(H₁) "," hyp(H₂) "," hyp(H₃) constr(E) :=
  applys_to H₁ E; applys_to H₂ E; applys_to H₃ E.
Tactic Notation "applys_to" hyp(H₁) "," hyp(H₂) "," hyp(H₃) "," hyp(H₄) constr(E) :=
  applys_to H₁ E; applys_to H₂ E; applys_to H₃ E; applys_to H₄ E.

constructors calls constructor or econstructor.

Tactic Notation "constructors" :=
first [ constructor | econstructor ]; unfold eq'.

Assertions

asserts H: T is another syntax for assert (H : T), which also works with introduction patterns. For instance, one can write: asserts \[x P\] (∃ n, n = 3), or asserts \[H|H\] (n = 0 ∨ n = 1).

Tactic Notation "asserts" simple_intropattern(I) ":" constr(T) :=
let H := fresh "TEMP" 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 "TEMP" 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 "TEMP" 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. ∀ x, P x → ∀ 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 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 (>> 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. Definitions in the conclusion are not unfolded in this case.

(* 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
    | ∀ _:?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.

(* Note: refuse to instantiate a dependent hypothesis with a proposition;
refuse to instantiate an argument of type Type with one that
does not have the type Type.
*)

newer version : support for typeclasses

Ltac app_typeclass t cont :=
let t' := constr:(t _) in
cont t'.

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;
      fast_rm_inside args
    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
  | _ ⇒ constr:(args)
  end.

lets H: (>> E₀ E₁ .. EN) will instantiate lemma E₀ 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, or empty. Syntax lets H: E₀ E₁ .. EN is also available. If the last argument EN is ___ (triple-underscore), then all arguments of H will be instantiated.

Ltac lets_build I Ei :=
  let args := list_boxer_of Ei in
  let args := args_unfold_head_if_not_product_but_params args in
(*    let Ei''' := args_unfold_head_if_not_product Ei'' in*)
  build_app args ltac:(fun R ⇒ lets_base I R).

Tactic Notation "lets" simple_intropattern(I) ":" constr(E) :=
  lets_build I E.
Tactic Notation "lets" ":" constr(E) :=
  let H := fresh in lets H: E.
Tactic Notation "lets" ":" constr(E₀)
constr(A₁) :=
  lets: (>> E₀ A₁).
Tactic Notation "lets" ":" constr(E₀)
constr(A₁) constr(A₂) :=
  lets: (>> E₀ A₁ A₂).
Tactic Notation "lets" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) :=
  lets: (>> E₀ A₁ A₂ A₃).
Tactic Notation "lets" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  lets: (>> E₀ A₁ A₂ A₃ A₄).
Tactic Notation "lets" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  lets: (>> E₀ A₁ A₂ A₃ A₄ A₅).

(* --todo: deprecated, do not use *)
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(I) ":" constr(E₀)
constr(A₁) :=
  lets I: (>> E₀ A₁).
Tactic Notation "lets" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) :=
  lets I: (>> E₀ A₁ A₂).
Tactic Notation "lets" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) :=
  lets I: (>> E₀ A₁ A₂ A₃).
Tactic Notation "lets" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  lets I: (>> E₀ A₁ A₂ A₃ A₄).
Tactic Notation "lets" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  lets I: (>> E₀ A₁ A₂ A₃ A₄ A₅).

Tactic Notation "lets" simple_intropattern(I₁) simple_intropattern(I₂) ":" constr(E₀)
constr(A₁) :=
  lets [I₁ I₂]: E₀ A₁.
Tactic Notation "lets" simple_intropattern(I₁) simple_intropattern(I₂) ":" constr(E₀)
constr(A₁) constr(A₂) :=
  lets [I₁ I₂]: E₀ A₁ A₂.
Tactic Notation "lets" simple_intropattern(I₁) simple_intropattern(I₂) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) :=
  lets [I₁ I₂]: E₀ A₁ A₂ A₃.
Tactic Notation "lets" simple_intropattern(I₁) simple_intropattern(I₂) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  lets [I₁ I₂]: E₀ A₁ A₂ A₃ A₄.
Tactic Notation "lets" simple_intropattern(I₁) simple_intropattern(I₂) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  lets [I₁ I₂]: E₀ A₁ A₂ A₃ A₄ A₅.

forwards H: (>> E₀ E₁ .. EN) is short for forwards H: (>> E₀ E₁ .. EN ___). The arguments Ei can be wildcards __ (except E₀). H may be an introduction pattern, or a sequence of introduction pattern, or empty. Syntax forwards H: E₀ E₁ .. EN is also available.

Ltac forwards_build_app_arg Ei :=
  let args := list_boxer_of Ei in
  let args := (eval simpl in (args ++ ((boxer ___)::nil ))) in
  let args := args_unfold_head_if_not_product args in
  args.

Ltac forwards_then Ei cont :=
  let args := forwards_build_app_arg Ei in
  let args := args_unfold_head_if_not_product_but_params args in
  build_app args cont.

Tactic Notation "forwards" simple_intropattern(I) ":" constr(Ei) :=
let args := forwards_build_app_arg Ei in
lets I: args.

Tactic Notation "forwards" ":" constr(E) :=
  let H := fresh in forwards H: E.
Tactic Notation "forwards" ":" constr(E₀)
constr(A₁) :=
  forwards: (>> E₀ A₁).
Tactic Notation "forwards" ":" constr(E₀)
constr(A₁) constr(A₂) :=
  forwards: (>> E₀ A₁ A₂).
Tactic Notation "forwards" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) :=
  forwards: (>> E₀ A₁ A₂ A₃).
Tactic Notation "forwards" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  forwards: (>> E₀ A₁ A₂ A₃ A₄).
Tactic Notation "forwards" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  forwards: (>> E₀ A₁ A₂ A₃ A₄ A₅).

(* --TODO: deprecated, do not use *)
Tactic Notation "forwards" simple_intropattern(I₁) simple_intropattern(I₂)
":" constr(E) :=
  forwards [I₁ I₂]: E.
Tactic Notation "forwards" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) ":" constr(E) :=
  forwards [I₁ [I₂ I₃]]: E.
Tactic Notation "forwards" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) ":" constr(E) :=
  forwards [I₁ [I₂ [I₃ I₄]]]: E.
Tactic Notation "forwards" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
":" constr(E) :=
  forwards [I₁ [I₂ [I₃ [I₄ I₅]]]]: E.

Tactic Notation "forwards" simple_intropattern(I) ":" constr(E₀)
constr(A₁) :=
  forwards I: (>> E₀ A₁).
Tactic Notation "forwards" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) :=
  forwards I: (>> E₀ A₁ A₂).
Tactic Notation "forwards" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) :=
  forwards I: (>> E₀ A₁ A₂ A₃).
Tactic Notation "forwards" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  forwards I: (>> E₀ A₁ A₂ A₃ A₄).
Tactic Notation "forwards" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  forwards I: (>> E₀ A₁ A₂ A₃ A₄ A₅).

forwards_nounfold I: E is like forwards I: E but does not unfold the head constant of E if there is no visible quantification or hypothesis in E. It is meant to be used mainly by tactics.

Tactic Notation "forwards_nounfold" simple_intropattern(I) ":" constr(Ei) :=
  let args := list_boxer_of Ei in
  let args := (eval simpl in (args ++ ((boxer ___)::nil ))) in
  build_app args ltac:(fun R ⇒ lets_base I R).

forwards_nounfold_then E ltac:(fun K ⇒ ..) is like forwards: E but it provides the resulting term to a continuation, under the name K.

Ltac forwards_nounfold_then Ei cont :=
  let args := list_boxer_of Ei in
  let args := (eval simpl in (args ++ ((boxer ___)::nil ))) in
  build_app args cont.

applys (>> E₀ E₁ .. EN) instantiates lemma E₀ on the arguments Ei (which may be wildcards __), and apply the resulting term to the current goal, using the tactic applys defined earlier on. applys E₀ E₁ E₂ .. EN is also available.

Ltac applys_build Ei :=
  let args := list_boxer_of Ei in
  let args := args_unfold_head_if_not_product_but_params args in
  build_app args ltac:(fun R ⇒
   first [ apply R | eapply R | rapply R ]).

Ltac applys_base E :=
  match type of E with
  | list Boxer ⇒ applys_build E
  | _ ⇒ first [ rapply E | applys_build E ]
  end; fast_rm_inside E.

Tactic Notation "applys" constr(E) :=
  applys_base E.
Tactic Notation "applys" constr(E₀) constr(A₁) :=
  applys (>> E₀ A₁).
Tactic Notation "applys" constr(E₀) constr(A₁) constr(A₂) :=
  applys (>> E₀ A₁ A₂).
Tactic Notation "applys" constr(E₀) constr(A₁) constr(A₂) constr(A₃) :=
  applys (>> E₀ A₁ A₂ A₃).
Tactic Notation "applys" constr(E₀) constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  applys (>> E₀ A₁ A₂ A₃ A₄).
Tactic Notation "applys" constr(E₀) constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  applys (>> E₀ A₁ A₂ A₃ A₄ A₅).

fapplys (>> E₀ E₁ .. EN) instantiates lemma E₀ on the arguments Ei and on the argument ___ meaning that all evars should be explicitly instantiated, and apply the resulting term to the current goal. fapplys E₀ E₁ E₂ .. EN is also available.

Ltac fapplys_build Ei :=
  let args := list_boxer_of Ei in
  let args := (eval simpl in (args ++ ((boxer ___)::nil ))) in
  let args := args_unfold_head_if_not_product_but_params args in
  build_app args ltac:(fun R ⇒ apply R).

Tactic Notation "fapplys" constr(E₀) := (* --todo: use the tactic for that*)
  match type of E₀ with
  | list Boxer ⇒ fapplys_build E₀
  | _ ⇒ fapplys_build (>> E₀)
  end.
Tactic Notation "fapplys" constr(E₀) constr(A₁) :=
  fapplys (>> E₀ A₁).
Tactic Notation "fapplys" constr(E₀) constr(A₁) constr(A₂) :=
  fapplys (>> E₀ A₁ A₂).
Tactic Notation "fapplys" constr(E₀) constr(A₁) constr(A₂) constr(A₃) :=
  fapplys (>> E₀ A₁ A₂ A₃).
Tactic Notation "fapplys" constr(E₀) constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  fapplys (>> E₀ A₁ A₂ A₃ A₄).
Tactic Notation "fapplys" constr(E₀) constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  fapplys (>> E₀ A₁ A₂ A₃ A₄ A₅).

specializes H (>> E₁ E₂ .. EN) will instantiate hypothesis H on the arguments Ei (which may be wildcards __). If the last argument EN is ___ (triple-underscore), then all arguments of H get instantiated.

Ltac specializes_build H Ei :=
  let H' := fresh "TEMP" in rename H into H';
  let args := list_boxer_of Ei in
  let args := constr:((boxer H')::args) in
  let args := args_unfold_head_if_not_product args in
  build_app args ltac:(fun R ⇒ lets H: R);
  clear H'.

Ltac specializes_base H Ei :=
specializes_build H Ei; fast_rm_inside Ei.

Tactic Notation "specializes" hyp(H) :=
  specializes_base H (___).
Tactic Notation "specializes" hyp(H) constr(A) :=
  specializes_base H A.
Tactic Notation "specializes" hyp(H) constr(A₁) constr(A₂) :=
  specializes H (>> A₁ A₂).
Tactic Notation "specializes" hyp(H) constr(A₁) constr(A₂) constr(A₃) :=
  specializes H (>> A₁ A₂ A₃).
Tactic Notation "specializes" hyp(H) constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  specializes H (>> A₁ A₂ A₃ A₄).
Tactic Notation "specializes" hyp(H) constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  specializes H (>> A₁ A₂ A₃ A₄ A₅).

specializes_vars H is equivalent to specializes H __ .. __ with as many double underscore as the number of dependent arguments visible from the type of H. Note that no unfolding is currently being performed (this behavior might change in the future). The current implementation is restricted to the case where H is an existing hypothesis -- TODO: generalize.

Ltac specializes_var_base H :=
  match type of H with
  | ?P → ?Q ⇒ fail 1
  | ∀ _:_, _ ⇒ specializes H __
  end.

Ltac specializes_vars_base H :=
repeat (specializes_var_base H).

Tactic Notation "specializes_var" hyp(H) :=
specializes_var_base H.

Tactic Notation "specializes_vars" hyp(H) :=
specializes_vars_base H.

Experimental Tactics for Application

fapply is a version of apply based on forwards.

sapply stands for "super apply". It tries apply, eapply, applys and fapply, and also tries to head-normalize the goal first.

Adding Assumptions

lets_simpl H: E is the same as lets H: E excepts that it calls simpl on the hypothesis H. lets_simpl: E is also provided.

Tactic Notation "lets_simpl" ident(H) ":" constr(E) :=
lets H: E; try simpl in H.

Tactic Notation "lets_simpl" ":" constr(T) :=
let H := fresh "TEMP" in lets_simpl H: T.

lets_hnf H: E is the same as lets H: E excepts that it calls hnf to set the definition in head normal form. lets_hnf: E is also provided.

Tactic Notation "lets_hnf" ident(H) ":" constr(E) :=
lets H: E; hnf in H.

Tactic Notation "lets_hnf" ":" constr(T) :=
let H := fresh "TEMP" in lets_hnf H: T.

puts X: E is a synonymous for pose (X := E). Alternative syntax is puts: E.

Tactic Notation "puts" ident(X) ":" constr(E) :=
pose (X := E).
Tactic Notation "puts" ":" constr(E) :=
let X := fresh "X" in pose (X := E).

Application of Tautologies

logic E, where E is a fact, is equivalent to assert H:E; [tauto | eapply H; clear H]. It is useful for instance to prove a conjunction [A ∧ B] by showing first [A] and then [A → B], through the command [logic (foral A B, A → (A → B) → A ∧ B)]

Ltac logic_base E cont :=
assert (H:E); [ cont tt | eapply H; clear H ].

Tactic Notation "logic" constr(E) :=
logic_base E ltac:(fun _ ⇒ tauto).

Application Modulo Equalities

The tactic equates replaces a goal of the form P x y z with a goal of the form P x ?a z and a subgoal ?a = y. The introduction of the evar ?a makes it possible to apply lemmas that would not apply to the original goal, for example a lemma of the form ∀ n m, P n n m, because x and y might be equal but not convertible.

Usage is equates i₁ ... ik, where the indices are the positions of the arguments to be replaced by evars, counting from the right-hand side. If 0 is given as argument, then the entire goal is replaced by an evar.

Section equatesLemma.
Variables (A₀ A₁ : Type).
Variables (A₂ : ∀ (x₁ : A₁), Type).
Variables (A₃ : ∀ (x₁ : A₁) (x₂ : A₂ x₁), Type).
Variables (A₄ : ∀ (x₁ : A₁) (x₂ : A₂ x₁) (x₃ : A₃ x₂), Type).
Variables (A₅ : ∀ (x₁ : A₁) (x₂ : A₂ x₁) (x₃ : A₃ x₂) (x₄ : A₄ x₃), Type).
Variables (A₆ : ∀ (x₁ : A₁) (x₂ : A₂ x₁) (x₃ : A₃ x₂) (x₄ : A₄ x₃) (x₅ : A₅ x₄), Type).

Lemma equates_0 : ∀ (P Q:Prop),
P → P = Q → Q.
Proof. intros. subst. auto. Qed.

Lemma equates_1 :
∀ (P:A₀ →Prop) x₁ y₁,
P y₁ → x₁ = y₁ → P x₁.
Proof. intros. subst. auto. Qed.

Lemma equates_2 :
∀ y₁ (P:A₀ →∀(x₁:A₁),Prop) x₁ x₂,
P y₁ x₂ → x₁ = y₁ → P x₁ x₂.
Proof. intros. subst. auto. Qed.

Lemma equates_3 :
∀ y₁ (P:A₀ →∀(x₁:A₁)(x₂:A₂ x₁),Prop) x₁ x₂ x₃,
P y₁ x₂ x₃ → x₁ = y₁ → P x₁ x₂ x₃.
Proof. intros. subst. auto. Qed.

Lemma equates_4 :
∀ y₁ (P:A₀ →∀(x₁:A₁)(x₂:A₂ x₁)(x₃:A₃ x₂),Prop) x₁ x₂ x₃ x₄,
P y₁ x₂ x₃ x₄ → x₁ = y₁ → P x₁ x₂ x₃ x₄.
Proof. intros. subst. auto. Qed.

Lemma equates_5 :
∀ y₁ (P:A₀ →∀(x₁:A₁)(x₂:A₂ x₁)(x₃:A₃ x₂)(x₄:A₄ x₃),Prop) x₁ x₂ x₃ x₄ x₅,
P y₁ x₂ x₃ x₄ x₅ → x₁ = y₁ → P x₁ x₂ x₃ x₄ x₅.
Proof. intros. subst. auto. Qed.

Lemma equates_6 :
  ∀ y₁ (P:A₀ →∀(x₁:A₁)(x₂:A₂ x₁)(x₃:A₃ x₂)(x₄:A₄ x₃)(x₅:A₅ x₄),Prop)
  x₁ x₂ x₃ x₄ x₅ x₆,
  P y₁ x₂ x₃ x₄ x₅ x₆ → x₁ = y₁ → P x₁ x₂ x₃ x₄ x₅ x₆.
Proof. intros. subst. auto. Qed.

End equatesLemma.

Ltac equates_one n :=
let L := equates_lemma n in
eapply L.

Ltac equates_several E cont :=
  let all_pos := match type of E with
    | List.list Boxer ⇒ constr:(E)
    | _ ⇒ constr:((boxer E)::nil)
    end in
  let rec go pos :=
     match pos with
     | nil ⇒ cont tt
     | (boxer ?n)::?pos' ⇒ equates_one n; [ instantiate; go pos' | ]
     end in
  go all_pos.

Tactic Notation "equates" constr(E) :=
  equates_several E ltac:(fun _ ⇒ idtac).
Tactic Notation "equates" constr(n₁) constr(n₂) :=
  equates (>> n₁ n₂).
Tactic Notation "equates" constr(n₁) constr(n₂) constr(n₃) :=
  equates (>> n₁ n₂ n₃).
Tactic Notation "equates" constr(n₁) constr(n₂) constr(n₃) constr(n₄) :=
  equates (>> n₁ n₂ n₃ n₄).

applys_eq H i₁ .. iK is the same as equates i₁ .. iK followed by apply H on the first subgoal.

Tactic Notation "applys_eq" constr(H) constr(E) :=
  equates_several E ltac:(fun _ ⇒ sapply H).
Tactic Notation "applys_eq" constr(H) constr(n₁) constr(n₂) :=
  applys_eq H (>> n₁ n₂).
Tactic Notation "applys_eq" constr(H) constr(n₁) constr(n₂) constr(n₃) :=
  applys_eq H (>> n₁ n₂ n₃).
Tactic Notation "applys_eq" constr(H) constr(n₁) constr(n₂) constr(n₃) constr(n₄) :=
  applys_eq H (>> n₁ n₂ n₃ n₄).

Absurd Goals

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 ].

false E tries to exploit lemma E to prove the goal false. false E₁ .. EN is equivalent to false (>> E₁ .. EN), which tries to apply applys (>> E₁ .. EN) and if it does not work then tries forwards H: (>> E₁ .. EN) followed with false

Ltac false_then E cont :=
  false_goal; first
  [ applys E; instantiate
  | forwards_then E ltac:(fun M ⇒
      pose M; jauto_set_hyps; intros; false) ];
  cont tt.
  (* --TODO: is cont needed? *)

Tactic Notation "false" constr(E) :=
  false_then E ltac:(fun _ ⇒ idtac).
Tactic Notation "false" constr(E) constr(E₁) :=
  false (>> E E₁).
Tactic Notation "false" constr(E) constr(E₁) constr(E₂) :=
  false (>> E E₁ E₂).
Tactic Notation "false" constr(E) constr(E₁) constr(E₂) constr(E₃) :=
  false (>> E E₁ E₂ E₃).
Tactic Notation "false" constr(E) constr(E₁) constr(E₂) constr(E₃) constr(E₄) :=
  false (>> E E₁ E₂ E₃ E₄).

false_invert H proves a goal if it absurd after calling inversion H and false

Ltac false_invert_for H :=
let M := fresh "TEMP" in pose (M := H); inversion H; false.

Tactic Notation "false_invert" constr(H) :=
try solve [ false_invert_for H | false ].

false_invert proves any goal provided there is at least one hypothesis H in the context (or as a universally quantified hypothesis visible at the head of the goal) that can be proved absurd by calling inversion H.

Ltac false_invert_iter :=
  match goal with H:_ ⊢ _ ⇒
    solve [ inversion H; false
          | clear H; false_invert_iter
          | fail 2 ] end.

Tactic Notation "false_invert" :=
intros; solve [ false_invert_iter | false ].

tryfalse_invert H and tryfalse_invert are like the above but leave the goal unchanged if they don't solve it.

Tactic Notation "tryfalse_invert" constr(H) :=
try (false_invert H).

Tactic Notation "tryfalse_invert" :=
try false_invert.

false_neq_self_hyp proves any goal if the context contains an hypothesis of the form E ≠ E. It is a restricted and optimized version of false. It is intended to be used by other tactics only.

Ltac false_neq_self_hyp :=
match goal with H: ?x ≠ ?x ⊢ _ ⇒
false_goal; apply H; reflexivity end.

Introduction and Generalization

Introduction

introv is used to name only non-dependent hypothesis.

If introv is called on a goal of the form ∀ x, H, it should introduce all the variables quantified with a ∀ at the head of the goal, but it does not introduce hypotheses that preceed an arrow constructor, like in P → Q.
If introv is called on a goal that is not of the form ∀ x, H nor P → Q, the tactic unfolds definitions until the goal takes the form ∀ x, H or P → Q. If unfolding definitions does not produces a goal of this form, then the tactic introv does nothing at all.

(* introv_rec introduces all visible variables.
It does not try to unfold any definition. *)

Ltac introv_rec :=
  match goal with
  | ⊢ ?P → ?Q ⇒ idtac
  | ⊢ ∀ _, _ ⇒ intro; introv_rec
  | ⊢ _ ⇒ idtac
  end.

(* introv_noarg forces the goal to be a ∀ or an →,
   and then calls introv_rec to introduces variables
   (possibly none, in which case introv is the same as hnf).
   If the goal is not a product, then it does not do anything. *)

  (* simpler yet perhaps less efficient imlementation *)
  Ltac introv_noarg_not_optimized :=
    intro; match goal with H:_⊢_ ⇒ revert H end; introv_rec.

(* introv_arg H introduces one non-dependent hypothesis
   under the name H, after introducing the variables
   quantified with a ∀ that preceeds this hypothesis.
   This tactic fails if there does not exist a hypothesis
   to be introduced. *)
  (* todo: __ in introv means "intros" *)

Ltac introv_arg H :=
  hnf; match goal with
  | ⊢ ?P → ?Q ⇒ intros H
  | ⊢ ∀ _, _ ⇒ intro; introv_arg H
  end.

(* introv I₁ .. IN iterates introv Ik *)

Tactic Notation "introv" :=
  introv_noarg.
Tactic Notation "introv" simple_intropattern(I₁) :=
  introv_arg I₁.
Tactic Notation "introv" simple_intropattern(I₁) simple_intropattern(I₂) :=
  introv I₁; introv I₂.
Tactic Notation "introv" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) :=
  introv I₁; introv I₂ I₃.
Tactic Notation "introv" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) :=
  introv I₁; introv I₂ I₃ I₄.
Tactic Notation "introv" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅) :=
  introv I₁; introv I₂ I₃ I₄ I₅.
Tactic Notation "introv" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
simple_intropattern(I₆) :=
  introv I₁; introv I₂ I₃ I₄ I₅ I₆.
Tactic Notation "introv" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
simple_intropattern(I₆) simple_intropattern(I₇) :=
  introv I₁; introv I₂ I₃ I₄ I₅ I₆ I₇.
Tactic Notation "introv" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
simple_intropattern(I₆) simple_intropattern(I₇) simple_intropattern(I₈) :=
  introv I₁; introv I₂ I₃ I₄ I₅ I₆ I₇ I₈.
Tactic Notation "introv" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
simple_intropattern(I₆) simple_intropattern(I₇) simple_intropattern(I₈)
simple_intropattern(I₉) :=
  introv I₁; introv I₂ I₃ I₄ I₅ I₆ I₇ I₈ I₉.
Tactic Notation "introv" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
simple_intropattern(I₆) simple_intropattern(I₇) simple_intropattern(I₈)
simple_intropattern(I₉) simple_intropattern(I₁₀) :=
  introv I₁; introv I₂ I₃ I₄ I₅ I₆ I₇ I₈ I₉ I₁₀.

intros_all repeats intro as long as possible. Contrary to intros, it unfolds any definition on the way. Remark that it also unfolds the definition of negation, so applying intros_all to a goal of the form ∀ x, P x → ¬Q will introduce x and P x and Q, and will leave False in the goal.

Tactic Notation "intros_all" :=
repeat intro.

intros_hnf introduces an hypothesis and sets in head normal form

Tactic Notation "intro_hnf" :=
intro; match goal with H: _ ⊢ _ ⇒ hnf in H end.

Introduction using ⇒ and =>>

(* ⇒ I₁ .. IN is the same as intros I₁ .. IN *)

Ltac ltac_intros_post := idtac.

Tactic Notation "=>" :=
  intros.
Tactic Notation "=>" simple_intropattern(I₁) :=
  intros I₁.
Tactic Notation "=>" simple_intropattern(I₁) simple_intropattern(I₂) :=
  intros I₁ I₂.
Tactic Notation "=>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) :=
  intros I₁ I₂ I₃.
Tactic Notation "=>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) :=
  intros I₁ I₂ I₃ I₄.
Tactic Notation "=>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅) :=
  intros I₁ I₂ I₃ I₄ I₅.
Tactic Notation "=>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
simple_intropattern(I₆) :=
  intros I₁ I₂ I₃ I₄ I₅ I₆.
Tactic Notation "=>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
simple_intropattern(I₆) simple_intropattern(I₇) :=
  intros I₁ I₂ I₃ I₄ I₅ I₆ I₇.
Tactic Notation "=>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
simple_intropattern(I₆) simple_intropattern(I₇) simple_intropattern(I₈) :=
  intros I₁ I₂ I₃ I₄ I₅ I₆ I₇ I₈.
Tactic Notation "=>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
simple_intropattern(I₆) simple_intropattern(I₇) simple_intropattern(I₈)
simple_intropattern(I₉) :=
  intros I₁ I₂ I₃ I₄ I₅ I₆ I₇ I₈ I₉.
Tactic Notation "=>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
simple_intropattern(I₆) simple_intropattern(I₇) simple_intropattern(I₈)
simple_intropattern(I₉) simple_intropattern(I₁₀) :=
  intros I₁ I₂ I₃ I₄ I₅ I₆ I₇ I₈ I₉ I₁₀.

(* =>> first introduces all non-dependent variables,
   then behaves as intros. It unfolds the head of the goal using hnf
   if there are not head visible quantifiers.

   Remark: instances of Inhab are treated as non-dependent and
   are introduced automatically. *)

(* NOTE: this tactic is later redefined for supporting Inhab *)
Ltac intro_nondeps_aux_special_intro G :=
fail.

Ltac intro_nondeps_aux is_already_hnf :=
  match goal with
  | ⊢ (?P → ?Q) ⇒ idtac
  | ⊢ ?G → _ ⇒ intro_nondeps_aux_special_intro G;
                  intro; intro_nondeps_aux true
  | ⊢ (∀ _,_) ⇒ intros ?; intro_nondeps_aux true
  | ⊢ _ ⇒
     match is_already_hnf with
     | true ⇒ idtac
     | false ⇒ hnf; intro_nondeps_aux true
     end
  end.

Ltac intro_nondeps tt := intro_nondeps_aux false.

Tactic Notation "=>>" :=
  intro_nondeps tt.
Tactic Notation "=>>" simple_intropattern(I₁) :=
  =>>; intros I₁.
Tactic Notation "=>>" simple_intropattern(I₁) simple_intropattern(I₂) :=
  =>>; intros I₁ I₂.
Tactic Notation "=>>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) :=
  =>>; intros I₁ I₂ I₃.
Tactic Notation "=>>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) :=
  =>>; intros I₁ I₂ I₃ I₄.
Tactic Notation "=>>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅) :=
  =>>; intros I₁ I₂ I₃ I₄ I₅.
Tactic Notation "=>>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
simple_intropattern(I₆) :=
  =>>; intros I₁ I₂ I₃ I₄ I₅ I₆.
Tactic Notation "=>>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
simple_intropattern(I₆) simple_intropattern(I₇) :=
  =>>; intros I₁ I₂ I₃ I₄ I₅ I₆ I₇.
Tactic Notation "=>>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
simple_intropattern(I₆) simple_intropattern(I₇) simple_intropattern(I₈) :=
  =>>; intros I₁ I₂ I₃ I₄ I₅ I₆ I₇ I₈.
Tactic Notation "=>>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
simple_intropattern(I₆) simple_intropattern(I₇) simple_intropattern(I₈)
simple_intropattern(I₉) :=
  =>>; intros I₁ I₂ I₃ I₄ I₅ I₆ I₇ I₈ I₉.
Tactic Notation "=>>" simple_intropattern(I₁) simple_intropattern(I₂)
simple_intropattern(I₃) simple_intropattern(I₄) simple_intropattern(I₅)
simple_intropattern(I₆) simple_intropattern(I₇) simple_intropattern(I₈)
simple_intropattern(I₉) simple_intropattern(I₁₀) :=
  =>>; intros I₁ I₂ I₃ I₄ I₅ I₆ I₇ I₈ I₉ I₁₀.

(* ---------------------------------------------------------------------- *)

Generalization

gen X₁ .. XN is a shorthand for calling generalize dependent successively on variables XN...X₁. Note that the variables are generalized in reverse order, following the convention of the generalize tactic: it means that X₁ will be the first quantified variable in the resulting goal.

Tactic Notation "gen" ident(X₁) :=
  generalize dependent X₁.
Tactic Notation "gen" ident(X₁) ident(X₂) :=
  gen X₂; gen X₁.
Tactic Notation "gen" ident(X₁) ident(X₂) ident(X₃) :=
  gen X₃; gen X₂; gen X₁.
Tactic Notation "gen" ident(X₁) ident(X₂) ident(X₃) ident(X₄) :=
  gen X₄; gen X₃; gen X₂; gen X₁.
Tactic Notation "gen" ident(X₁) ident(X₂) ident(X₃) ident(X₄) ident(X₅) :=
  gen X₅; gen X₄; gen X₃; gen X₂; gen X₁.
Tactic Notation "gen" ident(X₁) ident(X₂) ident(X₃) ident(X₄) ident(X₅)
ident(X₆) :=
  gen X₆; gen X₅; gen X₄; gen X₃; gen X₂; gen X₁.
Tactic Notation "gen" ident(X₁) ident(X₂) ident(X₃) ident(X₄) ident(X₅)
ident(X₆) ident(X₇) :=
  gen X₇; gen X₆; gen X₅; gen X₄; gen X₃; gen X₂; gen X₁.
Tactic Notation "gen" ident(X₁) ident(X₂) ident(X₃) ident(X₄) ident(X₅)
ident(X₆) ident(X₇) ident(X₈) :=
  gen X₈; gen X₇; gen X₆; gen X₅; gen X₄; gen X₃; gen X₂; gen X₁.
Tactic Notation "gen" ident(X₁) ident(X₂) ident(X₃) ident(X₄) ident(X₅)
ident(X₆) ident(X₇) ident(X₈) ident(X₉) :=
  gen X₉; gen X₈; gen X₇; gen X₆; gen X₅; gen X₄; gen X₃; gen X₂; gen X₁.
Tactic Notation "gen" ident(X₁) ident(X₂) ident(X₃) ident(X₄) ident(X₅)
ident(X₆) ident(X₇) ident(X₈) ident(X₉) ident(X₁₀) :=
  gen X₁₀; gen X₉; gen X₈; gen X₇; gen X₆; gen X₅; gen X₄; gen X₃; gen X₂; gen X₁.

generalizes X is a shorthand for calling generalize X; clear X. It is weaker than tactic gen X since it does not support dependencies. It is mainly intended for writing tactics.

Tactic Notation "generalizes" hyp(X) :=
  generalize X; clear X.
Tactic Notation "generalizes" hyp(X₁) hyp(X₂) :=
  generalizes X₁; generalizes X₂.
Tactic Notation "generalizes" hyp(X₁) hyp(X₂) hyp(X₃) :=
  generalizes X₁ X₂; generalizes X₃.
Tactic Notation "generalizes" hyp(X₁) hyp(X₂) hyp(X₃) hyp(X₄) :=
  generalizes X₁ X₂ X₃; generalizes X₄.

Naming

sets X: E is the same as set (X := E) in ×, that is, it replaces all occurences of E by a fresh meta-variable X whose definition is E.

Tactic Notation "sets" ident(X) ":" constr(E) :=
set (X := E) in ×.

def_to_eq E X H applies when X := E is a local definition. It adds an assumption H: X = E and then clears the definition of X. def_to_eq_sym is similar except that it generates the equality H: E = X.

Ltac def_to_eq X HX E :=
assert (HX : X = E) by reflexivity; clearbody X.
Ltac def_to_eq_sym X HX E :=
assert (HX : E = X) by reflexivity; clearbody X.

set_eq X H: E generates the equality H: X = E, for a fresh name X, and replaces E by X in the current goal. Syntaxes set_eq X: E and set_eq: E are also available. Similarly, set_eq <- X H: E generates the equality H: E = X.

sets_eq X HX: E does the same but replaces E by X everywhere in the goal. sets_eq X HX: E in H replaces in H. set_eq X HX: E in ⊢ performs no substitution at all.

Tactic Notation "set_eq" ident(X) ident(HX) ":" constr(E) :=
  set (X := E); def_to_eq X HX E.
Tactic Notation "set_eq" ident(X) ":" constr(E) :=
  let HX := fresh "EQ" X in set_eq X HX: E.
Tactic Notation "set_eq" ":" constr(E) :=
  let X := fresh "X" in set_eq X: E.

Tactic Notation "set_eq" "<-" ident(X) ident(HX) ":" constr(E) :=
  set (X := E); def_to_eq_sym X HX E.
Tactic Notation "set_eq" "<-" ident(X) ":" constr(E) :=
  let HX := fresh "EQ" X in set_eq <- X HX: E.
Tactic Notation "set_eq" "<-" ":" constr(E) :=
  let X := fresh "X" in set_eq <- X: E.

Tactic Notation "sets_eq" ident(X) ident(HX) ":" constr(E) :=
  set (X := E) in *; def_to_eq X HX E.
Tactic Notation "sets_eq" ident(X) ":" constr(E) :=
  let HX := fresh "EQ" X in sets_eq X HX: E.
Tactic Notation "sets_eq" ":" constr(E) :=
  let X := fresh "X" in sets_eq X: E.

Tactic Notation "sets_eq" "<-" ident(X) ident(HX) ":" constr(E) :=
  set (X := E) in *; def_to_eq_sym X HX E.
Tactic Notation "sets_eq" "<-" ident(X) ":" constr(E) :=
  let HX := fresh "EQ" X in sets_eq <- X HX: E.
Tactic Notation "sets_eq" "<-" ":" constr(E) :=
  let X := fresh "X" in sets_eq <- X: E.

Tactic Notation "set_eq" ident(X) ident(HX) ":" constr(E) "in" hyp(H) :=
  set (X := E) in H; def_to_eq X HX E.
Tactic Notation "set_eq" ident(X) ":" constr(E) "in" hyp(H) :=
  let HX := fresh "EQ" X in set_eq X HX: E in H.
Tactic Notation "set_eq" ":" constr(E) "in" hyp(H) :=
  let X := fresh "X" in set_eq X: E in H.

Tactic Notation "set_eq" "<-" ident(X) ident(HX) ":" constr(E) "in" hyp(H) :=
  set (X := E) in H; def_to_eq_sym X HX E.
Tactic Notation "set_eq" "<-" ident(X) ":" constr(E) "in" hyp(H) :=
  let HX := fresh "EQ" X in set_eq <- X HX: E in H.
Tactic Notation "set_eq" "<-" ":" constr(E) "in" hyp(H) :=
  let X := fresh "X" in set_eq <- X: E in H.

Tactic Notation "set_eq" ident(X) ident(HX) ":" constr(E) "in" "⊢" :=
  set (X := E) in ⊢; def_to_eq X HX E.
Tactic Notation "set_eq" ident(X) ":" constr(E) "in" "⊢" :=
  let HX := fresh "EQ" X in set_eq X HX: E in ⊢.
Tactic Notation "set_eq" ":" constr(E) "in" "⊢" :=
  let X := fresh "X" in set_eq X: E in ⊢.

Tactic Notation "set_eq" "<-" ident(X) ident(HX) ":" constr(E) "in" "⊢" :=
  set (X := E) in ⊢; def_to_eq_sym X HX E.
Tactic Notation "set_eq" "<-" ident(X) ":" constr(E) "in" "⊢" :=
  let HX := fresh "EQ" X in set_eq <- X HX: E in ⊢.
Tactic Notation "set_eq" "<-" ":" constr(E) "in" "⊢" :=
  let X := fresh "X" in set_eq <- X: E in ⊢.

gen_eq X: E is a tactic whose purpose is to introduce equalities so as to work around the limitation of the induction tactic which typically loses information. gen_eq E as X replaces all occurences of term E with a fresh variable X and the equality X = E as extra hypothesis to the current conclusion. In other words a conclusion C will be turned into (X = E) → C. gen_eq: E and gen_eq: E as X are also accepted.

Tactic Notation "gen_eq" ident(X) ":" constr(E) :=
  let EQ := fresh "EQ" X in sets_eq X EQ: E; revert EQ.
Tactic Notation "gen_eq" ":" constr(E) :=
  let X := fresh "X" in gen_eq X: E.
Tactic Notation "gen_eq" ":" constr(E) "as" ident(X) :=
  gen_eq X: E.
Tactic Notation "gen_eq" ident(X₁) ":" constr(E₁) ","
  ident(X₂) ":" constr(E₂) :=
  gen_eq X₂: E₂; gen_eq X₁: E₁.
Tactic Notation "gen_eq" ident(X₁) ":" constr(E₁) ","
  ident(X₂) ":" constr(E₂) "," ident(X₃) ":" constr(E₃) :=
  gen_eq X₃: E₃; gen_eq X₂: E₂; gen_eq X₁: E₁.

sets_let X finds the first let-expression in the goal and names its body X. sets_eq_let X is similar, except that it generates an explicit equality. Tactics sets_let X in H and sets_eq_let X in H allow specifying a particular hypothesis (by default, the first one that contains a let is considered).

Known limitation: it does not seem possible to support naming of multiple let-in constructs inside a term, from ltac.

Ltac sets_let_base tac :=
  match goal with
  | ⊢ context[let _ := ?E in _] ⇒ tac E; cbv zeta
  | H: context[let _ := ?E in _] ⊢ _ ⇒ tac E; cbv zeta in H
  end.

Ltac sets_let_in_base H tac :=
match type of H with context[let _ := ?E in _] ⇒
tac E; cbv zeta in H end.

Tactic Notation "sets_let" ident(X) :=
  sets_let_base ltac:(fun E ⇒ sets X: E).
Tactic Notation "sets_let" ident(X) "in" hyp(H) :=
  sets_let_in_base H ltac:(fun E ⇒ sets X: E).
Tactic Notation "sets_eq_let" ident(X) :=
  sets_let_base ltac:(fun E ⇒ sets_eq X: E).
Tactic Notation "sets_eq_let" ident(X) "in" hyp(H) :=
  sets_let_in_base H ltac:(fun E ⇒ sets_eq X: E).

Rewriting

rewrites E is similar to rewrite except that it supports the rm directives to clear hypotheses on the fly, and that it supports a list of arguments in the form rewrites (>> E₁ E₂ E₃) to indicate that forwards should be invoked first before rewrites is called.

Ltac rewrites_base E cont :=
  match type of E with
  | List.list Boxer ⇒ forwards_then E cont
  | _ ⇒ cont E; fast_rm_inside E
  end.

Tactic Notation "rewrites" constr(E) :=
  rewrites_base E ltac:(fun M ⇒ rewrite M ).
Tactic Notation "rewrites" constr(E) "in" hyp(H) :=
  rewrites_base E ltac:(fun M ⇒ rewrite M in H).
Tactic Notation "rewrites" constr(E) "in" "*" :=
  rewrites_base E ltac:(fun M ⇒ rewrite M in *).
Tactic Notation "rewrites" "<-" constr(E) :=
  rewrites_base E ltac:(fun M ⇒ rewrite <- M ).
Tactic Notation "rewrites" "<-" constr(E) "in" hyp(H) :=
  rewrites_base E ltac:(fun M ⇒ rewrite <- M in H).
Tactic Notation "rewrites" "<-" constr(E) "in" "*" :=
  rewrites_base E ltac:(fun M ⇒ rewrite <- M in *).

(* --TODO: extend tactics below to use rewrites *)

rewrite_all E iterates version of rewrite E as long as possible. Warning: this tactic can easily get into an infinite loop. Syntax for rewriting from right to left and/or into an hypothese is similar to the one of rewrite.

Tactic Notation "rewrite_all" constr(E) :=
  repeat rewrite E.
Tactic Notation "rewrite_all" "<-" constr(E) :=
  repeat rewrite <- E.
Tactic Notation "rewrite_all" constr(E) "in" ident(H) :=
  repeat rewrite E in H.
Tactic Notation "rewrite_all" "<-" constr(E) "in" ident(H) :=
  repeat rewrite <- E in H.
Tactic Notation "rewrite_all" constr(E) "in" "*" :=
  repeat rewrite E in ×.
Tactic Notation "rewrite_all" "<-" constr(E) "in" "*" :=
  repeat rewrite <- E in ×.

asserts_rewrite E asserts that an equality E holds (generating a corresponding subgoal) and rewrite it straight away in the current goal. It avoids giving a name to the equality and later clearing it. Syntax for rewriting from right to left and/or into an hypothese is similar to the one of rewrite. Note: the tactic replaces plays a similar role.

Ltac asserts_rewrite_tactic E action :=
let EQ := fresh "TEMP" in (assert (EQ : E);
[ idtac | action EQ; clear EQ ]).

Tactic Notation "asserts_rewrite" constr(E) :=
  asserts_rewrite_tactic E ltac:(fun EQ ⇒ rewrite EQ).
Tactic Notation "asserts_rewrite" "<-" constr(E) :=
  asserts_rewrite_tactic E ltac:(fun EQ ⇒ rewrite <- EQ).
Tactic Notation "asserts_rewrite" constr(E) "in" hyp(H) :=
  asserts_rewrite_tactic E ltac:(fun EQ ⇒ rewrite EQ in H).
Tactic Notation "asserts_rewrite" "<-" constr(E) "in" hyp(H) :=
  asserts_rewrite_tactic E ltac:(fun EQ ⇒ rewrite <- EQ in H).
Tactic Notation "asserts_rewrite" constr(E) "in" "*" :=
  asserts_rewrite_tactic E ltac:(fun EQ ⇒ rewrite EQ in *).
Tactic Notation "asserts_rewrite" "<-" constr(E) "in" "*" :=
  asserts_rewrite_tactic E ltac:(fun EQ ⇒ rewrite <- EQ in *).

cuts_rewrite E is the same as asserts_rewrite E except that subgoals are permuted.

Ltac cuts_rewrite_tactic E action :=
let EQ := fresh "TEMP" in (cuts EQ: E;
[ action EQ; clear EQ | idtac ]).

Tactic Notation "cuts_rewrite" constr(E) :=
  cuts_rewrite_tactic E ltac:(fun EQ ⇒ rewrite EQ).
Tactic Notation "cuts_rewrite" "<-" constr(E) :=
  cuts_rewrite_tactic E ltac:(fun EQ ⇒ rewrite <- EQ).
Tactic Notation "cuts_rewrite" constr(E) "in" hyp(H) :=
  cuts_rewrite_tactic E ltac:(fun EQ ⇒ rewrite EQ in H).
Tactic Notation "cuts_rewrite" "<-" constr(E) "in" hyp(H) :=
  cuts_rewrite_tactic E ltac:(fun EQ ⇒ rewrite <- EQ in H).

rewrite_except H EQ rewrites equality EQ everywhere but in hypothesis H. Mainly useful for other tactics.

Ltac rewrite_except H EQ :=
  let K := fresh "TEMP" in let T := type of H in
  set (K := T) in H;
  rewrite EQ in *; unfold K in H; clear K.

rewrites E at K applies when E is of the form T₁ = T₂ rewrites the equality E at the K-th occurence of T₁ in the current goal. Syntaxes rewrites <- E at K and rewrites E at K in H are also available.

Tactic Notation "rewrites" constr(E) "at" constr(K) :=
  match type of E with ?T₁ = ?T₂ ⇒
    ltac_action_at K of T₁ do (rewrites E) end.
Tactic Notation "rewrites" "<-" constr(E) "at" constr(K) :=
  match type of E with ?T₁ = ?T₂ ⇒
    ltac_action_at K of T₂ do (rewrites <- E) end.
Tactic Notation "rewrites" constr(E) "at" constr(K) "in" hyp(H) :=
  match type of E with ?T₁ = ?T₂ ⇒
    ltac_action_at K of T₁ in H do (rewrites E in H) end.
Tactic Notation "rewrites" "<-" constr(E) "at" constr(K) "in" hyp(H) :=
  match type of E with ?T₁ = ?T₂ ⇒
    ltac_action_at K of T₂ in H do (rewrites <- E in H) end.

Replace

replaces E with F is the same as replace E with F except that the equality E = F is generated as first subgoal. Syntax replaces E with F in H is also available. Note that contrary to replace, replaces does not try to solve the equality by assumption. Note: replaces E with F is similar to asserts_rewrite (E = F).

Tactic Notation "replaces" constr(E) "with" constr(F) :=
let T := fresh "TEMP" in assert (T: E = F); [ | replace E with F; clear T ].

Tactic Notation "replaces" constr(E) "with" constr(F) "in" hyp(H) :=
let T := fresh "TEMP" in assert (T: E = F); [ | replace E with F in H; clear T ].

replaces E at K with F replaces the K-th occurence of E with F in the current goal. Syntax replaces E at K with F in H is also available.

Tactic Notation "replaces" constr(E) "at" constr(K) "with" constr(F) :=
let T := fresh "TEMP" in assert (T: E = F); [ | rewrites T at K; clear T ].

Tactic Notation "replaces" constr(E) "at" constr(K) "with" constr(F) "in" hyp(H) :=
let T := fresh "TEMP" in assert (T: E = F); [ | rewrites T at K in H; clear T ].

Change

changes is like change except that it does not silently fail to perform its task. (Note that, changes is implemented using rewrite, meaning that it might perform additional beta-reductions compared with the original change tactic.

(* --TODO: support "changes (E₁ = E₂)" *)

Tactic Notation "changes" constr(E₁) "with" constr(E₂) "in" hyp(H) :=
asserts_rewrite (E₁ = E₂) in H; [ reflexivity | ].

Tactic Notation "changes" constr(E₁) "with" constr(E₂) :=
asserts_rewrite (E₁ = E₂); [ reflexivity | ].

Tactic Notation "changes" constr(E₁) "with" constr(E₂) "in" "*" :=
asserts_rewrite (E₁ = E₂) in *; [ reflexivity | ].

Renaming

renames X₁ to Y₁, ..., XN to YN is a shorthand for a sequence of renaming operations rename Xi into Yi.

Tactic Notation "renames" ident(X₁) "to" ident(Y₁) :=
  rename X₁ into Y₁.
Tactic Notation "renames" ident(X₁) "to" ident(Y₁) ","
ident(X₂) "to" ident(Y₂) :=
  renames X₁ to Y₁; renames X₂ to Y₂.
Tactic Notation "renames" ident(X₁) "to" ident(Y₁) ","
ident(X₂) "to" ident(Y₂) "," ident(X₃) "to" ident(Y₃) :=
  renames X₁ to Y₁; renames X₂ to Y₂, X₃ to Y₃.
Tactic Notation "renames" ident(X₁) "to" ident(Y₁) ","
ident(X₂) "to" ident(Y₂) "," ident(X₃) "to" ident(Y₃) ","
ident(X₄) "to" ident(Y₄) :=
  renames X₁ to Y₁; renames X₂ to Y₂, X₃ to Y₃, X₄ to Y₄.
Tactic Notation "renames" ident(X₁) "to" ident(Y₁) ","
ident(X₂) "to" ident(Y₂) "," ident(X₃) "to" ident(Y₃) ","
ident(X₄) "to" ident(Y₄) "," ident(X₅) "to" ident(Y₅) :=
  renames X₁ to Y₁; renames X₂ to Y₂, X₃ to Y₃, X₄ to Y₄, X₅ to Y₅.
Tactic Notation "renames" ident(X₁) "to" ident(Y₁) ","
ident(X₂) "to" ident(Y₂) "," ident(X₃) "to" ident(Y₃) ","
ident(X₄) "to" ident(Y₄) "," ident(X₅) "to" ident(Y₅) ","
ident(X₆) "to" ident(Y₆) :=
  renames X₁ to Y₁; renames X₂ to Y₂, X₃ to Y₃, X₄ to Y₄, X₅ to Y₅, X₆ to Y₆.

Unfolding

unfolds unfolds the head definition in the goal, i.e. if the goal has form P x₁ ... xN then it calls unfold P. If the goal is an equality, it tries to unfold the head constant on the left-hand side, and otherwise tries on the right-hand side. If the goal is a product, it calls intros first. warning: this tactic is overriden in LibReflect.

Ltac apply_to_head_of E cont :=
  let go E :=
    let P := get_head E in cont P in
  match E with
  | ∀ _,_ ⇒ intros; apply_to_head_of E cont
  | ?A = ?B ⇒ first [ go A | go B ]
  | ?A ⇒ go A
  end.

Ltac unfolds_base :=
match goal with ⊢ ?G ⇒
apply_to_head_of G ltac:(fun P ⇒ unfold P) end.

Tactic Notation "unfolds" :=
unfolds_base.

unfolds in H unfolds the head definition of hypothesis H, i.e. if H has type P x₁ ... xN then it calls unfold P in H.

Ltac unfolds_in_base H :=
match type of H with ?G ⇒
apply_to_head_of G ltac:(fun P ⇒ unfold P in H) end.

Tactic Notation "unfolds" "in" hyp(H) :=
unfolds_in_base H.

unfolds in H₁,H₂,..,HN allows unfolding the head constant in several hypotheses at once.

Tactic Notation "unfolds" "in" hyp(H₁) hyp(H₂) :=
  unfolds in H₁; unfolds in H₂.
Tactic Notation "unfolds" "in" hyp(H₁) hyp(H₂) hyp(H₃) :=
  unfolds in H₁; unfolds in H₂ H₃.
Tactic Notation "unfolds" "in" hyp(H₁) hyp(H₂) hyp(H₃) hyp(H₄) :=
  unfolds in H₁; unfolds in H₂ H₃ H₄.
Tactic Notation "unfolds" "in" hyp(H₁) hyp(H₂) hyp(H₃) hyp(H₄) hyp(H₅) :=
  unfolds in H₁; unfolds in H₂ H₃ H₄ H₅.

unfolds P₁,..,PN is a shortcut for unfold P₁,..,PN in ×.

Tactic Notation "unfolds" constr(F₁) :=
  unfold F₁ in ×.
Tactic Notation "unfolds" constr(F₁) "," constr(F₂) :=
  unfold F₁,F₂ in ×.
Tactic Notation "unfolds" constr(F₁) "," constr(F₂)
"," constr(F₃) :=
  unfold F₁,F₂,F₃ in ×.
Tactic Notation "unfolds" constr(F₁) "," constr(F₂)
"," constr(F₃) "," constr(F₄) :=
  unfold F₁,F₂,F₃,F₄ in ×.
Tactic Notation "unfolds" constr(F₁) "," constr(F₂)
"," constr(F₃) "," constr(F₄) "," constr(F₅) :=
  unfold F₁,F₂,F₃,F₄,F₅ in ×.
Tactic Notation "unfolds" constr(F₁) "," constr(F₂)
"," constr(F₃) "," constr(F₄) "," constr(F₅) "," constr(F₆) :=
  unfold F₁,F₂,F₃,F₄,F₅,F₆ in ×.
Tactic Notation "unfolds" constr(F₁) "," constr(F₂)
"," constr(F₃) "," constr(F₄) "," constr(F₅)
"," constr(F₆) "," constr(F₇) :=
  unfold F₁,F₂,F₃,F₄,F₅,F₆,F₇ in ×.
Tactic Notation "unfolds" constr(F₁) "," constr(F₂)
"," constr(F₃) "," constr(F₄) "," constr(F₅)
"," constr(F₆) "," constr(F₇) "," constr(F₈) :=
  unfold F₁,F₂,F₃,F₄,F₅,F₆,F₇,F₈ in ×.

folds P₁,..,PN is a shortcut for fold P₁ in *; ..; fold PN in ×.

Tactic Notation "folds" constr(H) :=
  fold H in ×.
Tactic Notation "folds" constr(H₁) "," constr(H₂) :=
  folds H₁; folds H₂.
Tactic Notation "folds" constr(H₁) "," constr(H₂) "," constr(H₃) :=
  folds H₁; folds H₂; folds H₃.
Tactic Notation "folds" constr(H₁) "," constr(H₂) "," constr(H₃)
"," constr(H₄) :=
  folds H₁; folds H₂; folds H₃; folds H₄.
Tactic Notation "folds" constr(H₁) "," constr(H₂) "," constr(H₃)
"," constr(H₄) "," constr(H₅) :=
  folds H₁; folds H₂; folds H₃; folds H₄; folds H₅.

Simplification

simpls is a shortcut for simpl in ×.

Tactic Notation "simpls" :=
simpl in ×.

simpls P₁,..,PN is a shortcut for simpl P₁ in *; ..; simpl PN in ×.

Tactic Notation "simpls" constr(F₁) :=
  simpl F₁ in ×.
Tactic Notation "simpls" constr(F₁) "," constr(F₂) :=
  simpls F₁; simpls F₂.
Tactic Notation "simpls" constr(F₁) "," constr(F₂)
"," constr(F₃) :=
  simpls F₁; simpls F₂; simpls F₃.
Tactic Notation "simpls" constr(F₁) "," constr(F₂)
"," constr(F₃) "," constr(F₄) :=
  simpls F₁; simpls F₂; simpls F₃; simpls F₄.

unsimpl E replaces all occurence of X by E, where X is the result which the tactic simpl would give when applied to E. It is useful to undo what simpl has simplified too far.

Tactic Notation "unsimpl" constr(E) :=
let F := (eval simpl in E) in change F with E.

unsimpl E in H is similar to unsimpl E but it applies inside a particular hypothesis H.

Tactic Notation "unsimpl" constr(E) "in" hyp(H) :=
let F := (eval simpl in E) in change F with E in H.

unsimpl E in × applies unsimpl E everywhere possible. unsimpls E is a synonymous.

Tactic Notation "unsimpl" constr(E) "in" "*" :=
let F := (eval simpl in E) in change F with E in ×.
Tactic Notation "unsimpls" constr(E) :=
unsimpl E in ×.

nosimpl t protects the Coq termt against some forms of simplification. See Gonthier's work for details on this trick.

Notation "'nosimpl' t" := (match tt with tt ⇒ t end)
(at level 10).

Reduction

Tactic Notation "hnfs" := hnf in ×.

Substitution

substs does the same as subst, except that it does not fail when there are circular equalities in the context.

Tactic Notation "substs" :=
repeat (match goal with H: ?x = ?y ⊢ _ ⇒
first [ subst x | subst y ] end).

Implementation of substs below, which allows to call subst on all the hypotheses that lie beyond a given position in the proof context.

Ltac substs_below limit :=
  match goal with H: ?T ⊢ _ ⇒
  match T with
  | limit ⇒ idtac
  | ?x = ?y ⇒
    first [ subst x; substs_below limit
          | subst y; substs_below limit
          | generalizes H; substs_below limit; intro ]
  end end.

substs below body E applies subst on all equalities that appear in the context below the first hypothesis whose body is E. If there is no such hypothesis in the context, it is equivalent to subst. For instance, if H is an hypothesis, then substs below H will substitute equalities below hypothesis H.

Tactic Notation "substs" "below" "body" constr(M) :=
substs_below M.

substs below H applies subst on all equalities that appear in the context below the hypothesis named H. Note that the current implementation is technically incorrect since it will confuse different hypotheses with the same body.

Tactic Notation "substs" "below" hyp(H) :=
match type of H with ?M ⇒ substs below body M end.

subst_hyp H substitutes the equality contained in the first hypothesis from the context.

Ltac intro_subst_hyp := fail. (* definition further on *)

subst_hyp H substitutes the equality contained in H.

Ltac subst_hyp_base H :=
  match type of H with
  | (_,_,_,_,_) = (_,_,_,_,_) ⇒ injection H; clear H; do 4 intro_subst_hyp
  | (_,_,_,_) = (_,_,_,_) ⇒ injection H; clear H; do 4 intro_subst_hyp
  | (_,_,_) = (_,_,_) ⇒ injection H; clear H; do 3 intro_subst_hyp
  | (_,_) = (_,_) ⇒ injection H; clear H; do 2 intro_subst_hyp
  | ?x = ?x ⇒ clear H
  | ?x = ?y ⇒ first [ subst x | subst y ]
  end.

Tactic Notation "subst_hyp" hyp(H) := subst_hyp_base H.

Ltac intro_subst_hyp ::=
let H := fresh "TEMP" in intros H; subst_hyp H.

intro_subst is a shorthand for intro H; subst_hyp H: it introduces and substitutes the equality at the head of the current goal.

Tactic Notation "intro_subst" :=
let H := fresh "TEMP" in intros H; subst_hyp H.

subst_local substitutes all local definition from the context

Ltac subst_local :=
repeat match goal with H:=_ ⊢ _ ⇒ subst H end.

subst_eq E takes an equality x = t and replace x with t everywhere in the goal

Ltac subst_eq_base E :=
let H := fresh "TEMP" in lets H: E; subst_hyp H.

Tactic Notation "subst_eq" constr(E) :=
subst_eq_base E.

Tactics to Work with Proof Irrelevance

Require Import ProofIrrelevance.

pi_rewrite E replaces E of type Prop with a fresh unification variable, and is thus a practical way to exploit proof irrelevance, without writing explicitly rewrite (proof_irrelevance E E'). Particularly useful when E' is a big expression.

Ltac pi_rewrite_base E rewrite_tac :=
let E' := fresh "TEMP" in let T := type of E in evar (E':T);
rewrite_tac (@proof_irrelevance _ E E'); subst E'.

Tactic Notation "pi_rewrite" constr(E) :=
pi_rewrite_base E ltac:(fun X ⇒ rewrite X).
Tactic Notation "pi_rewrite" constr(E) "in" hyp(H) :=
pi_rewrite_base E ltac:(fun X ⇒ rewrite X in H).

Proving Equalities

Note: current implementation only supports up to arity 5

fequal is a variation on f_equal which has a better behaviour on equalities between n-ary tuples.

Ltac fequal_base :=
  let go := f_equal; [ fequal_base | ] in
  match goal with
  | ⊢ (_,_,_) = (_,_,_) ⇒ go
  | ⊢ (_,_,_,_) = (_,_,_,_) ⇒ go
  | ⊢ (_,_,_,_,_) = (_,_,_,_,_) ⇒ go
  | ⊢ (_,_,_,_,_,_) = (_,_,_,_,_,_) ⇒ go
  | ⊢ _ ⇒ f_equal
  end.

Tactic Notation "fequal" :=
fequal_base.

fequals is the same as fequal except that it tries and solve all trivial subgoals, using reflexivity and congruence (as well as the proof-irrelevance principle). fequals applies to goals of the form f x₁ .. xN = f y₁ .. yN and produces some subgoals of the form xi = yi).

Ltac fequal_post :=
first [ reflexivity | congruence | apply proof_irrelevance | idtac ].

Tactic Notation "fequals" :=
fequal; fequal_post.

fequals_rec calls fequals recursively. It is equivalent to repeat (progress fequals).

Tactic Notation "fequals_rec" :=
repeat (progress fequals).

Inversion

Basic Inversion

invert keep H is same to inversion H except that it puts all the facts obtained in the goal. The keyword keep means that the hypothesis H should not be removed.

Tactic Notation "invert" "keep" hyp(H) :=
pose ltac_mark; inversion H; gen_until_mark.

invert keep H as X₁ .. XN is the same as inversion H as ... except that only hypotheses which are not variable need to be named explicitely, in a similar fashion as introv is used to name only hypotheses.

Tactic Notation "invert" "keep" hyp(H) "as" simple_intropattern(I₁) :=
  invert keep H; introv I₁.
Tactic Notation "invert" "keep" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) :=
  invert keep H; introv I₁ I₂.
Tactic Notation "invert" "keep" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) simple_intropattern(I₃) :=
  invert keep H; introv I₁ I₂ I₃.

invert H is same to inversion H except that it puts all the facts obtained in the goal and clears hypothesis H. In other words, it is equivalent to invert keep H; clear H.

Tactic Notation "invert" hyp(H) :=
invert keep H; clear H.

invert H as X₁ .. XN is the same as invert keep H as X₁ .. XN but it also clears hypothesis H.

Tactic Notation "invert_tactic" hyp(H) tactic(tac) :=
  let H' := fresh "TEMP" in rename H into H'; tac H'; clear H'.
Tactic Notation "invert" hyp(H) "as" simple_intropattern(I₁) :=
  invert_tactic H (fun H ⇒ invert keep H as I₁).
Tactic Notation "invert" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) :=
  invert_tactic H (fun H ⇒ invert keep H as I₁ I₂).
Tactic Notation "invert" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) simple_intropattern(I₃) :=
  invert_tactic H (fun H ⇒ invert keep H as I₁ I₂ I₃).

Inversion with Substitution

Our inversion tactics is able to get rid of dependent equalities generated by inversion, using proof irrelevance.

(* --we do not import Eqdep because it imports nasty hints automatically
From TLC Require Import Eqdep. *)

Axiom inj_pair2 : (* is in fact derivable from the axioms in LibAxiom.v *)
∀ (U : Type) (P : U → Type) (p : U) (x y : P p),
existT P p x = existT P p y → x = y.
(* Proof using. apply Eqdep.EqdepTheory.inj_pair2. Qed. *)

Ltac inverts_tactic H i₁ i₂ i₃ i₄ i₅ i₆ :=
  let rec go i₁ i₂ i₃ i₄ i₅ i₆ :=
    match goal with
    | ⊢ (ltac_Mark → _) ⇒ intros _
    | ⊢ (?x = ?y → _) ⇒ let H := fresh "TEMP" in intro H;
                           first [ subst x | subst y ];
                           go i₁ i₂ i₃ i₄ i₅ i₆
    | ⊢ (existT ?P ?p ?x = existT ?P ?p ?y → _) ⇒
         let H := fresh "TEMP" in intro H;
         generalize (@inj_pair2 _ P p x y H);
         clear H; go i₁ i₂ i₃ i₄ i₅ i₆
    | ⊢ (?P → ?Q) ⇒ i₁; go i₂ i₃ i₄ i₅ i₆ ltac:(intro)
    | ⊢ (∀ _, _) ⇒ intro; go i₁ i₂ i₃ i₄ i₅ i₆
    end in
  generalize ltac_mark; invert keep H; go i₁ i₂ i₃ i₄ i₅ i₆;
  unfold eq' in ×.

inverts keep H is same to invert keep H except that it applies subst to all the equalities generated by the inversion.

Tactic Notation "inverts" "keep" hyp(H) :=
inverts_tactic H ltac:(intro) ltac:(intro) ltac:(intro)
ltac:(intro) ltac:(intro) ltac:(intro).

inverts keep H as X₁ .. XN is the same as invert keep H as X₁ .. XN except that it applies subst to all the equalities generated by the inversion

Tactic Notation "inverts" "keep" hyp(H) "as" simple_intropattern(I₁) :=
  inverts_tactic H ltac:(intros I₁)
   ltac:(intro) ltac:(intro) ltac:(intro) ltac:(intro) ltac:(intro).
Tactic Notation "inverts" "keep" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) :=
  inverts_tactic H ltac:(intros I₁) ltac:(intros I₂)
   ltac:(intro) ltac:(intro) ltac:(intro) ltac:(intro).
Tactic Notation "inverts" "keep" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) simple_intropattern(I₃) :=
  inverts_tactic H ltac:(intros I₁) ltac:(intros I₂) ltac:(intros I₃)
   ltac:(intro) ltac:(intro) ltac:(intro).
Tactic Notation "inverts" "keep" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) simple_intropattern(I₃) simple_intropattern(I₄) :=
  inverts_tactic H ltac:(intros I₁) ltac:(intros I₂) ltac:(intros I₃)
   ltac:(intros I₄) ltac:(intro) ltac:(intro).
Tactic Notation "inverts" "keep" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) simple_intropattern(I₃) simple_intropattern(I₄)
simple_intropattern(I₅) :=
  inverts_tactic H ltac:(intros I₁) ltac:(intros I₂) ltac:(intros I₃)
   ltac:(intros I₄) ltac:(intros I₅) ltac:(intro).
Tactic Notation "inverts" "keep" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) simple_intropattern(I₃) simple_intropattern(I₄)
simple_intropattern(I₅) simple_intropattern(I₆) :=
  inverts_tactic H ltac:(intros I₁) ltac:(intros I₂) ltac:(intros I₃)
   ltac:(intros I₄) ltac:(intros I₅) ltac:(intros I₆).

inverts H is same to inverts keep H except that it clears hypothesis H.

Tactic Notation "inverts" hyp(H) :=
inverts keep H; try clear H.

inverts H as X₁ .. XN is the same as inverts keep H as X₁ .. XN but it also clears the hypothesis H.

Tactic Notation "inverts_tactic" hyp(H) tactic(tac) :=
  let H' := fresh "TEMP" in rename H into H'; tac H'; clear H'.
Tactic Notation "inverts" hyp(H) "as" simple_intropattern(I₁) :=
  invert_tactic H (fun H ⇒ inverts keep H as I₁).
Tactic Notation "inverts" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) :=
  invert_tactic H (fun H ⇒ inverts keep H as I₁ I₂).
Tactic Notation "inverts" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) simple_intropattern(I₃) :=
  invert_tactic H (fun H ⇒ inverts keep H as I₁ I₂ I₃).
Tactic Notation "inverts" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) simple_intropattern(I₃) simple_intropattern(I₄) :=
  invert_tactic H (fun H ⇒ inverts keep H as I₁ I₂ I₃ I₄).
Tactic Notation "inverts" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) simple_intropattern(I₃) simple_intropattern(I₄)
simple_intropattern(I₅) :=
  invert_tactic H (fun H ⇒ inverts keep H as I₁ I₂ I₃ I₄ I₅).
Tactic Notation "inverts" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) simple_intropattern(I₃) simple_intropattern(I₄)
simple_intropattern(I₅) simple_intropattern(I₆) :=
  invert_tactic H (fun H ⇒ inverts keep H as I₁ I₂ I₃ I₄ I₅ I₆).

inverts H as performs an inversion on hypothesis H, substitutes generated equalities, and put in the goal the other freshly-created hypotheses, for the user to name explicitly. inverts keep H as is the same except that it does not clear H. TODO: reimplement inverts above using this one

Ltac inverts_as_tactic H :=
  let rec go tt :=
    match goal with
    | ⊢ (ltac_Mark → _) ⇒ intros _
    | ⊢ (?x = ?y → _) ⇒ let H := fresh "TEMP" in intro H;
                           first [ subst x | subst y ];
                           go tt
    | ⊢ (existT ?P ?p ?x = existT ?P ?p ?y → _) ⇒
         let H := fresh "TEMP" in intro H;
         generalize (@inj_pair2 _ P p x y H);
         clear H; go tt
    | ⊢ (∀ _, _) ⇒
       intro; let H := get_last_hyp tt in mark_to_generalize H; go tt
    end in
  pose ltac_mark; inversion H;
  generalize ltac_mark; gen_until_mark;
  go tt; gen_to_generalize; unfolds ltac_to_generalize;
  unfold eq' in ×.

Tactic Notation "inverts" "keep" hyp(H) "as" :=
inverts_as_tactic H.

Tactic Notation "inverts" hyp(H) "as" :=
inverts_as_tactic H; clear H.

Tactic Notation "inverts" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) simple_intropattern(I₃) simple_intropattern(I₄)
simple_intropattern(I₅) simple_intropattern(I₆) simple_intropattern(I₇) :=
inverts H as; introv I₁ I₂ I₃ I₄ I₅ I₆ I₇.
Tactic Notation "inverts" hyp(H) "as" simple_intropattern(I₁)
simple_intropattern(I₂) simple_intropattern(I₃) simple_intropattern(I₄)
simple_intropattern(I₅) simple_intropattern(I₆) simple_intropattern(I₇)
simple_intropattern(I₈) :=
inverts H as; introv I₁ I₂ I₃ I₄ I₅ I₆ I₇ I₈.

lets_inverts E as I₁ .. IN is intuitively equivalent to inverts E, with the difference that it applies to any expression and not just to the name of an hypothesis.

Ltac lets_inverts_base E cont :=
let H := fresh "TEMP" in lets H: E; try cont H.

Tactic Notation "lets_inverts" constr(E) :=
  lets_inverts_base E ltac:(fun H ⇒ inverts H).
Tactic Notation "lets_inverts" constr(E) "as" simple_intropattern(I₁) :=
  lets_inverts_base E ltac:(fun H ⇒ inverts H as I₁).
Tactic Notation "lets_inverts" constr(E) "as" simple_intropattern(I₁)
simple_intropattern(I₂) :=
  lets_inverts_base E ltac:(fun H ⇒ inverts H as I₁ I₂).
Tactic Notation "lets_inverts" constr(E) "as" simple_intropattern(I₁)
simple_intropattern(I₂) simple_intropattern(I₃) :=
  lets_inverts_base E ltac:(fun H ⇒ inverts H as I₁ I₂ I₃).
Tactic Notation "lets_inverts" constr(E) "as" simple_intropattern(I₁)
simple_intropattern(I₂) simple_intropattern(I₃) simple_intropattern(I₄) :=
  lets_inverts_base E ltac:(fun H ⇒ inverts H as I₁ I₂ I₃ I₄).

Injection with Substitution

Underlying implementation of injects

Ltac injects_tactic H :=
  let rec go _ :=
    match goal with
    | ⊢ (ltac_Mark → _) ⇒ intros _
    | ⊢ (?x = ?y → _) ⇒ let H := fresh "TEMP" in intro H;
                           first [ subst x | subst y | idtac ];
                           go tt
    end in
  generalize ltac_mark; injection H; go tt.

injects keep H takes an hypothesis H of the form C a₁ .. aN = C b₁ .. bN and substitute all equalities ai = bi that have been generated.

Tactic Notation "injects" "keep" hyp(H) :=
injects_tactic H.

injects H is similar to injects keep H but clears the hypothesis H.

Tactic Notation "injects" hyp(H) :=
injects_tactic H; clear H.

inject H as X₁ .. XN is the same as injection followed by intros X₁ .. XN

Tactic Notation "inject" hyp(H) :=
  injection H.
Tactic Notation "inject" hyp(H) "as" ident(X₁) :=
  injection H; intros X₁.
Tactic Notation "inject" hyp(H) "as" ident(X₁) ident(X₂) :=
  injection H; intros X₁ X₂.
Tactic Notation "inject" hyp(H) "as" ident(X₁) ident(X₂) ident(X₃) :=
  injection H; intros X₁ X₂ X₃.
Tactic Notation "inject" hyp(H) "as" ident(X₁) ident(X₂) ident(X₃)
ident(X₄) :=
  injection H; intros X₁ X₂ X₃ X₄.
Tactic Notation "inject" hyp(H) "as" ident(X₁) ident(X₂) ident(X₃)
ident(X₄) ident(X₅) :=
  injection H; intros X₁ X₂ X₃ X₄ X₅.

Inversion and Injection with Substitution --rough implementation

The tactics inversions and injections provided in this section are similar to inverts and injects except that they perform substitution on all equalities from the context and not only the ones freshly generated. The counterpart is that they have simpler implementations.

DEPRECATED: these tactics should no longer be used.

inversions keep H is the same as inversions H but it does not clear hypothesis H.

Tactic Notation "inversions" "keep" hyp(H) :=
inversion H; subst.

inversions H is a shortcut for inversion H followed by subst and clear H. It is a rough implementation of inverts keep H which behave badly when the proof context already contains equalities. It is provided in case the better implementation turns out to be too slow.

Tactic Notation "inversions" hyp(H) :=
inversion H; subst; try clear H.

injections keep H is the same as injection H followed by intros and subst. It is a rough implementation of injects keep H which behave badly when the proof context already contains equalities, or when the goal starts with a forall or an implication.

Tactic Notation "injections" "keep" hyp(H) :=
injection H; intros; subst.

injections H is the same as injection H followed by clear H and intros and subst. It is a rough implementation of injects keep H which behave badly when the proof context already contains equalities, or when the goal starts with a forall or an implication.

Tactic Notation "injections" "keep" hyp(H) :=
injection H; clear H; intros; subst.

Case Analysis

cases is similar to case_eq E except that it generates the equality in the context and not in the goal, and generates the equality the other way round. The syntax cases E as H allows specifying the name H of that hypothesis.

Tactic Notation "cases" constr(E) "as" ident(H) :=
  let X := fresh "TEMP" in
  set (X := E) in *; def_to_eq_sym X H E;
  destruct X.

Tactic Notation "cases" constr(E) :=
let H := fresh "Eq" in cases E as H.

case_if_post H is to be defined later as a tactic to clean up hypothesis H and the goal. By defaults, it looks for obvious contradictions. Currently, this tactic is extended in LibReflect to clean up boolean propositions.

Ltac case_if_post H :=
tryfalse.

case_if looks for a pattern of the form if ?B then ?E₁ else ?E₂ in the goal, and perform a case analysis on B by calling destruct B. Subgoals containing a contradiction are discarded. case_if looks in the goal first, and otherwise in the first hypothesis that contains an if statement. case_if in H can be used to specify which hypothesis to consider. Syntaxes case_if as Eq and case_if in H as Eq allows to name the hypothesis coming from the case analysis.

Ltac case_if_on_tactic_core E Eq :=
  match type of E with
  | {_}+{_} ⇒ destruct E as [Eq | Eq]
  | _ ⇒ let X := fresh "TEMP" in
         sets_eq <- X Eq: E;
         destruct X
  end.

Ltac case_if_on_tactic E Eq :=
case_if_on_tactic_core E Eq; case_if_post Eq.

Tactic Notation "case_if_on" constr(E) "as" simple_intropattern(Eq) :=
case_if_on_tactic E Eq.

Tactic Notation "case_if" "as" simple_intropattern(Eq) :=
  match goal with
  | ⊢ context [if ?B then _ else _] ⇒ case_if_on B as Eq
  | K: context [if ?B then _ else _] ⊢ _ ⇒ case_if_on B as Eq
  end.

Tactic Notation "case_if" "in" hyp(H) "as" simple_intropattern(Eq) :=
match type of H with context [if ?B then _ else _] ⇒
case_if_on B as Eq end.

Tactic Notation "case_if" :=
let Eq := fresh "C" in case_if as Eq.

Tactic Notation "case_if" "in" hyp(H) :=
let Eq := fresh "C" in case_if in H as Eq.

cases_if is similar to case_if with two main differences: if it creates an equality of the form x = y and then substitutes it in the goal

Ltac cases_if_on_tactic_core E Eq :=
  match type of E with
  | {_}+{_} ⇒ destruct E as [Eq|Eq]; try subst_hyp Eq
  | _ ⇒ let X := fresh "TEMP" in
         sets_eq <- X Eq: E;
         destruct X
  end.

Ltac cases_if_on_tactic E Eq :=
cases_if_on_tactic_core E Eq; tryfalse; case_if_post Eq.

Tactic Notation "cases_if_on" constr(E) "as" simple_intropattern(Eq) :=
cases_if_on_tactic E Eq.

Tactic Notation "cases_if" "as" simple_intropattern(Eq) :=
  match goal with
  | ⊢ context [if ?B then _ else _] ⇒ cases_if_on B as Eq
  | K: context [if ?B then _ else _] ⊢ _ ⇒ cases_if_on B as Eq
  end.

Tactic Notation "cases_if" "in" hyp(H) "as" simple_intropattern(Eq) :=
match type of H with context [if ?B then _ else _] ⇒
cases_if_on B as Eq end.

Tactic Notation "cases_if" :=
let Eq := fresh "C" in cases_if as Eq.

Tactic Notation "cases_if" "in" hyp(H) :=
let Eq := fresh "C" in cases_if in H as Eq.

case_ifs is like repeat case_if

Ltac case_ifs_core :=
repeat case_if.

Tactic Notation "case_ifs" :=
case_ifs_core.

destruct_if looks for a pattern of the form if ?B then ?E₁ else ?E₂ in the goal, and perform a case analysis on B by calling destruct B. It looks in the goal first, and otherwise in the first hypothesis that contains an if statement.

Ltac destruct_if_post := tryfalse.

Tactic Notation "destruct_if"
"as" simple_intropattern(Eq₁) simple_intropattern(Eq₂) :=
  match goal with
  | ⊢ context [if ?B then _ else _] ⇒ destruct B as [Eq₁|Eq₂]
  | K: context [if ?B then _ else _] ⊢ _ ⇒ destruct B as [Eq₁|Eq₂]
  end;
  destruct_if_post.

Tactic Notation "destruct_if" "in" hyp(H)
"as" simple_intropattern(Eq₁) simple_intropattern(Eq₂) :=
  match type of H with context [if ?B then _ else _] ⇒
    destruct B as [Eq₁|Eq₂] end;
  destruct_if_post.

Tactic Notation "destruct_if" "as" simple_intropattern(Eq) :=
destruct_if as Eq Eq.
Tactic Notation "destruct_if" "in" hyp(H) "as" simple_intropattern(Eq) :=
destruct_if in H as Eq Eq.

Tactic Notation "destruct_if" :=
let Eq := fresh "C" in destruct_if as Eq Eq.
Tactic Notation "destruct_if" "in" hyp(H) :=
let Eq := fresh "C" in destruct_if in H as Eq Eq.

BROKEN since v₈.5beta2. TODO: cleanup.

destruct_head_match performs a case analysis on the argument of the head pattern matching when the goal has the form match ?E with ... or match ?E with ... = _ or _ = match ?E with .... Due to the limits of Ltac, this tactic will not fail if a match does not occur. Instead, it might perform a case analysis on an unspecified subterm from the goal. Warning: experimental.

Ltac find_head_match T :=
  match T with context [?E] ⇒
    match T with
    | E ⇒ fail 1
    | _ ⇒ constr:(E)
    end
  end.

Ltac destruct_head_match_core cont :=
  match goal with
  | ⊢ ?T₁ = ?T₂ ⇒ first [ let E := find_head_match T₁ in cont E
                          | let E := find_head_match T₂ in cont E ]
  | ⊢ ?T₁ ⇒ let E := find_head_match T₁ in cont E
  end;
  destruct_if_post.

Tactic Notation "destruct_head_match" "as" simple_intropattern(I) :=
destruct_head_match_core ltac:(fun E ⇒ destruct E as I).

Tactic Notation "destruct_head_match" :=
destruct_head_match_core ltac:(fun E ⇒ destruct E).

(**--provided for compatibility with remember *)

cases' E is similar to case_eq E except that it generates the equality in the context and not in the goal. The syntax cases' E as H allows specifying the name H of that hypothesis.

Tactic Notation "cases'" constr(E) "as" ident(H) :=
  let X := fresh "TEMP" in
  set (X := E) in *; def_to_eq X H E;
  destruct X.

Tactic Notation "cases'" constr(E) :=
let x := fresh "Eq" in cases' E as H.

cases_if' is similar to cases_if except that it generates the symmetric equality.

Ltac cases_if_on' E Eq :=
  match type of E with
  | {_}+{_} ⇒ destruct E as [Eq|Eq]; try subst_hyp Eq
  | _ ⇒ let X := fresh "TEMP" in
         sets_eq X Eq: E;
         destruct X
  end; case_if_post Eq.

Tactic Notation "cases_if'" "as" simple_intropattern(Eq) :=
  match goal with
  | ⊢ context [if ?B then _ else _] ⇒ cases_if_on' B Eq
  | K: context [if ?B then _ else _] ⊢ _ ⇒ cases_if_on' B Eq
  end.

Tactic Notation "cases_if'" :=
let Eq := fresh "C" in cases_if' as Eq.

Induction

inductions E is a shorthand for dependent induction E. inductions E gen X₁ .. XN is a shorthand for dependent induction E generalizing X₁ .. XN.

From Coq Require Import Program.Equality.

Ltac inductions_post :=
unfold eq' in ×.

Tactic Notation "inductions" ident(E) :=
  dependent induction E; inductions_post.
Tactic Notation "inductions" ident(E) "gen" ident(X₁) :=
  dependent induction E generalizing X₁; inductions_post.
Tactic Notation "inductions" ident(E) "gen" ident(X₁) ident(X₂) :=
  dependent induction E generalizing X₁ X₂; inductions_post.
Tactic Notation "inductions" ident(E) "gen" ident(X₁) ident(X₂)
ident(X₃) :=
  dependent induction E generalizing X₁ X₂ X₃; inductions_post.
Tactic Notation "inductions" ident(E) "gen" ident(X₁) ident(X₂)
ident(X₃) ident(X₄) :=
  dependent induction E generalizing X₁ X₂ X₃ X₄; inductions_post.
Tactic Notation "inductions" ident(E) "gen" ident(X₁) ident(X₂)
ident(X₃) ident(X₄) ident(X₅) :=
  dependent induction E generalizing X₁ X₂ X₃ X₄ X₅; inductions_post.
Tactic Notation "inductions" ident(E) "gen" ident(X₁) ident(X₂)
ident(X₃) ident(X₄) ident(X₅) ident(X₆) :=
  dependent induction E generalizing X₁ X₂ X₃ X₄ X₅ X₆; inductions_post.
Tactic Notation "inductions" ident(E) "gen" ident(X₁) ident(X₂)
ident(X₃) ident(X₄) ident(X₅) ident(X₆) ident(X₇) :=
  dependent induction E generalizing X₁ X₂ X₃ X₄ X₅ X₆ X₇; inductions_post.
Tactic Notation "inductions" ident(E) "gen" ident(X₁) ident(X₂)
ident(X₃) ident(X₄) ident(X₅) ident(X₆) ident(X₇) ident(X₈) :=
  dependent induction E generalizing X₁ X₂ X₃ X₄ X₅ X₆ X₇ X₈; inductions_post.

induction_wf IH: E X is used to apply the well-founded induction principle, for a given well-founded relation. It applies to a goal PX where PX is a proposition on X. First, it sets up the goal in the form (fun a ⇒ P a) X, using pattern X, and then it applies the well-founded induction principle instantiated on E.

Here E may be either:

a proof of wf R for R of type A→A→Prop
a binary relation of type A→A→Prop
a measure of type A → nat // only when LibWf is used

Syntaxes induction_wf: E X and induction_wf E X.

(* DEPRECATED
Tactic Notation "induction_wf" ident(IH) ":" constr(E) ident(X) :=
pattern X; apply (well_founded_ind E); clear X; intros X IH.
*)

(* Tactic is later extended in module LibWf *)
Ltac induction_wf_core_then IH E X cont :=
  let T := type of E in
  let T := eval hnf in T in
  let clearX tt :=
    first [ clear X | fail 3 "the variable on which the induction is done appears in the hypotheses" ] in
  match T with
  (* Support for measures from LibWf, add this:
  | ?A -> nat =>
     induction_wf_core_then IH (wf_measure E) X cont
  *)
  | ?A → ?A → Prop ⇒
     pattern X;
     first [
       applys well_founded_ind E;
       clearX tt;
       [ (* Support for wf from LibWf
         change well_founded with wf; auto with wf *)
       | intros X IH; cont tt ]
     | fail 2 ]
  | _ ⇒
    pattern X;
    applys well_founded_ind E;
    clearX tt;
    intros X IH;
    cont tt
  end.

Ltac induction_wf_core IH E X :=
induction_wf_core_then IH E X ltac:(fun _ ⇒ idtac).

Tactic Notation "induction_wf" ident(IH) ":" constr(E) ident(X) :=
  induction_wf_core IH E X.
Tactic Notation "induction_wf" ":" constr(E) ident(X) :=
  let IH := fresh "IH" in induction_wf IH: E X.
Tactic Notation "induction_wf" ":" constr(E) ident(X) :=
  induction_wf: E X.

Induction on the height of a derivation: the helper tactic induct_height helps proving the equivalence of the auxiliary judgment that includes a counter for the maximal height (see LibTacticsDemos for an example)

From Coq Require Import Arith.Compare_dec.
From Coq Require Import omega.Omega.

Lemma induct_height_max2 : ∀ n₁ n₂ : nat,
  ∃ n, n₁ < n ∧ n₂ < n.
Proof using.
  intros. destruct (lt_dec n₁ n₂).
  ∃ (S n₂). omega.
  ∃ (S n₁). omega.
Qed.

Ltac induct_height_step x :=
  match goal with
  | H: ∃ _, _ ⊢ _ ⇒
     let n := fresh "n" in let y := fresh "x" in
     destruct H as [n ?];
     forwards (y&?&?): induct_height_max2 n x;
     induct_height_step y
  | _ ⇒ ∃ (S x); eauto
end.

Ltac induct_height := induct_height_step O.

Coinduction

Tactic cofixs IH is like cofix IH except that the coinduction hypothesis is tagged in the form IH: COIND P instead of being just IH: P. This helps other tactics clearing the coinduction hypothesis using clear_coind

Definition COIND (P:Prop) := P.

Tactic Notation "cofixs" ident(IH) :=
cofix IH;
match type of IH with ?P ⇒ change P with (COIND P) in IH end.

Tactic clear_coind clears all the coinduction hypotheses, assuming that they have been tagged

Ltac clear_coind :=
repeat match goal with H: COIND _ ⊢ _ ⇒ clear H end.

Tactic abstracts tac is like abstract tac except that it clears the coinduction hypotheses so that the productivity check will be happy. For example, one can use abstracts omega to obtain the same behavior as omega but with an auxiliary lemma being generated.

Tactic Notation "abstracts" tactic(tac) :=
clear_coind; tac.

Decidable Equality

decides_equality is the same as decide equality excepts that it is able to unfold definitions at head of the current goal.

Ltac decides_equality_tactic :=
first [ decide equality | progress(unfolds); decides_equality_tactic ].

Tactic Notation "decides_equality" :=
decides_equality_tactic.

Equivalence

iff H can be used to prove an equivalence P ↔ Q and name H the hypothesis obtained in each case. The syntaxes iff and iff H₁ H₂ are also available to specify zero or two names. The tactic iff <- H swaps the two subgoals, i.e. produces (Q -> P) as first subgoal.

Lemma iff_intro_swap : ∀ (P Q : Prop),
(Q → P ) → (P → Q ) → (P ↔ Q ).
Proof using. intuition. Qed.

Tactic Notation "iff" simple_intropattern(H₁) simple_intropattern(H₂) :=
  split; [ intros H₁ | intros H₂ ].
Tactic Notation "iff" simple_intropattern(H) :=
  iff H H.
Tactic Notation "iff" :=
  let H := fresh "H" in iff H.

Tactic Notation "iff" "<-" simple_intropattern(H₁) simple_intropattern(H₂) :=
  apply iff_intro_swap; [ intros H₁ | intros H₂ ].
Tactic Notation "iff" "<-" simple_intropattern(H) :=
  iff <- H H.
Tactic Notation "iff" "<-" :=
  let H := fresh "H" in iff <- H.

N-ary Conjunctions and Disjunctions

N-ary Conjunctions Splitting in Goals

Underlying implementation of splits.

Ltac splits_tactic N :=
  match N with
  | O ⇒ fail
  | S O ⇒ idtac
  | S ?N' ⇒ split; [| splits_tactic N']
  end.

Ltac unfold_goal_until_conjunction :=
  match goal with
  | ⊢ _ ∧ _ ⇒ idtac
  | _ ⇒ progress(unfolds); unfold_goal_until_conjunction
  end.

Ltac get_term_conjunction_arity T :=
  match T with
  | _ ∧ _ ∧ _ ∧ _ ∧ _ ∧ _ ∧ _ ∧ _ ⇒ constr:(8)
  | _ ∧ _ ∧ _ ∧ _ ∧ _ ∧ _ ∧ _ ⇒ constr:(7)
  | _ ∧ _ ∧ _ ∧ _ ∧ _ ∧ _ ⇒ constr:(6)
  | _ ∧ _ ∧ _ ∧ _ ∧ _ ⇒ constr:(5)
  | _ ∧ _ ∧ _ ∧ _ ⇒ constr:(4)
  | _ ∧ _ ∧ _ ⇒ constr:(3)
  | _ ∧ _ ⇒ constr:(2)
  | _ → ?T' ⇒ get_term_conjunction_arity T'
  | _ ⇒ let P := get_head T in
         let T' := eval unfold P in T in
         match T' with
         | T ⇒ fail 1
         | _ ⇒ get_term_conjunction_arity T'
         end
         (* --TODO: warning this can loop... *)
  end.

Ltac get_goal_conjunction_arity :=
match goal with ⊢ ?T ⇒ get_term_conjunction_arity T end.

splits applies to a goal of the form (T₁ ∧ .. ∧ TN) and destruct it into N subgoals T₁ .. TN. If the goal is not a conjunction, then it unfolds the head definition.

Tactic Notation "splits" :=
  unfold_goal_until_conjunction;
  let N := get_goal_conjunction_arity in
  splits_tactic N.

splits N is similar to splits, except that it will unfold as many definitions as necessary to obtain an N-ary conjunction.

Tactic Notation "splits" constr(N) :=
let N := number_to_nat N in
splits_tactic N.

N-ary Conjunctions Deconstruction

Underlying implementation of destructs.

destructs T allows destructing a term T which is a N-ary conjunction. It is equivalent to destruct T as (H₁ .. HN), except that it does not require to manually specify N different names.

Tactic Notation "destructs" constr(T) :=
  let TT := type of T in
  let N := get_term_conjunction_arity TT in
  destructs_conjunction_tactic N T.

destructs N T is equivalent to destruct T as (H₁ .. HN), except that it does not require to manually specify N different names. Remark that it is not restricted to N-ary conjunctions.

Tactic Notation "destructs" constr(N) constr(T) :=
let N := number_to_nat N in
destructs_conjunction_tactic N T.

Proving goals which are N-ary disjunctions

Underlying implementation of branch.

Ltac unfold_goal_until_disjunction :=
  match goal with
  | ⊢ _ ∨ _ ⇒ idtac
  | _ ⇒ progress(unfolds); unfold_goal_until_disjunction
  end.

Ltac get_goal_disjunction_arity :=
match goal with ⊢ ?T ⇒ get_term_disjunction_arity T end.

branch N applies to a goal of the form P₁ ∨ ... ∨ PK ∨ ... ∨ PN and leaves the goal PK. It only able to unfold the head definition (if there is one), but for more complex unfolding one should use the tactic branch K of N.

Tactic Notation "branch" constr(K) :=
  let K := number_to_nat K in
  unfold_goal_until_disjunction;
  let N := get_goal_disjunction_arity in
  branch_tactic K N.

branch K of N is similar to branch K except that the arity of the disjunction N is given manually, and so this version of the tactic is able to unfold definitions. In other words, applies to a goal of the form P₁ ∨ ... ∨ PK ∨ ... ∨ PN and leaves the goal PK.

Tactic Notation "branch" constr(K) "of" constr(N) :=
  let N := number_to_nat N in
  let K := number_to_nat K in
  branch_tactic K N.

N-ary Disjunction Deconstruction

Underlying implementation of branches.

Ltac destructs_disjunction_tactic N T :=
  match N with
  | 2 ⇒ destruct T as [? | ?]
  | 3 ⇒ destruct T as [? | [? | ?]]
| 4 ⇒ destruct T as [? | [? | [? | ?]]]
| 5 ⇒ destruct T as [? | [? | [? | [? | ?]]]]
end.

branches T allows destructing a term T which is a N-ary disjunction. It is equivalent to destruct T as [ H₁ | .. | HN ] , and produces N subgoals corresponding to the N possible cases.

Tactic Notation "branches" constr(T) :=
  let TT := type of T in
  let N := get_term_disjunction_arity TT in
  destructs_disjunction_tactic N T.

branches N T is the same as branches T except that the arity is forced to N. This version is useful to unfold definitions on the fly.

Tactic Notation "branches" constr(N) constr(T) :=
let N := number_to_nat N in
destructs_disjunction_tactic N T.

branches automatically finds a hypothesis h that is a disjunction and destructs it.

Tactic Notation "branches" :=
match goal with h: _ ∨ _ ⊢ _ ⇒ branches h end.

(* ---------------------------------------------------------------------- *)

N-ary Existentials

(* Underlying implementation of ∃. *)

Ltac get_goal_existential_arity :=
match goal with ⊢ ?T ⇒ get_term_existential_arity T end.

∃ T₁ ... TN is a shorthand for ∃ T₁; ...; ∃ TN. It is intended to prove goals of the form exist X₁ .. XN, P. If an argument provided is __ (double underscore), then an evar is introduced. ∃ T₁ .. TN ___ is equivalent to ∃ T₁ .. TN __ __ __ with as many __ as possible.

Tactic Notation "exists_original" constr(T₁) :=
  ∃ T₁.
Tactic Notation "exists" constr(T₁) :=
  match T₁ with
  | ltac_wild ⇒ esplit
  | ltac_wilds ⇒ repeat esplit
  | _ ⇒ ∃ T₁
  end.
Tactic Notation "exists" constr(T₁) constr(T₂) :=
  ∃ T₁; ∃ T₂.
Tactic Notation "exists" constr(T₁) constr(T₂) constr(T₃) :=
  ∃ T₁; ∃ T₂; ∃ T₃.
Tactic Notation "exists" constr(T₁) constr(T₂) constr(T₃) constr(T₄) :=
  ∃ T₁; ∃ T₂; ∃ T₃; ∃ T₄.
Tactic Notation "exists" constr(T₁) constr(T₂) constr(T₃) constr(T₄)
constr(T₅) :=
  ∃ T₁; ∃ T₂; ∃ T₃; ∃ T₄; ∃ T₅.
Tactic Notation "exists" constr(T₁) constr(T₂) constr(T₃) constr(T₄)
constr(T₅) constr(T₆) :=
  ∃ T₁; ∃ T₂; ∃ T₃; ∃ T₄; ∃ T₅; ∃ T₆.

For compatibility with Coq syntax, ∃ T₁, .., TN is also provided.

Tactic Notation "exists" constr(T₁) "," constr(T₂) :=
  ∃ T₁ T₂.
Tactic Notation "exists" constr(T₁) "," constr(T₂) "," constr(T₃) :=
  ∃ T₁ T₂ T₃.
Tactic Notation "exists" constr(T₁) "," constr(T₂) "," constr(T₃) "," constr(T₄) :=
  ∃ T₁ T₂ T₃ T₄.
Tactic Notation "exists" constr(T₁) "," constr(T₂) "," constr(T₃) "," constr(T₄) ","
constr(T₅) :=
  ∃ T₁ T₂ T₃ T₄ T₅.
Tactic Notation "exists" constr(T₁) "," constr(T₂) "," constr(T₃) "," constr(T₄) ","
constr(T₅) "," constr(T₆) :=
  ∃ T₁ T₂ T₃ T₄ T₅ T₆.

(* The tactic exists___ N is short for ∃ __ ... __
   with N double-underscores. The tactic ∃ is equivalent
   to calling exists___ N, where the value of N is obtained
   by counting the number of existentials syntactically present
   at the head of the goal. The behaviour of ∃ differs
   from that of ∃ ___ is the case where the goal is a
   definition which yields an existential only after unfolding. *)

Tactic Notation "exists___" constr(N) :=
  let rec aux N :=
    match N with
    | 0 ⇒ idtac
    | S ?N' ⇒ esplit; aux N'
    end in
  let N := number_to_nat N in aux N.

  (* --TODO: deprecated *)
Tactic Notation "exists___" :=
  let N := get_goal_existential_arity in
  exists___ N.

(* --TODO: does not seem to work *)
Tactic Notation "exists" :=
exists___.

(* --TODO: exists_all is the new syntax for exists___ *)
Tactic Notation "exists_all" := exists___.

Existentials and conjunctions in hypotheses

unpack or unpack H destructs conjunctions and existentials in all or one hypothesis.

Ltac unpack_core :=
  repeat match goal with
  | H: _ ∧ _ ⊢ _ ⇒ destruct H
  | H: ∃ (varname: _), _ ⊢ _ ⇒
      (* kludge to preserve the name of the quantified variable *)
      let name := fresh varname in
      destruct H as [name ?]
  end.

Ltac unpack_hypothesis H :=
  try match type of H with
  | _ ∧ _ ⇒
      let h₁ := fresh "TEMP" in
      let h₂ := fresh "TEMP" in
      destruct H as [ h₁ h₂ ];
      unpack_hypothesis h₁;
      unpack_hypothesis h₂
  | ∃ (varname: _), _ ⇒
      (* kludge to preserve the name of the quantified variable *)
      let name := fresh varname in
      let body := fresh "TEMP" in
      destruct H as [name body];
      unpack_hypothesis body
  end.

Tactic Notation "unpack" :=
unpack_core.
Tactic Notation "unpack" constr(H) :=
unpack_hypothesis H.

Tactics to Prove Typeclass Instances

typeclass is an automation tactic specialized for finding typeclass instances.

Tactic Notation "typeclass" :=
let go _ := eauto with typeclass_instances in
solve [ go tt | constructor; go tt ].

solve_typeclass is a simpler version of typeclass, to use in hint tactics for resolving instances

Tactic Notation "solve_typeclass" :=
solve [ eauto with typeclass_instances ].

Tactics to Invoke Automation

Definitions for Parsing Compatibility

Tactic Notation "f_equal" :=
  f_equal.
Tactic Notation "constructor" :=
  constructor.
Tactic Notation "simple" :=
  simpl.

Tactic Notation "split" :=
split.

Tactic Notation "right" :=
right.
Tactic Notation "left" :=
left.

hint to Add Hints Local to a Lemma

hint E adds E as an hypothesis so that automation can use it. Syntax hint E₁,..,EN is available

Tactic Notation "hint" constr(E) :=
  let H := fresh "Hint" in lets H: E.
Tactic Notation "hint" constr(E₁) "," constr(E₂) :=
  hint E₁; hint E₂.
Tactic Notation "hint" constr(E₁) "," constr(E₂) "," constr(E₃) :=
  hint E₁; hint E₂; hint(E₃).
Tactic Notation "hint" constr(E₁) "," constr(E₂) "," constr(E₃) "," constr(E₄) :=
  hint E₁; hint E₂; hint(E₃); hint(E₄ ).

jauto, a New Automation Tactic

jauto is better at intuition eauto because it can open existentials from the context. In the same time, jauto can be faster than intuition eauto because it does not destruct disjunctions from the context. The strategy of jauto can be summarized as follows:

open all the existentials and conjunctions from the context
call esplit and split on the existentials and conjunctions in the goal
call eauto.

Tactic Notation "jauto" :=
try solve [ jauto_set; eauto ].

Tactic Notation "jauto_fast" :=
try solve [ auto | eauto | jauto ].

iauto is a shorthand for intuition eauto

Tactic Notation "iauto" := try solve [intuition eauto].

Definitions of Automation Tactics

The two following tactics defined the default behaviour of "light automation" and "strong automation". These tactics may be redefined at any time using the syntax Ltac .. ::= ...

auto_tilde is the tactic which will be called each time a symbol ¬ is used after a tactic.

Ltac auto_tilde_default := auto.
Ltac auto_tilde := auto_tilde_default.

auto_star is the tactic which will be called each time a symbol × is used after a tactic.

(* SPECIAL VERSION FOR SF*)
Ltac auto_star_default := try solve [ jauto ].
Ltac auto_star := auto_star_default.

autos¬ is a notation for tactic auto_tilde. It may be followed by lemmas (or proofs terms) which auto will be able to use for solving the goal. autos is an alias for autos¬

Tactic Notation "autos" :=
  auto_tilde.
Tactic Notation "autos" "~" :=
  auto_tilde.
Tactic Notation "autos" "~" constr(E₁) :=
  lets: E₁; auto_tilde.
Tactic Notation "autos" "~" constr(E₁) constr(E₂) :=
  lets: E₁; lets: E₂; auto_tilde.
Tactic Notation "autos" "~" constr(E₁) constr(E₂) constr(E₃) :=
  lets: E₁; lets: E₂; lets: E₃; auto_tilde.

autos× is a notation for tactic auto_star. It may be followed by lemmas (or proofs terms) which auto will be able to use for solving the goal.

Tactic Notation "autos" "*" :=
  auto_star.
Tactic Notation "autos" "*" constr(E₁) :=
  lets: E₁; auto_star.
Tactic Notation "autos" "*" constr(E₁) constr(E₂) :=
  lets: E₁; lets: E₂; auto_star.
Tactic Notation "autos" "*" constr(E₁) constr(E₂) constr(E₃) :=
  lets: E₁; lets: E₂; lets: E₃; auto_star.

auto_false is a version of auto able to spot some contradictions. There is an ad-hoc support for goals in ↔: split is called first. auto_false¬ and auto_false× are also available.

Ltac auto_false_base cont :=
  try solve [
    intros_all; try match goal with ⊢ _ ↔ _ ⇒ split end;
    solve [ cont tt | intros_all; false; cont tt ] ].

Tactic Notation "auto_false" :=
   auto_false_base ltac:(fun tt ⇒ auto).
Tactic Notation "auto_false" "~" :=
   auto_false_base ltac:(fun tt ⇒ auto_tilde).
Tactic Notation "auto_false" "*" :=
   auto_false_base ltac:(fun tt ⇒ auto_star).

Tactic Notation "dauto" :=
dintuition eauto.

Parsing for Light Automation

Any tactic followed by the symbol ¬ will have auto_tilde called on all of its subgoals. Three exceptions:

cuts and asserts only call auto on their first subgoal,
apply¬ relies on sapply rather than apply,
tryfalse¬ is defined as tryfalse by auto_tilde.

Some builtin tactics are not defined using tactic notations and thus cannot be extended, e.g., simpl and unfold. For these, notation such as simpl¬ will not be available.

Tactic Notation "equates" "~" constr(E) :=
   equates E; auto_tilde.
Tactic Notation "equates" "~" constr(n₁) constr(n₂) :=
  equates n₁ n₂; auto_tilde.
Tactic Notation "equates" "~" constr(n₁) constr(n₂) constr(n₃) :=
  equates n₁ n₂ n₃; auto_tilde.
Tactic Notation "equates" "~" constr(n₁) constr(n₂) constr(n₃) constr(n₄) :=
  equates n₁ n₂ n₃ n₄; auto_tilde.

Tactic Notation "applys_eq" "~" constr(H) constr(E) :=
  applys_eq H E; auto_tilde.
Tactic Notation "applys_eq" "~" constr(H) constr(n₁) constr(n₂) :=
  applys_eq H n₁ n₂; auto_tilde.
Tactic Notation "applys_eq" "~" constr(H) constr(n₁) constr(n₂) constr(n₃) :=
  applys_eq H n₁ n₂ n₃; auto_tilde.
Tactic Notation "applys_eq" "~" constr(H) constr(n₁) constr(n₂) constr(n₃) constr(n₄) :=
  applys_eq H n₁ n₂ n₃ n₄; auto_tilde.

Tactic Notation "apply" "~" constr(H) :=
sapply H; auto_tilde.

Tactic Notation "destruct" "~" constr(H) :=
  destruct H; auto_tilde.
Tactic Notation "destruct" "~" constr(H) "as" simple_intropattern(I) :=
  destruct H as I; auto_tilde.
Tactic Notation "f_equal" "~" :=
  f_equal; auto_tilde.
Tactic Notation "induction" "~" constr(H) :=
  induction H; auto_tilde.
Tactic Notation "inversion" "~" constr(H) :=
  inversion H; auto_tilde.
Tactic Notation "split" "~" :=
  split; auto_tilde.
Tactic Notation "subst" "~" :=
  subst; auto_tilde.
Tactic Notation "right" "~" :=
  right; auto_tilde.
Tactic Notation "left" "~" :=
  left; auto_tilde.
Tactic Notation "constructor" "~" :=
  constructor; auto_tilde.
Tactic Notation "constructors" "~" :=
  constructors; auto_tilde.

Tactic Notation "false" "~" :=
  false; auto_tilde.
Tactic Notation "false" "~" constr(E) :=
  false_then E ltac:(fun _ ⇒ auto_tilde).
Tactic Notation "false" "~" constr(E₀) constr(E₁) :=
  false¬(>> E₀ E₁).
Tactic Notation "false" "~" constr(E₀) constr(E₁) constr(E₂) :=
  false¬(>> E₀ E₁ E₂).
Tactic Notation "false" "~" constr(E₀) constr(E₁) constr(E₂) constr(E₃) :=
  false¬(>> E₀ E₁ E₂ E₃).
Tactic Notation "false" "~" constr(E₀) constr(E₁) constr(E₂) constr(E₃) constr(E₄) :=
  false¬(>> E₀ E₁ E₂ E₃ E₄).
Tactic Notation "tryfalse" "~" :=
  try solve [ false¬].

Tactic Notation "asserts" "~" simple_intropattern(H) ":" constr(E) :=
  asserts H: E; [ auto_tilde | idtac ].
Tactic Notation "asserts" "~" ":" constr(E) :=
  let H := fresh "H" in asserts¬H: E.
Tactic Notation "cuts" "~" simple_intropattern(H) ":" constr(E) :=
  cuts H: E; [ auto_tilde | idtac ].
Tactic Notation "cuts" "~" ":" constr(E) :=
  cuts: E; [ auto_tilde | idtac ].

Tactic Notation "lets" "~" simple_intropattern(I) ":" constr(E) :=
  lets I: E; auto_tilde.
Tactic Notation "lets" "~" simple_intropattern(I) ":" constr(E₀)
constr(A₁) :=
  lets I: E₀ A₁; auto_tilde.
Tactic Notation "lets" "~" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) :=
  lets I: E₀ A₁ A₂; auto_tilde.
Tactic Notation "lets" "~" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) :=
  lets I: E₀ A₁ A₂ A₃; auto_tilde.
Tactic Notation "lets" "~" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  lets I: E₀ A₁ A₂ A₃ A₄; auto_tilde.
Tactic Notation "lets" "~" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  lets I: E₀ A₁ A₂ A₃ A₄ A₅; auto_tilde.

Tactic Notation "lets" "~" ":" constr(E) :=
  lets: E; auto_tilde.
Tactic Notation "lets" "~" ":" constr(E₀)
constr(A₁) :=
  lets: E₀ A₁; auto_tilde.
Tactic Notation "lets" "~" ":" constr(E₀)
constr(A₁) constr(A₂) :=
  lets: E₀ A₁ A₂; auto_tilde.
Tactic Notation "lets" "~" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) :=
  lets: E₀ A₁ A₂ A₃; auto_tilde.
Tactic Notation "lets" "~" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  lets: E₀ A₁ A₂ A₃ A₄; auto_tilde.
Tactic Notation "lets" "~" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  lets: E₀ A₁ A₂ A₃ A₄ A₅; auto_tilde.

Tactic Notation "forwards" "~" simple_intropattern(I) ":" constr(E) :=
  forwards I: E; auto_tilde.
Tactic Notation "forwards" "~" simple_intropattern(I) ":" constr(E₀)
constr(A₁) :=
  forwards I: E₀ A₁; auto_tilde.
Tactic Notation "forwards" "~" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) :=
  forwards I: E₀ A₁ A₂; auto_tilde.
Tactic Notation "forwards" "~" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) :=
  forwards I: E₀ A₁ A₂ A₃; auto_tilde.
Tactic Notation "forwards" "~" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  forwards I: E₀ A₁ A₂ A₃ A₄; auto_tilde.
Tactic Notation "forwards" "~" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  forwards I: E₀ A₁ A₂ A₃ A₄ A₅; auto_tilde.

Tactic Notation "forwards" "~" ":" constr(E) :=
  forwards: E; auto_tilde.
Tactic Notation "forwards" "~" ":" constr(E₀)
constr(A₁) :=
  forwards: E₀ A₁; auto_tilde.
Tactic Notation "forwards" "~" ":" constr(E₀)
constr(A₁) constr(A₂) :=
  forwards: E₀ A₁ A₂; auto_tilde.
Tactic Notation "forwards" "~" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) :=
  forwards: E₀ A₁ A₂ A₃; auto_tilde.
Tactic Notation "forwards" "~" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  forwards: E₀ A₁ A₂ A₃ A₄; auto_tilde.
Tactic Notation "forwards" "~" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  forwards: E₀ A₁ A₂ A₃ A₄ A₅; auto_tilde.

Tactic Notation "applys" "~" constr(H) :=
  sapply H; auto_tilde. (*todo?*)
Tactic Notation "applys" "~" constr(E₀) constr(A₁) :=
  applys E₀ A₁; auto_tilde.
Tactic Notation "applys" "~" constr(E₀) constr(A₁) :=
  applys E₀ A₁; auto_tilde.
Tactic Notation "applys" "~" constr(E₀) constr(A₁) constr(A₂) :=
  applys E₀ A₁ A₂; auto_tilde.
Tactic Notation "applys" "~" constr(E₀) constr(A₁) constr(A₂) constr(A₃) :=
  applys E₀ A₁ A₂ A₃; auto_tilde.
Tactic Notation "applys" "~" constr(E₀) constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  applys E₀ A₁ A₂ A₃ A₄; auto_tilde.
Tactic Notation "applys" "~" constr(E₀) constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  applys E₀ A₁ A₂ A₃ A₄ A₅; auto_tilde.

Tactic Notation "specializes" "~" hyp(H) :=
  specializes H; auto_tilde.
Tactic Notation "specializes" "~" hyp(H) constr(A₁) :=
  specializes H A₁; auto_tilde.
Tactic Notation "specializes" hyp(H) constr(A₁) constr(A₂) :=
  specializes H A₁ A₂; auto_tilde.
Tactic Notation "specializes" hyp(H) constr(A₁) constr(A₂) constr(A₃) :=
  specializes H A₁ A₂ A₃; auto_tilde.
Tactic Notation "specializes" hyp(H) constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  specializes H A₁ A₂ A₃ A₄; auto_tilde.
Tactic Notation "specializes" hyp(H) constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  specializes H A₁ A₂ A₃ A₄ A₅; auto_tilde.

Tactic Notation "fapply" "~" constr(E) :=
fapply E; auto_tilde.
Tactic Notation "sapply" "~" constr(E) :=
sapply E; auto_tilde.

Tactic Notation "logic" "~" constr(E) :=
logic_base E ltac:(fun _ ⇒ auto_tilde).

Tactic Notation "intros_all" "~" :=
intros_all; auto_tilde.

Tactic Notation "unfolds" "~" :=
  unfolds; auto_tilde.
Tactic Notation "unfolds" "~" constr(F₁) :=
  unfolds F₁; auto_tilde.
Tactic Notation "unfolds" "~" constr(F₁) "," constr(F₂) :=
  unfolds F₁, F₂; auto_tilde.
Tactic Notation "unfolds" "~" constr(F₁) "," constr(F₂) "," constr(F₃) :=
  unfolds F₁, F₂, F₃; auto_tilde.
Tactic Notation "unfolds" "~" constr(F₁) "," constr(F₂) "," constr(F₃) ","
constr(F₄) :=
  unfolds F₁, F₂, F₃, F₄; auto_tilde.

Tactic Notation "simple" "~" :=
  simpl; auto_tilde.
Tactic Notation "simple" "~" "in" hyp(H) :=
  simpl in H; auto_tilde.
Tactic Notation "simpls" "~" :=
  simpls; auto_tilde.
Tactic Notation "hnfs" "~" :=
  hnfs; auto_tilde.
Tactic Notation "hnfs" "~" "in" hyp(H) :=
  hnf in H; auto_tilde.
Tactic Notation "substs" "~" :=
  substs; auto_tilde.
Tactic Notation "intro_hyp" "~" hyp(H) :=
  subst_hyp H; auto_tilde.
Tactic Notation "intro_subst" "~" :=
  intro_subst; auto_tilde.
Tactic Notation "subst_eq" "~" constr(E) :=
  subst_eq E; auto_tilde.

Tactic Notation "rewrite" "~" constr(E) :=
  rewrite E; auto_tilde.
Tactic Notation "rewrite" "~" "<-" constr(E) :=
  rewrite <- E; auto_tilde.
Tactic Notation "rewrite" "~" constr(E) "in" hyp(H) :=
  rewrite E in H; auto_tilde.
Tactic Notation "rewrite" "~" "<-" constr(E) "in" hyp(H) :=
  rewrite <- E in H; auto_tilde.

Tactic Notation "rewrites" "~" constr(E) :=
  rewrites E; auto_tilde.
Tactic Notation "rewrites" "~" constr(E) "in" hyp(H) :=
  rewrites E in H; auto_tilde.
Tactic Notation "rewrites" "~" constr(E) "in" "*" :=
  rewrites E in *; auto_tilde.
Tactic Notation "rewrites" "~" "<-" constr(E) :=
  rewrites <- E; auto_tilde.
Tactic Notation "rewrites" "~" "<-" constr(E) "in" hyp(H) :=
  rewrites <- E in H; auto_tilde.
Tactic Notation "rewrites" "~" "<-" constr(E) "in" "*" :=
  rewrites <- E in *; auto_tilde.

Tactic Notation "rewrite_all" "~" constr(E) :=
  rewrite_all E; auto_tilde.
Tactic Notation "rewrite_all" "~" "<-" constr(E) :=
  rewrite_all <- E; auto_tilde.
Tactic Notation "rewrite_all" "~" constr(E) "in" ident(H) :=
  rewrite_all E in H; auto_tilde.
Tactic Notation "rewrite_all" "~" "<-" constr(E) "in" ident(H) :=
  rewrite_all <- E in H; auto_tilde.
Tactic Notation "rewrite_all" "~" constr(E) "in" "*" :=
  rewrite_all E in *; auto_tilde.
Tactic Notation "rewrite_all" "~" "<-" constr(E) "in" "*" :=
  rewrite_all <- E in *; auto_tilde.

Tactic Notation "asserts_rewrite" "~" constr(E) :=
  asserts_rewrite E; auto_tilde.
Tactic Notation "asserts_rewrite" "~" "<-" constr(E) :=
  asserts_rewrite <- E; auto_tilde.
Tactic Notation "asserts_rewrite" "~" constr(E) "in" hyp(H) :=
  asserts_rewrite E in H; auto_tilde.
Tactic Notation "asserts_rewrite" "~" "<-" constr(E) "in" hyp(H) :=
  asserts_rewrite <- E in H; auto_tilde.
Tactic Notation "asserts_rewrite" "~" constr(E) "in" "*" :=
  asserts_rewrite E in *; auto_tilde.
Tactic Notation "asserts_rewrite" "~" "<-" constr(E) "in" "*" :=
  asserts_rewrite <- E in *; auto_tilde.

Tactic Notation "cuts_rewrite" "~" constr(E) :=
  cuts_rewrite E; auto_tilde.
Tactic Notation "cuts_rewrite" "~" "<-" constr(E) :=
  cuts_rewrite <- E; auto_tilde.
Tactic Notation "cuts_rewrite" "~" constr(E) "in" hyp(H) :=
  cuts_rewrite E in H; auto_tilde.
Tactic Notation "cuts_rewrite" "~" "<-" constr(E) "in" hyp(H) :=
  cuts_rewrite <- E in H; auto_tilde.

Tactic Notation "erewrite" "~" constr(E) :=
erewrite E; auto_tilde.

Tactic Notation "fequal" "~" :=
  fequal; auto_tilde.
Tactic Notation "fequals" "~" :=
  fequals; auto_tilde.
Tactic Notation "pi_rewrite" "~" constr(E) :=
  pi_rewrite E; auto_tilde.
Tactic Notation "pi_rewrite" "~" constr(E) "in" hyp(H) :=
  pi_rewrite E in H; auto_tilde.

Tactic Notation "invert" "~" hyp(H) :=
  invert H; auto_tilde.
Tactic Notation "inverts" "~" hyp(H) :=
  inverts H; auto_tilde.
Tactic Notation "inverts" "~" hyp(E) "as" :=
  inverts E as; auto_tilde.
Tactic Notation "injects" "~" hyp(H) :=
  injects H; auto_tilde.
Tactic Notation "inversions" "~" hyp(H) :=
  inversions H; auto_tilde.

Tactic Notation "cases" "~" constr(E) "as" ident(H) :=
  cases E as H; auto_tilde.
Tactic Notation "cases" "~" constr(E) :=
  cases E; auto_tilde.
Tactic Notation "case_if" "~" :=
  case_if; auto_tilde.
Tactic Notation "case_ifs" "~" :=
  case_ifs; auto_tilde.
Tactic Notation "case_if" "~" "in" hyp(H) :=
  case_if in H; auto_tilde.
Tactic Notation "cases_if" "~" :=
  cases_if; auto_tilde.
Tactic Notation "cases_if" "~" "in" hyp(H) :=
  cases_if in H; auto_tilde.
Tactic Notation "destruct_if" "~" :=
  destruct_if; auto_tilde.
Tactic Notation "destruct_if" "~" "in" hyp(H) :=
  destruct_if in H; auto_tilde.
Tactic Notation "destruct_head_match" "~" :=
  destruct_head_match; auto_tilde.

Tactic Notation "cases'" "~" constr(E) "as" ident(H) :=
  cases' E as H; auto_tilde.
Tactic Notation "cases'" "~" constr(E) :=
  cases' E; auto_tilde.
Tactic Notation "cases_if'" "~" "as" ident(H) :=
  cases_if' as H; auto_tilde.
Tactic Notation "cases_if'" "~" :=
  cases_if'; auto_tilde.

Tactic Notation "decides_equality" "~" :=
decides_equality; auto_tilde.

Tactic Notation "iff" "~" :=
  iff; auto_tilde.
Tactic Notation "iff" "~" simple_intropattern(I) :=
  iff I; auto_tilde.
Tactic Notation "splits" "~" :=
  splits; auto_tilde.
Tactic Notation "splits" "~" constr(N) :=
  splits N; auto_tilde.

Tactic Notation "destructs" "~" constr(T) :=
destructs T; auto_tilde.
Tactic Notation "destructs" "~" constr(N) constr(T) :=
destructs N T; auto_tilde.

Tactic Notation "branch" "~" constr(N) :=
branch N; auto_tilde.
Tactic Notation "branch" "~" constr(K) "of" constr(N) :=
branch K of N; auto_tilde.

Tactic Notation "branches" "~" :=
  branches; auto_tilde.
Tactic Notation "branches" "~" constr(T) :=
  branches T; auto_tilde.
Tactic Notation "branches" "~" constr(N) constr(T) :=
  branches N T; auto_tilde.

Tactic Notation "exists" "~" :=
  ∃; auto_tilde.
Tactic Notation "exists___" "~" :=
  exists___; auto_tilde.
Tactic Notation "exists" "~" constr(T₁) :=
  ∃ T₁; auto_tilde.
Tactic Notation "exists" "~" constr(T₁) constr(T₂) :=
  ∃ T₁ T₂; auto_tilde.
Tactic Notation "exists" "~" constr(T₁) constr(T₂) constr(T₃) :=
  ∃ T₁ T₂ T₃; auto_tilde.
Tactic Notation "exists" "~" constr(T₁) constr(T₂) constr(T₃) constr(T₄) :=
  ∃ T₁ T₂ T₃ T₄; auto_tilde.
Tactic Notation "exists" "~" constr(T₁) constr(T₂) constr(T₃) constr(T₄)
constr(T₅) :=
  ∃ T₁ T₂ T₃ T₄ T₅; auto_tilde.
Tactic Notation "exists" "~" constr(T₁) constr(T₂) constr(T₃) constr(T₄)
constr(T₅) constr(T₆) :=
  ∃ T₁ T₂ T₃ T₄ T₅ T₆; auto_tilde.

Tactic Notation "exists" "~" constr(T₁) "," constr(T₂) :=
  ∃ T₁ T₂; auto_tilde.
Tactic Notation "exists" "~" constr(T₁) "," constr(T₂) "," constr(T₃) :=
  ∃ T₁ T₂ T₃; auto_tilde.
Tactic Notation "exists" "~" constr(T₁) "," constr(T₂) "," constr(T₃) ","
constr(T₄) :=
  ∃ T₁ T₂ T₃ T₄; auto_tilde.
Tactic Notation "exists" "~" constr(T₁) "," constr(T₂) "," constr(T₃) ","
constr(T₄) "," constr(T₅) :=
  ∃ T₁ T₂ T₃ T₄ T₅; auto_tilde.
Tactic Notation "exists" "~" constr(T₁) "," constr(T₂) "," constr(T₃) ","
constr(T₄) "," constr(T₅) "," constr(T₆) :=
  ∃ T₁ T₂ T₃ T₄ T₅ T₆; auto_tilde.

Parsing for Strong Automation

Any tactic followed by the symbol × will have auto× called on all of its subgoals. The exceptions to these rules are the same as for light automation.

Exception: use subs× instead of subst× if you import the library Coq.Classes.Equivalence.

Tactic Notation "equates" "*" constr(E) :=
   equates E; auto_star.
Tactic Notation "equates" "*" constr(n₁) constr(n₂) :=
  equates n₁ n₂; auto_star.
Tactic Notation "equates" "*" constr(n₁) constr(n₂) constr(n₃) :=
  equates n₁ n₂ n₃; auto_star.
Tactic Notation "equates" "*" constr(n₁) constr(n₂) constr(n₃) constr(n₄) :=
  equates n₁ n₂ n₃ n₄; auto_star.

Tactic Notation "applys_eq" "*" constr(H) constr(E) :=
  applys_eq H E; auto_star.
Tactic Notation "applys_eq" "*" constr(H) constr(n₁) constr(n₂) :=
  applys_eq H n₁ n₂; auto_star.
Tactic Notation "applys_eq" "*" constr(H) constr(n₁) constr(n₂) constr(n₃) :=
  applys_eq H n₁ n₂ n₃; auto_star.
Tactic Notation "applys_eq" "*" constr(H) constr(n₁) constr(n₂) constr(n₃) constr(n₄) :=
  applys_eq H n₁ n₂ n₃ n₄; auto_star.

Tactic Notation "apply" "*" constr(H) :=
sapply H; auto_star.

Tactic Notation "destruct" "*" constr(H) :=
  destruct H; auto_star.
Tactic Notation "destruct" "*" constr(H) "as" simple_intropattern(I) :=
  destruct H as I; auto_star.
Tactic Notation "f_equal" "*" :=
  f_equal; auto_star.
Tactic Notation "induction" "*" constr(H) :=
  induction H; auto_star.
Tactic Notation "inversion" "*" constr(H) :=
  inversion H; auto_star.
Tactic Notation "split" "*" :=
  split; auto_star.
Tactic Notation "subs" "*" :=
  subst; auto_star.
Tactic Notation "subst" "*" :=
  subst; auto_star.
Tactic Notation "right" "*" :=
  right; auto_star.
Tactic Notation "left" "*" :=
  left; auto_star.
Tactic Notation "constructor" "*" :=
  constructor; auto_star.
Tactic Notation "constructors" "*" :=
  constructors; auto_star.

Tactic Notation "false" "*" :=
  false; auto_star.
Tactic Notation "false" "*" constr(E) :=
  false_then E ltac:(fun _ ⇒ auto_star).
Tactic Notation "false" "*" constr(E₀) constr(E₁) :=
  false× (>> E₀ E₁).
Tactic Notation "false" "*" constr(E₀) constr(E₁) constr(E₂) :=
  false× (>> E₀ E₁ E₂).
Tactic Notation "false" "*" constr(E₀) constr(E₁) constr(E₂) constr(E₃) :=
  false× (>> E₀ E₁ E₂ E₃).
Tactic Notation "false" "*" constr(E₀) constr(E₁) constr(E₂) constr(E₃) constr(E₄) :=
  false× (>> E₀ E₁ E₂ E₃ E₄).
Tactic Notation "tryfalse" "*" :=
  try solve [ false× ].

Tactic Notation "asserts" "*" simple_intropattern(H) ":" constr(E) :=
  asserts H: E; [ auto_star | idtac ].
Tactic Notation "asserts" "*" ":" constr(E) :=
  let H := fresh "H" in asserts× H: E.
Tactic Notation "cuts" "*" simple_intropattern(H) ":" constr(E) :=
  cuts H: E; [ auto_star | idtac ].
Tactic Notation "cuts" "*" ":" constr(E) :=
  cuts: E; [ auto_star | idtac ].

Tactic Notation "lets" "*" simple_intropattern(I) ":" constr(E) :=
  lets I: E; auto_star.
Tactic Notation "lets" "*" simple_intropattern(I) ":" constr(E₀)
constr(A₁) :=
  lets I: E₀ A₁; auto_star.
Tactic Notation "lets" "*" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) :=
  lets I: E₀ A₁ A₂; auto_star.
Tactic Notation "lets" "*" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) :=
  lets I: E₀ A₁ A₂ A₃; auto_star.
Tactic Notation "lets" "*" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  lets I: E₀ A₁ A₂ A₃ A₄; auto_star.
Tactic Notation "lets" "*" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  lets I: E₀ A₁ A₂ A₃ A₄ A₅; auto_star.

Tactic Notation "lets" "*" ":" constr(E) :=
  lets: E; auto_star.
Tactic Notation "lets" "*" ":" constr(E₀)
constr(A₁) :=
  lets: E₀ A₁; auto_star.
Tactic Notation "lets" "*" ":" constr(E₀)
constr(A₁) constr(A₂) :=
  lets: E₀ A₁ A₂; auto_star.
Tactic Notation "lets" "*" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) :=
  lets: E₀ A₁ A₂ A₃; auto_star.
Tactic Notation "lets" "*" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  lets: E₀ A₁ A₂ A₃ A₄; auto_star.
Tactic Notation "lets" "*" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  lets: E₀ A₁ A₂ A₃ A₄ A₅; auto_star.

Tactic Notation "forwards" "*" simple_intropattern(I) ":" constr(E) :=
  forwards I: E; auto_star.
Tactic Notation "forwards" "*" simple_intropattern(I) ":" constr(E₀)
constr(A₁) :=
  forwards I: E₀ A₁; auto_star.
Tactic Notation "forwards" "*" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) :=
  forwards I: E₀ A₁ A₂; auto_star.
Tactic Notation "forwards" "*" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) :=
  forwards I: E₀ A₁ A₂ A₃; auto_star.
Tactic Notation "forwards" "*" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  forwards I: E₀ A₁ A₂ A₃ A₄; auto_star.
Tactic Notation "forwards" "*" simple_intropattern(I) ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  forwards I: E₀ A₁ A₂ A₃ A₄ A₅; auto_star.

Tactic Notation "forwards" "*" ":" constr(E) :=
  forwards: E; auto_star.
Tactic Notation "forwards" "*" ":" constr(E₀)
constr(A₁) :=
  forwards: E₀ A₁; auto_star.
Tactic Notation "forwards" "*" ":" constr(E₀)
constr(A₁) constr(A₂) :=
  forwards: E₀ A₁ A₂; auto_star.
Tactic Notation "forwards" "*" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) :=
  forwards: E₀ A₁ A₂ A₃; auto_star.
Tactic Notation "forwards" "*" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  forwards: E₀ A₁ A₂ A₃ A₄; auto_star.
Tactic Notation "forwards" "*" ":" constr(E₀)
constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  forwards: E₀ A₁ A₂ A₃ A₄ A₅; auto_star.

Tactic Notation "applys" "*" constr(H) :=
  sapply H; auto_star. (*todo?*)
Tactic Notation "applys" "*" constr(E₀) constr(A₁) :=
  applys E₀ A₁; auto_star.
Tactic Notation "applys" "*" constr(E₀) constr(A₁) :=
  applys E₀ A₁; auto_star.
Tactic Notation "applys" "*" constr(E₀) constr(A₁) constr(A₂) :=
  applys E₀ A₁ A₂; auto_star.
Tactic Notation "applys" "*" constr(E₀) constr(A₁) constr(A₂) constr(A₃) :=
  applys E₀ A₁ A₂ A₃; auto_star.
Tactic Notation "applys" "*" constr(E₀) constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  applys E₀ A₁ A₂ A₃ A₄; auto_star.
Tactic Notation "applys" "*" constr(E₀) constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  applys E₀ A₁ A₂ A₃ A₄ A₅; auto_star.

Tactic Notation "specializes" "*" hyp(H) :=
  specializes H; auto_star.
Tactic Notation "specializes" "~" hyp(H) constr(A₁) :=
  specializes H A₁; auto_star.
Tactic Notation "specializes" hyp(H) constr(A₁) constr(A₂) :=
  specializes H A₁ A₂; auto_star.
Tactic Notation "specializes" hyp(H) constr(A₁) constr(A₂) constr(A₃) :=
  specializes H A₁ A₂ A₃; auto_star.
Tactic Notation "specializes" hyp(H) constr(A₁) constr(A₂) constr(A₃) constr(A₄) :=
  specializes H A₁ A₂ A₃ A₄; auto_star.
Tactic Notation "specializes" hyp(H) constr(A₁) constr(A₂) constr(A₃) constr(A₄) constr(A₅) :=
  specializes H A₁ A₂ A₃ A₄ A₅; auto_star.

Tactic Notation "fapply" "*" constr(E) :=
fapply E; auto_star.
Tactic Notation "sapply" "*" constr(E) :=
sapply E; auto_star.

Tactic Notation "logic" constr(E) :=
logic_base E ltac:(fun _ ⇒ auto_star).

Tactic Notation "intros_all" "*" :=
intros_all; auto_star.

Tactic Notation "unfolds" "*" :=
  unfolds; auto_star.
Tactic Notation "unfolds" "*" constr(F₁) :=
  unfolds F₁; auto_star.
Tactic Notation "unfolds" "*" constr(F₁) "," constr(F₂) :=
  unfolds F₁, F₂; auto_star.
Tactic Notation "unfolds" "*" constr(F₁) "," constr(F₂) "," constr(F₃) :=
  unfolds F₁, F₂, F₃; auto_star.
Tactic Notation "unfolds" "*" constr(F₁) "," constr(F₂) "," constr(F₃) ","
constr(F₄) :=
  unfolds F₁, F₂, F₃, F₄; auto_star.

Tactic Notation "simple" "*" :=
  simpl; auto_star.
Tactic Notation "simple" "*" "in" hyp(H) :=
  simpl in H; auto_star.
Tactic Notation "simpls" "*" :=
  simpls; auto_star.
Tactic Notation "hnfs" "*" :=
  hnfs; auto_star.
Tactic Notation "hnfs" "*" "in" hyp(H) :=
  hnf in H; auto_star.
Tactic Notation "substs" "*" :=
  substs; auto_star.
Tactic Notation "intro_hyp" "*" hyp(H) :=
  subst_hyp H; auto_star.
Tactic Notation "intro_subst" "*" :=
  intro_subst; auto_star.
Tactic Notation "subst_eq" "*" constr(E) :=
  subst_eq E; auto_star.

Tactic Notation "rewrite" "*" constr(E) :=
  rewrite E; auto_star.
Tactic Notation "rewrite" "*" "<-" constr(E) :=
  rewrite <- E; auto_star.
Tactic Notation "rewrite" "*" constr(E) "in" hyp(H) :=
  rewrite E in H; auto_star.
Tactic Notation "rewrite" "*" "<-" constr(E) "in" hyp(H) :=
  rewrite <- E in H; auto_star.

Tactic Notation "rewrites" "*" constr(E) :=
  rewrites E; auto_star.
Tactic Notation "rewrites" "*" constr(E) "in" hyp(H):=
  rewrites E in H; auto_star.
Tactic Notation "rewrites" "*" constr(E) "in" "*":=
  rewrites E in *; auto_star.
Tactic Notation "rewrites" "*" "<-" constr(E) :=
  rewrites <- E; auto_star.
Tactic Notation "rewrites" "*" "<-" constr(E) "in" hyp(H):=
  rewrites <- E in H; auto_star.
Tactic Notation "rewrites" "*" "<-" constr(E) "in" "*":=
  rewrites <- E in *; auto_star.

Tactic Notation "rewrite_all" "*" constr(E) :=
  rewrite_all E; auto_star.
Tactic Notation "rewrite_all" "*" "<-" constr(E) :=
  rewrite_all <- E; auto_star.
Tactic Notation "rewrite_all" "*" constr(E) "in" ident(H) :=
  rewrite_all E in H; auto_star.
Tactic Notation "rewrite_all" "*" "<-" constr(E) "in" ident(H) :=
  rewrite_all <- E in H; auto_star.
Tactic Notation "rewrite_all" "*" constr(E) "in" "*" :=
  rewrite_all E in *; auto_star.
Tactic Notation "rewrite_all" "*" "<-" constr(E) "in" "*" :=
  rewrite_all <- E in *; auto_star.

Tactic Notation "asserts_rewrite" "*" constr(E) :=
  asserts_rewrite E; auto_star.
Tactic Notation "asserts_rewrite" "*" "<-" constr(E) :=
  asserts_rewrite <- E; auto_star.
Tactic Notation "asserts_rewrite" "*" constr(E) "in" hyp(H) :=
  asserts_rewrite E; auto_star.
Tactic Notation "asserts_rewrite" "*" "<-" constr(E) "in" hyp(H) :=
  asserts_rewrite <- E; auto_star.
Tactic Notation "asserts_rewrite" "*" constr(E) "in" "*" :=
  asserts_rewrite E in *; auto_tilde.
Tactic Notation "asserts_rewrite" "*" "<-" constr(E) "in" "*" :=
  asserts_rewrite <- E in *; auto_tilde.

Tactic Notation "cuts_rewrite" "*" constr(E) :=
  cuts_rewrite E; auto_star.
Tactic Notation "cuts_rewrite" "*" "<-" constr(E) :=
  cuts_rewrite <- E; auto_star.
Tactic Notation "cuts_rewrite" "*" constr(E) "in" hyp(H) :=
  cuts_rewrite E in H; auto_star.
Tactic Notation "cuts_rewrite" "*" "<-" constr(E) "in" hyp(H) :=
  cuts_rewrite <- E in H; auto_star.

Tactic Notation "erewrite" "*" constr(E) :=
erewrite E; auto_star.

Tactic Notation "fequal" "*" :=
  fequal; auto_star.
Tactic Notation "fequals" "*" :=
  fequals; auto_star.
Tactic Notation "pi_rewrite" "*" constr(E) :=
  pi_rewrite E; auto_star.
Tactic Notation "pi_rewrite" "*" constr(E) "in" hyp(H) :=
  pi_rewrite E in H; auto_star.

Tactic Notation "invert" "*" hyp(H) :=
  invert H; auto_star.
Tactic Notation "inverts" "*" hyp(H) :=
  inverts H; auto_star.
Tactic Notation "inverts" "*" hyp(E) "as" :=
  inverts E as; auto_star.
Tactic Notation "injects" "*" hyp(H) :=
  injects H; auto_star.
Tactic Notation "inversions" "*" hyp(H) :=
  inversions H; auto_star.

Tactic Notation "cases" "*" constr(E) "as" ident(H) :=
  cases E as H; auto_star.
Tactic Notation "cases" "*" constr(E) :=
  cases E; auto_star.
Tactic Notation "case_if" "*" :=
  case_if; auto_star.
Tactic Notation "case_ifs" "*" :=
  case_ifs; auto_star.
Tactic Notation "case_if" "*" "in" hyp(H) :=
  case_if in H; auto_star.
Tactic Notation "cases_if" "*" :=
  cases_if; auto_star.
Tactic Notation "cases_if" "*" "in" hyp(H) :=
  cases_if in H; auto_star.
Tactic Notation "destruct_if" "*" :=
  destruct_if; auto_star.
Tactic Notation "destruct_if" "*" "in" hyp(H) :=
  destruct_if in H; auto_star.
Tactic Notation "destruct_head_match" "*" :=
  destruct_head_match; auto_star.

Tactic Notation "cases'" "*" constr(E) "as" ident(H) :=
  cases' E as H; auto_star.
Tactic Notation "cases'" "*" constr(E) :=
  cases' E; auto_star.
Tactic Notation "cases_if'" "*" "as" ident(H) :=
  cases_if' as H; auto_star.
Tactic Notation "cases_if'" "*" :=
  cases_if'; auto_star.

Tactic Notation "decides_equality" "*" :=
decides_equality; auto_star.

Tactic Notation "iff" "*" :=
  iff; auto_star.
Tactic Notation "iff" "*" simple_intropattern(I) :=
  iff I; auto_star.
Tactic Notation "splits" "*" :=
  splits; auto_star.
Tactic Notation "splits" "*" constr(N) :=
  splits N; auto_star.

Tactic Notation "destructs" "*" constr(T) :=
destructs T; auto_star.
Tactic Notation "destructs" "*" constr(N) constr(T) :=
destructs N T; auto_star.

Tactic Notation "branch" "*" constr(N) :=
branch N; auto_star.
Tactic Notation "branch" "*" constr(K) "of" constr(N) :=
branch K of N; auto_star.

Tactic Notation "branches" "*" constr(T) :=
branches T; auto_star.
Tactic Notation "branches" "*" constr(N) constr(T) :=
branches N T; auto_star.

Tactic Notation "exists" "*" :=
  ∃; auto_star.
Tactic Notation "exists___" "*" :=
  exists___; auto_star.
Tactic Notation "exists" "*" constr(T₁) :=
  ∃ T₁; auto_star.
Tactic Notation "exists" "*" constr(T₁) constr(T₂) :=
  ∃ T₁ T₂; auto_star.
Tactic Notation "exists" "*" constr(T₁) constr(T₂) constr(T₃) :=
  ∃ T₁ T₂ T₃; auto_star.
Tactic Notation "exists" "*" constr(T₁) constr(T₂) constr(T₃) constr(T₄) :=
  ∃ T₁ T₂ T₃ T₄; auto_star.
Tactic Notation "exists" "*" constr(T₁) constr(T₂) constr(T₃) constr(T₄)
constr(T₅) :=
  ∃ T₁ T₂ T₃ T₄ T₅; auto_star.
Tactic Notation "exists" "*" constr(T₁) constr(T₂) constr(T₃) constr(T₄)
constr(T₅) constr(T₆) :=
  ∃ T₁ T₂ T₃ T₄ T₅ T₆; auto_star.

Tactic Notation "exists" "*" constr(T₁) "," constr(T₂) :=
  ∃ T₁ T₂; auto_star.
Tactic Notation "exists" "*" constr(T₁) "," constr(T₂) "," constr(T₃) :=
  ∃ T₁ T₂ T₃; auto_star.
Tactic Notation "exists" "*" constr(T₁) "," constr(T₂) "," constr(T₃) ","
  constr(T₄) :=
  ∃ T₁ T₂ T₃ T₄; auto_star.
Tactic Notation "exists" "*" constr(T₁) "," constr(T₂) "," constr(T₃) ","
constr(T₄) "," constr(T₅) :=
  ∃ T₁ T₂ T₃ T₄ T₅; auto_star.
Tactic Notation "exists" "*" constr(T₁) "," constr(T₂) "," constr(T₃) ","
constr(T₄) "," constr(T₅) "," constr(T₆) :=
  ∃ T₁ T₂ T₃ T₄ T₅ T₆; auto_star.

Tactics to Sort Out the Proof Context

Hiding Hypotheses

(* Implementation *)

Definition ltac_something (P:Type) (e:P) := e.

Notation "'Something'" :=
(@ltac_something _ _).

Lemma ltac_something_eq : ∀ (e:Type),
e = (@ltac_something _ e ).
Proof using. auto. Qed.

Lemma ltac_something_hide : ∀ (e:Type),
e → (@ltac_something _ e ).
Proof using. auto. Qed.

Lemma ltac_something_show : ∀ (e:Type),
(@ltac_something _ e ) → e.
Proof using. auto. Qed.

hide_def x and show_def x can be used to hide/show the body of the definition x.

Tactic Notation "hide_def" hyp(x) :=
  let x' := constr:(x) in
  let T := eval unfold x in x' in
  change T with (@ltac_something _ T) in x.

Tactic Notation "show_def" hyp(x) :=
  let x' := constr:(x) in
  let U := eval unfold x in x' in
  match U with @ltac_something _ ?T ⇒
    change U with T in x end.

show_def unfolds Something in the goal

Tactic Notation "show_def" :=
  unfold ltac_something.
Tactic Notation "show_def" "in" hyp(H) :=
  unfold ltac_something in H.
Tactic Notation "show_def" "in" "*" :=
  unfold ltac_something in ×.

hide_defs and show_defs applies to all definitions

Tactic Notation "hide_defs" :=
  repeat match goal with H := ?T ⊢ _ ⇒
    match T with
    | @ltac_something _ _ ⇒ fail 1
    | _ ⇒ change T with (@ltac_something _ T) in H
    end
  end.

Tactic Notation "show_defs" :=
repeat match goal with H := (@ltac_something _ ?T) ⊢ _ ⇒
change (@ltac_something _ T) with T in H end.

hide_hyp H replaces the type of H with the notation Something and show_hyp H reveals the type of the hypothesis. Note that the hidden type of H remains convertible the real type of H.

Tactic Notation "show_hyp" hyp(H) :=
apply ltac_something_show in H.

Tactic Notation "hide_hyp" hyp(H) :=
apply ltac_something_hide in H.

hide_hyps and show_hyps can be used to hide/show all hypotheses of type Prop.

Tactic Notation "show_hyps" :=
repeat match goal with
H: @ltac_something _ _ ⊢ _ ⇒ show_hyp H end.

Tactic Notation "hide_hyps" :=
  repeat match goal with H: ?T ⊢ _ ⇒
    match type of T with
    | Prop ⇒
      match T with
      | @ltac_something _ _ ⇒ fail 2
      | _ ⇒ hide_hyp H
      end
    | _ ⇒ fail 1
    end
  end.

hide H and show H automatically select between hide_hyp or hide_def, and show_hyp or show_def. Similarly hide_all and show_all apply to all.

Tactic Notation "hide" hyp(H) :=
first [hide_def H | hide_hyp H].

Tactic Notation "show" hyp(H) :=
first [show_def H | show_hyp H].

Tactic Notation "hide_all" :=
hide_hyps; hide_defs.

Tactic Notation "show_all" :=
unfold ltac_something in ×.

hide_term E can be used to hide a term from the goal. show_term or show_term E can be used to reveal it. hide_term E in H can be used to specify an hypothesis.

Tactic Notation "hide_term" constr(E) :=
  change E with (@ltac_something _ E).
Tactic Notation "show_term" constr(E) :=
  change (@ltac_something _ E) with E.
Tactic Notation "show_term" :=
  unfold ltac_something.

Tactic Notation "hide_term" constr(E) "in" hyp(H) :=
  change E with (@ltac_something _ E) in H.
Tactic Notation "show_term" constr(E) "in" hyp(H) :=
  change (@ltac_something _ E) with E in H.
Tactic Notation "show_term" "in" hyp(H) :=
  unfold ltac_something in H.

show_unfold R unfolds the definition of R and reveals the hidden definition of R. --todo:test, and implement using unfold simply

(* --TODO: change "unfolds" *)

Tactic Notation "show_unfold" constr(R₁) :=
unfold R₁; show_def.
Tactic Notation "show_unfold" constr(R₁) "," constr(R₂) :=
unfold R₁, R₂; show_def.

Sorting Hypotheses

sort sorts out hypotheses from the context by moving all the propositions (hypotheses of type Prop) to the bottom of the context.

Ltac sort_tactic :=
  try match goal with H: ?T ⊢ _ ⇒
  match type of T with Prop ⇒
    generalizes H; (try sort_tactic); intro
  end end.

Tactic Notation "sort" :=
sort_tactic.

Clearing Hypotheses

clears X₁ ... XN is a variation on clear which clears the variables X₁..XN as well as all the hypotheses which depend on them. Contrary to clear, it never fails.

Tactic Notation "clears" ident(X₁) :=
  let rec doit _ :=
  match goal with
  | H:context[X₁] ⊢ _ ⇒ clear H; try (doit tt)
  | _ ⇒ clear X₁
  end in doit tt.
Tactic Notation "clears" ident(X₁) ident(X₂) :=
  clears X₁; clears X₂.
Tactic Notation "clears" ident(X₁) ident(X₂) ident(X₃) :=
  clears X₁; clears X₂; clears X₃.
Tactic Notation "clears" ident(X₁) ident(X₂) ident(X₃) ident(X₄) :=
  clears X₁; clears X₂; clears X₃; clears X₄.
Tactic Notation "clears" ident(X₁) ident(X₂) ident(X₃) ident(X₄)
ident(X₅) :=
  clears X₁; clears X₂; clears X₃; clears X₄; clears X₅.
Tactic Notation "clears" ident(X₁) ident(X₂) ident(X₃) ident(X₄)
ident(X₅) ident(X₆) :=
  clears X₁; clears X₂; clears X₃; clears X₄; clears X₅; clears X₆.

clears (without any argument) clears all the unused variables from the context. In other words, it removes any variable which is not a proposition (i.e. not of type Prop) and which does not appear in another hypothesis nor in the goal.

(* --TODO: rename to clears_var ? *)

Ltac clears_tactic :=
  match goal with H: ?T ⊢ _ ⇒
  match type of T with
  | Prop ⇒ generalizes H; (try clears_tactic); intro
  | ?TT ⇒ clear H; (try clears_tactic)
  | ?TT ⇒ generalizes H; (try clears_tactic); intro
  end end.

Tactic Notation "clears" :=
clears_tactic.

clears_all clears all the hypotheses from the context that can be cleared. It leaves only the hypotheses that are mentioned in the goal.

Ltac clears_or_generalizes_all_core :=
repeat match goal with H: _ ⊢ _ ⇒
first [ clear H | generalizes H] end.

Tactic Notation "clears_all" :=
  generalize ltac_mark;
  clears_or_generalizes_all_core;
  intro_until_mark.

clears_but H₁ H₂ .. HN clears all hypotheses except the one that are mentioned and those that cannot be cleared.

Ltac clears_but_core cont :=
  generalize ltac_mark;
  cont tt;
  clears_or_generalizes_all_core;
  intro_until_mark.

Tactic Notation "clears_but" :=
  clears_but_core ltac:(fun _ ⇒ idtac).
Tactic Notation "clears_but" ident(H₁) :=
  clears_but_core ltac:(fun _ ⇒ gen H₁).
Tactic Notation "clears_but" ident(H₁) ident(H₂) :=
  clears_but_core ltac:(fun _ ⇒ gen H₁ H₂).
Tactic Notation "clears_but" ident(H₁) ident(H₂) ident(H₃) :=
  clears_but_core ltac:(fun _ ⇒ gen H₁ H₂ H₃).
Tactic Notation "clears_but" ident(H₁) ident(H₂) ident(H₃) ident(H₄) :=
  clears_but_core ltac:(fun _ ⇒ gen H₁ H₂ H₃ H₄).
Tactic Notation "clears_but" ident(H₁) ident(H₂) ident(H₃) ident(H₄) ident(H₅) :=
  clears_but_core ltac:(fun _ ⇒ gen H₁ H₂ H₃ H₄ H₅).

Lemma demo_clears_all_and_clears_but :
  ∀ x y:nat, y < 2 → x = x → x ≥ 2 → x < 3 → True.
Proof using.
  introv M₁ M₂ M₃. dup 6.
  (* clears_all clears all hypotheses. *)
  clears_all. auto.
  (* clears_but H clears all but H *)
  clears_but M₃. auto.
  clears_but y. auto.
  clears_but x. auto.
  clears_but M₂ M₃. auto.
  clears_but x y. auto.
Qed.

clears_last clears the last hypothesis in the context. clears_last N clears the last N hypotheses in the context.

Tactic Notation "clears_last" :=
match goal with H: ?T ⊢ _ ⇒ clear H end.

Ltac clears_last_base N :=
  match number_to_nat N with
  | 0 ⇒ idtac
  | S ?p ⇒ clears_last; clears_last_base p
  end.

Tactic Notation "clears_last" constr(N) :=
clears_last_base N.

Tactics for Development Purposes

Skipping Subgoals

(* SF DOES NOT NEED an alternative implementation of the skip tactic *)

Tactic Notation "skip" :=
admit.

demo is like admit but it documents the fact that admit is intended

Tactic Notation "demo" :=
skip.

admits H: T adds an assumption named H of type T to the current context, blindly assuming that it is true. admit: T is another possible syntax. Note that H may be an intro pattern.

Tactic Notation "admits" simple_intropattern(I) ":" constr(T) :=
  asserts I: T; [ skip | ].
Tactic Notation "admits" ":" constr(T) :=
  let H := fresh "TEMP" in admits H: T.
Tactic Notation "admits" "~" ":" constr(T) :=
  admits: T; auto_tilde.
Tactic Notation "admits" "*" ":" constr(T) :=
  admits: T; auto_star.

admit_cuts T simply replaces the current goal with T.

Tactic Notation "admit_cuts" constr(T) :=
cuts: T; [ skip | ].

admit_goal H applies to any goal. It simply assumes the current goal to be true. The assumption is named "H". It is useful to set up proof by induction or coinduction. Syntax admit_goal is also accepted.

Tactic Notation "admit_goal" ident(H) :=
match goal with ⊢ ?G ⇒ admits H: G end.

Tactic Notation "admit_goal" :=
let IH := fresh "IH" in admit_goal IH.

admit_rewrite T can be applied when T is an equality. It blindly assumes this equality to be true, and rewrite it in the goal.

Tactic Notation "admit_rewrite" constr(T) :=
let M := fresh "TEMP" in admits M: T; rewrite M; clear M.

admit_rewrite T in H is similar as admit_rewrite, except that it rewrites in hypothesis H.

Tactic Notation "admit_rewrite" constr(T) "in" hyp(H) :=
let M := fresh "TEMP" in admits M: T; rewrite M in H; clear M.

admit_rewrites_all T is similar as admit_rewrite, except that it rewrites everywhere (goal and all hypotheses).

Tactic Notation "admit_rewrite_all" constr(T) :=
let M := fresh "TEMP" in admits M: T; rewrite_all M; clear M.

forwards_nounfold_admit_sides_then E ltac:(fun K ⇒ ..) is like forwards: E but it provides the resulting term to a continuation, under the name K, and it admits any side-condition produced by the instantiation of E, using the skip tactic.

Inductive ltac_goal_to_discard := ltac_goal_to_discard_intro.

Ltac forwards_nounfold_admit_sides_then S cont :=
  let MARK := fresh "TEMP" in
  generalize ltac_goal_to_discard_intro;
  intro MARK;
  forwards_nounfold_then S ltac:(fun K ⇒
    clear MARK;
    cont K);
  match goal with
  | MARK: ltac_goal_to_discard ⊢ _ ⇒ skip
  | _ ⇒ idtac
  end.

(* ********************************************************************** *)

Compatibility with standard library

The module Program contains definitions that conflict with the current module. If you import Program, either directly or indirectly (e.g., through Setoid or ZArith), you will need to import the compability definitions through the top-level command: Import LibTacticsCompatibility.

Module LibTacticsCompatibility.
  Tactic Notation "apply" "*" constr(H) :=
    sapply H; auto_star.
  Tactic Notation "subst" "*" :=
    subst; auto_star.
End LibTacticsCompatibility.

Open Scope nat_scope.

(* 2022-03-14 05:28:25 (UTC+00) *)