This Library provides several coq tactics and tacticals to deal with hypothesis during a proof.

Main page and documentation: https://github.com/Matafou/LibHyps

Demo file demo.v acts as a documentation.

Short description:

LibHyps provides utilities for hypothesis manipulations. In particular a new tactic especialize H and a set of tacticals to appy or iterate tactics either on all hypothesis of a goal or on "new' hypothesis after a tactic. It also provide syntax for a few predefined such iterators.

QUICK REF: especialize (BROKEN IN COQ-8.18)

This tactic is currently broken in coq v8.18. I am working on it. This may need some work on coq side.

• especialize H at n [as h]. Creates a subgoal to prove the nth dependent premise of H, creating necessary evars for non unifiable variables. Once proved the subgoal is used to remove the nth premise of H (or of a new created hypothesis if the as option is given).

• especialize H at * [as h]. Creates one subgoal for each dependent premise of H, creating necessary evars for non unifiable variables. Once proved the subgoal is used to remove the premises of H (or of a new createdd hypothesis if the as option is given).

• especialize H until n [as h]. Creates one subgoal for each n first dependent premises of H, creating necessary evars for non unifiable variables. Once proved the subgoal is used to remove the premises of H (or of a new created hypothesis if the as option is given).

QUICK REF: Pre-defined tacticals /s /n...

The most useful user-dedicated tacticals are the following

• tac /s try to apply subst on each new hyp.
• tac /r revert each new hyp.
• tac /n auto-rename each new hyp.
• tac /g group all non-Prop new hyp at the top of the goal.
• combine the above, as in tac /s/n/g.
• usual combinations have shortcuts: \sng, \sn,\ng,\sg...

Install

Quick install using opam

If you have not done it already add the coq platform repository to opam!

opam repo add coq-released https://coq.inria.fr/opam/released

and then:

opam install coq-libhyps

Quick install using github:

Clone the github repository:

git clone https://github.com/Matafou/LibHyps

then compile:

configure.sh
make
make install

Quick test:

Require Import LibHyps.LibHyps.

Demo files demo.v.

More information

Deprecation from 1.0.x to 2.0.x

• "!tac", "!!tac" etc are now only loaded if you do: Import LibHyps.LegacyNotations., the composable tacticals described above are preferred.
• "tac1 ;; tac2" remains, but you can also use "tac1; { tac2 }".
• "tac1 ;!; tac2" remains, but you can also use "tac1; {< tac2 }".

KNOWN BUGS

Due to Ltac limitation, it is difficult to define a tactic notation tac1 ; { tac2 } which delays tac1 and tac2 in all cases. Sometimes (rarely) you will have to write (idtac; tac1); {idtac; tac2}. You may then use tactic notation like: Tactic Notation tac1' := idtac; tac1..

Examples

Require Import LibHyps.LibHyps.

Lemma foo: forall x y z:nat,
    x = y -> forall  a b t : nat, a+1 = t+2 -> b + 5 = t - 7 ->  (forall u v, v+1 = 1 -> u+1 = 1 -> a+1 = z+2)  -> z = b + x-> True.
Proof.
  intros.
  (* ugly names *)
  Undo.
  (* Example of using the iterator on new hyps: this prints each new hyp name. *)
  intros; {fun h => idtac h}.
  Undo.
  (* This gives sensible names to each new hyp. *)
  intros ; { autorename }.
  Undo.
  (* short syntax: *)
  intros /n.
  Undo.
  (* same thing but use subst if possible, and group non prop hyps to the top. *)
  intros ; { substHyp }; { autorename}; {move_up_types}.
  Undo.
  (* short syntax: *)  
  intros /s/n/g.
  Undo.
  (* Even shorter: *)  
  intros /s/n/g.

  (* Let us instantiate the 2nd premis of h_all_eq_add_add without copying its type: *)
  (* BROKEN IN COQ 8.18*)
  (* especialize h_all_eq_add_add_ at 2.
  { apply Nat.add_0_l. }
  (* now h_all_eq_add_add is specialized *)
  Undo 6. *)
  Undo 2.
  intros until 1.
  (** The taticals apply after any tactic. Notice how H:x=y is not new
    and hence not substituted, whereas z = b + x is. *)
  destruct x eqn:heq;intros /sng.
  - apply I.
  - apply I.
Qed.

Short Documentation

The following explains how it works under the hood, for people willing to apply more generic iterators to their own tactics. See also the code.

Iterator on all hypothesis

• onAllHyps tac does tac H for each hypothesis H of the current goal.
• onAllHypsRev tac same as onAllHyps tac but in reverse order (good for reverting for instance).

Iterators on ALL NEW hypothesis (since LibHyps-1.2.0)

• tac1 ;{! tac2 } applies tac1 to current goal and then tac2 to the list of all new hypothesis in each subgoal (iteration: oldest first). The list is a term of type LibHyps.TacNewHyps.DList. See the code.
• tac1 ;{!< tac2 } is similar but the list of new hyps is reveresed.

Iterators on EACH NEW hypothesis

• tac1 ;{ tac2 } applies tac1 to current goal and then tac2 to each new hypothesis in each subgoal (iteration: older first).

• tac1 ;{< tac2 } is similar but applies tac2 on newer hyps first.

• tac1 ;; tac2 is a synonym of tac1; { tac2 }.

• tac1 ;!; tac2 is a synonym of tac1; {< tac2 }.

Customizable hypothesis auto naming system

Using previous taticals (in particular the ;!; tactical), some tactic allow to rename hypothesis automatically.

• autorename H rename H according to the current naming scheme (which is customizable, see below).

• rename_all_hyps applies autorename to all hypothesis.

• !tac applies tactic tac and then applies autorename to each new hypothesis. Shortcut for: (Tac ;!; revert_if_norename ;; autorename)..`

• !!tac same as !tac with lesser priority (less than ;) to apply renaming after a group of chained tactics.

How to cstomize the naming scheme

The naming engine analyzes the type of hypothesis and generates a name mimicking the first levels of term structure. At each level the customizable tactic rename_hyp is called. One can redefine it at will. It must be of the following form:

(** Redefining rename_hyp*)
(* First define a naming ltac. It takes the current level n and
   the sub-term th being looked at. It returns a "name". *)
Ltac rename_hyp_default n th :=
   match th with
   | (ind1 _ _) => name (`ind1`)
   | (ind1 _ _ ?x ?y) => name (`ind1` ++ x#(S n)x ++ y$n)
   | f1 _ ?x = ?y => name (`f1` ++ x#n ++ y#n)
   | _ => previously_defined_renaming_tac1 n th (* cumulative with previously defined renaming tactics *)
   | _ => previously_defined_renaming_tac2 n th
   end.

(* Then overwrite the definition of rename_hyp using the ::= operator. :*)
Ltac rename_hyp ::= my_rename_hyp.

Where:

• `id` to use the name id itself
• t$n to recursively call the naming engine on term t, n being the maximum depth allowed
• name ++ name to concatenate name parts.

How to define variants of these tacticals?

Some more example of tacticals performing cleaning and renaming on new hypothesis.

(* subst or revert *)
Tactic Notation (at level 4) "??" tactic4(tac1) :=
  (tac1 ;; substHyp ;!; revertHyp).
(* subst or rename or revert *)
Tactic Notation "!!!" tactic3(Tac) := (Tac ;; substHyp ;!; revert_if_norename ;; autorename).
(* subst or rename or revert + move up if in (Set or Type). *)
Tactic Notation (at level 4) "!!!!" tactic4(Tac) :=
      (Tac ;; substHyp ;!; revert_if_norename ;; autorename ;; move_up_types).