Library Coq.Init.Tactics


Require Import Notations.
Require Import Ltac.
Require Import Logic.
Require Import Specif.

Useful tactics

Ex falso quodlibet : a tactic for proving False instead of the current goal. This is just a nicer name for tactics such as cut False.

Ltac exfalso := Coq.Init.Ltac.exfalso.

A tactic for proof by contradiction. With contradict H,
  • H:~A |- B gives |- A
  • H:~A |- ~B gives H: B |- A
  • H: A |- B gives |- ~A
  • H: A |- ~B gives H: B |- ~A
  • H:False leads to a resolved subgoal.
Moreover, negations may be in unfolded forms, and A or B may live in Type

Ltac contradict H :=
  let save tac H := let x:=fresh in intro x; tac H; rename x into H
  in
  let negpos H := case H; clear H
  in
  let negneg H := save negpos H
  in
  let pospos H :=
    let A := type of H in (exfalso; revert H; try fold (~A))
  in
  let posneg H := save pospos H
  in
  let neg H := match goal with
   | |- (~_) => negneg H
   | |- (_->False) => negneg H
   | |- _ => negpos H
  end in
  let pos H := match goal with
   | |- (~_) => posneg H
   | |- (_->False) => posneg H
   | |- _ => pospos H
  end in
  match type of H with
   | (~_) => neg H
   | (_->False) => neg H
   | _ => (elim H;fail) || pos H
  end.


Ltac false_hyp H G :=
  let T := type of H in absurd T; [ apply G | assumption ].


Ltac case_eq x := generalize (eq_refl x); pattern x at -1; case x.


Ltac destr_eq H := discriminate H || (try (injection H as [= H])).


Tactic Notation "destruct_with_eqn" constr(x) :=
  destruct x eqn:?.
Tactic Notation "destruct_with_eqn" ident(n) :=
  try intros until n; destruct n eqn:?.
Tactic Notation "destruct_with_eqn" ":" ident(H) constr(x) :=
  destruct x eqn:H.
Tactic Notation "destruct_with_eqn" ":" ident(H) ident(n) :=
  try intros until n; destruct n eqn:H.

Break every hypothesis of a certain type

Ltac destruct_all t :=
 match goal with
  | x : t |- _ => destruct x; destruct_all t
  | _ => idtac
 end.


Tactic Notation "rewrite_all" constr(eq) := repeat rewrite eq in *.
Tactic Notation "rewrite_all" "<-" constr(eq) := repeat rewrite <- eq in *.

Tactics for applying equivalences.
The following code provides tactics "apply -> t", "apply <- t", "apply -> t in H" and "apply <- t in H". Here t is a term whose type consists of nested dependent and nondependent products with an equivalence A <-> B as the conclusion. The tactics with "->" in their names apply A -> B while those with "<-" in the name apply B -> A.


Ltac find_equiv H :=
let T := type of H in
lazymatch T with
| ?A -> ?B =>
  let H1 := fresh in
  let H2 := fresh in
  cut A;
  [intro H1; pose proof (H H1) as H2; clear H H1;
   rename H2 into H; find_equiv H |
   clear H]
| forall x : ?t, _ =>
  let a := fresh "a" in
  let H1 := fresh "H" in
    evar (a : t); pose proof (H a) as H1; unfold a in H1;
    clear a; clear H; rename H1 into H; find_equiv H
| ?A <-> ?B => idtac
| _ => fail "The given statement does not seem to end with an equivalence."
end.

Ltac bapply lemma todo :=
let H := fresh in
  pose proof lemma as H;
  find_equiv H; [todo H; clear H | .. ].

Tactic Notation "apply" "->" constr(lemma) :=
bapply lemma ltac:(fun H => destruct H as [H _]; apply H).

Tactic Notation "apply" "<-" constr(lemma) :=
bapply lemma ltac:(fun H => destruct H as [_ H]; apply H).

Tactic Notation "apply" "->" constr(lemma) "in" hyp(J) :=
bapply lemma ltac:(fun H => destruct H as [H _]; apply H in J).

Tactic Notation "apply" "<-" constr(lemma) "in" hyp(J) :=
bapply lemma ltac:(fun H => destruct H as [_ H]; apply H in J).

An experimental tactic simpler than auto that is useful for ending proofs "in one step"

Ltac easy :=
  let rec use_hyp H :=
    match type of H with
    | _ /\ _ => exact H || destruct_hyp H
    | _ => try solve [inversion H]
    end
  with do_intro := let H := fresh in intro H; use_hyp H
  with destruct_hyp H := case H; clear H; do_intro; do_intro in
  let rec use_hyps :=
    match goal with
    | H : _ /\ _ |- _ => exact H || (destruct_hyp H; use_hyps)
    | H : _ |- _ => solve [inversion H]
    | _ => idtac
    end in
  let do_atom :=
    solve [ trivial with eq_true | reflexivity | symmetry; trivial | contradiction ] in
  let rec do_ccl :=
    try do_atom;
    repeat (do_intro; try do_atom);
    solve [ split; do_ccl ] in
  solve [ do_atom | use_hyps; do_ccl ] ||
  fail "Cannot solve this goal".

Tactic Notation "now" tactic(t) := t; easy.

Slightly more than easy

Ltac easy' := repeat split; simpl; easy || now destruct 1.

A tactic to document or check what is proved at some point of a script

Ltac now_show c := change c.

Support for rewriting decidability statements

Set Implicit Arguments.

Lemma decide_left : forall (C:Prop) (decide:{C}+{~C}),
  C -> forall P:{C}+{~C}->Prop, (forall H:C, P (left _ H)) -> P decide.

Lemma decide_right : forall (C:Prop) (decide:{C}+{~C}),
  ~C -> forall P:{C}+{~C}->Prop, (forall H:~C, P (right _ H)) -> P decide.

Tactic Notation "decide" constr(lemma) "with" constr(H) :=
  let try_to_merge_hyps H :=
     try (clear H; intro H) ||
     (let H' := fresh H "bis" in intro H'; try clear H') ||
     (let H' := fresh in intro H'; try clear H') in
  match type of H with
  | ~ ?C => apply (decide_right lemma H); try_to_merge_hyps H
  | ?C -> False => apply (decide_right lemma H); try_to_merge_hyps H
  | _ => apply (decide_left lemma H); try_to_merge_hyps H
  end.

Clear an hypothesis and its dependencies

Tactic Notation "clear" "dependent" hyp(h) :=
 let rec depclear h :=
  clear h ||
  lazymatch goal with
   | H : context [ h ] |- _ => depclear H; depclear h
   | H := context [ h ] |- _ => depclear H; depclear h
  end ||
  fail "hypothesis to clear is used in the conclusion (maybe indirectly)"
 in depclear h.

Revert an hypothesis and its dependencies : this is actually generalize dependent...

Tactic Notation "revert" "dependent" hyp(h) :=
 generalize dependent h.

Provide an error message for dependent induction/dependent destruction that reports an import is required to use it. Importing Coq.Program.Equality will shadow this notation with the actual tactics.

Tactic Notation "dependent" "induction" ident(H) :=
  fail "To use dependent induction, first [Require Import Coq.Program.Equality.]".

Tactic Notation "dependent" "destruction" ident(H) :=
  fail "To use dependent destruction, first [Require Import Coq.Program.Equality.]".

inversion_sigma

The built-in inversion will frequently leave equalities of dependent pairs. When the first type in the pair is an hProp or otherwise simplifies, inversion_sigma is useful; it will replace the equality of pairs with a pair of equalities, one involving a term casted along the other. This might also prove useful for writing a version of inversion / dependent destruction which does not lose information, i.e., does not turn a goal which is provable into one which requires axiom K / UIP.

Ltac lookup_inversion_sigma_rect H :=
  lazymatch type of H with
  | ex_intro _ _ _ = ex_intro _ _ _
    => uconstr:(eq_ex_rect_ex_intro)
  | exist _ _ _ = exist _ _ _
    => uconstr:(eq_sig_rect_exist)
  | existT _ _ _ = existT _ _ _
    => uconstr:(eq_sigT_rect_existT)
  | _ = ex_intro _ _ _
    => uconstr:(eq_ex_rect_ex_intro_r)
  | _ = exist _ _ _
    => uconstr:(eq_sig_rect_exist_r)
  | _ = existT _ _ _
    => uconstr:(eq_sigT_rect_existT_r)
  | ex_intro _ _ _ = _
    => uconstr:(eq_ex_rect_ex_intro_l)
  | exist _ _ _ = _
    => uconstr:(eq_sig_rect_exist_l)
  | existT _ _ _ = _
    => uconstr:(eq_sigT_rect_existT_l)
  | ex_intro2 _ _ _ _ _ = ex_intro2 _ _ _ _ _
    => uconstr:(eq_ex2_rect_ex_intro2)
  | exist2 _ _ _ _ _ = exist2 _ _ _ _ _
    => uconstr:(eq_sig2_rect_exist2)
  | existT2 _ _ _ _ _ = existT2 _ _ _ _ _
    => uconstr:(eq_sigT2_rect_existT2)
  | _ = ex_intro2 _ _ _ _ _
    => uconstr:(eq_ex2_rect_ex_intro2_r)
  | _ = exist2 _ _ _ _ _
    => uconstr:(eq_sig2_rect_exist2_r)
  | _ = existT2 _ _ _ _ _
    => uconstr:(eq_sigT2_rect_existT2_r)
  | ex_intro2 _ _ _ _ _ = _
    => uconstr:(eq_ex2_rect_ex_intro2_l)
  | exist2 _ _ _ _ _ = _
    => uconstr:(eq_sig2_rect_exist2_l)
  | existT2 _ _ _ _ _ = _
    => uconstr:(eq_sigT2_rect_existT2_l)
  | _ = _ :> ?T
    => let sig := uconstr:(@sig) in
       let sig2 := uconstr:(@sig2) in
       let sigT := uconstr:(@sigT) in
       let sigT2 := uconstr:(@sigT2) in
       let ex := uconstr:(@ex) in
       let ex2 := uconstr:(@ex2) in
       fail 0 "Type" "of" H "is" "not" "an" "equality" "of" "recognized" "Σ" "types:" "expected" "one" "of" sig sig2 sigT sigT2 ex "or" ex2 "but" "got" T
  | _
    => fail 0 H "is" "not" "an" "equality" "of" "Σ" "types"
  end.
Ltac inversion_sigma_on_as H ip :=
  let rect := lookup_inversion_sigma_rect H in
  induction H as ip using rect.
Ltac inversion_sigma_on H := inversion_sigma_on_as H ipattern:([]).
Ltac inversion_sigma_step :=
  match goal with
  | [ H : _ |- _ ] => inversion_sigma_on H
  end.
Ltac inversion_sigma := repeat inversion_sigma_step.

Tactic Notation "inversion_sigma" := inversion_sigma.
Tactic Notation "inversion_sigma" hyp(H) := inversion_sigma_on H.
Tactic Notation "inversion_sigma" hyp(H) "as" simple_intropattern(ip) := inversion_sigma_on_as H ip.

A version of time that works for constrs

Ltac time_constr tac :=
  let eval_early := match goal with _ => restart_timer end in
  let ret := tac () in
  let eval_early := match goal with _ => finish_timing ( "Tactic evaluation" ) end in
  ret.

Useful combinators

Ltac assert_fails tac :=
  tryif (once tac) then gfail 0 tac "succeeds" else idtac.
Tactic Notation "assert_fails" tactic3(tac) :=
  assert_fails tac.

Create HintDb rewrite discriminated.
#[global]
Hint Variables Opaque : rewrite.
Create HintDb typeclass_instances discriminated.