The Coq Proof Assistant

# 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 elimtype False and other cut False.
Ltac exfalso := elimtype False.
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 absurd_hyp H :=
idtac "absurd_hyp is OBSOLETE: use contradict instead.";
let T := type of H in
absurd T.
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.