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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Permut.v 9245 2006-10-17 12:53:34Z notin $ i*)
(** Tactics related to (dependent) equality and proof irrelevance. *)
Require Export ProofIrrelevance.
Require Export JMeq.
(** Notation for heterogenous equality. *)
Notation " [ x : X ] = [ y : Y ] " := (@JMeq X x Y y) (at level 0, X at next level, Y at next level).
(** Do something on an heterogeneous equality appearing in the context. *)
Ltac on_JMeq tac :=
match goal with
| [ H : @JMeq ?x ?X ?y ?Y |- _ ] => tac H
end.
(** Try to apply [JMeq_eq] to get back a regular equality when the two types are equal. *)
Ltac simpl_one_JMeq :=
on_JMeq
ltac:(fun H => let H' := fresh "H" in
assert (H' := JMeq_eq H) ; clear H ; rename H' into H).
(** Repeat it for every possible hypothesis. *)
Ltac simpl_JMeq := repeat simpl_one_JMeq.
(** Just simplify an h.eq. without clearing it. *)
Ltac simpl_one_dep_JMeq :=
on_JMeq
ltac:(fun H => let H' := fresh "H" in
assert (H' := JMeq_eq H)).
Require Import Eqdep.
(** Tries to eliminate a call to [eq_rect] (the substitution principle) by any means available. *)
Ltac elim_eq_rect :=
match goal with
| [ |- ?t ] =>
match t with
| context [ @eq_rect _ _ _ _ _ ?p ] =>
let P := fresh "P" in
set (P := p); simpl in P ;
((case P ; clear P) || (clearbody P; rewrite (UIP_refl _ _ P); clear P))
| context [ @eq_rect _ _ _ _ _ ?p _ ] =>
let P := fresh "P" in
set (P := p); simpl in P ;
((case P ; clear P) || (clearbody P; rewrite (UIP_refl _ _ P); clear P))
end
end.
(** Rewrite using uniqueness of indentity proofs [H = refl_equal X]. *)
Ltac simpl_uip :=
match goal with
[ H : ?X = ?X |- _ ] => rewrite (UIP_refl _ _ H) in *; clear H
end.
(** Simplify equalities appearing in the context and goal. *)
Ltac simpl_eq := simpl ; repeat (elim_eq_rect ; simpl) ; repeat (simpl_uip ; simpl).
(** Try to abstract a proof of equality, if no proof of the same equality is present in the context. *)
Ltac abstract_eq_hyp H' p :=
let ty := type of p in
let tyred := eval simpl in ty in
match tyred with
?X = ?Y =>
match goal with
| [ H : X = Y |- _ ] => fail 1
| _ => set (H':=p) ; try (change p with H') ; clearbody H' ; simpl in H'
end
end.
(** Apply the tactic tac to proofs of equality appearing as coercion arguments.
Just redefine this tactic (using [Ltac on_coerce_proof tac ::=]) to handle custom coercion operators.
*)
Ltac on_coerce_proof tac T :=
match T with
| context [ eq_rect _ _ _ _ ?p ] => tac p
end.
Ltac on_coerce_proof_gl tac :=
match goal with
[ |- ?T ] => on_coerce_proof tac T
end.
(** Abstract proofs of equalities of coercions. *)
Ltac abstract_eq_proof := on_coerce_proof_gl ltac:(fun p => let H := fresh "eqH" in abstract_eq_hyp H p).
Ltac abstract_eq_proofs := repeat abstract_eq_proof.
(** Factorize proofs, by using proof irrelevance so that two proofs of the same equality
in the goal become convertible. *)
Ltac pi_eq_proof_hyp p :=
let ty := type of p in
let tyred := eval simpl in ty in
match tyred with
?X = ?Y =>
match goal with
| [ H : X = Y |- _ ] =>
match p with
| H => fail 2
| _ => rewrite (proof_irrelevance (X = Y) p H)
end
| _ => fail " No hypothesis with same type "
end
end.
(** Factorize proofs of equality appearing as coercion arguments. *)
Ltac pi_eq_proof := on_coerce_proof_gl pi_eq_proof_hyp.
Ltac pi_eq_proofs := repeat pi_eq_proof.
(** The two preceding tactics in sequence. *)
Ltac clear_eq_proofs :=
abstract_eq_proofs ; pi_eq_proofs.
Hint Rewrite <- eq_rect_eq : refl_id.
(** The refl_id database should be populated with lemmas of the form
[coerce_* t (refl_equal _) = t]. *)
Ltac rewrite_refl_id := autorewrite with refl_id.
(** Clear the context and goal of equality proofs. *)
Ltac clear_eq_ctx :=
rewrite_refl_id ; clear_eq_proofs.
(** Reapeated elimination of [eq_rect] applications.
Abstracting equalities makes it run much faster than an naive implementation. *)
Ltac simpl_eqs :=
repeat (elim_eq_rect ; simpl ; clear_eq_ctx).
(** Clear unused reflexivity proofs. *)
Ltac clear_refl_eq :=
match goal with [ H : ?X = ?X |- _ ] => clear H end.
Ltac clear_refl_eqs := repeat clear_refl_eq.
(** Clear unused equality proofs. *)
Ltac clear_eq :=
match goal with [ H : _ = _ |- _ ] => clear H end.
Ltac clear_eqs := repeat clear_eq.
(** Combine all the tactics to simplify goals containing coercions. *)
Ltac simplify_eqs :=
simpl ; simpl_eqs ; clear_eq_ctx ; clear_refl_eqs ;
try subst ; simpl ; repeat simpl_uip ; rewrite_refl_id.
(** A tactic to remove trivial equality guards in hypotheses. *)
Ltac simpl_IH_eq H :=
let tac H' := clear H ; rename H' into H in
let H' := fresh "H" in
match type of H with
| JMeq _ _ -> _ =>
assert (H' := H (JMeq_refl _)) ; tac H'
| _ = _ -> _ =>
assert (H' := H (refl_equal _)) ; tac H'
end.
Ltac simpl_IH_eqs H := repeat simpl_IH_eq H.
Ltac simpl_IHs_eqs :=
match goal with
| [ H : JMeq _ _ -> _ |- _ ] => simpl_IH_eqs H
| [ H : _ = _ -> _ |- _ ] => simpl_IH_eqs H
end.
Require Import Coq.Program.Tactics.
(** The following tactics allow to do induction on an already instantiated inductive predicate
by first generalizing it and adding the proper equalities to the context, in a maner similar to
the BasicElim tactic of "Elimination with a motive" by Conor McBride. *)
Tactic Notation "dependent" "induction" ident(H) :=
generalize_eqs H ; clear H ; (intros until 1 || intros until H) ;
induction H ; intros ; subst* ; try discriminates ; try simpl_IHs_eqs.
(** This tactic also generalizes the goal by the given variables before the induction. *)
Tactic Notation "dependent" "induction" ident(H) "generalizing" ne_hyp_list(l) :=
generalize_eqs H ; clear H ; (intros until 1 || intros until H) ;
generalize l ; clear l ; induction H ; intros ; subst* ; try discriminates ; try simpl_IHs_eqs.
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