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Commit c94ad7e0 authored by Pierre Letouzey's avatar Pierre Letouzey
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No more Alpha.v, indirect proofs (via nam2mix) instead

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(** * Properties of rename and alpha-equivalence for named formulas *)
(** The NatDed development, Pierre Letouzey, 2019.
This file is released under the CC0 License, see the LICENSE file *)
Require Import Defs NameProofs Nam.
Import ListNotations.
Import Nam.Form.
Local Open Scope bool_scope.
Local Open Scope lazy_bool_scope.
Local Open Scope eqb_scope.
Implicit Types x y z v : variable.
Local Coercion Var : variable >-> term.
(** TERM SUBSTITUTION *)
Lemma term_subst_notin x u t :
~Names.In x (Term.vars t) ->
Term.subst x u t = t.
Proof.
induction t as [v|f a IH] using term_ind'; cbn.
- case eqbspec; auto. namedec.
- intros NI. f_equal. apply map_id_iff.
intros t Ht. apply IH; auto. eapply unionmap_notin; eauto.
Qed.
Lemma term_vars_subst x u t :
Names.Subset (Term.vars (Term.subst x u t))
(Names.union (Names.remove x (Term.vars t)) (Term.vars u)).
Proof.
induction t using term_ind'; cbn.
- case eqbspec; cbn; auto with set.
- intros v. rewrite unionmap_map_in.
intros (a & IN & IN'). apply H in IN; trivial.
revert IN.
nameiff. intros [(U,V)|W]; auto. left; split; eauto with set.
Qed.
Lemma term_vars_subst_in x u t :
Names.In x (Term.vars t) ->
Names.Equal (Term.vars (Term.subst x u t))
(Names.union (Names.remove x (Term.vars t)) (Term.vars u)).
Proof.
revert t.
fix IH 1; destruct t; cbn.
- case eqbspec; cbn; auto with set.
- revert l.
fix IH' 1; destruct l; cbn.
+ namedec.
+ destruct (NamesP.In_dec x (Term.vars t)) as [E|E];
destruct (NamesP.In_dec x (Names.unionmap Term.vars l)) as [E'|E'].
* intros _.
fold (Term.subst x u t). rewrite (IH t E).
rewrite (IH' l E'). namedec.
* intros _.
assert (E'' : map (Term.subst x u) l = l).
{ apply map_id_iff. intros a Ha.
apply term_subst_notin. eapply unionmap_notin; eauto. }
rewrite E''.
fold (Term.subst x u t). rewrite (IH t E).
namedec.
* intros _.
fold (Term.subst x u t).
rewrite (term_subst_notin x u t) by trivial.
rewrite (IH' l E'). namedec.
* nameiff. tauto.
Qed.
Lemma term_vars_subst_in' x u t :
Names.In x (Term.vars t) ->
Names.Subset (Term.vars u) (Term.vars (Term.subst x u t)).
Proof.
induction t using term_ind'; cbn.
- case eqbspec; cbn; auto with set.
- intros IN v Hv. rewrite unionmap_map_in.
rewrite unionmap_in in IN. destruct IN as (a & Ha & IN).
exists a; split; auto. apply H; auto.
Qed.
(** Combining several term substitutions *)
Lemma term_subst_subst x u x' u' t :
x<>x' -> ~Names.In x (Term.vars u') ->
Term.subst x' u' (Term.subst x u t) =
Term.subst x (Term.subst x' u' u) (Term.subst x' u' t).
Proof.
intros NE NI.
induction t using term_ind'; cbn.
- case eqbspec; [intros ->|intros NE'].
+ rewrite <- eqb_neq in NE. rewrite NE. cbn.
now rewrite eqb_refl.
+ case eqbspec; [intros ->|intros NE''].
* cbn. rewrite eqb_refl.
rewrite term_subst_notin; auto.
* cbn. rewrite <-eqb_neq in NE',NE''. now rewrite NE', NE''.
- f_equal.
rewrite !map_map.
apply map_ext_in.
auto.
Qed.
Lemma term_subst_chain x y u t :
~Names.In y (Term.vars t) ->
Term.subst y u (Term.subst x (Var y) t) = Term.subst x u t.
Proof.
intros NI.
induction t using term_ind'; cbn.
- case eqbspec; [intros ->|intros NE].
+ simpl. now rewrite eqb_refl.
+ rewrite term_subst_notin; auto with set.
- simpl in *. rewrite unionmap_in in NI.
f_equal.
rewrite !map_map.
apply map_ext_in.
intros. apply H; auto. contradict NI. now exists a.
Qed.
Lemma term_subst_bis v z u t :
z <> v ->
Term.subst v u (Term.subst v (Var z) t) =
Term.subst v (Var z) t.
Proof.
induction t as [x | f l IH] using term_ind'; intro H; cbn.
- case eqbspec; cbn.
+ apply eqb_neq in H. now rewrite H.
+ intros H'. apply eqb_neq in H'. now rewrite H'.
- f_equal. rewrite map_map. apply map_ext_iff. intros.
apply IH; auto.
Qed.
(** A fresh "reconciliating" variable could be arbitrary *)
Lemma term_subst_rename x x' z z' t t':
~Names.In z (Names.union (Term.vars t) (Term.vars t')) ->
~Names.In z' (Names.union (Term.vars t) (Term.vars t')) ->
Term.subst x z t = Term.subst x' z t' ->
Term.subst x z' t = Term.subst x' z' t'.
Proof.
revert t t'.
fix IH 1. destruct t, t'; intros Hz Hz'; cbn in *; try easy.
- do 2 case eqbspec; intros H H'; subst; intros [=]; namedec.
- case eqbspec; intros _ [=].
- case eqbspec; intros _ [=].
- intros [= <- E]. f_equal. clear f.
revert l l0 Hz Hz' E.
fix IH' 1. destruct l, l0; intros Hz Hz'; cbn in *; try easy.
intros [= E E']. f_equal.
+ apply IH; auto. namedec. namedec.
+ apply IH'; auto. namedec. namedec.
Qed.
(** RENAMING *)
(** For [rename x y f] to be meaningful, we ask for
the following precondition. Actually, some simple
properties over [rename] may even hold without this
precondition. *)
Notation RenOk y f := (~Names.In y (allvars f)).
Lemma rename_height x y f :
height (rename x y f) = height f.
Proof.
induction f; cbn; auto. f_equal. destruct eqb; auto.
Qed.
Lemma rename_height_lt x y f h :
height f < h -> height (rename x y f) < h.
Proof.
now rewrite rename_height.
Qed.
Hint Resolve rename_height_lt.
(** No-op renaming *)
Lemma rename_notin x y f :
~Names.In x (freevars f) ->
rename x y f = f.
Proof.
induction f; cbn; intros NI; f_equal; auto with set.
- apply map_id_iff. intros a Ha. apply term_subst_notin.
eapply unionmap_notin; eauto.
- case eqbspec; cbn; auto with set.
Qed.
(** Free variables and substitutions *)
Lemma freevars_allvars f : Names.Subset (freevars f) (allvars f).
Proof.
induction f; cbn; auto with set.
Qed.
Lemma allvars_rename x y f :
Names.Subset (allvars (rename x y f)) (Names.add y (allvars f)).
Proof.
induction f; cbn; try namedec.
- generalize (term_vars_subst x y (Fun "" l)). cbn. namedec.
- destruct eqb; cbn; namedec.
Qed.
Lemma allvars_rename_2 x y f :
Names.In x (freevars f) ->
Names.In y (allvars (rename x y f)).
Proof.
induction f; cbn; try namedec.
- intros. apply (term_vars_subst_in' _ _ (Fun "" l)); auto with set.
- case eqbspec; cbn; namedec.
Qed.
Lemma freevars_rename_in x y f :
RenOk y f ->
Names.In x (freevars f) ->
Names.Equal (freevars (rename x y f))
(Names.add y (Names.remove x (freevars f))).
Proof.
induction f; cbn; intros NI IN.
- namedec.
- namedec.
- rewrite (term_vars_subst_in x y (Fun "" l)); simpl; auto with set.
- auto.
- rewrite Names.union_spec in NI. apply Decidable.not_or in NI.
rewrite Names.union_spec in IN.
destruct (NamesP.In_dec x (freevars f1)) as [IN1|IN1];
destruct (NamesP.In_dec x (freevars f2)) as [IN2|IN2];
try rewrite IHf1 by intuition;
try rewrite IHf2 by intuition;
try rewrite (rename_notin x y f2) by trivial;
try rewrite (rename_notin x y f1) by trivial;
namedec.
- case eqbspec; cbn. namedec.
intros. rewrite IHf; namedec.
Qed.
Lemma freevars_rename_subset x y f :
Names.Subset (freevars (rename x y f))
(Names.add y (Names.remove x (freevars f))).
Proof.
induction f; cbn.
- namedec.
- namedec.
- rewrite (term_vars_subst x y (Fun "" l)). simpl. namedec.
- auto.
- rewrite IHf1, IHf2. namedec.
- case eqbspec; cbn. namedec.
intros. rewrite IHf. namedec.
Qed.
Lemma freevars_rename_subset2 x y f :
Names.Subset (Names.remove x (freevars f))
(freevars (rename x y f)).
Proof.
induction f; cbn.
- namedec.
- namedec.
- destruct (NamesP.In_dec x (Term.vars (Fun "" l))) as [H|H]; simpl in *.
+ rewrite (term_vars_subst_in x y (Fun "" l)); auto. simpl. namedec.
+ replace (map (Term.subst x y) l) with l. namedec.
symmetry. apply map_id_iff.
intros. apply term_subst_notin. contradict H.
rewrite unionmap_in. now exists a.
- auto.
- namedec.
- case eqbspec; cbn. namedec.
intros. rewrite <- IHf. namedec.
Qed.
Lemma freevars_rename_subset3 x y f :
Names.Subset (freevars f)
(Names.add x (freevars (rename x y f))).
Proof.
rewrite <- freevars_rename_subset2. namedec.
Qed.
(** Renaming in chain *)
Lemma rename_chain x y z f :
RenOk y f ->
rename y z (rename x y f) = rename x z f.
Proof.
induction f; cbn; intros R; f_equal; auto.
- rewrite !map_map.
apply map_ext_in. intros. apply term_subst_chain.
rewrite unionmap_in in R. contradict R. now exists a.
- auto with set.
- auto with set.
- intros. f_equal. repeat case eqbspec; auto.
namedec.
intros. apply rename_notin. rewrite freevars_allvars. namedec.
intros. apply IHf. namedec.
Qed.
(** Renaming at independent locations *)
Lemma rename_rename x y x' y' f :
x<>y -> x<>y' -> x'<>y ->
rename y y' (rename x x' f) = rename x x' (rename y y' f).
Proof.
intros XY XY' YX'.
induction f; cbn; f_equal; auto.
- rewrite !map_map.
apply map_ext. intros.
rewrite term_subst_subst; simpl; auto with set.
rewrite <-eqb_neq in YX'. now rewrite YX'.
- intuition.
Qed.
(** Renaming twice at the same spot *)
Lemma rename_bis v z z' f :
z <> v ->
rename v z' (rename v z f) = rename v z f.
Proof.
induction f; simpl; intros; f_equal; auto.
- rewrite map_map. apply map_ext_iff. intros.
apply term_subst_bis; auto.
- case eqbspec; simpl; intuition.
Qed.
(** ALPHA EQUIVALENCE *)
(** A weaker but faster version of namedec *)
Ltac namedec0 :=
repeat (rewrite ?Names.add_spec, ?Names.union_spec,
?Names.remove_spec, ?Names.singleton_spec in *);
intuition auto.
Ltac simpl_fresh vars H :=
unfold vars in H; cbn in H;
rewrite ?Names.add_spec, ?Names.union_spec in H;
repeat (apply Decidable.not_or in H; destruct H as (?,H));
clear vars H.
Ltac simpl_height :=
match goal with
| H : S _ < S _ |- _ =>
apply Nat.succ_lt_mono in H; simpl_height
| H : Nat.max _ _ < _ |- _ =>
apply Nat.max_lub_lt_iff in H; destruct H; simpl_height
| _ => idtac
end.
Lemma height_ind (P : formula -> Prop) :
(forall h, (forall f, height f < h -> P f) ->
(forall f, height f < S h -> P f)) ->
forall f, P f.
Proof.
intros IH f.
set (h := S (height f)).
assert (LT : height f < h) by (cbn; auto with * ).
clearbody h. revert f LT.
induction h as [|h IHh]; [inversion 1|eauto].
Qed.
Lemma AEq_refl f :
AlphaEq f f.
Proof.
induction f as [h IH f LT] using height_ind.
destruct f; cbn in *; simpl_height; auto.
destruct (exist_fresh (allvars f)) as (z,Hz).
apply AEqQu with (z:=z); auto with set.
Qed.
Lemma AEq_sym f f' :
AlphaEq f f' -> AlphaEq f' f.
Proof.
revert f'.
induction f as [h IH f LT] using height_ind.
destruct f, f'; cbn in *; simpl_height;
try (inversion_clear 1; auto).
apply AEqQu with (z:=z); auto with set.
Qed.
Lemma AEq_freevars f f' :
AlphaEq f f' -> Names.Equal (freevars f) (freevars f').
Proof.
induction 1; cbn; auto with set.
revert IHAlphaEq.
destruct (NamesP.In_dec v (freevars f)) as [E|E];
destruct (NamesP.In_dec v' (freevars f')) as [E'|E'];
try (rewrite (rename_notin v z f E));
try (rewrite (rename_notin v' z f' E'));
repeat (rewrite freevars_rename_in by auto with set);
cbn;
rewrite <-!freevars_allvars in H.
- (* namedec should suffice but take ages ?! *)
intros EQ x. specialize (EQ x). namedec.
- intros EQ x. specialize (EQ x). namedec.
- intros EQ x. specialize (EQ x). namedec.
- namedec.
Qed.
(** An adhoc helper lemma for the next one. *)
Lemma AEq_rename_aux f f' x x' v z z0 :
~Names.In z (Names.union (Names.add x (allvars f))
(Names.add v (allvars f'))) ->
~Names.In z0 (Names.union (allvars f)
(allvars (rename x' z f'))) ->
AlphaEq (rename x z0 f) (rename v z0 (rename x' z f')) ->
~Names.In x' (freevars f').
Proof.
intros Hz Hz0 EQ IN.
assert (RenOk z0 f) by auto with set.
assert (RenOk z f') by auto with set.
assert (z0 <> z).
{ contradict Hz0. subst z0.
nameiff. right.
rewrite <- freevars_allvars.
rewrite freevars_rename_in; auto. namedec. }
assert (NI : ~Names.In z (freevars (rename x z0 f))).
{ rewrite freevars_rename_subset.
rewrite freevars_allvars. namedec. }
apply AEq_freevars in EQ.
rewrite EQ in NI.
contradict NI.
rewrite <- freevars_rename_subset2.
rewrite freevars_rename_in; namedec.
Qed.
(* The crucial lemma, utterly technical for something so obvious *)
Lemma AEq_rename_any f f' x x' z z' :
~Names.In z (Names.union (allvars f) (allvars f')) ->
~Names.In z' (Names.union (allvars f) (allvars f')) ->
AlphaEq (rename x z f) (rename x' z f') ->
AlphaEq (rename x z' f) (rename x' z' f').
Proof.
revert f' x x' z z'.
induction f as [h IH f LT] using height_ind.
destruct f, f'; intros x x' z z' Hz Hz'; cbn in *; simpl_height;
try (inversion 1; auto; fail).
- inversion 1; subst.
assert (E : map (Term.subst x z') l = map (Term.subst x' z') l0).
{ injection (term_subst_rename x x' z z' (Fun "" l) (Fun "" l0)).
auto. namedec0. namedec0. cbn; f_equal; auto. }
now rewrite E.
- inversion 1; eauto.
- inversion 1; subst; constructor; eapply IH; eauto; namedec0.
- repeat (case eqbspec; cbn).
+ trivial.
+ intros NE <-. inversion_clear 1.
assert (~Names.In x' (freevars f')).
{ eapply AEq_rename_aux; eauto. }
rewrite !(rename_notin x') in * by trivial.
apply AEqQu with z0; auto.
+ intros <- NE. inversion_clear 1.
assert (~Names.In x (freevars f)).
{ apply AEq_sym in H1.
eapply AEq_rename_aux; eauto; namedec0. }
rewrite !(rename_notin x) in * by trivial.
apply AEqQu with z0; auto.
+ intros NE NE'. inversion_clear 1.
(* Let's pick a suitably fresh variable z1 *)
set (vars :=
Names.add x (Names.add x' (Names.add z (Names.add z'
(Names.unions
[ allvars (rename x z f);
allvars (rename x' z f');
allvars f; allvars f']))))).
destruct (exist_fresh vars) as (z1,Hz1).
simpl_fresh vars Hz1.
(* Let's use z1 instead of z0 in the main hyp *)
apply IH with (z':=z1) in H1; [ | auto | auto | namedec0].
clear H0 z0.
(* Let's use z1 in the conclusion. *)
apply AEqQu with z1; auto.
rewrite !allvars_rename; cbn; namedec0.
revert H1.
(* We now need to swap the substitutions to
change z' with z thanks to the IH. *)
rewrite !(rename_rename x), !(rename_rename x');
auto; try (namedec0;fail).
apply IH; auto;
rewrite !allvars_rename; cbn; namedec0.
Qed.
Lemma AEq_trans f1 f2 f3 :
AlphaEq f1 f2 -> AlphaEq f2 f3 -> AlphaEq f1 f3.
Proof.
revert f2 f3.
induction f1 as [h IH f1 LT] using height_ind.
destruct f1, f2, f3; cbn in *; simpl_height;
inversion_clear 1; inversion_clear 1; eauto.
set (vars :=
Names.unions [ (allvars f1); (allvars f2); (allvars f3) ]).
destruct (exist_fresh vars) as (z1,Hz1). simpl_fresh vars Hz1.
apply AEq_rename_any with (z':=z1) in H1; [ |namedec|namedec].
apply AEq_rename_any with (z':=z1) in H3; [ |namedec|namedec].
apply AEqQu with z1; eauto. namedec.
Qed.
Instance AEq_equiv : Equivalence AlphaEq.
Proof.
split; [exact AEq_refl| exact AEq_sym | exact AEq_trans].
Qed.
Lemma AEqQu_iff q v v' f f' z :
~Names.In z (Names.union (allvars f) (allvars f')) ->
AlphaEq (Quant q v f) (Quant q v' f') <->
AlphaEq (rename v z f) (rename v' z f').
Proof.
intros NI.
split.
- inversion 1; subst. eapply AEq_rename_any; eauto.
- now apply AEqQu.
Qed.
Lemma AEq_rename f f' :
AlphaEq f f' ->
forall x y,
RenOk y f -> RenOk y f' ->
AlphaEq (rename x y f) (rename x y f').
Proof.
revert f'.
induction f as [h IH f LT] using height_ind.
destruct f, f'; intros EQ x y IS1 IS2; inversion_clear EQ;
cbn in *; simpl_height; auto.
- constructor; intuition.
- rename v0 into v'.
rename H0 into EQ.
clear q; rename q0 into q.
assert (RenOk z f) by auto with set.
assert (RenOk z f') by auto with set.
repeat (case eqbspec; cbn); intros; auto.
+ subst v; subst v'.
apply AEqQu with z; auto.
+ subst v. clear IS1.
rewrite rename_notin; [ apply AEqQu with z; auto | ].
{ case (eqbspec x z).
- intros ->; rewrite freevars_allvars; namedec.
- intros NE.
apply AEq_freevars in EQ.
rewrite (freevars_rename_subset3 v' z), <- EQ.
rewrite freevars_rename_subset. namedec. }
+ subst v'. clear IS2.
rewrite rename_notin; [ apply AEqQu with z; auto | ].
{ case (eqbspec x z).
- intros ->; rewrite freevars_allvars; namedec.
- intros NE.
apply AEq_freevars in EQ.
rewrite (freevars_rename_subset3 v z), EQ.
rewrite freevars_rename_subset. namedec. }
+ set (vars :=
Names.add x (Names.add v (Names.add v' (Names.add y
(Names.unions [allvars f; allvars f']))))).
destruct (exist_fresh vars) as (z',Hz').
simpl_fresh vars Hz'.
apply AEq_rename_any with (z' := z') in EQ; [ |namedec|namedec].
clear z H H0 H1.
apply AEqQu with z'; [rewrite 2 allvars_rename; namedec|].
rewrite 2 (rename_rename x) by auto with set.
apply IH; auto; rewrite allvars_rename; namedec.
Qed.
Lemma AEqQu_nosubst f f' q v :
AlphaEq f f' -> AlphaEq (Quant q v f) (Quant q v f').
Proof.
intros.
set (vars := Names.union (allvars f) (allvars f')).
destruct (exist_fresh vars) as (z,Hz).
apply AEqQu with z. exact Hz.
apply AEq_rename; auto with set.
Qed.
Lemma AEqQu_rename f q v z :
RenOk z f ->
AlphaEq (Quant q v f) (Quant q z (rename v z f)).
Proof.
intros R.
set (vars := Names.add z (Names.add v (allvars f))).
destruct (exist_fresh vars) as (z',Hz').
apply AEqQu with z'.
- rewrite allvars_rename. namedec.
- case (eqbspec v z).
+ intros ->.
rewrite (rename_notin z z); auto. reflexivity.
rewrite freevars_allvars; auto.