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fix(named): free variable preservation for the full metaterm reduction including RX (M5 metatheory)

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sneeker committed 2026-09-22 23:55:00 +02:00
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theory/NamedMeta.v
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@@ -240,3 +240,63 @@ Qed.
Print Assumptions nplug_fv_mono.
Print Assumptions nplug_fv_upper.
Print Assumptions nred_core_fv.
Lemma nplug_esub_fv : forall C X d x u y,
y <> x ->
In y (nfv (nplug C (NESub (NMVar X d) x u))) ->
In y (nfv (nplug C (NMVar X d))) \/ In y (nfv u).
Proof.
induction C; intros X d x u y Hne Hy; simpl in *.
- apply in_app_iff in Hy as [Hy|Hy].
+ left. apply in_filter_neq in Hy as [Hy _]. simpl. exact Hy.
+ right. exact Hy.
- apply in_app_iff in Hy as [Hy|Hy].
+ apply (IHC X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
* left. apply in_app_iff. left. exact Hy.
* right. exact Hy.
+ left. apply in_app_iff. right. exact Hy.
- apply in_app_iff in Hy as [Hy|Hy].
+ left. apply in_app_iff. left. exact Hy.
+ apply (IHC X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
* left. apply in_app_iff. right. exact Hy.
* right. exact Hy.
- apply in_filter_neq in Hy as [Hy Hz].
apply (IHC X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
+ left. apply filter_neq_in; [exact Hy | exact Hz].
+ right. exact Hy.
- apply in_app_iff in Hy as [Hy|Hy].
+ apply in_filter_neq in Hy as [Hy Hz].
apply (IHC X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
* left. apply in_app_iff. left. apply filter_neq_in; [exact Hy | exact Hz].
* right. exact Hy.
+ left. apply in_app_iff. right. exact Hy.
- apply in_app_iff in Hy as [Hy|Hy].
+ left. apply in_app_iff. left. exact Hy.
+ apply (IHC X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
* left. apply in_app_iff. right. exact Hy.
* right. exact Hy.
Qed.
Lemma nred_fv : forall t t', nred t t' -> incl (nfv t') (nfv t).
Proof.
intros t t' H. induction H; simpl; unfold incl in *.
- intros y Hy. exact Hy.
- intros y Hy. apply in_app_iff. left.
apply filter_neq_in; [exact Hy | intro E; apply H; subst; exact Hy].
- intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+ apply in_filter_neq in Hy as [Hy Hne].
apply (nplug_esub_fv C X d x u y Hne) in Hy. destruct Hy as [Hy|Hy].
* apply in_app_iff. left. apply filter_neq_in; [exact Hy | exact Hne].
* apply in_app_iff. right. exact Hy.
+ apply in_app_iff. right. exact Hy.
- intros y Hy. apply in_app_iff in Hy as [Hy|Hy].
+ apply in_filter_neq in Hy as [Hy Hne].
apply (nplug_fv_upper C u (NVar x)) in Hy. apply in_app_iff in Hy as [Hy|Hy].
* apply in_app_iff. left. apply filter_neq_in; [exact Hy | exact Hne].
* apply in_app_iff. right. exact Hy.
+ apply in_app_iff. right. exact Hy.
- apply nplug_fv_mono. exact IHnred.
Qed.
Print Assumptions nplug_esub_fv.
Print Assumptions nred_fv.