add(metaterm): free variable laws for lifting opening and closing on metaterms (M5 binding)

graphs/dependency.dot

@@ -32,6 +32,14 @@ digraph theorem_deps {
   enumerate_check [label="enumerate_check", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="enumerate_check theory/Enumerate.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   enumerate_sound [label="enumerate_sound", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="enumerate_sound theory/Enumerate.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   enumerate_complete [label="enumerate_complete", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="enumerate_complete theory/Enumerate.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  of_trm_fvs [label="of_trm_fvs", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="of_trm_fvs theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  of_trm_lift [label="of_trm_lift", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="of_trm_lift theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  of_trm_open [label="of_trm_open", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="of_trm_open theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  of_trm_close [label="of_trm_close", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="of_trm_close theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  of_trm_injective [label="of_trm_injective", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="of_trm_injective theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  mfvs_lift_m [label="mfvs_lift_m", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mfvs_lift_m theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  mfvs_close_m [label="mfvs_close_m", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mfvs_close_m theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  mfvs_open_m [label="mfvs_open_m", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mfvs_open_m theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   Plus_red1_context [label="Plus_red1_context", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Plus_red1_context theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   occurs_count_zero [label="occurs_count_zero", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="occurs_count_zero theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   occurs_count_false [label="occurs_count_false", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="occurs_count_false theory/Metatheory.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
@@ -128,6 +136,8 @@ digraph theorem_deps {
   check_all_true -> enumerate_check;
   steps_sound -> enumerate_sound;
   steps_complete -> enumerate_complete;
+  of_trm_lift -> of_trm_open;
+  mfvs_lift_m -> mfvs_open_m;
   occurs_count_zero -> occurs_count_pos;
   occurs_count_false -> occurs_count_zfill_plug;
   occurs_lift_self -> occurs_count_zfill_plug;

graphs/theorems.json

@@ -9,6 +9,7 @@
     "html": "https://arxiv.org/html/2312.13270v1"
   },
   "milestones": [
+    "M0",
     "M1",
     "M2",
     "M3",
@@ -288,6 +289,90 @@
         "steps_complete"
       ]
     },
+    {
+      "id": "of_trm_fvs",
+      "label": "of_trm_fvs",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "of_trm_lift",
+      "label": "of_trm_lift",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "of_trm_open",
+      "label": "of_trm_open",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": [
+        "of_trm_lift"
+      ]
+    },
+    {
+      "id": "of_trm_close",
+      "label": "of_trm_close",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "of_trm_injective",
+      "label": "of_trm_injective",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mfvs_lift_m",
+      "label": "mfvs_lift_m",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mfvs_close_m",
+      "label": "mfvs_close_m",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mfvs_open_m",
+      "label": "mfvs_open_m",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": [
+        "mfvs_lift_m"
+      ]
+    },
     {
       "id": "Plus_red1_context",
       "label": "Plus_red1_context",

theory/Metaterm.v

@@ -110,3 +110,54 @@ Print Assumptions of_trm_lift.
 Print Assumptions of_trm_open.
 Print Assumptions of_trm_close.
 Print Assumptions of_trm_injective.
+
+Lemma mfvs_lift_m : forall t k, mfvs (lift_m k t) = mfvs t.
+Proof.
+  induction t; intros k; simpl.
+  - destruct (Nat.ltb n k); reflexivity.
+  - reflexivity.
+  - reflexivity.
+  - rewrite IHt1, IHt2. reflexivity.
+  - rewrite IHt. reflexivity.
+  - rewrite IHt1, IHt2. reflexivity.
+Qed.
+
+Lemma mfvs_close_m : forall t x k y, In y (mfvs (close_rec_m x k t)) -> In y (mfvs t).
+Proof.
+  induction t; intros x k y H; simpl in *.
+  - destruct H.
+  - destruct (Nat.eqb a x) eqn:E.
+    + destruct H.
+    + destruct H as [Hy | Hf]. subst y. simpl. left. reflexivity. destruct Hf.
+  - exact H.
+  - apply in_app_iff in H as [H|H]; apply in_app_iff.
+    + left. apply (IHt1 x k y). exact H.
+    + right. apply (IHt2 x k y). exact H.
+  - apply (IHt x (S k) y). exact H.
+  - apply in_app_iff in H as [H|H]; apply in_app_iff.
+    + left. apply (IHt1 x (S k) y). exact H.
+    + right. apply (IHt2 x k y). exact H.
+Qed.
+
+Lemma mfvs_open_m : forall t k u y,
+  In y (mfvs (open_rec_m k u t)) -> In y (mfvs u) \/ In y (mfvs t).
+Proof.
+  induction t; intros k u y H; simpl in *.
+  - destruct (Nat.eqb n k) eqn:E.
+    + apply Nat.eqb_eq in E. subst n. simpl in H.
+      left. rewrite <- (mfvs_lift_m u k). exact H.
+    + simpl in H. destruct H.
+  - right. exact H.
+  - right. exact H.
+  - apply in_app_iff in H as [H|H].
+    + destruct (IHt1 k u y H) as [Hl|Hr]. left; exact Hl. right; apply in_app_iff; left; exact Hr.
+    + destruct (IHt2 k u y H) as [Hl|Hr]. left; exact Hl. right; apply in_app_iff; right; exact Hr.
+  - destruct (IHt (S k) u y H) as [Hl|Hr]. left; exact Hl. right; exact Hr.
+  - apply in_app_iff in H as [H|H].
+    + destruct (IHt1 (S k) u y H) as [Hl|Hr]. left; exact Hl. right; apply in_app_iff; left; exact Hr.
+    + destruct (IHt2 k u y H) as [Hl|Hr]. left; exact Hl. right; apply in_app_iff; right; exact Hr.
+Qed.
+
+Print Assumptions mfvs_lift_m.
+Print Assumptions mfvs_close_m.
+Print Assumptions mfvs_open_m.