add(named): congruence of the C equivalence and the R rule with capture avoidance (M5 named rules)

graphs/dependency.dot

@@ -49,6 +49,9 @@ digraph theorem_deps {
   mred_B_intro [label="mred_B_intro", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mred_B_intro theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   mred_gc_intro [label="mred_gc_intro", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mred_gc_intro theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   mred_r_intro [label="mred_r_intro", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mred_r_intro theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  Plus_mred_of_Plus_red1 [label="Plus_mred_of_Plus_red1", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Plus_mred_of_Plus_red1 theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  mred_full_comp_pure [label="mred_full_comp_pure", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mred_full_comp_pure theory/Metaterm.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  mred_plus_one [label="mred_plus_one", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="mred_plus_one 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];
@@ -70,6 +73,8 @@ digraph theorem_deps {
   isubst_meta_in [label="isubst_meta_in", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="isubst_meta_in theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   isubst_meta_notin [label="isubst_meta_notin", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="isubst_meta_notin theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   eqC_in_Es [label="eqC_in_Es", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="eqC_in_Es theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  Es_ctx_any [label="Es_ctx_any", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Es_ctx_any theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
+  nred_R_intro [label="nred_R_intro", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="nred_R_intro theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   Es_refl_any [label="Es_refl_any", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Es_refl_any theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   Es_sym_any [label="Es_sym_any", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Es_sym_any theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
   Es_trans_any [label="Es_trans_any", color="#38761d", fillcolor="#d9ead3", style="filled", fontcolor="#222222", tooltip="Es_trans_any theory/NamedMeta.v https://arxiv.org/html/2312.13270v1", shape=ellipse];
@@ -155,6 +160,8 @@ digraph theorem_deps {
   mfvs_lift_m -> mfvs_open_m;
   of_trm_plug -> mred_term_context;
   of_trm_moccurs -> of_trm_moccurs0;
+  Plus_mred_of_Plus_red1 -> mred_full_comp_pure;
+  lemma_2_2_full_comp -> mred_full_comp_pure;
   occurs_count_zero -> occurs_count_pos;
   occurs_count_false -> occurs_count_zfill_plug;
   occurs_lift_self -> occurs_count_zfill_plug;

graphs/theorems.json

@@ -467,6 +467,39 @@
       "file": "theory/Metaterm.v",
       "depends": []
     },
+    {
+      "id": "Plus_mred_of_Plus_red1",
+      "label": "Plus_mred_of_Plus_red1",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
+    {
+      "id": "mred_full_comp_pure",
+      "label": "mred_full_comp_pure",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": [
+        "Plus_mred_of_Plus_red1",
+        "lemma_2_2_full_comp"
+      ]
+    },
+    {
+      "id": "mred_plus_one",
+      "label": "mred_plus_one",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/Metaterm.v",
+      "depends": []
+    },
     {
       "id": "Plus_red1_context",
       "label": "Plus_red1_context",
@@ -706,6 +739,26 @@
       "file": "theory/NamedMeta.v",
       "depends": []
     },
+    {
+      "id": "Es_ctx_any",
+      "label": "Es_ctx_any",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
+    {
+      "id": "nred_R_intro",
+      "label": "nred_R_intro",
+      "kind": "infra",
+      "status": "proved",
+      "milestone": "M0",
+      "section": "",
+      "file": "theory/NamedMeta.v",
+      "depends": []
+    },
     {
       "id": "Es_refl_any",
       "label": "Es_refl_any",

theory/NamedMeta.v

@@ -70,7 +70,21 @@ Inductive Es : ntrm -> ntrm -> Prop :=
 | Es_trans : forall t u v, Es t u -> Es u v -> Es t v
 | Es_C : forall t x u y v,
     y <> x -> ~ In y (nfv u) -> ~ In x (nfv v) ->
-    Es (NESub (NESub t x u) y v) (NESub (NESub t y v) x u).
+    Es (NESub (NESub t x u) y v) (NESub (NESub t y v) x u)
+| Es_ctx : forall C t t', Es t t' -> Es (nplug C t) (nplug C t').
+
+Fixpoint cbinders (C : nctx) : list atom :=
+  match C with
+  | nHole => []
+  | nAppL C1 _ => cbinders C1
+  | nAppR _ C1 => cbinders C1
+  | nLam x C1 => x :: cbinders C1
+  | nESubL C1 _ _ => cbinders C1
+  | nESubR _ _ C1 => cbinders C1
+  end.
+
+Definition cavoid (C : nctx) (phi : list atom) : bool :=
+  forallb (fun y => negb (existsb (Nat.eqb y) phi)) (cbinders C).
 
 Inductive nred : ntrm -> ntrm -> Prop :=
 | nred_B : forall t x u, nred (NApp (NLam x t) u) (NESub t x u)
@@ -82,6 +96,11 @@ Inductive nred : ntrm -> ntrm -> Prop :=
     is_subst_chain C = false ->
     nred (NESub (nplug C (NMVar X d)) x u)
          (NESub (nplug C (NESub (NMVar X d) x u)) x u)
+| nred_R : forall C x u phi,
+    In x phi ->
+    (forall y, In y (nfv u) -> In y phi) ->
+    cavoid C phi = true ->
+    nred (NESub (nplug C (NVar x)) x u) (NESub (nplug C u) x u)
 | nred_ctx : forall C t t', nred t t' -> nred (nplug C t) (nplug C t').
 
 Lemma eqC_in_Es : forall t x u y v,
@@ -89,6 +108,20 @@ Lemma eqC_in_Es : forall t x u y v,
   Es (NESub (NESub t x u) y v) (NESub (NESub t y v) x u).
 Proof. intros. apply Es_C; assumption. Qed.
 
+Lemma Es_ctx_any : forall C t t', Es t t' -> Es (nplug C t) (nplug C t').
+Proof. intros. apply Es_ctx. exact H. Qed.
+
+Lemma nred_R_intro : forall C x u phi,
+  In x phi -> (forall y, In y (nfv u) -> In y phi) -> cavoid C phi = true ->
+  nred (NESub (nplug C (NVar x)) x u) (NESub (nplug C u) x u).
+Proof.
+  intros C x u phi Hx Hu Hc.
+  apply nred_R with (phi := phi).
+  - exact Hx.
+  - exact Hu.
+  - exact Hc.
+Qed.
+
 Lemma Es_refl_any : forall t, Es t t.
 Proof. intros. apply Es_refl. Qed.
 
@@ -100,6 +133,8 @@ Proof. intros. apply Es_trans with (u := u); assumption. Qed.
 
 Print Assumptions isubst_meta_in.
 Print Assumptions isubst_meta_notin.
+Print Assumptions Es_ctx_any.
+Print Assumptions nred_R_intro.
 Print Assumptions eqC_in_Es.
 Print Assumptions Es_refl_any.
 Print Assumptions Es_sym_any.