add(subsystem): occurrence count per explicit substitution and the absence of a critical overlap between R and Gc (M3 infrastructure)

theory/Subsystem.v

@@ -38,3 +38,27 @@ Proof. intros. apply red_sub_base. apply rs_r. exact H. Qed.
 Print Assumptions red_sub_root_to_red1.
 Print Assumptions red_sub_to_red1.
 Print Assumptions red_sub_context.
+
+Lemma occurs_zfill : forall C k, occurs k (zfill C k) = true.
+Proof.
+  induction C; intros k; simpl.
+  - rewrite Nat.eqb_refl. reflexivity.
+  - rewrite IHC. reflexivity.
+  - rewrite IHC. rewrite orb_true_r. reflexivity.
+  - rewrite (IHC (S k)). reflexivity.
+  - rewrite (IHC (S k)). reflexivity.
+  - rewrite IHC. rewrite orb_true_r. reflexivity.
+Qed.
+
+Lemma zfill_occurs0 : forall C body, zfill C 0 = body -> occurs0 body = true.
+Proof. intros C body H. rewrite <- H. apply occurs_zfill. Qed.
+
+Lemma R_Gc_disjoint : forall body, occurs0 body = false -> ~ (exists C, zfill C 0 = body).
+Proof.
+  intros body Hf [C HC]. apply zfill_occurs0 in HC.
+  rewrite Hf in HC. discriminate HC.
+Qed.
+
+Print Assumptions occurs_zfill.
+Print Assumptions zfill_occurs0.
+Print Assumptions R_Gc_disjoint.