edit membership registration section for clarity

standards/core/rln-contract.md

@@ -93,7 +93,7 @@ graph TD;
     GracePeriod -.-> |"time G passed"| Expired;
     GracePeriod --> |"erase"| ErasedAwaitsWithdrawal;
     Expired --> |"erase"| ErasedAwaitsWithdrawal;
-    Expired --> |"another membership reuses slot"| ErasedAwaitsWithdrawal;
+    Expired --> |"overwritten by a new membership"| ErasedAwaitsWithdrawal;
     ErasedAwaitsWithdrawal --> |"withdraw"| Erased;
 
 ```
@@ -137,32 +137,38 @@ Availability of membership-specific functionalities[^2] MUST be as follows:
 
 ### Register a membership
 
-Let us define the following aggregate rate limits:
-- `R_{active}` is the total rate limit of all _Active_ memberships;
-- `R_{grace_period}` is the total rate limit of all _GracePeriod_ memberships;
-- `R_{expired}` is the total rate limit of all _Expired_ memberships;
-- `r` is the requested rate limit of a new membership.
-
-Let `R_{free} = R_{max} - R_{active} - R_{grace_period} - R_{expired}` be the free rate limit that is available without overwriting _Expired_ slots.
-
 Membership registration is subject to the following requirements:
-- If `r <= R_{free}`, the new membership MUST be registered (assuming all other necessary conditions hold). The new membership MAY overwrite one or multiple _Expired_ memberships.
+- The keeper MUST specify the requested rate limit  `r` of a new membership at registration time[^3].
-- If `r > R_{free}`:
+- Registration MUST fail if `r < r_{min}` or `r > r_{max}`.
-	- if `r > R_{free} + R_{expired}`, registration MUST fail;
-	- if `r <= R_{free} + R_{expired}`, the new membership MUST be registered by overwriting some _Expired_ memberships.
-- The sender of the registration transaction MAY specify a list of _Expired_ memberships whose slots will be overwritten. If the list is not provided, the contract MAY use any criteria to select _Expired_ memberships to overwrite (see Implementation Suggestions).
-- If a new membership A overwrites an _Expired_ membership B:
-	- membership B MUST become _ErasedAwaitsWithdrawal_;
-	- the current total rate limit MUST be decremented by the rate limit of membership B;
-	- the contract MUST take all necessary steps to ensure that the keeper of membership B can withdraw their deposit later.
-- Registration MUST fail if the requested rate limit for a new membership is lower than `r_{min}` or higher than `r_{max}`.
 - The keeper MUST lock up a deposit to register a membership.
-- The keeper MUST specify the rate limit[^3] of a membership at registration time.
 - The size of the deposit MUST depend on the specified rate limit.
 - In case of a successful registration:
 	- the new membership MUST become _Active_;
 	- the new membership MUST have an expiration term `T` and a grace period duration `G`;
 	- the current total rate limit MUST be incremented by the rate limit of the new membership.
+#### Overwriting other memberships
+
+Let us define the following rate limits:
+- `R_{active}` is the total rate limit of all _Active_ memberships;
+- `R_{grace_period}` is the total rate limit of all _GracePeriod_ memberships;
+- `R_{expired}` is the total rate limit of all _Expired_ memberships.
+
+Let us define the free rate limit that is available without overwriting _Expired_ memberships as follows:
+
+```
+R_{free} = R_{max} - R_{active} - R_{grace_period} - R_{expired}
+```
+
+Membership registration is additionally subject to the following requirements:
+- If `r <= R_{free}`, the new membership MUST be registered (assuming all other necessary conditions hold). The new membership MAY overwrite one or multiple _Expired_ memberships.
+- If `r > R_{free}`:
+	- if `r > R_{free} + R_{expired}`, registration MUST fail;
+	- if `r <= R_{free} + R_{expired}`, the new membership MUST be registered by overwriting some _Expired_ memberships.
+- The sender of the registration transaction MAY specify a list of _Expired_ memberships to be overwritten. If the list is not provided, the contract MAY use any criteria to select _Expired_ memberships to overwrite (see Implementation Suggestions).
+- If a new membership A overwrites an _Expired_ membership B:
+	- membership B MUST become _ErasedAwaitsWithdrawal_;
+	- the current total rate limit MUST be decremented by the rate limit of membership B;
+	- the contract MUST take all necessary steps to ensure that the keeper of membership B can withdraw their deposit later.
 
 [^3]: A user-facing application SHOULD suggest default rate limits to the keeper (see Implementation Suggestions).