Restore Merkle examples and statements table

Author: tideofwords

book/src/merkletree.md

@@ -94,6 +94,15 @@ root = hash((L_root, R_root, 2)).
 The full `Dictionary` is then represented in the backend as `root` (four field elements in the Plonky2 backend).
 
 ### Example 2
+So for example, imagine we have 8 key-pairs, where the keys are just an enumeration from 0 to 7, then the tree leaves positions would look like:
+![](img/merkletree-example-1-a.png)
+
+Now let's change the key of the leaf `key=1`, and set it as `key=13`. Then, their respective leaf paths will be the same until they diverge in the 4-th bit:
+
+![](img/merkletree-example-1-b.png)
+
+
+### Example 3
 
 Suppose we have 4 key-values, where the first four bits of the hashes of the keys are `0000`, `0100`, `1010` and `1011`. The tree would look like:
 ![](img/merkletree-example-2-a.png)

book/src/statements.md

@@ -16,6 +16,30 @@ On the back end, a statement is identified by a unique numerical _identifier_.
 
 The POD system has several builtin statements. These statements are associated to a reserved set of statement IDs.
 
+### Backend statements
+
+<font color="red">TODO: update table of backend statements </font>
+
+A statement is a code (or, in the frontend, string identifier) followed by 0 or more arguments. These arguments may consist of up to three anchored keys and up to one POD value.
+
+The following table summarises the natively-supported statements, where we write `value_of(ak)` for 'the value anchored key `ak` maps to', which is of type `PODValue`, and `key_of(ak)` for the key part of `ak`:
+
+| Code | Identifier  | Args                | Meaning                                                           |
+|------|-------------|---------------------|-------------------------------------------------------------------|
+| 0    | `None`      |                     | no statement (useful for padding)                                 |
+| 1    | `ValueOf`   | `ak`, `value`       | `value_of(ak) = value`                                            |
+| 2    | `Eq`        | `ak1`, `ak2`        | `value_of(ak1) = value_of(ak2)`                                   |
+| 3    | `NEq`       | `ak1`, `ak2`        | `value_of(ak1) != value_of(ak2)`                                  |
+| 4    | `Gt`        | `ak1`, `ak2`        | `value_of(ak1) > value_of(ak2)`                                   |
+| 5    | `LEq`       | `ak1`, `ak2`        | `value_of(ak1) <= value_of(ak2)`                                  |
+| 6    | `Contains`  | `ak1`, `ak2`        | `(key_of(ak2), value_of(ak2)) ∈ value_of(ak1)` (Merkle inclusion) |
+| 7    | `Sintains`  | `ak1`, `ak2`        | `(key_of(ak2), value_of(ak2)) ∉ value_of(ak1)` (Merkle exclusion) |
+| 8    | `SumOf`     | `ak1`, `ak2`, `ak3` | `value_of(ak1) = value_of(ak2) + value_of(ak3)`                   |
+| 9    | `ProductOf` | `ak1`, `ak2`, `ak3` | `value_of(ak1) = value_of(ak2) * value_of(ak3)`                   |
+| 10   | `MaxOf`     | `ak1`, `ak2`, `ak3` | `value_of(ak1) = max(value_of(ak2), value_of(ak3))`               |
+
+### Frontend statements
+
 ```
 ValueOf(key: AnchoredKey, value: ScalarOrVec)