update example with @artwyman IsComposite suggestion & transitive_eq example

Author: arnaucube

book/src/simpleexample.md

@@ -149,52 +149,53 @@ eth_friend(?1, ?2, ?3, ?4) = and<
 ## Another perspective
 When working with statements and operations, it might be useful to see them from another perspective:
 
-- A *statement* can be seen as the *constraints* of a traditional zk-circuit,
-  which can be true or false.
 - A *predicate* is a relation formula, which when filled with values becomes a
   *statement*.
+- A *statement* can be seen as the *constraints* of a traditional zk-circuit,
+  which can be true or false.
 - An *operation* comprises the deduction rules, which are rules used to deduce
   new statements from previous statements or used to construct new statements
   from values.
 
 $$
-predicate \cong circuit/relation to be fulfilled\\
-statement \cong constraints filled with the witness\\
+predicate \cong circuit/relation~to~be~fulfilled\\
+statement \cong constraints~filled~with~the~witness\\
 operations \cong deduction~rules
 $$
 
 
 For example,
-- `Equal` for integers is a predicate.
-- `5 Equal 5` is a statement (true)
-- `5 Equal 4` is a statement (false)
+- `Equal` for integers is a *predicate*
+- `st_1 = Equal(A, B)` is a *statement*
+- `st_2 = Equal(B, C)` is a *statement*
+- `st_3 = TransitiveEqualFromStatements(st_1, st_2)` is an *operation*, which yields the statement `st_3 = Equal(A, C)`
+
 
 
 
-<div style="display:flex;margin-left:-60px;">
+<div style="display:flex;">
 <div style="padding:10px;max-width:50%; border-right:1px solid #ccc;">
 
 So, for example, for the given predicate:
 
 ```
-IsWood(item, private: ingreds,inputs,key)=AND(
-    ItemDef(item, ingreds, inputs, key)
-    Equal(inputs, {{}})
-    DictContains(ingreds, "blueprint", "wood")
-}
+IsComposite(n, private: a, b) = AND(
+  ProductOf(n, a, b)
+  GtFromEntries(a, 1)
+  GtFromEntries(b, 1)
+)
 ```
 
 </div>
 <div style="padding:10px;max-width:45%;">
 
 We can view it as:
 
-The *statement* `IsWood(item)` is `true` if and only if:<br>
- $\exists$ `item`, `ingreds`, `inputs`, `key`
+The *statement* `IsComposite(n)` is `true` if and only if $\exists$ `n`, `a`, `b`
  such that the following statements hold:
- - `ItemDef(0x1234, ingreds, inputs, key)`
- - `Equal(inputs, {})`
- - `DictContains(ingreds, "blueprint", "wood")`
+- $n = a \cdot b$
+- $a > 1$
+- $b > 1$
 
 </div>
 </div>