chore(docs): Document signed integers and integer overflow behavior

docs/docs/language_concepts/data_types/01_integers.md

@@ -1,37 +1,101 @@
 ---
 title: Integers
-description:
-  Explore the Integer data type in Noir. Learn about its methods, see real-world examples, and grasp how to efficiently use Integers in your Noir code.
-keywords:
-  [
-    noir,
-    integer types,
-    methods,
-    examples,
-    arithmetic,
-  ]
+description: Explore the Integer data type in Noir. Learn about its methods, see real-world examples, and grasp how to efficiently use Integers in your Noir code.
+keywords: [noir, integer types, methods, examples, arithmetic]
 ---
 
-An integer type is a range constrained field type. The Noir frontend currently supports unsigned,
-arbitrary-sized integer types.
+An integer type is a range constrained field type. The Noir frontend supports arbitrarily-sized, both unsigned and signed integer types.
 
 > **Note:** When an integer is defined in Noir without a specific type, it will default to `Field`. The one exception is for loop indices which default to `u64` since comparisons on `Field`s are not possible.
 
-An integer type is specified first with the letter `u`, indicating its unsigned nature, followed by
-its length in bits (e.g. `32`). For example, a `u32` variable can store a value in the range of
-$\\([0,2^{32}-1]\\)$.
+## Unsigned Integers
 
-> **Note:** The default proving backend supports both even (e.g. `u16`, `u48`) and odd (e.g. `u5`, `u3`)
-> sized integer types.
+An unsigned integer type is specified first with the letter `u` (indicating its unsigned nature) followed by its length in bits (e.g. `8`):
 
-Taking a look of how the type is used:
+```rust
+fn main() {
+    let x: u8 = 1;
+    let y: u8 = 1;
+    let z = x + y;
+    assert (z == 2);
+}
+```
+
+The length in bits determines the boundaries the integer type can store. For example, a `u8` variable can store a value in the range of 0 to 255 (i.e. $\\2^{8}-1\\$).
+
+## Signed Integers
+
+A signed integer type is specified first with the letter `i` (which stands for integer) followed by its length in bits (e.g. `8`):
 
 ```rust
-fn main(x : Field, y : u32) {
-    let z = x as u32 + y;
+fn main() {
+    let x: i8 = -1;
+    let y: i8 = -1;
+    let z = x + y;
+    assert (z == -2);
 }
 ```
 
-`x`, `y` and `z` are all private values in this example. However, `x` is a field while `y` and `z`
-are unsigned 32-bit integers. If `y` or `z` exceeds the range $\\([0,2^{32}-1]\\)$, proofs created
-will be rejected by the verifier.
+The bit count of the integer determines the maximum and minimum integers the type can store. For example, an `i8` variable can store a value in the range of -128 to 127 (i.e. $\\-2^{7}\\$ to $\\2^{7}-1\\$).
+
+## Overflows
+
+Computations that exceed the type boundaries will result in overflow errors. This happens with both signed and unsigned integers. For example, attempting to prove:
+
+```rust
+fn main(x: u8, y: u8) {
+    let z = x + y;
+}
+```
+
+With:
+
+```toml
+x = "255"
+y = "1"
+```
+
+Would result in:
+
+```
+$ nargo prove
+error: Assertion failed: 'attempt to add with overflow'
+┌─ ~/src/main.nr:9:13
+│
+│     let z = x + y;
+│             -----
+│
+= Call stack:
+  ...
+```
+
+A similar error would happen with signed integers, for example while trying to prove:
+
+```rust
+fn main() {
+    let x: i8 = -118;
+    let y: i8 = -11;
+    let z = x + y;
+}
+```
+
+> **Note:** The default proving backend supports both even (e.g. _u2_) and odd (e.g. _u3_) arbitrarily-sized integer types up to _u127_.
+
+### Wrapping methods
+
+Although integer overflow is expected to error, some use-cases rely on wrapping. For these use-cases, the standard library provides `wrapping` variants of certain common operations:
+
+````rust
+fn wrapping_add<T>(x: T, y: T) -> T;
+fn wrapping_sub<T>(x: T, y: T) -> T;
+fn wrapping_mul<T>(x: T, y: T) -> T;
+
+example usage:
+
+```rust
+use dep::std;
+
+fn main(x: u8, y: u8) -> pub u8 {
+    std::wrapping_add(x + y)
+}
+````