docs: add fibonacci documentation

.gitignore

@@ -1 +1,3 @@
 /target
+
+*_proof.json

README.md

@@ -78,7 +78,7 @@ brainfuck_prover prove --file my_program.bf --output my_program_proof.json
 Or if you built from source,
 
 ```shell
-./target/release/brainfuck_prover prove --file my_program.bf --output my_program_proof.json
+./target/release/brainfuck_prover prove --file ./brainfuck_programs/hello_kakarot.bf --output hello_kakarot_proof.json
 ```
 
 ### Verify
@@ -89,6 +89,30 @@ To verify a proof, the proof must be stored in a JSON file (`--output` flag from
 brainfuck_prover verify my_program_proof.json
 ```
 
+Or if you built from source and previously generated the proof of the 19th Fibonacci number:
+
+```shell
+./target/release/brainfuck_prover verify hello_kakarot.json
+```
+
+### Visualizing the memory
+
+To visualize the memory of the Brainfuck VM, use the `--memory` flag of the `brainfuck_prover`, and reduce the RAM size to avoid printing too much memory cells to your terminal with the `--ram-size` flag.
+
+Let's try it with a Brainfuck program that yields the 19th Fibonacci number. Note that it is a bit more resource intensive than the other example programs.
+
+```shell
+./target/release/brainfuck_prover prove --file ./brainfuck_programs/fib19.bf --output fib19_proof.json --memory --ram-size 20
+```
+
+You should be able to see:
+
+```shell
+[M31(0), M31(2584), M31(4181), M31(0), M31(0), M31(0), M31(0), M31(0), M31(0), M31(0), M31(0), M31(0), M31(0), M31(0), M31(0), M31(0), M31(0), M31(0), M31(0), M31(0)]
+```
+
+The third memory cell contains the desired output: `Fibonacci(19) = 4181`.
+
 ## Project Objectives
 
 - Capacity of generating and verifying a proof for arbitrary Brainfuck programs.

crates/brainfuck_vm/tests/integration.rs

@@ -89,12 +89,16 @@ fn test_hello_kakarot_world() {
 }
 
 #[test]
-fn test_fib_19() {
-    let path = "../../brainfuck_programs/fib.bf";
+fn test_fib19() {
+    let path = "../../brainfuck_programs/fib19.bf";
 
     let code = compile_from_path(path);
     let (mut machine, output) = create_test_machine(&code, &[]);
     machine.execute().unwrap();
+    // 19th Fibonacci number is 4181
+    // Output is misleading as the I/O of Brainfuck is only 1 byte,
+    // so the memory values are truncated
+    // (hence the output of 85, 4181 % 256 = 85)
     let expected_output = [85];
     assert_eq!(output.output(), expected_output);
 }