📝 Add an example based on Fibonacci

example/001-simple/fibonacci/main.spec.ts

@@ -0,0 +1,87 @@
+import fc from 'fast-check';
+import { fibo } from './src/fibonacci';
+
+// The complexity of the algorithm is O(n)
+// As a consequence we limit the value of n to 1000
+const MaxN = 1000;
+
+describe('fibonacci', () => {
+  it('should be equal to the sum of fibo(n-1) and fibo(n-2)', () => {
+    fc.assert(
+      fc.property(fc.integer(2, MaxN), n => {
+        expect(fibo(n)).toBe(fibo(n - 1) + fibo(n - 2));
+      })
+    );
+  });
+
+  // The following properties are listed on the Wikipedia page:
+  // https://fr.wikipedia.org/wiki/Suite_de_Fibonacci#Divisibilit%C3%A9_des_nombres_de_Fibonacci
+
+  it('should fulfill fibo(p)*fibo(q+1)+fibo(p-1)*fibo(q) = fibo(p+q)', () => {
+    fc.assert(
+      fc.property(fc.integer(1, MaxN), fc.integer(0, MaxN), (p, q) => {
+        expect(fibo(p + q)).toBe(fibo(p) * fibo(q + 1) + fibo(p - 1) * fibo(q));
+      })
+    );
+  });
+
+  it('should fulfill fibo(2p-1) = fibo²(p-1)+fibo²(p)', () => {
+    // Special case of the property above
+    fc.assert(
+      fc.property(fc.integer(1, MaxN), p => {
+        expect(fibo(2 * p - 1)).toBe(fibo(p - 1) * fibo(p - 1) + fibo(p) * fibo(p));
+      })
+    );
+  });
+
+  it('should fulfill Catalan identity', () => {
+    fc.assert(
+      fc.property(fc.integer(0, MaxN), fc.integer(0, MaxN), (a, b) => {
+        const [p, q] = a < b ? [b, a] : [a, b];
+        const sign = (p - q) % 2 === 0 ? 1n : -1n; // (-1)^(p-q)
+        expect(fibo(p) * fibo(p) - fibo(p - q) * fibo(p + q)).toBe(sign * fibo(q) * fibo(q));
+      })
+    );
+  });
+
+  it('should fulfill Cassini identity', () => {
+    fc.assert(
+      fc.property(fc.integer(1, MaxN), fc.integer(0, MaxN), p => {
+        const sign = p % 2 === 0 ? 1n : -1n; // (-1)^p
+        expect(fibo(p + 1) * fibo(p - 1) - fibo(p) * fibo(p)).toBe(sign);
+      })
+    );
+  });
+
+  it('should fibo(nk) divisible by fibo(n)', () => {
+    fc.assert(
+      fc.property(fc.integer(1, MaxN), fc.integer(0, 100), (n, k) => {
+        expect(fibo(n * k) % fibo(n)).toBe(0n);
+      })
+    );
+  });
+
+  it('should fulfill gcd(fibo(a), fibo(b)) = fibo(gcd(a,b))', () => {
+    fc.assert(
+      fc.property(fc.integer(1, MaxN), fc.integer(1, MaxN), (a, b) => {
+        const gcd = <T extends bigint | number>(a: T, b: T, zero: T): T => {
+          a = a < zero ? (-a as T) : a;
+          b = b < zero ? (-b as T) : b;
+          if (b > a) {
+            const temp = a;
+            a = b;
+            b = temp;
+          }
+          // eslint-disable-next-line no-constant-condition
+          while (true) {
+            if (b == zero) return a;
+            a = (a % b) as T;
+            if (a == zero) return b;
+            b = (b % a) as T;
+          }
+        };
+        expect(gcd(fibo(a), fibo(b), 0n)).toBe(fibo(gcd(a, b, 0)));
+      })
+    );
+  });
+});

example/001-simple/fibonacci/src/fibonacci.ts

@@ -0,0 +1,10 @@
+export function fibo(n: number) {
+  let a = 0n;
+  let b = 1n;
+  for (let i = 0; i !== n; ++i) {
+    const previousA = a;
+    a = b;
+    b = previousA + b;
+  }
+  return a;
+}

example/README.md

@@ -15,6 +15,7 @@ Following examples show how you could think about properties given algorithms ra
 101 for property based:
 
 - `decompPrime` - Returns the list of prime factors corresponding to the input value
+- `fibonacci` - Returns the item at a given position in the sequence of Fibonacci
 - `indexOf` - Returns the position of the first occurrence of `pattern` in `text`
 - `sort` - Returns a sorted copy of the input array
 

example/tsconfig.json

@@ -1,7 +1,7 @@
 {
     "compilerOptions": {
       "downlevelIteration": true,
-      "target": "es5",
+      "target": "es2020",
       "lib": ["dom", "dom.iterable", "esnext"],
       "allowJs": true,
       "skipLibCheck": true,