Inductive proof for Euclid's algorithm
Here is Euclid's algorithm.
The input is two integers $x \geq y \geq 1$.
While $x &gt; y$, the algorithm replaces $x,y$ with $y, x\bmod y$.
The final output is $x$.
In order to prove that this algorithm correctly computed the GCD of $x$ and $y$, you have to prove two things:

The algorithm always terminates.
If the algorithm terminates, then it outputs the GCD (partial correctness).

The first part is proved by showing that $x + y$ always decreases throughout the loop. The second part is proved by induction, using the equation $$ \mathit{gcd}(x,y) = \mathit{gcd}(y,x\bmod y). $$
Both parts also use the loop invariant $x \geq y \geq 1$.
What you call the &quot;deductive proof&quot; is only part of the proof of partial correctness. It forms the main technical part inside a proof by induction on the number of iterations.