Difference between Inductive hypothesis and inductive goal

Question

For example:

$\forall x: \forall y: \forall z:$ x * (y + z) = (x * y) + (x * z)

by induction on z, letting x and y be arbitrary.

What would be my inductive hypothesis and inductive goal in this case?

Answer 1

The principle of induction over natural numbers is:

If $P(n)$ is some proposition involving a natural number $n$ and we want to show that $P$ holds for all numbers $n$, we have to :

show that $P(0)$ holds;

show that, for any $k$, if $P(k)$ holds, then so does $P(S(k))$;

conclude that $P(n)$ holds for all $n$.

The goal is proving $P(n)$ for all $n$, which breaks into two separate subgoals:

• first showing $P(0)$ : the base case,

• and then showing $P(k) \to P(S(k))$ : the induction step. In proving this step, we have to assume the induction hypotheses : $P(k)$, and derive $P(S(k))$.