Second Principle of Recursive Definition

Theorem
Let $\N$ be the natural numbers.

Let $T$ be a set.

Let $a \in T$.

For each $n \in \N_{>0}$, let $G_n: T^n \to T$ be a mapping.

Then there exists exactly one mapping $f: \N \to T$ such that:


 * $\forall x \in \N: \map f x = \begin{cases}

a & : x = 0 \\ \map {G_n} {\map f 0, \ldots, \map f n} & : x = n + 1 \end{cases}$

Proof
Define $T^*$ to be the Kleene closure of $T$:


 * $T^* := \ds \bigcup_{i \mathop = 1}^\infty T^i$

Note that, for convenience, the empty sequence is excluded from $T^*$.

Now define a mapping $\GG: T^* \to T^*$ by:


 * $\map \GG {t_1, \ldots, t_n} = \tuple {t_1, \ldots, t_n, \map {G_n} {t_1, \ldots, t_n} }$

that is, extending each finite sequence $\tuple {t_1, \ldots, t_n}$ with the element $\map {G_n} {t_1, \ldots, t_n} \in T$.

By the Principle of Recursive Definition applied to $\GG$ and the finite sequence $\sequence a$, we obtain a unique mapping:


 * $\FF: \N \to T^*: \map \FF x = \begin{cases} \sequence a & : x = 0 \\ \map \GG {\map \FF n} & : x = n + 1 \end {cases}$

Next define $f: \N \to T$ by:


 * $\map f n = \text {the last element of $\map \FF n$}$

We claim that this $f$ has the sought properties, which will be proven by the Principle of Mathematical Induction.

We prove the following assertions by induction:


 * $\map \FF n = \tuple {\map f 0, \map f 1, \ldots, \map f {n - 1}, \map {G_n} {\map f 0, \ldots, \map f {n - 1} } }$
 * $\map f n = \map {G_n} {\map f 0, \ldots, \map f {n - 1} }$

For $n = 0$, these statements do not make sense, however it is immediate that $\map f 0 = \map {\operatorname {last} } {\sequence a} = a$.

For the base case, $n = 1$, we have:

Now assume that we have that:


 * $\map \FF n = \tuple {\map f 0, \map f 1, \ldots, \map f {n - 1}, \map {G_n} {\map f 0, \ldots, \map f {n - 1} } }$
 * $\map f n = \map {G_n} {\map f 0, \ldots, \map f {n - 1} }$

Then:

The result follows by the Principle of Mathematical Induction.

Also known as
Some authors go through considerable effort to define the sequence $\sequence {G_n}_n$ as a single mapping $G$.

The domain of such a mapping is then for example given as one of the following:


 * $\Dom G := \set {f: \N_{0} } = \ds \bigcup_{n \mathop\in \N} \paren {\N_{0} } T^n$

At we deem the presentation with separate $G_n$ to be more enlightening.

Also see

 * Principle of Recursive Definition
 * Transfinite Recursion