For this problem, do not assume that i^k has already been...

For this problem, do not assume that $i^k$ has already been defined for Nat. You may assume the results in Section 3.2. For $i, k \in N a t$, define $$ r(i, k)=[\text { if } k=0 \text { then } 1 \text { else } i r(i, k-1)], $$ where $i r(i, k-1)$ is the product of $i$ with $r(i, k-1)$. Notice that $r(0,0)=1$. Prove the following: 1. For $k \in N a t^{+}, r(0, k)=0$. 2. For any $i, r(i, 1)=i$. 3. For any $i, j, k \in N a t, r(i, j+k)=r(i, j) r(i, k)$. 4. For any $i, j, k, r(r(i, j), k)=r(i, j k)$. [Remark. This problem defines $i^k$ and establishes some of its basic properties. Intuitively, $i^k$ equals the result of multiplying $i$ by itself " $k$ times," although the case $k=0$ does not exactly fit this description. The case $0^0$ is curious. Compare this problem with the previous one.]

For this problem, do not assume that $i^k$ has already been defined for Nat. You may assume the results in Section 3.2. For $i, k \in N a t$, define
$$
r(i, k)=[\text { if } k=0 \text { then } 1 \text { else } i r(i, k-1)],
$$
where $i r(i, k-1)$ is the product of $i$ with $r(i, k-1)$. Notice that $r(0,0)=1$. Prove the following:
1. For $k \in N a t^{+}, r(0, k)=0$.
2. For any $i, r(i, 1)=i$.
3. For any $i, j, k \in N a t, r(i, j+k)=r(i, j) r(i, k)$.
4. For any $i, j, k, r(r(i, j), k)=r(i, j k)$.
[Remark. This problem defines $i^k$ and establishes some of its basic properties. Intuitively, $i^k$ equals the result of multiplying $i$ by itself " $k$ times," although the case $k=0$ does not exactly fit this description. The case $0^0$ is curious. Compare this problem with the previous one.]

Key Concepts

Recursive Definitions

This concept involves defining functions or sequences in terms of themselves, usually by specifying a base case and a recursive step. In this problem, exponentiation is defined recursively for natural numbers: r(i, 0) is defined as 1, and r(i, k) is defined as i multiplied by r(i, k-1), which mirrors how many fundamental operations in mathematics are built from simpler cases.

Exponentiation

Exponentiation is the operation of raising a number (the base) to the power of another number (the exponent), resulting in repeated multiplication of the base. The recursive definition given for r(i, k) formalizes this process for natural numbers and encapsulates the idea that i^k (or r(i, k) here) equals the product of i multiplied by itself k times.

Properties of Exponentiation

These properties include the identity property (i raised to the power 1 equals i), the product rule (i raised to the sum of two exponents equals the product of i raised to each exponent separately), and the power of a power rule (raising an exponential expression to another exponent results in the multiplication of exponents). These rules are essential for simplifying expressions and proving further identities involving exponents.

Mathematical Induction

This is a proof technique used to establish that a statement holds for all natural numbers by verifying the base case and proving that if it holds for an arbitrary natural number k, it then holds for k+1. Although not explicitly mentioned in the problem statement, induction is a natural method for proving the properties of exponentiation defined recursively.

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