Let λbe a list of real numbers. Define sumsqr (λ)=[ if λ=() then 0 else head (λ) head(λ)+su

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Let λbe a list of real numbers. Define sumsqr (λ)=[ if ...

Let $\lambda$ be a list of real numbers. Define $$ \text { sumsqr }(\lambda)=[\text { if } \lambda=() \text { then } 0 \text { else head }(\lambda) \operatorname{head}(\lambda)+\operatorname{sumsqr}(\operatorname{tail}(\lambda))] \text {, } $$ where head $(\lambda)$ head $(\lambda)$ is the numerical product of the head of $\lambda$ with itself. The sumsqr function returns the sum of the squares of all members of $\lambda$. Now let $\lambda=\left(x_1 \cdots x_k\right)$ be any nonempty list of real numbers. Prove that: if $\operatorname{sumsqr}(\lambda)=0$, then every member of $\lambda$ is 0 . In other words, for $i=1, \cdots, k, x_i=0$. Use induction on $k=\operatorname{length}(\lambda)$, and assume basic properties of the real numbers.

Let $\lambda$ be a list of real numbers. Define
$$
\text { sumsqr }(\lambda)=[\text { if } \lambda=() \text { then } 0 \text { else head }(\lambda) \operatorname{head}(\lambda)+\operatorname{sumsqr}(\operatorname{tail}(\lambda))] \text {, }
$$
where head $(\lambda)$ head $(\lambda)$ is the numerical product of the head of $\lambda$ with itself. The sumsqr function returns the sum of the squares of all members of $\lambda$. Now let $\lambda=\left(x_1 \cdots x_k\right)$ be any nonempty list of real numbers. Prove that: if $\operatorname{sumsqr}(\lambda)=0$, then every member of $\lambda$ is 0 . In other words, for $i=1, \cdots, k, x_i=0$. Use induction on $k=\operatorname{length}(\lambda)$, and assume basic properties of the real numbers.

Key Concepts

Mathematical Induction

Mathematical induction is a method of proving statements that are asserted about integers (typically natural numbers). The proof consists of a base case, where the statement is verified for an initial value (often 0 or 1), and an inductive step, where one assumes the statement for an arbitrary integer k and then shows it holds for k+1. This technique is essential for establishing properties that are defined recursively or that naturally extend over a countable set, such as the length of a list.

Recursive Definitions

Recursive definitions specify an object in terms of simpler, smaller instances of itself. In this context, the function sumsqr is defined using a base case for the empty list and a recursive case that breaks down the list into its head (first element) and tail (remaining elements). This definition allows properties of the function to be proven by analyzing how the recursive structure unfolds, often with induction.

Properties of Squared Real Numbers

Squaring a real number results in a non-negative value, and the only time a squared value is zero is when the original number itself is zero. This key property of real numbers underpins the argument that if a sum of squares (which are all non-negative) equals zero, then each individual term must be zero. This fact is crucial in proving that all elements of a list must be zero if the sumsqr function returns zero.

List Data Structures and Recursion

Lists are fundamental data structures that consist of ordered collections of elements, and operations on lists are often defined recursively (for example, viewing a list as a head element followed by a tail list). Understanding how to manipulate lists recursively is important in computer science and mathematics, particularly when applying induction to prove properties about functions defined on lists.

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