Question

The tail function strips off the first element of a list and returns a list of the remaining elements, so that $$ \text { length }(\operatorname{tail}(\lambda))=\text { length }(\lambda)-1 $$ for any nonempty list $\lambda$. Let $\lambda$ be any list with length $n \geq 1$. Let $1 \leq k \leq n$. Define the function $f$ as follows: $$ f(k, \lambda)=[\text { if } k=1 \text { then } \operatorname{tail}(\lambda) \text { else tail }(f(k-1, \lambda))] . $$ Prove that $k+$ length $(f(k, \lambda))=$ length $(\lambda)$.

   The tail function strips off the first element of a list and returns a list of the remaining elements, so that
$$
\text { length }(\operatorname{tail}(\lambda))=\text { length }(\lambda)-1
$$
for any nonempty list $\lambda$. Let $\lambda$ be any list with length $n \geq 1$. Let $1 \leq k \leq n$. Define the function $f$ as follows:
$$
f(k, \lambda)=[\text { if } k=1 \text { then } \operatorname{tail}(\lambda) \text { else tail }(f(k-1, \lambda))] .
$$
Prove that $k+$ length $(f(k, \lambda))=$ length $(\lambda)$.

Logic, sets, and recursion

Robert L. Causey 1st Edition

Chapter 3, Problem 46

Instant Answer

Step 1

In this case, we have $f(1, \lambda) = \text{tail}(\lambda)$. We can see that $k + \text{length}(f(k, \lambda)) = 1 + \text{length}(\text{tail}(\lambda)) = \text{length}(\lambda)$, which satisfies the given equation. Now, let's assume that the equation holds for Show more…

Show all steps

Thanks for your feedback!

The tail function strips off the first element of a list and returns a list of the remaining elements, so that $$ \text { length }(\operatorname{tail}(\lambda))=\text { length }(\lambda)-1 $$ for any nonempty list $\lambda$. Let $\lambda$ be any list with length $n \geq 1$. Let $1 \leq k \leq n$. Define the function $f$ as follows: $$ f(k, \lambda)=[\text { if } k=1 \text { then } \operatorname{tail}(\lambda) \text { else tail }(f(k-1, \lambda))] . $$ Prove that $k+$ length $(f(k, \lambda))=$ length $(\lambda)$.

Play audio

Feedback

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later

Please give Ace some feedback