Questions tagged [induction]

Questions about mathematical induction and inductive proofs.

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Failing to solve a recurrence by induction

My question is strongly related to the one asked here: How do I show T(n) = 2T(n-1) + k is O(2^n)? $$T(n)=2T(n-1)+1$$ Going with the steps, I reached the point where: $$c*2^{n}\geq c*2^{n}+1$$ which ...
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How can I prove the correctness of this exponentiation algorithm using induction?

I have the following algorithm. How could I prove it using induction that for every $n\ge 0$, Exp(n)${}= 2 ^ n$? ...
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Is Inductive Logic Programming approach applicable to general theories (not just sets of Horn clauses)?

Inductive Logic Programming (https://en.wikipedia.org/wiki/Inductive_logic_programming) find hypothesis theory H for background theory B and set of examples E. ILP algorithms and implementations ...
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1answer
20 views

How to show a function is primitive recursive by induction?

I know, loosely speaking, if we can define a function $f$ in term of \begin{align} &f(0,\vec{x})=g(\vec{x})\\ &f(n+1,\vec{x})=h(f(n),n,\vec{x}) \end{align} where functions $g,h$ are primitive ...
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Proving that grammar generates language with multi variable inequalities

I am struggling with trying to come up with the inductive proofs to prove that the following grammar is equal to $ L = \{0^m1^n | m \leq 2n, n \leq 2m \}$ The grammar is $ S → A \:|\: B $ $ A → 00A1 \:...
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1answer
26 views

Using inductive hypothesis on recurrence relation?

I have a recurrence relation as follows $$T(n) = 2T(\lfloor n/2\rfloor) + n\log(n)$$ Using the induction hypothesis how do I obtain a relation $T(n)\leq E$ such that $E$ contains neither $T$ nor floor ...
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1answer
33 views

Determining which recursive term is bigger if they share the same definition

We are given a recursive definition: $a_1 = x,\\a_2=y, \\a_n= c_1a_{n-1}+c_2a_{n-2} \text{ for }n\ge3 $ where $x,y,c_1,c_2,n$ are natural numbers we are to prove that $a_n \le c_3^n$ for all n The ...
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Is my understanding of strictness correct in this proof of a `foldl` rule?

Exercise G in Chapter 6 of Richard Bird's Thinking Functionally with Haskell asks the reader to prove foldl f e . concat = foldl (foldl f) e given the rule ...
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Why are mathematical proofs so hard?

I am an electrical engineer and trying to make a transition into machine learning. I read in multiple articles that I have to learn data structures and algorithms, before this I have to learn about ...
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Proving building a balanced BST out of sorted array is $\Theta(n)$

I'm having hard time proving building a balanced BST out of sorted array is $\Theta(n)$ I got the following formula: $$T(n)=2T(\frac{n}{2})+\Theta(1)$$ I tried to prove it by induction but got stuck ...
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1answer
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on coq: Why is the proof complete after proving only for one induction when we have more than one variable?

So I'm learning coq. And I came across the proof for associativity in addition forall (a b c : nat) Appearntly when we do ...
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1answer
30 views

Prove by induction that the recurrence form of bubble sort is $\Omega(n^2)$

The recurrence form of bubble sort is $T(n)=T(n-1)+ n- 1$ How can I prove by induction that this is $\Omega(n^2)$? I'm stuck with $T(n+1) \geq cn^2 + n = n(cn+1)$
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178 views

Given a tournament with $2^n$ vertices, show that there is a sub-tournament with at least $n + 1$ vertices that is acyclic

So a tournament is just a complete directed graph, I believe. I'm having trouble proving this problem. I know it is induction however. I was thinking the base case is $2^1$ vertices, and therefore ...
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Prove that a red-black tree with $n$ internal nodes has height at most $2\lg(n+1)$

I cannot understand the first paragraph of the proof, which comes from the known book Introduction to Algorithms, third-edition, and I consider it has some errors, could anyone help me check about it? ...
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1answer
89 views

When can the coinduction hypothesis be used?

We can use the induction hypothesis when we are proving a property for a structure that is well-ordered. I am aware that there is a proof for this. When it comes to coinduction, I'm confused. One of ...
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Induction pitfalls with O notation and recursion

I read the following in CLRS 3rd Ed: I'm not sure I understand exactly how to avoid this pitfall. How would one know that the $\mathcal{O}$ notation in this case grows with $n$ and is thus not ...
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1answer
50 views

Which inductive schemes can encode the following Agda definition?

Which induction schemes (e.g. induction-recursion by Dybjer and Setzer, "Irish" induction-recursion by McBride or induction-induction by Forsberg and Setzer or perhaps some simpler ones) allow one to ...
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solving a problem with induction

The original question is the following prove that $2·\sum_{i=0}^{n-1} 3^{i} = 3^n-1$ for all n $\geq$ 1 I know that I have to prove by induction and have successfully done the base case, my IH is ...
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1answer
78 views

Deriving recursive definition from function specification

Given this function specification, where name xs is bound to a list, # denotes its cardinality and ...
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33 views

Hanoi towers recursive expression for EVERY algorithm

What the recursive algorithm for moving $n$ disks says, is: If $n > 1$, move $n-1$ discs from A to B. Move the $n$th disk from A to C. If $n > 1$, move $n-1$ discs from B to C. Let $T_n$ be ...
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Induction proof given recurrence of algorithm

I am having trouble starting this proof and wanted some clarification. Here are the details given: ...
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Using induction prove that a K-SAT problem is NP-Complete

Using induction on k, how do I prove that the K-SAT problem is NP-complete? On wikipedia, it describes the Cook-Levin theorem to prove that K-SAT is NPC by reducting the K-SAT problem to a circuit-...
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Prove grade-school multiplication algorithm applied to binary numbers

I want to prove that the basic multiplication algorithm is correct when applied to binary numbers. I try to follow the steps described here and here but didn't succeed. The basic implementation ...
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Rules for consistency with mutual inductive families?

I'm trying to use a proof assistant to define a type and a relation that are mutually dependent on each other: ...
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How to prove a recursive's function Big-Theta without using repeated substitution, master theorem, or having the closed form?

I have a function defined: $V(j, k)$ where $j, k \in \mathbb{N}$ and $t > 0 \in \mathbb{N}$ and $1 \leq q \leq j - 1$. Note $\mathbb{N}$ includes $0$. $V(j, k) = \begin{cases} tj & k \leq 2 \\...
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1answer
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Proving that a property is k-inductive with an SMT solver (parametric resettable counter)

I'm following the slides at https://homepage.cs.uiowa.edu/~tinelli/talks/FT-11.pdf where Tinelli explains how k-induction works in the context of SMT based model checking. A parametric and resettable ...
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1answer
105 views

Proving a DFA recognizes a language using induction

The following DFA recognizes the language containing either the substring $101$ or $010$. I need to prove this by using induction. So far, I have managed to split each state up was follows: q0: ...
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Proof that this sorting algorithm sorts the input [duplicate]

I'm given this "sorting" algorithm and now I'm supposed to prove, that if given an array of integers of length $n$, sort(A,0,n-1) will sort it. ...
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1answer
231 views

Recursive definition for the length of a string?

I found a couple of answers online but I don't quite understand why the answer is right: The length of a string is: If a string has no characters, then its length is 0. Otherwise, the length of the ...
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94 views

proving DFA stuck

This DFA fulfills: Define a function $diff: \{0,1\}^*\to\Bbb Z$, for $w \in\{0,1\}^*$, $diff(w)=($# of 1's in $w)- ($# of 0's in $w$). Thus, $diff(\epsilon)=0$; $diff(0)=−1$; $diff(1)=1$. Let $L = ...
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dfa subtract multiple of 3 [duplicate]

Define a function 𝑑𝑖𝑓𝑓 ∈{0,1}→ℤ so: for everything w ∈{0,1}, diff w = # of 1's in w- # of 0's in w. Thus: 𝑑𝑖𝑓𝑓 𝜀=0; 𝑑𝑖𝑓𝑓 0=−1; 𝑑𝑖𝑓𝑓 0=−1; Let 𝐿 = {𝑤∈ {0,1} * | 𝑑𝑖𝑓𝑓 𝑤 = 3𝑚 ...
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2answers
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prove by induction that the complete recursion tree for computing the nth Fibonacci number has n leaves

I have referenced this similar question: Prove correctness of recursive Fibonacci algorithm, using proof by induction *Edit: my professor had a significant typo in this assignment, I have attempted ...
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Inductive proof on Quicksort with Explicit Stacking

Prove by induction that if Quicksort with Explicit Stacking is modified so that the end-points of the larger sublist are stacked, and the other sublist is sorted first, then the maximum stack size is $...
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1answer
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Is my proof of my greedy algorithm to find subsequence correct?

Credit to KleinBerg and Taros Book Some of your friends have gotten into the burgeoning field of time-series data mining, in which one looks for patterns in sequences of events that occur over time. ...
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568 views

Iterative Fibonacci algorithm correctness proof, finding loop invariants

The algorithm take in an integer $n$ and outputs the $n$th number in the Fibonacci sequence ($F_n$). The sequence starts with $F_0$. I am trying to prove the correctness assuming valid input: ...
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1answer
257 views

Induction on strings (words)

Given is an alphabet $\Sigma = \{ 0, 1, 2 \}$ and a function quer to calculate the cross sum of a word. $quer : \Sigma^*\to \Bbb N$ with: $$quer(w)=\begin{cases} 0, &\text{when } w=\epsilon\\ ...
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How does one know what statements in Coq require Induction?

I was trying to learn Coq using the famous book Software Foundations. In it I found the following: ...
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1answer
392 views

Induction to prove equivalence of a recursive and iterative algorithm for Towers of Hanoi

Using induction how do you prove that two algorithm implementations, one recursive and the other iterative, of the Towers of Hanoi perform identical move operations? The implementations are as follows....
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2answers
660 views

Proof of QuickSort algorithm correctness

Recently I’ve studied QuickSort and understood its general idea. Basically, we do the following: Pick an element from the array (no matter which one and how in this context) Rearrange elements in ...
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Is this a valid induction proof example ?

Learning induction proof now, found a "simple" example, which is a bit confusing to me (not sure if it is a valid example). If so, why the IH( suppose a root of rank k has at least $2^k$ vertices in ...
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119 views

How to prove that the predecessor of each node in Dijkstra form a tree?

Prove that the array prev[.] computed by Dijkstra’s algorithm, the edges (v, prev[v]) for all v ∈ V , form a tree In order to prove this I used induction. Lemma :...
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2answers
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Why does the denotational semantics for a while loop have a existence quantifier?

I was going through these notes and they have the following operator on partial functions: $$ \mathcal F^{k}(\bot)(\sigma) = \left\{ \begin{array}{ll} \alpha( [\![s]\!]\sigma ) & [\![b]\!]\...
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PL: How can I prove the type of something using “Inversion for Typing”?

I'm currently going through this book about programming languages, and in section 4.2, Lemma 4.2 it says this: Lemma 4.2 (Inversion for Typing). Suppose that $\Gamma \vdash e : \tau$. If $e = \texttt{...
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1answer
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How to prove even with structural induction, for expression

E::= zero|two|expression + expression|expression*expression, E element of expression how do i prove E to be even. I have no clue on how to go about tackling this.
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1answer
90 views

How to prove that a string is made up of subsequences occurring some arbitrary number of times using concatenation?

How to prove that a string, s is made up of n > 1 subsequences occurring some arbitrary number of times using concatenation ...
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1answer
147 views

Induction on typing derivation in refinement types system

From the text Principles of Type Refinement page 14: The author introduces in definition 2.2.7 the rule: $$ \dfrac{\Pi \vdash t : R \qquad R \le S}{\Pi \vdash t : S} $$ and gives the following ...
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644 views

Is Loop Invariant Proof a form of Induction?

As far as I see, what computer scientists refer to as loop invariant proofs are exact replicas of induction proof. Is it true? Can I state that loop invariant proof implies an induction? Is there a ...
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1answer
202 views

Proving time-complexity analysis for merge-sort-like algorihtm

I have this algorithm, which is exactly like merge-sort, but instead of halving the array each recursion, it actually splits it into $1/4$ and $3/4$ parts. Other then that, it does exactly the same ...