Questions tagged [induction]

Questions about mathematical induction and inductive proofs.

29 questions with no upvoted or accepted answers
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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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96 views

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 = \...
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29 views

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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Induction / (machine learning) of Resource Grammars for Grammatical Framework?

Grammatical Framework is based in Abstract Categorial Grammars. It is known that Combinatory Categorial Grammars have grammar induction/learning capabilities see e.g. https://link.springer.com/article/...
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50 views

Structural induction in non-local program transformation

Assume a functional language and a specialization operation (pulling out sub-expressions): let f x y = (h 23 x) + (g 42 y) becomes ...
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67 views

Self referential data

Can anyone show me step by step how to do a proof on self referential structures. For example, this example is in scala ...
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3k views

Insertion sort Proof by Induction

I am reading Algorithm design manual by Skiena. It gives proof of Insertion sort by Induction. I am giving the proof described in the below. Consider the correctness of insertion sort, which we ...
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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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60 views

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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65 views

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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108 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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46 views

Proving that $(A \cup B)^* = A^*(BA^*)^*$

I would like to prove that $(A \cup B)^* = A^*(BA^*)^*$, where * means the Kleene star. I would like to use induction to prove this equality but I do not how to proceed and how is the best way to set ...
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128 views

Program equivalence with Structural induction

I recently attended my first lecture on structural induction and the professor stated that structural induction can be used to to prove that two programming languages are equivalent. I am new to ...
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90 views

Proving that a BST with N>=1 nodes will have log(N+1) levels

I am trying to prove by induction the following theorem: Use Induction to prove the following fact: for every integer, $N\ge 1$ , a BST with $N$ nodes must have at least $\log( N + 1)$ levels. I've ...
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36 views

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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75 views

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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67 views

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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40 views

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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62 views

Is this proof of the map fusion law correct?

I am reading Functional Programming in Scala, where I have been asked to prove the map fusion law. Since Scala is what I am familiar with, I am using as my notation a kind of pseudo-Scala. Here is ...
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24 views

Structural induction a particular case of (co)induction rule of an inductive predicate

What is the relation between the principle of induction on terms: ...
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41 views

Proof that this sorting algorithm sorts the input

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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83 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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151 views

When proof by induction on length string is not possible?

I found out an exercise where you have to prove the correctness of the following CFG: Let $L=\{ 0^i 1^j|2i \leq j \leq 3i \}\:$ and $\: G: S\rightarrow 0S11 | 0S111| \epsilon$ claim: Every string $w ...
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73 views

Inductive proof that $n^2 + bn + d$ is $O(n^2)$ using definition of big O

Given that $T(n) = n^2 + bn + d$ then it's $O(n^2)$ if I can prove that: $O(n^2) = \{T(n): \text{there exist positive constants } c, n_0 \text{ such that } \forall n \geq n_0, 0 \leq T(n) \leq cn^2 \}$...
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216 views

How does the induction proof work in this example?

Refer to answer 1.1 of this file: http://www.dei.unipd.it/~geppo/AA/DOCS/NPC.pdf From my understanding and this thread, https://math.stackexchange.com/a/928412, we need 3 steps for that proof. ...
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41 views

How would one prove that the following scheme definition is an ordered stream of integers

How would one prove that the following scheme definition is an ordered stream of integers (define integers (cons-stream 1 (add-streams ones integers)))
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564 views

Describe the language generated by a given context free grammar

I had an exercise: Describe the language generated by the following given context free grammar and prove it by induction. $$\begin{align} S &\to SA \mid \epsilon \\ A &\to aS \mid bA \...
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2answers
331 views

proof using induction of automaton

How I can explain this. Consider the following automaton, $A$. Prove using the method of induction that every word/string $w\in L(A)$ contains an odd number(length) of $1$'s. Show that there are ...
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1answer
557 views

Proving correctness of a recursive algorithm using induction

For the program mean(A,n) if n = 1 then return A[n] else return A[n]/n+mean(A,n-1)*(n-1)/n end Show that if the recursive call to ...