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Questions tagged [theorem]

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms.

A theorem is a mathematical statement that has been proven from axioms and previously established theorems. Theorems are important results in mathematics which need to be proven and have widespread applications in all fields of mathematics and science, including computer science.

A statement that is believed to be true, but hasn't been proven is called a conjecture or hypothesis. An intermediate result that helps in proving a statement is called a lemma, and like theorems, they also have to be proved.

One of the most famous theorems is the Pythagorean theorem.

  • Given a right triangle with known side lengths a, b, meeting at a right angle, the square of the length of the hypotenuse c, opposite to the right angle, is equal to the square of the side length a and b. Mathematically, that can be stated as c^2 = a^2 + b^2, or c = sqrt(a^2 + b^2)
  • There has been over 300 different proofs, including the similar triangles proof and the proof given by Euclid.
  • It is commonplace in geometry and trigonometry. The Pythagorean trigonometric identity states that sin^2(x) + cos^2(x) == 1 for all x, and is a corollary of the Pythagorean theorem.

Another famous theorem is the Euclid's theorem.

  • There are infinitely many prime numbers.
  • The proof has been provided by Euclid's Elements in antiquity.

Theorem Proving automation

It is known that some theorems have been proven using computer assistance, including the famous four color theorem (where no geographical map requires five or more colors if the territories are all contiguous) and the Kepler conjecture, where the best sphere packing is the tie between face-centered cubic and the hexagonal close packing, at 74% filled space.

The field of automated theorem proving, aka ATP has been developed. These include E theorem prover, Isabelle, and ACL2.

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Has anyone tried proving Z3 with Z3 itself?

Has anyone tried proving Z3 with Z3 itself? Is it even possible, to prove that Z3 is correct, using Z3? More theoretical, is it possible to prove that tool X is correct, using X itself?
Longfei Zhu
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cleveref fails for theorem environments sharing the same counter

I want to number all theorems and corollaries on the same (section) counter. But when I do this, cleveref names them both 'theorem'. Here is a minimal example: \documentclass{amsart} \usepackage{amsthm,…
obco
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How to use inline R code in a bookdown theorem or example environment

I use bookdown to generate documents in both html and PDF. How could I use results from inline R code in theorem and example environments? Here is what I tried: --- title: "Test" output: bookdown::pdf_book: toc: false html_document: toc:…
QuantIbex
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Theorem Proof Using Prolog

How can I write theorem proofs using Prolog? I have tried to write it like this: parallel(X,Y) :- perpendicular(X,Z), perpendicular(Y,Z), X \== Y, !. perpendicular(X,Y) :- perpendicular(X,Z), parallel(Z,Y), !. Can you…
Aman
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Böhm-Jacopini theorem

According to Böhm-Jacopini theorem, an algorithm can be written using only three statements: sequence selection iteration Lots of teachers, assumes that theorem as an act of faith, and they teach to not use (goto, jump, break, multiple return,…
Stefano Balzarotti
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prolog first order logic

I'm trying to find a way to put the following first order logic expression into Prolog: (p(0) or p(1)) and not (p(0) and p(1)) This means that it should respond in the following way to queries: ?- p(0) Yes. ?- p(1) Yes. ?- p(0),p(1). No. I tried…
marczoid
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Algorithm cost using master theorem

Can anyone please help me with this question? T(n)=T(n^(1/2)) + theta (lg lg n) This is what I have done so far. Let: m = lg n s(m)=s(m/2) + theta (lg m) Applying the master theorem here a=1 b=2 m^log 2 (1) = m^0 =1 Now I'm stuck.
aneena
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Theorem and Proof Environment in Beamer

I am currently trying to use quarto beamer to making lecture slides. I would like to use the theorem environment in beamer, the qmd file however cannot render properly. Rendering stopped with latex error showing that Command \theorem already…
Ginger Shu-Ying Yeh
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Proving insertion sort algorithm using Isabelle

I did some implementation of the insert sort algorithm in Isabelle/HOL for the generation of code (ML, Python, among others). I'm sure the corresponding functions work fine, but I need to create some theorems to prove it and be super sure it works.…
He_slp13
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How to properly use keyword 'theorem' in Isabelle?

I obtained the following code from Isabelle's wikipedia page: theorem sqrt2_not_rational: "sqrt (real 2) ∉ ℚ" proof assume "sqrt (real 2) ∈ ℚ" then obtain m n :: nat where n_nonzero: "n ≠ 0" and sqrt_rat: "¦sqrt (real 2)¦ = real m / real…
Timothy Swan
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Create a beamer presentation by restating theorems of another tex file

I'm creating a Beamer presentation based on the content of another tex file (a report). To avoid copy-pasting and allow automatic updating of the content, I would like to simply call the specific theorem I want from the original file and restate it…
SGH
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Problem with theorem decimal numbering in LaTeX

Below is the beginning of my LaTeX code. When I compile, theorem numbers do not show the decimal point. For example, the output shows Theorem 11 instead of Theorem 1.1. How can I fix this ? \documentclass[smallcondensed]{svjour3} \smartqed …
user463498
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How to number nonconsecutive addenda in LaTeX without disrupting original Theorem numbering?

I am writing a thesis heavy with Definitions, Theorems, Lemmas and the like. I need to give a basic definition of a concept, and then later in the thesis expand on this definition as more parameters add to the complexity and thus to the definition.…
Rikke Hamilton
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Binomial Theorem - algorithm in C

I'm trying to find a solution(fix errors) in my programme which must count the binomial theorem from definition. Firstly I created the definition of "factorial" - "silnia". 1) The algorithm determines the value of SN1 (n,k) of the definition.…
Maciej Januszewski
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Can SMT solver like Z3 prove 1+2+...+n = n(n+1)/2?

The theorem 1+2+...+n = n(n+1)/2 seems to be provable by first translating it to CNFs and repeatedly applying resolution rules to them, as discussed in the introduction section of [1]. I believe this means the theorem is solvable by SMT solver like…
Jason
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