Proof theory is an area of logic that studies proof as formal mathematical objects. For questions asking how to write proofs or for checking an informal proof, please use the proof-writing or proof-strategy tags instead.

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Disequality in Type Theory

Is it possible to prove $0 \neq 1$ in (non-univalent) Martin-Löf type theory, where $0$ and $1$ are natural numbers (defined using the usual inductive type $0 : \mathbb{N}$, $S : \mathbb{N} \to ...
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Prove the Cauchy-Schwarz Inequality (missing a step)

during lecture notes I only caught most of the proof and couldnt write a step down fast enough, and I'm having a touch trouble seeing how to get from the previous step to the next. Here is what i have ...
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Hyperbolic line with isometry. [on hold]

I have been doing some research to help solve for models of hyperbolic geometry. For a function f(z) in the following conditions are equivalent: f sends two points of l(0) to points in l(0), f is ...
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Inconsistent theory with uniformly long refutation?

I understand that there are theorems in PA that necessarily require "very long" proofs; cmp. [1]. On the other hand it seems interesting to think about Life in an inconsistent world. So is it ...
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Why can't you prove the law of the excluded middle in intuitionistic logic (for layman)?

I am learning about the difference between booleans and classical logics in Coq, and why logical propositions are sort of a superset of booleans: Why are logical connectives and booleans separate in ...
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Can we extend NP-HARD proof of one problem to another?

I have tried to put my question in following diagram I have two domains A and B. I can formulate two optimizations problems 1 and 2. In domain A, efficient algorithms exist for solving these two ...
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prove number is an integer [closed]

So I have the following statement: $a$ is a positive integer and $x = \sqrt[n]{a}$ that has the charesteristic $x^n=a$. Show that $x$ is a rational number. I know that a rational number is on ...
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Why does the dependent product type need “forall”?

I feel stupid asking this question because it is so fundamental to logic and math. However, in my starting to learn proof theory and now type theory, I have not seen an explanation on why you need the ...
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Completeness for Infinitary Logic?

I have heard a rumor that there is a proof system for certain infinitary logics, given by Carol Karp (?) in her thesis, but I can't find a copy. The result that I'm told exists is the following: A ...
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Is it possible to nonconstructively prove that a statement can be proven or disproven within a formal system?

I've heard of many examples of statements that have been proven to be independent of a formal system, meaning that they can't be proven within that formal system (for example, the Continuum Hypothesis ...
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58 views

Natural deduction proof from falsehood

How does a natural proof of $⊥\rightarrow A$ (let $A$ be an arbitrary formula) look like in the classical calculus of natural deduction? Thanks
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Natural Deduction rules for $\lnot$ in classical and intuitionstic logic

Following the very useful answer by Peter Smith to my prevoius post , I'm still reflecting about the "imperfection" connected with the Intro- ans Elim-rules for $\lnot$ in Natural Deduction (I mean ...
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Pythagorian theorem in language of Hilbert's system of geometry

How can one formulate the Pythagorian theorem in the language of Hilbert's system of geometry? How can one speak about the length of the hypotenuse for example?
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Proving that $x^2 + 4$ is not divisible by $3$

I need to show the following: For any integer $x$, $x^2 + 4$ is not divisible by $3$. I was trying proof by contraposition, but I do not believe that is the most efficient way to go about this. ...
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WLOG and “by symmetry” arguments and the foundations of mathematics

John Harrison's paper Without Loss of Generality raises the interesting point that although "without loss of generality"/"by symmetry" arguments are a common proof technique, there is no corresponding ...
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Can all theorems be deduced directly from the ZFC axioms?

I stumbled upon a website called metamath that claims to be able to do this : http://us.metamath.org/mpegif/mmset.html
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Proof of cut elimination

I am reading Proofs and types and am blocked at the proof of cut elimination in sequent calculus (chap 13). I don't see either how the cuts are being pushed up above the preceding steps to the top of ...
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mathematical proof vs. first-order logic deductions

For a long time I thought that the standard mathematical proofs, only were an informal or imperfect way of writing the proof in the language of first-order logic. When I say standard mathematical ...
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Proving the dimension of the intersection of 2 subspaces

Assume that $U$ and $W$ are distinct subspaces $( U ≠ W )$ of a four-dimensional vector space $V$ and $\dim(U) = \dim(W) = 3$. Prove that $\dim ( U ∩ W ) = 2$.
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proof of validity of tautology in first order logic

Every first-order logic formula which has a tautological shape in propositional logic is a valid formula. Will it be possible to give a formal proof for the above ? Thanks and Regards.
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Proof by reflection and Homotopy Type Theory

I have been looking into various proof assistants, and came across the method of proof by reflection in Coq, which (from what I understand) allows one to verify a program provides the correct "answer" ...
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1answer
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How to show that the displaying numbers of a onto function is k!S(n,k)?

Let it be $A$,$B$ sets that $|A|$=$n$, $|B|$=$k$ and $|A|>|B|$. How to show that the displaying numbers of an onto function $f$:$A$ $\rightarrow$ $B$ is: $\begin{Bmatrix} n \\ k\end{Bmatrix}$$k!$ ...
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prove that this number contains two equal digits

We delete the first digit from the number $7^{1996}$ and then we add it to the remaining number, repeat this until we get a number consisting of $10$ digits, prove that, this number contains two equal ...
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Rigorous proof of '${\{A \Rightarrow B}\} \iff {\{\neg B \Rightarrow \neg A}\}$' for a high school student

One method to prove the statement 'If A, then B' is to prove that 'If not B, then not A'. First time that I saw this method it was not (and still isn't) obvious. So I used a more obvious example to ...
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Prove that to any three numbers positive integers [closed]

Prove that for any three positive integers, following equality holds $$\operatorname{lcm}(ab , bc , ca ) \cdot \gcd(a , b, c )=abc$$
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Proving a trigonometric identity with tangents [closed]

Prove that: $$\tan^227^\circ +2 \tan27^\circ \tan36^\circ=1$$ any help, I appreciate it.
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How do I know which of these are mathematical statements?

While reading this book called "How to Read and do Proofs" by Daniel Solow(Google) I found the following exercise at the end of the first chapter. So how do I know if something is a mathematical ...
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Is Fermat's Last theorem equivalent to $1 + 1 = 2$? [closed]

I got into a debate with someone concerning whether FLT is equivalent to $1 + 1 =2$. He said common sense tells us it isn't equivalent. However, I disagreed. Since both are provable statements, they ...
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1answer
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Can we see natural deduction rules as functions or even as formal grammars?

Is there a way of seeing natural deduction rules as functions or even as formal grammars, maybe context-free grammars or Lambek grammars? It seems quite "easy" to see the rules as functions which take ...
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68 views

Using Sequent Calculus to prove $\exists x_1 x_2 [ B ( x_1 , x_2 ) \rightarrow \forall y_1 y_2 B ( y_1 , y_2 ) ]$

I need to prove the validity of the following formula using the sequent calculus LK: $$ \exists x_1 x_2 [ B ( x_1 , x_2 ) \rightarrow \forall y_1 y_2 B ( y_1 , y_2 ) ] \text{.} $$ I already had a look ...
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Explanation on the symmetry between identity axiom and cut rule

In Proofs And Types at the beginning of 5.1.4 Girard says that the identity axiom is somewhat complementary to the cut rule, more specifically 'The identity axiom says that $C$ (on the left) is ...
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Weakening and Contraction

I saw this site saying weakening is a structural rule where the hypotheses or conclusion of a sequent may be extended with additional members and that contraction is a rule where two equal (or ...
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What does “rigorous proof” mean?

I have heard several times that some mathematician has given another and more rigorous for an established theorem, but I don't know what does it really mean and what differences makes it to be more ...
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Problem with a step in a proof in predictive control

I'm trying to follow a demontration written in an optimal control paper. In one of the steps, it states What I'm having troubles with is the last step, it states that because of the convexity of ...
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What are some arguments/counterarguments for Zeilberger's “proof certificates”?

Here is the quote I wish to ask about: "I speculate that similar developments will occur elsewhere in mathematics, and will 'trivialize' large parts of mathematics, by reducing mathematical ...
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Encyclopedia of Mathematical Proofs with no English

I was wondering if anyone is aware of a modern book that builds a subset of elementary number theory from Peano axioms preferably in a Principia Mathematica fashion? Or similarly an encyclopedia of ...
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Consistent, complete axiom system that proves its own consistency

Is there a consistent, complete axiom system that proves its own consistency? I know that this question isn't exact and I haven't defined when an axiom system proves its own consistency because ...
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Is it always possible to algebraically express a function defined by a set of rules?

Let's say you have an arbitrary function defined by a set of rules such that for example: Domain $\hspace{9mm}$ Range $\hspace{5mm}$ 1 $\hspace{23mm}$ 2 $\hspace{5mm}$ 2 $\hspace{23mm}$ 2 ...
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What kind of trivial statement still needs to be proven?

There are many statements that seem to be needless of a proof since they are ‘evident’ mainly because of our intuition. But some of them have proofs. For example, in C. Adams’ Introduction to ...
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Clever proofs that prove one identity is equal to another, without going through the original identity?

In a previous question I attempted to formalize the argument of going from one proof of an identity to another, which turned out to be harder than I thought. The thing is, while it may be impossible ...
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Is it always possible to go from one identity to another?

This question was inspired by this Quora question. I'm sure lots of you are familiar with the fact that we have many different representations of $\pi$, things like $$ \begin{align} \pi & = ...
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Prove $ \{(p \lor q) \land (p \implies r) \land (q \implies r) \} \implies r$ is a tautology using logical properties

I spent quite a bit of time on this and have little to no ideas on how to proceed. Using the conditional laws and De Morgan's law, I got to $$( \sim p \land \sim q) \lor (p \land \sim r) \lor(q ...
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Proof by Induction Divisibility.

$6^n-5n+4$ is divisible by 5 for all positive integers $n$. $n >=1$ Prove By Induction My attempt is as follows: $n=1$ $6^1-5(1) +4$ $=5$, Therefore 5 is divisible by 5 so $n=1$ is true ...
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183 views

What if 'proof by contradiction' is not a valid method of proof?

I've just been reading this question about the existence (or lack thereof) of contradictions in maths. I've been wondering: What if 'proof by contradiction' is not a valid method to (dis)prove a ...
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Isomorphic: Properties Proved for A are True for B?

To make this a little more concrete, consider vector spaces. An isomorphism between two spaces is an invertible linear transformation. It seems to then be commonly asserted or assumed that if A and ...
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Using the notion of provability only, how to show that $\Gamma \nvdash \varphi$?

For a practical example, suppose I want to show that $\{ P\} \nvdash Q$. From completeness, this is trivial: just find a model where $P$ is true and $Q$ false. But suppose I am stubborn and I don't ...
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If $T$ a consistent set of sentences and $a,b$ sentences such that $T\vdash (a\rightarrow b)$and $T\vdash (\lnot a\rightarrow b)$ Then $T\vdash b$ [closed]

I am stucked at this problem for a long time: Let $T$ be a consistent set of first-order sentences and let $\alpha,\beta$ be sentences. Prove that if $T\vdash( \alpha\rightarrow \beta)$ and ...
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Idea of a proof by contradiction

Is the idea of a proof by contradiction to prove that the desired conclusion is both true and false or can it be any derived statement that is true and false (not necessarily relating to the ...
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Proving that $\sum_{i=2}^n(5i-4)=\frac{n(5n-3)-2}{2}$ for all $n\geq 1$ by mathematical induction

I have this question: Show, using mathematical induction, that for all natural numbers $n$, $$6 + 11 + 16 + 21 + \cdots + (5n-4) = \frac{n(5n-3)-2}{2}$$ I am confused in that that question states ...
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Prove by Natural deduction that $\lnot\exists xP(x)\rightarrow\forall x\lnot P(x)$

I got this problem: Prove by Natural deduction in First Order Logic that $\lnot\exists xP(x)\rightarrow\forall x \lnot P(x)$ I tried to prove it using the Contradiction Theorem but I got ...