# What is Proof verification: Definition and 19 Discussions

In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer.

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1. ### I Question on proof ##\Lambda^{\perp}(AU) = U^{-1} \Lambda^{\perp}(A)##

Say we have as special lattice ## \Lambda^{\perp}(A) = \left\{z \in \mathbf{Z^m} : Az = 0 \in \mathbf{Z_q^n}\right\}##. We define ##U \in \mathbf{Z^{m \times m}}## as an invertible matrix then I want to proof the following fact: $$\Lambda^{\perp}(AU) = U^{-1} \Lambda^{\perp}(A)$$ My idea: Let...
2. ### I Prove that a triangle with lattice points cannot be equilateral

I assumed three points for a triangle P1 = (a, c), P2 = (c, d), P3 = (b, e) and of course: a, b, c, d, e∈Z Using the distance formula between each of the points and setting them equal: \sqrt { (b - a)^2 + (e - d)^2 } = \sqrt { (c - a)^2 + (d - d)^2 } = \sqrt { (b - c)^2 + (e - d)^2 }(e+d)2 =...
3. ### Proof of Subspace Topology Problem: Error Identification & Explanation

I have already seen proofs of this problem, but none of them match the one I did, therefore I would be glad if someone could indicate where is the mistake here. Thanks in advance.**My proof:** Take a limit point x of U that is not in U, but is in K (in other words x \in K \cap(\overline{U}-U))...
4. ### Indirect Proof Proof verification: sequence a_n=(−1)^n does not converge

Theorem: Show that the sequence ## a_n = (-1)^n ## for all ## n \in \mathbb{N}, ## does not converge. My Proof: Suppose that there exists a limit ##L## such that ## a_n \rightarrow L ##. Specifically, for ## \epsilon = 1 ## there exists ## n_0 ## s.t. for all ## n > n_0## then ##|(-1)^n-L|<1##...
5. ### Inequalities Since ε is arbitrarily small, do the inequalities hold?

#### If ##b \leq x_n \leq c## for all but a finite number of n, show that ##b \leq \operatorname{lim inf}_{n \to \infty} x_n## and ##\operatorname{lim sup}_{n \to \infty} x_n \leq c_n## (Buck, Advanced Calculus, Section 1.6, Exercise 24) Let ##\beta =\operatorname{lim inf}_{n \to \infty} x_n##...
6. ### Induction proof verification ##2^{n+2} < (n+1)## for all n ##\geq 6##

$2^{n+2} < (n+1)!$ for all n $\geq 6$ Step 1: For n = 6, $256 < 5040$. We assume $2^{k+2} < (k+1)!$ Induction step: $2 * 2^{k+2} < 2*(k+1)!$ By noting $2*(k+1)! < (k+2)!$ Then $2^{k+3} < (k+2)!$
7. ### Relation between components and path-components of ##X##

Homework Statement Theorem: If ##X## is a topological space, each path component of ##X## lies in a component of ##X##. If ##X## is locally path connected, then the components and the path components of ##X## are the same. I need help locating errors in my proof. Please help. Homework...
8. ### Prove that there exists a graph with these points such that....

Homework Statement Let us have ##n \geq 3## points in a square whose side length is ##1##. Prove that there exists a graph with these points such that ##G## is connected, and $$\sum_{\{v_i,v_j\} \in E(G)}{|v_i - v_j|} \leq 10\sqrt{n}$$ Prove also the ##10## in the inequality can't be replaced...
9. ### Solve $z^{1+i} = 4$: Homework

Homework Statement Find ##z## in ##z^{1+i}=4##. Is my solution correct Homework Equations ##\log(z_1 z_2)=\log(z_1)+\log(z_2)## such that ##z_1, z_2\in \{z\in\Bbb{C} : (z=x+iy) \land (x\in\Bbb{R}) \land -\infty \lt y \lt +\infty\}## ##re^{i\theta}=r(\cos\theta + i\sin\theta)## The Attempt at a...
10. ### Image of a f with a local minima at all points is countable.

Homework Statement Let ##f:\Bbb{R} \to \Bbb{R}## be a function such that ##f## has a local minimum for all ##x \in \Bbb{R}## (This means that for each ##x \in \Bbb{R}## there is an ##\epsilon \gt 0## where if ##\vert x-t\vert \lt \epsilon## then ##f(x) \leq f(t)##.). Then the image of ##f## is...
11. ### Number of indie vectors ##\leq ## cardinality of spanning set

Homework Statement In a finite-dimensional vector space, the length of every linearly independent list of vectors is less than or equal to the length of every spanning list. It's quite long :nb), hope you guys read through it. Thanks! :smile: Homework Equations N/A The Attempt at a Solution...
12. ### Discrete metric is a metric

Homework Statement Let ##x,y\in X## such that ##X## is a metric space. Let ##d(x,y)=0## if and only if ##x=y## and ##d(x,y)=1## if and only if ##x\neq y## Homework Equations N/A The Attempt at a Solution I have already seen various approaches in proving this. Although, I just want to know if...
13. ### Show that ##\frac{1}{x^2}## is not uniformly continuous on (0,∞).

Homework Statement Show that ##f(x)=\frac{1}{x^2}## is not uniformly continuous at ##(0,\infty)##. Homework Equations N/A The Attempt at a Solution Given ##\epsilon=1##. We want to show that we can compute for ##x## and ##y## such that ##\vert x-y\vert\lt\delta## and at the same time ##\vert...
14. ### I Zorn's Lemma: Need help finding errors in proof

Proposition(Zorn's Lemma): Let ##X\neq\emptyset## be of partial order with the property that ##\forall Y\subseteq X## such that ##Y## is of total-order then ##Y## has an upperbound, then ##X## contains a maximal element. Proof: Case 1: ##B\neq\emptyset## such that ##B##=##\{####b\in X##: ##b##...
15. ### Regarding Real numbers as limits of Cauchy sequences

Homework Statement Let ##x\in\Bbb{R}## such that ##x\neq 0##. Then ##x=LIM_{n\rightarrow\infty}a_n## for some Cauchy sequence ##(a_n)_{n=1}^{\infty}## which is bounded away from zero. 2. Relevant definitions and propositions: 3. The attempt at a proof: Proof:(by construction) Let...
16. M

### Show isomorphism under specific conditions

Homework Statement Let ##A,B## be subgroups of a finite abelian group ##G## Show that ##\langle g_1A \rangle \times \langle g_2A \rangle \cong \langle g_1,g_2 \rangle## where ##g_1,g_2 \in B## and ##A \cap B = \{e_G\}## where ##g_1 A, g_2 A \in G/A## (which makes sense since ##G## is abelian...
17. ### Proof verification of EF/MF phases

1. Statement: Prove that the EF and the MF are in phase far away from an oscillating electric dipole Homework Equations : The oscillatory motion equations for charges (q(t) = q_0sin(ωt) etc.)[/B] The Attempt at a Solution : Attached PDF file[/B]