Commit 06809460 by Andreas Lochbihler

### merge from afp-2021

 ... ... @@ -664,6 +664,39 @@ abstract = developed in the AFP entry on the transcendence of e.

[Hermite_Lindemann] title = The Hermite–Lindemann–Weierstraß Transcendence Theorem author = Manuel Eberl topic = Mathematics/Number theory date = 2021-03-03 notify = eberlm@in.tum.de abstract =

This article provides a formalisation of the Hermite-Lindemann-Weierstraß Theorem (also known as simply Hermite-Lindemann or Lindemann-Weierstraß). This theorem is one of the crowning achievements of 19th century number theory.

The theorem states that if $\alpha_1, \ldots, \alpha_n\in\mathbb{C}$ are algebraic numbers that are linearly independent over $\mathbb{Z}$, then $e^{\alpha_1},\ldots,e^{\alpha_n}$ are algebraically independent over $\mathbb{Q}$.

Like the previous formalisation in Coq by Bernard, I proceeded by formalising Baker's version of the theorem and proof and then deriving the original one from that. Baker's version states that for any algebraic numbers $\beta_1, \ldots, \beta_n\in\mathbb{C}$ and distinct algebraic numbers $\alpha_i, \ldots, \alpha_n\in\mathbb{C}$, we have $\beta_1 e^{\alpha_1} + \ldots + \beta_n e^{\alpha_n} = 0$ if and only if all the $\beta_i$ are zero.

This has a number of direct corollaries, e.g.:

• $e$ and $\pi$ are transcendental
• $e^z$, $\sin z$, $\tan z$, etc. are transcendental for algebraic $z\in\mathbb{C}\setminus\{0\}$
• $\ln z$ is transcendental for algebraic $z\in\mathbb{C}\setminus\{0, 1\}$
[DFS_Framework] title = A Framework for Verifying Depth-First Search Algorithms author = Peter Lammich , René Neumann ... ... @@ -10153,6 +10186,19 @@ abstract = algorithm, the Deutsch-Jozsa algorithm and the quantum Prisoner's Dilemma. [Projective_Measurements] title = Quantum projective measurements and the CHSH inequality author = Mnacho Echenim topic = Computer science/Algorithms/Quantum computing, Mathematics/Physics/Quantum information date = 2021-03-03 notify = mnacho.echenim@univ-grenoble-alpes.fr abstract = This work contains a formalization of quantum projective measurements, also known as von Neumann measurements, which are based on elements of spectral theory. We also formalized the CHSH inequality, an inequality involving expectations in a probability space that is violated by quantum measurements, thus proving that quantum mechanics cannot be modeled with an underlying local hidden-variable theory. [Finite-Map-Extras] title = Finite Map Extras author = Javier Díaz ... ... @@ -10309,3 +10355,47 @@ abstract = discussed in greater detail in the corresponding Bachelor's Thesis. [Sunflowers] title = The Sunflower Lemma of Erdős and Rado author = René Thiemann topic = Mathematics/Combinatorics date = 2021-02-25 notify = rene.thiemann@uibk.ac.at abstract = We formally define sunflowers and provide a formalization of the sunflower lemma of Erdős and Rado: whenever a set of size-k-sets has a larger cardinality than (r - 1)k · k!, then it contains a sunflower of cardinality r. [Mereology] title = Mereology author = Ben Blumson topic = Logic/Philosophical aspects date = 2021-03-01 notify = benblumson@gmail.com abstract = We use Isabelle/HOL to verify elementary theorems and alternative axiomatizations of classical extensional mereology. [Modular_arithmetic_LLL_and_HNF_algorithms] title = Two algorithms based on modular arithmetic: lattice basis reduction and Hermite normal form computation author = Ralph Bottesch <>, Jose Divasón , René Thiemann topic = Computer science/Algorithms/Mathematical date = 2021-03-12 notify = rene.thiemann@uibk.ac.at abstract = We verify two algorithms for which modular arithmetic plays an essential role: Storjohann's variant of the LLL lattice basis reduction algorithm and Kopparty's algorithm for computing the Hermite normal form of a matrix. To do this, we also formalize some facts about the modulo operation with symmetric range. Our implementations are based on the original papers, but are otherwise efficient. For basis reduction we formalize two versions: one that includes all of the optimizations/heuristics from Storjohann's paper, and one excluding a heuristic that we observed to often decrease efficiency. We also provide a fast, self-contained certifier for basis reduction, based on the efficient Hermite normal form algorithm.
 ... ... @@ -30,6 +30,8 @@ Isabelle. Older versions of archive entries will remain available.

Technische Universität München
• Gerwin Klein, Data61
• Andreas Lochbihler, Digital Asset
• Tobias Nipkow, Technische Universität München
• Larry Paulson, ... ...