## +24 ## Composing Ancient Mathematical Knowledge Into Powerful Bit-fiddling ## JAMIE POND ## TLDW; New insights from Ancient Egyptian Multiplication gives us the freedom of synthesising operations

Language (SDL) • Ditaa diagram • Diagrama de Gantt • Mathematic with AsciiMath or JLaTeXMath notation Los diagramas son definidos usando un lenguaje simple e intuitivo. ## 1 Diagrama de Secuencia robust keyword, depending on how you want them to be drawn. You define state change using the @ notation, and the is verb. @startuml robust "Web Browser" as WB concise "Web User" as WU start and ends at [Test prototype]'s @endgantt ## 11 Maths You can use AsciiMath or JLaTeXMath notation within PlantUML: @startuml :$int\_0^1f(x)dx$; :$x~2+y\_1+z\_12^34$; note

Mathematik in AsciiMath- oder JLaTeXMath-Notation • Entity Relationship diagram Diagramme werden in einfacher und intuitiver Sprache durch textuelle Notation beschrieben. ## 1 Sequenzdiagramm Die Erstellung p7_1.jpg) ### 1.8 Ändern der Pfeil Farbe Sie können die Farbe einzelner Pfeile mit folgender Notation ändern: @startuml Bob -[#red]> Alice : hello Alice -[#0000FF]->Bob : ok @enduml ![Image (inline style) You can change the color or style of individual arrows using the inline following notation: • #color;line.[bold|dashed|dotted];text:color @startuml actor foo foo --> (bar) : normal

Numeric Literal Coefficients 45 Syntax Conflicts 46 13.5 Literal zero and one 46 14 Mathematical Operations and Elementary Functions 49 14.1 Arithmetic Operators 49 14.2 Bitwise Operators 575 52.9 Utility Collections ..... 581 53 Mathematics ..... 583 53.1 Mathematical Operators ..... 583 53.2 Mathematical Functions ..... 604 54 Examples ..... 617 55 Numbers ..... 635 55.1 Python, but also supports general programming. To achieve this, Julia builds upon the lineage of mathematical programming languages, but also borrows much from popular dynamic languages, including Lisp, Perl

on Namespaces ..... 82 6.4 Attributes ..... 84 6.5 More on Implicit Arguments ..... 85 6.6 Notation ..... 87 6.7 Coercions ..... 89 6.8 Displaying Information ..... 89 6.9 Setting Options . 10.1 Type Classes and Instances ..... 141 10.2 Chaining Instances ..... 143 10.3 Inferring Notation ..... 144 10.4 Decidable Propositions ..... 145 10.5 Managing Type Class Inference ..... 147 computational methods to establish claims that are expressed in precise mathematical terms. These can include ordinary mathematical theorems, as well as claims that pieces of hardware or software, network

indicates what sort of object the expression denotes. For example, an expression may denote a mathematical object like a natural number, a data type, an assertion, or a proof. Lean has a small and carefully defines notation for the data type, as well as for zero and add. (In fact, Lean uses type classes, a very handy mechanism used by functional programming languages like Haskell, to share notation and properties properties across algebraic structures.) Lean uses the Unicode character N as alternative notation for the type nat. You can enter this in an editor by writing $ \mathtt{nat} $ . Of course, we can also

building mathematical libraries and stating and proving mathematical theorems. From that perspective, the point of Lean is to implement a formal axiomatic framework in which one can define mathematical objects for mathematics, the CIC is much more than a programming language. One can define all kinds of mathematical objects: number systems, ranging from the natural numbers to the complex numbers; algebraic structures types alongside other mathematical objects, and write programs alongside mathematical proofs. Terms in the Calculus of Inductive Constructions are therefore used to represent mathematical objects, programs

and Quantifiers 26 2.4 Reasoning about Equality 29 2.5 Rewriting Tactics 30 2.6 Proofs by Mathematical Induction 31 2.7 Induction Tactic 33 2.8 Cleanup Tactics 33 2.9 Summary of New Lean Constructs 172 11.5 Quotient Types ..... 175 11.6 Summary of New Lean Constructs ..... 181 12 Basic Mathematical Structures ..... 183 12.1 Type Classes over a Single Binary Operator ..... 183 12.2 Type Classes strength of proof assistants is that they help develop highly trustworthy, unambiguous proofs of mathematical statements, using a precise logic. They can be used to prove arbitrarily advanced results, and

Arbitrary Precision Arithmetic 22 4.4 Numeric Literal Coefficients 23 4.5 Literal zero and one 25 5 Mathematical Operations and Elementary Functions 27 5.1 Arithmetic Operators 27 5.2 Boolean Operators 28 5 Collections 711 42.8 Dequeues 717 42.9 Utility Collections 727 43 Mathematics 728 43.1 Mathematical Operators 728 43.2 Mathematical Functions 756 43.3 Customizable binary operators 797 44 Numbers 799 44.1 Standard Python, but also supports general programming. To achieve this, Julia builds upon the lineage of mathematical programming languages, but also borrows much from popular dynamic languages, including Lisp, Perl

Arbitrary Precision Arithmetic 20 4.4 Numeric Literal Coefficients 21 4.5 Literal zero and one 23 5 Mathematical Operations and Elementary Functions 24 5.1 Arithmetic Operators 24 5.2 Boolean Operators 25 5 Collections 649 42.8 Dequeues 655 42.9 Utility Collections 663 43 Mathematics 665 43.1 Mathematical Operators 665 43.2 Mathematical Functions 692 43.3 Customizable binary operators 730 44 Numbers 731 44.1 Standard Python, but also supports general programming. To achieve this, Julia builds upon the lineage of mathematical programming languages, but also borrows much from popular dynamic languages, including Lisp, Perl