Composing Ancient Mathematical Knowledge Into Powerful Bit-fiddling techniques Jamie Pond -- CppCon 2024TLDW; New insights from Ancient Egyptian Multiplication gives us the freedom of synthesising

Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.6 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 10.3 Inferring Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 10 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 \nat. Of course, we can also define non-recursive

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 sys- tems, ranging from the natural numbers to the complex numbers; algebraic 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

Rewriting Tactics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Proofs by Mathematical Induction . . . . . . . . . . . . . . . . . . . . . 31 2.7 Induction Tactic . . . . . . . . . 175 11.6 Summary of New Lean Constructs . . . . . . . . . . . . . . . . . . . . . 181 12 Basic Mathematical Structures 183 12.1 Type Classes over a Single Binary Operator . . . . . . . . . . . . . . . 183 strength of proof assistants is that they help develop highly trustwor- thy, unambiguous proofs of mathematical statements, using a precise logic. They can be used to prove arbitrarily advanced results, and

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 i ii CONTENTS 5 Mathematical Operations and Elementary Functions 25 5.1 Arithmetic Operators . . . . . . . . . . . . . . . . . . 605 43 Mathematics 607 43.1 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 43.2 Mathematical Functions . . . . . . . . . . . . . . . Python, but also supports general program- ming. To achieve this, Julia builds upon the lineage of mathematical programming languages, but also borrows much from popular dynamic languages, including Lisp,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 i ii CONTENTS 5 Mathematical Operations and Elementary Functions 25 5.1 Arithmetic Operators . . . . . . . . . . . . . . . . . . 609 43 Mathematics 611 43.1 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 43.2 Mathematical Functions . . . . . . . . . . . . . . . Python, but also supports general program- ming. To achieve this, Julia builds upon the lineage of mathematical programming languages, but also borrows much from popular dynamic languages, including Lisp,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 i ii CONTENTS 5 Mathematical Operations and Elementary Functions 25 5.1 Arithmetic Operators . . . . . . . . . . . . . . . . . . 603 43 Mathematics 605 43.1 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 43.2 Mathematical Functions . . . . . . . . . . . . . . . Python, but also supports general program- ming. To achieve this, Julia builds upon the lineage of mathematical programming languages, but also borrows much from popular dynamic languages, including Lisp,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 i ii CONTENTS 5 Mathematical Operations and Elementary Functions 25 5.1 Arithmetic Operators . . . . . . . . . . . . . . . . . . 605 43 Mathematics 607 43.1 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 43.2 Mathematical Functions . . . . . . . . . . . . . . . Python, but also supports general program- ming. To achieve this, Julia builds upon the lineage of mathematical programming languages, but also borrows much from popular dynamic languages, including Lisp,

and one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5 Mathematical Operations and Elementary Functions 22 5.1 Arithmetic Operators . . . . . . . . . . . . . . . . . . 583 43 Mathematics 584 43.1 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 43.2 Mathematical Functions . . . . . . . . . . . . . . . Python, but also supports general program- ming. To achieve this, Julia builds upon the lineage of mathematical programming languages, but also borrows much from popular dynamic languages, including Lisp,