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

lambda trick Comparison Binding Constraints Architecture Substitution Construction Conclusion Mathematical expressions Normal distribution PDF f = 1 √ 2πσ2 e− (x−µ)2 2σ2 2 + + 2 Normal distribution symbols Expression templates are stateful Stateless Expression Templates Formulas are stateless Data is injected in formulas Both live their lives in different world What changed since then? C++20 Conclusion Symbols with constraints It would be nice to be able to provide information on the mathematical entity it represents 1 symbol a; 2 symbol w; 3 symbol t; 4 symbol phi; 1 symbol a;

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

Theoretical foundations Outline of an implementation Conclusion Mathematical and software architecture Question How to abstract the problem? Mathematical abstractions Well-posed and abstracted in mathematics mathematics ⇒ ‘‘easy” to abstract in software Mathematical architecture ⇒ Software architecture Counter-example: units Intuitively understood in physics But very fuzzy foundations in mathematics ⇒ Incredibly DedicationIntroduction Firsts steps Context Theoretical foundations Outline of an implementation Conclusion The mathematical architecture of algorithms John C. Butcher Numerical Methods for Ordinary Differential Equations

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 and software systems in mathematical terms, at which point establishing claims as to their correctness becomes a form of theorem proving. Conversely, the proof of a mathematical theorem may require a lengthy verifying that the computation does what it is supposed to do. The gold standard for supporting a mathematical claim is to provide a proof, and twentieth-century developments in logic show most if not all

duplication: • Compiler for Haskell-like recursive equations, we can use it to write proofs. • Mathematical structures (e.g., Groups and Rings) are first-class citizens. • Some references: • In praise Dan Suciu University of Washington https://arxiv.org/pdf/1802.02229.pdf Mathlib The Lean mathematical library, mathlib, is a community-driven effort to build a unified library of mathematics formalized Mathematics? Tom Hales (University of Pittsburgh) “To develop software and services for transforming mathematical results as they appear in journal article abstracts into formally structured data that machines

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 i ii CONTENTS 5 Mathematical Operations and Elementary Functions 27 5.1 Arithmetic Operators . . . . . . . . . . . . . . . . . . 588 45 Mathematics 589 45.1 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 45.2 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,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 i ii CONTENTS 5 Mathematical Operations and Elementary Functions 27 5.1 Arithmetic Operators . . . . . . . . . . . . . . . . . . 588 45 Mathematics 589 45.1 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 45.2 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 12 Mathematical Operations and Elementary Functions 45 12.1 Arithmetic Operators . . . . . . . . . . . . . . . . . . 594 52 Mathematics 597 52.1 Mathematical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 52.2 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,