trick Comparison Binding Constraints Architecture Substitution Construction Conclusion Symbolic Calculus for High-Performance Computing from Scratch using C++23 Vincent Reverdy Laboratoire d’Annecy de people, you can do it yourself) What this talk is A tutorial so you can build your own symbolic calculus tools from scratch in modern C++ CppCon - Vincent Reverdy - October 4th, 2023 - Aurora, Colorado Comparison Binding Constraints Architecture Substitution Construction Conclusion Concepts: the constructions (expressions) y(t) = a × sin (ω × t + ϕ) Symbolic expressions term: “a mathematical object”:

to include all conventional mathematics. Lean’s underlying logical framework, the Calculus of Inductive Constructions, constitutes a surprisingly good programming language. It is expressive enough to define compilation in a future version of Lean. Viewed from a computational perspective, the Calculus of Inductive Constructions is an instance of a purely functional programming language. This means that a program natural ways. At the same, the system provides complementary mechanisms for proving claims, using inductive principles that capture the structure of the function definitions. One novel feature of Lean is

theory. Specifi- cally, it implements a version of dependent type theory known as the Calculus of Inductive Constructions. The CIC is a formal language with a small and precise set of rules that governs the the structure generated freely and inductively by a constant, zero, and a unary function succ: inductive nat : Type | zero : nat | succ : nat → nat If you copy this definition into the editor window at and metaprogramming. 2 Defining Objects in Lean As a foundational framework, the Calculus of Inductive Constructions, or CIC, is flexi- ble enough to define all kinds of mathematical objects. It can define

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7 Inductive Types 95 7.1 Enumerated Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.6 Tactics for Inductive Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.7 Inductive Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.9 Mutual and Nested Inductive Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7.10 Exercises

. . . . . . . . . 49 iii II Functional–Logic Programming 51 4 Functional Programming 53 4.1 Inductive Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 Structural Induction 71 4.10 Dependent Inductive Types . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.11 Summary of New Lean Constructs . . . . . . . . . . . . . . . . . . . . . 74 5 Inductive Predicates 75 5.1 Introductory Program Equivalence . . . . . . . . . . . . . . . . . . . . 162 10.9 A Simpler Approach Based on an Inductive Predicate . . . . . . . . . 164 IV Mathematics 165 11 Logical Foundations of Mathematics 167 11

uk/~csetzer/lectures/intertheo/07/] (CS__336), Anton Setzer, Swansea University, Lent 2008. Inductive and inductive-recursive definitions in Intuitionistic Type Theory [https://www.cs.uoregon.edu/researc The interval and path types Transport Partial elements Homogeneous composition Glue types Higher inductive types Cubical identity types and computational HoTT/UF Cubical Agda with erased glue References record Stream (A : Set) : Set where coinductive field hd : A tl : Stream A As opposed to inductive record types, we have to introduce the keyword coinductive before defining the fields that constitute

uk/~csetzer/lectures/intertheo/07/] (CS__336), Anton Setzer, Swansea University, Lent 2008. Inductive and inductive-recursive definitions in Intuitionistic Type Theory [https://www.cs.uoregon.edu/researc The interval and path types Transport Partial elements Homogeneous composition Glue types Higher inductive types Cubical identity types and computational HoTT/UF References Appendix: Cubical Agda primitives record Stream (A : Set) : Set where coinductive field hd : A tl : Stream A As opposed to inductive record types, we have to introduce the keyword coinductive before defining the fields that constitute

uk/~csetzer/lectures/intertheo/07/] (CS__336), Anton Setzer, Swansea University, Lent 2008. Inductive and inductive-recursive definitions in Intuitionistic Type Theory [https://www.cs.uoregon.edu/researc The interval and path types Transport Partial elements Homogeneous composition Glue types Higher inductive types Cubical identity types and computational HoTT/UF References Appendix: Cubical Agda primitives record Stream (A : Set) : Set where coinductive field hd : A tl : Stream A As opposed to inductive record types, we have to introduce the keyword coinductive before defining the fields that constitute

uk/~csetzer/lectures/intertheo/07/] (CS__336), Anton Setzer, Swansea University, Lent 2008. Inductive and inductive-recursive definitions in Intuitionistic Type Theory [https://www.cs.uoregon.edu/researc and path types Transport Partial elements Homogeneous composition Glue types Higher inductive types Indexed inductive types Cubical identity types and computational HoTT/UF Cubical Agda with erased Glue record Stream (A : Set) : Set where coinductive field hd : A tl : Stream A As opposed to inductive record types, we have to introduce the keyword coinductive before defining the fields that constitute

• Interactive Theorem proving (CS__336), Anton Setzer, Swansea University, Lent 2008. • Inductive and inductive-recursive definitions in Intuitionistic Type Theory, lectures by Peter Dybjer at the Ore- follows: record Stream (A : Set) : Set where coinductive field hd : A tl : Stream A As opposed to inductive record types, we have to introduce the keyword coinductive before defining the fields that constitute integrated with the check for size-change termination, thus allowing interesting combinations of inductive and coinductive types. We can for instance define the type of stream processors, along with some