Toward a new formalization of real numbers

Transcription

1 Toward a new formalization of real numbers Catherine Lelay Institute for Advanced Sciences September 24, / 9

2 Toward a new formalization of real numbers in Coq using Univalent Foundations Catherine Lelay Institute for Advanced Sciences September 24, / 9

3 What is Formal Proof? Formal proof A formal proof is a proof written in a language precise enough to be checked by a computer. 2 / 9

4 What is Formal Proof? Formal proof A formal proof is a proof written in a language precise enough to be checked by a computer. Proof assistant Software for writing and verifying formal proofs. Examples: ACL2, Coq, HOL Light, Isabelle HOL, Mizar, PVS 2 / 9

5 What is Formal Proof? Formal proof A formal proof is a proof written in a language precise enough to be checked by a computer. Proof assistant Software for writing and verifying formal proofs. Examples: ACL2, Coq, HOL Light, Isabelle HOL, Mizar, PVS + guaranteed result 2 / 9

6 What is Formal Proof? Formal proof A formal proof is a proof written in a language precise enough to be checked by a computer. Proof assistant Software for writing and verifying formal proofs. Examples: ACL2, Coq, HOL Light, Isabelle HOL, Mizar, PVS + guaranteed result tedious process 2 / 9

7 What is Formal Proof? Formal proof A formal proof is a proof written in a language precise enough to be checked by a computer. Proof assistant Software for writing and verifying formal proofs. Examples: ACL2, Coq, HOL Light, Isabelle HOL, Mizar, PVS + guaranteed result tedious process 2 / 9

8 Denition of R Denition The set of real numbers R is : a non-trivial abelian eld totally ordered archimedean complete 3 / 9

9 Denition of R Denition The set of real numbers R is : a non-trivial abelian eld totally ordered archimedean complete In proof assistants, R can be: Axiomatic (Coq standard library, PVS) Constructive Cauchy sequences (C-CoRN/MathClasses, Isabelle/HOL) Dedekind cuts (HOL4, Mizar) 3 / 9

10 Denition of R Denition The set of real numbers R is : a non-trivial abelian eld totally ordered archimedean complete In proof assistants, R can be: Axiomatic (Coq standard library, PVS) Constructive Cauchy sequences (C-CoRN/MathClasses, Isabelle/HOL) Dedekind cuts (HOL4, Mizar) 3 / 9

11 Denition of R Denition The set of real numbers R is : a non-trivial abelian eld totally ordered archimedean complete In proof assistants, R can be: Axiomatic (Coq standard library, PVS) Constructive Cauchy sequences (C-CoRN/MathClasses, Isabelle/HOL) Dedekind cuts (HOL4, Mizar) 3 / 9

12 Denition of R Denition The set of real numbers R is : a non-trivial abelian eld totally ordered archimedean complete In proof assistants, R can be: Axiomatic (Coq standard library, PVS) Constructive Cauchy sequences (C-CoRN/MathClasses, Isabelle/HOL) Dedekind cuts (HOL4, Mizar) 3 / 9

13 Denition of R Denition The set of real numbers R is : a non-trivial abelian eld totally ordered archimedean complete In proof assistants, R can be: Axiomatic (Coq standard library, PVS) Constructive Cauchy sequences (C-CoRN/MathClasses, Isabelle/HOL) Dedekind cuts (HOL4, Mizar) 3 / 9

14 Real numbers in Coq Reals (Coq standard library) C-CoRN/MathClasses 4 / 9

15 Real numbers in Coq Reals (Coq standard library) Axioms User-friendly library of real analysis about total functions (adding Coquelicot) Proofs cannot be extracted as programs C-CoRN/MathClasses 4 / 9

16 Real numbers in Coq Reals (Coq standard library) Axioms User-friendly library of real analysis about total functions (adding Coquelicot) Proofs cannot be extracted as programs C-CoRN/MathClasses Cauchy sequences Good library of real analysis about total and partial functions Not easy to use 4 / 9

17 Real numbers in Coq Reals (Coq standard library) Axioms User-friendly library of real analysis about total functions (adding Coquelicot) Proofs cannot be extracted as programs C-CoRN/MathClasses Cauchy sequences Good library of real analysis about total and partial functions Not easy to use Goal: Build a new library based on Dedekind cuts 4 / 9

18 Denition of Dedekind cuts x 5 / 9

19 Denition of Dedekind cuts x Denition (Dedekind cut) (L, U) Q Q where : x L, y U, and L U =, x L, y Q, y < x y L, x L, y Q, x < y y L, x U, y Q, x < y y U, x U, y Q, y < x y U, (r, x, y) Q 3, x < r < y x L y U. R = {Dedekind cuts} 5 / 9

20 A second denition of Dedekind cuts x Denition (Dedekind cut) L Q where : x L, M Q, x L, x M, x L, y Q, y < x y L, x L, y Q, x < y y L. R = {Dedekind cuts} 6 / 9

21 A second denition of Dedekind cuts x Denition (Dedekind cut) L Q where : x L, M Q, x L, x M, x L, y Q, y < x y L, x L, y Q, x < y y L. R = {Dedekind cuts} Easier to dene Diculties to dene multiplication 6 / 9

22 A better denition of Dedekind cuts x Denition (Dedekind cut) L {r Q 0 r} where : M Q, x L, x M, x L, y Q, 0 y < x y L, x L, y Q, x < y y L. [0; + ) = {Dedekind cuts} R = ([0; + ) [0; + )) / 7 / 9

23 A better denition of Dedekind cuts x Denition (Dedekind cut) L {r Q 0 r} where : M Q, x L, x M, x L, y Q, 0 y < x y L, x L, y Q, x < y y L. [0; + ) = {Dedekind cuts} R = ([0; + ) [0; + )) / Easier to dene Good denitions for operations 7 / 9

24 A better denition of Dedekind cuts x Denition (Dedekind cut) L {r Q 0 r} where : M Q, x L, x M, x L, y Q, 0 y < x y L, x L, y Q, x < y y L. [0; + ) = {Dedekind cuts} R = ([0; + ) [0; + )) / Easier to dene Good denitions for operations? Quotients in Coq proof assistant 7 / 9

25 UniMath A Coq library which aim to formalize a substantial body of mathematics using the univalent point of view. UniMath website 8 / 9

26 UniMath A Coq library which aim to formalize a substantial body of mathematics using the univalent point of view. UniMath website Extension of the underlying theory of Coq Univalent point of view allow a few classical constructions in Coq which are previously impossible (or realy hard) 8 / 9

27 UniMath A Coq library which aim to formalize a substantial body of mathematics using the univalent point of view. UniMath website Extension of the underlying theory of Coq Univalent point of view allow a few classical constructions in Coq which are previously impossible (or realy hard) For my work: New board Work with conceptors 8 / 9

28 Toward a new formalization of real numbers R + = [0; + ) 0, 1, +,, 1 R = (R + R + )/ 0, 1, +,,, 1 To go further: real analysis 9 / 9