Constructing Number Systems in Coq

Transcription

1 April 29, 2011

2 Table of contents 1 2 Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts 3 Definition Order Multiplication Addition 4

3 of my thesis Elegant construction of number systems in Coq N + Q + R + R Discuss the necessity of additional assumptions Excluded middle XM - R + Extensionality PE, FE, CE - R + Proof irrelevance PI - R + Strong excluded middle SXM - R

4 of my thesis Elegant construction of number systems in Coq N + Q + R + R Discuss the necessity of additional assumptions Excluded middle XM - R + Extensionality PE, FE, CE - R + Proof irrelevance PI - R + Strong excluded middle SXM - R

5 Landau s Grundlagen der Analysis Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts Natural Numbers, Peano Axioms Construction of Fractions, Rational, Real and Complex Numbers Basic theorems and their proofs (about 300)

6 Natural Numbers Æ Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts Æ = {1,2,3...} Inductive nat : Type := O : nat S : nat -> nat O as origin or one S as the successor function Coercion bool Prop, (leq : nat nat bool)

7 Fractions F Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts F = { x 1 x 2 : x 1,x 2 Æ} Definition (Equivalence of fractions) x 1,x 2,y 1,y 2 Æ : Definition (Order of fractions) x 1,x 2,y 1,y 2 Æ : x 1 x 2 y 1 y 2 : x 1 y 2 = y 1 x 2 x1 x2 < y1 y2 : x 1 y 2 < y 1 x 2

8 Positive Rational Numbers Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts Definition (Positive Rational Numbers) The (positive) Rational Numbers É + are defined as F modulo : É + := F/ In other words: Let X É + : x y. x X (y X x y)

9 Reducing a fraction Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts Let red : frac frac be the function that reduces a fraction Property (1) x. x red x Property (2) x y. x y red x = red y

10 Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts Definition (Positive Rational Numbers) The (positive) Rational Numbers É + are defined as: É + := {f F red f = f} Avoid the use of FE The theorems about the rationals reduce to the theorems about fractions.

11 Defining red Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts Different possibilities to define red (gcd, first) Prove the 2 properties of red Property (1) x. x red x Property (2) x y. x y red x = red y

12 Defining red Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts Preliminary note Function first : (nat bool) nat nat yields to a set p and an upper bound x for the minimum the least element in p first p x = minp We can represent y x. p y having type bool using first: p(first p x)

13 Defining red Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts Given a fraction x = x 1 x 2 we define N x := { y 1 y 2. x 1 x 2 y 1 y 2 } rednum x := minn x D x := { y 2 x 1 x 2 rednum x y 2 } redden x := mind x

14 Defining red Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts Given a fraction x = x 1 x 2 we define N x := { y 1 y 2. x 1 x 2 y 1 y 2 } rednum x := min N x minn x := first N x x 1 N x := { y 1 y 2 y 1 x 2. x 1 x 2 y 1 y 2 }

15 Defining red Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts Given a fraction x = x 1 x 2 we define N x := { y 1 y 2. x 1 x 2 y 1 y 2 } rednum x := min N x minn x := first N x x 1 N x := { y 1 y 2 y 1 x 2. x 1 x 2 y 1 y 2 }

16 Defining red Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts Given a fraction x = x 1 x 2 we define N x := { y 1 y 2. x 1 x 2 y 1 y 2 } rednum x := min N x minn x := first N x x 1 N x := { y 1 y 2 y 1 x 2. x 1 x 2 y 1 y 2 }

17 Defining red Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts Given a fraction x = x 1 x 2 we define D x := { y 2 x 1 x 2 rednum x y 2 } redden x := mind x mind x := first D x (rednum x x 2 )

18 Defining red Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts Given a fraction x = x 1 x 2 we define D x := { y 2 x 1 x 2 rednum x y 2 } redden x := mind x mind x := first D x (rednum x x 2 )

19 Dedekind Cuts Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts Definition (Dedekind Cut) A Dedekind Cut Θ is a set of positive Rational Numbers with the following properties: X. X Θ X. X / Θ X Y. Y Θ X < Y X Θ X. X Θ Y. X < Y Y Θ Intuition: Θ = (0,θ) É where θ Ê +.

20 Dedekind Cuts Landau s Grundlagen der Analysis Natural Numbers Æ Fractions Positive Rational Numbers Dedekind Cuts Definition (Cut Extensionality) Two Cuts Θ and Ξ are equal if they contain the same rational numbers. That is, CE := Θ Ξ. ( X. X Θ X Ξ) Θ = Ξ We can prove PE FE CE or SE CE. Note PE PI. Definition (Order) Given two Cuts Θ and Ξ we define Θ < Ξ : Θ Ξ Z. Z Ξ Z / Θ

21 Definition Definition Order Multiplication Addition Inductive real : Type := Z : real P : cut -> real N : cut -> real.

22 Order of Definition Order Multiplication Addition N Θ < N Ξ := Ξ < Θ N Θ < P Ξ := True. P Θ < Z := False P Θ < P Ξ := Θ < Ξ

23 Multiplication Definition Order Multiplication Addition Z η := Z ǫ Z := Z P Θ P Ξ := P (Θ Ξ) N Θ N Ξ := P (Θ Ξ) N Θ P Ξ := N (Θ Ξ) P Θ N Ξ := N (Θ Ξ) if ǫ Z

24 Addition Definition Order Multiplication Addition ǫ+z := ǫ Z +η := η N Θ+N Ξ := N (Θ+Ξ) P Θ+P Ξ := P (Θ+Ξ) P Θ+N Ξ := Z P Θ+N Ξ := N (Ξ Θ) P Θ+N Ξ := P (Θ Ξ) N Θ+P Ξ := P Ξ+N Θ if η Z if Θ = Ξ if Θ < Ξ if Θ > Ξ

25 Addition Definition Order Multiplication Addition ǫ+z := ǫ Z +η := η N Θ+N Ξ := N (Θ+Ξ) P Θ+P Ξ := P (Θ+Ξ) P Θ+N Ξ := Z P Θ+N Ξ := N (Ξ Θ) P Θ+N Ξ := P (Θ Ξ) N Θ+P Ξ := P Ξ+N Θ if η Z if Θ = Ξ if Θ < Ξ if Θ > Ξ

26 Strong Excluded Middle Definition Order Multiplication Addition XM := X : Prop. X X SXM := X : Prop. { X }+{ X } STR := Θ Ξ. { Θ < Ξ }+{ Θ = Ξ }+{ Ξ < Θ } SXM STR

27 Strong Excluded Middle Definition Order Multiplication Addition XM := X : Prop. X X SXM := X : Prop. { X }+{ X } STR := Θ Ξ. { Θ < Ξ }+{ Θ = Ξ }+{ Ξ < Θ } SXM STR

28 Definition Order Multiplication Addition Given two subsets P and Q of the real numbers we define P < Q := ǫ η. ǫ P η Q ǫ < η P := ǫ. ǫ P P Q = R := ǫ. ǫ P ǫ Q ub P η := ǫ P. ǫ η

29 Definition Order Multiplication Addition Theorem (Supremum Property) Let P be a nonempty subset of the real numbers that is bounded from above. That is, P and η. ub P η Then there is a (unique) least upper bound ζ. This is a real number ζ with the following property: ub P ζ and η. ub P η ζ η

30 Definition Order Multiplication Addition Theorem (Dedekind s Fundamental Theorem) Let P and Q be given with P < Q, P, Q, and P Q = R. Then there is a unique ζ such that ǫ.(ǫ < ζ ǫ P) (ζ < ǫ ǫ Q)

31 Definition Order Multiplication Addition Theorem (Fundamental Theorem) Let P and Q be given with P < Q, P and Q. Then there is a ζ such that ǫ.(ǫ < ζ ǫ / Q) (ζ < ǫ ǫ / P) If P Q = R, we can prove Dedekind s Fundamental Theorem.

32 Definition Order Multiplication Addition Let P and Q be given with P < Q, P and Q. If neither P contains a positive number nor Q contains a negative number, we set ζ to Z. If both P contains a positive number and Q contains a negative number, we have a contradiction. If P contains a positive number, we construct the cut Θ = {X ǫ P. X < ǫ} and set ζ to P Θ. (Analogous if Q contains a negative number.)

33 Definition Order Multiplication Addition Let P and Q be given with P < Q, P and Q. If neither P contains a positive number nor Q contains a negative number, we set ζ to Z. If both P contains a positive number and Q contains a negative number, we have a contradiction. If P contains a positive number, we construct the cut Θ = {X ǫ P. X < ǫ} and set ζ to P Θ. (Analogous if Q contains a negative number.)

34 Definition Order Multiplication Addition Let P and Q be given with P < Q, P and Q. If neither P contains a positive number nor Q contains a negative number, we set ζ to Z. If both P contains a positive number and Q contains a negative number, we have a contradiction. If P contains a positive number, we construct the cut Θ = {X ǫ P. X < ǫ} and set ζ to P Θ. (Analogous if Q contains a negative number.)

35 Definition Order Multiplication Addition Landau defines Θ in a different way. Define Θ to be the cut Θ = {X X P ǫ P. X < ǫ} If P Q = R we have Θ = Θ

36 Differences to Landau Proof of Peano axioms Definition of < independent from + Function first for Well-Ordering Principle Definition of Rational Numbers using red Third property of cuts More general formulation of Dedekind s Fundamental Theorem

37 Additional Assumptions Excluded middle Well-Ordering Principle WP XM Trichotomy for cuts TR XM Other assumptions Cut extensionality CE, SE CE or PE FE CE Strong excluded middle or strong trichotomy for cuts SXM STR

38 References E. Landau : Grundlagen der Analysis (1930) G. Smolka, C. E. Brown : Introduction to Computational Logic (Lecture Notes SS 2010) Y. Bertot, P. Castéran : Interactive Theorem Proving and Program Development: Coq Art: The Calculus of Inductive Constructions (2004) J. Harrison : Theorem Proving with the (1998)