Constructing Number Systems in Coq - uni-saarland.dehornung/BachelorThesis/finalTalk.pdf ·...

GoalWhat we already discussed

Real NumbersConclusion

Constructing Number Systems in Coq

Carsten Hornung

April 29, 2011

Carsten Hornung Constructing Number Systems in Coq



Table of contents

1 Goal

2 What we already discussedLandau’s Grundlagen der AnalysisNatural Numbers NFractionsPositive Rational NumbersDedekind Cuts

3 Real NumbersDefinitionOrderMultiplicationAdditionCompleteness

4 Conclusion




Goal 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




Goal 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




Landau’s Grundlagen der AnalysisNatural Numbers NFractionsPositive Rational NumbersDedekind Cuts

Landau’s Grundlagen der Analysis

Natural Numbers, Peano Axioms

Construction of

Fractions, Rational, Real and Complex Numbers

Basic theorems and their proofs (about 300)





Natural Numbers NN = {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)





Fractions F

F = {x1

x2: x1, x2 ∈ N}

Definition (Equivalence of fractions)

∀x1, x2, y1, y2 ∈ N :

x1x2

∼y1y2

:⇔ x1 · y2 = y1 · x2

Definition (Order of fractions)

∀x1, x2, y1, y2 ∈ N :

x1

x2<

y1

y2:⇔ x1 · y2 < y1 · x2





Positive Rational Numbers

Definition (Positive Rational Numbers)

The (positive) Rational Numbers Q+ are defined as F modulo ∼:Q+ := F/ ∼

In other words: Let X ∈ Q+:

∀x y . x ∈ X → (y ∈ X ↔ x ∼ y)





Reducing a fraction

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





Goal

Definition (Positive Rational Numbers)

The (positive) Rational Numbers Q+ are defined as:Q+ := {f ∈ F | red f = f }

Avoid the use of FE

The theorems about the rationals reduce to the theoremsabout fractions.





Defining red

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





Defining red

Preliminary note

Function first : (nat → bool) → nat → nat yields to a set pand an upper bound x for the minimum the least element in p

first p x = min p

We can represent ∃y ≤ x . p y having type bool using first:

p(first p x)





Defining red

Given a fraction x = x1

x2we define

Nx := { y1 | ∃y2.x1x2

∼y1y2

}

rednumx := minNx

Dx := { y2 |x1x2

∼rednumx

y2}

reddenx := minDx





Defining red


x2we define

Nx := { y1 | ∃y2.x1x2

∼y1y2

}

rednumx := min Nx

minNx := first Nx x1

Nx := { y1 | ∃y2 ≤ y1 · x2.x1x2

∼y1y2

}





Defining red


x2we define

Nx := { y1 | ∃y2.x1x2

∼y1y2

}

rednumx := min Nx


Nx := { y1 | ∃y2 ≤ y1 · x2.x1x2

∼y1y2

}





Defining red


x2we define

Nx := { y1 | ∃y2.x1x2

∼y1y2

}

rednumx := min Nx


Nx := { y1 | ∃y2 ≤ y1 · x2.x1x2

∼y1y2

}





Defining red


x2we define

Dx := { y2 |x1x2

∼rednumx

y2}

reddenx := minDx

minDx := first Dx (rednumx · x2)





Defining red


x2we define

Dx := { y2 |x1x2

∼rednumx

y2}

reddenx := minDx

minDx := first Dx (rednumx · x2)





Dedekind Cuts

Definition (Dedekind Cut)

A Dedekind Cut Θ is a set of positive Rational Numbers with thefollowing 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) ∩Q where θ ∈ R+.





Dedekind Cuts

Definition (Cut Extensionality)

Two Cuts Θ and Ξ are equal if they contain the same rationalnumbers. 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




DefinitionOrderMultiplicationAdditionCompleteness

Definition

Inductive real : Type :=

| Z : real

| P : cut -> real

| N : cut -> real.





Order of Real Numbers

N Θ < N Ξ := Ξ < Θ

N Θ < P Ξ := True

...

P Θ < Z := False

P Θ < P Ξ := Θ < Ξ





Multiplication

Z · η := Z

ǫ · Z := Z if ǫ 6= 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)





Addition

ǫ+ Z := ǫ

Z + η := η if η 6= Z

N Θ+N Ξ := N (Θ + Ξ)

P Θ+ P Ξ := P (Θ + Ξ)

P Θ+N Ξ := Z if Θ = Ξ

P Θ+N Ξ := N (Ξ −Θ) if Θ < Ξ

P Θ+N Ξ := P (Θ− Ξ) if Θ > Ξ

N Θ+ P Ξ := P Ξ +N Θ





Addition

ǫ+ Z := ǫ

Z + η := η if η 6= Z

N Θ+N Ξ := N (Θ + Ξ)

P Θ+ P Ξ := P (Θ + Ξ)

P Θ+N Ξ := Z if Θ = Ξ

P Θ+N Ξ := N (Ξ −Θ) if Θ < Ξ

P Θ+N Ξ := P (Θ− Ξ) if Θ > Ξ

N Θ+ P Ξ := P Ξ +N Θ





Strong Excluded Middle

XM := ∀X : Prop. X ∨ ¬X

SXM := ∀X : Prop. { X }+ { ¬X }

STR := ∀Θ Ξ. { Θ < Ξ }+ { Θ = Ξ }+ { Ξ < Θ }

SXM ↔ STR





Strong Excluded Middle

XM := ∀X : Prop. X ∨ ¬X

SXM := ∀X : Prop. { X }+ { ¬X }

STR := ∀Θ Ξ. { Θ < Ξ }+ { Θ = Ξ }+ { Ξ < Θ }

SXM ↔ STR





Completeness

Given two subsets P and Q of the real numbers we define

P < Q := ∀ǫ η. ǫ ∈ P → η ∈ Q → ǫ < η

P 6= ∅ := ∃ǫ. ǫ ∈ P

P ∪ Q = R := ∀ǫ. ǫ ∈ P ∨ ǫ ∈ Q

ub P η := ∀ǫ ∈ P . ǫ ≤ η





Completeness

Theorem (Supremum Property)

Let P be a nonempty subset of the real numbers that is boundedfrom above. That is,

P 6= ∅ and ∃η. ub P η

Then there is a (unique) least upper bound ζ. This is a realnumber ζ with the following property:

ub P ζ and ∀η. ub P η → ζ ≤ η





Completeness

Theorem (Dedekind’s Fundamental Theorem)

Let P and Q be given with P < Q, P 6= ∅, Q 6= ∅, andP ∪ Q = R. Then there is a unique ζ such that

∀ǫ.(ǫ < ζ → ǫ ∈ P) ∧ (ζ < ǫ → ǫ ∈ Q)





Completeness

Theorem (Fundamental Theorem)

Let P and Q be given with P < Q, P 6= ∅ and Q 6= ∅. Then thereis a ζ such that

∀ǫ.(ǫ < ζ → ǫ /∈ Q) ∧ (ζ < ǫ → ǫ /∈ P)

If P ∪ Q = R, we can prove Dedekind’s Fundamental Theorem.





Completeness

Let P and Q be given with P < Q, P 6= ∅ and Q 6= ∅.

If neither P contains a positive number nor Q contains anegative number, we set ζ to Z.

If both P contains a positive number and Q contains anegative 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 negativenumber.)





Completeness





Θ = {X | ∃ǫ ∈ P . X < ǫ}






Completeness





Θ = {X | ∃ǫ ∈ P . X < ǫ}






Completeness

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′




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 FundamentalTheorem




Additional Assumptions

Excluded middle

Well-Ordering Principle WP ↔ XMTrichotomy for cuts TR ↔ XM

Other assumptions

Cut extensionality CE , SE → CE or PE → FE → CEStrong excluded middle or strong trichotomy for cutsSXM ↔ STR




References

E. Landau : Grundlagen der Analysis (1930)

G. Smolka, C. E. Brown : Introduction to Computational

Logic (Lecture Notes SS 2010)

Y. Bertot, P. Casteran : Interactive Theorem Proving and

Program Development: Coq’Art: The Calculus of

Inductive Constructions (2004)

J. Harrison : Theorem Proving with the Real Numbers

(1998)


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