(*

This file is a part of IsarMathLib -

a library of formalized mathematics for Isabelle/Isar.

Copyright (C) 2005 - 2009 Slawomir Kolodynski

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*)

header{*\isaheader{Real\_ZF.thy}*}

theory Real_ZF imports Int_ZF_IML Ring_ZF_1

begin

text{*The goal of the @{text "Real_ZF"} series of theory files is

to provide a contruction of the set of

real numbers. There are several ways to construct real numbers.

Most common start from the rational numbers and use Dedekind cuts

or Cauchy sequences. @{text "Real_ZF_x.thy"} series formalizes

an alternative

approach that constructs real numbers directly from the group of integers.

Our formalization is mostly based on \cite{Arthan2004}.

Different variants of this contruction are also

described in \cite{Campo2003} and \cite{Street2003}.

I recommend to read these papers, but for the impatient here is a short

description: we take a set of maps $s : Z\rightarrow Z$ such that

the set $\{s(m+n)-s(m)-s(n)\}_{n,m \in Z}$ is finite

($Z$ means the integers here). We call these maps slopes.

Slopes form a group with the natural addition

$(s+r)(n) = s(n)+r(n)$. The maps such that the set $s(Z)$ is finite

(finite range functions) form a subgroup of slopes.

The additive group of real numbers is defined as the

quotient group of slopes by the (sub)group of finite range functions.

The multiplication is defined as the projection of the composition of slopes

into the resulting quotient (coset) space.

*}

section{*The definition of real numbers*}

text{*This section contains the construction of the ring of real numbers

as classes of slopes - integer almost homomorphisms. The real definitions

are in @{text "Group_ZF_2"} theory, here we just specialize the definitions

of almost homomorphisms, their equivalence and operations to the

additive group of integers from the general case of abelian groups considered

in @{text "Group_ZF_2"}.*}

text{*The set of slopes is defined as the set of almost homomorphisms

on the additive group of integers.*}

definition

"Slopes ≡ AlmostHoms(int,IntegerAddition)"

text{*The first operation on slopes (pointwise addition) is a special case

of the first operation on almost homomorphisms.*}

definition

"SlopeOp1 ≡ AlHomOp1(int,IntegerAddition)"

text{*The second operation on slopes (composition) is a special case

of the second operation on almost homomorphisms.*}

definition

"SlopeOp2 ≡ AlHomOp2(int,IntegerAddition)"

text{*Bounded integer maps are functions from integers

to integers that have finite range. They play a role of

zero in the set of real numbers we are constructing.*}

definition

"BoundedIntMaps ≡ FinRangeFunctions(int,int)"

text{*Bounded integer maps form a normal subgroup of slopes.

The equivalence relation on slopes is the (group) quotient

relation defined by this subgroup.

*}

definition

"SlopeEquivalenceRel ≡ QuotientGroupRel(Slopes,SlopeOp1,BoundedIntMaps)"

text{*The set of real numbers is the set of equivalence classes of

slopes.*}

definition

"RealNumbers ≡ Slopes//SlopeEquivalenceRel"

text{*The addition on real numbers is defined as the projection of

pointwise addition of slopes on the quotient. This means that

the additive group of real numbers is the quotient group:

the group of slopes (with pointwise addition) defined by the

normal subgroup of bounded integer maps.*}

definition

"RealAddition ≡ ProjFun2(Slopes,SlopeEquivalenceRel,SlopeOp1)"

text{*Multiplication is defined as the projection of composition

of slopes on the quotient. The fact that it works is probably the

most surprising part of the construction. *}

definition

"RealMultiplication ≡ ProjFun2(Slopes,SlopeEquivalenceRel,SlopeOp2)"

text{*We first show that we can use theorems proven in some proof contexts

(locales). The locale @{text "group1"} requires assumption that we deal with

an abelian group. The next lemma allows to use all theorems proven

in the context called @{text "group1"}.*}

lemma Real_ZF_1_L1: shows "group1(int,IntegerAddition)"

using group1_axioms.intro group1_def Int_ZF_1_T2 by simp;

text{*Real numbers form a ring. This is a special case of the theorem

proven in @{text "Ring_ZF_1.thy"}, where we show the same in general for

almost homomorphisms rather than slopes.*}

theorem Real_ZF_1_T1: shows "IsAring(RealNumbers,RealAddition,RealMultiplication)"

proof -

let ?AH = "AlmostHoms(int,IntegerAddition)"

let ?Op1 = "AlHomOp1(int,IntegerAddition)"

let ?FR = "FinRangeFunctions(int,int)"

let ?Op2 = "AlHomOp2(int,IntegerAddition)"

let ?R = "QuotientGroupRel(?AH,?Op1,?FR)"

let ?A = "ProjFun2(?AH,?R,?Op1)"

let ?M = "ProjFun2(?AH,?R,?Op2)"

have "IsAring(?AH//?R,?A,?M)" using Real_ZF_1_L1 group1.Ring_ZF_1_1_T1

by simp;

then show ?thesis using Slopes_def SlopeOp2_def SlopeOp1_def

BoundedIntMaps_def SlopeEquivalenceRel_def RealNumbers_def

RealAddition_def RealMultiplication_def by simp;

qed;

text{*We can use theorems proven in @{text "group0"} and @{text "group1"}

contexts applied to the group of real numbers.*}

lemma Real_ZF_1_L2: shows

"group0(RealNumbers,RealAddition)"

"RealAddition {is commutative on} RealNumbers"

"group1(RealNumbers,RealAddition)"

proof -

have

"IsAgroup(RealNumbers,RealAddition)"

"RealAddition {is commutative on} RealNumbers"

using Real_ZF_1_T1 IsAring_def by auto;

then show

"group0(RealNumbers,RealAddition)"

"RealAddition {is commutative on} RealNumbers"

"group1(RealNumbers,RealAddition)"

using group1_axioms.intro group0_def group1_def

by auto

qed;

text{*Let's define some notation.*}

locale real0 =

fixes real ("\<real>")

defines real_def [simp]: "\<real> ≡ RealNumbers"

fixes ra (infixl "\<ra>" 69)

defines ra_def [simp]: "a\<ra> b ≡ RealAddition`⟨a,b⟩"

fixes rminus ("\<rm> _" 72)

defines rminus_def [simp]:"\<rm>a ≡ GroupInv(\<real>,RealAddition)`(a)"

fixes rsub (infixl "\<rs>" 69)

defines rsub_def [simp]: "a\<rs>b ≡ a\<ra>(\<rm>b)"

fixes rm (infixl "·" 70)

defines rm_def [simp]: "a·b ≡ RealMultiplication`⟨a,b⟩"

fixes rzero ("\<zero>")

defines rzero_def [simp]:

"\<zero> ≡ TheNeutralElement(RealNumbers,RealAddition)"

fixes rone ("\<one>")

defines rone_def [simp]:

"\<one> ≡ TheNeutralElement(RealNumbers,RealMultiplication)"

fixes rtwo ("\<two>")

defines rtwo_def [simp]: "\<two> ≡ \<one>\<ra>\<one>"

fixes non_zero ("\<real>⇩_{0}")

defines non_zero_def[simp]: "\<real>⇩_{0}≡ \<real>-{\<zero>}"

fixes inv ("_¯ " [90] 91)

defines inv_def[simp]:

"a¯ ≡ GroupInv(\<real>⇩_{0},restrict(RealMultiplication,\<real>⇩_{0}×\<real>⇩_{0}))`(a)";

text{*In @{text "real0"} context all theorems proven in the @{text "ring0"},

context are valid.*}

lemma (in real0) Real_ZF_1_L3: shows

"ring0(\<real>,RealAddition,RealMultiplication)"

using Real_ZF_1_T1 ring0_def ring0.Ring_ZF_1_L1

by auto

text{*Lets try out our notation to see that zero and one are real numbers.*}

lemma (in real0) Real_ZF_1_L4: shows "\<zero>∈\<real>" "\<one>∈\<real>"

using Real_ZF_1_L3 ring0.Ring_ZF_1_L2 by auto;

text{*The lemma below lists some properties that

require one real number to state.*}

lemma (in real0) Real_ZF_1_L5: assumes A1: "a∈\<real>"

shows

"(\<rm>a) ∈ \<real>"

"(\<rm>(\<rm>a)) = a"

"a\<ra>\<zero> = a"

"\<zero>\<ra>a = a"

"a·\<one> = a"

"\<one>·a = a"

"a\<rs>a = \<zero>"

"a\<rs>\<zero> = a"

using assms Real_ZF_1_L3 ring0.Ring_ZF_1_L3 by auto;

text{*The lemma below lists some properties that

require two real numbers to state.*}

lemma (in real0) Real_ZF_1_L6: assumes "a∈\<real>" "b∈\<real>"

shows

"a\<ra>b ∈ \<real>"

"a\<rs>b ∈ \<real>"

"a·b ∈ \<real>"

"a\<ra>b = b\<ra>a"

"(\<rm>a)·b = \<rm>(a·b)"

"a·(\<rm>b) = \<rm>(a·b)"

using assms Real_ZF_1_L3 ring0.Ring_ZF_1_L4 ring0.Ring_ZF_1_L7

by auto;

text{*Multiplication of reals is associative.*}

lemma (in real0) Real_ZF_1_L6A: assumes "a∈\<real>" "b∈\<real>" "c∈\<real>"

shows "a·(b·c) = (a·b)·c"

using assms Real_ZF_1_L3 ring0.Ring_ZF_1_L11

by simp;

text{*Addition is distributive with respect to multiplication.*}

lemma (in real0) Real_ZF_1_L7: assumes "a∈\<real>" "b∈\<real>" "c∈\<real>"

shows

"a·(b\<ra>c) = a·b \<ra> a·c"

"(b\<ra>c)·a = b·a \<ra> c·a"

"a·(b\<rs>c) = a·b \<rs> a·c"

"(b\<rs>c)·a = b·a \<rs> c·a"

using assms Real_ZF_1_L3 ring0.ring_oper_distr ring0.Ring_ZF_1_L8

by auto;

text{*A simple rearrangement with four real numbers.*}

lemma (in real0) Real_ZF_1_L7A:

assumes "a∈\<real>" "b∈\<real>" "c∈\<real>" "d∈\<real>"

shows "a\<rs>b \<ra> (c\<rs>d) = a\<ra>c\<rs>b\<rs>d"

using assms Real_ZF_1_L2 group0.group0_4_L8A by simp;

text{* @{text "RealAddition"} is defined as the projection of the

first operation on slopes (that is, slope addition) on the quotient

(slopes divided by the "almost equal" relation. The next lemma plays with

definitions to show that this is the same as the operation induced on the

appriopriate quotient group. The names @{text "AH"}, @{text "Op1"}

and @{text "FR"} are used in @{text "group1"} context to denote almost

homomorphisms, the first operation on @{text "AH"} and finite range

functions resp.*}

lemma Real_ZF_1_L8: assumes

"AH = AlmostHoms(int,IntegerAddition)" and

"Op1 = AlHomOp1(int,IntegerAddition)" and

"FR = FinRangeFunctions(int,int)"

shows "RealAddition = QuotientGroupOp(AH,Op1,FR)"

using assms RealAddition_def SlopeEquivalenceRel_def

QuotientGroupOp_def Slopes_def SlopeOp1_def BoundedIntMaps_def

by simp;

text{*The symbol @{text "\<zero>"} in the @{text "real0"} context is defined

as the neutral element of real addition. The next lemma shows that this

is the same as the neutral element of the appriopriate quotient group.*}

lemma (in real0) Real_ZF_1_L9: assumes

"AH = AlmostHoms(int,IntegerAddition)" and

"Op1 = AlHomOp1(int,IntegerAddition)" and

"FR = FinRangeFunctions(int,int)" and

"r = QuotientGroupRel(AH,Op1,FR)"

shows

"TheNeutralElement(AH//r,QuotientGroupOp(AH,Op1,FR)) = \<zero>"

"SlopeEquivalenceRel = r"

using assms Slopes_def Real_ZF_1_L8 RealNumbers_def

SlopeEquivalenceRel_def SlopeOp1_def BoundedIntMaps_def

by auto;

text{*Zero is the class of any finite range function.*}

lemma (in real0) Real_ZF_1_L10:

assumes A1: "s ∈ Slopes"

shows "SlopeEquivalenceRel``{s} = \<zero> <-> s ∈ BoundedIntMaps"

proof -

let ?AH = "AlmostHoms(int,IntegerAddition)"

let ?Op1 = "AlHomOp1(int,IntegerAddition)"

let ?FR = "FinRangeFunctions(int,int)"

let ?r = "QuotientGroupRel(?AH,?Op1,?FR)"

let ?e = "TheNeutralElement(?AH//?r,QuotientGroupOp(?AH,?Op1,?FR))"

from A1 have

"group1(int,IntegerAddition)"

"s∈?AH"

using Real_ZF_1_L1 Slopes_def

by auto;

then have "?r``{s} = ?e <-> s ∈ ?FR"

using group1.Group_ZF_3_3_L5 by simp;

moreover have

"?r = SlopeEquivalenceRel"

"?e = \<zero>"

"?FR = BoundedIntMaps"

using SlopeEquivalenceRel_def Slopes_def SlopeOp1_def

BoundedIntMaps_def Real_ZF_1_L9 by auto;

ultimately show ?thesis by simp;

qed;

text{*We will need a couple of results from @{text "Group_ZF_3.thy"}

The first two that state that the definition

of addition and multiplication of real numbers are consistent,

that is the result

does not depend on the choice of the slopes representing the numbers.

The second one implies that what we call @{text "SlopeEquivalenceRel"}

is actually an equivalence relation on the set of slopes.

We also show that the neutral element of the multiplicative operation on

reals (in short number $1$) is the class of the identity function on

integers.*}

lemma Real_ZF_1_L11: shows

"Congruent2(SlopeEquivalenceRel,SlopeOp1)"

"Congruent2(SlopeEquivalenceRel,SlopeOp2)"

"SlopeEquivalenceRel ⊆ Slopes × Slopes"

"equiv(Slopes, SlopeEquivalenceRel)"

"SlopeEquivalenceRel``{id(int)} =

TheNeutralElement(RealNumbers,RealMultiplication)"

"BoundedIntMaps ⊆ Slopes"

proof -

let ?G = "int"

let ?f = "IntegerAddition"

let ?AH = "AlmostHoms(int,IntegerAddition)"

let ?Op1 = "AlHomOp1(int,IntegerAddition)"

let ?Op2 = "AlHomOp2(int,IntegerAddition)"

let ?FR = "FinRangeFunctions(int,int)"

let ?R = "QuotientGroupRel(?AH,?Op1,?FR)"

have

"Congruent2(?R,?Op1)"

"Congruent2(?R,?Op2)"

using Real_ZF_1_L1 group1.Group_ZF_3_4_L13A group1.Group_ZF_3_3_L4

by auto;

then show

"Congruent2(SlopeEquivalenceRel,SlopeOp1)"

"Congruent2(SlopeEquivalenceRel,SlopeOp2)"

using SlopeEquivalenceRel_def SlopeOp1_def Slopes_def

BoundedIntMaps_def SlopeOp2_def by auto;

have "equiv(?AH,?R)"

using Real_ZF_1_L1 group1.Group_ZF_3_3_L3 by simp;

then show "equiv(Slopes,SlopeEquivalenceRel)"

using BoundedIntMaps_def SlopeEquivalenceRel_def SlopeOp1_def Slopes_def

by simp;

then show "SlopeEquivalenceRel ⊆ Slopes × Slopes"

using equiv_type by simp;

have "?R``{id(int)} = TheNeutralElement(?AH//?R,ProjFun2(?AH,?R,?Op2))"

using Real_ZF_1_L1 group1.Group_ZF_3_4_T2 by simp;

then show "SlopeEquivalenceRel``{id(int)} =

TheNeutralElement(RealNumbers,RealMultiplication)"

using Slopes_def RealNumbers_def

SlopeEquivalenceRel_def SlopeOp1_def BoundedIntMaps_def

RealMultiplication_def SlopeOp2_def

by simp;

have "?FR ⊆ ?AH" using Real_ZF_1_L1 group1.Group_ZF_3_3_L1

by simp;

then show "BoundedIntMaps ⊆ Slopes"

using BoundedIntMaps_def Slopes_def by simp;

qed;

text{*A one-side implication of the equivalence from @{text "Real_ZF_1_L10"}:

the class of a bounded integer map is the real zero.*}

lemma (in real0) Real_ZF_1_L11A: assumes "s ∈ BoundedIntMaps"

shows "SlopeEquivalenceRel``{s} = \<zero>"

using assms Real_ZF_1_L11 Real_ZF_1_L10 by auto;

text{*The next lemma is rephrases the result from @{text "Group_ZF_3.thy"}

that says that the negative (the group inverse with respect to real

addition) of the class of a slope is the class of that slope

composed with the integer additive group inverse. The result and proof is not

very readable as we use mostly generic set theory notation with long names

here. @{text "Real_ZF_1.thy"} contains the same statement written in a more

readable notation: $[-s] = -[s]$.*}

lemma (in real0) Real_ZF_1_L12: assumes A1: "s ∈ Slopes" and

Dr: "r = QuotientGroupRel(Slopes,SlopeOp1,BoundedIntMaps)"

shows "r``{GroupInv(int,IntegerAddition) O s} = \<rm>(r``{s})"

proof -

let ?G = "int"

let ?f = "IntegerAddition"

let ?AH = "AlmostHoms(int,IntegerAddition)"

let ?Op1 = "AlHomOp1(int,IntegerAddition)"

let ?FR = "FinRangeFunctions(int,int)"

let ?F = "ProjFun2(Slopes,r,SlopeOp1)"

from A1 Dr have

"group1(?G, ?f)"

"s ∈ AlmostHoms(?G, ?f)"

"r = QuotientGroupRel(

AlmostHoms(?G, ?f), AlHomOp1(?G, ?f), FinRangeFunctions(?G, ?G))"

and "?F = ProjFun2(AlmostHoms(?G, ?f), r, AlHomOp1(?G, ?f))"

using Real_ZF_1_L1 Slopes_def SlopeOp1_def BoundedIntMaps_def

by auto;

then have

"r``{GroupInv(?G, ?f) O s} =

GroupInv(AlmostHoms(?G, ?f) // r, ?F)`(r `` {s})"

using group1.Group_ZF_3_3_L6 by simp;

with Dr show ?thesis

using RealNumbers_def Slopes_def SlopeEquivalenceRel_def RealAddition_def

by simp;

qed;

text{*Two classes are equal iff the slopes that represent them

are almost equal.*}

lemma Real_ZF_1_L13: assumes "s ∈ Slopes" "p ∈ Slopes"

and "r = SlopeEquivalenceRel"

shows "r``{s} = r``{p} <-> ⟨s,p⟩ ∈ r"

using assms Real_ZF_1_L11 eq_equiv_class equiv_class_eq

by blast;

text{*Identity function on integers is a slope.

Thislemma concludes the easy part of the construction that follows from

the fact that slope equivalence classes form a ring. It is easy to see

that multiplication of classes of almost homomorphisms is not

commutative in general.

The remaining properties of real numbers, like commutativity of

multiplication and the existence of multiplicative inverses have to be

proven using properties of the group of integers, rather that in general

setting of abelian groups.*}

lemma Real_ZF_1_L14: shows "id(int) ∈ Slopes"

proof -

have "id(int) ∈ AlmostHoms(int,IntegerAddition)"

using Real_ZF_1_L1 group1.Group_ZF_3_4_L15

by simp;

then show ?thesis using Slopes_def by simp;

qed;

end;