(*

This file is a part of IsarMathLib -

a library of formalized mathematics for Isabelle/Isar.

Copyright (C) 2005 - 2009 Slawomir Kolodynski

This program is free software; Redistribution and use in source and binary forms,

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*)

header{*\isaheader{Real\_ZF\_1.thy}*}

theory Real_ZF_1 imports Real_ZF Int_ZF_3 OrderedField_ZF

begin

text{*In this theory file we continue the construction of real numbers started

in @{text "Real_ZF"} to a succesful conclusion.

We put here those parts of the construction that

can not be done in the general settings of abelian groups and require

integers.*}

section{*Definitions and notation*}

text{*In this section we define notions and notation needed for the rest of

the construction.*}

text{*We define positive slopes as those that take an infinite number of

posititive values on the positive integers (see @{text "Int_ZF_2"}

for properties of positive slopes).*}

definition

"PositiveSlopes ≡ {s ∈ Slopes.

s``(PositiveIntegers) ∩ PositiveIntegers ∉ Fin(int)}"

text{*The order on the set of real numbers is constructed by

specifying the set of positive reals. This set is defined

as the projection of the set of positive slopes.*}

definition

"PositiveReals ≡ {SlopeEquivalenceRel``{s}. s ∈ PositiveSlopes}"

text{*The order relation on real numbers is constructed from the set of

positive elements in a standard way (see section

"Alternative definitions" in @{text "OrderedGroup_ZF"}.)

*}

definition

"OrderOnReals ≡ OrderFromPosSet(RealNumbers,RealAddition,PositiveReals)"

text{*The next locale extends the locale @{text "real0"} to define notation

specific to the construction of real numbers. The notation follows the

one defined in @{text "Int_ZF_2.thy"}.

If $m$ is an integer, then the real number which is the class of the slope

$n\mapsto m\cdot n$ is denoted @{text "m⇧^{R}"}.

For a real number $a$ notation $\lfloor a \rfloor$ means the largest integer

$m$ such that the real version of it (that is, $m^R$) is not greater

than $a$.

For an integer $m$ and a subset

of reals $S$ the expression $\Gamma(S,m)$ is defined as

$\max \{\lfloor p^R\cdot x\rfloor : x\in S\}$. This is plays a role in

the proof of completeness of real numbers.

We also reuse some notation defined in the @{text "int0"} context, like

@{text "\<int>⇩_{+}"} (the set of positive integers) and abs$(m)$ ( the absolute

value of an integer, and some defined in the @{text "int1"} context, like

the addition ( @{text "\<fp>"}) and composition (@{text "o"}

of slopes. *}

locale real1 = real0 +

fixes AlEq (infix "∼" 68)

defines AlEq_def[simp]: "s ∼ r ≡ ⟨s,r⟩ ∈ SlopeEquivalenceRel"

fixes slope_add (infix "\<fp>" 70)

defines slope_add_def[simp]:

"s \<fp> r ≡ SlopeOp1`⟨s,r⟩"

fixes slope_comp (infix "o" 71)

defines slope_comp_def[simp]: "s o r ≡ SlopeOp2`⟨s,r⟩"

fixes slopes ("\<S>")

defines slopes_def[simp]: "\<S> ≡ AlmostHoms(int,IntegerAddition)"

fixes posslopes ("\<S>⇩_{+}")

defines posslopes_def[simp]: "\<S>⇩_{+}≡ PositiveSlopes"

fixes slope_class ("[ _ ]")

defines slope_class_def[simp]: "[f] ≡ SlopeEquivalenceRel``{f}"

fixes slope_neg ("\<fm>_" [90] 91)

defines slope_neg_def[simp]: "\<fm>s ≡ GroupInv(int,IntegerAddition) O s"

fixes lesseqr (infix "\<lsq>" 60)

defines lesseqr_def[simp]: "a \<lsq> b ≡ ⟨a,b⟩ ∈ OrderOnReals"

fixes sless (infix "\<ls>" 60)

defines sless_def[simp]: "a \<ls> b ≡ a\<lsq>b ∧ a≠b"

fixes positivereals ("\<real>⇩_{+}")

defines positivereals_def[simp]: "\<real>⇩_{+}≡ PositiveSet(\<real>,RealAddition,OrderOnReals)"

fixes intembed ("_⇧^{R}" [90] 91)

defines intembed_def[simp]:

"m⇧^{R}≡ [{⟨n,IntegerMultiplication`⟨m,n⟩ ⟩. n ∈ int}]"

fixes floor ("⌊ _ ⌋")

defines floor_def[simp]:

"⌊a⌋ ≡ Maximum(IntegerOrder,{m ∈ int. m⇧^{R}\<lsq> a})"

fixes Γ

defines Γ_def[simp]: "Γ(S,p) ≡ Maximum(IntegerOrder,{⌊p⇧^{R}·x⌋. x∈S})"

fixes ia (infixl "\<za>" 69)

defines ia_def[simp]: "a\<za>b ≡ IntegerAddition`⟨ a,b⟩"

fixes iminus ("\<zm> _" 72)

defines iminus_def[simp]: "\<zm>a ≡ GroupInv(int,IntegerAddition)`(a)"

fixes isub (infixl "\<zs>" 69)

defines isub_def[simp]: "a\<zs>b ≡ a\<za> (\<zm> b)"

fixes intpositives ("\<int>⇩_{+}")

defines intpositives_def[simp]:

"\<int>⇩_{+}≡ PositiveSet(int,IntegerAddition,IntegerOrder)"

fixes zlesseq (infix "\<zlq>" 60)

defines lesseq_def[simp]: "m \<zlq> n ≡ ⟨m,n⟩ ∈ IntegerOrder"

fixes imult (infixl "\<zmu>" 70)

defines imult_def[simp]: "a\<zmu>b ≡ IntegerMultiplication`⟨ a,b⟩"

fixes izero ("\<zero>⇩_{Z}")

defines izero_def[simp]: "\<zero>⇩_{Z}≡ TheNeutralElement(int,IntegerAddition)"

fixes ione ("\<one>⇩_{Z}")

defines ione_def[simp]: "\<one>⇩_{Z}≡ TheNeutralElement(int,IntegerMultiplication)"

fixes itwo ("\<two>⇩_{Z}")

defines itwo_def[simp]: "\<two>⇩_{Z}≡ \<one>⇩_{Z}\<za>\<one>⇩_{Z}"

fixes abs

defines abs_def[simp]:

"abs(m) ≡ AbsoluteValue(int,IntegerAddition,IntegerOrder)`(m)"

fixes δ

defines δ_def[simp]: "δ(s,m,n) ≡ s`(m\<za>n)\<zs>s`(m)\<zs>s`(n)";

section{*Multiplication of real numbers*}

text{*Multiplication of real numbers is defined as a projection of

composition of slopes onto the space of equivalence classes of slopes.

Thus, the product of the real numbers given as classes of slopes $s$ and

$r$ is defined as the class of $s\circ r$. The goal of this section is to

show that multiplication defined this way is commutative.*}

text{*Let's recall a theorem from @{text "Int_ZF_2.thy"} that states that

if $f,g$ are slopes,

then $f\circ g$ is equivalent to $g\circ f$.

Here we conclude from that that

the classes of $f\circ g$ and $g\circ f$ are the same.*}

lemma (in real1) Real_ZF_1_1_L2: assumes A1: "f ∈ \<S>" "g ∈ \<S>"

shows "[fog] = [gof]"

proof -

from A1 have "fog ∼ gof"

using Slopes_def int1.Arthan_Th_9 SlopeOp1_def BoundedIntMaps_def

SlopeEquivalenceRel_def SlopeOp2_def by simp;

then show ?thesis using Real_ZF_1_L11 equiv_class_eq

by simp;

qed;

text{*Classes of slopes are real numbers.*}

lemma (in real1) Real_ZF_1_1_L3: assumes A1: "f ∈ \<S>"

shows "[f] ∈ \<real>"

proof -

from A1 have "[f] ∈ Slopes//SlopeEquivalenceRel"

using Slopes_def quotientI by simp;

then show "[f] ∈ \<real>" using RealNumbers_def by simp;

qed;

text{*Each real number is a class of a slope.*}

lemma (in real1) Real_ZF_1_1_L3A: assumes A1: "a∈\<real>"

shows "∃f∈\<S> . a = [f]"

proof -

from A1 have "a ∈ \<S>//SlopeEquivalenceRel"

using RealNumbers_def Slopes_def by simp

then show ?thesis using quotient_def

by simp;

qed;

text{*It is useful to have the definition of addition and multiplication

in the @{text "real1"} context notation.*}

lemma (in real1) Real_ZF_1_1_L4:

assumes A1: "f ∈ \<S>" "g ∈ \<S>"

shows

"[f] \<ra> [g] = [f\<fp>g]"

"[f] · [g] = [fog]"

proof -

let ?r = "SlopeEquivalenceRel"

have "[f]·[g] = ProjFun2(\<S>,?r,SlopeOp2)`⟨[f],[g]⟩"

using RealMultiplication_def Slopes_def by simp;

also from A1 have "… = [fog]"

using Real_ZF_1_L11 EquivClass_1_L10 Slopes_def

by simp;

finally show "[f] · [g] = [fog]" by simp;

have "[f] \<ra> [g] = ProjFun2(\<S>,?r,SlopeOp1)`⟨[f],[g]⟩"

using RealAddition_def Slopes_def by simp;

also from A1 have "… = [f\<fp>g]"

using Real_ZF_1_L11 EquivClass_1_L10 Slopes_def

by simp;

finally show "[f] \<ra> [g] = [f\<fp>g]" by simp;

qed;

text{*The next lemma is essentially the same as @{text " Real_ZF_1_L12"}, but

written in the notation defined in the @{text "real1"} context. It states

that if $f$ is a slope, then $-[f] = [-f]$.*}

lemma (in real1) Real_ZF_1_1_L4A: assumes "f ∈ \<S>"

shows "[\<fm>f] = \<rm>[f]"

using assms Slopes_def SlopeEquivalenceRel_def Real_ZF_1_L12

by simp;

text{*Subtracting real numbers correspods to adding the opposite slope.*}

lemma (in real1) Real_ZF_1_1_L4B: assumes A1: "f ∈ \<S>" "g ∈ \<S>"

shows "[f] \<rs> [g] = [f\<fp>(\<fm>g)]"

proof -

from A1 have "[f\<fp>(\<fm>g)] = [f] \<ra> [\<fm>g]"

using Slopes_def BoundedIntMaps_def int1.Int_ZF_2_1_L12

Real_ZF_1_1_L4 by simp;

with A1 show "[f] \<rs> [g] = [f\<fp>(\<fm>g)]"

using Real_ZF_1_1_L4A by simp;

qed;

text{*Multiplication of real numbers is commutative.*}

theorem (in real1) real_mult_commute: assumes A1: "a∈\<real>" "b∈\<real>"

shows "a·b = b·a"

proof -

from A1 have

"∃f∈\<S> . a = [f]"

"∃g∈\<S> . b = [g]"

using Real_ZF_1_1_L3A by auto;

then obtain f g where

"f ∈ \<S>" "g ∈ \<S>" and "a = [f]" "b = [g]"

by auto;

then show "a·b = b·a"

using Real_ZF_1_1_L4 Real_ZF_1_1_L2 by simp;

qed;

text{*Multiplication is commutative on reals.*}

lemma real_mult_commutative: shows

"RealMultiplication {is commutative on} RealNumbers"

using real1.real_mult_commute IsCommutative_def

by simp;

text{*The neutral element of multiplication of reals

(denoted as @{text "\<one>"} in the @{text "real1"} context)

is the class of identity function on integers. This is really shown

in @{text "Real_ZF_1_L11"}, here we only rewrite it in the notation used

in the @{text "real1"} context.*}

lemma (in real1) real_one_cl_identity: shows "[id(int)] = \<one>"

using Real_ZF_1_L11 by simp;

text{*If $f$ is bounded, then its class is the neutral element of additive

operation on reals (denoted as @{text "\<zero>"} in the @{text "real1"} context).*}

lemma (in real1) real_zero_cl_bounded_map:

assumes "f ∈ BoundedIntMaps" shows "[f] = \<zero>"

using assms Real_ZF_1_L11A by simp;

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

almost equal. This is proven in @{text "Real_ZF_1_L13"}, here we just

rewrite it in the notation used in the @{text "real1"} context.*}

lemma (in real1) Real_ZF_1_1_L5:

assumes "f ∈ \<S>" "g ∈ \<S>"

shows "[f] = [g] <-> f ∼ g"

using assms Slopes_def Real_ZF_1_L13 by simp;

text{*If the pair of function belongs to the slope equivalence relation, then

their classes are equal. This is convenient, because we don't need to assume

that $f,g$ are slopes (follows from the fact that $f\sim g$).*}

lemma (in real1) Real_ZF_1_1_L5A: assumes "f ∼ g"

shows "[f] = [g]"

using assms Real_ZF_1_L11 Slopes_def Real_ZF_1_1_L5

by auto;

text{*Identity function on integers is a slope.

This is proven in @{text "Real_ZF_1_L13"}, here we just

rewrite it in the notation used in the @{text "real1"} context.*}

lemma (in real1) id_on_int_is_slope: shows "id(int) ∈ \<S>"

using Real_ZF_1_L14 Slopes_def by simp;

text{*A result from @{text "Int_ZF_2.thy"}: the identity function on integers

is not almost equal to any bounded function.*}

lemma (in real1) Real_ZF_1_1_L7:

assumes A1: "f ∈ BoundedIntMaps"

shows "¬(id(int) ∼ f)"

using assms Slopes_def SlopeOp1_def BoundedIntMaps_def

SlopeEquivalenceRel_def BoundedIntMaps_def int1.Int_ZF_2_3_L12

by simp;

text{*Zero is not one.*}

lemma (in real1) real_zero_not_one: shows "\<one>≠\<zero>"

proof -

{ assume A1: "\<one>=\<zero>"

have "∃f ∈ \<S>. \<zero> = [f]"

using Real_ZF_1_L4 Real_ZF_1_1_L3A by simp;

with A1 have

"∃f ∈ \<S>. [id(int)] = [f] ∧ [f] = \<zero>"

using real_one_cl_identity by auto;

then have False using Real_ZF_1_1_L5 Slopes_def

Real_ZF_1_L10 Real_ZF_1_1_L7 id_on_int_is_slope

by auto;

} then show "\<one>≠\<zero>" by auto;

qed;

text{*Negative of a real number is a real number. Property of groups.*}

lemma (in real1) Real_ZF_1_1_L8: assumes "a∈\<real>" shows "(\<rm>a) ∈ \<real>"

using assms Real_ZF_1_L2 group0.inverse_in_group

by simp;

text{*An identity with three real numbers. *}

lemma (in real1) Real_ZF_1_1_L9: assumes "a∈\<real>" "b∈\<real>" "c∈\<real>"

shows "a·(b·c) = a·c·b"

using assms real_mult_commutative Real_ZF_1_L3 ring0.Ring_ZF_2_L4

by simp

section{*The order on reals*}

text{*In this section we show that the order relation defined by prescribing

the set of positive reals as the projection of the set of positive

slopes makes the ring of real numbers into an ordered ring. We also

collect the facts about ordered groups and rings that we use in

the construction.*}

text{*Positive slopes are slopes and positive reals are real.*}

lemma Real_ZF_1_2_L1: shows

"PositiveSlopes ⊆ Slopes"

"PositiveReals ⊆ RealNumbers"

proof -

have "PositiveSlopes =

{s ∈ Slopes. s``(PositiveIntegers) ∩ PositiveIntegers ∉ Fin(int)}"

using PositiveSlopes_def by simp;

then show "PositiveSlopes ⊆ Slopes" by (rule subset_with_property);

then have

"{SlopeEquivalenceRel``{s}. s ∈ PositiveSlopes } ⊆

Slopes//SlopeEquivalenceRel"

using EquivClass_1_L1A by simp;

then show "PositiveReals ⊆ RealNumbers"

using PositiveReals_def RealNumbers_def by simp;

qed;

text{*Positive reals are the same as classes of a positive slopes.*}

lemma (in real1) Real_ZF_1_2_L2:

shows "a ∈ PositiveReals <-> (∃f∈\<S>⇩_{+}. a = [f])"

proof

assume "a ∈ PositiveReals"

then have "a ∈ {([s]). s ∈ \<S>⇩_{+}}" using PositiveReals_def

by simp; (* it has to be ([a]), otherwise fails*)

then show "∃f∈\<S>⇩_{+}. a = [f]" by auto;

next assume "∃f∈\<S>⇩_{+}. a = [f]"

then have "a ∈ {([s]). s ∈ \<S>⇩_{+}}" by auto;

then show "a ∈ PositiveReals" using PositiveReals_def

by simp;

qed;

text{*Let's recall from @{text "Int_ZF_2.thy"} that the sum and composition

of positive slopes is a positive slope.*}

lemma (in real1) Real_ZF_1_2_L3:

assumes "f∈\<S>⇩_{+}" "g∈\<S>⇩_{+}"

shows

"f\<fp>g ∈ \<S>⇩_{+}"

"fog ∈ \<S>⇩_{+}"

using assms Slopes_def PositiveSlopes_def PositiveIntegers_def

SlopeOp1_def int1.sum_of_pos_sls_is_pos_sl

SlopeOp2_def int1.comp_of_pos_sls_is_pos_sl

by auto;

text{*Bounded integer maps are not positive slopes.*}

lemma (in real1) Real_ZF_1_2_L5:

assumes "f ∈ BoundedIntMaps"

shows "f ∉ \<S>⇩_{+}"

using assms BoundedIntMaps_def Slopes_def PositiveSlopes_def

PositiveIntegers_def int1.Int_ZF_2_3_L1B by simp;

text{*The set of positive reals is closed under addition and multiplication.

Zero (the neutral element of addition) is not a positive number. *}

lemma (in real1) Real_ZF_1_2_L6: shows

"PositiveReals {is closed under} RealAddition"

"PositiveReals {is closed under} RealMultiplication"

"\<zero> ∉ PositiveReals"

proof -

{ fix a fix b

assume "a ∈ PositiveReals" and "b ∈ PositiveReals"

then obtain f g where

I: "f ∈ \<S>⇩_{+}" "g ∈ \<S>⇩_{+}" and

II: "a = [f]" "b = [g]"

using Real_ZF_1_2_L2 by auto;

then have "f ∈ \<S>" "g ∈ \<S>" using Real_ZF_1_2_L1 Slopes_def

by auto;

with I II have

"a\<ra>b ∈ PositiveReals ∧ a·b ∈ PositiveReals"

using Real_ZF_1_1_L4 Real_ZF_1_2_L3 Real_ZF_1_2_L2

by auto;

} then show

"PositiveReals {is closed under} RealAddition"

"PositiveReals {is closed under} RealMultiplication"

using IsOpClosed_def

by auto;

{ assume "\<zero> ∈ PositiveReals"

then obtain f where "f ∈ \<S>⇩_{+}" and "\<zero> = [f]"

using Real_ZF_1_2_L2 by auto;

then have False

using Real_ZF_1_2_L1 Slopes_def Real_ZF_1_L10 Real_ZF_1_2_L5

by auto;

} then show "\<zero> ∉ PositiveReals" by auto;

qed;

text{*If a class of a slope $f$ is not zero, then either $f$ is

a positive slope or $-f$ is a positive slope. The real proof is in

@{text "Int_ZF_2.thy."}*}

lemma (in real1) Real_ZF_1_2_L7:

assumes A1: "f ∈ \<S>" and A2: "[f] ≠ \<zero>"

shows "(f ∈ \<S>⇩_{+}) Xor ((\<fm>f) ∈ \<S>⇩_{+})"

using assms Slopes_def SlopeEquivalenceRel_def BoundedIntMaps_def

PositiveSlopes_def PositiveIntegers_def

Real_ZF_1_L10 int1.Int_ZF_2_3_L8 by simp;

text{*The next lemma rephrases @{text " Int_ZF_2_3_L10"} in the notation

used in @{text "real1"} context.*}

lemma (in real1) Real_ZF_1_2_L8:

assumes A1: "f ∈ \<S>" "g ∈ \<S>"

and A2: "(f ∈ \<S>⇩_{+}) Xor (g ∈ \<S>⇩_{+})"

shows "([f] ∈ PositiveReals) Xor ([g] ∈ PositiveReals)"

using assms PositiveReals_def SlopeEquivalenceRel_def Slopes_def

SlopeOp1_def BoundedIntMaps_def PositiveSlopes_def PositiveIntegers_def

int1.Int_ZF_2_3_L10 by simp;

text{*The trichotomy law for the (potential) order on reals: if $a\neq 0$,

then either $a$ is positive or $-a$ is potitive.*}

lemma (in real1) Real_ZF_1_2_L9:

assumes A1: "a∈\<real>" and A2: "a≠\<zero>"

shows "(a ∈ PositiveReals) Xor ((\<rm>a) ∈ PositiveReals)"

proof -

from A1 obtain f where I: "f ∈ \<S>" "a = [f]"

using Real_ZF_1_1_L3A by auto;

with A2 have "([f] ∈ PositiveReals) Xor ([\<fm>f] ∈ PositiveReals)"

using Slopes_def BoundedIntMaps_def int1.Int_ZF_2_1_L12

Real_ZF_1_2_L7 Real_ZF_1_2_L8 by simp;

with I show "(a ∈ PositiveReals) Xor ((\<rm>a) ∈ PositiveReals)"

using Real_ZF_1_1_L4A by simp;

qed;

text{*Finally we are ready to prove that real numbers form an ordered ring

with no zero divisors.*}

theorem reals_are_ord_ring: shows

"IsAnOrdRing(RealNumbers,RealAddition,RealMultiplication,OrderOnReals)"

"OrderOnReals {is total on} RealNumbers"

"PositiveSet(RealNumbers,RealAddition,OrderOnReals) = PositiveReals"

"HasNoZeroDivs(RealNumbers,RealAddition,RealMultiplication)"

proof -

let ?R = "RealNumbers"

let ?A = "RealAddition"

let ?M = "RealMultiplication"

let ?P = "PositiveReals"

let ?r = "OrderOnReals"

let ?z = "TheNeutralElement(?R, ?A)"

have I:

"ring0(?R, ?A, ?M)"

"?M {is commutative on} ?R"

"?P ⊆ ?R"

"?P {is closed under} ?A"

"TheNeutralElement(?R, ?A) ∉ ?P"

"∀a∈?R. a ≠ ?z --> (a ∈ ?P) Xor (GroupInv(?R, ?A)`(a) ∈ ?P)"

"?P {is closed under} ?M"

"?r = OrderFromPosSet(?R, ?A, ?P)"

using real0.Real_ZF_1_L3 real_mult_commutative Real_ZF_1_2_L1

real1.Real_ZF_1_2_L6 real1.Real_ZF_1_2_L9 OrderOnReals_def

by auto;

then show "IsAnOrdRing(?R, ?A, ?M, ?r)"

by (rule ring0.ring_ord_by_positive_set)

from I show "?r {is total on} ?R"

by (rule ring0.ring_ord_by_positive_set)

from I show "PositiveSet(?R,?A,?r) = ?P"

by (rule ring0.ring_ord_by_positive_set)

from I show "HasNoZeroDivs(?R,?A,?M)"

by (rule ring0.ring_ord_by_positive_set);

qed;

text{*All theorems proven in the @{text "ring1"}

(about ordered rings), @{text "group3"} (about ordered groups) and

@{text "group1"} (about groups)

contexts are valid as applied to ordered real numbers with addition

and (real) order.*}

lemma Real_ZF_1_2_L10: shows

"ring1(RealNumbers,RealAddition,RealMultiplication,OrderOnReals)"

"IsAnOrdGroup(RealNumbers,RealAddition,OrderOnReals)"

"group3(RealNumbers,RealAddition,OrderOnReals)"

"OrderOnReals {is total on} RealNumbers"

proof -

show "ring1(RealNumbers,RealAddition,RealMultiplication,OrderOnReals)"

using reals_are_ord_ring OrdRing_ZF_1_L2 by simp;

then show

"IsAnOrdGroup(RealNumbers,RealAddition,OrderOnReals)"

"group3(RealNumbers,RealAddition,OrderOnReals)"

"OrderOnReals {is total on} RealNumbers"

using ring1.OrdRing_ZF_1_L4 by auto

qed;

text{*If $a=b$ or $b-a$ is positive, then $a$ is less or equal $b$.*}

lemma (in real1) Real_ZF_1_2_L11: assumes A1: "a∈\<real>" "b∈\<real>" and

A3: "a=b ∨ b\<rs>a ∈ PositiveReals"

shows "a\<lsq>b"

using assms reals_are_ord_ring Real_ZF_1_2_L10

group3.OrderedGroup_ZF_1_L30 by simp;

text{*A sufficient condition for two classes to be in the real order.*}

lemma (in real1) Real_ZF_1_2_L12: assumes A1: "f ∈ \<S>" "g ∈ \<S>" and

A2: "f∼g ∨ (g \<fp> (\<fm>f)) ∈ \<S>⇩_{+}"

shows "[f] \<lsq> [g]"

proof -

from A1 A2 have "[f] = [g] ∨ [g]\<rs>[f] ∈ PositiveReals"

using Real_ZF_1_1_L5A Real_ZF_1_2_L2 Real_ZF_1_1_L4B

by auto;

with A1 show "[f] \<lsq> [g]" using Real_ZF_1_1_L3 Real_ZF_1_2_L11

by simp;

qed;

text{*Taking negative on both sides reverses the inequality, a case with

an inverse on one side. Property of ordered groups.*}

lemma (in real1) Real_ZF_1_2_L13:

assumes A1: "a∈\<real>" and A2: "(\<rm>a) \<lsq> b"

shows "(\<rm>b) \<lsq> a"

using assms Real_ZF_1_2_L10 group3.OrderedGroup_ZF_1_L5AG

by simp;

text{*Real order is antisymmetric.*}

lemma (in real1) real_ord_antisym:

assumes A1: "a\<lsq>b" "b\<lsq>a" shows "a=b"

proof -

from A1 have

"group3(RealNumbers,RealAddition,OrderOnReals)"

"⟨a,b⟩ ∈ OrderOnReals" "⟨b,a⟩ ∈ OrderOnReals"

using Real_ZF_1_2_L10 by auto;

then show "a=b" by (rule group3.group_order_antisym);

qed;

text{*Real order is transitive.*}

lemma (in real1) real_ord_transitive: assumes A1: "a\<lsq>b" "b\<lsq>c"

shows "a\<lsq>c"

proof -

from A1 have

"group3(RealNumbers,RealAddition,OrderOnReals)"

"⟨a,b⟩ ∈ OrderOnReals" "⟨b,c⟩ ∈ OrderOnReals"

using Real_ZF_1_2_L10 by auto

then have "⟨a,c⟩ ∈ OrderOnReals"

by (rule group3.Group_order_transitive);

then show "a\<lsq>c" by simp;

qed;

text{*We can multiply both sides of an inequality

by a nonnegative real number.*}

lemma (in real1) Real_ZF_1_2_L14:

assumes "a\<lsq>b" and "\<zero>\<lsq>c"

shows

"a·c \<lsq> b·c"

"c·a \<lsq> c·b"

using assms Real_ZF_1_2_L10 ring1.OrdRing_ZF_1_L9

by auto;

text{*A special case of @{text "Real_ZF_1_2_L14"}: we can multiply

an inequality by a real number.*}

lemma (in real1) Real_ZF_1_2_L14A:

assumes A1: "a\<lsq>b" and A2: "c∈\<real>⇩_{+}"

shows "c·a \<lsq> c·b"

using assms Real_ZF_1_2_L10 ring1.OrdRing_ZF_1_L9A

by simp

text{*In the @{text "real1"} context notation $a\leq b$

implies that $a$ and $b$ are real numbers.*}

lemma (in real1) Real_ZF_1_2_L15: assumes "a\<lsq>b" shows "a∈\<real>" "b∈\<real>"

using assms Real_ZF_1_2_L10 group3.OrderedGroup_ZF_1_L4

by auto;

text{*$a\leq b$ implies that $0 \leq b -a$.*}

lemma (in real1) Real_ZF_1_2_L16: assumes "a\<lsq>b"

shows "\<zero> \<lsq> b\<rs>a"

using assms Real_ZF_1_2_L10 group3.OrderedGroup_ZF_1_L12A

by simp;

text{*A sum of nonnegative elements is nonnegative.*}

lemma (in real1) Real_ZF_1_2_L17: assumes "\<zero>\<lsq>a" "\<zero>\<lsq>b"

shows "\<zero> \<lsq> a\<ra>b"

using assms Real_ZF_1_2_L10 group3.OrderedGroup_ZF_1_L12

by simp;

text{*We can add sides of two inequalities*}

lemma (in real1) Real_ZF_1_2_L18: assumes "a\<lsq>b" "c\<lsq>d"

shows "a\<ra>c \<lsq> b\<ra>d"

using assms Real_ZF_1_2_L10 group3.OrderedGroup_ZF_1_L5B

by simp;

text{*The order on real is reflexive.*}

lemma (in real1) real_ord_refl: assumes "a∈\<real>" shows "a\<lsq>a"

using assms Real_ZF_1_2_L10 group3.OrderedGroup_ZF_1_L3

by simp;

text{*We can add a real number to both sides of an inequality.*}

lemma (in real1) add_num_to_ineq: assumes "a\<lsq>b" and "c∈\<real>"

shows "a\<ra>c \<lsq> b\<ra>c"

using assms Real_ZF_1_2_L10 IsAnOrdGroup_def by simp;

text{*We can put a number on the other side of an inequality,

changing its sign.*}

lemma (in real1) Real_ZF_1_2_L19:

assumes "a∈\<real>" "b∈\<real>" and "c \<lsq> a\<ra>b"

shows "c\<rs>b \<lsq> a"

using assms Real_ZF_1_2_L10 group3.OrderedGroup_ZF_1_L9C

by simp;

text{*What happens when one real number is not greater or equal than another?*}

lemma (in real1) Real_ZF_1_2_L20: assumes "a∈\<real>" "b∈\<real>" and "¬(a\<lsq>b)"

shows "b \<ls> a"

proof -

from assms have I:

"group3(\<real>,RealAddition,OrderOnReals)"

"OrderOnReals {is total on} \<real>"

"a∈\<real>" "b∈\<real>" "¬(⟨a,b⟩ ∈ OrderOnReals)"

using Real_ZF_1_2_L10 by auto;

then have "⟨b,a⟩ ∈ OrderOnReals"

by (rule group3.OrderedGroup_ZF_1_L8);

then have "b \<lsq> a" by simp;

moreover from I have "a≠b" by (rule group3.OrderedGroup_ZF_1_L8);

ultimately show "b \<ls> a" by auto;

qed;

text{*We can put a number on the other side of an inequality,

changing its sign, version with a minus.*}

lemma (in real1) Real_ZF_1_2_L21:

assumes "a∈\<real>" "b∈\<real>" and "c \<lsq> a\<rs>b"

shows "c\<ra>b \<lsq> a"

using assms Real_ZF_1_2_L10 group3.OrderedGroup_ZF_1_L5J

by simp;

text{*The order on reals is a relation on reals.*}

lemma (in real1) Real_ZF_1_2_L22: shows "OrderOnReals ⊆ \<real>×\<real>"

using Real_ZF_1_2_L10 IsAnOrdGroup_def

by simp;

text{*A set that is bounded above in the sense defined by order

on reals is a subset of real numbers.*}

lemma (in real1) Real_ZF_1_2_L23:

assumes A1: "IsBoundedAbove(A,OrderOnReals)"

shows "A ⊆ \<real>"

using A1 Real_ZF_1_2_L22 Order_ZF_3_L1A

by blast; (* A1 has to be here *)

text{*Properties of the maximum of three real numbers.*}

lemma (in real1) Real_ZF_1_2_L24:

assumes A1: "a∈\<real>" "b∈\<real>" "c∈\<real>"

shows

"Maximum(OrderOnReals,{a,b,c}) ∈ {a,b,c}"

"Maximum(OrderOnReals,{a,b,c}) ∈ \<real>"

"a \<lsq> Maximum(OrderOnReals,{a,b,c})"

"b \<lsq> Maximum(OrderOnReals,{a,b,c})"

"c \<lsq> Maximum(OrderOnReals,{a,b,c})"

proof -

have "IsLinOrder(\<real>,OrderOnReals)"

using Real_ZF_1_2_L10 group3.group_ord_total_is_lin

by simp;

with A1 show

"Maximum(OrderOnReals,{a,b,c}) ∈ {a,b,c}"

"Maximum(OrderOnReals,{a,b,c}) ∈ \<real>"

"a \<lsq> Maximum(OrderOnReals,{a,b,c})"

"b \<lsq> Maximum(OrderOnReals,{a,b,c})"

"c \<lsq> Maximum(OrderOnReals,{a,b,c})"

using Finite_ZF_1_L2A by auto;

qed;

text{*A form of transitivity for the order on reals.*}

lemma (in real1) real_strict_ord_transit:

assumes A1: "a\<lsq>b" and A2: "b\<ls>c"

shows "a\<ls>c"

proof -

from A1 A2 have I:

"group3(\<real>,RealAddition,OrderOnReals)"

"⟨a,b⟩ ∈ OrderOnReals" "⟨b,c⟩ ∈ OrderOnReals ∧ b≠c"

using Real_ZF_1_2_L10 by auto

then have "⟨a,c⟩ ∈ OrderOnReals ∧ a≠c" by (rule group3.group_strict_ord_transit)

then show "a\<ls>c" by simp

qed;

text{*We can multiply a right hand side of an inequality between

positive real numbers by a number that is greater than one.*}

lemma (in real1) Real_ZF_1_2_L25:

assumes "b ∈ \<real>⇩_{+}" and "a\<lsq>b" and "\<one>\<ls>c"

shows "a\<ls>b·c"

using assms reals_are_ord_ring Real_ZF_1_2_L10 ring1.OrdRing_ZF_3_L17

by simp;

text{*We can move a real number to the other side of a strict inequality,

changing its sign.*}

lemma (in real1) Real_ZF_1_2_L26:

assumes "a∈\<real>" "b∈\<real>" and "a\<rs>b \<ls> c"

shows "a \<ls> c\<ra>b"

using assms Real_ZF_1_2_L10 group3.OrderedGroup_ZF_1_L12B

by simp;

text{*Real order is translation invariant.*}

lemma (in real1) real_ord_transl_inv:

assumes "a\<lsq>b" and "c∈\<real>"

shows "c\<ra>a \<lsq> c\<ra>b"

using assms Real_ZF_1_2_L10 IsAnOrdGroup_def

by simp;

text{*It is convenient to have the transitivity of the order on integers

in the notation specific to @{text "real1"} context. This may be confusing

for the presentation readers: even though @{text "\<zlq>"} and

@{text "\<lsq>"} are printed in the same way, they are different symbols

in the source. In the @{text "real1"} context the former denotes

inequality between integers, and the latter denotes inequality between real

numbers (classes of slopes). The next lemma is about transitivity of the

order relation on integers.*}

lemma (in real1) int_order_transitive:

assumes A1: "a\<zlq>b" "b\<zlq>c"

shows "a\<zlq>c"

proof -

from A1 have

"⟨a,b⟩ ∈ IntegerOrder" and "⟨b,c⟩ ∈ IntegerOrder"

by auto

then have "⟨a,c⟩ ∈ IntegerOrder"

by (rule Int_ZF_2_L5);

then show "a\<zlq>c" by simp;

qed;

text{*A property of nonempty subsets of real numbers that don't

have a maximum: for any element we can find one that is (strictly) greater.*}

lemma (in real1) Real_ZF_1_2_L27:

assumes "A⊆\<real>" and "¬HasAmaximum(OrderOnReals,A)" and "x∈A"

shows "∃y∈A. x\<ls>y"

using assms Real_ZF_1_2_L10 group3.OrderedGroup_ZF_2_L2B

by simp;

text{*The next lemma shows what happens when one real number

is not greater or equal than another.*}

lemma (in real1) Real_ZF_1_2_L28:

assumes "a∈\<real>" "b∈\<real>" and "¬(a\<lsq>b)"

shows "b\<ls>a"

proof -

from assms have

"group3(\<real>,RealAddition,OrderOnReals)"

"OrderOnReals {is total on} \<real>"

"a∈\<real>" "b∈\<real>" "⟨a,b⟩ ∉ OrderOnReals"

using Real_ZF_1_2_L10 by auto

then have "⟨b,a⟩ ∈ OrderOnReals ∧ b≠a"

by (rule group3.OrderedGroup_ZF_1_L8)

then show "b\<ls>a" by simp

qed;

text{*If a real number is less than another, then the second one can not

be less or equal that the first.*}

lemma (in real1) Real_ZF_1_2_L29:

assumes "a\<ls>b" shows "¬(b\<lsq>a)"

proof -

from assms have

"group3(\<real>,RealAddition,OrderOnReals)"

"⟨a,b⟩ ∈ OrderOnReals" "a≠b"

using Real_ZF_1_2_L10 by auto;

then have "⟨b,a⟩ ∉ OrderOnReals"

by (rule group3.OrderedGroup_ZF_1_L8AA);

then show "¬(b\<lsq>a)" by simp;

qed;

section{*Inverting reals*}

text{*In this section we tackle the issue of existence of (multiplicative)

inverses of real numbers and show that real numbers form an

ordered field. We also restate here some facts specific to ordered fields

that we need for the construction. The actual proofs of most of these facts

can be found in @{text "Field_ZF.thy"} and @{text "OrderedField_ZF.thy"}*}

text{*We rewrite the theorem from @{text "Int_ZF_2.thy"} that shows

that for every positive slope we can find one that is almost equal and

has an inverse.*}

lemma (in real1) pos_slopes_have_inv: assumes "f ∈ \<S>⇩_{+}"

shows "∃g∈\<S>. f∼g ∧ (∃h∈\<S>. goh ∼ id(int))"

using assms PositiveSlopes_def Slopes_def PositiveIntegers_def

int1.pos_slope_has_inv SlopeOp1_def SlopeOp2_def

BoundedIntMaps_def SlopeEquivalenceRel_def

by simp;

text{*The set of real numbers we are constructing is an ordered field.*}

theorem (in real1) reals_are_ord_field: shows

"IsAnOrdField(RealNumbers,RealAddition,RealMultiplication,OrderOnReals)"

proof -

let ?R = "RealNumbers"

let ?A = "RealAddition"

let ?M = "RealMultiplication"

let ?r = "OrderOnReals"

have "ring1(?R,?A,?M,?r)" and "\<zero> ≠ \<one>"

using reals_are_ord_ring OrdRing_ZF_1_L2 real_zero_not_one

by auto;

moreover have "?M {is commutative on} ?R"

using real_mult_commutative by simp

moreover have

"∀a∈PositiveSet(?R,?A,?r). ∃b∈?R. a·b = \<one>"

proof

fix a assume "a ∈ PositiveSet(?R,?A,?r)"

then obtain f where I: "f∈\<S>⇩_{+}" and II: "a = [f]"

using reals_are_ord_ring Real_ZF_1_2_L2

by auto;

then have "∃g∈\<S>. f∼g ∧ (∃h∈\<S>. goh ∼ id(int))"

using pos_slopes_have_inv by simp;

then obtain g where

III: "g∈\<S>" and IV: "f∼g" and V: "∃h∈\<S>. goh ∼ id(int)"

by auto;

from V obtain h where VII: "h∈\<S>" and VIII: "goh ∼ id(int)"

by auto;

from I III IV have "[f] = [g]"

using Real_ZF_1_2_L1 Slopes_def Real_ZF_1_1_L5

by auto;

with II III VII VIII have "a·[h] = \<one>"

using Real_ZF_1_1_L4 Real_ZF_1_1_L5A real_one_cl_identity

by simp;

with VII show "∃b∈?R. a·b = \<one>" using Real_ZF_1_1_L3

by auto;

qed;

ultimately show ?thesis using ring1.OrdField_ZF_1_L4

by simp;

qed;

text{*Reals form a field.*}

lemma reals_are_field:

shows "IsAfield(RealNumbers,RealAddition,RealMultiplication)"

using real1.reals_are_ord_field OrdField_ZF_1_L1A

by simp;

text{*Theorem proven in @{text "field0"} and @{text "field1"} contexts

are valid as applied to real numbers.*}

lemma field_cntxts_ok: shows

"field0(RealNumbers,RealAddition,RealMultiplication)"

"field1(RealNumbers,RealAddition,RealMultiplication,OrderOnReals)"

using reals_are_field real1.reals_are_ord_field

field_field0 OrdField_ZF_1_L2 by auto;

text{*If $a$ is positive, then $a^{-1}$ is also positive.*}

lemma (in real1) Real_ZF_1_3_L1: assumes "a ∈ \<real>⇩_{+}"

shows "a¯ ∈ \<real>⇩_{+}" "a¯ ∈ \<real>"

using assms field_cntxts_ok field1.OrdField_ZF_1_L8 PositiveSet_def

by auto;

text{*A technical fact about multiplying strict inequality by the inverse

of one of the sides.*}

lemma (in real1) Real_ZF_1_3_L2:

assumes "a ∈ \<real>⇩_{+}" and "a¯ \<ls> b"

shows "\<one> \<ls> b·a"

using assms field_cntxts_ok field1.OrdField_ZF_2_L2

by simp;

text{*If $a$ is smaller than $b$, then $(b-a)^{-1}$ is positive.*}

lemma (in real1) Real_ZF_1_3_L3: assumes "a\<ls>b"

shows "(b\<rs>a)¯ ∈ \<real>⇩_{+}"

using assms field_cntxts_ok field1.OrdField_ZF_1_L9

by simp;

text{*We can put a positive factor on the other side of a strict

inequality, changing it to its inverse.*}

lemma (in real1) Real_ZF_1_3_L4:

assumes A1: "a∈\<real>" "b∈\<real>⇩_{+}" and A2: "a·b \<ls> c"

shows "a \<ls> c·b¯"

using assms field_cntxts_ok field1.OrdField_ZF_2_L6

by simp;

text{*We can put a positive factor on the other side of a strict

inequality, changing it to its inverse, version with the product

initially on the right hand side.*}

lemma (in real1) Real_ZF_1_3_L4A:

assumes A1: "b∈\<real>" "c∈\<real>⇩_{+}" and A2: "a \<ls> b·c"

shows "a·c¯ \<ls> b"

using assms field_cntxts_ok field1.OrdField_ZF_2_L6A

by simp;

text{*We can put a positive factor on the other side of an inequality,

changing it to its inverse, version with the product

initially on the right hand side.*}

lemma (in real1) Real_ZF_1_3_L4B:

assumes A1: "b∈\<real>" "c∈\<real>⇩_{+}" and A2: "a \<lsq> b·c"

shows "a·c¯ \<lsq> b"

using assms field_cntxts_ok field1.OrdField_ZF_2_L5A

by simp;

text{*We can put a positive factor on the other side of an inequality,

changing it to its inverse, version with the product

initially on the left hand side.*}

lemma (in real1) Real_ZF_1_3_L4C:

assumes A1: "a∈\<real>" "b∈\<real>⇩_{+}" and A2: "a·b \<lsq> c"

shows "a \<lsq> c·b¯"

using assms field_cntxts_ok field1.OrdField_ZF_2_L5

by simp;

text{*A technical lemma about solving a strict inequality with three

real numbers and inverse of a difference.*}

lemma (in real1) Real_ZF_1_3_L5:

assumes "a\<ls>b" and "(b\<rs>a)¯ \<ls> c"

shows "\<one> \<ra> a·c \<ls> b·c"

using assms field_cntxts_ok field1.OrdField_ZF_2_L9

by simp;

text{*We can multiply an inequality by the inverse of a positive number.*}

lemma (in real1) Real_ZF_1_3_L6:

assumes "a\<lsq>b" and "c∈\<real>⇩_{+}" shows "a·c¯ \<lsq> b·c¯"

using assms field_cntxts_ok field1.OrdField_ZF_2_L3

by simp;

text{*We can multiply a strict inequality by a positive number or its inverse.

*}

lemma (in real1) Real_ZF_1_3_L7:

assumes "a\<ls>b" and "c∈\<real>⇩_{+}" shows

"a·c \<ls> b·c"

"c·a \<ls> c·b"

"a·c¯ \<ls> b·c¯"

using assms field_cntxts_ok field1.OrdField_ZF_2_L4

by auto;

text{*An identity with three real numbers, inverse and cancelling.*}

lemma (in real1) Real_ZF_1_3_L8: assumes"a∈\<real>" "b∈\<real>" "b≠\<zero>" "c∈\<real>"

shows "a·b·(c·b¯) = a·c"

using assms field_cntxts_ok field0.Field_ZF_2_L6

by simp;

section{*Completeness*}

text{*This goal of this section is to show that the order on real numbers

is complete, that is every subset of reals that is bounded above

has a smallest upper bound.*}

text{*If $m$ is an integer, then @{text "m⇧^{R}"} is a real number.

Recall that in @{text "real1"} context @{text "m⇧^{R}"} denotes the class

of the slope $n\mapsto m\cdot n$.*}

lemma (in real1) real_int_is_real: assumes "m ∈ int"

shows "m⇧^{R}∈ \<real>"

using assms int1.Int_ZF_2_5_L1 Real_ZF_1_1_L3 by simp;

text{*The negative of the real embedding of an integer is the embedding

of the negative of the integer.*}

lemma (in real1) Real_ZF_1_4_L1: assumes "m ∈ int"

shows "(\<zm>m)⇧^{R}= \<rm>(m⇧^{R})"

using assms int1.Int_ZF_2_5_L3 int1.Int_ZF_2_5_L1 Real_ZF_1_1_L4A

by simp;

text{*The embedding of sum of integers is the sum of embeddings.*}

lemma (in real1) Real_ZF_1_4_L1A: assumes "m ∈ int" "k ∈ int"

shows "m⇧^{R}\<ra> k⇧^{R}= ((m\<za>k)⇧^{R})"

using assms int1.Int_ZF_2_5_L1 SlopeOp1_def int1.Int_ZF_2_5_L3A

Real_ZF_1_1_L4 by simp;

text{*The embedding of a difference of integers is the difference

of embeddings.*}

lemma (in real1) Real_ZF_1_4_L1B: assumes A1: "m ∈ int" "k ∈ int"

shows "m⇧^{R}\<rs> k⇧^{R}= (m\<zs>k)⇧^{R}"

proof -

from A1 have "(\<zm>k) ∈ int" using int0.Int_ZF_1_1_L4

by simp;

with A1 have "(m\<zs>k)⇧^{R}= m⇧^{R}\<ra> (\<zm>k)⇧^{R}"

using Real_ZF_1_4_L1A by simp;

with A1 show "m⇧^{R}\<rs> k⇧^{R}= (m\<zs>k)⇧^{R}"

using Real_ZF_1_4_L1 by simp;

qed;

text{*The embedding of the product of integers is the product of embeddings.*}

lemma (in real1) Real_ZF_1_4_L1C: assumes "m ∈ int" "k ∈ int"

shows "m⇧^{R}· k⇧^{R}= (m\<zmu>k)⇧^{R}"

using assms int1.Int_ZF_2_5_L1 SlopeOp2_def int1.Int_ZF_2_5_L3B

Real_ZF_1_1_L4 by simp;

text{*For any real numbers there is an integer whose real version is

greater or equal.*}

lemma (in real1) Real_ZF_1_4_L2: assumes A1: "a∈\<real>"

shows "∃m∈int. a \<lsq> m⇧^{R}"

proof -

from A1 obtain f where I: "f∈\<S>" and II: "a = [f]"

using Real_ZF_1_1_L3A by auto;

then have "∃m∈int. ∃g∈\<S>.

{⟨n,m\<zmu>n⟩ . n ∈ int} ∼ g ∧ (f∼g ∨ (g \<fp> (\<fm>f)) ∈ \<S>⇩_{+})"

using int1.Int_ZF_2_5_L2 Slopes_def SlopeOp1_def

BoundedIntMaps_def SlopeEquivalenceRel_def

PositiveIntegers_def PositiveSlopes_def

by simp;

then obtain m g where III: "m∈int" and IV: "g∈\<S>" and

"{⟨n,m\<zmu>n⟩ . n ∈ int} ∼ g ∧ (f∼g ∨ (g \<fp> (\<fm>f)) ∈ \<S>⇩_{+})"

by auto;

then have "m⇧^{R}= [g]" and "f ∼ g ∨ (g \<fp> (\<fm>f)) ∈ \<S>⇩_{+}"

using Real_ZF_1_1_L5A by auto;

with I II IV have "a \<lsq> m⇧^{R}" using Real_ZF_1_2_L12

by simp;

with III show "∃m∈int. a \<lsq> m⇧^{R}" by auto;

qed;

text{*For any real numbers there is an integer whose real version (embedding)

is less or equal.*}

lemma (in real1) Real_ZF_1_4_L3: assumes A1: "a∈\<real>"

shows "{m ∈ int. m⇧^{R}\<lsq> a} ≠ 0"

proof -

from A1 have "(\<rm>a) ∈ \<real>" using Real_ZF_1_1_L8

by simp;

then obtain m where I: "m∈int" and II: "(\<rm>a) \<lsq> m⇧^{R}"

using Real_ZF_1_4_L2 by auto;

let ?k = "GroupInv(int,IntegerAddition)`(m)"

from A1 I II have "?k ∈ int" and "?k⇧^{R}\<lsq> a"

using Real_ZF_1_2_L13 Real_ZF_1_4_L1 int0.Int_ZF_1_1_L4

by auto;

then show ?thesis by auto

qed;

text{*Embeddings of two integers are equal only if the integers are equal.*}

lemma (in real1) Real_ZF_1_4_L4:

assumes A1: "m ∈ int" "k ∈ int" and A2: "m⇧^{R}= k⇧^{R}"

shows "m=k"

proof -

let ?r = "{⟨n, IntegerMultiplication ` ⟨m, n⟩⟩ . n ∈ int}"

let ?s = "{⟨n, IntegerMultiplication ` ⟨k, n⟩⟩ . n ∈ int}"

from A1 A2 have "?r ∼ ?s"

using int1.Int_ZF_2_5_L1 AlmostHoms_def Real_ZF_1_1_L5

by simp;

with A1 have

"m ∈ int" "k ∈ int"

"⟨?r,?s⟩ ∈ QuotientGroupRel(AlmostHoms(int, IntegerAddition),

AlHomOp1(int, IntegerAddition),FinRangeFunctions(int, int))"

using SlopeEquivalenceRel_def Slopes_def SlopeOp1_def

BoundedIntMaps_def by auto;

then show "m=k" by (rule int1.Int_ZF_2_5_L6);

qed;

text{*The embedding of integers preserves the order.*}

lemma (in real1) Real_ZF_1_4_L5: assumes A1: "m\<zlq>k"

shows "m⇧^{R}\<lsq> k⇧^{R}"

proof -

let ?r = "{⟨n, m\<zmu>n⟩ . n ∈ int}"

let ?s = "{⟨n, k\<zmu>n⟩ . n ∈ int}"

from A1 have "?r ∈ \<S>" "?s ∈ \<S>"

using int0.Int_ZF_2_L1A int1.Int_ZF_2_5_L1 by auto;

moreover from A1 have "?r ∼ ?s ∨ ?s \<fp> (\<fm>?r) ∈ \<S>⇩_{+}"

using Slopes_def SlopeOp1_def BoundedIntMaps_def SlopeEquivalenceRel_def

PositiveIntegers_def PositiveSlopes_def

int1.Int_ZF_2_5_L4 by simp;

ultimately show "m⇧^{R}\<lsq> k⇧^{R}" using Real_ZF_1_2_L12

by simp;

qed;

text{*The embedding of integers preserves the strict order.*}

lemma (in real1) Real_ZF_1_4_L5A: assumes A1: "m\<zlq>k" "m≠k"

shows "m⇧^{R}\<ls> k⇧^{R}"

proof -

from A1 have "m⇧^{R}\<lsq> k⇧^{R}" using Real_ZF_1_4_L5

by simp;

moreover

from A1 have T: "m ∈ int" "k ∈ int"

using int0.Int_ZF_2_L1A by auto;

with A1 have "m⇧^{R}≠ k⇧^{R}" using Real_ZF_1_4_L4

by auto;

ultimately show "m⇧^{R}\<ls> k⇧^{R}" by simp

qed;

text{*For any real number there is a positive integer

whose real version is (strictly) greater.

This is Lemma 14 i) in \cite{Arthan2004}.*}

lemma (in real1) Arthan_Lemma14i: assumes A1: "a∈\<real>"

shows "∃n∈\<int>⇩_{+}. a \<ls> n⇧^{R}"

proof -

from A1 obtain m where I: "m∈int" and II: "a \<lsq> m⇧^{R}"

using Real_ZF_1_4_L2 by auto;

let ?n = "GreaterOf(IntegerOrder,\<one>⇩_{Z},m) \<za> \<one>⇩_{Z}"

from I have T: "?n ∈\<int>⇩_{+}" and "m \<zlq> ?n" "m≠?n"

using int0.Int_ZF_1_5_L7B by auto;

then have III: "m⇧^{R}\<ls> ?n⇧^{R}"

using Real_ZF_1_4_L5A by simp;

with II have "a \<ls> ?n⇧^{R}" by (rule real_strict_ord_transit);

with T show ?thesis by auto

qed;

text{*If one embedding is less or equal than another, then the integers

are also less or equal.*}

lemma (in real1) Real_ZF_1_4_L6:

assumes A1: "k ∈ int" "m ∈ int" and A2: "m⇧^{R}\<lsq> k⇧^{R}"

shows "m\<zlq>k"

proof -

{ assume A3: "⟨m,k⟩ ∉ IntegerOrder"

with A1 have "⟨k,m⟩ ∈ IntegerOrder"

by (rule int0.Int_ZF_2_L19);

then have "k⇧^{R}\<lsq> m⇧^{R}" using Real_ZF_1_4_L5

by simp;

with A2 have "m⇧^{R}= k⇧^{R}" by (rule real_ord_antisym);

with A1 have "k = m" using Real_ZF_1_4_L4

by auto;

moreover from A1 A3 have "k≠m" by (rule int0.Int_ZF_2_L19)

ultimately have False by simp;

} then show "m\<zlq>k" by auto;

qed;

text{*The floor function is well defined and has expected properties.*}

lemma (in real1) Real_ZF_1_4_L7: assumes A1: "a∈\<real>"

shows

"IsBoundedAbove({m ∈ int. m⇧^{R}\<lsq> a},IntegerOrder)"

"{m ∈ int. m⇧^{R}\<lsq> a} ≠ 0"

"⌊a⌋ ∈ int"

"⌊a⌋⇧^{R}\<lsq> a"

proof -

let ?A = "{m ∈ int. m⇧^{R}\<lsq> a}"

from A1 obtain K where I: "K∈int" and II: "a \<lsq> (K⇧^{R})"

using Real_ZF_1_4_L2 by auto;

{ fix n assume "n ∈ ?A"

then have III: "n ∈ int" and IV: "n⇧^{R}\<lsq> a"

by auto

from IV II have "(n⇧^{R}) \<lsq> (K⇧^{R})"

by (rule real_ord_transitive);

with I III have "n\<zlq>K" using Real_ZF_1_4_L6

by simp;

} then have "∀n∈?A. ⟨n,K⟩ ∈ IntegerOrder"

by simp;

then show "IsBoundedAbove(?A,IntegerOrder)"

by (rule Order_ZF_3_L10);

moreover from A1 show "?A ≠ 0" using Real_ZF_1_4_L3

by simp

ultimately have "Maximum(IntegerOrder,?A) ∈ ?A"

by (rule int0.int_bounded_above_has_max);

then show "⌊a⌋ ∈ int" "⌊a⌋⇧^{R}\<lsq> a" by auto

qed;

text{*Every integer whose embedding is less or equal a real number $a$

is less or equal than the floor of $a$.*}

lemma (in real1) Real_ZF_1_4_L8:

assumes A1: "m ∈ int" and A2: "m⇧^{R}\<lsq> a"

shows "m \<zlq> ⌊a⌋"

proof -

let ?A = "{m ∈ int. m⇧^{R}\<lsq> a}"

from A2 have "IsBoundedAbove(?A,IntegerOrder)" and "?A≠0"

using Real_ZF_1_2_L15 Real_ZF_1_4_L7 by auto;

then have "∀x∈?A. ⟨x,Maximum(IntegerOrder,?A)⟩ ∈ IntegerOrder"

by (rule int0.int_bounded_above_has_max);

with A1 A2 show "m \<zlq> ⌊a⌋" by simp;

qed;

text{*Integer zero and one embed as real zero and one.*}

lemma (in real1) int_0_1_are_real_zero_one:

shows "\<zero>⇩_{Z}⇧^{R}= \<zero>" "\<one>⇩_{Z}⇧^{R}= \<one>"

using int1.Int_ZF_2_5_L7 BoundedIntMaps_def

real_one_cl_identity real_zero_cl_bounded_map

by auto;

text{*Integer two embeds as the real two.*}

lemma (in real1) int_two_is_real_two: shows "\<two>⇩_{Z}⇧^{R}= \<two>"

proof -

have "\<two>⇩_{Z}⇧^{R}= \<one>⇩_{Z}⇧^{R}\<ra> \<one>⇩_{Z}⇧^{R}"

using int0.int_zero_one_are_int Real_ZF_1_4_L1A

by simp;

also have "… = \<two>" using int_0_1_are_real_zero_one

by simp

finally show "\<two>⇩_{Z}⇧^{R}= \<two>" by simp

qed;

text{*A positive integer embeds as a positive (hence nonnegative) real.*}

lemma (in real1) int_pos_is_real_pos: assumes A1: "p∈\<int>⇩_{+}"

shows

"p⇧^{R}∈ \<real>"

"\<zero> \<lsq> p⇧^{R}"

"p⇧^{R}∈ \<real>⇩_{+}"

proof -

from A1 have I: "p ∈ int" "\<zero>⇩_{Z}\<zlq> p" "\<zero>⇩_{Z}≠ p"

using PositiveSet_def by auto;

then have "p⇧^{R}∈ \<real>" "\<zero>⇩_{Z}⇧^{R}\<lsq> p⇧^{R}"

using real_int_is_real Real_ZF_1_4_L5 by auto;

then show "p⇧^{R}∈ \<real>" "\<zero> \<lsq> p⇧^{R}"

using int_0_1_are_real_zero_one by auto;

moreover have "\<zero> ≠ p⇧^{R}"

proof -

{ assume "\<zero> = p⇧^{R}"

with I have False using int_0_1_are_real_zero_one

int0.int_zero_one_are_int Real_ZF_1_4_L4 by auto;

} then show "\<zero> ≠ p⇧^{R}" by auto;

qed;

ultimately show "p⇧^{R}∈ \<real>⇩_{+}" using PositiveSet_def

by simp;

qed;

text{*The ordered field of reals we are constructing is archimedean, i.e.,

if $x,y$ are its elements with $y$ positive, then there is a positive

integer $M$ such that $x$ is smaller than $M^R y$. This is Lemma 14 ii) in \cite{Arthan2004}.*}

lemma (in real1) Arthan_Lemma14ii: assumes A1: "x∈\<real>" "y ∈ \<real>⇩_{+}"

shows "∃M∈\<int>⇩_{+}. x \<ls> M⇧^{R}·y"

proof -

from A1 have

"∃C∈\<int>⇩_{+}. x \<ls> C⇧^{R}" and "∃D∈\<int>⇩_{+}. y¯ \<ls> D⇧^{R}"

using Real_ZF_1_3_L1 Arthan_Lemma14i by auto;

then obtain C D where

I: "C∈\<int>⇩_{+}" and II: "x \<ls> C⇧^{R}" and

III: "D∈\<int>⇩_{+}" and IV: "y¯ \<ls> D⇧^{R}"

by auto;

let ?M = "C\<zmu>D"

from I III have

T: "?M ∈ \<int>⇩_{+}" "C⇧^{R}∈ \<real>" "D⇧^{R}∈ \<real>"

using int0.pos_int_closed_mul_unfold PositiveSet_def real_int_is_real

by auto;

with A1 I III have "C⇧^{R}·(D⇧^{R}·y) = ?M⇧^{R}·y"

using PositiveSet_def Real_ZF_1_L6A Real_ZF_1_4_L1C

by simp;

moreover from A1 I II IV have

"x \<ls> C⇧^{R}·(D⇧^{R}·y)"

using int_pos_is_real_pos Real_ZF_1_3_L2 Real_ZF_1_2_L25

by auto;

ultimately have "x \<ls> ?M⇧^{R}·y"

by auto;

with T show ?thesis by auto;

qed;

text{*Taking the floor function preserves the order.*}

lemma (in real1) Real_ZF_1_4_L9: assumes A1: "a\<lsq>b"

shows "⌊a⌋ \<zlq> ⌊b⌋"

proof -

from A1 have T: "a∈\<real>" using Real_ZF_1_2_L15

by simp;

with A1 have "⌊a⌋⇧^{R}\<lsq> a" and "a\<lsq>b"

using Real_ZF_1_4_L7 by auto;

then have "⌊a⌋⇧^{R}\<lsq> b" by (rule real_ord_transitive);

moreover from T have "⌊a⌋ ∈ int" using Real_ZF_1_4_L7

by simp;

ultimately show "⌊a⌋ \<zlq> ⌊b⌋" using Real_ZF_1_4_L8

by simp;

qed;

text{*If $S$ is bounded above and $p$ is a positive intereger, then

$\Gamma(S,p)$ is well defined. *}

lemma (in real1) Real_ZF_1_4_L10:

assumes A1: "IsBoundedAbove(S,OrderOnReals)" "S≠0" and A2: "p∈\<int>⇩_{+}"

shows

"IsBoundedAbove({⌊p⇧^{R}·x⌋. x∈S},IntegerOrder)"

"Γ(S,p) ∈ {⌊p⇧^{R}·x⌋. x∈S}"

"Γ(S,p) ∈ int"

proof -

let ?A = "{⌊p⇧^{R}·x⌋. x∈S}"

from A1 obtain X where I: "∀x∈S. x\<lsq>X"

using IsBoundedAbove_def by auto;

{ fix m assume "m ∈ ?A"

then obtain x where "x∈S" and II: "m = ⌊p⇧^{R}·x⌋"

by auto;

with I have "x\<lsq>X" by simp

moreover from A2 have "\<zero> \<lsq> p⇧^{R}" using int_pos_is_real_pos

by simp;

ultimately have "p⇧^{R}·x \<lsq> p⇧^{R}·X" using Real_ZF_1_2_L14

by simp;

with II have "m \<zlq> ⌊p⇧^{R}·X⌋" using Real_ZF_1_4_L9

by simp;

} then have "∀m∈?A. ⟨m,⌊p⇧^{R}·X⌋⟩ ∈ IntegerOrder"

by auto;

then show II: "IsBoundedAbove(?A,IntegerOrder)"

by (rule Order_ZF_3_L10)

moreover from A1 have III: "?A ≠ 0" by simp;

ultimately have "Maximum(IntegerOrder,?A) ∈ ?A"

by (rule int0.int_bounded_above_has_max)

moreover from II III have "Maximum(IntegerOrder,?A) ∈ int"

by (rule int0.int_bounded_above_has_max);

ultimately show "Γ(S,p) ∈ {⌊p⇧^{R}·x⌋. x∈S}" and "Γ(S,p) ∈ int"

by auto;

qed;

text{*If $p$ is a positive integer, then

for all $s\in S$ the floor of $p\cdot x$ is not greater that $\Gamma(S,p)$.*}

lemma (in real1) Real_ZF_1_4_L11:

assumes A1: "IsBoundedAbove(S,OrderOnReals)" and A2: "x∈S" and A3: "p∈\<int>⇩_{+}"

shows "⌊p⇧^{R}·x⌋ \<zlq> Γ(S,p)"

proof -

let ?A = "{⌊p⇧^{R}·x⌋. x∈S}"

from A2 have "S≠0" by auto;

with A1 A3 have "IsBoundedAbove(?A,IntegerOrder)" "?A ≠ 0"

using Real_ZF_1_4_L10 by auto;

then have "∀x∈?A. ⟨x,Maximum(IntegerOrder,?A)⟩ ∈ IntegerOrder"

by (rule int0.int_bounded_above_has_max);

with A2 show "⌊p⇧^{R}·x⌋ \<zlq> Γ(S,p)" by simp;

qed;

text{*The candidate for supremum is an integer mapping with values

given by $\Gamma$.*}

lemma (in real1) Real_ZF_1_4_L12:

assumes A1: "IsBoundedAbove(S,OrderOnReals)" "S≠0" and

A2: "g = {⟨p,Γ(S,p)⟩. p∈\<int>⇩_{+}}"

shows

"g : \<int>⇩_{+}->int"

"∀n∈\<int>⇩_{+}. g`(n) = Γ(S,n)"

proof -

from A1 have "∀n∈\<int>⇩_{+}. Γ(S,n) ∈ int" using Real_ZF_1_4_L10

by simp

with A2 show I: "g : \<int>⇩_{+}->int" using ZF_fun_from_total by simp

{ fix n assume "n∈\<int>⇩_{+}"

with A2 I have "g`(n) = Γ(S,n)" using ZF_fun_from_tot_val

by simp

} then show "∀n∈\<int>⇩_{+}. g`(n) = Γ(S,n)" by simp

qed;

text{*Every integer is equal to the floor of its embedding.*}

lemma (in real1) Real_ZF_1_4_L14: assumes A1: "m ∈ int"

shows "⌊m⇧^{R}⌋ = m"

proof -

let ?A = "{n ∈ int. n⇧^{R}\<lsq> m⇧^{R}}"

have "antisym(IntegerOrder)" using int0.Int_ZF_2_L4

by simp

moreover from A1 have "m ∈ ?A"

using real_int_is_real real_ord_refl by auto;

moreover from A1 have "∀n ∈ ?A. ⟨n,m⟩ ∈ IntegerOrder"

using Real_ZF_1_4_L6 by auto;

ultimately show "⌊m⇧^{R}⌋ = m" using Order_ZF_4_L14

by auto;

qed;

text{*Floor of (real) zero is (integer) zero.*}

lemma (in real1) floor_01_is_zero_one: shows

"⌊\<zero>⌋ = \<zero>⇩_{Z}" "⌊\<one>⌋ = \<one>⇩_{Z}"

proof -

have "⌊(\<zero>⇩_{Z})⇧^{R}⌋ = \<zero>⇩_{Z}" and "⌊(\<one>⇩_{Z})⇧^{R}⌋ = \<one>⇩_{Z}"

using int0.int_zero_one_are_int Real_ZF_1_4_L14

by auto

then show "⌊\<zero>⌋ = \<zero>⇩_{Z}" and "⌊\<one>⌋ = \<one>⇩_{Z}"

using int_0_1_are_real_zero_one

by auto

qed;

text{*Floor of (real) two is (integer) two.*}

lemma (in real1) floor_2_is_two: shows "⌊\<two>⌋ = \<two>⇩_{Z}"

proof -

have "⌊(\<two>⇩_{Z})⇧^{R}⌋ = \<two>⇩_{Z}"

using int0.int_two_three_are_int Real_ZF_1_4_L14

by simp

then show "⌊\<two>⌋ = \<two>⇩_{Z}" using int_two_is_real_two

by simp

qed;

text{*Floor of a product of embeddings of integers is equal to the

product of integers.*}

lemma (in real1) Real_ZF_1_4_L14A: assumes A1: "m ∈ int" "k ∈ int"

shows "⌊m⇧^{R}·k⇧^{R}⌋ = m\<zmu>k"

proof -

from A1 have T: "m\<zmu>k ∈ int"

using int0.Int_ZF_1_1_L5 by simp;

from A1 have "⌊m⇧^{R}·k⇧^{R}⌋ = ⌊(m\<zmu>k)⇧^{R}⌋" using Real_ZF_1_4_L1C

by simp;

with T show "⌊m⇧^{R}·k⇧^{R}⌋ = m\<zmu>k" using Real_ZF_1_4_L14

by simp;

qed;

text{*Floor of the sum of a number and the embedding of an

integer is the floor of the number plus the integer.*}

lemma (in real1) Real_ZF_1_4_L15: assumes A1: "x∈\<real>" and A2: "p ∈ int"

shows "⌊x \<ra> p⇧^{R}⌋ = ⌊x⌋ \<za> p"

proof -

let ?A = "{n ∈ int. n⇧^{R}\<lsq> x \<ra> p⇧^{R}}"

have "antisym(IntegerOrder)" using int0.Int_ZF_2_L4

by simp

moreover have "⌊x⌋ \<za> p ∈ ?A"

proof -

from A1 A2 have "⌊x⌋⇧^{R}\<lsq> x" and "p⇧^{R}∈ \<real>"

using Real_ZF_1_4_L7 real_int_is_real by auto;

then have "⌊x⌋⇧^{R}\<ra> p⇧^{R}\<lsq> x \<ra> p⇧^{R}"

using add_num_to_ineq by simp;

moreover from A1 A2 have "(⌊x⌋ \<za> p)⇧^{R}= ⌊x⌋⇧^{R}\<ra> p⇧^{R}"

using Real_ZF_1_4_L7 Real_ZF_1_4_L1A by simp;

ultimately have "(⌊x⌋ \<za> p)⇧^{R}\<lsq> x \<ra> p⇧^{R}"

by simp

moreover from A1 A2 have "⌊x⌋ \<za> p ∈ int"

using Real_ZF_1_4_L7 int0.Int_ZF_1_1_L5 by simp;

ultimately show "⌊x⌋ \<za> p ∈ ?A" by auto;

qed;

moreover have "∀n∈?A. n \<zlq> ⌊x⌋ \<za> p"

proof

fix n assume "n∈?A"

then have I: "n ∈ int" and "n⇧^{R}\<lsq> x \<ra> p⇧^{R}"

by auto

with A1 A2 have "n⇧^{R}\<rs> p⇧^{R}\<lsq> x"

using real_int_is_real Real_ZF_1_2_L19

by simp

with A2 I have "⌊(n\<zs>p)⇧^{R}⌋ \<zlq> ⌊x⌋"

using Real_ZF_1_4_L1B Real_ZF_1_4_L9

by simp;

moreover

from A2 I have "n\<zs>p ∈ int"

using int0.Int_ZF_1_1_L5 by simp;

then have "⌊(n\<zs>p)⇧^{R}⌋ = n\<zs>p"

using Real_ZF_1_4_L14 by simp;

ultimately have "n\<zs>p \<zlq> ⌊x⌋"

by simp;

with A2 I show "n \<zlq> ⌊x⌋ \<za> p"

using int0.Int_ZF_2_L9C by simp;

qed;

ultimately show "⌊x \<ra> p⇧^{R}⌋ = ⌊x⌋ \<za> p"

using Order_ZF_4_L14 by auto;

qed;

text{*Floor of the difference of a number and the embedding of an

integer is the floor of the number minus the integer.*}

lemma (in real1) Real_ZF_1_4_L16: assumes A1: "x∈\<real>" and A2: "p ∈ int"

shows "⌊x \<rs> p⇧^{R}⌋ = ⌊x⌋ \<zs> p"

proof -

from A2 have "⌊x \<rs> p⇧^{R}⌋ = ⌊x \<ra> (\<zm>p)⇧^{R}⌋"

using Real_ZF_1_4_L1 by simp;

with A1 A2 show "⌊x \<rs> p⇧^{R}⌋ = ⌊x⌋ \<zs> p"

using int0.Int_ZF_1_1_L4 Real_ZF_1_4_L15 by simp;

qed;

text{*The floor of sum of embeddings is the sum of the integers.*}

lemma (in real1) Real_ZF_1_4_L17: assumes "m ∈ int" "n ∈ int"

shows "⌊(m⇧^{R}) \<ra> n⇧^{R}⌋ = m \<za> n"

using assms real_int_is_real Real_ZF_1_4_L15 Real_ZF_1_4_L14

by simp;

text{*A lemma about adding one to floor.*}

lemma (in real1) Real_ZF_1_4_L17A: assumes A1: "a∈\<real>"

shows "\<one> \<ra> ⌊a⌋⇧^{R}= (\<one>⇩_{Z}\<za> ⌊a⌋)⇧^{R}"

proof -

have "\<one> \<ra> ⌊a⌋⇧^{R}= \<one>⇩_{Z}⇧^{R}\<ra> ⌊a⌋⇧^{R}"

using int_0_1_are_real_zero_one by simp;

with A1 show "\<one> \<ra> ⌊a⌋⇧^{R}= (\<one>⇩_{Z}\<za> ⌊a⌋)⇧^{R}"

using int0.int_zero_one_are_int Real_ZF_1_4_L7 Real_ZF_1_4_L1A

by simp;

qed;

text{*The difference between the a number and the embedding of its floor

is (strictly) less than one.*}

lemma (in real1) Real_ZF_1_4_L17B: assumes A1: "a∈\<real>"

shows

"a \<rs> ⌊a⌋⇧^{R}\<ls> \<one>"

"a \<ls> (\<one>⇩_{Z}\<za> ⌊a⌋)⇧^{R}"

proof -

from A1 have T1: "⌊a⌋ ∈ int" "⌊a⌋⇧^{R}∈ \<real>" and

T2: "\<one> ∈ \<real>" "a \<rs> ⌊a⌋⇧^{R}∈ \<real>"

using Real_ZF_1_4_L7 real_int_is_real Real_ZF_1_L6 Real_ZF_1_L4

by auto;

{ assume "\<one> \<lsq> a \<rs> ⌊a⌋⇧^{R}"

with A1 T1 have "⌊\<one>⇩_{Z}⇧^{R}\<ra> ⌊a⌋⇧^{R}⌋ \<zlq> ⌊a⌋"

using Real_ZF_1_2_L21 Real_ZF_1_4_L9 int_0_1_are_real_zero_one

by simp;

with T1 have False

using int0.int_zero_one_are_int Real_ZF_1_4_L17

int0.Int_ZF_1_2_L3AA by simp;

} then have I: "¬(\<one> \<lsq> a \<rs> ⌊a⌋⇧^{R})" by auto;

with T2 show II: "a \<rs> ⌊a⌋⇧^{R}\<ls> \<one>"

by (rule Real_ZF_1_2_L20);

with A1 T1 I II have

"a \<ls> \<one> \<ra> ⌊a⌋⇧^{R}"

using Real_ZF_1_2_L26 by simp;

with A1 show "a \<ls> (\<one>⇩_{Z}\<za> ⌊a⌋)⇧^{R}"

using Real_ZF_1_4_L17A by simp;

qed;

text{*The next lemma corresponds to Lemma 14 iii) in \cite{Arthan2004}.

It says that we can find a rational number between any two different

real numbers.*}

lemma (in real1) Arthan_Lemma14iii: assumes A1: "x\<ls>y"

shows "∃M∈int. ∃N∈\<int>⇩_{+}. x·N⇧^{R}\<ls> M⇧^{R}∧ M⇧^{R}\<ls> y·N⇧^{R}"

proof -;

from A1 have "(y\<rs>x)¯ ∈ \<real>⇩_{+}" using Real_ZF_1_3_L3

by simp

then have

"∃N∈\<int>⇩_{+}. (y\<rs>x)¯ \<ls> N⇧^{R}"

using Arthan_Lemma14i PositiveSet_def by simp;

then obtain N where I: "N∈\<int>⇩_{+}" and II: "(y\<rs>x)¯ \<ls> N⇧^{R}"

by auto;

let ?M = "\<one>⇩_{Z}\<za> ⌊x·N⇧^{R}⌋"

from A1 I have

T1: "x∈\<real>" "N⇧^{R}∈ \<real>" "N⇧^{R}∈ \<real>⇩_{+}" "x·N⇧^{R}∈ \<real>"

using Real_ZF_1_2_L15 PositiveSet_def real_int_is_real

Real_ZF_1_L6 int_pos_is_real_pos by auto;

then have T2: "?M ∈ int" using

int0.int_zero_one_are_int Real_ZF_1_4_L7 int0.Int_ZF_1_1_L5

by simp;

from T1 have III: "x·N⇧^{R}\<ls> ?M⇧^{R}"

using Real_ZF_1_4_L17B by simp;

from T1 have "(\<one> \<ra> ⌊x·N⇧^{R}⌋⇧^{R}) \<lsq> (\<one> \<ra> x·N⇧^{R})"

using Real_ZF_1_4_L7 Real_ZF_1_L4 real_ord_transl_inv

by simp;

with T1 have "?M⇧^{R}\<lsq> (\<one> \<ra> x·N⇧^{R})"

using Real_ZF_1_4_L17A by simp;

moreover from A1 II have "(\<one> \<ra> x·N⇧^{R}) \<ls> y·N⇧^{R}"

using Real_ZF_1_3_L5 by simp;

ultimately have "?M⇧^{R}\<ls> y·N⇧^{R}"

by (rule real_strict_ord_transit);

with I T2 III show ?thesis by auto;

qed;

text{*Some estimates for the homomorphism difference of the floor

function.*}

lemma (in real1) Real_ZF_1_4_L18: assumes A1: "x∈\<real>" "y∈\<real>"

shows

"abs(⌊x\<ra>y⌋ \<zs> ⌊x⌋ \<zs> ⌊y⌋) \<zlq> \<two>⇩_{Z}"

proof -

from A1 have T:

"⌊x⌋⇧^{R}∈ \<real>" "⌊y⌋⇧^{R}∈ \<real>"

"x\<ra>y \<rs> (⌊x⌋⇧^{R}) ∈ \<real>"

using Real_ZF_1_4_L7 real_int_is_real Real_ZF_1_L6

by auto;

from A1 have

"\<zero> \<lsq> x \<rs> (⌊x⌋⇧^{R}) \<ra> (y \<rs> (⌊y⌋⇧^{R}))"

"x \<rs> (⌊x⌋⇧^{R}) \<ra> (y \<rs> (⌊y⌋⇧^{R})) \<lsq> \<two>"

using Real_ZF_1_4_L7 Real_ZF_1_2_L16 Real_ZF_1_2_L17

Real_ZF_1_4_L17B Real_ZF_1_2_L18 by auto;

moreover from A1 T have

"x \<rs> (⌊x⌋⇧^{R}) \<ra> (y \<rs> (⌊y⌋⇧^{R})) = x\<ra>y \<rs> (⌊x⌋⇧^{R}) \<rs> (⌊y⌋⇧^{R})"

using Real_ZF_1_L7A by simp;

ultimately have

"\<zero> \<lsq> x\<ra>y \<rs> (⌊x⌋⇧^{R}) \<rs> (⌊y⌋⇧^{R})"

"x\<ra>y \<rs> (⌊x⌋⇧^{R}) \<rs> (⌊y⌋⇧^{R}) \<lsq> \<two>"

by auto;

then have

"⌊\<zero>⌋ \<zlq> ⌊x\<ra>y \<rs> (⌊x⌋⇧^{R}) \<rs> (⌊y⌋⇧^{R})⌋"

"⌊x\<ra>y \<rs> (⌊x⌋⇧^{R}) \<rs> (⌊y⌋⇧^{R})⌋ \<zlq> ⌊\<two>⌋"

using Real_ZF_1_4_L9 by auto;

then have

"\<zero>⇩_{Z}\<zlq> ⌊x\<ra>y \<rs> (⌊x⌋⇧^{R}) \<rs> (⌊y⌋⇧^{R})⌋"

"⌊x\<ra>y \<rs> (⌊x⌋⇧^{R}) \<rs> (⌊y⌋⇧^{R})⌋ \<zlq> \<two>⇩_{Z}"

using floor_01_is_zero_one floor_2_is_two by auto;

moreover from A1 have

"⌊x\<ra>y \<rs> (⌊x⌋⇧^{R}) \<rs> (⌊y⌋⇧^{R})⌋ = ⌊x\<ra>y⌋ \<zs> ⌊x⌋ \<zs> ⌊y⌋"

using Real_ZF_1_L6 Real_ZF_1_4_L7 real_int_is_real Real_ZF_1_4_L16

by simp;

ultimately have

"\<zero>⇩_{Z}\<zlq> ⌊x\<ra>y⌋ \<zs> ⌊x⌋ \<zs> ⌊y⌋"

"⌊x\<ra>y⌋ \<zs> ⌊x⌋ \<zs> ⌊y⌋ \<zlq> \<two>⇩_{Z}"

by auto;

then show "abs(⌊x\<ra>y⌋ \<zs> ⌊x⌋ \<zs> ⌊y⌋) \<zlq> \<two>⇩_{Z}"

using int0.Int_ZF_2_L16 by simp

qed;

text{*Suppose $S\neq \emptyset$ is bounded above and

$\Gamma(S,m) =\lfloor m^R\cdot x\rfloor$ for some positive integer $m$

and $x\in S$. Then if $y\in S, x\leq y$ we also have

$\Gamma(S,m) =\lfloor m^R\cdot y\rfloor$.*}

lemma (in real1) Real_ZF_1_4_L20:

assumes A1: "IsBoundedAbove(S,OrderOnReals)" "S≠0" and

A2: "n∈\<int>⇩_{+}" "x∈S" and

A3: "Γ(S,n) = ⌊n⇧^{R}·x⌋" and

A4: "y∈S" "x\<lsq>y"

shows "Γ(S,n) = ⌊n⇧^{R}·y⌋"

proof -

from A2 A4 have "⌊n⇧^{R}·x⌋ \<zlq> ⌊(n⇧^{R})·y⌋"

using int_pos_is_real_pos Real_ZF_1_2_L14 Real_ZF_1_4_L9

by simp;

with A3 have "⟨Γ(S,n),⌊(n⇧^{R})·y⌋⟩ ∈ IntegerOrder"

by simp

moreover from A1 A2 A4 have "⟨⌊n⇧^{R}·y⌋,Γ(S,n)⟩ ∈ IntegerOrder"

using Real_ZF_1_4_L11 by simp;

ultimately show "Γ(S,n) = ⌊n⇧^{R}·y⌋"

by (rule int0.Int_ZF_2_L3)

qed;

text{*The homomorphism difference of $n\mapsto \Gamma(S,n)$ is bounded

by $2$ on positive integers.*}

lemma (in real1) Real_ZF_1_4_L21:

assumes A1: "IsBoundedAbove(S,OrderOnReals)" "S≠0" and

A2: "m∈\<int>⇩_{+}" "n∈\<int>⇩_{+}"

shows "abs(Γ(S,m\<za>n) \<zs> Γ(S,m) \<zs> Γ(S,n)) \<zlq> \<two>⇩_{Z}"

proof -

from A2 have T: "m\<za>n ∈ \<int>⇩_{+}" using int0.pos_int_closed_add_unfolded

by simp;

with A1 A2 have

"Γ(S,m) ∈ {⌊m⇧^{R}·x⌋. x∈S}" and

"Γ(S,n) ∈ {⌊n⇧^{R}·x⌋. x∈S}" and

"Γ(S,m\<za>n) ∈ {⌊(m\<za>n)⇧^{R}·x⌋. x∈S}"

using Real_ZF_1_4_L10 by auto;

then obtain a b c where I: "a∈S" "b∈S" "c∈S"

and II:

"Γ(S,m) = ⌊m⇧^{R}·a⌋"

"Γ(S,n) = ⌊n⇧^{R}·b⌋"

"Γ(S,m\<za>n) = ⌊(m\<za>n)⇧^{R}·c⌋"

by auto

let ?d = "Maximum(OrderOnReals,{a,b,c})"

from A1 I have "a∈\<real>" "b∈\<real>" "c∈\<real>"

using Real_ZF_1_2_L23 by auto;

then have IV:

"?d ∈ {a,b,c}"

"?d ∈ \<real>"

"a \<lsq> ?d"

"b \<lsq> ?d"

"c \<lsq> ?d"

using Real_ZF_1_2_L24 by auto;

with I have V: "?d ∈ S" by auto;

from A1 T I II IV V have "Γ(S,m\<za>n) = ⌊(m\<za>n)⇧^{R}·?d⌋"

using Real_ZF_1_4_L20 by blast;

also from A2 have "… = ⌊((m⇧^{R})\<ra>(n⇧^{R}))·?d⌋"

using Real_ZF_1_4_L1A PositiveSet_def by simp;

also from A2 IV have "… = ⌊(m⇧^{R})·?d \<ra> (n⇧^{R})·?d⌋"

using PositiveSet_def real_int_is_real Real_ZF_1_L7

by simp;

finally have "Γ(S,m\<za>n) = ⌊(m⇧^{R})·?d \<ra> (n⇧^{R})·?d⌋"

by simp

moreover from A1 A2 I II IV V have "Γ(S,m) = ⌊m⇧^{R}·?d⌋"

using Real_ZF_1_4_L20 by blast;

moreover from A1 A2 I II IV V have "Γ(S,n) = ⌊n⇧^{R}·?d⌋"

using Real_ZF_1_4_L20 by blast;

moreover from A1 T I II IV V have "Γ(S,m\<za>n) = ⌊(m\<za>n)⇧^{R}·?d⌋"

using Real_ZF_1_4_L20 by blast;

ultimately have "abs(Γ(S,m\<za>n) \<zs> Γ(S,m) \<zs> Γ(S,n)) =

abs(⌊(m⇧^{R})·?d \<ra> (n⇧^{R})·?d⌋ \<zs> ⌊m⇧^{R}·?d⌋ \<zs> ⌊n⇧^{R}·?d⌋)"

by simp;

with A2 IV show

"abs(Γ(S,m\<za>n) \<zs> Γ(S,m) \<zs> Γ(S,n)) \<zlq> \<two>⇩_{Z}"

using PositiveSet_def real_int_is_real Real_ZF_1_L6

Real_ZF_1_4_L18 by simp;

qed

text{*The next lemma provides sufficient condition for an odd function

to be an almost homomorphism.

It says for odd functions we only need to check that

the homomorphism difference

(denoted @{text "δ"} in the @{text "real1"} context) is bounded on positive

integers. This is really proven in @{text "Int_ZF_2.thy"}, but we

restate it here for convenience. Recall from @{text "Group_ZF_3.thy"} that

@{text "OddExtension"} of a function defined on the set of positive elements

(of an ordered group) is the only odd function that is equal to the given

one when restricted to positive elements.*}

lemma (in real1) Real_ZF_1_4_L21A:

assumes A1: "f:\<int>⇩_{+}->int" "∀a∈\<int>⇩_{+}. ∀b∈\<int>⇩_{+}. abs(δ(f,a,b)) \<zlq> L"

shows "OddExtension(int,IntegerAddition,IntegerOrder,f) ∈ \<S>"

using A1 int1.Int_ZF_2_1_L24 by auto; (*A1 has to be here *)

text{*The candidate for (a representant of) the supremum of a

nonempty bounded above set is a slope.*}

lemma (in real1) Real_ZF_1_4_L22:

assumes A1: "IsBoundedAbove(S,OrderOnReals)" "S≠0" and

A2: "g = {⟨p,Γ(S,p)⟩. p∈\<int>⇩_{+}}"

shows "OddExtension(int,IntegerAddition,IntegerOrder,g) ∈ \<S>"

proof -

from A1 A2 have "g: \<int>⇩_{+}->int" by (rule Real_ZF_1_4_L12);

moreover have "∀m∈\<int>⇩_{+}. ∀n∈\<int>⇩_{+}. abs(δ(g,m,n)) \<zlq> \<two>⇩_{Z}"

proof -

{ fix m n assume A3: "m∈\<int>⇩_{+}" "n∈\<int>⇩_{+}"

then have "m\<za>n ∈ \<int>⇩_{+}" "m∈\<int>⇩_{+}" "n∈\<int>⇩_{+}"

using int0.pos_int_closed_add_unfolded

by auto

moreover from A1 A2 have "∀n∈\<int>⇩_{+}. g`(n) = Γ(S,n)"

by (rule Real_ZF_1_4_L12)

ultimately have "δ(g,m,n) = Γ(S,m\<za>n) \<zs> Γ(S,m) \<zs> Γ(S,n)"

by simp;

moreover from A1 A3 have

"abs(Γ(S,m\<za>n) \<zs> Γ(S,m) \<zs> Γ(S,n)) \<zlq> \<two>⇩_{Z}"

by (rule Real_ZF_1_4_L21)

ultimately have "abs(δ(g,m,n)) \<zlq> \<two>⇩_{Z}"

by simp

} then show "∀m∈\<int>⇩_{+}. ∀n∈\<int>⇩_{+}. abs(δ(g,m,n)) \<zlq> \<two>⇩_{Z}"

by simp

qed;

ultimately show ?thesis by (rule Real_ZF_1_4_L21A);

qed;

text{*A technical lemma used in the proof that all elements

of $S$ are less or equal than the candidate for supremum of $S$.*}

lemma (in real1) Real_ZF_1_4_L23:

assumes A1: "f ∈ \<S>" and A2: "N ∈ int" "M ∈ int" and

A3: "∀n∈\<int>⇩_{+}. M\<zmu>n \<zlq> f`(N\<zmu>n)"

shows "M⇧^{R}\<lsq> [f]·(N⇧^{R})"

proof -

let ?M⇩_{S}= "{⟨n, M\<zmu>n⟩ . n ∈ int}"

let ?N⇩_{S}= "{⟨n, N\<zmu>n⟩ . n ∈ int}"

from A1 A2 have T: "?M⇩_{S}∈ \<S>" "?N⇩_{S}∈ \<S>" "fo?N⇩_{S}∈ \<S>"

using int1.Int_ZF_2_5_L1 int1.Int_ZF_2_1_L11 SlopeOp2_def

by auto;

moreover from A1 A2 A3 have "?M⇩_{S}∼ fo?N⇩_{S}∨ fo?N⇩_{S}\<fp> (\<fm>?M⇩_{S}) ∈ \<S>⇩_{+}"

using int1.Int_ZF_2_5_L8 SlopeOp2_def SlopeOp1_def Slopes_def

BoundedIntMaps_def SlopeEquivalenceRel_def PositiveIntegers_def

PositiveSlopes_def by simp;

ultimately have "[?M⇩_{S}] \<lsq> [fo?N⇩_{S}]" using Real_ZF_1_2_L12

by simp;

with A1 T show "M⇧^{R}\<lsq> [f]·(N⇧^{R})" using Real_ZF_1_1_L4

by simp;

qed;

text{*A technical lemma aimed used in the proof the candidate for supremum of

$S$ is less or equal than any upper bound for $S$.*}

lemma (in real1) Real_ZF_1_4_L23A:

assumes A1: "f ∈ \<S>" and A2: "N ∈ int" "M ∈ int" and

A3: "∀n∈\<int>⇩_{+}. f`(N\<zmu>n) \<zlq> M\<zmu>n "

shows "[f]·(N⇧^{R}) \<lsq> M⇧^{R}"

proof -

let ?M⇩_{S}= "{⟨n, M\<zmu>n⟩ . n ∈ int}"

let ?N⇩_{S}= "{⟨n, N\<zmu>n⟩ . n ∈ int}"

from A1 A2 have T: "?M⇩_{S}∈ \<S>" "?N⇩_{S}∈ \<S>" "fo?N⇩_{S}∈ \<S>"

using int1.Int_ZF_2_5_L1 int1.Int_ZF_2_1_L11 SlopeOp2_def

by auto;

moreover from A1 A2 A3 have

"fo?N⇩_{S}∼ ?M⇩_{S}∨ ?M⇩_{S}\<fp> (\<fm>(fo?N⇩_{S})) ∈ \<S>⇩_{+}"

using int1.Int_ZF_2_5_L9 SlopeOp2_def SlopeOp1_def Slopes_def

BoundedIntMaps_def SlopeEquivalenceRel_def PositiveIntegers_def

PositiveSlopes_def by simp;

ultimately have "[fo?N⇩_{S}] \<lsq> [?M⇩_{S}]" using Real_ZF_1_2_L12

by simp;

with A1 T show " [f]·(N⇧^{R})\<lsq> M⇧^{R}" using Real_ZF_1_1_L4

by simp;

qed;

text{* The essential condition to claim that the candidate for supremum

of $S$ is greater or equal than all elements of $S$.*}

lemma (in real1) Real_ZF_1_4_L24:

assumes A1: "IsBoundedAbove(S,OrderOnReals)" and

A2: "x\<ls>y" "y∈S" and

A4: "N ∈ \<int>⇩_{+}" "M ∈ int" and

A5: "M⇧^{R}\<ls> y·N⇧^{R}" and A6: "p ∈ \<int>⇩_{+}"

shows "p\<zmu>M \<zlq> Γ(S,p\<zmu>N)"

proof -

from A2 A4 A6 have T1:

"N⇧^{R}∈ \<real>⇩_{+}" "y∈\<real>" "p⇧^{R}∈ \<real>⇩_{+}"

"p\<zmu>N ∈ \<int>⇩_{+}" "(p\<zmu>N)⇧^{R}∈ \<real>⇩_{+}"

using int_pos_is_real_pos Real_ZF_1_2_L15

int0.pos_int_closed_mul_unfold by auto;

with A4 A6 have T2:

"p ∈ int" "p⇧^{R}∈ \<real>" "N⇧^{R}∈ \<real>" "N⇧^{R}≠ \<zero>" "M⇧^{R}∈ \<real>"

using real_int_is_real PositiveSet_def by auto;

from T1 A5 have "⌊(p\<zmu>N)⇧^{R}·(M⇧^{R}·(N⇧^{R})¯)⌋ \<zlq> ⌊(p\<zmu>N)⇧^{R}·y⌋"

using Real_ZF_1_3_L4A Real_ZF_1_3_L7 Real_ZF_1_4_L9

by simp;

moreover from A1 A2 T1 have "⌊(p\<zmu>N)⇧^{R}·y⌋ \<zlq> Γ(S,p\<zmu>N)"

using Real_ZF_1_4_L11 by simp;

ultimately have I: "⌊(p\<zmu>N)⇧^{R}·(M⇧^{R}·(N⇧^{R})¯)⌋ \<zlq> Γ(S,p\<zmu>N)"

by (rule int_order_transitive);

from A4 A6 have "(p\<zmu>N)⇧^{R}·(M⇧^{R}·(N⇧^{R})¯) = p⇧^{R}·N⇧^{R}·(M⇧^{R}·(N⇧^{R})¯)"

using PositiveSet_def Real_ZF_1_4_L1C by simp;

with A4 T2 have "⌊(p\<zmu>N)⇧^{R}·(M⇧^{R}·(N⇧^{R})¯)⌋ = p\<zmu>M"

using Real_ZF_1_3_L8 Real_ZF_1_4_L14A by simp;

with I show "p\<zmu>M \<zlq> Γ(S,p\<zmu>N)" by simp;

qed;

text{*An obvious fact about odd extension

of a function $p\mapsto \Gamma(s,p)$ that is used a couple of times

in proofs.*}

lemma (in real1) Real_ZF_1_4_L24A:

assumes A1: "IsBoundedAbove(S,OrderOnReals)" "S≠0" and A2: "p ∈ \<int>⇩_{+}"

and A3:

"h = OddExtension(int,IntegerAddition,IntegerOrder,{⟨p,Γ(S,p)⟩. p∈\<int>⇩_{+}})"

shows "h`(p) = Γ(S,p)"

proof -

let ?g = "{⟨p,Γ(S,p)⟩. p∈\<int>⇩_{+}}"

from A1 have I: "?g : \<int>⇩_{+}->int" using Real_ZF_1_4_L12

by blast;

with A2 A3 show "h`(p) = Γ(S,p)"

using int0.Int_ZF_1_5_L11 ZF_fun_from_tot_val

by simp;

qed;

text{*The candidate for the supremum of $S$ is not smaller than

any element of $S$.*}

lemma (in real1) Real_ZF_1_4_L25:

assumes A1: "IsBoundedAbove(S,OrderOnReals)" and

A2: "¬HasAmaximum(OrderOnReals,S)" and

A3: "x∈S" and A4:

"h = OddExtension(int,IntegerAddition,IntegerOrder,{⟨p,Γ(S,p)⟩. p∈\<int>⇩_{+}})"

shows "x \<lsq> [h]"

proof -

from A1 A2 A3 have

"S ⊆ \<real>" "¬HasAmaximum(OrderOnReals,S)" "x∈S"

using Real_ZF_1_2_L23 by auto;

then have "∃y∈S. x\<ls>y" by (rule Real_ZF_1_2_L27);

then obtain y where I: "y∈S" and II: "x\<ls>y"

by auto;

from II have

"∃M∈int. ∃N∈\<int>⇩_{+}. x·N⇧^{R}\<ls> M⇧^{R}∧ M⇧^{R}\<ls> y·N⇧^{R}"

using Arthan_Lemma14iii by simp;

then obtain M N where III: "M ∈ int" "N∈\<int>⇩_{+}" and

IV: "x·N⇧^{R}\<ls> M⇧^{R}" "M⇧^{R}\<ls> y·N⇧^{R}"

by auto;

from II III IV have V: "x \<lsq> M⇧^{R}·(N⇧^{R})¯"

using int_pos_is_real_pos Real_ZF_1_2_L15 Real_ZF_1_3_L4

by auto;

from A3 have VI: "S≠0" by auto;

with A1 A4 have T1: "h ∈ \<S>" using Real_ZF_1_4_L22

by simp;

moreover from III have "N ∈ int" "M ∈ int"

using PositiveSet_def by auto;

moreover have "∀n∈\<int>⇩_{+}. M\<zmu>n \<zlq> h`(N\<zmu>n)"

proof

let ?g = "{⟨p,Γ(S,p)⟩. p∈\<int>⇩_{+}}"

fix n assume A5: "n∈\<int>⇩_{+}"

with III have T2: "N\<zmu>n ∈ \<int>⇩_{+}"

using int0.pos_int_closed_mul_unfold by simp;

from III A5 have

"N\<zmu>n = n\<zmu>N" and "n\<zmu>M = M\<zmu>n"

using PositiveSet_def int0.Int_ZF_1_1_L5 by auto;

moreover

from A1 I II III IV A5 have

"IsBoundedAbove(S,OrderOnReals)"

"x\<ls>y" "y∈S"

"N ∈ \<int>⇩_{+}" "M ∈ int"

"M⇧^{R}\<ls> y·N⇧^{R}" "n ∈ \<int>⇩_{+}"

by auto;

then have "n\<zmu>M \<zlq> Γ(S,n\<zmu>N)" by (rule Real_ZF_1_4_L24);

moreover from A1 A4 VI T2 have "h`(N\<zmu>n) = Γ(S,N\<zmu>n)"

using Real_ZF_1_4_L24A by simp;

ultimately show "M\<zmu>n \<zlq> h`(N\<zmu>n)" by auto;

qed

ultimately have "M⇧^{R}\<lsq> [h]·N⇧^{R}" using Real_ZF_1_4_L23

by simp;

with III T1 have "M⇧^{R}·(N⇧^{R})¯ \<lsq> [h]"

using int_pos_is_real_pos Real_ZF_1_1_L3 Real_ZF_1_3_L4B

by simp;

with V show "x \<lsq> [h]" by (rule real_ord_transitive);

qed;

text{* The essential condition to claim that the candidate for supremum

of $S$ is less or equal than any upper bound of $S$.*}

lemma (in real1) Real_ZF_1_4_L26:

assumes A1: "IsBoundedAbove(S,OrderOnReals)" and

A2: "x\<lsq>y" "x∈S" and

A4: "N ∈ \<int>⇩_{+}" "M ∈ int" and

A5: "y·N⇧^{R}\<ls> M⇧^{R}" and A6: "p ∈ \<int>⇩_{+}"

shows "⌊(N\<zmu>p)⇧^{R}·x⌋ \<zlq> M\<zmu>p"

proof -

from A2 A4 A6 have T:

"p\<zmu>N ∈ \<int>⇩_{+}" "p ∈ int" "N ∈ int"

"p⇧^{R}∈ \<real>⇩_{+}" "p⇧^{R}∈ \<real>" "N⇧^{R}∈ \<real>" "x ∈ \<real>" "y ∈ \<real>"

using int0.pos_int_closed_mul_unfold PositiveSet_def

real_int_is_real Real_ZF_1_2_L15 int_pos_is_real_pos

by auto;

with A2 have "(p\<zmu>N)⇧^{R}·x \<lsq> (p\<zmu>N)⇧^{R}·y"

using int_pos_is_real_pos Real_ZF_1_2_L14A

by simp;

moreover from A4 T have I:

"(p\<zmu>N)⇧^{R}= p⇧^{R}·N⇧^{R}"

"(p\<zmu>M)⇧^{R}= p⇧^{R}·M⇧^{R}"

using Real_ZF_1_4_L1C by auto;

ultimately have "(p\<zmu>N)⇧^{R}·x \<lsq> p⇧^{R}·N⇧^{R}·y"

by simp;

moreover

from A5 T I have "p⇧^{R}·(y·N⇧^{R}) \<ls> (p\<zmu>M)⇧^{R}"

using Real_ZF_1_3_L7 by simp;

with T have "p⇧^{R}·N⇧^{R}·y \<ls> (p\<zmu>M)⇧^{R}" using Real_ZF_1_1_L9

by simp;

ultimately have "(p\<zmu>N)⇧^{R}·x \<ls> (p\<zmu>M)⇧^{R}"

by (rule real_strict_ord_transit);

then have "⌊(p\<zmu>N)⇧^{R}·x⌋ \<zlq> ⌊(p\<zmu>M)⇧^{R}⌋"

using Real_ZF_1_4_L9 by simp;

moreover

from A4 T have "p\<zmu>M ∈ int" using int0.Int_ZF_1_1_L5

by simp;

then have "⌊(p\<zmu>M)⇧^{R}⌋ = p\<zmu>M" using Real_ZF_1_4_L14

by simp;

moreover from A4 A6 have "p\<zmu>N = N\<zmu>p" and "p\<zmu>M = M\<zmu>p"

using PositiveSet_def int0.Int_ZF_1_1_L5 by auto

ultimately show "⌊(N\<zmu>p)⇧^{R}·x⌋ \<zlq> M\<zmu>p" by simp;

qed;

text{*A piece of the proof of the fact that the candidate for the supremum

of $S$ is not greater than any upper bound of $S$, done separately for

clarity (of mind).*}

lemma (in real1) Real_ZF_1_4_L27:

assumes "IsBoundedAbove(S,OrderOnReals)" "S≠0" and

"h = OddExtension(int,IntegerAddition,IntegerOrder,{⟨p,Γ(S,p)⟩. p∈\<int>⇩_{+}})"

and "p ∈ \<int>⇩_{+}"

shows "∃x∈S. h`(p) = ⌊p⇧^{R}·x⌋"

using assms Real_ZF_1_4_L10 Real_ZF_1_4_L24A by auto;

text{*The candidate for the supremum of $S$ is not greater than

any upper bound of $S$.*}

lemma (in real1) Real_ZF_1_4_L28:

assumes A1: "IsBoundedAbove(S,OrderOnReals)" "S≠0"

and A2: "∀x∈S. x\<lsq>y" and A3:

"h = OddExtension(int,IntegerAddition,IntegerOrder,{⟨p,Γ(S,p)⟩. p∈\<int>⇩_{+}})"

shows "[h] \<lsq> y"

proof -

from A1 obtain a where "a∈S" by auto;

with A1 A2 A3 have T: "y∈\<real>" "h ∈ \<S>" "[h] ∈ \<real>"

using Real_ZF_1_2_L15 Real_ZF_1_4_L22 Real_ZF_1_1_L3

by auto;

{ assume "¬([h] \<lsq> y)"

with T have "y \<ls> [h]" using Real_ZF_1_2_L28

by blast;

then have "∃M∈int. ∃N∈\<int>⇩_{+}. y·N⇧^{R}\<ls> M⇧^{R}∧ M⇧^{R}\<ls> [h]·N⇧^{R}"

using Arthan_Lemma14iii by simp;

then obtain M N where I: "M∈int" "N∈\<int>⇩_{+}" and

II: "y·N⇧^{R}\<ls> M⇧^{R}" "M⇧^{R}\<ls> [h]·N⇧^{R}"

by auto;

from I have III: "N⇧^{R}∈ \<real>⇩_{+}" using int_pos_is_real_pos

by simp;

have "∀p∈\<int>⇩_{+}. h`(N\<zmu>p) \<zlq> M\<zmu>p"

proof

fix p assume A4: "p∈\<int>⇩_{+}"

with A1 A3 I have "∃x∈S. h`(N\<zmu>p) = ⌊(N\<zmu>p)⇧^{R}·x⌋"

using int0.pos_int_closed_mul_unfold Real_ZF_1_4_L27

by simp;

with A1 A2 I II A4 show "h`(N\<zmu>p) \<zlq> M\<zmu>p"

using Real_ZF_1_4_L26 by auto;

qed;

with T I have "[h]·N⇧^{R}\<lsq> M⇧^{R}"

using PositiveSet_def Real_ZF_1_4_L23A

by simp;

with T III have "[h] \<lsq> M⇧^{R}·(N⇧^{R})¯"

using Real_ZF_1_3_L4C by simp;

moreover from T II III have "M⇧^{R}·(N⇧^{R})¯ \<ls> [h]"

using Real_ZF_1_3_L4A by simp;

ultimately have False using Real_ZF_1_2_L29 by blast;

} then show "[h] \<lsq> y" by auto;

qed;

text{*Now we can prove that every nonempty subset of reals that is bounded

above has a supremum. Proof by considering two cases: when the set has a

maximum and when it does not.*}

lemma (in real1) real_order_complete:

assumes A1: "IsBoundedAbove(S,OrderOnReals)" "S≠0"

shows "HasAminimum(OrderOnReals,\<Inter>a∈S. OrderOnReals``{a})"

proof -

{ assume "HasAmaximum(OrderOnReals,S)"

with A1 have "HasAminimum(OrderOnReals,\<Inter>a∈S. OrderOnReals``{a})"

using Real_ZF_1_2_L10 IsAnOrdGroup_def IsPartOrder_def

Order_ZF_5_L6 by simp }

moreover

{ assume A2: "¬HasAmaximum(OrderOnReals,S)"

let ?h = "OddExtension(int,IntegerAddition,IntegerOrder,{⟨p,Γ(S,p)⟩. p∈\<int>⇩_{+}})"

let ?r = "OrderOnReals"

from A1 have "antisym(OrderOnReals)" "S≠0"

using Real_ZF_1_2_L10 IsAnOrdGroup_def IsPartOrder_def by auto;

moreover from A1 A2 have "∀x∈S. ⟨x,[?h]⟩ ∈ ?r"

using Real_ZF_1_4_L25 by simp;

moreover from A1 have "∀y. (∀x∈S. ⟨x,y⟩ ∈ ?r) --> ⟨[?h],y⟩ ∈ ?r"

using Real_ZF_1_4_L28 by simp;

ultimately have "HasAminimum(OrderOnReals,\<Inter>a∈S. OrderOnReals``{a})"

by (rule Order_ZF_5_L5) }

ultimately show ?thesis by blast

qed;

text{*Finally, we are ready to formulate the main result: that the

construction of real numbers from the additive group of integers

results in a complete ordered field.

This theorem completes the construction. It was fun.*}

theorem eudoxus_reals_are_reals: shows

"IsAmodelOfReals(RealNumbers,RealAddition,RealMultiplication,OrderOnReals)"

using real1.reals_are_ord_field real1.real_order_complete

IsComplete_def IsAmodelOfReals_def by simp;

end;