(* This file is a part of IsarMathLib -

a library of formalized mathematics for Isabelle/Isar.

Copyright (C) 2005, 2006 Slawomir Kolodynski

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

theory OrderedField_ZF imports OrderedRing_ZF Field_ZF

begin

text{*This theory covers basic facts about ordered fiels.*}

section{*Definition and basic properties*}

text{*Here we define ordered fields and proove their basic properties.*}

text{*Ordered field is a notrivial ordered ring such that all

non-zero elements have an inverse. We define the notion of being a ordered

field as

a statement about four sets. The first set, denoted @{text "K"} is the

carrier of the field. The second set, denoted @{text "A"} represents the

additive operation on @{text "K"} (recall that in ZF set theory functions

are sets). The third set @{text "M"} represents the multiplicative operation

on @{text "K"}. The fourth set @{text "r"} is the order

relation on @{text "K"}.*}

definition

"IsAnOrdField(K,A,M,r) ≡ (IsAnOrdRing(K,A,M,r) ∧

(M {is commutative on} K) ∧

TheNeutralElement(K,A) ≠ TheNeutralElement(K,M) ∧

(∀a∈K. a≠TheNeutralElement(K,A)-->

(∃b∈K. M`⟨a,b⟩ = TheNeutralElement(K,M))))"

text{*The next context (locale) defines notation used for ordered fields.

We do that by extending the notation defined in the

@{text "ring1"} context that is used for oredered rings and

adding some assumptions to make sure we are

talking about ordered fields in this context.

We should rename the carrier from $R$ used in the @{text "ring1"}

context to $K$, more appriopriate for fields. Theoretically the Isar locale

facility supports such renaming, but we experienced diffculties using

some lemmas from @{text "ring1"} locale after renaming.

*}

locale field1 = ring1 +

assumes mult_commute: "M {is commutative on} R"

assumes not_triv: "\<zero> ≠ \<one>"

assumes inv_exists: "∀a∈R. a≠\<zero> --> (∃b∈R. a·b = \<one>)"

fixes non_zero ("R⇩_{0}")

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

fixes inv ("_¯ " [96] 97)

defines inv_def[simp]: "a¯ ≡ GroupInv(R⇩_{0},restrict(M,R⇩_{0}×R⇩_{0}))`(a)"

text{*The next lemma assures us that we are talking fields

in the @{text "field1"} context.*}

lemma (in field1) OrdField_ZF_1_L1: shows "IsAnOrdField(R,A,M,r)"

using OrdRing_ZF_1_L1 mult_commute not_triv inv_exists IsAnOrdField_def

by simp;

text{*Ordered field is a field, of course.*}

lemma OrdField_ZF_1_L1A: assumes "IsAnOrdField(K,A,M,r)"

shows "IsAfield(K,A,M)"

using assms IsAnOrdField_def IsAnOrdRing_def IsAfield_def

by simp;

text{*Theorems proven in @{text "field0"} (about fields) context are valid

in the @{text "field1"} context (about ordered fields). *}

lemma (in field1) OrdField_ZF_1_L1B: shows "field0(R,A,M)"

using OrdField_ZF_1_L1 OrdField_ZF_1_L1A field_field0

by simp;

text{*We can use theorems proven in the @{text "field1"} context whenever we

talk about an ordered field.*}

lemma OrdField_ZF_1_L2: assumes "IsAnOrdField(K,A,M,r)"

shows "field1(K,A,M,r)"

using assms IsAnOrdField_def OrdRing_ZF_1_L2 ring1_def

IsAnOrdField_def field1_axioms_def field1_def

by auto;

text{*In ordered rings the existence of a right inverse for all positive

elements implies the existence of an inverse for all non zero elements.*}

lemma (in ring1) OrdField_ZF_1_L3:

assumes A1: "∀a∈R⇩_{+}. ∃b∈R. a·b = \<one>" and A2: "c∈R" "c≠\<zero>"

shows "∃b∈R. c·b = \<one>"

proof -

{ assume "c∈R⇩_{+}"

with A1 have "∃b∈R. c·b = \<one>" by simp }

moreover

{ assume "c∉R⇩_{+}"

with A2 have "(\<rm>c) ∈ R⇩_{+}"

using OrdRing_ZF_3_L2A by simp;

with A1 obtain b where "b∈R" and "(\<rm>c)·b = \<one>"

by auto;

with A2 have "(\<rm>b) ∈ R" "c·(\<rm>b) = \<one>"

using Ring_ZF_1_L3 Ring_ZF_1_L7 by auto;

then have "∃b∈R. c·b = \<one>" by auto }

ultimately show ?thesis by blast

qed;

text{*Ordered fields are easier to deal with, because it is sufficient

to show the existence of an inverse for the set of positive elements.*}

lemma (in ring1) OrdField_ZF_1_L4:

assumes "\<zero> ≠ \<one>" and "M {is commutative on} R"

and "∀a∈R⇩_{+}. ∃b∈R. a·b = \<one>"

shows "IsAnOrdField(R,A,M,r)"

using assms OrdRing_ZF_1_L1 OrdField_ZF_1_L3 IsAnOrdField_def

by simp;

text{*The set of positive field elements is closed under multiplication.*}

lemma (in field1) OrdField_ZF_1_L5: shows "R⇩_{+}{is closed under} M"

using OrdField_ZF_1_L1B field0.field_has_no_zero_divs OrdRing_ZF_3_L3

by simp;

text{*The set of positive field elements is closed under multiplication:

the explicit version.*}

lemma (in field1) pos_mul_closed:

assumes A1: "\<zero> \<ls> a" "\<zero> \<ls> b"

shows "\<zero> \<ls> a·b"

proof -

from A1 have "a ∈ R⇩_{+}" and "b ∈ R⇩_{+}"

using OrdRing_ZF_3_L14 by auto;

then show "\<zero> \<ls> a·b"

using OrdField_ZF_1_L5 IsOpClosed_def PositiveSet_def

by simp;

qed;

text{*In fields square of a nonzero element is positive. *}

lemma (in field1) OrdField_ZF_1_L6: assumes "a∈R" "a≠\<zero>"

shows "a⇧^{2}∈ R⇩_{+}"

using assms OrdField_ZF_1_L1B field0.field_has_no_zero_divs

OrdRing_ZF_3_L15 by simp;

text{*The next lemma restates the fact @{text "Field_ZF"} that out notation

for the field inverse means what it is supposed to mean.*}

lemma (in field1) OrdField_ZF_1_L7: assumes "a∈R" "a≠\<zero>"

shows "a·(a¯) = \<one>" "(a¯)·a = \<one>"

using assms OrdField_ZF_1_L1B field0.Field_ZF_1_L6

by auto;

text{*A simple lemma about multiplication and cancelling of a positive field

element.*}

lemma (in field1) OrdField_ZF_1_L7A:

assumes A1: "a∈R" "b ∈ R⇩_{+}"

shows

"a·b·b¯ = a"

"a·b¯·b = a"

proof -

from A1 have "b∈R" "b≠\<zero>" using PositiveSet_def

by auto

with A1 show "a·b·b¯ = a" and "a·b¯·b = a"

using OrdField_ZF_1_L1B field0.Field_ZF_1_L7

by auto;

qed;

text{*Some properties of the inverse of a positive element.*}

lemma (in field1) OrdField_ZF_1_L8: assumes A1: "a ∈ R⇩_{+}"

shows "a¯ ∈ R⇩_{+}" "a·(a¯) = \<one>" "(a¯)·a = \<one>"

proof -

from A1 have I: "a∈R" "a≠\<zero>" using PositiveSet_def

by auto;

with A1 have "a·(a¯)⇧^{2}∈ R⇩_{+}"

using OrdField_ZF_1_L1B field0.Field_ZF_1_L5 OrdField_ZF_1_L6

OrdField_ZF_1_L5 IsOpClosed_def by simp;

with I show "a¯ ∈ R⇩_{+}"

using OrdField_ZF_1_L1B field0.Field_ZF_2_L1

by simp;

from I show "a·(a¯) = \<one>" "(a¯)·a = \<one>"

using OrdField_ZF_1_L7 by auto

qed;

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

lemma (in field1) OrdField_ZF_1_L9: assumes "a\<ls>b"

shows "(b\<rs>a)¯ ∈ R⇩_{+}"

using assms OrdRing_ZF_1_L14 OrdField_ZF_1_L8

by simp;

text{*In ordered fields if at least one of $a,b$ is not zero, then

$a^2+b^2 > 0$, in particular $a^2+b^2\neq 0$ and exists the

(multiplicative) inverse of $a^2+b^2$. *}

lemma (in field1) OrdField_ZF_1_L10:

assumes A1: "a∈R" "b∈R" and A2: "a ≠ \<zero> ∨ b ≠ \<zero>"

shows "\<zero> \<ls> a⇧^{2}\<ra> b⇧^{2}" and "∃c∈R. (a⇧^{2}\<ra> b⇧^{2})·c = \<one>"

proof -

from A1 A2 show "\<zero> \<ls> a⇧^{2}\<ra> b⇧^{2}"

using OrdField_ZF_1_L1B field0.field_has_no_zero_divs

OrdRing_ZF_3_L19 by simp;

then have

"(a⇧^{2}\<ra> b⇧^{2})¯ ∈ R" and "(a⇧^{2}\<ra> b⇧^{2})·(a⇧^{2}\<ra> b⇧^{2})¯ = \<one>"

using OrdRing_ZF_1_L3 PositiveSet_def OrdField_ZF_1_L8

by auto;

then show "∃c∈R. (a⇧^{2}\<ra> b⇧^{2})·c = \<one>" by auto;

qed;

section{*Inequalities*}

text{*In this section we develop tools to deal inequalities in fields.*}

text{*We can multiply strict inequality by a positive element.*}

lemma (in field1) OrdField_ZF_2_L1:

assumes "a\<ls>b" and "c∈R⇩_{+}"

shows "a·c \<ls> b·c"

using assms OrdField_ZF_1_L1B field0.field_has_no_zero_divs

OrdRing_ZF_3_L13

by simp;

text{*A special case of @{text "OrdField_ZF_2_L1"} when we multiply

an inverse by an element.*}

lemma (in field1) OrdField_ZF_2_L2:

assumes A1: "a∈R⇩_{+}" and A2: "a¯ \<ls> b"

shows "\<one> \<ls> b·a"

proof -

from A1 A2 have "(a¯)·a \<ls> b·a"

using OrdField_ZF_2_L1 by simp;

with A1 show "\<one> \<ls> b·a"

using OrdField_ZF_1_L8 by simp

qed;

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

lemma (in field1) OrdField_ZF_2_L3:

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

using assms OrdField_ZF_1_L8 OrdRing_ZF_1_L9A

by simp;

text{*We can multiply a strict inequality by a positive element

or its inverse.*}

lemma (in field1) OrdField_ZF_2_L4:

assumes "a\<ls>b" and "c∈R⇩_{+}"

shows

"a·c \<ls> b·c"

"c·a \<ls> c·b"

"a·c¯ \<ls> b·c¯"

using assms OrdField_ZF_1_L1B field0.field_has_no_zero_divs

OrdField_ZF_1_L8 OrdRing_ZF_3_L13 by auto;

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

changing it to its inverse.*}

lemma (in field1) OrdField_ZF_2_L5:

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

shows "a \<lsq> c·b¯"

proof -

from A1 A2 have "a·b·b¯ \<lsq> c·b¯"

using OrdField_ZF_2_L3 by simp;

with A1 show "a \<lsq> c·b¯" using OrdField_ZF_1_L7A

by simp;

qed;

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

changing it to its inverse, version with a product initially on the

right hand side.*}

lemma (in field1) OrdField_ZF_2_L5A:

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

shows "a·c¯ \<lsq> b"

proof -

from A1 A2 have "a·c¯ \<lsq> b·c·c¯"

using OrdField_ZF_2_L3 by simp

with A1 show "a·c¯ \<lsq> b" using OrdField_ZF_1_L7A

by simp

qed;

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

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

initially on the left hand side.*}

lemma (in field1) OrdField_ZF_2_L6:

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

shows "a \<ls> c·b¯"

proof -

from A1 A2 have "a·b·b¯ \<ls> c·b¯"

using OrdField_ZF_2_L4 by simp

with A1 show "a \<ls> c·b¯" using OrdField_ZF_1_L7A

by simp;

qed;

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

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

initially on the right hand side.*}

lemma (in field1) OrdField_ZF_2_L6A:

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

shows "a·c¯ \<ls> b"

proof -

from A1 A2 have "a·c¯ \<ls> b·c·c¯"

using OrdField_ZF_2_L4 by simp

with A1 show "a·c¯ \<ls> b" using OrdField_ZF_1_L7A

by simp

qed;

text{*Sometimes we can reverse an inequality by taking inverse

on both sides.*}

lemma (in field1) OrdField_ZF_2_L7:

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

shows "b¯ \<lsq> a"

proof -

from A1 have "a¯ ∈ R⇩_{+}" using OrdField_ZF_1_L8

by simp;

with A2 have "b ∈ R⇩_{+}" using OrdRing_ZF_3_L7

by blast;

then have T: "b ∈ R⇩_{+}" "b¯ ∈ R⇩_{+}" using OrdField_ZF_1_L8

by auto

with A1 A2 have "b¯·a¯·a \<lsq> b¯·b·a"

using OrdRing_ZF_1_L9A by simp;

moreover

from A1 A2 T have

"b¯ ∈ R" "a∈R" "a≠\<zero>" "b∈R" "b≠\<zero>"

using PositiveSet_def OrdRing_ZF_1_L3 by auto;

then have "b¯·a¯·a = b¯" and "b¯·b·a = a"

using OrdField_ZF_1_L1B field0.Field_ZF_1_L7

field0.Field_ZF_1_L6 Ring_ZF_1_L3

by auto;

ultimately show "b¯ \<lsq> a" by simp;

qed;

text{*Sometimes we can reverse a strict inequality by taking inverse

on both sides.*}

lemma (in field1) OrdField_ZF_2_L8:

assumes A1: "a∈R⇩_{+}" and A2: "a¯ \<ls> b"

shows "b¯ \<ls> a"

proof -

from A1 A2 have "a¯ ∈ R⇩_{+}" "a¯ \<lsq>b"

using OrdField_ZF_1_L8 by auto;

then have "b ∈ R⇩_{+}" using OrdRing_ZF_3_L7

by blast;

then have "b∈R" "b≠\<zero>" using PositiveSet_def by auto;

with A2 have "b¯ ≠ a"

using OrdField_ZF_1_L1B field0.Field_ZF_2_L4

by simp;

with A1 A2 show "b¯ \<ls> a"

using OrdField_ZF_2_L7 by simp;

qed;

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

field elements and inverse of a difference.*}

lemma (in field1) OrdField_ZF_2_L9:

assumes A1: "a\<ls>b" and A2: "(b\<rs>a)¯ \<ls> c"

shows "\<one> \<ra> a·c \<ls> b·c"

proof -

from A1 A2 have "(b\<rs>a)¯ ∈ R⇩_{+}" "(b\<rs>a)¯ \<lsq> c"

using OrdField_ZF_1_L9 by auto;

then have T1: "c ∈ R⇩_{+}" using OrdRing_ZF_3_L7 by blast;

with A1 A2 have T2:

"a∈R" "b∈R" "c∈R" "c≠\<zero>" "c¯ ∈ R"

using OrdRing_ZF_1_L3 OrdField_ZF_1_L8 PositiveSet_def

by auto;

with A1 A2 have "c¯ \<ra> a \<ls> b\<rs>a \<ra> a"

using OrdRing_ZF_1_L14 OrdField_ZF_2_L8 ring_strict_ord_trans_inv

by simp;

with T1 T2 have "(c¯ \<ra> a)·c \<ls> b·c"

using Ring_ZF_2_L1A OrdField_ZF_2_L1 by simp;

with T1 T2 show "\<one> \<ra> a·c \<ls> b·c"

using ring_oper_distr OrdField_ZF_1_L8

by simp;

qed;

section{*Definition of real numbers*}

text{*The only purpose of this section is to define what does it mean

to be a model of real numbers.*}

text{*We define model of real numbers as any quadruple of sets $(K,A,M,r)$

such that $(K,A,M,r)$ is an ordered field and the order relation $r$

is complete, that is every set that is nonempty and bounded above in this

relation has a supremum. *}

definition

"IsAmodelOfReals(K,A,M,r) ≡ IsAnOrdField(K,A,M,r) ∧ (r {is complete})";

end