(* This file is a part of IsarMathLib - a library of formalized mathematics for Isabelle/Isar. Copyright (C) 2005, 2006 Slawomir Kolodynski This program is free software; Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. The name of the author may not be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. *) section ‹Ordered fields› theory OrderedField_ZF imports OrderedRing_ZF Field_ZF begin text‹This theory covers basic facts about ordered fiels.› subsection‹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 ‹K› is the carrier of the field. The second set, denoted ‹A› represents the additive operation on ‹K› (recall that in ZF set theory functions are sets). The third set ‹M› represents the multiplicative operation on ‹K›. The fourth set ‹r› is the order relation on ‹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 ‹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 ‹ring1› context to $K$, more appriopriate for fields. Theoretically the Isar locale facility supports such renaming, but we experienced diffculties using some lemmas from ‹ring1› locale after renaming. › locale field1 = ring1 + assumes mult_commute: "M {is commutative on} R" assumes not_triv: "𝟬 ≠ 𝟭" assumes inv_exists: "∀a∈R. a≠𝟬 ⟶ (∃b∈R. a⋅b = 𝟭)" fixes non_zero ("R⇩_{0}") defines non_zero_def[simp]: "R⇩_{0}≡ R-{𝟬}" 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 ‹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 ‹field0› (about fields) context are valid in the ‹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 ‹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 = 𝟭" and A2: "c∈R" "c≠𝟬" shows "∃b∈R. c⋅b = 𝟭" proof - { assume "c∈R⇩_{+}" with A1 have "∃b∈R. c⋅b = 𝟭" 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 = 𝟭" by auto with A2 have "(\<rm>b) ∈ R" "c⋅(\<rm>b) = 𝟭" using Ring_ZF_1_L3 Ring_ZF_1_L7 by auto then have "∃b∈R. c⋅b = 𝟭" 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 "𝟬 ≠ 𝟭" and "M {is commutative on} R" and "∀a∈R⇩_{+}. ∃b∈R. a⋅b = 𝟭" 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: "𝟬 \<ls> a" "𝟬 \<ls> b" shows "𝟬 \<ls> a⋅b" proof - from A1 have "a ∈ R⇩_{+}" and "b ∈ R⇩_{+}" using OrdRing_ZF_3_L14 by auto then show "𝟬 \<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≠𝟬" 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 ‹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≠𝟬" shows "a⋅(a¯) = 𝟭" "(a¯)⋅a = 𝟭" 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≠𝟬" 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¯) = 𝟭" "(a¯)⋅a = 𝟭" proof - from A1 have I: "a∈R" "a≠𝟬" 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¯) = 𝟭" "(a¯)⋅a = 𝟭" using OrdField_ZF_1_L7 by auto qed text‹If $a$ is smaller than $b$, 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 ≠ 𝟬 ∨ b ≠ 𝟬" shows "𝟬 \<ls> a⇧^{2}\<ra> b⇧^{2}" and "∃c∈R. (a⇧^{2}\<ra> b⇧^{2})⋅c = 𝟭" proof - from A1 A2 show "𝟬 \<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})¯ = 𝟭" using OrdRing_ZF_1_L3 PositiveSet_def OrdField_ZF_1_L8 by auto then show "∃c∈R. (a⇧^{2}\<ra> b⇧^{2})⋅c = 𝟭" by auto qed subsection‹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 ‹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 "𝟭 \<ls> b⋅a" proof - from A1 A2 have "(a¯)⋅a \<ls> b⋅a" using OrdField_ZF_2_L1 by simp with A1 show "𝟭 \<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≠𝟬" "b∈R" "b≠𝟬" 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≠𝟬" 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 "𝟭 \<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≠𝟬" "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 "𝟭 \<ra> a⋅c \<ls> b⋅c" using ring_oper_distr OrdField_ZF_1_L8 by simp qed subsection‹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