(* 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, 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. HIS 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 ‹Construction real numbers - the generic part› theory Real_ZF imports Int_ZF_IML Ring_ZF_1 begin text‹The goal of the ‹Real_ZF› series of theory files is to provide a contruction of the set of real numbers. There are several ways to construct real numbers. Most common start from the rational numbers and use Dedekind cuts or Cauchy sequences. ‹Real_ZF_x.thy› series formalizes an alternative approach that constructs real numbers directly from the group of integers. Our formalization is mostly based on \cite{Arthan2004}. Different variants of this contruction are also described in \cite{Campo2003} and \cite{Street2003}. I recommend to read these papers, but for the impatient here is a short description: we take a set of maps $s : Z\rightarrow Z$ such that the set $\{s(m+n)-s(m)-s(n)\}_{n,m \in Z}$ is finite ($Z$ means the integers here). We call these maps slopes. Slopes form a group with the natural addition $(s+r)(n) = s(n)+r(n)$. The maps such that the set $s(Z)$ is finite (finite range functions) form a subgroup of slopes. The additive group of real numbers is defined as the quotient group of slopes by the (sub)group of finite range functions. The multiplication is defined as the projection of the composition of slopes into the resulting quotient (coset) space. › subsection‹The definition of real numbers› text‹This section contains the construction of the ring of real numbers as classes of slopes - integer almost homomorphisms. The real definitions are in ‹Group_ZF_2› theory, here we just specialize the definitions of almost homomorphisms, their equivalence and operations to the additive group of integers from the general case of abelian groups considered in ‹Group_ZF_2›.› text‹The set of slopes is defined as the set of almost homomorphisms on the additive group of integers.› definition "Slopes ≡ AlmostHoms(int,IntegerAddition)" text‹The first operation on slopes (pointwise addition) is a special case of the first operation on almost homomorphisms.› definition "SlopeOp1 ≡ AlHomOp1(int,IntegerAddition)" text‹The second operation on slopes (composition) is a special case of the second operation on almost homomorphisms.› definition "SlopeOp2 ≡ AlHomOp2(int,IntegerAddition)" text‹Bounded integer maps are functions from integers to integers that have finite range. They play a role of zero in the set of real numbers we are constructing.› definition "BoundedIntMaps ≡ FinRangeFunctions(int,int)" text‹Bounded integer maps form a normal subgroup of slopes. The equivalence relation on slopes is the (group) quotient relation defined by this subgroup. › definition "SlopeEquivalenceRel ≡ QuotientGroupRel(Slopes,SlopeOp1,BoundedIntMaps)" text‹The set of real numbers is the set of equivalence classes of slopes.› definition "RealNumbers ≡ Slopes//SlopeEquivalenceRel" text‹The addition on real numbers is defined as the projection of pointwise addition of slopes on the quotient. This means that the additive group of real numbers is the quotient group: the group of slopes (with pointwise addition) defined by the normal subgroup of bounded integer maps.› definition "RealAddition ≡ ProjFun2(Slopes,SlopeEquivalenceRel,SlopeOp1)" text‹Multiplication is defined as the projection of composition of slopes on the quotient. The fact that it works is probably the most surprising part of the construction.› definition "RealMultiplication ≡ ProjFun2(Slopes,SlopeEquivalenceRel,SlopeOp2)" text‹We first show that we can use theorems proven in some proof contexts (locales). The locale ‹group1› requires assumption that we deal with an abelian group. The next lemma allows to use all theorems proven in the context called ‹group1›.› lemma Real_ZF_1_L1: shows "group1(int,IntegerAddition)" using group1_axioms.intro group1_def Int_ZF_1_T2 by simp text‹Real numbers form a ring. This is a special case of the theorem proven in ‹Ring_ZF_1.thy›, where we show the same in general for almost homomorphisms rather than slopes.› theorem Real_ZF_1_T1: shows "IsAring(RealNumbers,RealAddition,RealMultiplication)" proof - let ?AH = "AlmostHoms(int,IntegerAddition)" let ?Op1 = "AlHomOp1(int,IntegerAddition)" let ?FR = "FinRangeFunctions(int,int)" let ?Op2 = "AlHomOp2(int,IntegerAddition)" let ?R = "QuotientGroupRel(?AH,?Op1,?FR)" let ?A = "ProjFun2(?AH,?R,?Op1)" let ?M = "ProjFun2(?AH,?R,?Op2)" have "IsAring(?AH//?R,?A,?M)" using Real_ZF_1_L1 group1.Ring_ZF_1_1_T1 by simp then show ?thesis using Slopes_def SlopeOp2_def SlopeOp1_def BoundedIntMaps_def SlopeEquivalenceRel_def RealNumbers_def RealAddition_def RealMultiplication_def by simp qed text‹We can use theorems proven in ‹group0› and ‹group1› contexts applied to the group of real numbers.› lemma Real_ZF_1_L2: shows "group0(RealNumbers,RealAddition)" "RealAddition {is commutative on} RealNumbers" "group1(RealNumbers,RealAddition)" proof - have "IsAgroup(RealNumbers,RealAddition)" "RealAddition {is commutative on} RealNumbers" using Real_ZF_1_T1 IsAring_def by auto then show "group0(RealNumbers,RealAddition)" "RealAddition {is commutative on} RealNumbers" "group1(RealNumbers,RealAddition)" using group1_axioms.intro group0_def group1_def by auto qed text‹Let's define some notation.› locale real0 = fixes real ("ℝ") defines real_def [simp]: "ℝ ≡ RealNumbers" fixes ra (infixl "\<ra>" 69) defines ra_def [simp]: "a\<ra> b ≡ RealAddition`⟨a,b⟩" fixes rminus ("\<rm> _" 72) defines rminus_def [simp]:"\<rm>a ≡ GroupInv(ℝ,RealAddition)`(a)" fixes rsub (infixl "\<rs>" 69) defines rsub_def [simp]: "a\<rs>b ≡ a\<ra>(\<rm>b)" fixes rm (infixl "⋅" 70) defines rm_def [simp]: "a⋅b ≡ RealMultiplication`⟨a,b⟩" fixes rzero ("𝟬") defines rzero_def [simp]: "𝟬 ≡ TheNeutralElement(RealNumbers,RealAddition)" fixes rone ("𝟭") defines rone_def [simp]: "𝟭 ≡ TheNeutralElement(RealNumbers,RealMultiplication)" fixes rtwo ("𝟮") defines rtwo_def [simp]: "𝟮 ≡ 𝟭\<ra>𝟭" fixes non_zero ("ℝ⇩_{0}") defines non_zero_def[simp]: "ℝ⇩_{0}≡ ℝ-{𝟬}" fixes inv ("_¯ " [90] 91) defines inv_def[simp]: "a¯ ≡ GroupInv(ℝ⇩_{0},restrict(RealMultiplication,ℝ⇩_{0}×ℝ⇩_{0}))`(a)" text‹In ‹real0› context all theorems proven in the ‹ring0›, context are valid.› lemma (in real0) Real_ZF_1_L3: shows "ring0(ℝ,RealAddition,RealMultiplication)" using Real_ZF_1_T1 ring0_def ring0.Ring_ZF_1_L1 by auto text‹Lets try out our notation to see that zero and one are real numbers.› lemma (in real0) Real_ZF_1_L4: shows "𝟬∈ℝ" "𝟭∈ℝ" using Real_ZF_1_L3 ring0.Ring_ZF_1_L2 by auto text‹The lemma below lists some properties that require one real number to state.› lemma (in real0) Real_ZF_1_L5: assumes A1: "a∈ℝ" shows "(\<rm>a) ∈ ℝ" "(\<rm>(\<rm>a)) = a" "a\<ra>𝟬 = a" "𝟬\<ra>a = a" "a⋅𝟭 = a" "𝟭⋅a = a" "a\<rs>a = 𝟬" "a\<rs>𝟬 = a" using assms Real_ZF_1_L3 ring0.Ring_ZF_1_L3 by auto text‹The lemma below lists some properties that require two real numbers to state.› lemma (in real0) Real_ZF_1_L6: assumes "a∈ℝ" "b∈ℝ" shows "a\<ra>b ∈ ℝ" "a\<rs>b ∈ ℝ" "a⋅b ∈ ℝ" "a\<ra>b = b\<ra>a" "(\<rm>a)⋅b = \<rm>(a⋅b)" "a⋅(\<rm>b) = \<rm>(a⋅b)" using assms Real_ZF_1_L3 ring0.Ring_ZF_1_L4 ring0.Ring_ZF_1_L7 by auto text‹Multiplication of reals is associative.› lemma (in real0) Real_ZF_1_L6A: assumes "a∈ℝ" "b∈ℝ" "c∈ℝ" shows "a⋅(b⋅c) = (a⋅b)⋅c" using assms Real_ZF_1_L3 ring0.Ring_ZF_1_L11 by simp text‹Addition is distributive with respect to multiplication.› lemma (in real0) Real_ZF_1_L7: assumes "a∈ℝ" "b∈ℝ" "c∈ℝ" shows "a⋅(b\<ra>c) = a⋅b \<ra> a⋅c" "(b\<ra>c)⋅a = b⋅a \<ra> c⋅a" "a⋅(b\<rs>c) = a⋅b \<rs> a⋅c" "(b\<rs>c)⋅a = b⋅a \<rs> c⋅a" using assms Real_ZF_1_L3 ring0.ring_oper_distr ring0.Ring_ZF_1_L8 by auto text‹A simple rearrangement with four real numbers.› lemma (in real0) Real_ZF_1_L7A: assumes "a∈ℝ" "b∈ℝ" "c∈ℝ" "d∈ℝ" shows "a\<rs>b \<ra> (c\<rs>d) = a\<ra>c\<rs>b\<rs>d" using assms Real_ZF_1_L2 group0.group0_4_L8A by simp text‹‹RealAddition› is defined as the projection of the first operation on slopes (that is, slope addition) on the quotient (slopes divided by the "almost equal" relation. The next lemma plays with definitions to show that this is the same as the operation induced on the appriopriate quotient group. The names ‹AH›, ‹Op1› and ‹FR› are used in ‹group1› context to denote almost homomorphisms, the first operation on ‹AH› and finite range functions resp.› lemma Real_ZF_1_L8: assumes "AH = AlmostHoms(int,IntegerAddition)" and "Op1 = AlHomOp1(int,IntegerAddition)" and "FR = FinRangeFunctions(int,int)" shows "RealAddition = QuotientGroupOp(AH,Op1,FR)" using assms RealAddition_def SlopeEquivalenceRel_def QuotientGroupOp_def Slopes_def SlopeOp1_def BoundedIntMaps_def by simp text‹The symbol ‹𝟬› in the ‹real0› context is defined as the neutral element of real addition. The next lemma shows that this is the same as the neutral element of the appriopriate quotient group.› lemma (in real0) Real_ZF_1_L9: assumes "AH = AlmostHoms(int,IntegerAddition)" and "Op1 = AlHomOp1(int,IntegerAddition)" and "FR = FinRangeFunctions(int,int)" and "r = QuotientGroupRel(AH,Op1,FR)" shows "TheNeutralElement(AH//r,QuotientGroupOp(AH,Op1,FR)) = 𝟬" "SlopeEquivalenceRel = r" using assms Slopes_def Real_ZF_1_L8 RealNumbers_def SlopeEquivalenceRel_def SlopeOp1_def BoundedIntMaps_def by auto text‹Zero is the class of any finite range function.› lemma (in real0) Real_ZF_1_L10: assumes A1: "s ∈ Slopes" shows "SlopeEquivalenceRel``{s} = 𝟬 ⟷ s ∈ BoundedIntMaps" proof - let ?AH = "AlmostHoms(int,IntegerAddition)" let ?Op1 = "AlHomOp1(int,IntegerAddition)" let ?FR = "FinRangeFunctions(int,int)" let ?r = "QuotientGroupRel(?AH,?Op1,?FR)" let ?e = "TheNeutralElement(?AH//?r,QuotientGroupOp(?AH,?Op1,?FR))" from A1 have "group1(int,IntegerAddition)" "s∈?AH" using Real_ZF_1_L1 Slopes_def by auto then have "?r``{s} = ?e ⟷ s ∈ ?FR" using group1.Group_ZF_3_3_L5 by simp moreover have "?r = SlopeEquivalenceRel" "?e = 𝟬" "?FR = BoundedIntMaps" using SlopeEquivalenceRel_def Slopes_def SlopeOp1_def BoundedIntMaps_def Real_ZF_1_L9 by auto ultimately show ?thesis by simp qed text‹We will need a couple of results from ‹Group_ZF_3.thy› The first two that state that the definition of addition and multiplication of real numbers are consistent, that is the result does not depend on the choice of the slopes representing the numbers. The second one implies that what we call ‹SlopeEquivalenceRel› is actually an equivalence relation on the set of slopes. We also show that the neutral element of the multiplicative operation on reals (in short number $1$) is the class of the identity function on integers.› lemma Real_ZF_1_L11: shows "Congruent2(SlopeEquivalenceRel,SlopeOp1)" "Congruent2(SlopeEquivalenceRel,SlopeOp2)" "SlopeEquivalenceRel ⊆ Slopes × Slopes" "equiv(Slopes, SlopeEquivalenceRel)" "SlopeEquivalenceRel``{id(int)} = TheNeutralElement(RealNumbers,RealMultiplication)" "BoundedIntMaps ⊆ Slopes" proof - let ?G = "int" let ?f = "IntegerAddition" let ?AH = "AlmostHoms(int,IntegerAddition)" let ?Op1 = "AlHomOp1(int,IntegerAddition)" let ?Op2 = "AlHomOp2(int,IntegerAddition)" let ?FR = "FinRangeFunctions(int,int)" let ?R = "QuotientGroupRel(?AH,?Op1,?FR)" have "Congruent2(?R,?Op1)" "Congruent2(?R,?Op2)" using Real_ZF_1_L1 group1.Group_ZF_3_4_L13A group1.Group_ZF_3_3_L4 by auto then show "Congruent2(SlopeEquivalenceRel,SlopeOp1)" "Congruent2(SlopeEquivalenceRel,SlopeOp2)" using SlopeEquivalenceRel_def SlopeOp1_def Slopes_def BoundedIntMaps_def SlopeOp2_def by auto have "equiv(?AH,?R)" using Real_ZF_1_L1 group1.Group_ZF_3_3_L3 by simp then show "equiv(Slopes,SlopeEquivalenceRel)" using BoundedIntMaps_def SlopeEquivalenceRel_def SlopeOp1_def Slopes_def by simp then show "SlopeEquivalenceRel ⊆ Slopes × Slopes" using equiv_type by simp have "?R``{id(int)} = TheNeutralElement(?AH//?R,ProjFun2(?AH,?R,?Op2))" using Real_ZF_1_L1 group1.Group_ZF_3_4_T2 by simp then show "SlopeEquivalenceRel``{id(int)} = TheNeutralElement(RealNumbers,RealMultiplication)" using Slopes_def RealNumbers_def SlopeEquivalenceRel_def SlopeOp1_def BoundedIntMaps_def RealMultiplication_def SlopeOp2_def by simp have "?FR ⊆ ?AH" using Real_ZF_1_L1 group1.Group_ZF_3_3_L1 by simp then show "BoundedIntMaps ⊆ Slopes" using BoundedIntMaps_def Slopes_def by simp qed text‹A one-side implication of the equivalence from ‹Real_ZF_1_L10›: the class of a bounded integer map is the real zero.› lemma (in real0) Real_ZF_1_L11A: assumes "s ∈ BoundedIntMaps" shows "SlopeEquivalenceRel``{s} = 𝟬" using assms Real_ZF_1_L11 Real_ZF_1_L10 by auto text‹The next lemma is rephrases the result from ‹Group_ZF_3.thy› that says that the negative (the group inverse with respect to real addition) of the class of a slope is the class of that slope composed with the integer additive group inverse. The result and proof is not very readable as we use mostly generic set theory notation with long names here. ‹Real_ZF_1.thy› contains the same statement written in a more readable notation: $[-s] = -[s]$.› lemma (in real0) Real_ZF_1_L12: assumes A1: "s ∈ Slopes" and Dr: "r = QuotientGroupRel(Slopes,SlopeOp1,BoundedIntMaps)" shows "r``{GroupInv(int,IntegerAddition) O s} = \<rm>(r``{s})" proof - let ?G = "int" let ?f = "IntegerAddition" let ?AH = "AlmostHoms(int,IntegerAddition)" let ?Op1 = "AlHomOp1(int,IntegerAddition)" let ?FR = "FinRangeFunctions(int,int)" let ?F = "ProjFun2(Slopes,r,SlopeOp1)" from A1 Dr have "group1(?G, ?f)" "s ∈ AlmostHoms(?G, ?f)" "r = QuotientGroupRel( AlmostHoms(?G, ?f), AlHomOp1(?G, ?f), FinRangeFunctions(?G, ?G))" and "?F = ProjFun2(AlmostHoms(?G, ?f), r, AlHomOp1(?G, ?f))" using Real_ZF_1_L1 Slopes_def SlopeOp1_def BoundedIntMaps_def by auto then have "r``{GroupInv(?G, ?f) O s} = GroupInv(AlmostHoms(?G, ?f) // r, ?F)`(r `` {s})" using group1.Group_ZF_3_3_L6 by simp with Dr show ?thesis using RealNumbers_def Slopes_def SlopeEquivalenceRel_def RealAddition_def by simp qed text‹Two classes are equal iff the slopes that represent them are almost equal.› lemma Real_ZF_1_L13: assumes "s ∈ Slopes" "p ∈ Slopes" and "r = SlopeEquivalenceRel" shows "r``{s} = r``{p} ⟷ ⟨s,p⟩ ∈ r" using assms Real_ZF_1_L11 eq_equiv_class equiv_class_eq by blast text‹Identity function on integers is a slope. This lemma concludes the easy part of the construction that follows from the fact that slope equivalence classes form a ring. It is easy to see that multiplication of classes of almost homomorphisms is not commutative in general. The remaining properties of real numbers, like commutativity of multiplication and the existence of multiplicative inverses have to be proven using properties of the group of integers, rather that in general setting of abelian groups. This is done in the ‹Real_ZF_1› theory.› lemma Real_ZF_1_L14: shows "id(int) ∈ Slopes" proof - have "id(int) ∈ AlmostHoms(int,IntegerAddition)" using Real_ZF_1_L1 group1.Group_ZF_3_4_L15 by simp then show ?thesis using Slopes_def by simp qed end