Theory Introduction

theory Introduction
imports equalities
(*     This file is a part of IsarMathLib -     a library of formalized mathematics for Isabelle/Isar.    Copyright (C) 2008  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.*)header{*\isaheader{Introduction.thy}*}theory Introduction imports equalitiesbegintext{*This theory does not contain any formalized mathematics used in   other theories, but is an introduction to IsarMathLib project. *}section{*How to read IsarMathLib proofs - a tutorial*}text{*Isar (the Isabelle's formal proof language) was designed to be similar  to the standard language of mathematics. Any person able to read proofs in  a typical mathematical paper should be able to read and understand  Isar proofs without having to learn a special proof language.   However, Isar is a formal proof language and as such it does contain a   couple of constructs whose meaning is hard to guess. In this tutorial   we will define a notion and prove an example theorem about that notion,  explaining Isar syntax along the way. This tutorial may also serve as a   style guide for IsarMathLib contributors. Note that this tutorial  aims to help in reading the presentation of the Isar language that is used  in IsarMathLib proof document and HTML rendering on the   FormalMath.org site, but does not teach how to write proofs that can be  verified by Isabelle. This presentation is different than the source   processed by Isabelle (the concept that the source and presentation   look different should be familiar to any LaTeX user). To learn  how to write Isar proofs one needs to study the source of this tutorial as well.  *}text{*The first thing that mathematicians typically do is to define  notions. In Isar this is done with the @{text "definition"} keyword.  In our case we define a notion of two   sets being disjoint. We will use the infix notation, i.e. the string  @{text "{is disjoint with}"} put between two sets to denote our notion   of disjointness.   The left side of the @{text "≡"} symbol is the notion   being defined, the right side says how we define it. In Isabelle @{text "0"}  is used to denote both zero (of natural numbers) and the empty set, which is  not surprising as those two things are the same in set theory.*}definition   AreDisjoint (infix "{is disjoint with}" 90) where  "A {is disjoint with} B ≡ A ∩ B = 0"text{*We are ready to prove a theorem. Here we show that the relation  of being disjoint is symmetric. We start with one of the keywords  ''theorem'', ''lemma'' or ''corollary''. In Isar they are synonymous.  Then we provide a name for the theorem. In standard mathematics   theorems are numbered. In Isar we can do that too, but it is  considered better to give theorems meaningful names.  After the ''shows'' keyword we give the statement to show. The   @{text "<->"} symbol denotes the equivalence in Isabelle/ZF. Here  we want to show that "A is disjoint with B iff and only if B is disjoint   with A". To prove this fact we show two implications - the first one that   @{text "A {is disjoint with} B"} implies @{text "B {is disjoint with} A"}  and then the converse one. Each of these implications is formulated  as a statement to be proved and then proved in   a subproof like a mini-theorem.  Each subproof uses a proof block to show the implication. Proof blocks  are delimited with curly brackets in Isar.   Proof block is one of the constructs that  does not exist in informal mathematics, so it may be confusing.   When reading a proof containing a proof block I suggest to focus first   on what is that we are proving in it. This can be done by looking  at the first line or two of the block and then at the last statement.   In our case the block starts with   "assume @{text "A {is disjoint with} B"} and the last statement  is "then have @{text "B {is disjoint with} A"}". It is a typical pattern   when someone needs to prove an implication: one assumes the antecedent  and then shows that the consequent follows from this assumption.  Implications are denoted with the   @{text "-->"} symbol in Isabelle.   After we prove both implications we collect them   using the ''moreover'' construct. The keyword ''ultimately''  indicates that what follows is the conclusion of the statements   collected with ''moreover''. The ''show'' keyword is like ''have'',  except that it indicates that we have arrived at the claim of the   theorem (or a subproof).  *}theorem disjointness_symmetric:   shows "A {is disjoint with} B <-> B {is disjoint with} A"proof -  have "A {is disjoint with} B --> B {is disjoint with} A"  proof -    { assume "A {is disjoint with} B"      then have "A ∩ B = 0" using AreDisjoint_def by simp      hence "B ∩ A = 0" by auto      then have  "B {is disjoint with} A"        using AreDisjoint_def by simp    } thus ?thesis by simp  qed  moreover have "B {is disjoint with} A --> A {is disjoint with} B"  proof -    { assume "B {is disjoint with} A"      then have "B ∩ A = 0" using AreDisjoint_def by simp      hence "A ∩ B = 0" by auto      then have  "A {is disjoint with} B"        using AreDisjoint_def by simp    } thus ?thesis by simp  qed  ultimately show ?thesis by blastqedsection{*Overview of the project*}text{*   The  @{text "Fol1"}, @{text " ZF1"} and @{text "Nat_ZF_IML"} theory   files contain some background material that is needed for   the remaining theories.  @{text "Order_ZF"} and @{text "Order_ZF_1a"} reformulate   material from standard Isabelle's   @{text "Order"} theory in terms of non-strict (less-or-equal)   order relations.  @{text "Order_ZF_1"} on the other hand directly continues the @{text "Order"}  theory file using strict order relations (less and not equal). This is useful  for translating theorems from Metamath.  In @{text "NatOrder_ZF"} we prove that the usual order on natural numbers  is linear.    The @{text "func1"} theory provides basic facts about functions.  @{text "func_ZF"} continues this development with more advanced  topics that relate to algebraic properties of binary operations,   like lifting a binary operation to a function space,  associative, commutative and distributive operations and properties  of functions related to order relations. @{text "func_ZF_1"} is   about properties of functions related to order relations.  The standard Isabelle's @{text "Finite"} theory defines the finite  powerset of a set as a certain "datatype" (?) with some recursive  properties. IsarMathLib's @{text "Finite1"}   and  @{text "Finite_ZF_1"} theories develop more facts about this notion.   These two theories are obsolete now.   They will be gradually replaced by an approach based on set theory  rather than tools specific to Isabelle. This approach is presented  in @{text "Finite_ZF"} theory file.  In @{text "FinOrd_ZF"} we talk about ordered finite sets.  The @{text "EquivClass1"} theory file is a reformulation of   the material in the standard  Isabelle's @{text "EquivClass"} theory in the spirit of ZF set theory.    @{text "FiniteSeq_ZF"} discusses the notion of finite sequences   (a.k.a. lists).  @{text "InductiveSeq_ZF"} provides the definition and properties of  (what is known in basic calculus as) sequences defined by induction,   i. e. by a formula of the form $a_0 = x,\ a_{n+1} = f(a_n)$.  @{text "Fold_ZF"} shows how the familiar from functional   programming notion of fold can be interpreted in set theory.  @{text "Partitions_ZF"} is about splitting a set into non-overlapping  subsets. This is a common trick in proofs.  @{text "Semigroup_ZF"} treats the expressions of the form   $a_0\cdot a_1\cdot .. \cdot a_n$, (i.e. products of finite sequences),   where "$\cdot$" is an associative binary operation.  @{text "CommutativeSemigroup_ZF"} is another take on a similar subject.  This time we consider the case when the operation is commutative  and the result of depends only on the set of elements we are  summing (additively speaking), but not the order.  The @{text "Topology_ZF"} series covers basics of general topology:   interior, closure, boundary, compact sets, separation axioms and   continuous functions.    @{text "Group_ZF"}, @{text "Group_ZF_1"}, @{text "Group_ZF_1b"}   and @{text "Group_ZF_2"}  provide basic facts of the group theory. @{text "Group_ZF_3"}   considers the notion of almost homomorphisms that is nedeed for the   real numbers construction in @{text "Real_ZF"}.  The @{text "TopologicalGroup"} connects the @{text "Topology_ZF"} and   @{text "Group_ZF"} series and starts the subject of topological groups  with some basic definitions and facts.  In @{text "DirectProduct_ZF"} we define direct product of groups and show  some its basic properties.  The @{text "OrderedGroup_ZF"} theory treats ordered groups.   This is a suprisingly large theory for such relatively obscure topic.    @{text "Ring_ZF"} defines rings. @{text "Ring_ZF_1"} covers   the properties of  rings that are specific to the real numbers construction   in @{text "Real_ZF"}.  The @{text "OrderedRing_ZF"} theory looks at the consequences of adding  a linear order to the ring algebraic structure.    @{text "Field_ZF"} and @{text "OrderedField_ZF"} contain basic facts  about (you guessed it) fields and ordered fields.     @{text "Int_ZF_IML"} theory considers the integers   as a monoid (multiplication) and an abelian ordered group (addition).   In @{text "Int_ZF_1"} we show that integers form a commutative ring.  @{text "Int_ZF_2"} contains some facts about slopes (almost homomorphisms   on integers) needed for real numbers construction,   used in @{text "Real_ZF_1"}.  In the @{text "IntDiv_ZF_IML"} theory translates some properties of the   integer quotient and reminder functions studied in the standard Isabelle's  @{text "IntDiv_ZF"} theory to the notation used in IsarMathLib.    The @{text "Real_ZF"} and @{text "Real_ZF_1"} theories   contain the construction of real numbers based on the paper \cite{Arthan2004}  by R. D. Arthan (not Cauchy sequences, not Dedekind sections).   The heavy lifting  is done mostly in @{text "Group_ZF_3"}, @{text "Ring_ZF_1"}   and @{text "Int_ZF_2"}. @{text "Real_ZF"} contains   the part of the construction that can be done  starting from generic abelian groups (rather than additive group of integers).  This allows to show that real numbers form a ring.   @{text "Real_ZF_1"} continues the construction using properties specific  to the integers and showing that real numbers constructed this way  form a complete ordered field.  In @{text "Complex_ZF"} we construct complex numbers starting from  a complete ordered field (a model of real numbers). We also define   the notation for writing about complex numbers and prove that the   structure of complex numbers constructed there satisfies the axioms  of complex numbers used in Metamath.  @{text "MMI_prelude"} defines the @{text "mmisar0"} context in which   most theorems translated from Metamath are proven. It also contains a   chapter explaining how the translation works.  In the @{text "Metamath_interface"} theory we prove a theorem  that the @{text "mmisar0"} context is valid (can be used)   in the @{text "complex0"} context.   All theories using the translated results will import the  @{text "Metamath_interface"} theory. The @{text "Metamath_sampler"}  theory provides some examples of using the translated theorems  in the @{text "complex0"} context.  The theories @{text "MMI_logic_and_sets"}, @{text "MMI_Complex"},   @{text "MMI_Complex_1"} and @{text "MMI_Complex_2"}  contain the theorems imported from the  Metamath's set.mm database. As the translated proofs are rather verbose  these theories are not printed in this proof document.  The full list of translated facts can be found in the   @{text "Metamath_theorems.txt"} file included in the   IsarMathLib distribution.  The @{text "MMI_examples"} provides some theorems imported from Metamath  that are printed in this proof document as examples of how translated  proofs look like.*};end