(*<*)

theory Paper

imports "Quotient"

        "LaTeXsugar"

        "../Nominal/FSet"

begin

notation (latex output)

  rel_conj ("_ OOO _" [53, 53] 52)

and

  fun_map ("_ ---> _" [51, 51] 50)

and

  fun_rel ("_ ===> _" [51, 51] 50)

and

  list_eq (infix "\<approx>" 50) (* Not sure if we want this notation...? *)

ML {*

fun nth_conj n (_, r) = nth (HOLogic.dest_conj r) n;

fun style_lhs_rhs proj = Scan.succeed (fn ctxt => fn t =>

  let

    val concl =

      Object_Logic.drop_judgment (ProofContext.theory_of ctxt) (Logic.strip_imp_concl t)

  in

    case concl of (_ $ l $ r) => proj (l, r)

    | _ => error ("Binary operator expected in term: " ^ Syntax.string_of_term ctxt concl)

  end);

*}

setup {*

  Term_Style.setup "rhs1" (style_lhs_rhs (nth_conj 0)) #>

  Term_Style.setup "rhs2" (style_lhs_rhs (nth_conj 1)) #>

  Term_Style.setup "rhs3" (style_lhs_rhs (nth_conj 2))

*}

(*>*)

section {* Introduction *}

text {* 

   \begin{flushright}

  {\em ``Not using a [quotient] package has its advantages: we do not have to\\ 

    collect all the theorems we shall ever want into one giant list;''}\\

    Paulson \cite{Paulson06}

  \end{flushright}\smallskip

  \noindent

  Isabelle is a generic theorem prover in which many logics can be implemented. 

  The most widely used one, however, is

  Higher-Order Logic (HOL). This logic consists of a small number of 

  axioms and inference

  rules over a simply-typed term-language. Safe reasoning in HOL is ensured by two very restricted 

  mechanisms for extending the logic: one is the definition of new constants

  in terms of existing ones; the other is the introduction of new types

  by identifying non-empty subsets in existing types. It is well understood 

  to use both mechanism for dealing with quotient constructions in HOL (cite Larry).

  For example the integers in Isabelle/HOL are constructed by a quotient construction over 

  the type @{typ "nat \<times> nat"} and the equivalence relation

% I would avoid substraction for natural numbers.

  @{text [display] "(n\<^isub>1, n\<^isub>2) \<approx> (m\<^isub>1, m\<^isub>2) \<equiv> n\<^isub>1 - n \<^isub>2 = m\<^isub>1 - m \<^isub>2"}

  \noindent

  Similarly one can construct the type of finite sets by quotienting lists

  according to the equivalence relation

  @{text [display] "xs \<approx> ys \<equiv> (\<forall>x. x \<in> xs \<longleftrightarrow> x \<in> ys)"}

  \noindent

  where @{text "\<in>"} stands for membership in a list.

  The problem is that in order to start reasoning about, for example integers, 

  definitions and theorems need to be transferred, or \emph{lifted}, 

  from the ``raw'' type @{typ "nat \<times> nat"} to the quotient type @{typ int}. 

  This lifting usually requires a lot of tedious reasoning effort.

  The purpose of a \emph{quotient package} is to ease the lifting and automate

  the reasoning involved as much as possible. Such a package is a central

  component of the new version of Nominal Isabelle where representations 

  of alpha-equated terms are constructed according to specifications given by

  the user. 

  In the context of HOL, there have been several quotient packages (...). The

  most notable is the one by Homeier (...) implemented in HOL4. However, what is

  surprising, none of them can deal compositions of quotients, for example with 

  lifting theorems about @{text "concat"}:

  @{thm [display] concat.simps(1)}

  @{thm [display] concat.simps(2)[no_vars]}

  \noindent

  One would like to lift this definition to the operation:

  @{thm [display] fconcat_empty[no_vars]}

  @{thm [display] fconcat_insert[no_vars]}

  \noindent

  What is special about this operation is that we have as input

  lists of lists which after lifting turn into finite sets of finite

  sets.

*}

subsection {* Contributions *}

text {*

  We present the detailed lifting procedure, which was not shown before.

  The quotient package presented in this paper has the following

  advantages over existing packages:

  \begin{itemize}

  \item We define quotient composition, function map composition and

    relation map composition. This lets lifting polymorphic types with

    subtypes quotiented as well. We extend the notions of

    respectfulness and preservation to cope with quotient

    composition.

  \item We allow lifting only some occurrences of quotiented

    types. Rsp/Prs extended. (used in nominal)

  \item The quotient package is very modular. Definitions can be added

  \item Can be used both manually (attribute, separate tactics,

    rsp/prs databases) and programatically (automated definition of

    lifted constants, the rsp proof obligations and theorem statement

    translation according to given quotients).

  \end{itemize}

*}

section {* Quotient Type*}

text {*

  In this section we present the definitions of a quotient that follow

  those by Homeier, the proofs can be found there.

  \begin{definition}[Quotient]

  A relation $R$ with an abstraction function $Abs$

  and a representation function $Rep$ is a \emph{quotient}

  if and only if:

  \begin{enumerate}

  \item @{thm (rhs1) Quotient_def[of "R", no_vars]}

  \item @{thm (rhs2) Quotient_def[of "R", no_vars]}

  \item @{thm (rhs3) Quotient_def[of "R", no_vars]}

  \end{enumerate}

  \end{definition}

  \begin{definition}[Relation map and function map]\\

  @{thm fun_rel_def[of "R1" "R2", no_vars]}\\

  @{thm fun_map_def[no_vars]}

  \end{definition}

  The main theorems for building higher order quotients is:

  \begin{lemma}[Function Quotient]

  If @{thm (prem 1) fun_quotient[no_vars]} and @{thm (prem 2) fun_quotient[no_vars]}

  then @{thm (concl) fun_quotient[no_vars]}

  \end{lemma}

*}

subsection {* Higher Order Logic *}

text {*

  Types:

  \begin{eqnarray}\nonumber

  @{text "\<sigma> ::="} & @{text "\<alpha>"} & \textrm{(type variable)} \\ \nonumber

      @{text "|"} & @{text "(\<sigma>,\<dots>,\<sigma>)\<kappa>"} & \textrm{(type construction)}

  \end{eqnarray}

  Terms:

  \begin{eqnarray}\nonumber

  @{text "t ::="} & @{text "x\<^isup>\<sigma>"} & \textrm{(variable)} \\ \nonumber

      @{text "|"} & @{text "c\<^isup>\<sigma>"} & \textrm{(constant)} \\ \nonumber

      @{text "|"} & @{text "t t"} & \textrm{(application)} \\ \nonumber

      @{text "|"} & @{text "\<lambda>x\<^isup>\<sigma>. t"} & \textrm{(abstraction)} \\ \nonumber

  \end{eqnarray}

*}

section {* Constants *}

(* Say more about containers? *)

text {*

  To define a constant on the lifted type, an aggregate abstraction

  function is applied to the raw constant. Below we describe the operation

  that generates

  an aggregate @{term "Abs"} or @{term "Rep"} function given the

  compound raw type and the compound quotient type.

  This operation will also be used in translations of theorem statements

  and in the lifting procedure.

  The operation is additionally able to descend into types for which

  maps are known. Such maps for most common types (list, pair, sum,

  option, \ldots) are described in Homeier, and we assume that @{text "map"}

  is the function that returns a map for a given type. Then REP/ABS is defined

  as follows:

  \begin{itemize}

  \item @{text "ABS(\<alpha>\<^isub>1, \<alpha>\<^isub>2)"} = @{text "id"}

  \item @{text "REP(\<alpha>\<^isub>1, \<alpha>\<^isub>2)"}  =  @{text "id"}

  \item @{text "ABS(\<sigma>, \<sigma>)"}  =  @{text "id"}

  \item @{text "REP(\<sigma>, \<sigma>)"}  =  @{text "id"}

  \item @{text "ABS(\<sigma>\<^isub>1\<rightarrow>\<sigma>\<^isub>2,\<tau>\<^isub>1\<rightarrow>\<tau>\<^isub>2)"}  =  @{text "REP(\<sigma>\<^isub>1,\<tau>\<^isub>1) ---> ABS(\<sigma>\<^isub>2,\<tau>\<^isub>2)"}

  \item @{text "REP(\<sigma>\<^isub>1\<rightarrow>\<sigma>\<^isub>2,\<tau>\<^isub>1\<rightarrow>\<tau>\<^isub>2)"}  =  @{text "ABS(\<sigma>\<^isub>1,\<tau>\<^isub>1) ---> REP(\<sigma>\<^isub>2,\<tau>\<^isub>2)"}

  \item @{text "ABS((\<sigma>\<^isub>1,\<dots>,\<sigma>\<^isub>n))\<kappa>, (\<tau>\<^isub>1,\<dots>,\<tau>\<^isub>n))\<kappa>)"}  =  @{text "(map \<kappa>) (ABS(\<sigma>\<^isub>1,\<tau>\<^isub>1)) \<dots> (ABS(\<sigma>\<^isub>n,\<tau>\<^isub>n))"}

  \item @{text "REP((\<sigma>\<^isub>1,\<dots>,\<sigma>\<^isub>n))\<kappa>, (\<tau>\<^isub>1,\<dots>,\<tau>\<^isub>n))\<kappa>)"}  =  @{text "(map \<kappa>) (REP(\<sigma>\<^isub>1,\<tau>\<^isub>1)) \<dots> (REP(\<sigma>\<^isub>n,\<tau>\<^isub>n))"}

  \item @{text "ABS((\<sigma>\<^isub>1,\<dots>,\<sigma>\<^isub>n))\<kappa>\<^isub>1, (\<tau>\<^isub>1,\<dots>,\<tau>\<^isub>m))\<kappa>\<^isub>2)"}  =  @{text "Abs_\<kappa>\<^isub>2 \<circ> (map \<kappa>\<^isub>1) (ABS(\<rho>\<^isub>1,\<nu>\<^isub>1) \<dots> (ABS(\<rho>\<^isub>p,\<nu>\<^isub>p)"} provided @{text "\<eta> \<kappa>\<^isub>2 = (\<alpha>\<^isub>1\<dots>\<alpha>\<^isub>p)\<kappa>\<^isub>1 \<and> \<exists>s. s(\<sigma>s\<kappa>\<^isub>1)=\<rho>s\<kappa>\<^isub>1 \<and> s(\<tau>s\<kappa>\<^isub>2)=\<nu>s\<kappa>\<^isub>2"}

  \item @{text "REP((\<sigma>\<^isub>1,\<dots>,\<sigma>\<^isub>n))\<kappa>\<^isub>1, (\<tau>\<^isub>1,\<dots>,\<tau>\<^isub>m))\<kappa>\<^isub>2)"}  =  @{text "(map \<kappa>\<^isub>1) (REP(\<rho>\<^isub>1,\<nu>\<^isub>1) \<dots> (REP(\<rho>\<^isub>p,\<nu>\<^isub>p) \<circ> Rep_\<kappa>\<^isub>2"} provided @{text "\<eta> \<kappa>\<^isub>2 = (\<alpha>\<^isub>1\<dots>\<alpha>\<^isub>p)\<kappa>\<^isub>1 \<and> \<exists>s. s(\<sigma>s\<kappa>\<^isub>1)=\<rho>s\<kappa>\<^isub>1 \<and> s(\<tau>s\<kappa>\<^isub>2)=\<nu>s\<kappa>\<^isub>2"}

  \end{itemize}

  Apart from the last 2 points the definition is same as the one implemented in

  in Homeier's HOL package, below is the definition of @{term fconcat}

  that shows the last points:

  @{thm fconcat_def[no_vars]}

  The aggregate @{term Abs} function takes a finite set of finite sets

  and applies @{term "map rep_fset"} composed with @{term rep_fset} to

  its input, obtaining a list of lists, passes the result to @{term concat}

  obtaining a list and applies @{term abs_fset} obtaining the composed

  finite set.

*}

subsection {* Respectfulness *}

text {*

  A respectfulness lemma for a constant states that the equivalence

  class returned by this constant depends only on the equivalence

  classes of the arguments applied to the constant. This can be

  expressed in terms of an aggregate relation between the constant

  and itself, for example the respectfullness for @{term "append"}

  can be stated as:

  @{thm [display] append_rsp[no_vars]}

  \noindent

  Which is equivalent to:

  @{thm [display] append_rsp_unfolded[no_vars]}

  Below we show the algorithm for finding the aggregate relation.

  This algorithm uses

  the relation composition which we define as:

  \begin{definition}[Composition of Relations]

  @{abbrev "rel_conj R1 R2"} where @{text OO} is the predicate

  composition @{thm pred_compI[no_vars]}

  \end{definition}

  Given an aggregate raw type and quotient type:

  \begin{itemize}

  \item For equal types or free type variables return equality

  \item For equal type constructors use the appropriate rel

    function applied to the results for the argument pairs

  \item For unequal type constructors, look in the quotients information

    for a quotient type that matches the type constructor, and instantiate

    the type appropriately getting back an instantiation environment. We

    apply the environment to the arguments and recurse composing it with

    the aggregate relation function.

  \end{itemize}

  Again, the the behaviour of our algorithm in the last situation is

  novel, so lets look at the example of respectfullness for @{term concat}.

  The statement as computed by the algorithm above is:

  @{thm [display] concat_rsp[no_vars]}

  \noindent

  By unfolding the definition of relation composition and relation map

  we can see the equivalent statement just using the primitive list

  equivalence relation:

  @{thm [display] concat_rsp_unfolded[of "a" "a'" "b'" "b", no_vars]}

  The statement reads that, for any lists of lists @{term a} and @{term b}

  if there exist intermediate lists of lists @{term "a'"} and @{term "b'"}

  such that each element of @{term a} is in the relation with an appropriate

  element of @{term a'}, @{term a'} is in relation with @{term b'} and each

  element of @{term b'} is in relation with the appropriate element of

  @{term b}.

*}

subsection {* Preservation *}

text {*

  To be able to lift theorems that talk about constants that are not

  lifted but whose type changes when lifting is performed additionally

  preservation theorems are needed.

  To lift theorems that talk about insertion in lists of lifted types

  we need to know that for any quotient type with the abstraction and

  representation functions @{text "Abs"} and @{text Rep} we have:

  @{thm [display] (concl) cons_prs[no_vars]}

  This is not enough to lift theorems that talk about quotient compositions.

  For some constants (for example empty list) it is possible to show a

  general compositional theorem, but for @{term "op #"} it is necessary

  to show that it respects the particular quotient type:

  @{thm [display] insert_preserve2[no_vars]}

*}

subsection {* Composition of Quotient theorems *}

text {*

  Given two quotients, one of which quotients a container, and the

  other quotients the type in the container, we can write the

  composition of those quotients. To compose two quotient theorems

  we compose the relations with relation composition

  and the abstraction and relation functions with function composition.

  The @{term "Rep"} and @{term "Abs"} functions that we obtain are

  the same as the ones created by in the aggregate functions and the

  relation is the same as the one given by aggregate relations.

  This becomes especially interesting

  when we compose the quotient with itself, as there is no simple

  intermediate step.

  Lets take again the example of @{term concat}. To be able to lift

  theorems that talk about it we will first prove the composition

  quotient theorems, which then lets us perform the lifting procedure

  in an unchanged way:

  @{thm [display] quotient_compose_list[no_vars]}

*}

section {* Lifting Theorems *}

text {*

  The core of the quotient package takes an original theorem that

  talks about the raw types, and the statement of the theorem that

  it is supposed to produce. This is different from other existing

  quotient packages, where only the raw theorems was necessary.

  We notice that in some cases only some occurrences of the raw

  types need to be lifted. This is for example the case in the

  new Nominal package, where a raw datatype that talks about

  pairs of natural numbers or strings (being lists of characters)

  should not be changed to a quotient datatype with constructors

  taking integers or finite sets of characters. To simplify the

  use of the quotient package we additionally provide an automated

  statement translation mechanism that replaces occurrences of

  types that match given quotients by appropriate lifted types.

  Lifting the theorems is performed in three steps. In the following

  we call these steps \emph{regularization}, \emph{injection} and

  \emph{cleaning} following the names used in Homeier's HOL

  implementation.

  We first define the statement of the regularized theorem based

  on the original theorem and the goal theorem. Then we define

  the statement of the injected theorem, based on the regularized

  theorem and the goal. We then show the 3 proofs, and all three

  can be performed independently from each other.

*}

subsection {* Regularization and Injection statements *}

text {*

  The function that gives the statement of the regularized theorem

  takes the statement of the raw theorem (a term) and the statement

  of the lifted theorem. The intuition behind the procedure is that

  it replaces quantifiers and abstractions involving raw types

  by bounded ones, and equalities involving raw types are replaced

  by appropriate aggregate relations. It is defined as follows:

  \begin{itemize}

  \item @{text "REG (\<lambda>x : \<sigma>. t, \<lambda>x : \<sigma>. s) = \<lambda>x : \<sigma>. REG (t, s)"}

  \item @{text "REG (\<lambda>x : \<sigma>. t, \<lambda>x : \<tau>. s) = \<lambda>x : \<sigma> \<in> Res (REL (\<sigma>, \<tau>)). REG (t, s)"}

  \item @{text "REG (\<forall>x : \<sigma>. t, \<forall>x : \<sigma>. s) = \<forall>x : \<sigma>. REG (t, s)"}

  \item @{text "REG (\<forall>x : \<sigma>. t, \<forall>x : \<tau>. s) = \<forall>x : \<sigma> \<in> Res (REL (\<sigma>, \<tau>)). REG (t, s)"}

  \item @{text "REG ((op =) : \<sigma>, (op =) : \<sigma>) = (op =) : \<sigma>"}

  \item @{text "REG ((op =) : \<sigma>, (op =) : \<tau>) = REL (\<sigma>, \<tau>) : \<sigma>"}

  \item @{text "REG (t\<^isub>1 t\<^isub>2, s\<^isub>1 s\<^isub>2) = REG (t\<^isub>1, s\<^isub>1) REG (t\<^isub>2, s\<^isub>2)"}

  \item @{text "REG (v\<^isub>1, v\<^isub>2) = v\<^isub>1"}

  \item @{text "REG (c\<^isub>1, c\<^isub>2) = c\<^isub>1"}

  \end{itemize}

  Existential quantifiers and unique existential quantifiers are defined

  similarily to the universal one.

  The function that gives the statment of the injected theorem

  takes the statement of the regularized theorems and the statement

  of the lifted theorem both as terms.

  \begin{itemize}

  \item @{text "INJ ((\<lambda>x. t) : \<sigma>, (\<lambda>x. s) : \<sigma>) = \<lambda>x. (INJ (t, s)"}

  \item @{text "INJ ((\<lambda>x. t) : \<sigma>, (\<lambda>x. s) : \<tau>) = REP(\<sigma>,\<tau>) (ABS (\<sigma>,\<tau>) (\<lambda>x. (INJ (t, s))))"}

  \item @{text "INJ ((\<lambda>x \<in> R. t) : \<sigma>, (\<lambda>x. s) : \<tau>) = REP(\<sigma>,\<tau>) (ABS (\<sigma>,\<tau>) (\<lambda>x \<in> R. (INJ (t, s))))"}

  \item @{text "INJ (\<forall> t, \<forall> s) = \<forall> (INJ (t, s)"}

  \item @{text "INJ (\<forall> t \<in> R, \<forall> s) = \<forall> (INJ (t, s) \<in> R"}

  \item @{text "INJ (t\<^isub>1 t\<^isub>2, s\<^isub>1 s\<^isub>2) = INJ (t\<^isub>1, s\<^isub>1) INJ (t\<^isub>2, s\<^isub>2)"}

  \item @{text "INJ (v\<^isub>1 : \<sigma>, v\<^isub>2 : \<sigma>) = v\<^isub>1"}

  \item @{text "INJ (v\<^isub>1 : \<sigma>, v\<^isub>2 : \<tau>) = REP(\<sigma>,\<tau>) (ABS (\<sigma>,\<tau>) (v\<^isub>1))"}

  \item @{text "INJ (c\<^isub>1 : \<sigma>, c\<^isub>2 : \<sigma>) = c\<^isub>1"}

  \item @{text "INJ (c\<^isub>1 : \<sigma>, c\<^isub>2 : \<tau>) = REP(\<sigma>,\<tau>) (ABS (\<sigma>,\<tau>) (c\<^isub>1))"}

  \end{itemize}

  For existential quantifiers and unique existential quantifiers it is

  defined similarily to the universal one.

*}

subsection {* Proof of Regularization *}

text {*

  Example of non-regularizable theorem ($0 = 1$).

  Separtion of regularization from injection thanks to the following 2 lemmas:

  \begin{lemma}

  If @{term R2} is an equivalence relation, then:

  \begin{eqnarray}

  @{thm (rhs) ball_reg_eqv_range[no_vars]} & = & @{thm (lhs) ball_reg_eqv_range[no_vars]}\\

  @{thm (rhs) bex_reg_eqv_range[no_vars]} & = & @{thm (lhs) bex_reg_eqv_range[no_vars]}

  \end{eqnarray}

  \end{lemma}

  Other lemmas used in regularization:

  @{thm [display] ball_reg_eqv[no_vars]}

  @{thm [display] babs_reg_eqv[no_vars]}

  @{thm [display] babs_simp[no_vars]}

  @{thm [display] ball_reg_right[no_vars]}

  @{thm [display] bex_reg_left[no_vars]}

  @{thm [display] bex1_bexeq_reg[no_vars]}

subsection {* Injection *}

text {*

  The 2 key lemmas are:

  @{thm [display] apply_rsp[no_vars]}

  @{thm [display] rep_abs_rsp[no_vars]}

*}

subsection {* Cleaning *}

text {* Preservation of quantifiers, abstractions, relations, quotient-constants

  (definitions) and user given constant preservation lemmas *}

section {* Examples *}

section {* Related Work *}

text {*

  \begin{itemize}

  \item Peter Homeier's package~\cite{Homeier05} (and related work from there)

  \item John Harrison's one~\cite{harrison-thesis} is the first one to lift theorems

    but only first order.

  \item PVS~\cite{PVS:Interpretations}

  \item MetaPRL~\cite{Nogin02}

  \item Manually defined quotients in Isabelle/HOL Library (Markus's Quotient\_Type,