(*<*)

theory Paper

imports "Quotient" 

        "LaTeXsugar"

begin

notation (latex output)

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

(*>*)

section {* Introduction *}

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

    respectfullness and preservation to cope with quotient

    composition.

  \item We allow lifting only some occurrences of quotiented

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

  \item We regularize more thanks to new lemmas. (inductions in

    nominal).

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

    separately, rsp and prs can be proved separately and theorems can

    be lifted on a need basis. (useful with type-classes).

  \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 {*

  Defintion of quotient,

  Equivalence,

  Relation map and function map

*}

section {* Constants *}

text {*

  Rep and Abs, Rsp and Prs

*}

section {* Lifting Theorems *}

text {* TBD *}

text {* Why providing a statement to prove is necessary is some cases *}

subsection {* Regularization *}

text {*

Transformation of the theorem statement:

\begin{itemize}

\item Quantifiers and abstractions involving raw types replaced by bounded ones.

\item Equalities involving raw types replaced by bounded ones.

\end{itemize}

The procedure.

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

New regularization 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}

*}

subsection {* Injection *}

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 (and related work from there), John

    Harrison's one.

  \item Manually defined quotients in Isabelle/HOL Library (Larry's

    quotients, Markus's Quotient\_Type, Dixon's FSet, \ldots)

  \item Oscar Slotosch defines quotient-type automatically but no

    lifting.

  \item PER. And how to avoid it.

  \item Necessity of Hilbert Choice op.

  \item Setoids in Coq

  \end{itemize}

*}

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end

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