25 section {* A Brief Overview about the Nominal Logic Work *}

    32 section {* A Short Review of the Nominal Logic Work *}

    33 

    34 text {*

    35   At its core, Nominal Isabelle is based on the nominal logic work by Pitts

    36   \cite{Pitts03}. The central notions in this work are sorted atoms and

    37   permutations of atoms. The sorted atoms represent different

    38   kinds of variables, such as term- and type-variables in Core-Haskell, and it

    39   is assumed that there is an infinite supply of atoms for each sort. 

    40   However, in order to simplify the description of our work, we shall

    41   assume in this paper that there is only a single sort of atoms.

    42 

    43   Permutations are bijective functions from atoms to atoms that are 

    44   the identity everywhere except on a finite number of atoms. There is a 

    45   two-place permutation operation written

    46 

    47   @{text [display,indent=5] "_ \<bullet> _  ::  (\<alpha> \<times> \<alpha>) list \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}

    48 

    49   \noindent 

    50   with a generic type in which @{text "\<alpha>"} stands for the type of atoms 

    51   and @{text "\<beta>"} for the type of the objects on which the permutation 

    52   acts. In Nominal Isabelle the identity permutation is written as @{term "0::perm"},

    53   the composition of two permutations @{term p} and @{term q} as \mbox{@{term "p + q"}} 

    54   and the inverse permutation @{term p} as @{text "- p"}. The permutation

    55   operation is defined for products, lists, sets, functions, booleans etc 

    56   (see \cite{HuffmanUrban10}).

    57 

    58   The most original aspect of the nominal logic work of Pitts et al is a general

    59   definition for ``the set of free variables of an object @{text "x"}''.  This

    60   definition is general in the sense that it applies not only to lambda-terms,

    61   but also to lists, products, sets and even functions. The definition depends

    62   only on the permutation operation and on the notion of equality defined for

    63   the type of @{text x}, namely:

    64 

    65   @{thm [display,indent=5] supp_def[no_vars, THEN eq_reflection]}

    66 

    67   \noindent

    68   There is also the derived notion for when an atom @{text a} is \emph{fresh}

    69   for an @{text x}, defined as

    70   

    71   @{thm [display,indent=5] fresh_def[no_vars]}

    72 

    73   \noindent

    74   A striking consequence of these definitions is that we can prove

    75   without knowing anything about the structure of @{term x} that

    76   swapping two fresh atoms, say @{text a} and @{text b}, leave 

    77   @{text x} unchanged. For the proof we use the following lemma 

    78   about swappings applied to an @{text x}:

    79 

    80 *}

    81 

    34   Acknowledgements: We thank Andrew Pitts for the many discussions

    92 

    93   \noindent

    94   {\bf Acknowledgements:} We thank Andrew Pitts for the many discussions