header {* Basic group theory *}
theory Group imports Main begin
text {*
\medskip\noindent The meta-level type system of Isabelle supports
\emph{intersections} and \emph{inclusions} of type classes. These
directly correspond to intersections and inclusions of type
predicates in a purely set theoretic sense. This is sufficient as a
means to describe simple hierarchies of structures. As an
illustration, we use the well-known example of semigroups, monoids,
general groups and Abelian groups.
*}
subsection {* Monoids and Groups *}
text {*
First we declare some polymorphic constants required later for the
signature parts of our structures.
*}
consts
times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<odot>" 70)
invers :: "'a \<Rightarrow> 'a" ("(_\<inv>)" [1000] 999)
one :: 'a ("\<one>")
text {*
\noindent Next we define class @{text monoid} of monoids with
operations @{text \<odot>} and @{text \<one>}. Note that multiple class
axioms are allowed for user convenience --- they simply represent
the conjunction of their respective universal closures.
*}
axclass monoid \<subseteq> type
assoc: "(x \<odot> y) \<odot> z = x \<odot> (y \<odot> z)"
left_unit: "\<one> \<odot> x = x"
right_unit: "x \<odot> \<one> = x"
text {*
\noindent So class @{text monoid} contains exactly those types
@{text \<tau>} where @{text "\<odot> \<Colon> \<tau> \<Rightarrow> \<tau> \<Rightarrow> \<tau>"} and @{text "\<one> \<Colon> \<tau>"}
are specified appropriately, such that @{text \<odot>} is associative and
@{text \<one>} is a left and right unit element for the @{text \<odot>}
operation.
*}
text {*
\medskip Independently of @{text monoid}, we now define a linear
hierarchy of semigroups, general groups and Abelian groups. Note
that the names of class axioms are automatically qualified with each
class name, so we may re-use common names such as @{text assoc}.
*}
axclass semigroup \<subseteq> type
assoc: "(x \<odot> y) \<odot> z = x \<odot> (y \<odot> z)"
axclass group \<subseteq> semigroup
left_unit: "\<one> \<odot> x = x"
left_inverse: "x\<inv> \<odot> x = \<one>"
axclass agroup \<subseteq> group
commute: "x \<odot> y = y \<odot> x"
text {*
\noindent Class @{text group} inherits associativity of @{text \<odot>}
from @{text semigroup} and adds two further group axioms. Similarly,
@{text agroup} is defined as the subset of @{text group} such that
for all of its elements @{text \<tau>}, the operation @{text "\<odot> \<Colon> \<tau> \<Rightarrow> \<tau> \<Rightarrow>
\<tau>"} is even commutative.
*}
subsection {* Abstract reasoning *}
text {*
In a sense, axiomatic type classes may be viewed as \emph{abstract
theories}. Above class definitions gives rise to abstract axioms
@{text assoc}, @{text left_unit}, @{text left_inverse}, @{text
commute}, where any of these contain a type variable @{text "'a \<Colon>
c"} that is restricted to types of the corresponding class @{text
c}. \emph{Sort constraints} like this express a logical
precondition for the whole formula. For example, @{text assoc}
states that for all @{text \<tau>}, provided that @{text "\<tau> \<Colon>
semigroup"}, the operation @{text "\<odot> \<Colon> \<tau> \<Rightarrow> \<tau> \<Rightarrow> \<tau>"} is associative.
\medskip From a technical point of view, abstract axioms are just
ordinary Isabelle theorems, which may be used in proofs without
special treatment. Such ``abstract proofs'' usually yield new
``abstract theorems''. For example, we may now derive the following
well-known laws of general groups.
*}
theorem group_right_inverse: "x \<odot> x\<inv> = (\<one>\<Colon>'a\<Colon>group)"
proof -
have "x \<odot> x\<inv> = \<one> \<odot> (x \<odot> x\<inv>)"
by (simp only: group_class.left_unit)
also have "... = \<one> \<odot> x \<odot> x\<inv>"
by (simp only: semigroup_class.assoc)
also have "... = (x\<inv>)\<inv> \<odot> x\<inv> \<odot> x \<odot> x\<inv>"
by (simp only: group_class.left_inverse)
also have "... = (x\<inv>)\<inv> \<odot> (x\<inv> \<odot> x) \<odot> x\<inv>"
by (simp only: semigroup_class.assoc)
also have "... = (x\<inv>)\<inv> \<odot> \<one> \<odot> x\<inv>"
by (simp only: group_class.left_inverse)
also have "... = (x\<inv>)\<inv> \<odot> (\<one> \<odot> x\<inv>)"
by (simp only: semigroup_class.assoc)
also have "... = (x\<inv>)\<inv> \<odot> x\<inv>"
by (simp only: group_class.left_unit)
also have "... = \<one>"
by (simp only: group_class.left_inverse)
finally show ?thesis .
qed
text {*
\noindent With @{text group_right_inverse} already available, @{text
group_right_unit}\label{thm:group-right-unit} is now established
much easier.
*}
theorem group_right_unit: "x \<odot> \<one> = (x\<Colon>'a\<Colon>group)"
proof -
have "x \<odot> \<one> = x \<odot> (x\<inv> \<odot> x)"
by (simp only: group_class.left_inverse)
also have "... = x \<odot> x\<inv> \<odot> x"
by (simp only: semigroup_class.assoc)
also have "... = \<one> \<odot> x"
by (simp only: group_right_inverse)
also have "... = x"
by (simp only: group_class.left_unit)
finally show ?thesis .
qed
text {*
\medskip Abstract theorems may be instantiated to only those types
@{text \<tau>} where the appropriate class membership @{text "\<tau> \<Colon> c"} is
known at Isabelle's type signature level. Since we have @{text
"agroup \<subseteq> group \<subseteq> semigroup"} by definition, all theorems of @{text
semigroup} and @{text group} are automatically inherited by @{text
group} and @{text agroup}.
*}
subsection {* Abstract instantiation *}
text {*
From the definition, the @{text monoid} and @{text group} classes
have been independent. Note that for monoids, @{text right_unit}
had to be included as an axiom, but for groups both @{text
right_unit} and @{text right_inverse} are derivable from the other
axioms. With @{text group_right_unit} derived as a theorem of group
theory (see page~\pageref{thm:group-right-unit}), we may now
instantiate @{text "monoid \<subseteq> semigroup"} and @{text "group \<subseteq>
monoid"} properly as follows (cf.\ \figref{fig:monoid-group}).
\begin{figure}[htbp]
\begin{center}
\small
\unitlength 0.6mm
\begin{picture}(65,90)(0,-10)
\put(15,10){\line(0,1){10}} \put(15,30){\line(0,1){10}}
\put(15,50){\line(1,1){10}} \put(35,60){\line(1,-1){10}}
\put(15,5){\makebox(0,0){@{text agroup}}}
\put(15,25){\makebox(0,0){@{text group}}}
\put(15,45){\makebox(0,0){@{text semigroup}}}
\put(30,65){\makebox(0,0){@{text type}}} \put(50,45){\makebox(0,0){@{text monoid}}}
\end{picture}
\hspace{4em}
\begin{picture}(30,90)(0,0)
\put(15,10){\line(0,1){10}} \put(15,30){\line(0,1){10}}
\put(15,50){\line(0,1){10}} \put(15,70){\line(0,1){10}}
\put(15,5){\makebox(0,0){@{text agroup}}}
\put(15,25){\makebox(0,0){@{text group}}}
\put(15,45){\makebox(0,0){@{text monoid}}}
\put(15,65){\makebox(0,0){@{text semigroup}}}
\put(15,85){\makebox(0,0){@{text type}}}
\end{picture}
\caption{Monoids and groups: according to definition, and by proof}
\label{fig:monoid-group}
\end{center}
\end{figure}
*}
instance monoid \<subseteq> semigroup
proof
fix x y z :: "'a\<Colon>monoid"
show "x \<odot> y \<odot> z = x \<odot> (y \<odot> z)"
by (rule monoid_class.assoc)
qed
instance group \<subseteq> monoid
proof
fix x y z :: "'a\<Colon>group"
show "x \<odot> y \<odot> z = x \<odot> (y \<odot> z)"
by (rule semigroup_class.assoc)
show "\<one> \<odot> x = x"
by (rule group_class.left_unit)
show "x \<odot> \<one> = x"
by (rule group_right_unit)
qed
text {*
\medskip The $\INSTANCE$ command sets up an appropriate goal that
represents the class inclusion (or type arity, see
\secref{sec:inst-arity}) to be proven (see also
\cite{isabelle-isar-ref}). The initial proof step causes
back-chaining of class membership statements wrt.\ the hierarchy of
any classes defined in the current theory; the effect is to reduce
to the initial statement to a number of goals that directly
correspond to any class axioms encountered on the path upwards
through the class hierarchy.
*}
subsection {* Concrete instantiation \label{sec:inst-arity} *}
text {*
So far we have covered the case of the form $\INSTANCE$~@{text
"c\<^sub>1 \<subseteq> c\<^sub>2"}, namely \emph{abstract instantiation} ---
$c@1$ is more special than @{text "c\<^sub>1"} and thus an instance
of @{text "c\<^sub>2"}. Even more interesting for practical
applications are \emph{concrete instantiations} of axiomatic type
classes. That is, certain simple schemes @{text "(\<alpha>\<^sub>1, \<dots>,
\<alpha>\<^sub>n) t \<Colon> c"} of class membership may be established at the
logical level and then transferred to Isabelle's type signature
level.
\medskip As a typical example, we show that type @{typ bool} with
exclusive-or as @{text \<odot>} operation, identity as @{text \<inv>}, and
@{term False} as @{text \<one>} forms an Abelian group.
*}
defs (overloaded)
times_bool_def: "x \<odot> y \<equiv> x \<noteq> (y\<Colon>bool)"
inverse_bool_def: "x\<inv> \<equiv> x\<Colon>bool"
unit_bool_def: "\<one> \<equiv> False"
text {*
\medskip It is important to note that above $\DEFS$ are just
overloaded meta-level constant definitions, where type classes are
not yet involved at all. This form of constant definition with
overloading (and optional recursion over the syntactic structure of
simple types) are admissible as definitional extensions of plain HOL
\cite{Wenzel:1997:TPHOL}. The Haskell-style type system is not
required for overloading. Nevertheless, overloaded definitions are
best applied in the context of type classes.
\medskip Since we have chosen above $\DEFS$ of the generic group
operations on type @{typ bool} appropriately, the class membership
@{text "bool \<Colon> agroup"} may be now derived as follows.
*}
instance bool :: agroup
proof (intro_classes,
unfold times_bool_def inverse_bool_def unit_bool_def)
fix x y z
show "((x \<noteq> y) \<noteq> z) = (x \<noteq> (y \<noteq> z))" by blast
show "(False \<noteq> x) = x" by blast
show "(x \<noteq> x) = False" by blast
show "(x \<noteq> y) = (y \<noteq> x)" by blast
qed
text {*
The result of an $\INSTANCE$ statement is both expressed as a
theorem of Isabelle's meta-logic, and as a type arity of the type
signature. The latter enables type-inference system to take care of
this new instance automatically.
\medskip We could now also instantiate our group theory classes to
many other concrete types. For example, @{text "int \<Colon> agroup"}
(e.g.\ by defining @{text \<odot>} as addition, @{text \<inv>} as negation
and @{text \<one>} as zero) or @{text "list \<Colon> (type) semigroup"}
(e.g.\ if @{text \<odot>} is defined as list append). Thus, the
characteristic constants @{text \<odot>}, @{text \<inv>}, @{text \<one>}
really become overloaded, i.e.\ have different meanings on different
types.
*}
subsection {* Lifting and Functors *}
text {*
As already mentioned above, overloading in the simply-typed HOL
systems may include recursion over the syntactic structure of types.
That is, definitional equations @{text "c\<^sup>\<tau> \<equiv> t"} may also
contain constants of name @{text c} on the right-hand side --- if
these have types that are structurally simpler than @{text \<tau>}.
This feature enables us to \emph{lift operations}, say to Cartesian
products, direct sums or function spaces. Subsequently we lift
@{text \<odot>} component-wise to binary products @{typ "'a \<times> 'b"}.
*}
defs (overloaded)
times_prod_def: "p \<odot> q \<equiv> (fst p \<odot> fst q, snd p \<odot> snd q)"
text {*
It is very easy to see that associativity of @{text \<odot>} on @{typ 'a}
and @{text \<odot>} on @{typ 'b} transfers to @{text \<odot>} on @{typ "'a \<times>
'b"}. Hence the binary type constructor @{text \<odot>} maps semigroups
to semigroups. This may be established formally as follows.
*}
instance * :: (semigroup, semigroup) semigroup
proof (intro_classes, unfold times_prod_def)
fix p q r :: "'a\<Colon>semigroup \<times> 'b\<Colon>semigroup"
show
"(fst (fst p \<odot> fst q, snd p \<odot> snd q) \<odot> fst r,
snd (fst p \<odot> fst q, snd p \<odot> snd q) \<odot> snd r) =
(fst p \<odot> fst (fst q \<odot> fst r, snd q \<odot> snd r),
snd p \<odot> snd (fst q \<odot> fst r, snd q \<odot> snd r))"
by (simp add: semigroup_class.assoc)
qed
text {*
Thus, if we view class instances as ``structures'', then overloaded
constant definitions with recursion over types indirectly provide
some kind of ``functors'' --- i.e.\ mappings between abstract
theories.
*}
end