(* Title: HOL/Isar_examples/Group.thy
ID: $Id$
Author: Markus Wenzel, TU Muenchen
*)
header {* Basic group theory *}
theory Group imports Main begin
subsection {* Groups and calculational reasoning *}
text {*
Groups over signature $({\times} :: \alpha \To \alpha \To \alpha,
\idt{one} :: \alpha, \idt{inverse} :: \alpha \To \alpha)$ are defined
as an axiomatic type class as follows. Note that the parent class
$\idt{times}$ is provided by the basic HOL theory.
*}
consts
one :: "'a"
inverse :: "'a => 'a"
axclass
group < times
group_assoc: "(x * y) * z = x * (y * z)"
group_left_one: "one * x = x"
group_left_inverse: "inverse x * x = one"
text {*
The group axioms only state the properties of left one and inverse,
the right versions may be derived as follows.
*}
theorem group_right_inverse: "x * inverse x = (one::'a::group)"
proof -
have "x * inverse x = one * (x * inverse x)"
by (simp only: group_left_one)
also have "... = one * x * inverse x"
by (simp only: group_assoc)
also have "... = inverse (inverse x) * inverse x * x * inverse x"
by (simp only: group_left_inverse)
also have "... = inverse (inverse x) * (inverse x * x) * inverse x"
by (simp only: group_assoc)
also have "... = inverse (inverse x) * one * inverse x"
by (simp only: group_left_inverse)
also have "... = inverse (inverse x) * (one * inverse x)"
by (simp only: group_assoc)
also have "... = inverse (inverse x) * inverse x"
by (simp only: group_left_one)
also have "... = one"
by (simp only: group_left_inverse)
finally show ?thesis .
qed
text {*
With \name{group-right-inverse} already available,
\name{group-right-one}\label{thm:group-right-one} is now established
much easier.
*}
theorem group_right_one: "x * one = (x::'a::group)"
proof -
have "x * one = x * (inverse x * x)"
by (simp only: group_left_inverse)
also have "... = x * inverse x * x"
by (simp only: group_assoc)
also have "... = one * x"
by (simp only: group_right_inverse)
also have "... = x"
by (simp only: group_left_one)
finally show ?thesis .
qed
text {*
\medskip The calculational proof style above follows typical
presentations given in any introductory course on algebra. The basic
technique is to form a transitive chain of equations, which in turn
are established by simplifying with appropriate rules. The low-level
logical details of equational reasoning are left implicit.
Note that ``$\dots$'' is just a special term variable that is bound
automatically to the argument\footnote{The argument of a curried
infix expression happens to be its right-hand side.} of the last fact
achieved by any local assumption or proven statement. In contrast to
$\var{thesis}$, the ``$\dots$'' variable is bound \emph{after} the
proof is finished, though.
There are only two separate Isar language elements for calculational
proofs: ``\isakeyword{also}'' for initial or intermediate
calculational steps, and ``\isakeyword{finally}'' for exhibiting the
result of a calculation. These constructs are not hardwired into
Isabelle/Isar, but defined on top of the basic Isar/VM interpreter.
Expanding the \isakeyword{also} and \isakeyword{finally} derived
language elements, calculations may be simulated by hand as
demonstrated below.
*}
theorem "x * one = (x::'a::group)"
proof -
have "x * one = x * (inverse x * x)"
by (simp only: group_left_inverse)
note calculation = this
-- {* first calculational step: init calculation register *}
have "... = x * inverse x * x"
by (simp only: group_assoc)
note calculation = trans [OF calculation this]
-- {* general calculational step: compose with transitivity rule *}
have "... = one * x"
by (simp only: group_right_inverse)
note calculation = trans [OF calculation this]
-- {* general calculational step: compose with transitivity rule *}
have "... = x"
by (simp only: group_left_one)
note calculation = trans [OF calculation this]
-- {* final calculational step: compose with transitivity rule ... *}
from calculation
-- {* ... and pick up the final result *}
show ?thesis .
qed
text {*
Note that this scheme of calculations is not restricted to plain
transitivity. Rules like anti-symmetry, or even forward and backward
substitution work as well. For the actual implementation of
\isacommand{also} and \isacommand{finally}, Isabelle/Isar maintains
separate context information of ``transitivity'' rules. Rule
selection takes place automatically by higher-order unification.
*}
subsection {* Groups as monoids *}
text {*
Monoids over signature $({\times} :: \alpha \To \alpha \To \alpha,
\idt{one} :: \alpha)$ are defined like this.
*}
axclass monoid < times
monoid_assoc: "(x * y) * z = x * (y * z)"
monoid_left_one: "one * x = x"
monoid_right_one: "x * one = x"
text {*
Groups are \emph{not} yet monoids directly from the definition. For
monoids, \name{right-one} had to be included as an axiom, but for
groups both \name{right-one} and \name{right-inverse} are derivable
from the other axioms. With \name{group-right-one} derived as a
theorem of group theory (see page~\pageref{thm:group-right-one}), we
may still instantiate $\idt{group} \subseteq \idt{monoid}$ properly
as follows.
*}
instance group < monoid
by (intro_classes,
rule group_assoc,
rule group_left_one,
rule group_right_one)
text {*
The \isacommand{instance} command actually is a version of
\isacommand{theorem}, setting up a goal that reflects the intended
class relation (or type constructor arity). Thus any Isar proof
language element may be involved to establish this statement. When
concluding the proof, the result is transformed into the intended
type signature extension behind the scenes.
*}
subsection {* More theorems of group theory *}
text {*
The one element is already uniquely determined by preserving an
\emph{arbitrary} group element.
*}
theorem group_one_equality: "e * x = x ==> one = (e::'a::group)"
proof -
assume eq: "e * x = x"
have "one = x * inverse x"
by (simp only: group_right_inverse)
also have "... = (e * x) * inverse x"
by (simp only: eq)
also have "... = e * (x * inverse x)"
by (simp only: group_assoc)
also have "... = e * one"
by (simp only: group_right_inverse)
also have "... = e"
by (simp only: group_right_one)
finally show ?thesis .
qed
text {*
Likewise, the inverse is already determined by the cancel property.
*}
theorem group_inverse_equality:
"x' * x = one ==> inverse x = (x'::'a::group)"
proof -
assume eq: "x' * x = one"
have "inverse x = one * inverse x"
by (simp only: group_left_one)
also have "... = (x' * x) * inverse x"
by (simp only: eq)
also have "... = x' * (x * inverse x)"
by (simp only: group_assoc)
also have "... = x' * one"
by (simp only: group_right_inverse)
also have "... = x'"
by (simp only: group_right_one)
finally show ?thesis .
qed
text {*
The inverse operation has some further characteristic properties.
*}
theorem group_inverse_times:
"inverse (x * y) = inverse y * inverse (x::'a::group)"
proof (rule group_inverse_equality)
show "(inverse y * inverse x) * (x * y) = one"
proof -
have "(inverse y * inverse x) * (x * y) =
(inverse y * (inverse x * x)) * y"
by (simp only: group_assoc)
also have "... = (inverse y * one) * y"
by (simp only: group_left_inverse)
also have "... = inverse y * y"
by (simp only: group_right_one)
also have "... = one"
by (simp only: group_left_inverse)
finally show ?thesis .
qed
qed
theorem inverse_inverse: "inverse (inverse x) = (x::'a::group)"
proof (rule group_inverse_equality)
show "x * inverse x = one"
by (simp only: group_right_inverse)
qed
theorem inverse_inject: "inverse x = inverse y ==> x = (y::'a::group)"
proof -
assume eq: "inverse x = inverse y"
have "x = x * one"
by (simp only: group_right_one)
also have "... = x * (inverse y * y)"
by (simp only: group_left_inverse)
also have "... = x * (inverse x * y)"
by (simp only: eq)
also have "... = (x * inverse x) * y"
by (simp only: group_assoc)
also have "... = one * y"
by (simp only: group_right_inverse)
also have "... = y"
by (simp only: group_left_one)
finally show ?thesis .
qed
end