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% appendix.tex %
% Principal lemmas %
% ----------------------------------------------------------------------------
\section{Principal lemmas}
\label{PrincipalLemmas}
Binomial coefficients, \verb@CHOOSE@:
\begin{session}
\begin{verbatim}
CHOOSE_LESS = |- !n k. n < k ==> (CHOOSE n k = 0)
CHOOSE_SAME = |- !n. CHOOSE n n = 1
\end{verbatim}
\end{session}
Algebraic laws:
\begin{session}
\begin{verbatim}
RIGHT_CANCELLATION =
|- !plus. Group plus ==> (!a b c. (plus b a = plus c a) ==> (b = c))
RING_0 =
|- !plus times.
RING(plus,times) ==>
(!a. times(Id plus)a = Id plus) /\ (!a. times a(Id plus) = Id plus)
\end{verbatim}
\end{session}
\newpage
Scalar powers:
\begin{session}
\begin{verbatim}
POWER_1 = |- !plus. MONOID plus ==> (POWER plus 1 a = a)
POWER_DISTRIB =
|- !plus times.
RING(plus,times) ==>
(!a b n. times a(POWER plus n b) = POWER plus n(times a b))
POWER_ADD =
|- !plus.
MONOID plus ==>
(!m n a. POWER plus(m + n)a = plus(POWER plus m a)(POWER plus n a))
\end{verbatim}
\end{session}
Reduction of lists:
\begin{session}
\begin{verbatim}
REDUCE_APPEND =
|- !plus.
MONOID plus ==>
(!as bs.
REDUCE plus(APPEND as bs) = plus(REDUCE plus as)(REDUCE plus bs))
REDUCE_MAP_MULT =
|- !plus times.
RING(plus,times) ==>
(!a bs. REDUCE plus(MAP(times a)bs) = times a(REDUCE plus bs))
REDUCE_MAP =
|- !plus.
MONOID plus /\ COMMUTATIVE plus ==>
(!f g as.
REDUCE plus(MAP(\k. plus(f k)(g k))as) =
plus(REDUCE plus(MAP f as))(REDUCE plus(MAP g as)))
\end{verbatim}
\end{session}
Subranges of the natural numbers:
\begin{session}
\begin{verbatim}
RANGE_TOP = |- !n m. RANGE m(SUC n) = APPEND(RANGE m n)[m + n]
RANGE_SHIFT = |- !n m. RANGE(SUC m)n = MAP SUC(RANGE m n)
\end{verbatim}
\end{session}
\newpage
$\Sigma$-notation:
\begin{session}
\begin{verbatim}
SIGMA_ID = |- !plus m f. SIGMA(plus,m,0)f = Id plus
SIGMA_LEFT_SPLIT =
|- !plus m n f. SIGMA(plus,m,SUC n)f = plus(f m)(SIGMA(plus,SUC m,n)f)
SIGMA_RIGHT_SPLIT =
|- !plus.
MONOID plus ==>
(!m n f. SIGMA(plus,m,SUC n)f = plus(SIGMA(plus,m,n)f)(f(m + n)))
SIGMA_SHIFT =
|- !plus m n f. SIGMA(plus,SUC m,n)f = SIGMA(plus,m,n)(f o SUC)
SIGMA_MULT_COMM =
|- !plus times.
RING(plus,times) ==>
(!m n a f.
times a(SIGMA(plus,m,n)f) = SIGMA(plus,m,n)((times a) o f))
SIGMA_PLUS_COMM =
|- !plus.
MONOID plus /\ COMMUTATIVE plus ==>
(!f g m n.
plus(SIGMA(plus,m,n)f)(SIGMA(plus,m,n)g) =
SIGMA(plus,m,n)(\k. plus(f k)(g k)))
SIGMA_EXT =
|- !plus.
MONOID plus ==>
(!f g m n.
(!k. k < n ==> (f(m + k) = g(m + k))) ==>
(SIGMA(plus,m,n)f = SIGMA(plus,m,n)g))
\end{verbatim}
\end{session}
Terms from the Binomial Theorem:
\begin{session}
\begin{verbatim}
BINTERM_0 =
|- !plus times.
RING(plus,times) ==>
(!a b n. BINTERM plus times a b n 0 = POWER times n a)
BINTERM_n =
|- !plus times.
RING(plus,times) ==>
(!a b n. BINTERM plus times a b n n = POWER times n b)
\end{verbatim}
\end{session}
|
