From 80c3bb9e9695ef6fae0d95a474bbf1fbe2bc3329 Mon Sep 17 00:00:00 2001 From: Christoph Fuerst Date: Sun, 2 Apr 2017 20:37:32 +0200 Subject: [PATCH] Divided into seperate files --- integerroot.txt | 29 ++++++++++++++++++++++ lefttoright.txt | 40 ++++++++++++++++++++++++++++++ report/formal.pdf | Bin 368967 -> 297554 bytes report/formal.tex | 72 ++++++++++++++++++++++++++++++++---------------------- 4 files changed, 112 insertions(+), 29 deletions(-) create mode 100644 integerroot.txt create mode 100644 lefttoright.txt diff --git a/integerroot.txt b/integerroot.txt new file mode 100644 index 0000000..e372df0 --- /dev/null +++ b/integerroot.txt @@ -0,0 +1,29 @@ +// +// Computing the Integer-Square-Root of a non-negative integer a +// Author: Christoph Fuerst (2017) + +val K: ℕ; + +type nat = ℕ[K]; + +//////////////////////////////////////////////////////////////////// +// Algorithm: Integer-Square-Root +//////////////////////////////////////////////////////////////////// +proc integerroot(a:nat): nat + ensures result^2 ≤ a ∧ a < (result+1)^2; +{ + var x:nat = 0; + var y:ℕ[(K+1)^2] = 1; + var z:ℕ[(K+1)^2] = 1; + + while y ≤ a do + invariant x^2 ≤ a ∧ y = x^2+z ∧ z = 2*x+1; + { + x=x+1; + z=z+2; + y=y+z; + } + + return x; +} + diff --git a/lefttoright.txt b/lefttoright.txt new file mode 100644 index 0000000..285ec7b --- /dev/null +++ b/lefttoright.txt @@ -0,0 +1,40 @@ +// +// Computing Powers by Left-To-Right Exponentation +// Author: Christoph Fuerst (2017) + +val N: ℕ; +val M: ℕ; + +type Base = ℕ[N]; +type Exp = ℕ[M]; +type Result = ℕ[N^M]; + +//////////////////////////////////////////////////////////////////// +// Algorithm: Calculate x^n by Left-To-Right Exponentation +//////////////////////////////////////////////////////////////////// +proc LeftToRightExponentation(x:Base,n:Exp): Result + ensures result = x^n; +{ + var res:Result = 1; + var locn:Exp = n; + var locx:ℕ[N^(2*M)] = x; + while locn > 0 do + invariant (n ≥ locn) ∧ (locn ≥ 0) ∧ + (x^n = res*(locx^locn)) ∧ + (old_locx ≤ locx) ∧ + (res = 0 ∨ old_res ≤ res) ∧ + (((locn%2 = 1) ⇒ (old_res < res)) ∨ + ((locn%2 = 0) ⇒ (old_res = res))); + decreases locn; + { + if locn%2 = 1 then + { + res = locx*res; + } + locx = locx^2; + locn = locn/2; + } + return res; +} + + diff --git a/report/formal.pdf b/report/formal.pdf index 8af4496cbfe420f645c0ace6377fc36c20f18919..a3781aa42e03167d2f86b82137b4d2825acae24d 100644 GIT binary patch delta 106994 zcmV()K;OT|!WPn$5|AVTFqh#W1Sx;rSW9!;HW0q&SMUhUgbZE;$)oGqtvha~V{6(& z(gRJy5@#qvLqe&YU!Pqps8^IY$t91A2rhOPyWhS*@nn(kq-Yws3AymR=v+fQy3N zKCO`JZkEEY>)M9)&)sUh@e^`Dc*Apkb9H+TPeGE`ZT~J<5TcSRfX!r53bNpAJ+vK( z%9kV{xhM`m+JPlZQe~BqhOiPM6M$Q)A| zDFExY=+=}1Gr=ZbfIDOXu+-hc#pnmRU)DbE4DqDRXzyt=gdDfl>#hZ2z&;p_24_{C z$Efa70u6jZv$I*MwP5F6J&uHM&d{bE>qI>;Nood;dfK5tXfV>!VpYQue%c0HMr-(? zA?Ud6znvIw1SUxcRC>y`DS8O57657=7V5x%-0+U*qwC(HgpFjdAPNkTbl;B6W`uA7 z*rv?cdG?dzenh&$&o0bl`=J*{FYAkER!_TztM;u~$VUE99F$Wp$+ 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$$ hence we have shown $x^2\leq a < (x+1)^2$, and taking square roots shows the claim. +\newpage +\subsection{The RISCAL Formalization} +%{\scriptsize \verbatiminput{../integerroot.txt}} +{\scriptsize \verbatimboxed{../integerroot.txt}} \appendix -\section{Listing of the developed Theory} -{\scriptsize \verbatiminput{../numbertheory.txt}} - - -\section{Listing of Implementations} -\lstset{language=C++, - backgroundcolor=\color{black!5}, % set backgroundcolor - basicstyle=\ttfamily\tiny,% basic font setting - breakatwhitespace=false, % sets if automatic breaks should only happen at whitespace - breaklines=true, % sets automatic line breaking - captionpos=b, % sets the caption-position to bottom - deletekeywords={...}, % if you want to delete keywords from the given language - escapeinside={\%*}{*)}, % if you want to add LaTeX within your code - extendedchars=true, % lets you use non-ASCII characters; for 8-bits encodings only, does not work with UTF-8 - frame=single, % adds a frame around the code - keepspaces=true, % keeps spaces in text, useful for keeping indentation of code (possibly needs columns=flexible) - language=Octave, % the language of the code - morekeywords={*,...}, % if you want to add more keywords to the set - numbers=left, % where to put the line-numbers; possible values are (none, left, right) - numbersep=5pt, % how far the line-numbers are from the code - showspaces=false, % show spaces everywhere adding particular underscores; it overrides 'showstringspaces' - showstringspaces=false, % underline spaces within strings only - showtabs=false, % show tabs within strings adding particular underscores - stepnumber=1, % the step between two line-numbers. If it's 1, each line will be numbered - tabsize=2, % sets default tabsize to 2 spaces -} - -{\tiny \lstinputlisting{../src/primefactors.cpp}} + +% \section{Listing of Implementations} +% \lstset{language=C++, +% backgroundcolor=\color{black!5}, % set backgroundcolor +% basicstyle=\ttfamily\tiny,% basic font setting +% breakatwhitespace=false, % sets if automatic breaks should only happen at whitespace +% breaklines=true, % sets automatic line breaking +% captionpos=b, % sets the caption-position to bottom +% deletekeywords={...}, % if you want to delete keywords from the given language +% escapeinside={\%*}{*)}, % if you want to add LaTeX within your code +% extendedchars=true, % lets you use non-ASCII characters; for 8-bits encodings only, does not work with UTF-8 +% frame=single, % adds a frame around the code +% keepspaces=true, % keeps spaces in text, useful for keeping indentation of code (possibly needs columns=flexible) +% language=Octave, % the language of the code +% morekeywords={*,...}, % if you want to add more keywords to the set +% numbers=left, % where to put the line-numbers; possible values are (none, left, right) +% numbersep=5pt, % how far the line-numbers are from the code +% showspaces=false, % show spaces everywhere adding particular underscores; it overrides 'showstringspaces' +% showstringspaces=false, % underline spaces within strings only +% showtabs=false, % show tabs within strings adding particular underscores +% stepnumber=1, % the step between two line-numbers. If it's 1, each line will be numbered +% tabsize=2, % sets default tabsize to 2 spaces +% } +% +% {\tiny \lstinputlisting{../src/primefactors.cpp}} \begin{thebibliography}{9} \bibitem{hardylittlewood} -- 2.1.4