File:Inductive proofs of properties of add, mult from recursive definitions.pdf
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| Description |
English: Shows recursive definitions of addition (+) and multiplication (*) on natural numbers and inductive proofs of commutativity, associativity, distributivity by Peano induction; also indicates which property is used in the proof of which other one. |
| Date | |
| Source | Own work |
| Author | Jochen Burghardt |
| Other versions | File:Inductive proofs of properties of add, mult from recursive definitions svg.svg,File:Inductive proofs of properties of add, mult from recursive definitions (exercise version).pdf |
| LaTeX source code |
|---|
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\definecolor{fLb} {rgb}{0.70,0.50,0.50} % label
\definecolor{fCj} {rgb}{0.00,0.00,0.00} % conjecture
\definecolor{fPr} {rgb}{0.50,0.70,0.50} % proof
\definecolor{fRf} {rgb}{0.50,0.50,0.70} % reference
\definecolor{fEq} {rgb}{0.50,0.50,0.50} % proof equality
\definecolor{fLn} {rgb}{0.99,0.00,0.00} % "uses"-line
\definecolor{fLg} {rgb}{0.70,0.70,0.50} % legend
\newcommand{\lm}[1]{% % lemma
\begin{array}{r@{\;}ll}%
#1%
\end{array}%
}
\newcommand{\lb}[1]{% % lemma label
\multicolumn{3}{l}{\mbox{\textcolor{fLb}{\bf Lemma #1:}}}\\[1ex]%
}
\newcommand{\df}[1]{% % definition label
\multicolumn{3}{l}{\mbox{\textcolor{fLb}{\bf Definition #1:}}}\\[1ex]%
}
\newcommand{\cj}[2]{% % conjecture
& \multicolumn{2}{l}{\color{fCj}\mbox{\Huge $\mathbf{#1}$}}\\[1ex]
\multicolumn{1}{l}{\color{fCj}\mbox{\Huge $\mathbf{=}$}}
& \multicolumn{2}{l}{\color{fCj}\mbox{\Huge $\mathbf{#2}$}}\\[1ex]
}
\newcommand{\pr}[1]{% % proof
\multicolumn{3}{l}{%
\mbox{\textcolor{fPr}{Proof by induction on $#1$:}}}\\%
}
\newcommand{\bc}{% % base case
\multicolumn{3}{l}{\mbox{\textcolor{fPr}{Base case:}}}\\%
}
\newcommand{\ic}{% % inductive case
\multicolumn{3}{l}{\mbox{\textcolor{fPr}{Inductive case:}}}\\%
}
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\mbox{\textcolor{fRf}{ by #1}}%
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$\begin{array}[b]{ccccccc}
\rule{65mm}{0mm}
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& \rule{65mm}{0mm}
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%
\lm{
\df{1}
\cj{x+0}{x}
}
%
&
&
%
\lm{
\df{2}
\cj{x+Sy}{S(x+y)}
}
%
&
&
%
\lm{
\df{3}
\cj{x \cdot 0}{0}
}
%
&
&
%
\lm{
\df{4}
\cj{x \cdot Sy}{x \cdot y+x}
}
%
\\
&
&
&
&
&
&
\\[50mm]
%
\lm{
\lb{5}
\cj{0+x}{x}
\pr{x}
\bc
& 0+0 & \\
= & 0 & \rs{Def.\ 1} \\
\ic
& 0+Sx & \\
= & S(0+x) & \rs{Def.\ 2} \\
= & Sx & \rs{I.H.} \\
}
%
&
&
%
\lm{
\lb{6}
\cj{Sx+y}{S(x+y)}
\pr{y}
\bc
& Sx+0 & \\
= & Sx & \rs{Def.\ 1} \\
= & S(x+0) & \rs{Def.\ 1} \\
\ic
& Sx+Sy & \\
= & S(Sx+y) & \rs{Def.\ 2} \\
= & ss(x+y) & \rs{I.H.} \\
= & S(x+Sy) & \rs{Def.\ 2} \\
}
%
&
&
%
\lm{
\lb{8}
\cj{(x+y)+z}{x+(y+z)}
\pr{z}
\bc
& (x+y)+0 & \\
= & x+y & \rs{Def.\ 1} \\
= & x+(y+0) & \rs{Def.\ 1} \\
\ic
& (x+y)+sz & \\
= & S((x+y)+z) & \rs{Def.\ 2} \\
= & S(x+(y+z)) & \rs{I.H.} \\
= & x+S(y+z) & \rs{Def.\ 2} \\
= & x+(y+sz) & \rs{Def.\ 2} \\
}
%
&
&
%
\lm{
\lb{10}
\cj{0 \cdot x}{0}
\pr{x}
\bc
& 0 \cdot 0 & \\
= & 0 & \rs{Def.\ 3} \\
\ic
& 0 \cdot Sx & \\
= & 0 \cdot x+0 & \rs{Def.\ 4} \\
= & 0+0 & \rs{I.H.} \\
= & 0 & \rs{Def.\ 1} \\
}
%
\\
&
&
&
&
&
&
\\[50mm]
%
\lm{
\lb{7}
\cj{x+y}{y+x}
\pr{y}
\bc
& x+0 & \\
= & x & \rs{Def.\ 1} \\
= & 0+x & \rs{Lem.\ 5} \\
\ic
& x+Sy & \\
= & S(x+y) & \rs{Def.\ 2} \\
= & S(y+x) & \rs{I.H.} \\
= & Sy+x & \rs{Lem.\ 6} \\
}
%
&
&
&
&
%
\lm{
\lb{9}
\cj{x \cdot (y+z)}{x \cdot y+x \cdot z}
\pr{z}
\bc
& x \cdot (y+0) & \\
= & x \cdot y & \rs{Def.\ 1} \\
= & x \cdot y+0 & \rs{Def.\ 1} \\
= & x \cdot y+x \cdot 0 & \rs{Def.\ 3} \\
\ic
& x \cdot (y+sz) & \\
= & x \cdot S(y+z) & \rs{Def.\ 2} \\
= & x \cdot (y+z)+x & \rs{Def.\ 4} \\
= & (x \cdot y+x \cdot z)+x & \rs{I.H.} \\
= & x \cdot y+(x \cdot z+x) & \rs{Lem.\ 8} \\
= & x \cdot y+x \cdot sz & \rs{Def.\ 4} \\
}
%
&
&
\\
&
&
&
&
&
&
\\[50mm]
&
&
%
\lm{
\lb{11}
\cj{Sx \cdot y}{x \cdot y+y}
\pr{y}
\bc
& Sx \cdot 0 & \\
= & 0 & \rs{Def.\ 3} \\
= & 0+0 & \rs{Def.\ 1} \\
= & x \cdot 0+0 & \rs{Def.\ 4} \\
\ic
& Sx \cdot Sy & \\
= & Sx \cdot y+Sx & \rs{Def.\ 4} \\
= & (x \cdot y+y)+Sx & \rs{I.H.} \\
= & S((x \cdot y+y)+x)& \rs{Def.\ 2} \\
= & S(x \cdot y+(y+x))& \rs{Lem.\ 8} \\
= & S(x \cdot y+(x+y))& \rs{Lem.\ 7} \\
= & S((x \cdot y+x)+y)& \rs{Lem.\ 8} \\
= & (x \cdot y+x)+Sy & \rs{Def.\ 2} \\
= & x \cdot Sy+Sy & \rs{Def.\ 4} \\
}
%
&
&
%
\lm{
\lb{13}
\cj{(x \cdot y) \cdot z}{x \cdot (y \cdot z)}
\pr{z}
\bc
& (x \cdot y) \cdot 0 & \\
= & 0 & \rs{Def.\ 3} \\
= & x \cdot 0 & \rs{Def.\ 3} \\
= & x \cdot (y \cdot 0) & \rs{Def.\ 3} \\
\ic
& (x \cdot y) \cdot sz & \\
= & (x \cdot y) \cdot z+x \cdot y & \rs{Def.\ 4} \\
= & x \cdot (y \cdot z)+x \cdot y & \rs{I.H.} \\
= & x \cdot (y \cdot z+y) & \rs{Lem.\ 9} \\
= & x \cdot (y \cdot sz) & \rs{Def.\ 4} \\
}
%
&&
\\
&
&
&
&
&
&
\\[50mm]
\color{fLg}
\begin{tabular}{ll|}
\hline
\multicolumn{2}{l|}{\bf Legend:} \\
$S(x)$ & Successor of $x$ \\
Def. & Definition \\
Lem. & Lemma \\
I.H. & Induction Hypothesis \\
\multicolumn{2}{l|}{\bf Binding Priorities:} \\
%\multicolumn{2}{l}{$S$ , $ \cdot $ , $+$} \\
\multicolumn{2}{l|}{$Sx \cdot y+z$ denotes $((S(x)) \cdot y)+z$} \\
\multicolumn{2}{l|}{\bf Used Induction Scheme:} \\
If & $P(0)$ \\
and & $P(x)$ always implies $P(Sx)$, \\
then & always $P(x)$. \\
&\\
\multicolumn{2}{l|}{Red arrow: use of lemma} \\
\multicolumn{2}{l|}{Definition-uses omitted} \\
\end{tabular}
&
&
&
&
%
\lm{
\lb{12}
\cj{x \cdot y}{y \cdot x}
\pr{y}
\bc
& x \cdot 0 & \\
= & 0 & \rs{Def.\ 3} \\
= & 0 \cdot x & \rs{Lem.\ 10} \\
\ic
& x \cdot Sy & \\
= & x \cdot y+x & \rs{Def.\ 4} \\
= & y \cdot x+x & \rs{I.H.} \\
= & Sy \cdot x & \rs{Lem.\ 11} \\
}
%
&
&
\\
\rule{0cm}{0cm}
\\
\end{array}$
\end{document}
|
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 18:11, 7 November 2013 |  | 2,862 × 3,247 (39 KB) | Jochen Burghardt (talk | contribs) | User created page with UploadWizard |
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