Original file (2,862 × 3,247 pixels, file size: 39 KB, MIME type: application/pdf)
This is a file from the
Wikimedia Commons. Information from its
description page there is shown below. Commons is a freely licensed media file repository. You can help. |
DescriptionInductive proofs of properties of add, mult from recursive definitions.pdf |
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 |
---|
\documentclass[10pt]{article}
\usepackage[pdftex]{color}
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\usepackage{amssymb}
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\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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\color{fEq}
$\begin{array}b{ccccccc}
\rule{65mm}{0mm}
& \rule{65mm}{0mm}
& \rule{65mm}{0mm}
& \rule{65mm}{0mm}
& \rule{65mm}{0mm}
& \rule{65mm}{0mm}
& \rule{65mm}{0mm} \\
%
\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 | |
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current | 18:11, 7 November 2013 | 2,862 × 3,247 (39 KB) | Jochen Burghardt | User created page with UploadWizard |
This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.
If the file has been modified from its original state, some details may not fully reflect the modified file.
Software used | TeX |
---|---|
Conversion program | pdfTeX-1.40.3 |
Encrypted | no |
Page size | 1374.8 x 1559.06 pts |
Version of PDF format | 1.4 |
Original file (2,862 × 3,247 pixels, file size: 39 KB, MIME type: application/pdf)
This is a file from the
Wikimedia Commons. Information from its
description page there is shown below. Commons is a freely licensed media file repository. You can help. |
DescriptionInductive proofs of properties of add, mult from recursive definitions.pdf |
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 |
---|
\documentclass[10pt]{article}
\usepackage[pdftex]{color}
\usepackage[paperwidth=485mm,paperheight=550mm]{geometry}
\usepackage{amssymb}
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{485mm}
\setlength{\textheight}{550mm}
\setlength{\evensidemargin}{0cm}
\setlength{\oddsidemargin}{-23mm}
\setlength{\parindent}{0cm}
\setlength{\parskip}{1ex}
\setlength{\unitlength}{1mm}
\sloppy
\begin{document}
\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:}}}\\%
}
\newcommand{\rs}[1]{% % reason
\mbox{\textcolor{fRf}{ by #1}}%
}
\color{fLn}
\begin{picture}(0,0)%
\thicklines%
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\color{fEq}
$\begin{array}b{ccccccc}
\rule{65mm}{0mm}
& \rule{65mm}{0mm}
& \rule{65mm}{0mm}
& \rule{65mm}{0mm}
& \rule{65mm}{0mm}
& \rule{65mm}{0mm}
& \rule{65mm}{0mm} \\
%
\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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