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English: Proof of the induction step for commutativity of + from Peano axioms, guided by rippling. The usual definition of addition is employed:
1:   u + 0 = u
2:   u + Suc(v) = Suc(u+v)

Moreover the following lemma, which is in turn provable by induction, is employed:

3:   Suc(u) + v = Suc(u+v)
Date
Source Own work
Author Jochen Burghardt
LaTeX source code 
\documentclass[12pt]{article}

\setlength{\unitlength}{10mm}
\usepackage[pdftex]{color}
\usepackage{amssymb}
\usepackage[paperwidth=19.5\unitlength,paperheight=6.5\unitlength]{geometry}
\setlength{\topmargin}{-36mm}
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\setlength{\textheight}{6.5\unitlength}
\setlength{\oddsidemargin}{-23mm}
\setlength{\parindent}{0cm}
\renewcommand{\arraystretch}{1.9}
\pagestyle{empty}

% colors
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\definecolor{bGoal}     {rgb}{0.90,0.90,0.99}

\newcommand{\tree}[1]{}

\newcommand{\Frame}[2]{%
        \begin{picture}(19,6)%
        \put(7,6){\makebox(0,0)[tl]{%
                $\begin{array}{lllll}%
                \multicolumn{4}{l}{\mbox{\sf Justification:}}   \\
                #1%
                \end{array}$%
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\newcommand{\front}[2]{\fbox{\textcolor{fFront}{$#1$}}^{#2}}
\newcommand{\hole}[1]{\textcolor{black}{\underline{#1}}}



\begin{document}



% reasoning lines

\newcommand{\ra}{%
        \textcolor{white}{\Leftarrow}
        & \front{Suc(\hole{x})}{\uparrow} + y
        & =
        & y + \front{Suc(\hole{x})}{\uparrow}
        &
        \\
}

\newcommand{\rb}{%
        \Leftrightarrow
        & \front{Suc(\hole{x+y})}{\uparrow}
        & =
        & y + \front{Suc(\hole{x})}{\uparrow}
        & \mbox{ by a Lemma}
        \\
}

\newcommand{\rc}{%
        \Leftrightarrow
        & \front{Suc(\hole{x+y})}{\uparrow}
        & =
        & \front{Suc(\hole{y+x})}{\uparrow}
        & \mbox{ by Definition of } +
        \\
}

\newcommand{\rd}{%
        \Leftarrow
        & x+y
        & =
        & y+x
        & \mbox{ by Substitutivity}
        \\
}

\newcommand{\re}{%
        \Leftrightarrow
        & \it true
        &
        &
        & \mbox{ by Induction hypothesis}
        \\
}



% page frames



\Frame{}{%
%!2}textree g fs 1 wd 1.5 | texcalc oy 0.5
\tree{$=$}
        {\tree{$+$}
                {\tree{\textcolor{fFront}{$Suc$}}
                        {\tree{$x$}
                        }
                }
                {\tree{$y$}
                }
        }
        {\tree{$+$}
                {\tree{$y$}
                }
                {\tree{\textcolor{fFront}{$Suc$}}
                        {\tree{$x$}
                        }
                }
        }

\put(0.750,0.700){\makebox(0.000,0.000)[t]{$x$}}%aaa
\put(0.750,1.900){\makebox(0.000,0.000)[t]{\textcolor{fFront}{$Suc$}}}%aa
\put(0.750,1.500){\line(0,-1){0.600}}%aa-a
\put(2.250,1.900){\makebox(0.000,0.000)[t]{$y$}}%ab
\put(1.500,3.100){\makebox(0.000,0.000)[t]{$+$}}%a
\put(1.500,2.700){\line(-5,-4){0.750}}%a-a
\put(1.500,2.700){\line(5,-4){0.750}}%a-b
\put(3.750,1.900){\makebox(0.000,0.000)[t]{$y$}}%ba
\put(5.250,0.700){\makebox(0.000,0.000)[t]{$x$}}%bba
\put(5.250,1.900){\makebox(0.000,0.000)[t]{\textcolor{fFront}{$Suc$}}}%bb
\put(5.250,1.500){\line(0,-1){0.600}}%bb-a
\put(4.500,3.100){\makebox(0.000,0.000)[t]{$+$}}%b
\put(4.500,2.700){\line(-5,-4){0.750}}%b-a
\put(4.500,2.700){\line(5,-4){0.750}}%b-b
\put(3.000,4.300){\makebox(0.000,0.000)[t]{$=$}}%
\put(3.000,3.900){\line(-5,-2){1.500}}%-a
\put(3.000,3.900){\line(5,-2){1.500}}%-b

}



\Frame{\ra\rb}{%
%!2}textree g fs 1 wd 1.5 | texcalc oy 0.5
\tree{$=$}
        {\tree{\textcolor{fFront}{$Suc$}}
                {\tree{$+$}
                        {\tree{$x$}
                        }
                        {\tree{$y$}
                        }
                }
        }
        {\tree{$+$}
                {\tree{$y$}
                }
                {\tree{\textcolor{fFront}{$Suc$}}
                        {\tree{$x$}
                        }
                }
        }

\put(0.750,0.700){\makebox(0.000,0.000)[t]{$x$}}%aaa
\put(2.250,0.700){\makebox(0.000,0.000)[t]{$y$}}%aab
\put(1.500,1.900){\makebox(0.000,0.000)[t]{$+$}}%aa
\put(1.500,1.500){\line(-5,-4){0.750}}%aa-a
\put(1.500,1.500){\line(5,-4){0.750}}%aa-b
\put(1.500,3.100){\makebox(0.000,0.000)[t]{\textcolor{fFront}{$Suc$}}}%a
\put(1.500,2.700){\line(0,-1){0.600}}%a-a
\put(3.750,1.900){\makebox(0.000,0.000)[t]{$y$}}%ba
\put(5.250,0.700){\makebox(0.000,0.000)[t]{$x$}}%bba
\put(5.250,1.900){\makebox(0.000,0.000)[t]{\textcolor{fFront}{$Suc$}}}%bb
\put(5.250,1.500){\line(0,-1){0.600}}%bb-a
\put(4.500,3.100){\makebox(0.000,0.000)[t]{$+$}}%b
\put(4.500,2.700){\line(-5,-4){0.750}}%b-a
\put(4.500,2.700){\line(5,-4){0.750}}%b-b
\put(3.000,4.300){\makebox(0.000,0.000)[t]{$=$}}%
\put(3.000,3.900){\line(-5,-2){1.500}}%-a
\put(3.000,3.900){\line(5,-2){1.500}}%-b

}



\Frame{\ra\rb\rc}{%
%!2}textree g fs 1 wd 1.5 | texcalc oy 0.5
\tree{$=$}
        {\tree{\textcolor{fFront}{$Suc$}}
                {\tree{$+$}
                        {\tree{$x$}
                        }
                        {\tree{$y$}
                        }
                }
        }
        {\tree{\textcolor{fFront}{$Suc$}}
                {\tree{$+$}
                        {\tree{$y$}
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                        {\tree{$x$}
                        }
                }
        }

\put(0.750,0.700){\makebox(0.000,0.000)[t]{$x$}}%aaa
\put(2.250,0.700){\makebox(0.000,0.000)[t]{$y$}}%aab
\put(1.500,1.900){\makebox(0.000,0.000)[t]{$+$}}%aa
\put(1.500,1.500){\line(-5,-4){0.750}}%aa-a
\put(1.500,1.500){\line(5,-4){0.750}}%aa-b
\put(1.500,3.100){\makebox(0.000,0.000)[t]{\textcolor{fFront}{$Suc$}}}%a
\put(1.500,2.700){\line(0,-1){0.600}}%a-a
\put(3.750,0.700){\makebox(0.000,0.000)[t]{$y$}}%baa
\put(5.250,0.700){\makebox(0.000,0.000)[t]{$x$}}%bab
\put(4.500,1.900){\makebox(0.000,0.000)[t]{$+$}}%ba
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\put(4.500,3.100){\makebox(0.000,0.000)[t]{\textcolor{fFront}{$Suc$}}}%b
\put(4.500,2.700){\line(0,-1){0.600}}%b-a
\put(3.000,4.300){\makebox(0.000,0.000)[t]{$=$}}%
\put(3.000,3.900){\line(-5,-2){1.500}}%-a
\put(3.000,3.900){\line(5,-2){1.500}}%-b

}



\Frame{\ra\rb\rc\rd}{%
%!2}textree g fs 1 wd 1.5 | texcalc oy 1.7
\tree{$=$}
        {\tree{$+$}
                {\tree{$x$}
                }
                {\tree{$y$}
                }
        }
        {\tree{$+$}
                {\tree{$y$}
                }
                {\tree{$x$}
                }
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\put(0.750,1.900){\makebox(0.000,0.000)[t]{$x$}}%aa
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\put(4.500,2.700){\line(5,-4){0.750}}%b-b
\put(3.000,4.300){\makebox(0.000,0.000)[t]{$=$}}%
\put(3.000,3.900){\line(-5,-2){1.500}}%-a
\put(3.000,3.900){\line(5,-2){1.500}}%-b

}



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\tree{\sf done}

\put(3.000,4.300){\makebox(0.000,0.000)[t]{\sf done}}%

}

\end{document}

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Date/TimeThumbnailDimensionsUserComment
current10:59, 16 October 2020Thumbnail for version as of 10:59, 16 October 20201,150 × 383, 5 pages (60 KB)Jochen Burghardt (talk | contribs)Uploaded own work with UploadWizard

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