File:Inductive proofs of properties of add, mult from recursive definitions (exercise version).pdf
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Summary
DescriptionInductive proofs of properties of add, mult from recursive definitions (exercise version).pdf |
English: English: Shows recursive definitions of addition (+) and multiplication (*) on natural numbers and inductive proofs of commutativity, associativity, distributivity by Peano induction; some of the later ones are omitted as exercises. Also indicates which property is used in the proof of which other one. |
Date | |
Source | Adapted from File:Inductive proofs of properties of add, mult from recursive definitions.pdf |
Author | Adapted by me; the original is by User:Jochen_Burghardt. |
LaTeX source code |
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& 0+0 & \\
= & 0 & \rs{Def.\ 1} \\
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& 0+Sx & \\
= & S(0+x) & \rs{Def.\ 2} \\
= & Sx & \rs{I.H.} \\
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%
&
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& Sx+0 & \\
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= & S(x+0) & \rs{Def.\ 1} \\
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& Sx+Sy & \\
= & S(Sx+y) & \rs{Def.\ 2} \\
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& (x+y)+0 & \\
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= & x+S(y+z) & \rs{Def.\ 2} \\
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& x+0 & \\
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= & 0+x & \rs{Lem.\ 5} \\
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& x+Sy & \\
= & S(x+y) & \rs{Def.\ 2} \\
= & S(y+x) & \rs{I.H.} \\
= & Sy+x & \rs{Lem.\ 6} \\
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%
&
&
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%
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& x \cdot (y+0) & \\
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= & x \cdot y+0 & \rs{Def.\ 1} \\
= & x \cdot y+x \cdot 0 & \rs{Def.\ 3} \\
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& x \cdot (y+sz) & \\
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&
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\\
\\
\\
\\
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%
&&
\\
&
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&
&
&
\\[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}
&
&
&
&
%
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\end{document}
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 21:44, 6 February 2022 | 2,862 × 3,247 (76 KB) | Felix QW | Uploaded a work by Adapted by me; the original is by User:Jochen_Burghardt. from Adapted from File:Inductive proofs of properties of add, mult from recursive definitions.pdf with UploadWizard |
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Conversion program | pdfTeX-1.40.23 |
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Page size | 1374.8 x 1559.06 pts |
Version of PDF format | 1.5 |