Natural deduction like proof

Question

I'm not sure what the right word is for this kind of proof (in the picture):

enter image description here

It is some kind of combination of natural deduction and normal math proofs. I'm wondering how I could do this in LaTeX. I doesn't really need to look exactly like this. In the picture, I've put statement 6-11 to the right for example, but this is not really necessary. More generally I would like something as:

Given
1. statement
2. statement
Ⅰ   Statement to prove
3. hypothese
   4. statement (justification)
5. statement (justification)
II   Other statement to prove
etc. etc.

Any suggestion for doing something like this in LaTeX, maybe some packages that I could use ?

take a look at http://tex.stackexchange.com/questions/147063/lemmon-style-proof-for-logic/147065#147065 — d-cmst, Jan 12 '14 at 15:54
possibly related: Which packages/practices are relevant for writing Structured Derivations? (similar to Dijkstra's calculational style of proofs) — cmhughes, Jan 12 '14 at 16:15
You might find http://www.logicmatters.net/latex-for-logicians/nd/ helpful (scroll down to Fitch-style layouts which are formatted in the way I think you have in mind for natural deduction). There might be something helpful there even though you are not doing exactly the same thing. — cfr, Jan 12 '14 at 17:10
unfortunately, the visual isn't really intelligible to me. but you may be looking for something like what is described in this tugboat article: ProofCheck: Writing and checking complete proofs in LaTeX by Bob Neveln and Bob Alps. — barbara beeton, Jan 12 '14 at 18:10
@barbarabeeton http://snag.gy/Iv7sU.jpg Here I did a proof in english, with this kind of style. — Kasper, Jan 12 '14 at 20:59
@Kasper -- thank you for the link to the example. this does look more like the finch-style proof layout than what was described in the tugboat article. — barbara beeton, Jan 12 '14 at 21:21
@Kasper - the example you posted for barbarabeeton is more fitch-like than the first. If that format is good, I definitely think you should look at the packages for formatting fitch-style natural deduction proofs. — cfr, Jan 13 '14 at 02:18

cfr · Accepted Answer · 2014-01-15T22:58:01.820

I'm having a lot of problems accessing the site right now but perhaps this will be useful. It does not look quite like the image you posted but it does maintain the basic structure. It supplements lplfitch with two new commands, \given and \toprove{}, to help keep the formatting of your variation consistent. Note that the commands beginning \l... are from the package. You need not use them if you prefer to type the full mathematical symbols directly or if their names are counter-intuitive from a mathematical perspective. Anyway, I hope it may be useful in some way:

\documentclass{article}

\usepackage[T1]{fontenc}
\usepackage{textcomp}
\usepackage{lmodern}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{lplfitch}
\renewcommand{\formula}[1]{\ensuremath{#1}}
\newcommand*{\fitchline}{\cline{1-1}\\[-2ex]}
\newcounter{thingstoprove}
\newcommand*{\given}{%
  \setcounter{thingstoprove}{0}%
   & {\bfseries Given}\\\fitchline}
\newcommand*{\toprove}[1]{%
  \stepcounter{thingstoprove}%
  \\[-2ex] & {\bfseries \Roman{thingstoprove}}\formula{\; #1} & To show\\\fitchline}
\setlength{\fitchprfwidth}{.7\textwidth}

\begin{document}

\noindent\fitchprf{%
  \given
  \pline[1.]{\lall a,b \in \mathcal{Z}\ ( a|b \liff \lis c \in \mathcal{Z}\ ac = b )}[Pr.]\\
  \pline[2.]{\lall a,b \in \mathcal{Z}, p \in \mathcal{P}\ ( p|ab \lif p|a \lor p|b )}[Pr.]\\
  \pline[3.]{\lall x \in \mathcal{Q}\ ( \lis m,n \in \mathcal{Z},  n \neq 0 : x = \frac{m}{n}, gcd(m,n) = 1 )}[Pr.]}{%
  \toprove{\sqrt{2} \notin \mathcal{Q}}
  \subproof{\pline[4.]{\sqrt{2} \in \mathcal{Q}}[Ass.\ for \emph{reductio}]}{%
    \pline[5.]{\sqrt{2} = \frac{m}{n} \text{ for some } m, n \in \mathcal{Z}, gcd (m,n) = 1}[3,4]\\
    \pline[6.]{2n^2 = m^2}[multiply by 2; square]\\
    \pline[7.]{2|m^2}[1,6]\\
    \pline[8.]{2|m}[2,7]\\
    \pline[9.]{m = 2c \text{ for some } c \in \mathcal{Z}}[1,8]\\
    \pline[10.]{2n^2 = 4c^2}[6,9]\\
    \pline[11.]{n^2 = 2c^2}[divide by 2]\\
    \pline[12.]{2|n^2}[1,11]\\
    \pline[13.]{2|n}[2,12]\\
    \pline[14.]{gcd(m,n) \geq 2}[8,13]\\
    \pline[15.]{\lfalse}[\lfalsei{5}{14}]
    }
  \pline[16.]{\sqrt{2} \notin \mathcal{Q}}[\lnoti{4--15}]
}\bigskip

\noindent\fitchprf{%
  \given
  \pline[1.]{statement}\\
  \pline[2.]{statement}}
  {%
    \toprove{\text{Statement to prove}}
    \subproof{\pline[3.]{hypothesis}[Hyp.]}{%
      \pline[4.]{statement}[justification]
      }
    \pline[5.]{statement}[justification]\\
    \toprove{\text{Other statement to prove}}
    \ellipsesline
}

\end{document}

fitch-style proofs with lplfitch

If you like to live dangerously, the following should give precisely the same output with automatic management of line numbering for proof lines:

\documentclass{article}

\usepackage[T1]{fontenc}
\usepackage{textcomp}
\usepackage{lmodern}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{lplfitch}
\usepackage{etoolbox}
\renewcommand{\formula}[1]{\ensuremath{#1}}
\newcommand*{\fitchline}{\cline{1-1}\\[-2ex]}
\newcounter{thingstoprove}
\newcommand*{\given}{%
  \setcounter{thingstoprove}{0}%
   & {\bfseries Given}\\\fitchline}
\newcommand*{\toprove}[1]{%
  \stepcounter{thingstoprove}%
  \\[-2ex] & {\bfseries \Roman{thingstoprove}}\formula{\; #1} & To show\\\fitchline}
\setlength{\fitchprfwidth}{.7\textwidth}
\newcounter{plineno}
\newcommand*{\reallyresetplineno}{\setcounter{plineno}{0}}
\let\resetplineno\reallyresetplineno
\pretocmd{\fitchprf}{\resetplineno}{\GenericWarning{Hackery}{Successfully prepended code to fitchprf.}}{\GenericWarning{Hackery}{Failed to patch fitchprf. Sorry, you will need to manage line numbering yourself.}}
\apptocmd{\fitchprf}{\global\let\resetplineno\reallyresetplineno}{\GenericWarning{Hackery}{Successfully appended code to fitchprf.}}{\GenericWarning{Hackery}{Failed to patch fitchprf. Sorry, you will need to manage line numbering yourself.}}
\pretocmd{\subproof}{\global\let\resetplineno\relax}{\GenericWarning{Hackery}{Successfully prepended code to subproof.}}{\GenericWarning{Hackery}{Failed to patch subproof. Sorry, you will need to manage line numbering yourself.}}
\makeatletter
  \patchcmd{\pline}{\@empty}{\stepcounter{plineno}\arabic{plineno}.}{\GenericWarning{Hackery}{Successfully patched pline.}}{\GenericWarning{Hackery}{Failed to patch pline. Sorry, you will need to manage line numbering yourself.}}
\makeatother

\begin{document}

\noindent\fitchprf{%
  \given
  \pline{\lall a,b \in \mathcal{Z}\ ( a|b \liff \lis c \in \mathcal{Z}\ ac = b )}[Pr.]\\
  \pline{\lall a,b \in \mathcal{Z}, p \in \mathcal{P}\ ( p|ab \lif p|a \lor p|b )}[Pr.]\\
  \pline{\lall x \in \mathcal{Q}\ ( \lis m,n \in \mathcal{Z},  n \neq 0 : x = \frac{m}{n}, gcd(m,n) = 1 )}[Pr.]}{%
  \toprove{\sqrt{2} \notin \mathcal{Q}}
  \subproof{\pline{\sqrt{2} \in \mathcal{Q}}[Ass.\ for \emph{reductio}]}{%
    \pline{\sqrt{2} = \frac{m}{n} \text{ for some } m, n \in \mathcal{Z}, gcd (m,n) = 1}[3,4]\\
    \pline{2n^2 = m^2}[multiply by 2; square]\\
    \pline{2|m^2}[1,6]\\
    \pline{2|m}[2,7]\\
    \pline{m = 2c \text{ for some } c \in \mathcal{Z}}[1,8]\\
    \pline{2n^2 = 4c^2}[6,9]\\
    \pline{n^2 = 2c^2}[divide by 2]\\
    \pline{2|n^2}[1,11]\\
    \pline{2|n}[2,12]\\
    \pline{gcd(m,n) \geq 2}[8,13]\\
    \pline{\lfalse}[\lfalsei{5}{14}]
    }
  \pline{\sqrt{2} \notin \mathcal{Q}}[\lnoti{4--15}]
}\bigskip

\noindent\fitchprf{%
  \given
  \pline{statement}\\
  \pline{statement}}
  {%
    \toprove{\text{Statement to prove}}
    \subproof{\pline{hypothesis}[Hyp.]}{%
      \pline{statement}[justification]
      }
    \pline{statement}[justification]\\
    \toprove{\text{Other statement to prove}}
    \ellipsesline
}

\end{document}

Is it possible to define a counter for \pline? Is seems that typing line numbers manually is not a very efficient (nor robust) thing to do. — Herr K., Jan 15 '14 at 21:42
@KevinC See the addition to my answer. This gives just the same output but manages number of proof lines automagically. Since this patches the commands of lplfitch rather more seriously than the original code (it uses the facilities of etoolbox), this is likely to be more fragile. But it does seem to work with the examples I used, at least. — cfr, Jan 15 '14 at 23:00
@Kasper No problem. It was interesting - I've never used this package or the etoolbox patching commands before. But I've wanted an excuse to play with the latter! — cfr, Jan 16 '14 at 14:55

Natural deduction like proof

1 Answers1