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buzzproof

🐝 LaTeX style for Linear Style Natural Deduction proofs similar to way found in COMP1120 at UoM

Contents

  1. Preamble
  2. Setup
  3. Notation
  4. Documentation
  5. Customization

Preamble

This style is based on the very popular bussproofs package which allows for the construction of proof trees in the style of the sequent calculus and many other proof systems, however this package allows one to construction linear proofs.

Setup

LaTeX style for Formal Grammars and operations

In order to load the package, use the following code:

\usepackage{buzzproof}

Once you have done this you can now create your proof by creating a linearproof enviroment and passing it asequence of inference rules

\begin{linearproof}
    \axiom{p}{p}
    \implicationintro{}{p \rightarrow p}{1}
    \weakening{}{p \rightarrow p}{2}
    \conjunctionintro{}{p\land (p \rightarrow p)}{1,3}
\end{linearproof}

Will output as the following:

Notation

A judgement is a list of propositions called the antecedents followed by a single proposition known as the consequent. A judgement has the form, Γ⊢ϕ where everything to the left of the turnstile are the antecedents and to the left the consequent.

Documentation

Commands

\axiom

  • Description: Adds a proof line using the axiom rule.
  • Usage: \axiom{antecedents}{consequent}
  • Example:
\axiom{p}{p}

\weakening

  • Description: Adds a proof line using the weakening rule.
  • Usage: \weakening{antecedents}{consequent}{lines-infered-from}
  • Example:
\weakening{p,q}{p}{1}

\conjunctionintro

  • Description: Adds a proof line using the conjunction introduction rule.
  • Usage: \conjunctionintro{antecedents}{consequent}{lines-infered-from}
  • Example:
\conjunctionintro{p,q}{p \land q}{1,2}

\disjunctionintro

  • Description: Adds a proof line using the disjunction introduction rule.
  • Usage: \disjunctionintro{antecedents}{consequent}{lines-infered-from}
  • Example:
\disjunctionintro{p}{p \lor q}{1}

\implicationintro

  • Description: Adds a proof line using the implication introduction rule.
  • Usage: \implicationintro{antecedents}{consequent}{lines-infered-from}
  • Example:
\implicationintro{q}{p \rightarrow q}{2}

\negationintro

  • Description: Adds a proof line using the negation introduction rule.
  • Usage: \negationintro{antecedents}{consequent}{lines-infered-from}
  • Example:
 \negationintro{p}{p}{1}

\doublenegationintro

  • Description: Adds a proof line using the double negation introduction rule.
  • Usage: \doublenegationintro{antecedents}{consequent}{lines-infered-from}
  • Example:
 \doublenegationintro{p}{\neg\neg p}{1}

\conjunctionelim

  • Description: Adds a proof line using the conjunction elimination rule.
  • Usage: \conjunctionelim{antecedents}{consequent}{lines-infered-from}
  • Example:
\conjunctionelim{p\landq}{p}{1}

\disjunctionelim

  • Description: Adds a proof line using the disjunction elimination rule.
  • Usage: \disjunctionelim{antecedents}{consequent}{lines-infered-from}
  • Example:
\disjunctionelim{\Gamma}{\phi}{1,2,3}

\implicationelim

  • Description: Adds a proof line using the implication elimination rule.
  • Usage: \implicationelim{antecedents}{consequent}{lines-infered-from}
  • Example:
\implicationelim{p, p \rightarrow q}{q}{1,2}

\negationelim

  • Description: Adds a proof line using the negation elimination rule.
  • Usage: \negationelim{antecedents}{consequent}{lines-infered-from}
  • Example:
 \negationelim{p, \neg p}{\bot}{1,2}

\doublenegationelim

  • Description: Adds a proof line using the double negation elimination rule.
  • Usage: \doublenegationelim{antecedents}{consequent}{lines-infered-from}
  • Example:
 \doublenegationelim{\neg\negp}{p}{1}

Enviroments

linearproof

  • Description: Enviroment for passing inference rules into to render a formatted proof
  • Usage: \begin{linearproof} {inferences} \end{linearproof}
  • Example:
   \begin{linearproof}
       \axiom{p}{p}
       \implicationintro{}{p \rightarrow p}{1}
       \weakening{}{p \rightarrow p}{2}
       \conjunctionintro{}{p\land (p \rightarrow p)}{1,3}
   \end{linearproof}

Customization

You can easily define your own rules using the base functions:

  • baseRuleWithNoPremises
  • baseRuleWithPremises

Where if you have assumptions that have no premises (similar to the axiom) rule use baseRuleWithNoPremises and baseRuleWithPremises otherwise. However, if your functions are simply introduction or elimination rules which involve some premises, consider using the following for the appropriate usage:

  • introductionRule
  • eliminationRule

An example would be if we wanted to exnted to FOL we would want the Universal Quantifier Introduction rule, which could be implemented as:

\newcommand{\universalquantifierintro}[3]{
    \baseRuleWithPremises{#1}{#2}{#3}{I_{\forall}}
}
% or
\newcommand{\universalquantifierintro}[3]{
    \introductionRule{#1}{#2}{#3}{\forall}
}
\begin{linearproof}
  ...
  \universalquantifierintro{\Gamma}{\forall{x}.A}{2}
\end{linearproof}