nadia-proof 0.1.3

NADIA is a proof assistant for teaching natural deduction to computer science students. NADIA allows students to write their proofs and automatically checks whether the proofs are correct and, if not, displays any errors found.

Natural Deduction Proof Assistant (NADIA)

The NADIA is a tool written in Python that can be used in desktop, or in a web platform. The main idea is that the students can write their proofs as similar as possible to what is available in the textbooks and to what the students would usually write on paper. NADIA allows the students to automatically check whether a proof in the natural deduction is valid. If the proof is not correct, NADIA will display the errors of the proof. So, the students may make mistakes and learn from the errors. The web interface is very easy-to-use and has:

• An area for editing the proof in plain text. The students should write a proof in the Fitch-style.
• A message area to display whether the proof is valid, the countermodel, or the errors on the proof.
• And the following links:
  • Check, to check the correctness of the proof;
  • Manual, to view a document with the inference rules and examples;
  • Fitch, to generate the LaTeX code in a Fitch-style of a valid proof. Use the logicproof package in your LaTeX code;
  • Gentzen, to generate the LaTeX code in a Gentzen-style of a valid proof. Use the proof package in your LaTeX code;
  • Fitch LaTeX in Overleaf, to open the proof source code directly in Overleaf that is a collaborative platform for editing LaTeX

To facilitate the writing of the proofs, we made the following conventions in NADIA:

• The Atoms are written in capital letters (e.g. A, B, H(x));
• Variables are written with the first letter in lowercase, followed by letters and numbers (e.g. x, x0, xP0);
• Formulas with $\forall x$ and $\exists x$ are represented by $Ax$ and $Ex$ ('A' and 'E' followed by the variable x). For instance, Ax(H(x)->M(x)) represents $\forall x~(H(x)\rightarrow M(x))$.
• Table below shows the equivalence of logic symbols and those used in NADIA.
• The order of precedence of quantifiers and logical connectives is defined by $\lnot,\forall,\exists,\wedge,\vee,\rightarrow$ with right alignment. For example:
  • Formula ~A&B -> C represents formula $(((\lnot A)\land B)\rightarrow C)$;
  • The theorem ~A|B |- A->C represents $((\lnot A)\vee B)\vdash (A\rightarrow B)$.
• Each inference rule will be named by its respective connective and i (introduction) or e (elimination). For example, ->e represents the elemination and rule.
• The justifications for the premises use the reserved word pre.

Symbol	$\lnot$	$\land$	$\lor$	$\rightarrow$	$\forall x$	$\exists x$	$\bot$	box	$\vdash$
LaTeX	$\backslash\textrm{lnot}$	$\backslash\textrm{land}$	$\backslash\textrm{lor}$	$\backslash\textrm{rightarrow}$	$\backslash\textrm{forall x}$	$\backslash\textrm{exists x}$	$\backslash\textrm{bot}$	$[.~]$	$\backslash\textrm{vdash}$
NADIA	~	&	$\mid$	->	Ax	Ex	@	{ }	|-

License

NADIA is avalible by a MIT License

Copyright (c) 2022-2022 Davi Romero de Vasconcelos

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Requirements:

• You must install the rply 0.7.8 package

NADIA in your code

You can import NADIA in your code (basic usage)

from nadia.nadia_pt_fo import check_proof

print(check_proof('''1. A|B              pre
2. A->C             pre
3. B->C             pre
4. {    A           hip
5.      C           ->e 4,2
   } 
6. {    B           hip
7.      C           ->e 6,3
   }
8. C                |e 1, 4-5, 6-7'''))