Copyright © 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc. http://fsf.org/ Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.
The purpose of this License is to make a manual, textbook, or other functional and useful document free in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others.
This License is a kind of “copyleft”, which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software.
We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference.
This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions stated herein. The “Document”, below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as “you”. You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law.
A “Modified Version” of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language.
A “Secondary Section” is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document's overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them.
The “Invariant Sections” are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant. The Document may contain zero Invariant Sections. If the Document does not identify any Invariant Sections then there are none.
The “Cover Texts” are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License. A Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words.
A “Transparent” copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, that is suitable for revising the document straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup, or absence of markup, has been arranged to thwart or discourage subsequent modification by readers is not Transparent. An image format is not Transparent if used for any substantial amount of text. A copy that is not “Transparent” is called “Opaque”.
Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo input format, LaTeX input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML, PostScript or PDF designed for human modification. Examples of transparent image formats include PNG, XCF and JPG. Opaque formats include proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML, PostScript or PDF produced by some word processors for output purposes only.
The “Title Page” means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, “Title Page” means the text near the most prominent appearance of the work's title, preceding the beginning of the body of the text.
The “publisher” means any person or entity that distributes copies of the Document to the public.
A section “Entitled XYZ” means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language. (Here XYZ stands for a specific section name mentioned below, such as “Acknowledgements”, “Dedications”, “Endorsements”, or “History”.) To “Preserve the Title” of such a section when you modify the Document means that it remains a section “Entitled XYZ” according to this definition.
The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document. These Warranty Disclaimers are considered to be included by reference in this License, but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License.
You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3.
You may also lend copies, under the same conditions stated above, and you may publicly display copies.
If you publish printed copies (or copies in media that commonly have printed covers) of the Document, numbering more than 100, and the Document's license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also clearly and legibly identify you as the publisher of these copies. The front cover must present the full title with all words of the title equally prominent and visible. You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects.
If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages.
If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a computer-network location from which the general network-using public has access to download using public-standard network protocols a complete Transparent copy of the Document, free of added material. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public.
It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document.
You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version:
If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version's license notice. These titles must be distinct from any other section titles.
You may add a section Entitled “Endorsements”, provided it contains nothing but endorsements of your Modified Version by various parties—for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard.
You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one.
The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version.
You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers.
The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work.
In the combination, you must combine any sections Entitled “History” in the various original documents, forming one section Entitled “History”; likewise combine any sections Entitled “Acknowledgements”, and any sections Entitled “Dedications”. You must delete all sections Entitled “Endorsements.”
You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects.
You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.
A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an “aggregate” if the copyright resulting from the compilation is not used to limit the legal rights of the compilation's users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document.
If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document's Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate.
Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail.
If a section in the Document is Entitled “Acknowledgements”, “Dedications”, or “History”, the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title.
You may not copy, modify, sublicense, or distribute the Document except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense, or distribute it is void, and will automatically terminate your rights under this License.
However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation.
Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice.
Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, receipt of a copy of some or all of the same material does not give you any rights to use it.
The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.
Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License “or any later version” applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. If the Document specifies that a proxy can decide which future versions of this License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Document.
“Massive Multiauthor Collaboration Site” (or “MMC Site”) means any World Wide Web server that publishes copyrightable works and also provides prominent facilities for anybody to edit those works. A public wiki that anybody can edit is an example of such a server. A “Massive Multiauthor Collaboration” (or “MMC”) contained in the site means any set of copyrightable works thus published on the MMC site.
“CC-BY-SA” means the Creative Commons Attribution-Share Alike 3.0 license published by Creative Commons Corporation, a not-for-profit corporation with a principal place of business in San Francisco, California, as well as future copyleft versions of that license published by that same organization.
“Incorporate” means to publish or republish a Document, in whole or in part, as part of another Document.
An MMC is “eligible for relicensing” if it is licensed under this License, and if all works that were first published under this License somewhere other than this MMC, and subsequently incorporated in whole or in part into the MMC, (1) had no cover texts or invariant sections, and (2) were thus incorporated prior to November 1, 2008.
The operator of an MMC Site may republish an MMC contained in the site under CC-BY-SA on the same site at any time before August 1, 2009, provided the MMC is eligible for relicensing.
To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page:
Copyright (C) year your name. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled ``GNU Free Documentation License''.
If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the “with...Texts.” line with this:
with the Invariant Sections being list their titles, with the Front-Cover Texts being list, and with the Back-Cover Texts being list.
If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation.
If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.
This manual is for GNU Aris, the logical proof program.
Copyright (C) 2012 Ian Dunn
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”.
This manual is for GNU Aris, the logical proof program.
This is edition 0.1, for
aris version 0.1
This manual is for GNU Aris, a sequential proof program, designed to assist anyone interested in solving logical proofs. Aris supports both propositional and predicate logic, as well as Boolean algebra and arithmetical logic in the form of abstract sequences. It uses a predefined set of both inference and equivalence rules, however gives the user options to use older proofs as lemmas, including Isabelle's Isar proofs.
This chapter describes the basic usage of GNU Aris.
When Aris is loaded up, you will see a few things. You will see the rules window and a proof window. The rules window contains the different rules. A rule will not be selected if a premise is in focus.
Aris has several connectives (see Connectives). When the keyboard command for the desired connective is activated, the desired connective will be inserted at the current cursor point, overwriting selected text.
Hitting Ctrl+J adds a conclusion to the proof. A conlusion is always added after the current line, or, if the line is a premise, then the conclusion is added after the last premise. If the current line is a conclusion, then it is highlighted in cyan. Hitting Ctrl+P adds a premise to the proof. A premise will always be added after the last premise. Pressing Ctrl+B adds a subproof to the proof. When in a subproof, a conclusion will always be added within the subproof. The first line of a subproof does not require a rule, but instead acts as a premise. Pressing Ctrl+D ends the current subproof. This creates a new conclusion just after the subproof.
Holing CTRL, and left-clicking on a sentence will select a sentence as a reference sentence. The current line's reference sentences are highlighted in violet. Only a line before the current line can be selected as a reference. In addition, if the sentence is in a different subproof than the current line, then the sentence can not be selected as a reference. Each rule requires a specific amount of references (see Rules Index).
Holding SHIFT and left-clicking on a sentence will select the sentence. The sentence will be highlighted in red-orange. Multiple sentences can be selected this way, however when another action is taken, all of them will be de-selected. Pressing CTRL+K will kill (cut) the selected lines, and CTRL+G will copy the selected lines. If no lines are selected, then the current line will be used.
The rules are divided into five categories: Inference, Equivalence, Predicate, Boolean, and Miscellaneous.
The premises of any of these rules can be in any order.
One of the basic rules of logic, modus ponens say that ‘if P happens, then Q must happen. P happened, so Q must happen’.
For example, if it is known that ‘If the dog begins to bark, then someone is at the door’, and it is also known that ‘the dog has begun to bark’, then modus ponens says that ‘someone must be at the door’.
Modus Ponens requires exactly two references.
What addition says is that something is already known, so it must be true that that something or something else, or something else, etc. must also be true.
For example, if it is known that ‘The sky is blue’, then addition says that it can be inferred that ‘The sky is blue, or the sky is yellow, or the sky is pink’, since only one of those statements has to be true.
Addition requires exactly one reference.
Simplification says that if it is known that P and Q and R, etc. is known to be true, then P is true.
For instance, if it is known that ‘It is cloudy, and it is raining’, then simplification allows the inference of ‘It is cloudy’ and ‘It is raining’.
Simplification requires exactly one reference.
What conjunction is saying is the exact opposite of simplification. If P is known, and Q is known, and R is know, etc. then P and Q and R, etc. is also known.
Take for example, that it is known that ‘I don't like green eggs and ham’, and ‘I would not eat them in a house’, and ‘I would not eat them with a mouse’. Conjunction allows us to infer that ‘I don't like green egss and ham, and I would not eat them in a house, and I would not eat them with a mouse.’.
Conjunction requires at least two references.
Also referred to as the chain rule, hypothetical syllogism states that if one knows that 'if P then Q', and 'if R then S', then one can infer 'if P then S'. For example, if it is known ‘if it is raining, then it is cloudy’, and ‘if it is cloudy, then it is not sunny’, and ‘if it is not sunny, then it is cold’, then hypothetical syllogism allows us to infer that ‘if it is raining, then it is cold’. This works with any number of conditional statements, as long as they all follow this pattern.
Hypothetical Syllogism requires at least two references.
Disjunctive syllogism is commonly used when disjunctions are present. It claims that if one knows that 'P or Q or R', and 'P is false', and 'R is false', then Q must be true. This works with any number of disjuncts.
A law of logic, excluded middle asserts that something is either true, or it is not true.
Excluded middle requires zero references.
Constructive Dilemma requires at least three references.
Equivalence rules operate on any valid part of the sentence, and work both ways. Each equivalence rule requires one reference.
Implication uses the definition of the conditional. It is also valid to claim something such as -(-P v Q) v (-R v S) <=> (P → Q) → (R → S), because implication is recursive.
A note to users, typically association is used as P ^ (Q ^ R) <=> (P ^ Q) ^ R. While Aris will allow you to prove that this is equivalent, association allows the removal of one pair of parentheses at a time. (P ^ Q) ^ (R ^ S) <=> P ^ Q ^ R ^ S is also valid in Aris, because association allows recursion, but only when removing several sets of parentheses or adding several sets of parentheses.
Just like addition and multiplication, conjunctions and disjunctions are commutative. This of course means that ‘I would like some pie and I would like some cake’ is the same as saying ‘I would like some cake and I would like some pie’.
Idempotence claims that ‘I like blue and I like blue’ is the same as saying ‘I like blue’.
Equivalence uses the definition of the biconditional. Claiming that ‘P if and only if Q’ is exactly the same as claiming ‘if P then Q’ and ‘if Q then P’. Equivalence is the only rule that works with biconditionals explicitly, and is thus used any time a biconditional is seen.
You probably learned in english class that saying ‘I would not like to disagree’ is the same thing as saying ‘I would like to agree’. That's what double negation claims.
For the convenience of the user, the equivalence rules work recursively. For example
This is an example of using implication recursively. Recursion only works if the rule is being used the same way. For example, removing multiple parentheses with association is fine, however adding and removing parentheses with association is not.
Commutatvitity and idempotence work differently than the others when it comes to recursion. If commutativity is applied to a connective, then no parts of that connective, or parts of those parts, and so on, can be used in commutativity. However, other parts from the sentence can be rearranged. The same goes for idempotence.
The predicate rules are the rules that work specifically with predicate logic.
Universal Generalization claims that if a property ‘P’ is true for some arbitrary object, then it is true for all objects. A symbol is arbitrary if nothing is known about, or rather if it was not introduced through a premise or using existential instantiation.
Universal Generalization claims that if a property ‘P’ is true for all objects, then it must be true for an object ‘a’.
Existential Generalization claims that if ‘P’ is true for some object, then there exists an object for which ‘P’ is true.
Existential Instantiation claims that if there exists an object for which property ‘P’ is true, then it can be claimed that some unused object has this property. In this case, ‘a’ becomes a placeholder for the object.
Bound Variable allows the user to substitute any bound variable for another bound variable, given that the second bound variable does not appear anywhere in the scope of the quantifier of the first bound variable. For example, if it is known that Vx(Vy(P(x) ^ P(y))), an invalid use of bound variable would be to state that Vx(Vx(P(x) ^ P(x))).
Identity asserts that any variable is identical to itself.
Free variable allows the user to substitute a free variable for another free variable, given that the two are identical.
Aris can be set to use 'boolean mode', a mode used for boolean algebra. In boolean mode, only equivalence rules and boolean rules can be used.
This handy little rule allows one to use proofs one has already done. The premises don't have to match exactly, but they must be of the same form. Aris will check for each symbol it recognizes (connectives, quantifiers, parentheses, comma, and identity). These symbols must match exactly. Aris will then check that the sentences match the correct form, or rather that they appear in the correct order.
For example, if you already did a proof of the form:
And want to reuse it, then your reference sentences must be in the form ‘A’ <-> ‘B’, and ‘A’. They do not have to be in that order, however. Then, your conclusion must be the second half of the biconditional.
This is where Isar interoperability comes in. Instead of selecting a previous Aris proof, a .thy file can be used. Aris will attempt to translate it into a form that it can use, using most of the keywords as references, and the lemmas and theorems as goals. These are the sentences that can be proved. For more information, see Isabelle/Isar.
Given a subproof with premise 'P' and conclusion (the LAST sentence) 'Q', one can infer from subproof 'P → Q'.
This introduces a new sequence given a function. The sequence introduced this way must not have been used, and the final argument of the given function must be the bound variable of the sentence.
This rule is how Aris implements mathematical induction. ‘P(z(x))’ is the base case, and the inductive step is ‘P(x)’ → ‘P(s(x))’.
Sequence Logic, often abbreviated 'seqlog', is an alternative arithmetical representation system from the standard Peano Axioms.
Seqlog uses the symbols 's' (the sucessor function), 'z' (the zero function), 'n' (the sequence of natural numbers), 'v' (the value function), '0' (the number zero), and '\0' (null object).
Sequence Logic, often abbreviated 'seqlog', uses the following six axioms:
The first axiom states that no sucessor is the zero object, or, to put it differently, that the zero object is the first object. The second axiom states that no two different objects have the same sucessor. Using these two axioms, a 'Universal Sequence' can be defined, in a way similar to how the Peano Axioms define the natural numbers. The third axiom is the definition of a sequence, stating that the value under a given sequence 'S' of every object 'x' can be determined by a function. The rule 'sq' introduces such a sequence (see sq).
The fourth axiom defines zero, and introduces the natural numbers. It states that the first element in the natural numbers is zero. The fifth axiom defines the null object, and states that the natural numbers are defined on all inputs. The sixth axiom states that the natural numbers are one-to-one, or that if two numbers are equal, then the objects that they represent are equal.
Mathematical induction requires a base case, and an inductive step. In Aris, this is used in conjunction (ha see cn) with seqlog. For seqlog, the induction scheme is:
In addition to everything else Aris can do, Aris can also use other proofs from other systems with the lemma rule (see lm).
Aris will scan an Isar proof, which is a proof done using Isabelle, and look for certain keywords.
Standard definition of a function in seqlog.
Lemmas and theorems are treated the same. Lemmas end up as the goals of the proofs that Aris creates, and are the actual sentences that can be deduced. It takes the 'if-then' form of each lemma.