This is aris.info, produced by makeinfo version 5.1 from aris.texi.

This manual is for GNU Aris, the logical proof program.

   Copyright (C) 2013 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".
INFO-DIR-SECTION Mathematics
START-INFO-DIR-ENTRY
* GNU Aris: (aris).    The logical proof interface.
END-INFO-DIR-ENTRY

                     Version 1.3, 3 November 2008

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File: aris.info,  Node: Top,  Next: Introduction,  Up: (dir)

Aris
****

This manual is for GNU Aris, the logical proof program.

   Copyright (C) 2013 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 1.2, for 'aris' version 2.1

* Menu:

* Introduction::         Overview of Aris.
* Terms::                Terms used in this manual.
* Options::              The command-line options used for Aris.
* Basic Usage::          A quick walkthrough of Aris usage.
* Menu Options::         The keyboard shortcuts used in Aris.
* Rules Index::          The logical rules used in Aris.
* Customization::        Customizing GNU Aris.
* Submission::           Submitting proofs for grading.
* Sequence Logic::       An overview of Sequence Logic.
* Interoperability::     An explanation of interoperability.


File: aris.info,  Node: Introduction,  Next: Terms,  Prev: Top,  Up: Top

1 Introduction
**************

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.


File: aris.info,  Node: Terms,  Next: Options,  Prev: Introduction,  Up: Top

2 Terms
*******

Biconditional
     A biconditional is a connective that connects two sentences,
     denoted by '<->'.  A biconditional claims that 'sentence a if and
     only if sentence b' is a new sentence.  A biconditional can be
     inserted in Aris using the key combination 'CTRL+5' (*note Other
     Key Shortcuts::).

Conclusion
     A conclusion is a sentence that is derived from a combination of
     other sentences and a rule.  A focused conclusion will be
     highlighted in cyan.  A conclusion has a set of references
     associated with it, which are highlighted in violet.  Both of these
     colors can be changed using customization (*note Customization::).

Conditional
     A conditional is a connective that connects two sentences, denoted
     by '->'.  A conditional claims that 'if sentence a, then sentence
     b' is a new sentence.  A conditional can be inserted in Aris using
     the key combination 'CTRL+4' (*note Other Key Shortcuts::).

Conjunction
     A conjunction is a connective that connects two or more sentences,
     denoted by '^'.  A conjunction claims that 'sentence a and sentence
     b' is a new sentence.  A conjunction can be inserted in Aris using
     the key combination 'CTRL+7' (*note Other Key Shortcuts::).

Connective
     A connective is a logical symbol that connects one or more
     sentences.  The connectives used in system PSI are conjuction
     ('^'), disjunction ('v'), negation, ('~'), conditional ('->'), and
     biconditional ('<->').  In addition, system PSI recognizes the
     one-place connectives of the tautology ('T') and the contradiction
     ('!').

Contradiction
     A contradiction is a zero-place connective that stands on its own,
     denoted by '!'.  A contradiction represents something that is
     always false.  A contradiction is only used with the boolean rules
     (*note Boolean Rules::), and can be inserted using the key
     combination 'CTRL+6'.

Disjunction
     A disjunction is a connective that connects two or more sentences,
     denoted by 'v'.  A disjunction claims that 'sentence a or sentence
     b' is a new sentence.  A disjunction can be inserted in Aris using
     the key combination 'CTRL+\' (*note Other Key Shortcuts::).

Evaluate
     To evaluate a sentence means different things depending on the type
     of sentence.  At the very least, evaluation checks the sentence for
     text errors, i.e.  mis-matched parenthesis, etc.  Evaluating a
     conclusion checks that the sentence's text logically follows from
     the given references and its rule.  Evaluating a goal means
     checking the corresponding proof for a sentence with this exact
     same text.  To evaluate a sentence use the key combination 'CTRL+E'
     (*note Proof Windows::).

Evaluation Value
     An evaluation value is the value that appears to the right of a
     sentence's text entry.  It can either be a square, a light green
     check, an X, an X polygon, a pointer box, or a dark green check
     icon.  The square means that the sentence is awaiting evaluation.
     The light gren check means that either the conclusion logically
     follows from its references and rule, or that the goal has been
     found in the corresponding proof.  In the case of premises, this
     means that the premise has no syntactic errors.  The red X means
     that either the conclusion does not logically follow from its
     references and rule, or that the goal has not been found in the
     corresponding proof.  The X polygon means that there is a text
     error with this sentence.  The green check means that one of the
     conclusion's references has a text error.  The pointer box means
     that the conclusion is missing a rule.

Existential
     An existential is a quantifier that precedes the rest of sentence,
     denoted by '3'.  An existential claims that 'there exists at least
     on item that has property property P' is a new sentence, assuming
     that 'P' is a valid predicate.  An existential quantifier can be
     inserted into Aris using the key combination 'CTRL+3'.

Function Symbol
     A function symbol maps one object to another object.  These are
     always lower case.  Examples of a function symbol is seqlog's 'z'
     and 's' symbols (*note Sequence Logic::).

Goal
     A goal is a sentence that the user is looking to meet in a certain
     proof.  The goal window contains all of these sentences, and can be
     toggled by the key combination 'CTRL+L' (*note Proof Windows::).
     When a sentence in the proof matches a goal, the proof sentence's
     line number is highlighted in red, while the goal's line number is
     changed to match the proof sentence's line number.

Negation
     A negation is a connective that is inserted in front of a sentence,
     denoted by '~'.  A neagion claims that the opposite of the negation
     is true.  A negation can be inserted into Aris using the key
     combination 'CTRL+`' (*note Other Key Shortcuts::).

Null Object
     A null object is an object, denoted by 'nil'.  In Aris it resembles
     a null byte, '\0', and represents an undefined object in a
     sequence.  A null object can be inserted by using the key
     combination 'CTRL+.'.

Predicate
     A predicate is a type of logical symbol that denotes a property of
     or relation between one or more objects.  These always begin with
     capital letters, and generally use prefix notation.  Exceptions of
     this are the identity predicate ('='), the less than predicate
     ('<'), and the element of predicate.

Premise
     A premise is a sentence that is given.  A premise has no rule
     associated with it, nor does it have an evaluation value, unless
     there is an error in it.  Any variables introduced in a premise are
     not considered arbitrary.

Proof
     A proof is a set of sentences, beginning with a set of premises and
     ending with a set of conclusions, that the user is trying to derive
     something from.  The proof window is the main window that appears
     when the user opens up a proof.

Quantifier
     A quantifier is a type of logical symbol that claims something
     about the amount, or quantity, of an object that holds a specific
     property.  The quantifiers used in system PSI are the universal
     ('V'), and the existential ('3').

Reference
     A reference is a sentence that is being used to derive a
     conclusion.  A reference is highlighted in violet, and can be added
     or removed from the current sentence by holding down 'CTRL', and
     left-clicking on the desired reference.

Rules
     The rules in Aris are a combination of inference rules, equivalence
     rules, and predicate rules that the user can use to derive
     sentences.  The rules window (also referred to as the rules tablet)
     is shared amongst all of the proofs in Aris.  It can be toggled by
     the key combination 'CTRL+R' (*note Proof Windows::) from any proof
     window.  For the list of rules, *note Rules Index::.

Sentence
     A sentence is a line in Aris.  A sentence always consists of a text
     entry, a line number, and an evaluation value.

Subproof
     A proof within a proof.  To begin a subproof in Aris, use the key
     combination 'CTRL+B', and to exit one, use the key combination
     'CTRL+D'.

Tautology
     A tautology is a zero-place connective that stands on its own,
     denoted by 'T'.  A tautology represents something that is always
     true.  A tautology is only used with the boolean rules (*note
     Boolean Rules::), and can be inserted using the key combination
     'CTRL+1'.

Universal
     A universal is a quantifier that precedes the rest of sentence,
     denoted by 'V'.  A universal claims that 'for all items, they have
     property P' is a new sentence, assuming that 'P' is a valid
     predicate.  A universal quantifier can be inserted into Aris using
     the key combination 'CTRL+2'.

Variable
     A variable represents an object within a proof.  A variable is
     introduced when it is first used.  If the line that it is
     introduced in is a premise, the start of a subproof, or a line
     using existential instantiation (*note ei::), then it is *not*
     considered arbitrary.  Otherwise, it is.  Only the variables from
     lines that can be selected from the current line are taken into
     account when processing it.  This means that after a subproof is
     ended, then lines after it don't worry about variables introduced
     within it.


File: aris.info,  Node: Options,  Next: Basic Usage,  Prev: Terms,  Up: Top

3 Options
*********

'-a VARIABLE'
'--variable=VARIABLE'
     Use VARIABLE as a known variable in evaluation mode.  Prepend an
     '*' to specifiy that the variable is arbitrary.

'-b'
'--boolean'
     Start Aris in boolean mode.

'-c CONCLUSION'
'--conclusion=CONCLUSION'
     Use CONCLUSION as a conclusion in evaluation mode.  This flag can
     only be specified once.

'-e'
'--evaluate'
     Run Aris in evaluation mode.  This means that no GUI will be
     loaded.

'-f FILE'
'--file=FILE'
     Evaluate FILE if running Aris in evaluation mode, otherwise load
     FILE in Aris.  This flag can be specified multiple times.

'-g'
'--grade'
     Grades a file specified by the file flag.  This flag is ignored if
     used more than once.

'-l'
'--list'
     List the rules available in Aris, and exit.

'-p PREMISE'
'--premise=PREMISE'
     Use PREMISE as a premise in evalution mode.  This flag can be
     specified multiple times.

'-r RULE'
'--rule=RULE'
     Use RULE as a rule in evaluation mode.  This flag can only be
     specified once.

'-t TEXT'
'--text=TEXT'
     Simply check the correctness of TEXT in evaluation mode.

'-v'
'--verbose'
     Run Aris verbosely, printing status and error messages.

'-x'
'--latex=FILE'
     Convert FILE to a LaTeX proof file in evaluation mode.

'--version'
     Print the version of Aris and exit.

'-h'
'--help'
     Print a help message and exit.


File: aris.info,  Node: Basic Usage,  Next: Menu Options,  Prev: Options,  Up: Top

4 Basic Usage
*************

This chapter describes the basic usage of GNU Aris.

* Menu:

* Startup::     When the program is started.
* Connectives:: Using connectives.
* Adding::      Adding sentences to the proof.
* Selecting::   Selecting sentences.
* Syntax::      The syntax that Aris uses.


File: aris.info,  Node: Startup,  Next: Connectives,  Up: Basic Usage

4.1 Startup
===========

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.

   The initial layout of Aris is a single sentence.  From left to right,
the items of a sentence are: its line number, its text entry, its
evaluation value, and its rule.  The rule will not be initially visible,
since no rule has been selected.  In addition, premises do not have
rules, and thus the rule will not appear.


File: aris.info,  Node: Connectives,  Next: Adding,  Prev: Startup,  Up: Basic Usage

4.2 Connectives
===============

Aris has several connectives (*note Connectives: Terms.) When the
keyboard command for the desired connective is activated, the desired
connective will be inserted at the current cursor point, overwriting
selected text.

   For example, pressing 'Ctrl+7' inserts a conjunction.  This is the
character that looks like an upside-down 'v'.  This is also called a
'logical and'.


File: aris.info,  Node: Adding,  Next: Selecting,  Prev: Connectives,  Up: Basic Usage

4.3 Adding Sentences
====================

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.


File: aris.info,  Node: Selecting,  Next: Syntax,  Prev: Adding,  Up: Basic Usage

4.4 Selecting Sentences
=======================

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 (*note
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.


File: aris.info,  Node: Syntax,  Prev: Selecting,  Up: Basic Usage

4.5 Aris Syntax
===============

Aris expects a certain form of syntax.  Most of the connectives are
infix, which means that they are placed in between their arguments.  The
negation is one exception to this.

   Aris expects that all predicates start with an uppercase character.
Aris also expects that all function symbols begin with a lower case
character.  After this, Aris will except any combination of upper case
letters, lower case letters, numbers, or '_'.


File: aris.info,  Node: Menu Options,  Next: Rules Index,  Prev: Basic Usage,  Up: Top

5 Menu Options
**************

The main GUIs for Aris all have menu bars.  Each of the three types of
GUIs have different menu bars, and the options for each of these are
described in this section.

* Menu:

* Proof Windows::        The window for proofs.
* Rules Table::          The rules table that floats around.
* Other Key Shortcuts::  Miscellaneous keyboard shortcuts

   The keyboard shortcuts here (with the exception of the connectives)
can be changed using customization *Note Customization::.  The shortcuts
listed here are the default shortcuts for Aris.


File: aris.info,  Node: Proof Windows,  Next: Rules Table,  Up: Menu Options

5.1 Proof Windows
=================

These are the menu options for the main proof windows.  Each one can be
assigned a key command.

'New'
'Ctrl+N'
     Start a new proof.  A new window is opened for this proof.

'Open'
'CTRL+O'
     Open an existing proof in a new window.

'Save'
'CTRL+S'
     Save the current proof.

'Save As'
'CTRL+SHIFT+S'
     Save the current proof under a different name.

'Close'
'CTRL+W'
     Close the current proof.

'Quit'
'CTRL+Q'
     Exit Aris.  But since logic is so much fun, I doubt you'll ever
     want to use this one.

'Add Premise'
'CTRL+P'
     Insert a new premise at the end of the other premises.

'Add Conclusion'
'CTRL+J'
     Insert a new conclusion after the current line if it is a
     conclusion, or at the start of the conclusions if it is a premise.

'Add Subproof'
'CTRL+B'
     Begin a new subproof after the current line if it is a conclusion,
     or at the start of the conclusions if it is a premise.  This is
     unavailable in boolean mode, since subproofs can't be used.

'End Subproof'
'CTRL+D'
     End the current subproof, if there is one.  Otherwise, this doesn't
     do anything.  This is unavailable in boolean mode, since subproofs
     can't be used.

'Copy Line'
'CTRL+G'
     Copy the current line.

'Kill Line'
'CTRL+K'
     Kill, or cut, the current line.

'Insert Line'
'CTRL+I'
     Insert the copied/killed line.

'Evaluate Line'
'CTRL+E'
     Evaluate the logical validity of the current line.

'Evaluate Proof'
'CTRL+F'
     Evaluate the logical validity of the current proof.  This evaluates
     each line of the proof.

'Toggle Goals...'
'CTRL+L'
     Toggle the goal window for the current proof.

'Toggle Boolean Mode'
'CTRL+M'
     Toggle boolean mode for the current proof *Note Boolean Rules::.

'Import Proof'
     Import another proof into this one.  This will merge the premises
     of the other proof into the current one, and insert the goals of
     the other proof as conclusions.  In addition, it sets the
     conclusions' references as the premises, and sets them all to use
     the lemma rule *note lm::.

'Toggle Rules'
'CTRL+R'
     Toggle the rules window.

'Small'
'CTRL+-'
     Change the font size to small (8pt).

'Medium'
'CTRL+0'
     Change the font size to medium (12pt).

'Large'
'CTRL+='
     Change the font size to large (16pt).

'Custom'
     Change the font size to a custom size.  This menu option opens a
     dialog box with a numerical entry.

'Contents'
'F1'
     Display Aris help.  This is the only key command that cannot be
     modified.

'About GNU Aris'
     Displays information about GNU Aris.


File: aris.info,  Node: Rules Table,  Next: Other Key Shortcuts,  Prev: Proof Windows,  Up: Menu Options

5.2 Rules Table
===============

These are the commands for the rules window.  Many of them are the same
as for the main proof window.

'New'
'Ctrl+N'
     Start a new proof.  A new window is opened for this proof.

'Open'
'CTRL+O'
     Open an existing proof in a new window.

'Submit Proofs...'
     Submits all open proofs for grading.  There is more on this in the
     Submission session in this manual *Note Submission::.

'Quit'
'CTRL+Q'
     Exit Aris.

'Small'
'CTRL+-'
     Change the font size to small (8pt).

'Medium'
'CTRL+0'
     Change the font size to medium (12pt).

'Large'
'CTRL+='
     Change the font size to large (16pt).

'Custom'
     Change the font size to a custom size.  This menu option opens a
     dialog box with a numerical entry.

'Contents'
'F1'
     Display Aris help.  This is the only key command that cannot be
     modified.

'Customize...'
     Opens the customization dialog.  For more information on this, see
     *Note Customization::.

'About GNU Aris'
     Displays information about GNU Aris.


File: aris.info,  Node: Other Key Shortcuts,  Prev: Rules Table,  Up: Menu Options

5.3 Other Key Shortcuts
=======================

These are the keyboard shortcuts for each of the connectives.  Unlike
most of the other keyboard shortcuts, these cannot be modified.

'CTRL+7'
     Insert a conjunction ('^') into Aris.

'CTRL+\'
     Insert a disjunction ('v') into Aris.

'CTRL+`'
     Insert a negation ('-') into Aris.

'CTRL+4'
     Insert a conditional ('->') into Aris.

'CTRL+5'
     Insert a biconditional ('<->') into Aris.

'CTRL+2'
     Insert a universal ('V') into Aris.

'CTRL+3'
     Insert an existential ('3') into Aris.

'CTRL+6'
     Insert a tautology ('T') into Aris.

'CTRL+1'
     Insert a contradiction ('!') into Aris.

'CTRL+;'
     Insert an 'element of' predicate into Aris.

'CTRL+.'
     Insert a null object ('nil') into Aris.


File: aris.info,  Node: Rules Index,  Next: Customization,  Prev: Menu Options,  Up: Top

6 Rules Index
*************

The rules are divided into five categories: Inference, Equivalence,
Predicate, Boolean, and Miscellaneous.

* Menu:

* Inference Rules::     Rules that work on basic logic.
* Equivalence Rules::   Rules that work on sentence parts.
* Predicate Rules::     Rules that work with predicates and quantifiers.
* Boolean Rules::       Rules that handle Boolean Algebra
* Miscellaneous Rules:: Rules that don't fit in anywhere else.


File: aris.info,  Node: Inference Rules,  Next: Equivalence Rules,  Up: Rules Index

6.1 Inference Rules
===================

The premises of any of these rules can be in any order.

* Menu:

* mp::  Modus Ponens
* ad::  Addition
* sm::  Simplification
* cn::  Conjunction
* hs::  Hypothetical Syllogism
* ds::  Disjunctive Syllogism
* ex::  Excluded Middle
* cd::  Constructive Dilemma


File: aris.info,  Node: mp,  Next: ad,  Up: Inference Rules

6.1.1 Modus Ponens
------------------

     P -> Q
     P
     -----
     Q

   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.


File: aris.info,  Node: ad,  Next: sm,  Prev: mp,  Up: Inference Rules

6.1.2 Addition
--------------

     P
     -----
     P v Q v R v ...

   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.


File: aris.info,  Node: sm,  Next: cn,  Prev: ad,  Up: Inference Rules

6.1.3 Simplification
--------------------

     P ^ Q ^ R ^ ...
     -----
     P (or Q, or R, or ...)

   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.


File: aris.info,  Node: cn,  Next: hs,  Prev: sm,  Up: Inference Rules

6.1.4 Conjunction
-----------------

     P
     Q
     R
     -----
     P ^ Q ^ R

   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.


File: aris.info,  Node: hs,  Next: ds,  Prev: cn,  Up: Inference Rules

6.1.5 Hypothetical Syllogism
----------------------------

     P -> Q
     R -> S
     Q -> R
     -----
     P -> S

   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.


File: aris.info,  Node: ds,  Next: ex,  Prev: hs,  Up: Inference Rules

6.1.6 Disjunctive Syllogism
---------------------------

     ~P
     P v Q v R
     ~R
     -----
     Q

   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.


File: aris.info,  Node: ex,  Next: cd,  Prev: ds,  Up: Inference Rules

6.1.7 Excluded Middle
---------------------

     -----
     P v ~P

   A law of logic, excluded middle asserts that something is either
true, or it is not true.

   Excluded middle requires zero references.


File: aris.info,  Node: cd,  Prev: ex,  Up: Inference Rules

6.1.8 Constructive Dilemma
--------------------------

     P -> R
     P v Q
     Q -> S
     -----
     R v S

   Constructive Dilemma requires at least three references.


File: aris.info,  Node: Equivalence Rules,  Next: Predicate Rules,  Prev: Inference Rules,  Up: Rules Index

6.2 Equivalence Rules
=====================

Equivalence rules operate on any valid part of the sentence, and work
both ways.  Each equivalence rule requires one reference.

* Menu:

* im::  Implication
* dm::  DeMorgan
* as::  Association
* co::  Commutativity
* id::  Idempotence
* dt::  Distribution
* eq::  Equivalence
* dn::  Double Negation
* ep::  Exportation
* sb::  Subsumption
* Recursion:: Recursion in the equivalence rules


File: aris.info,  Node: im,  Next: dm,  Up: Equivalence Rules

6.2.1 Implication
-----------------

     P -> Q <=> ~P v Q

   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.


File: aris.info,  Node: dm,  Next: as,  Prev: im,  Up: Equivalence Rules

6.2.2 DeMorgan
--------------

     ~(P ^ Q) <=> ~P v ~Q
     ~(P v Q) <=> ~P ^ ~Q
     ~3x(P(x)) <=> Vx(~P(x))
     ~Vx(P(x)) <=> 3x(~P(x))

   DeMorgan's Laws.


File: aris.info,  Node: as,  Next: co,  Prev: dm,  Up: Equivalence Rules

6.2.3 Association
-----------------

     P ^ (Q ^ R) <=> P ^ Q ^ R
     P v (Q v R) <=> P v Q v R

   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.


File: aris.info,  Node: co,  Next: id,  Prev: as,  Up: Equivalence Rules

6.2.4 Commutativity
-------------------

     P ^ Q ^ R <=> Q ^ R ^ P
     P v Q v R <=> Q v R v P

   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'.


File: aris.info,  Node: id,  Next: dt,  Prev: co,  Up: Equivalence Rules

6.2.5 Idempotence
-----------------

     P ^ P ^ Q ^ R ^ R ^ R <=> P ^ Q ^ R
     P v P v Q v R v R v R <=> P v Q v R

   Idempotence claims that 'I like blue and I like blue' is the same as
saying 'I like blue'.


File: aris.info,  Node: dt,  Next: eq,  Prev: id,  Up: Equivalence Rules

6.2.6 Distribution
------------------

     P ^ (Q0 v Q1 v ...  v Qn) <=> (P ^ Q0) v (P ^ Q1) v (P ^ Q2) v ...
     v (P ^ Qn)
     P v (Q0 v Q1 ^ ...  ^ Qn) <=> (P v Q0) ^ (P v Q1) ^ (P v Q2) ^ ...
     ^ (P v Qn)
     3x(P(x) v Q(x)) <=> 3x(P(x)) v 3x(Q(x))
     Vx(P(x) ^ Q(x)) <=> Vx(P(x)) ^ Vx(Q(x))


File: aris.info,  Node: eq,  Next: dn,  Prev: dt,  Up: Equivalence Rules

6.2.7 Equivalence
-----------------

     P <-> Q <=> (P -> Q) ^ (Q -> R)

   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.


File: aris.info,  Node: dn,  Next: ep,  Prev: eq,  Up: Equivalence Rules

6.2.8 Double Negation
---------------------

     ~~P <=> P

   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.


File: aris.info,  Node: ep,  Next: sb,  Prev: dn,  Up: Equivalence Rules

6.2.9 Exportation
-----------------

     (P ^ Q) -> R <=> P -> (Q -> R)

   This is one of the few equivalence rules that deals with
conditionals.


File: aris.info,  Node: sb,  Next: Recursion,  Prev: ep,  Up: Equivalence Rules

6.2.10 Subsumption
------------------

     P ^ (P v Q) <=> P
     P v (P ^ Q) <=> P

   Also called absorption.  This rule can be used in Boolean mode.


File: aris.info,  Node: Recursion,  Prev: sb,  Up: Equivalence Rules

6.2.11 Recursion in the Equivalence Rules
-----------------------------------------

For the convenience of the user, the equivalence rules work recursively.
For example

     ~(~P v Q) v (~R v S)
     -----
     (P -> Q) -> (R -> S)

   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.


File: aris.info,  Node: Predicate Rules,  Next: Boolean Rules,  Prev: Equivalence Rules,  Up: Rules Index

6.3 Predicate Rules
===================

The predicate rules are the rules that work specifically with predicate
logic.

* Menu:

* ug:: Universal Generalization
* ui:: Universal Instantiation
* eg:: Existential Generalization
* ei:: Existential Instantiation
* bv:: Bound Variable
* nq:: Null Quantifier
* pr:: Prenex
* ii:: Identity
* fv:: Free Variable


File: aris.info,  Node: ug,  Next: ui,  Up: Predicate Rules

6.3.1 Universal Generalization
------------------------------

     P(a) ; a is arbitrary
     -----
     Vx(P(x))

   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.  *note Variables:
Terms.

   Universal Generalization uses exactly one reference.


File: aris.info,  Node: ui,  Next: eg,  Prev: ug,  Up: Predicate Rules

6.3.2 Universal Instantiation
-----------------------------

     Vx(P(x))
     -----
     P(a)

   Universal Generalization claims that if a property 'P' is true for
all objects, then it must be true for an object 'a'.

   Universal Generalization uses exactly one reference.


File: aris.info,  Node: eg,  Next: ei,  Prev: ui,  Up: Predicate Rules

6.3.3 Existential Generalization
--------------------------------

     P(a)
     -----
     3x(P(x))

   Existential Generalization claims that if 'P' is true for some
object, then there exists an object for which 'P' is true.

   Existential Generalization uses exactly one reference.


File: aris.info,  Node: ei,  Next: bv,  Prev: eg,  Up: Predicate Rules

6.3.4 Existential Instantiation.
--------------------------------

     3x(P(x))
     -----
     P(a) ; a must not have been used before

   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.  *note Variables: Terms.

   Existential Instantiation uses exactly one reference.


File: aris.info,  Node: bv,  Next: nq,  Prev: ei,  Up: Predicate Rules

6.3.5 Bound Variable
--------------------

     Vx(P(x)) <=> Vy(P(y))
     3x(P(x)) <=> 3y(P(y))

   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))).

   As an equivalence rule, bound variable uses only one reference, and
can work on any part of the sentence.


File: aris.info,  Node: nq,  Next: pr,  Prev: bv,  Up: Predicate Rules

6.3.6 Null Quantifier
---------------------

     Vx(P(a)) <=> P(a)

   If a quantifier's bound variable does not appear in its scope, then
the quantifier is said to be null, and can be removed using Null
Quantifier.

   As an equivalence rule, null quantifier can be used on any part of
the sentence, and only uses one reference.


File: aris.info,  Node: pr,  Next: ii,  Prev: nq,  Up: Predicate Rules

6.3.7 Prenex
------------

     3x(P(x) ^ Q(a)) <=> 3x(P(x)) ^ Q(a)
     Vx(P(x) ^ Q(a)) <=> Vx(P(x)) ^ Q(a)
     3x(P(x) v Q(a)) <=> 3x(P(x)) v Q(a)
     Vx(P(x) v Q(a)) <=> Vx(P(x)) v Q(a)

   The Prenex Laws are used to move quantifiers to the start of the
sentence.

   Prenex uses only one reference, and, being an equivalence rule, can
be used on any part of the sentence.


File: aris.info,  Node: ii,  Next: fv,  Prev: pr,  Up: Predicate Rules

6.3.8 Identity
--------------

     -----
     a = a

   Identity asserts that any variable is identical to itself.

   Identity does not use any references.


File: aris.info,  Node: fv,  Prev: ii,  Up: Predicate Rules

6.3.9 Free Variable
-------------------

     a = b
     P(a)
     -----
     P(b)

   Free Variable allows the user to substitute a free variable for
another free variable, given that the two are identical.

   Free Variable uses exactly two references.


File: aris.info,  Node: Boolean Rules,  Next: Miscellaneous Rules,  Prev: Predicate Rules,  Up: Rules Index

6.4 Boolean Rules
=================

Aris can be set to use 'boolean mode', a mode used for boolean algebra.
In boolean mode, only equivalence rules that handle negations,
conjunctions, or disjunctions and boolean rules can be used.  In
standard mode, the boolean rules can still be used, however.

* Menu:

* bi::  Boolean Identity
* bn::  Boolean Negation
* bd::  Boolean Dominance
* sn::  Symbol Negation


File: aris.info,  Node: bi,  Next: bn,  Up: Boolean Rules

6.4.1 Boolean Identity
----------------------

     A ^ T <=> A
     A v !  <=> A

   Boolean Identity claims that the conjunction of a sentence with a
tautology is logically equivalent to the sentence.  It also claims that
the disjunction of a sentence an a contradiction is logically equivalent
to the sentence.


File: aris.info,  Node: bn,  Next: bd,  Prev: bi,  Up: Boolean Rules

6.4.2 Boolean Negation
----------------------

     A ^ ~A <=> !
     A v ~A <=> T

   Boolean Negation claims that the conjunction of a setentence and its
contradiction is a contradiction, and the disjunction of a sentence and
its negation is a tautology.


File: aris.info,  Node: bd,  Next: sn,  Prev: bn,  Up: Boolean Rules

6.4.3 Boolean Dominance
-----------------------

     A ^ !  <=> !
     A v T <=> T

   Boolean Dominance claims that the conjunction of a sentence and a
contradiction is logically equivalent to a contradiction.  It also
claims that the disjunction of a sentence and a tautology is logically
equivalent to a tautology.


File: aris.info,  Node: sn,  Prev: bd,  Up: Boolean Rules

6.4.4 Symbol Negation
---------------------

     ~T <=> !
     ~!  <=> T

   Symbol Negation claims that a tautology is the opposite of a
contradiction.


File: aris.info,  Node: Miscellaneous Rules,  Prev: Boolean Rules,  Up: Rules Index

6.5 Miscellaneous Rules
=======================

* Menu:

* lm:: Lemma
* sp:: Subproof
* sq:: Sequence
* in:: Induction


File: aris.info,  Node: lm,  Next: sp,  Up: Miscellaneous Rules

6.5.1 Lemma
-----------

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:

     A <-> B
     A
     -----
     B

   And want to reuse it, then your reference sentences must be in the
form 'A' <-> 'B', and 'A'.  In general, 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 *note
Isabelle/Isar::.


File: aris.info,  Node: sp,  Next: sq,  Prev: lm,  Up: Miscellaneous Rules

6.5.2 Subproof
--------------

Given a subproof with premise 'P' and conclusion (the LAST sentence)
'Q', one can infer from subproof 'P -> Q'. In some circles, this is
called conditional introduction.


File: aris.info,  Node: sq,  Next: in,  Prev: sp,  Up: Miscellaneous Rules

6.5.3 Sequence
--------------

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.


File: aris.info,  Node: in,  Prev: sq,  Up: Miscellaneous Rules

6.5.4 Induction
---------------

     Vx(P(z(x)) ^ (P(x) -> P(s(x))))
     -----
     Vx(P(x))

   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))'.


File: aris.info,  Node: Customization,  Next: Submission,  Prev: Rules Index,  Up: Top

7 Customization
***************

GNU Aris can be customized using the customization dialog, or manually
through the customization file.  The customization dialog can be
accessed through the rules table, under 'Help'.

* Menu:

* Dialog::      The customization dialog.
* Config File:: The customization file format.


File: aris.info,  Node: Dialog,  Next: Config File,  Up: Customization

7.1 Customization Dialog
========================

When the dialog appears, there are several tabs.  These are explained as
follows:

Main Keys
     This tab allows the user to customize the keyboard shortcuts that
     Aris uses with the proof and rules table menus.  Each entry
     corresponds to a menu item.  The only option that can not be edited
     is the 'Contents' menu keyboard shortcut.

Goal Keys
     This tab allows the user to customize the keyboard shortcuts that
     Aris uses with the goal menus.

Display
     This tab allows the user to customize the display settings.
     Included in here are the font size presets, which will be set to
     the indicated size when activated; the default font size, which
     Aris will be in when loaded; and the color preferences, which will
     change the colors that Aris hilights different objects in.

Grade Server
     This tab allows the user to set preferences specific to the grade
     server.  The two options here are for the IP address of the grading
     server, and the password used to authenticate into the server.
     (*note Submission::)

   For a description of the format of key commands, see *note Config
File::.


File: aris.info,  Node: Config File,  Prev: Dialog,  Up: Customization

7.2 Customization File
======================

The config file uses s-expressions to store the customization file.  It
is stored under the home directory, and called '.aris'.  There are
several key words that it recognizes:

'key-cmd'
'(key-cmd 'cmd' 'key')'

     Assigns menu 'cmd' to keyboard shortcut 'key'.  The keyboard
     shortcuts are all in the same format, which is either 's' or 'c', a
     plus sign, then a letter.  A 'c' before the letter means 'Hold
     control, and press the key', and 's' means the same except with the
     shift key.

'font-size'
'(font-size 'type' size)'

     Assigns 'size' to font type 'type'.  The 'type' key word can be
     either 'Small', which means set the small font preset, 'Medium',
     which means set the medium font preset, 'Large', which means set
     the large font preset, or 'Default', which means set the font size
     that Aris loads up with intially.

'color-pref'
'(color-pref 'type' color)'

     Assigns 'color' to color preference 'type'.  The 'color' key word
     is in hexidecimal.

'grade'
'(grade 'key' 'value')'

     This allows customization of the grade server's information (*note
     Submission::).  The two options for 'key' are 'ip' and 'pass'.  The
     'ip' key sets the grade server's IP address, and the 'pass' sets up
     the password GNU Aris will use for authentication.


File: aris.info,  Node: Submission,  Next: Sequence Logic,  Prev: Customization,  Up: Top

8 Proof Submission
******************

GNU Aris allows users to submit their proofs to a grading server, which
allows instructors to use Aris in their classes.  Aris submits proofs
through FTP, and allows users to indicate an email for themselves and an
optional email for their instructors.

   Submission allows for all open proofs to be submitted.  This is done
by specifying a problem designation ('11.10', 'nats', etc.).  Only those
that have designations are submitted.  The designation is changed by
editing the text box next to each file name in the submission dialog
box.

   Grading runs by checking the correctness of the entire proof.  It is
the responsibility of the grading server to confirm that the proof is
the correct proof.

   When the files are submitted, Aris submits them to the server, along
with a directive file.  The directive file will be named
'USER.directive', where 'USER' is the base name of the email address
specified.  The files submitted will be renamed to be 'BASE-USER.tle',
where 'BASE' is the original basename of the file.  This prevents
filename conflicts on the FTP server.  The grading server will then run
Aris in grade mode (*note Options::), and use the email provided to
email the results back to the user.  If the user specified an
instructor's email address, then Aris will CC the instructor.

   Sample scripts are included with GNU Aris.  These are the files
'doc/collect.sh' and 'doc/collect.el'.  The grade server would run
'collect.sh', which will call GNU Emacs in batch mode while loading
'collect.el'.  It is 'collect.el' that handles the emails.


File: aris.info,  Node: Sequence Logic,  Next: Interoperability,  Prev: Submission,  Up: Top

9 Sequence Logic
****************

Sequence Logic, often abbreviated 'seqlog', is an alternative
arithmetical representation system from the standard Peano Axioms in
First-Order Logic.  Seqlog's original purpose was allowing more natural
definition of recursive functions in FOL.

   Seqlog uses the symbols 's' (the sucessor function), 'z' (the zero
function), 'v' (the value function), and '\0' (null object).

* Menu:

* Axioms::      The axioms used by seqlog.
* Induction::   An overview of the induction scheme.


File: aris.info,  Node: Axioms,  Next: Induction,  Up: Sequence Logic

9.1 Axioms
==========

Sequence Logic, often abbreviated 'seqlog', uses the following six
axioms:

   * VxVy(~s(x) = z(y))
   * VxVy(s(x) = s(y) -> x = y)
   * Vx(v(S,x) = f_S(x))
   * Vx(v(\0,x) = \0)

   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
(*note sq::).  The fourth axiom defines the nil object.  This is a lot
like 'NULL' in C, or 'nil' in lisp.

   The natural numbers are defined as a sequence.  For example, Vx(x =
nat -> VyVz(v(nat,y) = v(nat,z) -> y = z) ^ Vy(~v(nat,y) = \0)).  This
will define an infinite sequence (2nd part), that is one-to-one (1st
part).  Then, to define zero, one simply states Vx(v(nat,z(x)) = 0).
This means that the zero'th element of the 'nat' sequence is the object
'0'.


File: aris.info,  Node: Induction,  Prev: Axioms,  Up: Sequence Logic

9.2 Induction
=============

Mathematical induction requires a base case, and an inductive step.  In
Aris, this is used in conjunction with seqlog.  For seqlog, the
induction scheme is:

     Vx(P(z(x)) ^ (P(x) -> P(s(x)))) -> Vx(P(x))


File: aris.info,  Node: Interoperability,  Prev: Sequence Logic,  Up: Top

10 Interoperability
*******************

In addition to everything else Aris can do, Aris can also use other
proofs from other systems with the lemma rule (*note lm::).

* Menu:

* Isabelle/Isar::        The proof system Isar.


File: aris.info,  Node: Isabelle/Isar,  Up: Interoperability

10.1 Isar Interoperability
==========================

Aris will scan an Isar proof, which is a proof done using Isabelle, and
look for certain keywords.  This is still being tested, and doesn't work
fully yet.  This section will be updated as more of this is implemented.

* Menu:

* fun::                 The 'fun' keyword.
* type_synonym::        The 'type_synonym' keyword.
* lemma/theorem::       The 'lemma' keyword.
* case::                The 'case' keyword.
* primrec::             The 'primrec' keyword.
* definition::          The 'definiton' keyword.
* datatype::            The 'datatype' keyword.
* class::               The 'class' keyword.
* instance::            The 'instance' keyword.
* other::               Everything else.


File: aris.info,  Node: fun,  Next: type_synonym,  Up: Isabelle/Isar

10.1.1 fun keyword.
-------------------

Standard definition of a function in seqlog.


File: aris.info,  Node: type_synonym,  Next: lemma/theorem,  Prev: fun,  Up: Isabelle/Isar

10.1.2 type_synonym keyword
---------------------------


File: aris.info,  Node: lemma/theorem,  Next: case,  Prev: type_synonym,  Up: Isabelle/Isar

10.1.3 lemma and theorem keywords
---------------------------------

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.


File: aris.info,  Node: case,  Next: primrec,  Prev: lemma/theorem,  Up: Isabelle/Isar

10.1.4 case keyword
-------------------


File: aris.info,  Node: primrec,  Next: definition,  Prev: case,  Up: Isabelle/Isar

10.1.5 primrec keyword
----------------------


File: aris.info,  Node: definition,  Next: datatype,  Prev: primrec,  Up: Isabelle/Isar

10.1.6 definition keyword
-------------------------


File: aris.info,  Node: datatype,  Next: class,  Prev: definition,  Up: Isabelle/Isar

10.1.7 datatype keyword
-----------------------


File: aris.info,  Node: class,  Next: instance,  Prev: datatype,  Up: Isabelle/Isar

10.1.8 class keyword
--------------------


File: aris.info,  Node: instance,  Next: other,  Prev: class,  Up: Isabelle/Isar

10.1.9 instance keyword
-----------------------


File: aris.info,  Node: other,  Prev: instance,  Up: Isabelle/Isar

10.1.10 everything else
-----------------------



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