                     Version 1.3, 3 November 2008

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====================================================

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

   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".


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

Aris
****

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 1.0, for `aris' version 1.7

* 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.
* Keyboard Shortcuts::   The keyboard shortcuts used in Aris.
* Rules Index::          The logical rules used in Aris.
* 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
     Keyboard 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.

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 Keyboard 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 Keyboard
     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 Keyboard 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 Keyboard Shortcuts::).

Evaluation Value
     An evaluation value is the value that appears to the right of a
     sentence's text entry.  It can either be ` ', `T', `F', `*', `#',
     or `?'.  ` ' means that the sentence is awaiting evaluation,
     except in the case of premises, where this could also mean that
     there are no errors with it.  `T' means that either the conclusion
     logically follows from its references and rule, or that the goal
     has been found in the corresponding proof.  `F' 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.  `*' means that there is a text error with this sentence.
     `?' means that one of the conclusion's references has a text
     error.  `#' 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 Keyboard
     Shortcuts::).  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 Keyboard 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 (`='), and the less than 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.

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 Keyboard
     Shortcuts::) 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'.



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.

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

`-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.

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

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

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



File: aris.info,  Node: Basic Usage,  Next: Keyboard Shortcuts,  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.


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.


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.


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,  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: Keyboard Shortcuts,  Next: Rules Index,  Prev: Basic Usage,  Up: Top

5 Keyboard Shortcuts
********************

`CTRL+N'
     Start a new proof.

`CTRL+O'
     Open an existing proof.

`CTRL+S'
     Save the current proof.

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

`CTRL+W'
     Close the current proof.

`CTRL+Q'
     Exit Aris.

`CTRL+P'
     Insert a new premise after the current line.

`CTRL+J'
     Insert a new conclusion after the current line.

`CTRL+B'
     Begin a new subproof after the current line.

`CTRL+D'
     End the current subproof, if there is one.

`CTRL+G'
     Copy the current line.

`CTRL+K'
     Kill the current line.

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

`CTRL+E'
     Evaluate the current line.

`CTRL+F'
     Evaluate the current proof.

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

`CTRL+M'
     Toggle boolean mode for the current proof.

`CTRL+R'
     Toggle the rules window.

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

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

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

`F1'
     Display Aris help.

`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: Sequence Logic,  Prev: Keyboard Shortcuts,  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)


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


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.


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'.


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.


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.


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))).


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

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

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


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)


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.


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.


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 and boolean rules can be used.

* 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


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

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

     A ^ ~A <=> !

     A v ~A <=> T


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

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

     A ^ ! <=> !

     A v T <=> T


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

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

     ~T <=> !

     ~! <=> T


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'.  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'.


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: Sequence Logic,  Next: Interoperability,  Prev: Rules Index,  Up: Top

7 Sequence Logic
****************

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).

* Menu:

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


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

7.1 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(n,z(x)) = 0)

   * Vx(~v(n,x) = \0)

   * VxVy(v(n,x) = v(n,y) -> x = y)

   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 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.


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

7.2 Induction
=============

Mathematical induction requires a base case, and an inductive step.  In
Aris, this is used in conjunction (ha *note cn::) 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

8 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

8.1 Isar Interoperability
=========================

Aris will scan an Isar proof, which is a proof done using Isabelle, and
look for certain keywords.

* 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

8.1.1 fun keyword.
------------------

Standard definition of a function in seqlog.


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

8.1.2 type_synonym keyword
--------------------------


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

8.1.3 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

8.1.4 case keyword
------------------


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

8.1.5 primrec keyword
---------------------


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

8.1.6 definition keyword
------------------------


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

8.1.7 datatype keyword
----------------------


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

8.1.8 class keyword
-------------------


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

8.1.9 instance keyword
----------------------


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

8.1.10 everything else
----------------------



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