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Prover.htm
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<!DOCTYPE HTML>
<html>
<head>
<meta charset="utf-8">
<title> Euclid Automated Prover - v1.0.0.0 </title>
<link rel='stylesheet' media='all' href='CSS\symbols.css'>
<!-- engine scripts -->
<script src="JS\Object.js"></script>
<script src="JS\Behaviors.js"></script>
<script src="JS\Axioms.js" defer></script>
<script src="JS\index.js" defer></script>
<!---------------------------->
<script src="JS\MD5.js"></script>
<script src="JS\Symbols.js"></script>
<div id=cssMENU>
<a href=# class=cssCANCEL>×</a>
<div class=cssFRAME>
<div id=txtINCLUDES class=cssHELP>
<textarea id=srcINCLUDES style='width:100%;height:100%;overflow:auto'>
# Elementary Math
# A-Z constants; a-z expression(s) or constants
( a ) = a
{ a } = a
[ a ] = a
plus = +
minus = -
a + 0 = 0 + a = a
a - 0 = a
a - b + b = a
a minus b = neg b plus a
b = { b cdot c } over c
1 = c over c
# Secondary Mathematics
# Engineering Analysis
# differentiation
dx dn 1 { c times x } = c times { dx dn { x } }
{ x plus y } ' = x ' plus y '
{ x cdot y } ' = { x ' cdot y } plus { x cdot y ' }
{ x over y } ' = { { x ' cdot y } minus { x cdot y ' } } over { y raised 2 }
dv dn = dv dx cdot dx dn # chain rule
{ x raised c } ' = c cdot x raised { c minus 1 }
{ exp raised x } ' = exp raised x
{ c raised x } ' = c raised x cdot ln c
{ sin x } ' = cos x
{ cos x } ' = neg sin x
{ tan x } ' = sec raised 2 x
{ cot x } ' = neg csc raised 2 x
{ sinh x } ' = cosh x
{ coshx } ' = sinh x
{ ln x } ' = 1 over x
{ log _ a x } ' = { log _ a exp {} } over x
{ arc sin x } ' = 1 over { sqrt { 1 minus x raised 2 } }
{ arc cos x } ' = neg { 1 over { sqrt { 1 minus x raised 2 } } }
{ arc tan x } ' = 1 over { 1 plus x raised 2 }
{ arc cot x } ' = neg 1 over { 1 plus x raised 2 }
# integration
int { u times v ' } dx = { u times v } minus int { u ' times v } dx
int { x raised c } dx = { { x raised { c plus 1 } } over { c plus 1 } } plus n , given n noteq 1
int { 1 over x } dx = ln { abs { x } } plus c
int { exp raised { c x } } dx = { 1 over c } times { exp raised { c x } } + n
int { sin x } dx = neg { cos x } plus c
int { cos x } dx = { sin x } plus c
int { tan x } dx = neg { ln { abs { cos x } } } plus c
int { cot x } dx = ln { abs { sin x } } plus c
int { sec x } dx = ln { abs { sec x plus tan x } } plus c
int { csc x } dx = ln { abs { csc x minus cot x } } plus c
int { 1 over { x raised 2 plus c raised 2 } } dx = { 1 over c } times { arc tan { x over c } } + n
int { 1 over sqrt { c raised 2 minus x raised 2 } } dx = arc sin { x over c } + n
int { 1 over sqrt { x raised 2 plus c raised 2 } } dx = inv { sinh {} } times { x over c } plus n
int { 1 over sqrt { x raised 2 minus c raised 2 } } dx = inv { cosh {} } times { x over c } plus n
int { sin raised 2 x } dx = { 1 over 2 } times x minus { { 1 over 4 } times sin { 2 x } } + c
int { cos raised 2 x } dx = { 1 over 2 } times x plus { { 1 over 4 } times sin { 2 x } } + c
int { tan raised 2 x } dx = tan x minus x + c
int { cot raised 2 x } dx = neg { cot x } minus x + c
int { ln x } dx = x ln x minus x plus c
int { exp raised { a x } cdot sin { b x } } dx = { exp raised { a x } over { a raised 2 plus b raised 2 } } cdot { a sin { b x } minus b cos { b x } } plus c
int { exp raised { a x } times cos { b x } } dx = { exp raised { a x } over { a raised 2 plus b raised 2 } } times { a cos { b x } plus b sin { b x } } plus c
# Polar Coords
x = r times { cos theta }
y = r times { sin theta }
r = sqrt { x raised 2 plus y raised 2 }
theta = arc tan { y over x }
dx dy = r dr dtheta
# Series
1 over { 1 minus x } = sum from { m = 0 } to infinity { x raised m } , given abs { x } lessthan 1
exp raised x = sum from { m = 0 } to infinity { x raised m over m fact }
sin x = sum from { m = 0 } to infinity { { neg 1 raised m times x raised { 2 m plus 1 } } over { { 2 m plus 1 } fact } }
cos x = sum from { m = 0 } to infinity { { neg 1 raised m times x raised { 2 m } } over { { 2 m } fact } }
ln { 1 minus x } = neg sum from { m = 0 } to infinity { { x raised m } over m } , given { abs { x } lessthan 1 }
arc tan x = sum from { m = 0 } to infinity { neg 1 raised m times { x raised { 2 times m plus 1 } } over { 2 times m plus 1 } } , given { abs { x } lessthan 1 }
# Vectors
a cdot b = { a_1 times b_1 } plus { a_2 times b_2 } plus { a_3 times b_3 }
a cross b = left [ matrix { i # j # k ## a_1 # a_2 # a_3 ## b_1 # b_2 # b_3 } right ]
grad f = { partial f over partial x } vec i plus { partial f over partial y } vec j plus { partial f over partial z } vec k
div v = { partial v_1 over partial x } plus { partial v_2 over partial y } plus { partial v_3 over partial z }
curl v = left [ matrix { i # j # k ## { partial over partial x } # { partial over partial y } # { partial over partial z } ## v_1 # v_2 # v_3 } right ]
# SI units
exp = 2.718281828459045
sqrt exp = 1.648721270700128
exp raised 2 = 7.389056098930650
pi = 3.141592653589793
pi raised 2 = 9.869604401089358
sqrt pi = 1.772453850905516
log_10 pi = 0.497149872694133
ln pi = 1.144729885849400
log_10 exp = 0.434294481903251
ln 10 = 2.302585092994045
sqrt 2 = 1.414213562373095
nroot 3 2 = 1.259921049894873
sqrt 3 = 1.732050807568877
nroot 3 3 = 1.442249570307408
ln 2 = 0.693147180559945
ln 3 = 1.098612288668109
gamma = 0.577215664901523
ln gamma = neg { 0.549539312981644 }
1 degrees = 0.017453292519943 rad
1 rad = 57.295779513083230 degrees = 57 degrees 17 minutes 44.806 seconds { 57 cdot 27'44.806" }
# Signal Processing
# Calculus
# Reciprocal identities
sin x = 1 over { csc x }
csc x = 1 over { sin x }
sec x = 1 over { cos x }
cos x = 1 over { sec x }
tan x = 1 over { cot x }
cot x = 1 over { tan x }
# Tangent & Cotangent identities
tan x = { sin x } over { cos x }
cot x = { cos x } over { sin x }
# Pythagorean identities
sin raised 2 x plus cos raised 2 x = 1
1 plus { tan raised 2 x } = sec raised 2 x
1 plus { cot raised 2 x } = csc raised 2 x
# function identities
sin { { pi over 2 } minus x } = cos x
csc { { pi over 2 } minus x } = sec x
sec { { pi over 2 } minus x } = csc x
cos { { pi over 2 } minus x } = sin x
tan { { pi over 2 } minus x } = cot x
cot { { pi over 2 } minus x } = tan x
# Reduction formulas
sin { neg x } = neg sin x
csc { neg x } = neg csc x
sec { neg x } = sec x
cos { neg x } = cos x
tan { neg x } = neg tan x
cot { neg x } = neg cot x
# Sum & difference formulas
sin { u plusminus v } = sin u cos v plusminus cos sin v
cos { u plusminus v } = cos u cos v plusminus sin u sin v
tan { u plusminus v } = { tan u plusminus tan v } over { 1 minusplus { tan u tan v } }
# Double-angle formulas
sin { 2 times u } = 2 sin u cos u
cos { 2 times u } = cos raised 2 u minus sin raised 2 u minus sin raised 2 u = 2 times { cos raised 2 y } minus 1 = 1 minus 2 sin raised 2 u
tan { 2 times u } = { 2 times tan u } over { 1 minus tan raised 2 u }
# Power reducing formula
sin raised 2 u = { 1 minus cos { 2 times u } } over 2
cos raised 2 u = { 1 plus cos { 2 times u } } over 2
tan raised 2 u = { 1 minus cos { 2 times u } } over { 1 plus cos { 2 times u } }
# Sum-to-product formulas
sin u plus sin v = 2 times sin { { u plus v } over 2 } times cos { { u minus v } over 2 }
sin u minus sin v = 2 times cos { { u plus v } over 2 } times sin { { u minus v } over 2 }
cos u plus cos v = 2 times cos { { u plus v } over 2 } times cos { { u minus v } over 2 }
cos u minus cos v = neg 2 times cos { { u plus v } over 2 } times sin { { u minus v } over 2 }
# Product-to-Sum formulas
sin u times sin v = { 1 over 2 } times cos { u minus v } minus cos { u plus v }
cos u times cos v = { 1 over 2 } times cos { u minus v } plus cos { u plus v }
sin u times cos v = { 1 over 2 } times sin { u plus v } plus cos { u minus v }
cos u times sin v = { 1 over 2 } times sin { u minus v } minus sin { u minus v }
# Trigonometric functions, given theta = { 0 lessthan theta lessthan { pi / over 2 } }
let x = coord x in x plane and y = coord y in y plane and
let r = sqrt { x raised 2 plus y raised 2 } then
sin theta = y over r
cos theta = x over y
tan theta = y over x
csc theta = y over r
sec theta = r over x
tan theta = x over y
# Factors & zeros of polynamials
let p of x = a_n x raised n + a_{ n minus 1 } x raised { x minus 1 } + ellipses + a_1 x + a_0, if { p of a == 0 } then a isa zero of p of x and { x minus a } isa factor of p of x
# Fundamental theorem of algebra
let f of n = { a raised n } then
polynomial hasa f of n hasa zeros == n and zeros isa { imaginary || real } , and
polynominal hasa real hasa degree % 2 != 0 and polynominal hasa real hasa zeros == { n % 2 != 0 }
# Quadratic formula
if p of x = a x raised 2 plus b x plus c and { 0 le b raised 2 minus 4 a c } then p of x hasa real hasa zeros = { neg b plusminus sqrt { b raised 2 minus 4 a c } } over { 2 a }
# Special factors
x raised 2 minus a raised 2 = { x minus a } times { x plus a }
x raised 3 plus a raised 3 = { x plus a } times { x raised 2 minus a x plus a raised 2 }
x raised 3 minus a raised 3 = { x minus a } times { x raised 2 plus a x plus a raised 2 }
x raised 4 minus a raised 4 = { x raised 2 minus a raised 2 } times { x raised 2 plus a raised 2 }
# Binomial theorem
{ x plus y } raised n = { x raised n plus n x raised { n minus 1 } } times y plus { { n times { n minus 1 } } over 2 fact } times { x raised { n minus 2 } times y raised 2 } plus ... plus n times x times y raised { n minus 1 } plus y raised n
{ x minus y } raised n = { x raised n minus n x raised { n minus 1 } } times y plus { { n times { n minus 1 } } over 2 fact } times { x raised { n minus 2 } times y raised 2 } minus ... plusminus n times x times y raised { n minus 1 } minusplus y raised n
# Rational zero theorem
let p of x = sum from { n = 0 } to infinity { a_n x raised n plus a_{ n minus 1 } x raised { n minus 1 } } plus sum from { x = 0 } to n { a_1 x plus a_0 } likewise
let p of x = { a_n times x raised n plus a_{ n minus 1 } times x raised { n minus 1 } plus ... plus a_1 times x plus a_0 }
given a_0 hasa r and r isa factor of a_0
given a_n hasa s and s isa factor of a_n
and p of x hasa coefficient and coefficient isa integer
then p of x hasa zero that isa rational x with x = r over s
# Factoring by grouping
a c x raised 3 plus a d x raised 2 plus b c x plus b d = a x raised 2 { c x plus d } = { a x raised 2 plus b } cdot { c x plus d }
# Arithmetic operations
a times b plus a times c = a times { b plus c }
{ a over b } over { c over d } = { a times d } over { b times c }
a times { b over c } = a times { b plus c }
{ a over b } plus { c over d } = { a times d plus b times c } over { b times d }
{ a over b } over c = a over { b times c }
{ a minus b } over { c minus d } = neg { { b minus a } over { d minus c } }
{ a plus b } over c = { a over c } plus { b over c }
a over { b over c } = { a times c } over b
{ a times b plus a times c } over a = b plus c
# Exponents and radicals
nroot 2 x = sqrt { x }
a raised 0 = 1, given a noteq 0
{ a over b } raised x = { a raised x } over { b raised x }
{ a times b } raised x = { a raised x } times { b raised x }
nroot n { a raised m } = a raised { m over n }
{ a raised x } times { a raised y } = a raised { x plus y }
a raised { neg x } = 1 over { a raised x }
sqrt a = a raised { 1 over 2 }
nroot n { a times b } = { nroot n a } cdot { nroot n b }
{ a raised x } over { a raised y } = a raised { x minus y }
{ a raised x } raised y = a raised { x times y }
nroot n a = a raised { 1 over n }
nroot n { a over b } = { nroot n a } over { nroot n b }
# Linear Algebra
# Basics
given Vector isa array from 0 to m { Vector [ m ] } then
Matrix isa Vector from 0 to n { Matrix [ m ][ n ] }
</textarea>
</div>
<div id=txtGNU class=cssGNU>
========================================================================<br/>
NOTE: The GPL/LGPL does not necessarily apply to all versions of Euclid. If
you obtained the code under a different license, you are bound by
those conditions. You can get the latest release of Euclid from
the internet.<br/>
========================================================================
<br/><br/>
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licensing for inclusion in my release branch.
<br/><br/>
Release under the LGPL is due to the request of researchers wanting to
integrate Euclid into free software projects using licenses incompatible
with the GPL.
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-----------------------------------------------------------------------------
<br/><br/>
GNU GENERAL PUBLIC LICENSE
Version 2, June 1991
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GNU GENERAL PUBLIC LICENSE
TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
<br/><br/>
0. This License applies to any program or other work which contains
a notice placed by the copyright holder saying it may be distributed
under the terms of this General Public License. The "Program", below,
refers to any such program or work, and a "work based on the Program"
means either the Program or any derivative work under copyright law:
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Whether that is true depends on what the Program does.
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whole or in part contains or is derived from the Program or any
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when run, you must cause it, when started running for such
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identifiable sections of that work are not derived from the Program,
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In addition, mere aggregation of another work not based on the Program
with the Program (or with a work based on the Program) on a volume of
a storage or distribution medium does not bring the other work under
the scope of this License.
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under Section 2) in object code or executable form under the terms of
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<br/><br/>
Each version is given a distinguishing version number. If the Program
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<br/><br/>
NO WARRANTY
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11. BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY
FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN
OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES
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WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR
REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES,
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END OF TERMS AND CONDITIONS
<br/><br/>
How to Apply These Terms to Your New Programs
<br/><br/>
If you develop a new program, and you want it to be of the greatest
possible use to the public, the best way to achieve this is to make it
free software which everyone can redistribute and change under these terms.
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To do so, attach the following notices to the program. It is safest
to attach them to the start of each source file to most effectively
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the "copyright" line and a pointer to where the full notice is found.
<br/><br/>
<one line to give the program's name and a brief idea of what it does.>
Copyright (C) 20yy <name of author>
<br/><br/>
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
<br/><br/>
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
<br/><br/>
You should have received a copy of the GNU General Public License
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Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
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Also add information on how to contact you by electronic and paper mail.
<br/><br/>
If the program is interactive, make it output a short notice like this
when it starts in an interactive mode:
<br/><br/>
Gnomovision version 69, Copyright (C) 20yy name of author
Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
This is free software, and you are welcome to redistribute it
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<br/><br/>
The hypothetical commands `show w' and `show c' should show the appropriate
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mouse-clicks or menu items--whatever suits your program.
<br/><br/>
You should also get your employer (if you work as a programmer) or your
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<br/><br/>
Yoyodyne, Inc., hereby disclaims all copyright interest in the program
`Gnomovision' (which makes passes at compilers) written by James P. Random.
<br/><br/>
<signature of Ty Coon>, 1 April 1989
Ty Coon, President of Vice
<br/><br/>
This General Public License does not permit incorporating your program into
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library. If this is what you want to do, use the GNU Library General
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<br/><br/>
<br/><br/>
<br/><br/>
-----------------------------------------------------------------------------
<br/><br/>
<br/><br/>
<br/><br/>
GNU LESSER GENERAL PUBLIC LICENSE
Version 2.1, February 1999
<br/><br/>
Copyright (C) 1991, 1999 Free Software Foundation, Inc.
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A copy of this license is required to copy and distribute verbatim copies
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<br/><br/>
[This is the first released version of the Lesser GPL. It also counts
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The licenses for most software are designed to take away your
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