Inhomogeneous electromagnetic wave equation - Revision history

99.249.166.31: /* E and B fields */

2014-04-10T03:57:19Z

E and B fields

← Older revision		Revision as of 05:57, 10 April 2014
Line 1:		Line 1:
	~~'''HOL Light''' is a member of the [[HOL theorem prover family]]~~. ~~Like~~ the ~~other members~~, it is ~~a [[proof assistant]] for classical [[higher order logic]]~~. ~~Compared with other HOL systems, HOL Light~~ is ~~intended to have relatively simple foundations. HOL Light is authored and maintained by~~ the ~~mathematician and computer scientist [[John Harrison (mathematician)\|John Harrison]]~~. ~~HOL Light~~ is ~~released under the [[BSD licenses#2-clause\|simplified BSD license]]~~.~~<ref>http://code.google.com/p/hol-light/</ref>~~		Hi there. Let me begin by introducing the author, her name is Sophia. To perform lacross is the thing I adore most of all. Office supervising is my occupation. For years he's been residing in Alaska and he doesn't plan on altering it.<br><br>Visit my webpage; [http://m-card.co.kr/xe/mcard_2013_promote01/29877 free psychic readings]

	~~==Logical foundations==~~

	~~HOL Light is based on a formulation of [[type theory]] with equality~~
	~~as the only [[primitive notion]]. The primitive rules of inference~~
	~~are the following:~~
	~~{\| class="wikitable"~~
	\|-
	~~\| style="text-align: center;" \| <math> \cfrac{\qquad }{ \vdash t = t}</math>~~
	~~\| style="text-align: center;" \| REFL~~
	~~\| reflexivity of equality~~
	\|-
	~~\| style="text-align: center;" \| <math> \cfrac{\Gamma \vdash~~ s = t ~~\qquad \Delta \vdash t = u}~~
	~~{\Gamma \cup \Delta \vdash s = u}~~
	~~</math>~~
	~~\| style="text-align: center;" \| TRANS~~
	~~\| transitivity of equality~~
	\|-
	~~\| style="text-align: center;" \| <math> \cfrac{\Gamma \vdash f = g \qquad \Delta \vdash x = y}~~
	~~{\Gamma \cup \Delta \vdash f(x) = g(y)}~~
	~~</math>~~
	~~\| style="text-align: center;" \| MK_COMB~~
	~~\| congruence of equality~~
	\|-
	~~\| style="text-align: center;" \| <math> \cfrac{\Gamma \vdash s = t}{\Gamma \vdash (\lambda x. s) = (\lambda x~~. ~~t)}~~
	~~</math>~~
	~~\| style="text-align: center;" \| ABS~~
	~~\| abstraction of equality (<math>x</math> must not be free in <math>\Gamma</math>)~~
	\|-
	~~\| style="text-align: center;" \| <math>\cfrac{\qquad}{\vdash (\lambda x. t) x = t}~~
	~~</math>~~
	~~\| style="text-align: center;" \| BETA~~
	~~\| connection of abstraction and function application~~
	\|-
	~~\| style="text-align: center;" \| <math> \cfrac{\qquad }{ \{p\} \vdash p}~~
	~~</math>~~
	~~\| style="text-align: center;" \| ASSUME~~
	~~\| assuming <math>p</math>, prove <math>p</math>~~
	\|-
	~~\| style="text-align: center;" \| <math> \cfrac{\Gamma \vdash p = q \qquad \Delta \vdash p}~~
	~~{\Gamma \cup \Delta \vdash q}~~
	~~</math>~~
	~~\| style="text-align: center;" \| EQ_MP~~
	~~\| relation of equality and deduction~~
	\|-
	~~\| style="text-align: center;" \| <math> \cfrac{\Gamma \vdash p \qquad \Delta \vdash q}~~
	~~{(\Gamma - \{q\}) \cup (\Delta - \{p\}) \vdash p = q}~~
	~~</math>~~
	~~\| style="text-align: center;" \| DEDUCT_ANTISYM_RULE~~
	~~\| deduce equality from 2-way deducibility~~
	\|-
	~~\| style="text-align: center;" \| <math> \cfrac{\Gamma[x_1,\ldots,x_n] \vdash p[x_1,\ldots,x_n]}~~
	~~{\Gamma[t_1,\ldots,t_n] \vdash p[t_1,\ldots,t_n]}~~
	<~~/math~~>
	~~\| style="text-align: center;" \| INST~~
	~~\| instantiate variables in assumptions and conclusion of theorem~~
	\|-
	~~\| style="text-align: center;" \|~~ <~~math~~> ~~\cfrac{\Gamma[\alpha_1,\ldots,\alpha_n] \vdash p[\alpha_1,\ldots,\alpha_n]}~~
	~~{\Gamma[\tau_1,\ldots,\tau_n] \vdash p[\tau_1,\ldots,\tau_n]}~~
	~~</math>~~
	~~\| style="text-align: center;" \| INST_TYPE~~
	~~\| instantiate type variables in assumptions and conclusion of theorem~~
	\|}

	~~This formulation of type theory is very close to the one described in~~
	~~section II.2 of {{Harvtxt\|Lambek\|Scott\|1986}}.~~

	~~==References==~~
	~~{{reflist}}~~
	*{{Citation
	~~\| last = Lambek~~
	~~\| first = J~~
	~~\| coauthors = P. J. Scott~~
	~~\| title = Introduction to Higher Order Categorical logic~~
	~~\| publisher = Cambridge University Press~~
	~~\| year = 1986~~
	}}

	~~==Further reading==~~

	*{{Citation
	~~\| author = Freek Wiedijk~~
	~~\| title = Formal Proof &mdash~~; ~~Getting Started~~
	~~\| journal =~~ [~~[Notices of the American Mathematical Society]]~~
	~~\|date=December 2008~~
	~~\| volume = 55~~
	~~\| issue = 11~~
	~~\| pages = 1408–1414~~
	~~\| url =~~ http://~~www.ams.org/notices/200811/tx081101408p.pdf~~
	~~\| accessdate = 2008-12~~-14
	}}

	~~==External links==~~
	* [http://www.cl.~~cam.ac.uk~~/~~users~~/~~jrh~~/~~hol-light/ HOL Light]~~

	~~[[Category:Free theorem provers]]~~
	~~[[Category:Proof assistants]]~~
	~~[[Category:OCaml software]~~]

en>Jackfork: Reverted 1 edit by 137.150.101.180 (talk) to last revision by Yintan. (TW)

2013-11-08T02:03:20Z

Reverted 1 edit by 137.150.101.180 (talk) to last revision by Yintan. (TW)

← Older revision		Revision as of 04:03, 8 November 2013
Line 1:		Line 1:
	~~Oscar~~ is ~~how he's known as and he totally enjoys this name~~. ~~To do aerobics~~ is a ~~thing that I'm completely addicted~~ to. ~~California~~ is ~~where her house~~ is ~~but she needs to move because~~ of ~~her family members~~. ~~Supervising is my profession~~.<br><br>~~Have a look at my web page~~: [http://www.~~youronlinepublishers~~.~~com~~/~~authWiki~~/~~AudreaocMalmrw at home std test~~]		'''HOL Light''' is a member of the [[HOL theorem prover family]]. Like the other members, it is a [[proof assistant]] for classical [[higher order logic]]. Compared with other HOL systems, HOL Light is intended to have relatively simple foundations. HOL Light is authored and maintained by the mathematician and computer scientist [[John Harrison (mathematician)\|John Harrison]]. HOL Light is released under the [[BSD licenses#2-clause\|simplified BSD license]].<ref>http://code.google.com/p/hol-light/</ref>

			==Logical foundations==

			HOL Light is based on a formulation of [[type theory]] with equality
			as the only [[primitive notion]]. The primitive rules of inference
			are the following:
			{\| class="wikitable"
			\|-
			\| style="text-align: center;" \| <math> \cfrac{\qquad }{ \vdash t = t}</math>
			\| style="text-align: center;" \| REFL
			\| reflexivity of equality
			\|-
			\| style="text-align: center;" \| <math> \cfrac{\Gamma \vdash s = t \qquad \Delta \vdash t = u}
			{\Gamma \cup \Delta \vdash s = u}
			</math>
			\| style="text-align: center;" \| TRANS
			\| transitivity of equality
			\|-
			\| style="text-align: center;" \| <math> \cfrac{\Gamma \vdash f = g \qquad \Delta \vdash x = y}
			{\Gamma \cup \Delta \vdash f(x) = g(y)}
			</math>
			\| style="text-align: center;" \| MK_COMB
			\| congruence of equality
			\|-
			\| style="text-align: center;" \| <math> \cfrac{\Gamma \vdash s = t}{\Gamma \vdash (\lambda x. s) = (\lambda x. t)}
			</math>
			\| style="text-align: center;" \| ABS
			\| abstraction of equality (<math>x</math> must not be free in <math>\Gamma</math>)
			\|-
			\| style="text-align: center;" \| <math>\cfrac{\qquad}{\vdash (\lambda x. t) x = t}
			</math>
			\| style="text-align: center;" \| BETA
			\| connection of abstraction and function application
			\|-
			\| style="text-align: center;" \| <math> \cfrac{\qquad }{ \{p\} \vdash p}
			</math>
			\| style="text-align: center;" \| ASSUME
			\| assuming <math>p</math>, prove <math>p</math>
			\|-
			\| style="text-align: center;" \| <math> \cfrac{\Gamma \vdash p = q \qquad \Delta \vdash p}
			{\Gamma \cup \Delta \vdash q}
			</math>
			\| style="text-align: center;" \| EQ_MP
			\| relation of equality and deduction
			\|-
			\| style="text-align: center;" \| <math> \cfrac{\Gamma \vdash p \qquad \Delta \vdash q}
			{(\Gamma - \{q\}) \cup (\Delta - \{p\}) \vdash p = q}
			</math>
			\| style="text-align: center;" \| DEDUCT_ANTISYM_RULE
			\| deduce equality from 2-way deducibility
			\|-
			\| style="text-align: center;" \| <math> \cfrac{\Gamma[x_1,\ldots,x_n] \vdash p[x_1,\ldots,x_n]}
			{\Gamma[t_1,\ldots,t_n] \vdash p[t_1,\ldots,t_n]}
			</math>
			\| style="text-align: center;" \| INST
			\| instantiate variables in assumptions and conclusion of theorem
			\|-
			\| style="text-align: center;" \| <math> \cfrac{\Gamma[\alpha_1,\ldots,\alpha_n] \vdash p[\alpha_1,\ldots,\alpha_n]}
			{\Gamma[\tau_1,\ldots,\tau_n] \vdash p[\tau_1,\ldots,\tau_n]}
			</math>
			\| style="text-align: center;" \| INST_TYPE
			\| instantiate type variables in assumptions and conclusion of theorem
			\|}

			This formulation of type theory is very close to the one described in
			section II.2 of {{Harvtxt\|Lambek\|Scott\|1986}}.

			==References==
			{{reflist}}
			*{{Citation
			\| last = Lambek
			\| first = J
			\| coauthors = P. J. Scott
			\| title = Introduction to Higher Order Categorical logic
			\| publisher = Cambridge University Press
			\| year = 1986
			}}

			==Further reading==

			*{{Citation
			\| author = Freek Wiedijk
			\| title = Formal Proof — Getting Started
			\| journal = [[Notices of the American Mathematical Society]]
			\|date=December 2008
			\| volume = 55
			\| issue = 11
			\| pages = 1408–1414
			\| url = http://www.ams.org/notices/200811/tx081101408p.pdf
			\| accessdate = 2008-12-14
			}}

			==External links==
			* [http://www.cl.cam.ac.uk/users/jrh/hol-light/ HOL Light]

			[[Category:Free theorem provers]]
			[[Category:Proof assistants]]
			[[Category:OCaml software]]

en>Headbomb: Help needed: Covariance and contravariance

2012-04-12T02:16:38Z

Help needed: Covariance and contravariance

New page

Oscar is how he's known as and he totally enjoys this name. To do aerobics is a thing that I'm completely addicted to. California is where her house is but she needs to move because of her family members. Supervising is my profession.<br><br>Have a look at my web page: [http://www.youronlinepublishers.com/authWiki/AudreaocMalmrw at home std test]