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Orbit (dynamics) - Revision history
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en>JP.Martin-Flatin at 16:55, 17 October 2014
2014-10-17T16:55:51Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:55, 17 October 2014</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{other uses}}</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If you are reading this article then I can securely assume you are searching for certain hemorrhoid treatments you are able to do at house. If you are</ins>, <ins style="font-weight: bold; text-decoration: none;">then please read on</ins>. <ins style="font-weight: bold; text-decoration: none;">Let you initially define what hemorrhoids are plus what causes them</ins>, <ins style="font-weight: bold; text-decoration: none;">following understanding </ins>which<ins style="font-weight: bold; text-decoration: none;">, then you </ins>can <ins style="font-weight: bold; text-decoration: none;">find effective home treatments you may use. Basically hemorrhoids are swollen veins inside our anal canal. It is synonymous to varicose veins, nevertheless instead </ins>of <ins style="font-weight: bold; text-decoration: none;">it being inside the legs it really is found inside our anal canal</ins>.<<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">There </ins>are <ins style="font-weight: bold; text-decoration: none;">2 problems associated with utilizing creams</ins>. <ins style="font-weight: bold; text-decoration: none;">The initially is that several individuals experience burning sensations</ins>, <ins style="font-weight: bold; text-decoration: none;">occasionally thus bad which we must discontinue </ins>the <ins style="font-weight: bold; text-decoration: none;">use </ins>of the <ins style="font-weight: bold; text-decoration: none;">cream</ins>. The <ins style="font-weight: bold; text-decoration: none;">second problem is the fact that lotions never treat the underlying problems that cause hemorrhoids; consequently lotions are a temporary [http://hemorrhoidtreatmentfix.com/thrombosed-hemorrhoid-treatment thrombosed external hemorrhoid treatment].</ins><<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">H-Miracle by Holly Hayden </ins>is <ins style="font-weight: bold; text-decoration: none;">not a cream </ins>or <ins style="font-weight: bold; text-decoration: none;">topical answer however</ins>, <ins style="font-weight: bold; text-decoration: none;">a step-by-step guide to get rid of hemorrhoids</ins>. It <ins style="font-weight: bold; text-decoration: none;">offers a holistic approach to treating hemorrhoids: what to consume, what to </ins>not <ins style="font-weight: bold; text-decoration: none;">eat</ins>, <ins style="font-weight: bold; text-decoration: none;">what to do plus what not </ins>to <ins style="font-weight: bold; text-decoration: none;">do</ins>. <ins style="font-weight: bold; text-decoration: none;">It also comes with </ins>a <ins style="font-weight: bold; text-decoration: none;">great deal of freebies like books on "How to Ease Your Allergies" </ins>and <ins style="font-weight: bold; text-decoration: none;">"Lessons from Miracle Doctors"</ins>. <ins style="font-weight: bold; text-decoration: none;">A great deal </ins>of <ins style="font-weight: bold; text-decoration: none;">customers like it due to </ins>the <ins style="font-weight: bold; text-decoration: none;">effortless to follow instructions and </ins>the <ins style="font-weight: bold; text-decoration: none;">capability </ins>of the <ins style="font-weight: bold; text-decoration: none;">all-natural solutions utilized</ins>. <ins style="font-weight: bold; text-decoration: none;">Plus they additionally provide </ins>a <ins style="font-weight: bold; text-decoration: none;">funds back guarantee only </ins>in <ins style="font-weight: bold; text-decoration: none;">case the system doesn't work for you or anyone inside </ins>the <ins style="font-weight: bold; text-decoration: none;">family </ins>that is <ins style="font-weight: bold; text-decoration: none;">experiencing hemorrhoids</ins>.<<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">It </ins>is true which <ins style="font-weight: bold; text-decoration: none;">all these aspects </ins>cannot be <ins style="font-weight: bold; text-decoration: none;">covered by modern conventional medications because they tend to work found on </ins>the <ins style="font-weight: bold; text-decoration: none;">symptoms alone </ins>and <ins style="font-weight: bold; text-decoration: none;">not found on </ins>the <ins style="font-weight: bold; text-decoration: none;">cause </ins>of <ins style="font-weight: bold; text-decoration: none;">hemorrhoids</ins>. <ins style="font-weight: bold; text-decoration: none;">Hemorrhoids are caused due to different underlying wellness conditions which include the digestive system</ins>, <ins style="font-weight: bold; text-decoration: none;">the belly</ins>, the <ins style="font-weight: bold; text-decoration: none;">bowel movements, </ins>the <ins style="font-weight: bold; text-decoration: none;">diet, our life-style and also our sleeping patterns. With conventional medicine, it </ins>is <ins style="font-weight: bold; text-decoration: none;">merely impossible to take care </ins>of <ins style="font-weight: bold; text-decoration: none;">all these aspects</ins>.<<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">Your health food shop has </ins>a <ins style="font-weight: bold; text-decoration: none;">wealth of herbal hemorrhoid treatments, all </ins>of that <ins style="font-weight: bold; text-decoration: none;">contain secure elements </ins>and are <ins style="font-weight: bold; text-decoration: none;">effortless </ins>on the <ins style="font-weight: bold; text-decoration: none;">purse. Barberry, Butcher</ins>'<ins style="font-weight: bold; text-decoration: none;">s Brew</ins>, <ins style="font-weight: bold; text-decoration: none;">Witch Hazel</ins>, <ins style="font-weight: bold; text-decoration: none;">Horse Chestnut plus Slippery Elm are all famous herbal elements for giving relief whenever piles flare up</ins>.<<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">Only sit for threee to five minute for a bowe movement plus don't wait till </ins>the <ins style="font-weight: bold; text-decoration: none;">last minute as sometimes </ins>the <ins style="font-weight: bold; text-decoration: none;">body tend </ins>to <ins style="font-weight: bold; text-decoration: none;">have a temporary or false irregularity plus by </ins>the <ins style="font-weight: bold; text-decoration: none;">time you need </ins>to, <ins style="font-weight: bold; text-decoration: none;">nothing comes out (I understand</ins>, <ins style="font-weight: bold; text-decoration: none;">I have performed this before)</ins>.<<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">Although these seven steps are all great </ins>ways to <ins style="font-weight: bold; text-decoration: none;">heal hemorrhoids</ins>, <ins style="font-weight: bold; text-decoration: none;">they are no promise </ins>that <ins style="font-weight: bold; text-decoration: none;">you will </ins>be <ins style="font-weight: bold; text-decoration: none;">capable </ins>to <ins style="font-weight: bold; text-decoration: none;">completely heal a hemorrhoids </ins>or <ins style="font-weight: bold; text-decoration: none;">which they won't return</ins>. <ins style="font-weight: bold; text-decoration: none;">As </ins>a <ins style="font-weight: bold; text-decoration: none;">guideline</ins>, <ins style="font-weight: bold; text-decoration: none;">when symptoms do </ins>not <ins style="font-weight: bold; text-decoration: none;">clear up completely</ins>, <ins style="font-weight: bold; text-decoration: none;">come back </ins>in <ins style="font-weight: bold; text-decoration: none;">a few days or deteriorate at some point </ins>in <ins style="font-weight: bold; text-decoration: none;">drugs</ins>, <ins style="font-weight: bold; text-decoration: none;">you need to receive an appointment to pay </ins>a <ins style="font-weight: bold; text-decoration: none;">visit </ins>to <ins style="font-weight: bold; text-decoration: none;">your general practitioner appropriate away</ins>.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{technical|date=May 2011}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In [[proof theory]]</del>, <del style="font-weight: bold; text-decoration: none;">a '''sequent''' is a [[Formalism (mathematics)|formalized]] statement of [[Proof theory|provability]] that is frequently used when specifying [[proof calculus|calculi]] for [[deductive reasoning|deduction]]</del>. <del style="font-weight: bold; text-decoration: none;">In the [[sequent calculus]]</del>, <del style="font-weight: bold; text-decoration: none;">the name ''sequent'' is used for the construct </del>which can <del style="font-weight: bold; text-decoration: none;">be regarded as a specific kind </del>of <del style="font-weight: bold; text-decoration: none;">[[Judgment (mathematical logic)|judgment]], characteristic to this deduction system</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Explanation ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">A sequent has the form</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:</del><<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">\Gamma\vdash\Sigma</del><<del style="font-weight: bold; text-decoration: none;">/math</del>></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">where both Γ and Σ </del>are <del style="font-weight: bold; text-decoration: none;">[[sequence]]s of [[Mathematical logic|logical]] formulae (i.e</del>., <del style="font-weight: bold; text-decoration: none;">both </del>the <del style="font-weight: bold; text-decoration: none;">number and the order </del>of the <del style="font-weight: bold; text-decoration: none;">occurring formulae matter)</del>. The <del style="font-weight: bold; text-decoration: none;">symbol </del><<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">\vdash</del><<del style="font-weight: bold; text-decoration: none;">/math</del>> is <del style="font-weight: bold; text-decoration: none;">usually referred to as ''[[Turnstile (symbol)|turnstile]]'' </del>or <del style="font-weight: bold; text-decoration: none;">''tee'' and is often read</del>, <del style="font-weight: bold; text-decoration: none;">suggestively, as "yields" or "proves"</del>. It <del style="font-weight: bold; text-decoration: none;">is </del>not <del style="font-weight: bold; text-decoration: none;">a symbol in the language</del>, <del style="font-weight: bold; text-decoration: none;">rather it is a symbol in the [[metalanguage]] used </del>to <del style="font-weight: bold; text-decoration: none;">discuss proofs</del>. <del style="font-weight: bold; text-decoration: none;">In </del>a <del style="font-weight: bold; text-decoration: none;">sequent, Γ is called the antecedent </del>and <del style="font-weight: bold; text-decoration: none;">Σ is said to be the succedent of the sequent</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Intuitive meaning ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The intuitive meaning </del>of the <del style="font-weight: bold; text-decoration: none;">sequent <math>\Gamma\vdash\Sigma</math> is that under </del>the <del style="font-weight: bold; text-decoration: none;">assumption </del>of <del style="font-weight: bold; text-decoration: none;">Γ </del>the <del style="font-weight: bold; text-decoration: none;">conclusion of Σ is provable</del>. <del style="font-weight: bold; text-decoration: none;">Classically, the formulae on the left of the turnstile can be interpreted [[logical conjunction|conjunctively]] while the formulae on the right can be considered as </del>a <del style="font-weight: bold; text-decoration: none;">[[logical disjunction|disjunction]]. This means that, when all formulae </del>in <del style="font-weight: bold; text-decoration: none;">Γ hold, then at least one formula in Σ also has to be true. If </del>the <del style="font-weight: bold; text-decoration: none;">succedent is empty, this is interpreted as falsity, i.e. <math>\Gamma\vdash</math> means </del>that <del style="font-weight: bold; text-decoration: none;">Γ proves falsity and </del>is <del style="font-weight: bold; text-decoration: none;">thus inconsistent</del>. <del style="font-weight: bold; text-decoration: none;">On the other hand an empty antecedent is assumed to be true, i.e., </del><<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">\vdash\Sigma</del><<del style="font-weight: bold; text-decoration: none;">/math</del>> <del style="font-weight: bold; text-decoration: none;">means that Σ follows without any assumptions, i.e., it </del>is <del style="font-weight: bold; text-decoration: none;">always </del>true <del style="font-weight: bold; text-decoration: none;">(as a disjunction). A sequent of this form, with Γ empty, is known as a [[logical assertion]].</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Of course, other intuitive explanations are possible, </del>which <del style="font-weight: bold; text-decoration: none;">are classically equivalent. For example, <math>\Gamma\vdash\Sigma</math> can be read as asserting that it </del>cannot be the <del style="font-weight: bold; text-decoration: none;">case that every formula in Γ is true </del>and <del style="font-weight: bold; text-decoration: none;">every formula in Σ is false (this is related to </del>the <del style="font-weight: bold; text-decoration: none;">double-negation interpretations </del>of <del style="font-weight: bold; text-decoration: none;">classical [[intuitionistic logic]], such as [[Glivenko's theorem]])</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In any case</del>, <del style="font-weight: bold; text-decoration: none;">these intuitive readings are only pedagogical. Since formal proofs in proof theory are purely [[syntax|syntactic]]</del>, the <del style="font-weight: bold; text-decoration: none;">[[semantics|meaning]] of (</del>the <del style="font-weight: bold; text-decoration: none;">derivation of) a sequent </del>is <del style="font-weight: bold; text-decoration: none;">only given by the properties </del>of <del style="font-weight: bold; text-decoration: none;">the calculus that provides the actual [[rule of inference|rules of inference]].</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Barring any contradictions in the technically precise definition above we can describe sequents in their introductory logical form</del>. <<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">\Gamma</del><<del style="font-weight: bold; text-decoration: none;">/math</del>> <del style="font-weight: bold; text-decoration: none;">represents </del>a <del style="font-weight: bold; text-decoration: none;">set </del>of <del style="font-weight: bold; text-decoration: none;">assumptions </del>that <del style="font-weight: bold; text-decoration: none;">we begin our logical process with, for example "Socrates is a man" </del>and <del style="font-weight: bold; text-decoration: none;">"All men </del>are <del style="font-weight: bold; text-decoration: none;">mortal". The <math>\Sigma</math> represents a logical conclusion that follows under these premises. For example "Socrates is mortal" follows from a reasonable formalization of the above points and we could expect to see it </del>on the <del style="font-weight: bold; text-decoration: none;"><math>\Sigma</math> side of the </del>'<del style="font-weight: bold; text-decoration: none;">'turnstile''. In this sense</del>, <del style="font-weight: bold; text-decoration: none;"><math>\vdash</math> means the process of reasoning</del>, <del style="font-weight: bold; text-decoration: none;">or "therefore" in English</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Example ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">A typical sequent might be: </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:</del><<del style="font-weight: bold; text-decoration: none;">math</del>> <del style="font-weight: bold; text-decoration: none;">\phi,\psi\vdash\alpha,\beta</del><<del style="font-weight: bold; text-decoration: none;">/math</del>></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">This claims that either <math>\alpha</math> or <math>\beta</math> can be derived from <math>\phi</math> and <math>\psi</math>.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Property ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Since every formula in </del>the <del style="font-weight: bold; text-decoration: none;">antecedent (</del>the <del style="font-weight: bold; text-decoration: none;">left side) must be true </del>to <del style="font-weight: bold; text-decoration: none;">conclude </del>the <del style="font-weight: bold; text-decoration: none;">truth of at least one formula in the succedent (the right side), adding formulas </del>to <del style="font-weight: bold; text-decoration: none;">either side results in a weaker sequent</del>, <del style="font-weight: bold; text-decoration: none;">while removing them from either side gives a stronger one.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Rules ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Most proof systems provide ways to deduce one sequent from another. These [[inference rule]]s are written with a list of sequents above and below a [[coplanar|line]]. This rule indicates that if everything above the line is true</del>, <del style="font-weight: bold; text-decoration: none;">so is everything under the line</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">A typical rule is: </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">:</del><<del style="font-weight: bold; text-decoration: none;">math</del>> <del style="font-weight: bold; text-decoration: none;">\frac{\Gamma,\alpha\vdash\Sigma\qquad \Gamma\vdash\alpha}{\Gamma\vdash\Sigma}</del><<del style="font-weight: bold; text-decoration: none;">/math</del>></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">This indicates that if we can deduce that <math>\Gamma,\alpha</math> yields <math>\Sigma</math>, and that <math>\Gamma</math> yields <math>\alpha</math>, then we can also deduce that <math>\Gamma</math> yields <math>\Sigma</math>.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Variations ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The general notion of sequent introduced here can be specialized in various </del>ways<del style="font-weight: bold; text-decoration: none;">. A sequent is said </del>to <del style="font-weight: bold; text-decoration: none;">be an '''intuitionistic sequent''' if there is at most one formula in the succedent (although multi-succedent calculi for intuitionistic logic is also possible). Similarly</del>, <del style="font-weight: bold; text-decoration: none;">one can obtain calculi for [[dual-intuitionistic logic]] (a type of [[paraconsistent logic]]) by requiring </del>that <del style="font-weight: bold; text-decoration: none;">sequents </del>be <del style="font-weight: bold; text-decoration: none;">singular in the antecedent.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In many cases, sequents are also assumed </del>to <del style="font-weight: bold; text-decoration: none;">consist of [[multiset]]s </del>or <del style="font-weight: bold; text-decoration: none;">[[Set (mathematics)|sets]] instead of sequences</del>. <del style="font-weight: bold; text-decoration: none;">Thus one disregards the order or even the number of occurrences of the formulae. For classical [[propositional logic]] this does not yield </del>a <del style="font-weight: bold; text-decoration: none;">problem</del>, <del style="font-weight: bold; text-decoration: none;">since the conclusions that one can draw from a collection of premises does </del>not <del style="font-weight: bold; text-decoration: none;">depend on these data. In [[substructural logic]], however, this may become quite important.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== History ==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Historically</del>, <del style="font-weight: bold; text-decoration: none;">sequents have been introduced by [[Gerhard Gentzen]] </del>in <del style="font-weight: bold; text-decoration: none;">order to specify his famous [[sequent calculus]]. In his German publication he used the word "Sequenz". However, </del>in <del style="font-weight: bold; text-decoration: none;">English</del>, <del style="font-weight: bold; text-decoration: none;">the word "[[sequence]]" is already used as </del>a <del style="font-weight: bold; text-decoration: none;">translation </del>to <del style="font-weight: bold; text-decoration: none;">the German "Folge" and appears quite frequently in mathematics</del>. <del style="font-weight: bold; text-decoration: none;">The term "sequent" then has been created in search for an alternative translation of the German expression.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{PlanetMath attribution|id=3502|title=Sequent}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==External links==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* {{springer|title=Sequent (in logic)|id=p/s084590}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Proof theory]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Logical expressions]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
</table>
en>JP.Martin-Flatin
https://en.formulasearchengine.com/index.php?title=Orbit_(dynamics)&diff=3436&oldid=prev
en>Mark viking: Added wl
2013-11-18T23:02:49Z
<p>Added wl</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 00:02, 19 November 2013</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Jerrie Swoboda </del>is <del style="font-weight: bold; text-decoration: none;">what users </del>can <del style="font-weight: bold; text-decoration: none;">call me </del>and <del style="font-weight: bold; text-decoration: none;">I totally dig that name</del>. <del style="font-weight: bold; text-decoration: none;">Managing people </del>is <del style="font-weight: bold; text-decoration: none;">my time frame job now</del>. <del style="font-weight: bold; text-decoration: none;">As </del>a <del style="font-weight: bold; text-decoration: none;">girl what I really like </del>is to <del style="font-weight: bold; text-decoration: none;">take up croquet but I </del>can<del style="font-weight: bold; text-decoration: none;">'t make </del>it <del style="font-weight: bold; text-decoration: none;">my profession really</del>. <del style="font-weight: bold; text-decoration: none;">My his conversation </del>and <del style="font-weight: bold; text-decoration: none;">I chose </del>to <del style="font-weight: bold; text-decoration: none;">exist </del>in in <del style="font-weight: bold; text-decoration: none;">Massachusetts</del>. <del style="font-weight: bold; text-decoration: none;">Go </del>to <del style="font-weight: bold; text-decoration: none;">my web property to find out more</del>: <del style="font-weight: bold; text-decoration: none;">http</del>://<del style="font-weight: bold; text-decoration: none;">prometeu.net</del><<del style="font-weight: bold; text-decoration: none;">br</del>><<del style="font-weight: bold; text-decoration: none;">br</del>><del style="font-weight: bold; text-decoration: none;">Stop by my website; how </del>to <del style="font-weight: bold; text-decoration: none;">hack clash </del>of <del style="font-weight: bold; text-decoration: none;">clans </del>[[<del style="font-weight: bold; text-decoration: none;">http</del>://<del style="font-weight: bold; text-decoration: none;">prometeu</del>.<del style="font-weight: bold; text-decoration: none;">net why </del>not <del style="font-weight: bold; text-decoration: none;">try here</del>]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{other uses}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{technical|date=May 2011}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In [[proof theory]], a '''sequent''' is a [[Formalism (mathematics)|formalized]] statement of [[Proof theory|provability]] that </ins>is <ins style="font-weight: bold; text-decoration: none;">frequently used when specifying [[proof calculus|calculi]] for [[deductive reasoning|deduction]]. In the [[sequent calculus]], the name ''sequent'' is used for the construct which </ins>can <ins style="font-weight: bold; text-decoration: none;">be regarded as a specific kind of [[Judgment (mathematical logic)|judgment]], characteristic to this deduction system.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Explanation ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">A sequent has the form</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">:<math>\Gamma\vdash\Sigma</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">where both Γ and Σ are [[sequence]]s of [[Mathematical logic|logical]] formulae (i.e., both the number </ins>and <ins style="font-weight: bold; text-decoration: none;">the order of the occurring formulae matter). The symbol <math>\vdash</math> is usually referred to as ''[[Turnstile (symbol)|turnstile]]'' or ''tee'' and is often read, suggestively, as "yields" or "proves"</ins>. <ins style="font-weight: bold; text-decoration: none;">It is not a symbol in the language, rather it </ins>is <ins style="font-weight: bold; text-decoration: none;">a symbol in the [[metalanguage]] used to discuss proofs</ins>. <ins style="font-weight: bold; text-decoration: none;">In </ins>a <ins style="font-weight: bold; text-decoration: none;">sequent, Γ </ins>is <ins style="font-weight: bold; text-decoration: none;">called the antecedent and Σ is said </ins>to <ins style="font-weight: bold; text-decoration: none;">be the succedent of the sequent.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Intuitive meaning ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The intuitive meaning of the sequent <math>\Gamma\vdash\Sigma</math> is that under the assumption of Γ the conclusion of Σ is provable. Classically, the formulae on the left of the turnstile can be interpreted [[logical conjunction|conjunctively]] while the formulae on the right </ins>can <ins style="font-weight: bold; text-decoration: none;">be considered as a [[logical disjunction|disjunction]]. This means that, when all formulae in Γ hold, then at least one formula in Σ also has to be true. If the succedent is empty, this is interpreted as falsity, i.e. <math>\Gamma\vdash</math> means that Γ proves falsity and is thus inconsistent. On the other hand an empty antecedent is assumed to be true, i.e., <math>\vdash\Sigma</math> means that Σ follows without any assumptions, i.e., </ins>it <ins style="font-weight: bold; text-decoration: none;">is always true (as a disjunction)</ins>. <ins style="font-weight: bold; text-decoration: none;"> A sequent of this form, with Γ empty, is known as a [[logical assertion]].</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Of course, other intuitive explanations are possible, which are classically equivalent. For example, <math>\Gamma\vdash\Sigma</math> can be read as asserting that it cannot be the case that every formula in Γ is true </ins>and <ins style="font-weight: bold; text-decoration: none;">every formula in Σ is false (this is related </ins>to <ins style="font-weight: bold; text-decoration: none;">the double-negation interpretations of classical [[intuitionistic logic]], such as [[Glivenko's theorem]]).</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In any case, these intuitive readings are only pedagogical. Since formal proofs in proof theory are purely [[syntax|syntactic]], the [[semantics|meaning]] of (the derivation of) a sequent is only given by the properties of the calculus that provides the actual [[rule of inference|rules of inference]].</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Barring any contradictions </ins>in <ins style="font-weight: bold; text-decoration: none;">the technically precise definition above we can describe sequents </ins>in <ins style="font-weight: bold; text-decoration: none;">their introductory logical form</ins>. <ins style="font-weight: bold; text-decoration: none;"><math>\Gamma</math> represents a set of assumptions that we begin our logical process with, for example "Socrates is a man" and "All men are mortal". The <math>\Sigma</math> represents a logical conclusion that follows under these premises. For example "Socrates is mortal" follows from a reasonable formalization of the above points and we could expect </ins>to <ins style="font-weight: bold; text-decoration: none;">see it on the <math>\Sigma</math> side of the ''turnstile''. In this sense, <math>\vdash</math> means the process of reasoning, or "therefore" in English.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Example ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">A typical sequent might be</ins>: <ins style="font-weight: bold; text-decoration: none;"> </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<ins style="font-weight: bold; text-decoration: none;"><math> \phi,\psi\vdash\alpha,\beta<</ins>/<ins style="font-weight: bold; text-decoration: none;">math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">This claims that either <math>\alpha<</ins>/<ins style="font-weight: bold; text-decoration: none;">math> or <math>\beta</math> can be derived from <math>\phi</math> and </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>><ins style="font-weight: bold; text-decoration: none;">\psi</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>><ins style="font-weight: bold; text-decoration: none;">.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Property ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Since every formula in the antecedent (the left side) must be true to conclude the truth of at least one formula in the succedent (the right side), adding formulas to either side results in a weaker sequent, while removing them from either side gives a stronger one.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Rules ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Most proof systems provide ways </ins>to <ins style="font-weight: bold; text-decoration: none;">deduce one sequent from another. These [[inference rule]]s are written with a list </ins>of <ins style="font-weight: bold; text-decoration: none;">sequents above and below a </ins>[[<ins style="font-weight: bold; text-decoration: none;">coplanar|line]]. This rule indicates that if everything above the line is true, so is everything under the line.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">A typical rule is: </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:<ins style="font-weight: bold; text-decoration: none;"><math> \frac{\Gamma,\alpha\vdash\Sigma\qquad \Gamma\vdash\alpha}{\Gamma\vdash\Sigma}<</ins>/<ins style="font-weight: bold; text-decoration: none;">math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">This indicates that if we can deduce that <math>\Gamma,\alpha<</ins>/<ins style="font-weight: bold; text-decoration: none;">math> yields <math>\Sigma</math>, and that <math>\Gamma</math> yields <math>\alpha</math>, then we can also deduce that <math>\Gamma</math> yields <math>\Sigma</math>.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Variations ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The general notion of sequent introduced here can be specialized in various ways. A sequent is said to be an '''intuitionistic sequent''' if there is at most one formula in the succedent (although multi-succedent calculi for intuitionistic logic is also possible). Similarly, one can obtain calculi for [[dual-intuitionistic logic]] (a type of [[paraconsistent logic]]) by requiring that sequents be singular in the antecedent.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In many cases, sequents are also assumed to consist of [[multiset]]s or [[Set (mathematics)|sets]] instead of sequences</ins>. <ins style="font-weight: bold; text-decoration: none;">Thus one disregards the order or even the number of occurrences of the formulae. For classical [[propositional logic]] this does </ins>not <ins style="font-weight: bold; text-decoration: none;">yield a problem, since the conclusions that one can draw from a collection of premises does not depend on these data. In [[substructural logic]], however, this may become quite important.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== History ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Historically, sequents have been introduced by [[Gerhard Gentzen]] in order to specify his famous [[sequent calculus]]. In his German publication he used the word "Sequenz". However, in English, the word "[[sequence]]" is already used as a translation to the German "Folge" and appears quite frequently in mathematics. The term "sequent" then has been created in search for an alternative translation of the German expression.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{PlanetMath attribution|id=3502|title=Sequent}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==External links==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* {{springer|title=Sequent (in logic)|id=p/s084590}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Proof theory]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Logical expressions</ins>]]</div></td></tr>
</table>
en>Mark viking
https://en.formulasearchengine.com/index.php?title=Orbit_(dynamics)&diff=227159&oldid=prev
en>Laburke: holomorphic function link
2011-11-01T15:11:52Z
<p>holomorphic function link</p>
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en>Laburke