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Handshaking lemma with 3 people?


Handshaking lemma with 3 people?

By : tiendung ngo
Date : November 20 2020, 11:01 PM
To fix this issue
Does the handshaking lemma hold true because 0 is even and there are zero vertices of odd degree?
code :


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Using Ogden’s Lemma versus regular Pumping Lemma for Context-Free Grammars

Using Ogden’s Lemma versus regular Pumping Lemma for Context-Free Grammars


By : Illya Shykyrinsky
Date : March 29 2020, 07:55 AM
hop of those help? One important stumbling issue here is that "being able to pump" does not imply context free, rather "not being able to pump" shows it is not context free. Similarly, being grey does not imply you're an elephant, but being an elephant does imply you're grey...
code :
Grammar context free        => Pumping Lemma is definitely satisfied  
Grammar not context free    => Pumping Lemma *may* be satisfied
Pumping Lemma satisfied     => Grammar *may* be context free
Pumping Lemma not satisfied => Grammar definitely not context free
# (we can write exactly the same for Ogden's Lemma)
# Here "=>" should be read as implies
how to make an example to test the rev_app immediately after lemma proved. an starting example for custom lemma

how to make an example to test the rev_app immediately after lemma proved. an starting example for custom lemma


By : user3630847
Date : March 29 2020, 07:55 AM
it helps some times Firstly, you need the syntax for list enumeration (I just picked it up in the src/HOL/List.thy file):
code :
syntax
  -- {* list Enumeration *}
  "_list" :: "args => 'a list"    ("[(_)]")

translations
  "[x, xs]" == "x#[xs]"
  "[x]" == "x#[]"
lemma example1: "rev [a, b] = [b, a]"
  by simp
value "rev [a, b]" (* return "[b, a]" *)
value "rev [a, b] = [b, a]" (* returns "True" *)
lemma rev_app [simp]: "rev(xs @ ys) = (rev ys) @ (rev xs)"
  apply (induct_tac xs)
  by simp_all
how to discover new lemma or guess or search the next lemma which want to be proved in Isabelle

how to discover new lemma or guess or search the next lemma which want to be proved in Isabelle


By : user3724273
Date : March 29 2020, 07:55 AM
will be helpful for those in need Johansson et al. have recently presented a system for theory exploration, that is, coming up with lemmas based on your definitions. You can find their implementation on GitHub and the paper on arXiv. In the paper, you will also find a lot of examples. The only drawback is that, as far as I can tell, their implementation only works with Isabelle2013-2.
Johansson, Moa, et al. "Hipster: Integrating Theory Exploration in a Proof Assistant." Intelligent Computer Mathematics. Springer International Publishing, 2014. 108-122.
Isabelle auto prover works on lemma, hangs on special case of the lemma

Isabelle auto prover works on lemma, hangs on special case of the lemma


By : Paweł Kaczorowski
Date : March 29 2020, 07:55 AM
will be helpful for those in need First of all: it is inconvenient to state goals with the HOL universal quantifier ∀. Free variables in goals are implicitly universally quantified anyway, so you can simply leave out the ∀. You will, however, tell the induction command to universally quantify these variables in the induction step using arbitrary:
code :
lemma ListSumTAux_1 : "ListSumTAux xs (a+b) = a + ListSumTAux xs b"
  apply (induct xs arbitrary: a b)
  apply (auto)
done  
⋀a. ListSumTAux xs a = a + ListSumTAux xs 0
ListSumTAux xs a → a + ListSumTAux xs 0 → a + (0 + ListSumTAux xs 0) → 
    a + (0 + (0 + ListSumTAux xs 0))
Use of Handshaking Lemma to find number of subarrays with even Sum

Use of Handshaking Lemma to find number of subarrays with even Sum


By : Greg McClement
Date : March 29 2020, 07:55 AM
fixed the issue. Will look into that further I was attempting to do practice questions and came across a solution I do not understand the reasoning behind. , First we have an array of numbers, for example:
code :
[1,3,5,2,10,7]
[1,4,9,11,21,28]
2*E = odd*(odd - 1) + even*(even - 1) => E = odd*(odd - 1)/ 2 + even*(even - 1)/2
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