Hello,

I don't want to leave a false statement here. Constraints 3 and 4 are 
NOT enough to restrict the value set. The induction base as well as the 
definition of value range and definition range is missing for this proof 
to be complete. I wrote a working induction proof, if anyone is 
interested...

Regards
Christoph

Am 24.11.2017 um 00:01 schrieb Christoph Naber pentium120mhz@xxxxxxxxx:
> Conditions 1-3 wouldn't work on their own.
> i.e. one could introduce items that do not belong to the original set. 
> So one additional constraint has to check that.
> every $s in $sequences[item] satisfies (
> B B B  every $item in $s/item satisfies (
> B B B  B B B  some $i in $set satisfies deep-equal($i, $item)))
>
> On top of that the deep-equal uniqueness check for sequences would be 
> necessary if you drop constraint 4.
> every $s in $sequences, $t in ($sequences except $s) satisfies 
> not(deep-equal($s, $t))
>
> Instead I think Rogers' forth condition is _absolutly marvelous_. 
> Overall it enforces some kind of induction constraint on the sequences.
> "if you have one sequence that isn't full length, there have to be $n 
> bigger sequences that have been extended with each of the items from 
> the set"
>
> I would be surprised if it would be necessary to have more than just 
> the constraints 3 and 4 to decide the combinatorical question. I 
> played around, but "lengthy" (see below) gave always the same result 
> as "compressed"
>
> <xsl:stylesheet version="2.0"
> B B B  xmlns:xsl="http://www.w3.org/1999/XSL/Transform"
> B B B  xmlns:math="http://www.w3.org/2005/xpath-functions/math"
> B B B  exclude-result-prefixes="math">
>
> B B B  <xsl:output method="xml" encoding="UTF-8" indent="yes"/>
>
> B B B  <xsl:template match="/" >
> B B B  B B B  <xsl:variable name="set">
> B B B  B B B  B B B  <item>A</item>
> B B B  B B B  B B B  <item>B</item>
> B B B  B B B  </xsl:variable>
>
> B B B  B B B  <xsl:variable name="sequences">
> B B B  B B B  B B B  <sequence />
> B B B  B B B  B B B  <sequence>
> B B B  B B B  B B B  B B B  <item>A</item>
> B B B  B B B  B B B  </sequence>
> B B B  B B B  B B B  <sequence>
> B B B  B B B  B B B  B B B  <item>B</item>
> B B B  B B B  B B B  </sequence>
> B B B  B B B  B B B  <sequence>
> B B B  B B B  B B B  B B B  <item>A</item>
> B B B  B B B  B B B  B B B  <item>A</item>
> B B B  B B B  B B B  </sequence>
> B B B  B B B  B B B  <sequence>
> B B B  B B B  B B B  B B B  <item>A</item>
> B B B  B B B  B B B  B B B  <item>B</item>
> B B B  B B B  B B B  </sequence>
> B B B  B B B  B B B  <sequence>
> B B B  B B B  B B B  B B B  <item>B</item>
> B B B  B B B  B B B  B B B  <item>A</item>
> B B B  B B B  B B B  </sequence>
> B B B  B B B  B B B  <sequence>
> B B B  B B B  B B B  B B B  <item>B</item>
> B B B  B B B  B B B  B B B  <item>B</item>
> B B B  B B B  B B B  </sequence>
> B B B  B B B  </xsl:variable>
>
> B B B  B B B  <xsl:variable name="n" select="count($set/item)" />
>
>
> B B B  B B B  <evaluation n="{$n}">
> B B B  B B B  B B B  <compressed>
> B B B  B B B  B B B  B B B  <xsl:comment>The total number of sequences is sum(n^k) 
> for k = 0 to n</xsl:comment>
> B B B  B B B  B B B  B B B  <constraint no="1">
> B B B  B B B  B B B  B B B  B B B  <xsl:value-of select="count($sequences/sequence) 
> eq sum(for $k in 0 to $n return math:pow($n, $k))" />
> B B B  B B B  B B B  B B B  </constraint>
> B B B  B B B  B B B  B B B  <xsl:comment>Inductional proof</xsl:comment>
> B B B  B B B  B B B  B B B  <constraint no="2">
> B B B  B B B  B B B  B B B  B B B  <xsl:value-of select="
> B B B  B B B  B B B  B B B  every $s in $sequences/sequence[count(item) lt $n] 
> satisfies
> B B B  B B B  B B B  B B B  B B B  every $i in $set/item satisfies
> B B B  B B B  B B B  B B B  B B B  B B B  some $sequence-extended in $sequences/sequence 
> satisfies
> B B B B B B B B B B B B B B B B B B B B B B B B B B B B  deep-equal($sequence-extended/item, 
> ($s/item, $i))" />
> B B B  B B B  B B B  B B B  </constraint>
> B B B  B B B  B B B  </compressed>
>
> B B B  B B B  B B B  <lengthy>
> <xsl:comment>sequence[empty(item)]</xsl:comment>
> B B B  B B B  B B B  B B B  <constraint no="1">
> B B B  B B B  B B B  B B B  B B B  <xsl:value-of select="every $s in 
> $sequences/sequence satisfies count($s/item) le $n" />
> B B B  B B B  B B B  B B B  </constraint>
> B B B  B B B  B B B  B B B  <xsl:comment>All sequences have a length less than or 
> equal to count($items)</xsl:comment>
> B B B  B B B  B B B  B B B  <constraint no="2">
> B B B  B B B  B B B  B B B  B B B  <xsl:value-of select="every $s in 
> $sequences/sequence satisfies count($s/item) le $n" />
> B B B  B B B  B B B  B B B  </constraint>
>
> B B B  B B B  B B B  B B B  <xsl:comment>The total number of sequences is sum(n^k) 
> for k = 0 to n</xsl:comment>
> B B B  B B B  B B B  B B B  <constraint no="3">
> B B B  B B B  B B B  B B B  B B B  <xsl:value-of select="count($sequences/sequence) 
> eq sum(for $k in 0 to $n return math:pow($n, $k))" />
> B B B  B B B  B B B  B B B  </constraint>
>
> B B B  B B B  B B B  B B B  <xsl:comment>All sequences are unique with respect to 
> item-type and -order</xsl:comment>
> B B B  B B B  B B B  B B B  <constraint no="4">
> B B B  B B B  B B B  B B B  B B B  <xsl:value-of select="every $s in 
> $sequences/sequence, $t in ($sequences/sequence except $s) satisfies 
> not(deep-equal($s, $t))" />
> B B B  B B B  B B B  B B B  </constraint>
>
> B B B  B B B  B B B  B B B  <xsl:comment>All non-empty sequences must only contain 
> items from the set</xsl:comment>
> B B B  B B B  B B B  B B B  <constraint no="5">
> B B B  B B B  B B B  B B B  B B B  <xsl:value-of select="
> B B B  B B B  B B B  B B B  B B B  B B B  every $s in $sequences/sequence[item] satisfies (
> B B B  B B B  B B B  B B B  B B B  B B B  B B B  every $item in $s/item satisfies (
> B B B  B B B  B B B  B B B  B B B  B B B  B B B  B B B  some $i in $set/item satisfies 
> deep-equal($i, $item)
> B B B  B B B  B B B  B B B  B B B  B B B  B B B  )
> B B B  B B B  B B B  B B B  B B B  B B B  )" />
> B B B  B B B  B B B  B B B  </constraint>
> B B B  B B B  B B B  </lengthy>
>
> B B B  B B B  </evaluation>
> B B B  </xsl:template>
> </xsl:stylesheet>
>
>
> Best regards
> Christoph Naber
>
> Am 22.11.2017 um 20:38 schrieb David Carlisle d.p.carlisle@xxxxxxxxx:
> > your condition 4 is the most complicated and I don't think you need > it, given conditions 1-3 you just need to say that no two of your > 
> sequences are deep-equal. > > David > > On 22 November 2017 at 18:53, 
> Costello, Roger L. costello@xxxxxxxxx > 
> <xsl-list-service@xxxxxxxxxxxxxxxxxxxxxx> wrote: >> Hi Folks, >> >> 
> Thank you for your help this past week in answering my question >> 
> about sequences. Below is a description of the sequences, the >> 
> constraints they must satisfy, and XPath expressions for >> 
> implementing the constraints. /Roger >> >> Problem: Sometimes you want 
> all possible sequences of elements of a >> set with lengths from 0 to 
> n, where n is the number of elements in >> the set. Such sequences are 
> called n-tuples, or, permutations with >> repetition. >> 
> (https://en.wikipedia.org/wiki/Permutation#Permutations_with_repetition) 
> >> >> >> Let's look at some examples.
> >> >> Here is a set: {A, B} >> >> Here are the valid sequences: (), (A), 
> (B), (A, A), (A, B), (B, A), >> (B, B) >> >> Notice it has: - The 
> empty sequence. - Two sequences, one for each >> element in the set. - 
> Four sequences, each consisting of two >> elements, in all 
> permutations. >> >> Total number of sequences: 7 >> >> Suppose the set 
> contains three elements: {A, B, C} >> >> Then here are the valid 
> sequences: >> >> (), (A), (B), (C), (A, A), (A, B), (A, C), (B, A), 
> (B, B), (B, C), >> (C, A), (C, B), (C, C), (A, A, A), (A, A, B), (A, 
> A, C), (A, B, A), >> (A, B, B), (A, B, C), (A, C, A), (A, C, B), (A, 
> C, C), (B, A, A), >> (B, A, B), (B, A, C), (B, B, A), (B, B, B), (B, 
> B, C), (B, C, A), >> (B, C, B), (B, C, C), (C, A, A), (C, A, B), (C, 
> A, C), (C, B, A), >> (C, B, B), (C, B, C), (C, C, A), (C, C, B), (C, 
> C, C) >> >> Notice it has: - The empty sequence. - Three sequences, 
> one for >> each element in the set. - Nine sequences, each consisting 
> of two >> elements, in all permutations. - Twenty-seven sequences, 
> each >> consisting of three elements, in all permutations. >> >> Total 
> number of sequences: 40 >> >> If the set has 4 elements, then the 
> total number of sequences is >> 341. >> >> The number of sequences 
> grows rapidly as the number of elements in >> the set increases. >> >> 
> Of all possible sequences in the universe, only certain sequences >> 
> are valid (i.e., have the desired properties). What are the >> 
> constraints that sequences must satisfy to be valid? >> >> Here are 
> the constraints that sequences must satisfy: >> >> Constraint 1. There 
> must be an empty sequence. Constraint 2. All >> sequences have a 
> length less than or equal to n (the length of the >> set). Constraint 
> 3. The total number of sequences is sum(n^k) for k >> = 0 to n. 
> Constraint 4. For every sequence s that does not already >> have the 
> maximum length, there is, for every item i in the set, an >> 
> (extended) sequence s' whose items are the same as s plus item i. >> 
> >> Let's look at an XML representation of the sequences and how to >> 
> express the constraints using XPath. >> >> Here is an XML 
> representation of a set containing two elements: >> >> <set> 
> <item>A</item> <item>B</item> </set> >> >> Here is an XML 
> representation of the sequences for that set: >> >> <sequences> 
> <sequence/> <sequence> <item>A</item> </sequence> >> <sequence> 
> <item>B</item> </sequence> <sequence> <item>A</item> >> <item>A</item> 
> </sequence> <sequence> <item>A</item> >> <item>B</item> </sequence> 
> <sequence> <item>B</item> >> <item>A</item> </sequence> <sequence> 
> <item>B</item> >> <item>B</item> </sequence> </sequences> >> >> The 
> empty sequence () is represented by: >> >> <sequence/> >> >> The 
> sequence (A) is represented by: >> >> <sequence> <item>A</item> 
> </sequence> >> >> The sequence (A, B) is represented by: >> >> 
> <sequence> <item>A</item> <item>B</item> </sequence> >> >> And so 
> forth. >> >> Below are XPath expressions for each of the constraints. 
> >> >> Note: assume the root element <sequences> is the context node, 
> $set >> is a variable holding the set XML document, and $n is a 
> variable >> holding the number of elements in the set. >> >> 
> Constraint 1. There must be an empty sequence. >> >> 
> sequence[empty(item)] >> >> Constraint 2. All sequences have a length 
> less than or equal to n. >> >> every $sequence in sequence satisfies 
> count($sequence/item) le $n >> >> Constraint 3. The total number of 
> sequences is sum(n^k) for k = 0 >> to n. >> >> count(sequence) = 
> sum(for $i in 0 to $n return math:pow($n, $i)) >> >> Constraint 4. For 
> every sequence s that does not already have the >> maximum length, 
> there is, for every item i in the set, an >> (extended) sequence s' 
> whose items are the same as s plus item i. >> >> every $sequence in 
> sequence[count(item) lt $n] satisfies every >> $item in $set//item 
> satisfies some $sequence-extended in sequence >> satisfies 
> deep-equal($sequence-extended/item, ($sequence/item, >> $item)) >> >> 
> Acknowledgement >> >> Thank you to the following people for their 
> fantastic help with >> creating the XPath expressions: >> >> - David 
> Carlisle - Michael Kay - Christoph Naber >> >
>
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