Show Rado Graphs Are Stable Under Edge and Vertex Removal
$begingroup$
Assume $e$ is any edge in a graph $R$ and $v$ is any vertex in a graph $R$.
- Show that $R-e$ is a Rado Graph if and only if $R$ is a Rado Graph.
- Show that $R-v$ is a Rado Graph if $R$ is a Rado Graph
Do these proofs look good?
(1)
For the removal of an edge $e$, the proof follows from considering the Rado property that $forall$ 2 finite sets of vertices $V, W in R$, $exists$ a vertex $v$ that is connected by edges to every vertex in $V$ and not connected to any vertex in $W$.
Assume that $e$ was one of these edges joining a vertex $x in V$ to a vertex $v in (V cup W) backslash R$ satisfying the Rado property for the given $V, W$. I may also choose $V'$ and $W'$ to be sets s.t. $x in W$ and $V' cup W' = V cup W$. Then still $exists$ $v' in (V cup W) backslash R$ satisfying the Rado property for the given $V', W' in R$. Now, in the graph $R-e$, $v'$ will satisfy the Rado property for $V, W$, and $v$ will satisfy the Rado property for $V', W'$. The argument for the only if part of this condition is the same but consider adding an edge to the graph $R-e$. $square$
(2)
If I remove a vertex $x$ from the graph $R$, I may well have first removed $deg(x)$ edges from $x$. The Rado property will hold under removal of $deg(x)$ edges from a vertex $x in R$ by repeated application of (1). Now I want to remove $x$ which should cause no issue given that x could only be the vertex satisfying the Rado property for $R$ given $V=phi$ and $W$ finite with $W in R-v$, but $R$ was a Rado graph before removing all the edges from $v$ (which would have connected to some vertices possibly in $W$), so there is a vertex $v neq x in R$ that will satisfy the Rado property in this case. Thus, $R-v$ is necessarily Rado. $square$
proof-verification graph-theory
edited Dec 3 '18 at 5:45
Alex Ravsky
39.7k32281
asked Dec 2 '18 at 5:09
rjm27trekkierjm27trekkie
1219
$endgroup$
add a comment |
$begingroup$
Assume $e$ is any edge in a graph $R$ and $v$ is any vertex in a graph $R$.
- Show that $R-e$ is a Rado Graph if and only if $R$ is a Rado Graph.
- Show that $R-v$ is a Rado Graph if $R$ is a Rado Graph
Do these proofs look good?
(1)
For the removal of an edge $e$, the proof follows from considering the Rado property that $forall$ 2 finite sets of vertices $V, W in R$, $exists$ a vertex $v$ that is connected by edges to every vertex in $V$ and not connected to any vertex in $W$.
Assume that $e$ was one of these edges joining a vertex $x in V$ to a vertex $v in (V cup W) backslash R$ satisfying the Rado property for the given $V, W$. I may also choose $V'$ and $W'$ to be sets s.t. $x in W$ and $V' cup W' = V cup W$. Then still $exists$ $v' in (V cup W) backslash R$ satisfying the Rado property for the given $V', W' in R$. Now, in the graph $R-e$, $v'$ will satisfy the Rado property for $V, W$, and $v$ will satisfy the Rado property for $V', W'$. The argument for the only if part of this condition is the same but consider adding an edge to the graph $R-e$. $square$
(2)
If I remove a vertex $x$ from the graph $R$, I may well have first removed $deg(x)$ edges from $x$. The Rado property will hold under removal of $deg(x)$ edges from a vertex $x in R$ by repeated application of (1). Now I want to remove $x$ which should cause no issue given that x could only be the vertex satisfying the Rado property for $R$ given $V=phi$ and $W$ finite with $W in R-v$, but $R$ was a Rado graph before removing all the edges from $v$ (which would have connected to some vertices possibly in $W$), so there is a vertex $v neq x in R$ that will satisfy the Rado property in this case. Thus, $R-v$ is necessarily Rado. $square$
proof-verification graph-theory
edited Dec 3 '18 at 5:45
Alex Ravsky
39.7k32281
asked Dec 2 '18 at 5:09
rjm27trekkierjm27trekkie
1219
$endgroup$
1
$begingroup$
Does it have to be that complicated? Don't we know that, if $V$ and $W$ are disjoint finite sets of vertices in the Rado graph, than there is not just one but an infinite number of vertices $x$ joined to everything in $V$ and nothing in $W$? And isn't it pretty clear than adding or subtracting an edge, or subtracting a vertex, can't spoil more than $1$ of those infinitely many $x$'s?
$endgroup$
– bof
Dec 2 '18 at 11:55
$begingroup$
If we suppose that the Rado property is satisfied by more than one vertex for any 2 finite sets then yes, but the definition doesn't tell me i get 2, just at least 1. So yes, i should have to care about there possibly being only one vertex satisfying the property for given V and W
$endgroup$
– rjm27trekkie
Dec 2 '18 at 16:45
$begingroup$
Then why don't you first prove as a lemma that you can find more than one vertex for given $V$ and $W$? (Hint: what if you take the first vertex $x$ that you got, add it to $V$, and use the Rado property again?)
$endgroup$
– bof
Dec 3 '18 at 11:25
$begingroup$
By the way, I believe the usual definition of the Rado property says that there is a vertex $x$ such that $x$ is joined to every vertex in $V$ and to no vertex in $W$, and $xnotin W$. You left out the $xnotin W$ part. Your version works too, but makes slightly more work. You might start by proving that your version is equivalent to the usual one.
$endgroup$
– bof
Dec 3 '18 at 11:28
add a comment |
$begingroup$
Assume $e$ is any edge in a graph $R$ and $v$ is any vertex in a graph $R$.
- Show that $R-e$ is a Rado Graph if and only if $R$ is a Rado Graph.
- Show that $R-v$ is a Rado Graph if $R$ is a Rado Graph
Do these proofs look good?
(1)
For the removal of an edge $e$, the proof follows from considering the Rado property that $forall$ 2 finite sets of vertices $V, W in R$, $exists$ a vertex $v$ that is connected by edges to every vertex in $V$ and not connected to any vertex in $W$.
Assume that $e$ was one of these edges joining a vertex $x in V$ to a vertex $v in (V cup W) backslash R$ satisfying the Rado property for the given $V, W$. I may also choose $V'$ and $W'$ to be sets s.t. $x in W$ and $V' cup W' = V cup W$. Then still $exists$ $v' in (V cup W) backslash R$ satisfying the Rado property for the given $V', W' in R$. Now, in the graph $R-e$, $v'$ will satisfy the Rado property for $V, W$, and $v$ will satisfy the Rado property for $V', W'$. The argument for the only if part of this condition is the same but consider adding an edge to the graph $R-e$. $square$
(2)
If I remove a vertex $x$ from the graph $R$, I may well have first removed $deg(x)$ edges from $x$. The Rado property will hold under removal of $deg(x)$ edges from a vertex $x in R$ by repeated application of (1). Now I want to remove $x$ which should cause no issue given that x could only be the vertex satisfying the Rado property for $R$ given $V=phi$ and $W$ finite with $W in R-v$, but $R$ was a Rado graph before removing all the edges from $v$ (which would have connected to some vertices possibly in $W$), so there is a vertex $v neq x in R$ that will satisfy the Rado property in this case. Thus, $R-v$ is necessarily Rado. $square$
proof-verification graph-theory
edited Dec 3 '18 at 5:45
Alex Ravsky
39.7k32281
asked Dec 2 '18 at 5:09
rjm27trekkierjm27trekkie
1219
$endgroup$
Assume $e$ is any edge in a graph $R$ and $v$ is any vertex in a graph $R$.
- Show that $R-e$ is a Rado Graph if and only if $R$ is a Rado Graph.
- Show that $R-v$ is a Rado Graph if $R$ is a Rado Graph
Do these proofs look good?
(1)
For the removal of an edge $e$, the proof follows from considering the Rado property that $forall$ 2 finite sets of vertices $V, W in R$, $exists$ a vertex $v$ that is connected by edges to every vertex in $V$ and not connected to any vertex in $W$.
Assume that $e$ was one of these edges joining a vertex $x in V$ to a vertex $v in (V cup W) backslash R$ satisfying the Rado property for the given $V, W$. I may also choose $V'$ and $W'$ to be sets s.t. $x in W$ and $V' cup W' = V cup W$. Then still $exists$ $v' in (V cup W) backslash R$ satisfying the Rado property for the given $V', W' in R$. Now, in the graph $R-e$, $v'$ will satisfy the Rado property for $V, W$, and $v$ will satisfy the Rado property for $V', W'$. The argument for the only if part of this condition is the same but consider adding an edge to the graph $R-e$. $square$
(2)
If I remove a vertex $x$ from the graph $R$, I may well have first removed $deg(x)$ edges from $x$. The Rado property will hold under removal of $deg(x)$ edges from a vertex $x in R$ by repeated application of (1). Now I want to remove $x$ which should cause no issue given that x could only be the vertex satisfying the Rado property for $R$ given $V=phi$ and $W$ finite with $W in R-v$, but $R$ was a Rado graph before removing all the edges from $v$ (which would have connected to some vertices possibly in $W$), so there is a vertex $v neq x in R$ that will satisfy the Rado property in this case. Thus, $R-v$ is necessarily Rado. $square$
proof-verification graph-theory
proof-verification graph-theory
edited Dec 3 '18 at 5:45
Alex Ravsky
39.7k32281
asked Dec 2 '18 at 5:09
rjm27trekkierjm27trekkie
1219
edited Dec 3 '18 at 5:45
Alex Ravsky
39.7k32281
asked Dec 2 '18 at 5:09
rjm27trekkierjm27trekkie
1219
edited Dec 3 '18 at 5:45
Alex Ravsky
39.7k32281
edited Dec 3 '18 at 5:45
Alex Ravsky
39.7k32281
edited Dec 3 '18 at 5:45
Alex Ravsky
39.7k32281
39.7k32281
asked Dec 2 '18 at 5:09
rjm27trekkierjm27trekkie
1219
asked Dec 2 '18 at 5:09
rjm27trekkierjm27trekkie
1219
asked Dec 2 '18 at 5:09
rjm27trekkierjm27trekkie
1219
1219
1
$begingroup$
Does it have to be that complicated? Don't we know that, if $V$ and $W$ are disjoint finite sets of vertices in the Rado graph, than there is not just one but an infinite number of vertices $x$ joined to everything in $V$ and nothing in $W$? And isn't it pretty clear than adding or subtracting an edge, or subtracting a vertex, can't spoil more than $1$ of those infinitely many $x$'s?
$endgroup$
– bof
Dec 2 '18 at 11:55
$begingroup$
If we suppose that the Rado property is satisfied by more than one vertex for any 2 finite sets then yes, but the definition doesn't tell me i get 2, just at least 1. So yes, i should have to care about there possibly being only one vertex satisfying the property for given V and W
$endgroup$
– rjm27trekkie
Dec 2 '18 at 16:45
$begingroup$
Then why don't you first prove as a lemma that you can find more than one vertex for given $V$ and $W$? (Hint: what if you take the first vertex $x$ that you got, add it to $V$, and use the Rado property again?)
$endgroup$
– bof
Dec 3 '18 at 11:25
$begingroup$
By the way, I believe the usual definition of the Rado property says that there is a vertex $x$ such that $x$ is joined to every vertex in $V$ and to no vertex in $W$, and $xnotin W$. You left out the $xnotin W$ part. Your version works too, but makes slightly more work. You might start by proving that your version is equivalent to the usual one.
$endgroup$
– bof
Dec 3 '18 at 11:28
add a comment |
1
$begingroup$
Does it have to be that complicated? Don't we know that, if $V$ and $W$ are disjoint finite sets of vertices in the Rado graph, than there is not just one but an infinite number of vertices $x$ joined to everything in $V$ and nothing in $W$? And isn't it pretty clear than adding or subtracting an edge, or subtracting a vertex, can't spoil more than $1$ of those infinitely many $x$'s?
$endgroup$
– bof
Dec 2 '18 at 11:55
$begingroup$
If we suppose that the Rado property is satisfied by more than one vertex for any 2 finite sets then yes, but the definition doesn't tell me i get 2, just at least 1. So yes, i should have to care about there possibly being only one vertex satisfying the property for given V and W
$endgroup$
– rjm27trekkie
Dec 2 '18 at 16:45
$begingroup$
Then why don't you first prove as a lemma that you can find more than one vertex for given $V$ and $W$? (Hint: what if you take the first vertex $x$ that you got, add it to $V$, and use the Rado property again?)
$endgroup$
– bof
Dec 3 '18 at 11:25
$begingroup$
By the way, I believe the usual definition of the Rado property says that there is a vertex $x$ such that $x$ is joined to every vertex in $V$ and to no vertex in $W$, and $xnotin W$. You left out the $xnotin W$ part. Your version works too, but makes slightly more work. You might start by proving that your version is equivalent to the usual one.
$endgroup$
– bof
Dec 3 '18 at 11:28
1
1
$begingroup$
Does it have to be that complicated? Don't we know that, if $V$ and $W$ are disjoint finite sets of vertices in the Rado graph, than there is not just one but an infinite number of vertices $x$ joined to everything in $V$ and nothing in $W$? And isn't it pretty clear than adding or subtracting an edge, or subtracting a vertex, can't spoil more than $1$ of those infinitely many $x$'s?
$endgroup$
– bof
Dec 2 '18 at 11:55
$begingroup$
Does it have to be that complicated? Don't we know that, if $V$ and $W$ are disjoint finite sets of vertices in the Rado graph, than there is not just one but an infinite number of vertices $x$ joined to everything in $V$ and nothing in $W$? And isn't it pretty clear than adding or subtracting an edge, or subtracting a vertex, can't spoil more than $1$ of those infinitely many $x$'s?
$endgroup$
– bof
Dec 2 '18 at 11:55
$begingroup$
If we suppose that the Rado property is satisfied by more than one vertex for any 2 finite sets then yes, but the definition doesn't tell me i get 2, just at least 1. So yes, i should have to care about there possibly being only one vertex satisfying the property for given V and W
$endgroup$
– rjm27trekkie
Dec 2 '18 at 16:45
$begingroup$
If we suppose that the Rado property is satisfied by more than one vertex for any 2 finite sets then yes, but the definition doesn't tell me i get 2, just at least 1. So yes, i should have to care about there possibly being only one vertex satisfying the property for given V and W
$endgroup$
– rjm27trekkie
Dec 2 '18 at 16:45
$begingroup$
Then why don't you first prove as a lemma that you can find more than one vertex for given $V$ and $W$? (Hint: what if you take the first vertex $x$ that you got, add it to $V$, and use the Rado property again?)
$endgroup$
– bof
Dec 3 '18 at 11:25
$begingroup$
Then why don't you first prove as a lemma that you can find more than one vertex for given $V$ and $W$? (Hint: what if you take the first vertex $x$ that you got, add it to $V$, and use the Rado property again?)
$endgroup$
– bof
Dec 3 '18 at 11:25
$begingroup$
By the way, I believe the usual definition of the Rado property says that there is a vertex $x$ such that $x$ is joined to every vertex in $V$ and to no vertex in $W$, and $xnotin W$. You left out the $xnotin W$ part. Your version works too, but makes slightly more work. You might start by proving that your version is equivalent to the usual one.
$endgroup$
– bof
Dec 3 '18 at 11:28
$begingroup$
By the way, I believe the usual definition of the Rado property says that there is a vertex $x$ such that $x$ is joined to every vertex in $V$ and to no vertex in $W$, and $xnotin W$. You left out the $xnotin W$ part. Your version works too, but makes slightly more work. You might start by proving that your version is equivalent to the usual one.
$endgroup$
– bof
Dec 3 '18 at 11:28
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I disagree with part of your first proof. A vertex $v in (V cup W) backslash R$ makes no sense, since $V$ and $W$ are both subsets of R, thus $(V cup W) backslash R = phi$. A definition of $R backslash (V cup W)$ makes more sense to me, especially given how you defined the Rado property (each vertex $v$ which fulfills the property for some $V$ and $W$ cannot be a member of either $V$ nor $W$).
answered Dec 3 '18 at 9:46
theasianpianisttheasianpianist
31
$endgroup$
$begingroup$
This is the definition. $V subset R, W subset R$ but $V cup W neq R$. They are not spanning subsets of $R$ because they are FINITE while $R$ is countably infinite. (V cup W) | R describes all these vertices not in V or W (2 finite subsets of R) that are also in R (that's what the relative set complement under R ( operator) means
$endgroup$
– rjm27trekkie
Dec 3 '18 at 15:38
$begingroup$
en.wikipedia.org/wiki/Complement_(set_theory) I stand by the statement that $(V cup W) backslash R$ means all members of $V cup W$ which are not in $R$.
$endgroup$
– theasianpianist
Dec 3 '18 at 15:54
add a comment |
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1 Answer
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$begingroup$
I disagree with part of your first proof. A vertex $v in (V cup W) backslash R$ makes no sense, since $V$ and $W$ are both subsets of R, thus $(V cup W) backslash R = phi$. A definition of $R backslash (V cup W)$ makes more sense to me, especially given how you defined the Rado property (each vertex $v$ which fulfills the property for some $V$ and $W$ cannot be a member of either $V$ nor $W$).
answered Dec 3 '18 at 9:46
theasianpianisttheasianpianist
31
$endgroup$
$begingroup$
This is the definition. $V subset R, W subset R$ but $V cup W neq R$. They are not spanning subsets of $R$ because they are FINITE while $R$ is countably infinite. (V cup W) | R describes all these vertices not in V or W (2 finite subsets of R) that are also in R (that's what the relative set complement under R ( operator) means
$endgroup$
– rjm27trekkie
Dec 3 '18 at 15:38
$begingroup$
en.wikipedia.org/wiki/Complement_(set_theory) I stand by the statement that $(V cup W) backslash R$ means all members of $V cup W$ which are not in $R$.
$endgroup$
– theasianpianist
Dec 3 '18 at 15:54
add a comment |
$begingroup$
I disagree with part of your first proof. A vertex $v in (V cup W) backslash R$ makes no sense, since $V$ and $W$ are both subsets of R, thus $(V cup W) backslash R = phi$. A definition of $R backslash (V cup W)$ makes more sense to me, especially given how you defined the Rado property (each vertex $v$ which fulfills the property for some $V$ and $W$ cannot be a member of either $V$ nor $W$).
answered Dec 3 '18 at 9:46
theasianpianisttheasianpianist
31
$endgroup$
$begingroup$
This is the definition. $V subset R, W subset R$ but $V cup W neq R$. They are not spanning subsets of $R$ because they are FINITE while $R$ is countably infinite. (V cup W) | R describes all these vertices not in V or W (2 finite subsets of R) that are also in R (that's what the relative set complement under R ( operator) means
$endgroup$
– rjm27trekkie
Dec 3 '18 at 15:38
$begingroup$
en.wikipedia.org/wiki/Complement_(set_theory) I stand by the statement that $(V cup W) backslash R$ means all members of $V cup W$ which are not in $R$.
$endgroup$
– theasianpianist
Dec 3 '18 at 15:54
add a comment |
$begingroup$
I disagree with part of your first proof. A vertex $v in (V cup W) backslash R$ makes no sense, since $V$ and $W$ are both subsets of R, thus $(V cup W) backslash R = phi$. A definition of $R backslash (V cup W)$ makes more sense to me, especially given how you defined the Rado property (each vertex $v$ which fulfills the property for some $V$ and $W$ cannot be a member of either $V$ nor $W$).
answered Dec 3 '18 at 9:46
theasianpianisttheasianpianist
31
$endgroup$
I disagree with part of your first proof. A vertex $v in (V cup W) backslash R$ makes no sense, since $V$ and $W$ are both subsets of R, thus $(V cup W) backslash R = phi$. A definition of $R backslash (V cup W)$ makes more sense to me, especially given how you defined the Rado property (each vertex $v$ which fulfills the property for some $V$ and $W$ cannot be a member of either $V$ nor $W$).
answered Dec 3 '18 at 9:46
theasianpianisttheasianpianist
31
answered Dec 3 '18 at 9:46
theasianpianisttheasianpianist
31
answered Dec 3 '18 at 9:46
theasianpianisttheasianpianist
31
answered Dec 3 '18 at 9:46
theasianpianisttheasianpianist
31
31
$begingroup$
This is the definition. $V subset R, W subset R$ but $V cup W neq R$. They are not spanning subsets of $R$ because they are FINITE while $R$ is countably infinite. (V cup W) | R describes all these vertices not in V or W (2 finite subsets of R) that are also in R (that's what the relative set complement under R ( operator) means
$endgroup$
– rjm27trekkie
Dec 3 '18 at 15:38
$begingroup$
en.wikipedia.org/wiki/Complement_(set_theory) I stand by the statement that $(V cup W) backslash R$ means all members of $V cup W$ which are not in $R$.
$endgroup$
– theasianpianist
Dec 3 '18 at 15:54
add a comment |
$begingroup$
This is the definition. $V subset R, W subset R$ but $V cup W neq R$. They are not spanning subsets of $R$ because they are FINITE while $R$ is countably infinite. (V cup W) | R describes all these vertices not in V or W (2 finite subsets of R) that are also in R (that's what the relative set complement under R ( operator) means
$endgroup$
– rjm27trekkie
Dec 3 '18 at 15:38
$begingroup$
en.wikipedia.org/wiki/Complement_(set_theory) I stand by the statement that $(V cup W) backslash R$ means all members of $V cup W$ which are not in $R$.
$endgroup$
– theasianpianist
Dec 3 '18 at 15:54
$begingroup$
This is the definition. $V subset R, W subset R$ but $V cup W neq R$. They are not spanning subsets of $R$ because they are FINITE while $R$ is countably infinite. (V cup W) | R describes all these vertices not in V or W (2 finite subsets of R) that are also in R (that's what the relative set complement under R ( operator) means
$endgroup$
– rjm27trekkie
Dec 3 '18 at 15:38
$begingroup$
This is the definition. $V subset R, W subset R$ but $V cup W neq R$. They are not spanning subsets of $R$ because they are FINITE while $R$ is countably infinite. (V cup W) | R describes all these vertices not in V or W (2 finite subsets of R) that are also in R (that's what the relative set complement under R ( operator) means
$endgroup$
– rjm27trekkie
Dec 3 '18 at 15:38
$begingroup$
en.wikipedia.org/wiki/Complement_(set_theory) I stand by the statement that $(V cup W) backslash R$ means all members of $V cup W$ which are not in $R$.
$endgroup$
– theasianpianist
Dec 3 '18 at 15:54
$begingroup$
en.wikipedia.org/wiki/Complement_(set_theory) I stand by the statement that $(V cup W) backslash R$ means all members of $V cup W$ which are not in $R$.
$endgroup$
– theasianpianist
Dec 3 '18 at 15:54
add a comment |
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1
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Does it have to be that complicated? Don't we know that, if $V$ and $W$ are disjoint finite sets of vertices in the Rado graph, than there is not just one but an infinite number of vertices $x$ joined to everything in $V$ and nothing in $W$? And isn't it pretty clear than adding or subtracting an edge, or subtracting a vertex, can't spoil more than $1$ of those infinitely many $x$'s?
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– bof
Dec 2 '18 at 11:55
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If we suppose that the Rado property is satisfied by more than one vertex for any 2 finite sets then yes, but the definition doesn't tell me i get 2, just at least 1. So yes, i should have to care about there possibly being only one vertex satisfying the property for given V and W
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– rjm27trekkie
Dec 2 '18 at 16:45
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Then why don't you first prove as a lemma that you can find more than one vertex for given $V$ and $W$? (Hint: what if you take the first vertex $x$ that you got, add it to $V$, and use the Rado property again?)
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– bof
Dec 3 '18 at 11:25
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By the way, I believe the usual definition of the Rado property says that there is a vertex $x$ such that $x$ is joined to every vertex in $V$ and to no vertex in $W$, and $xnotin W$. You left out the $xnotin W$ part. Your version works too, but makes slightly more work. You might start by proving that your version is equivalent to the usual one.
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– bof
Dec 3 '18 at 11:28