diff --git a/info4/kapitel-0/Mengentheorie.ipynb b/info4/kapitel-0/Mengentheorie.ipynb
index 88f92cb38714c0903ae75e4a0a94c67020ea92a6..37a6a1f985ec57be2c846475ad8d7443d1036136 100644
--- a/info4/kapitel-0/Mengentheorie.ipynb
+++ b/info4/kapitel-0/Mengentheorie.ipynb
@@ -6,7 +6,7 @@
"source": [
"# Theoretische Informatik - Vorlesung 0 - Teil 2 Mengentheorie\n",
"\n",
- "* April 2020\n",
+ "* April 2023\n",
"* Michael Leuschel\n",
"* Lehrstuhl Softwaretechnik und Programmiersprachen\n",
"* Heinrich-Heine Universität Düsseldorf\n",
@@ -18,7 +18,7 @@
"Auswertung der Formeln erfolgt mit dem [Jupyter Kernel](https://gitlab.cs.uni-duesseldorf.de/general/stups/prob2-jupyter-kernel) für [ProB](https://prob.hhu.de/w/)\n",
"\n",
"\n",
- "Um dieses Notebook zu starten kann man entweder selber Jupyter und den [ProB Kernel](https://gitlab.cs.uni-duesseldorf.de/general/stups/prob2-jupyter-kernel) installieren.\n",
+ "Um dieses Notebook zu starten kann man entweder selber Jupyter und den [ProB-Kernel](https://gitlab.cs.uni-duesseldorf.de/general/stups/prob2-jupyter-kernel) installieren.\n",
"Man kann aber auch die Notebooks vom Browser aus mit Binder starten (das dauert besonders beim ersten Mal etwas länger):\n",
"\n",
"[](https://mybinder.org/v2/git/https%3A%2F%2Fgitlab.cs.uni-duesseldorf.de%2Fgeneral%2Fstups%2Fprob-teaching-notebooks.git/HEAD?urlpath=%2Ftree%2Finfo4%2Fkapitel-0%2FMengentheorie.ipynb)"
@@ -33,22 +33,20 @@
"Fundamentale Idee der Mengentheorie: \n",
"\n",
"* _\"Unter einer ‚Menge‘ verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten m unserer Anschauung oder unseres Denkens (welche die ‚Elemente‘ von M genannt werden) zu einem Ganzen.\"_ Georg Cantor (siehe [Textstelle als Bild in Wikipedia](https://de.wikipedia.org/wiki/Datei:Textstelle_mit_der_Mengendefinition_von_Georg_Cantor.png))\n",
- "* _\"The ability to regard any collection of objects as a single entity (i.e., as a set).\"_ (Keith Devlin. Joy of Sets. Springer-Verlag.)\n"
+ "* _\"The ability to regard any collection of objects as a single entity (i.e., as a set).\"_ (Keith Devlin. Joy of Sets. Springer-Verlag.)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
- "\n",
"In der Regel gibt es eine Domäne an \"Objekten\" mit denen man Mengen bauen kann.\n",
"Was genau diese Objekte sind interessiert uns in der Mengentheorie nicht.\n",
"\n",
"Fundamental sind diese beiden Symbole:\n",
"* wenn $a$ ein Objekt ist und $x$ eine Menge, dann \n",
- " * ist $a \\in x$ wahr, wenn $a$ ein Element von $x$ ist\n",
- " * ist $a \\not\\in x$ wahr, wenn $a$ **kein** Element von $x$ ist.\n",
- "\n",
+ " * ist $a \\in x$ wahr, wenn $a$ ein Element von $x$ ist\n",
+ " * ist $a \\not\\in x$ wahr, wenn $a$ **kein** Element von $x$ ist.\n",
"\n",
"$\\in$ und $\\not\\in$ sind Prädikate, verbunden durch die Eigenschaft:\n",
"* $\\forall(a,x).(a\\not\\in x \\Leftrightarrow \\neg(a \\in x))$"
@@ -62,10 +60,10 @@
"\n",
"Eine besondere Menge ist die leere Menge $\\emptyset$.\n",
"Sie hat keine Elemente:\n",
- " * $z = \\emptyset \\Leftrightarrow \\forall(a).(a\\not\\in z)$\n",
+ "* $z = \\emptyset \\Leftrightarrow \\forall(a).(a\\not\\in z)$\n",
"\n",
"Zwei Mengen $x$ und $y$ sind gleich gdw sie die gleichen Elemente haben:\n",
- " * $\\forall(x,y).(x=y \\Leftrightarrow \\forall(a).(a\\in x \\Leftrightarrow a \\in y))$"
+ "* $\\forall(x,y).(x=y \\Leftrightarrow \\forall(a).(a\\in x \\Leftrightarrow a \\in y))$"
]
},
{
@@ -261,8 +259,8 @@
"\n",
"Drei wichtige Operationen auf Mengen sind wie folgt.\n",
"\n",
- "Vereinigung von Mengen $\\cup$ :\n",
- " * $z = x\\cup y \\Leftrightarrow \\forall(a).(a\\in z \\Leftrightarrow (a\\in x \\vee a \\in y))$\n",
+ "Vereinigung von Mengen $\\cup$:\n",
+ "* $z = x\\cup y \\Leftrightarrow \\forall(a).(a\\in z \\Leftrightarrow (a\\in x \\vee a \\in y))$\n",
"\n",
"Schnitt von Mengen $\\cap$:\n",
"* $z = x\\cap y \\Leftrightarrow \\forall(a).(a\\in z \\Leftrightarrow (a\\in x \\wedge a \\in y))$\n",
@@ -314,7 +312,7 @@
}
],
"source": [
- "{2,3,5}∩{5,7}"
+ "{2,3,5} ∩ {5,7}"
]
},
{
@@ -337,7 +335,7 @@
}
],
"source": [
- "{2,3,5}−{5,7}"
+ "{2,3,5} \\ {5,7}"
]
},
{
@@ -348,7 +346,7 @@
" \n",
"Man kann die Elemente einer Menge auch durch ein Prädikat beschreiben:$\\{a \\mid P(a)\\}$.\n",
"Es gilt:\n",
- "* $z = \\{a \\mid P(a)\\} \\Leftrightarrow \\forall(a).(a\\in z \\Leftrightarrow P(a))$\n"
+ "* $z = \\{a \\mid P(a)\\} \\Leftrightarrow \\forall(a).(a\\in z \\Leftrightarrow P(a))$"
]
},
{
@@ -371,7 +369,7 @@
}
],
"source": [
- "{a | a>1 & a<6 & a≠4}"
+ "{a | a∈ℕ ∧ a>1 ∧ a<6 ∧ a≠4}"
]
},
{
@@ -389,10 +387,10 @@
{
"data": {
"text/markdown": [
- "$\\{\\mathit{a}\\mid \\mathit{a} > 10\\}$"
+ "$\\newcommand{\\nat}{\\mathord{\\mathbb N}}\\{\\mathit{a}\\mid\\mathit{a} \\in \\nat \\land \\mathit{a} > 10\\}$"
],
"text/plain": [
- "{a∣a > 10}"
+ "{a∣a ∈ ℕ ∧ a > 10}"
]
},
"execution_count": 12,
@@ -401,7 +399,7 @@
}
],
"source": [
- "{a | a>10}"
+ "{a | a∈ℕ ∧ a>10}"
]
},
{
@@ -424,7 +422,7 @@
}
],
"source": [
- "{a | a mod 2 = 0} ∩ {2,3,5}"
+ "{a | a∈ℕ ∧ a mod 2 = 0} ∩ {2,3,5}"
]
},
{
@@ -468,7 +466,7 @@
}
],
"source": [
- "x = {2,3} & y = {2,5} & xy = {a| a∈x ∨ a∈y}"
+ "x = {2,3} ∧ y = {2,5} ∧ xy = {a | a∈x ∨ a∈y}"
]
},
{
@@ -501,7 +499,7 @@
}
],
"source": [
- "x = {2,3} & y = {2,5} & xy = {a| a∈x ∧ a∈y}"
+ "x = {2,3} ∧ y = {2,5} ∧ xy = {a | a∈x ∧ a∈y}"
]
},
{
@@ -534,14 +532,14 @@
}
],
"source": [
- "x = {2,3} & y = {2,5} & xy = {a| a∈x ∧ a∉y}"
+ "x = {2,3} ∧ y = {2,5} ∧ xy = {a | a∈x ∧ a∉y}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
- "Als Kürzel führen wir auch die Notation $a..b$ für $\\{x \\mid x \\geq a \\wedge x \\leq b\\}$ ein."
+ "Als Kürzel führen wir auch die Notation $a..b$ für $\\{x \\mid x \\in \\mathbb{Z} \\wedge x \\geq a \\wedge x \\leq b\\}$ ein."
]
},
{
@@ -979,7 +977,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Achtung, die Kardinalität kann auch unendlich sein: je nach Formalismus, ist folgender Ausdruck entweder unendlich oder nicht wohl definiert: $\\mid \\{x \\mid x>0\\} \\mid$"
+ "Achtung, die Kardinalität kann auch unendlich sein: je nach Formalismus, ist folgender Ausdruck entweder unendlich oder nicht wohl definiert: $|\\{x \\mid x \\in \\mathbb{Z} \\wedge x>0\\}|$"
]
},
{
@@ -989,16 +987,16 @@
"outputs": [
{
"ename": "CommandExecutionException",
- "evalue": ":eval: NOT-WELL-DEFINED: \ncard applied to very large set, cardinality not representable in ProB: closure([x],[integer],b(greater(b(identifier(...),integer,[...]),b(value(...),integer,[...])),pred,[nodeid(pos(...))]))\n\n",
+ "evalue": ":eval: NOT-WELL-DEFINED: \ncard applied to very large set, cardinality not representable in ProB: closure([x],[integer],b(greater(b(identifier(...),integer,[...]),b(value(...),integer,[...])),pred,[nodeid(pos(...)),prob_annotation('DO_NOT_ENUMERATE'(...)),removed_typing]))\n\n",
"output_type": "error",
"traceback": [
"\u001b[1m\u001b[31m:eval: NOT-WELL-DEFINED: \u001b[0m",
- "\u001b[1m\u001b[31mcard applied to very large set, cardinality not representable in ProB: closure([x],[integer],b(greater(b(identifier(...),integer,[...]),b(value(...),integer,[...])),pred,[nodeid(pos(...))]))\u001b[0m"
+ "\u001b[1m\u001b[31mcard applied to very large set, cardinality not representable in ProB: closure([x],[integer],b(greater(b(identifier(...),integer,[...]),b(value(...),integer,[...])),pred,[nodeid(pos(...)),prob_annotation('DO_NOT_ENUMERATE'(...)),removed_typing]))\u001b[0m"
]
}
],
"source": [
- "card({x|x>0})"
+ "card({x|x∈ℤ ∧ x>0})"
]
},
{
@@ -1222,9 +1220,9 @@
],
"source": [
"{S,E,N,D,M,O,R,Y} ⊆ 0..9 ∧ card({S,E,N,D,M,O,R,Y}) = 8 ∧ S ≠ 0 & M ≠ 0 ∧\n",
- " S*1000 + E*100 + N*10 + D +\n",
- " M*1000 + O*100 + R*10 + E =\n",
- " M*10000 + O*1000 + N*100 + E*10 + Y"
+ " S*1000 + E*100 + N*10 + D +\n",
+ " M*1000 + O*100 + R*10 + E =\n",
+ " M*10000 + O*1000 + N*100 + E*10 + Y"
]
},
{
@@ -1254,12 +1252,13 @@
}
],
"source": [
- " {S,E,N,D, M,O,R, Y |\n",
- " {S,E,N,D, M,O,R, Y} ⊆ 0..9 ∧ S >0 ∧ M >0 ∧ \n",
- " card({S,E,N,D, M,O,R, Y}) = 8 ∧ \n",
- " S*1000 + E*100 + N*10 + D +\n",
- " M*1000 + O*100 + R*10 + E =\n",
- " M*10000 + O*1000 + N*100 + E*10 + Y }"
+ "{S,E,N,D, M,O,R, Y |\n",
+ " {S,E,N,D, M,O,R, Y} ⊆ 0..9 ∧ S >0 ∧ M >0 ∧\n",
+ " card({S,E,N,D, M,O,R, Y}) = 8 ∧\n",
+ " S*1000 + E*100 + N*10 + D +\n",
+ " M*1000 + O*100 + R*10 + E =\n",
+ " M*10000 + O*1000 + N*100 + E*10 + Y\n",
+ "}"
]
},
{
@@ -1272,7 +1271,7 @@
"text/markdown": [
"|S|E|N|D|M|O|R|Y|\n",
"|---|---|---|---|---|---|---|---|\n",
- "|$9$|$5$|$6$|$7$|$1$|$0$|$8$|$2$|\n"
+ "|9|5|6|7|1|0|8|2|\n"
],
"text/plain": [
"S\tE\tN\tD\tM\tO\tR\tY\n",
@@ -1285,12 +1284,13 @@
}
],
"source": [
- ":table {S,E,N,D, M,O,R, Y |\n",
- " {S,E,N,D, M,O,R, Y} ⊆ 0..9 ∧ S >0 ∧ M >0 ∧ \n",
- " card({S,E,N,D, M,O,R, Y}) = 8 ∧ \n",
- " S*1000 + E*100 + N*10 + D +\n",
- " M*1000 + O*100 + R*10 + E =\n",
- " M*10000 + O*1000 + N*100 + E*10 + Y }"
+ ":table {S,E,N,D, M,O,R, Y |\n",
+ " {S,E,N,D, M,O,R, Y} ⊆ 0..9 ∧ S >0 ∧ M >0 ∧\n",
+ " card({S,E,N,D, M,O,R, Y}) = 8 ∧\n",
+ " S*1000 + E*100 + N*10 + D +\n",
+ " M*1000 + O*100 + R*10 + E =\n",
+ " M*10000 + O*1000 + N*100 + E*10 + Y\n",
+ "}"
]
},
{
@@ -1324,7 +1324,7 @@
"text/markdown": [
"|K|I|S|P|A|O|N|\n",
"|---|---|---|---|---|---|---|\n",
- "|$2$|$0$|$3$|$4$|$1$|$8$|$9$|\n"
+ "|2|0|3|4|1|8|9|\n"
],
"text/plain": [
"K\tI\tS\tP\tA\tO\tN\n",
@@ -1341,7 +1341,7 @@
" {K,P} ⊆ 1..9 ∧\n",
" {I,S,A,O,N} ⊆ 0..9 ∧\n",
" (1000*K+100*I+10*S+S) * (1000*K+100*I+10*S+S) \n",
- " = 1000000*P+100000*A+10000*S+1000*S+100*I+10*O+N &\n",
+ " = 1000000*P+100000*A+10000*S+1000*S+100*I+10*O+N &\n",
" card({K, I, S, P, A, O, N}) = 7}"
]
},
@@ -1360,9 +1360,9 @@
"* $\\{2,3,4\\}$ \n",
"\n",
"Wir haben zum Beispiel:\n",
- "* $\\mathit{card}(\\{\\{2,3,4\\}\\}) = 1$\n",
- "* $\\mathit{card}(\\{\\{2\\},\\{3,4\\}\\}) = 2$\n",
- "* $\\mathit{card}(\\{2,3,4\\}) = 3$\n",
+ "* $|\\{\\{2,3,4\\}\\}| = 1$\n",
+ "* $|\\{\\{2\\},\\{3,4\\}\\}| = 2$\n",
+ "* $|\\{2,3,4\\}| = 3$\n",
"* $\\{2\\} \\in \\{\\{2\\},\\{3,4\\}\\} $\n",
"* $\\{2\\} \\not\\in \\{\\{2,3\\},\\{4\\}\\} $"
]
@@ -1717,14 +1717,14 @@
"text/markdown": [
"|Elements|\n",
"|---|\n",
- "|$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$|\n",
- "|$\\{1\\}$|\n",
- "|$\\{1,2\\}$|\n",
- "|$\\{1,3\\}$|\n",
- "|$\\{2\\}$|\n",
- "|$\\{1,2,3\\}$|\n",
- "|$\\{2,3\\}$|\n",
- "|$\\{3\\}$|\n"
+ "|∅|\n",
+ "|{1}|\n",
+ "|{1,2}|\n",
+ "|{1,3}|\n",
+ "|{2}|\n",
+ "|{1,2,3}|\n",
+ "|{2,3}|\n",
+ "|{3}|\n"
],
"text/plain": [
"Elements\n",
@@ -1757,10 +1757,10 @@
"text/markdown": [
"|Elements|\n",
"|---|\n",
- "|$\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\renewcommand{\\emptyset}{\\mathord\\varnothing}\\emptyset$|\n",
- "|$\\{\\emptyset\\}$|\n",
- "|$\\{\\emptyset,\\{1\\}\\}$|\n",
- "|$\\{\\{1\\}\\}$|\n"
+ "|∅|\n",
+ "|{∅}|\n",
+ "|{∅,{1}}|\n",
+ "|{{1}}|\n"
],
"text/plain": [
"Elements\n",
@@ -1902,12 +1902,12 @@
"* Was ist eine Relation?\n",
"* Wie kann man Relationen in Mengentheorie und Logik abbilden?\n",
"\n",
- "* https://en.wikipedia.org/wiki/Finitary\\_relation}:\n",
- " * _In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple._\n",
+ "* https://en.wikipedia.org/wiki/Finitary_relation:\n",
+ " > In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple.\n",
"\n",
- "* k-Tupel: $(1,2)$ (k=2, Paar)\n",
+ "* k-Tupel: $(1,2)$ (k = 2, Paar)\n",
"* Eine Relation weist k-Tupeln Wahrheitswerte zu: \n",
- " * Beispiel: $(1,2) \\mapsto TRUE$, $(2,1) \\mapsto FALSE$.\n",
+ " * Beispiel: $(1,2) \\mapsto \\mathit{TRUE}$, $(2,1) \\mapsto \\mathit{FALSE}$.\n",
"\n",
"## Unäre Relationen\n",
"\n",
@@ -1915,12 +1915,12 @@
"* die Menge an Werten für die die Relation wahr ist.\n",
"\n",
"Beispiele:\n",
- "* Relation ``Ziffer'' über die ganzen Zahlen ist\n",
- " * $0 .. 9 \\subseteq$ ℤ\n",
- "* Relation ``Gt0'' über die ganzen Zahlen ist\n",
- " * $\\{ \\mathit{x}|\\mathit{x} \\in $ℤ $\\wedge \\mathit{x} > 0\\} \\subseteq$ ℤ.\n",
- "* Relation ``Gerade'' über die ganzen Zahlen ist\n",
- " * $\\{ \\mathit{x}|\\mathit{x} \\in $ ℤ $\\wedge \\mathit{x} \\mod 2 = 0\\} \\subseteq$ ℤ.\n"
+ "* Relation \"Ziffer\" über die ganzen Zahlen ist\n",
+ " * $0 .. 9 \\subseteq \\mathbb{Z}$\n",
+ "* Relation \"Gt0\" über die ganzen Zahlen ist\n",
+ " * $\\{ \\mathit{x}|\\mathit{x} \\in \\mathbb{Z} \\wedge \\mathit{x} > 0\\} \\subseteq \\mathbb{Z}$.\n",
+ "* Relation \"Gerade\" über die ganzen Zahlen ist\n",
+ " * $\\{ \\mathit{x}|\\mathit{x} \\in \\mathbb{Z} \\wedge \\mathit{x} \\mod 2 = 0\\} \\subseteq \\mathbb{Z}$."
]
},
{
@@ -1932,10 +1932,10 @@
"Eine Relation ist die explizite Darstellung eines Prädikats als Menge:\n",
" \n",
"* Prädikate können mit logischen Junktoren und Quantoren verarbeitet werden:\n",
- " * $\\exists x. is\\_rich(x)$, $\\forall x.(is\\_poor(x) \\Rightarrow \\neg is\\_rich(x))$\n",
+ " * $\\exists x. \\mathrm{is\\_rich}(x)$, $\\forall x.(\\mathrm{is\\_poor}(x) \\Rightarrow \\neg \\mathrm{is\\_rich}(x))$\n",
" \n",
"* Relationen können mit mengentheoretischen Operationen verarbeitet werden:\n",
- " * $rich \\neq \\emptyset$, $rich \\cap poor = \\emptyset$."
+ " * $\\mathrm{rich} \\neq \\emptyset$, $\\mathrm{rich} \\cap \\mathrm{poor} = \\emptyset$."
]
},
{
@@ -2006,7 +2006,7 @@
{
"data": {
"text/markdown": [
- "$\\{(1\\mapsto \\mathit{FALSE}),(1\\mapsto \\mathit{TRUE}),(2\\mapsto \\mathit{FALSE}),(2\\mapsto \\mathit{TRUE})\\}$"
+ "$\\{(1\\mapsto\\mathit{FALSE}),(1\\mapsto\\mathit{TRUE}),(2\\mapsto\\mathit{FALSE}),(2\\mapsto\\mathit{TRUE})\\}$"
],
"text/plain": [
"{(1↦FALSE),(1↦TRUE),(2↦FALSE),(2↦TRUE)}"
@@ -2125,7 +2125,9 @@
"metadata": {},
"source": [
"Es gilt:\n",
+ "\n",
"* $A \\times B = \\emptyset \\equiv (A = \\emptyset \\vee B=\\emptyset)$.\n",
+ "\n",
"Das Kartesische Produkt wird auch Kreuzmenge oder Produktmenge genannt."
]
},
@@ -2137,7 +2139,7 @@
"\n",
"Eine binäre Relation über $x$ und $y$ ist eine Untermenge des kartesischen Produkts $x \\times y$ zweier Mengen. Die Mengen $x$ und $y$ können identisch sein.\n",
"\n",
- "Beispiel: die Relation ``kleiner'' über die Ziffern $0..9$:\n"
+ "Beispiel: die Relation \"kleiner\" über die Ziffern $0..9$:"
]
},
{
@@ -2168,7 +2170,7 @@
"metadata": {},
"source": [
"Diese Relation ist eine Untermenge von ℤ×ℤ.\n",
- "Ein anderes Beispiel ist die Relation ``halb'' über die ganzen Zahlen $1..10$:"
+ "Ein anderes Beispiel ist die Relation \"halb\" über die ganzen Zahlen $1..10$:"
]
},
{
@@ -2212,14 +2214,14 @@
"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ "<!-- Generated by graphviz version 7.1.0 (20230121.1956)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
- "<svg width=\"210pt\" height=\"305pt\"\n",
- " viewBox=\"0.00 0.00 210.00 305.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+ "<svg width=\"206pt\" height=\"305pt\"\n",
+ " viewBox=\"0.00 0.00 206.00 305.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 301)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-301 206,-301 206,4 -4,4\"/>\n",
+ "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-301 202,-301 202,4 -4,4\"/>\n",
"<!-- 10 -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>10</title>\n",
@@ -2235,9 +2237,9 @@
"<!-- 10->5 -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>10->5</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-260.8C27,-249.16 27,-233.55 27,-220.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-220.18 27,-210.18 23.5,-220.18 30.5,-220.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"42.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">halb</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-260.8C27,-249.58 27,-234.67 27,-221.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-221.98 27,-211.98 23.5,-221.98 30.5,-221.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"39\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">halb</text>\n",
"</g>\n",
"<!-- 8 -->\n",
"<g id=\"node3\" class=\"node\">\n",
@@ -2254,9 +2256,9 @@
"<!-- 8->4 -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>8->4</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-260.8C99,-249.16 99,-233.55 99,-220.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-220.18 99,-210.18 95.5,-220.18 102.5,-220.18\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-221.98 99,-211.98 95.5,-221.98 102.5,-221.98\"/>\n",
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@@ -2267,9 +2269,9 @@
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@@ -2286,9 +2288,9 @@
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"</g>\n",
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- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [(\"halb\",{a,b|a:1..10 & b:1..10 & b*2=a})]>"
+ "<Dot visualization: expr_as_graph [(\"halb\",{a,b| a∈1..10 ∧ b∈1..10 & b*2=a})]>"
]
},
"execution_count": 69,
@@ -2324,7 +2326,7 @@
"metadata": {},
"source": [
"Die Typen der Elemente kann abweichen.\n",
- "Zum Beispiel die Relation ``durch drei teilbar'' über die ganzen Zahlen $1..7$ zum Datentypen $STRING$"
+ "Zum Beispiel die Relation \"durch drei teilbar\" über die ganzen Zahlen $1..7$ zum Datentypen $\\mathit{STRING}$"
]
},
{
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"</g>\n",
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- "<text text-anchor=\"middle\" x=\"82\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">teilbar</text>\n",
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"</g>\n",
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+ "<text text-anchor=\"middle\" x=\"149.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">teilbar</text>\n",
"</g>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M451.71,-86.8C446.87,-75.35 440.39,-60.06 434.81,-46.89\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"438.11,-45.7 430.99,-37.86 431.67,-48.43 438.11,-45.7\"/>\n",
+ "<text text-anchor=\"middle\" x=\"460.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">teilbar</text>\n",
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@@ -2455,9 +2457,9 @@
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- "<text text-anchor=\"middle\" x=\"253\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">teilbar</text>\n",
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@@ -2468,15 +2470,15 @@
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"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
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- "<Dot visualization: expr_as_graph [(\"teilbar\",{x,y|x:1..7 & (x mod 3=0 => y=\"ja\") & (x mod 3>0 => y=\"nein\")})]>"
+ "<Dot visualization: expr_as_graph [(\"teilbar\",{x,y| x:1..7 ∧ (x mod 3 =0 => y = \"ja\") ∧ (x mod 3 >0 => y= \"nein\")})]>"
]
},
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@@ -2526,14 +2528,14 @@
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- "<!-- Generated by graphviz version 2.44.1 (0)\n",
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+ "<svg width=\"278pt\" height=\"235pt\"\n",
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"<title>state</title>\n",
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@@ -2556,9 +2558,9 @@
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@@ -2569,9 +2571,9 @@
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M316.74,-27.74C326.53,-27.89 334.5,-25.64 334.5,-21 334.5,-18.39 331.98,-16.53 328.03,-15.44\"/>\n",
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+ "<text text-anchor=\"middle\" x=\"346.5\" y=\"-17.3\" font-family=\"Times,serif\" font-size=\"14.00\">K10</text>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"399.34,-186.76 408.38,-192.27 404.75,-182.32 399.34,-186.76\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"350.91,-77.84 361.2,-80.4 354.75,-71.99 350.91,-77.84\"/>\n",
+ "<text text-anchor=\"middle\" x=\"317.39\" y=\"-61.09\" font-family=\"Times,serif\" font-size=\"14.00\">K10</text>\n",
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+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"192.57,-39 138.57,-39 138.57,-3 192.57,-3 192.57,-39\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.35,-85.84 92.56,-89.87 102.73,-92.83 102.35,-85.84\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M262.31,-15.3C245.19,-14.14 222.85,-13.94 203.95,-14.7\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"204.01,-11.19 194.21,-15.22 204.39,-18.18 204.01,-11.19\"/>\n",
+ "<text text-anchor=\"middle\" x=\"245.13\" y=\"-3.8\" font-family=\"Times,serif\" font-size=\"14.00\">K10</text>\n",
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+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"92.3,-111.85 38.3,-111.85 38.3,-75.85 92.3,-75.85 92.3,-111.85\"/>\n",
+ "<text text-anchor=\"middle\" x=\"65.3\" y=\"-90.15\" font-family=\"Times,serif\" font-size=\"14.00\">6</text>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M262.4,-25.08C222.67,-35.14 148.5,-58.96 103.33,-75.92\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.29,-72.57 94.2,-79.42 104.79,-79.11 102.29,-72.57\"/>\n",
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+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-229.72 0,-229.72 0,-193.72 54,-193.72 54,-229.72\"/>\n",
+ "<text text-anchor=\"middle\" x=\"27\" y=\"-208.02\" font-family=\"Times,serif\" font-size=\"14.00\">5</text>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M262.37,-35.59C212.1,-67.97 105.21,-145.48 54.6,-186.1\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"52.55,-183.26 47,-192.28 56.97,-188.69 52.55,-183.26\"/>\n",
+ "<text text-anchor=\"middle\" x=\"146.48\" y=\"-99.64\" font-family=\"Times,serif\" font-size=\"14.00\">K10</text>\n",
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+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"92.3,-347.59 38.3,-347.59 38.3,-311.59 92.3,-311.59 92.3,-347.59\"/>\n",
+ "<text text-anchor=\"middle\" x=\"65.3\" y=\"-325.89\" font-family=\"Times,serif\" font-size=\"14.00\">4</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"177.32,-373.72 174.9,-384.04 183.23,-377.48 177.32,-373.72\"/>\n",
- "<text text-anchor=\"middle\" x=\"261.84\" y=\"-247.41\" font-family=\"Times,serif\" font-size=\"14.00\">K10</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M271.91,-39.15C229.17,-90.96 121.18,-239.39 80.74,-301.64\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"77.88,-299.62 75.45,-309.93 83.78,-303.39 77.88,-299.62\"/>\n",
+ "<text text-anchor=\"middle\" x=\"164.32\" y=\"-174.2\" font-family=\"Times,serif\" font-size=\"14.00\">K10</text>\n",
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+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"192.57,-420.44 138.57,-420.44 138.57,-384.44 192.57,-384.44 192.57,-420.44\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M280.16,-39.45C255.4,-100.87 190.97,-298.6 171.08,-373.41\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"167.75,-372.3 168.66,-382.85 174.53,-374.04 167.75,-372.3\"/>\n",
+ "<text text-anchor=\"middle\" x=\"213.62\" y=\"-195.23\" font-family=\"Times,serif\" font-size=\"14.00\">K10</text>\n",
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+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"316.5,-420.44 262.5,-420.44 262.5,-384.44 316.5,-384.44 316.5,-420.44\"/>\n",
+ "<text text-anchor=\"middle\" x=\"289.5\" y=\"-398.74\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M286.39,-39.39C281.79,-100.58 281.55,-297.57 285.68,-372.83\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"282.18,-372.96 286.31,-382.71 289.17,-372.51 282.18,-372.96\"/>\n",
+ "<text text-anchor=\"middle\" x=\"274.04\" y=\"-209.91\" font-family=\"Times,serif\" font-size=\"14.00\">K10</text>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M291.9,-39.05C304.13,-90.55 351.86,-238.41 376.08,-301.03\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"372.68,-301.94 379.62,-309.94 379.18,-299.36 372.68,-301.94\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M419.69,-193.47C399.24,-159.69 347.31,-87.57 315.47,-48.1\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"318.24,-45.95 309.19,-40.44 312.83,-50.39 318.24,-45.95\"/>\n",
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+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"690.81,-112.6 636.81,-112.6 636.81,-76.6 690.81,-76.6 690.81,-112.6\"/>\n",
+ "<text text-anchor=\"middle\" x=\"663.81\" y=\"-90.9\" font-family=\"Times,serif\" font-size=\"14.00\">17</text>\n",
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"<title>20->17</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"585.63,-60.08 575.87,-55.96 580.94,-65.27 585.63,-60.08\"/>\n",
- "<text text-anchor=\"middle\" x=\"675.34\" y=\"-173.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M741.54,-165.02C730.82,-152.16 714.14,-134.87 698.95,-120.61\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"701.56,-118.25 691.82,-114.07 696.82,-123.41 701.56,-118.25\"/>\n",
+ "<text text-anchor=\"middle\" x=\"708.25\" y=\"-146.62\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>16</title>\n",
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- "<text text-anchor=\"middle\" x=\"428.07\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">16</text>\n",
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+ "<text text-anchor=\"middle\" x=\"552.01\" y=\"-33.93\" font-family=\"Times,serif\" font-size=\"14.00\">16</text>\n",
"</g>\n",
"<!-- 20->16 -->\n",
"<g id=\"edge5\" class=\"edge\">\n",
"<title>20->16</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M790.14,-276.95C729.46,-227.52 542.82,-91.63 463.87,-39.03\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.51,-35.92 455.24,-33.33 461.65,-41.76 465.51,-35.92\"/>\n",
- "<text text-anchor=\"middle\" x=\"612.5\" y=\"-161.79\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M734.72,-165.1C702.09,-137.74 632.36,-86.69 589.08,-58.11\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"591.03,-55.21 580.74,-52.69 587.21,-61.08 591.03,-55.21\"/>\n",
+ "<text text-anchor=\"middle\" x=\"649.9\" y=\"-115.4\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>15</title>\n",
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- "<text text-anchor=\"middle\" x=\"304.13\" y=\"-33.93\" font-family=\"Times,serif\" font-size=\"14.00\">15</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"455.07,-36 401.07,-36 401.07,0 455.07,0 455.07,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"428.07\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">15</text>\n",
"</g>\n",
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"<title>20->15</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M782.41,-277.58C697.52,-229.33 436.9,-96.43 340.48,-52.07\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.69,-48.78 331.14,-47.82 338.79,-55.15 341.69,-48.78\"/>\n",
- "<text text-anchor=\"middle\" x=\"546.94\" y=\"-153.63\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M725.31,-164.89C667.35,-131.47 531.83,-62.29 465.69,-32.21\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"467.25,-29.08 456.69,-28.18 464.39,-35.47 467.25,-29.08\"/>\n",
+ "<text text-anchor=\"middle\" x=\"583.5\" y=\"-87.35\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>14</title>\n",
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- "<text text-anchor=\"middle\" x=\"192.33\" y=\"-90.9\" font-family=\"Times,serif\" font-size=\"14.00\">14</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"331.13,-55.63 277.13,-55.63 277.13,-19.63 331.13,-19.63 331.13,-55.63\"/>\n",
+ "<text text-anchor=\"middle\" x=\"304.13\" y=\"-33.93\" font-family=\"Times,serif\" font-size=\"14.00\">14</text>\n",
"</g>\n",
"<!-- 20->14 -->\n",
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"<title>20->14</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"230,-99.62 219.43,-100.32 228.13,-106.37 230,-99.62\"/>\n",
- "<text text-anchor=\"middle\" x=\"491.35\" y=\"-181.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M725.24,-170.77C648.51,-141.44 430.92,-70.6 342.34,-45.92\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"343.49,-42.6 332.92,-43.34 341.64,-49.36 343.49,-42.6\"/>\n",
+ "<text text-anchor=\"middle\" x=\"521.79\" y=\"-97.14\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>13</title>\n",
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- "<text text-anchor=\"middle\" x=\"103.6\" y=\"-179.63\" font-family=\"Times,serif\" font-size=\"14.00\">13</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"219.33,-112.6 165.33,-112.6 165.33,-76.6 219.33,-76.6 219.33,-112.6\"/>\n",
+ "<text text-anchor=\"middle\" x=\"192.33\" y=\"-90.9\" font-family=\"Times,serif\" font-size=\"14.00\">13</text>\n",
"</g>\n",
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"<title>20->13</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"141.09,-182.48 130.76,-184.8 140.29,-189.43 141.09,-182.48\"/>\n",
- "<text text-anchor=\"middle\" x=\"447.16\" y=\"-225.77\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M725.25,-175.82C633.73,-156.62 337.47,-109.54 230.93,-97.05\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"231.61,-93.61 221.28,-95.97 230.82,-100.56 231.61,-93.61\"/>\n",
+ "<text text-anchor=\"middle\" x=\"466.09\" y=\"-125.24\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>12</title>\n",
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+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"130.6,-201.33 76.6,-201.33 76.6,-165.33 130.6,-165.33 130.6,-201.33\"/>\n",
+ "<text text-anchor=\"middle\" x=\"103.6\" y=\"-179.63\" font-family=\"Times,serif\" font-size=\"14.00\">12</text>\n",
"</g>\n",
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"<title>20->12</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"83.7,-288.51 73.85,-292.42 83.99,-295.5 83.7,-288.51\"/>\n",
- "<text text-anchor=\"middle\" x=\"418.66\" y=\"-281.01\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M725.07,-180.39C622.57,-175.47 262.17,-175.31 142.53,-179.9\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"142.43,-176.4 132.59,-180.33 142.73,-183.4 142.43,-176.4\"/>\n",
+ "<text text-anchor=\"middle\" x=\"421.8\" y=\"-168.95\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>11</title>\n",
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- "<text text-anchor=\"middle\" x=\"27\" y=\"-415.37\" font-family=\"Times,serif\" font-size=\"14.00\">11</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"73.63,-313.13 19.63,-313.13 19.63,-277.13 73.63,-277.13 73.63,-313.13\"/>\n",
+ "<text text-anchor=\"middle\" x=\"46.63\" y=\"-291.43\" font-family=\"Times,serif\" font-size=\"14.00\">11</text>\n",
"</g>\n",
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"<title>20->11</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M782.16,-296.76C664.25,-310.08 199.36,-383.57 64.1,-410.05\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"63.33,-406.64 54.21,-412.02 64.7,-413.5 63.33,-406.64\"/>\n",
- "<text text-anchor=\"middle\" x=\"408.63\" y=\"-342.21\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M725.3,-184.81C616.31,-196.91 211.56,-260.85 84.82,-285.75\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"84.38,-282.27 75.27,-287.67 85.76,-289.13 84.38,-282.27\"/>\n",
+ "<text text-anchor=\"middle\" x=\"393.06\" y=\"-224.08\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<title>10</title>\n",
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+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"54,-437.07 0,-437.07 0,-401.07 54,-401.07 54,-437.07\"/>\n",
+ "<text text-anchor=\"middle\" x=\"27\" y=\"-415.37\" font-family=\"Times,serif\" font-size=\"14.00\">10</text>\n",
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"<title>20->10</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"82.1,-524.5 73.92,-531.24 84.52,-531.07 82.1,-524.5\"/>\n",
- "<text text-anchor=\"middle\" x=\"447.53\" y=\"-403.2\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M725.45,-189.26C614.37,-219.91 193.73,-356.4 64.8,-403.37\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"63.87,-399.98 55.7,-406.72 66.29,-406.55 63.87,-399.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"383.12\" y=\"-285.11\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"73.63,-561.01 19.63,-561.01 19.63,-525.01 73.63,-525.01 73.63,-561.01\"/>\n",
+ "<text text-anchor=\"middle\" x=\"46.63\" y=\"-539.31\" font-family=\"Times,serif\" font-size=\"14.00\">9</text>\n",
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"<title>20->9</title>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M725.33,-194.1C616.06,-244.01 209.22,-451.1 83.72,-520.42\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"82.17,-517.28 75.14,-525.2 85.58,-523.39 82.17,-517.28\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M725.44,-199.52C620.53,-269.4 241.94,-544.35 133.05,-629.57\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M730.34,-201.66C640.54,-284.01 305.35,-619.08 214.89,-716.76\"/>\n",
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"<title>20->4</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"665.87,-714.84 666.75,-725.4 672.65,-716.6 665.87,-714.84\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M743.9,-201.69C709.81,-289.05 588.23,-662.69 558.39,-770.99\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"555.03,-770 555.82,-780.56 561.79,-771.81 555.03,-770\"/>\n",
+ "<text text-anchor=\"middle\" x=\"639.14\" y=\"-490.14\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<title>20->3</title>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M746.99,-201.56C728.76,-282.83 676.34,-612.59 665.26,-713.84\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"661.78,-713.47 664.25,-723.77 668.74,-714.18 661.78,-713.47\"/>\n",
+ "<text text-anchor=\"middle\" x=\"694.13\" y=\"-461.5\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"</g>\n",
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"<title>20->2</title>\n",
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- "<text text-anchor=\"middle\" x=\"790.9\" y=\"-402.82\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M749.71,-201.81C744.72,-273.85 744.5,-536.17 749.05,-625.1\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"745.54,-625.17 749.63,-634.95 752.53,-624.76 745.54,-625.17\"/>\n",
+ "<text text-anchor=\"middle\" x=\"737.38\" y=\"-417.25\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>1</title>\n",
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+ "<text text-anchor=\"middle\" x=\"809.51\" y=\"-539.31\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M752.26,-201.66C756.93,-260.1 785.54,-442.37 800.91,-513.84\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"797.41,-514.24 803.02,-523.23 804.24,-512.71 797.41,-514.24\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"823.22,-322.29 817.49,-313.38 816.42,-323.92 823.22,-322.29\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M827.06,-400.58C817.63,-359.69 786.04,-261.44 766.91,-212.3\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"770.22,-211.15 763.24,-203.17 763.72,-213.75 770.22,-211.15\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"866.66,-409.74 856.38,-412.33 866.03,-416.71 866.66,-409.74\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M856.38,-425.81C866.16,-425.96 874.14,-423.71 874.14,-419.07 874.14,-416.46 871.62,-414.61 867.67,-413.51\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"868.2,-410.05 857.89,-412.48 867.47,-417.01 868.2,-410.05\"/>\n",
+ "<text text-anchor=\"middle\" x=\"886.14\" y=\"-415.37\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M831.18,-400.81C830.12,-380.74 825.32,-348.19 820.13,-324.52\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"823.56,-323.77 817.84,-314.85 816.74,-325.39 823.56,-323.77\"/>\n",
+ "<text text-anchor=\"middle\" x=\"813.66\" y=\"-366.46\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M823.36,-400.62C800.65,-346.4 720.05,-187.55 682.73,-122.66\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"685.96,-121.26 677.89,-114.39 679.92,-124.79 685.96,-121.26\"/>\n",
+ "<text text-anchor=\"middle\" x=\"741.04\" y=\"-265.44\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"460.43,-40.29 450.69,-36.12 455.7,-45.45 460.43,-40.29\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M819.46,-400.69C780.33,-338.77 634.87,-138.09 576.42,-64.74\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"579.36,-62.82 570.36,-57.23 573.91,-67.21 579.36,-62.82\"/>\n",
+ "<text text-anchor=\"middle\" x=\"685.94\" y=\"-236.51\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.57,-56.17 331.31,-53.53 337.69,-61.99 341.57,-56.17\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M814.95,-400.82C757.32,-336.28 540.83,-119.41 459.31,-44.02\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"461.8,-41.55 452.06,-37.38 457.07,-46.71 461.8,-41.55\"/>\n",
+ "<text text-anchor=\"middle\" x=\"625.13\" y=\"-226.22\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M808.96,-400.81C730.24,-337.58 443.52,-128.99 340.92,-59.94\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"343.1,-57.19 332.84,-54.55 339.21,-63.01 343.1,-57.19\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"231.59,-106.92 221.05,-105.92 228.67,-113.28 231.59,-106.92\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"142.85,-189.04 132.27,-189.7 140.95,-195.78 142.85,-189.04\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M801.72,-411.98C684.02,-388.03 221.53,-314.64 84.86,-298.04\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"85.6,-294.6 75.26,-296.91 84.78,-301.55 85.6,-294.6\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"65.51,-412.46 55.65,-416.36 65.79,-419.46 65.51,-412.46\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"447.43,-790.96 443.31,-800.72 452.62,-795.66 447.43,-790.96\"/>\n",
+ "<text text-anchor=\"middle\" x=\"615.9\" y=\"-619.43\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>19->4</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"670.24,-714.75 669.5,-725.32 676.67,-717.52 670.24,-714.75\"/>\n",
- "<text text-anchor=\"middle\" x=\"730.29\" y=\"-580.34\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M811.81,-437.4C761.43,-499.01 616.38,-698.44 567.88,-772.52\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"564.99,-770.54 562.52,-780.84 570.88,-774.33 564.99,-770.54\"/>\n",
+ "<text text-anchor=\"middle\" x=\"677.85\" y=\"-608.76\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>19->3</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"753.51,-626.14 754.54,-636.69 760.31,-627.8 753.51,-626.14\"/>\n",
- "<text text-anchor=\"middle\" x=\"773.55\" y=\"-520.76\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M815.97,-437.31C783.77,-491.19 702.85,-649.66 673.97,-714.96\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"670.85,-713.36 670.1,-723.93 677.28,-716.13 670.85,-713.36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"732.97\" y=\"-579.94\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>19->2</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"804.71,-514.62 807.46,-524.85 811.69,-515.13 804.71,-514.62\"/>\n",
- "<text text-anchor=\"middle\" x=\"800.21\" y=\"-479.68\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M818.99,-437.54C802.06,-478.27 770.09,-575.97 757.34,-625.29\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"753.95,-624.43 754.95,-634.97 760.74,-626.1 753.95,-624.43\"/>\n",
+ "<text text-anchor=\"middle\" x=\"776.16\" y=\"-520.22\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>19->1</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"769.56,-209.48 762.58,-201.52 763.07,-212.1 769.56,-209.48\"/>\n",
- "<text text-anchor=\"middle\" x=\"782.24\" y=\"-309.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M821.14,-437.38C815.98,-457.3 810.62,-489.47 808.36,-513.08\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"804.87,-512.79 807.6,-523.02 811.85,-513.32 804.87,-512.79\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M805.62,-276.64C798.74,-258.87 785.41,-231.85 773.72,-211.43\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"776.85,-209.85 768.75,-203.03 770.83,-213.42 776.85,-209.85\"/>\n",
+ "<text text-anchor=\"middle\" x=\"801.67\" y=\"-247.84\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"807.99,-394.13 816.07,-400.99 814.03,-390.59 807.99,-394.13\"/>\n",
- "<text text-anchor=\"middle\" x=\"725.77\" y=\"-241.45\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M807.47,-313.39C808.53,-333.47 813.34,-366.01 818.52,-389.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"815.1,-390.43 820.81,-399.35 821.91,-388.82 815.1,-390.43\"/>\n",
+ "<text text-anchor=\"middle\" x=\"800.99\" y=\"-340.34\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>18->18</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"701.33,-85.27 691.05,-87.85 700.7,-92.24 701.33,-85.27\"/>\n",
- "<text text-anchor=\"middle\" x=\"723.31\" y=\"-90.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M836.75,-301.88C846.54,-302.02 854.51,-299.77 854.51,-295.13 854.51,-292.52 851.99,-290.67 848.04,-289.57\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"848.57,-286.11 838.26,-288.55 847.84,-293.07 848.57,-286.11\"/>\n",
+ "<text text-anchor=\"middle\" x=\"866.51\" y=\"-291.43\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"589.66,-45.89 579.1,-45.04 586.82,-52.29 589.66,-45.89\"/>\n",
- "<text text-anchor=\"middle\" x=\"600.22\" y=\"-51.65\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M800.97,-276.78C779.38,-241.25 723.12,-163.18 689.65,-121.68\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"692.48,-119.62 683.43,-114.1 687.07,-124.05 692.48,-119.62\"/>\n",
+ "<text text-anchor=\"middle\" x=\"733.31\" y=\"-203.03\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.65,-21.31 455.08,-22.15 463.87,-28.08 465.65,-21.31\"/>\n",
- "<text text-anchor=\"middle\" x=\"536.4\" y=\"-56.68\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M796.23,-276.97C756.91,-231.95 641.05,-115.61 584.15,-63.47\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"586.75,-61.1 577,-56.98 582.05,-66.29 586.75,-61.1\"/>\n",
+ "<text text-anchor=\"middle\" x=\"678.19\" y=\"-174.02\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>18->15</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.77,-35.72 331.46,-38.16 341.04,-42.68 341.77,-35.72\"/>\n",
- "<text text-anchor=\"middle\" x=\"474.66\" y=\"-51.64\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M789.87,-276.74C729.09,-227.27 543.79,-92.34 464.56,-39.5\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"466.98,-36.9 456.71,-34.3 463.12,-42.74 466.98,-36.9\"/>\n",
+ "<text text-anchor=\"middle\" x=\"615.22\" y=\"-161.92\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>18->14</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.37,-87.25 219.55,-91.23 229.71,-94.24 229.37,-87.25\"/>\n",
- "<text text-anchor=\"middle\" x=\"418.78\" y=\"-79.79\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M782.05,-277.37C697.12,-229.15 438.81,-97.41 341.62,-52.6\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"343.31,-49.53 332.76,-48.56 340.41,-55.9 343.31,-49.53\"/>\n",
+ "<text text-anchor=\"middle\" x=\"549.83\" y=\"-153.79\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 18->13 -->\n",
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"<title>18->13</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"140,-170.38 130.9,-175.81 141.4,-177.24 140,-170.38\"/>\n",
- "<text text-anchor=\"middle\" x=\"374.23\" y=\"-123.58\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M782.33,-283.11C683.84,-246.05 345.61,-136.01 230.8,-103.48\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"231.81,-100.13 221.24,-100.82 229.93,-106.87 231.81,-100.13\"/>\n",
+ "<text text-anchor=\"middle\" x=\"494.56\" y=\"-182.1\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<title>18->12</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"82.08,-276.31 73.92,-283.07 84.52,-282.87 82.08,-276.31\"/>\n",
- "<text text-anchor=\"middle\" x=\"345.67\" y=\"-178.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M782.28,-287.95C673.43,-265.58 269.36,-201.43 142.18,-186.13\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"142.6,-182.65 132.26,-184.98 141.79,-189.61 142.6,-182.65\"/>\n",
+ "<text text-anchor=\"middle\" x=\"450.23\" y=\"-225.84\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"61.24,-393.89 54.21,-401.82 64.65,-400 61.24,-393.89\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M782.36,-292.42C667.32,-287.2 219.82,-287.04 85.17,-291.96\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"85.22,-288.45 75.37,-292.36 85.5,-295.45 85.22,-288.45\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M782.16,-296.76C664.61,-310.04 202.15,-383.12 65.33,-409.81\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"64.82,-406.34 55.69,-411.73 66.19,-413.21 64.82,-406.34\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M782.43,-301.12C667.27,-333.01 218.24,-478.73 84.44,-527.37\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"83.52,-523.98 75.34,-530.72 85.94,-530.55 83.52,-523.98\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"139.14,-629.08 132.11,-637 142.55,-635.2 139.14,-629.08\"/>\n",
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"</g>\n",
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"<title>18->2</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"793.66,-516.89 800.56,-524.94 800.18,-514.35 793.66,-516.89\"/>\n",
- "<text text-anchor=\"middle\" x=\"717.34\" y=\"-317.99\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M803.34,-313.46C789.58,-371.79 760.58,-553.46 753.29,-625.23\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"749.82,-624.75 752.39,-635.02 756.79,-625.39 749.82,-624.75\"/>\n",
+ "<text text-anchor=\"middle\" x=\"766.32\" y=\"-473.14\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>18->1</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"716.27,-161.04 726.01,-165.2 720.99,-155.87 716.27,-161.04\"/>\n",
- "<text text-anchor=\"middle\" x=\"682.19\" y=\"-139.46\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M805.77,-313.36C802.06,-355.8 801.78,-461.53 804.94,-513.38\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"801.44,-513.53 805.66,-523.24 808.43,-513.02 801.44,-513.53\"/>\n",
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"</g>\n",
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"<title>17->20</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"775.88,-272.68 785.65,-276.8 780.58,-267.49 775.88,-272.68\"/>\n",
- "<text text-anchor=\"middle\" x=\"657.18\" y=\"-151.66\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M674.81,-112.9C685.54,-125.77 702.21,-143.05 717.41,-157.32\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"714.8,-159.67 724.53,-163.86 719.53,-154.52 714.8,-159.67\"/>\n",
+ "<text text-anchor=\"middle\" x=\"684.11\" y=\"-138.91\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"803.07,-395.48 812.07,-401.06 808.51,-391.08 803.07,-395.48\"/>\n",
- "<text text-anchor=\"middle\" x=\"668.99\" y=\"-213.11\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M669.6,-113.05C692.31,-167.27 772.91,-326.12 810.23,-391.01\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"806.99,-392.41 815.06,-399.28 813.04,-388.88 806.99,-392.41\"/>\n",
+ "<text text-anchor=\"middle\" x=\"727.91\" y=\"-240.83\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<text text-anchor=\"middle\" x=\"586.6\" y=\"-73.18\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M672.35,-112.95C693.95,-148.48 750.21,-226.55 783.67,-268.05\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"780.84,-270.12 789.89,-275.63 786.26,-265.68 780.84,-270.12\"/>\n",
+ "<text text-anchor=\"middle\" x=\"716.01\" y=\"-179.3\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 17->17 -->\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"589.52,-28.3 579.25,-30.89 588.89,-35.27 589.52,-28.3\"/>\n",
- "<text text-anchor=\"middle\" x=\"611.51\" y=\"-33.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M691.05,-101.34C700.84,-101.49 708.81,-99.24 708.81,-94.6 708.81,-91.99 706.29,-90.13 702.34,-89.04\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"702.87,-85.57 692.56,-88.01 702.14,-92.54 702.87,-85.57\"/>\n",
+ "<text text-anchor=\"middle\" x=\"720.81\" y=\"-90.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>17->16</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.57,-14.13 455.27,-16.6 464.87,-21.09 465.57,-14.13\"/>\n",
- "<text text-anchor=\"middle\" x=\"509.61\" y=\"-26.36\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M640.52,-76.22C625.72,-67.48 606.28,-57.38 589.37,-49.61\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"591.18,-46.58 580.62,-45.71 588.34,-52.98 591.18,-46.58\"/>\n",
+ "<text text-anchor=\"middle\" x=\"602.94\" y=\"-51.71\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"341.14,-29.15 331.35,-33.21 341.54,-36.14 341.14,-29.15\"/>\n",
- "<text text-anchor=\"middle\" x=\"447.5\" y=\"-36.72\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.33,-80.82C594.34,-64.34 514.12,-38.12 466.42,-25.14\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"467.35,-21.77 456.79,-22.6 465.56,-28.53 467.35,-21.77\"/>\n",
+ "<text text-anchor=\"middle\" x=\"539.38\" y=\"-56.78\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"228.75,-80.9 219.69,-86.39 230.2,-87.75 228.75,-80.9\"/>\n",
- "<text text-anchor=\"middle\" x=\"362.88\" y=\"-49.99\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.49,-86.4C573.12,-72.6 416.02,-47.54 342.82,-39.35\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"343.22,-35.88 332.91,-38.31 342.48,-42.84 343.22,-35.88\"/>\n",
+ "<text text-anchor=\"middle\" x=\"477.65\" y=\"-51.68\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"138.76,-164.07 130.63,-170.87 141.23,-170.62 138.76,-164.07\"/>\n",
- "<text text-anchor=\"middle\" x=\"318.04\" y=\"-93.88\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.43,-91.22C556.59,-86.91 323.91,-86.73 231.16,-90.68\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"231.12,-87.17 221.3,-91.15 231.46,-94.16 231.12,-87.17\"/>\n",
+ "<text text-anchor=\"middle\" x=\"421.79\" y=\"-79.75\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>17->12</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"80.91,-269.52 73.91,-277.48 84.34,-275.62 80.91,-269.52\"/>\n",
- "<text text-anchor=\"middle\" x=\"289.25\" y=\"-149.03\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.69,-95.77C545.31,-105.5 248.52,-152.31 142.05,-173.54\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"141.48,-170.08 132.38,-175.51 142.88,-176.94 141.48,-170.08\"/>\n",
+ "<text text-anchor=\"middle\" x=\"377.37\" y=\"-123.45\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"52.74,-391.91 47.1,-400.88 57.1,-397.39 52.74,-391.91\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.38,-100.41C537.41,-127.44 198.63,-237.32 84.5,-279.15\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"83.44,-275.81 75.28,-282.57 85.88,-282.37 83.44,-275.81\"/>\n",
+ "<text text-anchor=\"middle\" x=\"348.44\" y=\"-178.58\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>17->10</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"65.19,-515.19 61.04,-524.94 70.36,-519.91 65.19,-515.19\"/>\n",
- "<text text-anchor=\"middle\" x=\"284.38\" y=\"-290.38\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.35,-105.34C534.7,-151.57 179.98,-332.11 63.99,-396.36\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"62.5,-393.19 55.48,-401.12 65.91,-399.3 62.5,-393.19\"/>\n",
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"</g>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"116.42,-626.34 113.85,-636.62 122.27,-630.18 116.42,-626.34\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M636.37,-110.94C534.76,-178.55 180.77,-435.63 76.01,-517.73\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"74.09,-514.79 68.41,-523.74 78.43,-520.28 74.09,-514.79\"/>\n",
+ "<text text-anchor=\"middle\" x=\"344.19\" y=\"-303.13\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M641.61,-112.93C551.81,-195.28 216.62,-530.35 126.16,-628.03\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"123.62,-625.62 119.44,-635.35 128.78,-630.35 123.62,-625.62\"/>\n",
+ "<text text-anchor=\"middle\" x=\"395.89\" y=\"-374.28\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<text text-anchor=\"middle\" x=\"441.54\" y=\"-417.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M647.4,-112.89C574.54,-203.78 281.99,-606.26 208.87,-715.91\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"206.09,-713.76 203.51,-724.04 211.94,-717.62 206.09,-713.76\"/>\n",
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- "<text text-anchor=\"middle\" x=\"502.81\" y=\"-427.78\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"312.45,-770.25 311.53,-780.8 318.83,-773.13 312.45,-770.25\"/>\n",
+ "<text text-anchor=\"middle\" x=\"471.73\" y=\"-431.09\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"431.24,-789.89 432,-800.46 438,-791.72 431.24,-789.89\"/>\n",
+ "<text text-anchor=\"middle\" x=\"532.98\" y=\"-455.75\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M658.55,-112.75C637.69,-208.18 567.22,-652 553.74,-771.06\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"550.3,-770.33 552.72,-780.64 557.26,-771.07 550.3,-770.33\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M661.37,-112.97C655.83,-203.43 655.64,-601.57 660.78,-713.93\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"772.07,-277.21 782.34,-279.8 775.93,-271.37 772.07,-277.21\"/>\n",
- "<text text-anchor=\"middle\" x=\"596.08\" y=\"-143.94\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M569.83,-55.86C602.46,-83.21 672.19,-134.26 715.48,-162.84\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"713.52,-165.75 723.81,-168.27 717.34,-159.88 713.52,-165.75\"/>\n",
+ "<text text-anchor=\"middle\" x=\"630.65\" y=\"-98.15\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->19 -->\n",
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"<title>16->19</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"796.78,-396.78 806.52,-400.95 801.51,-391.62 796.78,-396.78\"/>\n",
- "<text text-anchor=\"middle\" x=\"605.97\" y=\"-203.81\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M561.69,-56.01C600.82,-117.93 746.28,-318.61 804.73,-391.96\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"801.79,-393.88 810.79,-399.47 807.24,-389.49 801.79,-393.88\"/>\n",
+ "<text text-anchor=\"middle\" x=\"671.21\" y=\"-212.79\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->18 -->\n",
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"<title>16->18</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"626.24,-91.29 636.8,-90.45 628.02,-84.52 626.24,-91.29\"/>\n",
- "<text text-anchor=\"middle\" x=\"555.49\" y=\"-63.52\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M565.29,-55.79C604.61,-100.81 720.47,-217.15 777.37,-269.3\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"774.76,-271.66 784.52,-275.79 779.47,-266.48 774.76,-271.66\"/>\n",
+ "<text text-anchor=\"middle\" x=\"659.33\" y=\"-151.34\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->17 -->\n",
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"<title>16->17</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"514.51,-41.5 524.81,-39.03 515.21,-34.54 514.51,-41.5\"/>\n",
- "<text text-anchor=\"middle\" x=\"499.46\" y=\"-36.87\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M575.31,-56.01C590.1,-64.75 609.54,-74.84 626.45,-82.62\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"624.64,-85.64 635.2,-86.51 627.49,-79.25 624.64,-85.64\"/>\n",
+ "<text text-anchor=\"middle\" x=\"612.88\" y=\"-73.11\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->16 -->\n",
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"<title>16->16</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.59,-8.67 455.31,-11.26 464.96,-15.64 465.59,-8.67\"/>\n",
- "<text text-anchor=\"middle\" x=\"487.57\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M579.25,-44.37C589.03,-44.52 597.01,-42.27 597.01,-37.63 597.01,-35.02 594.48,-33.17 590.53,-32.07\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"591.06,-28.61 580.75,-31.04 590.33,-35.57 591.06,-28.61\"/>\n",
+ "<text text-anchor=\"middle\" x=\"609.01\" y=\"-33.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>16->15</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"340.26,-22.02 331.2,-27.52 341.71,-28.87 340.26,-22.02\"/>\n",
- "<text text-anchor=\"middle\" x=\"356.52\" y=\"-9.8\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M524.79,-27.49C507.69,-23.64 485.39,-19.94 466.51,-17.75\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"467.08,-14.29 456.78,-16.75 466.37,-21.26 467.08,-14.29\"/>\n",
+ "<text text-anchor=\"middle\" x=\"507.65\" y=\"-26.42\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>16->14</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"227.56,-74.09 219.48,-80.95 230.07,-80.62 227.56,-74.09\"/>\n",
- "<text text-anchor=\"middle\" x=\"329.57\" y=\"-53.5\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M524.53,-33.2C480.31,-30.33 393.19,-30.12 342.74,-32.57\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"342.65,-29.07 332.86,-33.13 343.04,-36.06 342.65,-29.07\"/>\n",
+ "<text text-anchor=\"middle\" x=\"445.64\" y=\"-36.69\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->13 -->\n",
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"<title>16->13</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"137.27,-157.19 130.32,-165.18 140.74,-163.27 137.27,-157.19\"/>\n",
- "<text text-anchor=\"middle\" x=\"284.45\" y=\"-97.68\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M524.68,-38.16C461.19,-44.4 303.62,-69.14 230.6,-84.09\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"230.18,-80.6 221.11,-86.09 231.62,-87.45 230.18,-80.6\"/>\n",
+ "<text text-anchor=\"middle\" x=\"365.64\" y=\"-49.92\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->12 -->\n",
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"<title>16->12</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"71.55,-268 65.95,-276.99 75.94,-273.45 71.55,-268\"/>\n",
- "<text text-anchor=\"middle\" x=\"222.95\" y=\"-140.73\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M524.58,-42.97C447.65,-63.49 229.77,-134.07 141.46,-166.8\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"140.41,-163.46 132.28,-170.25 142.87,-170.01 140.41,-163.46\"/>\n",
+ "<text text-anchor=\"middle\" x=\"321.02\" y=\"-93.69\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>16->11</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"45.38,-390.97 41.26,-400.74 50.57,-395.67 45.38,-390.97\"/>\n",
- "<text text-anchor=\"middle\" x=\"212.29\" y=\"-218.43\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M524.8,-47.91C440.07,-86 180.95,-217.8 83.86,-271.88\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"82.23,-268.78 75.23,-276.74 85.66,-274.89 82.23,-268.78\"/>\n",
+ "<text text-anchor=\"middle\" x=\"292.33\" y=\"-148.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>16->10</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M411.32,-36.3C347.87,-114.83 123.86,-422.99 62.29,-516.2\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"59.22,-514.49 56.69,-524.78 65.09,-518.31 59.22,-514.49\"/>\n",
- "<text text-anchor=\"middle\" x=\"222.31\" y=\"-280.05\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M524.86,-53.51C435.13,-112.83 148.9,-320.66 56.22,-393.62\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"54.21,-390.75 48.56,-399.71 58.57,-396.23 54.21,-390.75\"/>\n",
+ "<text text-anchor=\"middle\" x=\"278.54\" y=\"-212.37\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->9 -->\n",
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"<title>16->9</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M415.93,-36.26C364.06,-125.93 163.46,-519.37 114.35,-627.34\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"111.09,-626.04 110.2,-636.6 117.48,-628.9 111.09,-626.04\"/>\n",
- "<text text-anchor=\"middle\" x=\"250.64\" y=\"-335.6\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M529.54,-56.04C445.7,-132.68 152.31,-425.96 68.86,-516.37\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"66.42,-513.85 62.26,-523.6 71.59,-518.57 66.42,-513.85\"/>\n",
+ "<text text-anchor=\"middle\" x=\"287.2\" y=\"-290\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 16->8 -->\n",
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"<title>16->8</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M419.64,-36.3C381.81,-134.27 231.51,-596.39 198.52,-715.49\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"195.1,-714.75 195.86,-725.32 201.85,-716.58 195.1,-714.75\"/>\n",
- "<text text-anchor=\"middle\" x=\"294.58\" y=\"-379.7\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M535.5,-55.94C465.03,-143.7 190.97,-520.71 120.24,-626.91\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"117.45,-624.78 114.88,-635.06 123.3,-628.63 117.45,-624.78\"/>\n",
+ "<text text-anchor=\"middle\" x=\"315.87\" y=\"-330.23\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>16->7</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"302.35,-772.14 304.76,-782.45 309.31,-772.89 302.35,-772.14\"/>\n",
- "<text text-anchor=\"middle\" x=\"349.87\" y=\"-408.15\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M540.06,-55.79C484.96,-151.48 257.84,-596.91 203.78,-714.85\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"200.65,-713.28 199.72,-723.84 207.03,-716.16 200.65,-713.28\"/>\n",
+ "<text text-anchor=\"middle\" x=\"359.92\" y=\"-389.12\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<title>16->6</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"421.88,-792.09 425.86,-801.91 428.87,-791.75 421.88,-792.09\"/>\n",
- "<text text-anchor=\"middle\" x=\"411.11\" y=\"-417.79\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M543.7,-55.78C504.91,-156.46 345.62,-646.2 310.7,-771.35\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"307.37,-770.27 308.12,-780.84 314.12,-772.11 307.37,-770.27\"/>\n",
+ "<text text-anchor=\"middle\" x=\"415.2\" y=\"-402.36\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<title>16->5</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"541.39,-773.31 546.86,-782.39 548.25,-771.88 541.39,-773.31\"/>\n",
- "<text text-anchor=\"middle\" x=\"501.25\" y=\"-408.08\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M546.84,-55.85C524.72,-158.3 444.68,-662.54 429.95,-790.55\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"426.5,-789.92 428.9,-800.24 433.46,-790.67 426.5,-789.92\"/>\n",
+ "<text text-anchor=\"middle\" x=\"476.4\" y=\"-412\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"<title>16->4</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"648.59,-717.17 655.4,-725.28 655.13,-714.69 648.59,-717.17\"/>\n",
- "<text text-anchor=\"middle\" x=\"527.22\" y=\"-379.84\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M549.74,-56.11C543.95,-157.41 543.77,-645.32 549.19,-770.96\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"545.68,-770.95 549.67,-780.76 552.68,-770.6 545.68,-770.95\"/>\n",
+ "<text text-anchor=\"middle\" x=\"537.46\" y=\"-417.33\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"732.51,-629.78 740.51,-636.73 738.59,-626.31 732.51,-629.78\"/>\n",
- "<text text-anchor=\"middle\" x=\"570.52\" y=\"-335.76\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M552.54,-55.79C561.89,-151.27 631.97,-595.06 656.18,-714.1\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"652.72,-714.64 658.2,-723.7 659.57,-713.2 652.72,-714.64\"/>\n",
+ "<text text-anchor=\"middle\" x=\"592.36\" y=\"-388.75\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M555.36,-55.86C577.92,-143.11 699.52,-518.1 739.83,-625.8\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"736.54,-626.97 743.37,-635.06 743.08,-624.47 736.54,-626.97\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M558.36,-55.94C591.24,-131.93 739.03,-422.53 791.31,-514.76\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"713.36,-172.25 723.92,-173.14 716.22,-165.86 713.36,-172.25\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"770.6,-276.24 780.88,-278.83 774.46,-270.4 770.6,-276.24\"/>\n",
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- "<text text-anchor=\"middle\" x=\"408.64\" y=\"-46.14\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M455.55,-31.78C497.54,-48.25 577.77,-74.48 625.47,-87.46\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"624.53,-90.83 635.1,-90 626.32,-84.06 624.53,-90.83\"/>\n",
+ "<text text-anchor=\"middle\" x=\"552.51\" y=\"-63.42\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<text text-anchor=\"middle\" x=\"346.69\" y=\"-23.42\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M455.29,-28.14C472.39,-31.99 494.69,-35.69 513.57,-37.88\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"513,-41.34 523.3,-38.88 513.71,-34.37 513,-41.34\"/>\n",
+ "<text text-anchor=\"middle\" x=\"472.43\" y=\"-36.81\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M455.31,-24.74C465.09,-24.89 473.07,-22.64 473.07,-18 473.07,-15.39 470.55,-13.54 466.6,-12.44\"/>\n",
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+ "<text text-anchor=\"middle\" x=\"485.07\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M400.6,-16.62C383.57,-18.13 361.44,-21.41 342.67,-25.11\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"342.05,-21.66 332.98,-27.14 343.48,-28.51 342.05,-21.66\"/>\n",
+ "<text text-anchor=\"middle\" x=\"359.64\" y=\"-9.66\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<text text-anchor=\"middle\" x=\"217.6\" y=\"-108.95\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M400.57,-22.27C358.44,-33.01 277.85,-58.93 230.22,-76.83\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.22,-73.46 221.13,-80.31 231.72,-80 229.22,-73.46\"/>\n",
+ "<text text-anchor=\"middle\" x=\"327.39\" y=\"-53.35\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<text text-anchor=\"middle\" x=\"159\" y=\"-151.43\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M400.59,-27.67C342.33,-53.32 206.51,-122.25 140.66,-159.3\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"138.99,-156.22 132.03,-164.21 142.46,-162.3 138.99,-156.22\"/>\n",
+ "<text text-anchor=\"middle\" x=\"282.62\" y=\"-97.28\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<text text-anchor=\"middle\" x=\"150.17\" y=\"-227.62\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M400.64,-33.5C330.7,-79.23 146.46,-212.95 75.24,-269.53\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"73.14,-266.73 67.54,-275.72 77.52,-272.19 73.14,-266.73\"/>\n",
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+ "<text text-anchor=\"middle\" x=\"214.83\" y=\"-218.36\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M411.32,-36.3C348.07,-114.59 125.21,-421.12 62.85,-515.34\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"60.06,-513.22 57.52,-523.51 65.92,-517.05 60.06,-513.22\"/>\n",
+ "<text text-anchor=\"middle\" x=\"225.09\" y=\"-279.62\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"190.42,-715.19 192.85,-725.5 197.38,-715.92 190.42,-715.19\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"111.72,-624.66 110.82,-635.22 118.11,-627.53 111.72,-624.66\"/>\n",
+ "<text text-anchor=\"middle\" x=\"253.36\" y=\"-335.11\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"297.88,-772.42 301.87,-782.24 304.87,-772.07 297.88,-772.42\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"195.5,-713.29 196.26,-723.86 202.25,-715.13 195.5,-713.29\"/>\n",
+ "<text text-anchor=\"middle\" x=\"297.23\" y=\"-379.16\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"302.56,-770.29 304.96,-780.61 309.52,-771.04 302.56,-770.29\"/>\n",
+ "<text text-anchor=\"middle\" x=\"352.46\" y=\"-407.37\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"536.82,-774.07 543.63,-782.18 543.37,-771.59 536.82,-774.07\"/>\n",
- "<text text-anchor=\"middle\" x=\"438.36\" y=\"-402.99\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"421.81,-790.58 425.79,-800.4 428.8,-790.24 421.81,-790.58\"/>\n",
+ "<text text-anchor=\"middle\" x=\"437.59\" y=\"-402.21\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"541.02,-771.48 546.48,-780.56 547.88,-770.06 541.02,-771.48\"/>\n",
+ "<text text-anchor=\"middle\" x=\"474.59\" y=\"-392.3\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M431.62,-36.28C457.31,-133.97 606.69,-594.39 651.41,-714.72\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"648.06,-715.75 654.87,-723.86 654.6,-713.27 648.06,-715.75\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"777.59,-520.65 787.31,-524.85 782.33,-515.5 777.59,-520.65\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M434.78,-36.45C474.3,-126.28 673.39,-517.5 734.7,-626.53\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"731.61,-628.19 739.6,-635.15 737.7,-624.73 731.61,-628.19\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M438.24,-36.4C488.75,-114.93 711.45,-421.86 785.55,-515.66\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"782.72,-517.74 791.7,-523.37 788.2,-513.37 782.72,-517.74\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"713.19,-178.35 723.76,-177.61 715.03,-171.6 713.19,-178.35\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"790.18,-399.51 800.44,-402.15 794.06,-393.69 790.18,-399.51\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"770.33,-283.24 780.88,-284.2 773.23,-276.87 770.33,-283.24\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"624.73,-96.35 635.04,-93.92 625.46,-89.39 624.73,-96.35\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"390.16,-33.97 399.22,-28.49 388.72,-27.12 390.16,-33.97\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.84,-85.27 219.57,-87.85 229.21,-92.24 229.84,-85.27\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M331.37,-44.37C341.16,-44.52 349.13,-42.27 349.13,-37.63 349.13,-35.02 346.61,-33.17 342.66,-32.07\"/>\n",
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+ "<text text-anchor=\"middle\" x=\"361.13\" y=\"-33.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"223.85,-67.47 216.92,-75.48 227.33,-73.54 223.85,-67.47\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M276.83,-51.76C238.24,-76.42 168.34,-127.07 130.16,-157.92\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"128.29,-154.92 122.79,-163.97 132.73,-160.33 128.29,-154.92\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M280.29,-55.94C230.21,-100.77 115.13,-215.75 67.62,-268.31\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"65.13,-265.85 61.09,-275.65 70.35,-270.51 65.13,-265.85\"/>\n",
+ "<text text-anchor=\"middle\" x=\"161.96\" y=\"-150.92\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<text text-anchor=\"middle\" x=\"103.25\" y=\"-317.64\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M286.8,-55.96C236.42,-117.57 91.37,-317 42.87,-391.08\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"39.99,-389.1 37.51,-399.4 45.87,-392.89 39.99,-389.1\"/>\n",
+ "<text text-anchor=\"middle\" x=\"152.84\" y=\"-227.32\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M291.69,-55.85C247.36,-131.63 99.32,-421.84 57.52,-514.45\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"54.43,-512.79 53.58,-523.35 60.83,-515.63 54.43,-512.79\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M295.49,-55.99C261.4,-143.35 139.81,-516.99 109.98,-625.29\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"106.62,-624.3 107.41,-634.86 113.38,-626.11 106.62,-624.3\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"190.62,-713.36 193.04,-723.67 197.58,-714.1 190.62,-713.36\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"297.81,-770.95 301.8,-780.76 304.8,-770.6 297.81,-770.95\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"532.14,-775.49 540.12,-782.46 538.23,-772.04 532.14,-775.49\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M304.77,-55.79C315.05,-158.08 394.71,-662.3 420.53,-790.47\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"417.08,-791.11 422.54,-800.19 423.94,-789.69 417.08,-791.11\"/>\n",
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- "<text text-anchor=\"middle\" x=\"407.32\" y=\"-403.77\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M307.77,-56.05C334.59,-157.43 493.48,-647.09 539.66,-771.65\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"536.31,-772.68 543.11,-780.81 542.86,-770.21 536.31,-772.68\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"720.74,-632.36 730.46,-636.58 725.49,-627.22 720.74,-632.36\"/>\n",
- "<text text-anchor=\"middle\" x=\"479.47\" y=\"-360.01\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M310.98,-56.12C353.72,-152.6 580.4,-597.95 646.09,-715.2\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"643.04,-716.92 651.02,-723.89 649.13,-713.46 643.04,-716.92\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"726.15,-629.67 735.1,-635.33 731.63,-625.32 726.15,-629.67\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M318.82,-56C388.16,-132.8 682.52,-427.5 778.89,-517.08\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"776.23,-519.39 785.95,-523.6 780.97,-514.24 776.23,-519.39\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M219.62,-102.11C311.14,-121.3 607.4,-168.38 713.94,-180.87\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"713.26,-184.32 723.59,-181.96 714.05,-177.36 713.26,-184.32\"/>\n",
+ "<text text-anchor=\"middle\" x=\"454.78\" y=\"-145.29\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"789.87,-406.75 800.42,-407.75 792.8,-400.39 789.87,-406.75\"/>\n",
+ "<text text-anchor=\"middle\" x=\"493.57\" y=\"-261.57\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"770.03,-289.6 780.6,-288.91 771.91,-282.86 770.03,-289.6\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"516.85,-56.89 524.97,-50.08 514.38,-50.34 516.85,-56.89\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M219.71,-97.97C299.55,-102.28 532.23,-102.46 624.98,-98.52\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"625.02,-102.02 634.84,-98.05 624.68,-95.03 625.02,-102.02\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"514.16,-51.63 523.22,-46.14 512.71,-44.78 514.16,-51.63\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"119.43,-154.09 115.53,-163.95 124.72,-158.68 119.43,-154.09\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"34.04,-388.89 33.29,-399.46 40.46,-391.66 34.04,-388.89\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"49.07,-512.88 49.94,-523.44 55.85,-514.64 49.07,-512.88\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"101.56,-624.74 104.03,-635.04 108.53,-625.45 101.56,-624.74\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"185.79,-714.01 189.81,-723.82 192.78,-713.65 185.79,-714.01\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"293.03,-771.6 298.51,-780.67 299.89,-770.17 293.03,-771.6\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"531.23,-773.89 539.21,-780.86 537.32,-770.43 531.23,-773.89\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"637.37,-718.23 646.32,-723.9 642.86,-713.88 637.37,-718.23\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M207.03,-112.82C281.76,-195.25 618.82,-532.63 722.29,-629.03\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"719.63,-631.33 729.34,-635.55 724.38,-626.19 719.63,-631.33\"/>\n",
+ "<text text-anchor=\"middle\" x=\"476.66\" y=\"-359.73\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"770.64,-523.22 780.9,-525.89 774.54,-517.41 770.64,-523.22\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"713.71,-190.25 723.55,-186.33 713.41,-183.26 713.71,-190.25\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"770.51,-295.81 780.85,-293.48 771.32,-288.85 770.51,-295.81\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"625.93,-107.84 635.03,-102.41 624.53,-100.98 625.93,-107.84\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"515.2,-57.5 523.33,-50.7 512.74,-50.95 515.2,-57.5\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"392.67,-45.11 399.63,-37.12 389.21,-39.03 392.67,-45.11\"/>\n",
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+ "<text text-anchor=\"middle\" x=\"59.49\" y=\"-237.81\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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"</g>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"96.6,-625.17 100.68,-634.95 103.59,-624.76 96.6,-625.17\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"180.91,-714.65 186.43,-723.69 187.75,-713.18 180.91,-714.65\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"288.12,-772.66 294.96,-780.76 294.67,-770.17 288.12,-772.66\"/>\n",
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- "<text text-anchor=\"middle\" x=\"277.14\" y=\"-548.09\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M110.31,-201.78C149.82,-291.6 348.91,-682.83 410.22,-791.86\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"407.14,-793.52 415.13,-800.48 413.22,-790.06 407.14,-793.52\"/>\n",
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- "<text text-anchor=\"middle\" x=\"333.15\" y=\"-521.34\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"525.61,-775.37 534.57,-781.02 531.09,-771.02 525.61,-775.37\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M118.3,-201.55C193.03,-283.98 530.09,-621.36 633.56,-717.76\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"630.9,-720.06 640.62,-724.28 635.65,-714.92 630.9,-720.06\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"713.67,-634.93 723.92,-637.6 717.57,-629.12 713.67,-634.93\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"770.1,-530.45 780.64,-531.47 773.04,-524.1 770.1,-530.45\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"714.79,-196.19 723.9,-190.79 713.41,-189.33 714.79,-196.19\"/>\n",
+ "<text text-anchor=\"middle\" x=\"382.11\" y=\"-246.98\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"790.17,-419.6 800.51,-417.29 790.99,-412.65 790.17,-419.6\"/>\n",
+ "<text text-anchor=\"middle\" x=\"420.48\" y=\"-362.99\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"629.58,-119.78 636.6,-111.84 626.16,-113.67 629.58,-119.78\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M73.78,-297.84C188.82,-303.07 636.32,-303.22 770.97,-298.31\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"770.93,-301.81 780.78,-297.91 770.64,-294.82 770.93,-301.81\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"627,-113.92 635.16,-107.16 624.57,-107.36 627,-113.92\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"409.69,-46.1 413.81,-36.33 404.5,-41.4 409.69,-46.1\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"516.41,-63.98 523.41,-56.03 512.98,-57.88 516.41,-63.98\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"291.95,-66.37 294.42,-56.06 286.06,-62.58 291.95,-66.37\"/>\n",
- "<text text-anchor=\"middle\" x=\"180.97\" y=\"-236.68\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"401.56,-46.4 407.16,-37.41 397.18,-40.95 401.56,-46.4\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"185.9,-123.39 186.64,-112.82 179.47,-120.62 185.9,-123.39\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"285.64,-66.91 289.67,-57.12 280.41,-62.26 285.64,-66.91\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"171.18,-718.61 179.26,-725.46 177.21,-715.06 171.18,-718.61\"/>\n",
- "<text text-anchor=\"middle\" x=\"88.96\" y=\"-580.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M46.35,-313.46C51.02,-371.91 79.62,-554.18 95,-625.65\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"91.5,-626.04 97.1,-635.04 98.33,-624.51 91.5,-626.04\"/>\n",
+ "<text text-anchor=\"middle\" x=\"58.67\" y=\"-473.35\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
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"<title>11->7</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"278.06,-776.92 287.07,-782.5 283.51,-772.52 278.06,-776.92\"/>\n",
- "<text text-anchor=\"middle\" x=\"143.98\" y=\"-609.55\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M49.65,-313.59C67.04,-383.16 146.9,-629.77 179.15,-714.62\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"175.86,-715.8 182.74,-723.85 182.38,-713.27 175.86,-715.8\"/>\n",
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"</g>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M52.98,-313.44C85.86,-389.43 233.65,-680.03 285.93,-772.26\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"282.76,-773.78 290.78,-780.71 288.83,-770.3 282.76,-773.78\"/>\n",
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"<title>11->5</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"514.57,-781.97 524.83,-784.61 518.45,-776.15 514.57,-781.97\"/>\n",
- "<text text-anchor=\"middle\" x=\"267.14\" y=\"-611.84\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M56.79,-313.53C107.31,-392.07 330.01,-698.99 404.1,-792.8\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"401.28,-794.87 410.26,-800.51 406.75,-790.51 401.28,-794.87\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"518.72,-776.89 528.45,-781.1 523.47,-771.75 518.72,-776.89\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M67.35,-313.57C156.93,-385.01 511.08,-642.57 626.95,-720.88\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"624.95,-723.75 635.2,-726.42 628.85,-717.94 624.95,-723.75\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M73.87,-312.24C182.98,-373.51 588.41,-580.21 714.73,-639.14\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"713.13,-642.26 723.68,-643.28 716.07,-635.9 713.13,-642.26\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"715.67,-202.42 723.84,-195.68 713.25,-195.85 715.67,-202.42\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"790.63,-425.68 800.49,-421.78 790.36,-418.68 790.63,-425.68\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.35,-417.44C171.91,-404.16 634.36,-331.08 771.18,-304.39\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"771.69,-307.86 780.82,-302.47 770.33,-300.99 771.69,-307.86\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.46,-408.32C156.11,-362.09 510.83,-181.56 626.82,-117.3\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"628.31,-120.48 635.34,-112.55 624.9,-114.37 628.31,-120.48\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"524.8,-65.95 530.45,-56.99 520.44,-60.47 524.8,-65.95\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"297.07,-66.2 297.91,-55.64 290.67,-63.37 297.07,-66.2\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"408.71,-47.18 412.83,-37.42 403.52,-42.48 408.71,-47.18\"/>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"190.27,-123.3 189.39,-112.74 183.49,-121.54 190.27,-123.3\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"291.15,-67.6 293.62,-57.3 285.26,-63.81 291.15,-67.6\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"185.29,-124.78 186.04,-114.21 178.86,-122.01 185.29,-124.78\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.2,-213.71 101.19,-203.17 95.4,-212.04 102.2,-213.71\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M73.82,-555.03C172.3,-592.09 510.53,-702.14 625.34,-734.66\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"624.33,-738.01 634.9,-737.32 626.21,-731.27 624.33,-738.01\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"194.36,-124.67 191.89,-114.37 187.4,-123.96 194.36,-124.67\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"110.6,-212.97 106.51,-203.19 103.61,-213.38 110.6,-212.97\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"231.61,-747.31 221.04,-748.14 229.82,-754.08 231.61,-747.31\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"343.14,-796.8 332.84,-799.27 342.43,-803.77 343.14,-796.8\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"514.09,-816.48 523.16,-811 512.66,-809.63 514.09,-816.48\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M455.57,-815.87C497.7,-805.13 578.29,-779.21 625.92,-761.32\"/>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"465.98,-804.17 456.92,-809.65 467.42,-811.02 465.98,-804.17\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M782.08,-548.82C683.11,-575.85 344.33,-685.73 230.2,-727.56\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"229.14,-724.22 220.98,-730.98 231.58,-730.78 229.14,-724.22\"/>\n",
+ "<text text-anchor=\"middle\" x=\"494.14\" y=\"-626.99\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->6 -->\n",
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"<title>1->6</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M740.4,-201.59C688.53,-291.26 487.93,-684.7 438.82,-792.67\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"435.57,-791.37 434.67,-801.93 441.96,-794.23 435.57,-791.37\"/>\n",
- "<text text-anchor=\"middle\" x=\"575.11\" y=\"-485.93\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M782.31,-553.29C697.58,-591.38 438.45,-723.18 341.36,-777.26\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"339.73,-774.16 332.73,-782.11 343.16,-780.26 339.73,-774.16\"/>\n",
+ "<text text-anchor=\"middle\" x=\"549.83\" y=\"-654.08\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->5 -->\n",
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"<title>1->5</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M744.03,-201.35C709.95,-288.52 587.05,-666.28 557.95,-772.59\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"554.56,-771.72 555.36,-782.28 561.32,-773.53 554.56,-771.72\"/>\n",
- "<text text-anchor=\"middle\" x=\"636.49\" y=\"-475.77\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M782.09,-558.51C712.14,-604.23 527.9,-737.96 456.68,-794.54\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"454.58,-791.74 448.98,-800.73 458.96,-797.19 454.58,-791.74\"/>\n",
+ "<text text-anchor=\"middle\" x=\"607.38\" y=\"-665.32\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->4 -->\n",
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"<title>1->4</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M746.99,-201.56C728.69,-283.16 675.91,-615.28 665.13,-715.06\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"661.62,-714.98 664.1,-725.28 668.58,-715.68 661.62,-714.98\"/>\n",
- "<text text-anchor=\"middle\" x=\"691.56\" y=\"-447.11\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M785.67,-561.31C735.58,-606.15 620.51,-721.13 573,-773.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"570.5,-771.23 566.47,-781.02 575.73,-775.88 570.5,-771.23\"/>\n",
+ "<text text-anchor=\"middle\" x=\"667.34\" y=\"-656.3\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->3 -->\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"745.63,-626.9 749.73,-636.67 752.62,-626.48 745.63,-626.9\"/>\n",
- "<text text-anchor=\"middle\" x=\"734.93\" y=\"-402.9\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M790.98,-561.18C761.51,-596.41 705.06,-673.91 678.31,-715.68\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"675.47,-713.62 673.13,-723.95 681.4,-717.33 675.47,-713.62\"/>\n",
+ "<text text-anchor=\"middle\" x=\"746.65\" y=\"-627.23\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->2 -->\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"797.73,-515.68 803.34,-524.67 804.56,-514.14 797.73,-515.68\"/>\n",
- "<text text-anchor=\"middle\" x=\"762.16\" y=\"-346.83\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M794.13,-561.33C783.31,-578.91 769.53,-605.65 760.67,-626.05\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"757.46,-624.66 756.89,-635.24 763.93,-627.32 757.46,-624.66\"/>\n",
+ "<text text-anchor=\"middle\" x=\"789.4\" y=\"-582.49\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"<!-- 1->1 -->\n",
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"<title>1->1</title>\n",
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- "<text text-anchor=\"middle\" x=\"812.04\" y=\"-179.63\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M836.75,-549.75C846.54,-549.9 854.51,-547.65 854.51,-543.01 854.51,-540.4 851.99,-538.54 848.04,-537.45\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"848.57,-533.98 838.26,-536.42 847.84,-540.95 848.57,-533.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"866.51\" y=\"-539.31\" font-family=\"Times,serif\" font-size=\"14.00\">K20</text>\n",
"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [(\"K20\",(1..20)*(1..20))]>"
+ "<Dot visualization: expr_as_graph [(\"K20\",(1..20)×(1..20))]>"
]
},
"execution_count": 76,
@@ -6552,11 +6554,10 @@
"# Definitions- und Wertebereich\n",
"\n",
"Definitionsbereich (Domain in Englisch):\n",
- "* $dom(r) = \\{a \\mid \\exists b.((a,b)\\in r)\\}$\n",
+ "* $\\mathrm{dom}(r) = \\{a \\mid \\exists b.((a,b)\\in r)\\}$\n",
"\n",
"Wertebereich (Bildmenge, Range in Englisch):\n",
- "* $ran(r) = \\{b \\mid \\exists a.((a,b)\\in r)\\}$\n",
- " "
+ "* $\\mathrm{ran}(r) = \\{b \\mid \\exists a.((a,b)\\in r)\\}$"
]
},
{
@@ -6685,14 +6686,14 @@
"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ "<!-- Generated by graphviz version 7.1.0 (20230121.1956)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
"<svg width=\"206pt\" height=\"305pt\"\n",
" viewBox=\"0.00 0.00 206.00 305.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 301)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-301 202,-301 202,4 -4,4\"/>\n",
+ "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-301 202,-301 202,4 -4,4\"/>\n",
"<!-- 10 -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>10</title>\n",
@@ -6708,9 +6709,9 @@
"<!-- 10->5 -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>10->5</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-260.8C27,-249.16 27,-233.55 27,-220.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-220.18 27,-210.18 23.5,-220.18 30.5,-220.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"32\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-260.8C27,-249.58 27,-234.67 27,-221.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-221.98 27,-211.98 23.5,-221.98 30.5,-221.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"30.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 8 -->\n",
"<g id=\"node3\" class=\"node\">\n",
@@ -6727,9 +6728,9 @@
"<!-- 8->4 -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>8->4</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-260.8C99,-249.16 99,-233.55 99,-220.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-220.18 99,-210.18 95.5,-220.18 102.5,-220.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"104\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-260.8C99,-249.58 99,-234.67 99,-221.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-221.98 99,-211.98 95.5,-221.98 102.5,-221.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"102.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 2 -->\n",
"<g id=\"node7\" class=\"node\">\n",
@@ -6740,9 +6741,9 @@
"<!-- 4->2 -->\n",
"<g id=\"edge4\" class=\"edge\">\n",
"<title>4->2</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-173.8C99,-162.16 99,-146.55 99,-133.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-133.18 99,-123.18 95.5,-133.18 102.5,-133.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"104\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-173.8C99,-162.58 99,-147.67 99,-134.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-134.98 99,-124.98 95.5,-134.98 102.5,-134.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"102.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 6 -->\n",
"<g id=\"node5\" class=\"node\">\n",
@@ -6759,9 +6760,9 @@
"<!-- 6->3 -->\n",
"<g id=\"edge3\" class=\"edge\">\n",
"<title>6->3</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M171,-260.8C171,-249.16 171,-233.55 171,-220.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"174.5,-220.18 171,-210.18 167.5,-220.18 174.5,-220.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"176\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M171,-260.8C171,-249.58 171,-234.67 171,-221.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"174.5,-221.98 171,-211.98 167.5,-221.98 174.5,-221.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"174.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 1 -->\n",
"<g id=\"node8\" class=\"node\">\n",
@@ -6772,15 +6773,17 @@
"<!-- 2->1 -->\n",
"<g id=\"edge5\" class=\"edge\">\n",
"<title>2->1</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-86.8C99,-75.16 99,-59.55 99,-46.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-46.18 99,-36.18 95.5,-46.18 102.5,-46.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"104\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-86.8C99,-75.58 99,-60.67 99,-47.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-47.98 99,-37.98 95.5,-47.98 102.5,-47.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"102.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [hh={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"h\",h)]>"
+ "<Dot visualization: expr_as_graph [LET h BE h={(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)} IN(\n",
+ "(\"h\",h)\n",
+ ")END]>"
]
},
"execution_count": 83,
@@ -6872,14 +6875,14 @@
"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ "<!-- Generated by graphviz version 7.1.0 (20230121.1956)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
"<svg width=\"494pt\" height=\"131pt\"\n",
" viewBox=\"0.00 0.00 494.00 131.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 127)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-127 490,-127 490,4 -4,4\"/>\n",
+ "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-127 490,-127 490,4 -4,4\"/>\n",
"<!-- 7 -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>7</title>\n",
@@ -6889,15 +6892,15 @@
"<!-- "nein" -->\n",
"<g id=\"node2\" class=\"node\">\n",
"<title>"nein"</title>\n",
- "<polygon fill=\"#fff68f\" stroke=\"#fff68f\" points=\"201.5,-36 140.5,-36 140.5,0 201.5,0 201.5,-36\"/>\n",
+ "<polygon fill=\"#fff68f\" stroke=\"#fff68f\" points=\"198,-36 144,-36 144,0 198,0 198,-36\"/>\n",
"<text text-anchor=\"middle\" x=\"171\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">"nein"</text>\n",
"</g>\n",
"<!-- 7->"nein" -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>7->"nein"</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.09,-88.01C76.43,-74.82 108.4,-55.95 133.25,-41.28\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"135.23,-44.18 142.06,-36.08 131.67,-38.15 135.23,-44.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"113\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.42,-87.81C76.94,-74.52 109.06,-55.56 133.88,-40.91\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"135.54,-43.99 142.37,-35.9 131.98,-37.97 135.54,-43.99\"/>\n",
+ "<text text-anchor=\"middle\" x=\"111.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
"</g>\n",
"<!-- 6 -->\n",
"<g id=\"node3\" class=\"node\">\n",
@@ -6914,9 +6917,9 @@
"<!-- 6->"ja" -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>6->"ja"</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"415.05,-46.75 415.73,-36.18 408.6,-44.02 415.05,-46.75\"/>\n",
- "<text text-anchor=\"middle\" x=\"412\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M394.29,-86.8C399.13,-75.35 405.61,-60.06 411.19,-46.89\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"414.33,-48.43 415.01,-37.86 407.89,-45.7 414.33,-48.43\"/>\n",
+ "<text text-anchor=\"middle\" x=\"410.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
"</g>\n",
"<!-- 5 -->\n",
"<g id=\"node5\" class=\"node\">\n",
@@ -6927,9 +6930,9 @@
"<!-- 5->"nein" -->\n",
"<g id=\"edge3\" class=\"edge\">\n",
"<title>5->"nein"</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M113.57,-86.8C124.01,-74.47 138.24,-57.68 149.92,-43.89\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"152.66,-46.07 156.45,-36.18 147.32,-41.54 152.66,-46.07\"/>\n",
- "<text text-anchor=\"middle\" x=\"145\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M113.57,-86.8C123.75,-74.78 137.52,-58.52 149.03,-44.94\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"151.43,-47.51 155.23,-37.62 146.09,-42.99 151.43,-47.51\"/>\n",
+ "<text text-anchor=\"middle\" x=\"143.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
"</g>\n",
"<!-- 4 -->\n",
"<g id=\"node6\" class=\"node\">\n",
@@ -6940,9 +6943,9 @@
"<!-- 4->"nein" -->\n",
"<g id=\"edge4\" class=\"edge\">\n",
"<title>4->"nein"</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M171,-86.8C171,-75.16 171,-59.55 171,-46.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"174.5,-46.18 171,-36.18 167.5,-46.18 174.5,-46.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"176\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M171,-86.8C171,-75.58 171,-60.67 171,-47.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"174.5,-47.98 171,-37.98 167.5,-47.98 174.5,-47.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"174.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
"</g>\n",
"<!-- 3 -->\n",
"<g id=\"node7\" class=\"node\">\n",
@@ -6953,9 +6956,9 @@
"<!-- 3->"ja" -->\n",
"<g id=\"edge5\" class=\"edge\">\n",
"<title>3->"ja"</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"437.4,-44.02 430.27,-36.18 430.95,-46.75 437.4,-44.02\"/>\n",
- "<text text-anchor=\"middle\" x=\"448\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M451.71,-86.8C446.87,-75.35 440.39,-60.06 434.81,-46.89\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"438.11,-45.7 430.99,-37.86 431.67,-48.43 438.11,-45.7\"/>\n",
+ "<text text-anchor=\"middle\" x=\"446.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
"</g>\n",
"<!-- 2 -->\n",
"<g id=\"node8\" class=\"node\">\n",
@@ -6966,9 +6969,9 @@
"<!-- 2->"nein" -->\n",
"<g id=\"edge6\" class=\"edge\">\n",
"<title>2->"nein"</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M228.43,-86.8C217.99,-74.47 203.76,-57.68 192.08,-43.89\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"194.68,-41.54 185.55,-36.18 189.34,-46.07 194.68,-41.54\"/>\n",
- "<text text-anchor=\"middle\" x=\"217\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M228.43,-86.8C218.25,-74.78 204.48,-58.52 192.97,-44.94\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"195.91,-42.99 186.77,-37.62 190.57,-47.51 195.91,-42.99\"/>\n",
+ "<text text-anchor=\"middle\" x=\"215.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
"</g>\n",
"<!-- 1 -->\n",
"<g id=\"node9\" class=\"node\">\n",
@@ -6979,15 +6982,17 @@
"<!-- 1->"nein" -->\n",
"<g id=\"edge7\" class=\"edge\">\n",
"<title>1->"nein"</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M287.91,-88.01C265.57,-74.82 233.6,-55.95 208.75,-41.28\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"210.33,-38.15 199.94,-36.08 206.77,-44.18 210.33,-38.15\"/>\n",
- "<text text-anchor=\"middle\" x=\"257\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M287.58,-87.81C265.06,-74.52 232.94,-55.56 208.12,-40.91\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"210.02,-37.97 199.63,-35.9 206.46,-43.99 210.02,-37.97\"/>\n",
+ "<text text-anchor=\"middle\" x=\"255.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d</text>\n",
"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [dhd={(1,\"nein\"),(2,\"nein\"),(3,\"ja\"),(4,\"nein\"),(5,\"nein\"),(6,\"ja\"),(7,\"nein\")} & h={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"d\",d)]>"
+ "<Dot visualization: expr_as_graph [LET d,h BE d={(1↦\"nein\"),(2↦\"nein\"),(3↦\"ja\"),(4↦\"nein\"),(5↦\"nein\"),(6↦\"ja\"),(7↦\"nein\")} & h={(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)} IN(\n",
+ "(\"d\",d)\n",
+ ")END]>"
]
},
"execution_count": 87,
@@ -7009,9 +7014,7 @@
" * $r[A] = \\{b \\mid \\exists a.((a,b)\\in r \\wedge a\\in A)\\}$\n",
"\n",
"Umkehrrelation:\n",
- " * $r^{-1} = \\{(b,a) \\mid (a,b)\\in r \\}$\n",
- "\n",
- "Im Notebook muss für die Umkehrrelation leider Tilde (~) verwendet werden: "
+ " * $r^{-1} = \\{(b,a) \\mid (a,b)\\in r \\}$"
]
},
{
@@ -7103,7 +7106,7 @@
}
],
"source": [
- "d~"
+ "d⁻¹"
]
},
{
@@ -7117,18 +7120,18 @@
"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ "<!-- Generated by graphviz version 7.1.0 (20230121.1956)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
"<svg width=\"494pt\" height=\"131pt\"\n",
" viewBox=\"0.00 0.00 494.00 131.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 127)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-127 490,-127 490,4 -4,4\"/>\n",
+ "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-127 490,-127 490,4 -4,4\"/>\n",
"<!-- "nein" -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>"nein"</title>\n",
- "<polygon fill=\"#fff68f\" stroke=\"#fff68f\" points=\"201.5,-123 140.5,-123 140.5,-87 201.5,-87 201.5,-123\"/>\n",
+ "<polygon fill=\"#fff68f\" stroke=\"#fff68f\" points=\"198,-123 144,-123 144,-87 198,-87 198,-123\"/>\n",
"<text text-anchor=\"middle\" x=\"171\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">"nein"</text>\n",
"</g>\n",
"<!-- 7 -->\n",
@@ -7140,9 +7143,9 @@
"<!-- "nein"->7 -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>"nein"->7</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M140.48,-88.49C129.22,-82.59 116.42,-75.67 105,-69 90.42,-60.48 74.68,-50.54 61.18,-41.77\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"62.95,-38.75 52.67,-36.21 59.12,-44.61 62.95,-38.75\"/>\n",
- "<text text-anchor=\"middle\" x=\"116\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d~</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M143.72,-88.86C132.87,-82.77 120.3,-75.63 109,-69 94.19,-60.31 78.02,-50.52 63.99,-41.92\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"65.84,-38.94 55.48,-36.69 62.17,-44.91 65.84,-38.94\"/>\n",
+ "<text text-anchor=\"middle\" x=\"118\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d⁻¹</text>\n",
"</g>\n",
"<!-- 5 -->\n",
"<g id=\"node3\" class=\"node\">\n",
@@ -7153,9 +7156,9 @@
"<!-- "nein"->5 -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>"nein"->5</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M156.43,-86.8C145.99,-74.47 131.76,-57.68 120.08,-43.89\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"122.68,-41.54 113.55,-36.18 117.34,-46.07 122.68,-41.54\"/>\n",
- "<text text-anchor=\"middle\" x=\"151\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d~</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M156.43,-86.8C146.25,-74.78 132.48,-58.52 120.97,-44.94\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"123.91,-42.99 114.77,-37.62 118.57,-47.51 123.91,-42.99\"/>\n",
+ "<text text-anchor=\"middle\" x=\"149\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d⁻¹</text>\n",
"</g>\n",
"<!-- 4 -->\n",
"<g id=\"node4\" class=\"node\">\n",
@@ -7166,9 +7169,9 @@
"<!-- "nein"->4 -->\n",
"<g id=\"edge3\" class=\"edge\">\n",
"<title>"nein"->4</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M171,-86.8C171,-75.16 171,-59.55 171,-46.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"174.5,-46.18 171,-36.18 167.5,-46.18 174.5,-46.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"182\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d~</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M171,-86.8C171,-75.58 171,-60.67 171,-47.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"174.5,-47.98 171,-37.98 167.5,-47.98 174.5,-47.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"180\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d⁻¹</text>\n",
"</g>\n",
"<!-- 2 -->\n",
"<g id=\"node5\" class=\"node\">\n",
@@ -7179,9 +7182,9 @@
"<!-- "nein"->2 -->\n",
"<g id=\"edge4\" class=\"edge\">\n",
"<title>"nein"->2</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M185.57,-86.8C196.01,-74.47 210.24,-57.68 221.92,-43.89\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"224.66,-46.07 228.45,-36.18 219.32,-41.54 224.66,-46.07\"/>\n",
- "<text text-anchor=\"middle\" x=\"223\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d~</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M185.57,-86.8C195.75,-74.78 209.52,-58.52 221.03,-44.94\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"223.43,-47.51 227.23,-37.62 218.09,-42.99 223.43,-47.51\"/>\n",
+ "<text text-anchor=\"middle\" x=\"221\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d⁻¹</text>\n",
"</g>\n",
"<!-- 1 -->\n",
"<g id=\"node6\" class=\"node\">\n",
@@ -7192,9 +7195,9 @@
"<!-- "nein"->1 -->\n",
"<g id=\"edge5\" class=\"edge\">\n",
"<title>"nein"->1</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M201.68,-88.74C213.2,-82.82 226.33,-75.82 238,-69 252.56,-60.49 268.24,-50.48 281.63,-41.65\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"283.67,-44.5 290.06,-36.05 279.79,-38.67 283.67,-44.5\"/>\n",
- "<text text-anchor=\"middle\" x=\"273\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d~</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M198.44,-89.11C209.55,-82.99 222.45,-75.77 234,-69 248.57,-60.46 264.42,-50.77 278.2,-42.21\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"279.9,-45.27 286.54,-37.01 276.2,-39.33 279.9,-45.27\"/>\n",
+ "<text text-anchor=\"middle\" x=\"267\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d⁻¹</text>\n",
"</g>\n",
"<!-- "ja" -->\n",
"<g id=\"node7\" class=\"node\">\n",
@@ -7211,9 +7214,9 @@
"<!-- "ja"->6 -->\n",
"<g id=\"edge6\" class=\"edge\">\n",
"<title>"ja"->6</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M415.71,-86.8C410.69,-74.93 403.91,-58.93 398.2,-45.45\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"401.4,-44.02 394.27,-36.18 394.95,-46.75 401.4,-44.02\"/>\n",
- "<text text-anchor=\"middle\" x=\"419\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d~</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M415.71,-86.8C410.87,-75.35 404.39,-60.06 398.81,-46.89\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"402.11,-45.7 394.99,-37.86 395.67,-48.43 402.11,-45.7\"/>\n",
+ "<text text-anchor=\"middle\" x=\"416\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d⁻¹</text>\n",
"</g>\n",
"<!-- 3 -->\n",
"<g id=\"node9\" class=\"node\">\n",
@@ -7224,15 +7227,17 @@
"<!-- "ja"->3 -->\n",
"<g id=\"edge7\" class=\"edge\">\n",
"<title>"ja"->3</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M430.29,-86.8C435.31,-74.93 442.09,-58.93 447.8,-45.45\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"451.05,-46.75 451.73,-36.18 444.6,-44.02 451.05,-46.75\"/>\n",
- "<text text-anchor=\"middle\" x=\"454\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d~</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M430.29,-86.8C435.13,-75.35 441.61,-60.06 447.19,-46.89\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"450.33,-48.43 451.01,-37.86 443.89,-45.7 450.33,-48.43\"/>\n",
+ "<text text-anchor=\"middle\" x=\"452\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">d⁻¹</text>\n",
"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [dhd={(1,\"nein\"),(2,\"nein\"),(3,\"ja\"),(4,\"nein\"),(5,\"nein\"),(6,\"ja\"),(7,\"nein\")} & h={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"d~\",d~)]>"
+ "<Dot visualization: expr_as_graph [LET d,h BE d={(1↦\"nein\"),(2↦\"nein\"),(3↦\"ja\"),(4↦\"nein\"),(5↦\"nein\"),(6↦\"ja\"),(7↦\"nein\")} & h={(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)} IN(\n",
+ "(\"d⁻¹\",d⁻¹)\n",
+ ")END]>"
]
},
"execution_count": 92,
@@ -7241,7 +7246,7 @@
}
],
"source": [
- ":dot expr_as_graph (\"d~\",d~)"
+ ":dot expr_as_graph (\"d⁻¹\",d⁻¹)"
]
},
{
@@ -7271,7 +7276,7 @@
}
],
"source": [
- "d~[{\"ja\"}]"
+ "d⁻¹[{\"ja\"}]"
]
},
{
@@ -7289,9 +7294,8 @@
"source": [
"# Binäre Relationen - Verknüpfung\n",
" \n",
- "Wir können Relationen $r_1$ und $r_2$ mit dem Operator ``;'' verknüpfen:\n",
- "* $(r_1 ; r_2)$ = $\\{(a,c) \\mid \\exists b. ( (a,b)\\in r_1 \\wedge (b,c)\\in r_2)\\}$.\n",
- "\n"
+ "Wir können Relationen $r_1$ und $r_2$ mit dem Operator \";\" verknüpfen:\n",
+ "* $(r_1 ; r_2)$ = $\\{(a,c) \\mid \\exists b. ( (a,b)\\in r_1 \\wedge (b,c)\\in r_2)\\}$."
]
},
{
@@ -7360,19 +7364,19 @@
"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ "<!-- Generated by graphviz version 7.1.0 (20230121.1956)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
- "<svg width=\"269pt\" height=\"305pt\"\n",
- " viewBox=\"0.00 0.00 269.00 305.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+ "<svg width=\"265pt\" height=\"305pt\"\n",
+ " viewBox=\"0.00 0.00 265.00 305.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 301)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-301 265,-301 265,4 -4,4\"/>\n",
+ "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-301 261,-301 261,4 -4,4\"/>\n",
"<!-- 8 -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>8</title>\n",
- "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"78,-297 24,-297 24,-261 78,-261 78,-297\"/>\n",
- "<text text-anchor=\"middle\" x=\"51\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">8</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"77,-297 23,-297 23,-261 77,-261 77,-297\"/>\n",
+ "<text text-anchor=\"middle\" x=\"50\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">8</text>\n",
"</g>\n",
"<!-- 2 -->\n",
"<g id=\"node2\" class=\"node\">\n",
@@ -7383,93 +7387,95 @@
"<!-- 8->2 -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>8->2</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M46.18,-260.82C42.54,-247.15 37.71,-227.5 35,-210 30.98,-184.09 28.95,-154.23 27.94,-133.17\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"31.43,-132.86 27.5,-123.03 24.44,-133.17 31.43,-132.86\"/>\n",
- "<text text-anchor=\"middle\" x=\"44.5\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">hh</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M45.87,-260.77C42.73,-247.08 38.54,-227.41 36,-210 32.31,-184.66 29.95,-155.62 28.58,-134.68\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"32.08,-134.65 27.97,-124.88 25.1,-135.08 32.08,-134.65\"/>\n",
+ "<text text-anchor=\"middle\" x=\"43\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">hh</text>\n",
"</g>\n",
"<!-- 4 -->\n",
"<g id=\"node3\" class=\"node\">\n",
"<title>4</title>\n",
- "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"117,-210 63,-210 63,-174 117,-174 117,-210\"/>\n",
- "<text text-anchor=\"middle\" x=\"90\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">4</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"113,-210 59,-210 59,-174 113,-174 113,-210\"/>\n",
+ "<text text-anchor=\"middle\" x=\"86\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">4</text>\n",
"</g>\n",
"<!-- 8->4 -->\n",
"<g id=\"edge4\" class=\"edge\">\n",
"<title>8->4</title>\n",
- "<path fill=\"none\" stroke=\"sienna\" d=\"M58.89,-260.8C64.34,-248.93 71.68,-232.93 77.86,-219.45\"/>\n",
- "<polygon fill=\"sienna\" stroke=\"sienna\" points=\"81.13,-220.72 82.12,-210.18 74.77,-217.8 81.13,-220.72\"/>\n",
- "<text text-anchor=\"middle\" x=\"78\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "<path fill=\"none\" stroke=\"sienna\" d=\"M57.29,-260.8C62.13,-249.35 68.61,-234.06 74.19,-220.89\"/>\n",
+ "<polygon fill=\"sienna\" stroke=\"sienna\" points=\"77.33,-222.43 78.01,-211.86 70.89,-219.7 77.33,-222.43\"/>\n",
+ "<text text-anchor=\"middle\" x=\"74.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 1 -->\n",
"<g id=\"node4\" class=\"node\">\n",
"<title>1</title>\n",
- "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"85,-36 31,-36 31,0 85,0 85,-36\"/>\n",
- "<text text-anchor=\"middle\" x=\"58\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"86,-36 32,-36 32,0 86,0 86,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"59\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
"</g>\n",
"<!-- 2->1 -->\n",
"<g id=\"edge7\" class=\"edge\">\n",
"<title>2->1</title>\n",
- "<path fill=\"none\" stroke=\"sienna\" d=\"M33.27,-86.8C37.56,-75.05 43.32,-59.24 48.21,-45.84\"/>\n",
- "<polygon fill=\"sienna\" stroke=\"sienna\" points=\"51.6,-46.77 51.74,-36.18 45.02,-44.37 51.6,-46.77\"/>\n",
- "<text text-anchor=\"middle\" x=\"50\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "<path fill=\"none\" stroke=\"sienna\" d=\"M33.48,-86.8C37.78,-75.35 43.54,-60.06 48.5,-46.89\"/>\n",
+ "<polygon fill=\"sienna\" stroke=\"sienna\" points=\"51.64,-48.48 51.89,-37.88 45.09,-46.01 51.64,-48.48\"/>\n",
+ "<text text-anchor=\"middle\" x=\"48.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 4->2 -->\n",
"<g id=\"edge6\" class=\"edge\">\n",
"<title>4->2</title>\n",
- "<path fill=\"none\" stroke=\"sienna\" d=\"M77.25,-173.8C68.2,-161.59 55.9,-144.99 45.74,-131.28\"/>\n",
- "<polygon fill=\"sienna\" stroke=\"sienna\" points=\"48.5,-129.13 39.73,-123.18 42.87,-133.29 48.5,-129.13\"/>\n",
- "<text text-anchor=\"middle\" x=\"68\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "<path fill=\"none\" stroke=\"sienna\" d=\"M74.06,-173.8C65.88,-162.01 54.86,-146.14 45.54,-132.72\"/>\n",
+ "<polygon fill=\"sienna\" stroke=\"sienna\" points=\"48.56,-130.93 39.99,-124.71 42.81,-134.92 48.56,-130.93\"/>\n",
+ "<text text-anchor=\"middle\" x=\"64.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 4->1 -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>4->1</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M86.81,-173.88C81.26,-144 69.74,-82.11 63.07,-46.27\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"66.44,-45.24 61.17,-36.05 59.56,-46.52 66.44,-45.24\"/>\n",
- "<text text-anchor=\"middle\" x=\"87.5\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">hh</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M83.26,-173.56C78.61,-143.94 69.13,-83.54 63.5,-47.64\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"66.99,-47.32 61.98,-37.98 60.07,-48.4 66.99,-47.32\"/>\n",
+ "<text text-anchor=\"middle\" x=\"83\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">hh</text>\n",
"</g>\n",
"<!-- 10 -->\n",
"<g id=\"node5\" class=\"node\">\n",
"<title>10</title>\n",
- "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"189,-297 135,-297 135,-261 189,-261 189,-297\"/>\n",
- "<text text-anchor=\"middle\" x=\"162\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">10</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"185,-297 131,-297 131,-261 185,-261 185,-297\"/>\n",
+ "<text text-anchor=\"middle\" x=\"158\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">10</text>\n",
"</g>\n",
"<!-- 5 -->\n",
"<g id=\"node6\" class=\"node\">\n",
"<title>5</title>\n",
- "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"189,-210 135,-210 135,-174 189,-174 189,-210\"/>\n",
- "<text text-anchor=\"middle\" x=\"162\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">5</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"185,-210 131,-210 131,-174 185,-174 185,-210\"/>\n",
+ "<text text-anchor=\"middle\" x=\"158\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">5</text>\n",
"</g>\n",
"<!-- 10->5 -->\n",
"<g id=\"edge3\" class=\"edge\">\n",
"<title>10->5</title>\n",
- "<path fill=\"none\" stroke=\"sienna\" d=\"M162,-260.8C162,-249.16 162,-233.55 162,-220.24\"/>\n",
- "<polygon fill=\"sienna\" stroke=\"sienna\" points=\"165.5,-220.18 162,-210.18 158.5,-220.18 165.5,-220.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"167\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "<path fill=\"none\" stroke=\"sienna\" d=\"M158,-260.8C158,-249.58 158,-234.67 158,-221.69\"/>\n",
+ "<polygon fill=\"sienna\" stroke=\"sienna\" points=\"161.5,-221.98 158,-211.98 154.5,-221.98 161.5,-221.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"161.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 6 -->\n",
"<g id=\"node7\" class=\"node\">\n",
"<title>6</title>\n",
- "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"261,-297 207,-297 207,-261 261,-261 261,-297\"/>\n",
- "<text text-anchor=\"middle\" x=\"234\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">6</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"257,-297 203,-297 203,-261 257,-261 257,-297\"/>\n",
+ "<text text-anchor=\"middle\" x=\"230\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">6</text>\n",
"</g>\n",
"<!-- 3 -->\n",
"<g id=\"node8\" class=\"node\">\n",
"<title>3</title>\n",
- "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"261,-210 207,-210 207,-174 261,-174 261,-210\"/>\n",
- "<text text-anchor=\"middle\" x=\"234\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"257,-210 203,-210 203,-174 257,-174 257,-210\"/>\n",
+ "<text text-anchor=\"middle\" x=\"230\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
"</g>\n",
"<!-- 6->3 -->\n",
"<g id=\"edge5\" class=\"edge\">\n",
"<title>6->3</title>\n",
- "<path fill=\"none\" stroke=\"sienna\" d=\"M234,-260.8C234,-249.16 234,-233.55 234,-220.24\"/>\n",
- "<polygon fill=\"sienna\" stroke=\"sienna\" points=\"237.5,-220.18 234,-210.18 230.5,-220.18 237.5,-220.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"239\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "<path fill=\"none\" stroke=\"sienna\" d=\"M230,-260.8C230,-249.58 230,-234.67 230,-221.69\"/>\n",
+ "<polygon fill=\"sienna\" stroke=\"sienna\" points=\"233.5,-221.98 230,-211.98 226.5,-221.98 233.5,-221.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"233.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [hh={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"h\",h)]>"
+ "<Dot visualization: expr_as_graph [LET h BE h={(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)} IN(\n",
+ "(\"h\",h,\"hh\",(h;h))\n",
+ ")END]>"
]
},
"execution_count": 97,
@@ -7492,14 +7498,14 @@
"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ "<!-- Generated by graphviz version 7.1.0 (20230121.1956)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
"<svg width=\"134pt\" height=\"131pt\"\n",
" viewBox=\"0.00 0.00 134.00 131.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 127)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-127 130,-127 130,4 -4,4\"/>\n",
+ "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-127 130,-127 130,4 -4,4\"/>\n",
"<!-- 8 -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>8</title>\n",
@@ -7515,9 +7521,9 @@
"<!-- 8->2 -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>8->2</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-86.8C27,-75.16 27,-59.55 27,-46.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-46.18 27,-36.18 23.5,-46.18 30.5,-46.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"36.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">hh</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-86.8C27,-75.58 27,-60.67 27,-47.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-47.98 27,-37.98 23.5,-47.98 30.5,-47.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"34\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">hh</text>\n",
"</g>\n",
"<!-- 4 -->\n",
"<g id=\"node3\" class=\"node\">\n",
@@ -7534,15 +7540,17 @@
"<!-- 4->1 -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>4->1</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-86.8C99,-75.16 99,-59.55 99,-46.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-46.18 99,-36.18 95.5,-46.18 102.5,-46.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"108.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">hh</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-86.8C99,-75.58 99,-60.67 99,-47.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-47.98 99,-37.98 95.5,-47.98 102.5,-47.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"106\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">hh</text>\n",
"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [hh={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"hh\",(h;h))]>"
+ "<Dot visualization: expr_as_graph [LET h BE h={(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)} IN(\n",
+ "(\"hh\",(h;h))\n",
+ ")END]>"
]
},
"execution_count": 98,
@@ -7565,14 +7573,14 @@
"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ "<!-- Generated by graphviz version 7.1.0 (20230121.1956)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
"<svg width=\"206pt\" height=\"305pt\"\n",
" viewBox=\"0.00 0.00 206.00 305.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 301)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-301 202,-301 202,4 -4,4\"/>\n",
+ "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-301 202,-301 202,4 -4,4\"/>\n",
"<!-- 10 -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>10</title>\n",
@@ -7588,9 +7596,9 @@
"<!-- 10->5 -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>10->5</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-260.8C27,-249.16 27,-233.55 27,-220.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-220.18 27,-210.18 23.5,-220.18 30.5,-220.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"32\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-260.8C27,-249.58 27,-234.67 27,-221.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-221.98 27,-211.98 23.5,-221.98 30.5,-221.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"30.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 8 -->\n",
"<g id=\"node3\" class=\"node\">\n",
@@ -7607,9 +7615,9 @@
"<!-- 8->4 -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>8->4</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-260.8C99,-249.16 99,-233.55 99,-220.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-220.18 99,-210.18 95.5,-220.18 102.5,-220.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"104\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-260.8C99,-249.58 99,-234.67 99,-221.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-221.98 99,-211.98 95.5,-221.98 102.5,-221.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"102.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 2 -->\n",
"<g id=\"node7\" class=\"node\">\n",
@@ -7620,9 +7628,9 @@
"<!-- 4->2 -->\n",
"<g id=\"edge4\" class=\"edge\">\n",
"<title>4->2</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-133.18 99,-123.18 95.5,-133.18 102.5,-133.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"104\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-173.8C99,-162.58 99,-147.67 99,-134.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-134.98 99,-124.98 95.5,-134.98 102.5,-134.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"102.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 6 -->\n",
"<g id=\"node5\" class=\"node\">\n",
@@ -7639,9 +7647,9 @@
"<!-- 6->3 -->\n",
"<g id=\"edge3\" class=\"edge\">\n",
"<title>6->3</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M171,-260.8C171,-249.16 171,-233.55 171,-220.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"174.5,-220.18 171,-210.18 167.5,-220.18 174.5,-220.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"176\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M171,-260.8C171,-249.58 171,-234.67 171,-221.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"174.5,-221.98 171,-211.98 167.5,-221.98 174.5,-221.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"174.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"<!-- 1 -->\n",
"<g id=\"node8\" class=\"node\">\n",
@@ -7652,15 +7660,17 @@
"<!-- 2->1 -->\n",
"<g id=\"edge5\" class=\"edge\">\n",
"<title>2->1</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-46.18 99,-36.18 95.5,-46.18 102.5,-46.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"104\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M99,-86.8C99,-75.58 99,-60.67 99,-47.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"102.5,-47.98 99,-37.98 95.5,-47.98 102.5,-47.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"102.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">h</text>\n",
"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [hh={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"h\",h)]>"
+ "<Dot visualization: expr_as_graph [LET h BE h={(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)} IN(\n",
+ "(\"h\",(h))\n",
+ ")END]>"
]
},
"execution_count": 99,
@@ -7682,7 +7692,7 @@
"* $r^1 = r$\n",
"* $r^k = (r^{k-1} ; r) = (r ; r^{k-1})$\n",
"\n",
- "* $r^+$ = $\\bigcup_{i\\geq 1} r^i$"
+ "* $r^+ = \\bigcup_{i\\geq 1} r^i$"
]
},
{
@@ -7795,14 +7805,14 @@
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"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ "<!-- Generated by graphviz version 7.1.0 (20230121.1956)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
- "<svg width=\"303pt\" height=\"305pt\"\n",
- " viewBox=\"0.00 0.00 303.00 305.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+ "<svg width=\"297pt\" height=\"305pt\"\n",
+ " viewBox=\"0.00 0.00 297.00 305.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 301)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-301 299,-301 299,4 -4,4\"/>\n",
+ "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-301 293,-301 293,4 -4,4\"/>\n",
"<!-- 10 -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>10</title>\n",
@@ -7818,9 +7828,9 @@
"<!-- 10->5 -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>10->5</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-220.18 27,-210.18 23.5,-220.18 30.5,-220.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"38\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h+</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-260.8C27,-249.58 27,-234.67 27,-221.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-221.98 27,-211.98 23.5,-221.98 30.5,-221.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"33.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h⁺</text>\n",
"</g>\n",
"<!-- 8 -->\n",
"<g id=\"node3\" class=\"node\">\n",
@@ -7837,81 +7847,83 @@
"<!-- 8->4 -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>8->4</title>\n",
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- "<text text-anchor=\"middle\" x=\"137\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h+</text>\n",
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+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"120.74,-217.55 112.07,-211.46 115.05,-221.62 120.74,-217.55\"/>\n",
+ "<text text-anchor=\"middle\" x=\"141.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h⁺</text>\n",
"</g>\n",
"<!-- 2 -->\n",
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"<title>2</title>\n",
- "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"185,-123 131,-123 131,-87 185,-87 185,-123\"/>\n",
- "<text text-anchor=\"middle\" x=\"158\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"179,-123 125,-123 125,-87 179,-87 179,-123\"/>\n",
+ "<text text-anchor=\"middle\" x=\"152\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
"</g>\n",
"<!-- 8->2 -->\n",
"<g id=\"edge3\" class=\"edge\">\n",
"<title>8->2</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"162.13,-132.97 158.4,-123.05 155.13,-133.13 162.13,-132.97\"/>\n",
- "<text text-anchor=\"middle\" x=\"172\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">h+</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M160.99,-260.56C159.26,-230.94 155.75,-170.54 153.67,-134.64\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"157.18,-134.78 153.1,-125 150.19,-135.18 157.18,-134.78\"/>\n",
+ "<text text-anchor=\"middle\" x=\"164.5\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">h⁺</text>\n",
"</g>\n",
"<!-- 1 -->\n",
"<g id=\"node6\" class=\"node\">\n",
"<title>1</title>\n",
- "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"185,-36 131,-36 131,0 185,0 185,-36\"/>\n",
- "<text text-anchor=\"middle\" x=\"158\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"179,-36 125,-36 125,0 179,0 179,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"152\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">1</text>\n",
"</g>\n",
"<!-- 8->1 -->\n",
"<g id=\"edge4\" class=\"edge\">\n",
"<title>8->1</title>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M168.68,-260.64C180.77,-226.81 203.61,-149.85 188,-87 184.46,-72.76 177.34,-58.22 170.35,-46.27\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"173.35,-44.47 165.13,-37.78 167.39,-48.13 173.35,-44.47\"/>\n",
+ "<text text-anchor=\"middle\" x=\"199.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">h⁺</text>\n",
"</g>\n",
"<!-- 4->2 -->\n",
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"<title>4->2</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M107.01,-173.72C111.93,-163.79 118.68,-151.29 126,-141 128.47,-137.53 131.25,-134.03 134.11,-130.66\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M109.73,-173.8C117,-162.12 126.78,-146.45 135.1,-133.11\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"137.99,-135.09 140.31,-124.75 132.05,-131.39 137.99,-135.09\"/>\n",
+ "<text text-anchor=\"middle\" x=\"135.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">h⁺</text>\n",
"</g>\n",
"<!-- 4->1 -->\n",
"<g id=\"edge7\" class=\"edge\">\n",
"<title>4->1</title>\n",
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- "<text text-anchor=\"middle\" x=\"111\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">h+</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M96.67,-173.51C94.52,-152.38 93,-116.05 103,-87 108.28,-71.65 118.27,-56.76 127.85,-44.85\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"130.26,-47.42 134.04,-37.52 124.92,-42.91 130.26,-47.42\"/>\n",
+ "<text text-anchor=\"middle\" x=\"109.5\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">h⁺</text>\n",
"</g>\n",
"<!-- 2->1 -->\n",
"<g id=\"edge8\" class=\"edge\">\n",
"<title>2->1</title>\n",
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- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"161.5,-46.18 158,-36.18 154.5,-46.18 161.5,-46.18\"/>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M152,-86.8C152,-75.58 152,-60.67 152,-47.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"155.5,-47.98 152,-37.98 148.5,-47.98 155.5,-47.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"158.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">h⁺</text>\n",
"</g>\n",
"<!-- 6 -->\n",
"<g id=\"node7\" class=\"node\">\n",
"<title>6</title>\n",
- "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"295,-297 241,-297 241,-261 295,-261 295,-297\"/>\n",
- "<text text-anchor=\"middle\" x=\"268\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">6</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"289,-297 235,-297 235,-261 289,-261 289,-297\"/>\n",
+ "<text text-anchor=\"middle\" x=\"262\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">6</text>\n",
"</g>\n",
"<!-- 3 -->\n",
"<g id=\"node8\" class=\"node\">\n",
"<title>3</title>\n",
- "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"295,-210 241,-210 241,-174 295,-174 295,-210\"/>\n",
- "<text text-anchor=\"middle\" x=\"268\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"289,-210 235,-210 235,-174 289,-174 289,-210\"/>\n",
+ "<text text-anchor=\"middle\" x=\"262\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
"</g>\n",
"<!-- 6->3 -->\n",
"<g id=\"edge5\" class=\"edge\">\n",
"<title>6->3</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M268,-260.8C268,-249.16 268,-233.55 268,-220.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"271.5,-220.18 268,-210.18 264.5,-220.18 271.5,-220.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"279\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h+</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M262,-260.8C262,-249.58 262,-234.67 262,-221.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"265.5,-221.98 262,-211.98 258.5,-221.98 265.5,-221.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"268.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">h⁺</text>\n",
"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [hh={(2,1),(4,2),(6,3),(8,4),(10,5)}(\"h+\",closure1(h))]>"
+ "<Dot visualization: expr_as_graph [LET h BE h={(2↦1),(4↦2),(6↦3),(8↦4),(10↦5)} IN(\n",
+ "(\"h⁺\",closure1(h))\n",
+ ")END]>"
]
},
"execution_count": 104,
@@ -7920,7 +7932,7 @@
}
],
"source": [
- ":dot expr_as_graph (\"h+\",closure1(h))"
+ ":dot expr_as_graph (\"h⁺\",closure1(h))"
]
},
{
@@ -7993,151 +8005,153 @@
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" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ "<!-- Generated by graphviz version 7.1.0 (20230121.1956)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
- "<svg width=\"315pt\" height=\"305pt\"\n",
- " viewBox=\"0.00 0.00 315.17 305.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+ "<svg width=\"259pt\" height=\"305pt\"\n",
+ " viewBox=\"0.00 0.00 259.38 305.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 301)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-301 311.17,-301 311.17,4 -4,4\"/>\n",
+ "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-301 255.38,-301 255.38,4 -4,4\"/>\n",
"<!-- \\{3\\} -->\n",
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"<title>\\{3\\}</title>\n",
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- "<text text-anchor=\"middle\" x=\"60.17\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">{3}</text>\n",
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+ "<text text-anchor=\"middle\" x=\"50.38\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">{3}</text>\n",
"</g>\n",
"<!-- \\{2,3\\} -->\n",
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"<title>\\{2,3\\}</title>\n",
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- "<text text-anchor=\"middle\" x=\"147.17\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">{2,3}</text>\n",
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"</g>\n",
"<!-- \\{1,3\\} -->\n",
"<g id=\"node3\" class=\"node\">\n",
"<title>\\{1,3\\}</title>\n",
- "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"100.67,-123 43.67,-123 43.67,-87 100.67,-87 100.67,-123\"/>\n",
- "<text text-anchor=\"middle\" x=\"72.17\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1,3}</text>\n",
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"</g>\n",
"<!-- \\{3\\}->\\{1,3\\} -->\n",
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"<title>\\{3\\}->\\{1,3\\}</title>\n",
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- "<text text-anchor=\"middle\" x=\"20.67\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
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"</g>\n",
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"</g>\n",
"<!-- \\{2,3\\}->\\{1,2,3\\} -->\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M123.38,-86.8C123.38,-75.58 123.38,-60.67 123.38,-47.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"126.88,-47.98 123.38,-37.98 119.88,-47.98 126.88,-47.98\"/>\n",
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"</g>\n",
"<!-- \\{1,3\\}->\\{1,2,3\\} -->\n",
"<g id=\"edge6\" class=\"edge\">\n",
"<title>\\{1,3\\}->\\{1,2,3\\}</title>\n",
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"</g>\n",
"<!-- \\{2\\} -->\n",
"<g id=\"node5\" class=\"node\">\n",
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- "<text text-anchor=\"middle\" x=\"234.17\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">{2}</text>\n",
+ "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"224.38,-210 170.38,-210 170.38,-174 224.38,-174 224.38,-210\"/>\n",
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"</g>\n",
"<!-- \\{2\\}->\\{2,3\\} -->\n",
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- "<text text-anchor=\"middle\" x=\"239.67\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M192.55,-173.93C189.01,-163.6 183.41,-150.6 175.38,-141 171.05,-135.81 165.75,-131.08 160.22,-126.89\"/>\n",
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+ "<text text-anchor=\"middle\" x=\"196.38\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{1,2\\} -->\n",
"<g id=\"node6\" class=\"node\">\n",
"<title>\\{1,2\\}</title>\n",
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+ "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"222.38,-123 168.38,-123 168.38,-87 222.38,-87 222.38,-123\"/>\n",
+ "<text text-anchor=\"middle\" x=\"195.38\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1,2}</text>\n",
"</g>\n",
"<!-- \\{2\\}->\\{1,2\\} -->\n",
"<g id=\"edge5\" class=\"edge\">\n",
"<title>\\{2\\}->\\{1,2\\}</title>\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M215.19,-173.55C222.53,-164.21 228.47,-152.32 224.38,-141 223.34,-138.11 221.94,-135.28 220.32,-132.58\"/>\n",
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"<!-- \\{1\\} -->\n",
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"<!-- \\{1\\}->\\{1,3\\} -->\n",
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+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [sub1sub1={({},{1}),({},{2}),({},{3}),({1},{1,2}),({1},{1,3}),({1,2},{1,2,3}),({1,3},{1,2,3}),({2},{1,2}),({2},{2,3}),({2,3},{1,2,3}),({3},{1,3}),({3},{2,3})}(\"sub1\",sub1)]>"
+ "<Dot visualization: expr_as_graph [LET sub1 BE sub1={(∅↦{1}),(∅↦{2}),(∅↦{3}),({1}↦{1,2}),({1}↦{1,3}),({1,2}↦{1,2,3}),({1,3}↦{1,2,3}),({2}↦{1,2}),({2}↦{2,3}),({2,3}↦{1,2,3}),({3}↦{1,3}),({3}↦{2,3})} IN(\n",
+ "(\"sub1\",sub1)\n",
+ ")END]>"
]
},
"execution_count": 108,
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@@ -8236,124 +8250,126 @@
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+ "<text text-anchor=\"middle\" x=\"285\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\} -->\n",
"<g id=\"node8\" class=\"node\">\n",
"<title>\\{\\}</title>\n",
- "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"283,-297 229,-297 229,-261 283,-261 283,-297\"/>\n",
- "<text text-anchor=\"middle\" x=\"256\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">{}</text>\n",
+ "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"265,-297 211,-297 211,-261 265,-261 265,-297\"/>\n",
+ "<text text-anchor=\"middle\" x=\"238\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">{}</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{3\\} -->\n",
"<g id=\"edge13\" class=\"edge\">\n",
"<title>\\{\\}->\\{3\\}</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M228.8,-262.2C221.08,-256.81 213.12,-250.28 207,-243 201.27,-236.19 196.55,-227.78 192.86,-219.81\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"196.01,-218.27 188.87,-210.45 189.57,-221.02 196.01,-218.27\"/>\n",
- "<text text-anchor=\"middle\" x=\"224.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M215.54,-260.71C209.44,-255.41 203.11,-249.31 198,-243 192.4,-236.09 187.3,-227.91 183.03,-220.19\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"186.19,-218.69 178.45,-211.45 179.99,-221.94 186.19,-218.69\"/>\n",
+ "<text text-anchor=\"middle\" x=\"211\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{2,3\\} -->\n",
"<g id=\"edge14\" class=\"edge\">\n",
"<title>\\{\\}->\\{2,3\\}</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M228.96,-273.34C195.64,-266.07 139.98,-248.62 112,-210 95.96,-187.87 94.27,-155.9 95.61,-133.26\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"99.1,-133.54 96.43,-123.28 92.12,-132.96 99.1,-133.54\"/>\n",
- "<text text-anchor=\"middle\" x=\"129.5\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M210.52,-271.77C180.07,-263.32 132.08,-245.06 108,-210 92.89,-188.01 91.17,-157.03 92.44,-134.62\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"95.93,-134.92 93.27,-124.66 88.95,-134.34 95.93,-134.92\"/>\n",
+ "<text text-anchor=\"middle\" x=\"121\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{1,2,3\\} -->\n",
"<g id=\"edge15\" class=\"edge\">\n",
"<title>\\{\\}->\\{1,2,3\\}</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M283.15,-278.74C329.94,-278.34 424.61,-269.75 465,-210 495.62,-164.7 495.45,-133.69 467,-87 449.6,-58.46 358.68,-37.34 302.02,-26.72\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"302.33,-23.22 291.86,-24.86 301.07,-30.1 302.33,-23.22\"/>\n",
- "<text text-anchor=\"middle\" x=\"505.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M265.49,-276.36C305.53,-272.33 378.73,-258.37 412,-210 443.13,-164.75 449.57,-135.61 424,-87 396.98,-35.63 324.72,-22.44 278.63,-19.42\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"278.91,-15.93 268.75,-18.91 278.56,-22.92 278.91,-15.93\"/>\n",
+ "<text text-anchor=\"middle\" x=\"452\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{1,3\\} -->\n",
"<g id=\"edge17\" class=\"edge\">\n",
"<title>\\{\\}->\\{1,3\\}</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M283.09,-273.23C316.56,-265.84 372.64,-248.27 402,-210 421.51,-184.57 437.38,-169.65 423,-141 420.49,-136 416.89,-131.53 412.78,-127.6\"/>\n",
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- "<text text-anchor=\"middle\" x=\"443.5\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M265.27,-271.16C294.68,-262.27 340.44,-243.73 364,-210 381.97,-184.27 389.89,-170.43 379,-141 377.92,-138.08 376.47,-135.23 374.79,-132.51\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"377.62,-130.44 368.86,-124.48 371.99,-134.6 377.62,-130.44\"/>\n",
+ "<text text-anchor=\"middle\" x=\"395\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{2\\} -->\n",
"<g id=\"edge16\" class=\"edge\">\n",
"<title>\\{\\}->\\{2\\}</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M228.69,-277.12C177.15,-274.9 68.28,-267.39 41,-243 34.5,-237.19 30.92,-228.64 28.97,-220.24\"/>\n",
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- "<text text-anchor=\"middle\" x=\"58.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M210.83,-275.99C164.95,-272.12 74.68,-262.23 50,-243 42.95,-237.51 37.98,-229.36 34.52,-221.23\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"37.82,-220.07 31.17,-211.82 31.23,-222.42 37.82,-220.07\"/>\n",
+ "<text text-anchor=\"middle\" x=\"63\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{1,2\\} -->\n",
"<g id=\"edge18\" class=\"edge\">\n",
"<title>\\{\\}->\\{1,2\\}</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M256,-260.88C256,-231 256,-169.11 256,-133.27\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"259.5,-133.05 256,-123.05 252.5,-133.05 259.5,-133.05\"/>\n",
- "<text text-anchor=\"middle\" x=\"273.5\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M238,-260.56C238,-230.94 238,-170.54 238,-134.64\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"241.5,-135 238,-125 234.5,-135 241.5,-135\"/>\n",
+ "<text text-anchor=\"middle\" x=\"251\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{1\\} -->\n",
"<g id=\"edge19\" class=\"edge\">\n",
"<title>\\{\\}->\\{1\\}</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M278.26,-260.8C294.81,-248.01 317.58,-230.42 335.77,-216.36\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"338,-219.06 343.77,-210.18 333.72,-213.52 338,-219.06\"/>\n",
- "<text text-anchor=\"middle\" x=\"335.5\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M256.21,-260.8C269.18,-248.55 286.81,-231.9 301.36,-218.16\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"303.54,-220.92 308.4,-211.51 298.73,-215.83 303.54,-220.92\"/>\n",
+ "<text text-anchor=\"middle\" x=\"301\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [sub1sub1={({},{1}),({},{2}),({},{3}),({1},{1,2}),({1},{1,3}),({1,2},{1,2,3}),({1,3},{1,2,3}),({2},{1,2}),({2},{2,3}),({2,3},{1,2,3}),({3},{1,3}),({3},{2,3})}(\"sub1\",closure1(sub1))]>"
+ "<Dot visualization: expr_as_graph [LET sub1 BE sub1={(∅↦{1}),(∅↦{2}),(∅↦{3}),({1}↦{1,2}),({1}↦{1,3}),({1,2}↦{1,2,3}),({1,3}↦{1,2,3}),({2}↦{1,2}),({2}↦{2,3}),({2,3}↦{1,2,3}),({3}↦{1,3}),({3}↦{2,3})} IN(\n",
+ "(\"sub1\",closure1(sub1))\n",
+ ")END]>"
]
},
"execution_count": 109,
@@ -8375,7 +8391,7 @@
"* $r^0 = \\{(a,a) \\mid a\\in A\\}$\n",
"* $r^1 = r$\n",
"* $r^k = (r^{k-1} ; r) = (r ; r^{k-1})$\n",
- "* $r^*$ = $\\bigcup_{i\\geq 0} r^i$\n"
+ "* $r^* = \\bigcup_{i\\geq 0} r^i$\n"
]
},
{
@@ -8435,14 +8451,14 @@
"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ "<!-- Generated by graphviz version 7.1.0 (20230121.1956)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
"<svg width=\"62pt\" height=\"218pt\"\n",
" viewBox=\"0.00 0.00 62.00 218.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 214)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-214 58,-214 58,4 -4,4\"/>\n",
+ "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-214 58,-214 58,4 -4,4\"/>\n",
"<!-- 2 -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>2</title>\n",
@@ -8458,9 +8474,9 @@
"<!-- 2->3 -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>2->3</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-86.8C27,-75.16 27,-59.55 27,-46.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-46.18 27,-36.18 23.5,-46.18 30.5,-46.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"30.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">r</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-86.8C27,-75.58 27,-60.67 27,-47.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-47.98 27,-37.98 23.5,-47.98 30.5,-47.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"29.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">r</text>\n",
"</g>\n",
"<!-- 1 -->\n",
"<g id=\"node3\" class=\"node\">\n",
@@ -8471,15 +8487,17 @@
"<!-- 1->2 -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>1->2</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-173.8C27,-162.16 27,-146.55 27,-133.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-133.18 27,-123.18 23.5,-133.18 30.5,-133.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"30.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">r</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-173.8C27,-162.58 27,-147.67 27,-134.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-134.98 27,-124.98 23.5,-134.98 30.5,-134.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"29.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">r</text>\n",
"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [Arsub1A={1,2,3} & r={(1,2),(2,3)} & sub1={({},{1}),({},{2}),({},{3}),({1},{1,2}),({1},{1,3}),({1,2},{1,2,3}),({1,3},{1,2,3}),({2},{1,2}),({2},{2,3}),({2,3},{1,2,3}),({3},{1,3}),({3},{2,3})}(\"r\",r)]>"
+ "<Dot visualization: expr_as_graph [LET A,r,sub1 BE A={1,2,3} & r={(1↦2),(2↦3)} & sub1={(∅↦{1}),(∅↦{2}),(∅↦{3}),({1}↦{1,2}),({1}↦{1,3}),({1,2}↦{1,2,3}),({1,3}↦{1,2,3}),({2}↦{1,2}),({2}↦{2,3}),({2,3}↦{1,2,3}),({3}↦{1,3}),({3}↦{2,3})} IN(\n",
+ "(\"r\",r)\n",
+ ")END]>"
]
},
"execution_count": 112,
@@ -8585,46 +8603,46 @@
"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ "<!-- Generated by graphviz version 7.1.0 (20230121.1956)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
- "<svg width=\"137pt\" height=\"218pt\"\n",
- " viewBox=\"0.00 0.00 137.00 218.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+ "<svg width=\"133pt\" height=\"218pt\"\n",
+ " viewBox=\"0.00 0.00 133.00 218.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 214)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-214 133,-214 133,4 -4,4\"/>\n",
+ "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-214 129,-214 129,4 -4,4\"/>\n",
"<!-- 3 -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>3</title>\n",
- "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"61,-36 7,-36 7,0 61,0 61,-36\"/>\n",
- "<text text-anchor=\"middle\" x=\"34\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"60,-36 6,-36 6,0 60,0 60,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"33\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n",
"</g>\n",
"<!-- 3->3 -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>3->3</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M61.24,-25.87C71.02,-26.04 79,-23.41 79,-18 79,-14.7 76.04,-12.44 71.51,-11.21\"/>\n",
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- "<text text-anchor=\"middle\" x=\"86\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M60.24,-26.15C70.02,-26.32 78,-23.61 78,-18 78,-14.85 75.48,-12.61 71.53,-11.28\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"72.1,-7.83 61.74,-10.04 71.22,-14.77 72.1,-7.83\"/>\n",
+ "<text text-anchor=\"middle\" x=\"84\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
"</g>\n",
"<!-- 2 -->\n",
"<g id=\"node2\" class=\"node\">\n",
"<title>2</title>\n",
- "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"97,-123 43,-123 43,-87 97,-87 97,-123\"/>\n",
- "<text text-anchor=\"middle\" x=\"70\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
+ "<polygon fill=\"#cdba96\" stroke=\"#cdba96\" points=\"95,-123 41,-123 41,-87 95,-87 95,-123\"/>\n",
+ "<text text-anchor=\"middle\" x=\"68\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">2</text>\n",
"</g>\n",
"<!-- 2->3 -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>2->3</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M62.71,-86.8C57.69,-74.93 50.91,-58.93 45.2,-45.45\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"48.4,-44.02 41.27,-36.18 41.95,-46.75 48.4,-44.02\"/>\n",
- "<text text-anchor=\"middle\" x=\"61\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M60.92,-86.8C56.2,-75.35 49.91,-60.06 44.49,-46.89\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"47.81,-45.78 40.77,-37.87 41.34,-48.45 47.81,-45.78\"/>\n",
+ "<text text-anchor=\"middle\" x=\"58\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
"</g>\n",
"<!-- 2->2 -->\n",
"<g id=\"edge3\" class=\"edge\">\n",
"<title>2->2</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M97.24,-112.87C107.02,-113.04 115,-110.41 115,-105 115,-101.7 112.04,-99.44 107.51,-98.21\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"107.55,-94.7 97.24,-97.13 106.82,-101.66 107.55,-94.7\"/>\n",
- "<text text-anchor=\"middle\" x=\"122\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M95.24,-113.15C105.02,-113.32 113,-110.61 113,-105 113,-101.85 110.48,-99.61 106.53,-98.28\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"107.1,-94.83 96.74,-97.04 106.22,-101.77 107.1,-94.83\"/>\n",
+ "<text text-anchor=\"middle\" x=\"119\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
"</g>\n",
"<!-- 1 -->\n",
"<g id=\"node3\" class=\"node\">\n",
@@ -8635,29 +8653,31 @@
"<!-- 1->3 -->\n",
"<g id=\"edge4\" class=\"edge\">\n",
"<title>1->3</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M24.05,-173.67C20.94,-153 16.9,-117.43 20,-87 21.39,-73.32 24.32,-58.33 27.15,-45.92\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.55,-46.72 29.45,-36.18 23.74,-45.1 30.55,-46.72\"/>\n",
- "<text text-anchor=\"middle\" x=\"27\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M24.12,-173.68C21.09,-153.02 17.12,-117.45 20,-87 21.23,-73.99 23.76,-59.76 26.27,-47.7\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"29.68,-48.48 28.39,-37.96 22.84,-46.98 29.68,-48.48\"/>\n",
+ "<text text-anchor=\"middle\" x=\"26\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
"</g>\n",
"<!-- 1->2 -->\n",
"<g id=\"edge5\" class=\"edge\">\n",
"<title>1->2</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M35.7,-173.8C41.7,-161.93 49.8,-145.93 56.62,-132.45\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"59.92,-133.68 61.31,-123.18 53.67,-130.52 59.92,-133.68\"/>\n",
- "<text text-anchor=\"middle\" x=\"58\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M35.3,-173.8C40.87,-162.24 48.34,-146.75 54.74,-133.5\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"57.72,-135.36 58.92,-124.83 51.42,-132.32 57.72,-135.36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"55\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
"</g>\n",
"<!-- 1->1 -->\n",
"<g id=\"edge6\" class=\"edge\">\n",
"<title>1->1</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.24,-199.87C64.02,-200.04 72,-197.41 72,-192 72,-188.7 69.04,-186.44 64.51,-185.21\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"64.55,-181.7 54.24,-184.13 63.82,-188.66 64.55,-181.7\"/>\n",
- "<text text-anchor=\"middle\" x=\"79\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M54.24,-200.15C64.02,-200.32 72,-197.61 72,-192 72,-188.85 69.48,-186.61 65.53,-185.28\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"66.1,-181.83 55.74,-184.04 65.22,-188.77 66.1,-181.83\"/>\n",
+ "<text text-anchor=\"middle\" x=\"78\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">r*</text>\n",
"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [Arsub1A={1,2,3} & r={(1,2),(2,3)} & sub1={({},{1}),({},{2}),({},{3}),({1},{1,2}),({1},{1,3}),({1,2},{1,2,3}),({1,3},{1,2,3}),({2},{1,2}),({2},{2,3}),({2,3},{1,2,3}),({3},{1,3}),({3},{2,3})}(\"r*\",id(A)\\/r\\/(r;r)\\/((r;r);r)\\/(((r;r);r);r))]>"
+ "<Dot visualization: expr_as_graph [LET A,r,sub1 BE A={1,2,3} & r={(1↦2),(2↦3)} & sub1={(∅↦{1}),(∅↦{2}),(∅↦{3}),({1}↦{1,2}),({1}↦{1,3}),({1,2}↦{1,2,3}),({1,3}↦{1,2,3}),({2}↦{1,2}),({2}↦{2,3}),({2,3}↦{1,2,3}),({3}↦{1,3}),({3}↦{2,3})} IN(\n",
+ "(\"r*\",id(A) \\/ r \\/ (r;r) \\/ (r;r;r) \\/ (r;r;r;r))\n",
+ ")END]>"
]
},
"execution_count": 116,
@@ -8710,14 +8730,14 @@
"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ "<!-- Generated by graphviz version 7.1.0 (20230121.1956)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
- "<svg width=\"97pt\" height=\"218pt\"\n",
- " viewBox=\"0.00 0.00 97.00 218.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+ "<svg width=\"94pt\" height=\"218pt\"\n",
+ " viewBox=\"0.00 0.00 94.00 218.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 214)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-214 93,-214 93,4 -4,4\"/>\n",
+ "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-214 90,-214 90,4 -4,4\"/>\n",
"<!-- 2 -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>2</title>\n",
@@ -8733,9 +8753,9 @@
"<!-- 2->3 -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>2->3</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M33.48,-86.8C37.9,-75.05 43.85,-59.24 48.89,-45.84\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"52.29,-46.77 52.53,-36.18 45.74,-44.3 52.29,-46.77\"/>\n",
- "<text text-anchor=\"middle\" x=\"53.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">r+</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M33.48,-86.8C37.78,-75.35 43.54,-60.06 48.5,-46.89\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"51.64,-48.48 51.89,-37.88 45.09,-46.01 51.64,-48.48\"/>\n",
+ "<text text-anchor=\"middle\" x=\"49.5\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">r⁺</text>\n",
"</g>\n",
"<!-- 1 -->\n",
"<g id=\"node3\" class=\"node\">\n",
@@ -8746,22 +8766,24 @@
"<!-- 1->2 -->\n",
"<g id=\"edge3\" class=\"edge\">\n",
"<title>1->2</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M52.52,-173.8C48.1,-162.05 42.15,-146.24 37.11,-132.84\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"40.26,-131.3 33.47,-123.18 33.71,-133.77 40.26,-131.3\"/>\n",
- "<text text-anchor=\"middle\" x=\"53.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">r+</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M52.52,-173.8C48.22,-162.35 42.46,-147.06 37.5,-133.89\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"40.91,-133.01 34.11,-124.88 34.36,-135.48 40.91,-133.01\"/>\n",
+ "<text text-anchor=\"middle\" x=\"49.5\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">r⁺</text>\n",
"</g>\n",
"<!-- 1->3 -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>1->3</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M63.77,-173.69C65.09,-168.09 66.34,-161.82 67,-156 72.08,-110.95 72.08,-99.05 67,-54 66.71,-51.45 66.31,-48.82 65.84,-46.2\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"69.25,-45.38 63.77,-36.31 62.39,-46.82 69.25,-45.38\"/>\n",
- "<text text-anchor=\"middle\" x=\"79.5\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">r+</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M60.3,-173.68C61.26,-159.95 62.49,-140.26 63,-123 63.47,-107.01 63.47,-102.99 63,-87 62.62,-74.06 61.83,-59.74 61.06,-47.6\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"64.56,-47.57 60.4,-37.83 57.58,-48.04 64.56,-47.57\"/>\n",
+ "<text text-anchor=\"middle\" x=\"68.5\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">r⁺</text>\n",
"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [Arsub1A={1,2,3} & r={(1,2),(2,3)} & sub1={({},{1}),({},{2}),({},{3}),({1},{1,2}),({1},{1,3}),({1,2},{1,2,3}),({1,3},{1,2,3}),({2},{1,2}),({2},{2,3}),({2,3},{1,2,3}),({3},{1,3}),({3},{2,3})}(\"r+\",closure1(r))]>"
+ "<Dot visualization: expr_as_graph [LET A,r,sub1 BE A={1,2,3} & r={(1↦2),(2↦3)} & sub1={(∅↦{1}),(∅↦{2}),(∅↦{3}),({1}↦{1,2}),({1}↦{1,3}),({1,2}↦{1,2,3}),({1,3}↦{1,2,3}),({2}↦{1,2}),({2}↦{2,3}),({2,3}↦{1,2,3}),({3}↦{1,3}),({3}↦{2,3})} IN(\n",
+ "(\"r⁺\",closure1(r))\n",
+ ")END]>"
]
},
"execution_count": 118,
@@ -8770,7 +8792,7 @@
}
],
"source": [
- ":dot expr_as_graph (\"r+\",closure1(r))"
+ ":dot expr_as_graph (\"r⁺\",closure1(r))"
]
},
{
@@ -8800,7 +8822,7 @@
}
],
"source": [
- "closure1(r)~[{3}]"
+ "closure1(r)⁻¹[{3}]"
]
},
{
@@ -8818,7 +8840,7 @@
{
"data": {
"text/markdown": [
- "$/*@symbolic*/ \\{\\mathit{z\\_},\\mathit{z\\_\\_}\\mid \\mathit{z\\_} \\mapsto \\mathit{z\\_\\_} \\in \\{(1\\mapsto 2),(1\\mapsto 3),(2\\mapsto 3)\\} \\lor \\mathit{z\\_} = \\mathit{z\\_\\_}\\}$"
+ "$/*@\\mathit{symbolic}*/ \\{\\mathit{z\\_},\\mathit{z\\_\\_}\\mid\\mathit{z\\_} \\mapsto \\mathit{z\\_\\_} \\in \\{(1\\mapsto 2),(1\\mapsto 3),(2\\mapsto 3)\\} \\lor \\mathit{z\\_} = \\mathit{z\\_\\_}\\}$"
],
"text/plain": [
"/*@symbolic*/ {z_,z__∣z_ ↦ z__ ∈ {(1↦2),(1↦3),(2↦3)} ∨ z_ = z__}"
@@ -8853,7 +8875,7 @@
}
],
"source": [
- "closure(r)~[{3}]"
+ "closure(r)⁻¹[{3}]"
]
},
{
@@ -8921,151 +8943,153 @@
"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ "<!-- Generated by graphviz version 7.1.0 (20230121.1956)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
- "<svg width=\"315pt\" height=\"305pt\"\n",
- " viewBox=\"0.00 0.00 315.17 305.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
+ "<svg width=\"259pt\" height=\"305pt\"\n",
+ " viewBox=\"0.00 0.00 259.38 305.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 301)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-301 311.17,-301 311.17,4 -4,4\"/>\n",
+ "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-301 255.38,-301 255.38,4 -4,4\"/>\n",
"<!-- \\{3\\} -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>\\{3\\}</title>\n",
- "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"87.17,-210 33.17,-210 33.17,-174 87.17,-174 87.17,-210\"/>\n",
- "<text text-anchor=\"middle\" x=\"60.17\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">{3}</text>\n",
+ "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"77.38,-210 23.38,-210 23.38,-174 77.38,-174 77.38,-210\"/>\n",
+ "<text text-anchor=\"middle\" x=\"50.38\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">{3}</text>\n",
"</g>\n",
"<!-- \\{2,3\\} -->\n",
"<g id=\"node2\" class=\"node\">\n",
"<title>\\{2,3\\}</title>\n",
- "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"175.67,-123 118.67,-123 118.67,-87 175.67,-87 175.67,-123\"/>\n",
- "<text text-anchor=\"middle\" x=\"147.17\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">{2,3}</text>\n",
+ "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"150.38,-123 96.38,-123 96.38,-87 150.38,-87 150.38,-123\"/>\n",
+ "<text text-anchor=\"middle\" x=\"123.38\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">{2,3}</text>\n",
"</g>\n",
"<!-- \\{3\\}->\\{2,3\\} -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>\\{3\\}->\\{2,3\\}</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M58.53,-173.62C58.47,-163.17 60.11,-150.16 67.17,-141 78.24,-126.64 88.33,-130 108.99,-123\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"110.49,-126.17 118.53,-119.28 107.94,-119.65 110.49,-126.17\"/>\n",
- "<text text-anchor=\"middle\" x=\"84.67\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M46.51,-173.97C45.14,-163.65 45.21,-150.65 51.38,-141 51.96,-140.09 68.8,-131.89 86.04,-123.65\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"87.23,-126.96 94.76,-119.5 84.22,-120.64 87.23,-126.96\"/>\n",
+ "<text text-anchor=\"middle\" x=\"64.38\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{1,3\\} -->\n",
"<g id=\"node3\" class=\"node\">\n",
"<title>\\{1,3\\}</title>\n",
- "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"100.67,-123 43.67,-123 43.67,-87 100.67,-87 100.67,-123\"/>\n",
- "<text text-anchor=\"middle\" x=\"72.17\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1,3}</text>\n",
+ "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"78.38,-123 24.38,-123 24.38,-87 78.38,-87 78.38,-123\"/>\n",
+ "<text text-anchor=\"middle\" x=\"51.38\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1,3}</text>\n",
"</g>\n",
"<!-- \\{3\\}->\\{1,3\\} -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>\\{3\\}->\\{1,3\\}</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M32.85,-180.68C13.62,-171.72 -7.02,-157.6 3.17,-141 10.1,-129.71 21.94,-121.92 33.86,-116.62\"/>\n",
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- "<text text-anchor=\"middle\" x=\"20.67\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M23.04,-177.53C8.5,-168.36 -4.99,-155.34 2.38,-141 5.42,-135.1 9.96,-129.98 15.07,-125.61\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"16.88,-128.63 22.83,-119.86 12.71,-123 16.88,-128.63\"/>\n",
+ "<text text-anchor=\"middle\" x=\"15.38\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{1,2,3\\} -->\n",
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"<title>\\{1,2,3\\}</title>\n",
- "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"182.67,-36 111.67,-36 111.67,0 182.67,0 182.67,-36\"/>\n",
- "<text text-anchor=\"middle\" x=\"147.17\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1,2,3}</text>\n",
+ "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"152.38,-36 94.38,-36 94.38,0 152.38,0 152.38,-36\"/>\n",
+ "<text text-anchor=\"middle\" x=\"123.38\" y=\"-14.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1,2,3}</text>\n",
"</g>\n",
"<!-- \\{2,3\\}->\\{1,2,3\\} -->\n",
"<g id=\"edge3\" class=\"edge\">\n",
"<title>\\{2,3\\}->\\{1,2,3\\}</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M147.17,-86.8C147.17,-75.16 147.17,-59.55 147.17,-46.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"150.67,-46.18 147.17,-36.18 143.67,-46.18 150.67,-46.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"164.67\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M123.38,-86.8C123.38,-75.58 123.38,-60.67 123.38,-47.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"126.88,-47.98 123.38,-37.98 119.88,-47.98 126.88,-47.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"136.38\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{1,3\\}->\\{1,2,3\\} -->\n",
"<g id=\"edge6\" class=\"edge\">\n",
"<title>\\{1,3\\}->\\{1,2,3\\}</title>\n",
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- "<text text-anchor=\"middle\" x=\"125.67\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M65.73,-86.84C73.82,-77.17 84.14,-64.89 93.38,-54 95.82,-51.13 98.37,-48.14 100.91,-45.16\"/>\n",
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+ "<text text-anchor=\"middle\" x=\"106.38\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{2\\} -->\n",
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"<title>\\{2\\}</title>\n",
- "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"261.17,-210 207.17,-210 207.17,-174 261.17,-174 261.17,-210\"/>\n",
- "<text text-anchor=\"middle\" x=\"234.17\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">{2}</text>\n",
+ "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"224.38,-210 170.38,-210 170.38,-174 224.38,-174 224.38,-210\"/>\n",
+ "<text text-anchor=\"middle\" x=\"197.38\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">{2}</text>\n",
"</g>\n",
"<!-- \\{2\\}->\\{2,3\\} -->\n",
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"<title>\\{2\\}->\\{2,3\\}</title>\n",
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- "<text text-anchor=\"middle\" x=\"239.67\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M192.55,-173.93C189.01,-163.6 183.41,-150.6 175.38,-141 171.05,-135.81 165.75,-131.08 160.22,-126.89\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"162.26,-124.05 152.05,-121.22 158.27,-129.8 162.26,-124.05\"/>\n",
+ "<text text-anchor=\"middle\" x=\"196.38\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{1,2\\} -->\n",
"<g id=\"node6\" class=\"node\">\n",
"<title>\\{1,2\\}</title>\n",
- "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"250.67,-123 193.67,-123 193.67,-87 250.67,-87 250.67,-123\"/>\n",
- "<text text-anchor=\"middle\" x=\"222.17\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1,2}</text>\n",
+ "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"222.38,-123 168.38,-123 168.38,-87 222.38,-87 222.38,-123\"/>\n",
+ "<text text-anchor=\"middle\" x=\"195.38\" y=\"-101.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1,2}</text>\n",
"</g>\n",
"<!-- \\{2\\}->\\{1,2\\} -->\n",
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- "<text text-anchor=\"middle\" x=\"289.67\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M215.19,-173.55C222.53,-164.21 228.47,-152.32 224.38,-141 223.34,-138.11 221.94,-135.28 220.32,-132.58\"/>\n",
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"<!-- \\{1,2\\}->\\{1,2,3\\} -->\n",
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+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M181.04,-86.84C172.95,-77.17 162.63,-64.89 153.38,-54 150.95,-51.13 148.4,-48.14 145.86,-45.16\"/>\n",
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"<!-- \\{1\\} -->\n",
"<g id=\"node7\" class=\"node\">\n",
"<title>\\{1\\}</title>\n",
- "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"178.17,-210 124.17,-210 124.17,-174 178.17,-174 178.17,-210\"/>\n",
- "<text text-anchor=\"middle\" x=\"151.17\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1}</text>\n",
+ "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"151.38,-210 97.38,-210 97.38,-174 151.38,-174 151.38,-210\"/>\n",
+ "<text text-anchor=\"middle\" x=\"124.38\" y=\"-188.3\" font-family=\"Times,serif\" font-size=\"14.00\">{1}</text>\n",
"</g>\n",
"<!-- \\{1\\}->\\{1,3\\} -->\n",
"<g id=\"edge8\" class=\"edge\">\n",
"<title>\\{1\\}->\\{1,3\\}</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M135.18,-173.8C123.62,-161.36 107.83,-144.36 94.94,-130.5\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"97.5,-128.12 88.13,-123.18 92.38,-132.88 97.5,-128.12\"/>\n",
- "<text text-anchor=\"middle\" x=\"133.67\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M109.61,-173.8C99.29,-161.78 85.33,-145.52 73.66,-131.94\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"76.54,-129.92 67.37,-124.62 71.23,-134.48 76.54,-129.92\"/>\n",
+ "<text text-anchor=\"middle\" x=\"104.38\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{1\\}->\\{1,2\\} -->\n",
"<g id=\"edge9\" class=\"edge\">\n",
"<title>\\{1\\}->\\{1,2\\}</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M157.95,-173.69C162.45,-163.51 169.03,-150.77 177.17,-141 180.49,-137.02 184.34,-133.19 188.35,-129.64\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"190.72,-132.22 196.2,-123.15 186.26,-126.83 190.72,-132.22\"/>\n",
- "<text text-anchor=\"middle\" x=\"194.67\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M130.74,-173.65C135.01,-163.46 141.33,-150.71 149.38,-141 152.41,-137.35 155.89,-133.85 159.54,-130.58\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"161.62,-133.4 167.11,-124.34 157.17,-128 161.62,-133.4\"/>\n",
+ "<text text-anchor=\"middle\" x=\"162.38\" y=\"-144.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\} -->\n",
"<g id=\"node8\" class=\"node\">\n",
"<title>\\{\\}</title>\n",
- "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"178.17,-297 124.17,-297 124.17,-261 178.17,-261 178.17,-297\"/>\n",
- "<text text-anchor=\"middle\" x=\"151.17\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">{}</text>\n",
+ "<polygon fill=\"#cae1ff\" stroke=\"#cae1ff\" points=\"151.38,-297 97.38,-297 97.38,-261 151.38,-261 151.38,-297\"/>\n",
+ "<text text-anchor=\"middle\" x=\"124.38\" y=\"-275.3\" font-family=\"Times,serif\" font-size=\"14.00\">{}</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{3\\} -->\n",
"<g id=\"edge10\" class=\"edge\">\n",
"<title>\\{\\}->\\{3\\}</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M131.61,-260.68C125.36,-255.09 118.44,-248.82 112.17,-243 103.32,-234.78 93.75,-225.65 85.27,-217.47\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"87.46,-214.72 77.83,-210.28 82.59,-219.75 87.46,-214.72\"/>\n",
- "<text text-anchor=\"middle\" x=\"129.67\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M109.41,-260.8C98.95,-248.78 84.79,-232.52 72.96,-218.94\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"75.79,-216.85 66.58,-211.61 70.51,-221.45 75.79,-216.85\"/>\n",
+ "<text text-anchor=\"middle\" x=\"104.38\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{2\\} -->\n",
"<g id=\"edge11\" class=\"edge\">\n",
"<title>\\{\\}->\\{2\\}</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M171.03,-260.99C177.3,-255.41 184.15,-249.08 190.17,-243 197.98,-235.11 206.12,-226.11 213.25,-217.94\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"216.11,-219.97 219.98,-210.11 210.8,-215.41 216.11,-219.97\"/>\n",
- "<text text-anchor=\"middle\" x=\"220.67\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M139.16,-260.8C149.48,-248.78 163.44,-232.52 175.11,-218.94\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"177.54,-221.48 181.4,-211.62 172.23,-216.92 177.54,-221.48\"/>\n",
+ "<text text-anchor=\"middle\" x=\"177.38\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"<!-- \\{\\}->\\{1\\} -->\n",
"<g id=\"edge12\" class=\"edge\">\n",
"<title>\\{\\}->\\{1\\}</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M151.17,-260.8C151.17,-249.16 151.17,-233.55 151.17,-220.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"154.67,-220.18 151.17,-210.18 147.67,-220.18 154.67,-220.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"168.67\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M124.38,-260.8C124.38,-249.58 124.38,-234.67 124.38,-221.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"127.88,-221.98 124.38,-211.98 120.88,-221.98 127.88,-221.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"137.38\" y=\"-231.8\" font-family=\"Times,serif\" font-size=\"14.00\">sub1</text>\n",
"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [Asub1A={1,2,3} & sub1={({},{1}),({},{2}),({},{3}),({1},{1,2}),({1},{1,3}),({1,2},{1,2,3}),({1,3},{1,2,3}),({2},{1,2}),({2},{2,3}),({2,3},{1,2,3}),({3},{1,3}),({3},{2,3})}(\"sub1\",sub1)]>"
+ "<Dot visualization: expr_as_graph [LET A,sub1 BE A={1,2,3} & sub1={(∅↦{1}),(∅↦{2}),(∅↦{3}),({1}↦{1,2}),({1}↦{1,3}),({1,2}↦{1,2,3}),({1,3}↦{1,2,3}),({2}↦{1,2}),({2}↦{2,3}),({2,3}↦{1,2,3}),({3}↦{1,3}),({3}↦{2,3})} IN(\n",
+ "(\"sub1\",sub1)\n",
+ ")END]>"
]
},
"execution_count": 124,
@@ -9205,18 +9229,10 @@
{
"data": {
"text/markdown": [
- "$\\mathit{TRUE}$\n",
- "\n",
- "**Solution:**\n",
- "* $\\mathit{A} = \\{1,2,3\\}$\n",
- "* $\\mathit{i10} = \\{(1\\mapsto 2),(2\\mapsto 3),(3\\mapsto 4),(4\\mapsto 5),(5\\mapsto 6),(6\\mapsto 7),(7\\mapsto 8),(8\\mapsto 9),(9\\mapsto 10),(10\\mapsto 11)\\}$"
+ "$\\mathit{TRUE}$"
],
"text/plain": [
- "TRUE\n",
- "\n",
- "Solution:\n",
- "\tA = {1,2,3}\n",
- "\ti10 = {(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
+ "TRUE"
]
},
"execution_count": 130,
@@ -9236,18 +9252,10 @@
{
"data": {
"text/markdown": [
- "$\\mathit{TRUE}$\n",
- "\n",
- "**Solution:**\n",
- "* $\\mathit{A} = \\{1,2,3\\}$\n",
- "* $\\mathit{i10} = \\{(1\\mapsto 2),(2\\mapsto 3),(3\\mapsto 4),(4\\mapsto 5),(5\\mapsto 6),(6\\mapsto 7),(7\\mapsto 8),(8\\mapsto 9),(9\\mapsto 10),(10\\mapsto 11)\\}$"
+ "$\\mathit{TRUE}$"
],
"text/plain": [
- "TRUE\n",
- "\n",
- "Solution:\n",
- "\tA = {1,2,3}\n",
- "\ti10 = {(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
+ "TRUE"
]
},
"execution_count": 131,
@@ -9274,18 +9282,10 @@
{
"data": {
"text/markdown": [
- "$\\mathit{TRUE}$\n",
- "\n",
- "**Solution:**\n",
- "* $\\mathit{A} = \\{1,2,3\\}$\n",
- "* $\\mathit{i10} = \\{(1\\mapsto 2),(2\\mapsto 3),(3\\mapsto 4),(4\\mapsto 5),(5\\mapsto 6),(6\\mapsto 7),(7\\mapsto 8),(8\\mapsto 9),(9\\mapsto 10),(10\\mapsto 11)\\}$"
+ "$\\mathit{TRUE}$"
],
"text/plain": [
- "TRUE\n",
- "\n",
- "Solution:\n",
- "\tA = {1,2,3}\n",
- "\ti10 = {(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
+ "TRUE"
]
},
"execution_count": 132,
@@ -9391,16 +9391,12 @@
"$\\mathit{TRUE}$\n",
"\n",
"**Solution:**\n",
- "* $\\mathit{A} = \\{1,2,3\\}$\n",
- "* $\\mathit{i10} = \\{(1\\mapsto 2),(2\\mapsto 3),(3\\mapsto 4),(4\\mapsto 5),(5\\mapsto 6),(6\\mapsto 7),(7\\mapsto 8),(8\\mapsto 9),(9\\mapsto 10),(10\\mapsto 11)\\}$\n",
"* $\\mathit{z} = 5$"
],
"text/plain": [
"TRUE\n",
"\n",
"Solution:\n",
- "\tA = {1,2,3}\n",
- "\ti10 = {(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}\n",
"\tz = 5"
]
},
@@ -9525,7 +9521,7 @@
}
],
"source": [
- "i10~[{4}]"
+ "i10⁻¹[{4}]"
]
},
{
@@ -9758,12 +9754,12 @@
"metadata": {},
"source": [
"Es gibt auch noch andere Arten an Funktionen:\n",
- "* partielle Funktionen, +->\n",
- "* injektive (umkehrbare) Funktionen >->\n",
- "* surjektive Funktionen die den Wertebereich komplett abdecken -->>\n",
- "* injektive und surjektive Funktionen >->>, auch Bijektionen gennant\n",
+ "* partielle Funktionen, `+->`\n",
+ "* injektive (umkehrbare) Funktionen `>->`\n",
+ "* surjektive Funktionen die den Wertebereich komplett abdecken `-->>`\n",
+ "* injektive und surjektive Funktionen `>->>`, auch Bijektionen gennant\n",
"\n",
- "(Die Pfeilsymbole sind sehr spezifisch für die B Sprache. Wir werden diese nicht im Skript verwenden.)"
+ "(Die Pfeilsymbole sind sehr spezifisch für die B-Sprache. Wir werden diese nicht im Skript verwenden.)"
]
},
{
@@ -9797,18 +9793,10 @@
{
"data": {
"text/markdown": [
- "$\\mathit{TRUE}$\n",
- "\n",
- "**Solution:**\n",
- "* $\\mathit{A} = \\{1,2,3\\}$\n",
- "* $\\mathit{i10} = \\{(1\\mapsto 2),(2\\mapsto 3),(3\\mapsto 4),(4\\mapsto 5),(5\\mapsto 6),(6\\mapsto 7),(7\\mapsto 8),(8\\mapsto 9),(9\\mapsto 10),(10\\mapsto 11)\\}$"
+ "$\\mathit{TRUE}$"
],
"text/plain": [
- "TRUE\n",
- "\n",
- "Solution:\n",
- "\tA = {1,2,3}\n",
- "\ti10 = {(1↦2),(2↦3),(3↦4),(4↦5),(5↦6),(6↦7),(7↦8),(8↦9),(9↦10),(10↦11)}"
+ "TRUE"
]
},
"execution_count": 152,
@@ -10041,14 +10029,14 @@
"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n",
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
" \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
- "<!-- Generated by graphviz version 2.44.1 (0)\n",
+ "<!-- Generated by graphviz version 7.1.0 (20230121.1956)\n",
" -->\n",
"<!-- Title: state Pages: 1 -->\n",
"<svg width=\"206pt\" height=\"131pt\"\n",
" viewBox=\"0.00 0.00 206.00 131.00\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
"<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 127)\">\n",
"<title>state</title>\n",
- "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-127 202,-127 202,4 -4,4\"/>\n",
+ "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-127 202,-127 202,4 -4,4\"/>\n",
"<!-- 3 -->\n",
"<g id=\"node1\" class=\"node\">\n",
"<title>3</title>\n",
@@ -10064,9 +10052,9 @@
"<!-- 3->33 -->\n",
"<g id=\"edge1\" class=\"edge\">\n",
"<title>3->33</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-86.8C27,-75.16 27,-59.55 27,-46.24\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-46.18 27,-36.18 23.5,-46.18 30.5,-46.18\"/>\n",
- "<text text-anchor=\"middle\" x=\"33\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">G</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M27,-86.8C27,-75.58 27,-60.67 27,-47.69\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"30.5,-47.98 27,-37.98 23.5,-47.98 30.5,-47.98\"/>\n",
+ "<text text-anchor=\"middle\" x=\"32\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">G</text>\n",
"</g>\n",
"<!-- 2 -->\n",
"<g id=\"node3\" class=\"node\">\n",
@@ -10083,9 +10071,9 @@
"<!-- 2->22 -->\n",
"<g id=\"edge2\" class=\"edge\">\n",
"<title>2->22</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M106.29,-86.8C111.31,-74.93 118.09,-58.93 123.8,-45.45\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"127.05,-46.75 127.73,-36.18 120.6,-44.02 127.05,-46.75\"/>\n",
- "<text text-anchor=\"middle\" x=\"125\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">G</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M106.29,-86.8C111.13,-75.35 117.61,-60.06 123.19,-46.89\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"126.33,-48.43 127.01,-37.86 119.89,-45.7 126.33,-48.43\"/>\n",
+ "<text text-anchor=\"middle\" x=\"124\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">G</text>\n",
"</g>\n",
"<!-- 1 -->\n",
"<g id=\"node5\" class=\"node\">\n",
@@ -10096,15 +10084,17 @@
"<!-- 1->22 -->\n",
"<g id=\"edge3\" class=\"edge\">\n",
"<title>1->22</title>\n",
- "<path fill=\"none\" stroke=\"firebrick\" d=\"M163.71,-86.8C158.69,-74.93 151.91,-58.93 146.2,-45.45\"/>\n",
- "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"149.4,-44.02 142.27,-36.18 142.95,-46.75 149.4,-44.02\"/>\n",
- "<text text-anchor=\"middle\" x=\"161\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">G</text>\n",
+ "<path fill=\"none\" stroke=\"firebrick\" d=\"M163.71,-86.8C158.87,-75.35 152.39,-60.06 146.81,-46.89\"/>\n",
+ "<polygon fill=\"firebrick\" stroke=\"firebrick\" points=\"150.11,-45.7 142.99,-37.86 143.67,-48.43 150.11,-45.7\"/>\n",
+ "<text text-anchor=\"middle\" x=\"160\" y=\"-57.8\" font-family=\"Times,serif\" font-size=\"14.00\">G</text>\n",
"</g>\n",
"</g>\n",
- "</svg>"
+ "</svg>\n"
],
"text/plain": [
- "<Dot visualization: expr_as_graph [AGA={1,2,3} & G={(1,22),(2,22),(3,33)}(\"G\",G)]>"
+ "<Dot visualization: expr_as_graph [LET A,G BE A={1,2,3} & G={(1↦22),(2↦22),(3↦33)} IN(\n",
+ "(\"G\",G)\n",
+ ")END]>"
]
},
"execution_count": 160,
@@ -10123,8 +10113,8 @@
"Es gilt also\n",
"* $G = [22,22,33] = \\{(1\\mapsto 22),(2\\mapsto 22),(3\\mapsto 33)\\}$\n",
" \n",
- "Wir schreiben $\\mathit{G} \\in \\mathit{seq}$(ℤ)\n",
- " oder aber auch $G \\in ℤ^*$ (mehr dazu später im Skript)\n",
+ "Wir schreiben $\\mathit{G} \\in \\mathrm{seq}(\\mathbb{Z})$\n",
+ " oder aber auch $G \\in \\mathbb{Z}^*$ (mehr dazu später im Skript)\n",
" \n",
"Das n-te Element einer Folge $G$ ist einfach $G(n)$:"
]
@@ -10278,7 +10268,7 @@
}
],
"source": [
- "G~[{22}]"
+ "G⁻¹[{22}]"
]
},
{
@@ -10295,14 +10285,14 @@
"metadata": {},
"source": [
"# Zusammenfassung Mengentheorie\n",
- " \n",
+ "\n",
"* Mengen, Notationen (per Prädikat)\n",
- "* Potenzmenge, Menge von Mengen, $\\phi \\neq \\{ \\phi\\}$\n",
+ "* Potenzmenge, Menge von Mengen, $\\phi \\neq \\{\\phi\\}$\n",
"* kartesisches Produkt, Relationen als Menge von Paaren/Tupeln\n",
"* Definitionsbereich, Wertebereich, Abbild, Umkehrrelation\n",
"* Transitive Hülle\n",
"* Funktionen\n",
- "* Endliche Folgen als Funktion von $ℤ$ nach Wertebereich"
+ "* Endliche Folgen als Funktion von $\\mathbb{Z}$ nach Wertebereich"
]
},
{
@@ -10310,19 +10300,13 @@
"metadata": {},
"source": [
"# Lernziele von Kapitel 0\n",
+ "\n",
"* logische Formeln verstehen und schreiben können\n",
"* logische Beweise verstehen: Wahrheitstabelle, Widerspruch, Deduktiver Beweis, Äuivalenzbeweis\n",
- "* Mengenausdrücke verstehen und nach Logik übbersetzen können\n",
+ "* Mengenausdrücke verstehen und nach Logik übersetzen können\n",
"* Problemstellungen nach Logik und Mengentheorie übersetzen können\n",
"* Relationen, Funktionen und Folgen in Mengendarstellung bearbeiten können"
]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
}
],
"metadata": {