afleidingsrij 2! toegevoegd

diff --git a/an_faculteit.tex b/an_faculteit.tex
index a8a92c7..364a5f9 100644 (file)
--- a/an_faculteit.tex
+++ b/an_faculteit.tex
@@ -130,7 +130,7 @@ $
 \langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll, [\:], [\:], [1,3,1,1,1]\rangle \Rightarrow\\
 \langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1, [\:], [\:], [1,1,1]\rangle \Rightarrow\\
 \langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;, [\:], [\:], [1,1,1,1]\rangle \Rightarrow\\
-\langle pointer; not;greater;subtract;push\:1;push \: 1;not, [\:], [\:], [1,1,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not, [\:], [\:], [0,1,1]\rangle \Rightarrow\\
 \langle pointer; not;greater;subtract;push\:1;push \: 1, [\:], [\:], [1,1,1]\rangle \Rightarrow\\
 \langle pointer; not;greater;subtract;push\:1, [\:], [\:], [1,1,1,1]\rangle \Rightarrow\\
 \langle pointer; not;greater;subtract, [\:], [\:], [1,1,1,1,1]\rangle \Rightarrow\\
@@ -149,3 +149,92 @@ achter elkaar uitgevoerd op de stack, waarbij het resultaat [1,1] is.\\
 Bij de voorbeelden staan ook nog de programma's tweefaculteit en 
 vijffaculteit die respectievelijk 2! en 5! uitrekenen.
 \end{landscape}
+
+\newpage
+
+\begin{landscape}
+De afleidingsrij voor 2! ziet er als volgt uit:\\
+$
+\langle R_1; R_2; R_3; greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1;push \: 2;push \: 1;push \: 2, [\:], [\:], [\:]\rangle \Rightarrow
+\langle R_1; R_2; R_3; greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1;push \: 2;push \: 1, [\:], [\:], [2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1;push \: 2, [\:], [\:], [1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1, [\:], [\:], [2,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll, [\:], [\:], [1,2,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup, [\:], [\:], [2,1]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3, [\:], [\:], [2,2,1]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1, [\:], [\:], [3,2,2,1]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll, [\:], [\:], [1,3,2,2,1]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1, [\:], [\:], [2,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not;subtract;, [\:], [\:], [1,2,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1;not, [\:], [\:], [1,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1;push \: 1, [\:], [\:], [0,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract;push\:1, [\:], [\:], [1,0,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater;subtract, [\:], [\:], [1,1,0,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3;greater, [\:], [\:], [0,0,1,2]\rangle \Rightarrow\\
+\langle R_1; R_2; R_3, [\:], [\:], [1,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup; roll; push \: 2; push \: 3; dup; pointer; not, [\:], [\:], [1,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup; roll; push \: 2; push \: 3; dup; pointer, [\:], [\:], [0,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup; roll; push \: 2; push \: 3; dup, [\:], [\:], [1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup; roll; push \: 2; push \: 3, [\:], [\:], [1,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup; roll; push \: 2, [\:], [\:], [3,1,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup; roll, [\:], [\:], [2,3,1,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup, [\:], [\:], [2,1,1]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4, [\:], [\:], [2,2,1,1]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll; push \: 1, [\:], [\:], [4,2,2,1,1]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul; roll, [\:], [\:], [1,4,2,2,1,1]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2; mul, [\:], [\:], [2,1,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1; push \: 2, [\:], [\:], [2,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll; push \: 1, [\:], [\:], [2,2,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop; roll, [\:], [\:], [1,2,2,1,2]\rangle \Rightarrow\\
+\langle R_1;R_2; pop, [\:], [\:], [1,2,2]\rangle \Rightarrow\\
+\langle R_1;R_2, [\:], [\:], [2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll; push \: 1; push \: 3; dup; roll; push \: 1; push \: 2, [\:], [\:], [2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll; push \: 1; push \: 3; dup; roll; push \: 1, [\:], [\:], [2,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll; push \: 1; push \: 3; dup; roll, [\:], [\:], [1,2,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll; push \: 1; push \: 3; dup, [\:], [\:], [2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll; push \: 1; push \: 3, [\:], [\:], [2,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll; push \: 1, [\:], [\:], [3,2,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll, [\:], [\:], [1,3,2,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1, [\:], [\:], [2,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub, [\:], [\:], [1,2,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3, [\:], [\:], [1,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll; push \: 2, [\:], [\:], [3,1,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop; roll, [\:], [\:], [2,3,1,2,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2; pop, [\:], [\:], [2,1,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1; push \: 2, [\:], [\:], [1,2]\rangle \Rightarrow\\
+\langle R_1; roll; push \: 1, [\:], [\:], [2,1,2]\rangle \Rightarrow\\
+\langle R_1; roll, [\:], [\:], [1,2,1,2]\rangle \Rightarrow\\
+\langle R_1, [\:], [\:], [2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1;push \: 2, [\:], [\:], [2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1, [\:], [\:], [2,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll, [\:], [\:], [1,2,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup, [\:], [\:], [1,2]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3, [\:], [\:], [1,1,2]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1, [\:], [\:], [3,1,1,2]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll, [\:], [\:], [1,3,1,1,2]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1, [\:], [\:], [1,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;, [\:], [\:], [1,1,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1;not, [\:], [\:], [0,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1;push \: 1, [\:], [\:], [1,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract;push\:1, [\:], [\:], [1,1,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater;subtract, [\:], [\:], [1,1,1,2,1]\rangle \Rightarrow\\
+\langle pointer; not;greater, [\:], [\:], [0,1,2,1]\rangle \Rightarrow\\
+\langle pointer; not, [\:], [\:], [1,2,1]\rangle \Rightarrow\\
+\langle pointer [\:], [\:], [0,2,1]\rangle \Rightarrow\\
+\langle [\:], [\:], [2,1]\rangle\\
+$
+
+In deze afleidingsrij zijn de volgende afkortingen gebruikt:\\
+$
+R_1: pointer; not;greater;subtract;push\:1;push \: 1;not;subtract;push \: 1;roll;push \: 1;push \: 3;dup;roll;push \: 1;push \: 2\\
+R_2: roll; push \: 1; push \: 2; pop; roll; push \: 2; push \: 3; sub; push \: 1; roll; push \: 1; push \: 3; dup; roll; push \: 1; push \: 2\\
+R_3: pop; roll; push \: 1; push \: 2; mul; roll; push \: 1; push \: 4; dup; roll; push \: 2; push \: 3; dup; pointer; not\\
+$
+
+Eerst worden x en y op de stack gestopt zodat de stack [y,x] = [1,2]. 
+De stack in de termineerende configuratie is [y',x'] = [2,1] en 2 = 2! dus x! = y' .
+Als bewijs dat de wiskundige operaties op de stack goed zijn uitgevoerd staat in 
+de appendix onder het kopje voorbeelden een stukje code.  In het programma 
+tweefaculteit worden alle wiskundige operaties die bij de commando's horen 
+achter elkaar uitgevoerd op de stack, waarbij het resultaat [2,1] is.\\
+\end{landscape}
\ No newline at end of file