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<BR><B>Left Termination</B> of the query pattern
log2_in_2(a, g)
w.r.t. the given <I>Prolog program</I> could not be shown:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>log2(X, Y)&#160;:-&#160;log2(X, 0, s(0), Y).<BR>log2(s(s(X)), Half, Acc, Y)&#160;:-&#160;log2(X, s(Half), Acc, Y).<BR>log2(X, s(s(Half)), Acc, Y)&#160;:-&#160;','(small(X), log2(Half, s(0), s(Acc), Y)).<BR>log2(X, Half, Y, Y)&#160;:-&#160;','(small(X), small(Half)).<BR>small(0).<BR>small(s(0)).<BR><BR>Queries:<BR><BR>log2(a,g).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>log2_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>log2_in</font>: (f,b,b,b) (b,b,b,b)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_a</font><BR>
<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_g</font><BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_gggg</font><BR>
<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_a</font><BR>
<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_g</font><BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_gggg</font><BR>
<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR><BR>Using Dependency Pairs [1,30] we result in the following initial DP problem:<BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_A</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>Half</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_A</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>Half</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_a</font><BR>
<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_g</font><BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_gggg</font><BR>
<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGGG</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>SMALL_IN_A</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>SMALL_IN_A</font><BR>
<FONT COLOR=#0000cc>U2_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AGGG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_A</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>Half</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_A</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>Half</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_a</font><BR>
<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_g</font><BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_gggg</font><BR>
<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGGG</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>SMALL_IN_A</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>SMALL_IN_A</font><BR>
<FONT COLOR=#0000cc>U2_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AGGG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 2 SCCs with 18 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_a</font><BR>
<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_g</font><BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_gggg</font><BR>
<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_g</font><BR>
<FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 RuleRemovalProof</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
<BR>Strictly oriented dependency pairs:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>Strictly oriented rules of the TRS R:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font></BLOCKQUOTE><BR>Used ordering: POLO with Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>0</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB> + x<SUB>3</SUB> + x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_GGGG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB> + x<SUB>3</SUB> + x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>small_in_g</font>(x<SUB>1</SUB>)</B>) = 2 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>small_out_g</font></B>) = 2<sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 RuleRemovalProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 DependencyGraphProof</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_a</font><BR>
<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_g</font><BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_gggg</font><BR>
<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 Instantiation</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Instantiation</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 Instantiation</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Instantiation</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Instantiation</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 NonTerminationProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>We used the non-termination processor [17] to show that the DP problem is infinite.<BR>Found a loop by semiunifying a rule from P directly.<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:none<BR><BR><BR>s = <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) evaluates to  t =<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>)<BR><BR>Thus s starts an infinite chain as s semiunifies with t with the following substitutions:<BR><UL><LI> Matcher: [<FONT COLOR=#cc0000>z0</font> / <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)]</LI>
<LI> Semiunifier: [ ]</LI></UL><BR><BR><hr><BR><B>Rewriting sequence</B><BR><BR>The DP semiunifies directly so there is only one rewrite step from LOG2_IN_AGGG(s(s(z0)), z1, z2) to LOG2_IN_AGGG(s(s(s(z0))), z1, z2).<BR><BR><BR><BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>log2_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>log2_in</font>: (f,b,b,b) (b,b,b,b)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_a</font><BR>
<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_a</font><BR>
<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR><BR>Using Dependency Pairs [1,30] we result in the following initial DP problem:<BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_A</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>Half</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_A</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>Half</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_a</font><BR>
<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>SMALL_IN_A</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>SMALL_IN_A</font><BR>
<FONT COLOR=#0000cc>U2_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_A</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>Half</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>SMALL_IN_A</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>Half</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_a</font><BR>
<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>SMALL_IN_A</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>SMALL_IN_A</font><BR>
<FONT COLOR=#0000cc>U2_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>SMALL_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 2 SCCs with 18 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_a</font><BR>
<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))</BLOCKQUOTE><BR>Pi is empty.<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 RuleRemovalProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
<BR>Strictly oriented dependency pairs:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR><BR>Used ordering: POLO with Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>0</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 2&middot;x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB> + x<SUB>3</SUB> + x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_GGGG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 1 + x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB> + x<SUB>3</SUB> + x<SUB>4</SUB> + x<SUB>5</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>small_in_g</font>(x<SUB>1</SUB>)</B>) = 2 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>small_out_g</font>(x<SUB>1</SUB>)</B>) = 2 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 RuleRemovalProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 DependencyGraphProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LOG2_IN_GGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Acc</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>)), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>Half</font>))
<BR><FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>Half</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_a</font><BR>
<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_a</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_in_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>small_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_gggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_aggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>log2_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 Instantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#cc0000>Half</font>, <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Half</font>), <FONT COLOR=#cc0000>Acc</font>, <FONT COLOR=#cc0000>Y</font>) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Instantiation</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 Instantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Instantiation</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Instantiation</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 NonTerminationProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>We used the non-termination processor [17] to show that the DP problem is infinite.<BR>Found a loop by semiunifying a rule from P directly.<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) &#8594; <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:none<BR><BR><BR>s = <FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>) evaluates to  t =<FONT COLOR=#0000cc>LOG2_IN_AGGG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))), <FONT COLOR=#cc0000>z1</font>, <FONT COLOR=#cc0000>z2</font>)<BR><BR>Thus s starts an infinite chain as s semiunifies with t with the following substitutions:<BR><UL><LI> Matcher: [<FONT COLOR=#cc0000>z0</font> / <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)]</LI>
<LI> Semiunifier: [ ]</LI></UL><BR><BR><hr><BR><B>Rewriting sequence</B><BR><BR>The DP semiunifies directly so there is only one rewrite step from LOG2_IN_AGGG(s(s(z0)), z1, z2) to LOG2_IN_AGGG(s(s(s(z0))), z1, z2).<BR><BR><BR><BR><BR><BR></body>

