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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="aofa_i00.html#V2" title="AOFA_I00:attr.2">Euclidean</a> <a href="aofa_000.html#NM1" title="AOFA_000:NM.1">preIfWhileAlgebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">X</font><br/> for <font color="Olive" title="b3">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Olive" title="b1">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b2">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b4">n</font>, <font color="Olive" title="b5">x</font>, <font color="Olive" title="b6">y</font>, <font color="Olive" title="b7">z</font>, <font color="Olive" title="b8">i</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Olive" title="b3">g</font>  st  ex <font color="Olive" title="b9">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Olive" title="b9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <font color="Olive" title="b9">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 ) holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K17" title="AOFA_000:func.17">for-do</a> <span class="p2">(<span class="default"><font color="Olive" title="b8">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b8">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b4">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b8">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b7">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b5">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b5">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b6">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b6">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Olive" title="b7">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Olive" title="b3">g</font></span><br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b2">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Olive" title="b1">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">y</font>, <font color="Olive" title="b6">z</font>, <font color="Olive" title="b7">i</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Olive" title="b2">g</font>  st  ex <font color="Olive" title="b8">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b8">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b8">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b8">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b8">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Olive" title="b8">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <font color="Olive" title="b8">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 ) holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K17" title="AOFA_000:func.17">for-do</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b6">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b4">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b5">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b5">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Olive" title="b6">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Olive" title="b2">g</font></span><br/><b>let </b><font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <br/> for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font>, <font color="Olive" title="b5">z</font>, <font color="Olive" title="b6">i</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Olive" title="b1">g</font>  st  ex <font color="Olive" title="b7">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Olive" title="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <font color="Olive" title="b7">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 ) holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K17" title="AOFA_000:func.17">for-do</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b6">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b6">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b5">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Olive" title="b5">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Olive" title="b1">g</font></span><br/><b>let </b><font color="Maroon" title="c4">g</font> be  <a href="aofa_i00.html#V1" title="AOFA_I00:attr.1">Euclidean</a>  <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font>, <font color="Olive" title="b4">z</font>, <font color="Olive" title="b5">i</font> being   <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Maroon" title="c4">g</font>  st  ex <font color="Olive" title="b6">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Olive" title="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <font color="Olive" title="b6">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 ) holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K17" title="AOFA_000:func.17">for-do</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b4">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Olive" title="b3">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Olive" title="b4">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Maroon" title="c4">g</font></span><br/>







<b>set </b><font color="Maroon" title="c5">S</font> =  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">T</font> = <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>let </b><font color="Maroon" title="c7">n</font>, <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font>, <font color="Maroon" title="c10">z</font>, <font color="Maroon" title="c11">i</font> be    <a href="aofa_i00.html#M1" title="AOFA_I00:mode.1">Variable</a> of <font color="Maroon" title="c4">g</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">d</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <font color="Olive" title="b1">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 ) implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K17" title="AOFA_000:func.17">for-do</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Maroon" title="c4">g</font> )</span><br/>

<b>given </b><font color="Maroon" title="c12">d</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><a NAME="E2:129"/><i><font color="Green" title="E67">00</font></i>: 
( <font color="Maroon" title="c12">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c12">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c12">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c12">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c12">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 &amp; <font color="Maroon" title="c12">d</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><a href="aofa_000.html#K17" title="AOFA_000:func.17">for-do</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Maroon" title="c4">g</font></span><br/>

<a NAME="E3:129"/><i><font color="Green" title="E68">D1</font></i>: 
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E2:129"><i><font color="Green" title="E67">00</font></i></a></i>;<br/>
<a NAME="E4:129"/><i><font color="Green" title="E69">D2</font></i>: 
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">z</font> &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">y</font> &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">z</font> )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E2:129"><i><font color="Green" title="E67">00</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c13">s</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D37" title="AOFA_000:def.37">AOFA_000:def 37</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><a href="aofa_000.html#K17" title="AOFA_000:func.17">for-do</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K21" title="AOFA_000:func.21">TerminatingPrograms</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c4">g</font></span><br/>

<b>set </b><font color="Maroon" title="c14">s2</font> = <font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c13">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c15">I1</font> = <font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font>;<br/>
<b>set </b><font color="Maroon" title="c16">I2</font> = <font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font>;<br/>
<b>set </b><font color="Maroon" title="c17">I3</font> = <font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font>;<br/>
<b>set </b><font color="Maroon" title="c18">I</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span>;<br/>
<a NAME="E5:129"/><i><font color="Green" title="E70">02</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K21" title="AOFA_000:func.21">TerminatingPrograms</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="ref" href="aofa_000.html#D36" title="AOFA_000:def.36">AOFA_000:def 36</a></i>;<br/>
<a NAME="E6:129"/><i><font color="Green" title="E71">03</font></i>: 
for <font color="Olive" title="b1">q</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Olive" title="b1">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">q</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> )</span><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="txt" href="aofa_i00.html#E4:129"><i><font color="Green" title="E69">D2</font></i></a>, <a class="ref" href="aofa_i00.html#T30" target="_self" title="AOFA_I00:th.30">Th212</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="txt" href="aofa_i00.html#E4:129"><i><font color="Green" title="E69">D2</font></i></a>, <a class="ref" href="aofa_i00.html#T27" target="_self" title="AOFA_I00:th.27">Th211</a></i>
<br/>.= 
<font color="Maroon" title="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="txt" href="aofa_i00.html#E4:129"><i><font color="Green" title="E69">D2</font></i></a>, <a class="ref" href="aofa_i00.html#T27" target="_self" title="AOFA_I00:th.27">Th211</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font></span><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font>
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font>
<b>by </b><i><a class="txt" href="aofa_i00.html#E4:129"><i><font color="Green" title="E69">D2</font></i></a>, <a class="ref" href="aofa_i00.html#T30" target="_self" title="AOFA_I00:th.30">Th212</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font>
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c19">q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font>
<b>by </b><i><a class="txt" href="aofa_i00.html#E4:129"><i><font color="Green" title="E69">D2</font></i></a>, <a class="ref" href="aofa_i00.html#T27" target="_self" title="AOFA_I00:th.27">Th211</a></i>
<br/>.= 
<font color="Maroon" title="c19">q</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">i</font>
<b>by </b><i><a class="txt" href="aofa_i00.html#E4:129"><i><font color="Green" title="E69">D2</font></i></a>, <a class="ref" href="aofa_i00.html#T27" target="_self" title="AOFA_I00:th.27">Th211</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:129"/>
 ex <font color="Olive" title="b1">d'</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">d'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">d'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">d'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E3:129"><i><font color="Green" title="E68">D1</font></i></a>, <a class="ref" href="aofa_i00.html#T1" target="_self" title="AOFA_I00:th.1">Lem3</a></i>;<br/>
<a NAME="E8:129"/><b>then </b>
 <a href="aofa_000.html#K17" title="AOFA_000:func.17">for-do</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="ref" href="aofa_000.html#T104" title="AOFA_000:th.104">AOFA_000:104</a>, <a class="txt" href="aofa_i00.html#E6:129"><i><font color="Green" title="E71">03</font></i></a>, <a class="ref" href="aofa_i00.html#T54" target="_self" title="AOFA_I00:th.54">ThT4</a></i>;<br/>
<a NAME="E9:129"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="aofa_i00.html#K89" title="AOFA_I00:func.89">.</a> <font color="Maroon" title="c13">s</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="aofa_000.html#K17" title="AOFA_000:func.17">for-do</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K21" title="AOFA_000:func.21">TerminatingPrograms</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="ref" href="aofa_000.html#D37" title="AOFA_000:def.37">AOFA_000:def 37</a></i>;<br/>
<b>hence </b><a NAME="E10:129"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><a href="aofa_000.html#K17" title="AOFA_000:func.17">for-do</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K51" title="AOFA_I00:func.51">:=</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon" title="c7">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="aofa_i00.html#K48" title="AOFA_I00:func.48">:=</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="aofa_i00.html#K81" title="AOFA_I00:func.81">+=</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K21" title="AOFA_000:func.21">TerminatingPrograms</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c2">X</font>,<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon" title="c3">b</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="aofa_i00.html#E5:129"><i><font color="Green" title="E70">02</font></i></a>, <a class="ref" href="aofa_000.html#D35" title="AOFA_000:def.35">AOFA_000:def 35</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
