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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="aofa_000.html#NM1" title="AOFA_000:NM.1">preIfWhileAlgebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">T</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">f</font> being    <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">T</font><br/>  for <font color="Olive" title="b4">I</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b4">I</font> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Olive" title="b3">f</font> holds <br/> for <font color="Olive" title="b5">P</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Olive" title="b4">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b5">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">f</font> being    <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">T</font><br/>  for <font color="Olive" title="b3">I</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b3">I</font> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Olive" title="b2">f</font> holds <br/> for <font color="Olive" title="b4">P</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Olive" title="b3">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b4">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">T</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being    <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font><br/>  for <font color="Olive" title="b2">I</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">I</font> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Olive" title="b1">f</font> holds <br/> for <font color="Olive" title="b3">P</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Olive" title="b2">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be    <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">I</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">I</font> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font> holds <br/> for <font color="Olive" title="b2">P</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Olive" title="b1">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b2">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">I</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">I</font> <a href="aofa_000.html#R5" title="AOFA_000:pred.5">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font> implies  for <font color="Olive" title="b1">P</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon" title="c5">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">P</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:176"/><span class="lab"><font color="Green" title="E108">A1</font></span>: 
 for <font color="Olive" title="b1">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> holds  <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Maroon" title="c5">I</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">f</font>)
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="aofa_000.html#D37" target="_self" title="AOFA_000:def.37">AOFA_000:def 37</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon" title="c5">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">P</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">P</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="aofa_000.html#D38" target="_self" title="AOFA_000:def.38">AOFA_000:def 38</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c5">I</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">f</font>) )</span><br/>



<span class="kw">thus </span><a NAME="E2:176"/>
( <font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c5">I</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">f</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:176"><span class="lab"><font color="Green" title="E108">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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