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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b4">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="b5">P</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b6">C</font>, <font color="Olive" title="b7">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="b6">C</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Olive" title="b4">f</font>,<font color="Olive" title="b5">P</font> &amp; <font color="Olive" title="b7">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Olive" title="b4">f</font>,<font color="Olive" title="b5">P</font> &amp; <font color="Olive" title="b5">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Olive" title="b6">C</font>,<font color="Olive" title="b4">f</font> &amp; <font color="Olive" title="b5">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Olive" title="b7">I</font>,<font color="Olive" title="b4">f</font> &amp; <font color="Olive" title="b4">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b7">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b6">C</font>,<font color="Olive" title="b4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b3">s</font>,<font color="Olive" title="b6">C</font> &amp; <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">P</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b6">C</font>,<font color="Olive" title="b7">I</font></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>,<font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">T</font>,<font color="Olive" title="b4">f</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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="Maroon" title="c2">S</font>,<font color="Olive" title="b1">T</font><br/> for <font color="Olive" title="b4">P</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b5">C</font>, <font color="Olive" title="b6">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="b5">C</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="b4">P</font> &amp; <font color="Olive" title="b6">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="b4">P</font> &amp; <font color="Olive" title="b4">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Olive" title="b5">C</font>,<font color="Olive" title="b3">f</font> &amp; <font color="Olive" title="b4">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Olive" title="b6">I</font>,<font color="Olive" title="b3">f</font> &amp; <font color="Olive" title="b3">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b6">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b5">C</font>,<font color="Olive" title="b3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">s</font>,<font color="Olive" title="b5">C</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">P</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b5">C</font>,<font color="Olive" title="b6">I</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">T</font>,<font color="Olive" title="b3">f</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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><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="Maroon" title="c3">T</font><br/> for <font color="Olive" title="b3">P</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">C</font>, <font color="Olive" title="b5">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">C</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="b3">P</font> &amp; <font color="Olive" title="b5">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="b3">P</font> &amp; <font color="Olive" title="b3">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Olive" title="b4">C</font>,<font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b3">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Olive" title="b5">I</font>,<font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b5">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b4">C</font>,<font color="Olive" title="b2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<font color="Olive" title="b4">C</font> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">P</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b4">C</font>,<font color="Olive" title="b5">I</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Olive" title="b2">f</font></span><br/><b>let </b><font color="Maroon" title="c4">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</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">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">P</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">C</font>, <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="b3">C</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="b2">P</font> &amp; <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="b1">f</font>,<font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b2">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Olive" title="b3">C</font>,<font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b2">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Olive" title="b4">I</font>,<font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b4">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b3">C</font>,<font color="Olive" title="b1">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">s</font>,<font color="Olive" title="b3">C</font> &amp; <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">P</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b3">C</font>,<font color="Olive" title="b4">I</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c5">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">C</font>, <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="b2">C</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c5">f</font>,<font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b3">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c5">f</font>,<font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Olive" title="b2">C</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Olive" title="b1">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Olive" title="b3">I</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c5">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b3">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">C</font>,<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">s</font>,<font color="Olive" title="b2">C</font> &amp; <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b2">C</font>,<font color="Olive" title="b3">I</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c5">f</font></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C</font>, <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="b1">C</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">P</font> &amp; <font color="Olive" title="b2">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c6">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c6">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Olive" title="b2">I</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c5">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b2">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b1">C</font>,<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">s</font>,<font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">I</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c5">f</font></span><br/><b>let </b><font color="Maroon" title="c7">C</font>, <font color="Maroon" title="c8">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">C</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c8">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c6">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c6">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c5">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon" title="c8">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c7">C</font> &amp; <font color="Maroon" title="c4">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="c4">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c8">I</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c5">f</font> )</span><br/>













<b>assume </b><b>that </b><br/><a NAME="E1:184"/><i><font color="Green" title="E109">A1</font></i>: 
( <font color="Maroon" title="c7">C</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c8">I</font> <a href="aofa_000.html#R6" title="AOFA_000:pred.6">is_terminating_wrt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">P</font> )
 <b>and </b><br/><a NAME="E2:184"/><i><font color="Green" title="E110">A2</font></i>: 
( <font color="Maroon" title="c6">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c6">P</font> <a href="aofa_000.html#R7" title="AOFA_000:pred.7">is_invariant_wrt</a> <font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c5">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Maroon" title="c8">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c7">C</font> or  not <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c8">I</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c5">f</font> )</span><br/>

<b>given </b><font color="Maroon" title="c9">r</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">S</font><b> such that </b><a NAME="E4:184"/><i><font color="Green" title="E111">A3</font></i>: 
( <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c7">C</font> &amp; <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c3">T</font> )
 <b>and </b><br/><a NAME="E5:184"/><i><font color="Green" title="E112">A4</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">r</font> holds <br/>( <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c7">C</font></span>)</span> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="aofa_000.html#D33" target="_self" title="AOFA_000:def.33">AOFA_000:def 33</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c8">I</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c5">f</font> )</span><br/>

<b>assume </b><a NAME="E6:184"/><i><font color="Green" title="E113">A5</font></i>: 
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 ; <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="c4">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c8">I</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c5">f</font></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">r</font> implies <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> );<br/>
<a NAME="E7:184"/><i><font color="Green" title="E114">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="aofa_000.html#E4:184"><i><font color="Green" title="E111">A3</font></i></a>, <a class="txt" href="aofa_000.html#E2:184"><i><font color="Green" title="E110">A2</font></i></a>, <a class="txt" href="aofa_000.html#E6:184"><i><font color="Green" title="E113">A5</font></i></a>, <a class="ref" href="aofa_000.html#D39" target="_self" title="AOFA_000:def.39">Def39</a></i>;<br/>
<a NAME="E8:184"/><i><font color="Green" title="E115">A7</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="184_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:184_1"/><i><font color="Green" title="E116">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">i</font>]
 <b>and </b><br/><a NAME="E2:184_1"/><i><font color="Green" title="E117">A9</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font></span><br/>

<a NAME="E3:184_1"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E4:184_1"/><b>then </b><i><font color="Green" title="E118">A10</font></i>: 
( <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c7">C</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="aofa_000.html#E5:184"><i><font color="Green" title="E112">A4</font></i></a>, <a class="txt" href="aofa_000.html#E2:184_1"><i><font color="Green" title="E117">A9</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E6:184_1"/>
<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c8">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E2:184"><i><font color="Green" title="E110">A2</font></i></a>, <a class="txt" href="aofa_000.html#E1:184_1"><i><font color="Green" title="E116">A8</font></i></a>, <a class="ref" href="aofa_000.html#D39" target="_self" title="AOFA_000:def.39">Def39</a>, <a class="txt" href="aofa_000.html#E2:184_1"><i><font color="Green" title="E117">A9</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:184_1"/><b>then </b>
<font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c8">I</font></span>)</span>,<font color="Maroon" title="c7">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E2:184"><i><font color="Green" title="E110">A2</font></i></a>, <a class="ref" href="aofa_000.html#D39" target="_self" title="AOFA_000:def.39">Def39</a></i>;<br/>
<b>hence </b><a NAME="E8:184_1"/>
<font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E4:184_1"><i><font color="Green" title="E118">A10</font></i></a>, <a class="ref" href="aofa_000.html#D29" target="_self" title="AOFA_000:def.29">Def29</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="E9:184"/><i><font color="Green" title="E116">A11</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="aofa_000.html#E7:184"><i><font color="Green" title="E114">A6</font></i></a>, <a class="txt" href="aofa_000.html#E8:184"><i><font color="Green" title="E115">A7</font></i></a>);<br/></i>
<div><i><font color="Green" title="E117">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="184_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">r</font> implies ( <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> &amp; <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="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c7">C</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c7">C</font></span>)</span> ) )</span><br/><b>assume </b><a NAME="E1:184_2"/><i><font color="Green" title="E118">A13</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">r</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> &amp; <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="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c7">C</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c7">C</font></span>)</span> )</span><br/><b>hence </b><a NAME="E2:184_2"/>
<font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E5:184"><i><font color="Green" title="E112">A4</font></i></a></i>; <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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c7">C</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c7">C</font></span>)</span> )</span><br/><b>then reconsider </b><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">S</font> ;<br/><a NAME="E4:184_2"/>
<font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/><a NAME="E5:184_2"/><b>then </b>
( <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">r</font> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:184_2"><i><font color="Green" title="E118">A13</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:184_2"/><b>then </b>
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E9:184"><i><font color="Green" title="E116">A11</font></i></a></i>;<br/><a NAME="E7:184_2"/><b>then </b><i><font color="Green" title="E119">A14</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c8">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="c5">f</font> &amp; <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c8">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:184"><i><font color="Green" title="E109">A1</font></i></a>, <a class="txt" href="aofa_000.html#E2:184"><i><font color="Green" title="E110">A2</font></i></a>, <a class="ref" href="aofa_000.html#D38" target="_self" title="AOFA_000:def.38">Def38</a>, <a class="ref" href="aofa_000.html#D39" target="_self" title="AOFA_000:def.39">Def39</a></i>;<br/><a NAME="E8:184_2"/><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="c5">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c8">I</font></span>)</span>,<font color="Maroon" title="c7">C</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="c5">f</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:184"><i><font color="Green" title="E109">A1</font></i></a>, <a class="ref" href="aofa_000.html#D38" target="_self" title="AOFA_000:def.38">Def38</a></i>;<br/><b>hence </b><a NAME="E9:184_2"/>
( <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="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c7">C</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Maroon" title="c7">C</font></span>)</span> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E5:184"><i><font color="Green" title="E112">A4</font></i></a>, <a class="txt" href="aofa_000.html#E1:184_2"><i><font color="Green" title="E118">A13</font></i></a>, <a class="txt" href="aofa_000.html#E7:184_2"><i><font color="Green" title="E119">A14</font></i></a>, <a class="ref" href="aofa_000.html#D35" target="_self" title="AOFA_000:def.35">Def35</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="E11:184"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c7">C</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="c5">f</font>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E1:184"><i><font color="Green" title="E109">A1</font></i></a>, <a class="txt" href="aofa_000.html#E6:184"><i><font color="Green" title="E113">A5</font></i></a>, <a class="ref" href="aofa_000.html#D38" target="_self" title="AOFA_000:def.38">Def38</a></i>;<br/>
<b>hence </b><a NAME="E12:184"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">s</font>,<span class="p2">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c8">I</font></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>,<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E4:184"><i><font color="Green" title="E111">A3</font></i></a>, <a class="txt" href="aofa_000.html#E10:184"><i><font color="Green" title="E117">A12</font></i></a>, <a class="ref" href="aofa_000.html#D35" target="_self" title="AOFA_000:def.35">Def35</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>
