docs/htmldoc/mathcomp.ssreflect.choice.html


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<h1 class="libtitle">Library mathcomp.ssreflect.choice</h1>

<div class="code">
<span class="comment">(*&nbsp;(c)&nbsp;Copyright&nbsp;2006-2016&nbsp;Microsoft&nbsp;Corporation&nbsp;and&nbsp;Inria.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br/>
&nbsp;Distributed&nbsp;under&nbsp;the&nbsp;terms&nbsp;of&nbsp;CeCILL-B.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;*)</span><br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="mathcomp.ssreflect.ssreflect.html#"><span class="id" title="library">mathcomp.ssreflect.ssreflect</span></a>.<br/>

<br/>
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<div class="doc">
 This file contains the definitions of:                                     
   choiceType == interface for types with a choice operator.                
    countType == interface for countable types (implies choiceType).        
 subCountType == interface for types that are both subType and countType.   
  xchoose exP == a standard x such that P x, given exP : exists x : T, P x  
                 when T is a choiceType. The choice depends only on the     
                 extent of P (in particular, it is independent of exP).     
   choose P x0 == if P x0, a standard x such that P x.                      
      pickle x == a nat encoding the value x : T, where T is a countType.   
    unpickle n == a partial inverse to pickle: unpickle (pickle x) = Some x 
  pickle_inv n == a sharp partial inverse to pickle pickle_inv n = Some x   
                  if and only if pickle x = n.                              
 [choiceType of T for cT] == clone for T of the choiceType cT.              
        [choiceType of T] == clone for T of the choiceType inferred for T.  
  [countType of T for cT] == clone for T of the countType cT.               
        [count Type of T] == clone for T of the countType inferred for T.   
 [choiceMixin of T by &lt;: ] == Choice mixin for T when T has a subType p      
                        structure with p : pred cT and cT has a Choice      
                        structure; the corresponding structure is Canonical.
  [countMixin of T by &lt;: ] == Count mixin for a subType T of a countType.    
  PcanChoiceMixin fK == Choice mixin for T, given f : T -&gt; cT where cT has  
                        a Choice structure, a left inverse partial function 
                        g and fK : pcancel f g.                             
   CanChoiceMixin fK == Choice mixin for T, given f : T -&gt; cT, g and        
                        fK : cancel f g.                                    
   PcanCountMixin fK == Count mixin for T, given f : T -&gt; cT where cT has   
                        a Countable structure, a left inverse partial       
                        function g and fK : pcancel f g.                    
    CanCountMixin fK == Count mixin for T, given f : T -&gt; cT, g and         
                        fK : cancel f g.                                    
      GenTree.tree T == generic n-ary tree type with nat-labeled nodes and  
                        T-labeled leaves, for example GenTree.Leaf (x : T), 
                        GenTree.Node 5 [:: t; t' ]. GenTree.tree is equipped 
                        with canonical eqType, choiceType, and countType    
                        instances, and so simple datatypes can be similarly 
                        equipped by encoding into GenTree.tree and using    
                        the mixins above.                                   
        CodeSeq.code == bijection from seq nat to nat.                      
      CodeSeq.decode == bijection inverse to CodeSeq.code.                  
 In addition to the lemmas relevant to these definitions, this file also    
 contains definitions of a Canonical choiceType and countType instances for 
 all basic datatypes (e.g., nat, bool, subTypes, pairs, sums, etc.).         
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<div class="code">

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<span class="id" title="keyword">Set Implicit Arguments</span>.<br/>

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<div class="doc">
 Technical definitions about coding and decoding of nat sequences, which    
 are used below to define various Canonical instances of the choice and     
 countable interfaces.                                                       
</div>
<div class="code">

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<span class="id" title="keyword">Module</span> <a name="CodeSeq"><span class="id" title="module">CodeSeq</span></a>.<br/>

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<div class="doc">
 Goedel-style one-to-one encoding of seq nat into nat.                      
 The code for [:: n1; ...; nk] has binary representation                    
          1 0 ... 0 1 ... 1 0 ... 0 1 0 ... 0                               
            &lt;-----&gt;         &lt;-----&gt;   &lt;-----&gt;                               
             nk 0s           n2 0s     n1 0s                                 
</div>
<div class="code">

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<span class="id" title="keyword">Definition</span> <a name="CodeSeq.code"><span class="id" title="definition">code</span></a> := <a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">n</span> <span class="id" title="var">m</span> ⇒ 2 <a class="idref" href="mathcomp.ssreflect.ssrnat.html#4c362bcf0e947e2792a2e6989b44aeb0"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#f460b977ac49dd1a229be682bc38c411"><span class="id" title="notation">.*2</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a>) 0.<br/>

<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="CodeSeq.decode_rec"><span class="id" title="definition">decode_rec</span></a> (<span class="id" title="var">v</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>) {<span class="id" title="keyword">struct</span> <span class="id" title="var">q</span>} :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.ssreflect.choice.html#q"><span class="id" title="variable">q</span></a>, <a class="idref" href="mathcomp.ssreflect.choice.html#r"><span class="id" title="variable">r</span></a> <span class="id" title="keyword">with</span><br/>
&nbsp;&nbsp;| 0, <span class="id" title="var">_</span>         ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#36229928b54642a4a7da943ccf8f9612"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#v"><span class="id" title="variable">v</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#36229928b54642a4a7da943ccf8f9612"><span class="id" title="notation">]</span></a><br/>
&nbsp;&nbsp;| <span class="id" title="var">q'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a>, 0     ⇒ <a class="idref" href="mathcomp.ssreflect.choice.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">rec</span></a> 0<a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">,</span></a> <span class="id" title="var">q'</span><a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">,</span></a> <span class="id" title="var">q'</span><a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">]</span></a><br/>
&nbsp;&nbsp;| <span class="id" title="var">q'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a>, 1     ⇒ <a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">rec</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#v"><span class="id" title="variable">v</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">,</span></a> <span class="id" title="var">q'</span><a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">,</span></a> <span class="id" title="var">q'</span><a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">]</span></a><br/>
&nbsp;&nbsp;| <span class="id" title="var">q'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a>, <span class="id" title="var">r'</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#793e3499c36e4c6595d810e871a5acdd"><span class="id" title="notation">.+2</span></a> ⇒ <a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">rec</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#v"><span class="id" title="variable">v</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">,</span></a> <span class="id" title="var">q'</span><a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">,</span></a> <span class="id" title="var">r'</span><a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">]</span></a><br/>
&nbsp;&nbsp;<span class="id" title="keyword">end</span> <span class="id" title="keyword">where</span> <a name="c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">&quot;</span></a>[ 'rec' v , q , r ]" := (<a class="idref" href="mathcomp.ssreflect.choice.html#decode_rec"><span class="id" title="definition">decode_rec</span></a> <span class="id" title="var">v</span> <span class="id" title="var">q</span> <span class="id" title="var">r</span>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="CodeSeq.decode"><span class="id" title="definition">decode</span></a> <span class="id" title="var">n</span> := <span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.choice.html#n"><span class="id" title="variable">n</span></a> <span class="id" title="keyword">is</span> 0 <span class="id" title="keyword">then</span> <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">rec</span></a> 0<a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">.-1</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="mathcomp.ssreflect.ssrnat.html#1d63841e595f2805afd872744cbb1cce"><span class="id" title="notation">.-1</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#c3b3d63d167c5615b729c64355a588ab"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="CodeSeq.decodeK"><span class="id" title="lemma">decodeK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#CodeSeq.decode"><span class="id" title="definition">decode</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#CodeSeq.code"><span class="id" title="definition">code</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="CodeSeq.codeK"><span class="id" title="lemma">codeK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#CodeSeq.code"><span class="id" title="definition">code</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#CodeSeq.decode"><span class="id" title="definition">decode</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="CodeSeq.ltn_code"><span class="id" title="lemma">ltn_code</span></a> <span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">j</span> ⇒ <a class="idref" href="mathcomp.ssreflect.choice.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation">&lt;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#CodeSeq.code"><span class="id" title="definition">code</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#s"><span class="id" title="variable">s</span></a>) <a class="idref" href="mathcomp.ssreflect.choice.html#s"><span class="id" title="variable">s</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="CodeSeq.gtn_decode"><span class="id" title="lemma">gtn_decode</span></a> <span class="id" title="var">n</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#all"><span class="id" title="definition">all</span></a> (<a class="idref" href="mathcomp.ssreflect.ssrnat.html#ltn"><span class="id" title="definition">ltn</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#n"><span class="id" title="variable">n</span></a>) (<a class="idref" href="mathcomp.ssreflect.choice.html#CodeSeq.decode"><span class="id" title="definition">decode</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#n"><span class="id" title="variable">n</span></a>).<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#CodeSeq"><span class="id" title="module">CodeSeq</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="OtherEncodings"><span class="id" title="section">OtherEncodings</span></a>.<br/>
</div>

<div class="doc">
 Miscellaneous encodings: option T -c-&gt; seq T, T1 * T2 -c-&gt; {i : T1 &amp; T2}   
 T1 + T2 -c-&gt; option T1 * option T2, unit -c-&gt; bool; bool -c-&gt; nat is       
 already covered in ssrnat by the nat_of_bool coercion, the odd predicate,  
 and their "cancellation" lemma oddb. We use these encodings to propagate   
 canonical structures through these type constructors so that ultimately    
 all Choice and Countable instanced derive from nat and the seq and sigT    
 constructors.                                                               
</div>
<div class="code">

<br/>
<span class="id" title="keyword">Variables</span> <a name="OtherEncodings.T"><span class="id" title="variable">T</span></a> <a name="OtherEncodings.T1"><span class="id" title="variable">T1</span></a> <a name="OtherEncodings.T2"><span class="id" title="variable">T2</span></a> : <span class="id" title="keyword">Type</span>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="seq_of_opt"><span class="id" title="definition">seq_of_opt</span></a> := @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#oapp"><span class="id" title="abbreviation">oapp</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#OtherEncodings.T"><span class="id" title="variable">T</span></a> <span class="id" title="var">_</span> (<a class="idref" href="mathcomp.ssreflect.seq.html#nseq"><span class="id" title="definition">nseq</span></a> 1) <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a>.<br/>
<span class="id" title="keyword">Lemma</span> <a name="seq_of_optK"><span class="id" title="lemma">seq_of_optK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#seq_of_opt"><span class="id" title="definition">seq_of_opt</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#ohead"><span class="id" title="definition">ohead</span></a>.  <br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="tag_of_pair"><span class="id" title="definition">tag_of_pair</span></a> (<span class="id" title="var">p</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#OtherEncodings.T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#OtherEncodings.T2"><span class="id" title="variable">T2</span></a>) := @<a class="idref" href="mathcomp.ssreflect.eqtype.html#Tagged"><span class="id" title="definition">Tagged</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#OtherEncodings.T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="mathcomp.ssreflect.choice.html#OtherEncodings.T2"><span class="id" title="variable">T2</span></a>) <a class="idref" href="mathcomp.ssreflect.choice.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="pair_of_tag"><span class="id" title="definition">pair_of_tag</span></a> (<span class="id" title="var">u</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">{</span></a><span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#OtherEncodings.T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#OtherEncodings.T2"><span class="id" title="variable">T2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">}</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#tag"><span class="id" title="definition">tag</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#tagged"><span class="id" title="definition">tagged</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/>
<span class="id" title="keyword">Lemma</span> <a name="tag_of_pairK"><span class="id" title="lemma">tag_of_pairK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#tag_of_pair"><span class="id" title="definition">tag_of_pair</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#pair_of_tag"><span class="id" title="definition">pair_of_tag</span></a>.  <br/>
<span class="id" title="keyword">Lemma</span> <a name="pair_of_tagK"><span class="id" title="lemma">pair_of_tagK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#pair_of_tag"><span class="id" title="definition">pair_of_tag</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#tag_of_pair"><span class="id" title="definition">tag_of_pair</span></a>.  <br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="opair_of_sum"><span class="id" title="definition">opair_of_sum</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#OtherEncodings.T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#3dcaec3b772747610227247939f96b01"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#OtherEncodings.T2"><span class="id" title="variable">T2</span></a>) :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.ssreflect.choice.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">with</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <span class="id" title="var">x</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> | <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <span class="id" title="var">y</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <a name="sum_of_opair"><span class="id" title="definition">sum_of_opair</span></a> <span class="id" title="var">p</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#oapp"><span class="id" title="abbreviation">oapp</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#some"><span class="id" title="abbreviation">some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">o</span></a> @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#OtherEncodings.T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#OtherEncodings.T2"><span class="id" title="variable">T2</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#omap"><span class="id" title="abbreviation">omap</span></a> (@<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.ssreflect.choice.html#OtherEncodings.T2"><span class="id" title="variable">T2</span></a>) <a class="idref" href="mathcomp.ssreflect.choice.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a>) <a class="idref" href="mathcomp.ssreflect.choice.html#p"><span class="id" title="variable">p</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a>.<br/>
<span class="id" title="keyword">Lemma</span> <a name="opair_of_sumK"><span class="id" title="lemma">opair_of_sumK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#pcancel"><span class="id" title="definition">pcancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#opair_of_sum"><span class="id" title="definition">opair_of_sum</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#sum_of_opair"><span class="id" title="definition">sum_of_opair</span></a>.  <br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="bool_of_unitK"><span class="id" title="lemma">bool_of_unitK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>) (<span class="id" title="keyword">fun</span> <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a>).<br/>
 
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#OtherEncodings"><span class="id" title="section">OtherEncodings</span></a>.<br/>

<br/>
</div>

<div class="doc">
 Generic variable-arity tree type, providing an encoding target for         
 miscellaneous user datatypes. The GenTree.tree type can be combined with   
 a sigT type to model multi-sorted concrete datatypes.                       
</div>
<div class="code">
<span class="id" title="keyword">Module</span> <a name="GenTree"><span class="id" title="module">GenTree</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="GenTree.Def"><span class="id" title="section">Def</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="GenTree.Def.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>.<br/>

<br/>
<span class="id" title="keyword">Inductive</span> <a name="GenTree.tree"><span class="id" title="inductive">tree</span></a> := <a name="GenTree.Leaf"><span class="id" title="constructor">Leaf</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.Def.T"><span class="id" title="variable">T</span></a> | <a name="GenTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> &amp; <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#tree"><span class="id" title="inductive">tree</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="GenTree.tree_rect"><span class="id" title="definition">tree_rect</span></a> <span class="id" title="var">K</span> <span class="id" title="var">IH_leaf</span> <span class="id" title="var">IH_node</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">fix</span> <span class="id" title="var">loop</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#t"><span class="id" title="variable">t</span></a> := <span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.ssreflect.choice.html#t"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
&nbsp;&nbsp;| <a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.Leaf"><span class="id" title="constructor">Leaf</span></a> <span class="id" title="var">x</span> ⇒ <a class="idref" href="mathcomp.ssreflect.choice.html#IH_leaf"><span class="id" title="variable">IH_leaf</span></a> <span class="id" title="var">x</span><br/>
&nbsp;&nbsp;| <a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">n</span> <span class="id" title="var">f0</span> ⇒<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="keyword">fix</span> <span class="id" title="var">iter_pair</span> <span class="id" title="var">f</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">t</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#prod"><span class="id" title="inductive">prod</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#t"><span class="id" title="variable">t</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#f"><span class="id" title="variable">f</span></a> :=<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.choice.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">t</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">f'</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#loop"><span class="id" title="variable">loop</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#t"><span class="id" title="variable">t</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#iter_pair"><span class="id" title="variable">iter_pair</span></a> <span class="id" title="var">f'</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#tt"><span class="id" title="constructor">tt</span></a> <span class="id" title="tactic">in</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.choice.html#IH_node"><span class="id" title="variable">IH_node</span></a> <span class="id" title="var">n</span> <span class="id" title="var">f0</span> (<a class="idref" href="mathcomp.ssreflect.choice.html#iter_pair"><span class="id" title="variable">iter_pair</span></a> <span class="id" title="var">f0</span>)<br/>
&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
<span class="id" title="keyword">Definition</span> <a name="GenTree.tree_rec"><span class="id" title="definition">tree_rec</span></a> (<span class="id" title="var">K</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.tree"><span class="id" title="inductive">tree</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Set</span>) := @<a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.tree_rect"><span class="id" title="definition">tree_rect</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#K"><span class="id" title="variable">K</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="GenTree.tree_ind"><span class="id" title="definition">tree_ind</span></a> <span class="id" title="var">K</span> <span class="id" title="var">IH_leaf</span> <span class="id" title="var">IH_node</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">fix</span> <span class="id" title="var">loop</span> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0fffdc558ce71ab561d36c8a8094dbe5"><span class="id" title="notation">:</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0fffdc558ce71ab561d36c8a8094dbe5"><span class="id" title="notation">Prop</span></a> := <span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.ssreflect.choice.html#t"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
&nbsp;&nbsp;| <a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.Leaf"><span class="id" title="constructor">Leaf</span></a> <span class="id" title="var">x</span> ⇒ <a class="idref" href="mathcomp.ssreflect.choice.html#IH_leaf"><span class="id" title="variable">IH_leaf</span></a> <span class="id" title="var">x</span><br/>
&nbsp;&nbsp;| <a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">n</span> <span class="id" title="var">f0</span> ⇒<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">let</span> <span class="id" title="keyword">fix</span> <span class="id" title="var">iter_conj</span> <span class="id" title="var">f</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">t</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#and"><span class="id" title="inductive">and</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#t"><span class="id" title="variable">t</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#True"><span class="id" title="inductive">True</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#f"><span class="id" title="variable">f</span></a> :=<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.choice.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="keyword">is</span> <span class="id" title="var">t</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <span class="id" title="var">f'</span> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#conj"><span class="id" title="constructor">conj</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#loop"><span class="id" title="variable">loop</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#t"><span class="id" title="variable">t</span></a>) (<a class="idref" href="mathcomp.ssreflect.choice.html#iter_conj"><span class="id" title="variable">iter_conj</span></a> <span class="id" title="var">f'</span>) <span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#I"><span class="id" title="constructor">Logic.I</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#IH_node"><span class="id" title="variable">IH_node</span></a> <span class="id" title="var">n</span> <span class="id" title="var">f0</span> (<a class="idref" href="mathcomp.ssreflect.choice.html#iter_conj"><span class="id" title="variable">iter_conj</span></a> <span class="id" title="var">f0</span>)<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>

<br/>
<span class="id" title="keyword">Fixpoint</span> <a name="GenTree.encode"><span class="id" title="definition">encode</span></a> <span class="id" title="var">t</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#3dcaec3b772747610227247939f96b01"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.Def.T"><span class="id" title="variable">T</span></a>) :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.ssreflect.choice.html#t"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">with</span><br/>
&nbsp;&nbsp;| <a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.Leaf"><span class="id" title="constructor">Leaf</span></a> <span class="id" title="var">x</span> ⇒ <a class="idref" href="mathcomp.ssreflect.seq.html#36229928b54642a4a7da943ccf8f9612"><span class="id" title="notation">[::</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <span class="id" title="var">_</span> <span class="id" title="var">x</span><a class="idref" href="mathcomp.ssreflect.seq.html#36229928b54642a4a7da943ccf8f9612"><span class="id" title="notation">]</span></a><br/>
&nbsp;&nbsp;| <a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">n</span> <span class="id" title="var">f</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <span class="id" title="var">_</span> <span class="id" title="var">n</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#rcons"><span class="id" title="definition">rcons</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#flatten"><span class="id" title="definition">flatten</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#encode"><span class="id" title="definition">encode</span></a> <span class="id" title="var">f</span>)) (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <span class="id" title="var">_</span> 0)<br/>
&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="GenTree.decode_step"><span class="id" title="definition">decode_step</span></a> <span class="id" title="var">c</span> <span class="id" title="var">fs</span> := <br/>
&nbsp;&nbsp;<span class="id" title="keyword">match</span> <a class="idref" href="mathcomp.ssreflect.choice.html#c"><span class="id" title="variable">c</span></a> <span class="id" title="keyword">with</span><br/>
&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#inr"><span class="id" title="constructor">inr</span></a> <span class="id" title="var">x</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.Leaf"><span class="id" title="constructor">Leaf</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#fs"><span class="id" title="variable">fs</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#fs"><span class="id" title="variable">fs</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><br/>
&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> 0 ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#fs"><span class="id" title="variable">fs</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#fs"><span class="id" title="variable">fs</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><br/>
&nbsp;&nbsp;| <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#inl"><span class="id" title="constructor">inl</span></a> <span class="id" title="var">n</span><a class="idref" href="mathcomp.ssreflect.ssrnat.html#361454269931ea8643f7b402f2ab7222"><span class="id" title="notation">.+1</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.Node"><span class="id" title="constructor">Node</span></a> <span class="id" title="var">n</span> <a class="idref" href="mathcomp.ssreflect.choice.html#fs"><span class="id" title="variable">fs</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#d7fed0909a58e41c49e3ee117361b0a5"><span class="id" title="notation">::</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#head"><span class="id" title="definition">head</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#fs"><span class="id" title="variable">fs</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#behead"><span class="id" title="definition">behead</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#fs"><span class="id" title="variable">fs</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><br/>
&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="GenTree.decode"><span class="id" title="definition">decode</span></a> <span class="id" title="var">c</span> := <a class="idref" href="mathcomp.ssreflect.seq.html#ohead"><span class="id" title="definition">ohead</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#foldr"><span class="id" title="definition">foldr</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.decode_step"><span class="id" title="definition">decode_step</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#747e2b5d553b2dfe76e024e1f8fb39d1"><span class="id" title="notation">[::]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#c"><span class="id" title="variable">c</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">).1</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="GenTree.codeK"><span class="id" title="lemma">codeK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#pcancel"><span class="id" title="definition">pcancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.encode"><span class="id" title="definition">encode</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.decode"><span class="id" title="definition">decode</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#GenTree.Def"><span class="id" title="section">Def</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#GenTree"><span class="id" title="module">GenTree</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="tree_eqMixin"><span class="id" title="definition">tree_eqMixin</span></a> (<span class="id" title="var">T</span> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) := <a class="idref" href="mathcomp.ssreflect.eqtype.html#PcanEqMixin"><span class="id" title="definition">PcanEqMixin</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#codeK"><span class="id" title="lemma">GenTree.codeK</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">tree_eqType</span> (<span class="id" title="var">T</span> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) := <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.EqType"><span class="id" title="abbreviation">EqType</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#tree"><span class="id" title="inductive">GenTree.tree</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>) (<a class="idref" href="mathcomp.ssreflect.choice.html#tree_eqMixin"><span class="id" title="definition">tree_eqMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>).<br/>

<br/>
</div>

<div class="doc">
 Structures for Types with a choice function, and for Types with countably  
 many elements. The two concepts are closely linked: we indeed make         
 Countable a subclass of Choice, as countable choice is valid in CiC. This  
 apparent redundancy is needed to ensure the consistency of the Canonical   
 inference, as the canonical Choice for a given type may differ from the    
 countable choice for its canonical Countable structure, e.g., for options. 
   The Choice interface exposes two choice functions; for T : choiceType    
 and P : pred T, we provide:                                                
    xchoose : (exists x, P x) -&gt; T                                          
    choose : pred T -&gt; T -&gt; T                                               
   While P (xchoose exP) will always hold, P (choose P x0) will be true if  
 and only if P x0 holds. Both xchoose and choose are extensional in P and   
 do not depend on the witness exP or x0 (provided P x0 holds). Note that    
 xchoose is slightly more powerful, but less convenient to use.             
   However, neither choose nor xchoose are composable: it would not be      
 be possible to extend the Choice structure to arbitrary pairs using only   
 these functions, for instance. Internally, the interfaces provides a       
 subtly stronger operation, Choice.InternalTheory.find, which performs a    
 limited search using an integer parameter only rather than a full value as 
 [x]choose does. This is not a restriction in a constructive theory, where  
 all types are concrete and hence countable. In the case of an axiomatic    
 theory, such as that of the Coq reals library, postulating a suitable      
 axiom of choice suppresses the need for guidance. Nevertheless this        
 operation is just what is needed to make the Choice interface compose.     
   The Countable interface provides three functions; for T : countType we   
 get pickle : T -&gt; nat, and unpickle, pickle_inv : nat -&gt; option T.         
 The functions provide an effective embedding of T in nat: unpickle is a    
 left inverse to pickle, which satisfies pcancel pickle unpickle, i.e.,     
 unpickle \o pickle =1 some; pickle_inv is a more precise inverse for which 
 we also have ocancel pickle_inv pickle. Both unpickle and pickle need to   
 be partial functions to allow for possibly empty types such as {x | P x}.  
   The names of these functions underline the correspondence with the       
 notion of "Serializable" types in programming languages.                   
   Finally, we need to provide a join class to let type inference unify     
 subType and countType class constraints, e.g., for a countable subType of  
 an uncountable choiceType (the issue does not arise earlier with eqType or 
 choiceType because in practice the base type of an Equality/Choice subType 
 is always an Equality/Choice Type).                                         
</div>
<div class="code">

<br/>
<span class="id" title="keyword">Module</span> <a name="Choice"><span class="id" title="module">Choice</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="Choice.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/>

<br/>
<span class="id" title="keyword">Record</span> <a name="Choice.mixin_of"><span class="id" title="record">mixin_of</span></a> <span class="id" title="var">T</span> := <a name="Choice.Mixin"><span class="id" title="constructor">Mixin</span></a> {<br/>
&nbsp;&nbsp;<a name="Choice.find"><span class="id" title="projection">find</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>;<br/>
&nbsp;&nbsp;<span class="id" title="var">_</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">P</span> <span class="id" title="var">n</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.ssreflect.choice.html#find"><span class="id" title="method">find</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a>;<br/>
&nbsp;&nbsp;<span class="id" title="var">_</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#find"><span class="id" title="method">find</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#n"><span class="id" title="variable">n</span></a>;<br/>
&nbsp;&nbsp;<span class="id" title="var">_</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">P</span> <span class="id" title="var">Q</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>, <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Q"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#find"><span class="id" title="method">find</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#find"><span class="id" title="method">find</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Q"><span class="id" title="variable">Q</span></a><br/>
}.<br/>

<br/>
<span class="id" title="keyword">Record</span> <a name="Choice.class_of"><span class="id" title="record">class_of</span></a> <span class="id" title="var">T</span> := <a name="Choice.Class"><span class="id" title="constructor">Class</span></a> {<a name="Choice.base"><span class="id" title="projection">base</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.class_of"><span class="id" title="abbreviation">Equality.class_of</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>; <a name="Choice.mixin"><span class="id" title="projection">mixin</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.mixin_of"><span class="id" title="record">mixin_of</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>}.<br/>

<br/>
<span class="id" title="keyword">Structure</span> <a name="Choice.type"><span class="id" title="record">type</span></a> := <a name="Choice.Pack"><span class="id" title="constructor">Pack</span></a> {<a name="Choice.sort"><span class="id" title="projection">sort</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#sort"><span class="id" title="method">sort</span></a>; <span class="id" title="var">_</span> : <span class="id" title="keyword">Type</span>}.<br/>
<span class="id" title="keyword">Variables</span> (<a name="Choice.ClassDef.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Choice.ClassDef.cT"><span class="id" title="variable">cT</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.type"><span class="id" title="record">type</span></a>).<br/>
<span class="id" title="keyword">Definition</span> <a name="Choice.class"><span class="id" title="definition">class</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">_</span> <span class="id" title="var">c</span> <span class="id" title="var">_</span> <span class="id" title="keyword">as</span> <span class="id" title="var">cT'</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#cT'"><span class="id" title="variable">cT'</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">c</span>.<br/>
<span class="id" title="keyword">Definition</span> <a name="Choice.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">c</span> <span class="id" title="keyword">of</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.class"><span class="id" title="definition">class</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#c"><span class="id" title="variable">c</span></a> := @<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Pack"><span class="id" title="constructor">Pack</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/>
<span class="id" title="keyword">Let</span> <a name="Choice.ClassDef.xT"><span class="id" title="variable">xT</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">T</span>.<br/>
<span class="id" title="keyword">Notation</span> <a name="Choice.xclass"><span class="id" title="abbreviation">xclass</span></a> := (<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.class"><span class="id" title="definition">class</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.ClassDef.xT"><span class="id" title="variable">xT</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="Choice.pack"><span class="id" title="definition">pack</span></a> <span class="id" title="var">m</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <span class="id" title="var">b</span> <span class="id" title="var">bT</span> &amp; <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.class"><span class="id" title="definition">Equality.class</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#bT"><span class="id" title="variable">bT</span></a>) <a class="idref" href="mathcomp.ssreflect.choice.html#b"><span class="id" title="variable">b</span></a> ⇒ <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Pack"><span class="id" title="constructor">Pack</span></a> (@<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Class"><span class="id" title="constructor">Class</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/>

<br/>
</div>

<div class="doc">
 Inheritance  
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a name="Choice.eqType"><span class="id" title="definition">eqType</span></a> := @<a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Pack"><span class="id" title="constructor">Equality.Pack</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/>

<br/>
<span class="id" title="keyword">Module</span> <span class="id" title="keyword">Import</span> <a name="Choice.Exports"><span class="id" title="module">Exports</span></a>.<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.base"><span class="id" title="projection">base</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.base"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.base"><span class="id" title="projection">class_of</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.base"><span class="id" title="projection">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.base"><span class="id" title="projection">Equality.class_of</span></a>.<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.sort"><span class="id" title="projection">sort</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.sort"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.sort"><span class="id" title="projection">type</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.sort"><span class="id" title="projection">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.sort"><span class="id" title="projection">Sortclass</span></a>.<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.eqType"><span class="id" title="definition">eqType</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.eqType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.eqType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.eqType"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.eqType"><span class="id" title="definition">Equality.type</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">eqType</span>.<br/>
<span class="id" title="keyword">Notation</span> <a name="Choice.Exports.choiceType"><span class="id" title="abbreviation">choiceType</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.type"><span class="id" title="record">type</span></a>.<br/>
<span class="id" title="keyword">Notation</span> <a name="Choice.Exports.choiceMixin"><span class="id" title="abbreviation">choiceMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.mixin_of"><span class="id" title="record">mixin_of</span></a>.<br/>
<span class="id" title="keyword">Notation</span> <a name="Choice.Exports.ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> <span class="id" title="var">T</span> <span class="id" title="var">m</span> := (@<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.pack"><span class="id" title="definition">pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">m</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>).<br/>
<span class="id" title="keyword">Notation</span> <a name="3c4bcf103732e5c03bac4c26d0c2ac12"><span class="id" title="notation">&quot;</span></a>[ 'choiceType' 'of' T 'for' cT ]" :=  (@<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">T</span> <span class="id" title="var">cT</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#idfun"><span class="id" title="abbreviation">idfun</span></a>)<br/>
&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'choiceType'  'of'  T  'for'  cT ]") : <span class="id" title="var">form_scope</span>.<br/>
<span class="id" title="keyword">Notation</span> <a name="1731a28227324c9e5fc49499029635b3"><span class="id" title="notation">&quot;</span></a>[ 'choiceType' 'of' T ]" := (@<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>)<br/>
&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'choiceType'  'of'  T ]") : <span class="id" title="var">form_scope</span>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Exports"><span class="id" title="module">Exports</span></a>.<br/>

<br/>
<span class="id" title="keyword">Module</span> <a name="Choice.InternalTheory"><span class="id" title="module">InternalTheory</span></a>.<br/>
<span class="id" title="keyword">Section</span> <a name="Choice.InternalTheory.InternalTheory"><span class="id" title="section">InternalTheory</span></a>.<br/>
</div>

<div class="doc">
 Inner choice function.  
</div>
<div class="code">
<span class="id" title="keyword">Definition</span> <a name="Choice.InternalTheory.find"><span class="id" title="definition">find</span></a> <span class="id" title="var">T</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.find"><span class="id" title="projection">find</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.mixin"><span class="id" title="projection">mixin</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.class"><span class="id" title="definition">class</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>)).<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="Choice.InternalTheory.InternalTheory.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Exports.choiceType"><span class="id" title="abbreviation">choiceType</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">P</span> <span class="id" title="var">Q</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.InternalTheory.InternalTheory.T"><span class="id" title="variable">T</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="Choice.InternalTheory.correct"><span class="id" title="lemma">correct</span></a> <span class="id" title="var">P</span> <span class="id" title="var">n</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.InternalTheory.find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="Choice.InternalTheory.complete"><span class="id" title="lemma">complete</span></a> <span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">n</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.InternalTheory.find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="Choice.InternalTheory.extensional"><span class="id" title="lemma">extensional</span></a> <span class="id" title="var">P</span> <span class="id" title="var">Q</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Q"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.InternalTheory.find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.InternalTheory.find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Q"><span class="id" title="variable">Q</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Fact</span> <a name="Choice.InternalTheory.xchoose_subproof"><span class="id" title="lemma">xchoose_subproof</span></a> <span class="id" title="var">P</span> <span class="id" title="var">exP</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.InternalTheory.find"><span class="id" title="definition">find</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="mathcomp.ssreflect.ssrnat.html#ex_minn"><span class="id" title="definition">ex_minn</span></a> (@<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.InternalTheory.complete"><span class="id" title="lemma">complete</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#exP"><span class="id" title="variable">exP</span></a>)) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.InternalTheory.InternalTheory"><span class="id" title="section">InternalTheory</span></a>.<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.InternalTheory"><span class="id" title="module">InternalTheory</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Choice"><span class="id" title="module">Choice</span></a>.<br/>
<span class="id" title="keyword">Export</span> <span class="id" title="var">Choice.Exports</span>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="ChoiceTheory"><span class="id" title="section">ChoiceTheory</span></a>.<br/>

<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">T</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#choiceType"><span class="id" title="abbreviation">choiceType</span></a>.<br/>
<span class="id" title="keyword">Import</span> <span class="id" title="var">Choice.InternalTheory</span> <span class="id" title="var">CodeSeq</span>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="ChoiceTheory.OneType"><span class="id" title="section">OneType</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="ChoiceTheory.OneType.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#choiceType"><span class="id" title="abbreviation">choiceType</span></a>.<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">P</span> <span class="id" title="var">Q</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.T"><span class="id" title="variable">T</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="xchoose"><span class="id" title="definition">xchoose</span></a> <span class="id" title="var">P</span> <span class="id" title="var">exP</span> := <a class="idref" href="mathcomp.ssreflect.eqtype.html#sval"><span class="id" title="abbreviation">sval</span></a> (@<a class="idref" href="mathcomp.ssreflect.choice.html#xchoose_subproof"><span class="id" title="lemma">xchoose_subproof</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#exP"><span class="id" title="variable">exP</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="xchooseP"><span class="id" title="lemma">xchooseP</span></a> <span class="id" title="var">P</span> <span class="id" title="var">exP</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> (@<a class="idref" href="mathcomp.ssreflect.choice.html#xchoose"><span class="id" title="definition">xchoose</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#exP"><span class="id" title="variable">exP</span></a>).<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="eq_xchoose"><span class="id" title="lemma">eq_xchoose</span></a> <span class="id" title="var">P</span> <span class="id" title="var">Q</span> <span class="id" title="var">exP</span> <span class="id" title="var">exQ</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Q"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> @<a class="idref" href="mathcomp.ssreflect.choice.html#xchoose"><span class="id" title="definition">xchoose</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#exP"><span class="id" title="variable">exP</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> @<a class="idref" href="mathcomp.ssreflect.choice.html#xchoose"><span class="id" title="definition">xchoose</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Q"><span class="id" title="variable">Q</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#exQ"><span class="id" title="variable">exQ</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="sigW"><span class="id" title="lemma">sigW</span></a> <span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="sig2W"><span class="id" title="lemma">sig2W</span></a> <span class="id" title="var">P</span> <span class="id" title="var">Q</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Q"><span class="id" title="variable">Q</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">{</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Q"><span class="id" title="variable">Q</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="sig_eqW"><span class="id" title="lemma">sig_eqW</span></a> (<span class="id" title="var">vT</span> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<span class="id" title="var">lhs</span> <span class="id" title="var">rhs</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#vT"><span class="id" title="variable">vT</span></a>) :<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#lhs"><span class="id" title="variable">lhs</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#rhs"><span class="id" title="variable">rhs</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#lhs"><span class="id" title="variable">lhs</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#rhs"><span class="id" title="variable">rhs</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="sig2_eqW"><span class="id" title="lemma">sig2_eqW</span></a> (<span class="id" title="var">vT</span> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.T"><span class="id" title="variable">T</span></a>) (<span class="id" title="var">lhs</span> <span class="id" title="var">rhs</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#vT"><span class="id" title="variable">vT</span></a>) :<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">x</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#lhs"><span class="id" title="variable">lhs</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#rhs"><span class="id" title="variable">rhs</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">{</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#lhs"><span class="id" title="variable">lhs</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#rhs"><span class="id" title="variable">rhs</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#f5350ad671d3ce0e1e463e298917cf6e"><span class="id" title="notation">}</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="choose"><span class="id" title="definition">choose</span></a> <span class="id" title="var">P</span> <span class="id" title="var">x0</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#insub"><span class="id" title="definition">insub</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x0"><span class="id" title="variable">x0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#addfbab002f749a52fe38aa5d236c15e"><span class="id" title="notation">{?</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#addfbab002f749a52fe38aa5d236c15e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#addfbab002f749a52fe38aa5d236c15e"><span class="id" title="notation">}</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">x</span> <span class="id" title="var">Px</span>) <span class="id" title="keyword">then</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.choice.html#xchoose"><span class="id" title="definition">xchoose</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#ex_intro"><span class="id" title="constructor">ex_intro</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3f6d75f5a9cb81b61e8423a7aac22056"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3f6d75f5a9cb81b61e8423a7aac22056"><span class="id" title="notation">eta</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3f6d75f5a9cb81b61e8423a7aac22056"><span class="id" title="notation">]</span></a> <span class="id" title="var">x</span> <span class="id" title="var">Px</span>)<br/>
&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="mathcomp.ssreflect.choice.html#x0"><span class="id" title="variable">x0</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="chooseP"><span class="id" title="lemma">chooseP</span></a> <span class="id" title="var">P</span> <span class="id" title="var">x0</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x0"><span class="id" title="variable">x0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x0"><span class="id" title="variable">x0</span></a>).<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="choose_id"><span class="id" title="lemma">choose_id</span></a> <span class="id" title="var">P</span> <span class="id" title="var">x0</span> <span class="id" title="var">y0</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x0"><span class="id" title="variable">x0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#y0"><span class="id" title="variable">y0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x0"><span class="id" title="variable">x0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#y0"><span class="id" title="variable">y0</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="eq_choose"><span class="id" title="lemma">eq_choose</span></a> <span class="id" title="var">P</span> <span class="id" title="var">Q</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Q"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#choose"><span class="id" title="definition">choose</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Q"><span class="id" title="variable">Q</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="ChoiceTheory.OneType.CanChoice"><span class="id" title="section">CanChoice</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="ChoiceTheory.OneType.CanChoice.sT"><span class="id" title="variable">sT</span></a> : <span class="id" title="keyword">Type</span>) (<a name="ChoiceTheory.OneType.CanChoice.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#sT"><span class="id" title="variable">sT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.T"><span class="id" title="variable">T</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="PcanChoiceMixin"><span class="id" title="lemma">PcanChoiceMixin</span></a> <span class="id" title="var">f'</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#pcancel"><span class="id" title="definition">pcancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.CanChoice.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#f'"><span class="id" title="variable">f'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#choiceMixin"><span class="id" title="abbreviation">choiceMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.CanChoice.sT"><span class="id" title="variable">sT</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="CanChoiceMixin"><span class="id" title="definition">CanChoiceMixin</span></a> <span class="id" title="var">f'</span> (<span class="id" title="var">fK</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.CanChoice.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#f'"><span class="id" title="variable">f'</span></a>) :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.choice.html#PcanChoiceMixin"><span class="id" title="lemma">PcanChoiceMixin</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#can_pcan"><span class="id" title="lemma">can_pcan</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#fK"><span class="id" title="variable">fK</span></a>).<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.CanChoice"><span class="id" title="section">CanChoice</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="ChoiceTheory.OneType.SubChoice"><span class="id" title="section">SubChoice</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="ChoiceTheory.OneType.SubChoice.P"><span class="id" title="variable">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.T"><span class="id" title="variable">T</span></a>) (<a name="ChoiceTheory.OneType.SubChoice.sT"><span class="id" title="variable">sT</span></a> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#subType"><span class="id" title="record">subType</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="sub_choiceMixin"><span class="id" title="definition">sub_choiceMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#PcanChoiceMixin"><span class="id" title="lemma">PcanChoiceMixin</span></a> (@<a class="idref" href="mathcomp.ssreflect.eqtype.html#valK"><span class="id" title="lemma">valK</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.SubChoice.P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.SubChoice.sT"><span class="id" title="variable">sT</span></a>).<br/>
<span class="id" title="keyword">Definition</span> <a name="sub_choiceClass"><span class="id" title="definition">sub_choiceClass</span></a> := @<a class="idref" href="mathcomp.ssreflect.choice.html#Class"><span class="id" title="constructor">Choice.Class</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.SubChoice.sT"><span class="id" title="variable">sT</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#sub_eqMixin"><span class="id" title="definition">sub_eqMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.SubChoice.sT"><span class="id" title="variable">sT</span></a>) <a class="idref" href="mathcomp.ssreflect.choice.html#sub_choiceMixin"><span class="id" title="definition">sub_choiceMixin</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sub_choiceType</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#Pack"><span class="id" title="constructor">Choice.Pack</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#sub_choiceClass"><span class="id" title="definition">sub_choiceClass</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.SubChoice.sT"><span class="id" title="variable">sT</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.SubChoice"><span class="id" title="section">SubChoice</span></a>.<br/>

<br/>
<span class="id" title="keyword">Fact</span> <a name="seq_choiceMixin"><span class="id" title="lemma">seq_choiceMixin</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#choiceMixin"><span class="id" title="abbreviation">choiceMixin</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.T"><span class="id" title="variable">T</span></a>).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">seq_choiceType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType.T"><span class="id" title="variable">T</span></a>) <a class="idref" href="mathcomp.ssreflect.choice.html#seq_choiceMixin"><span class="id" title="lemma">seq_choiceMixin</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.OneType"><span class="id" title="section">OneType</span></a>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="ChoiceTheory.TagChoice"><span class="id" title="section">TagChoice</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="ChoiceTheory.TagChoice.I"><span class="id" title="variable">I</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#choiceType"><span class="id" title="abbreviation">choiceType</span></a>) (<a name="ChoiceTheory.TagChoice.T_"><span class="id" title="variable">T_</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#choiceType"><span class="id" title="abbreviation">choiceType</span></a>).<br/>

<br/>
<span class="id" title="keyword">Fact</span> <a name="tagged_choiceMixin"><span class="id" title="lemma">tagged_choiceMixin</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#choiceMixin"><span class="id" title="abbreviation">choiceMixin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">{</span></a><span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.TagChoice.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.TagChoice.T_"><span class="id" title="variable">T_</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">}</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">tagged_choiceType</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">{</span></a><span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.TagChoice.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.TagChoice.T_"><span class="id" title="variable">T_</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">}</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#tagged_choiceMixin"><span class="id" title="lemma">tagged_choiceMixin</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory.TagChoice"><span class="id" title="section">TagChoice</span></a>.<br/>

<br/>
<span class="id" title="keyword">Fact</span> <a name="nat_choiceMixin"><span class="id" title="lemma">nat_choiceMixin</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#choiceMixin"><span class="id" title="abbreviation">choiceMixin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">nat_choiceType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#nat_choiceMixin"><span class="id" title="lemma">nat_choiceMixin</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="bool_choiceMixin"><span class="id" title="definition">bool_choiceMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#CanChoiceMixin"><span class="id" title="definition">CanChoiceMixin</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#oddb"><span class="id" title="lemma">oddb</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">bool_choiceType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#bool_choiceMixin"><span class="id" title="definition">bool_choiceMixin</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">bitseq_choiceType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#1731a28227324c9e5fc49499029635b3"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#1731a28227324c9e5fc49499029635b3"><span class="id" title="notation">choiceType</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#1731a28227324c9e5fc49499029635b3"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bitseq"><span class="id" title="definition">bitseq</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#1731a28227324c9e5fc49499029635b3"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="unit_choiceMixin"><span class="id" title="definition">unit_choiceMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#CanChoiceMixin"><span class="id" title="definition">CanChoiceMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#bool_of_unitK"><span class="id" title="lemma">bool_of_unitK</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">unit_choiceType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#unit_choiceMixin"><span class="id" title="definition">unit_choiceMixin</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="option_choiceMixin"><span class="id" title="definition">option_choiceMixin</span></a> <span class="id" title="var">T</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#CanChoiceMixin"><span class="id" title="definition">CanChoiceMixin</span></a> (@<a class="idref" href="mathcomp.ssreflect.choice.html#seq_of_optK"><span class="id" title="lemma">seq_of_optK</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">option_choiceType</span> <span class="id" title="var">T</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>) (<a class="idref" href="mathcomp.ssreflect.choice.html#option_choiceMixin"><span class="id" title="definition">option_choiceMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="sig_choiceMixin"><span class="id" title="definition">sig_choiceMixin</span></a> <span class="id" title="var">T</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>) : <a class="idref" href="mathcomp.ssreflect.choice.html#choiceMixin"><span class="id" title="abbreviation">choiceMixin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a> :=<br/>
&nbsp;&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.choice.html#sub_choiceMixin"><span class="id" title="definition">sub_choiceMixin</span></a> <span class="id" title="var">_</span>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sig_choiceType</span> <span class="id" title="var">T</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>) :=<br/>
&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#sig_choiceMixin"><span class="id" title="definition">sig_choiceMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="prod_choiceMixin"><span class="id" title="definition">prod_choiceMixin</span></a> <span class="id" title="var">T1</span> <span class="id" title="var">T2</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#CanChoiceMixin"><span class="id" title="definition">CanChoiceMixin</span></a> (@<a class="idref" href="mathcomp.ssreflect.choice.html#tag_of_pairK"><span class="id" title="lemma">tag_of_pairK</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T2"><span class="id" title="variable">T2</span></a>).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">prod_choiceType</span> <span class="id" title="var">T1</span> <span class="id" title="var">T2</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T2"><span class="id" title="variable">T2</span></a>) (<a class="idref" href="mathcomp.ssreflect.choice.html#prod_choiceMixin"><span class="id" title="definition">prod_choiceMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T2"><span class="id" title="variable">T2</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="sum_choiceMixin"><span class="id" title="definition">sum_choiceMixin</span></a> <span class="id" title="var">T1</span> <span class="id" title="var">T2</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#PcanChoiceMixin"><span class="id" title="lemma">PcanChoiceMixin</span></a> (@<a class="idref" href="mathcomp.ssreflect.choice.html#opair_of_sumK"><span class="id" title="lemma">opair_of_sumK</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T2"><span class="id" title="variable">T2</span></a>).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sum_choiceType</span> <span class="id" title="var">T1</span> <span class="id" title="var">T2</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#3dcaec3b772747610227247939f96b01"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T2"><span class="id" title="variable">T2</span></a>) (<a class="idref" href="mathcomp.ssreflect.choice.html#sum_choiceMixin"><span class="id" title="definition">sum_choiceMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T2"><span class="id" title="variable">T2</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="tree_choiceMixin"><span class="id" title="definition">tree_choiceMixin</span></a> <span class="id" title="var">T</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#PcanChoiceMixin"><span class="id" title="lemma">PcanChoiceMixin</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#codeK"><span class="id" title="lemma">GenTree.codeK</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">tree_choiceType</span> <span class="id" title="var">T</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#tree"><span class="id" title="inductive">GenTree.tree</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>) (<a class="idref" href="mathcomp.ssreflect.choice.html#tree_choiceMixin"><span class="id" title="definition">tree_choiceMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>).<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#ChoiceTheory"><span class="id" title="section">ChoiceTheory</span></a>.<br/>

<br/>
<span class="id" title="keyword">Notation</span> <a name="6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation">&quot;</span></a>[ 'choiceMixin' 'of' T 'by' &lt;: ]" :=<br/>
&nbsp;&nbsp;(<a class="idref" href="mathcomp.ssreflect.choice.html#sub_choiceMixin"><span class="id" title="definition">sub_choiceMixin</span></a> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#choiceMixin"><span class="id" title="abbreviation">choiceMixin</span></a> <span class="id" title="var">T</span>)<br/>
&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'choiceMixin'  'of'  T  'by'  &lt;: ]") : <span class="id" title="var">form_scope</span>.<br/>

<br/>
<span class="id" title="keyword">Module</span> <a name="Countable"><span class="id" title="module">Countable</span></a>.<br/>

<br/>
<span class="id" title="keyword">Record</span> <a name="Countable.mixin_of"><span class="id" title="record">mixin_of</span></a> (<span class="id" title="var">T</span> : <span class="id" title="keyword">Type</span>) : <span class="id" title="keyword">Type</span> := <a name="Countable.Mixin"><span class="id" title="constructor">Mixin</span></a> {<br/>
&nbsp;&nbsp;<a name="Countable.pickle"><span class="id" title="projection">pickle</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>;<br/>
&nbsp;&nbsp;<a name="Countable.unpickle"><span class="id" title="projection">unpickle</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>;<br/>
&nbsp;&nbsp;<a name="Countable.pickleK"><span class="id" title="projection">pickleK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#pcancel"><span class="id" title="definition">pcancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#pickle"><span class="id" title="method">pickle</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#unpickle"><span class="id" title="method">unpickle</span></a><br/>
}.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="Countable.EqMixin"><span class="id" title="definition">EqMixin</span></a> <span class="id" title="var">T</span> <span class="id" title="var">m</span> := <a class="idref" href="mathcomp.ssreflect.eqtype.html#PcanEqMixin"><span class="id" title="definition">PcanEqMixin</span></a> (@<a class="idref" href="mathcomp.ssreflect.choice.html#Countable.pickleK"><span class="id" title="projection">pickleK</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="keyword">Definition</span> <a name="Countable.ChoiceMixin"><span class="id" title="definition">ChoiceMixin</span></a> <span class="id" title="var">T</span> <span class="id" title="var">m</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#PcanChoiceMixin"><span class="id" title="lemma">PcanChoiceMixin</span></a> (@<a class="idref" href="mathcomp.ssreflect.choice.html#Countable.pickleK"><span class="id" title="projection">pickleK</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#m"><span class="id" title="variable">m</span></a>).<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="Countable.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/>

<br/>
<span class="id" title="keyword">Record</span> <a name="Countable.class_of"><span class="id" title="record">class_of</span></a> <span class="id" title="var">T</span> := <a name="Countable.Class"><span class="id" title="constructor">Class</span></a> { <a name="Countable.base"><span class="id" title="projection">base</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.class_of"><span class="id" title="record">Choice.class_of</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>; <a name="Countable.mixin"><span class="id" title="projection">mixin</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.mixin_of"><span class="id" title="record">mixin_of</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a> }.<br/>

<br/>
<span class="id" title="keyword">Structure</span> <a name="Countable.type"><span class="id" title="record">type</span></a> : <span class="id" title="keyword">Type</span> := <a name="Countable.Pack"><span class="id" title="constructor">Pack</span></a> {<a name="Countable.sort"><span class="id" title="projection">sort</span></a> : <span class="id" title="keyword">Type</span>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#sort"><span class="id" title="method">sort</span></a>; <span class="id" title="var">_</span> : <span class="id" title="keyword">Type</span>}.<br/>
<span class="id" title="keyword">Variables</span> (<a name="Countable.ClassDef.T"><span class="id" title="variable">T</span></a> : <span class="id" title="keyword">Type</span>) (<a name="Countable.ClassDef.cT"><span class="id" title="variable">cT</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.type"><span class="id" title="record">type</span></a>).<br/>
<span class="id" title="keyword">Definition</span> <a name="Countable.class"><span class="id" title="definition">class</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">_</span> <span class="id" title="var">c</span> <span class="id" title="var">_</span> <span class="id" title="keyword">as</span> <span class="id" title="var">cT'</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#cT'"><span class="id" title="variable">cT'</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">c</span>.<br/>
<span class="id" title="keyword">Definition</span> <a name="Countable.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">c</span> <span class="id" title="keyword">of</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.class"><span class="id" title="definition">class</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#c"><span class="id" title="variable">c</span></a> := @<a class="idref" href="mathcomp.ssreflect.choice.html#Countable.Pack"><span class="id" title="constructor">Pack</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/>
<span class="id" title="keyword">Let</span> <a name="Countable.ClassDef.xT"><span class="id" title="variable">xT</span></a> := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.Pack"><span class="id" title="constructor">Pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.ClassDef.cT"><span class="id" title="variable">cT</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">T</span>.<br/>
<span class="id" title="keyword">Notation</span> <a name="Countable.xclass"><span class="id" title="abbreviation">xclass</span></a> := (<a class="idref" href="mathcomp.ssreflect.choice.html#Countable.class"><span class="id" title="definition">class</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.class_of"><span class="id" title="record">class_of</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.ClassDef.xT"><span class="id" title="variable">xT</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="Countable.pack"><span class="id" title="definition">pack</span></a> <span class="id" title="var">m</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <span class="id" title="var">bT</span> <span class="id" title="var">b</span> &amp; <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.class"><span class="id" title="definition">Choice.class</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#bT"><span class="id" title="variable">bT</span></a>) <a class="idref" href="mathcomp.ssreflect.choice.html#b"><span class="id" title="variable">b</span></a> ⇒ <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.Pack"><span class="id" title="constructor">Pack</span></a> (@<a class="idref" href="mathcomp.ssreflect.choice.html#Countable.Class"><span class="id" title="constructor">Class</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.ClassDef.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.ClassDef.T"><span class="id" title="variable">T</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="Countable.eqType"><span class="id" title="definition">eqType</span></a> := @<a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Pack"><span class="id" title="constructor">Equality.Pack</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="Countable.choiceType"><span class="id" title="definition">choiceType</span></a> := @<a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Pack"><span class="id" title="constructor">Choice.Pack</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.ClassDef.cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.xclass"><span class="id" title="abbreviation">xclass</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.ClassDef.xT"><span class="id" title="variable">xT</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.ClassDef"><span class="id" title="section">ClassDef</span></a>.<br/>

<br/>
<span class="id" title="keyword">Module</span> <a name="Countable.Exports"><span class="id" title="module">Exports</span></a>.<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.base"><span class="id" title="projection">base</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.base"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.base"><span class="id" title="projection">class_of</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.base"><span class="id" title="projection">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.base"><span class="id" title="projection">Choice.class_of</span></a>.<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.mixin"><span class="id" title="projection">mixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.mixin"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.mixin"><span class="id" title="projection">class_of</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.mixin"><span class="id" title="projection">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.mixin"><span class="id" title="projection">mixin_of</span></a>.<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.sort"><span class="id" title="projection">sort</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.sort"><span class="id" title="projection">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.sort"><span class="id" title="projection">type</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.sort"><span class="id" title="projection">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.sort"><span class="id" title="projection">Sortclass</span></a>.<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.eqType"><span class="id" title="definition">eqType</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.eqType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.eqType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.eqType"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.eqType"><span class="id" title="definition">Equality.type</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">eqType</span>.<br/>
<span class="id" title="keyword">Coercion</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.choiceType"><span class="id" title="definition">choiceType</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.choiceType"><span class="id" title="definition">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.choiceType"><span class="id" title="definition">type</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.choiceType"><span class="id" title="definition">&gt;-&gt;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.choiceType"><span class="id" title="definition">Choice.type</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">choiceType</span>.<br/>
<span class="id" title="keyword">Notation</span> <a name="Countable.Exports.countType"><span class="id" title="abbreviation">countType</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.type"><span class="id" title="record">type</span></a>.<br/>
<span class="id" title="keyword">Notation</span> <a name="Countable.Exports.CountType"><span class="id" title="abbreviation">CountType</span></a> <span class="id" title="var">T</span> <span class="id" title="var">m</span> := (@<a class="idref" href="mathcomp.ssreflect.choice.html#Countable.pack"><span class="id" title="definition">pack</span></a> <span class="id" title="var">T</span> <span class="id" title="var">m</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>).<br/>
<span class="id" title="keyword">Notation</span> <a name="Countable.Exports.CountMixin"><span class="id" title="abbreviation">CountMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.Mixin"><span class="id" title="constructor">Mixin</span></a>.<br/>
<span class="id" title="keyword">Notation</span> <a name="Countable.Exports.CountChoiceMixin"><span class="id" title="abbreviation">CountChoiceMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.ChoiceMixin"><span class="id" title="definition">ChoiceMixin</span></a>.<br/>
<span class="id" title="keyword">Notation</span> <a name="45eb30de99675fc3a25ff0fa8a76c46a"><span class="id" title="notation">&quot;</span></a>[ 'countType' 'of' T 'for' cT ]" := (@<a class="idref" href="mathcomp.ssreflect.choice.html#Countable.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">T</span> <span class="id" title="var">cT</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#idfun"><span class="id" title="abbreviation">idfun</span></a>)<br/>
&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'countType'  'of'  T  'for'  cT ]") : <span class="id" title="var">form_scope</span>.<br/>
<span class="id" title="keyword">Notation</span> <a name="be2fb44e85835140c455bc256ce18d4c"><span class="id" title="notation">&quot;</span></a>[ 'countType' 'of' T ]" := (@<a class="idref" href="mathcomp.ssreflect.choice.html#Countable.clone"><span class="id" title="definition">clone</span></a> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>)<br/>
&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'countType'  'of'  T ]") : <span class="id" title="var">form_scope</span>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.Exports"><span class="id" title="module">Exports</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#Countable"><span class="id" title="module">Countable</span></a>.<br/>
<span class="id" title="keyword">Export</span> <span class="id" title="var">Countable.Exports</span>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="unpickle"><span class="id" title="definition">unpickle</span></a> <span class="id" title="var">T</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#unpickle"><span class="id" title="projection">Countable.unpickle</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#class"><span class="id" title="definition">Countable.class</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>).<br/>
<span class="id" title="keyword">Definition</span> <a name="pickle"><span class="id" title="definition">pickle</span></a> <span class="id" title="var">T</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#pickle"><span class="id" title="projection">Countable.pickle</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#class"><span class="id" title="definition">Countable.class</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>).<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="CountableTheory"><span class="id" title="section">CountableTheory</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variable</span> <a name="CountableTheory.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#countType"><span class="id" title="abbreviation">countType</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="pickleK"><span class="id" title="lemma">pickleK</span></a> : @<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#pcancel"><span class="id" title="definition">pcancel</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#CountableTheory.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#pickle"><span class="id" title="definition">pickle</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#unpickle"><span class="id" title="definition">unpickle</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Definition</span> <a name="pickle_inv"><span class="id" title="definition">pickle_inv</span></a> <span class="id" title="var">n</span> :=<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#obind"><span class="id" title="abbreviation">obind</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#CountableTheory.T"><span class="id" title="variable">T</span></a> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#pickle"><span class="id" title="definition">pickle</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>) (<a class="idref" href="mathcomp.ssreflect.choice.html#unpickle"><span class="id" title="definition">unpickle</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#n"><span class="id" title="variable">n</span></a>).<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="pickle_invK"><span class="id" title="lemma">pickle_invK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#ocancel"><span class="id" title="definition">ocancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#pickle_inv"><span class="id" title="definition">pickle_inv</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#pickle"><span class="id" title="definition">pickle</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="pickleK_inv"><span class="id" title="lemma">pickleK_inv</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#pcancel"><span class="id" title="definition">pcancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#pickle"><span class="id" title="definition">pickle</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#pickle_inv"><span class="id" title="definition">pickle_inv</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Lemma</span> <a name="pcan_pickleK"><span class="id" title="lemma">pcan_pickleK</span></a> <span class="id" title="var">sT</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span> :<br/>
&nbsp;&nbsp;@<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#pcancel"><span class="id" title="definition">pcancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#CountableTheory.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#sT"><span class="id" title="variable">sT</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#f'"><span class="id" title="variable">f'</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#pcancel"><span class="id" title="definition">pcancel</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#pickle"><span class="id" title="definition">pickle</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">o</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#f"><span class="id" title="variable">f</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#pcomp"><span class="id" title="definition">pcomp</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#f'"><span class="id" title="variable">f'</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#unpickle"><span class="id" title="definition">unpickle</span></a>).<br/>
 
<br/>
<span class="id" title="keyword">Definition</span> <a name="PcanCountMixin"><span class="id" title="definition">PcanCountMixin</span></a> <span class="id" title="var">sT</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span> (<span class="id" title="var">fK</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#pcancel"><span class="id" title="definition">pcancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#f'"><span class="id" title="variable">f'</span></a>) :=<br/>
&nbsp;&nbsp;@<a class="idref" href="mathcomp.ssreflect.choice.html#CountMixin"><span class="id" title="abbreviation">CountMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#sT"><span class="id" title="variable">sT</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> (<a class="idref" href="mathcomp.ssreflect.choice.html#pcan_pickleK"><span class="id" title="lemma">pcan_pickleK</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#fK"><span class="id" title="variable">fK</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="CanCountMixin"><span class="id" title="definition">CanCountMixin</span></a> <span class="id" title="var">sT</span> <span class="id" title="var">f</span> <span class="id" title="var">f'</span> (<span class="id" title="var">fK</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#f'"><span class="id" title="variable">f'</span></a>) :=<br/>
&nbsp;&nbsp;@<a class="idref" href="mathcomp.ssreflect.choice.html#PcanCountMixin"><span class="id" title="definition">PcanCountMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#sT"><span class="id" title="variable">sT</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#can_pcan"><span class="id" title="lemma">can_pcan</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#fK"><span class="id" title="variable">fK</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="sub_countMixin"><span class="id" title="definition">sub_countMixin</span></a> <span class="id" title="var">P</span> <span class="id" title="var">sT</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#PcanCountMixin"><span class="id" title="definition">PcanCountMixin</span></a> (@<a class="idref" href="mathcomp.ssreflect.eqtype.html#valK"><span class="id" title="lemma">valK</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#CountableTheory.T"><span class="id" title="variable">T</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#sT"><span class="id" title="variable">sT</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="pickle_seq"><span class="id" title="definition">pickle_seq</span></a> <span class="id" title="var">s</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#code"><span class="id" title="definition">CodeSeq.code</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#map"><span class="id" title="definition">map</span></a> (@<a class="idref" href="mathcomp.ssreflect.choice.html#pickle"><span class="id" title="definition">pickle</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#CountableTheory.T"><span class="id" title="variable">T</span></a>) <a class="idref" href="mathcomp.ssreflect.choice.html#s"><span class="id" title="variable">s</span></a>).<br/>
<span class="id" title="keyword">Definition</span> <a name="unpickle_seq"><span class="id" title="definition">unpickle_seq</span></a> <span class="id" title="var">n</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#pmap"><span class="id" title="definition">pmap</span></a> (@<a class="idref" href="mathcomp.ssreflect.choice.html#unpickle"><span class="id" title="definition">unpickle</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#CountableTheory.T"><span class="id" title="variable">T</span></a>) (<a class="idref" href="mathcomp.ssreflect.choice.html#decode"><span class="id" title="definition">CodeSeq.decode</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#n"><span class="id" title="variable">n</span></a>)).<br/>
<span class="id" title="keyword">Lemma</span> <a name="pickle_seqK"><span class="id" title="lemma">pickle_seqK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#pcancel"><span class="id" title="definition">pcancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#pickle_seq"><span class="id" title="definition">pickle_seq</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#unpickle_seq"><span class="id" title="definition">unpickle_seq</span></a>.<br/>
 
<br/>
<span class="id" title="keyword">Definition</span> <a name="seq_countMixin"><span class="id" title="definition">seq_countMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#CountMixin"><span class="id" title="abbreviation">CountMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#pickle_seqK"><span class="id" title="lemma">pickle_seqK</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">seq_countType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#CountType"><span class="id" title="abbreviation">CountType</span></a> (<a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#CountableTheory.T"><span class="id" title="variable">T</span></a>) <a class="idref" href="mathcomp.ssreflect.choice.html#seq_countMixin"><span class="id" title="definition">seq_countMixin</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#CountableTheory"><span class="id" title="section">CountableTheory</span></a>.<br/>

<br/>
<span class="id" title="keyword">Notation</span> <a name="99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation">&quot;</span></a>[ 'countMixin' 'of' T 'by' &lt;: ]" :=<br/>
&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="mathcomp.ssreflect.choice.html#sub_countMixin"><span class="id" title="definition">sub_countMixin</span></a> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#mixin_of"><span class="id" title="record">Countable.mixin_of</span></a> <span class="id" title="var">T</span>)<br/>
&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'countMixin'  'of'  T  'by'  &lt;: ]") : <span class="id" title="var">form_scope</span>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="SubCountType"><span class="id" title="section">SubCountType</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="SubCountType.T"><span class="id" title="variable">T</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#choiceType"><span class="id" title="abbreviation">choiceType</span></a>) (<a name="SubCountType.P"><span class="id" title="variable">P</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>).<br/>
<span class="id" title="keyword">Import</span> <span class="id" title="var">Countable</span>.<br/>

<br/>
<span class="id" title="keyword">Structure</span> <a name="subCountType"><span class="id" title="record">subCountType</span></a> : <span class="id" title="keyword">Type</span> :=<br/>
&nbsp;&nbsp;<a name="SubCountType"><span class="id" title="constructor">SubCountType</span></a> {<a name="subCount_sort"><span class="id" title="projection">subCount_sort</span></a> :&gt; <a class="idref" href="mathcomp.ssreflect.eqtype.html#subType"><span class="id" title="record">subType</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#SubCountType.P"><span class="id" title="variable">P</span></a>; <span class="id" title="var">_</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#mixin_of"><span class="id" title="record">mixin_of</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#subCount_sort"><span class="id" title="method">subCount_sort</span></a>}.<br/>

<br/>
<span class="id" title="keyword">Coercion</span> <span class="id" title="var">sub_countType</span> (<span class="id" title="var">sT</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#subCountType"><span class="id" title="record">subCountType</span></a>) :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#pack"><span class="id" title="definition">pack</span></a> (<span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.ssreflect.choice.html#SubCountType"><span class="id" title="constructor">SubCountType</span></a> <span class="id" title="var">_</span> <span class="id" title="var">m</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#sT"><span class="id" title="variable">sT</span></a> <span class="id" title="keyword">return</span> <a class="idref" href="mathcomp.ssreflect.choice.html#mixin_of"><span class="id" title="record">mixin_of</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#sT"><span class="id" title="variable">sT</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">m</span>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sub_countType</span>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="pack_subCountType"><span class="id" title="definition">pack_subCountType</span></a> <span class="id" title="var">U</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <span class="id" title="var">sT</span> <span class="id" title="var">cT</span> &amp; <a class="idref" href="mathcomp.ssreflect.eqtype.html#sub_sort"><span class="id" title="projection">sub_sort</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#sT"><span class="id" title="variable">sT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#sort"><span class="id" title="projection">sort</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#cT"><span class="id" title="variable">cT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#U"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#U"><span class="id" title="variable">U</span></a> ⇒<br/>
&nbsp;&nbsp;<span class="id" title="keyword">fun</span> <span class="id" title="var">b</span> <span class="id" title="var">m</span>   &amp; <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#phant_id"><span class="id" title="definition">phant_id</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#Class"><span class="id" title="constructor">Class</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#m"><span class="id" title="variable">m</span></a>) (<a class="idref" href="mathcomp.ssreflect.choice.html#class"><span class="id" title="definition">class</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#cT"><span class="id" title="variable">cT</span></a>) ⇒ @<a class="idref" href="mathcomp.ssreflect.choice.html#SubCountType"><span class="id" title="constructor">SubCountType</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#sT"><span class="id" title="variable">sT</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#m"><span class="id" title="variable">m</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#SubCountType"><span class="id" title="section">SubCountType</span></a>.<br/>

<br/>
</div>

<div class="doc">
 This assumes that T has both countType and subType structures.  
</div>
<div class="code">
<span class="id" title="keyword">Notation</span> <a name="8a7192fa64a42310658fd5be07ae4fcc"><span class="id" title="notation">&quot;</span></a>[ 'subCountType' 'of' T ]" :=<br/>
&nbsp;&nbsp;&nbsp;&nbsp;(@<a class="idref" href="mathcomp.ssreflect.choice.html#pack_subCountType"><span class="id" title="definition">pack_subCountType</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">T</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a>)<br/>
&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'subCountType'  'of'  T ]") : <span class="id" title="var">form_scope</span>.<br/>

<br/>
<span class="id" title="keyword">Section</span> <a name="TagCountType"><span class="id" title="section">TagCountType</span></a>.<br/>

<br/>
<span class="id" title="keyword">Variables</span> (<a name="TagCountType.I"><span class="id" title="variable">I</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#countType"><span class="id" title="abbreviation">countType</span></a>) (<a name="TagCountType.T_"><span class="id" title="variable">T_</span></a> : <a class="idref" href="mathcomp.ssreflect.choice.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#countType"><span class="id" title="abbreviation">countType</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="pickle_tagged"><span class="id" title="definition">pickle_tagged</span></a> (<span class="id" title="var">u</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">{</span></a><span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#TagCountType.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#TagCountType.T_"><span class="id" title="variable">T_</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">}</span></a>) :=<br/>
&nbsp;&nbsp;<a class="idref" href="mathcomp.ssreflect.choice.html#code"><span class="id" title="definition">CodeSeq.code</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">[::</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#pickle"><span class="id" title="definition">pickle</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#tag"><span class="id" title="definition">tag</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#u"><span class="id" title="variable">u</span></a>)<a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#pickle"><span class="id" title="definition">pickle</span></a> (<a class="idref" href="mathcomp.ssreflect.eqtype.html#tagged"><span class="id" title="definition">tagged</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#u"><span class="id" title="variable">u</span></a>)<a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">]</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="unpickle_tagged"><span class="id" title="definition">unpickle_tagged</span></a> <span class="id" title="var">s</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">if</span> <a class="idref" href="mathcomp.ssreflect.choice.html#decode"><span class="id" title="definition">CodeSeq.decode</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">is</span> <a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">[::</span></a> <span class="id" title="var">ni</span><a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">;</span></a> <span class="id" title="var">nx</span><a class="idref" href="mathcomp.ssreflect.seq.html#b2d6f6eec274c9f9919a378a42b5b183"><span class="id" title="notation">]</span></a> <span class="id" title="keyword">then</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#obind"><span class="id" title="abbreviation">obind</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">i</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#omap"><span class="id" title="abbreviation">omap</span></a> (@<a class="idref" href="mathcomp.ssreflect.eqtype.html#Tagged"><span class="id" title="definition">Tagged</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#TagCountType.I"><span class="id" title="variable">I</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#TagCountType.T_"><span class="id" title="variable">T_</span></a>) (<a class="idref" href="mathcomp.ssreflect.choice.html#unpickle"><span class="id" title="definition">unpickle</span></a> <span class="id" title="var">nx</span>)) (<a class="idref" href="mathcomp.ssreflect.choice.html#unpickle"><span class="id" title="definition">unpickle</span></a> <span class="id" title="var">ni</span>)<br/>
&nbsp;&nbsp;<span class="id" title="keyword">else</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#None"><span class="id" title="constructor">None</span></a>.<br/>
<span class="id" title="keyword">Lemma</span> <a name="pickle_taggedK"><span class="id" title="lemma">pickle_taggedK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#pcancel"><span class="id" title="definition">pcancel</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#pickle_tagged"><span class="id" title="definition">pickle_tagged</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#unpickle_tagged"><span class="id" title="definition">unpickle_tagged</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="tag_countMixin"><span class="id" title="definition">tag_countMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#CountMixin"><span class="id" title="abbreviation">CountMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#pickle_taggedK"><span class="id" title="lemma">pickle_taggedK</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">tag_countType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#CountType"><span class="id" title="abbreviation">CountType</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">{</span></a><span class="id" title="var">i</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#TagCountType.I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">&amp;</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#TagCountType.T_"><span class="id" title="variable">T_</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">}</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#tag_countMixin"><span class="id" title="definition">tag_countMixin</span></a>.<br/>

<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#TagCountType"><span class="id" title="section">TagCountType</span></a>.<br/>

<br/>
</div>

<div class="doc">
 The remaining Canonicals for standard datatypes.  
</div>
<div class="code">
<span class="id" title="keyword">Section</span> <a name="CountableDataTypes"><span class="id" title="section">CountableDataTypes</span></a>.<br/>

<br/>
<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">T</span> : <a class="idref" href="mathcomp.ssreflect.choice.html#countType"><span class="id" title="abbreviation">countType</span></a>.<br/>

<br/>
<span class="id" title="keyword">Lemma</span> <a name="nat_pickleK"><span class="id" title="lemma">nat_pickleK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#pcancel"><span class="id" title="definition">pcancel</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#id"><span class="id" title="abbreviation">id</span></a> (@<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#Some"><span class="id" title="constructor">Some</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>).  <br/>
<span class="id" title="keyword">Definition</span> <a name="nat_countMixin"><span class="id" title="definition">nat_countMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#CountMixin"><span class="id" title="abbreviation">CountMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#nat_pickleK"><span class="id" title="lemma">nat_pickleK</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">nat_countType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#CountType"><span class="id" title="abbreviation">CountType</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#nat_countMixin"><span class="id" title="definition">nat_countMixin</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="bool_countMixin"><span class="id" title="definition">bool_countMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#CanCountMixin"><span class="id" title="definition">CanCountMixin</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#oddb"><span class="id" title="lemma">oddb</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">bool_countType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#CountType"><span class="id" title="abbreviation">CountType</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#bool_countMixin"><span class="id" title="definition">bool_countMixin</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">bitseq_countType</span> :=  <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#be2fb44e85835140c455bc256ce18d4c"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#be2fb44e85835140c455bc256ce18d4c"><span class="id" title="notation">countType</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#be2fb44e85835140c455bc256ce18d4c"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.ssreflect.seq.html#bitseq"><span class="id" title="definition">bitseq</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#be2fb44e85835140c455bc256ce18d4c"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="unit_countMixin"><span class="id" title="definition">unit_countMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#CanCountMixin"><span class="id" title="definition">CanCountMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#bool_of_unitK"><span class="id" title="lemma">bool_of_unitK</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">unit_countType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#CountType"><span class="id" title="abbreviation">CountType</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#unit"><span class="id" title="inductive">unit</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#unit_countMixin"><span class="id" title="definition">unit_countMixin</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="option_countMixin"><span class="id" title="definition">option_countMixin</span></a> <span class="id" title="var">T</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#CanCountMixin"><span class="id" title="definition">CanCountMixin</span></a> (@<a class="idref" href="mathcomp.ssreflect.choice.html#seq_of_optK"><span class="id" title="lemma">seq_of_optK</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">option_countType</span> <span class="id" title="var">T</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#CountType"><span class="id" title="abbreviation">CountType</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#option"><span class="id" title="inductive">option</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>) (<a class="idref" href="mathcomp.ssreflect.choice.html#option_countMixin"><span class="id" title="definition">option_countMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="sig_countMixin"><span class="id" title="definition">sig_countMixin</span></a> <span class="id" title="var">T</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>) := <a class="idref" href="mathcomp.ssreflect.choice.html#99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation">countMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation">of</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation">&lt;:]</span></a>.<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sig_countType</span> <span class="id" title="var">T</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>) :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#CountType"><span class="id" title="abbreviation">CountType</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#sig_countMixin"><span class="id" title="definition">sig_countMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a>).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sig_subCountType</span> <span class="id" title="var">T</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>) :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#8a7192fa64a42310658fd5be07ae4fcc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#8a7192fa64a42310658fd5be07ae4fcc"><span class="id" title="notation">subCountType</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#8a7192fa64a42310658fd5be07ae4fcc"><span class="id" title="notation">of</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">{</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#5b63cb9ed0fed82566685c66e56592e4"><span class="id" title="notation">}</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#8a7192fa64a42310658fd5be07ae4fcc"><span class="id" title="notation">]</span></a>.<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="prod_countMixin"><span class="id" title="definition">prod_countMixin</span></a> <span class="id" title="var">T1</span> <span class="id" title="var">T2</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#CanCountMixin"><span class="id" title="definition">CanCountMixin</span></a> (@<a class="idref" href="mathcomp.ssreflect.choice.html#tag_of_pairK"><span class="id" title="lemma">tag_of_pairK</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T2"><span class="id" title="variable">T2</span></a>).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">prod_countType</span> <span class="id" title="var">T1</span> <span class="id" title="var">T2</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#CountType"><span class="id" title="abbreviation">CountType</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T2"><span class="id" title="variable">T2</span></a>) (<a class="idref" href="mathcomp.ssreflect.choice.html#prod_countMixin"><span class="id" title="definition">prod_countMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T2"><span class="id" title="variable">T2</span></a>).<br/>

<br/>
<span class="id" title="keyword">Definition</span> <a name="sum_countMixin"><span class="id" title="definition">sum_countMixin</span></a> <span class="id" title="var">T1</span> <span class="id" title="var">T2</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#PcanCountMixin"><span class="id" title="definition">PcanCountMixin</span></a> (@<a class="idref" href="mathcomp.ssreflect.choice.html#opair_of_sumK"><span class="id" title="lemma">opair_of_sumK</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T2"><span class="id" title="variable">T2</span></a>).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sum_countType</span> <span class="id" title="var">T1</span> <span class="id" title="var">T2</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#CountType"><span class="id" title="abbreviation">CountType</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#3dcaec3b772747610227247939f96b01"><span class="id" title="notation">+</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T2"><span class="id" title="variable">T2</span></a>) (<a class="idref" href="mathcomp.ssreflect.choice.html#sum_countMixin"><span class="id" title="definition">sum_countMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T1"><span class="id" title="variable">T1</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T2"><span class="id" title="variable">T2</span></a>).<br/>

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<span class="id" title="keyword">Definition</span> <a name="tree_countMixin"><span class="id" title="definition">tree_countMixin</span></a> <span class="id" title="var">T</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#PcanCountMixin"><span class="id" title="definition">PcanCountMixin</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#codeK"><span class="id" title="lemma">GenTree.codeK</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>).<br/>
<span class="id" title="keyword">Canonical</span> <span class="id" title="var">tree_countType</span> <span class="id" title="var">T</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#CountType"><span class="id" title="abbreviation">CountType</span></a> (<a class="idref" href="mathcomp.ssreflect.choice.html#tree"><span class="id" title="inductive">GenTree.tree</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>) (<a class="idref" href="mathcomp.ssreflect.choice.html#tree_countMixin"><span class="id" title="definition">tree_countMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#T"><span class="id" title="variable">T</span></a>).<br/>

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<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.ssreflect.choice.html#CountableDataTypes"><span class="id" title="section">CountableDataTypes</span></a>.<br/>
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