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<!DOCTYPE html>
<html lang="en-US">
<head>
<meta charset="UTF-8">
<title>Squirrel Prover - Squirrel Prover</title>
<link rel="stylesheet" href="style.css">
</head>
<body onkeydown="key(event)">
<span style="display: none;"><span class="squirrel-step" id="step0">
<span class="input-line" id="in0">(* Full model of LAK with pairs and tags, for authentication only. *)<br><br>(* R --> T: nr *)<br>(* T --> R: nT, h(<nR, nT, tag1>, k) *)<br>(* R --> T: h(<h(<nR, nT, tag1>, k), nr, tag2>, k)*)<br><br>set postQuantumSound = true.</span>
<span class="output-line" id="out0"></span>
</span>
<span class="squirrel-step" id="step1">
<span class="input-line" id="in1"><br><br>hash h<br><br>abstract ok:message<br>abstract ko:message<br><br>abstract tag1:message<br>abstract tag2:message<br><br>name key : index->message<br><br>channel cT<br>channel cR<br><br>process tag(i:index) =<br> new nT;<br> in(cR,nR);<br> out(cT,<nT,h(<<nR,nT>,tag1>,key(i))>);<br> in(cR,m3);<br> if m3 = h(<<h(<<nR,nT>,tag1>,key(i)),nR>,tag2>,key(i)) then<br> out(cT,ok)<br> else<br> out(cT,ko)<br><br>process reader =<br> new nR;<br> out(cR,nR);<br> in(cT,x);<br> try find i such that snd(x) = h(<<nR,fst(x)>,tag1>,key(i)) in<br> out(cR,h(<<snd(x),nR>,tag2>,key(i)))<br> else<br> out(cR,ko)<br><br>system ((!_k R: reader) | (!_i !_j T: tag(i))).</span>
<span class="output-line" id="out1">System before processing:<br> <br> (!_k R: <span class="pn" style="font-weight:bold; color: #0000AA">reader</span> ) | (!_i !_j T: <span class="pn" style="font-weight:bold; color: #0000AA">tag</span> i)<br><br>System after processing:<br> <br> (!_k<br> R:<br> <span class="pio" style="font-weight: bold">out</span>(<span class="pc">cR</span>,nR(k));<br> <span class="pio" style="font-weight: bold">in</span>(<span class="pc">cT</span>,<span class="pv" style="font-weight: bold; color: #AA00AA">x</span>);<br> <span class="pc" style="text-decoration: underline; color: #AA0000">find</span> (i) <span class="pc" style="text-decoration: underline; color: #AA0000">such that</span> (snd(x) = h(pair(pair(nR(k),fst(x)),tag1),key(i))) <span class="pc" style="text-decoration: underline; color: #AA0000">in</span><br> R1: <span class="pio" style="font-weight: bold">out</span>(<span class="pc">cR</span>,h(pair(pair(snd(x),nR(k)),tag2),key(i))); <span class="pn" style="font-weight:bold; color: #0000AA">null</span> <span class="pc" style="text-decoration: underline; color: #AA0000">else</span><br> R2: <span class="pio" style="font-weight: bold">out</span>(<span class="pc">cR</span>,ko); <span class="pn" style="font-weight:bold; color: #0000AA">null</span>) |<br> (!_i<br> !_j<br> <span class="pio" style="font-weight: bold">in</span>(<span class="pc">cR</span>,<span class="pv" style="font-weight: bold; color: #AA00AA">nR</span>);<br> T:<br> <span class="pio" style="font-weight: bold">out</span>(<span class="pc">cT</span>,pair(nT(i,j),h(pair(pair(nR,nT(i,j)),tag1),key(i))));<br> <span class="pio" style="font-weight: bold">in</span>(<span class="pc">cR</span>,<span class="pv" style="font-weight: bold; color: #AA00AA">m3</span>);<br> <span class="pc" style="text-decoration: underline; color: #AA0000">if</span> (m3 =<br> h(pair(pair(h(pair(pair(nR,nT(i,j)),tag1),key(i)),nR),tag2),key(i))) <span class="pc" style="text-decoration: underline; color: #AA0000">then</span><br> T1: <span class="pio" style="font-weight: bold">out</span>(<span class="pc">cT</span>,ok); <span class="pn" style="font-weight:bold; color: #0000AA">null</span> <span class="pc" style="text-decoration: underline; color: #AA0000">else</span> T2: <span class="pio" style="font-weight: bold">out</span>(<span class="pc">cT</span>,ko); <span class="pn" style="font-weight:bold; color: #0000AA">null</span>)<br><br>System default registered with actions (init,R,R1,R2,T,T1,T2).<br></span>
</span>
<span class="squirrel-step" id="step2">
<span class="input-line" id="in2"><br><br>axiom tags_neq : tag1 <> tag2.</span>
<span class="output-line" id="out2"></span>
</span>
<span class="squirrel-step" id="step3">
<span class="input-line" id="in3"><br><br>goal wa_R:<br> forall (k:index, i:index),<br> happens(R1(k,i)) =><br> cond@R1(k,i) =><br> exists (j:index),<br> T(i,j) < R1(k,i) &&<br> fst(input@R1(k,i)) = fst(output@T(i,j)) &&<br> snd(input@R1(k,i)) = snd(output@T(i,j)) &&<br> R(k) < T(i,j) &&<br> output@R(k) = input@T(i,j).</span>
<span class="output-line" id="out3">Goal wa_R :<br> forall (k,i:index),<br> <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">R1(k,i)</span>) =><br> <span class="gm" style="font-weight: bold; color: #AA00AA">cond</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> =><br> exists (j:index),<br> <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br> <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br> <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br> <span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br></span>
</span>
<span class="squirrel-step" id="step4">
<span class="input-line" id="in4"><br>Proof.</span>
<span class="output-line" id="out4">[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br><span class="sep" style="font-weight: bold">----------------------------------------</span><br>forall (k,i:index),<br> <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">R1(k,i)</span>) =><br> <span class="gm" style="font-weight: bold; color: #AA00AA">cond</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> =><br> exists (j:index),<br> <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br> <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br> <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br> <span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step5">
<span class="input-line" id="in5"><br> intro k i _ Hc.</span>
<span class="output-line" id="out5">[> Line 56: (intro k i _ Hc) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,k:index<br>Hc: <span class="gm" style="font-weight: bold; color: #AA00AA">cond</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br>_: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">R1(k,i)</span>)<br><span class="sep" style="font-weight: bold">----------------------------------------</span><br>exists (j:index),<br> <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br> <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br> <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br> <span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step6">
<span class="input-line" id="in6"><br> rewrite /cond in Hc.</span>
<span class="output-line" id="out6">[> Line 57: (rewrite ... in Hc) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,k:index<br>Hc: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>_: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">R1(k,i)</span>)<br><span class="sep" style="font-weight: bold">----------------------------------------</span><br>exists (j:index),<br> <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br> <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br> <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br> <span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step7">
<span class="input-line" id="in7"> euf Hc => // _ _ _.</span>
<span class="output-line" id="out7">[> Line 57: ((euf Hc);(intro // _ _ _)) <br>[goal> Focused goal (1/2):<br>System: left:default/left, right:default/right<br>Variables: i,k,k0:index<br>Hc: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>_: <<<span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k0,i)</span>),<span class="gn" style="color: #AA5500">nR</span>(k0)>,<span class="gf" style="font-weight: bold">tag2</span>> = <<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>><br>_: <span class="ga" style="color: #00AA00">R1(k0,i)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span><br>_: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">R1(k,i)</span>)<br><span class="sep" style="font-weight: bold">----------------------------------------</span><br>exists (j:index),<br> <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br> <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br> <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br> <span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step8">
<span class="input-line" id="in8"><br> + by use tags_neq.</span>
<span class="output-line" id="out8">[> Line 58: by (have ... as ) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,j,k:index<br>Hc: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>_: <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>><br>_: <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span><br>_: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">R1(k,i)</span>)<br><span class="sep" style="font-weight: bold">----------------------------------------</span><br>exists (j:index),<br> <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br> <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br> <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br> <span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step9">
<span class="input-line" id="in9"><br> + exists j.</span>
<span class="output-line" id="out9">[> Line 59: (exists j) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,j,k:index<br>Hc: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>_: <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>><br>_: <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span><br>_: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">R1(k,i)</span>)<br><span class="sep" style="font-weight: bold">----------------------------------------</span><br><span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br><span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br><span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br><span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step10">
<span class="input-line" id="in10"><br> assert (nR(k) = input@T(i,j)) as Mfresh by auto.</span>
<span class="output-line" id="out10">[> Line 60: ((have (nR(k) = input@T(i,j)), Mfresh); 1: by auto) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,j,k:index<br>Hc: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Mfresh: <span class="gn" style="color: #AA5500">nR</span>(k) = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br>_: <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>><br>_: <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span><br>_: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">R1(k,i)</span>)<br><span class="sep" style="font-weight: bold">----------------------------------------</span><br><span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br><span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br><span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br><span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step11">
<span class="input-line" id="in11"><br> fresh Mfresh => [Hfresh | Hfresh | [i' Hfresh]] //.</span>
<span class="output-line" id="out11">[> Line 61: ((fresh Mfresh);(intro [Hfresh|Hfresh|[i' Hfresh]] //)) <br>[goal> Focused goal (1/2):<br>System: left:default/left, right:default/right<br>Variables: i,j,k:index<br>Hc: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hfresh: <span class="ga" style="color: #00AA00">R2(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span><br>Mfresh: <span class="gn" style="color: #AA5500">nR</span>(k) = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br>_: <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>><br>_: <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span><br>_: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">R1(k,i)</span>)<br><span class="sep" style="font-weight: bold">----------------------------------------</span><br><span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br><span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br><span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br><span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step12">
<span class="input-line" id="in12"><br> - by depends R(k), R2(k).</span>
<span class="output-line" id="out12">[> Line 62: by (depends R(k), R2(k)) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,i',j,k:index<br>Hc: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hfresh: <span class="ga" style="color: #00AA00">R1(k,i')</span> < <span class="ga" style="color: #00AA00">T(i,j)</span><br>Mfresh: <span class="gn" style="color: #AA5500">nR</span>(k) = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br>_: <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>><br>_: <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span><br>_: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">R1(k,i)</span>)<br><span class="sep" style="font-weight: bold">----------------------------------------</span><br><span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br><span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br><span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) &&<br><span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step13">
<span class="input-line" id="in13"><br> - by depends R(k), R1(k,i').</span>
<span class="output-line" id="out13">[> Line 63: by (depends R(k), R1(k,i')) <br>[goal> Goal wa_R is proved <br></span>
</span>
<span class="squirrel-step" id="step14">
<span class="input-line" id="in14"><br>Qed.</span>
<span class="output-line" id="out14">Exiting proof mode.<br><br><br></span>
</span>
<span class="squirrel-step" id="step15">
<span class="input-line" id="in15"><br><br>goal executable_R1 (t:timestamp) (k,i:index) :<br> happens(t) => exec@t => R1(k,i)<=t => exec@R1(k,i) && cond@R1(k,i).</span>
<span class="output-line" id="out15">Goal executable_R1 :<br> <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(t) => <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@t => <span class="ga" style="color: #00AA00">R1(k,i)</span> <= t => <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">cond</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br></span>
</span>
<span class="squirrel-step" id="step16">
<span class="input-line" id="in16"><br>Proof.</span>
<span class="output-line" id="out16">[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,k:index,t:timestamp<br><span class="sep" style="font-weight: bold">----------------------------------------</span><br><span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(t) => <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@t => <span class="ga" style="color: #00AA00">R1(k,i)</span> <= t => <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">cond</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br><br></span>
</span>
<span class="squirrel-step" id="step17">
<span class="input-line" id="in17"><br> intro _ _ _.</span>
<span class="output-line" id="out17">[> Line 69: (intro _ _ _) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,k:index,t:timestamp<br>_: <span class="ga" style="color: #00AA00">R1(k,i)</span> <= t<br>_: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@t<br>_: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(t)<br><span class="sep" style="font-weight: bold">----------------------------------------</span><br><span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">cond</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br><br></span>
</span>
<span class="squirrel-step" id="step18">
<span class="input-line" id="in18"><br> executable t => // He.</span>
<span class="output-line" id="out18">[> Line 70: ((executable t);(intro // He)) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,k:index,t:timestamp<br>He: forall (t0:timestamp), t0 <= t => <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@t0<br>_: <span class="ga" style="color: #00AA00">R1(k,i)</span> <= t<br>_: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@t<br>_: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(t)<br><span class="sep" style="font-weight: bold">----------------------------------------</span><br><span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">cond</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br><br></span>
</span>
<span class="squirrel-step" id="step19">
<span class="input-line" id="in19"><br> by use He with R1(k,i).</span>
<span class="output-line" id="out19">[> Line 71: by (have ... as ) <br>[goal> Goal executable_R1 is proved <br></span>
</span>
<span class="squirrel-step" id="step20">
<span class="input-line" id="in20"><br>Qed.</span>
<span class="output-line" id="out20">Exiting proof mode.<br><br><br></span>
</span>
<span class="squirrel-step" id="step21">
<span class="input-line" id="in21"><br><br>goal wa_T:<br><br> forall (i:index, j:index),<br> happens(T1(i,j)) =><br> exec@T1(i,j) =><br><br> exists (k:index),<br> R1(k,i) < T1(i,j) &&<br> output@R1(k,i) = input@T1(i,j) &&<br> T(i,j) < R1(k,i) &&<br> fst(output@T(i,j)) = fst(input@R1(k,i)) &&<br> snd(output@T(i,j)) = snd(input@R1(k,i)) &&<br> R(k) < T(i,j) &&<br> output@R(k) = input@T(i,j).</span>
<span class="output-line" id="out21">Goal wa_T :<br> forall (i,j:index),<br> <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>) =><br> <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> =><br> exists (k:index),<br> <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br> <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br></span>
</span>
<span class="squirrel-step" id="step22">
<span class="input-line" id="in22"><br><br>Proof.</span>
<span class="output-line" id="out22">[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br><span class="sep" style="font-weight: bold">----------------------------------------</span><br>forall (i,j:index),<br> <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>) =><br> <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> =><br> exists (k:index),<br> <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br> <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step23">
<span class="input-line" id="in23"><br> intro i j Hh He.</span>
<span class="output-line" id="out23">[> Line 90: (intro i j Hh He) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,j:index<br>He: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>Hh: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>)<br><span class="sep" style="font-weight: bold">----------------------------------------</span><br>exists (k:index),<br> <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br> <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step24">
<span class="input-line" id="in24"><br> assert cond@T1(i,j) as Hc by auto.</span>
<span class="output-line" id="out24">[> Line 91: ((have cond@T1(i,j), Hc); 1: by auto) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,j:index<br>Hc: <span class="gm" style="font-weight: bold; color: #AA00AA">cond</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>He: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>Hh: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>)<br><span class="sep" style="font-weight: bold">----------------------------------------</span><br>exists (k:index),<br> <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br> <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step25">
<span class="input-line" id="in25"><br> use tags_neq as _.</span>
<span class="output-line" id="out25">[> Line 92: (have ... as _) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,j:index<br>Hc: <span class="gm" style="font-weight: bold; color: #AA00AA">cond</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>He: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>Hh: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>)<br>_: <span class="gf" style="font-weight: bold">tag1</span> <> <span class="gf" style="font-weight: bold">tag2</span><br><span class="sep" style="font-weight: bold">----------------------------------------</span><br>exists (k:index),<br> <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br> <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step26">
<span class="input-line" id="in26"><br> rewrite /cond in Hc; euf Hc => // H1t H1m _.</span>
<span class="output-line" id="out26">[> Line 93: (((rewrite ... in Hc);(euf Hc));(intro // H1t H1m _)) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,j,k:index<br>H1m: <<<span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>),<span class="gn" style="color: #AA5500">nR</span>(k)>,<span class="gf" style="font-weight: bold">tag2</span>> =<br> <<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>><br>H1t: <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> || <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span><br>Hc: <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>He: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>Hh: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>)<br>_: <span class="gf" style="font-weight: bold">tag1</span> <> <span class="gf" style="font-weight: bold">tag2</span><br><span class="sep" style="font-weight: bold">----------------------------------------</span><br>exists (k:index),<br> <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br> <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step27">
<span class="input-line" id="in27"><br> assert (snd(input@R1(k,i)) = h(<<input@T(i,j),nT(i,j)>,tag1>,key(i)))<br> as Heuf by auto.</span>
<span class="output-line" id="out27">[> Line 95: ((have<br> (snd(input@R1(k,i)) =<br> h(pair(pair(input@T(i,j),nT(i,j)),tag1),key(i))),<br> Heuf);<br> 1: by auto)<br><br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,j,k:index<br>H1m: <<<span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>),<span class="gn" style="color: #AA5500">nR</span>(k)>,<span class="gf" style="font-weight: bold">tag2</span>> =<br> <<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>><br>H1t: <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> || <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span><br>Hc: <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>He: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>Heuf: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hh: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>)<br>_: <span class="gf" style="font-weight: bold">tag1</span> <> <span class="gf" style="font-weight: bold">tag2</span><br><span class="sep" style="font-weight: bold">----------------------------------------</span><br>exists (k:index),<br> <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br> <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step28">
<span class="input-line" id="in28"><br> euf Heuf => // H2t H2m _.</span>
<span class="output-line" id="out28">[> Line 96: ((euf Heuf);(intro // H2t H2m _)) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,j,j0,k:index<br>H1m: <<<span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>),<span class="gn" style="color: #AA5500">nR</span>(k)>,<span class="gf" style="font-weight: bold">tag2</span>> =<br> <<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>><br>H1t: <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> || <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span><br>H2m: <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j0)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j0)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>><br>H2t: <span class="ga" style="color: #00AA00">T(i,j0)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> || <span class="ga" style="color: #00AA00">T(i,j0)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span><br>Hc: <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>He: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>Heuf: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hh: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>)<br>_: <span class="gf" style="font-weight: bold">tag1</span> <> <span class="gf" style="font-weight: bold">tag2</span><br><span class="sep" style="font-weight: bold">----------------------------------------</span><br>exists (k:index),<br> <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br> <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step29">
<span class="input-line" id="in29"><br> case H1t; case H2t; try auto.</span>
<span class="output-line" id="out29">[> Line 97: ((case H1t);((case H2t);(try auto))) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,j,j0,k:index<br>H1m: <<<span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>),<span class="gn" style="color: #AA5500">nR</span>(k)>,<span class="gf" style="font-weight: bold">tag2</span>> =<br> <<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>><br>H1t: <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span><br>H2m: <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j0)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j0)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>><br>H2t: <span class="ga" style="color: #00AA00">T(i,j0)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span><br>Hc: <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>He: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>Heuf: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hh: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>)<br>_: <span class="gf" style="font-weight: bold">tag1</span> <> <span class="gf" style="font-weight: bold">tag2</span><br><span class="sep" style="font-weight: bold">----------------------------------------</span><br>exists (k:index),<br> <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br> <span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br> <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br> <span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step30">
<span class="input-line" id="in30"><br> exists k.</span>
<span class="output-line" id="out30">[> Line 98: (exists k) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,j,j0,k:index<br>H1m: <<<span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>),<span class="gn" style="color: #AA5500">nR</span>(k)>,<span class="gf" style="font-weight: bold">tag2</span>> =<br> <<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>><br>H1t: <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span><br>H2m: <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j0)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j0)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>><br>H2t: <span class="ga" style="color: #00AA00">T(i,j0)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span><br>Hc: <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>He: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>Heuf: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hh: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>)<br>_: <span class="gf" style="font-weight: bold">tag1</span> <> <span class="gf" style="font-weight: bold">tag2</span><br><span class="sep" style="font-weight: bold">----------------------------------------</span><br><span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br><span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br><span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br><span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br><span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br><span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step31">
<span class="input-line" id="in31"><br> use executable_R1 with T1(i,j),k,i as [He' Hc'] => //.</span>
<span class="output-line" id="out31">[> Line 99: ((have ... as [He' Hc']);(intro //)) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,j,j0,k:index<br>H1m: <<<span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>),<span class="gn" style="color: #AA5500">nR</span>(k)>,<span class="gf" style="font-weight: bold">tag2</span>> =<br> <<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>><br>H1t: <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span><br>H2m: <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j0)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j0)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>><br>H2t: <span class="ga" style="color: #00AA00">T(i,j0)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span><br>Hc: <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hc': <span class="gm" style="font-weight: bold; color: #AA00AA">cond</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br>He: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>He': <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br>Heuf: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hh: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>)<br>_: <span class="gf" style="font-weight: bold">tag1</span> <> <span class="gf" style="font-weight: bold">tag2</span><br><span class="sep" style="font-weight: bold">----------------------------------------</span><br><span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br><span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br><span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br><span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br><span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br><span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step32">
<span class="input-line" id="in32"><br> assert h(<<nR(k),fst(input@R1(k,i))>,tag1>,key(i)) =<br> h(<<input@T(i,j),nT(i,j)>,tag1>,key(i))<br> as Hcoll by auto.</span>
<span class="output-line" id="out32">[> Line 102: ((have<br> (h(pair(pair(nR(k),fst(input@R1(k,i))),tag1),key(i)) =<br> h(pair(pair(input@T(i,j),nT(i,j)),tag1),key(i))),<br> Hcoll);<br> 1: by auto)<br><br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,j,j0,k:index<br>H1m: <<<span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>),<span class="gn" style="color: #AA5500">nR</span>(k)>,<span class="gf" style="font-weight: bold">tag2</span>> =<br> <<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>><br>H1t: <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span><br>H2m: <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j0)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j0)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>><br>H2t: <span class="ga" style="color: #00AA00">T(i,j0)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span><br>Hc: <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hc': <span class="gm" style="font-weight: bold; color: #AA00AA">cond</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br>Hcoll: <span class="gf" style="font-weight: bold">h</span>(<<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)) =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>He: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>He': <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br>Heuf: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hh: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>)<br>_: <span class="gf" style="font-weight: bold">tag1</span> <> <span class="gf" style="font-weight: bold">tag2</span><br><span class="sep" style="font-weight: bold">----------------------------------------</span><br><span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br><span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br><span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br><span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br><span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br><span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step33">
<span class="input-line" id="in33"><br> collision Hcoll => Hcoll'.</span>
<span class="output-line" id="out33">[> Line 103: ((collision Hcoll);(intro Hcoll')) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,j,j0,k:index<br>H1m: <<<span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>),<span class="gn" style="color: #AA5500">nR</span>(k)>,<span class="gf" style="font-weight: bold">tag2</span>> =<br> <<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>><br>H1t: <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span><br>H2m: <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j0)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j0)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>><br>H2t: <span class="ga" style="color: #00AA00">T(i,j0)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span><br>Hc: <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hc': <span class="gm" style="font-weight: bold; color: #AA00AA">cond</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br>Hcoll: <span class="gf" style="font-weight: bold">h</span>(<<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)) =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hcoll': <<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>><br>He: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>He': <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br>Heuf: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hh: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>)<br>_: <span class="gf" style="font-weight: bold">tag1</span> <> <span class="gf" style="font-weight: bold">tag2</span><br><span class="sep" style="font-weight: bold">----------------------------------------</span><br><span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br><span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br><span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br><span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br><span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br><span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step34">
<span class="input-line" id="in34"><br> assert (input@T(i,j) = nR(k)) as Hfresh by auto.</span>
<span class="output-line" id="out34">[> Line 104: ((have (input@T(i,j) = nR(k)), Hfresh); 1: by auto) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,j,j0,k:index<br>H1m: <<<span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>),<span class="gn" style="color: #AA5500">nR</span>(k)>,<span class="gf" style="font-weight: bold">tag2</span>> =<br> <<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>><br>H1t: <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span><br>H2m: <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j0)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j0)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>><br>H2t: <span class="ga" style="color: #00AA00">T(i,j0)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span><br>Hc: <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hc': <span class="gm" style="font-weight: bold; color: #AA00AA">cond</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br>Hcoll: <span class="gf" style="font-weight: bold">h</span>(<<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)) =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hcoll': <<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>><br>He: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>He': <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br>Heuf: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hfresh: <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span> = <span class="gn" style="color: #AA5500">nR</span>(k)<br>Hh: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>)<br>_: <span class="gf" style="font-weight: bold">tag1</span> <> <span class="gf" style="font-weight: bold">tag2</span><br><span class="sep" style="font-weight: bold">----------------------------------------</span><br><span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br><span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br><span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br><span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br><span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br><span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step35">
<span class="input-line" id="in35"><br> fresh Hfresh => [Hfresh' | Hfresh' | [i0 Hfresh']] //.</span>
<span class="output-line" id="out35">[> Line 105: ((fresh Hfresh);(intro [Hfresh'|Hfresh'|[i0 Hfresh']] //)) <br>[goal> Focused goal (1/2):<br>System: left:default/left, right:default/right<br>Variables: i,j,j0,k:index<br>H1m: <<<span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>),<span class="gn" style="color: #AA5500">nR</span>(k)>,<span class="gf" style="font-weight: bold">tag2</span>> =<br> <<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>><br>H1t: <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span><br>H2m: <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j0)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j0)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>><br>H2t: <span class="ga" style="color: #00AA00">T(i,j0)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span><br>Hc: <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hc': <span class="gm" style="font-weight: bold; color: #AA00AA">cond</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br>Hcoll: <span class="gf" style="font-weight: bold">h</span>(<<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)) =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hcoll': <<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>><br>He: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>He': <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br>Heuf: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hfresh: <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span> = <span class="gn" style="color: #AA5500">nR</span>(k)<br>Hfresh': <span class="ga" style="color: #00AA00">R2(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span><br>Hh: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>)<br>_: <span class="gf" style="font-weight: bold">tag1</span> <> <span class="gf" style="font-weight: bold">tag2</span><br><span class="sep" style="font-weight: bold">----------------------------------------</span><br><span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br><span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br><span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br><span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br><span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br><span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step36">
<span class="input-line" id="in36"><br> + by depends R(k), R2(k).</span>
<span class="output-line" id="out36">[> Line 106: by (depends R(k), R2(k)) <br>[goal> Focused goal (1/1):<br>System: left:default/left, right:default/right<br>Variables: i,i0,j,j0,k:index<br>H1m: <<<span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>),<span class="gn" style="color: #AA5500">nR</span>(k)>,<span class="gf" style="font-weight: bold">tag2</span>> =<br> <<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>><br>H1t: <span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span><br>H2m: <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j0)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j0)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>><br>H2t: <span class="ga" style="color: #00AA00">T(i,j0)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span><br>Hc: <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)),<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>>,<span class="gf" style="font-weight: bold">tag2</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hc': <span class="gm" style="font-weight: bold; color: #AA00AA">cond</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br>Hcoll: <span class="gf" style="font-weight: bold">h</span>(<<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i)) =<br> <span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hcoll': <<<span class="gn" style="color: #AA5500">nR</span>(k),<span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>)>,<span class="gf" style="font-weight: bold">tag1</span>> = <<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>><br>He: <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span><br>He': <span class="gm" style="font-weight: bold; color: #AA00AA">exec</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span><br>Heuf: <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) = <span class="gf" style="font-weight: bold">h</span>(<<<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>,<span class="gn" style="color: #AA5500">nT</span>(i,j)>,<span class="gf" style="font-weight: bold">tag1</span>>,<span class="gn" style="color: #AA5500">key</span>(i))<br>Hfresh: <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span> = <span class="gn" style="color: #AA5500">nR</span>(k)<br>Hfresh': <span class="ga" style="color: #00AA00">R1(k,i0)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span><br>Hh: <span class="gm" style="font-weight: bold; color: #AA00AA">happens</span>(<span class="ga" style="color: #00AA00">T1(i,j)</span>)<br>_: <span class="gf" style="font-weight: bold">tag1</span> <> <span class="gf" style="font-weight: bold">tag2</span><br><span class="sep" style="font-weight: bold">----------------------------------------</span><br><span class="ga" style="color: #00AA00">R1(k,i)</span> < <span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br><span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T1(i,j)</span> &&<br><span class="ga" style="color: #00AA00">T(i,j)</span> < <span class="ga" style="color: #00AA00">R1(k,i)</span> &&<br><span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">fst</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br><span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">T(i,j)</span>) = <span class="gf" style="font-weight: bold">snd</span>(<span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">R1(k,i)</span>) &&<br><span class="ga" style="color: #00AA00">R(k)</span> < <span class="ga" style="color: #00AA00">T(i,j)</span> && <span class="gm" style="font-weight: bold; color: #AA00AA">output</span>@<span class="ga" style="color: #00AA00">R(k)</span> = <span class="gm" style="font-weight: bold; color: #AA00AA">input</span>@<span class="ga" style="color: #00AA00">T(i,j)</span><br><br></span>
</span>
<span class="squirrel-step" id="step37">
<span class="input-line" id="in37"><br> + by depends R(k), R1(k,i0).</span>
<span class="output-line" id="out37">[> Line 107: by (depends R(k), R1(k,i0)) <br>[goal> Goal wa_T is proved <br></span>
</span>
<span class="squirrel-step" id="step38">
<span class="input-line" id="in38"><br>Qed.</span>
<span class="output-line" id="out38">Exiting proof mode.<br><br><br></span>
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