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1) Multiply-clocked overlapping implication. </p>
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Let c be the incoming outer clock. Then the clocking of m |-> q is equivalent to the clocking of @(c) m |-> q </p>
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In the presence of the incoming outer clock, m has a well-defined ending clock, and there is a welldefined clock that flows across |->. The multiply-clocked overlapped implication m |-> q is legal for incoming clock c if and only if the following two conditions are met: </p>
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a) Every explicit semantic leading clock of q is identical to the ending clock of m. </p>
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b) If inherited is a semantic leading clock of q, then the ending clock of m is equal to the clock that flows across |->. </p>
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</p>
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For example </p><div class="Element170">
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<div class="Element13"><div class="Element12" id="code00778"><pre class="Element12">@(c) s |-> p1 or @(c2) p2</pre></div></div>
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is not legal because the ending clock of the antecedent is c, while the consequent has c2 as an explicit semantic leading clock. </p>
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Also, </p><div class="Element170">
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<div class="Element13"><div class="Element12" id="code00779"><pre class="Element12">@(c) s ## (@(c1) s1) |-> p</pre></div></div>
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is not legal because the set of semantic leading clocks of p is {inherited}, the ending clock of the antecedent is c1, and the clock that flows across |-> and is inherited by p is c. </p>
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On the other hand, </p><div class="Element170">
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<div class="Element13"><div class="Element12" id="code00780"><pre class="Element12">@(c) s |-> p1 or @(c) p2</pre></div></div>
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and </p><div class="Element170">
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<div class="Element13"><div class="Element12" id="code00781"><pre class="Element12">@(c) s ## @(c1) s1 |-> p1 or @(c1) p2</pre></div></div>
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are both legal. </p>
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</p>
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2) Multiply-clocked if/if...else </p>
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Let c be the incoming outer clock. Then the clocking of if (b) q1 [ else q2 ] is equivalent to the clocking </p>
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of </p><div class="Element170">
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<div class="Element13"><div class="Element12" id="code00782"><pre class="Element12">@(c) if (b) q1 [ else q2 ]</pre></div></div>
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The boolean condition b is clocked by c, so the multiply-clocked if/if...else if (b) q1 [ else q2 ] is legal for incoming clock c if and only if the following condition is met:
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<li class="Element602">Every explicit semantic leading clock of q1 [ or q2 ] is identical to c.</li>
</ul>For example, </p><div class="Element170">
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<div class="Element13"><div class="Element12" id="code00783"><pre class="Element12">@(c) if (b) p1 else @(c) p2</pre></div></div>
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is legal, but </p><div class="Element170">
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<div class="Element13"><div class="Element12" id="code00784"><pre class="Element12">@(c) if (b) @(c) (p1 and @(c2) p2)</pre></div></div>
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is not.</p></div>
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