<html><head><meta http-equiv="Content-Type" content="text/html charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class="">I’ve updated the proposal with comments from the discussions and opened a pull request. Please let me know if you have any feedback before it’s merged:<div class=""><br class=""></div><div class=""><h1 style="box-sizing: border-box; font-size: 2.25em; margin-right: 0px; margin-bottom: 16px; margin-left: 0px; line-height: 1.2; padding-bottom: 0.3em; border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: rgb(238, 238, 238); color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; background-color: rgb(255, 255, 255); margin-top: 0px !important;" class="">More Powerful Constraints for Associated Types</h1><ul style="box-sizing: border-box; padding-left: 2em; margin-top: 0px; margin-bottom: 16px; color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; font-size: 16px; background-color: rgb(255, 255, 255);" class=""><li style="box-sizing: border-box;" class="">Proposal: <a href="https://github.com/apple/swift-evolution/blob/master/proposals/XXXX-associated-types-constraints.md" style="box-sizing: border-box; background-color: transparent; -webkit-text-decoration-skip: objects; color: rgb(64, 120, 192); text-decoration: none;" class="">SE-XXXX</a></li><li style="box-sizing: border-box;" class="">Author(s): <a href="http://github.com/hartbit" style="box-sizing: border-box; background-color: transparent; -webkit-text-decoration-skip: objects; color: rgb(64, 120, 192); text-decoration: none;" class="">David Hart</a>, <a href="https://github.com/hartbit/swift-evolution/blob/associated-types-constraints/proposals/jtbandes@gmail.com" style="box-sizing: border-box; background-color: transparent; -webkit-text-decoration-skip: objects; color: rgb(64, 120, 192); text-decoration: none;" class="">Jacob Bandes-Storch</a>, <a href="https://github.com/hartbit/swift-evolution/blob/associated-types-constraints/proposals/dgregor@apple.com" style="box-sizing: border-box; background-color: transparent; -webkit-text-decoration-skip: objects; color: rgb(64, 120, 192); text-decoration: none;" class="">Douglas Gregor</a></li><li style="box-sizing: border-box;" class="">Status: TBD</li><li style="box-sizing: border-box;" class="">Review manager: TBD</li></ul><h2 style="box-sizing: border-box; margin-top: 1em; margin-bottom: 16px; line-height: 1.225; font-size: 1.75em; padding-bottom: 0.3em; border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: rgb(238, 238, 238); color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; background-color: rgb(255, 255, 255);" class=""><a id="user-content-introduction" class="anchor" href="https://github.com/hartbit/swift-evolution/blob/associated-types-constraints/proposals/XXXX-associated-types-constraints.md#introduction" aria-hidden="true" style="box-sizing: border-box; background-color: transparent; -webkit-text-decoration-skip: objects; color: rgb(64, 120, 192); text-decoration: none; display: inline-block; padding-right: 2px; margin-left: -18px; line-height: 1;"><svg aria-hidden="true" class="octicon octicon-link" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path d="M4 9h1v1h-1c-1.5 0-3-1.69-3-3.5s1.55-3.5 3-3.5h4c1.45 0 3 1.69 3 3.5 0 1.41-0.91 2.72-2 3.25v-1.16c0.58-0.45 1-1.27 1-2.09 0-1.28-1.02-2.5-2-2.5H4c-0.98 0-2 1.22-2 2.5s1 2.5 2 2.5z m9-3h-1v1h1c1 0 2 1.22 2 2.5s-1.02 2.5-2 2.5H9c-0.98 0-2-1.22-2-2.5 0-0.83 0.42-1.64 1-2.09v-1.16c-1.09 0.53-2 1.84-2 3.25 0 1.81 1.55 3.5 3 3.5h4c1.45 0 3-1.69 3-3.5s-1.5-3.5-3-3.5z"></path></svg></a>Introduction</h2><p style="box-sizing: border-box; margin-top: 0px; margin-bottom: 16px; color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; font-size: 16px; background-color: rgb(255, 255, 255);" class="">This proposal seeks to introduce a <code style="box-sizing: border-box; font-family: Consolas, 'Liberation Mono', Menlo, Courier, monospace; font-size: 14px; padding: 0.2em 0px; margin: 0px; background-color: rgba(0, 0, 0, 0.0392157); border-top-left-radius: 3px; border-top-right-radius: 3px; border-bottom-right-radius: 3px; border-bottom-left-radius: 3px;" class="">where</code> clause to associated types declarations to bring the same expressive power as generic type constraints.</p><p style="box-sizing: border-box; margin-top: 0px; margin-bottom: 16px; color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; font-size: 16px; background-color: rgb(255, 255, 255);" class="">This proposal was discussed twice on the Swift Evolution list in the following threads:</p><ul style="box-sizing: border-box; padding-left: 2em; margin-top: 0px; margin-bottom: 16px; color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; font-size: 16px; background-color: rgb(255, 255, 255);" class=""><li style="box-sizing: border-box;" class=""><a href="http://thread.gmane.org/gmane.comp.lang.swift.evolution/14243" style="box-sizing: border-box; background-color: transparent; -webkit-text-decoration-skip: objects; color: rgb(64, 120, 192); text-decoration: none;" class="">[Completing Generics] Arbitrary requirements in protocols</a></li><li style="box-sizing: border-box;" class=""><a href="http://thread.gmane.org/gmane.comp.lang.swift.evolution/15201" style="box-sizing: border-box; background-color: transparent; -webkit-text-decoration-skip: objects; color: rgb(64, 120, 192); text-decoration: none;" class="">More Powerful Constraints for Associated Types</a></li></ul><h2 style="box-sizing: border-box; margin-top: 1em; margin-bottom: 16px; line-height: 1.225; font-size: 1.75em; padding-bottom: 0.3em; border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: rgb(238, 238, 238); color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; background-color: rgb(255, 255, 255);" class=""><a id="user-content-motivation" class="anchor" href="https://github.com/hartbit/swift-evolution/blob/associated-types-constraints/proposals/XXXX-associated-types-constraints.md#motivation" aria-hidden="true" style="box-sizing: border-box; background-color: transparent; -webkit-text-decoration-skip: objects; color: rgb(64, 120, 192); text-decoration: none; display: inline-block; padding-right: 2px; margin-left: -18px; line-height: 1;"><svg aria-hidden="true" class="octicon octicon-link" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path d="M4 9h1v1h-1c-1.5 0-3-1.69-3-3.5s1.55-3.5 3-3.5h4c1.45 0 3 1.69 3 3.5 0 1.41-0.91 2.72-2 3.25v-1.16c0.58-0.45 1-1.27 1-2.09 0-1.28-1.02-2.5-2-2.5H4c-0.98 0-2 1.22-2 2.5s1 2.5 2 2.5z m9-3h-1v1h1c1 0 2 1.22 2 2.5s-1.02 2.5-2 2.5H9c-0.98 0-2-1.22-2-2.5 0-0.83 0.42-1.64 1-2.09v-1.16c-1.09 0.53-2 1.84-2 3.25 0 1.81 1.55 3.5 3 3.5h4c1.45 0 3-1.69 3-3.5s-1.5-3.5-3-3.5z"></path></svg></a>Motivation</h2><p style="box-sizing: border-box; margin-top: 0px; margin-bottom: 16px; color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; font-size: 16px; background-color: rgb(255, 255, 255);" class="">Currently, associated type declarations can only express simple inheritance constraints and not the more sophisticated constraints available to generic types with the <code style="box-sizing: border-box; font-family: Consolas, 'Liberation Mono', Menlo, Courier, monospace; font-size: 14px; padding: 0.2em 0px; margin: 0px; background-color: rgba(0, 0, 0, 0.0392157); border-top-left-radius: 3px; border-top-right-radius: 3px; border-bottom-right-radius: 3px; border-bottom-left-radius: 3px;" class="">where</code> clause. Some designs, including many in the Standard Library, require more powerful constraints for associated types to be truly elegant. For example, the <code style="box-sizing: border-box; font-family: Consolas, 'Liberation Mono', Menlo, Courier, monospace; font-size: 14px; padding: 0.2em 0px; margin: 0px; background-color: rgba(0, 0, 0, 0.0392157); border-top-left-radius: 3px; border-top-right-radius: 3px; border-bottom-right-radius: 3px; border-bottom-left-radius: 3px;" class="">SequenceType</code> protocol can be declared as follows:</p><div class="highlight highlight-source-swift" style="box-sizing: border-box; margin-bottom: 16px; color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; font-size: 16px; background-color: rgb(255, 255, 255);"><pre style="box-sizing: border-box; font-family: Consolas, 'Liberation Mono', Menlo, Courier, monospace; font-size: 14px; margin-top: 0px; margin-bottom: 0px; line-height: 1.45; padding: 16px; overflow: auto; background-color: rgb(247, 247, 247); border-top-left-radius: 3px; border-top-right-radius: 3px; border-bottom-right-radius: 3px; border-bottom-left-radius: 3px; word-wrap: normal; word-break: normal;" class=""><span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">protocol</span> Sequence {
associatedtype Iterator <span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">:</span> IteratorProtocol
associatedtype SubSequence <span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">:</span> Sequence <span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">where</span> SubSequence<span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">.</span>Iterator<span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">.</span>Element <span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">==</span> Iterator<span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">.</span>Element
<span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">...</span>
}</pre></div><h2 style="box-sizing: border-box; margin-top: 1em; margin-bottom: 16px; line-height: 1.225; font-size: 1.75em; padding-bottom: 0.3em; border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: rgb(238, 238, 238); color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; background-color: rgb(255, 255, 255);" class=""><a id="user-content-detail-design" class="anchor" href="https://github.com/hartbit/swift-evolution/blob/associated-types-constraints/proposals/XXXX-associated-types-constraints.md#detail-design" aria-hidden="true" style="box-sizing: border-box; background-color: transparent; -webkit-text-decoration-skip: objects; color: rgb(64, 120, 192); text-decoration: none; display: inline-block; padding-right: 2px; margin-left: -18px; line-height: 1;"><svg aria-hidden="true" class="octicon octicon-link" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path d="M4 9h1v1h-1c-1.5 0-3-1.69-3-3.5s1.55-3.5 3-3.5h4c1.45 0 3 1.69 3 3.5 0 1.41-0.91 2.72-2 3.25v-1.16c0.58-0.45 1-1.27 1-2.09 0-1.28-1.02-2.5-2-2.5H4c-0.98 0-2 1.22-2 2.5s1 2.5 2 2.5z m9-3h-1v1h1c1 0 2 1.22 2 2.5s-1.02 2.5-2 2.5H9c-0.98 0-2-1.22-2-2.5 0-0.83 0.42-1.64 1-2.09v-1.16c-1.09 0.53-2 1.84-2 3.25 0 1.81 1.55 3.5 3 3.5h4c1.45 0 3-1.69 3-3.5s-1.5-3.5-3-3.5z"></path></svg></a>Detail Design</h2><p style="box-sizing: border-box; margin-top: 0px; margin-bottom: 16px; color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; font-size: 16px; background-color: rgb(255, 255, 255);" class="">With this proposal, the grammar for protocols associated types would be modified to:</p><p style="box-sizing: border-box; margin-top: 0px; margin-bottom: 16px; color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; font-size: 16px; background-color: rgb(255, 255, 255);" class=""><em style="box-sizing: border-box;" class="">protocol-associated-type-declaration</em> → <em style="box-sizing: border-box;" class="">attributes<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline; bottom: -0.25em;" class="">opt</span></em> <em style="box-sizing: border-box;" class="">access-level-modifier<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline; bottom: -0.25em;" class="">opt</span></em> <span style="box-sizing: border-box;" class="">associatedtype</span> <em style="box-sizing: border-box;" class="">typealias-name</em> <em style="box-sizing: border-box;" class="">type-inheritance-clause<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline; bottom: -0.25em;" class="">opt</span></em> <em style="box-sizing: border-box;" class="">typealias-assignment<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline; bottom: -0.25em;" class="">opt</span></em> <em style="box-sizing: border-box;" class="">requirement-clause<span style="box-sizing: border-box; font-size: 12px; line-height: 0; position: relative; vertical-align: baseline; bottom: -0.25em;" class="">opt</span></em></p><p style="box-sizing: border-box; margin-top: 0px; margin-bottom: 16px; color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; font-size: 16px; background-color: rgb(255, 255, 255);" class="">The new requirement-clause is then used by the compiler to validate the associated types of conforming types.</p><p style="box-sizing: border-box; margin-top: 0px; margin-bottom: 16px; color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; font-size: 16px; background-color: rgb(255, 255, 255);" class="">The proposal also allows protocols to use the associated types of their conforming protocols in their declaration where clause as below:</p><div class="highlight highlight-source-swift" style="box-sizing: border-box; margin-bottom: 16px; color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; font-size: 16px; background-color: rgb(255, 255, 255);"><pre style="box-sizing: border-box; font-family: Consolas, 'Liberation Mono', Menlo, Courier, monospace; font-size: 14px; margin-top: 0px; margin-bottom: 0px; line-height: 1.45; padding: 16px; overflow: auto; background-color: rgb(247, 247, 247); border-top-left-radius: 3px; border-top-right-radius: 3px; border-bottom-right-radius: 3px; border-bottom-left-radius: 3px; word-wrap: normal; word-break: normal;" class=""><span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">protocol</span> IntSequence <span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">:</span> Sequence <span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">where</span> Iterator<span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">.</span>Element <span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">==</span> <span class="pl-c1" style="box-sizing: border-box; color: rgb(0, 134, 179);">Int</span> {
<span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">...</span>
}</pre></div><h2 style="box-sizing: border-box; margin-top: 1em; margin-bottom: 16px; line-height: 1.225; font-size: 1.75em; padding-bottom: 0.3em; border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: rgb(238, 238, 238); color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; background-color: rgb(255, 255, 255);" class=""><a id="user-content-alternatives" class="anchor" href="https://github.com/hartbit/swift-evolution/blob/associated-types-constraints/proposals/XXXX-associated-types-constraints.md#alternatives" aria-hidden="true" style="box-sizing: border-box; background-color: transparent; -webkit-text-decoration-skip: objects; color: rgb(64, 120, 192); text-decoration: none; display: inline-block; padding-right: 2px; margin-left: -18px; line-height: 1;"><svg aria-hidden="true" class="octicon octicon-link" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path d="M4 9h1v1h-1c-1.5 0-3-1.69-3-3.5s1.55-3.5 3-3.5h4c1.45 0 3 1.69 3 3.5 0 1.41-0.91 2.72-2 3.25v-1.16c0.58-0.45 1-1.27 1-2.09 0-1.28-1.02-2.5-2-2.5H4c-0.98 0-2 1.22-2 2.5s1 2.5 2 2.5z m9-3h-1v1h1c1 0 2 1.22 2 2.5s-1.02 2.5-2 2.5H9c-0.98 0-2-1.22-2-2.5 0-0.83 0.42-1.64 1-2.09v-1.16c-1.09 0.53-2 1.84-2 3.25 0 1.81 1.55 3.5 3 3.5h4c1.45 0 3-1.69 3-3.5s-1.5-3.5-3-3.5z"></path></svg></a>Alternatives</h2><p style="box-sizing: border-box; margin-top: 0px; margin-bottom: 16px; color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; font-size: 16px; background-color: rgb(255, 255, 255);" class="">Douglas Gregor argues that the proposed syntax is redundant when adding new constraints to an associated type declared in a parent protocol and proposes another syntax: </p><div class="highlight highlight-source-swift" style="box-sizing: border-box; margin-bottom: 16px; color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; font-size: 16px; background-color: rgb(255, 255, 255);"><pre style="box-sizing: border-box; font-family: Consolas, 'Liberation Mono', Menlo, Courier, monospace; font-size: 14px; margin-top: 0px; margin-bottom: 0px; line-height: 1.45; padding: 16px; overflow: auto; background-color: rgb(247, 247, 247); border-top-left-radius: 3px; border-top-right-radius: 3px; border-bottom-right-radius: 3px; border-bottom-left-radius: 3px; word-wrap: normal; word-break: normal;" class=""><span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">protocol</span> Collection <span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">:</span> Sequence {
<span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">where</span> SubSequence <span class="pl-k" style="box-sizing: border-box; color: rgb(167, 29, 93);">:</span> Collection
}</pre></div><p style="box-sizing: border-box; margin-top: 0px; margin-bottom: 16px; color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; font-size: 16px; background-color: rgb(255, 255, 255);" class="">But as Douglas notes himself, that syntax will become ambiguous if we adopt the generic where clause at the end of declarations like discussed in proposal <a href="https://github.com/apple/swift-evolution/blob/master/proposals/0081-move-where-expression.md" style="box-sizing: border-box; background-color: transparent; -webkit-text-decoration-skip: objects; color: rgb(64, 120, 192); text-decoration: none;" class="">SE-0081: Move where clause to end of declaration</a>. For those reasons, it might be wiser not to introduce the shorthand syntax.</p><h2 style="box-sizing: border-box; margin-top: 1em; margin-bottom: 16px; line-height: 1.225; font-size: 1.75em; padding-bottom: 0.3em; border-bottom-width: 1px; border-bottom-style: solid; border-bottom-color: rgb(238, 238, 238); color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; background-color: rgb(255, 255, 255);" class=""><a id="user-content-acknowledgements" class="anchor" href="https://github.com/hartbit/swift-evolution/blob/associated-types-constraints/proposals/XXXX-associated-types-constraints.md#acknowledgements" aria-hidden="true" style="box-sizing: border-box; background-color: transparent; -webkit-text-decoration-skip: objects; color: rgb(64, 120, 192); text-decoration: none; display: inline-block; padding-right: 2px; margin-left: -18px; line-height: 1;"><svg aria-hidden="true" class="octicon octicon-link" height="16" version="1.1" viewBox="0 0 16 16" width="16"><path d="M4 9h1v1h-1c-1.5 0-3-1.69-3-3.5s1.55-3.5 3-3.5h4c1.45 0 3 1.69 3 3.5 0 1.41-0.91 2.72-2 3.25v-1.16c0.58-0.45 1-1.27 1-2.09 0-1.28-1.02-2.5-2-2.5H4c-0.98 0-2 1.22-2 2.5s1 2.5 2 2.5z m9-3h-1v1h1c1 0 2 1.22 2 2.5s-1.02 2.5-2 2.5H9c-0.98 0-2-1.22-2-2.5 0-0.83 0.42-1.64 1-2.09v-1.16c-1.09 0.53-2 1.84-2 3.25 0 1.81 1.55 3.5 3 3.5h4c1.45 0 3-1.69 3-3.5s-1.5-3.5-3-3.5z"></path></svg></a>Acknowledgements</h2><div style="box-sizing: border-box; margin-top: 0px; color: rgb(51, 51, 51); font-family: 'Helvetica Neue', Helvetica, 'Segoe UI', Arial, freesans, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; font-size: 16px; background-color: rgb(255, 255, 255); margin-bottom: 0px !important;" class="">Thanks to Dave Abrahams and Douglas Gregor for taking the time to help me through this proposal.</div></div></body></html>