<div dir="ltr">On Fri, Jul 22, 2016 at 9:19 PM, Matthew Johnson <span dir="ltr"><<a href="mailto:matthew@anandabits.com" target="_blank">matthew@anandabits.com</a>></span> wrote:<br><div class="gmail_extra"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div style="word-wrap:break-word"><br><div><span class=""><blockquote type="cite"><div>On Jul 22, 2016, at 9:12 PM, Xiaodi Wu <<a href="mailto:xiaodi.wu@gmail.com" target="_blank">xiaodi.wu@gmail.com</a>> wrote:</div><br><div><div dir="ltr">On Fri, Jul 22, 2016 at 9:09 PM, Jaden Geller <span dir="ltr"><<a href="mailto:jaden.geller@gmail.com" target="_blank">jaden.geller@gmail.com</a>></span> wrote:<br><div class="gmail_extra"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div style="word-wrap:break-word"><span><blockquote type="cite"><div dir="ltr"><div class="gmail_extra"><div class="gmail_quote"><div>1. NaN != NaN and +0 == -0 [what the traditional comparison operators are constrained to do]</div><div>2. NaN == NaN, +0 == -0, and the same number encoded different ways compare equal</div><div>3. NaN == NaN, +0 != -0, and the same number encoded different ways compare not equal</div></div></div></div></blockquote></span><div><div dir="ltr"><div class="gmail_extra"><div class="gmail_quote"><div><br></div><div>Though it seems super confusing that a language have THREE ways to compare values, that does almost seem necessary here. Do we actually need an operator that performs #3? I understand that that is equality under total ordering, but couldn't users just write `(a <=> b) == .same` if they want that?</div></div></div></div></div></div></blockquote><div><br></div><div>For floating point types, I think `===` shouldn't be #3. From a practical standpoint, no one ever wants that definition unless they are ordering things. Whereas you'd want #2 for things like `.index(of:)` and #1 for the traditional comparison operators.</div></div></div></div></div></blockquote><div><br></div></span><div>However, we have to introduce a new notion of identity for floating point types if `===` isn’t #3. Floating points are tricky enough already. Is that really a good thing?</div><div><br></div><div>Further, it encodes three separate meanings of equality in the protocols. We should avoid that if we can.</div><div><br></div><div>It feels like maybe the right solution is floating point specific algorithm overloads. It doesn’t seem like too big a surprise that this is the case when you really dig into the details.</div></div></div></blockquote><div><br></div><div><br></div><div>I see what you're getting at here. But I like your other alternative better, which is to define identity in a generically useful way for floating point types, and preserve IEEE semantics in its own method for floating point types.</div><div><br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div style="word-wrap:break-word"><div><div><div class="h5"><blockquote type="cite"><div><div dir="ltr"><div class="gmail_extra"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div style="word-wrap:break-word"><div><div><div><blockquote type="cite"><div>On Jul 22, 2016, at 7:04 PM, Xiaodi Wu <<a href="mailto:xiaodi.wu@gmail.com" target="_blank">xiaodi.wu@gmail.com</a>> wrote:</div><br><div><div dir="ltr" style="font-family:Helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px">On Fri, Jul 22, 2016 at 8:57 PM, Matthew Johnson<span> </span><span dir="ltr"><<a href="mailto:matthew@anandabits.com" target="_blank">matthew@anandabits.com</a>></span><span> </span>wrote:<br><div class="gmail_extra"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-color:rgb(204,204,204);border-left-style:solid;padding-left:1ex"><div style="word-wrap:break-word"><br><div><span><blockquote type="cite"><div>On Jul 22, 2016, at 8:54 PM, Xiaodi Wu via swift-evolution <<a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a>> wrote:</div><br><div><div dir="ltr">On Fri, Jul 22, 2016 at 8:52 PM, Jaden Geller via swift-evolution<span> </span><span dir="ltr"><<a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a>></span><span> </span>wrote:<br><div class="gmail_extra"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-color:rgb(204,204,204);border-left-style:solid;padding-left:1ex"><div style="word-wrap:break-word">"The totalOrder predicate will order these cases, and it also distinguishes between different representations of NaNs and between the same decimal floating point number encoded in different ways."<div>- [Wikipedia](<a href="https://en.wikipedia.org/wiki/IEEE_floating_point#Total-ordering_predicate" target="_blank">https://en.wikipedia.org/wiki/IEEE_floating_point#Total-ordering_predicate</a>)<br><div><br></div><div>Sounds like `===` should not return `true` for zeros of different signs, then.</div></div></div></blockquote><div><br></div><div>Fair enough; the result of that will be, as Pyry noted above, that:</div><div><br></div><div>```</div><div><span style="font-size:13px">[-0.0, 1.0, .nan, 0.0].firstIndex(of: 0.0)</span> <span style="font-size:13px">//=> 3, not 0</span></div><div><span style="font-size:13px">```</span></div></div></div></div></div></blockquote><div><br></div></span><div>Maybe we need floating point specific implementations of some algorithms to resolve this problem?</div><div><br></div><div>It doesn’t seem like there is a way to provide the semantics required by generic algorithms and still provide the expected behavior for floating point values. </div></div></div></blockquote><div><br></div><div>Well, what I'm trying to say is that generic algorithms such as `index(of:)` require only an equivalence relation. For floating point types, there are three ways to slice it:</div><div><br></div><div>1. NaN != NaN and +0 == -0 [what the traditional comparison operators are constrained to do]</div><div>2. NaN == NaN, +0 == -0, and the same number encoded different ways compare equal</div><div>3. NaN == NaN, +0 != -0, and the same number encoded different ways compare not equal</div><div><br></div><div>Both #2 and #3 can fall out of valid equivalence relations; if `===` behaved like #2 for FloatingPoint types, then generic algorithms work just fine. If we insist on using a total ordering defined by `<=>` all the time, then we've got problems.</div><div><br></div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-color:rgb(204,204,204);border-left-style:solid;padding-left:1ex"><div style="word-wrap:break-word"><div><div><div><br><blockquote type="cite"><div><div dir="ltr"><div class="gmail_extra"><div class="gmail_quote"><div><span style="font-size:13px"><br></span></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-color:rgb(204,204,204);border-left-style:solid;padding-left:1ex"><div style="word-wrap:break-word"><div><div><div><div><div><blockquote type="cite"><div>On Jul 22, 2016, at 6:48 PM, Dave Abrahams via swift-evolution <<a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a>> wrote:</div><br><div><br style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><span style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;float:none;display:inline">on Fri Jul 22 2016, Jaden Geller <</span><a href="mailto:swift-evolution@swift.org" style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px" target="_blank">swift-evolution@swift.org</a><span style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;float:none;display:inline">> wrote:</span><br style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><br style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><blockquote type="cite" style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><blockquote type="cite">For floating point, I'd hope that `a === b` if `(a <=> b) == .same`<br>*but not iff*. This is to satisfy IEEE 754: "Comparisons shall<br>ignore the sign of zero (so +0 = −0)".<br></blockquote><br>I don't see why both `(+0) === (-0)` and `(+0) <=> (-0)` can't return<br>`true` and `.same`, respectively. This doesn't break the total<br>ordering of values. `===` doesn't do raw memory comparison. They're<br>"identical", so it ought to return `true`.<br></blockquote><br style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><span style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;float:none;display:inline">It ought to do whatever IEEE-754 specifies that its total ordering test</span><br style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><span style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;float:none;display:inline">does. That is, IEEE-754 gets to decide whether the difference between</span><br style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><span style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;float:none;display:inline">+0 and -0 is “essential” to IEEE-754 floating point types, or not.</span><br style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><br style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><blockquote type="cite" style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><br><blockquote type="cite">On Jul 22, 2016, at 6:37 PM, Xiaodi Wu via swift-evolution<br><<a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a>> wrote:<br><br>On Fri, Jul 22, 2016 at 8:20 PM, Dave Abrahams via swift-evolution<br><<a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a>>><br>wrote:<br><br>on Fri Jul 22 2016, Daniel Duan <<a href="http://daniel-at-duan.org/" target="_blank">daniel-AT-duan.org</a>> wrote:<br><br><blockquote type="cite"><blockquote type="cite">On Jul 22, 2016, at 3:00 PM, Dave Abrahams via swift-evolution<br><<a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a>>><br>wrote:<br><br><br>on Fri Jul 22 2016, Daniel Duan<br><<a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a>><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a>>>><br>wrote:<br><br></blockquote><br><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">On Jul 22, 2016, at 11:05 AM, Dave Abrahams via swift-evolution<br><<a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a>><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a>>>><br>wrote:<br><br><br>on Thu Jul 21 2016, Duan<br></blockquote><br><blockquote type="cite"><<a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a>><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a>>><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a>><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a>>>>><br>wrote:<br><br><blockquote type="cite">Great proposal. I want to second that areSame may mislead user to<br>think this is about identity.<br><br>I like areEquivalent() but there may be better names.<br></blockquote><br>It really *is* about identity as I posted in a previous message. But<br>that doesn't change the fact that areEquivalent might be a better name.<br>It's one of the things we considered; it just seemed long for no real<br>benefit.<br><br></blockquote><br>If the addresses of the arguments aren’t being used, then we don’t consider<br>them part of their *identity*. I can follow this logic. My fear is most users<br>won’t make this leap on their own and get the same initial impression as I did.<br>It's entirely possible this fear is unfounded. Some educated bikesheding<br>wouldn't hurt here IMO :)<br></blockquote><br>Well, it's still a very real question whether we ought to have the<br>additional API surface implied by areSame, or wether we should collapse<br>it with ===.<br><br></blockquote><br>To spell this out (because I had to think about it for a second): === will be derived from<br><=>,<br>but also becomes default implementation for ==, which remains open for<br>customization.<br></blockquote><br>I was imagining roughly this (untested):<br><br> /// Two references are identical if they refer to the same<br> /// instance.<br> ///<br> /// - Note: Classes with a more-refined notion of “identical”<br> /// should conform to `Identifiable` and implement `===`.<br> func ===(lhs: AnyObject, rhs: AnyObject) -> Bool {<br> ObjectIdentifier(lhs) == ObjectIdentifier(rhs)<br> }<br><br> /// Supports testing that two values of `Self` are identical<br> ///<br> /// If `a` and `b` are of type `Self`, `a === b` means that<br> /// `a` and `b` are interchangeable in most code. A conforming<br> /// type can document that specific observable characteristics<br> /// (such as the `capacity` of an `Array`) are inessential and<br> /// thus not to be considered as part of the interchangeability<br> /// guarantee.<br> ///<br> /// - Requires: `===` induces an equivalence relation over<br> /// instances.<br> /// - Note: conforming types will gain an `==` operator that<br> /// forwards to `===`.<br> /// - Note: Types that require domain-specific `==`<br> /// implementations with different semantics (e.g. floating<br> /// point) should define a more-specific overload of `==`,<br> /// which will be used in contexts where the static type is<br> /// known to the compiler.<br> /// - Note: Generic code should usually use `==` to compare<br> /// conforming instances; that will always dispatch to `===`<br> /// and will be unaffected by more specific overloads of<br> /// `==`.<br> protocol Identifiable { // née Equatable name is negotiable<br> func ===(_: Self, _: aSelf) -> Bool<br> }<br><br> /// Default definition of `==` for Identifiable types.<br> func ==<T: Identifiable>(lhs: T, rhs: T) -> Bool {<br> return lhs === rhs<br> }<br><br> /// Conforming types have a default total ordering.<br> ///<br> /// If `a` and `b` are of type `Self`, `a <=> b` means that<br> /// `a` and `b` are interchangeable in most code. A conforming<br> /// type can document that specific observable characteristics<br> /// (such as the `capacity` of an `Array`) are inessential and<br> /// thus not to be considered as part of the interchangeability<br> /// guarantee.<br> ///<br> /// - Requires: `<=>` induces a total ordering over<br> /// instances.<br> /// - Requires: the semantics of `<=>` are consistent with<br> /// those of `===`. That is, `(a <=> b) == .equivalent`<br> /// iff `a === b`.<br><br>For floating point, I'd hope that `a === b` if `(a <=> b) == .same` *but not iff*. This is to satisfy IEEE 754: "Comparisons shall ignore the sign of zero (so +0 = −0)".<br><br> /// - Note: conforming types will gain `<`, `<=`, `>`, and `>=`<br> /// operators defined in terms of `<=>`.<br> /// - Note: Types that require domain-specific `<`, etc.<br> /// implementations with different semantics (e.g. floating<br> /// point) should define more-specific overloads of those<br> /// operators, which will be used in contexts where the<br> /// static type is known to the compiler.<br> /// - Note: Generic code can freely use `<=>` or the traditional<br> /// comparison operators to compare conforming instances;<br> /// the result will always be supplied by `<=>`<br> /// and will be unaffected by more specific overloads of<br> /// the other operators.<br> protocol Comparable : Identifiable {<br> func <=> (lhs: Self, rhs: Self) -> Ordering<br> }<br><br> /// Default implementations of `<`, `<=`, `>`, and `>=`.<br> extension Comparable {<br> static func <(lhs: Self, rhs: Self) -> Bool {<br> return (lhs <=> rhs) == .ascending<br> }<br> static func <=(lhs: Self, rhs: Self) -> Bool {<br> return (rhs <=> lhs) != .ascending<br> }<br> static func >(lhs: Self, rhs: Self) -> Bool {<br> return (lhs <=> rhs) == .descending<br> }<br> static func >=(lhs: Self, rhs: Self) -> Bool {<br> return (rhs <=> lhs) != .descending<br> }<br> }<br><br><blockquote type="cite">I like this idea. If we keep === as a separate thing, now users have 3 “opportunities” to define<br>equality. The must be few, if any, use cases for this.<br><br>Would love to see if anyone on the list can give us an example. Otherwise we should make<br>areSame === again™!<br><br><blockquote type="cite"><blockquote type="cite"><br><blockquote type="cite"><blockquote type="cite">Daniel Duan<br>Sent from my iPhone<br><br><blockquote type="cite">On Jul 21, 2016, at 6:32 PM, Robert Widmann via swift-evolution<br><<a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a>>><br>wrote:<br><br><br><blockquote type="cite">On Jul 21, 2016, at 6:19 PM, Xiaodi Wu<br><<a href="mailto:xiaodi.wu@gmail.com" target="_blank">xiaodi.wu@gmail.com</a><br><<a href="mailto:xiaodi.wu@gmail.com" target="_blank">mailto:xiaodi.wu@gmail.com</a>>><br>wrote:<br><br>This is nice. Is `areSame()` being proposed because static `==` is<br>the status quo and you're trying to make the point that `==` in the<br>future need not guarantee the same semantics?<br></blockquote><br>Yep! Equivalence and equality are strictly very different things.<br><br><blockquote type="cite"><br>Nit: I think the more common term in stdlib would be<br>`areEquivalent()`. Do you think `same` in that context (independent<br>of the word "ordering") might erroneously suggest identity?<br></blockquote><br>There is room for improvement here. Keep ‘em coming.<br><br><blockquote type="cite"><br><br><blockquote type="cite">On Thu, Jul 21, 2016 at 8:11 PM, Robert Widmann via<br>swift-evolution<br><<a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a>>><br>wrote:<br>Hello Swift Community,<br><br>Harlan Haskins, Jaden Geller, and I have been working on a<br>proposal to clean up the semantics of ordering relations in the<br>standard library. We have a draft that you can get as a gist.<br>Any feedback you might have about this proposal helps - though<br>please keeps your comments on Swift-Evolution and not on the gist.<br><br>Cheers,<br><br>~Robert Widmann<br><br><br><br><br><br>_______________________________________________<br>swift-evolution mailing list<br><a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a>><br><a href="https://lists.swift.org/mailman/listinfo/swift-evolution" target="_blank">https://lists.swift.org/mailman/listinfo/swift-evolution</a><br><<a href="https://lists.swift.org/mailman/listinfo/swift-evolution" target="_blank">https://lists.swift.org/mailman/listinfo/swift-evolution</a>><br><br></blockquote><br></blockquote><br>_______________________________________________<br>swift-evolution mailing list<br><a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a>><br><a href="https://lists.swift.org/mailman/listinfo/swift-evolution" target="_blank">https://lists.swift.org/mailman/listinfo/swift-evolution</a><br><<a href="https://lists.swift.org/mailman/listinfo/swift-evolution" target="_blank">https://lists.swift.org/mailman/listinfo/swift-evolution</a>><br></blockquote>_______________________________________________<br>swift-evolution mailing list<br><a href="mailto:swift-evolution@swift.org" target="_blank">swift-evolution@swift.org</a><br><<a href="mailto:swift-evolution@swift.org" target="_blank">mailto:swift-evolution@swift.org</a>><br><a href="https://lists.swift.org/mailman/listinfo/swift-evolution" target="_blank">https://lists.swift.org/mailman/listinfo/swift-evolution</a><br><<a href="https://lists.swift.org/mailman/listinfo/swift-evolution" 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target="_blank">https://lists.swift.org/mailman/listinfo/swift-evolution</a><br><br></blockquote><br style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><span style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;float:none;display:inline">--<span> </span></span><br style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><span style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;float:none;display:inline">Dave</span><br style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><br style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><span style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;float:none;display:inline">_______________________________________________</span><br style="font-family:helvetica;font-size:12px;font-style:normal;font-weight:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><span 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