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< ol class = "chapter" > < li class = "chapter-item expanded affix " > < a href = "../index.html" > halo2< / a > < / li > < li class = "chapter-item expanded " > < a href = "../concepts.html" > < strong aria-hidden = "true" > 1.< / strong > Concepts< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../concepts/proofs.html" > < strong aria-hidden = "true" > 1.1.< / strong > Proof systems< / a > < / li > < li class = "chapter-item expanded " > < a href = "../concepts/arithmetization.html" > < strong aria-hidden = "true" > 1.2.< / strong > PLONKish Arithmetization< / a > < / li > < li class = "chapter-item expanded " > < a href = "../concepts/chips.html" > < strong aria-hidden = "true" > 1.3.< / strong > Chips< / a > < / li > < li class = "chapter-item expanded " > < a href = "../concepts/gadgets.html" > < strong aria-hidden = "true" > 1.4.< / strong > Gadgets< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../user.html" > < strong aria-hidden = "true" > 2.< / strong > User Documentation< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../user/dev-tools.html" > < strong aria-hidden = "true" > 2.1.< / strong > Developer tools< / a > < / li > < li class = "chapter-item expanded " > < a href = "../user/simple-example.html" > < strong aria-hidden = "true" > 2.2.< / strong > A simple example< / a > < / li > < li class = "chapter-item expanded " > < a href = "../user/lookup-tables.html" > < strong aria-hidden = "true" > 2.3.< / strong > Lookup tables< / a > < / li > < li class = "chapter-item expanded " > < a href = "../user/gadgets.html" > < strong aria-hidden = "true" > 2.4.< / strong > Gadgets< / a > < / li > < li class = "chapter-item expanded " > < a href = "../user/tips-and-tricks.html" class = "active" > < strong aria-hidden = "true" > 2.5.< / strong > Tips and tricks< / a > < / li > < li class = "chapter-item expanded " > < a href = "../user/wasm-port.html" > < strong aria-hidden = "true" > 2.6.< / strong > WASM Guide< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../dev.html" > < strong aria-hidden = "true" > 3.< / strong > Developer Documentation< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../dev/features.html" > < strong aria-hidden = "true" > 3.1.< / strong > Feature development< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../design.html" > < strong aria-hidden = "true" > 4.< / strong > Design< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../design/proving-system.html" > < strong aria-hidden = "true" > 4.1.< / strong > Proving system< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../design/proving-system/lookup.html" > < strong aria-hidden = "true" > 4.1.1.< / strong > Lookup argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/proving-system/permutation.html" > < strong aria-hidden = "true" > 4.1.2.< / strong > Permutation argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/proving-system/circuit-commitments.html" > < strong aria-hidden = "true" > 4.1.3.< / strong > Circuit commitments< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/proving-system/vanishing.html" > < strong aria-hidden = "true" > 4.1.4.< / strong > Vanishing argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/proving-system/multipoint-opening.html" > < strong aria-hidden = "true" > 4.1.5.< / strong > Multipoint opening argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/proving-system/inner-product.html" > < strong aria-hidden = "true" > 4.1.6.< / strong > Inner product argument< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/proving-system/comparison.html" > < strong aria-hidden = "true" > 4.1.7.< / strong > Comparison to other work< / a > < / li > < / ol > < / li > < li class = "chapter-item expanded " > < a href = "../design/protocol.html" > < strong aria-hidden = "true" > 4.2.< / strong > Protocol Description< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/implementation.html" > < strong aria-hidden = "true" > 4.3.< / strong > Implementation< / a > < / li > < li > < ol class = "section" > < li class = "chapter-item expanded " > < a href = "../design/implementation/proofs.html" > < strong aria-hidden = "true" > 4.3.1.< / strong > Proofs< / a > < / li > < li class = "chapter-item expanded " > < a href = "../design/implementation/fields.html" > < strong aria-hidden = "true" > 4.3.2.< / strong > Fields< / a > < / li > < li cla
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< h1 id = "tips-and-tricks" > < a class = "header" href = "#tips-and-tricks" > Tips and tricks< / a > < / h1 >
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< p > This section contains various ideas and snippets that you might find useful while writing
halo2 circuits.< / p >
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< h2 id = "small-range-constraints" > < a class = "header" href = "#small-range-constraints" > Small range constraints< / a > < / h2 >
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< p > A common constraint used in R1CS circuits is the boolean constraint: < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6944em;" > < / span > < span class = "mord mathnormal" > b< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ∗ < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > (< / span > < span class = "mord" > 1< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > b< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.6444em;" > < / span > < span class = "mord" > 0< / span > < / span > < / span > < / span > .
This constraint can only be satisfied by < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6944em;" > < / span > < span class = "mord mathnormal" > b< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.6444em;" > < / span > < span class = "mord" > 0< / span > < / span > < / span > < / span > or < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6944em;" > < / span > < span class = "mord mathnormal" > b< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.6444em;" > < / span > < span class = "mord" > 1< / span > < / span > < / span > < / span > .< / p >
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< p > In halo2 circuits, you can similarly constrain a cell to have one of a small set of
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values. For example, to constrain < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.4306em;" > < / span > < span class = "mord mathnormal" > a< / span > < / span > < / span > < / span > to the range < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > [< / span > < span class = "mord" > 0..5< / span > < span class = "mclose" > ]< / span > < / span > < / span > < / span > , you would create a gate of
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the form:< / p >
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< p > < span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.4445em;" > < / span > < span class = "mord mathnormal" > a< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > (< / span > < span class = "mord" > 1< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > a< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > (< / span > < span class = "mord" > 2< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > a< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > (< / span > < span class = "mord" > 3< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > a< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > (< / span > < span class = "mord" > 4< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > a< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.6444em;" > < / span > < span class = "mord" > 0< / span > < / span > < / span > < / span > < / span > < / p >
< p > while to constrain < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.4306em;" > < / span > < span class = "mord mathnormal" > c< / span > < / span > < / span > < / span > to be either 7 or 13, you would use:< / p >
< p > < span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > (< / span > < span class = "mord" > 7< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > c< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > (< / span > < span class = "mord" > 13< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > c< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.6444em;" > < / span > < span class = "mord" > 0< / span > < / span > < / span > < / span > < / span > < / p >
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< blockquote >
< p > The underlying principle here is that we create a polynomial constraint with roots at
each value in the set of possible values we want to allow. In R1CS circuits, the maximum
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supported polynomial degree is 2 (due to all constraints being of the form < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.4653em;" > < / span > < span class = "mord mathnormal" > a< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ∗ < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.6944em;" > < / span > < span class = "mord mathnormal" > b< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.4306em;" > < / span > < span class = "mord mathnormal" > c< / span > < / span > < / span > < / span > ).
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In halo2 circuits, you can use arbitrary-degree polynomials - with the proviso that
higher-degree constraints are more expensive to use.< / p >
< / blockquote >
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< p > Note that the roots don't have to be constants; for example < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > a< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" > x< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > a< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > y< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" > a< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.04398em;" > z< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.6444em;" > < / span > < span class = "mord" > 0< / span > < / span > < / span > < / span > will constrain < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.4306em;" > < / span > < span class = "mord mathnormal" > a< / span > < / span > < / span > < / span > to be equal to one of < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > {< / span > < span class = "mord mathnormal" > x< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.1667em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > y< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.1667em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.04398em;" > z< / span > < span class = "mclose" > }< / span > < / span > < / span > < / span > where the latter can be arbitrary polynomials, as long as the whole expression stays within the maximum degree bound.< / p >
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< h2 id = "small-set-interpolation" > < a class = "header" href = "#small-set-interpolation" > Small set interpolation< / a > < / h2 >
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< p > We can use Lagrange interpolation to create a polynomial constraint that maps
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< span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.6833em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.22222em;" > Y< / span > < / span > < / span > < / span > for small sets of < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.7224em;vertical-align:-0.0391em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > ∈< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > {< / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3117em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mathnormal mtight" > i< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mclose" > }< / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.1667em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.22222em;" > Y< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > ∈< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:1em;vertical-align:-0.25em;" > < / span > < span class = "mopen" > {< / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.03588em;" > y< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3117em;" > < span style = "top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mathnormal mtight" > i< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mclose" > }< / span > < / span > < / span > < / span > . < / p >
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< p > For instance, say we want to map a 2-bit value to a " spread" version interleaved
with zeros. We first precompute the evaluations at each point:< / p >
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< p > < span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:4.8em;vertical-align:-2.15em;" > < / span > < span class = "mord" > < span class = "mtable" > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "col-align-r" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:2.65em;" > < span style = "top:-4.81em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > 00< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > →< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mord" > 0000< / span > < / span > < / span > < span style = "top:-3.61em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > 01< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > →< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mord" > 0001< / span > < / span > < / span > < span style = "top:-2.41em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > 10< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > →< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mord" > 0100< / span > < / span > < / span > < span style = "top:-1.21em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > 11< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > →< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mord" > 0101< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:2.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "col-align-c" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:2.65em;" > < span style = "top:-4.81em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > ⟹< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < / span > < span style = "top:-3.61em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > ⟹< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < / span > < span style = "top:-2.41em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > ⟹< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < / span > < span style = "top:-1.21em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > ⟹< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:2.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "col-align-l" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:2.65em;" > < span style = "top:-4.81em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > 0< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > →< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mord" > 0< / span > < / span > < / span > < span style = "top:-3.61em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > 1< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > →< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mord" > 1< / span > < / span > < / span > < span style = "top:-2.41em;" > < span class =
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< p > Then, we construct the Lagrange basis polynomial for each point using the
identity:
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< span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1.0361em;vertical-align:-0.2861em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.01968em;" > l< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3117em;" > < span style = "top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mathnormal mtight" style = "margin-right:0.05724em;" > j< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2861em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:2.6986em;vertical-align:-1.4382em;" > < / span > < span class = "mop op-limits" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:1.05em;" > < span style = "top:-1.8479em;margin-left:0em;" > < span class = "pstrut" style = "height:3.05em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > < span class = "mord mtight" > 0< / span > < span class = "mrel mtight" > ≤< / span > < span class = "mord mathnormal mtight" > m< / span > < span class = "mrel mtight" > < < / span > < span class = "mord mathnormal mtight" style = "margin-right:0.03148em;" > k< / span > < span class = "mpunct mtight" > ,< / span > < span class = "mspace mtight" style = "margin-right:0.3253em;" > < / span > < span class = "mord mathnormal mtight" > m< / span > < span class = "mrel mtight" > < span class = "mrel mtight" > < span class = "mord vbox mtight" > < span class = "thinbox mtight" > < span class = "rlap mtight" > < span class = "strut" style = "height:0.8889em;vertical-align:-0.1944em;" > < / span > < span class = "inner" > < span class = "mord mtight" > < span class = "mrel mtight" > < / span > < / span > < / span > < span class = "fix" > < / span > < / span > < / span > < / span > < / span > < span class = "mrel mtight" > =< / span > < / span > < span class = "mord mathnormal mtight" style = "margin-right:0.05724em;" > j< / span > < / span > < / span > < / span > < span style = "top:-3.05em;" > < span class = "pstrut" style = "height:3.05em;" > < / span > < span > < span class = "mop op-symbol large-op" > ∏< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:1.4382em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "mspace" style = "margin-right:0.1667em;" > < / span > < span class = "mord" > < span class = "mopen nulldelimiter" > < / span > < span class = "mfrac" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:1.2603em;" > < span style = "top:-2.314em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3117em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mathnormal mtight" style = "margin-right:0.05724em;" > j< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2861em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > − < / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.1514em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mathnormal mtig
where < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6944em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.03148em;" > k< / span > < / span > < / span > < / span > is the number of data points. (< span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6944em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.03148em;" > k< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.6444em;" > < / span > < span class = "mord" > 4< / span > < / span > < / span > < / span > in our example above.)< / p >
< p > Recall that the Lagrange basis polynomial < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:1.0361em;vertical-align:-0.2861em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.01968em;" > l< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3117em;" > < span style = "top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mathnormal mtight" style = "margin-right:0.05724em;" > j< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2861em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > evaluates to < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6444em;" > < / span > < span class = "mord" > 1< / span > < / span > < / span > < / span > at
< span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6833em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > =< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.7167em;vertical-align:-0.2861em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3117em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mathnormal mtight" style = "margin-right:0.05724em;" > j< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.2861em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > < / span > and < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.6444em;" > < / span > < span class = "mord" > 0< / span > < / span > < / span > < / span > at all other < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:0.8889em;vertical-align:-0.1944em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" > x< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3117em;" > < span style = "top:-2.55em;margin-left:0em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mathnormal mtight" > i< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mpunct" > ,< / span > < span class = "mspace" style = "margin-right:0.1667em;" > < / span > < span class = "mord mathnormal" style = "margin-right:0.05724em;" > j< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > < span class = "mrel" > < span class = "mord vbox" > < span class = "thinbox" > < span class = "rlap" > < span class = "strut" style = "height:0.8889em;vertical-align:-0.1944em;" > < / span > < span class = "inner" > < span class = "mord" > < span class = "mrel" > < / span > < / span > < / span > < span class = "fix" > < / span > < / span > < / span > < / span > < / span > < span class = "mrel" > =< / span > < / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < span class = "base" > < span class = "strut" style = "height:0.6595em;" > < / span > < span class = "mord mathnormal" > i< / span > < span class = "mord" > .< / span > < / span > < / span > < / span > < / p >
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< p > Continuing our example, we get four Lagrange basis polynomials:< / p >
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< p > < span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:6.933em;vertical-align:-3.2165em;" > < / span > < span class = "mord" > < span class = "mtable" > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "col-align-c" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:3.7165em;" > < span style = "top:-5.7165em;" > < span class = "pstrut" style = "height:3.01em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.01968em;" > l< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < span style = "top:-3.9155em;" > < span class = "pstrut" style = "height:3.01em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.01968em;" > l< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 1< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < span style = "top:-2.1145em;" > < span class = "pstrut" style = "height:3.01em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.01968em;" > l< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 2< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < span style = "top:-0.3135em;" > < span class = "pstrut" style = "height:3.01em;" > < / span > < span class = "mord" > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.01968em;" > l< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 3< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:3.2165em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "col-align-c" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:3.7165em;" > < span style = "top:
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< p > Our polynomial constraint is then< / p >
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< p > < span class = "katex-display" > < span class = "katex" > < span class = "katex-html" aria-hidden = "true" > < span class = "base" > < span class = "strut" style = "height:2.4em;vertical-align:-0.95em;" > < / span > < span class = "mord" > < span class = "mtable" > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "col-align-c" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:1.45em;" > < span style = "top:-3.61em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < / span > < / span > < span style = "top:-2.41em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < span class = "mrel" > ⟹< / span > < span class = "mspace" style = "margin-right:0.2778em;" > < / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.95em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "col-align-c" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:1.45em;" > < span style = "top:-3.61em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.10764em;" > f< / span > < span class = "mopen" > (< / span > < span class = "mord" > 0< / span > < span class = "mclose" > )< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.01968em;" > l< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < span style = "top:-2.41em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > 0< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mbin" > ⋅< / span > < span class = "mspace" style = "margin-right:0.2222em;" > < / span > < span class = "mord" > < span class = "mord mathnormal" style = "margin-right:0.01968em;" > l< / span > < span class = "msupsub" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:0.3011em;" > < span style = "top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;" > < span class = "pstrut" style = "height:2.7em;" > < / span > < span class = "sizing reset-size6 size3 mtight" > < span class = "mord mtight" > 0< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.15em;" > < span > < / span > < / span > < / span > < / span > < / span > < / span > < span class = "mopen" > (< / span > < span class = "mord mathnormal" style = "margin-right:0.07847em;" > X< / span > < span class = "mclose" > )< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.95em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "arraycolsep" style = "width:0.5em;" > < / span > < span class = "col-align-c" > < span class = "vlist-t vlist-t2" > < span class = "vlist-r" > < span class = "vlist" style = "height:1.45em;" > < span style = "top:-3.61em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > +< / span > < / span > < / span > < span style = "top:-2.41em;" > < span class = "pstrut" style = "height:3em;" > < / span > < span class = "mord" > < span class = "mord" > +< / span > < / span > < / span > < / span > < span class = "vlist-s" > < / span > < / span > < span class = "vlist-r" > < span class = "vlist" style = "height:0.95em;" > < span > < / span > < / span > < / span > < / span > < / span > < span class = "arraycolsep" style = "widt
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