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<!DOCTYPE html>
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<title>An Introduction to Refinement Types</title>
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<div id='toc' class='mobile-hidden'>
<ul class='chapter'>
<li><a href='01-intro.html'><b>1.</b>Whats this?</a></li>
<ul class='section'>
<li><a href='01-intro.html#the-first-bug'><b>1.1.</b> The First <em>Bug</em></a></li>
<li><a href='01-intro.html#fast-forward-to-present-day'><b>1.2.</b> Fast forward to Present Day</a></li>
<li><a href='01-intro.html#fast-forward-to-present-day-1'><b>1.3.</b> Fast forward to Present Day</a></li>
<li><a href='01-intro.html#fast-forward-to-present-day-2'><b>1.4.</b> Fast forward to Present Day</a></li>
<li><a href='01-intro.html#programming-languages-research'><b>1.5.</b> Programming Languages Research</a></li>
<li><a href='01-intro.html#modern-languages'><b>1.6.</b> Modern Languages</a></li>
<li><a href='01-intro.html#modern-languages-1'><b>1.7.</b> Modern Languages</a></li>
<li><a href='01-intro.html#modern-languages-2'><b>1.8.</b> Modern Languages</a></li>
<li><a href='01-intro.html#modern-languages-3'><b>1.9.</b> Modern Languages?</a></li>
<li><a href='01-intro.html#well-typed-programs-can-go-very-wrong'><b>1.10.</b> Well-typed programs can go very wrong!</a></li>
<li><a href='01-intro.html#well-typed-programs-can-go-very-wrong-1'><b>1.11.</b> Well-typed programs can go very wrong!</a></li>
<li><a href='01-intro.html#well-typed-programs-can-go-very-wrong-2'><b>1.12.</b> Well-typed programs can go very wrong!</a></li>
<li><a href='01-intro.html#goal-algorithmic-software-verification'><b>1.13.</b> Goal: Algorithmic Software Verification</a></li>
<li><a href='01-intro.html#goal-algorithmic-software-verification-1'><b>1.14.</b> Goal: Algorithmic Software Verification</a></li>
<li><a href='01-intro.html#outline'><b>1.15.</b> Outline</a></li>
<li><a href='01-intro.html#outline-1'><b>1.16.</b> Outline</a></li>
<li><a href='01-intro.html#outline-2'><b>1.17.</b> Outline</a></li>
<li><a href='01-intro.html#outline-3'><b>1.18.</b> Outline</a></li>
<li><a href='01-intro.html#outline-4'><b>1.19.</b> Outline</a></li>
</ul>
<li><a href='02-refinements.html'><b>2.</b>Simple Refinement Types</a></li>
<ul class='section'>
<li><a href='02-refinements.html#types'><b>2.1.</b> Types</a></li>
<li><a href='02-refinements.html#predicates'><b>2.2.</b> Predicates</a></li>
<li><a href='02-refinements.html#predicates-1'><b>2.3.</b> Predicates</a></li>
<li><a href='02-refinements.html#predicates-2'><b>2.4.</b> Predicates</a></li>
<li><a href='02-refinements.html#predicates-3'><b>2.5.</b> Predicates</a></li>
<li><a href='02-refinements.html#predicates-4'><b>2.6.</b> Predicates</a></li>
<li><a href='02-refinements.html#example-singletons'><b>2.7.</b> Example: Singletons</a></li>
<li><a href='02-refinements.html#example-natural-numbers'><b>2.8.</b> Example: Natural Numbers</a></li>
<li><a href='02-refinements.html#a-term-can-have-many-types'><b>2.9.</b> A Term Can Have <em>Many</em> Types</a></li>
<li><a href='02-refinements.html#predicate-subtyping-nuprl-pvs'><b>2.10.</b> 1. Predicate Subtyping <a href="http://pvs.csl.sri.com/papers/subtypes98/tse98.pdf">[NUPRL, PVS]</a></a></li>
<li><a href='02-refinements.html#predicate-subtyping-nuprl-pvs-1'><b>2.11.</b> 1. Predicate Subtyping <a href="http://pvs.csl.sri.com/papers/subtypes98/tse98.pdf">[NUPRL, PVS]</a></a></li>
<li><a href='02-refinements.html#predicate-subtyping-nuprl-pvs-2'><b>2.12.</b> 1. Predicate Subtyping <a href="http://pvs.csl.sri.com/papers/subtypes98/tse98.pdf">[NUPRL, PVS]</a></a></li>
<li><a href='02-refinements.html#example-natural-numbers-1'><b>2.13.</b> Example: Natural Numbers</a></li>
<li><a href='02-refinements.html#example-natural-numbers-2'><b>2.14.</b> Example: Natural Numbers</a></li>
<li><a href='02-refinements.html#typing-applications-function-calls'><b>2.15.</b> 2. Typing Applications (Function Calls)</a></li>
<li><a href='02-refinements.html#typing-applications-function-calls-1'><b>2.16.</b> 2. Typing Applications (Function Calls)</a></li>
<li><a href='02-refinements.html#typing-applications-function-calls-2'><b>2.17.</b> 2. Typing Applications (Function Calls)</a></li>
<li><a href='02-refinements.html#typing-applications-function-calls-3'><b>2.18.</b> 2. Typing Applications (Function Calls)</a></li>
<li><a href='02-refinements.html#recap-refinement-types-101'><b>2.19.</b> Recap: Refinement Types 101</a></li>
<li><a href='02-refinements.html#recap-refinement-types-101-1'><b>2.20.</b> Recap: Refinement Types 101</a></li>
<li><a href='02-refinements.html#recap-refinement-types-101-2'><b>2.21.</b> Recap: Refinement Types 101</a></li>
</ul>
<li><a href='03-examples.html'><b>3.</b>Refinement Types by Example</a></li>
<ul class='section'>
<li><a href='03-examples.html#refinement-types-by-example-1'><b>3.1.</b> Refinement Types by Example</a></li>
<li><a href='03-examples.html#refinement-types-by-example-2'><b>3.2.</b> Refinement Types by Example</a></li>
<li><a href='03-examples.html#refinement-types-by-example-3'><b>3.3.</b> Refinement Types by Example</a></li>
<li><a href='03-examples.html#specifications-pre-conditions'><b>3.4.</b> Specifications: Pre-Conditions</a></li>
<li><a href='03-examples.html#specifications-post-conditions'><b>3.5.</b> Specifications: Post-Conditions</a></li>
<li><a href='03-examples.html#refinement-types-by-example-4'><b>3.6.</b> Refinement Types by Example</a></li>
<li><a href='03-examples.html#verification-vector-sum'><b>3.7.</b> Verification: Vector Sum</a></li>
<li><a href='03-examples.html#verification-vector-sum-1'><b>3.8.</b> Verification: Vector Sum</a></li>
<li><a href='03-examples.html#verification-vector-sum-2'><b>3.9.</b> Verification: Vector Sum</a></li>
<li><a href='03-examples.html#refinement-types-by-example-5'><b>3.10.</b> Refinement Types by Example</a></li>
<li><a href='03-examples.html#inference'><b>3.11.</b> Inference</a></li>
<li><a href='03-examples.html#inference-vector-sum'><b>3.12.</b> Inference: Vector Sum</a></li>
<li><a href='03-examples.html#inference-vector-sum-1'><b>3.13.</b> Inference: Vector Sum</a></li>
<li><a href='03-examples.html#inference-vector-sum-2'><b>3.14.</b> Inference: Vector Sum</a></li>
<li><a href='03-examples.html#inference-vector-sum-3'><b>3.15.</b> Inference: Vector Sum</a></li>
<li><a href='03-examples.html#inference-vector-sum-4'><b>3.16.</b> Inference: Vector Sum</a></li>
<li><a href='03-examples.html#inference-vector-sum-5'><b>3.17.</b> Inference: Vector Sum</a></li>
<li><a href='03-examples.html#refinement-types-by-example-6'><b>3.18.</b> Refinement Types by Example</a></li>
<li><a href='03-examples.html#collections-higher-order-functions'><b>3.19.</b> Collections & Higher-Order Functions</a></li>
<li><a href='03-examples.html#collections-higher-order-functions-1'><b>3.20.</b> Collections & Higher-Order Functions</a></li>
<li><a href='03-examples.html#collections-higher-order-functions-2'><b>3.21.</b> Collections & Higher-Order Functions</a></li>
<li><a href='03-examples.html#collections-higher-order-functions-3'><b>3.22.</b> Collections & Higher-Order Functions</a></li>
<li><a href='03-examples.html#refinement-types-by-example-7'><b>3.23.</b> Refinement Types by Example</a></li>
<li><a href='03-examples.html#example-list-average'><b>3.24.</b> Example: List <code>average</code></a></li>
<li><a href='03-examples.html#refinements-for-datatypes'><b>3.25.</b> Refinements for Datatypes</a></li>
<li><a href='03-examples.html#measures-yield-refined-constructors'><b>3.26.</b> Measures Yield Refined Constructors</a></li>
<li><a href='03-examples.html#example-map-over-lists'><b>3.27.</b> Example: <code>map</code> over Lists</a></li>
<li><a href='03-examples.html#refinements-for-datatypes-1'><b>3.28.</b> Refinements for Datatypes</a></li>
<li><a href='03-examples.html#refinements-for-datatypes-2'><b>3.29.</b> Refinements for Datatypes</a></li>
<li><a href='03-examples.html#refinements-for-datatypes-3'><b>3.30.</b> Refinements for Datatypes</a></li>
<li><a href='03-examples.html#refinement-types-by-example-8'><b>3.31.</b> Refinement Types by Example</a></li>
</ul>
<li><a href='04-abstracting.html'><b>4.</b>Invariants In Constructors</a></li>
<ul class='section'>
<li><a href='04-abstracting.html#invariants-in-constructors-1'><b>4.1.</b> Invariants In Constructors</a></li>
<li><a href='04-abstracting.html#invariants-in-constructors-2'><b>4.2.</b> Invariants In Constructors</a></li>
<li><a href='04-abstracting.html#invariants-in-constructors-3'><b>4.3.</b> Invariants In Constructors</a></li>
<li><a href='04-abstracting.html#abstracting-refinements'><b>4.4.</b> Abstracting Refinements</a></li>
<li><a href='04-abstracting.html#abstracting-refinements-1'><b>4.5.</b> Abstracting Refinements</a></li>
<li><a href='04-abstracting.html#abstracting-refinements-2'><b>4.6.</b> Abstracting Refinements</a></li>
<li><a href='04-abstracting.html#using-abstract-refinements'><b>4.7.</b> Using Abstract Refinements</a></li>
<li><a href='04-abstracting.html#recap'><b>4.8.</b> Recap</a></li>
<li><a href='04-abstracting.html#recap-1'><b>4.9.</b> Recap</a></li>
<li><a href='04-abstracting.html#bounded-refinements-induction-as-a-type'><b>4.10.</b> Bounded Refinements: Induction as a Type</a></li>
<li><a href='04-abstracting.html#bounded-refinements-induction-as-a-type-1'><b>4.11.</b> Bounded Refinements: Induction as a Type</a></li>
<li><a href='04-abstracting.html#bounded-refinements-induction-as-a-type-2'><b>4.12.</b> Bounded Refinements: Induction as a Type</a></li>
<li><a href='04-abstracting.html#bounded-refinements-induction-as-a-type-3'><b>4.13.</b> Bounded Refinements: Induction as a Type</a></li>
</ul>
<li><a href='05-concl.html'><b>5.</b>LiquidHaskell</a></li>
<ul class='section'>
<li><a href='05-concl.html#liquidhaskell-1'><b>5.1.</b> LiquidHaskell</a></li>
<li><a href='05-concl.html#liquidhaskell-2'><b>5.2.</b> LiquidHaskell</a></li>
<li><a href='05-concl.html#evaluation'><b>5.3.</b> Evaluation</a></li>
<li><a href='05-concl.html#recap-refinement-types'><b>5.4.</b> Recap: Refinement Types</a></li>
<li><a href='05-concl.html#recap-refinement-types-1'><b>5.5.</b> Recap: Refinement Types</a></li>
<li><a href='05-concl.html#recap-refinement-types-2'><b>5.6.</b> Recap: Refinement Types</a></li>
<li><a href='05-concl.html#recap-refinement-types-3'><b>5.7.</b> Recap: Refinement Types</a></li>
<li><a href='05-concl.html#many-friends-many-directions'><b>5.8.</b> Many Friends, Many Directions</a></li>
<li><a href='05-concl.html#thank-you'><b>5.9.</b> Thank You!</a></li>
</ul>
<li><a href='05-termination.html'><b>6.</b>Avoiding Infinite Loops</a></li>
<ul class='section'>
<li><a href='05-termination.html#example-proving-termination-of-sum'><b>6.1.</b> Example: Proving Termination of <code>sum</code></a></li>
<li><a href='05-termination.html#proving-termination'><b>6.2.</b> Proving Termination</a></li>
<li><a href='05-termination.html#proving-termination-1'><b>6.3.</b> Proving Termination</a></li>
<li><a href='05-termination.html#proving-termination-2'><b>6.4.</b> Proving Termination</a></li>
<li><a href='05-termination.html#example-proving-termination-of-sum-1'><b>6.5.</b> Example: Proving Termination of <code>sum</code></a></li>
<li><a href='05-termination.html#proving-termination-3'><b>6.6.</b> Proving Termination</a></li>
<li><a href='05-termination.html#user-specified-termination-metrics'><b>6.7.</b> User Specified Termination Metrics</a></li>
<li><a href='05-termination.html#user-specified-termination-metrics-1'><b>6.8.</b> User Specified Termination Metrics</a></li>
<li><a href='05-termination.html#proving-termination-4'><b>6.9.</b> Proving Termination</a></li>
<li><a href='05-termination.html#lexicographic-termination'><b>6.10.</b> Lexicographic Termination</a></li>
<li><a href='05-termination.html#how-about-data-types'><b>6.11.</b> How About Data Types?</a></li>
<li><a href='05-termination.html#user-specified-metrics-on-adts'><b>6.12.</b> User specified metrics on ADTs</a></li>
<li><a href='05-termination.html#proving-termination-5'><b>6.13.</b> Proving Termination</a></li>
<li><a href='05-termination.html#termination-is-easy-in-practice'><b>6.14.</b> Termination is Easy in Practice</a></li>
<li><a href='05-termination.html#termination-is-easy-in-practice-1'><b>6.15.</b> Termination is Easy in Practice</a></li>
<li><a href='05-termination.html#recap'><b>6.16.</b> Recap</a></li>
<li><a href='05-termination.html#what-properties-can-be-expressed-in-the-logic'><b>6.17.</b> What properties can be expressed in the logic?</a></li>
</ul>
<li><a href='06-reflection.html'><b>7.</b>Refinement Reflection</a></li>
<ul class='section'>
<li><a href='06-reflection.html#theorems-about-haskell-functions'><b>7.1.</b> Theorems about Haskell functions</a></li>
<li><a href='06-reflection.html#theorems-about-haskell-functions-1'><b>7.2.</b> Theorems about Haskell functions</a></li>
<li><a href='06-reflection.html#types-as-theorems'><b>7.3.</b> Types As Theorems</a></li>
<li><a href='06-reflection.html#make-the-theorems-pretty'><b>7.4.</b> Make the theorems pretty!</a></li>
<li><a href='06-reflection.html#make-the-theorems-even-prettier'><b>7.5.</b> Make the theorems even prettier!</a></li>
<li><a href='06-reflection.html#use-more-smt-knowledge'><b>7.6.</b> Use more SMT knowledge</a></li>
<li><a href='06-reflection.html#theorems-about-haskell-functions-2'><b>7.7.</b> Theorems about Haskell functions</a></li>
<li><a href='06-reflection.html#refinement-reflection-1'><b>7.8.</b> Refinement Reflection</a></li>
<li><a href='06-reflection.html#fib-is-an-uninterpreted-function'><b>7.9.</b> <code>fib</code> is an uninterpreted function</a></li>
<li><a href='06-reflection.html#reflect-function-into-output-type'><b>7.10.</b> Reflect Function into Output Type</a></li>
<li><a href='06-reflection.html#reflection-at-result-type'><b>7.11.</b> Reflection at Result Type</a></li>
<li><a href='06-reflection.html#structuring-proofs'><b>7.12.</b> Structuring Proofs</a></li>
<li><a href='06-reflection.html#reusing-proofs-the-because-operator'><b>7.13.</b> Reusing Proofs: The "because" operator</a></li>
<li><a href='06-reflection.html#paper-pencil-style-proofs'><b>7.14.</b> Paper & Pencil style Proofs</a></li>
<li><a href='06-reflection.html#another-paper-pencil-proof'><b>7.15.</b> Another "Paper & Pencil" Proof</a></li>
<li><a href='06-reflection.html#generalizing-monotonicity-proof'><b>7.16.</b> Generalizing monotonicity proof</a></li>
<li><a href='06-reflection.html#reusing-theorems-by-application'><b>7.17.</b> Reusing Theorems by Application</a></li>
<li><a href='06-reflection.html#recap'><b>7.18.</b> Recap</a></li>
</ul>
<li><a href='07-structural-induction.html'><b>8.</b>Structural Induction</a></li>
<ul class='section'>
<li><a href='07-structural-induction.html#the-list-data-type'><b>8.1.</b> The list data type</a></li>
<li><a href='07-structural-induction.html#reflection-of-adts-into-the-logic'><b>8.2.</b> Reflection of ADTs into the logic</a></li>
<li><a href='07-structural-induction.html#reflection-of-structural-inductive-functions'><b>8.3.</b> Reflection of Structural Inductive Functions</a></li>
<li><a href='07-structural-induction.html#prove-fancy-lists-properties'><b>8.4.</b> Prove fancy lists properties</a></li>
<li><a href='07-structural-induction.html#recap'><b>8.5.</b> Recap</a></li>
<li><a href='07-structural-induction.html#appendix-reflection-of-non-recursive-functions'><b>8.6.</b> Appendix: Reflection of Non Recursive Functions</a></li>
<li><a href='07-structural-induction.html#proving-map-identity'><b>8.7.</b> Proving Map-Identity</a></li>
<li><a href='07-structural-induction.html#automation-proving-map-identity'><b>8.8.</b> Automation: Proving Map-Identity</a></li>
<li><a href='07-structural-induction.html#proving-map-fusion'><b>8.9.</b> Proving Map-Fusion</a></li>
<li><a href='07-structural-induction.html#onto-monoid-laws'><b>8.10.</b> Onto Monoid Laws</a></li>
<li><a href='07-structural-induction.html#monoid-laws-left-identity'><b>8.11.</b> Monoid Laws: Left Identity</a></li>
<li><a href='07-structural-induction.html#monoid-laws-right-identity'><b>8.12.</b> Monoid Laws: Right Identity</a></li>
<li><a href='07-structural-induction.html#monoid-laws-associativity'><b>8.13.</b> Monoid Laws: Associativity</a></li>
<li><a href='07-structural-induction.html#onto-monad-laws'><b>8.14.</b> Onto Monad Laws!</a></li>
<li><a href='07-structural-induction.html#monoid-laws-left-identity-1'><b>8.15.</b> Monoid Laws: Left Identity</a></li>
<li><a href='07-structural-induction.html#monoid-laws-right-identity-1'><b>8.16.</b> Monoid Laws: Right Identity</a></li>
<li><a href='07-structural-induction.html#monoid-laws-associativity-1'><b>8.17.</b> Monoid Laws: Associativity</a></li>
<li><a href='07-structural-induction.html#monoid-laws-associativity-2'><b>8.18.</b> Monoid Laws: Associativity</a></li>
</ul>
<li><a href='08-case-study-map-reduce.html'><b>9.</b>Case Study: MapReduce</a></li>
<ul class='section'>
<li><a href='08-case-study-map-reduce.html#mapreduce-library'><b>9.1.</b> MapReduce "Library"</a></li>
<li><a href='08-case-study-map-reduce.html#mapreduce-client-summing-list'><b>9.2.</b> MapReduce "Client": Summing List</a></li>
<li><a href='08-case-study-map-reduce.html#proving-code-equivalence'><b>9.3.</b> Proving Code Equivalence</a></li>
<li><a href='08-case-study-map-reduce.html#right-identity-of-plus'><b>9.4.</b> Right Identity of <code>plus</code></a></li>
<li><a href='08-case-study-map-reduce.html#distributivity-of-sum'><b>9.5.</b> Distributivity of <code>sum</code></a></li>
<li><a href='08-case-study-map-reduce.html#higher-order-map-reduce-theorem'><b>9.6.</b> Higher Order Map Reduce Theorem</a></li>
<li><a href='08-case-study-map-reduce.html#right-identity-of-plus-1'><b>9.7.</b> Right Identity of <code>plus</code></a></li>
<li><a href='08-case-study-map-reduce.html#warmup-associativity-of-append'><b>9.8.</b> Warmup: Associativity of Append</a></li>
<li><a href='08-case-study-map-reduce.html#proof-automation-associativity-of-append'><b>9.9.</b> Proof Automation: Associativity of Append</a></li>
<li><a href='08-case-study-map-reduce.html#distributivity-of-sum-1'><b>9.10.</b> Distributivity of <code>sum</code></a></li>
<li><a href='08-case-study-map-reduce.html#recap'><b>9.11.</b> Recap</a></li>
<li><a href='08-case-study-map-reduce.html#recap-1'><b>9.12.</b> Recap</a></li>
<li><a href='08-case-study-map-reduce.html#appendix-proof-of-mrtheorem'><b>9.13.</b> Appendix: Proof of <code>mRTheorem</code></a></li>
<li><a href='08-case-study-map-reduce.html#append-of-take-and-drop'><b>9.14.</b> Append of Take and Drop</a></li>
<li><a href='08-case-study-map-reduce.html#list-definition'><b>9.15.</b> List Definition</a></li>
<li><a href='08-case-study-map-reduce.html#list-manipulation'><b>9.16.</b> List Manipulation</a></li>
</ul>
</ul>
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<h2 id="liquidhaskell">LiquidHaskell</h2>
<p><br></p>
</div>
<div id="slide-1" class="slide">
<h2 id="liquidhaskell-1">LiquidHaskell</h2>
<p><br></p>
<p>Diverse Code Bases</p>
<p>10KLoc / 56 Modules</p>
</div>
<div id="slide-2" class="slide">
<h2 id="liquidhaskell-2">LiquidHaskell</h2>
<p><br></p>
<p>Memory Safety</p>
<p>Totality</p>
<p>Termination</p>
<p>Functional Correctness</p>
<p><strong>Inference is Crucial</strong></p>
</div>
<div id="slide-3" class="slide">
<h2 id="evaluation">Evaluation</h2>
<p><img src="img/code-spec-indiv.png" height=200px></p>
<p><strong>Specifications:</strong> 1 / 10 LOC (<em>ok</em>)</p>
<p><strong>Compile Time:</strong> 1s / 10 LOC (<em>not ok!</em>)</p>
</div>
<div id="slide-4" class="slide">
<h2 id="recap-refinement-types">Recap: Refinement Types</h2>
<p><br></p>
<p><strong>Expressive</strong></p>
<p>Verify <em>program specific</em> properties via <em>domain specific</em> analysis</p>
<p><br></p>
<p><strong>Automatic</strong></p>
<p>Rapid feedback to <em>influence design</em>, not only post-facto validation</p>
</div>
<div id="slide-5" class="slide">
<h2 id="recap-refinement-types-1">Recap: Refinement Types</h2>
<p><br></p>
<p><img src="img/tension3.png" height=200px></p>
</div>
<div id="slide-6" class="slide">
<h2 id="recap-refinement-types-2">Recap: Refinement Types</h2>
<p><br></p>
<p><img src="img/tension5.png" height=200px></p>
</div>
<div id="slide-7" class="slide">
<h2 id="recap-refinement-types-3">Recap: Refinement Types</h2>
<p><br></p>
<p><strong>SMT-Automated Dependent Typing</strong></p>
<table>
<tbody>
<tr class="odd">
<td style="text-align: left;"><strong>Properties:</strong></td>
<td style="text-align: left;">Predicates <em>+ Types</em></td>
</tr>
<tr class="even">
<td style="text-align: left;"><strong>Proofs:</strong></td>
<td style="text-align: left;">SMT Solvers <em>+ Subtyping</em></td>
</tr>
<tr class="odd">
<td style="text-align: left;"><strong>Inference:</strong></td>
<td style="text-align: left;">Abstr. Interp. <em>+ Hindley-Milner</em></td>
</tr>
</tbody>
</table>
</div>
<div id="slide-8" class="slide">
<h2 id="many-friends-many-directions">Many Friends, Many Directions</h2>
<p><br></p>
<table>
<tbody>
<tr class="odd">
<td style="text-align: left;"><a href="https://github.com/ats-lang/ats-lang.github.io">ATS</a></td>
<td style="text-align: left;">BU</td>
</tr>
<tr class="even">
<td style="text-align: left;"><a href="http://gowthamk.github.io/docs/icfp77-kaki.pdf">Catalyst</a></td>
<td style="text-align: left;">Purdue</td>
</tr>
<tr class="odd">
<td style="text-align: left;"><a href="http://www.mpi-sws.org/~joshua/type-refinements.info/">DML/Stardust</a></td>
<td style="text-align: left;">CMU/UBC</td>
</tr>
<tr class="even">
<td style="text-align: left;"><a href="http://fstar-lang.org">F7/F*</a></td>
<td style="text-align: left;">MSR/INRIA</td>
</tr>
<tr class="odd">
<td style="text-align: left;"><a href="http://arxiv.org/abs/1511.07033">RefinedRacket</a></td>
<td style="text-align: left;">IU</td>
</tr>
<tr class="even">
<td style="text-align: left;"><a href="http://arxiv.org/pdf/1510.08419.pdf">Synquid</a></td>
<td style="text-align: left;">MIT</td>
</tr>
</tbody>
</table>
</div>
<div id="slide-9" class="slide">
<h2 id="thank-you">Thank You!</h2>
<p><br> <br> <br></p>
<h4 style="border-bottom:none">
<a href="http://www.refinement-types.org"><code>http://www.refinement-types.org</code></a>
</h4>
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