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05-reflection.html
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<div id='toc' class='mobile-hidden'>
<ul class='chapter'>
<li><a href='01-intro.html'><b>1.</b>Follow Along at This URL</a></li>
<ul class='section'>
<li><a href='01-intro.html#whats-this'><b>1.1.</b> Whats this?</a></li>
<li><a href='01-intro.html#the-first-bug'><b>1.2.</b> The First <em>Bug</em></a></li>
<li><a href='01-intro.html#fast-forward-to-present-day'><b>1.3.</b> Fast forward to Present Day</a></li>
<li><a href='01-intro.html#fast-forward-to-present-day-1'><b>1.4.</b> Fast forward to Present Day</a></li>
<li><a href='01-intro.html#fast-forward-to-present-day-2'><b>1.5.</b> Fast forward to Present Day</a></li>
<li><a href='01-intro.html#programming-languages-research'><b>1.6.</b> Programming Languages Research</a></li>
<li><a href='01-intro.html#modern-languages'><b>1.7.</b> Modern Languages</a></li>
<li><a href='01-intro.html#modern-languages-1'><b>1.8.</b> Modern Languages</a></li>
<li><a href='01-intro.html#modern-languages-2'><b>1.9.</b> Modern Languages</a></li>
<li><a href='01-intro.html#modern-languages-3'><b>1.10.</b> Modern Languages?</a></li>
<li><a href='01-intro.html#well-typed-programs-can-go-very-wrong'><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-1'><b>1.12.</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>1.13.</b> Well-typed programs can go very wrong!</a></li>
<li><a href='01-intro.html#goal-language-integrated-verification'><b>1.14.</b> Goal: Language-Integrated Verification</a></li>
<li><a href='01-intro.html#goal-language-integrated-verification-1'><b>1.15.</b> Goal: Language-Integrated Verification</a></li>
<li><a href='01-intro.html#outline'><b>1.16.</b> Outline</a></li>
<li><a href='01-intro.html#outline-1'><b>1.17.</b> Outline</a></li>
<li><a href='01-intro.html#outline-2'><b>1.18.</b> Outline</a></li>
<li><a href='01-intro.html#outline-3'><b>1.19.</b> Outline</a></li>
<li><a href='01-intro.html#outline-4'><b>1.20.</b> Outline</a></li>
<li><a href='01-intro.html#outline-5'><b>1.21.</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>
<li><a href='02-refinements.html#outline'><b>2.22.</b> Outline</a></li>
<li><a href='02-refinements.html#outline-1'><b>2.23.</b> Outline</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>
<li><a href='03-examples.html#outline'><b>3.32.</b> Outline</a></li>
<li><a href='03-examples.html#outline-1'><b>3.33.</b> Outline</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='04-termination.html'><b>5.</b>Avoiding Infinite Loops</a></li>
<ul class='section'>
<li><a href='04-termination.html#example-proving-termination-of-sum'><b>5.1.</b> Example: Proving Termination of <code>sum</code></a></li>
<li><a href='04-termination.html#proving-termination'><b>5.2.</b> Proving Termination</a></li>
<li><a href='04-termination.html#proving-termination-1'><b>5.3.</b> Proving Termination</a></li>
<li><a href='04-termination.html#proving-termination-2'><b>5.4.</b> Proving Termination</a></li>
<li><a href='04-termination.html#proving-termination-3'><b>5.5.</b> Proving Termination</a></li>
<li><a href='04-termination.html#example-proving-termination-of-sum-1'><b>5.6.</b> Example: Proving Termination of <code>sum</code></a></li>
<li><a href='04-termination.html#user-specified-termination-metrics'><b>5.7.</b> User Specified Termination Metrics</a></li>
<li><a href='04-termination.html#user-specified-termination-metrics-1'><b>5.8.</b> User Specified Termination Metrics</a></li>
<li><a href='04-termination.html#user-specified-termination-metrics-2'><b>5.9.</b> User Specified Termination Metrics</a></li>
<li><a href='04-termination.html#proving-termination-4'><b>5.10.</b> Proving Termination</a></li>
<li><a href='04-termination.html#lexicographic-termination'><b>5.11.</b> Lexicographic Termination</a></li>
<li><a href='04-termination.html#how-about-data-types'><b>5.12.</b> How About Data Types?</a></li>
<li><a href='04-termination.html#user-specified-metrics-on-data-types'><b>5.13.</b> User specified metrics on Data Types</a></li>
<li><a href='04-termination.html#termination-can-be-tricky'><b>5.14.</b> Termination Can be Tricky</a></li>
<li><a href='04-termination.html#avoiding-infinite-loops-1'><b>5.15.</b> Avoiding Infinite Loops</a></li>
<li><a href='04-termination.html#avoiding-infinite-loops-2'><b>5.16.</b> Avoiding Infinite Loops</a></li>
<li><a href='04-termination.html#avoiding-infinite-loops-3'><b>5.17.</b> Avoiding Infinite Loops</a></li>
<li><a href='04-termination.html#avoiding-infinite-loops-4'><b>5.18.</b> Avoiding Infinite Loops</a></li>
<li><a href='04-termination.html#avoiding-infinite-loops-is-easy-in-practice'><b>5.19.</b> Avoiding Infinite Loops is Easy (in Practice)</a></li>
<li><a href='04-termination.html#avoiding-infinite-loops-is-easy-in-practice-1'><b>5.20.</b> Avoiding Infinite Loops is Easy (in Practice)</a></li>
<li><a href='04-termination.html#avoiding-infinite-loops-is-easy-in-practice-2'><b>5.21.</b> Avoiding Infinite Loops is Easy (in Practice)</a></li>
<li><a href='04-termination.html#avoiding-infinite-loops-is-easy-in-practice-3'><b>5.22.</b> Avoiding Infinite Loops is Easy (in Practice)</a></li>
<li><a href='04-termination.html#outline'><b>5.23.</b> Outline</a></li>
<li><a href='04-termination.html#outline-1'><b>5.24.</b> Outline</a></li>
</ul>
<li><a href='05-reflection.html'><b>6.</b>Types as Theorems, Programs as Proofs</a></li>
<ul class='section'>
<li><a href='05-reflection.html#types-as-theorems'><b>6.1.</b> Types As Theorems</a></li>
<li><a href='05-reflection.html#types-as-theorems-example'><b>6.2.</b> Types as Theorems: Example</a></li>
<li><a href='05-reflection.html#programs-as-proofs-example'><b>6.3.</b> Programs As Proofs: Example</a></li>
<li><a href='05-reflection.html#types-as-theorems-1'><b>6.4.</b> Types As Theorems</a></li>
<li><a href='05-reflection.html#types-as-theorems-example-1'><b>6.5.</b> Types as Theorems: Example</a></li>
<li><a href='05-reflection.html#programs-as-proofs-example-1'><b>6.6.</b> Programs As Proofs: Example</a></li>
<li><a href='05-reflection.html#types-as-theorems-programs-as-proofs-1'><b>6.7.</b> Types as Theorems, Programs as Proofs</a></li>
<li><a href='05-reflection.html#those-proofs-were-boring'><b>6.8.</b> Those Proofs were Boring</a></li>
<li><a href='05-reflection.html#those-proofs-were-boring-1'><b>6.9.</b> Those Proofs were Boring</a></li>
<li><a href='05-reflection.html#theorems-about-functions'><b>6.10.</b> Theorems about Functions</a></li>
<li><a href='05-reflection.html#refinement-reflection'><b>6.11.</b> Refinement Reflection</a></li>
<li><a href='05-reflection.html#reflect-function-into-output-type'><b>6.12.</b> Reflect Function into Output Type</a></li>
<li><a href='05-reflection.html#reflect-function-into-output-type-1'><b>6.13.</b> Reflect Function into Output Type</a></li>
<li><a href='05-reflection.html#reflection-at-result-type'><b>6.14.</b> Reflection at Result Type</a></li>
<li><a href='05-reflection.html#structuring-proofs-as-calculations'><b>6.15.</b> Structuring Proofs as Calculations</a></li>
<li><a href='05-reflection.html#types-as-theorems-programs-as-proofs-2'><b>6.16.</b> Types as Theorems, Programs as Proofs</a></li>
<li><a href='05-reflection.html#reusing-proofs-functions-as-lemmas'><b>6.17.</b> Reusing Proofs: Functions as Lemmas</a></li>
<li><a href='05-reflection.html#reusing-proofs-functions-as-lemmas-1'><b>6.18.</b> Reusing Proofs: Functions as Lemmas</a></li>
<li><a href='05-reflection.html#reusing-proofs-functions-as-lemmas-2'><b>6.19.</b> Reusing Proofs: Functions as Lemmas</a></li>
<li><a href='05-reflection.html#types-as-theorems-programs-as-proofs-3'><b>6.20.</b> Types as Theorems, Programs as Proofs</a></li>
<li><a href='05-reflection.html#types-as-theorems-programs-as-proofs-4'><b>6.21.</b> Types as Theorems, Programs as Proofs</a></li>
<li><a href='05-reflection.html#proof-by-logical-evaluation'><b>6.22.</b> Proof by Logical Evaluation</a></li>
<li><a href='05-reflection.html#proof-by-logical-evaluation-1'><b>6.23.</b> Proof by Logical Evaluation</a></li>
<li><a href='05-reflection.html#proof-by-logical-evaluation-2'><b>6.24.</b> Proof by Logical Evaluation</a></li>
<li><a href='05-reflection.html#proof-by-induction'><b>6.25.</b> Proof by Induction</a></li>
<li><a href='05-reflection.html#proof-by-induction-1'><b>6.26.</b> Proof by Induction</a></li>
<li><a href='05-reflection.html#proof-by-induction-2'><b>6.27.</b> Proof by Induction</a></li>
<li><a href='05-reflection.html#types-as-theorems-programs-as-proofs-5'><b>6.28.</b> Types as Theorems, Programs as Proofs</a></li>
<li><a href='05-reflection.html#types-as-theorems-programs-as-proofs-6'><b>6.29.</b> Types as Theorems, Programs as Proofs</a></li>
<li><a href='05-reflection.html#types-as-theorems-programs-as-proofs-7'><b>6.30.</b> Types as Theorems, Programs as Proofs</a></li>
<li><a href='05-reflection.html#theorems-about-data'><b>6.31.</b> Theorems about Data</a></li>
<li><a href='05-reflection.html#theorems-about-data-associativity-of-append'><b>6.32.</b> Theorems about Data: Associativity of <code>append</code></a></li>
<li><a href='05-reflection.html#case-study-mapreduce'><b>6.33.</b> Case Study: MapReduce</a></li>
<li><a href='05-reflection.html#case-study-mapreduce-1'><b>6.34.</b> Case Study: MapReduce</a></li>
<li><a href='05-reflection.html#reduce-theorem'><b>6.35.</b> Reduce Theorem</a></li>
<li><a href='05-reflection.html#types-as-theorems-programs-as-proofs-8'><b>6.36.</b> Types as Theorems, Programs as Proofs</a></li>
<li><a href='05-reflection.html#outline'><b>6.37.</b> Outline</a></li>
<li><a href='05-reflection.html#outline-1'><b>6.38.</b> Outline</a></li>
</ul>
<li><a href='06-concl.html'><b>7.</b>LiquidHaskell</a></li>
<ul class='section'>
<li><a href='06-concl.html#liquidhaskell-1'><b>7.1.</b> LiquidHaskell</a></li>
<li><a href='06-concl.html#liquidhaskell-2'><b>7.2.</b> LiquidHaskell</a></li>
<li><a href='06-concl.html#evaluation'><b>7.3.</b> Evaluation</a></li>
<li><a href='06-concl.html#recap-refinement-types'><b>7.4.</b> Recap: Refinement Types</a></li>
<li><a href='06-concl.html#recap-refinement-types-1'><b>7.5.</b> Recap: Refinement Types</a></li>
<li><a href='06-concl.html#recap-refinement-types-2'><b>7.6.</b> Recap: Refinement Types</a></li>
<li><a href='06-concl.html#recap-refinement-types-3'><b>7.7.</b> Recap: Refinement Types</a></li>
<li><a href='06-concl.html#many-friends-many-directions'><b>7.8.</b> Many Friends, Many Directions</a></li>
<li><a href='06-concl.html#thank-you'><b>7.9.</b> Thank You!</a></li>
</ul>
</ul>
</div>
<div id="checker-status">
<!-- Verifying ... -->
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ng-show="isChecking" ng-click="verifySource()">
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</button>
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type="button"
id="errorblockdropdown"
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<div id='page-wrapper'>
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<div class="hidden">
<div id="program-pane-0" class="welleditor" style="background:#fff; position:relative">
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<div id="program-0" class="programbox">{-# LANGUAGE TupleSections #-}
{-@ LIQUID "--no-warnings" @-}
{-@ LIQUID "--short-names" @-}
{-@ LIQUID "--diff" @-}
{-@ LIQUID "--exact-data-cons" @-}
{-@ LIQUID "--higherorder" @-}
{-@ LIQUID "--automatic-instances=liquidinstances" @-}
-- Hidden code
module Reflection where
import Language.Haskell.Liquid.ProofCombinators
import Prelude hiding (sum, (++))
sillyProof :: Int -> Int -> Proof
appendPf :: [a] -> [a] -> [a] -> ()
{-@ infix ++ @-}
{-@ reflect ++ @-}
by = (?)
{-
thm :: [a] -> [a] -> [a] -> ()
thm [] ys zs = ([] ++ ys) ++ zs
==. ys ++ zs
==. [] ++ (ys ++ zs)
*** QED
thm (x:xs) ys zs = ((x:xs) ++ ys) ++ zs
==. (x : (xs ++ ys)) ++ zs
==. x : ((xs ++ ys) ++ zs)
==. x : (xs ++ (ys ++ zs)) ? thm xs ys zs
==. (x : xs) ++ (ys ++ zs)
*** QED
-}</div>
</div>
</div>
<div id="slide-0" class="slide">
<h2 id="types-as-theorems-programs-as-proofs">Types as Theorems, Programs as Proofs</h2>
<p align="center">
<img src="img/escher-drawing-hands.jpg" height=300px/>
</p>
<p><strong>Classic Idea</strong> <a href="https://en.wikipedia.org/wiki/Curry–Howard_correspondence">Curry-Howard</a>, <a href="http://homepages.inf.ed.ac.uk/wadler/papers/propositions-as-types/propositions-as-types.pdf">Propositions As Types</a></p>
</div>
<div id="slide-1" class="slide">
<h2 id="types-as-theorems">Types As Theorems</h2>
<p><br></p>
<p>The Refined <em>Type</em> ...</p>
<p><span class="math display">\[\reft{v}{b}{P}\]</span></p>
<p>... corresponds to the <em>Theorem</em></p>
<p><span class="math display">\[\mathit{P}\]</span></p>
</div>
<div id="slide-2" class="slide">
<h2 id="types-as-theorems-example">Types as Theorems: Example</h2>
<p><br></p>
<p>The Refined <em>Type</em> ...</p>
<div id="program-pane-1" class="welleditor" style="background:#fff; position:relative">
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<div id="program-1" class="programbox">{-@ type OnePlusOneEqualsTwo = {v:() | 1 + 1 = 2} @-}</div>
</div>
<p>... corresponds to the <em>Theorem</em></p>
<p><span class="math display">\[1 + 1 = 2\]</span></p>
</div>
<div id="slide-3" class="slide">
<h2 id="programs-as-proofs-example">Programs As Proofs: Example</h2>
<p><br></p>
<p>The <em>Program</em> ...</p>
<div id="program-pane-2" class="welleditor" style="background:#fff; position:relative">
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<div id="program-2" class="programbox">{-@ easyProof :: OnePlusOneEqualsTwo @-}
easyProof = ()</div>
</div>
<p>... corresponds to a <em>Proof</em> that</p>
<p><span class="math display">\[1 + 1 = 2\]</span></p>
</div>
<div id="slide-4" class="slide">
<h2 id="types-as-theorems-1">Types As Theorems</h2>
<p><br></p>
<p>The Refined <em>Type</em> ...</p>
<p><span class="math display">\[\bindx{n}{\Nat} \rightarrow \reft{v}{b}{P(n)}\]</span></p>
<p>... corresponds to the <em>Theorem</em></p>
<p><span class="math display">\[\forall n \in \Nat.\ P(n)\]</span></p>
</div>
<div id="slide-5" class="slide">
<h2 id="types-as-theorems-example-1">Types as Theorems: Example</h2>
<p><br></p>
<p>The Refined <em>Type</em> ...</p>
<div id="program-pane-3" class="welleditor" style="background:#fff; position:relative">
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<div id="program-3" class="programbox">{-@ type PlusCommutes = n:Int -> m:Int -> { n + m = m + n } @-}</div>
</div>
<p>... corresponds to the <em>Theorem</em></p>
<p><span class="math display">\[\forall n, m.\ n + m = m + n\]</span></p>
</div>
<div id="slide-6" class="slide">
<h2 id="programs-as-proofs-example-1">Programs As Proofs: Example</h2>
<p><br></p>
<p>The <em>Program</em> ...</p>
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<div id="program-4" class="programbox">{-@ sillyProof :: PlusCommutes @-}
sillyProof n m = ()</div>
</div>
<p>... corresponds to a <em>Proof</em> that</p>
<p><span class="math display">\[\forall n, m.\ n + m = m + n\]</span></p>
</div>
<div id="slide-7" class="slide">
<h2 id="types-as-theorems-programs-as-proofs-1">Types as Theorems, Programs as Proofs</h2>
<p><br></p>
<p align="center">
<table>
<thead>
<tr class="header">
<th style="text-align: right;"><strong>Code</strong></th>
<th style="text-align: center;"></th>
<th style="text-align: left;"><strong>Math</strong></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">Types</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Theorems</td>
</tr>
<tr class="even">
<td style="text-align: right;">Programs</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Proofs</td>
</tr>
</tbody>
</table>
</p>
</div>
<div id="slide-8" class="slide">
<h2 id="those-proofs-were-boring">Those Proofs were Boring</h2>
<p><br></p>
<p><strong>Simple Arithmetic</strong></p>
<p>Automatically proved by SMT Solver</p>
</div>
<div id="slide-9" class="slide">
<h2 id="those-proofs-were-boring-1">Those Proofs were Boring</h2>
<p><br></p>
<p><strong>Simple Arithmetic</strong></p>
<p>Automatically proved by SMT Solver</p>
<p><br></p>
<p><strong>How about proofs about user-defined functions?</strong></p>
<p>Beyond automatic SMT, but the user can <em>write proofs</em></p>
</div>
<div id="slide-10" class="slide">
<h2 id="theorems-about-functions">Theorems about Functions</h2>
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<div id="program-5" class="programbox">{-@ reflect sum @-}
sum :: Int -> Int
sum n
| n <= 0 = 0
| otherwise = n + sum (n - 1)</div>
</div>
<p><strong>How can we prove the the theorems</strong></p>
<p><span class="math inline">\(\mathit{sum}(1) = 1\)</span>,</p>
<p><span class="math inline">\(\mathit{sum}(2) = 3\)</span>,</p>
<p><span class="math inline">\(\mathit{sum}(3) = 6\)</span>.</p>
</div>
<div id="slide-11" class="slide">
<h2 id="refinement-reflection">Refinement Reflection</h2>
<p><br></p>
<p>The <em>annotation</em></p>
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<div id="program-6" class="programbox">{-@ reflect sum @-}</div>
</div>
<p>Automatically gives <code>sum</code> the <em>type</em></p>
<pre class="spec"><code>sum :: n:Int -> {v:Int | v = if n == 0 then 0 else n + sum (n-1)}
</code></pre>
</div>
<div id="slide-12" class="slide">
<h2 id="reflect-function-into-output-type">Reflect Function into Output Type</h2>
<p><br></p>
<p>The type of <code>sum</code> connects <em>implementation</em> and <em>specification</em></p>
<pre class="spec"><code>sum :: n:Int -> {v:Int | v = if n == 0 then 0 else n + sum (n-1)}
</code></pre>
</div>
<div id="slide-13" class="slide">
<h2 id="reflect-function-into-output-type-1">Reflect Function into Output Type</h2>
<p><br></p>
<p>The type of <code>sum</code> connects <em>implementation</em> and <em>specification</em></p>
<pre class="spec"><code>sum :: n:Int -> {v:Int | v = if n == 0 then 0 else n + sum (n-1)}
</code></pre>
<p><br></p>
<p><strong>Key Idea</strong></p>
<p>Calling <code>sum n</code> <em>reveals</em> definition at <code>n</code> to refinement logic!</p>
</div>
<div id="slide-14" class="slide">
<h2 id="reflection-at-result-type">Reflection at Result Type</h2>
<br>
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<div id="program-7" class="programbox">{-@ sum3 :: _ -> { sum 3 == 6 } @-}
sum3 _ = let s0 = sum 0 -- s0 :: {sum 0 = 0 }
s1 = sum 1 -- s1 :: {sum 1 = 1 + sum 0}
s2 = sum 2 -- s2 :: {sum 2 = 2 + sum 1}
s3 = sum 3 -- s3 :: {sum 3 = 3 + sum 2}
in () -- SMT connects the dots.</div>
</div>
<p><strong>Key Idea</strong></p>
<p>Calling <code>sum n</code> <em>reveals</em> definition at <code>n</code> to refinement logic!</p>
</div>
<div id="slide-15" class="slide">
<h2 id="structuring-proofs-as-calculations">Structuring Proofs as Calculations</h2>
<p><br></p>
<p>Using combinators from <a href="https://github.com/ucsd-progsys/liquidhaskell/blob/develop/include/Language/Haskell/Liquid/ProofCombinators.hs"><code>ProofCombinators</code></a>!</p>
<br>
<div id="program-pane-8" class="welleditor" style="background:#fff; position:relative">
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<div id="program-8" class="programbox">{-@ sum3' :: _ -> { sum 3 = 6 } @-}
sum3' _ = sum 3
==. 3 + sum 2
==. 3 + 2 + sum 1
==. 3 + 2 + 1 + sum 0
==. 6
*** QED</div>
</div>
</div>
<div id="slide-16" class="slide">
<h2 id="types-as-theorems-programs-as-proofs-2">Types as Theorems, Programs as Proofs</h2>
<p><br></p>
<p align="center">
<table>
<thead>
<tr class="header">
<th style="text-align: right;"><strong>Code</strong></th>
<th style="text-align: center;"></th>
<th style="text-align: left;"><strong>Math</strong></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">Types</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Theorems</td>
</tr>
<tr class="even">
<td style="text-align: right;">Programs</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Proofs</td>
</tr>
</tbody>
</table>
</p>
</div>
<div id="slide-17" class="slide">
<h2 id="reusing-proofs-functions-as-lemmas">Reusing Proofs: Functions as Lemmas</h2>
<p><br></p>
<p><strong>Proofs are functions</strong></p>
</div>
<div id="slide-18" class="slide">
<h2 id="reusing-proofs-functions-as-lemmas-1">Reusing Proofs: Functions as Lemmas</h2>
<p><br></p>
<p><strong>Proofs are functions</strong></p>
<p>Reuse by <em>calling</em> the function</p>
</div>
<div id="slide-19" class="slide">
<h2 id="reusing-proofs-functions-as-lemmas-2">Reusing Proofs: Functions as Lemmas</h2>
<p><br></p>
<p><strong>Proofs are functions</strong></p>
<p>Reuse by <em>calling</em> the function</p>
<div id="program-pane-9" class="welleditor" style="background:#fff; position:relative">
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<div id="program-9" class="programbox">{-@ sum4 :: _ -> { sum 4 = 10 } @-}
sum4 _ = sum 4
==. 4 + sum 3
==. 4 + 6 ? sum3 () -- lemma { sum 3 == 6 }
==. 10
*** QED</div>
</div>
<p><code>?</code> is a library operator (read ``because'')</p>
</div>
<div id="slide-20" class="slide">
<h2 id="types-as-theorems-programs-as-proofs-3">Types as Theorems, Programs as Proofs</h2>
<p><br></p>
<p align="center">
<table>
<thead>
<tr class="header">
<th style="text-align: right;"><strong>Code</strong></th>
<th style="text-align: center;"></th>
<th style="text-align: left;"><strong>Math</strong></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">Types</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Theorems</td>
</tr>
<tr class="even">
<td style="text-align: right;">Programs</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Proofs</td>
</tr>
</tbody>
</table>
</p>
</div>
<div id="slide-21" class="slide">
<h2 id="types-as-theorems-programs-as-proofs-4">Types as Theorems, Programs as Proofs</h2>
<p><br></p>
<p align="center">
<table>
<thead>
<tr class="header">
<th style="text-align: right;"><strong>Code</strong></th>
<th style="text-align: center;"></th>
<th style="text-align: left;"><strong>Math</strong></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">Types</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Theorems</td>
</tr>
<tr class="even">
<td style="text-align: right;">Programs</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Proofs</td>
</tr>
<tr class="odd">
<td style="text-align: right;">Functions</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Lemmas</td>
</tr>
</tbody>
</table>
</p>
</div>
<div id="slide-22" class="slide">
<h2 id="proof-by-logical-evaluation">Proof by Logical Evaluation</h2>
<p><br></p>
<p><strong>Long chains of calculations are tedious</strong></p>
</div>
<div id="slide-23" class="slide">
<h2 id="proof-by-logical-evaluation-1">Proof by Logical Evaluation</h2>
<p><br></p>
<p><strong>Long chains of calculations are tedious</strong></p>
<p>Make the machine do the hard work!</p>
<p>A new algorithm to <a href="https://arxiv.org/abs/1711.03842">emulate computation in SMT logic</a></p>
</div>
<div id="slide-24" class="slide">
<h2 id="proof-by-logical-evaluation-2">Proof by Logical Evaluation</h2>
<p><br></p>
<p><strong>Long chains of calculations are tedious</strong></p>
<p>Make the machine do the hard work!</p>
<p>A new algorithm to <a href="https://arxiv.org/abs/1711.03842">emulate computation in SMT logic</a></p>
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<div id="program-10" class="programbox">{-@ sum3auto :: _ -> { sum 3 = 6 } @-}
sum3auto _ = ()
{-@ sum4auto :: _ -> { sum 4 = 10 } @-}
sum4auto _ = ()</div>
</div>
</div>
<div id="slide-25" class="slide">
<h2 id="proof-by-induction">Proof by Induction</h2>
<p><br></p>
<p>Lets prove the theorem</p>
<p><span class="math display">\[\forall n.\ \sum_{i = 0}^n i = \frac{n \times (n + 1)}{2}\]</span></p>
</div>
<div id="slide-26" class="slide">
<h2 id="proof-by-induction-1">Proof by Induction</h2>
<p><br></p>
<p>Lets prove the theorem</p>
<p><span class="math display">\[\forall n.\ \sum_{i = 0}^n i = \frac{n \times (n + 1)}{2}\]</span></p>
<p>that is</p>
<p><span class="math display">\[\forall n \in \Nat.\ 2 \times \mathit{sum}(n) = n \times (n + 1)\]</span></p>
</div>
<div id="slide-27" class="slide">
<h2 id="proof-by-induction-2">Proof by Induction</h2>
<p><span class="math display">\[\forall n \in \Nat.\ 2 \times \mathit{sum}(n) = n \times (n + 1)\]</span></p>
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<div id="program-11" class="programbox">{-@ sumPf :: n:Nat -> { 2 * sum n == n * (n + 1) } @-}
sumPf :: Int -> ()
sumPf 0 = 2 * sum 0
==. 0
*** QED
sumPf n = 2 * sum n
==. 2 * (n + sum (n-1))
==. 2 * (n + ((n - 1) * n)) ? sumPf (n - 1)
==. n * (n + 1)
*** QED</div>
</div>
<p><strong>Q:</strong> What happens if we use the wrong induction?</p>
</div>
<div id="slide-28" class="slide">
<h2 id="types-as-theorems-programs-as-proofs-5">Types as Theorems, Programs as Proofs</h2>
<p><br></p>
<p align="center">
<table>
<thead>
<tr class="header">
<th style="text-align: right;"><strong>Code</strong></th>
<th style="text-align: center;"></th>
<th style="text-align: left;"><strong>Math</strong></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">Types</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Theorems</td>
</tr>
<tr class="even">
<td style="text-align: right;">Programs</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Proofs</td>
</tr>
<tr class="odd">
<td style="text-align: right;">Functions</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Lemmas</td>
</tr>
</tbody>
</table>
</p>
</div>
<div id="slide-29" class="slide">
<h2 id="types-as-theorems-programs-as-proofs-6">Types as Theorems, Programs as Proofs</h2>
<p><br></p>
<p align="center">
<table>
<thead>
<tr class="header">
<th style="text-align: right;"><strong>Code</strong></th>
<th style="text-align: center;"></th>
<th style="text-align: left;"><strong>Math</strong></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">Types</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Theorems</td>
</tr>
<tr class="even">
<td style="text-align: right;">Programs</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Proofs</td>
</tr>
<tr class="odd">
<td style="text-align: right;">Functions</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Lemmas</td>
</tr>
<tr class="even">
<td style="text-align: right;">Branches</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Case-Splits</td>
</tr>
</tbody>
</table>
</p>
</div>
<div id="slide-30" class="slide">
<h2 id="types-as-theorems-programs-as-proofs-7">Types as Theorems, Programs as Proofs</h2>
<p><br></p>
<p align="center">
<table>
<thead>
<tr class="header">
<th style="text-align: right;"><strong>Code</strong></th>
<th style="text-align: center;"></th>
<th style="text-align: left;"><strong>Math</strong></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">Types</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Theorems</td>
</tr>
<tr class="even">
<td style="text-align: right;">Programs</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Proofs</td>
</tr>
<tr class="odd">
<td style="text-align: right;">Functions</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Lemmas</td>
</tr>
<tr class="even">
<td style="text-align: right;">Branches</td>
<td style="text-align: center;">are</td>
<td style="text-align: left;">Case-Splits</td>
</tr>
<tr class="odd">
<td style="text-align: right;">Recursion</td>
<td style="text-align: center;">is</td>
<td style="text-align: left;">Induction</td>
</tr>
</tbody>
</table>
</p>
</div>
<div id="slide-31" class="slide">
<h2 id="theorems-about-data">Theorems about Data</h2>
<p>Recall the list append function:</p>
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<div id="program-12" class="programbox">(++) :: [a] -> [a] -> [a]
[] ++ ys = ys
(x:xs) ++ ys = x : (xs ++ ys)</div>
</div>
<p>Lets prove that the operator is <em>associative</em></p>
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<div id="program-13" class="programbox">{-@ type AppendAssoc a = xs:[a] -> ys:[a] -> zs:[a]
-> { (xs ++ ys) ++ zs = xs ++ (ys ++ zs) }
@-}</div>
</div>
</div>
<div id="slide-32" class="slide">
<h2 id="theorems-about-data-associativity-of-append">Theorems about Data: Associativity of <code>append</code></h2>
<p><br></p>
<p>Lets write fill in a <em>calculational</em> proof:</p>
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<div id="program-14" class="programbox">{-@ appendPf :: AppendAssoc a @-}
appendPf xs ys zs = () -- Q: Can you help me fill this in?</div>
</div>
</div>
<div id="slide-33" class="slide">
<h2 id="case-study-mapreduce">Case Study: MapReduce</h2>
<p><br> <em>Chunk</em> inputs, <em>Map</em> operation in parallel, and <em>Reduce</em> the results. <br></p>
</div>
<div id="slide-34" class="slide">
<h2 id="case-study-mapreduce-1">Case Study: MapReduce</h2>
<p><br> <em>Chunk</em> inputs, <em>Map</em> operation in parallel, and <em>Reduce</em> the results. <br></p>
<p align="center">
<img src="img/map-reduce.jpg" height=250px/>
</p>
</div>
<div id="slide-35" class="slide">
<h2 id="reduce-theorem">Reduce Theorem</h2>
<p><strong>Description</strong></p>
<p>If <code>op</code> is associative then <code>reduce op xs == parallelReduce op xs</code></p>
<p><strong>Theorem</strong></p>
<pre class="spec"><code>reduceTheorem
:: op:(a -> a -> a) -- for any op-erator
-> xs:[a] -- for any collection xs