liquidhaskell-0.8.0.2: docs/slides/plpv14/lhs/04_AbstractRefinements.lhs.markdown
{#abstractrefinements}
========================
<div class="hidden">
<pre><span class=hs-linenum>7: </span><span class='hs-keyword'>module</span> <span class='hs-conid'>AbstractRefinements</span> <span class='hs-keyword'>where</span>
<span class=hs-linenum>8: </span>
<span class=hs-linenum>9: </span><span class='hs-keyword'>import</span> <span class='hs-conid'>Prelude</span>
<span class=hs-linenum>10: </span><span class='hs-keyword'>import</span> <span class='hs-conid'>Language</span><span class='hs-varop'>.</span><span class='hs-conid'>Haskell</span><span class='hs-varop'>.</span><span class='hs-conid'>Liquid</span><span class='hs-varop'>.</span><span class='hs-conid'>Prelude</span>
<span class=hs-linenum>11: </span><span class='hs-keyword'>{-@</span> <span class='hs-conid'>LIQUID</span> <span class='hs-str'>"--no-termination"</span> <span class='hs-keyword'>@-}</span>
<span class=hs-linenum>12: </span>
<span class=hs-linenum>13: </span><span class='hs-definition'>o</span><span class='hs-layout'>,</span> <span class='hs-varid'>no</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span>
<span class=hs-linenum>14: </span><span class='hs-definition'>maxInt</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span>
</pre>
</div>
Abstract Refinements
--------------------
Abstract Refinements
====================
Two Problems
------------
<div class="fragment">
**Problem 1:**
How do we specify *both* [increasing and decreasing lists](http://web.cecs.pdx.edu/~sheard/Code/QSort.html)?
</div>
<br>
<div class="fragment">
**Problem 2:**
How do we specify *iteration-dependence* in higher-order functions?
</div>
Problem Is Pervasive
--------------------
Lets distill it to a simple example...
<div class="fragment">
<br>
(First, a few aliases)
<br>
<pre><span class=hs-linenum>64: </span><span class='hs-keyword'>{-@</span> <span class='hs-keyword'>type</span> <span class='hs-conid'>Odd</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>{</span><span class='hs-varid'>v</span><span class='hs-conop'>:</span><span class='hs-conid'>Int</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>v</span> <span class='hs-varid'>mod</span> <span class='hs-num'>2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>1</span><span class='hs-layout'>}</span> <span class='hs-keyword'>@-}</span>
<span class=hs-linenum>65: </span><span class='hs-keyword'>{-@</span> <span class='hs-keyword'>type</span> <span class='hs-conid'>Even</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>{</span><span class='hs-varid'>v</span><span class='hs-conop'>:</span><span class='hs-conid'>Int</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-varid'>v</span> <span class='hs-varid'>mod</span> <span class='hs-num'>2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-num'>0</span><span class='hs-layout'>}</span> <span class='hs-keyword'>@-}</span>
</pre>
</div>
Example: `maxInt`
-----------------
Compute the larger of two `Int`s:
<br>
<pre><span class=hs-linenum>77: </span><span class='hs-definition'>maxInt</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span>
<span class=hs-linenum>78: </span><span class='hs-definition'>maxInt</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>y</span> <span class='hs-varop'><=</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>y</span>
</pre>
Example: `maxInt`
-----------------
Has *many incomparable* refinement types
<br>
<pre><span class=hs-linenum>89: </span><span class='hs-definition'>maxInt</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Nat</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Nat</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Nat</span>
<span class=hs-linenum>90: </span><span class='hs-definition'>maxInt</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Even</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Even</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Even</span>
<span class=hs-linenum>91: </span><span class='hs-definition'>maxInt</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Odd</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Odd</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Odd</span>
</pre>
<br>
<div class="fragment">Yikes. **Which** one should we use?</div>
Refinement Polymorphism
-----------------------
`maxInt` returns *one of* its two inputs `x` and `y`
<div align="center">
<br>
--------- --- -------------------------------------------
**If** : the *inputs* satisfy a property
**Then** : the *output* satisfies that property
--------- --- -------------------------------------------
<br>
</div>
<div class="fragment">Above holds *for all* properties!</div>
<br>
<div class="fragment">
**Need to abstract refinements over types**
</div>
By Type Polymorphism?
---------------------
<br>
<pre><span class=hs-linenum>133: </span><span class='hs-definition'>max</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>α</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>α</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>α</span>
<span class=hs-linenum>134: </span><span class='hs-definition'>max</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>y</span> <span class='hs-varop'><=</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>y</span>
</pre>
<div class="fragment">
Instantiate `α` at callsites
<pre><span class=hs-linenum>142: </span><span class='hs-keyword'>{-@</span> <span class='hs-varid'>o</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Odd</span> <span class='hs-keyword'>@-}</span>
<span class=hs-linenum>143: </span><a class=annot href="#"><span class=annottext>{VV : (Int) | ((VV mod 2) == 1)}</span><span class='hs-definition'>o</span></a> <span class='hs-keyglyph'>=</span> <a class=annot href="#"><span class=annottext>forall <p :: (GHC.Types.Int)-> Bool>.
{x3 : (Int)<p> | true}
-> {x2 : (Int)<p> | true} -> {x1 : (Int)<p> | true}</span><span class='hs-varid'>maxInt</span></a> <a class=annot href="#"><span class=annottext>{x2 : (Int) | (x2 == (3 : int))}</span><span class='hs-num'>3</span></a> <a class=annot href="#"><span class=annottext>{x2 : (Int) | (x2 == (7 : int))}</span><span class='hs-num'>7</span></a> <span class='hs-comment'>-- α := Odd</span>
<span class=hs-linenum>144: </span>
<span class=hs-linenum>145: </span><span class='hs-keyword'>{-@</span> <span class='hs-varid'>e</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Even</span> <span class='hs-keyword'>@-}</span>
<span class=hs-linenum>146: </span><a class=annot href="#"><span class=annottext>{VV : (Int) | ((VV mod 2) == 0)}</span><span class='hs-definition'>e</span></a> <span class='hs-keyglyph'>=</span> <a class=annot href="#"><span class=annottext>forall <p :: (GHC.Types.Int)-> Bool>.
{x3 : (Int)<p> | true}
-> {x2 : (Int)<p> | true} -> {x1 : (Int)<p> | true}</span><span class='hs-varid'>maxInt</span></a> <a class=annot href="#"><span class=annottext>{x2 : (Int) | (x2 == (2 : int))}</span><span class='hs-num'>2</span></a> <a class=annot href="#"><span class=annottext>{x2 : (Int) | (x2 == (8 : int))}</span><span class='hs-num'>8</span></a> <span class='hs-comment'>-- α := Even</span>
</pre>
</div>
By Type Polymorphism?
---------------------
<br>
<pre><span class=hs-linenum>155: </span><span class='hs-definition'>max</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>α</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>α</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>α</span>
<span class=hs-linenum>156: </span><span class='hs-definition'>max</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>y</span> <span class='hs-varop'><=</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>y</span>
</pre>
<br>
But there is a fly in the ointment ...
Polymorphic `max` in Haskell
----------------------------
In Haskell the type of max is
<pre><span class=hs-linenum>167: </span><span class='hs-definition'>max</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>α</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>α</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>α</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>α</span>
</pre>
<br>
Could *ignore* the class constraints, instantiate as before...
<pre><span class=hs-linenum>173: </span><span class='hs-keyword'>{-@</span> <span class='hs-varid'>o</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Odd</span> <span class='hs-keyword'>@-}</span>
<span class=hs-linenum>174: </span><span class='hs-definition'>o</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>max</span> <span class='hs-num'>3</span> <span class='hs-num'>7</span> <span class='hs-comment'>-- α := Odd </span>
</pre>
Polymorphic `(+)` in Haskell
----------------------------
... but this is *unsound*!
<pre><span class=hs-linenum>182: </span><span class='hs-definition'>max</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ord</span> <span class='hs-varid'>α</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>α</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>α</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>α</span>
<span class=hs-linenum>183: </span><span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Num</span> <span class='hs-varid'>α</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>α</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>α</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>α</span>
</pre>
<br>
<div class="fragment">
*Ignoring* class constraints would let us "prove":
<pre><span class=hs-linenum>193: </span><span class='hs-keyword'>{-@</span> <span class='hs-varid'>no</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Odd</span> <span class='hs-keyword'>@-}</span>
<span class=hs-linenum>194: </span><a class=annot href="#"><span class=annottext>{VV : (Int) | ((VV mod 2) == 1)}</span><span class='hs-definition'>no</span></a> <span class='hs-keyglyph'>=</span> <span class=hs-error><a class=annot href="#"><span class=annottext>{x2 : (Int) | (x2 == (3 : int))}</span><span class='hs-num'>3</span></a></span><span class=hs-error> </span><span class=hs-error><a class=annot href="#"><span class=annottext>x1:(Int) -> x2:(Int) -> {x4 : (Int) | (x4 == (x1 + x2))}</span><span class='hs-varop'>+</span></a></span><span class=hs-error> </span><span class=hs-error><a class=annot href="#"><span class=annottext>{x2 : (Int) | (x2 == (7 : int))}</span><span class='hs-num'>7</span></a></span> <span class='hs-comment'>-- α := Odd !</span>
</pre>
</div>
Type Polymorphism? No.
----------------------
<div class="fragment">Need to try a bit harder...</div>
By Parametric Refinements!
--------------------------
That is, enable *quantification over refinements*...
Parametric Refinements
----------------------
<pre><span class=hs-linenum>213: </span><span class='hs-keyword'>{-@</span> <span class='hs-varid'>maxInt</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyword'>forall</span> <span class='hs-varop'><</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Prop</span><span class='hs-varop'>>.</span>
<span class=hs-linenum>214: </span> <span class='hs-conid'>Int</span><span class='hs-varop'><</span><span class='hs-varid'>p</span><span class='hs-varop'>></span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span><span class='hs-varop'><</span><span class='hs-varid'>p</span><span class='hs-varop'>></span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span><span class='hs-varop'><</span><span class='hs-varid'>p</span><span class='hs-varop'>></span> <span class='hs-keyword'>@-}</span>
<span class=hs-linenum>215: </span>
<span class=hs-linenum>216: </span><a class=annot href="#"><span class=annottext>forall <p :: (GHC.Types.Int)-> Bool>.
{VV : (Int)<p> | true}
-> {VV : (Int)<p> | true} -> {VV : (Int)<p> | true}</span><span class='hs-definition'>maxInt</span></a> <a class=annot href="#"><span class=annottext>{VV : (Int) | ((papp1 p VV))}</span><span class='hs-varid'>x</span></a> <a class=annot href="#"><span class=annottext>{VV : (Int) | ((papp1 p VV))}</span><span class='hs-varid'>y</span></a> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <a class=annot href="#"><span class=annottext>{x3 : (Int) | ((papp1 p x3)) && (x3 == x)}</span><span class='hs-varid'>x</span></a> <a class=annot href="#"><span class=annottext>x1:{x6 : (Int) | ((papp1 p x6))}
-> x2:{x6 : (Int) | ((papp1 p x6))}
-> {x2 : (Bool) | (((Prop x2)) <=> (x1 <= x2))}</span><span class='hs-varop'><=</span></a> <a class=annot href="#"><span class=annottext>{x3 : (Int) | ((papp1 p x3)) && (x3 == y)}</span><span class='hs-varid'>y</span></a> <span class='hs-keyword'>then</span> <a class=annot href="#"><span class=annottext>{x3 : (Int) | ((papp1 p x3)) && (x3 == y)}</span><span class='hs-varid'>y</span></a> <span class='hs-keyword'>else</span> <a class=annot href="#"><span class=annottext>{x3 : (Int) | ((papp1 p x3)) && (x3 == x)}</span><span class='hs-varid'>x</span></a>
</pre>
<br>
<div class="fragment">Type says: **for any** `p` that is a property of `Int`, </div>
- <div class="fragment">`max` **takes** two `Int`s that satisfy `p`,</div>
- <div class="fragment">`max` **returns** an `Int` that satisfies `p`.</div>
Parametric Refinements
----------------------
<br>
<pre><span class=hs-linenum>232: </span><span class='hs-keyword'>{-@</span> <span class='hs-varid'>maxInt</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyword'>forall</span> <span class='hs-varop'><</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Prop</span><span class='hs-varop'>>.</span>
<span class=hs-linenum>233: </span> <span class='hs-conid'>Int</span><span class='hs-varop'><</span><span class='hs-varid'>p</span><span class='hs-varop'>></span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span><span class='hs-varop'><</span><span class='hs-varid'>p</span><span class='hs-varop'>></span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span><span class='hs-varop'><</span><span class='hs-varid'>p</span><span class='hs-varop'>></span> <span class='hs-keyword'>@-}</span>
<span class=hs-linenum>234: </span>
<span class=hs-linenum>235: </span><span class='hs-definition'>maxInt</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>x</span> <span class='hs-varop'><=</span> <span class='hs-varid'>y</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>y</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>x</span>
</pre>
<br>
[Key idea: ](http://goto.ucsd.edu/~rjhala/papers/abstract_refinement_types.html) `Int<p>` is `{v:Int | (p v)}`
<br>
<div class="fragment">So, Abstract Refinement is an *uninterpreted function* in SMT logic</div>
Parametric Refinements
----------------------
<br>
<pre><span class=hs-linenum>250: </span><span class='hs-keyword'>{-@</span> <span class='hs-varid'>maxInt</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyword'>forall</span> <span class='hs-varop'><</span><span class='hs-varid'>p</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Prop</span><span class='hs-varop'>>.</span>
<span class=hs-linenum>251: </span> <span class='hs-conid'>Int</span><span class='hs-varop'><</span><span class='hs-varid'>p</span><span class='hs-varop'>></span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span><span class='hs-varop'><</span><span class='hs-varid'>p</span><span class='hs-varop'>></span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span><span class='hs-varop'><</span><span class='hs-varid'>p</span><span class='hs-varop'>></span> <span class='hs-keyword'>@-}</span>
<span class=hs-linenum>252: </span>
<span class=hs-linenum>253: </span><span class='hs-definition'>maxInt</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>x</span> <span class='hs-varop'><=</span> <span class='hs-varid'>y</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>y</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>x</span>
</pre>
<br>
**Check** and **Instantiate** type using *SMT & predicate abstraction*
Using Abstract Refinements
--------------------------
- <div class="fragment">**When** we call `maxInt` with args with some refinement,</div>
- <div class="fragment">**Then** `p` instantiated with *same* refinement,</div>
- <div class="fragment">**Result** of call will also have concrete refinement.</div>
<div class="fragment">
<pre><span class=hs-linenum>272: </span><span class='hs-keyword'>{-@</span> <span class='hs-varid'>o'</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Odd</span> <span class='hs-keyword'>@-}</span>
<span class=hs-linenum>273: </span><a class=annot href="#"><span class=annottext>{VV : (Int) | ((VV mod 2) == 1)}</span><span class='hs-definition'>o'</span></a> <span class='hs-keyglyph'>=</span> <a class=annot href="#"><span class=annottext>forall <p :: (GHC.Types.Int)-> Bool>.
{x3 : (Int)<p> | true}
-> {x2 : (Int)<p> | true} -> {x1 : (Int)<p> | true}</span><span class='hs-varid'>maxInt</span></a> <a class=annot href="#"><span class=annottext>{x2 : (Int) | (x2 == (3 : int))}</span><span class='hs-num'>3</span></a> <a class=annot href="#"><span class=annottext>{x2 : (Int) | (x2 == (7 : int))}</span><span class='hs-num'>7</span></a> <span class='hs-comment'>-- p := \v -> Odd v </span>
<span class=hs-linenum>274: </span>
<span class=hs-linenum>275: </span><span class='hs-keyword'>{-@</span> <span class='hs-varid'>e'</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Even</span> <span class='hs-keyword'>@-}</span>
<span class=hs-linenum>276: </span><a class=annot href="#"><span class=annottext>{VV : (Int) | ((VV mod 2) == 0)}</span><span class='hs-definition'>e'</span></a> <span class='hs-keyglyph'>=</span> <a class=annot href="#"><span class=annottext>forall <p :: (GHC.Types.Int)-> Bool>.
{x3 : (Int)<p> | true}
-> {x2 : (Int)<p> | true} -> {x1 : (Int)<p> | true}</span><span class='hs-varid'>maxInt</span></a> <a class=annot href="#"><span class=annottext>{x2 : (Int) | (x2 == (2 : int))}</span><span class='hs-num'>2</span></a> <a class=annot href="#"><span class=annottext>{x2 : (Int) | (x2 == (8 : int))}</span><span class='hs-num'>8</span></a> <span class='hs-comment'>-- p := \v -> Even v </span>
</pre>
</div>
Using Abstract Refinements
--------------------------
Or any other property
<br>
<pre><span class=hs-linenum>289: </span><span class='hs-keyword'>{-@</span> <span class='hs-keyword'>type</span> <span class='hs-conid'>RGB</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>{</span><span class='hs-varid'>v</span><span class='hs-conop'>:</span><span class='hs-keyword'>_</span> <span class='hs-keyglyph'>|</span> <span class='hs-layout'>(</span><span class='hs-num'>0</span> <span class='hs-varop'><=</span> <span class='hs-varid'>v</span> <span class='hs-varop'>&&</span> <span class='hs-varid'>v</span> <span class='hs-varop'><</span> <span class='hs-num'>256</span><span class='hs-layout'>)</span><span class='hs-layout'>}</span> <span class='hs-keyword'>@-}</span>
<span class=hs-linenum>290: </span>
<span class=hs-linenum>291: </span><span class='hs-keyword'>{-@</span> <span class='hs-varid'>rgb</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>RGB</span> <span class='hs-keyword'>@-}</span>
<span class=hs-linenum>292: </span><a class=annot href="#"><span class=annottext>{v : (Int) | (v < 256) && (0 <= v)}</span><span class='hs-definition'>rgb</span></a> <span class='hs-keyglyph'>=</span> <a class=annot href="#"><span class=annottext>forall <p :: (GHC.Types.Int)-> Bool>.
{x3 : (Int)<p> | true}
-> {x2 : (Int)<p> | true} -> {x1 : (Int)<p> | true}</span><span class='hs-varid'>maxInt</span></a> <a class=annot href="#"><span class=annottext>{x2 : (Int) | (x2 == (56 : int))}</span><span class='hs-num'>56</span></a> <a class=annot href="#"><span class=annottext>{x2 : (Int) | (x2 == (8 : int))}</span><span class='hs-num'>8</span></a>
</pre>
Recap
-----
1. **Refinements:** Types + Predicates
2. **Subtyping:** SMT Implication
3. **Measures:** Strengthened Constructors
4. <div class="fragment">**Abstract:** Refinements over Type Signatures</div>