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finished first draft of Stlc

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Philip Wadler 2017-07-17 18:09:37 +01:00
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@ -32,7 +32,8 @@ much difference, though, because we've been careful to name our
own definitions and theorems the same as their counterparts in the
standard library, wherever they overlap.
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="1490" class="Keyword"
>open</a
><a name="1494"
@ -217,7 +218,8 @@ standard library, wherever they overlap.
><a name="1800" class="Symbol"
>)</a
>
{% endraw %}</pre>
</pre>
Documentation for the standard library can be found at
<https://agda.github.io/agda-stdlib/>.
@ -229,7 +231,8 @@ maps. For this purpose, we again use the type `Id` from the
[Lists](sf/Lists.html) chapter. To make this chapter self contained,
we repeat its definition here.
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="2166" class="Keyword"
>data</a
><a name="2170"
@ -270,11 +273,13 @@ we repeat its definition here.
><a name="2197" href="Maps.html#2171" class="Datatype"
>Id</a
>
{% endraw %}</pre>
</pre>
We recall a standard fact of logic.
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="2262" href="Maps.html#2262" class="Function"
>contrapositive</a
><a name="2276"
@ -423,12 +428,14 @@ We recall a standard fact of logic.
><a name="2374" class="Symbol"
>)</a
>
{% endraw %}</pre>
</pre>
Using the above, we can decide equality of two identifiers
by deciding equality on the underlying strings.
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="2509" href="Maps.html#2509" class="Function Operator"
>_&#8799;_</a
><a name="2512"
@ -706,7 +713,8 @@ by deciding equality on the underlying strings.
><a name="2737" href="https://agda.github.io/agda-stdlib/Agda.Builtin.Equality.html#140" class="InductiveConstructor"
>refl</a
>
{% endraw %}</pre>
</pre>
## Total Maps
@ -727,7 +735,8 @@ We build partial maps in two steps. First, we define a type of
_total maps_ that return a default value when we look up a key
that is not present in the map.
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="3573" href="Maps.html#3573" class="Function"
>TotalMap</a
><a name="3581"
@ -772,12 +781,14 @@ that is not present in the map.
><a name="3612" href="Maps.html#3603" class="Bound"
>A</a
>
{% endraw %}</pre>
</pre>
Intuitively, a total map over anfi element type `A` _is_ just a
function that can be used to look up ids, yielding `A`s.
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="3760" class="Keyword"
>module</a
><a name="3766"
@ -789,13 +800,15 @@ function that can be used to look up ids, yielding `A`s.
><a name="3776" class="Keyword"
>where</a
>
{% endraw %}</pre>
</pre>
The function `always` yields a total map given a
default element; this map always returns the default element when
applied to any id.
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="3944" href="Maps.html#3944" class="Function"
>always</a
><a name="3950"
@ -856,13 +869,15 @@ applied to any id.
><a name="3991" href="Maps.html#3985" class="Bound"
>v</a
>
{% endraw %}</pre>
</pre>
More interesting is the update function, which (as before) takes
a map `ρ`, a key `x`, and a value `v` and returns a new map that
takes `x` to `v` and takes every other key to whatever `ρ` does.
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="4216" class="Keyword"
>infixl</a
><a name="4222"
@ -1035,7 +1050,8 @@ takes `x` to `v` and takes every other key to whatever `ρ` does.
><a name="4353" href="Maps.html#4300" class="Bound"
>y</a
>
{% endraw %}</pre>
</pre>
This definition is a nice example of higher-order programming.
The update function takes a _function_ `ρ` and yields a new
@ -1044,7 +1060,8 @@ function that behaves like the desired map.
For example, we can build a map taking ids to naturals, where `x`
maps to 42, `y` maps to 69, and every other key maps to 0, as follows:
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="4688" class="Keyword"
>module</a
><a name="4694"
@ -1313,7 +1330,8 @@ maps to 42, `y` maps to 69, and every other key maps to 0, as follows:
><a name="4935" href="https://agda.github.io/agda-stdlib/Agda.Builtin.Equality.html#140" class="InductiveConstructor"
>refl</a
>
{% endraw %}</pre>
</pre>
This completes the definition of total maps. Note that we don't
need to define a `find` operation because it is just function
@ -1327,7 +1345,8 @@ the lemmas!
#### Exercise: 1 star, optional (apply-always)
The `always` map returns its default element for all keys:
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="5411" class="Keyword"
>postulate</a
><a name="5420"
@ -1408,10 +1427,12 @@ The `always` map returns its default element for all keys:
><a name="5478" href="Maps.html#5447" class="Bound"
>v</a
>
{% endraw %}</pre>
</pre>
<div class="hidden">
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="5528" href="Maps.html#5528" class="Function"
>apply-always&#8242;</a
><a name="5541"
@ -1508,7 +1529,8 @@ The `always` map returns its default element for all keys:
><a name="5606" href="https://agda.github.io/agda-stdlib/Agda.Builtin.Equality.html#140" class="InductiveConstructor"
>refl</a
>
{% endraw %}</pre>
</pre>
</div>
#### Exercise: 2 stars, optional (update-eq)
@ -1516,7 +1538,8 @@ Next, if we update a map `ρ` at a key `x` with a new value `v`
and then look up `x` in the map resulting from the update, we get
back `v`:
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="5830" class="Keyword"
>postulate</a
><a name="5839"
@ -1634,10 +1657,12 @@ back `v`:
><a name="5928" href="Maps.html#5889" class="Bound"
>v</a
>
{% endraw %}</pre>
</pre>
<div class="hidden">
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="5978" href="Maps.html#5978" class="Function"
>update-eq&#8242;</a
><a name="5988"
@ -1845,7 +1870,8 @@ back `v`:
><a name="6151" class="Symbol"
>)</a
>
{% endraw %}</pre>
</pre>
</div>
#### Exercise: 2 stars, optional (update-neq)
@ -1853,7 +1879,8 @@ On the other hand, if we update a map `m` at a key `x` and
then look up a _different_ key `y` in the resulting map, we get
the same result that `m` would have given:
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="6400" href="Maps.html#6400" class="Function"
>update-neq</a
><a name="6410"
@ -2105,12 +2132,14 @@ the same result that `m` would have given:
><a name="6594" href="https://agda.github.io/agda-stdlib/Agda.Builtin.Equality.html#140" class="InductiveConstructor"
>refl</a
>
{% endraw %}</pre>
</pre>
For the following lemmas, since maps are represented by functions, to
show two maps equal we will need to postulate extensionality.
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="6759" class="Keyword"
>postulate</a
><a name="6768"
@ -2221,7 +2250,8 @@ show two maps equal we will need to postulate extensionality.
><a name="6849" href="Maps.html#6805" class="Bound"
>&#961;&#8242;</a
>
{% endraw %}</pre>
</pre>
#### Exercise: 2 stars, optional (update-shadow)
If we update a map `ρ` at a key `x` with a value `v` and then
@ -2230,7 +2260,8 @@ resulting map behaves the same (gives the same result when applied
to any key) as the simpler map obtained by performing just
the second update on `ρ`:
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="7205" class="Keyword"
>postulate</a
><a name="7214"
@ -2384,10 +2415,12 @@ the second update on `ρ`:
><a name="7330" class="Symbol"
>)</a
>
{% endraw %}</pre>
</pre>
<div class="hidden">
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="7380" href="Maps.html#7380" class="Function"
>update-shadow&#8242;</a
><a name="7394"
@ -2770,7 +2803,8 @@ the second update on `ρ`:
><a name="7702" href="Maps.html#7676" class="Bound"
>x&#8802;y</a
>
{% endraw %}</pre>
</pre>
</div>
#### Exercise: 2 stars (update-same)
@ -2778,7 +2812,8 @@ Prove the following theorem, which states that if we update a map `ρ` to
assign key `x` the same value as it already has in `ρ`, then the
result is equal to `ρ`:
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="7940" class="Keyword"
>postulate</a
><a name="7949"
@ -2879,10 +2914,12 @@ result is equal to `ρ`:
><a name="8018" href="Maps.html#7975" class="Bound"
>&#961;</a
>
{% endraw %}</pre>
</pre>
<div class="hidden">
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="8068" href="Maps.html#8068" class="Function"
>update-same&#8242;</a
><a name="8080"
@ -3148,7 +3185,8 @@ result is equal to `ρ`:
><a name="8299" href="https://agda.github.io/agda-stdlib/Agda.Builtin.Equality.html#140" class="InductiveConstructor"
>refl</a
>
{% endraw %}</pre>
</pre>
</div>
#### Exercise: 3 stars, recommended (update-permute)
@ -3156,7 +3194,8 @@ Prove one final property of the `update` function: If we update a map
`m` at two distinct keys, it doesn't matter in which order we do the
updates.
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="8540" class="Keyword"
>postulate</a
><a name="8549"
@ -3370,10 +3409,12 @@ updates.
><a name="8697" class="Symbol"
>)</a
>
{% endraw %}</pre>
</pre>
<div class="hidden">
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="8747" href="Maps.html#8747" class="Function"
>update-permute&#8242;</a
><a name="8762"
@ -4044,11 +4085,13 @@ updates.
><a name="9312" class="Symbol"
>))</a
>
{% endraw %}</pre>
</pre>
And a slightly different version of the same proof.
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="9399" href="Maps.html#9399" class="Function"
>update-permute&#8242;&#8242;</a
><a name="9415"
@ -4640,7 +4683,8 @@ And a slightly different version of the same proof.
><a name="9895" class="Symbol"
>))</a
>
{% endraw %}</pre>
</pre>
</div>
## Partial maps
@ -4649,7 +4693,8 @@ Finally, we define _partial maps_ on top of total maps. A partial
map with elements of type `A` is simply a total map with elements
of type `Maybe A` and default element `nothing`.
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="10132" href="Maps.html#10132" class="Function"
>PartialMap</a
><a name="10142"
@ -4698,9 +4743,11 @@ of type `Maybe A` and default element `nothing`.
><a name="10187" class="Symbol"
>)</a
>
{% endraw %}</pre>
<pre class="Agda">{% raw %}
</pre>
<pre class="Agda">
<a name="10214" class="Keyword"
>module</a
><a name="10220"
@ -4712,9 +4759,11 @@ of type `Maybe A` and default element `nothing`.
><a name="10232" class="Keyword"
>where</a
>
{% endraw %}</pre>
<pre class="Agda">{% raw %}
</pre>
<pre class="Agda">
<a name="10265" href="Maps.html#10265" class="Function"
>&#8709;</a
><a name="10266"
@ -4763,9 +4812,11 @@ of type `Maybe A` and default element `nothing`.
><a name="10312" href="https://agda.github.io/agda-stdlib/Data.Maybe.Base.html#403" class="InductiveConstructor"
>nothing</a
>
{% endraw %}</pre>
<pre class="Agda">{% raw %}
</pre>
<pre class="Agda">
<a name="10347" class="Keyword"
>infixl</a
><a name="10353"
@ -4912,11 +4963,13 @@ of type `Maybe A` and default element `nothing`.
><a name="10473" class="Symbol"
>)</a
>
{% endraw %}</pre>
</pre>
We now lift all of the basic lemmas about total maps to partial maps.
<pre class="Agda">{% raw %}
<pre class="Agda">
<a name="10573" href="Maps.html#10573" class="Function"
>apply-&#8709;</a
><a name="10580"
@ -5013,9 +5066,11 @@ We now lift all of the basic lemmas about total maps to partial maps.
><a name="10667" href="Maps.html#10632" class="Bound"
>x</a
>
{% endraw %}</pre>
<pre class="Agda">{% raw %}
</pre>
<pre class="Agda">
<a name="10696" href="Maps.html#10696" class="Function"
>update-eq</a
><a name="10705"
@ -5177,9 +5232,11 @@ We now lift all of the basic lemmas about total maps to partial maps.
><a name="10837" class="Symbol"
>)</a
>
{% endraw %}</pre>
<pre class="Agda">{% raw %}
</pre>
<pre class="Agda">
<a name="10866" href="Maps.html#10866" class="Function"
>update-neq</a
><a name="10876"
@ -5389,9 +5446,11 @@ We now lift all of the basic lemmas about total maps to partial maps.
><a name="11035" href="Maps.html#10994" class="Bound"
>x&#8802;y</a
>
{% endraw %}</pre>
<pre class="Agda">{% raw %}
</pre>
<pre class="Agda">
<a name="11066" href="Maps.html#11066" class="Function"
>update-shadow</a
><a name="11079"
@ -5601,9 +5660,11 @@ We now lift all of the basic lemmas about total maps to partial maps.
><a name="11248" class="Symbol"
>)</a
>
{% endraw %}</pre>
<pre class="Agda">{% raw %}
</pre>
<pre class="Agda">
<a name="11277" href="Maps.html#11277" class="Function"
>update-same</a
><a name="11288"
@ -5786,9 +5847,11 @@ We now lift all of the basic lemmas about total maps to partial maps.
><a name="11462" href="Maps.html#11411" class="Bound"
>x</a
>
{% endraw %}</pre>
<pre class="Agda">{% raw %}
</pre>
<pre class="Agda">
<a name="11491" href="Maps.html#11491" class="Function"
>update-permute</a
><a name="11505"
@ -6074,4 +6137,259 @@ We now lift all of the basic lemmas about total maps to partial maps.
><a name="11718" href="Maps.html#11664" class="Bound"
>x&#8802;y</a
>
{% endraw %}</pre>
</pre>
We will also need the following basic facts about the `Maybe` type.
<pre class="Agda">
<a name="11818" href="Maps.html#11818" class="Function"
>just&#8802;nothing</a
><a name="11830"
> </a
><a name="11831" class="Symbol"
>:</a
><a name="11832"
> </a
><a name="11833" class="Symbol"
>&#8704;</a
><a name="11834"
> </a
><a name="11835" class="Symbol"
>{</a
><a name="11836" href="Maps.html#11836" class="Bound"
>X</a
><a name="11837"
> </a
><a name="11838" class="Symbol"
>:</a
><a name="11839"
> </a
><a name="11840" class="PrimitiveType"
>Set</a
><a name="11843" class="Symbol"
>}</a
><a name="11844"
> </a
><a name="11845" class="Symbol"
>&#8594;</a
><a name="11846"
> </a
><a name="11847" class="Symbol"
>&#8704;</a
><a name="11848"
> </a
><a name="11849" class="Symbol"
>{</a
><a name="11850" href="Maps.html#11850" class="Bound"
>x</a
><a name="11851"
> </a
><a name="11852" class="Symbol"
>:</a
><a name="11853"
> </a
><a name="11854" href="Maps.html#11836" class="Bound"
>X</a
><a name="11855" class="Symbol"
>}</a
><a name="11856"
> </a
><a name="11857" class="Symbol"
>&#8594;</a
><a name="11858"
> </a
><a name="11859" href="https://agda.github.io/agda-stdlib/Relation.Nullary.html#414" class="Function Operator"
>&#172;</a
><a name="11860"
> </a
><a name="11861" class="Symbol"
>(</a
><a name="11862" href="https://agda.github.io/agda-stdlib/Agda.Builtin.Equality.html#83" class="Datatype Operator"
>_&#8801;_</a
><a name="11865"
> </a
><a name="11866" class="Symbol"
>{</a
><a name="11867" class="Argument"
>A</a
><a name="11868"
> </a
><a name="11869" class="Symbol"
>=</a
><a name="11870"
> </a
><a name="11871" href="https://agda.github.io/agda-stdlib/Data.Maybe.Base.html#335" class="Datatype"
>Maybe</a
><a name="11876"
> </a
><a name="11877" href="Maps.html#11836" class="Bound"
>X</a
><a name="11878" class="Symbol"
>}</a
><a name="11879"
> </a
><a name="11880" class="Symbol"
>(</a
><a name="11881" href="https://agda.github.io/agda-stdlib/Data.Maybe.Base.html#373" class="InductiveConstructor"
>just</a
><a name="11885"
> </a
><a name="11886" href="Maps.html#11850" class="Bound"
>x</a
><a name="11887" class="Symbol"
>)</a
><a name="11888"
> </a
><a name="11889" href="https://agda.github.io/agda-stdlib/Data.Maybe.Base.html#403" class="InductiveConstructor"
>nothing</a
><a name="11896" class="Symbol"
>)</a
><a name="11897"
>
</a
><a name="11900" href="Maps.html#11818" class="Function"
>just&#8802;nothing</a
><a name="11912"
> </a
><a name="11913" class="Symbol"
>()</a
><a name="11915"
>
</a
><a name="11919" href="Maps.html#11919" class="Function"
>just-injective</a
><a name="11933"
> </a
><a name="11934" class="Symbol"
>:</a
><a name="11935"
> </a
><a name="11936" class="Symbol"
>&#8704;</a
><a name="11937"
> </a
><a name="11938" class="Symbol"
>{</a
><a name="11939" href="Maps.html#11939" class="Bound"
>X</a
><a name="11940"
> </a
><a name="11941" class="Symbol"
>:</a
><a name="11942"
> </a
><a name="11943" class="PrimitiveType"
>Set</a
><a name="11946" class="Symbol"
>}</a
><a name="11947"
> </a
><a name="11948" class="Symbol"
>{</a
><a name="11949" href="Maps.html#11949" class="Bound"
>x</a
><a name="11950"
> </a
><a name="11951" href="Maps.html#11951" class="Bound"
>y</a
><a name="11952"
> </a
><a name="11953" class="Symbol"
>:</a
><a name="11954"
> </a
><a name="11955" href="Maps.html#11939" class="Bound"
>X</a
><a name="11956" class="Symbol"
>}</a
><a name="11957"
> </a
><a name="11958" class="Symbol"
>&#8594;</a
><a name="11959"
> </a
><a name="11960" href="https://agda.github.io/agda-stdlib/Agda.Builtin.Equality.html#83" class="Datatype Operator"
>_&#8801;_</a
><a name="11963"
> </a
><a name="11964" class="Symbol"
>{</a
><a name="11965" class="Argument"
>A</a
><a name="11966"
> </a
><a name="11967" class="Symbol"
>=</a
><a name="11968"
> </a
><a name="11969" href="https://agda.github.io/agda-stdlib/Data.Maybe.Base.html#335" class="Datatype"
>Maybe</a
><a name="11974"
> </a
><a name="11975" href="Maps.html#11939" class="Bound"
>X</a
><a name="11976" class="Symbol"
>}</a
><a name="11977"
> </a
><a name="11978" class="Symbol"
>(</a
><a name="11979" href="https://agda.github.io/agda-stdlib/Data.Maybe.Base.html#373" class="InductiveConstructor"
>just</a
><a name="11983"
> </a
><a name="11984" href="Maps.html#11949" class="Bound"
>x</a
><a name="11985" class="Symbol"
>)</a
><a name="11986"
> </a
><a name="11987" class="Symbol"
>(</a
><a name="11988" href="https://agda.github.io/agda-stdlib/Data.Maybe.Base.html#373" class="InductiveConstructor"
>just</a
><a name="11992"
> </a
><a name="11993" href="Maps.html#11951" class="Bound"
>y</a
><a name="11994" class="Symbol"
>)</a
><a name="11995"
> </a
><a name="11996" class="Symbol"
>&#8594;</a
><a name="11997"
> </a
><a name="11998" href="Maps.html#11949" class="Bound"
>x</a
><a name="11999"
> </a
><a name="12000" href="https://agda.github.io/agda-stdlib/Agda.Builtin.Equality.html#83" class="Datatype Operator"
>&#8801;</a
><a name="12001"
> </a
><a name="12002" href="Maps.html#11951" class="Bound"
>y</a
><a name="12003"
>
</a
><a name="12006" href="Maps.html#11919" class="Function"
>just-injective</a
><a name="12020"
> </a
><a name="12021" href="https://agda.github.io/agda-stdlib/Agda.Builtin.Equality.html#140" class="InductiveConstructor"
>refl</a
><a name="12025"
> </a
><a name="12026" class="Symbol"
>=</a
><a name="12027"
> </a
><a name="12028" href="https://agda.github.io/agda-stdlib/Agda.Builtin.Equality.html#140" class="InductiveConstructor"
>refl</a
>
</pre>

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@ -373,3 +373,13 @@ We now lift all of the basic lemmas about total maps to partial maps.
→ x ≢ y → (ρ , x ↦ v , y ↦ w) ≡ (ρ , y ↦ w , x ↦ v)
update-permute ρ x v y w x≢y = TotalMap.update-permute ρ x (just v) y (just w) x≢y
\end{code}
We will also need the following basic facts about the `Maybe` type.
\begin{code}
just≢nothing : ∀ {X : Set} → ∀ {x : X} → ¬ (_≡_ {A = Maybe X} (just x) nothing)
just≢nothing ()
just-injective : ∀ {X : Set} {x y : X} → _≡_ {A = Maybe X} (just x) (just y) → x ≡ y
just-injective refl = refl
\end{code}

157
src/Stlc.lagda
View file

@ -7,7 +7,7 @@ permalink : /Stlc
The _lambda-calculus_, first published by the logician Alonzo Church in
1932, is a core calculus with only three syntactic constructs:
variables, abstraction, and application. It embodies the concept of
_functional abstraction_, which shows up in almsot every programming
_functional abstraction_, which shows up in almost every programming
language in some form (as functions, procedures, or methods).
The _simply-typed lambda calculus_ (or STLC) is a variant of the
lambda calculus published by Church in 1940. It has just the three
@ -38,10 +38,10 @@ lists, records, subtyping, and mutable state.
\begin{code}
open import Maps using (Id; id; _≟_; PartialMap; module PartialMap)
open PartialMap using (∅) renaming (_,_↦_ to _,_∶_)
open PartialMap using (∅; just-injective) renaming (_,_↦_ to _,_∶_)
open import Data.Nat using (ℕ)
open import Data.Maybe using (Maybe; just; nothing)
open import Relation.Nullary using (Dec; yes; no)
open import Relation.Nullary using (Dec; yes; no; ¬_)
open import Relation.Binary.PropositionalEquality using (_≡_; _≢_; refl)
\end{code}
@ -86,7 +86,7 @@ and three are for the base type, booleans:
Abstraction is also called lambda abstraction, and is the construct
from which the calculus takes its name.
With the exception of variables, each construct either constructs
With the exception of variables, each term form either constructs
a value of a given type (abstractions yield functions, true and
false yield booleans) or deconstructs it (applications use functions,
conditionals use booleans). We will see this again when we come
@ -177,21 +177,12 @@ two = λ[ f ∶ 𝔹 ⇒ 𝔹 ] λ[ x ∶ 𝔹 ] ` f · (` f · ` x)
In an abstraction `λ[ x ∶ A ] N` we call `x` the _bound_ variable
and `N` the _body_ of the abstraction. One of the most important
aspects of lambda calculus is that names of bound variables are
irrelevant. Thus the two terms
irrelevant. Thus the four terms
λ[ f ∶ 𝔹 ⇒ 𝔹 ] λ[ x ∶ 𝔹 ] ` f · (` f · ` x)
and
λ[ g ∶ 𝔹 ⇒ 𝔹 ] λ[ y ∶ 𝔹 ] ` g · (` g · ` y)
and
λ[ fred ∶ 𝔹 ⇒ 𝔹 ] λ[ xander ∶ 𝔹 ] ` fred · (` fred · ` xander)
and even
λ[ x ∶ 𝔹 ⇒ 𝔹 ] λ[ f ∶ 𝔹 ] ` x · (` x · ` f)
* `` λ[ f ∶ 𝔹 ⇒ 𝔹 ] λ[ x ∶ 𝔹 ] ` f · (` f · ` x) ``
* `` λ[ g ∶ 𝔹 ⇒ 𝔹 ] λ[ y ∶ 𝔹 ] ` g · (` g · ` y) ``
* `` λ[ fred ∶ 𝔹 ⇒ 𝔹 ] λ[ xander ∶ 𝔹 ] ` fred · (` fred · ` xander) ``
* `` λ[ x ∶ 𝔹 ⇒ 𝔹 ] λ[ f ∶ 𝔹 ] ` x · (` x · ` f) ``
are all considered equivalent. This equivalence relation
is sometimes called _alpha renaming_.
@ -460,8 +451,8 @@ conditional, we first reduce the condition until it becomes a value;
if the condition is true the conditional reduces to the first
branch and if false it reduces to the second branch.a
In an informal presentation of the formal semantics, the rules
are written as follows.
In an informal presentation of the formal semantics,
the rules for reduction are written as follows.
L ⟹ L′
--------------- ξ·₁
@ -501,6 +492,8 @@ a deconstructor, in our case `λ` and `·`, or
We give them names starting with the Greek letter beta, `β`,
and indeed such rules are traditionally called beta rules.
Here are the above rules formalised in Agda.
\begin{code}
infix 10 _⟹_
@ -659,20 +652,62 @@ In general, we use typing _judgements_ of the form
Γ ⊢ M ∶ A
which asserts in type environment `Γ` that term `M` has type `A`.
Here `Γ` provides types for all the free variables in `M`.
Environment `Γ` provides types for all the free variables in `M`.
Here are three examples.
* `` ∅ , f ∶ 𝔹 ⇒ 𝔹 , x ∶ 𝔹 ⊢ ` f · (` f · ` x) ∶ 𝔹 ``
* `` ∅ , f ∶ 𝔹 ⇒ 𝔹 ⊢ (λ[ x ∶ 𝔹 ] ` f · (` f · ` x)) ∶ 𝔹 ⇒ 𝔹 ``
* `` ∅ ⊢ (λ[ f ∶ 𝔹 ⇒ 𝔹 ] λ[ x ∶ 𝔹 ] ` f · (` f · ` x)) ∶ (𝔹 ⇒ 𝔹) ⇒ 𝔹 ⇒ 𝔹 ``
* `` ∅ , f ∶ 𝔹 ⇒ 𝔹 ⊢ (λ[ x ∶ 𝔹 ] ` f · (` f · ` x)) ∶ 𝔹 ⇒ 𝔹 ``
* `` ∅ , f ∶ 𝔹 ⇒ 𝔹 , x ∶ 𝔹 ⊢ ` f · (` f · ` x) ∶ 𝔹 ``
Environments are maps from free variables to types, built using `∅`
Environments are partial maps from identifiers to types, built using `∅`
for the empty map, and `Γ , x ∶ A` for the map that extends
environment `Γ` by mapping variable `x` to type `A`.
In an informal presentation of the formal semantics,
the rules for typing are written as follows.
Γ x ≡ A
----------- Ax
Γ ⊢ ` x ∶ A
Γ , x ∶ A ⊢ N ∶ B
------------------------ ⇒-I
Γ ⊢ λ[ x ∶ A ] N ∶ A ⇒ B
Γ ⊢ L ∶ A ⇒ B
Γ ⊢ M ∶ A
-------------- ⇒-E
Γ ⊢ L · M ∶ B
------------- 𝔹-I₁
Γ ⊢ true ∶ 𝔹
-------------- 𝔹-I₂
Γ ⊢ false ∶ 𝔹
Γ ⊢ L : 𝔹
Γ ⊢ M ∶ A
Γ ⊢ N ∶ A
-------------------------- 𝔹-E
Γ ⊢ if L then M else N ∶ A
As we will show later, the rules are deterministic, in that
at most one rule applies to every term.
The proof rules come in pairs, with rules to introduce and to
eliminate each connective, labeled `-I` and `-E`, respectively. As we
read the rules from top to bottom, introduction and elimination rules
do what they say on the tin: the first _introduces_ a formula for the
connective, which appears in the conclusion but not in the premises;
while the second _eliminates_ a formula for the connective, which appears in
a premise but not in the conclusion. An introduction rule describes
how to construct a value of the type (abstractions yield functions,
true and false yield booleans), while an elimination rule describes
how to deconstruct a value of the given type (applications use
functions, conditionals use booleans).
Here are the above rules formalised in Agda.
\begin{code}
Context : Set
@ -702,7 +737,38 @@ data _⊢_∶_ : Context → Term → Type → Set where
Γ ⊢ if L then M else N ∶ A
\end{code}
## Example type derivations
#### Example type derivations
Here are a couple of typing examples. First, here is how
they would be written in an informal description of the
formal semantics.
Derivation of `not`:
------------ Ax ------------- 𝔹-I₂ ------------- 𝔹-I₁
Γ₀ ⊢ ` x ∶ 𝔹 Γ₀ ⊢ false ∶ 𝔹 Γ₀ ⊢ true ∶ 𝔹
------------------------------------------------------ 𝔹-E
Γ₀ ⊢ if ` x then false else true ∶ 𝔹
--------------------------------------------------- ⇒-I
∅ ⊢ λ[ x ∶ 𝔹 ] if ` x then false else true ∶ 𝔹 ⇒ 𝔹
where `Γ₀ = ∅ , x ∶ 𝔹`.
Derivation of `two`:
----------------- Ax ------------ Ax
Γ₂ ⊢ ` f ∶ 𝔹 ⇒ 𝔹 Γ₂ ⊢ ` x ∶ 𝔹
----------------- Ax ------------------------------------- ⇒-E
Γ₂ ⊢ ` f ∶ 𝔹 ⇒ 𝔹 Γ₂ ⊢ ` f · ` x ∶ 𝔹
------------------------------------------- ⇒-E
Γ₂ ⊢ ` f · (` f · ` x) ∶ 𝔹
------------------------------------------ ⇒-I
Γ₁ ⊢ λ[ x ∶ 𝔹 ] ` f · (` f · ` x) ∶ 𝔹 ⇒ 𝔹
---------------------------------------------------------- ⇒-I
∅ ⊢ λ[ f ∶ 𝔹 ⇒ 𝔹 ] λ[ x ∶ 𝔹 ] ` f · (` f · ` x) ∶ 𝔹 ⇒ 𝔹
where `Γ₁ = ∅ , f ∶ 𝔹 ⇒ 𝔹` and `Γ₂ = ∅ , f ∶ 𝔹 ⇒ 𝔹 , x ∶ 𝔹`.
Here are the above derivations formalised in Agda.
\begin{code}
typing₁ : ∅ ⊢ not ∶ 𝔹 ⇒ 𝔹
@ -712,8 +778,10 @@ typing₂ : ∅ ⊢ two ∶ (𝔹 ⇒ 𝔹) ⇒ 𝔹 ⇒ 𝔹
typing₂ = ⇒-I (⇒-I (⇒-E (Ax refl) (⇒-E (Ax refl) (Ax refl))))
\end{code}
## Interaction with Agda
Construction of a type derivation is best done interactively.
We start with the declaration:
Start with the declaration:
typing₁ : ∅ ⊢ not ∶ 𝔹 ⇒ 𝔹
typing₁ = ?
@ -747,4 +815,37 @@ hole. After filling in all holes, the term is as above.
The entire process can be automated using Agsy, invoked with C-c C-a.
#### Non-examples
We can also show that terms are _not_ typeable.
For example, here is a formal proof that it is not possible
to type the term `` λ[ x ∶ 𝔹 ] λ[ y ∶ 𝔹 ] ` x · ` y ``.
In other words, no type `A` is the type of this term.
\begin{code}
contradiction : ∀ {A B} → ¬ (𝔹 ≡ A ⇒ B)
contradiction ()
notyping : ∀ {A} → ¬ (∅ ⊢ λ[ x ∶ 𝔹 ] λ[ y ∶ 𝔹 ] ` x · ` y ∶ A)
notyping (⇒-I (⇒-I (⇒-E (Ax Γx) (Ax Γy)))) = contradiction (just-injective Γx)
\end{code}
#### Quiz
For each of the following, given a type `A` for which it is derivable,
or explain why there is no such `A`.
1. `` ∅ , y ∶ A ⊢ λ[ x ∶ 𝔹 ] ` x ∶ 𝔹 ⇒ 𝔹 ``
2. `` ∅ ⊢ λ[ y ∶ 𝔹 ⇒ 𝔹 ] λ[ x ∶ 𝔹 ] ` y · ` x ∶ A ``
3. `` ∅ ⊢ λ[ y ∶ 𝔹 ⇒ 𝔹 ] λ[ x ∶ 𝔹 ] ` x · ` y ∶ A ``
4. `` ∅ , x ∶ A ⊢ λ[ y : 𝔹 ⇒ 𝔹 ] `y · `x : A ``
For each of the following, give type `A`, `B`, and `C` for which it is derivable,
or explain why there are no such types.
1. `` ∅ ⊢ λ[ y ∶ 𝔹 ⇒ 𝔹 ⇒ 𝔹 ] λ[ x ∶ 𝔹 ] ` y · ` x ∶ A ``
2. `` ∅ , x ∶ A ⊢ x · x ∶ B ``
3. `` ∅ , x ∶ A , y ∶ B ⊢ λ[ z ∶ C ] ` x · (` y · ` z) ∶ D ``

10
src/StlcProp.lagda
View file

@ -19,7 +19,8 @@ open import Data.Sum using (_⊎_; inj₁; inj₂)
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Relation.Binary.PropositionalEquality using (_≡_; _≢_; refl; trans; sym)
open import Maps using (Id; _≟_; PartialMap)
open Maps.PartialMap using (∅; apply-∅; update-permute) renaming (_,_↦_ to _,_∶_)
open Maps.PartialMap using (∅; apply-∅; update-permute; just≢nothing; just-injective)
renaming (_,_↦_ to _,_∶_)
open import Stlc
import Data.Nat using (ℕ)
\end{code}
@ -320,12 +321,9 @@ postulate
<div class="hidden">
\begin{code}
contradiction : ∀ {X : Set} → ∀ {x : X} → ¬ (_≡_ {A = Maybe X} (just x) nothing)
contradiction ()
∅⊢-closed′ : ∀ {M A} → ∅ ⊢ M ∶ A → closed M
∅⊢-closed′ {M} {A} ⊢M {x} x∈M with free-lemma x∈M ⊢M
... | (B , ∅x≡B) = contradiction (trans (sym ∅x≡B) (apply-∅ x))
... | (B , ∅x≡B) = just≢nothing (trans (sym ∅x≡B) (apply-∅ x))
\end{code}
</div>
@ -504,8 +502,6 @@ weaken-closed {V} {A} {Γ} ⊢V = context-lemma Γ~Γ′ ⊢V
x∉V : ¬ (x ∈ V)
x∉V = ∅⊢-closed ⊢V {x}
just-injective : ∀ {X : Set} {x y : X} → _≡_ {A = Maybe X} (just x) (just y) → x ≡ y
just-injective refl = refl
\end{code}
\begin{code}