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renamed dom and cod

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Jeremy Siek 2020-03-11 15:55:44 -04:00
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93
src/plfa/part3/Denotational.lagda.md
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@ -953,59 +953,59 @@ function, we want to show that `v ↦ w ⊑ u` implies that `u` includes
a set of function values such that the join of their domains is less
than `v` and the join of their codomains is greater than `w`.
To this end we define the following dom and cod functions. Given some
value `u` (that represents a set of entries), `dom u` returns the join of
their domains and `cod u` returns the join of their codomains.
To this end we define the following `⨆dom` and `⨆cod` functions. Given some
value `u` (that represents a set of entries), `⨆dom u` returns the join of
their domains and `⨆cod u` returns the join of their codomains.
```
dom : (u : Value) → Value
dom ⊥ = ⊥
dom (v ↦ w) = v
dom (u ⊔ u′) = dom u ⊔ dom u′
⨆dom : (u : Value) → Value
⨆dom ⊥ = ⊥
⨆dom (v ↦ w) = v
⨆dom (u ⊔ u′) = ⨆dom u ⊔ ⨆dom u′
cod : (u : Value) → Value
cod ⊥ = ⊥
cod (v ↦ w) = w
cod (u ⊔ u′) = cod u ⊔ cod u′
⨆cod : (u : Value) → Value
⨆cod ⊥ = ⊥
⨆cod (v ↦ w) = w
⨆cod (u ⊔ u′) = ⨆cod u ⊔ ⨆cod u′
```
We need just one property each for `dom` and `cod`. Given a collection of
We need just one property each for `⨆dom` and `⨆cod`. Given a collection of
functions represented by value `u`, and an entry `v ↦ w ∈ u`, we know
that `v` is included in the domain of `v`.
```
↦∈→⊆dom : ∀{u v w : Value}
↦∈→⊆⨆dom : ∀{u v w : Value}
→ all-funs u → (v ↦ w) ∈ u
----------------------
→ v ⊆ dom u
↦∈→⊆dom {⊥} fg () u∈v
↦∈→⊆dom {v ↦ w} fg refl u∈v = u∈v
↦∈→⊆dom {u ⊔ u′} fg (inj₁ v↦w∈u) u∈v =
let ih = ↦∈→⊆dom (λ z → fg (inj₁ z)) v↦w∈u in
→ v ⊆ ⨆dom u
↦∈→⊆⨆dom {⊥} fg () u∈v
↦∈→⊆⨆dom {v ↦ w} fg refl u∈v = u∈v
↦∈→⊆⨆dom {u ⊔ u′} fg (inj₁ v↦w∈u) u∈v =
let ih = ↦∈→⊆⨆dom (λ z → fg (inj₁ z)) v↦w∈u in
inj₁ (ih u∈v)
↦∈→⊆dom {u ⊔ u′} fg (inj₂ v↦w∈u′) u∈v =
let ih = ↦∈→⊆dom (λ z → fg (inj₂ z)) v↦w∈u′ in
↦∈→⊆⨆dom {u ⊔ u′} fg (inj₂ v↦w∈u′) u∈v =
let ih = ↦∈→⊆⨆dom (λ z → fg (inj₂ z)) v↦w∈u′ in
inj₂ (ih u∈v)
```
Regarding `cod`, suppose we have a collection of functions represented
by `u`, but all of them are just copies of `v ↦ w`. Then the `cod u` is
Regarding `⨆cod`, suppose we have a collection of functions represented
by `u`, but all of them are just copies of `v ↦ w`. Then the `⨆cod u` is
included in `w`.
```
⊆↦→cod⊆ : ∀{u v w : Value}
⊆↦→⨆cod⊆ : ∀{u v w : Value}
→ u ⊆ v ↦ w
---------
→ cod u ⊆ w
⊆↦→cod⊆ {⊥} s refl with s {⊥} refl
→ ⨆cod u ⊆ w
⊆↦→⨆cod⊆ {⊥} s refl with s {⊥} refl
... | ()
⊆↦→cod⊆ {C ↦ C′} s m with s {C ↦ C′} refl
⊆↦→⨆cod⊆ {C ↦ C′} s m with s {C ↦ C′} refl
... | refl = m
⊆↦→cod⊆ {u ⊔ u′} s (inj₁ x) = ⊆↦→cod⊆ (λ {C} z → s (inj₁ z)) x
⊆↦→cod⊆ {u ⊔ u′} s (inj₂ y) = ⊆↦→cod⊆ (λ {C} z → s (inj₂ z)) y
⊆↦→⨆cod⊆ {u ⊔ u′} s (inj₁ x) = ⊆↦→⨆cod⊆ (λ {C} z → s (inj₁ z)) x
⊆↦→⨆cod⊆ {u ⊔ u′} s (inj₂ y) = ⊆↦→⨆cod⊆ (λ {C} z → s (inj₂ z)) y
```
With the `dom` and `cod` functions in hand, we can make precise the
With the `⨆dom` and `⨆cod` functions in hand, we can make precise the
conclusion of the inversion principle for functions, which we package
into the following predicate named `factor`. We say that `v ↦ w`
_factors_ `u` into `u′` if `u′` is a included in `u`, if `u′` contains only
@ -1014,7 +1014,7 @@ than `w`.
```
factor : (u : Value) → (u′ : Value) → (v : Value) → (w : Value) → Set
factor u u′ v w = all-funs u′ × u′ ⊆ u × dom u′ ⊑ v × w ⊑ cod u′
factor u u′ v w = all-funs u′ × u′ ⊆ u × ⨆dom u′ ⊑ v × w ⊑ ⨆cod u′
```
We prove the inversion principle for functions by induction on the
@ -1038,7 +1038,7 @@ so we have `all-funs u′` and `u′ ⊆ u`.
By the induction hypothesis for `u ⊑ u₂`, we know
that for any `v′ ↦ w′ ∈ u`, `v′ ↦ w′` factors `u₂` into `u₃`.
With these facts in hand, we proceed by induction on `u′`
to prove that `(dom u′) ↦ (cod u′)` factors `u₂` into `u₃`.
to prove that `(⨆dom u′) ↦ (⨆cod u′)` factors `u₂` into `u₃`.
We discuss each case of the proof in the text below.
```
@ -1046,7 +1046,7 @@ sub-inv-trans : ∀{u′ u₂ u : Value}
→ all-funs u′ → u′ ⊆ u
→ (∀{v′ w′} → v′ ↦ w′ ∈ u → Σ[ u₃ ∈ Value ] factor u₂ u₃ v′ w′)
---------------------------------------------------------------
→ Σ[ u₃ ∈ Value ] factor u₂ u₃ (dom u′) (cod u′)
→ Σ[ u₃ ∈ Value ] factor u₂ u₃ (⨆dom u′) (⨆cod u′)
sub-inv-trans {⊥} {u₂} {u} fu′ u′⊆u IH =
⊥-elim (contradiction (fu′ refl) ¬Fun⊥)
sub-inv-trans {u₁′ ↦ u₂′} {u₂} {u} fg u′⊆u IH = IH (↦⊆→∈ u′⊆u)
@ -1074,18 +1074,18 @@ sub-inv-trans {u₁′ ⊔ u₂′} {u₂} {u} fg u′⊆u IH
* Suppose `u′ ≡ u₁′ ↦ u₂′`. Then `u₁′ ↦ u₂′ ∈ u` and we can apply the
premise (the induction hypothesis from `u ⊑ u₂`) to obtain that
`u₁′ ↦ u₂′` factors of `u₂ into u₂′`. This case is complete because
`dom u′ ≡ u₁′` and `cod u′ ≡ u₂′`.
`⨆dom u′ ≡ u₁′` and `⨆cod u′ ≡ u₂′`.
* Suppose `u′ ≡ u₁′ ⊔ u₂′`. Then we have `u₁′ ⊆ u` and `u₂′ ⊆ u`. We also
have `all-funs u₁′` and `all-funs u₂′`, so we can apply the induction hypothesis
for both `u₁′` and `u₂′`. So there exists values `u₃₁` and `u₃₂` such that
`(dom u₁′) ↦ (cod u₁′)` factors `u` into `u₃₁` and
`(dom u₂′) ↦ (cod u₂′)` factors `u` into `u₃₂`.
We will show that `(dom u) ↦ (cod u)` factors `u` into `u₃₁ ⊔ u₃₂`.
`(⨆dom u₁′) ↦ (⨆cod u₁′)` factors `u` into `u₃₁` and
`(⨆dom u₂′) ↦ (⨆cod u₂′)` factors `u` into `u₃₂`.
We will show that `(⨆dom u) ↦ (⨆cod u)` factors `u` into `u₃₁ ⊔ u₃₂`.
So we need to show that
dom (u₃₁ ⊔ u₃₂) ⊑ dom (u₁′ ⊔ u₂′)
cod (u₁′ ⊔ u₂′) ⊑ cod (u₃₁ ⊔ u₃₂)
⨆dom (u₃₁ ⊔ u₃₂) ⊑ ⨆dom (u₁′ ⊔ u₂′)
⨆cod (u₁′ ⊔ u₂′) ⊑ ⨆cod (u₃₁ ⊔ u₃₂)
But those both follow directly from the factoring of
`u` into `u₃₁` and `u₃₂`, using the monotonicity of `⊔` with respect to `⊑`.
@ -1183,10 +1183,10 @@ Let `v` and `w` be arbitrary values.
By the induction hypothesis for `u ⊑ u₂`, we know
that for any `v′ ↦ w′ ∈ u`, `v′ ↦ w′` factors `u₂`.
Now we apply the lemma sub-inv-trans, which gives us
some `u₃` such that `(dom u′) ↦ (cod u′)` factors `u₂` into `u₃`.
some `u₃` such that `(⨆dom u′) ↦ (⨆cod u′)` factors `u₂` into `u₃`.
We show that `v ↦ w` also factors `u₂` into `u₃`.
From `dom u₃ ⊑ dom u′` and `dom u′ ⊑ v`, we have `dom u₃ ⊑ v`.
From `w ⊑ cod u′` and `cod u′ ⊑ cod u₃`, we have `w ⊑ cod u₃`,
From `⨆dom u₃ ⊑ ⨆dom u′` and `⨆dom u′ ⊑ v`, we have `⨆dom u₃ ⊑ v`.
From `w ⊑ ⨆cod u′` and `⨆cod u′ ⊑ ⨆cod u₃`, we have `w ⊑ ⨆cod u₃`,
and this case is complete.
* Case `⊑-fun`.
@ -1197,7 +1197,7 @@ Let `v` and `w` be arbitrary values.
Given that `v ↦ w ∈ u₁₁ ↦ u₁₂`, we have `v ≡ u₁₁` and `w ≡ u₁₂`.
We show that `u₁₁ ↦ u₁₂` factors `u₂₁ ↦ u₂₂` into itself.
We need to show that `dom (u₂₁ ↦ u₂₂) ⊑ u₁₁` and `u₁₂ ⊑ cod (u₂₁ ↦ u₂₂)`,
We need to show that `⨆dom (u₂₁ ↦ u₂₂) ⊑ u₁₁` and `u₁₂ ⊑ ⨆cod (u₂₁ ↦ u₂₂)`,
but that is equivalent to our premises `u₂₁ ⊑ u₁₁` and `u₁₂ ⊑ u₂₂`.
* Case `⊑-dist`.
@ -1223,13 +1223,13 @@ sub-inv-fun : ∀{v w u₁ : Value}
→ (v ↦ w) ⊑ u₁
-----------------------------------------------------
→ Σ[ u₂ ∈ Value ] all-funs u₂ × u₂ ⊆ u₁
× (∀{v′ w′} → (v′ ↦ w′) ∈ u₂ → v′ ⊑ v) × w ⊑ cod u₂
× (∀{v′ w′} → (v′ ↦ w′) ∈ u₂ → v′ ⊑ v) × w ⊑ ⨆cod u₂
sub-inv-fun{v}{w}{u₁} abc
with sub-inv abc {v}{w} refl
... | ⟨ u₂ , ⟨ f , ⟨ u₂⊆u₁ , ⟨ db , cc ⟩ ⟩ ⟩ ⟩ =
⟨ u₂ , ⟨ f , ⟨ u₂⊆u₁ , ⟨ G , cc ⟩ ⟩ ⟩ ⟩
where G : ∀{D E} → (D ↦ E) ∈ u₂ → D ⊑ v
G{D}{E} m = ⊑-trans (⊆→⊑ (↦∈→⊆dom f m)) db
G{D}{E} m = ⊑-trans (⊆→⊑ (↦∈→⊆⨆dom f m)) db
```
The second corollary is the inversion rule that one would expect for
@ -1247,8 +1247,8 @@ less-than with functions on the left and right-hand sides.
... | ⟨ u , ⟨ u′ , u↦u′∈Γ ⟩ ⟩
with Γ⊆v34 u↦u′∈Γ
... | refl =
let codΓ⊆w′ = ⊆↦→cod⊆ Γ⊆v34 in
⟨ lt1 u↦u′∈Γ , ⊑-trans lt2 (⊆→⊑ codΓ⊆w′) ⟩
let ⨆codΓ⊆w′ = ⊆↦→⨆cod⊆ Γ⊆v34 in
⟨ lt1 u↦u′∈Γ , ⊑-trans lt2 (⊆→⊑ ⨆codΓ⊆w′) ⟩
```
@ -1356,6 +1356,7 @@ This chapter uses the following unicode:
↦ U+21A6 RIGHTWARDS ARROW FROM BAR (\mapsto)
⊔ U+2294 SQUARE CUP (\lub)
⊑ U+2291 SQUARE IMAGE OF OR EQUAL TO (\sqsubseteq)
⨆ U+2A06 N-ARY SQUARE UNION OPERATOR (\Lub)
⊢ U+22A2 RIGHT TACK (\|- or \vdash)
↓ U+2193 DOWNWARDS ARROW (\d)
ᶜ U+1D9C MODIFIER LETTER SMALL C (\^c)