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Jeremy Siek 2020-03-11 16:08:04 -04:00
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18
src/plfa/part3/Denotational.lagda.md
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@ -1017,12 +1017,22 @@ 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′
```
So the inversion principle for functions can be stated as
v ↦ w ⊑ u
---------------
→ factor u u′ v w
We prove the inversion principle for functions by induction on the
derivation of the less-than relation. To make the induction hypothesis
stronger, we broaden the premise to `u₁ ⊑ u₂` (instead of `v ↦ w ⊑
u`), and strengthen the conclusion to say that for _every_ function
value `v ↦ w ∈ u₁`, we have that `v ↦ w` factors `u₂` into some
value `u₃`.
stronger, we broaden the premise `v ↦ w ⊑ u` to `u₁ ⊑ u₂`, and
strengthen the conclusion to say that for _every_ function value
`v ↦ w ∈ u₁`, we have that `v ↦ w` factors `u₂` into some value `u₃`.
→ u₁ ⊑ u₂
------------------------------------------------------
→ ∀{v w} → v ↦ w ∈ u₁ → Σ[ u₃ ∈ Value ] factor u₂ u₃ v w
### Inversion of less-than for functions, the case for ⊑-trans