5.2 KiB
Auxiliary operations:
index family of a constructor Idx(K) := [0; arity(K)[
head of an ordered family (we write x for any object here, value, pattern etc.) head((xᵢ)i ∈ I) = x_min(I)
tail of an ordered family tail((xᵢ)i ∈ I) := (xᵢ)i ≠ min(I)
head constructor of a value or pattern constr(K(xᵢ)ⁱ) = K constr(_) = ⊥ constr(x) = ⊥
first non-⊥ element of an ordered family First((xᵢ)ⁱ) := ⊥ if ∀i, xᵢ = ⊥ First((xᵢ)ⁱ) := x_min{i | xᵢ ≠ ⊥} if ∃i, xᵢ ≠ ⊥
Definition of running a pattern row against a (same-size) value vector (pᵢ)ⁱ(vᵢ)ⁱ
∅ (∅) := []
(_, tail(pᵢ)ⁱ) (vᵢ) := tail(pᵢ)ⁱ(tail(vᵢ)ⁱ)
(x, tail(pᵢ)ⁱ) (vᵢ) := σ[x↦v₀] if tail(pᵢ)ⁱ(tail(vᵢ)ⁱ) = σ
(K(qⱼ)ʲ, tail(pᵢ)ⁱ) (K(v'ⱼ)ʲ,tail(vⱼ)ʲ) := ((qⱼ)ʲ + tail(pᵢ)ⁱ)((v'ⱼ)ʲ + tail(vᵢ)ⁱ)
(K(qⱼ)ʲ, tail(pᵢ)ⁱ) (K'(v'ₗ)ˡ,tail(vⱼ)ʲ) := ⊥ if K ≠ K'
(q₁|q₂, tail(pᵢ)ⁱ) (vᵢ)ⁱ := First((q₁,tail(pᵢ)ⁱ)(vᵢ)ⁱ, (q₂,tail(pᵢ)ⁱ)(vᵢ)ⁱ)
Definition of group decomposition:
let constrs((pᵢ)i ∈ I) = { K | K = constr(pᵢ), i ∈ I }
let Groups(m) where m = ((aᵢ)ⁱ ((pᵢⱼ)ⁱ → eⱼ)ⁱʲ) =
let (Kₖ)ᵏ = constrs(pᵢ₀)ⁱ in
( Kₖ →
((a₀.ₗ)ˡ + tail(aᵢ)ⁱ)
(
if pₒⱼ is Kₖ(qₗ) then
(qₗ)ˡ + tail(pᵢⱼ)ⁱ → eⱼ
if pₒⱼ is _ then
(_)ˡ + tail(pᵢⱼ)ⁱ → eⱼ
else ⊥
)ʲ
), (
tail(aᵢ)ⁱ, (tail(pᵢⱼ)ⁱ → eⱼ if p₀ⱼ is _ else ⊥)ʲ
)
Groups(m) is an auxiliary function that decomposes a matrix m into submatrices, according to the head constructor of their first pattern. Groups(m) returns one submatrix m_r for each head constructor K that occurs on the first row of m, plus one "wildcard submatrix" mwild that matches on all values that do not start with one of those head constructors.
Intuitively, m is equivalent to its decomposition in the following sense: if the first pattern of an input vector (v_i)^i starts with one of the head constructors Kₖ, then running (v_i)^i against m is the same as running it against the submatrix mKₖ; otherwise (its head constructor is none of the Kₖ) it is equivalent to running it against the wildcard submatrix.
We formalize this intuition as follows:
Lemma (Groups): Let $m$ be a matrix with $Groups(m) = (kᵣ → mᵣ)ᵏ, m_{wild}$. For any value vector $(vᵢ)ᴵ$ such that $v₀ = K(v'ₗ)ˡ$ for some constructor K, we have: \[ if K = Kₖ for some k then m(vᵢ)ⁱ = mₖ((v'ₗ)ˡ +++ (vᵢ)^{i∈I\{0}}) otherwise m(vᵢ)ⁱ = m_{wild}(vᵢ)^{i∈I\{0}} \]
Proof:
Let $m$ be a matrix ((aᵢ)ⁱ, ((pᵢⱼ)ⁱ → eⱼ)ʲ) with $Groups(m) = (Kₖ → mₖ)ᵏ, m_{wild}$.
Below we are going to assume that m is a simplified matrix:
the first row does not with an or-pattern or a variable
I think it would be simpler to not consider them at all in the formalization. If we do want them, the discrepancy can be fixed later by defining a simplification step that explodes head-or-patterns and binds variable, and showing that it preserves the matrix semantics.
Let (vᵢ)ⁱ be an input matrix with v₀ = Kᵥ(v'ₗ)ˡ for some constructor Kᵥ.
We have to show that:
- if Kₖ = Kᵥ for some Kₖ ∈ constrs(p₀ⱼ)ʲ, then
m(vᵢ)ⁱ = mₖ((v'ₗ)ˡ
+tail(vᵢ)ⁱ) - otherwise m(vᵢ)ⁱ = mwild(tail(vᵢ)ⁱ)
Let us call (rₖⱼ) the j-th row of the submatrix mₖ, and rⱼwild the j-th row of the wildcard submatrix mwild.
Our goal contains same-behavior equalities between matrices, for a fixed input vector (vᵢ)ⁱ. It suffices to show same-behavior equalities between each row of the matrices for this input vector. We show that for any j,
- if Kₖ = Kᵥ for some Kₖ ∈ constrs(p₀ⱼ)ʲ, then
(pᵢⱼ)ⁱ(vᵢ)ⁱ = rₖⱼ((v'ₗ)ˡ
+tail(vᵢ)ⁱ - otherwise (pᵢⱼ)ⁱ(vᵢ)ⁱ = rⱼwild tail(vᵢ)ⁱ
In the first case (Kᵥ is Kₖ for some Kₖ ∈ constrs(p₀ⱼ)ʲ), we have to prove that
(pᵢⱼ)ⁱ(vᵢ)ⁱ = rₖⱼ((v'ₗ)ˡ + tail(vᵢ)ⁱ
By definition of mₖ we know that rₖⱼ is equal to
if pₒⱼ is Kₖ(qₗ) then
(qₗ)ˡ + tail(pᵢⱼ)ⁱ → eⱼ
if pₒⱼ is _ then
(_)ˡ + tail(pᵢⱼ)ⁱ → eⱼ
else ⊥
By definition of (pᵢ)ⁱ(vᵢ)ⁱ we know that (pᵢⱼ)ⁱ(vᵢ) is equal to
(K(qⱼ)ʲ, tail(pᵢⱼ)ⁱ) (K(v'ₗ)ˡ,tail(vᵢ)ⁱ) := ((qⱼ)ʲ + tail(pᵢⱼ)ⁱ)((v'ₗ)ˡ + tail(vᵢ)ⁱ)
(_, tail(pᵢⱼ)ⁱ) (vᵢ) := tail(pᵢⱼ)ⁱ(tail(vᵢ)ⁱ)
(K(qⱼ)ʲ, tail(pᵢⱼ)ⁱ) (K'(v'ₗ)ˡ,tail(vⱼ)ʲ) := ⊥ if K ≠ K'
We prove this first case by a second case analysis on p₀ⱼ.
TODO
In the second case (Kᵥ is distinct from Kₖ for all Kₖ ∈ constrs(pₒⱼ)ʲ), we have to prove that
(pᵢⱼ)ⁱ(vᵢ)ⁱ = rⱼwild tail(vᵢ)ⁱ
TODO