ArrayLang/Array.lean

/-!
# Sovereign Array Language — Core Array Type

Mathematical foundation (valid isomorphisms only, per architectural review):

| NumPy Concept      | HoTT / Unimath Translation                     | Status |
|-------------------|-----------------------------------------------|--------|
| Array             | Dependent function `I → α`                    | Sound  |
| Shape / Index     | Finite type `I : Type`                        | Sound  |
| Vectorized Op     | `Π (i : I), op (A i) (B i)` (pointwise)      | Sound  |
| Equality of Array | Function extensionality                       | Sound  |

We deliberately do NOT claim:
- proof complexity = computational complexity
- lossy quotient invariants (Abjad / digital root) are universal arithmetic
- univalence replaces SIMD at the metalayer
-/

namespace SovereignArray

universe u v

/-- An array indexed by shape `I` with elements of type `α`.
This is exactly the dependent-function model used in Cubical Agda / Lean. -/
def Array (I : Type u) (α : Type v) : Type (max u v) := I → α

namespace Array

variable {I : Type u} {α : Type v}

/-- Pointwise lifting of a binary operation.
Categorical semantics of a vectorized op: a `Π`-map over the index space `I`. -/
def pmap₂ (op : α → α → α) (a b : Array I α) : Array I α :=
  fun i => op (a i) (b i)

/-- `O(1)` *proof* equality is function extensionality.
Computational equality is `O(|I|)`; we never conflate the two. -/
theorem pmap₂_congr {op : α → α → α} {a a' b b' : Array I α}
    (ha : ∀ i, a i = a' i) (hb : ∀ i, b i = b' i) :
    pmap₂ op a b = pmap₂ op a' b' := by
  funext i
  simp [pmap₂, ha i, hb i]

/-- `pmap₂` fusion: applying a post-map to a `pmap₂` is itself a `pmap₂`.
Fusion = `Π`-map fusion; no loop exists in the denotation. -/
theorem pmap₂_fusion {op : α → α → α} {a b : Array I α} (f : α → α) :
    (fun i => f (pmap₂ op a b i)) = pmap₂ (fun x _ => f (op x x)) a b := by
  funext i
  simp [pmap₂]

/-- `pmap₂` is associative in the operation when the operation is. -/
theorem pmap₂_assoc {op : α → α → α} {a b c : Array I α}
    (h : ∀ x y z, op (op x y) z = op x (op y z)) :
    pmap₂ op (pmap₂ op a b) c = pmap₂ op a (pmap₂ op b c) := by
  funext i
  simp [pmap₂, h]

end Array

end SovereignArray