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# Broadcasting as Pullback
Broadcasting = pullback along projection `π : J → I`.
`broadcast(f, π) = f ∘ π` is the categorical semantics of broadcasting.
-/
import ArrayLang.Array
namespace SovereignArray
/-- General pullback along a projection. `pullback π f = f ∘ π`. -/
def pullback {I J : Type*} (π : J → I) (f : I → α) : J → α := f ∘ π
/-- Broadcasting: align `v` (indexed by `I`) to `J` via `π`, then add `w` (indexed by `J`).
This is the `Π`-map `fun j => v (π j) + w j`. -/
def broadcast {α : Type*} [Add α] {I J : Type*} (π : J → I)
(v : I → α) (w : J → α) : J → α :=
fun j => v (π j) + w j
/-- The definition is literally the pullback-plus-add form. -/
theorem broadcast_is_pullback {α : Type*} [Add α] {I J : Type*} (π : J → I) :
(fun (v : I → α) (w : J → α) => broadcast π v w) =
(fun v w j => v (π j) + w j) := rfl
/-- `broadcast` is `pullback π v` added pointwise to `w`. -/
theorem broadcast_eq_pullback {α : Type*} [Add α] {I J : Type*} (π : J → I)
(v : I → α) (w : J → α) :
broadcast π v w = fun j => pullback π v j + w j := rfl
/-- Two successive broadcasts along `π₂ ∘ π₁` fuse into one pullback. -/
theorem broadcast_comp {α : Type*} [Add α] {I J K : Type*}
(π₁ : J → I) (π₂ : K → J) (v : I → α) (w : K → α) :
broadcast π₂ (pullback π₁ v) w = broadcast (π₁ ∘ π₂) v w := rfl
end SovereignArray
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