```------------------------------------------------------------------------
-- Lexicographic induction
------------------------------------------------------------------------

{-# OPTIONS --universe-polymorphism #-}

module Induction.Lexicographic where

open import Induction
open import Data.Product

-- The structure of lexicographic induction.

Σ-Rec : ∀ {ℓ} {A : Set ℓ} {B : A → Set ℓ} →
RecStruct A → (∀ x → RecStruct (B x)) → RecStruct (Σ A B)
Σ-Rec RecA RecB P (x , y) =
-- Either x is constant and y is "smaller", ...
RecB x (λ y' → P (x , y')) y
×
-- ...or x is "smaller" and y is arbitrary.
RecA (λ x' → ∀ y' → P (x' , y')) x

_⊗_ : ∀ {ℓ} {A B : Set ℓ} →
RecStruct A → RecStruct B → RecStruct (A × B)
RecA ⊗ RecB = Σ-Rec RecA (λ _ → RecB)

-- Constructs a recursor builder for lexicographic induction.

Σ-rec-builder :
∀ {ℓ} {A : Set ℓ} {B : A → Set ℓ}
{RecA : RecStruct A}
{RecB : ∀ x → RecStruct (B x)} →
RecursorBuilder RecA → (∀ x → RecursorBuilder (RecB x)) →
RecursorBuilder (Σ-Rec RecA RecB)
Σ-rec-builder {RecA = RecA} {RecB = RecB} recA recB P f (x , y) =
(p₁ x y p₂x , p₂x)
where
p₁ : ∀ x y →
RecA (λ x' → ∀ y' → P (x' , y')) x →
RecB x (λ y' → P (x , y')) y
p₁ x y x-rec = recB x
(λ y' → P (x , y'))
(λ y y-rec → f (x , y) (y-rec , x-rec))
y

p₂ : ∀ x → RecA (λ x' → ∀ y' → P (x' , y')) x
p₂ = recA (λ x → ∀ y → P (x , y))
(λ x x-rec y → f (x , y) (p₁ x y x-rec , x-rec))

p₂x = p₂ x

[_⊗_] : ∀ {ℓ} {A B : Set ℓ} {RecA : RecStruct A} {RecB : RecStruct B} →
RecursorBuilder RecA → RecursorBuilder RecB →
RecursorBuilder (RecA ⊗ RecB)
[ recA ⊗ recB ] = Σ-rec-builder recA (λ _ → recB)

------------------------------------------------------------------------
-- Example

private

open import Data.Nat
open import Induction.Nat as N

-- The Ackermann function à la Rózsa Péter.

ackermann : ℕ → ℕ → ℕ
ackermann m n = build [ N.rec-builder ⊗ N.rec-builder ]
AckPred ack (m , n)
where
AckPred : ℕ × ℕ → Set
AckPred _ = ℕ

ack : ∀ p → (N.Rec ⊗ N.Rec) AckPred p → AckPred p
ack (zero  , n)     _                   = 1 + n
ack (suc m , zero)  (_         , ackm•) = ackm• 1
ack (suc m , suc n) (ack[1+m]n , ackm•) = ackm• ack[1+m]n
```