module Chapter8.EnumU.Pi where open import Level renaming ( zero to zeroL ; suc to sucL ) open import Function open import Data.Unit open import Data.Product open import Data.Nat open import Data.Fin hiding (lift) open import Chapter2.Logic open import Chapter5.Desc open import Chapter5.Desc.Fixpoint open import Chapter5.Desc.Lifting open import Chapter5.Desc.Induction open import Chapter8.EnumU private record ⟨π_$_⟩ (E : EnumU {sucL zeroL})(P : EnumT E → Set) : Set₁ where constructor return field call : Set open ⟨π_$_⟩ public πh : (E : EnumU)(P : EnumT E → Set) → ⟨π E $ P ⟩ πh = induction EnumUD (λ E → (P : EnumT E → Set) → ⟨π E $ P ⟩) (λ {xs} → α {xs}) where α : {E : ⟦ EnumUD ⟧ EnumU} → [_]^ {k = zeroL} (EnumUD {sucL zeroL}) (λ E → (P : EnumT E → Set) → ⟨π E $ P ⟩) E → (P : EnumT ⟨ E ⟩ → Set) → ⟨π ⟨ E ⟩ $ P ⟩ α {lift zero , lift tt} (lift tt) = λ P → return ⊤ α {lift (suc zero) , (a , e , lift tt)} (ih , lift tt) = λ P → return (P ze × call (ih (P ∘ su))) α {lift (suc (suc ())) , _} _ π : (E : EnumU)(P : EnumT E → Set) → Set π E P = call (πh E P)