{-# OPTIONS --cubical --safe --exact-split #-} module Cubical.Structures.Set.Mon.Free where open import Cubical.Foundations.Everything open import Cubical.Data.Sigma open import Cubical.Data.List open import Cubical.Data.Nat open import Cubical.Data.Nat.Order import Cubical.Data.Empty as ⊥ import Cubical.Structures.Set.Mon.Desc as M import Cubical.Structures.Free as F open import Cubical.Structures.Sig open import Cubical.Structures.Str public open import Cubical.Structures.Tree open import Cubical.Structures.Eq open import Cubical.Structures.Arity data FreeMon {ℓ : Level} (A : Type ℓ) : Type ℓ where η : (a : A) -> FreeMon A e : FreeMon A _⊕_ : FreeMon A -> FreeMon A -> FreeMon A unitl : ∀ m -> e ⊕ m ≡ m unitr : ∀ m -> m ⊕ e ≡ m assocr : ∀ m n o -> (m ⊕ n) ⊕ o ≡ m ⊕ (n ⊕ o) trunc : isSet (FreeMon A) module elimFreeMonSet {p n : Level} {A : Type n} (P : FreeMon A -> Type p) (η* : (a : A) -> P (η a)) (e* : P e) (_⊕*_ : {m n : FreeMon A} -> P m -> P n -> P (m ⊕ n)) (unitl* : {m : FreeMon A} (m* : P m) -> PathP (λ i → P (unitl m i)) (e* ⊕* m*) m*) (unitr* : {m : FreeMon A} (m* : P m) -> PathP (λ i → P (unitr m i)) (m* ⊕* e*) m*) (assocr* : {m n o : FreeMon A} (m* : P m) -> (n* : P n) -> (o* : P o) -> PathP (λ i → P (assocr m n o i)) ((m* ⊕* n*) ⊕* o*) (m* ⊕* (n* ⊕* o*))) (trunc* : {xs : FreeMon A} -> isSet (P xs)) where f : (x : FreeMon A) -> P x f (η a) = η* a f e = e* f (x ⊕ y) = f x ⊕* f y f (unitl x i) = unitl* (f x) i f (unitr x i) = unitr* (f x) i f (assocr x y z i) = assocr* (f x) (f y) (f z) i f (trunc xs ys p q i j) = isOfHLevel→isOfHLevelDep 2 (\xs -> trunc* {xs = xs}) (f xs) (f ys) (cong f p) (cong f q) (trunc xs ys p q) i j module recFreeMonSet {p n : Level} {A : Type n} (P : Type p) (η* : (a : A) -> P) (e* : P) (_⊕*_ : P -> P -> P) (unitl* : (m* : P) -> PathP (λ i → P) (e* ⊕* m*) m*) (unitr* : (m* : P) -> PathP (λ i → P) (m* ⊕* e*) m*) (assocr* : (m* : P) -> (n* : P) -> (o* : P) -> PathP (λ i → P) ((m* ⊕* n*) ⊕* o*) (m* ⊕* (n* ⊕* o*))) (trunc* : isSet P) where f : (x : FreeMon A) -> P f = elimFreeMonSet.f (\_ -> P) η* e* _⊕*_ unitl* unitr* assocr* trunc* module elimFreeMonProp {p n : Level} {A : Type n} (P : FreeMon A -> Type p) (η* : (a : A) -> P (η a)) (e* : P e) (_⊕*_ : {m n : FreeMon A} -> P m -> P n -> P (m ⊕ n)) (trunc* : {xs : FreeMon A} -> isProp (P xs)) where f : (x : FreeMon A) -> P x f = elimFreeMonSet.f P η* e* _⊕*_ unitl* unitr* assocr* (isProp→isSet trunc*) where abstract unitl* : {m : FreeMon A} (m* : P m) -> PathP (λ i → P (unitl m i)) (e* ⊕* m*) m* unitl* {m} m* = toPathP (trunc* (transp (λ i -> P (unitl m i)) i0 (e* ⊕* m*)) m*) unitr* : {m : FreeMon A} (m* : P m) -> PathP (λ i → P (unitr m i)) (m* ⊕* e*) m* unitr* {m} m* = toPathP (trunc* (transp (λ i -> P (unitr m i)) i0 (m* ⊕* e*)) m*) assocr* : {m n o : FreeMon A} (m* : P m) -> (n* : P n) -> (o* : P o) -> PathP (λ i → P (assocr m n o i)) ((m* ⊕* n*) ⊕* o*) (m* ⊕* (n* ⊕* o*)) assocr* {m} {n} {o} m* n* o* = toPathP (trunc* (transp (λ i -> P (assocr m n o i)) i0 ((m* ⊕* n*) ⊕* o*)) (m* ⊕* (n* ⊕* o*))) freeMon-α : ∀ {n : Level} {X : Type n} -> sig M.MonSig (FreeMon X) -> FreeMon X freeMon-α (M.`e , i) = e freeMon-α (M.`⊕ , i) = i fzero ⊕ i fone module Free {x y : Level} {A : Type x} {𝔜 : struct y M.MonSig} (isSet𝔜 : isSet (𝔜 .car)) (𝔜-monoid : 𝔜 ⊨ M.MonSEq) where module 𝔜 = M.MonSEq 𝔜 𝔜-monoid 𝔉 : struct x M.MonSig 𝔉 = < FreeMon A , freeMon-α > module _ (f : A -> 𝔜 .car) where _♯ : FreeMon A -> 𝔜 .car (η a) ♯ = f a e ♯ = 𝔜.e (m ⊕ n) ♯ = (m ♯) 𝔜.⊕ (n ♯) (unitl m i) ♯ = 𝔜.unitl (m ♯) i (unitr m i) ♯ = 𝔜.unitr (m ♯) i (assocr m n o i) ♯ = 𝔜.assocr (m ♯) (n ♯) (o ♯) i (trunc m n p q i j) ♯ = isSet𝔜 (m ♯) (n ♯) (cong _♯ p) (cong _♯ q) i j ♯-isMonHom : structHom 𝔉 𝔜 fst ♯-isMonHom = _♯ snd ♯-isMonHom M.`e i = 𝔜.e-eta snd ♯-isMonHom M.`⊕ i = 𝔜.⊕-eta i _♯ private freeMonEquivLemma : (g : structHom 𝔉 𝔜) -> (x : FreeMon A) -> g .fst x ≡ ((g .fst ∘ η) ♯) x freeMonEquivLemma (g , homMonWit) = elimFreeMonProp.f (λ x -> g x ≡ ((g ∘ η) ♯) x) (λ _ -> refl) (sym (homMonWit M.`e (lookup [])) ∙ 𝔜.e-eta) (λ {m} {n} p q -> g (m ⊕ n) ≡⟨ sym (homMonWit M.`⊕ (lookup (m ∷ n ∷ []))) ⟩ 𝔜 .alg (M.`⊕ , (λ w -> g (lookup (m ∷ n ∷ []) w))) ≡⟨ 𝔜.⊕-eta (lookup (m ∷ n ∷ [])) g ⟩ g m 𝔜.⊕ g n ≡⟨ cong₂ 𝔜._⊕_ p q ⟩ _ ∎ ) (isSet𝔜 _ _) freeMonEquivLemma-β : (g : structHom 𝔉 𝔜) -> g ≡ ♯-isMonHom (g .fst ∘ η) freeMonEquivLemma-β g = structHom≡ 𝔉 𝔜 g (♯-isMonHom (g .fst ∘ η)) isSet𝔜 (funExt (freeMonEquivLemma g)) freeMonEquiv : structHom 𝔉 𝔜 ≃ (A -> 𝔜 .car) freeMonEquiv = isoToEquiv (iso (λ g -> g .fst ∘ η) ♯-isMonHom (λ _ -> refl) (sym ∘ freeMonEquivLemma-β)) module FreeMonDef = F.Definition M.MonSig M.MonEqSig M.MonSEq freeMon-sat : ∀ {n} {X : Type n} -> < FreeMon X , freeMon-α > ⊨ M.MonSEq freeMon-sat M.`unitl ρ = unitl (ρ fzero) freeMon-sat M.`unitr ρ = unitr (ρ fzero) freeMon-sat M.`assocr ρ = assocr (ρ fzero) (ρ fone) (ρ ftwo) freeMonDef : ∀ {ℓ ℓ'} -> FreeMonDef.Free ℓ ℓ' 2 F.Definition.Free.F freeMonDef = FreeMon F.Definition.Free.η freeMonDef = η F.Definition.Free.α freeMonDef = freeMon-α F.Definition.Free.sat freeMonDef = freeMon-sat F.Definition.Free.isFree freeMonDef isSet𝔜 satMon = (Free.freeMonEquiv isSet𝔜 satMon) .snd