{-# OPTIONS --cubical --safe --exact-split #-} module Cubical.Structures.Set.CMon.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.Set.Mon.Free as FM import Cubical.Structures.Set.CMon.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 FreeCMon {ℓ : Level} (A : Type ℓ) : Type ℓ where η : (a : A) -> FreeCMon A e : FreeCMon A _⊕_ : FreeCMon A -> FreeCMon A -> FreeCMon A unitl : ∀ m -> e ⊕ m ≡ m unitr : ∀ m -> m ⊕ e ≡ m assocr : ∀ m n o -> (m ⊕ n) ⊕ o ≡ m ⊕ (n ⊕ o) comm : ∀ m n -> m ⊕ n ≡ n ⊕ m trunc : isSet (FreeCMon A) module elimFreeCMonSet {p n : Level} {A : Type n} (P : FreeCMon A -> Type p) (η* : (a : A) -> P (η a)) (e* : P e) (_⊕*_ : {m n : FreeCMon A} -> P m -> P n -> P (m ⊕ n)) (unitl* : {m : FreeCMon A} (m* : P m) -> PathP (λ i → P (unitl m i)) (e* ⊕* m*) m*) (unitr* : {m : FreeCMon A} (m* : P m) -> PathP (λ i → P (unitr m i)) (m* ⊕* e*) m*) (assocr* : {m n o : FreeCMon 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*))) (comm* : {m n : FreeCMon A} (m* : P m) -> (n* : P n) -> PathP (λ i → P (comm m n i)) (m* ⊕* n*) (n* ⊕* m*)) (trunc* : {xs : FreeCMon A} -> isSet (P xs)) where f : (x : FreeCMon 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 (comm x y i) = comm* (f x) (f y) 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 elimFreeCMonProp {p n : Level} {A : Type n} (P : FreeCMon A -> Type p) (η* : (a : A) -> P (η a)) (e* : P e) (_⊕*_ : {m n : FreeCMon A} -> P m -> P n -> P (m ⊕ n)) (trunc* : {xs : FreeCMon A} -> isProp (P xs)) where f : (x : FreeCMon A) -> P x f = elimFreeCMonSet.f P η* e* _⊕*_ unitl* unitr* assocr* comm* (isProp→isSet trunc*) where abstract unitl* : {m : FreeCMon 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 : FreeCMon 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 : FreeCMon 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*))) comm* : {m n : FreeCMon A} (m* : P m) (n* : P n) -> PathP (λ i → P (comm m n i)) (m* ⊕* n*) (n* ⊕* m*) comm* {m} {n} m* n* = toPathP (trunc* (transp (λ i -> P (comm m n i)) i0 (m* ⊕* n*)) (n* ⊕* m*)) freeCMon-α : ∀ {ℓ} {X : Type ℓ} -> sig M.MonSig (FreeCMon X) -> FreeCMon X freeCMon-α (M.`e , _) = e freeCMon-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-cmon : 𝔜 ⊨ M.CMonSEq) where module 𝔜 = M.CMonSEq 𝔜 𝔜-cmon 𝔉 : struct x M.MonSig 𝔉 = < FreeCMon A , freeCMon-α > module _ (f : A -> 𝔜 .car) where _♯ : FreeCMon 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 comm m n i ♯ = 𝔜.comm (m ♯) (n ♯) 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 freeCMonEquivLemma : (g : structHom 𝔉 𝔜) -> (x : FreeCMon A) -> g .fst x ≡ ((g .fst ∘ η) ♯) x freeCMonEquivLemma (g , homMonWit) = elimFreeCMonProp.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 _ _) freeCMonEquivLemma-β : (g : structHom 𝔉 𝔜) -> g ≡ ♯-isMonHom (g .fst ∘ η) freeCMonEquivLemma-β g = structHom≡ 𝔉 𝔜 g (♯-isMonHom (g .fst ∘ η)) isSet𝔜 (funExt (freeCMonEquivLemma g)) freeCMonEquiv : structHom 𝔉 𝔜 ≃ (A -> 𝔜 .car) freeCMonEquiv = isoToEquiv (iso (λ g -> g .fst ∘ η) ♯-isMonHom (λ _ -> refl) (sym ∘ freeCMonEquivLemma-β)) module FreeCMonDef = F.Definition M.MonSig M.CMonEqSig M.CMonSEq freeCMon-sat : ∀ {n} {X : Type n} -> < FreeCMon X , freeCMon-α > ⊨ M.CMonSEq freeCMon-sat (M.`mon M.`unitl) ρ = unitl (ρ fzero) freeCMon-sat (M.`mon M.`unitr) ρ = unitr (ρ fzero) freeCMon-sat (M.`mon M.`assocr) ρ = assocr (ρ fzero) (ρ fone) (ρ ftwo) freeCMon-sat M.`comm ρ = comm (ρ fzero) (ρ fone) freeMonDef : ∀ {ℓ ℓ'} -> FreeCMonDef.Free ℓ ℓ' 2 F.Definition.Free.F freeMonDef = FreeCMon F.Definition.Free.η freeMonDef = η F.Definition.Free.α freeMonDef = freeCMon-α F.Definition.Free.sat freeMonDef = freeCMon-sat F.Definition.Free.isFree freeMonDef isSet𝔜 satMon = (Free.freeCMonEquiv isSet𝔜 satMon) .snd