module Group.Syntax where
open import SOAS.Common
open import SOAS.Context
open import SOAS.Variable
open import SOAS.Families.Core
open import SOAS.Construction.Structure
open import SOAS.ContextMaps.Inductive
open import SOAS.Metatheory.Syntax
open import Group.Signature
private
variable
Γ Δ Π : Ctx
α : *T
𝔛 : Familyₛ
module G:Terms (𝔛 : Familyₛ) where
data G : Familyₛ where
var : ℐ ⇾̣ G
mvar : 𝔛 α Π → Sub G Π Γ → G α Γ
ε : G * Γ
_⊕_ : G * Γ → G * Γ → G * Γ
⊖_ : G * Γ → G * Γ
infixl 20 _⊕_
infixr 40 ⊖_
open import SOAS.Metatheory.MetaAlgebra ⅀F 𝔛
Gᵃ : MetaAlg G
Gᵃ = record
{ 𝑎𝑙𝑔 = λ where
(unitₒ ⅋ _) → ε
(addₒ ⅋ a , b) → _⊕_ a b
(negₒ ⅋ a) → ⊖_ a
; 𝑣𝑎𝑟 = var ; 𝑚𝑣𝑎𝑟 = λ 𝔪 mε → mvar 𝔪 (tabulate mε) }
module Gᵃ = MetaAlg Gᵃ
module _ {𝒜 : Familyₛ}(𝒜ᵃ : MetaAlg 𝒜) where
open MetaAlg 𝒜ᵃ
𝕤𝕖𝕞 : G ⇾̣ 𝒜
𝕊 : Sub G Π Γ → Π ~[ 𝒜 ]↝ Γ
𝕊 (t ◂ σ) new = 𝕤𝕖𝕞 t
𝕊 (t ◂ σ) (old v) = 𝕊 σ v
𝕤𝕖𝕞 (mvar 𝔪 mε) = 𝑚𝑣𝑎𝑟 𝔪 (𝕊 mε)
𝕤𝕖𝕞 (var v) = 𝑣𝑎𝑟 v
𝕤𝕖𝕞 ε = 𝑎𝑙𝑔 (unitₒ ⅋ tt)
𝕤𝕖𝕞 (_⊕_ a b) = 𝑎𝑙𝑔 (addₒ ⅋ 𝕤𝕖𝕞 a , 𝕤𝕖𝕞 b)
𝕤𝕖𝕞 (⊖_ a) = 𝑎𝑙𝑔 (negₒ ⅋ 𝕤𝕖𝕞 a)
𝕤𝕖𝕞ᵃ⇒ : MetaAlg⇒ Gᵃ 𝒜ᵃ 𝕤𝕖𝕞
𝕤𝕖𝕞ᵃ⇒ = record
{ ⟨𝑎𝑙𝑔⟩ = λ{ {t = t} → ⟨𝑎𝑙𝑔⟩ t }
; ⟨𝑣𝑎𝑟⟩ = refl
; ⟨𝑚𝑣𝑎𝑟⟩ = λ{ {𝔪 = 𝔪}{mε} → cong (𝑚𝑣𝑎𝑟 𝔪) (dext (𝕊-tab mε)) } }
where
open ≡-Reasoning
⟨𝑎𝑙𝑔⟩ : (t : ⅀ G α Γ) → 𝕤𝕖𝕞 (Gᵃ.𝑎𝑙𝑔 t) ≡ 𝑎𝑙𝑔 (⅀₁ 𝕤𝕖𝕞 t)
⟨𝑎𝑙𝑔⟩ (unitₒ ⅋ _) = refl
⟨𝑎𝑙𝑔⟩ (addₒ ⅋ _) = refl
⟨𝑎𝑙𝑔⟩ (negₒ ⅋ _) = refl
𝕊-tab : (mε : Π ~[ G ]↝ Γ)(v : ℐ α Π) → 𝕊 (tabulate mε) v ≡ 𝕤𝕖𝕞 (mε v)
𝕊-tab mε new = refl
𝕊-tab mε (old v) = 𝕊-tab (mε ∘ old) v
module _ (g : G ⇾̣ 𝒜)(gᵃ⇒ : MetaAlg⇒ Gᵃ 𝒜ᵃ g) where
open MetaAlg⇒ gᵃ⇒
𝕤𝕖𝕞! : (t : G α Γ) → 𝕤𝕖𝕞 t ≡ g t
𝕊-ix : (mε : Sub G Π Γ)(v : ℐ α Π) → 𝕊 mε v ≡ g (index mε v)
𝕊-ix (x ◂ mε) new = 𝕤𝕖𝕞! x
𝕊-ix (x ◂ mε) (old v) = 𝕊-ix mε v
𝕤𝕖𝕞! (mvar 𝔪 mε) rewrite cong (𝑚𝑣𝑎𝑟 𝔪) (dext (𝕊-ix mε))
= trans (sym ⟨𝑚𝑣𝑎𝑟⟩) (cong (g ∘ mvar 𝔪) (tab∘ix≈id mε))
𝕤𝕖𝕞! (var v) = sym ⟨𝑣𝑎𝑟⟩
𝕤𝕖𝕞! ε = sym ⟨𝑎𝑙𝑔⟩
𝕤𝕖𝕞! (_⊕_ a b) rewrite 𝕤𝕖𝕞! a | 𝕤𝕖𝕞! b = sym ⟨𝑎𝑙𝑔⟩
𝕤𝕖𝕞! (⊖_ a) rewrite 𝕤𝕖𝕞! a = sym ⟨𝑎𝑙𝑔⟩
G:Syn : Syntax
G:Syn = record
{ ⅀F = ⅀F
; ⅀:CS = ⅀:CompatStr
; mvarᵢ = G:Terms.mvar
; 𝕋:Init = λ 𝔛 → let open G:Terms 𝔛 in record
{ ⊥ = G ⋉ Gᵃ
; ⊥-is-initial = record { ! = λ{ {𝒜 ⋉ 𝒜ᵃ} → 𝕤𝕖𝕞 𝒜ᵃ ⋉ 𝕤𝕖𝕞ᵃ⇒ 𝒜ᵃ }
; !-unique = λ{ {𝒜 ⋉ 𝒜ᵃ} (f ⋉ fᵃ⇒) {x = t} → 𝕤𝕖𝕞! 𝒜ᵃ f fᵃ⇒ t } } } }
open Syntax G:Syn public
open G:Terms public
open import SOAS.Families.Build public
open import SOAS.Syntax.Shorthands Gᵃ public
open import SOAS.Metatheory G:Syn public