module TotalParserCombinators.Derivative.LeftInverse where
open import Codata.Musical.Notation
open import Data.Maybe hiding (_>>=_)
open import Data.Product
open import Relation.Binary.PropositionalEquality
open import Relation.Binary.HeterogeneousEquality using (refl)
open import TotalParserCombinators.Derivative.Definition
open import TotalParserCombinators.Derivative.SoundComplete
import TotalParserCombinators.InitialBag as I
open import TotalParserCombinators.Lib
open import TotalParserCombinators.Parser
open import TotalParserCombinators.Semantics
complete∘sound : ∀ {Tok R xs x s t}
(p : Parser Tok R xs) (x∈ : x ∈ D t p · s) →
complete (sound p x∈) ≡ x∈
complete∘sound token return = refl
complete∘sound (p₁ ∣ p₂) (∣-left x∈p₁) rewrite complete∘sound p₁ x∈p₁ = refl
complete∘sound (p₁ ∣ p₂) (∣-right ._ x∈p₂) rewrite complete∘sound p₂ x∈p₂ = refl
complete∘sound (f <$> p) (<$> x∈p) rewrite complete∘sound p x∈p = refl
complete∘sound (_⊛_ {fs = nothing} {xs = just _} p₁ p₂) (f∈p₁′ ⊛ x∈p₂) rewrite complete∘sound p₁ f∈p₁′ = refl
complete∘sound (_⊛_ {fs = just _} {xs = just _} p₁ p₂) (∣-left (f∈p₁′ ⊛ x∈p₂)) rewrite complete∘sound p₁ f∈p₁′ = refl
complete∘sound (_⊛_ {fs = just fs} {xs = just xs} p₁ p₂) (∣-right ._ (f∈ret⋆ ⊛ x∈p₂′))
with (refl , f∈fs) ← Return⋆.sound fs f∈ret⋆
| refl ← Return⋆.complete∘sound fs f∈ret⋆
rewrite I.complete∘sound p₁ f∈fs | complete∘sound p₂ x∈p₂′ = refl
complete∘sound (_⊛_ {fs = nothing} {xs = nothing} p₁ p₂) (f∈p₁′ ⊛ x∈p₂) rewrite complete∘sound (♭ p₁) f∈p₁′ = refl
complete∘sound (_⊛_ {fs = just fs} {xs = nothing} p₁ p₂) (∣-left (f∈p₁′ ⊛ x∈p₂)) rewrite complete∘sound (♭ p₁) f∈p₁′ = refl
complete∘sound (_⊛_ {fs = just fs} {xs = nothing} p₁ p₂) (∣-right ._ (f∈ret⋆ ⊛ x∈p₂′))
with (refl , f∈fs) ← Return⋆.sound fs f∈ret⋆
| refl ← Return⋆.complete∘sound fs f∈ret⋆
rewrite I.complete∘sound (♭ p₁) f∈fs | complete∘sound p₂ x∈p₂′ = refl
complete∘sound (_>>=_ {xs = nothing} {f = just _} p₁ p₂) (x∈p₁′ >>= y∈p₂x) rewrite complete∘sound p₁ x∈p₁′ = refl
complete∘sound (_>>=_ {xs = just _} {f = just _} p₁ p₂) (∣-left (x∈p₁′ >>= y∈p₂x)) rewrite complete∘sound p₁ x∈p₁′ = refl
complete∘sound (_>>=_ {xs = just xs} {f = just _} p₁ p₂) (∣-right ._ (y∈ret⋆ >>= z∈p₂′y))
with (refl , y∈xs) ← Return⋆.sound xs y∈ret⋆
| refl ← Return⋆.complete∘sound xs y∈ret⋆
rewrite I.complete∘sound p₁ y∈xs | complete∘sound (p₂ _) z∈p₂′y = refl
complete∘sound (_>>=_ {xs = nothing} {f = nothing} p₁ p₂) (x∈p₁′ >>= y∈p₂x) rewrite complete∘sound (♭ p₁) x∈p₁′ = refl
complete∘sound (_>>=_ {xs = just _} {f = nothing} p₁ p₂) (∣-left (x∈p₁′ >>= y∈p₂x)) rewrite complete∘sound (♭ p₁) x∈p₁′ = refl
complete∘sound (_>>=_ {xs = just xs} {f = nothing} p₁ p₂) (∣-right ._ (y∈ret⋆ >>= z∈p₂′y))
with (refl , y∈xs) ← Return⋆.sound xs y∈ret⋆
| refl ← Return⋆.complete∘sound xs y∈ret⋆
rewrite I.complete∘sound (♭ p₁) y∈xs | complete∘sound (p₂ _) z∈p₂′y = refl
complete∘sound (nonempty p) x∈p = complete∘sound p x∈p
complete∘sound (cast _ p) x∈p = complete∘sound p x∈p
complete∘sound (return _) ()
complete∘sound fail ()