```------------------------------------------------------------------------
-- The derivative operator does not remove any ambiguity
------------------------------------------------------------------------

module TotalParserCombinators.Derivative.RightInverse where

open import Data.List
open import Data.Maybe
open import Relation.Binary.HeterogeneousEquality as H using (_≅_; refl)
open import Relation.Binary.PropositionalEquality

open import TotalParserCombinators.Derivative.SoundComplete
import TotalParserCombinators.InitialBag as I
open import TotalParserCombinators.Lib
open import TotalParserCombinators.Parser
open import TotalParserCombinators.Semantics hiding (_≅_)

mutual

sound∘complete : ∀ {Tok R xs x s t} {p : Parser Tok R xs} →
(x∈p : x ∈ p · t ∷ s) →
sound p (complete x∈p) ≡ x∈p
sound∘complete x∈p = H.≅-to-≡ (sound∘complete′ x∈p refl)

sound∘complete′ :
∀ {Tok R xs x s′ s t} {p : Parser Tok R xs}
(x∈p : x ∈ p · s′) (eq : s′ ≡ t ∷ s) →
sound p (complete′ p x∈p eq) ≅ x∈p
sound∘complete′ token       refl = refl

sound∘complete′ (∣-left    x∈p₁) refl rewrite sound∘complete x∈p₁ = refl
sound∘complete′ (∣-right _ x∈p₂) refl rewrite sound∘complete x∈p₂ = refl

sound∘complete′ (<\$> x∈p)   refl rewrite sound∘complete x∈p  = refl

sound∘complete′       (_⊛_ {s₁ = _ ∷ _} {fs = nothing} {xs = just _}  f∈p₁ x∈p₂) refl
rewrite sound∘complete f∈p₁ = refl
sound∘complete′       (_⊛_ {s₁ = _ ∷ _} {fs = just _}  {xs = just _}  f∈p₁ x∈p₂) refl
rewrite sound∘complete f∈p₁ = refl
sound∘complete′ {Tok} (_⊛_ {s₁ = []}    {fs = just _}  {xs = just _}  f∈p₁ x∈p₂) refl
rewrite Return⋆.sound∘complete {Tok = Tok} (I.complete f∈p₁)
| I.sound∘complete f∈p₁
| sound∘complete x∈p₂ = refl

sound∘complete′       (_⊛_ {s₁ = _ ∷ _} {fs = nothing} {xs = nothing} f∈p₁ x∈p₂) refl
rewrite sound∘complete f∈p₁ = refl
sound∘complete′       (_⊛_ {s₁ = _ ∷ _} {fs = just _}  {xs = nothing} f∈p₁ x∈p₂) refl
rewrite sound∘complete f∈p₁ = refl
sound∘complete′ {Tok} (_⊛_ {s₁ = []}    {fs = just _}  {xs = nothing} f∈p₁ x∈p₂) refl
rewrite Return⋆.sound∘complete {Tok = Tok} (I.complete f∈p₁)
| I.sound∘complete f∈p₁
| sound∘complete x∈p₂ = refl

sound∘complete′       (_>>=_ {s₁ = _ ∷ _} {xs = nothing} {f = just _}  x∈p₁ y∈p₂x) refl
rewrite sound∘complete x∈p₁ = refl
sound∘complete′       (_>>=_ {s₁ = _ ∷ _} {xs = just _}  {f = just _}  x∈p₁ y∈p₂x) refl
rewrite sound∘complete x∈p₁ = refl
sound∘complete′ {Tok} (_>>=_ {s₁ = []}    {xs = just _}  {f = just _}  x∈p₁ y∈p₂x) refl
rewrite Return⋆.sound∘complete {Tok = Tok} (I.complete x∈p₁)
| I.sound∘complete x∈p₁
| sound∘complete y∈p₂x = refl

sound∘complete′       (_>>=_ {s₁ = _ ∷ _} {xs = nothing} {f = nothing} x∈p₁ y∈p₂x) refl
rewrite sound∘complete x∈p₁ = refl
sound∘complete′       (_>>=_ {s₁ = _ ∷ _} {xs = just _}  {f = nothing} x∈p₁ y∈p₂x) refl
rewrite sound∘complete x∈p₁ = refl
sound∘complete′ {Tok} (_>>=_ {s₁ = []}    {xs = just _}  {f = nothing} x∈p₁ y∈p₂x) refl
rewrite Return⋆.sound∘complete {Tok = Tok} (I.complete x∈p₁)
| I.sound∘complete x∈p₁
| sound∘complete y∈p₂x = refl

sound∘complete′ (nonempty x∈p) refl rewrite sound∘complete x∈p = refl

sound∘complete′ (cast x∈p)     refl rewrite sound∘complete x∈p = refl

sound∘complete′ (_⊛_   {s₁ = []} {fs = nothing} f∈p₁ _) _ with I.complete f∈p₁
... | ()
sound∘complete′ (_>>=_ {s₁ = []} {xs = nothing} x∈p₁ _) _ with I.complete x∈p₁
... | ()

sound∘complete′ return ()
```