```------------------------------------------------------------------------
-- The backend does not remove any ambiguity
------------------------------------------------------------------------

-- This module contains a proof showing that
-- This implies that the (finite) type x ∈ parse p s contains at least
-- as many proofs as x ∈ p · s. In other words, if the output of
-- parse p s contains n copies of x, then there are at most n distinct
-- parse trees in x ∈ p · s.

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

mutual

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

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

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

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

D-sound∘D-complete′       (_⊛_ {s₁ = _ ∷ _} {fs = nothing} {xs = just _}  f∈p₁ x∈p₂) refl
rewrite D-sound∘D-complete f∈p₁ = refl
D-sound∘D-complete′       (_⊛_ {s₁ = _ ∷ _} {fs = just _}  {xs = just _}  f∈p₁ x∈p₂) refl
rewrite D-sound∘D-complete f∈p₁ = refl
D-sound∘D-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
| D-sound∘D-complete x∈p₂ = refl

D-sound∘D-complete′       (_⊛_ {s₁ = _ ∷ _} {fs = nothing} {xs = nothing} f∈p₁ x∈p₂) refl
rewrite D-sound∘D-complete f∈p₁ = refl
D-sound∘D-complete′       (_⊛_ {s₁ = _ ∷ _} {fs = just _}  {xs = nothing} f∈p₁ x∈p₂) refl
rewrite D-sound∘D-complete f∈p₁ = refl
D-sound∘D-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
| D-sound∘D-complete x∈p₂ = refl

D-sound∘D-complete′       (_>>=_ {s₁ = _ ∷ _} {xs = nothing} {f = just _}  x∈p₁ y∈p₂x) refl
rewrite D-sound∘D-complete x∈p₁ = refl
D-sound∘D-complete′       (_>>=_ {s₁ = _ ∷ _} {xs = just _}  {f = just _}  x∈p₁ y∈p₂x) refl
rewrite D-sound∘D-complete x∈p₁ = refl
D-sound∘D-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
| D-sound∘D-complete y∈p₂x = refl

D-sound∘D-complete′       (_>>=_ {s₁ = _ ∷ _} {xs = nothing} {f = nothing} x∈p₁ y∈p₂x) refl
rewrite D-sound∘D-complete x∈p₁ = refl
D-sound∘D-complete′       (_>>=_ {s₁ = _ ∷ _} {xs = just _}  {f = nothing} x∈p₁ y∈p₂x) refl
rewrite D-sound∘D-complete x∈p₁ = refl
D-sound∘D-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
| D-sound∘D-complete y∈p₂x = refl

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

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

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

D-sound∘D-complete′ return ()

sound∘complete : ∀ {Tok R xs x} {p : Parser Tok R xs}
(s : List Tok) (x∈p : x ∈ p · s) →
sound s (complete s x∈p) ≡ x∈p
sound∘complete []      x∈p = I.sound∘complete x∈p
sound∘complete (t ∷ s) x∈p
rewrite sound∘complete s (D-complete x∈p) = D-sound∘D-complete x∈p
```