```------------------------------------------------------------------------
-- The backend does not introduce any unneeded ambiguity
------------------------------------------------------------------------

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

open import Coinduction
open import Data.List
import Data.List.Any as Any
open import Data.Maybe
open import Data.Product
open import Relation.Binary.PropositionalEquality
open import Relation.Binary.HeterogeneousEquality using (refl)

open Any.Membership-≡

open import TotalParserCombinators.Lib
import TotalParserCombinators.InitialBag as I
open import TotalParserCombinators.Parser
open import TotalParserCombinators.Semantics

D-complete∘D-sound : ∀ {Tok R xs x s t}
(p : Parser Tok R xs) (x∈ : x ∈ D t p · s) →
D-complete (D-sound p x∈) ≡ x∈
D-complete∘D-sound token     return            = refl
D-complete∘D-sound (p₁ ∣ p₂) (∣-left     x∈p₁) rewrite D-complete∘D-sound p₁ x∈p₁ = refl
D-complete∘D-sound (p₁ ∣ p₂) (∣-right ._ x∈p₂) rewrite D-complete∘D-sound p₂ x∈p₂ = refl
D-complete∘D-sound (f <\$> p) (<\$> x∈p)         rewrite D-complete∘D-sound p  x∈p  = refl

D-complete∘D-sound (_⊛_ {fs = nothing} {xs = just _}  p₁ p₂)             (f∈p₁′  ⊛ x∈p₂)  rewrite D-complete∘D-sound p₁ f∈p₁′ = refl
D-complete∘D-sound (_⊛_ {fs = just _}  {xs = just _}  p₁ p₂) (∣-left     (f∈p₁′  ⊛ x∈p₂)) rewrite D-complete∘D-sound p₁ f∈p₁′ = refl
D-complete∘D-sound (_⊛_ {fs = just fs} {xs = just xs} p₁ p₂) (∣-right ._ (f∈ret⋆ ⊛ x∈p₂′))
with          Return⋆.sound fs f∈ret⋆
| Return⋆.complete∘sound fs f∈ret⋆
D-complete∘D-sound (_⊛_ {fs = just fs} {xs = just xs} p₁ p₂) (∣-right ._ (.(Return⋆.complete f∈fs) ⊛ x∈p₂′)) | (refl , f∈fs) | refl
rewrite I.complete∘sound p₁ f∈fs | D-complete∘D-sound p₂ x∈p₂′ = refl
D-complete∘D-sound (_⊛_ {fs = nothing} {xs = nothing} p₁ p₂)             (f∈p₁′  ⊛ x∈p₂)  rewrite D-complete∘D-sound (♭ p₁) f∈p₁′ = refl
D-complete∘D-sound (_⊛_ {fs = just fs} {xs = nothing} p₁ p₂) (∣-left     (f∈p₁′  ⊛ x∈p₂)) rewrite D-complete∘D-sound (♭ p₁) f∈p₁′ = refl
D-complete∘D-sound (_⊛_ {fs = just fs} {xs = nothing} p₁ p₂) (∣-right ._ (f∈ret⋆ ⊛ x∈p₂′))
with          Return⋆.sound fs f∈ret⋆
| Return⋆.complete∘sound fs f∈ret⋆
D-complete∘D-sound (_⊛_ {fs = just fs} {xs = nothing} p₁ p₂) (∣-right ._ (.(Return⋆.complete f∈fs) ⊛ x∈p₂′)) | (refl , f∈fs) | refl
rewrite I.complete∘sound (♭ p₁) f∈fs | D-complete∘D-sound p₂ x∈p₂′ = refl

D-complete∘D-sound (_>>=_ {xs = nothing} {f = just _} p₁ p₂)             (x∈p₁′  >>= y∈p₂x)  rewrite D-complete∘D-sound p₁ x∈p₁′ = refl
D-complete∘D-sound (_>>=_ {xs = just _}  {f = just _} p₁ p₂) (∣-left     (x∈p₁′  >>= y∈p₂x)) rewrite D-complete∘D-sound p₁ x∈p₁′ = refl
D-complete∘D-sound (_>>=_ {xs = just xs} {f = just _} p₁ p₂) (∣-right ._ (y∈ret⋆ >>= z∈p₂′y))
with          Return⋆.sound xs y∈ret⋆
| Return⋆.complete∘sound xs y∈ret⋆
D-complete∘D-sound (_>>=_ {xs = just xs} {f = just _} p₁ p₂) (∣-right ._ (.(Return⋆.complete y∈xs) >>= z∈p₂′y)) | (refl , y∈xs) | refl
rewrite I.complete∘sound p₁ y∈xs | D-complete∘D-sound (p₂ _) z∈p₂′y = refl
D-complete∘D-sound (_>>=_ {xs = nothing} {f = nothing} p₁ p₂)             (x∈p₁′  >>= y∈p₂x)  rewrite D-complete∘D-sound (♭ p₁) x∈p₁′ =
refl
D-complete∘D-sound (_>>=_ {xs = just _}  {f = nothing} p₁ p₂) (∣-left     (x∈p₁′  >>= y∈p₂x)) rewrite D-complete∘D-sound (♭ p₁) x∈p₁′ =
refl
D-complete∘D-sound (_>>=_ {xs = just xs} {f = nothing} p₁ p₂) (∣-right ._ (y∈ret⋆ >>= z∈p₂′y))
with          Return⋆.sound xs y∈ret⋆
| Return⋆.complete∘sound xs y∈ret⋆
D-complete∘D-sound (_>>=_ {xs = just xs} {f = nothing} p₁ p₂) (∣-right ._ (.(Return⋆.complete y∈xs) >>= z∈p₂′y)) | (refl , y∈xs) | refl
rewrite I.complete∘sound (♭ p₁) y∈xs | D-complete∘D-sound (p₂ _) z∈p₂′y = refl

D-complete∘D-sound (nonempty p) x∈p = D-complete∘D-sound p x∈p
D-complete∘D-sound (cast _ p)   x∈p = D-complete∘D-sound p x∈p

D-complete∘D-sound (return _) ()
D-complete∘D-sound fail       ()

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