------------------------------------------------------------------------
-- The Agda standard library
--
-- A data structure which keeps track of an upper bound on the number
-- of elements /not/ in a given list
------------------------------------------------------------------------

import Level
open import Relation.Binary

module Data.List.Countdown (D : DecSetoid Level.zero Level.zero) where

open import Data.Empty
open import Data.Fin using (Fin; zero; suc; punchOut)
open import Data.Fin.Properties
  using (suc-injective; punchOut-injective)
open import Function
open import Function.Equality using (_⟨$⟩_)
open import Function.Injection
  using (Injection; module Injection)
open import Data.List
open import Data.List.Any as Any using (here; there)
open import Data.Nat.Base using (; zero; suc)
open import Data.Product
open import Data.Sum
open import Relation.Nullary
open import Relation.Binary.PropositionalEquality as PropEq
  using (_≡_; _≢_; refl; cong)
open PropEq.≡-Reasoning

private
  open module D = DecSetoid D
    hiding (refl) renaming (Carrier to Elem)
  open import Data.List.Any.Membership D.setoid

------------------------------------------------------------------------
-- Helper functions

private

  drop-inj₂ :  {A B : Set} {x y} 
              inj₂ {A = A} {B = B} x  inj₂ y  x  y
  drop-inj₂ refl = refl

  -- The /first/ occurrence of x in xs.

  first-occurrence :  {xs} x  x  xs  x  xs
  first-occurrence x (here x≈y)           = here x≈y
  first-occurrence x (there {x = y} x∈xs) with x  y
  ... | yes x≈y = here x≈y
  ... | no  _   = there $ first-occurrence x x∈xs

  -- The index of the first occurrence of x in xs.

  first-index :  {xs} x  x  xs  Fin (length xs)
  first-index x x∈xs = Any.index $ first-occurrence x x∈xs

  -- first-index preserves equality of its first argument.

  first-index-cong :  {x₁ x₂ xs} (x₁∈xs : x₁  xs) (x₂∈xs : x₂  xs) 
                     x₁  x₂  first-index x₁ x₁∈xs  first-index x₂ x₂∈xs
  first-index-cong {x₁} {x₂} x₁∈xs x₂∈xs x₁≈x₂ = helper x₁∈xs x₂∈xs
    where
    helper :  {xs} (x₁∈xs : x₁  xs) (x₂∈xs : x₂  xs) 
             first-index x₁ x₁∈xs  first-index x₂ x₂∈xs
    helper (here x₁≈x) (here x₂≈x)           = refl
    helper (here x₁≈x) (there {x = x} x₂∈xs) with x₂  x
    ... | yes x₂≈x = refl
    ... | no  x₂≉x = ⊥-elim (x₂≉x (trans (sym x₁≈x₂) x₁≈x))
    helper (there {x = x} x₁∈xs) (here x₂≈x) with x₁  x
    ... | yes x₁≈x = refl
    ... | no  x₁≉x = ⊥-elim (x₁≉x (trans x₁≈x₂ x₂≈x))
    helper (there {x = x} x₁∈xs) (there x₂∈xs) with x₁  x | x₂  x
    ... | yes x₁≈x | yes x₂≈x = refl
    ... | yes x₁≈x | no  x₂≉x = ⊥-elim (x₂≉x (trans (sym x₁≈x₂) x₁≈x))
    ... | no  x₁≉x | yes x₂≈x = ⊥-elim (x₁≉x (trans x₁≈x₂ x₂≈x))
    ... | no  x₁≉x | no  x₂≉x = cong suc $ helper x₁∈xs x₂∈xs

  -- first-index is injective in its first argument.

  first-index-injective
    :  {x₁ x₂ xs} (x₁∈xs : x₁  xs) (x₂∈xs : x₂  xs) 
      first-index x₁ x₁∈xs  first-index x₂ x₂∈xs  x₁  x₂
  first-index-injective {x₁} {x₂} = helper
    where
    helper :  {xs} (x₁∈xs : x₁  xs) (x₂∈xs : x₂  xs) 
             first-index x₁ x₁∈xs  first-index x₂ x₂∈xs  x₁  x₂
    helper (here x₁≈x) (here x₂≈x)             _  = trans x₁≈x (sym x₂≈x)
    helper (here x₁≈x) (there {x = x} x₂∈xs)   _  with x₂  x
    helper (here x₁≈x) (there {x = x} x₂∈xs)   _  | yes x₂≈x = trans x₁≈x (sym x₂≈x)
    helper (here x₁≈x) (there {x = x} x₂∈xs)   () | no  x₂≉x
    helper (there {x = x} x₁∈xs) (here x₂≈x)   _  with x₁  x
    helper (there {x = x} x₁∈xs) (here x₂≈x)   _  | yes x₁≈x = trans x₁≈x (sym x₂≈x)
    helper (there {x = x} x₁∈xs) (here x₂≈x)   () | no  x₁≉x
    helper (there {x = x} x₁∈xs) (there x₂∈xs) _  with x₁  x | x₂  x
    helper (there {x = x} x₁∈xs) (there x₂∈xs) _  | yes x₁≈x | yes x₂≈x = trans x₁≈x (sym x₂≈x)
    helper (there {x = x} x₁∈xs) (there x₂∈xs) () | yes x₁≈x | no  x₂≉x
    helper (there {x = x} x₁∈xs) (there x₂∈xs) () | no  x₁≉x | yes x₂≈x
    helper (there {x = x} x₁∈xs) (there x₂∈xs) eq | no  x₁≉x | no  x₂≉x =
      helper x₁∈xs x₂∈xs (suc-injective eq)

------------------------------------------------------------------------
-- The countdown data structure

-- If counted ⊕ n is inhabited then there are at most n values of type
-- Elem which are not members of counted (up to _≈_). You can read the
-- symbol _⊕_ as partitioning Elem into two parts: counted and
-- uncounted.

infix 4 _⊕_

record _⊕_ (counted : List Elem) (n : ) : Set where
  field
    -- An element can be of two kinds:
    -- ⑴ It is provably in counted.
    -- ⑵ It is one of at most n elements which may or may not be in
    --   counted. The "at most n" part is guaranteed by the field
    --   "injective".

    kind      :  x  x  counted  Fin n
    injective :  {x y i}  kind x  inj₂ i  kind y  inj₂ i  x  y

-- A countdown can be initialised by proving that Elem is finite.

empty :  {n}  Injection D.setoid (PropEq.setoid (Fin n))  []  n
empty inj =
  record { kind      = inj₂  _⟨$⟩_ to
         ; injective = λ {x} {y} {i} eq₁ eq₂  injective (begin
             to ⟨$⟩ x  ≡⟨ drop-inj₂ eq₁ 
             i         ≡⟨ PropEq.sym $ drop-inj₂ eq₂ 
             to ⟨$⟩ y  )
         }
  where open Injection inj

-- A countdown can also be initialised by proving that Elem is finite.

emptyFromList : (counted : List Elem)  (∀ x  x  counted) 
                []  length counted
emptyFromList counted complete = empty record
  { to = record
    { _⟨$⟩_ = λ x  first-index x (complete x)
    ; cong  = first-index-cong (complete _) (complete _)
    }
  ; injective = first-index-injective (complete _) (complete _)
  }

-- Finds out if an element has been counted yet.

lookup :  {counted n}  counted  n   x  Dec (x  counted)
lookup {counted} _ x = Any.any (_≟_ x) counted

-- When no element remains to be counted all elements have been
-- counted.

lookup! :  {counted}  counted  zero   x  x  counted
lookup! counted⊕0 x with _⊕_.kind counted⊕0 x
... | inj₁ x∈counted = x∈counted
... | inj₂ ()

private

  -- A variant of lookup!.

  lookup‼ :  {m counted} 
            counted  m   x  x  counted   λ n  m  suc n
  lookup‼ {suc m} counted⊕n x x∉counted = (m , refl)
  lookup‼ {zero}  counted⊕n x x∉counted =
    ⊥-elim (x∉counted $ lookup! counted⊕n x)

-- Counts a previously uncounted element.

insert :  {counted n} 
         counted  suc n   x  x  counted  x  counted  n
insert {counted} {n} counted⊕1+n x x∉counted =
  record { kind = kind′; injective = inj }
  where
  open _⊕_ counted⊕1+n

  helper :  x y i {j} 
           kind x  inj₂ i  kind y  inj₂ j  i  j  x  y
  helper _ _ _ eq₁ eq₂ refl = injective eq₁ eq₂

  kind′ :  y  y  x  counted  Fin n
  kind′  y with y  x | kind x | kind y | helper x y
  kind′  y | yes y≈x | _              | _              | _   = inj₁ (here y≈x)
  kind′  y | _       | inj₁ x∈counted | _              | _   = ⊥-elim (x∉counted x∈counted)
  kind′  y | _       | _              | inj₁ y∈counted | _   = inj₁ (there y∈counted)
  kind′  y | no  y≉x | inj₂ i         | inj₂ j         | hlp =
    inj₂ (punchOut (y≉x  sym  hlp _ refl refl))

  inj :  {y z i}  kind′ y  inj₂ i  kind′ z  inj₂ i  y  z
  inj {y} {z} eq₁ eq₂ with y  x | z  x | kind x | kind y | kind z
                         | helper x y | helper x z | helper y z
  inj ()  _   | yes _ | _     | _              | _      | _      | _ | _ | _
  inj _   ()  | _     | yes _ | _              | _      | _      | _ | _ | _
  inj _   _   | no  _ | no  _ | inj₁ x∈counted | _      | _      | _ | _ | _ = ⊥-elim (x∉counted x∈counted)
  inj ()  _   | no  _ | no  _ | inj₂ _         | inj₁ _ | _      | _ | _ | _
  inj _   ()  | no  _ | no  _ | inj₂ _         | _      | inj₁ _ | _ | _ | _
  inj eq₁ eq₂ | no  _ | no  _ | inj₂ i         | inj₂ _ | inj₂ _ | _ | _ | hlp =
    hlp _ refl refl $
      punchOut-injective {i = i} _ _ $
        (PropEq.trans (drop-inj₂ eq₁) (PropEq.sym (drop-inj₂ eq₂)))

-- Counts an element if it has not already been counted.

lookupOrInsert :  {counted m} 
                 counted  m 
                  x  x  counted 
                        λ n  m  suc n × x  counted  n
lookupOrInsert counted⊕n x with lookup counted⊕n x
... | yes x∈counted = inj₁ x∈counted
... | no  x∉counted with lookup‼ counted⊕n x x∉counted
...   | (n , refl) = inj₂ (n , refl , insert counted⊕n x x∉counted)