------------------------------------------------------------------------
-- The Agda standard library
--
-- Decision procedures for finite sets and subsets of finite sets
------------------------------------------------------------------------

module Data.Fin.Dec where

open import Function
import Data.Bool as Bool
open import Data.Nat.Base hiding (_<_)
open import Data.Vec hiding (_∈_)
open import Data.Vec.Relation.Equality.DecPropositional Bool._≟_
open import Data.Fin
open import Data.Fin.Subset
open import Data.Fin.Subset.Properties
open import Data.Product as Prod
open import Data.Empty
open import Function
import Function.Equivalence as Eq
open import Relation.Binary as B
open import Relation.Nullary
import Relation.Nullary.Decidable as Dec
open import Relation.Unary as U using (Pred)

infix  _∈?_

_∈?_ : ∀ {n} x (p : Subset n) → Dec (x ∈ p)
zero  ∈? inside  ∷ p = yes here
zero  ∈? outside ∷ p = no  λ()
suc n ∈? s ∷ p       with n ∈? p
... | yes n∈p = yes (there n∈p)
... | no  n∉p = no  (n∉p ∘ drop-there)

private

  restrictP : ∀ {p n} → (Fin (suc n) → Set p) → (Fin n → Set p)
  restrictP P f = P (suc f)

  restrict : ∀ {p n} {P : Fin (suc n) → Set p} →
             U.Decidable P → U.Decidable (restrictP P)
  restrict dec f = dec (suc f)

any? : ∀ {n p} {P : Fin n → Set p} →
       U.Decidable P → Dec (∃ P)
any? {zero}          dec = no λ { (() , _) }
any? {suc n} {_} {P} dec with dec zero | any? (restrict dec)
...                  | yes p | _            = yes (_ , p)
...                  | _     | yes (_ , p') = yes (_ , p')
...                  | no ¬p | no ¬p'       = no helper
  where
  helper : ∄ P
  helper (zero  , p)  = ¬p p
  helper (suc f , p') = ¬p' (_ , p')

nonempty? : ∀ {n} (p : Subset n) → Dec (Nonempty p)
nonempty? p = any? (λ x → x ∈? p)

private

  restrict∈ : ∀ {p q n}
              (P : Fin (suc n) → Set p) {Q : Fin (suc n) → Set q} →
              (∀ {f} → Q f → Dec (P f)) →
              (∀ {f} → restrictP Q f → Dec (restrictP P f))
  restrict∈ _ dec {f} Qf = dec {suc f} Qf

decFinSubset : ∀ {p q n} {P : Fin n → Set p} {Q : Fin n → Set q} →
               U.Decidable Q →
               (∀ {f} → Q f → Dec (P f)) →
               Dec (∀ {f} → Q f → P f)
decFinSubset {n = zero}          _    _    = yes λ{}
decFinSubset {n = suc n} {P} {Q} decQ decP = helper
  where
  helper : Dec (∀ {f} → Q f → P f)
  helper with decFinSubset (restrict decQ) (restrict∈ P decP)
  helper | no ¬q⟶p = no (λ q⟶p → ¬q⟶p (λ {f} q → q⟶p {suc f} q))
  helper | yes q⟶p with decQ zero
  helper | yes q⟶p | yes q₀ with decP q₀
  helper | yes q⟶p | yes q₀ | no ¬p₀ = no (λ q⟶p → ¬p₀ (q⟶p {zero} q₀))
  helper | yes q⟶p | yes q₀ | yes p₀ = yes (λ {_} → hlpr _)
    where
    hlpr : ∀ f → Q f → P f
    hlpr zero    _  = p₀
    hlpr (suc f) qf = q⟶p qf
  helper | yes q⟶p | no ¬q₀ = yes (λ {_} → hlpr _)
    where
    hlpr : ∀ f → Q f → P f
    hlpr zero    q₀ = ⊥-elim (¬q₀ q₀)
    hlpr (suc f) qf = q⟶p qf

all∈? : ∀ {n p} {P : Fin n → Set p} {q} →
        (∀ {f} → f ∈ q → Dec (P f)) →
        Dec (∀ {f} → f ∈ q → P f)
all∈? {q = q} dec = decFinSubset (λ f → f ∈? q) dec

all? : ∀ {n p} {P : Fin n → Set p} →
       U.Decidable P → Dec (∀ f → P f)
all? dec with all∈? {q = ⊤} (λ {f} _ → dec f)
...      | yes ∀p = yes (λ f → ∀p ∈⊤)
...      | no ¬∀p = no  (λ ∀p → ¬∀p (λ {f} _ → ∀p f))

decLift : ∀ {n p} {P : Fin n → Set p} →
          U.Decidable P → U.Decidable (Lift P)
decLift dec p = all∈? (λ {x} _ → dec x)

private

  restrictSP : ∀ {n p} → Side → (Subset (suc n) → Set p) → (Subset n → Set p)
  restrictSP s P p = P (s ∷ p)

  restrictS : ∀ {n p} {P : Subset (suc n) → Set p} →
              (s : Side) → U.Decidable P → U.Decidable (restrictSP s P)
  restrictS s dec p = dec (s ∷ p)

anySubset? : ∀ {n p} {P : Subset n → Set p} →
             U.Decidable P → Dec (∃ P)
anySubset? {zero} {_} {P} dec with dec []
... | yes P[] = yes (_ , P[])
... | no ¬P[] = no helper
  where
  helper : ∄ P
  helper ([] , P[]) = ¬P[] P[]
anySubset? {suc n} {_} {P} dec with anySubset? (restrictS inside  dec)
                                  | anySubset? (restrictS outside dec)
... | yes (_ , Pp) | _            = yes (_ , Pp)
... | _            | yes (_ , Pp) = yes (_ , Pp)
... | no ¬Pp       | no ¬Pp'      = no helper
    where
    helper : ∄ P
    helper (inside  ∷ p , Pp)  = ¬Pp  (_ , Pp)
    helper (outside ∷ p , Pp') = ¬Pp' (_ , Pp')

-- If a decidable predicate P over a finite set is sometimes false,
-- then we can find the smallest value for which this is the case.

¬∀⟶∃¬-smallest :
  ∀ n {p} (P : Fin n → Set p) → U.Decidable P →
  ¬ (∀ i → P i) → ∃ λ i → ¬ P i × ((j : Fin′ i) → P (inject j))
¬∀⟶∃¬-smallest zero    P dec ¬∀iPi = ⊥-elim (¬∀iPi (λ()))
¬∀⟶∃¬-smallest (suc n) P dec ¬∀iPi with dec zero
¬∀⟶∃¬-smallest (suc n) P dec ¬∀iPi | no ¬P0 = (zero , ¬P0 , λ ())
¬∀⟶∃¬-smallest (suc n) P dec ¬∀iPi | yes P0 =
  Prod.map suc (Prod.map id extend′) $
    ¬∀⟶∃¬-smallest n (λ n → P (suc n)) (dec ∘ suc) (¬∀iPi ∘ extend)
  where
  extend : (∀ i → P (suc i)) → (∀ i → P i)
  extend ∀iP[1+i] zero    = P0
  extend ∀iP[1+i] (suc i) = ∀iP[1+i] i

  extend′ : ∀ {i : Fin n} →
            ((j : Fin′ i) → P (suc (inject j))) →
            ((j : Fin′ (suc i)) → P (inject j))
  extend′ g zero    = P0
  extend′ g (suc j) = g j


-- When P is a decidable predicate over a finite set the following
-- lemma can be proved.

¬∀⟶∃¬ : ∀ n {p} (P : Fin n → Set p) → U.Decidable P →
        ¬ (∀ i → P i) → ∃ λ i → ¬ P i
¬∀⟶∃¬ n P dec ¬P = Prod.map id proj₁ $ ¬∀⟶∃¬-smallest n P dec ¬P

-- Decision procedure for _⊆_ (obtained via the natural lattice
-- order).

infix  _⊆?_

_⊆?_ : ∀ {n} → B.Decidable (_⊆_ {n = n})
[]          ⊆? []          = yes id
outside ∷ p ⊆? y ∷ q with p ⊆? q
... | yes p⊆q = yes λ { (there v∈p) → there (p⊆q v∈p)}
... | no  p⊈q = no (p⊈q ∘ drop-∷-⊆)
inside  ∷ p ⊆? outside ∷ q = no (λ p⊆q → case (p⊆q here) of λ())
inside  ∷ p ⊆? inside  ∷ q with p ⊆? q
... | yes p⊆q = yes λ { here → here ; (there v) → there (p⊆q v)}
... | no  p⊈q = no (p⊈q ∘ drop-∷-⊆)