module regular-language where

open import Level renaming ( suc to Suc ; zero to Zero )
open import Data.List 
open import Data.Nat hiding ( _≟_ )
open import Data.Fin hiding ( _+_ )
open import Data.Empty 
open import Data.Product
-- open import Data.Maybe
open import  Relation.Nullary
open import  Relation.Binary.PropositionalEquality hiding ( [_] )
open import logic
open import nat
open import automaton
open import finiteSet

language : { Σ : Set } → Set
language {Σ} = List Σ → Bool

language-L : { Σ : Set } → Set
language-L {Σ} = List (List Σ)

open Automaton

record RegularLanguage ( Σ : Set ) : Set (Suc Zero) where
   field
      states : Set 
      astart : states 
      aℕ : ℕ
      afin : FiniteSet states {aℕ}
      automaton : Automaton states Σ
   contain : List Σ → Bool
   contain x = accept automaton astart x

Union : {Σ : Set} → ( A B : language {Σ} ) → language {Σ}
Union {Σ} A B x = (A x ) \/ (B x)

split : {Σ : Set} → (List Σ → Bool)
      → ( List Σ → Bool) → List Σ → Bool
split x y  [] = x [] /\ y []
split x y (h  ∷ t) = (x [] /\ y (h  ∷ t)) \/
  split (λ t1 → x (  h ∷ t1 ))  (λ t2 → y t2 ) t

Concat : {Σ : Set} → ( A B : language {Σ} ) → language {Σ}
Concat {Σ} A B = split A B

{-# TERMINATING #-}
Star : {Σ : Set} → ( A : language {Σ} ) → language {Σ}
Star {Σ} A = split A ( Star {Σ} A )

test-AB→split : {Σ : Set} → {A B : List In2 → Bool} → split A B ( i0 ∷ i1 ∷ i0 ∷ [] ) ≡ (
       ( A [] /\ B ( i0 ∷ i1 ∷ i0 ∷ [] ) ) \/ 
       ( A ( i0 ∷ [] ) /\ B ( i1 ∷ i0 ∷ [] ) ) \/ 
       ( A ( i0 ∷ i1 ∷ [] ) /\ B ( i0 ∷ [] ) ) \/
       ( A ( i0 ∷ i1 ∷ i0 ∷ [] ) /\ B  []  ) 
   )
test-AB→split {_} {A} {B} = refl

open RegularLanguage 
isRegular : {Σ : Set} → (A : language {Σ} ) → ( x : List Σ ) → (r : RegularLanguage Σ ) → Set
isRegular A x r = A x ≡ contain r x 

postulate 
   fin-× : {A B : Set} → { a b : ℕ } → FiniteSet A {a} → FiniteSet B {b} → FiniteSet (A × B) {a * b}

M-Union : {Σ : Set} → (A B : RegularLanguage Σ ) → RegularLanguage Σ
M-Union {Σ} A B = record {
       states =  states A × states B
     ; astart = ( astart A , astart B )
     ; aℕ = aℕ A * aℕ B
     ; afin = fin-× (afin A) (afin B)
     ; automaton = record {
             δ = λ q x → ( δ (automaton A) (proj₁ q) x , δ (automaton B) (proj₂ q) x )
           ; aend = λ q → ( aend (automaton A) (proj₁ q) \/ aend (automaton B) (proj₂ q) )
        }
   } 

closed-in-union :  {Σ : Set} → (A B : RegularLanguage Σ ) → ( x : List Σ ) → isRegular (Union (contain A) (contain B)) x ( M-Union A B )
closed-in-union A B [] = lemma where
   lemma : aend (automaton A) (astart A) \/ aend (automaton B) (astart B) ≡
           aend (automaton A) (astart A) \/ aend (automaton B) (astart B)
   lemma = refl
closed-in-union {Σ} A B ( h ∷ t ) = lemma1 t ((δ (automaton A) (astart A) h)) ((δ (automaton B) (astart B) h)) where
   lemma1 : (t : List Σ) → (qa : states A ) → (qb : states B ) → 
     accept (automaton A) qa t \/ accept (automaton B) qb  t
       ≡ accept (automaton (M-Union A B)) (qa , qb) t
   lemma1 [] qa qb = refl
   lemma1 (h ∷ t ) qa qb = lemma1 t ((δ (automaton A) qa h)) ((δ (automaton B) qb h))

-- M-Concat : {Σ : Set} → (A B : RegularLanguage Σ ) → RegularLanguage Σ
-- M-Concat {Σ} A B = record {
--        states =  states A ∨ states B
--      ; astart = case1 (astart A )
--      ; automaton = record {
--              δ = {!!}
--            ; aend = {!!}
--         }
--    } 
-- 
-- closed-in-concat :  {Σ : Set} → (A B : RegularLanguage Σ ) → ( x : List Σ ) → isRegular (Concat (contain A) (contain B)) x ( M-Concat A B )
-- closed-in-concat = {!!}

open import nfa
open import sbconst2
open FiniteSet
open import Data.Nat.Properties hiding ( _≟_ )
open import Relation.Binary as B hiding (Decidable)

postulate 
   fin-∨ : {A B : Set} → { a b : ℕ } → FiniteSet A {a} → FiniteSet B {b} → FiniteSet (A ∨ B) {a + b}
   fin→ : {A : Set} → { a : ℕ } → FiniteSet A {a} → FiniteSet (A → Bool ) {exp 2 a}

Concat-NFA :  {Σ : Set} → (A B : RegularLanguage Σ ) → NAutomaton (states A ∨ states B) Σ 
Concat-NFA {Σ} A B = record { Nδ = δnfa ; Nend = nend } 
   module Concat-NFA where
       δnfa : states A ∨ states B → Σ → states A ∨ states B → Bool
       δnfa (case1 q) i (case1 q₁) = equal? (afin A) (δ (automaton A) q i) q₁
       δnfa (case1 qa) i (case2 qb) = (aend (automaton A) qa ) /\
           (equal? (afin B) qb (δ (automaton B) (astart B) i) )
       δnfa (case2 q) i (case2 q₁) = equal? (afin B) (δ (automaton B) q i) q₁
       δnfa _ i _ = false
       nend : states A ∨ states B → Bool
       nend (case2 q) = aend (automaton B) q
       nend (case1 q) = aend (automaton A) q /\ aend (automaton B) (astart B) -- empty B case

Concat-NFA-start :  {Σ : Set} → (A B : RegularLanguage Σ ) → states A ∨ states B → Bool
Concat-NFA-start A B q = equal? (fin-∨ (afin A) (afin B)) (case1 (astart A)) q

M-Concat : {Σ : Set} → (A B : RegularLanguage Σ ) → RegularLanguage Σ
M-Concat {Σ} A B = record {
       states = states A ∨ states B → Bool
     ; astart = Concat-NFA-start A B
     ; aℕ = finℕ finf
     ; afin = finf
     ; automaton = subset-construction fin (Concat-NFA A B) (case1 (astart A))
   } where
       fin : FiniteSet (states A ∨ states B ) {aℕ A + aℕ B}
       fin = fin-∨ (afin A) (afin B)
       finf : FiniteSet (states A ∨ states B → Bool ) 
       finf = fin→ fin 
       
record Split {Σ : Set} (A : List Σ → Bool ) ( B : List Σ → Bool ) (x :  List Σ ) : Set where
    field
        sp0 : List Σ
        sp1 : List Σ
        sp-concat : sp0 ++ sp1 ≡ x
        prop0 : A sp0 ≡ true
        prop1 : B sp1 ≡ true

open Split

list-empty++ : {Σ : Set} → (x y : List Σ) → x ++ y ≡ [] → (x ≡ [] ) ∧ (y ≡ [] )
list-empty++ [] [] refl = record { proj1 = refl ; proj2 = refl }
list-empty++ [] (x ∷ y) ()
list-empty++ (x ∷ x₁) y ()

open _∧_

open import Relation.Binary.PropositionalEquality hiding ( [_] )

c-split-lemma : {Σ : Set} → (A B : List Σ → Bool ) → (h : Σ) → ( t : List Σ ) → split A B (h ∷ t ) ≡ true
   → ( ¬ (A [] ≡ true )) ∨ ( ¬ (B ( h ∷ t ) ≡ true) )
   → split (λ t1 → A (h ∷ t1)) B t ≡ true
c-split-lemma {Σ} A B h t eq (case1 ¬p ) = sym ( begin
      true
  ≡⟨  sym eq  ⟩
      split A B (h ∷ t ) 
  ≡⟨⟩
      A [] /\ B (h ∷ t) \/ split (λ t1 → A (h ∷ t1)) B t
  ≡⟨  cong ( λ k → k \/ split (λ t1 → A (h ∷ t1)) B t ) (bool-and-1 (¬-bool-t ¬p)) ⟩
      false \/ split (λ t1 → A (h ∷ t1)) B t
  ≡⟨ bool-or-1 refl ⟩
      split (λ t1 → A (h ∷ t1)) B t
  ∎ ) where open ≡-Reasoning
c-split-lemma {Σ} A B h t eq (case2 ¬p ) = sym ( begin
      true
  ≡⟨  sym eq  ⟩
      split A B (h ∷ t ) 
  ≡⟨⟩
      A [] /\ B (h ∷ t) \/ split (λ t1 → A (h ∷ t1)) B t
  ≡⟨  cong ( λ k → k \/ split (λ t1 → A (h ∷ t1)) B t ) (bool-and-2 (¬-bool-t ¬p)) ⟩
      false \/ split (λ t1 → A (h ∷ t1)) B t
  ≡⟨ bool-or-1 refl ⟩
      split (λ t1 → A (h ∷ t1)) B t
  ∎ ) where open ≡-Reasoning

split→AB :  {Σ : Set} → (A B : List Σ → Bool ) → ( x : List Σ ) → split A B x ≡ true → Split A B x
split→AB {Σ} A B [] eq with bool-≡-? (A []) true | bool-≡-? (B []) true 
split→AB {Σ} A B [] eq | yes eqa | yes eqb = 
    record { sp0 = [] ; sp1 = [] ; sp-concat = refl ; prop0 = eqa ; prop1 = eqb }
split→AB {Σ} A B [] eq | yes p | no ¬p = ⊥-elim (lemma-∧-1 eq (¬-bool-t ¬p ))
split→AB {Σ} A B [] eq | no ¬p | t = ⊥-elim (lemma-∧-0 eq (¬-bool-t ¬p ))
split→AB {Σ} A B (h ∷ t ) eq with bool-≡-? (A []) true | bool-≡-? (B (h ∷ t )) true
... | yes px | yes py = record { sp0 = [] ; sp1 = h ∷ t ; sp-concat = refl ; prop0 = px ; prop1 = py }
... | no px | _ with split→AB (λ t1 → A ( h ∷ t1 )) B t (c-split-lemma A B h t eq (case1 px) )
... | S = record { sp0 = h ∷ sp0 S  ; sp1 = sp1 S ; sp-concat = cong ( λ k → h ∷ k) (sp-concat S) ; prop0 = prop0 S ; prop1 = prop1 S }
split→AB {Σ} A B (h ∷ t ) eq  | _ | no px with split→AB (λ t1 → A ( h ∷ t1 )) B t (c-split-lemma A B h t eq (case2 px) )
... | S = record { sp0 = h ∷ sp0 S  ; sp1 = sp1 S ; sp-concat = cong ( λ k → h ∷ k) (sp-concat S) ; prop0 = prop0 S ; prop1 = prop1 S }

AB→split :  {Σ : Set} → (A B : List Σ → Bool ) → ( x y : List Σ ) → A x ≡ true → B y ≡ true → split A B (x ++ y ) ≡ true
AB→split {Σ} A B [] [] eqa eqb = begin
       split A B [] 
     ≡⟨⟩
       A [] /\ B []
     ≡⟨ cong₂ (λ j k → j /\ k ) eqa eqb ⟩
      true
     ∎  where open ≡-Reasoning
AB→split {Σ} A B [] (h ∷ y ) eqa eqb = begin
      split A B (h ∷ y )
     ≡⟨⟩
      A [] /\ B (h ∷ y) \/ split (λ t1 → A (h ∷ t1)) B y
     ≡⟨ cong₂ (λ j k → j /\ k \/ split (λ t1 → A (h ∷ t1)) B y ) eqa eqb ⟩
      true /\ true \/ split (λ t1 → A (h ∷ t1)) B y
     ≡⟨⟩
      true \/ split (λ t1 → A (h ∷ t1)) B y
     ≡⟨⟩
      true
     ∎  where open ≡-Reasoning
AB→split {Σ} A B (h ∷ t) y eqa eqb = begin
       split A B ((h ∷ t) ++ y)  
     ≡⟨⟩
       A [] /\ B (h ∷ t ++ y) \/ split (λ t1 → A (h ∷ t1)) B (t ++ y)
     ≡⟨ cong ( λ k →  A [] /\ B (h ∷ t ++ y) \/ k ) ( begin
          split (λ t1 → A (h ∷ t1)) B (t ++ y) 
        ≡⟨ AB→split {Σ} (λ t1 → A (h ∷ t1)) B t y eqa eqb ⟩
          true 
        ∎ ) ⟩
       A [] /\ B (h ∷ t ++ y) \/ true
     ≡⟨ bool-or-3 ⟩
      true
     ∎  where open ≡-Reasoning

open NAutomaton
open import Data.List.Properties

closed-in-concat :  {Σ : Set} → (A B : RegularLanguage Σ ) → ( x : List Σ ) → isRegular (Concat (contain A) (contain B)) x ( M-Concat A B )
closed-in-concat {Σ} A B x = ≡-Bool-func closed-in-concat→ closed-in-concat← where
    finab = (fin-∨ (afin A) (afin B))
    NFA = (Concat-NFA A B)
    abmove : (q : states A ∨ states B) → (h : Σ ) → states A ∨ states B
    abmove (case1 q) h = case1 (δ (automaton A) q h)
    abmove (case2 q) h = case2 (δ (automaton B) q h)
    lemma-nmove-ab : (q : states A ∨ states B) → (h : Σ ) → Nδ NFA q h (abmove q h) ≡ true
    lemma-nmove-ab (case1 q) _ = equal?-refl (afin A)
    lemma-nmove-ab (case2 q) _ = equal?-refl (afin B)
    nmove : (q : states A ∨ states B) (nq : states A ∨ states B → Bool ) → (nq q ≡ true) → ( h : Σ ) →
       exists finab (λ qn → nq qn /\ Nδ NFA qn h (abmove q h)) ≡ true
    nmove (case1 q) nq nqt h = found finab (case1 q) ( bool-and-tt nqt (lemma-nmove-ab (case1 q)  h) )  
    nmove (case2 q) nq nqt h = found finab (case2 q) ( bool-and-tt nqt (lemma-nmove-ab (case2 q) h) ) where
    acceptB : (z : List Σ) → (q : states B) → (nq : states A ∨ states B → Bool ) → (nq (case2 q) ≡ true) → ( accept (automaton B) q z ≡ true ) 
        → Naccept NFA finab nq z  ≡ true
    acceptB [] q nq nqt fb = lemma8 where
        lemma8 : exists finab ( λ q → nq q /\ Nend NFA q ) ≡ true
        lemma8 = found finab (case2 q) ( bool-and-tt nqt fb )
    acceptB (h ∷ t ) q nq nq=q fb = acceptB t (δ (automaton B) q h) (Nmoves NFA finab nq h) (nmove (case2 q) nq nq=q h) fb 
    acceptA : (y z : List Σ) → (q : states A) → (nq : states A ∨ states B → Bool ) → (nq (case1 q) ≡ true)
        → ( accept (automaton A) q y ≡ true ) → ( accept (automaton B) (astart B) z ≡ true ) 
        → Naccept NFA finab nq (y ++ z)  ≡ true
    acceptA [] [] q nq nqt fa fb = found finab (case1 q) (bool-and-tt nqt (bool-and-tt fa fb )) 
    acceptA [] (h ∷ z)  q nq nq=q fa fb = acceptB z nextb (Nmoves NFA finab nq h) lemma70 fb where
         nextb : states B
         nextb = δ (automaton B) (astart B) h
         lemma70 : exists finab (λ qn → nq qn /\ Nδ NFA qn h (case2 nextb)) ≡ true
         lemma70 = found finab (case1 q) ( bool-and-tt nq=q (bool-and-tt fa (lemma-nmove-ab (case2 (astart B)) h) ))
    acceptA (h ∷ t) z q nq nq=q fa fb = acceptA t z (δ (automaton A) q h) (Nmoves NFA finab nq h) (nmove (case1 q) nq nq=q h)  fa fb where
    acceptAB : Split (contain A) (contain B) x
        → Naccept NFA finab (equal? finab (case1 (astart A))) x  ≡ true
    acceptAB S = subst ( λ k → Naccept NFA finab (equal? finab (case1 (astart A))) k  ≡ true  ) ( sp-concat S )
        (acceptA (sp0 S) (sp1 S)  (astart A) (equal? finab (case1 (astart A))) (equal?-refl finab) (prop0 S) (prop1 S) )

    closed-in-concat→ : Concat (contain A) (contain B) x ≡ true → contain (M-Concat A B) x ≡ true
    closed-in-concat→ concat with split→AB (contain A) (contain B) x concat
    ... | S = begin
          accept (subset-construction finab NFA (case1 (astart A))) (Concat-NFA-start A B ) x 
       ≡⟨ ≡-Bool-func (subset-construction-lemma← finab NFA (case1 (astart A)) x ) 
          (subset-construction-lemma→ finab NFA (case1 (astart A)) x ) ⟩
          Naccept NFA finab (equal? finab (case1 (astart A))) x
       ≡⟨ acceptAB S ⟩
         true
       ∎  where open ≡-Reasoning

    open Found

    contain-B : ( h : Σ) → ( t : List Σ ) → ( nq : states A ∨ states B → Bool ) → (qb : states B) → ( nq (case2 qb) ≡ true ) →
       Naccept NFA finab nq (h ∷ t) ≡ true → accept (automaton B) (δ (automaton B) (astart B) h) t ≡ true
    contain-B = {!!}

    condition-A : (nq :  states A ∨ states B → Bool ) → (qa : states A) → (q : states A ∨ states B) → Set
    condition-A nq qa q =  ( nq q  ≡ true → q ≡ case1 qa ) ∨ ( (nq (case1 qa) /\ aend (automaton A) qa /\ nq (case2 (astart B))) ≡ true )

    contain-A : (x : List Σ) → (nq : states A ∨ states B → Bool ) → Naccept NFA finab nq x ≡ true
          → (qa : states A )  → (nq (case1 qa) ≡ true) → ( (q :  states A ∨ states B ) → condition-A nq qa q )
          → split (accept (automaton A) qa ) (contain B) x ≡ true
    contain-A [] nq fn qa qat cond with found← finab fn 
    ... | S with cond (found-q S)
    ... | case2 end = {!!}
    ... | case1 qa=q with qa=q (bool-∧→tt-0 (found-p S))
    ... | refl = bool-∧→tt-1 (found-p S)
    contain-A (h ∷ t) nq fn qa qat cond with bool-≡-? ((aend (automaton A) qa) /\  accept (automaton B) (δ (automaton B) (astart B) h) t ) true
    ... | yes eq = bool-or-41 eq
    ... | no ne = bool-or-31 (contain-A t (Nmoves NFA finab nq h) fn (δ (automaton A) qa h) (nmove (case1 qa) nq qat h) lemma11-1 ) where
        lemma11-1 : (q : states A ∨ states B)  → condition-A (λ q → exists finab (λ qn → nq qn /\ Nδ NFA qn h q)) (δ (automaton A) qa h) q
        lemma11-1 q with cond q
        ... | case2 end = {!!}
        ... | case1 qa=q = case1 (lemma11 q) where
         lemma11 : (q : states A ∨ states B)
            → exists finab (λ qn → nq qn /\ Nδ NFA qn h q) ≡ true
            → q ≡ case1 (δ (automaton A) qa h)
         lemma11 q ex = lemma17 lemma19 where
            lemma17 : exists finab (λ qn → nq qn /\ Nδ NFA (case1 qa) h q) ≡ true → q ≡ case1 (δ (automaton A) qa h)
            lemma17 en with found← finab en 
            ... | S with found-q S | found-p S | {!!} -- qa=q  (bool-∧→tt-0 ( found-p S))(found-q S)
            ... | (case1 qq) | e1 | eq =  lemma21 q refl where
                 lemma20 : nq (case1 qa) /\ Concat-NFA.δnfa A B (case1 qa) h q ≡ true
                 lemma20 = {!!}
                 lemma21 : (qq : states A ∨ states B) → qq ≡ q → q ≡ case1 (δ (automaton A) qa h)
                 lemma21 (case1 x) refl = cong ( λ k → case1 k ) (sym lemma23 ) where
                     lemma23 : (δ (automaton A) qa h) ≡ x
                     lemma23 = equal→refl (afin A) ( bool-∧→tt-1 e1)
                 lemma21 (case2 x) refl with bool-≡-? (aend (automaton A) qa) true
                 ... | no not1 = ⊥-elim ( not1 ( bool-∧→tt-0 ( bool-∧→tt-1 e1)))
                 ... | yes a-end = ⊥-elim ( ne (bool-and-tt a-end lemma25 ) ) where
                     lemma24 : aend (automaton A) qa /\ (equal? (afin B) x (δ (automaton B) (astart B) h)) ≡ true
                     lemma24 = bool-∧→tt-1 e1
                     lemma25 : accept (automaton B) (δ (automaton B) (astart B) h) t ≡ true
                     lemma25 =  contain-B h t nq (astart B) {!!} fn
            lemma18 : (qn :  states A ∨ states B) → nq qn /\ Nδ NFA qn h q ≡ true → nq qn /\ Nδ NFA (case1 qa) h q ≡ true
            lemma18 qn eq = begin
                  nq qn /\ Nδ NFA (case1 qa) h q
               ≡⟨ cong ( λ k → nq qn /\ Nδ NFA k h q ) (sym {!!}) ⟩
                  nq qn /\ Nδ NFA qn h q
               ≡⟨  eq ⟩
                  true
               ∎  where open ≡-Reasoning
            lemma19 : exists finab (λ qn → nq qn /\ Nδ NFA (case1 qa) h q) ≡ true
            lemma19 = found finab fq ( (lemma18 fq (found-p ( found← finab ex )))) where
                fq = found-q ( found← finab ex )

        lemma14 : split (λ t1 → accept (automaton A) qa (h ∷ t1)) (contain B) t ≡ true
        lemma14 = contain-A t (Nmoves NFA finab nq h) fn (δ (automaton A) qa h) (nmove (case1 qa) nq qat h) lemma11-1

    lemma10 : Naccept NFA finab (equal? finab (case1 (astart A))) x  ≡ true → split (contain A) (contain B) x ≡ true
    lemma10 CC = contain-A x (Concat-NFA-start A B ) CC (astart A) (equal?-refl finab) lemma12-1 where
        lemma12-1 : (q : states A ∨ states B) → condition-A (Concat-NFA-start A B) (astart A) q
        lemma12-1 q = case1 (λ eq → sym ( equal→refl finab eq ) )

    closed-in-concat← : contain (M-Concat A B) x ≡ true → Concat (contain A) (contain B) x ≡ true
    closed-in-concat← C with subset-construction-lemma← finab NFA (case1 (astart A)) x C
    ... | CC = lemma10 CC