open import Level hiding ( suc ; zero )
open import Algebra
module sym3n where

open import Symmetric 
open import Data.Unit
open import Function.Inverse as Inverse using (_↔_; Inverse; _InverseOf_)
open import Function
open import Data.Nat hiding (_⊔_) -- using (ℕ; suc; zero)
open import Relation.Nullary
open import Data.Empty
open import Data.Product

open import Gutil 
open import Putil 
open import Solvable using (solvable)
open import  Relation.Binary.PropositionalEquality hiding ( [_] )

open import Data.Fin
open import Data.Fin.Permutation hiding (_∘ₚ_)

infixr  200 _∘ₚ_
_∘ₚ_ = Data.Fin.Permutation._∘ₚ_


sym3solvable : solvable (Symmetric 3)
solvable.dervied-length sym3solvable = 2
solvable.end sym3solvable x d = solved1 x d where

   open import Data.List using ( List ; [] ; _∷_ )

   open Solvable (Symmetric 3)
   open import FLutil
   open import Data.List.Fresh hiding ([_])
   open import Relation.Nary using (⌊_⌋)

   p0id :  FL→perm ((# 0) :: ((# 0) :: ((# 0 ) :: f0))) =p= pid
   p0id = pleq _ _ refl

   t1  : FList 3 →  FList 3
   t1  x =  tl2 x x [] where
       tl3 : (FL 3) → ( z : FList 3) → FList 3 → FList 3
       tl3 h [] w = w
       tl3 h (x ∷# z) w = tl3 h z (FLinsert ( perm→FL [ FL→perm h , FL→perm x ] ) w )
       tl2 : ( x z : FList 3) → FList 3 →  FList 3
       tl2 [] _ x = x
       tl2 (h ∷# x) z w = tl2 x z (tl3 h z w)

   stage10  :  FList 3
   stage10  =  {!!} -- t1 (Flist (fmax ))

   p0 =  FL→perm ((# 0) :: ((# 0) :: ((# 0 ) :: f0))) 
   p1 =  FL→perm ((# 0) :: ((# 1) :: ((# 0 ) :: f0))) 
   p2 =  FL→perm ((# 1) :: ((# 0) :: ((# 0 ) :: f0))) 
   p3 =  FL→perm ((# 1) :: ((# 1) :: ((# 0 ) :: f0))) 
   p4 =  FL→perm ((# 2) :: ((# 0) :: ((# 0 ) :: f0))) 
   p5 =  FL→perm ((# 2) :: ((# 1) :: ((# 0 ) :: f0))) 
   t0  =  plist p0 ∷ plist p1 ∷  plist p2 ∷ plist p3 ∷ plist p4 ∷  plist p5 ∷ [] 

   tt4  = plist [ p0 , p0 ] ∷ plist [ p1 , p0 ] ∷  plist [ p2 , p0 ] ∷ plist [ p3 , p0 ] ∷ plist [ p4 , p0 ] ∷  plist [ p5 , p1 ] ∷ 
          plist [ p0 , p1 ] ∷ plist [ p1 , p1 ] ∷  plist [ p2 , p1 ] ∷ plist [ p3 , p1 ] ∷ plist [ p4 , p1 ] ∷  plist [ p5 , p1 ] ∷ 
          plist [ p0 , p2 ] ∷ plist [ p1 , p2 ] ∷  plist [ p2 , p2 ] ∷ plist [ p3 , p2 ] ∷ plist [ p4 , p2 ] ∷  plist [ p5 , p2 ] ∷ 
          plist [ p0 , p3 ] ∷ plist [ p1 , p3 ] ∷  plist [ p3 , p3 ] ∷ plist [ p3 , p3 ] ∷ plist [ p4 , p3 ] ∷  plist [ p5 , p3 ] ∷ 
          plist [ p0 , p4 ] ∷ plist [ p1 , p4 ] ∷  plist [ p3 , p4 ] ∷ plist [ p3 , p4 ] ∷ plist [ p4 , p4 ] ∷  plist [ p5 , p4 ] ∷ 
          plist [ p0 , p5 ] ∷ plist [ p1 , p5 ] ∷  plist [ p3 , p5 ] ∷ plist [ p3 , p5 ] ∷ plist [ p4 , p4 ] ∷  plist [ p5 , p5 ] ∷ 
          []

   open _=p=_
   
   stage1 :  (x : Permutation 3 3) →  Set (Level.suc Level.zero)
   stage1 x = Commutator (λ x₂ → Lift (Level.suc Level.zero) ⊤)  x 

   open import logic

   pFL : ( g : Permutation 3 3) → { x : FL 3 } →  perm→FL g ≡ x → g =p=  FL→perm x
   pFL g {x} refl = ptrans (psym (FL←iso g)) ( FL-inject refl ) 

   open ≡-Reasoning

--   st01 : ( x y : Permutation 3 3) →   x =p= p3 →  y =p= p3 → x ∘ₚ  y =p= p4 
--   st01 x y s t = record { peq = λ q → ( begin
--         (x ∘ₚ y) ⟨$⟩ʳ q
--       ≡⟨ peq ( presp s t ) q ⟩
--          ( p3  ∘ₚ p3 ) ⟨$⟩ʳ q
--       ≡⟨ peq  p33=4 q  ⟩
--         p4 ⟨$⟩ʳ q
--       ∎ ) }

   st00 = perm→FL [ FL→perm ((suc zero) :: (suc zero :: (zero :: f0 ))) , FL→perm  ((suc (suc zero)) :: (suc zero) :: (zero :: f0))  ]

   
   stage12  :  (x : Permutation 3 3) → stage1 x →  ( x =p= pid ) ∨ ( x =p= p3 ) ∨ ( x =p= p4 )
   stage12 = {!!}


   solved1 :  (x : Permutation 3 3) →  Commutator (λ x₁ → Commutator (λ x₂ → Lift (Level.suc Level.zero) ⊤) x₁) x → x =p= pid
   solved1 _ uni = prefl
   solved1 x (gen {f} {g} d d₁) with solved1 f d | solved1 g d₁
   ... | record { peq = f=e } | record { peq = g=e } = record { peq = λ q → genlem q } where
      genlem : ( q : Fin 3 ) → g ⟨$⟩ʳ ( f ⟨$⟩ʳ q ) ≡ q
      genlem q = begin
             g ⟨$⟩ʳ ( f ⟨$⟩ʳ q )
          ≡⟨ g=e ( f ⟨$⟩ʳ q ) ⟩
             f ⟨$⟩ʳ q 
          ≡⟨ f=e q ⟩
             q
          ∎ 
   solved1 x (ccong {f} {g} (record {peq = f=g}) d) with solved1 f d
   ... | record { peq = f=e }  =  record  { peq = λ q → cc q } where
      cc : ( q : Fin 3 ) → x ⟨$⟩ʳ q ≡ q
      cc q = begin
             x ⟨$⟩ʳ q
          ≡⟨ sym (f=g q) ⟩
             f ⟨$⟩ʳ q
          ≡⟨ f=e q ⟩
             q
          ∎ 
   solved1 _ (comm {g} {h} x y) = {!!}