open import Level hiding ( suc ; zero )
open import Algebra
module sym4 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_

sym4solvable : solvable (Symmetric 4)
solvable.dervied-length sym4solvable = 3
solvable.end sym4solvable x d = solved1 x {!!} where

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

   open Solvable (Symmetric 4)
   -- open Group (Symmetric 2) using (_⁻¹)

   open _=p=_

   -- Klien
   --
   --  1                     (1,2),(3,4)           (1,3),(2,4)           (1,4),(2,3)
   --  0 ∷ 1 ∷ 2 ∷ 3 ∷ [] ,  1 ∷ 0 ∷ 3 ∷ 2 ∷ [] ,  2 ∷ 3 ∷ 0 ∷ 1 ∷ [] ,  3 ∷ 2 ∷ 1 ∷ 0 ∷ [] ,  


   data Klein : (x : Permutation 4 4 ) → Set where
       kid : Klein pid
       ka  : Klein (pswap (pswap pid))
       kb  : Klein (pid {4} ∘ₚ pins (n≤ 3) ∘ₚ pins (n≤ 3 ) )
       kc  : Klein (pins (n≤ 3)  ∘ₚ  pins (n≤ 2) ∘ₚ pswap (pid {2}))

   a0 =  pid {4}
   a1 =  pswap (pswap (pid {0}))
   a2 =  pid {4} ∘ₚ pins (n≤ 3) ∘ₚ pins (n≤ 3 ) 
   a3 =  pins (n≤ 3)  ∘ₚ  pins (n≤ 2) ∘ₚ pswap (pid {2})

   --   1 0  
   --   2 1 0 
   --   3 2 1 0

   k1 : { x :  Permutation 4 4 } → Klein x → List ℕ
   k1 {x} kid = plist x
   k1 {x} ka = plist x
   k1 {x} kb = plist x
   k1 {x} kc = plist x

   k2 = k1 kid ∷ k1 ka ∷ k1 kb ∷ k1 kc ∷ []
   k3 = plist  (a1  ∘ₚ a2 ) ∷ plist (a1 ∘ₚ a3)  ∷ plist (a2 ∘ₚ a1 ) ∷  []

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

   p0 =  FL→perm ((# 0) :: ((# 0) :: ((# 0) :: ((# 0 ) :: f0))))
   p1 =  FL→perm ((# 0) :: ((# 0) :: ((# 1) :: ((# 0 ) :: f0))))
   p2 =  FL→perm ((# 0) :: ((# 1) :: ((# 0) :: ((# 0 ) :: f0))))
   p3 =  FL→perm ((# 0) :: ((# 1) :: ((# 1) :: ((# 0 ) :: f0))))
   p4 =  FL→perm ((# 0) :: ((# 2) :: ((# 0) :: ((# 0 ) :: f0))))
   p5 =  FL→perm ((# 0) :: ((# 2) :: ((# 1) :: ((# 0 ) :: f0))))
   p6 =  FL→perm ((# 1) :: ((# 0) :: ((# 0) :: ((# 0 ) :: f0))))
   p7 =  FL→perm ((# 1) :: ((# 0) :: ((# 1) :: ((# 0 ) :: f0))))
   p8 =  FL→perm ((# 1) :: ((# 1) :: ((# 0) :: ((# 0 ) :: f0))))
   p9 =  FL→perm ((# 1) :: ((# 1) :: ((# 1) :: ((# 0 ) :: f0))))
   pa =  FL→perm ((# 1) :: ((# 2) :: ((# 0) :: ((# 0 ) :: f0))))
   pb =  FL→perm ((# 1) :: ((# 2) :: ((# 1) :: ((# 0 ) :: f0))))
   t0  =  plist p0 ∷ plist p1 ∷  plist p2 ∷ plist p3 ∷ plist p4 ∷  plist p5 ∷ []

   
   solved1 :  (x : Permutation 4 4) →  Commutator (λ x₁ → Commutator (λ x₂ → Lift (Level.suc Level.zero) ⊤) x₁) x → x =p= pid
   solved1 = {!!}