open import Level hiding ( suc ; zero )
open import Algebra
module sym3 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

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)

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

   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 ∷ [] 

   open _=p=_
   
   solved1 :  (x : Permutation 3 3) →  Commutator (λ x₁ → Commutator (λ x₂ → Lift (Level.suc Level.zero) ⊤) x₁) x → x =p= pid
   solved1 = {!!}