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1 open import Level hiding ( suc ; zero )
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2 open import Algebra
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3 module sym4 where
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4
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5 open import Symmetric
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6 open import Data.Unit
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7 open import Function.Inverse as Inverse using (_↔_; Inverse; _InverseOf_)
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8 open import Function
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9 open import Data.Nat hiding (_⊔_) -- using (ℕ; suc; zero)
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10 open import Relation.Nullary
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11 open import Data.Empty
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12 open import Data.Product
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13
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14 open import Gutil
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15 open import Putil
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16 open import Solvable using (solvable)
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17 open import Relation.Binary.PropositionalEquality hiding ( [_] )
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18
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19 open import Data.Fin
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20 open import Data.Fin.Permutation hiding (_∘ₚ_)
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21
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22 infixr 200 _∘ₚ_
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23 _∘ₚ_ = Data.Fin.Permutation._∘ₚ_
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24
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25 sym4solvable : solvable (Symmetric 4)
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26 solvable.dervied-length sym4solvable = 3
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27 solvable.end sym4solvable x d = solved1 x {!!} where
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28
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29 open import Data.List using ( List ; [] ; _∷_ )
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30
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31 open Solvable (Symmetric 4)
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32 -- open Group (Symmetric 2) using (_⁻¹)
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33
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34 p0 : FL→perm ((# 0) :: ((# 0) :: ((# 0) :: ((# 0 ) :: f0)))) =p= pid
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35 p0 = {!!} -- record { peq = p00 } where
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36
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37 open _=p=_
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38
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39 -- Klien
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40 --
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41 -- 1 (1,2),(3,4) (1,3),(2,4) (1,4),(2,3)
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42 -- 0 ∷ 1 ∷ 2 ∷ 3 ∷ [] , 1 ∷ 0 ∷ 3 ∷ 2 ∷ [] , 2 ∷ 3 ∷ 0 ∷ 1 ∷ [] , 3 ∷ 2 ∷ 1 ∷ 0 ∷ [] ,
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43
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44
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45 data Klein : (x : Permutation 4 4 ) → Set where
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46 kid : Klein pid
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47 ka : Klein (pswap (pswap pid))
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48 kb : Klein (pid {4} ∘ₚ pins (n≤ 3) ∘ₚ pins (n≤ 3 ) )
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49 kc : Klein (pins (n≤ 3) ∘ₚ pins (n≤ 2) ∘ₚ pswap (pid {2}))
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50
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51 a0 = pid {4}
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52 a1 = pswap (pswap (pid {0}))
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53 a2 = pid {4} ∘ₚ pins (n≤ 3) ∘ₚ pins (n≤ 3 )
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54 a3 = pins (n≤ 3) ∘ₚ pins (n≤ 2) ∘ₚ pswap (pid {2})
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55
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56 -- 1 0
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57 -- 2 1 0
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58 -- 3 2 1 0
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59
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60 k1 : { x : Permutation 4 4 } → Klein x → List ℕ
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61 k1 {x} kid = plist x
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62 k1 {x} ka = plist x
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63 k1 {x} kb = plist x
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64 k1 {x} kc = plist x
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65
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66 k2 = k1 kid ∷ k1 ka ∷ k1 kb ∷ k1 kc ∷ []
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67 k3 = plist (a1 ∘ₚ a2 ) ∷ plist (a1 ∘ₚ a3) ∷ plist (a2 ∘ₚ a1 ) ∷ []
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69 solved1 : (x : Permutation 4 4) → Commutator (λ x₁ → Commutator (λ x₂ → Lift (Level.suc Level.zero) ⊤) x₁) x → x =p= pid
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70 solved1 = {!!}
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