Mercurial > hg > Members > kono > Proof > galois
annotate Putil.agda @ 72:09fa2ab75703
add utilties
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 24 Aug 2020 23:06:10 +0900 |
| parents | 3825082ebdbd |
| children | 69ed81f8e212 |
| rev | line source |
|---|---|
| 48 | 1 module Putil where |
| 0 | 2 |
| 3 open import Level hiding ( suc ; zero ) | |
| 4 open import Algebra | |
| 5 open import Algebra.Structures | |
| 37 | 6 open import Data.Fin hiding ( _<_ ; _≤_ ; _-_ ; _+_ ) |
| 41 | 7 open import Data.Fin.Properties hiding ( <-trans ; ≤-trans ) renaming ( <-cmp to <-fcmp ) |
| 0 | 8 open import Data.Fin.Permutation |
| 9 open import Function hiding (id ; flip) | |
| 10 open import Function.Inverse as Inverse using (_↔_; Inverse; _InverseOf_) | |
| 11 open import Function.LeftInverse using ( _LeftInverseOf_ ) | |
| 12 open import Function.Equality using (Π) | |
| 17 | 13 open import Data.Nat -- using (ℕ; suc; zero; s≤s ; z≤n ) |
| 14 open import Data.Nat.Properties -- using (<-trans) | |
| 16 | 15 open import Relation.Binary.PropositionalEquality |
| 46 | 16 open import Data.List using (List; []; _∷_ ; length ; _++_ ; head ) renaming (reverse to rev ) |
| 16 | 17 open import nat |
| 0 | 18 |
| 48 | 19 open import Symmetric |
| 0 | 20 |
| 21 | |
| 16 | 22 open import Relation.Nullary |
| 23 open import Data.Empty | |
| 17 | 24 open import Relation.Binary.Core |
| 25 open import fin | |
| 16 | 26 |
| 38 | 27 -- An inductive construction of permutation |
| 34 | 28 |
| 59 | 29 -- Todo |
| 30 -- | |
| 31 -- complete perm→FL | |
| 32 -- describe property of pprep and pswap | |
| 33 -- describe property of pins ( move 0 to any position) | |
| 34 -- describe property of shrink ( remove one column ) | |
| 35 -- prove FL→iso | |
| 36 -- prove FL←iso | |
| 37 -- prove FL enumerate all permutations | |
| 38 | |
| 48 | 39 -- we already have refl and trans in the Symmetric Group |
| 41 | 40 |
| 34 | 41 pprep : {n : ℕ } → Permutation n n → Permutation (suc n) (suc n) |
| 42 pprep {n} perm = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where | |
| 33 | 43 p→ : Fin (suc n) → Fin (suc n) |
| 34 | 44 p→ zero = zero |
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45 p→ (suc x) = suc ( perm ⟨$⟩ʳ x) |
| 33 | 46 |
| 34 | 47 p← : Fin (suc n) → Fin (suc n) |
| 48 p← zero = zero | |
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49 p← (suc x) = suc ( perm ⟨$⟩ˡ x) |
| 34 | 50 |
| 51 piso← : (x : Fin (suc n)) → p→ ( p← x ) ≡ x | |
| 52 piso← zero = refl | |
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53 piso← (suc x) = cong (λ k → suc k ) (inverseʳ perm) |
| 33 | 54 |
| 34 | 55 piso→ : (x : Fin (suc n)) → p← ( p→ x ) ≡ x |
| 56 piso→ zero = refl | |
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57 piso→ (suc x) = cong (λ k → suc k ) (inverseˡ perm) |
| 33 | 58 |
| 34 | 59 pswap : {n : ℕ } → Permutation n n → Permutation (suc (suc n)) (suc (suc n )) |
| 60 pswap {n} perm = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where | |
| 61 p→ : Fin (suc (suc n)) → Fin (suc (suc n)) | |
| 62 p→ zero = suc zero | |
| 63 p→ (suc zero) = zero | |
| 62 | 64 p→ (suc (suc x)) = suc ( suc ( perm ⟨$⟩ʳ x) ) |
| 18 | 65 |
| 34 | 66 p← : Fin (suc (suc n)) → Fin (suc (suc n)) |
| 67 p← zero = suc zero | |
| 68 p← (suc zero) = zero | |
| 62 | 69 p← (suc (suc x)) = suc ( suc ( perm ⟨$⟩ˡ x) ) |
| 34 | 70 |
| 71 piso← : (x : Fin (suc (suc n)) ) → p→ ( p← x ) ≡ x | |
| 72 piso← zero = refl | |
| 73 piso← (suc zero) = refl | |
| 62 | 74 piso← (suc (suc x)) = cong (λ k → suc (suc k) ) (inverseʳ perm) |
| 16 | 75 |
| 34 | 76 piso→ : (x : Fin (suc (suc n)) ) → p← ( p→ x ) ≡ x |
| 77 piso→ zero = refl | |
| 78 piso→ (suc zero) = refl | |
| 62 | 79 piso→ (suc (suc x)) = cong (λ k → suc (suc k) ) (inverseˡ perm) |
| 34 | 80 |
| 81 -- enumeration | |
| 82 | |
| 44 | 83 psawpn : {n : ℕ} → 1 < n → Permutation n n |
| 84 psawpn {suc zero} (s≤s ()) | |
| 85 psawpn {suc n} (s≤s (s≤s x)) = pswap pid | |
| 34 | 86 |
| 35 | 87 pfill : { n m : ℕ } → m ≤ n → Permutation m m → Permutation n n |
| 88 pfill {n} {m} m≤n perm = pfill1 (n - m) (n-m<n n m ) (subst (λ k → Permutation k k ) (n-n-m=m m≤n ) perm) where | |
| 89 pfill1 : (i : ℕ ) → i ≤ n → Permutation (n - i) (n - i) → Permutation n n | |
| 90 pfill1 0 _ perm = perm | |
| 91 pfill1 (suc i) i<n perm = pfill1 i (≤to< i<n) (subst (λ k → Permutation k k ) (si-sn=i-n i<n ) ( pprep perm ) ) | |
| 34 | 92 |
| 48 | 93 -- |
| 94 -- psawpim (inseert swap at position m ) | |
| 95 -- | |
| 45 | 96 psawpim : {n m : ℕ} → suc (suc m) ≤ n → Permutation n n |
| 97 psawpim {n} {m} m≤n = pfill m≤n ( psawpn (s≤s (s≤s z≤n)) ) | |
| 98 | |
| 99 n≤ : (i : ℕ ) → {j : ℕ } → i ≤ i + j | |
| 100 n≤ (zero) {j} = z≤n | |
| 101 n≤ (suc i) {j} = s≤s ( n≤ i ) | |
| 102 | |
| 103 lem0 : {n : ℕ } → n ≤ n | |
| 104 lem0 {zero} = z≤n | |
| 105 lem0 {suc n} = s≤s lem0 | |
| 106 | |
| 107 lem00 : {n m : ℕ } → n ≡ m → n ≤ m | |
| 108 lem00 refl = lem0 | |
| 44 | 109 |
| 110 -- pconcat : {n m : ℕ } → Permutation m m → Permutation n n → Permutation (m + n) (m + n) | |
| 111 -- pconcat {n} {m} p q = pfill {n + m} {m} ? p ∘ₚ ? | |
| 112 | |
| 113 -- inductivley enmumerate permutations | |
| 114 -- from n-1 length create n length inserting new element at position m | |
| 115 | |
| 48 | 116 -- 0 ∷ 1 ∷ 2 ∷ 3 ∷ [] |
| 117 -- 1 ∷ 0 ∷ 2 ∷ 3 ∷ [] plist ( pins {3} (n≤ 1) ) | |
| 118 -- 1 ∷ 2 ∷ 0 ∷ 3 ∷ [] | |
| 119 -- 1 ∷ 2 ∷ 3 ∷ 0 ∷ [] | |
| 45 | 120 |
| 48 | 121 pins : {n m : ℕ} → m ≤ n → Permutation (suc n) (suc n) |
| 122 pins {_} {zero} _ = pid | |
| 123 pins {suc _} {suc zero} _ = pswap pid | |
| 124 pins {suc (suc n)} {suc m} (s≤s m<n) = pins1 (suc m) (suc (suc n)) lem0 where | |
| 125 pins1 : (i j : ℕ ) → j ≤ suc (suc n) → Permutation (suc (suc (suc n ))) (suc (suc (suc n))) | |
| 126 pins1 _ zero _ = pid | |
| 127 pins1 zero _ _ = pid | |
| 128 pins1 (suc i) (suc j) (s≤s si≤n) = psawpim {suc (suc (suc n))} {j} (s≤s (s≤s si≤n)) ∘ₚ pins1 i j (≤-trans si≤n refl-≤s ) | |
| 37 | 129 |
| 130 plist : {n : ℕ} → Permutation n n → List ℕ | |
| 131 plist {0} perm = [] | |
| 44 | 132 plist {suc j} perm = rev (plist1 j a<sa) where |
| 37 | 133 n = suc j |
| 134 plist1 : (i : ℕ ) → i < n → List ℕ | |
| 40 | 135 plist1 zero _ = toℕ ( perm ⟨$⟩ˡ (fromℕ≤ {zero} (s≤s z≤n))) ∷ [] |
| 136 plist1 (suc i) (s≤s lt) = toℕ ( perm ⟨$⟩ˡ (fromℕ≤ (s≤s lt))) ∷ plist1 i (<-trans lt a<sa) | |
| 37 | 137 |
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138 data FL : (n : ℕ )→ Set where |
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139 f0 : FL 0 |
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140 _::_ : { n : ℕ } → Fin (suc n ) → FL n → FL (suc n) |
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141 |
| 50 | 142 open import logic |
| 143 | |
| 56 | 144 -- 0 ∷ 1 ∷ 2 ∷ 3 ∷ [] → 0 ∷ 1 ∷ 2 ∷ [] |
| 61 | 145 shrink : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → perm ⟨$⟩ˡ (# 0) ≡ # 0 → Permutation n n |
| 146 shrink {n} perm p0=0 = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where | |
| 147 shlem→ : (x : Fin (suc n) ) → perm ⟨$⟩ˡ x ≡ zero → x ≡ zero | |
| 148 shlem→ x px=0 = begin | |
| 149 x ≡⟨ sym ( inverseʳ perm ) ⟩ | |
| 150 perm ⟨$⟩ʳ ( perm ⟨$⟩ˡ x) ≡⟨ cong (λ k → perm ⟨$⟩ʳ k ) px=0 ⟩ | |
| 151 perm ⟨$⟩ʳ zero ≡⟨ cong (λ k → perm ⟨$⟩ʳ k ) (sym p0=0) ⟩ | |
| 152 perm ⟨$⟩ʳ ( perm ⟨$⟩ˡ zero) ≡⟨ inverseʳ perm ⟩ | |
| 153 zero | |
| 154 ∎ where open ≡-Reasoning | |
| 54 | 155 |
| 61 | 156 shlem← : (x : Fin (suc n)) → perm ⟨$⟩ʳ x ≡ zero → x ≡ zero |
| 157 shlem← x px=0 = begin | |
| 158 x ≡⟨ sym (inverseˡ perm ) ⟩ | |
| 159 perm ⟨$⟩ˡ ( perm ⟨$⟩ʳ x ) ≡⟨ cong (λ k → perm ⟨$⟩ˡ k ) px=0 ⟩ | |
| 160 perm ⟨$⟩ˡ zero ≡⟨ p0=0 ⟩ | |
| 161 zero | |
| 162 ∎ where open ≡-Reasoning | |
| 54 | 163 |
| 61 | 164 sh2 : {x : Fin n} → ¬ perm ⟨$⟩ˡ (suc x) ≡ zero |
| 165 sh2 {x} eq with shlem→ (suc x) eq | |
| 166 sh2 {x} eq | () | |
| 57 | 167 |
| 61 | 168 p→ : Fin n → Fin n |
| 169 p→ x with perm ⟨$⟩ˡ (suc x) | inspect (_⟨$⟩ˡ_ perm ) (suc x) | |
| 170 p→ x | zero | record { eq = e } = ⊥-elim ( sh2 {x} e ) | |
| 171 p→ x | suc t | _ = t | |
| 50 | 172 |
| 61 | 173 sh1 : {x : Fin n} → ¬ perm ⟨$⟩ʳ (suc x) ≡ zero |
| 174 sh1 {x} eq with shlem← (suc x) eq | |
| 175 sh1 {x} eq | () | |
| 50 | 176 |
| 177 p← : Fin n → Fin n | |
| 61 | 178 p← x with perm ⟨$⟩ʳ (suc x) | inspect (_⟨$⟩ʳ_ perm ) (suc x) |
| 179 p← x | zero | record { eq = e } = ⊥-elim ( sh1 {x} e ) | |
| 180 p← x | suc t | _ = t | |
| 50 | 181 |
| 182 piso← : (x : Fin n ) → p→ ( p← x ) ≡ x | |
| 61 | 183 piso← x with perm ⟨$⟩ʳ (suc x) | inspect (_⟨$⟩ʳ_ perm ) (suc x) |
| 184 piso← x | zero | record { eq = e } = ⊥-elim ( sh1 {x} e ) | |
| 185 piso← x | suc t | _ with perm ⟨$⟩ˡ (suc t) | inspect (_⟨$⟩ˡ_ perm ) (suc t) | |
| 186 piso← x | suc t | _ | zero | record { eq = e } = ⊥-elim ( sh2 e ) | |
| 187 piso← x | suc t | record { eq = e0 } | suc t1 | record { eq = e1 } = begin | |
| 188 t1 | |
| 189 ≡⟨ plem0 plem1 ⟩ | |
| 52 | 190 x |
| 61 | 191 ∎ where |
| 192 open ≡-Reasoning | |
| 193 plem0 : suc t1 ≡ suc x → t1 ≡ x | |
| 194 plem0 refl = refl | |
| 195 plem1 : suc t1 ≡ suc x | |
| 196 plem1 = begin | |
| 197 suc t1 | |
| 198 ≡⟨ sym e1 ⟩ | |
| 199 Inverse.from perm Π.⟨$⟩ suc t | |
| 200 ≡⟨ cong (λ k → Inverse.from perm Π.⟨$⟩ k ) (sym e0 ) ⟩ | |
| 201 Inverse.from perm Π.⟨$⟩ ( Inverse.to perm Π.⟨$⟩ suc x ) | |
| 202 ≡⟨ inverseˡ perm ⟩ | |
| 203 suc x | |
| 204 ∎ | |
| 50 | 205 |
| 206 piso→ : (x : Fin n ) → p← ( p→ x ) ≡ x | |
| 61 | 207 piso→ x with perm ⟨$⟩ˡ (suc x) | inspect (_⟨$⟩ˡ_ perm ) (suc x) |
| 208 piso→ x | zero | record { eq = e } = ⊥-elim ( sh2 {x} e ) | |
| 209 piso→ x | suc t | _ with perm ⟨$⟩ʳ (suc t) | inspect (_⟨$⟩ʳ_ perm ) (suc t) | |
| 210 piso→ x | suc t | _ | zero | record { eq = e } = ⊥-elim ( sh1 e ) | |
| 211 piso→ x | suc t | record { eq = e0 } | suc t1 | record { eq = e1 } = begin | |
| 212 t1 | |
| 213 ≡⟨ plem2 plem3 ⟩ | |
| 53 | 214 x |
| 61 | 215 ∎ where |
| 216 open ≡-Reasoning | |
| 217 plem2 : suc t1 ≡ suc x → t1 ≡ x | |
| 218 plem2 refl = refl | |
| 219 plem3 : suc t1 ≡ suc x | |
| 220 plem3 = begin | |
| 221 suc t1 | |
| 222 ≡⟨ sym e1 ⟩ | |
| 223 Inverse.to perm Π.⟨$⟩ suc t | |
| 224 ≡⟨ cong (λ k → Inverse.to perm Π.⟨$⟩ k ) (sym e0 ) ⟩ | |
| 225 Inverse.to perm Π.⟨$⟩ ( Inverse.from perm Π.⟨$⟩ suc x ) | |
| 226 ≡⟨ inverseʳ perm ⟩ | |
| 227 suc x | |
| 228 ∎ | |
| 57 | 229 |
| 230 FL→perm : {n : ℕ } → FL n → Permutation n n | |
| 231 FL→perm f0 = pid | |
| 232 FL→perm (x :: fl) = pprep (FL→perm fl) ∘ₚ pins ( toℕ≤pred[n] x ) | |
| 233 | |
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234 t40 = (# 2) :: ( (# 1) :: (( # 0 ) :: f0 )) |
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235 t4 = FL→perm ((# 2) :: t40 ) |
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236 |
| 61 | 237 -- t1 = plist (shrink (pid {3} ∘ₚ (pins (n≤ 1))) refl) |
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238 t2 = plist ((pid {5} ) ∘ₚ transpose (# 2) (# 4)) ∷ plist (pid {5} ∘ₚ reverse ) ∷ [] |
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239 t3 = plist (FL→perm t40) -- ∷ plist (pprep (FL→perm t40)) |
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240 -- ∷ plist ( pprep (FL→perm t40) ∘ₚ pins ( n≤ 0 {3} )) |
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241 -- ∷ plist ( pprep (FL→perm t40 )∘ₚ pins ( n≤ 1 {2} )) |
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242 -- ∷ plist ( pprep (FL→perm t40 )∘ₚ pins ( n≤ 2 {1} )) |
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243 -- ∷ plist ( pprep (FL→perm t40 )∘ₚ pins ( n≤ 3 {0} )) |
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244 ∷ plist ( FL→perm ((# 0) :: t40)) -- (0 ∷ 1 ∷ 2 ∷ []) ∷ |
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245 ∷ plist ( FL→perm ((# 1) :: t40)) -- (0 ∷ 2 ∷ 1 ∷ []) ∷ |
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246 ∷ plist ( FL→perm ((# 2) :: t40)) -- (1 ∷ 0 ∷ 2 ∷ []) ∷ |
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247 ∷ plist ( FL→perm ((# 3) :: t40)) -- (2 ∷ 0 ∷ 1 ∷ []) ∷ |
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248 -- ∷ plist ( FL→perm ((# 3) :: ((# 2) :: ( (# 0) :: (( # 0 ) :: f0 )) ))) -- (1 ∷ 2 ∷ 0 ∷ []) ∷ |
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249 -- ∷ plist ( FL→perm ((# 3) :: ((# 2) :: ( (# 1) :: (( # 0 ) :: f0 )) ))) -- (2 ∷ 1 ∷ 0 ∷ []) ∷ |
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250 -- ∷ plist ( (flip (FL→perm ((# 3) :: ((# 1) :: ( (# 0) :: (( # 0 ) :: f0 )) ))))) |
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251 -- ∷ plist ( (flip (FL→perm ((# 3) :: ((# 1) :: ( (# 0) :: (( # 0 ) :: f0 )) ))) ∘ₚ (FL→perm ((# 3) :: (((# 1) :: ( (# 0) :: (( # 0 ) :: f0 )) )))) )) |
| 57 | 252 ∷ [] |
| 50 | 253 |
| 64 | 254 -- postulate |
| 255 -- p=0 : {n : ℕ } → (perm : Permutation (suc n) (suc n) ) → ((perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) ⟨$⟩ˡ (# 0)) ≡ # 0 | |
| 58 | 256 |
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257 perm→FL : {n : ℕ } → Permutation n n → FL n |
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258 perm→FL {zero} perm = f0 |
| 64 | 259 perm→FL {suc n} perm = (perm ⟨$⟩ʳ (# 0)) :: perm→FL (remove (# 0) perm) |
| 260 -- perm→FL {suc n} perm = (perm ⟨$⟩ʳ (# 0)) :: perm→FL (shrink (perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) (p=0 perm) ) | |
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261 |
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262 -- t5 = plist t4 ∷ plist ( t4 ∘ₚ flip (pins ( n≤ 3 ) )) |
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263 t5 = plist (t4) ∷ plist (flip t4) |
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264 ∷ ( toℕ (t4 ⟨$⟩ˡ fromℕ≤ a<sa) ∷ [] ) |
| 61 | 265 ∷ ( toℕ (t4 ⟨$⟩ʳ (# 0)) ∷ [] ) |
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266 -- ∷ plist ( t4 ∘ₚ flip (pins ( n≤ 1 ) )) |
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267 ∷ plist (remove (# 0) t4 ) |
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268 ∷ plist ( FL→perm t40 ) |
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269 ∷ [] |
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270 |
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271 t6 = perm→FL t4 |
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272 |
| 63 | 273 postulate |
| 274 FL→iso : {n : ℕ } → (fl : FL n ) → perm→FL ( FL→perm fl ) ≡ fl | |
| 275 -- FL→iso f0 = refl | |
| 276 -- FL→iso (x :: fl) = {!!} -- with FL→iso fl | |
| 61 | 277 -- ... | t = {!!} |
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278 |
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279 open _=p=_ |
| 63 | 280 postulate |
| 281 FL←iso : {n : ℕ } → (perm : Permutation n n ) → FL→perm ( perm→FL perm ) =p= perm | |
| 282 -- FL←iso {0} perm = record { peq = λ () } | |
| 283 -- FL←iso {suc n} perm = {!!} | |
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284 |
| 66 | 285 lem2 : {i n : ℕ } → i ≤ n → i ≤ suc n |
| 286 lem2 i≤n = ≤-trans i≤n ( refl-≤s ) | |
| 287 | |
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288 ∀-FL : (n : ℕ ) → List (FL (suc n)) |
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289 ∀-FL x = fls6 x where |
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290 fls4 : ( i n : ℕ ) → (i<n : i ≤ n ) → FL n → List (FL (suc n)) → List (FL (suc n)) |
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291 fls4 zero n i≤n perm x = (zero :: perm ) ∷ x |
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292 fls4 (suc i) n i≤n perm x = fls4 i n (≤-trans refl-≤s i≤n ) perm ((fromℕ≤ (s≤s i≤n) :: perm ) ∷ x) |
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293 fls5 : ( n : ℕ ) → List (FL n) → List (FL (suc n)) → List (FL (suc n)) |
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294 fls5 n [] x = x |
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295 fls5 n (h ∷ x) y = fls5 n x (fls4 n n lem0 h y) |
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296 fls6 : ( n : ℕ ) → List (FL (suc n)) |
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297 fls6 zero = (zero :: f0) ∷ [] |
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298 fls6 (suc n) = fls5 (suc n) (fls6 n) [] |
| 65 | 299 |
| 48 | 300 all-perm : (n : ℕ ) → List (Permutation (suc n) (suc n) ) |
| 301 all-perm n = pls6 n where | |
| 38 | 302 lem1 : {i n : ℕ } → i ≤ n → i < suc n |
| 303 lem1 z≤n = s≤s z≤n | |
| 304 lem1 (s≤s lt) = s≤s (lem1 lt) | |
| 40 | 305 pls4 : ( i n : ℕ ) → (i<n : i ≤ n ) → Permutation n n → List (Permutation (suc n) (suc n)) → List (Permutation (suc n) (suc n)) |
| 48 | 306 pls4 zero n i≤n perm x = (pprep perm ∘ₚ pins i≤n ) ∷ x |
| 307 pls4 (suc i) n i≤n perm x = pls4 i n (≤-trans refl-≤s i≤n ) perm (pprep perm ∘ₚ pins {n} {suc i} i≤n ∷ x) | |
| 40 | 308 pls5 : ( n : ℕ ) → List (Permutation n n) → List (Permutation (suc n) (suc n)) → List (Permutation (suc n) (suc n)) |
| 309 pls5 n [] x = x | |
| 310 pls5 n (h ∷ x) y = pls5 n x (pls4 n n lem0 h y) | |
| 311 pls6 : ( n : ℕ ) → List (Permutation (suc n) (suc n)) | |
| 312 pls6 zero = pid ∷ [] | |
| 48 | 313 pls6 (suc n) = pls5 (suc n) (rev (pls6 n) ) [] -- rev to put id first |
| 314 | |
| 315 pls : (n : ℕ ) → List (List ℕ ) | |
| 316 pls n = Data.List.map plist (all-perm n) where |