Mercurial > hg > Members > kono > Proof > galois
annotate Putil.agda @ 105:e435dbe2e7a6
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 01 Sep 2020 17:45:33 +0900 |
| parents | 2d0738a38ac9 |
| children | 02f54eab9205 |
| rev | line source |
|---|---|
| 90 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 48 | 2 module Putil where |
| 0 | 3 |
| 4 open import Level hiding ( suc ; zero ) | |
| 5 open import Algebra | |
| 6 open import Algebra.Structures | |
| 37 | 7 open import Data.Fin hiding ( _<_ ; _≤_ ; _-_ ; _+_ ) |
| 90 | 8 open import Data.Fin.Properties hiding ( <-trans ; ≤-trans ; ≤-irrelevant ) renaming ( <-cmp to <-fcmp ) |
| 0 | 9 open import Data.Fin.Permutation |
| 10 open import Function hiding (id ; flip) | |
| 11 open import Function.Inverse as Inverse using (_↔_; Inverse; _InverseOf_) | |
| 12 open import Function.LeftInverse using ( _LeftInverseOf_ ) | |
| 13 open import Function.Equality using (Π) | |
| 17 | 14 open import Data.Nat -- using (ℕ; suc; zero; s≤s ; z≤n ) |
| 15 open import Data.Nat.Properties -- using (<-trans) | |
| 16 | 16 open import Relation.Binary.PropositionalEquality |
| 80 | 17 open import Data.List using (List; []; _∷_ ; length ; _++_ ; head ; tail ) renaming (reverse to rev ) |
| 16 | 18 open import nat |
| 0 | 19 |
| 48 | 20 open import Symmetric |
| 0 | 21 |
| 22 | |
| 16 | 23 open import Relation.Nullary |
| 24 open import Data.Empty | |
| 17 | 25 open import Relation.Binary.Core |
| 80 | 26 open import Relation.Binary.Definitions |
| 17 | 27 open import fin |
| 16 | 28 |
| 38 | 29 -- An inductive construction of permutation |
| 34 | 30 |
| 59 | 31 -- Todo |
| 32 -- | |
| 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 | |
| 48 | 38 -- we already have refl and trans in the Symmetric Group |
| 41 | 39 |
| 34 | 40 pprep : {n : ℕ } → Permutation n n → Permutation (suc n) (suc n) |
| 41 pprep {n} perm = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where | |
| 33 | 42 p→ : Fin (suc n) → Fin (suc n) |
| 34 | 43 p→ zero = zero |
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48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
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44 p→ (suc x) = suc ( perm ⟨$⟩ʳ x) |
| 33 | 45 |
| 34 | 46 p← : Fin (suc n) → Fin (suc n) |
| 47 p← zero = zero | |
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48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
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48 p← (suc x) = suc ( perm ⟨$⟩ˡ x) |
| 34 | 49 |
| 50 piso← : (x : Fin (suc n)) → p→ ( p← x ) ≡ x | |
| 51 piso← zero = refl | |
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60
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
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52 piso← (suc x) = cong (λ k → suc k ) (inverseʳ perm) |
| 33 | 53 |
| 34 | 54 piso→ : (x : Fin (suc n)) → p← ( p→ x ) ≡ x |
| 55 piso→ zero = refl | |
|
60
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
56 piso→ (suc x) = cong (λ k → suc k ) (inverseˡ perm) |
| 33 | 57 |
| 34 | 58 pswap : {n : ℕ } → Permutation n n → Permutation (suc (suc n)) (suc (suc n )) |
| 59 pswap {n} perm = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where | |
| 60 p→ : Fin (suc (suc n)) → Fin (suc (suc n)) | |
| 61 p→ zero = suc zero | |
| 62 p→ (suc zero) = zero | |
| 62 | 63 p→ (suc (suc x)) = suc ( suc ( perm ⟨$⟩ʳ x) ) |
| 18 | 64 |
| 34 | 65 p← : Fin (suc (suc n)) → Fin (suc (suc n)) |
| 66 p← zero = suc zero | |
| 67 p← (suc zero) = zero | |
| 62 | 68 p← (suc (suc x)) = suc ( suc ( perm ⟨$⟩ˡ x) ) |
| 34 | 69 |
| 70 piso← : (x : Fin (suc (suc n)) ) → p→ ( p← x ) ≡ x | |
| 71 piso← zero = refl | |
| 72 piso← (suc zero) = refl | |
| 62 | 73 piso← (suc (suc x)) = cong (λ k → suc (suc k) ) (inverseʳ perm) |
| 16 | 74 |
| 34 | 75 piso→ : (x : Fin (suc (suc n)) ) → p← ( p→ x ) ≡ x |
| 76 piso→ zero = refl | |
| 77 piso→ (suc zero) = refl | |
| 62 | 78 piso→ (suc (suc x)) = cong (λ k → suc (suc k) ) (inverseˡ perm) |
| 34 | 79 |
| 80 -- enumeration | |
| 81 | |
| 44 | 82 psawpn : {n : ℕ} → 1 < n → Permutation n n |
| 83 psawpn {suc zero} (s≤s ()) | |
| 84 psawpn {suc n} (s≤s (s≤s x)) = pswap pid | |
| 34 | 85 |
| 35 | 86 pfill : { n m : ℕ } → m ≤ n → Permutation m m → Permutation n n |
| 87 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 | |
| 88 pfill1 : (i : ℕ ) → i ≤ n → Permutation (n - i) (n - i) → Permutation n n | |
| 89 pfill1 0 _ perm = perm | |
| 90 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 | 91 |
| 48 | 92 -- |
| 93 -- psawpim (inseert swap at position m ) | |
| 94 -- | |
| 45 | 95 psawpim : {n m : ℕ} → suc (suc m) ≤ n → Permutation n n |
| 96 psawpim {n} {m} m≤n = pfill m≤n ( psawpn (s≤s (s≤s z≤n)) ) | |
| 97 | |
| 98 n≤ : (i : ℕ ) → {j : ℕ } → i ≤ i + j | |
| 99 n≤ (zero) {j} = z≤n | |
| 100 n≤ (suc i) {j} = s≤s ( n≤ i ) | |
| 101 | |
| 102 lem0 : {n : ℕ } → n ≤ n | |
| 103 lem0 {zero} = z≤n | |
| 104 lem0 {suc n} = s≤s lem0 | |
| 105 | |
| 106 lem00 : {n m : ℕ } → n ≡ m → n ≤ m | |
| 107 lem00 refl = lem0 | |
| 44 | 108 |
| 80 | 109 plist1 : {n : ℕ} → Permutation (suc n) (suc n) → (i : ℕ ) → i < suc n → List ℕ |
| 110 plist1 {n} perm zero _ = toℕ ( perm ⟨$⟩ˡ (fromℕ< {zero} (s≤s z≤n))) ∷ [] | |
| 111 plist1 {n} perm (suc i) (s≤s lt) = toℕ ( perm ⟨$⟩ˡ (fromℕ< (s≤s lt))) ∷ plist1 perm i (<-trans lt a<sa) | |
| 112 | |
| 37 | 113 plist : {n : ℕ} → Permutation n n → List ℕ |
| 114 plist {0} perm = [] | |
| 80 | 115 plist {suc n} perm = rev (plist1 perm n a<sa) |
| 116 | |
| 89 | 117 -- pconcat : {n m : ℕ } → Permutation m m → Permutation n n → Permutation (m + n) (m + n) |
| 118 -- pconcat {n} {m} p q = pfill {n + m} {m} ? p ∘ₚ ? | |
| 119 | |
| 120 -- inductivley enmumerate permutations | |
| 121 -- from n-1 length create n length inserting new element at position m | |
| 122 | |
| 123 -- 0 ∷ 1 ∷ 2 ∷ 3 ∷ [] -- 0 ∷ 1 ∷ 2 ∷ 3 ∷ [] | |
| 124 -- 1 ∷ 0 ∷ 2 ∷ 3 ∷ [] plist ( pins {3} (n≤ 1) ) 1 ∷ 0 ∷ 2 ∷ 3 ∷ [] | |
| 125 -- 1 ∷ 2 ∷ 0 ∷ 3 ∷ [] plist ( pins {3} (n≤ 2) ) 2 ∷ 0 ∷ 1 ∷ 3 ∷ [] | |
| 126 -- 1 ∷ 2 ∷ 3 ∷ 0 ∷ [] plist ( pins {3} (n≤ 3) ) 3 ∷ 0 ∷ 1 ∷ 2 ∷ [] | |
| 94 | 127 -- pins : {n m : ℕ} → m ≤ n → Permutation (suc n) (suc n) |
| 128 -- pins {_} {zero} _ = pid | |
| 129 -- pins {suc _} {suc zero} _ = pswap pid | |
| 130 -- pins {suc (suc n)} {suc m} (s≤s m<n) = pins1 (suc m) (suc (suc n)) lem0 where | |
| 131 -- pins1 : (i j : ℕ ) → j ≤ suc (suc n) → Permutation (suc (suc (suc n ))) (suc (suc (suc n))) | |
| 132 -- pins1 _ zero _ = pid | |
| 133 -- pins1 zero _ _ = pid | |
| 134 -- 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 a≤sa ) | |
| 89 | 135 |
| 90 | 136 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
| 91 | 137 open ≡-Reasoning |
| 90 | 138 |
| 94 | 139 pins : {n m : ℕ} → m ≤ n → Permutation (suc n) (suc n) |
| 140 pins {_} {zero} _ = pid | |
| 141 pins {suc n} {suc m} (s≤s m≤n) = permutation p← p→ record { left-inverse-of = piso← ; right-inverse-of = piso→ } where | |
| 89 | 142 |
| 143 next : Fin (suc (suc n)) → Fin (suc (suc n)) | |
| 144 next zero = suc zero | |
| 91 | 145 next (suc x) = fromℕ< (≤-trans (fin<n {_} {x} ) a≤sa ) |
| 89 | 146 |
| 147 p→ : Fin (suc (suc n)) → Fin (suc (suc n)) | |
| 148 p→ x with <-cmp (toℕ x) (suc m) | |
| 90 | 149 ... | tri< a ¬b ¬c = fromℕ< (≤-trans (s≤s a) (s≤s (s≤s m≤n) )) |
| 89 | 150 ... | tri≈ ¬a b ¬c = zero |
| 151 ... | tri> ¬a ¬b c = x | |
| 152 | |
| 153 p← : Fin (suc (suc n)) → Fin (suc (suc n)) | |
| 154 p← zero = fromℕ< (s≤s (s≤s m≤n)) | |
| 155 p← (suc x) with <-cmp (toℕ x) (suc m) | |
| 91 | 156 ... | tri< a ¬b ¬c = fromℕ< (≤-trans (fin<n {_} {x}) a≤sa ) |
| 89 | 157 ... | tri≈ ¬a b ¬c = suc x |
| 158 ... | tri> ¬a ¬b c = suc x | |
| 90 | 159 |
| 160 mm : toℕ (fromℕ< {suc m} {suc (suc n)} (s≤s (s≤s m≤n))) ≡ suc m | |
| 161 mm = toℕ-fromℕ< (s≤s (s≤s m≤n)) | |
| 162 | |
| 91 | 163 mma : (x : Fin (suc n) ) → suc (toℕ x) ≤ suc m → toℕ ( fromℕ< (≤-trans (fin<n {_} {x}) a≤sa ) ) ≤ m |
| 164 mma x (s≤s x<sm) = subst (λ k → k ≤ m) (sym (toℕ-fromℕ< (≤-trans fin<n a≤sa ) )) x<sm | |
| 89 | 165 |
| 91 | 166 p3 : (x : Fin (suc n) ) → toℕ (fromℕ< (≤-trans (fin<n {_} {suc x} ) (s≤s a≤sa))) ≡ suc (toℕ x) |
| 167 p3 x = begin | |
| 168 toℕ (fromℕ< (≤-trans (fin<n {_} {suc x} ) (s≤s a≤sa))) | |
| 169 ≡⟨ toℕ-fromℕ< ( s≤s ( ≤-trans fin<n a≤sa ) ) ⟩ | |
| 170 suc (toℕ x) | |
| 171 ∎ | |
| 92 | 172 |
| 173 piso→ : (x : Fin (suc (suc n)) ) → p← ( p→ x ) ≡ x | |
| 174 piso→ zero with <-cmp (toℕ (fromℕ< (≤-trans (s≤s z≤n) (s≤s (s≤s m≤n) )))) (suc m) | |
| 175 ... | tri< a ¬b ¬c = refl | |
| 176 piso→ (suc x) with <-cmp (toℕ (suc x)) (suc m) | |
| 94 | 177 ... | tri≈ ¬a refl ¬c = p13 where |
| 178 p13 : fromℕ< (s≤s (s≤s m≤n)) ≡ suc x | |
| 179 p13 = cong (λ k → suc k ) (fromℕ<-toℕ _ (s≤s m≤n) ) | |
| 95 | 180 ... | tri> ¬a ¬b c = p16 (suc x) refl where |
| 181 p16 : (y : Fin (suc (suc n))) → y ≡ suc x → p← y ≡ suc x | |
| 182 p16 zero eq = ⊥-elim ( nat-≡< (cong (λ k → suc (toℕ k) ) eq) (s≤s (s≤s (z≤n)))) | |
| 183 p16 (suc y) eq with <-cmp (toℕ y) (suc m) -- suc (suc m) < toℕ (suc x) | |
| 96 | 184 ... | tri< a ¬b ¬c = ⊥-elim ( nat-≡< refl ( ≤-trans c (subst (λ k → k < suc m) p17 a )) ) where |
| 185 -- x = suc m case, c : suc (suc m) ≤ suc (toℕ x), a : suc (toℕ y) ≤ suc m, suc y ≡ suc x | |
| 186 p17 : toℕ y ≡ toℕ x | |
| 187 p17 with <-cmp (toℕ y) (toℕ x) | cong toℕ eq | |
| 188 ... | tri< a ¬b ¬c | seq = ⊥-elim ( nat-≡< seq (s≤s a) ) | |
| 189 ... | tri≈ ¬a b ¬c | seq = b | |
| 190 ... | tri> ¬a ¬b c | seq = ⊥-elim ( nat-≡< (sym seq) (s≤s c)) | |
| 95 | 191 ... | tri≈ ¬a b ¬c = eq |
| 192 ... | tri> ¬a ¬b c₁ = eq | |
| 92 | 193 ... | tri< a ¬b ¬c = p10 (fromℕ< (≤-trans (s≤s a) (s≤s (s≤s m≤n) ))) refl where |
| 194 p10 : (y : Fin (suc (suc n)) ) → y ≡ fromℕ< (≤-trans (s≤s a) (s≤s (s≤s m≤n) )) → p← y ≡ suc x | |
| 195 p10 zero () | |
| 93 | 196 p10 (suc y) eq = p15 where |
| 197 p12 : toℕ y ≡ suc (toℕ x) | |
| 198 p12 = begin | |
| 199 toℕ y | |
| 200 ≡⟨ cong (λ k → Data.Nat.pred (toℕ k)) eq ⟩ | |
| 201 toℕ (fromℕ< (≤-trans a (s≤s m≤n))) | |
| 202 ≡⟨ toℕ-fromℕ< {suc (toℕ x)} {suc n} (≤-trans a (s≤s m≤n)) ⟩ | |
| 203 suc (toℕ x) | |
| 92 | 204 ∎ |
| 93 | 205 p15 : p← (suc y) ≡ suc x |
| 206 p15 with <-cmp (toℕ y) (suc m) -- eq : suc y ≡ suc (suc (fromℕ< (≤-pred (≤-trans a (s≤s m≤n))))) , a : suc x < suc m | |
| 207 ... | tri< a₁ ¬b ¬c = p11 where | |
| 208 p11 : fromℕ< (≤-trans (fin<n {_} {y}) a≤sa ) ≡ suc x | |
| 209 p11 = begin | |
| 210 fromℕ< (≤-trans (fin<n {_} {y}) a≤sa ) | |
| 211 ≡⟨ fromℕ<-irrelevant _ _ p12 _ (s≤s (fin<n {suc n})) ⟩ | |
| 212 suc (fromℕ< (fin<n {suc n} {x} )) | |
| 213 ≡⟨ cong suc (fromℕ<-toℕ x _ ) ⟩ | |
| 214 suc x | |
| 215 ∎ | |
| 216 ... | tri≈ ¬a b ¬c = ⊥-elim ( nat-≡< b (subst (λ k → k < suc m) (sym p12) a )) -- suc x < suc m -> y = suc x → toℕ y < suc m | |
| 217 ... | tri> ¬a ¬b c = ⊥-elim ( nat-<> c (subst (λ k → k < suc m) (sym p12) a )) | |
| 92 | 218 |
| 89 | 219 piso← : (x : Fin (suc (suc n)) ) → p→ ( p← x ) ≡ x |
| 90 | 220 piso← zero with <-cmp (toℕ (fromℕ< (s≤s (s≤s m≤n)))) (suc m) | mm |
| 221 ... | tri< a ¬b ¬c | t = ⊥-elim ( ¬b t ) | |
| 222 ... | tri≈ ¬a b ¬c | t = refl | |
| 223 ... | tri> ¬a ¬b c | t = ⊥-elim ( ¬b t ) | |
| 224 piso← (suc x) with <-cmp (toℕ x) (suc m) | |
| 91 | 225 ... | tri> ¬a ¬b c with <-cmp (toℕ (suc x)) (suc m) |
| 226 ... | tri< a ¬b₁ ¬c = ⊥-elim ( nat-<> a (<-trans c a<sa ) ) | |
| 227 ... | tri≈ ¬a₁ b ¬c = ⊥-elim ( nat-≡< (sym b) (<-trans c a<sa )) | |
| 228 ... | tri> ¬a₁ ¬b₁ c₁ = refl | |
| 229 piso← (suc x) | tri≈ ¬a b ¬c with <-cmp (toℕ (suc x)) (suc m) | |
| 230 ... | tri< a ¬b ¬c₁ = ⊥-elim ( nat-≡< b (<-trans a<sa a) ) | |
| 231 ... | tri≈ ¬a₁ refl ¬c₁ = ⊥-elim ( nat-≡< b a<sa ) | |
| 232 ... | tri> ¬a₁ ¬b c = refl | |
| 233 piso← (suc x) | tri< a ¬b ¬c with <-cmp (toℕ ( fromℕ< (≤-trans (fin<n {_} {x}) a≤sa ) )) (suc m) | |
| 90 | 234 ... | tri≈ ¬a b ¬c₁ = ⊥-elim ( ¬a (s≤s (mma x a))) |
| 235 ... | tri> ¬a ¬b₁ c = ⊥-elim ( ¬a (s≤s (mma x a))) | |
| 236 ... | tri< a₁ ¬b₁ ¬c₁ = p0 where | |
| 237 p2 : suc (suc (toℕ x)) ≤ suc (suc n) | |
| 238 p2 = s≤s (fin<n {suc n} {x}) | |
| 91 | 239 p6 : suc (toℕ (fromℕ< (≤-trans (fin<n {_} {suc x}) (s≤s a≤sa)))) ≤ suc (suc n) |
| 90 | 240 p6 = s≤s (≤-trans a₁ (s≤s m≤n)) |
| 241 p0 : fromℕ< (≤-trans (s≤s a₁) (s≤s (s≤s m≤n) )) ≡ suc x | |
| 242 p0 = begin | |
| 243 fromℕ< (≤-trans (s≤s a₁) (s≤s (s≤s m≤n) )) | |
| 244 ≡⟨⟩ | |
| 245 fromℕ< (s≤s (≤-trans a₁ (s≤s m≤n))) | |
| 91 | 246 ≡⟨ lemma10 (p3 x) {p6} {p2} ⟩ |
| 90 | 247 fromℕ< ( s≤s (fin<n {suc n} {x}) ) |
| 91 | 248 ≡⟨⟩ |
| 90 | 249 suc (fromℕ< (fin<n {suc n} {x} )) |
| 91 | 250 ≡⟨ cong suc (fromℕ<-toℕ x _ ) ⟩ |
| 90 | 251 suc x |
| 91 | 252 ∎ |
| 90 | 253 |
| 94 | 254 t7 = plist (pins {3} (n≤ 3)) ∷ plist (flip ( pins {3} (n≤ 3) )) ∷ plist ( pins {3} (n≤ 3) ∘ₚ flip ( pins {3} (n≤ 3))) ∷ [] |
| 255 -- t8 = {!!} | |
| 89 | 256 |
| 97 | 257 open import logic |
| 258 | |
| 259 open _∧_ | |
| 260 | |
| 261 perm1 : {perm : Permutation 1 1 } {q : Fin 1} → (perm ⟨$⟩ʳ q ≡ # 0) ∧ ((perm ⟨$⟩ˡ q ≡ # 0)) | |
| 262 perm1 {p} {q} = ⟪ perm01 _ _ , perm00 _ _ ⟫ where | |
| 263 perm01 : (x y : Fin 1) → (p ⟨$⟩ʳ x) ≡ y | |
| 264 perm01 x y with p ⟨$⟩ʳ x | |
| 265 perm01 zero zero | zero = refl | |
| 266 perm00 : (x y : Fin 1) → (p ⟨$⟩ˡ x) ≡ y | |
| 267 perm00 x y with p ⟨$⟩ˡ x | |
| 268 perm00 zero zero | zero = refl | |
| 269 | |
| 270 | |
| 271 p=0 : {n : ℕ } → (perm : Permutation (suc n) (suc n) ) → ((perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) ⟨$⟩ˡ (# 0)) ≡ # 0 | |
| 272 p=0 {zero} perm with ((perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) ⟨$⟩ˡ (# 0)) | |
| 273 ... | zero = refl | |
| 274 p=0 {suc n} perm with perm ⟨$⟩ʳ (# 0) | inspect (_⟨$⟩ʳ_ perm ) (# 0)| toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0)) | inspect toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0)) | |
| 275 ... | zero | record { eq = e} | m<n | _ = p001 where | |
| 276 p001 : perm ⟨$⟩ˡ ( pins m<n ⟨$⟩ʳ zero) ≡ zero | |
| 277 p001 = subst (λ k → perm ⟨$⟩ˡ k ≡ zero ) e (inverseˡ perm) | |
| 278 ... | suc t | record { eq = e } | m<n | record { eq = e1 } = p002 where -- m<n : suc (toℕ t) ≤ suc n | |
| 279 p002 : perm ⟨$⟩ˡ ( pins m<n ⟨$⟩ʳ zero) ≡ zero | |
| 280 p002 = p005 zero (toℕ t) refl m<n refl where -- suc (toℕ t) ≤ suc n | |
| 281 p003 : (s : Fin (suc (suc n))) → s ≡ (perm ⟨$⟩ʳ (# 0)) → perm ⟨$⟩ˡ s ≡ # 0 | |
| 282 p003 s eq = subst (λ k → perm ⟨$⟩ˡ k ≡ zero ) (sym eq) (inverseˡ perm) | |
| 283 p005 : (x : Fin (suc (suc n))) → (m : ℕ ) → x ≡ zero → (m≤n : suc m ≤ suc n ) → m ≡ toℕ t → perm ⟨$⟩ˡ ( pins m≤n ⟨$⟩ʳ zero) ≡ zero | |
| 284 p005 zero m eq (s≤s m≤n) meq = p004 where | |
| 285 p004 : perm ⟨$⟩ˡ (fromℕ< (s≤s (s≤s m≤n))) ≡ zero | |
| 286 p004 = p003 (fromℕ< (s≤s (s≤s m≤n))) ( | |
| 287 begin | |
| 288 fromℕ< (s≤s (s≤s m≤n)) | |
| 289 ≡⟨ fromℕ<-irrelevant _ _ (cong suc meq) (s≤s (s≤s m≤n)) (subst (λ k → suc k < suc (suc n)) meq (s≤s (s≤s m≤n)) ) ⟩ | |
| 290 fromℕ< (subst (λ k → suc k < suc (suc n)) meq (s≤s (s≤s m≤n)) ) | |
| 291 ≡⟨ fromℕ<-toℕ {suc (suc n)} (suc t) (subst (λ k → suc k < suc (suc n)) meq (s≤s (s≤s m≤n)) ) ⟩ | |
| 292 suc t | |
| 293 ≡⟨ sym e ⟩ | |
| 294 (perm ⟨$⟩ʳ (# 0)) | |
| 295 ∎ ) | |
| 296 | |
| 101 | 297 px=x : {n : ℕ } → (x : Fin (suc n)) → pins ( toℕ≤pred[n] x ) ⟨$⟩ʳ (# 0) ≡ x |
| 298 px=x {n} zero = refl | |
| 103 | 299 px=x {suc n} (suc x) = p001 where |
| 300 p002 : fromℕ< (s≤s (toℕ≤pred[n] x)) ≡ x | |
| 301 p002 = fromℕ<-toℕ x (s≤s (toℕ≤pred[n] x)) | |
| 302 p001 : (pins (toℕ≤pred[n] (suc x)) ⟨$⟩ʳ (# 0)) ≡ suc x | |
| 303 p001 with <-cmp 0 ((toℕ x)) | |
| 304 ... | tri< a ¬b ¬c = cong suc p002 | |
| 305 ... | tri≈ ¬a b ¬c = cong suc p002 | |
| 97 | 306 |
| 307 -- pp : {n : ℕ } → (perm : Permutation (suc n) (suc n) ) → Fin (suc n) | |
| 308 -- pp perm → (( perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) ⟨$⟩ˡ (# 0)) | |
| 309 | |
| 80 | 310 plist2 : {n : ℕ} → Permutation (suc n) (suc n) → (i : ℕ ) → i < suc n → List ℕ |
| 311 plist2 {n} perm zero _ = toℕ ( perm ⟨$⟩ʳ (fromℕ< {zero} (s≤s z≤n))) ∷ [] | |
| 312 plist2 {n} perm (suc i) (s≤s lt) = toℕ ( perm ⟨$⟩ʳ (fromℕ< (s≤s lt))) ∷ plist2 perm i (<-trans lt a<sa) | |
| 313 | |
| 314 plist0 : {n : ℕ} → Permutation n n → List ℕ | |
| 315 plist0 {0} perm = [] | |
| 316 plist0 {suc n} perm = plist2 perm n a<sa | |
| 317 | |
| 85 | 318 open _=p=_ |
| 319 | |
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320 -- |
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321 -- plist cong |
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322 -- |
| 85 | 323 ←pleq : {n : ℕ} → (x y : Permutation n n ) → x =p= y → plist0 x ≡ plist0 y |
| 324 ←pleq {zero} x y eq = refl | |
| 325 ←pleq {suc n} x y eq = ←pleq1 n a<sa where | |
| 326 ←pleq1 : (i : ℕ ) → (i<sn : i < suc n ) → plist2 x i i<sn ≡ plist2 y i i<sn | |
| 327 ←pleq1 zero _ = cong ( λ k → toℕ k ∷ [] ) ( peq eq (fromℕ< {zero} (s≤s z≤n))) | |
| 328 ←pleq1 (suc i) (s≤s lt) = cong₂ ( λ j k → toℕ j ∷ k ) ( peq eq (fromℕ< (s≤s lt))) ( ←pleq1 i (<-trans lt a<sa) ) | |
| 80 | 329 |
| 330 headeq : {A : Set } → {x y : A } → {xt yt : List A } → (x ∷ xt) ≡ (y ∷ yt) → x ≡ y | |
| 331 headeq refl = refl | |
| 332 | |
| 333 taileq : {A : Set } → {x y : A } → {xt yt : List A } → (x ∷ xt) ≡ (y ∷ yt) → xt ≡ yt | |
| 334 taileq refl = refl | |
| 335 | |
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336 -- |
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337 -- plist equalizer |
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338 -- |
| 80 | 339 pleq : {n : ℕ} → (x y : Permutation n n ) → plist0 x ≡ plist0 y → x =p= y |
| 340 pleq {0} x y refl = record { peq = λ q → pleq0 q } where | |
| 341 pleq0 : (q : Fin 0 ) → (x ⟨$⟩ʳ q) ≡ (y ⟨$⟩ʳ q) | |
| 342 pleq0 () | |
| 343 pleq {suc n} x y eq = record { peq = λ q → pleq1 n a<sa eq q fin<n } where | |
| 344 pleq1 : (i : ℕ ) → (i<sn : i < suc n ) → plist2 x i i<sn ≡ plist2 y i i<sn → (q : Fin (suc n)) → toℕ q < suc i → x ⟨$⟩ʳ q ≡ y ⟨$⟩ʳ q | |
| 81 | 345 pleq1 zero i<sn eq q q<i with <-cmp (toℕ q) zero |
| 346 ... | tri< () ¬b ¬c | |
| 347 ... | tri> ¬a ¬b c = ⊥-elim (nat-≤> c q<i ) | |
| 348 ... | tri≈ ¬a b ¬c = begin | |
| 349 x ⟨$⟩ʳ q | |
| 350 ≡⟨ cong ( λ k → x ⟨$⟩ʳ k ) (toℕ-injective b )⟩ | |
| 351 x ⟨$⟩ʳ zero | |
| 352 ≡⟨ toℕ-injective (headeq eq) ⟩ | |
| 353 y ⟨$⟩ʳ zero | |
| 354 ≡⟨ cong ( λ k → y ⟨$⟩ʳ k ) (sym (toℕ-injective b )) ⟩ | |
| 355 y ⟨$⟩ʳ q | |
| 91 | 356 ∎ |
| 80 | 357 pleq1 (suc i) (s≤s i<sn) eq q q<i with <-cmp (toℕ q) (suc i) |
| 358 ... | tri< a ¬b ¬c = pleq1 i (<-trans i<sn a<sa ) (taileq eq) q a | |
| 81 | 359 ... | tri> ¬a ¬b c = ⊥-elim (nat-≤> c q<i ) |
| 80 | 360 ... | tri≈ ¬a b ¬c = begin |
| 361 x ⟨$⟩ʳ q | |
| 362 ≡⟨ cong (λ k → x ⟨$⟩ʳ k) (pleq3 b) ⟩ | |
| 363 x ⟨$⟩ʳ (suc (fromℕ< i<sn)) | |
| 364 ≡⟨ toℕ-injective pleq2 ⟩ | |
| 365 y ⟨$⟩ʳ (suc (fromℕ< i<sn)) | |
| 366 ≡⟨ cong (λ k → y ⟨$⟩ʳ k) (sym (pleq3 b)) ⟩ | |
| 367 y ⟨$⟩ʳ q | |
| 368 ∎ where | |
| 369 pleq3 : toℕ q ≡ suc i → q ≡ suc (fromℕ< i<sn) | |
| 370 pleq3 tq=si = toℕ-injective ( begin | |
| 371 toℕ q | |
| 372 ≡⟨ b ⟩ | |
| 373 suc i | |
| 374 ≡⟨ sym (toℕ-fromℕ< (s≤s i<sn)) ⟩ | |
| 375 toℕ (fromℕ< (s≤s i<sn)) | |
| 376 ≡⟨⟩ | |
| 377 toℕ (suc (fromℕ< i<sn)) | |
| 91 | 378 ∎ ) |
| 80 | 379 pleq2 : toℕ ( x ⟨$⟩ʳ (suc (fromℕ< i<sn)) ) ≡ toℕ ( y ⟨$⟩ʳ (suc (fromℕ< i<sn)) ) |
| 380 pleq2 = headeq eq | |
| 37 | 381 |
| 87 | 382 pprep-cong : {n : ℕ} → {x y : Permutation n n } → x =p= y → pprep x =p= pprep y |
| 383 pprep-cong {n} {x} {y} x=y = record { peq = pprep-cong1 } where | |
| 384 pprep-cong1 : (q : Fin (suc n)) → (pprep x ⟨$⟩ʳ q) ≡ (pprep y ⟨$⟩ʳ q) | |
| 385 pprep-cong1 zero = refl | |
| 386 pprep-cong1 (suc q) = begin | |
| 387 pprep x ⟨$⟩ʳ suc q | |
| 388 ≡⟨⟩ | |
| 389 suc ( x ⟨$⟩ʳ q ) | |
| 390 ≡⟨ cong ( λ k → suc k ) ( peq x=y q ) ⟩ | |
| 391 suc ( y ⟨$⟩ʳ q ) | |
| 392 ≡⟨⟩ | |
| 393 pprep y ⟨$⟩ʳ suc q | |
| 91 | 394 ∎ |
| 87 | 395 |
| 396 pprep-dist : {n : ℕ} → {x y : Permutation n n } → pprep (x ∘ₚ y) =p= (pprep x ∘ₚ pprep y) | |
| 397 pprep-dist {n} {x} {y} = record { peq = pprep-dist1 } where | |
| 398 pprep-dist1 : (q : Fin (suc n)) → (pprep (x ∘ₚ y) ⟨$⟩ʳ q) ≡ ((pprep x ∘ₚ pprep y) ⟨$⟩ʳ q) | |
| 399 pprep-dist1 zero = refl | |
| 400 pprep-dist1 (suc q) = cong ( λ k → suc k ) refl | |
| 401 | |
| 402 pswap-cong : {n : ℕ} → {x y : Permutation n n } → x =p= y → pswap x =p= pswap y | |
| 403 pswap-cong {n} {x} {y} x=y = record { peq = pswap-cong1 } where | |
| 404 pswap-cong1 : (q : Fin (suc (suc n))) → (pswap x ⟨$⟩ʳ q) ≡ (pswap y ⟨$⟩ʳ q) | |
| 405 pswap-cong1 zero = refl | |
| 406 pswap-cong1 (suc zero) = refl | |
| 407 pswap-cong1 (suc (suc q)) = begin | |
| 408 pswap x ⟨$⟩ʳ suc (suc q) | |
| 409 ≡⟨⟩ | |
| 410 suc (suc (x ⟨$⟩ʳ q)) | |
| 411 ≡⟨ cong ( λ k → suc (suc k) ) ( peq x=y q ) ⟩ | |
| 412 suc (suc (y ⟨$⟩ʳ q)) | |
| 413 ≡⟨⟩ | |
| 414 pswap y ⟨$⟩ʳ suc (suc q) | |
| 91 | 415 ∎ |
| 87 | 416 |
| 417 pswap-dist : {n : ℕ} → {x y : Permutation n n } → pprep (pprep (x ∘ₚ y)) =p= (pswap x ∘ₚ pswap y) | |
| 418 pswap-dist {n} {x} {y} = record { peq = pswap-dist1 } where | |
| 419 pswap-dist1 : (q : Fin (suc (suc n))) → ((pprep (pprep (x ∘ₚ y))) ⟨$⟩ʳ q) ≡ ((pswap x ∘ₚ pswap y) ⟨$⟩ʳ q) | |
| 420 pswap-dist1 zero = refl | |
| 421 pswap-dist1 (suc zero) = refl | |
| 422 pswap-dist1 (suc (suc q)) = cong ( λ k → suc (suc k) ) refl | |
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423 |
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424 data FL : (n : ℕ )→ Set where |
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425 f0 : FL 0 |
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426 _::_ : { n : ℕ } → Fin (suc n ) → FL n → FL (suc n) |
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427 |
| 88 | 428 shlem→ : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → (p0=0 : perm ⟨$⟩ˡ (# 0) ≡ # 0 ) → (x : Fin (suc n) ) → perm ⟨$⟩ˡ x ≡ zero → x ≡ zero |
| 429 shlem→ perm p0=0 x px=0 = begin | |
| 61 | 430 x ≡⟨ sym ( inverseʳ perm ) ⟩ |
| 431 perm ⟨$⟩ʳ ( perm ⟨$⟩ˡ x) ≡⟨ cong (λ k → perm ⟨$⟩ʳ k ) px=0 ⟩ | |
| 432 perm ⟨$⟩ʳ zero ≡⟨ cong (λ k → perm ⟨$⟩ʳ k ) (sym p0=0) ⟩ | |
| 433 perm ⟨$⟩ʳ ( perm ⟨$⟩ˡ zero) ≡⟨ inverseʳ perm ⟩ | |
| 434 zero | |
| 435 ∎ where open ≡-Reasoning | |
| 54 | 436 |
| 88 | 437 shlem← : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → (p0=0 : perm ⟨$⟩ˡ (# 0) ≡ # 0 ) → (x : Fin (suc n)) → perm ⟨$⟩ʳ x ≡ zero → x ≡ zero |
| 438 shlem← perm p0=0 x px=0 = begin | |
| 61 | 439 x ≡⟨ sym (inverseˡ perm ) ⟩ |
| 440 perm ⟨$⟩ˡ ( perm ⟨$⟩ʳ x ) ≡⟨ cong (λ k → perm ⟨$⟩ˡ k ) px=0 ⟩ | |
| 441 perm ⟨$⟩ˡ zero ≡⟨ p0=0 ⟩ | |
| 442 zero | |
| 443 ∎ where open ≡-Reasoning | |
| 54 | 444 |
| 88 | 445 sh2 : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → (p0=0 : perm ⟨$⟩ˡ (# 0) ≡ # 0 ) → {x : Fin n} → ¬ perm ⟨$⟩ˡ (suc x) ≡ zero |
| 446 sh2 perm p0=0 {x} eq with shlem→ perm p0=0 (suc x) eq | |
| 447 sh2 perm p0=0 {x} eq | () | |
| 448 | |
| 449 sh1 : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → (p0=0 : perm ⟨$⟩ˡ (# 0) ≡ # 0 ) → {x : Fin n} → ¬ perm ⟨$⟩ʳ (suc x) ≡ zero | |
| 450 sh1 perm p0=0 {x} eq with shlem← perm p0=0 (suc x) eq | |
| 451 sh1 perm p0=0 {x} eq | () | |
| 452 | |
| 453 | |
| 454 -- 0 ∷ 1 ∷ 2 ∷ 3 ∷ [] → 0 ∷ 1 ∷ 2 ∷ [] | |
| 455 shrink : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → perm ⟨$⟩ˡ (# 0) ≡ # 0 → Permutation n n | |
| 456 shrink {n} perm p0=0 = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where | |
| 57 | 457 |
| 61 | 458 p→ : Fin n → Fin n |
| 88 | 459 p→ x with perm ⟨$⟩ʳ (suc x) | inspect (_⟨$⟩ʳ_ perm ) (suc x) |
| 460 p→ x | zero | record { eq = e } = ⊥-elim ( sh1 perm p0=0 {x} e ) | |
| 61 | 461 p→ x | suc t | _ = t |
| 50 | 462 |
| 463 p← : Fin n → Fin n | |
| 88 | 464 p← x with perm ⟨$⟩ˡ (suc x) | inspect (_⟨$⟩ˡ_ perm ) (suc x) |
| 465 p← x | zero | record { eq = e } = ⊥-elim ( sh2 perm p0=0 {x} e ) | |
| 61 | 466 p← x | suc t | _ = t |
| 50 | 467 |
| 468 piso← : (x : Fin n ) → p→ ( p← x ) ≡ x | |
| 88 | 469 piso← x with perm ⟨$⟩ˡ (suc x) | inspect (_⟨$⟩ˡ_ perm ) (suc x) |
| 470 piso← x | zero | record { eq = e } = ⊥-elim ( sh2 perm p0=0 {x} e ) | |
| 471 piso← x | suc t | _ with perm ⟨$⟩ʳ (suc t) | inspect (_⟨$⟩ʳ_ perm ) (suc t) | |
| 472 piso← x | suc t | _ | zero | record { eq = e } = ⊥-elim ( sh1 perm p0=0 e ) | |
| 61 | 473 piso← x | suc t | record { eq = e0 } | suc t1 | record { eq = e1 } = begin |
| 474 t1 | |
| 475 ≡⟨ plem0 plem1 ⟩ | |
| 52 | 476 x |
| 61 | 477 ∎ where |
| 478 open ≡-Reasoning | |
| 479 plem0 : suc t1 ≡ suc x → t1 ≡ x | |
| 480 plem0 refl = refl | |
| 481 plem1 : suc t1 ≡ suc x | |
| 482 plem1 = begin | |
| 483 suc t1 | |
| 484 ≡⟨ sym e1 ⟩ | |
| 88 | 485 Inverse.to perm Π.⟨$⟩ suc t |
| 486 ≡⟨ cong (λ k → Inverse.to perm Π.⟨$⟩ k ) (sym e0) ⟩ | |
| 487 Inverse.to perm Π.⟨$⟩ ( Inverse.from perm Π.⟨$⟩ suc x ) | |
| 488 ≡⟨ inverseʳ perm ⟩ | |
| 61 | 489 suc x |
| 490 ∎ | |
| 50 | 491 |
| 492 piso→ : (x : Fin n ) → p← ( p→ x ) ≡ x | |
| 88 | 493 piso→ x with perm ⟨$⟩ʳ (suc x) | inspect (_⟨$⟩ʳ_ perm ) (suc x) |
| 494 piso→ x | zero | record { eq = e } = ⊥-elim ( sh1 perm p0=0 {x} e ) | |
| 495 piso→ x | suc t | _ with perm ⟨$⟩ˡ (suc t) | inspect (_⟨$⟩ˡ_ perm ) (suc t) | |
| 496 piso→ x | suc t | _ | zero | record { eq = e } = ⊥-elim ( sh2 perm p0=0 e ) | |
| 61 | 497 piso→ x | suc t | record { eq = e0 } | suc t1 | record { eq = e1 } = begin |
| 498 t1 | |
| 499 ≡⟨ plem2 plem3 ⟩ | |
| 53 | 500 x |
| 61 | 501 ∎ where |
| 502 open ≡-Reasoning | |
| 503 plem2 : suc t1 ≡ suc x → t1 ≡ x | |
| 504 plem2 refl = refl | |
| 505 plem3 : suc t1 ≡ suc x | |
| 506 plem3 = begin | |
| 507 suc t1 | |
| 508 ≡⟨ sym e1 ⟩ | |
| 88 | 509 Inverse.from perm Π.⟨$⟩ suc t |
| 510 ≡⟨ cong (λ k → Inverse.from perm Π.⟨$⟩ k ) (sym e0 ) ⟩ | |
| 511 Inverse.from perm Π.⟨$⟩ ( Inverse.to perm Π.⟨$⟩ suc x ) | |
| 512 ≡⟨ inverseˡ perm ⟩ | |
| 61 | 513 suc x |
| 514 ∎ | |
| 57 | 515 |
| 88 | 516 shrink-iso : { n : ℕ } → {perm : Permutation n n} → shrink (pprep perm) refl =p= perm |
| 517 shrink-iso {n} {perm} = record { peq = λ q → refl } | |
| 518 | |
| 98 | 519 shrink-cong : { n : ℕ } → {x y : Permutation (suc n) (suc n)} |
| 520 → x =p= y | |
| 521 → (x=0 : x ⟨$⟩ˡ (# 0) ≡ # 0 ) → (y=0 : y ⟨$⟩ˡ (# 0) ≡ # 0 ) → shrink x x=0 =p= shrink y y=0 | |
| 99 | 522 shrink-cong {n} {x} {y} x=y x=0 y=0 = record { peq = p002 } where |
| 523 p002 : (q : Fin n) → (shrink x x=0 ⟨$⟩ʳ q) ≡ (shrink y y=0 ⟨$⟩ʳ q) | |
| 524 p002 q with x ⟨$⟩ʳ (suc q) | inspect (_⟨$⟩ʳ_ x ) (suc q) | y ⟨$⟩ʳ (suc q) | inspect (_⟨$⟩ʳ_ y ) (suc q) | |
| 525 p002 q | zero | record { eq = ex } | zero | ey = ⊥-elim ( sh1 x x=0 ex ) | |
| 526 p002 q | zero | record { eq = ex } | suc py | ey = ⊥-elim ( sh1 x x=0 ex ) | |
| 527 p002 q | suc px | ex | zero | record { eq = ey } = ⊥-elim ( sh1 y y=0 ey ) | |
| 528 p002 q | suc px | record { eq = ex } | suc py | record { eq = ey } = p003 ( begin | |
| 529 suc px | |
| 530 ≡⟨ sym ex ⟩ | |
| 531 x ⟨$⟩ʳ (suc q) | |
| 532 ≡⟨ peq x=y (suc q) ⟩ | |
| 533 y ⟨$⟩ʳ (suc q) | |
| 534 ≡⟨ ey ⟩ | |
| 535 suc py | |
| 536 ∎ ) where | |
| 537 p003 : suc px ≡ suc py → px ≡ py | |
| 538 p003 refl = refl | |
| 98 | 539 |
| 57 | 540 FL→perm : {n : ℕ } → FL n → Permutation n n |
| 541 FL→perm f0 = pid | |
| 542 FL→perm (x :: fl) = pprep (FL→perm fl) ∘ₚ pins ( toℕ≤pred[n] x ) | |
| 543 | |
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544 t40 = (# 2) :: ( (# 1) :: (( # 0 ) :: f0 )) |
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545 t4 = FL→perm ((# 2) :: t40 ) |
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546 |
| 61 | 547 -- t1 = plist (shrink (pid {3} ∘ₚ (pins (n≤ 1))) refl) |
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548 t2 = plist ((pid {5} ) ∘ₚ transpose (# 2) (# 4)) ∷ plist (pid {5} ∘ₚ reverse ) ∷ [] |
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549 t3 = plist (FL→perm t40) -- ∷ plist (pprep (FL→perm t40)) |
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550 -- ∷ plist ( pprep (FL→perm t40) ∘ₚ pins ( n≤ 0 {3} )) |
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551 -- ∷ plist ( pprep (FL→perm t40 )∘ₚ pins ( n≤ 1 {2} )) |
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552 -- ∷ plist ( pprep (FL→perm t40 )∘ₚ pins ( n≤ 2 {1} )) |
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553 -- ∷ plist ( pprep (FL→perm t40 )∘ₚ pins ( n≤ 3 {0} )) |
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554 ∷ plist ( FL→perm ((# 0) :: t40)) -- (0 ∷ 1 ∷ 2 ∷ []) ∷ |
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555 ∷ plist ( FL→perm ((# 1) :: t40)) -- (0 ∷ 2 ∷ 1 ∷ []) ∷ |
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556 ∷ plist ( FL→perm ((# 2) :: t40)) -- (1 ∷ 0 ∷ 2 ∷ []) ∷ |
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557 ∷ plist ( FL→perm ((# 3) :: t40)) -- (2 ∷ 0 ∷ 1 ∷ []) ∷ |
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558 -- ∷ plist ( FL→perm ((# 3) :: ((# 2) :: ( (# 0) :: (( # 0 ) :: f0 )) ))) -- (1 ∷ 2 ∷ 0 ∷ []) ∷ |
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559 -- ∷ plist ( FL→perm ((# 3) :: ((# 2) :: ( (# 1) :: (( # 0 ) :: f0 )) ))) -- (2 ∷ 1 ∷ 0 ∷ []) ∷ |
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560 -- ∷ plist ( (flip (FL→perm ((# 3) :: ((# 1) :: ( (# 0) :: (( # 0 ) :: f0 )) ))))) |
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561 -- ∷ plist ( (flip (FL→perm ((# 3) :: ((# 1) :: ( (# 0) :: (( # 0 ) :: f0 )) ))) ∘ₚ (FL→perm ((# 3) :: (((# 1) :: ( (# 0) :: (( # 0 ) :: f0 )) )))) )) |
| 57 | 562 ∷ [] |
| 50 | 563 |
| 58 | 564 |
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565 perm→FL : {n : ℕ } → Permutation n n → FL n |
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566 perm→FL {zero} perm = f0 |
| 98 | 567 perm→FL {suc n} perm = (perm ⟨$⟩ʳ (# 0)) :: perm→FL (shrink (perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) (p=0 perm) ) |
| 568 -- perm→FL {suc n} perm = (perm ⟨$⟩ʳ (# 0)) :: perm→FL (remove (# 0) perm) | |
| 569 | |
| 570 pcong-pF : {n : ℕ } → {x y : Permutation n n} → x =p= y → perm→FL x ≡ perm→FL y | |
| 99 | 571 pcong-pF {zero} eq = refl |
| 100 | 572 pcong-pF {suc n} {x} {y} eq = cong₂ (λ j k → j :: k ) ( peq eq (# 0)) (pcong-pF (shrink-cong (presp eq p001 ) (p=0 x) (p=0 y))) where |
| 573 p002 : x ⟨$⟩ʳ (# 0) ≡ y ⟨$⟩ʳ (# 0) | |
| 574 p002 = peq eq (# 0) | |
| 575 p001 : flip (pins (toℕ≤pred[n] (x ⟨$⟩ʳ (# 0)))) =p= flip (pins (toℕ≤pred[n] (y ⟨$⟩ʳ (# 0)))) | |
| 576 p001 = subst ( λ k → flip (pins (toℕ≤pred[n] (x ⟨$⟩ʳ (# 0)))) =p= flip (pins (toℕ≤pred[n] k ))) p002 prefl | |
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577 |
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578 -- t5 = plist t4 ∷ plist ( t4 ∘ₚ flip (pins ( n≤ 3 ) )) |
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579 t5 = plist (t4) ∷ plist (flip t4) |
| 74 | 580 ∷ ( toℕ (t4 ⟨$⟩ˡ fromℕ< a<sa) ∷ [] ) |
| 61 | 581 ∷ ( toℕ (t4 ⟨$⟩ʳ (# 0)) ∷ [] ) |
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582 -- ∷ plist ( t4 ∘ₚ flip (pins ( n≤ 1 ) )) |
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583 ∷ plist (remove (# 0) t4 ) |
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584 ∷ plist ( FL→perm t40 ) |
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585 ∷ [] |
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586 |
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587 t6 = perm→FL t4 |
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588 |
| 105 | 589 ---- |
| 590 -- if n is fixed, perm→FL ( FL→perm fl ) ≡ fl is refl for each concrete fl | |
| 591 -- so we may prove this easily by co-induction | |
| 592 -- | |
| 98 | 593 FL→iso : {n : ℕ } → (fl : FL n ) → perm→FL ( FL→perm fl ) ≡ fl |
| 594 FL→iso f0 = refl | |
| 100 | 595 FL→iso {suc n} (x :: fl) = cong₂ ( λ j k → j :: k ) f001 f002 where |
| 98 | 596 perm = pprep (FL→perm fl) ∘ₚ pins ( toℕ≤pred[n] x ) |
| 597 f001 : perm ⟨$⟩ʳ (# 0) ≡ x | |
| 100 | 598 f001 = begin |
| 599 (pprep (FL→perm fl) ∘ₚ pins ( toℕ≤pred[n] x )) ⟨$⟩ʳ (# 0) | |
| 101 | 600 ≡⟨⟩ |
| 601 pins ( toℕ≤pred[n] x ) ⟨$⟩ʳ (# 0) | |
| 602 ≡⟨ px=x x ⟩ | |
| 100 | 603 x |
| 604 ∎ | |
| 98 | 605 x=0 : (perm ∘ₚ flip (pins (toℕ≤pred[n] x))) ⟨$⟩ˡ (# 0) ≡ # 0 |
| 100 | 606 x=0 = subst ( λ k → (perm ∘ₚ flip (pins (toℕ≤pred[n] k))) ⟨$⟩ˡ (# 0) ≡ # 0 ) f001 (p=0 perm) |
| 98 | 607 x=0' : (pprep (FL→perm fl) ∘ₚ pid) ⟨$⟩ˡ (# 0) ≡ # 0 |
| 100 | 608 x=0' = refl |
| 103 | 609 f003 : (q : Fin (suc n)) → |
| 610 ((perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) ⟨$⟩ʳ q) ≡ | |
| 611 ((perm ∘ₚ flip (pins (toℕ≤pred[n] x))) ⟨$⟩ʳ q) | |
| 612 f003 q = cong (λ k → (perm ∘ₚ flip (pins (toℕ≤pred[n] k))) ⟨$⟩ʳ q ) f001 | |
| 98 | 613 f002 : perm→FL (shrink (perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) (p=0 perm) ) ≡ fl |
| 614 f002 = begin | |
| 615 perm→FL (shrink (perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) (p=0 perm) ) | |
| 103 | 616 ≡⟨ pcong-pF (shrink-cong {n} {perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))} {perm ∘ₚ flip (pins (toℕ≤pred[n] x))} record {peq = f003 } (p=0 perm) x=0) ⟩ |
| 98 | 617 perm→FL (shrink (perm ∘ₚ flip (pins (toℕ≤pred[n] x))) x=0 ) |
| 618 ≡⟨⟩ | |
| 619 perm→FL (shrink ((pprep (FL→perm fl) ∘ₚ pins ( toℕ≤pred[n] x )) ∘ₚ flip (pins (toℕ≤pred[n] x))) x=0 ) | |
| 103 | 620 ≡⟨ pcong-pF (shrink-cong (passoc (pprep (FL→perm fl)) (pins ( toℕ≤pred[n] x )) (flip (pins (toℕ≤pred[n] x))) ) x=0 x=0) ⟩ |
| 98 | 621 perm→FL (shrink (pprep (FL→perm fl) ∘ₚ (pins ( toℕ≤pred[n] x ) ∘ₚ flip (pins (toℕ≤pred[n] x)))) x=0 ) |
| 103 | 622 ≡⟨ pcong-pF (shrink-cong {n} {pprep (FL→perm fl) ∘ₚ (pins ( toℕ≤pred[n] x ) ∘ₚ flip (pins (toℕ≤pred[n] x)))} {pprep (FL→perm fl) ∘ₚ pid} |
| 623 ( presp {suc n} {pprep (FL→perm fl) } {_} {(pins ( toℕ≤pred[n] x ) ∘ₚ flip (pins (toℕ≤pred[n] x)))} {pid} prefl | |
| 624 record { peq = λ q → inverseˡ (pins ( toℕ≤pred[n] x )) } ) x=0 x=0') ⟩ | |
| 98 | 625 perm→FL (shrink (pprep (FL→perm fl) ∘ₚ pid) x=0' ) |
| 103 | 626 ≡⟨ pcong-pF (shrink-cong {n} {pprep (FL→perm fl) ∘ₚ pid} {pprep (FL→perm fl)} record {peq = λ q → refl } x=0' x=0') ⟩ -- prefl won't work |
| 98 | 627 perm→FL (shrink (pprep (FL→perm fl)) x=0' ) |
| 628 ≡⟨ pcong-pF shrink-iso ⟩ | |
| 629 perm→FL ( FL→perm fl ) | |
| 630 ≡⟨ FL→iso fl ⟩ | |
| 631 fl | |
| 632 ∎ | |
| 633 | |
| 104 | 634 pcong-Fp : {n : ℕ } → {x y : FL n} → x ≡ y → FL→perm x =p= FL→perm y |
| 635 pcong-Fp {n} {x} {x} refl = prefl | |
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636 |
| 104 | 637 FL←iso : {n : ℕ } → (perm : Permutation n n ) → FL→perm ( perm→FL perm ) =p= perm |
| 638 FL←iso {0} perm = record { peq = λ () } | |
| 639 FL←iso {suc n} perm with perm→FL perm | inspect perm→FL perm | |
| 640 ... | x :: fl | record { eq = e } = ptrans (pcong-Fp e ) f004 where | |
| 105 | 641 f003 : perm→FL (shrink (perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) (p=0 perm)) ≡ fl |
| 642 f003 = {!!} | |
| 643 f004 : FL→perm ( x :: fl ) =p= perm | |
| 104 | 644 f004 = record { peq = λ q → ( begin |
| 645 (pprep (FL→perm fl) ∘ₚ pins ( toℕ≤pred[n] x )) ⟨$⟩ʳ q | |
| 105 | 646 ≡⟨ cong (λ k → (pprep (FL→perm k) ∘ₚ pins ( toℕ≤pred[n] x )) ⟨$⟩ʳ q ) (sym f003 ) ⟩ |
| 104 | 647 (pprep (FL→perm (perm→FL (shrink (perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) (p=0 perm) ))) ∘ₚ pins ( toℕ≤pred[n] x )) ⟨$⟩ʳ q |
| 105 | 648 ≡⟨ peq (presp (pprep-cong (FL←iso _ ) ) prefl ) q ⟩ |
| 104 | 649 (pprep (shrink (perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) (p=0 perm)) ∘ₚ pins ( toℕ≤pred[n] x )) ⟨$⟩ʳ q |
| 650 ≡⟨ {!!} ⟩ | |
| 651 perm ⟨$⟩ʳ q | |
| 652 ∎ ) } | |
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653 |
| 66 | 654 lem2 : {i n : ℕ } → i ≤ n → i ≤ suc n |
| 91 | 655 lem2 i≤n = ≤-trans i≤n ( a≤sa ) |
| 66 | 656 |
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657 ∀-FL : (n : ℕ ) → List (FL (suc n)) |
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658 ∀-FL x = fls6 x where |
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659 fls4 : ( i n : ℕ ) → (i<n : i ≤ n ) → FL n → List (FL (suc n)) → List (FL (suc n)) |
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660 fls4 zero n i≤n perm x = (zero :: perm ) ∷ x |
| 91 | 661 fls4 (suc i) n i≤n perm x = fls4 i n (≤-trans a≤sa i≤n ) perm ((fromℕ< (s≤s i≤n) :: perm ) ∷ x) |
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662 fls5 : ( n : ℕ ) → List (FL n) → List (FL (suc n)) → List (FL (suc n)) |
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663 fls5 n [] x = x |
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664 fls5 n (h ∷ x) y = fls5 n x (fls4 n n lem0 h y) |
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665 fls6 : ( n : ℕ ) → List (FL (suc n)) |
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666 fls6 zero = (zero :: f0) ∷ [] |
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667 fls6 (suc n) = fls5 (suc n) (fls6 n) [] |
| 65 | 668 |
| 97 | 669 tf1 = ∀-FL 4 |
| 670 tf2 = Data.List.map (λ k → ⟪ plist (FL→perm k ) , k ⟫ ) tf1 | |
| 671 | |
| 48 | 672 all-perm : (n : ℕ ) → List (Permutation (suc n) (suc n) ) |
| 673 all-perm n = pls6 n where | |
| 38 | 674 lem1 : {i n : ℕ } → i ≤ n → i < suc n |
| 675 lem1 z≤n = s≤s z≤n | |
| 676 lem1 (s≤s lt) = s≤s (lem1 lt) | |
| 40 | 677 pls4 : ( i n : ℕ ) → (i<n : i ≤ n ) → Permutation n n → List (Permutation (suc n) (suc n)) → List (Permutation (suc n) (suc n)) |
| 48 | 678 pls4 zero n i≤n perm x = (pprep perm ∘ₚ pins i≤n ) ∷ x |
| 91 | 679 pls4 (suc i) n i≤n perm x = pls4 i n (≤-trans a≤sa i≤n ) perm (pprep perm ∘ₚ pins {n} {suc i} i≤n ∷ x) |
| 40 | 680 pls5 : ( n : ℕ ) → List (Permutation n n) → List (Permutation (suc n) (suc n)) → List (Permutation (suc n) (suc n)) |
| 681 pls5 n [] x = x | |
| 682 pls5 n (h ∷ x) y = pls5 n x (pls4 n n lem0 h y) | |
| 683 pls6 : ( n : ℕ ) → List (Permutation (suc n) (suc n)) | |
| 684 pls6 zero = pid ∷ [] | |
| 48 | 685 pls6 (suc n) = pls5 (suc n) (rev (pls6 n) ) [] -- rev to put id first |
| 686 | |
| 687 pls : (n : ℕ ) → List (List ℕ ) | |
| 75 | 688 pls n = Data.List.map plist (all-perm n) |