Mercurial > hg > Members > kono > Proof > galois
annotate Putil.agda @ 87:c68956f6c3ad
tc fix
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 27 Aug 2020 11:44:58 +0900 |
| parents | c5329963c583 |
| children | 405c1f727ffe |
| 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 |
| 80 | 16 open import Data.List using (List; []; _∷_ ; length ; _++_ ; head ; tail ) 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 |
| 80 | 25 open import Relation.Binary.Definitions |
| 17 | 26 open import fin |
| 16 | 27 |
| 38 | 28 -- An inductive construction of permutation |
| 34 | 29 |
| 59 | 30 -- Todo |
| 31 -- | |
| 32 -- complete perm→FL | |
| 33 -- describe property of pprep and pswap | |
| 34 -- describe property of pins ( move 0 to any position) | |
| 35 -- describe property of shrink ( remove one column ) | |
| 36 -- prove FL→iso | |
| 37 -- prove FL←iso | |
| 38 -- prove FL enumerate all permutations | |
| 39 | |
| 48 | 40 -- we already have refl and trans in the Symmetric Group |
| 41 | 41 |
| 34 | 42 pprep : {n : ℕ } → Permutation n n → Permutation (suc n) (suc n) |
| 43 pprep {n} perm = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where | |
| 33 | 44 p→ : Fin (suc n) → Fin (suc n) |
| 34 | 45 p→ zero = zero |
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46 p→ (suc x) = suc ( perm ⟨$⟩ʳ x) |
| 33 | 47 |
| 34 | 48 p← : Fin (suc n) → Fin (suc n) |
| 49 p← zero = zero | |
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50 p← (suc x) = suc ( perm ⟨$⟩ˡ x) |
| 34 | 51 |
| 52 piso← : (x : Fin (suc n)) → p→ ( p← x ) ≡ x | |
| 53 piso← zero = refl | |
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54 piso← (suc x) = cong (λ k → suc k ) (inverseʳ perm) |
| 33 | 55 |
| 34 | 56 piso→ : (x : Fin (suc n)) → p← ( p→ x ) ≡ x |
| 57 piso→ zero = refl | |
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58 piso→ (suc x) = cong (λ k → suc k ) (inverseˡ perm) |
| 33 | 59 |
| 34 | 60 pswap : {n : ℕ } → Permutation n n → Permutation (suc (suc n)) (suc (suc n )) |
| 61 pswap {n} perm = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where | |
| 62 p→ : Fin (suc (suc n)) → Fin (suc (suc n)) | |
| 63 p→ zero = suc zero | |
| 64 p→ (suc zero) = zero | |
| 62 | 65 p→ (suc (suc x)) = suc ( suc ( perm ⟨$⟩ʳ x) ) |
| 18 | 66 |
| 34 | 67 p← : Fin (suc (suc n)) → Fin (suc (suc n)) |
| 68 p← zero = suc zero | |
| 69 p← (suc zero) = zero | |
| 62 | 70 p← (suc (suc x)) = suc ( suc ( perm ⟨$⟩ˡ x) ) |
| 34 | 71 |
| 72 piso← : (x : Fin (suc (suc n)) ) → p→ ( p← x ) ≡ x | |
| 73 piso← zero = refl | |
| 74 piso← (suc zero) = refl | |
| 62 | 75 piso← (suc (suc x)) = cong (λ k → suc (suc k) ) (inverseʳ perm) |
| 16 | 76 |
| 34 | 77 piso→ : (x : Fin (suc (suc n)) ) → p← ( p→ x ) ≡ x |
| 78 piso→ zero = refl | |
| 79 piso→ (suc zero) = refl | |
| 62 | 80 piso→ (suc (suc x)) = cong (λ k → suc (suc k) ) (inverseˡ perm) |
| 34 | 81 |
| 82 -- enumeration | |
| 83 | |
| 44 | 84 psawpn : {n : ℕ} → 1 < n → Permutation n n |
| 85 psawpn {suc zero} (s≤s ()) | |
| 86 psawpn {suc n} (s≤s (s≤s x)) = pswap pid | |
| 34 | 87 |
| 35 | 88 pfill : { n m : ℕ } → m ≤ n → Permutation m m → Permutation n n |
| 89 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 | |
| 90 pfill1 : (i : ℕ ) → i ≤ n → Permutation (n - i) (n - i) → Permutation n n | |
| 91 pfill1 0 _ perm = perm | |
| 92 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 | 93 |
| 48 | 94 -- |
| 95 -- psawpim (inseert swap at position m ) | |
| 96 -- | |
| 45 | 97 psawpim : {n m : ℕ} → suc (suc m) ≤ n → Permutation n n |
| 98 psawpim {n} {m} m≤n = pfill m≤n ( psawpn (s≤s (s≤s z≤n)) ) | |
| 99 | |
| 100 n≤ : (i : ℕ ) → {j : ℕ } → i ≤ i + j | |
| 101 n≤ (zero) {j} = z≤n | |
| 102 n≤ (suc i) {j} = s≤s ( n≤ i ) | |
| 103 | |
| 104 lem0 : {n : ℕ } → n ≤ n | |
| 105 lem0 {zero} = z≤n | |
| 106 lem0 {suc n} = s≤s lem0 | |
| 107 | |
| 108 lem00 : {n m : ℕ } → n ≡ m → n ≤ m | |
| 109 lem00 refl = lem0 | |
| 44 | 110 |
| 111 -- pconcat : {n m : ℕ } → Permutation m m → Permutation n n → Permutation (m + n) (m + n) | |
| 112 -- pconcat {n} {m} p q = pfill {n + m} {m} ? p ∘ₚ ? | |
| 113 | |
| 114 -- inductivley enmumerate permutations | |
| 115 -- from n-1 length create n length inserting new element at position m | |
| 116 | |
| 48 | 117 -- 0 ∷ 1 ∷ 2 ∷ 3 ∷ [] |
| 118 -- 1 ∷ 0 ∷ 2 ∷ 3 ∷ [] plist ( pins {3} (n≤ 1) ) | |
| 119 -- 1 ∷ 2 ∷ 0 ∷ 3 ∷ [] | |
| 120 -- 1 ∷ 2 ∷ 3 ∷ 0 ∷ [] | |
| 45 | 121 |
| 48 | 122 pins : {n m : ℕ} → m ≤ n → Permutation (suc n) (suc n) |
| 123 pins {_} {zero} _ = pid | |
| 124 pins {suc _} {suc zero} _ = pswap pid | |
| 125 pins {suc (suc n)} {suc m} (s≤s m<n) = pins1 (suc m) (suc (suc n)) lem0 where | |
| 126 pins1 : (i j : ℕ ) → j ≤ suc (suc n) → Permutation (suc (suc (suc n ))) (suc (suc (suc n))) | |
| 127 pins1 _ zero _ = pid | |
| 128 pins1 zero _ _ = pid | |
| 129 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 | 130 |
| 80 | 131 plist1 : {n : ℕ} → Permutation (suc n) (suc n) → (i : ℕ ) → i < suc n → List ℕ |
| 132 plist1 {n} perm zero _ = toℕ ( perm ⟨$⟩ˡ (fromℕ< {zero} (s≤s z≤n))) ∷ [] | |
| 133 plist1 {n} perm (suc i) (s≤s lt) = toℕ ( perm ⟨$⟩ˡ (fromℕ< (s≤s lt))) ∷ plist1 perm i (<-trans lt a<sa) | |
| 134 | |
| 37 | 135 plist : {n : ℕ} → Permutation n n → List ℕ |
| 136 plist {0} perm = [] | |
| 80 | 137 plist {suc n} perm = rev (plist1 perm n a<sa) |
| 138 | |
| 139 plist2 : {n : ℕ} → Permutation (suc n) (suc n) → (i : ℕ ) → i < suc n → List ℕ | |
| 140 plist2 {n} perm zero _ = toℕ ( perm ⟨$⟩ʳ (fromℕ< {zero} (s≤s z≤n))) ∷ [] | |
| 141 plist2 {n} perm (suc i) (s≤s lt) = toℕ ( perm ⟨$⟩ʳ (fromℕ< (s≤s lt))) ∷ plist2 perm i (<-trans lt a<sa) | |
| 142 | |
| 143 plist0 : {n : ℕ} → Permutation n n → List ℕ | |
| 144 plist0 {0} perm = [] | |
| 145 plist0 {suc n} perm = plist2 perm n a<sa | |
| 146 | |
| 85 | 147 open _=p=_ |
| 148 | |
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149 -- |
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150 -- plist cong |
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151 -- |
| 85 | 152 ←pleq : {n : ℕ} → (x y : Permutation n n ) → x =p= y → plist0 x ≡ plist0 y |
| 153 ←pleq {zero} x y eq = refl | |
| 154 ←pleq {suc n} x y eq = ←pleq1 n a<sa where | |
| 155 ←pleq1 : (i : ℕ ) → (i<sn : i < suc n ) → plist2 x i i<sn ≡ plist2 y i i<sn | |
| 156 ←pleq1 zero _ = cong ( λ k → toℕ k ∷ [] ) ( peq eq (fromℕ< {zero} (s≤s z≤n))) | |
| 157 ←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 | 158 |
| 159 headeq : {A : Set } → {x y : A } → {xt yt : List A } → (x ∷ xt) ≡ (y ∷ yt) → x ≡ y | |
| 160 headeq refl = refl | |
| 161 | |
| 162 taileq : {A : Set } → {x y : A } → {xt yt : List A } → (x ∷ xt) ≡ (y ∷ yt) → xt ≡ yt | |
| 163 taileq refl = refl | |
| 164 | |
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165 -- |
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166 -- plist equalizer |
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167 -- |
| 80 | 168 pleq : {n : ℕ} → (x y : Permutation n n ) → plist0 x ≡ plist0 y → x =p= y |
| 169 pleq {0} x y refl = record { peq = λ q → pleq0 q } where | |
| 170 pleq0 : (q : Fin 0 ) → (x ⟨$⟩ʳ q) ≡ (y ⟨$⟩ʳ q) | |
| 171 pleq0 () | |
| 172 pleq {suc n} x y eq = record { peq = λ q → pleq1 n a<sa eq q fin<n } where | |
| 173 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 | 174 pleq1 zero i<sn eq q q<i with <-cmp (toℕ q) zero |
| 175 ... | tri< () ¬b ¬c | |
| 176 ... | tri> ¬a ¬b c = ⊥-elim (nat-≤> c q<i ) | |
| 177 ... | tri≈ ¬a b ¬c = begin | |
| 178 x ⟨$⟩ʳ q | |
| 179 ≡⟨ cong ( λ k → x ⟨$⟩ʳ k ) (toℕ-injective b )⟩ | |
| 180 x ⟨$⟩ʳ zero | |
| 181 ≡⟨ toℕ-injective (headeq eq) ⟩ | |
| 182 y ⟨$⟩ʳ zero | |
| 183 ≡⟨ cong ( λ k → y ⟨$⟩ʳ k ) (sym (toℕ-injective b )) ⟩ | |
| 184 y ⟨$⟩ʳ q | |
| 185 ∎ where | |
| 186 open ≡-Reasoning | |
| 80 | 187 pleq1 (suc i) (s≤s i<sn) eq q q<i with <-cmp (toℕ q) (suc i) |
| 188 ... | tri< a ¬b ¬c = pleq1 i (<-trans i<sn a<sa ) (taileq eq) q a | |
| 81 | 189 ... | tri> ¬a ¬b c = ⊥-elim (nat-≤> c q<i ) |
| 80 | 190 ... | tri≈ ¬a b ¬c = begin |
| 191 x ⟨$⟩ʳ q | |
| 192 ≡⟨ cong (λ k → x ⟨$⟩ʳ k) (pleq3 b) ⟩ | |
| 193 x ⟨$⟩ʳ (suc (fromℕ< i<sn)) | |
| 194 ≡⟨ toℕ-injective pleq2 ⟩ | |
| 195 y ⟨$⟩ʳ (suc (fromℕ< i<sn)) | |
| 196 ≡⟨ cong (λ k → y ⟨$⟩ʳ k) (sym (pleq3 b)) ⟩ | |
| 197 y ⟨$⟩ʳ q | |
| 198 ∎ where | |
| 199 open ≡-Reasoning | |
| 200 pleq3 : toℕ q ≡ suc i → q ≡ suc (fromℕ< i<sn) | |
| 201 pleq3 tq=si = toℕ-injective ( begin | |
| 202 toℕ q | |
| 203 ≡⟨ b ⟩ | |
| 204 suc i | |
| 205 ≡⟨ sym (toℕ-fromℕ< (s≤s i<sn)) ⟩ | |
| 206 toℕ (fromℕ< (s≤s i<sn)) | |
| 207 ≡⟨⟩ | |
| 208 toℕ (suc (fromℕ< i<sn)) | |
| 209 ∎ ) where open ≡-Reasoning | |
| 210 pleq2 : toℕ ( x ⟨$⟩ʳ (suc (fromℕ< i<sn)) ) ≡ toℕ ( y ⟨$⟩ʳ (suc (fromℕ< i<sn)) ) | |
| 211 pleq2 = headeq eq | |
| 37 | 212 |
| 87 | 213 pprep-cong : {n : ℕ} → {x y : Permutation n n } → x =p= y → pprep x =p= pprep y |
| 214 pprep-cong {n} {x} {y} x=y = record { peq = pprep-cong1 } where | |
| 215 pprep-cong1 : (q : Fin (suc n)) → (pprep x ⟨$⟩ʳ q) ≡ (pprep y ⟨$⟩ʳ q) | |
| 216 pprep-cong1 zero = refl | |
| 217 pprep-cong1 (suc q) = begin | |
| 218 pprep x ⟨$⟩ʳ suc q | |
| 219 ≡⟨⟩ | |
| 220 suc ( x ⟨$⟩ʳ q ) | |
| 221 ≡⟨ cong ( λ k → suc k ) ( peq x=y q ) ⟩ | |
| 222 suc ( y ⟨$⟩ʳ q ) | |
| 223 ≡⟨⟩ | |
| 224 pprep y ⟨$⟩ʳ suc q | |
| 225 ∎ where open ≡-Reasoning | |
| 226 | |
| 227 pprep-dist : {n : ℕ} → {x y : Permutation n n } → pprep (x ∘ₚ y) =p= (pprep x ∘ₚ pprep y) | |
| 228 pprep-dist {n} {x} {y} = record { peq = pprep-dist1 } where | |
| 229 pprep-dist1 : (q : Fin (suc n)) → (pprep (x ∘ₚ y) ⟨$⟩ʳ q) ≡ ((pprep x ∘ₚ pprep y) ⟨$⟩ʳ q) | |
| 230 pprep-dist1 zero = refl | |
| 231 pprep-dist1 (suc q) = cong ( λ k → suc k ) refl | |
| 232 | |
| 233 pswap-cong : {n : ℕ} → {x y : Permutation n n } → x =p= y → pswap x =p= pswap y | |
| 234 pswap-cong {n} {x} {y} x=y = record { peq = pswap-cong1 } where | |
| 235 pswap-cong1 : (q : Fin (suc (suc n))) → (pswap x ⟨$⟩ʳ q) ≡ (pswap y ⟨$⟩ʳ q) | |
| 236 pswap-cong1 zero = refl | |
| 237 pswap-cong1 (suc zero) = refl | |
| 238 pswap-cong1 (suc (suc q)) = begin | |
| 239 pswap x ⟨$⟩ʳ suc (suc q) | |
| 240 ≡⟨⟩ | |
| 241 suc (suc (x ⟨$⟩ʳ q)) | |
| 242 ≡⟨ cong ( λ k → suc (suc k) ) ( peq x=y q ) ⟩ | |
| 243 suc (suc (y ⟨$⟩ʳ q)) | |
| 244 ≡⟨⟩ | |
| 245 pswap y ⟨$⟩ʳ suc (suc q) | |
| 246 ∎ where open ≡-Reasoning | |
| 247 | |
| 248 pswap-dist : {n : ℕ} → {x y : Permutation n n } → pprep (pprep (x ∘ₚ y)) =p= (pswap x ∘ₚ pswap y) | |
| 249 pswap-dist {n} {x} {y} = record { peq = pswap-dist1 } where | |
| 250 pswap-dist1 : (q : Fin (suc (suc n))) → ((pprep (pprep (x ∘ₚ y))) ⟨$⟩ʳ q) ≡ ((pswap x ∘ₚ pswap y) ⟨$⟩ʳ q) | |
| 251 pswap-dist1 zero = refl | |
| 252 pswap-dist1 (suc zero) = refl | |
| 253 pswap-dist1 (suc (suc q)) = cong ( λ k → suc (suc k) ) refl | |
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254 |
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255 data FL : (n : ℕ )→ Set where |
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256 f0 : FL 0 |
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257 _::_ : { n : ℕ } → Fin (suc n ) → FL n → FL (suc n) |
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258 |
| 50 | 259 open import logic |
| 260 | |
| 56 | 261 -- 0 ∷ 1 ∷ 2 ∷ 3 ∷ [] → 0 ∷ 1 ∷ 2 ∷ [] |
| 61 | 262 shrink : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → perm ⟨$⟩ˡ (# 0) ≡ # 0 → Permutation n n |
| 263 shrink {n} perm p0=0 = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where | |
| 264 shlem→ : (x : Fin (suc n) ) → perm ⟨$⟩ˡ x ≡ zero → x ≡ zero | |
| 265 shlem→ x px=0 = begin | |
| 266 x ≡⟨ sym ( inverseʳ perm ) ⟩ | |
| 267 perm ⟨$⟩ʳ ( perm ⟨$⟩ˡ x) ≡⟨ cong (λ k → perm ⟨$⟩ʳ k ) px=0 ⟩ | |
| 268 perm ⟨$⟩ʳ zero ≡⟨ cong (λ k → perm ⟨$⟩ʳ k ) (sym p0=0) ⟩ | |
| 269 perm ⟨$⟩ʳ ( perm ⟨$⟩ˡ zero) ≡⟨ inverseʳ perm ⟩ | |
| 270 zero | |
| 271 ∎ where open ≡-Reasoning | |
| 54 | 272 |
| 61 | 273 shlem← : (x : Fin (suc n)) → perm ⟨$⟩ʳ x ≡ zero → x ≡ zero |
| 274 shlem← x px=0 = begin | |
| 275 x ≡⟨ sym (inverseˡ perm ) ⟩ | |
| 276 perm ⟨$⟩ˡ ( perm ⟨$⟩ʳ x ) ≡⟨ cong (λ k → perm ⟨$⟩ˡ k ) px=0 ⟩ | |
| 277 perm ⟨$⟩ˡ zero ≡⟨ p0=0 ⟩ | |
| 278 zero | |
| 279 ∎ where open ≡-Reasoning | |
| 54 | 280 |
| 61 | 281 sh2 : {x : Fin n} → ¬ perm ⟨$⟩ˡ (suc x) ≡ zero |
| 282 sh2 {x} eq with shlem→ (suc x) eq | |
| 283 sh2 {x} eq | () | |
| 57 | 284 |
| 61 | 285 p→ : Fin n → Fin n |
| 286 p→ x with perm ⟨$⟩ˡ (suc x) | inspect (_⟨$⟩ˡ_ perm ) (suc x) | |
| 287 p→ x | zero | record { eq = e } = ⊥-elim ( sh2 {x} e ) | |
| 288 p→ x | suc t | _ = t | |
| 50 | 289 |
| 61 | 290 sh1 : {x : Fin n} → ¬ perm ⟨$⟩ʳ (suc x) ≡ zero |
| 291 sh1 {x} eq with shlem← (suc x) eq | |
| 292 sh1 {x} eq | () | |
| 50 | 293 |
| 294 p← : Fin n → Fin n | |
| 61 | 295 p← x with perm ⟨$⟩ʳ (suc x) | inspect (_⟨$⟩ʳ_ perm ) (suc x) |
| 296 p← x | zero | record { eq = e } = ⊥-elim ( sh1 {x} e ) | |
| 297 p← x | suc t | _ = t | |
| 50 | 298 |
| 299 piso← : (x : Fin n ) → p→ ( p← x ) ≡ x | |
| 61 | 300 piso← x with perm ⟨$⟩ʳ (suc x) | inspect (_⟨$⟩ʳ_ perm ) (suc x) |
| 301 piso← x | zero | record { eq = e } = ⊥-elim ( sh1 {x} e ) | |
| 302 piso← x | suc t | _ with perm ⟨$⟩ˡ (suc t) | inspect (_⟨$⟩ˡ_ perm ) (suc t) | |
| 303 piso← x | suc t | _ | zero | record { eq = e } = ⊥-elim ( sh2 e ) | |
| 304 piso← x | suc t | record { eq = e0 } | suc t1 | record { eq = e1 } = begin | |
| 305 t1 | |
| 306 ≡⟨ plem0 plem1 ⟩ | |
| 52 | 307 x |
| 61 | 308 ∎ where |
| 309 open ≡-Reasoning | |
| 310 plem0 : suc t1 ≡ suc x → t1 ≡ x | |
| 311 plem0 refl = refl | |
| 312 plem1 : suc t1 ≡ suc x | |
| 313 plem1 = begin | |
| 314 suc t1 | |
| 315 ≡⟨ sym e1 ⟩ | |
| 316 Inverse.from perm Π.⟨$⟩ suc t | |
| 317 ≡⟨ cong (λ k → Inverse.from perm Π.⟨$⟩ k ) (sym e0 ) ⟩ | |
| 318 Inverse.from perm Π.⟨$⟩ ( Inverse.to perm Π.⟨$⟩ suc x ) | |
| 319 ≡⟨ inverseˡ perm ⟩ | |
| 320 suc x | |
| 321 ∎ | |
| 50 | 322 |
| 323 piso→ : (x : Fin n ) → p← ( p→ x ) ≡ x | |
| 61 | 324 piso→ x with perm ⟨$⟩ˡ (suc x) | inspect (_⟨$⟩ˡ_ perm ) (suc x) |
| 325 piso→ x | zero | record { eq = e } = ⊥-elim ( sh2 {x} e ) | |
| 326 piso→ x | suc t | _ with perm ⟨$⟩ʳ (suc t) | inspect (_⟨$⟩ʳ_ perm ) (suc t) | |
| 327 piso→ x | suc t | _ | zero | record { eq = e } = ⊥-elim ( sh1 e ) | |
| 328 piso→ x | suc t | record { eq = e0 } | suc t1 | record { eq = e1 } = begin | |
| 329 t1 | |
| 330 ≡⟨ plem2 plem3 ⟩ | |
| 53 | 331 x |
| 61 | 332 ∎ where |
| 333 open ≡-Reasoning | |
| 334 plem2 : suc t1 ≡ suc x → t1 ≡ x | |
| 335 plem2 refl = refl | |
| 336 plem3 : suc t1 ≡ suc x | |
| 337 plem3 = begin | |
| 338 suc t1 | |
| 339 ≡⟨ sym e1 ⟩ | |
| 340 Inverse.to perm Π.⟨$⟩ suc t | |
| 341 ≡⟨ cong (λ k → Inverse.to perm Π.⟨$⟩ k ) (sym e0 ) ⟩ | |
| 342 Inverse.to perm Π.⟨$⟩ ( Inverse.from perm Π.⟨$⟩ suc x ) | |
| 343 ≡⟨ inverseʳ perm ⟩ | |
| 344 suc x | |
| 345 ∎ | |
| 57 | 346 |
| 347 FL→perm : {n : ℕ } → FL n → Permutation n n | |
| 348 FL→perm f0 = pid | |
| 349 FL→perm (x :: fl) = pprep (FL→perm fl) ∘ₚ pins ( toℕ≤pred[n] x ) | |
| 350 | |
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351 t40 = (# 2) :: ( (# 1) :: (( # 0 ) :: f0 )) |
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352 t4 = FL→perm ((# 2) :: t40 ) |
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353 |
| 61 | 354 -- t1 = plist (shrink (pid {3} ∘ₚ (pins (n≤ 1))) refl) |
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355 t2 = plist ((pid {5} ) ∘ₚ transpose (# 2) (# 4)) ∷ plist (pid {5} ∘ₚ reverse ) ∷ [] |
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356 t3 = plist (FL→perm t40) -- ∷ plist (pprep (FL→perm t40)) |
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357 -- ∷ plist ( pprep (FL→perm t40) ∘ₚ pins ( n≤ 0 {3} )) |
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358 -- ∷ plist ( pprep (FL→perm t40 )∘ₚ pins ( n≤ 1 {2} )) |
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359 -- ∷ plist ( pprep (FL→perm t40 )∘ₚ pins ( n≤ 2 {1} )) |
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360 -- ∷ plist ( pprep (FL→perm t40 )∘ₚ pins ( n≤ 3 {0} )) |
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361 ∷ plist ( FL→perm ((# 0) :: t40)) -- (0 ∷ 1 ∷ 2 ∷ []) ∷ |
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362 ∷ plist ( FL→perm ((# 1) :: t40)) -- (0 ∷ 2 ∷ 1 ∷ []) ∷ |
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363 ∷ plist ( FL→perm ((# 2) :: t40)) -- (1 ∷ 0 ∷ 2 ∷ []) ∷ |
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364 ∷ plist ( FL→perm ((# 3) :: t40)) -- (2 ∷ 0 ∷ 1 ∷ []) ∷ |
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365 -- ∷ plist ( FL→perm ((# 3) :: ((# 2) :: ( (# 0) :: (( # 0 ) :: f0 )) ))) -- (1 ∷ 2 ∷ 0 ∷ []) ∷ |
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366 -- ∷ plist ( FL→perm ((# 3) :: ((# 2) :: ( (# 1) :: (( # 0 ) :: f0 )) ))) -- (2 ∷ 1 ∷ 0 ∷ []) ∷ |
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367 -- ∷ plist ( (flip (FL→perm ((# 3) :: ((# 1) :: ( (# 0) :: (( # 0 ) :: f0 )) ))))) |
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368 -- ∷ plist ( (flip (FL→perm ((# 3) :: ((# 1) :: ( (# 0) :: (( # 0 ) :: f0 )) ))) ∘ₚ (FL→perm ((# 3) :: (((# 1) :: ( (# 0) :: (( # 0 ) :: f0 )) )))) )) |
| 57 | 369 ∷ [] |
| 50 | 370 |
| 64 | 371 -- postulate |
| 372 -- p=0 : {n : ℕ } → (perm : Permutation (suc n) (suc n) ) → ((perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) ⟨$⟩ˡ (# 0)) ≡ # 0 | |
| 58 | 373 |
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374 perm→FL : {n : ℕ } → Permutation n n → FL n |
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375 perm→FL {zero} perm = f0 |
| 64 | 376 perm→FL {suc n} perm = (perm ⟨$⟩ʳ (# 0)) :: perm→FL (remove (# 0) perm) |
| 377 -- perm→FL {suc n} perm = (perm ⟨$⟩ʳ (# 0)) :: perm→FL (shrink (perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) (p=0 perm) ) | |
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378 |
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379 -- t5 = plist t4 ∷ plist ( t4 ∘ₚ flip (pins ( n≤ 3 ) )) |
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380 t5 = plist (t4) ∷ plist (flip t4) |
| 74 | 381 ∷ ( toℕ (t4 ⟨$⟩ˡ fromℕ< a<sa) ∷ [] ) |
| 61 | 382 ∷ ( toℕ (t4 ⟨$⟩ʳ (# 0)) ∷ [] ) |
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383 -- ∷ plist ( t4 ∘ₚ flip (pins ( n≤ 1 ) )) |
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384 ∷ plist (remove (# 0) t4 ) |
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385 ∷ plist ( FL→perm t40 ) |
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386 ∷ [] |
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387 |
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388 t6 = perm→FL t4 |
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389 |
| 63 | 390 postulate |
| 391 FL→iso : {n : ℕ } → (fl : FL n ) → perm→FL ( FL→perm fl ) ≡ fl | |
| 392 -- FL→iso f0 = refl | |
| 393 -- FL→iso (x :: fl) = {!!} -- with FL→iso fl | |
| 61 | 394 -- ... | t = {!!} |
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395 |
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396 open _=p=_ |
| 63 | 397 postulate |
| 398 FL←iso : {n : ℕ } → (perm : Permutation n n ) → FL→perm ( perm→FL perm ) =p= perm | |
| 399 -- FL←iso {0} perm = record { peq = λ () } | |
| 400 -- FL←iso {suc n} perm = {!!} | |
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401 |
| 66 | 402 lem2 : {i n : ℕ } → i ≤ n → i ≤ suc n |
| 403 lem2 i≤n = ≤-trans i≤n ( refl-≤s ) | |
| 404 | |
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405 ∀-FL : (n : ℕ ) → List (FL (suc n)) |
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406 ∀-FL x = fls6 x where |
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407 fls4 : ( i n : ℕ ) → (i<n : i ≤ n ) → FL n → List (FL (suc n)) → List (FL (suc n)) |
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408 fls4 zero n i≤n perm x = (zero :: perm ) ∷ x |
| 74 | 409 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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410 fls5 : ( n : ℕ ) → List (FL n) → List (FL (suc n)) → List (FL (suc n)) |
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411 fls5 n [] x = x |
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412 fls5 n (h ∷ x) y = fls5 n x (fls4 n n lem0 h y) |
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413 fls6 : ( n : ℕ ) → List (FL (suc n)) |
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414 fls6 zero = (zero :: f0) ∷ [] |
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415 fls6 (suc n) = fls5 (suc n) (fls6 n) [] |
| 65 | 416 |
| 48 | 417 all-perm : (n : ℕ ) → List (Permutation (suc n) (suc n) ) |
| 418 all-perm n = pls6 n where | |
| 38 | 419 lem1 : {i n : ℕ } → i ≤ n → i < suc n |
| 420 lem1 z≤n = s≤s z≤n | |
| 421 lem1 (s≤s lt) = s≤s (lem1 lt) | |
| 40 | 422 pls4 : ( i n : ℕ ) → (i<n : i ≤ n ) → Permutation n n → List (Permutation (suc n) (suc n)) → List (Permutation (suc n) (suc n)) |
| 48 | 423 pls4 zero n i≤n perm x = (pprep perm ∘ₚ pins i≤n ) ∷ x |
| 424 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 | 425 pls5 : ( n : ℕ ) → List (Permutation n n) → List (Permutation (suc n) (suc n)) → List (Permutation (suc n) (suc n)) |
| 426 pls5 n [] x = x | |
| 427 pls5 n (h ∷ x) y = pls5 n x (pls4 n n lem0 h y) | |
| 428 pls6 : ( n : ℕ ) → List (Permutation (suc n) (suc n)) | |
| 429 pls6 zero = pid ∷ [] | |
| 48 | 430 pls6 (suc n) = pls5 (suc n) (rev (pls6 n) ) [] -- rev to put id first |
| 431 | |
| 432 pls : (n : ℕ ) → List (List ℕ ) | |
| 75 | 433 pls n = Data.List.map plist (all-perm n) |