Mercurial > hg > Members > kono > Proof > galois
annotate Putil.agda @ 89:dcb4450680ab
...
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 28 Aug 2020 13:33:35 +0900 |
| parents | 405c1f727ffe |
| children | bb8c5b366219 |
| 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 -- describe property of pins ( move 0 to any position) | |
| 33 -- describe property of shrink ( remove one column ) | |
| 34 -- prove FL→iso | |
| 35 -- prove FL←iso | |
| 36 | |
| 48 | 37 -- we already have refl and trans in the Symmetric Group |
| 41 | 38 |
| 34 | 39 pprep : {n : ℕ } → Permutation n n → Permutation (suc n) (suc n) |
| 40 pprep {n} perm = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where | |
| 33 | 41 p→ : Fin (suc n) → Fin (suc n) |
| 34 | 42 p→ zero = zero |
|
60
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
43 p→ (suc x) = suc ( perm ⟨$⟩ʳ x) |
| 33 | 44 |
| 34 | 45 p← : Fin (suc n) → Fin (suc n) |
| 46 p← zero = zero | |
|
60
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
47 p← (suc x) = suc ( perm ⟨$⟩ˡ x) |
| 34 | 48 |
| 49 piso← : (x : Fin (suc n)) → p→ ( p← x ) ≡ x | |
| 50 piso← zero = refl | |
|
60
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
51 piso← (suc x) = cong (λ k → suc k ) (inverseʳ perm) |
| 33 | 52 |
| 34 | 53 piso→ : (x : Fin (suc n)) → p← ( p→ x ) ≡ x |
| 54 piso→ zero = refl | |
|
60
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
55 piso→ (suc x) = cong (λ k → suc k ) (inverseˡ perm) |
| 33 | 56 |
| 34 | 57 pswap : {n : ℕ } → Permutation n n → Permutation (suc (suc n)) (suc (suc n )) |
| 58 pswap {n} perm = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where | |
| 59 p→ : Fin (suc (suc n)) → Fin (suc (suc n)) | |
| 60 p→ zero = suc zero | |
| 61 p→ (suc zero) = zero | |
| 62 | 62 p→ (suc (suc x)) = suc ( suc ( perm ⟨$⟩ʳ x) ) |
| 18 | 63 |
| 34 | 64 p← : Fin (suc (suc n)) → Fin (suc (suc n)) |
| 65 p← zero = suc zero | |
| 66 p← (suc zero) = zero | |
| 62 | 67 p← (suc (suc x)) = suc ( suc ( perm ⟨$⟩ˡ x) ) |
| 34 | 68 |
| 69 piso← : (x : Fin (suc (suc n)) ) → p→ ( p← x ) ≡ x | |
| 70 piso← zero = refl | |
| 71 piso← (suc zero) = refl | |
| 62 | 72 piso← (suc (suc x)) = cong (λ k → suc (suc k) ) (inverseʳ perm) |
| 16 | 73 |
| 34 | 74 piso→ : (x : Fin (suc (suc n)) ) → p← ( p→ x ) ≡ x |
| 75 piso→ zero = refl | |
| 76 piso→ (suc zero) = refl | |
| 62 | 77 piso→ (suc (suc x)) = cong (λ k → suc (suc k) ) (inverseˡ perm) |
| 34 | 78 |
| 79 -- enumeration | |
| 80 | |
| 44 | 81 psawpn : {n : ℕ} → 1 < n → Permutation n n |
| 82 psawpn {suc zero} (s≤s ()) | |
| 83 psawpn {suc n} (s≤s (s≤s x)) = pswap pid | |
| 34 | 84 |
| 35 | 85 pfill : { n m : ℕ } → m ≤ n → Permutation m m → Permutation n n |
| 86 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 | |
| 87 pfill1 : (i : ℕ ) → i ≤ n → Permutation (n - i) (n - i) → Permutation n n | |
| 88 pfill1 0 _ perm = perm | |
| 89 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 | 90 |
| 48 | 91 -- |
| 92 -- psawpim (inseert swap at position m ) | |
| 93 -- | |
| 45 | 94 psawpim : {n m : ℕ} → suc (suc m) ≤ n → Permutation n n |
| 95 psawpim {n} {m} m≤n = pfill m≤n ( psawpn (s≤s (s≤s z≤n)) ) | |
| 96 | |
| 97 n≤ : (i : ℕ ) → {j : ℕ } → i ≤ i + j | |
| 98 n≤ (zero) {j} = z≤n | |
| 99 n≤ (suc i) {j} = s≤s ( n≤ i ) | |
| 100 | |
| 101 lem0 : {n : ℕ } → n ≤ n | |
| 102 lem0 {zero} = z≤n | |
| 103 lem0 {suc n} = s≤s lem0 | |
| 104 | |
| 105 lem00 : {n m : ℕ } → n ≡ m → n ≤ m | |
| 106 lem00 refl = lem0 | |
| 44 | 107 |
| 80 | 108 plist1 : {n : ℕ} → Permutation (suc n) (suc n) → (i : ℕ ) → i < suc n → List ℕ |
| 109 plist1 {n} perm zero _ = toℕ ( perm ⟨$⟩ˡ (fromℕ< {zero} (s≤s z≤n))) ∷ [] | |
| 110 plist1 {n} perm (suc i) (s≤s lt) = toℕ ( perm ⟨$⟩ˡ (fromℕ< (s≤s lt))) ∷ plist1 perm i (<-trans lt a<sa) | |
| 111 | |
| 37 | 112 plist : {n : ℕ} → Permutation n n → List ℕ |
| 113 plist {0} perm = [] | |
| 80 | 114 plist {suc n} perm = rev (plist1 perm n a<sa) |
| 115 | |
| 89 | 116 -- pconcat : {n m : ℕ } → Permutation m m → Permutation n n → Permutation (m + n) (m + n) |
| 117 -- pconcat {n} {m} p q = pfill {n + m} {m} ? p ∘ₚ ? | |
| 118 | |
| 119 -- inductivley enmumerate permutations | |
| 120 -- from n-1 length create n length inserting new element at position m | |
| 121 | |
| 122 -- 0 ∷ 1 ∷ 2 ∷ 3 ∷ [] -- 0 ∷ 1 ∷ 2 ∷ 3 ∷ [] | |
| 123 -- 1 ∷ 0 ∷ 2 ∷ 3 ∷ [] plist ( pins {3} (n≤ 1) ) 1 ∷ 0 ∷ 2 ∷ 3 ∷ [] | |
| 124 -- 1 ∷ 2 ∷ 0 ∷ 3 ∷ [] plist ( pins {3} (n≤ 2) ) 2 ∷ 0 ∷ 1 ∷ 3 ∷ [] | |
| 125 -- 1 ∷ 2 ∷ 3 ∷ 0 ∷ [] plist ( pins {3} (n≤ 3) ) 3 ∷ 0 ∷ 1 ∷ 2 ∷ [] | |
| 126 pins : {n m : ℕ} → m ≤ n → Permutation (suc n) (suc n) | |
| 127 pins {_} {zero} _ = pid | |
| 128 pins {suc _} {suc zero} _ = pswap pid | |
| 129 pins {suc (suc n)} {suc m} (s≤s m<n) = pins1 (suc m) (suc (suc n)) lem0 where | |
| 130 pins1 : (i j : ℕ ) → j ≤ suc (suc n) → Permutation (suc (suc (suc n ))) (suc (suc (suc n))) | |
| 131 pins1 _ zero _ = pid | |
| 132 pins1 zero _ _ = pid | |
| 133 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 ) | |
| 134 | |
| 135 pins' : {n m : ℕ} → m ≤ n → Permutation (suc n) (suc n) | |
| 136 pins' {_} {zero} _ = pid | |
| 137 pins' {suc n} {suc m} (s≤s m≤n) = permutation p← p→ record { left-inverse-of = piso← ; right-inverse-of = piso→ } where | |
| 138 | |
| 139 next : Fin (suc (suc n)) → Fin (suc (suc n)) | |
| 140 next zero = suc zero | |
| 141 next (suc x) = fromℕ< (≤-trans (fin<n {_} {x} ) refl-≤s ) | |
| 142 | |
| 143 p→ : Fin (suc (suc n)) → Fin (suc (suc n)) | |
| 144 p→ x with <-cmp (toℕ x) (suc m) | |
| 145 ... | tri< a ¬b ¬c = fromℕ< (≤-trans (s≤s a) (≤-trans (s≤s (s≤s m≤n) ) lem0 )) | |
| 146 ... | tri≈ ¬a b ¬c = zero | |
| 147 ... | tri> ¬a ¬b c = x | |
| 148 | |
| 149 p← : Fin (suc (suc n)) → Fin (suc (suc n)) | |
| 150 p← zero = fromℕ< (s≤s (s≤s m≤n)) | |
| 151 p← (suc x) with <-cmp (toℕ x) (suc m) | |
| 152 ... | tri< a ¬b ¬c = fromℕ< (≤-trans (fin<n {_} {x}) refl-≤s ) | |
| 153 ... | tri≈ ¬a b ¬c = suc x | |
| 154 ... | tri> ¬a ¬b c = suc x | |
| 155 | |
| 156 piso← : (x : Fin (suc (suc n)) ) → p→ ( p← x ) ≡ x | |
| 157 piso← = {!!} | |
| 158 | |
| 159 piso→ : (x : Fin (suc (suc n)) ) → p← ( p→ x ) ≡ x | |
| 160 piso→ = {!!} | |
| 161 | |
| 162 t7 = plist (pins' {3} (n≤ 3)) ∷ plist (flip ( pins' {3} (n≤ 3) )) ∷ plist ( pins' {3} (n≤ 3) ∘ₚ flip ( pins' {3} (n≤ 3))) ∷ [] | |
| 163 t8 = {!!} | |
| 164 | |
| 165 | |
| 80 | 166 plist2 : {n : ℕ} → Permutation (suc n) (suc n) → (i : ℕ ) → i < suc n → List ℕ |
| 167 plist2 {n} perm zero _ = toℕ ( perm ⟨$⟩ʳ (fromℕ< {zero} (s≤s z≤n))) ∷ [] | |
| 168 plist2 {n} perm (suc i) (s≤s lt) = toℕ ( perm ⟨$⟩ʳ (fromℕ< (s≤s lt))) ∷ plist2 perm i (<-trans lt a<sa) | |
| 169 | |
| 170 plist0 : {n : ℕ} → Permutation n n → List ℕ | |
| 171 plist0 {0} perm = [] | |
| 172 plist0 {suc n} perm = plist2 perm n a<sa | |
| 173 | |
| 85 | 174 open _=p=_ |
| 175 | |
|
86
c5329963c583
(x : Permutation 1 1 ) → x =p= pid
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
85
diff
changeset
|
176 -- |
|
c5329963c583
(x : Permutation 1 1 ) → x =p= pid
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
85
diff
changeset
|
177 -- plist cong |
|
c5329963c583
(x : Permutation 1 1 ) → x =p= pid
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
85
diff
changeset
|
178 -- |
| 85 | 179 ←pleq : {n : ℕ} → (x y : Permutation n n ) → x =p= y → plist0 x ≡ plist0 y |
| 180 ←pleq {zero} x y eq = refl | |
| 181 ←pleq {suc n} x y eq = ←pleq1 n a<sa where | |
| 182 ←pleq1 : (i : ℕ ) → (i<sn : i < suc n ) → plist2 x i i<sn ≡ plist2 y i i<sn | |
| 183 ←pleq1 zero _ = cong ( λ k → toℕ k ∷ [] ) ( peq eq (fromℕ< {zero} (s≤s z≤n))) | |
| 184 ←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 | 185 |
| 186 headeq : {A : Set } → {x y : A } → {xt yt : List A } → (x ∷ xt) ≡ (y ∷ yt) → x ≡ y | |
| 187 headeq refl = refl | |
| 188 | |
| 189 taileq : {A : Set } → {x y : A } → {xt yt : List A } → (x ∷ xt) ≡ (y ∷ yt) → xt ≡ yt | |
| 190 taileq refl = refl | |
| 191 | |
|
86
c5329963c583
(x : Permutation 1 1 ) → x =p= pid
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
85
diff
changeset
|
192 -- |
|
c5329963c583
(x : Permutation 1 1 ) → x =p= pid
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
85
diff
changeset
|
193 -- plist equalizer |
|
c5329963c583
(x : Permutation 1 1 ) → x =p= pid
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
85
diff
changeset
|
194 -- |
| 80 | 195 pleq : {n : ℕ} → (x y : Permutation n n ) → plist0 x ≡ plist0 y → x =p= y |
| 196 pleq {0} x y refl = record { peq = λ q → pleq0 q } where | |
| 197 pleq0 : (q : Fin 0 ) → (x ⟨$⟩ʳ q) ≡ (y ⟨$⟩ʳ q) | |
| 198 pleq0 () | |
| 199 pleq {suc n} x y eq = record { peq = λ q → pleq1 n a<sa eq q fin<n } where | |
| 200 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 | 201 pleq1 zero i<sn eq q q<i with <-cmp (toℕ q) zero |
| 202 ... | tri< () ¬b ¬c | |
| 203 ... | tri> ¬a ¬b c = ⊥-elim (nat-≤> c q<i ) | |
| 204 ... | tri≈ ¬a b ¬c = begin | |
| 205 x ⟨$⟩ʳ q | |
| 206 ≡⟨ cong ( λ k → x ⟨$⟩ʳ k ) (toℕ-injective b )⟩ | |
| 207 x ⟨$⟩ʳ zero | |
| 208 ≡⟨ toℕ-injective (headeq eq) ⟩ | |
| 209 y ⟨$⟩ʳ zero | |
| 210 ≡⟨ cong ( λ k → y ⟨$⟩ʳ k ) (sym (toℕ-injective b )) ⟩ | |
| 211 y ⟨$⟩ʳ q | |
| 212 ∎ where | |
| 213 open ≡-Reasoning | |
| 80 | 214 pleq1 (suc i) (s≤s i<sn) eq q q<i with <-cmp (toℕ q) (suc i) |
| 215 ... | tri< a ¬b ¬c = pleq1 i (<-trans i<sn a<sa ) (taileq eq) q a | |
| 81 | 216 ... | tri> ¬a ¬b c = ⊥-elim (nat-≤> c q<i ) |
| 80 | 217 ... | tri≈ ¬a b ¬c = begin |
| 218 x ⟨$⟩ʳ q | |
| 219 ≡⟨ cong (λ k → x ⟨$⟩ʳ k) (pleq3 b) ⟩ | |
| 220 x ⟨$⟩ʳ (suc (fromℕ< i<sn)) | |
| 221 ≡⟨ toℕ-injective pleq2 ⟩ | |
| 222 y ⟨$⟩ʳ (suc (fromℕ< i<sn)) | |
| 223 ≡⟨ cong (λ k → y ⟨$⟩ʳ k) (sym (pleq3 b)) ⟩ | |
| 224 y ⟨$⟩ʳ q | |
| 225 ∎ where | |
| 226 open ≡-Reasoning | |
| 227 pleq3 : toℕ q ≡ suc i → q ≡ suc (fromℕ< i<sn) | |
| 228 pleq3 tq=si = toℕ-injective ( begin | |
| 229 toℕ q | |
| 230 ≡⟨ b ⟩ | |
| 231 suc i | |
| 232 ≡⟨ sym (toℕ-fromℕ< (s≤s i<sn)) ⟩ | |
| 233 toℕ (fromℕ< (s≤s i<sn)) | |
| 234 ≡⟨⟩ | |
| 235 toℕ (suc (fromℕ< i<sn)) | |
| 236 ∎ ) where open ≡-Reasoning | |
| 237 pleq2 : toℕ ( x ⟨$⟩ʳ (suc (fromℕ< i<sn)) ) ≡ toℕ ( y ⟨$⟩ʳ (suc (fromℕ< i<sn)) ) | |
| 238 pleq2 = headeq eq | |
| 37 | 239 |
| 87 | 240 pprep-cong : {n : ℕ} → {x y : Permutation n n } → x =p= y → pprep x =p= pprep y |
| 241 pprep-cong {n} {x} {y} x=y = record { peq = pprep-cong1 } where | |
| 242 pprep-cong1 : (q : Fin (suc n)) → (pprep x ⟨$⟩ʳ q) ≡ (pprep y ⟨$⟩ʳ q) | |
| 243 pprep-cong1 zero = refl | |
| 244 pprep-cong1 (suc q) = begin | |
| 245 pprep x ⟨$⟩ʳ suc q | |
| 246 ≡⟨⟩ | |
| 247 suc ( x ⟨$⟩ʳ q ) | |
| 248 ≡⟨ cong ( λ k → suc k ) ( peq x=y q ) ⟩ | |
| 249 suc ( y ⟨$⟩ʳ q ) | |
| 250 ≡⟨⟩ | |
| 251 pprep y ⟨$⟩ʳ suc q | |
| 252 ∎ where open ≡-Reasoning | |
| 253 | |
| 254 pprep-dist : {n : ℕ} → {x y : Permutation n n } → pprep (x ∘ₚ y) =p= (pprep x ∘ₚ pprep y) | |
| 255 pprep-dist {n} {x} {y} = record { peq = pprep-dist1 } where | |
| 256 pprep-dist1 : (q : Fin (suc n)) → (pprep (x ∘ₚ y) ⟨$⟩ʳ q) ≡ ((pprep x ∘ₚ pprep y) ⟨$⟩ʳ q) | |
| 257 pprep-dist1 zero = refl | |
| 258 pprep-dist1 (suc q) = cong ( λ k → suc k ) refl | |
| 259 | |
| 260 pswap-cong : {n : ℕ} → {x y : Permutation n n } → x =p= y → pswap x =p= pswap y | |
| 261 pswap-cong {n} {x} {y} x=y = record { peq = pswap-cong1 } where | |
| 262 pswap-cong1 : (q : Fin (suc (suc n))) → (pswap x ⟨$⟩ʳ q) ≡ (pswap y ⟨$⟩ʳ q) | |
| 263 pswap-cong1 zero = refl | |
| 264 pswap-cong1 (suc zero) = refl | |
| 265 pswap-cong1 (suc (suc q)) = begin | |
| 266 pswap x ⟨$⟩ʳ suc (suc q) | |
| 267 ≡⟨⟩ | |
| 268 suc (suc (x ⟨$⟩ʳ q)) | |
| 269 ≡⟨ cong ( λ k → suc (suc k) ) ( peq x=y q ) ⟩ | |
| 270 suc (suc (y ⟨$⟩ʳ q)) | |
| 271 ≡⟨⟩ | |
| 272 pswap y ⟨$⟩ʳ suc (suc q) | |
| 273 ∎ where open ≡-Reasoning | |
| 274 | |
| 275 pswap-dist : {n : ℕ} → {x y : Permutation n n } → pprep (pprep (x ∘ₚ y)) =p= (pswap x ∘ₚ pswap y) | |
| 276 pswap-dist {n} {x} {y} = record { peq = pswap-dist1 } where | |
| 277 pswap-dist1 : (q : Fin (suc (suc n))) → ((pprep (pprep (x ∘ₚ y))) ⟨$⟩ʳ q) ≡ ((pswap x ∘ₚ pswap y) ⟨$⟩ʳ q) | |
| 278 pswap-dist1 zero = refl | |
| 279 pswap-dist1 (suc zero) = refl | |
| 280 pswap-dist1 (suc (suc q)) = cong ( λ k → suc (suc k) ) refl | |
|
86
c5329963c583
(x : Permutation 1 1 ) → x =p= pid
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
85
diff
changeset
|
281 |
|
49
8b3b95362ca9
remove (fromℕ≤ a<sa) perm is no good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
48
diff
changeset
|
282 data FL : (n : ℕ )→ Set where |
|
8b3b95362ca9
remove (fromℕ≤ a<sa) perm is no good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
48
diff
changeset
|
283 f0 : FL 0 |
|
8b3b95362ca9
remove (fromℕ≤ a<sa) perm is no good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
48
diff
changeset
|
284 _::_ : { n : ℕ } → Fin (suc n ) → FL n → FL (suc n) |
|
8b3b95362ca9
remove (fromℕ≤ a<sa) perm is no good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
48
diff
changeset
|
285 |
| 50 | 286 open import logic |
| 287 | |
| 88 | 288 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 |
| 289 shlem→ perm p0=0 x px=0 = begin | |
| 61 | 290 x ≡⟨ sym ( inverseʳ perm ) ⟩ |
| 291 perm ⟨$⟩ʳ ( perm ⟨$⟩ˡ x) ≡⟨ cong (λ k → perm ⟨$⟩ʳ k ) px=0 ⟩ | |
| 292 perm ⟨$⟩ʳ zero ≡⟨ cong (λ k → perm ⟨$⟩ʳ k ) (sym p0=0) ⟩ | |
| 293 perm ⟨$⟩ʳ ( perm ⟨$⟩ˡ zero) ≡⟨ inverseʳ perm ⟩ | |
| 294 zero | |
| 295 ∎ where open ≡-Reasoning | |
| 54 | 296 |
| 88 | 297 shlem← : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → (p0=0 : perm ⟨$⟩ˡ (# 0) ≡ # 0 ) → (x : Fin (suc n)) → perm ⟨$⟩ʳ x ≡ zero → x ≡ zero |
| 298 shlem← perm p0=0 x px=0 = begin | |
| 61 | 299 x ≡⟨ sym (inverseˡ perm ) ⟩ |
| 300 perm ⟨$⟩ˡ ( perm ⟨$⟩ʳ x ) ≡⟨ cong (λ k → perm ⟨$⟩ˡ k ) px=0 ⟩ | |
| 301 perm ⟨$⟩ˡ zero ≡⟨ p0=0 ⟩ | |
| 302 zero | |
| 303 ∎ where open ≡-Reasoning | |
| 54 | 304 |
| 88 | 305 sh2 : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → (p0=0 : perm ⟨$⟩ˡ (# 0) ≡ # 0 ) → {x : Fin n} → ¬ perm ⟨$⟩ˡ (suc x) ≡ zero |
| 306 sh2 perm p0=0 {x} eq with shlem→ perm p0=0 (suc x) eq | |
| 307 sh2 perm p0=0 {x} eq | () | |
| 308 | |
| 309 sh1 : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → (p0=0 : perm ⟨$⟩ˡ (# 0) ≡ # 0 ) → {x : Fin n} → ¬ perm ⟨$⟩ʳ (suc x) ≡ zero | |
| 310 sh1 perm p0=0 {x} eq with shlem← perm p0=0 (suc x) eq | |
| 311 sh1 perm p0=0 {x} eq | () | |
| 312 | |
| 313 | |
| 314 -- 0 ∷ 1 ∷ 2 ∷ 3 ∷ [] → 0 ∷ 1 ∷ 2 ∷ [] | |
| 315 shrink : {n : ℕ} → (perm : Permutation (suc n) (suc n) ) → perm ⟨$⟩ˡ (# 0) ≡ # 0 → Permutation n n | |
| 316 shrink {n} perm p0=0 = permutation p→ p← record { left-inverse-of = piso→ ; right-inverse-of = piso← } where | |
| 57 | 317 |
| 61 | 318 p→ : Fin n → Fin n |
| 88 | 319 p→ x with perm ⟨$⟩ʳ (suc x) | inspect (_⟨$⟩ʳ_ perm ) (suc x) |
| 320 p→ x | zero | record { eq = e } = ⊥-elim ( sh1 perm p0=0 {x} e ) | |
| 61 | 321 p→ x | suc t | _ = t |
| 50 | 322 |
| 323 p← : Fin n → Fin n | |
| 88 | 324 p← x with perm ⟨$⟩ˡ (suc x) | inspect (_⟨$⟩ˡ_ perm ) (suc x) |
| 325 p← x | zero | record { eq = e } = ⊥-elim ( sh2 perm p0=0 {x} e ) | |
| 61 | 326 p← x | suc t | _ = t |
| 50 | 327 |
| 328 piso← : (x : Fin n ) → p→ ( p← x ) ≡ x | |
| 88 | 329 piso← x with perm ⟨$⟩ˡ (suc x) | inspect (_⟨$⟩ˡ_ perm ) (suc x) |
| 330 piso← x | zero | record { eq = e } = ⊥-elim ( sh2 perm p0=0 {x} e ) | |
| 331 piso← x | suc t | _ with perm ⟨$⟩ʳ (suc t) | inspect (_⟨$⟩ʳ_ perm ) (suc t) | |
| 332 piso← x | suc t | _ | zero | record { eq = e } = ⊥-elim ( sh1 perm p0=0 e ) | |
| 61 | 333 piso← x | suc t | record { eq = e0 } | suc t1 | record { eq = e1 } = begin |
| 334 t1 | |
| 335 ≡⟨ plem0 plem1 ⟩ | |
| 52 | 336 x |
| 61 | 337 ∎ where |
| 338 open ≡-Reasoning | |
| 339 plem0 : suc t1 ≡ suc x → t1 ≡ x | |
| 340 plem0 refl = refl | |
| 341 plem1 : suc t1 ≡ suc x | |
| 342 plem1 = begin | |
| 343 suc t1 | |
| 344 ≡⟨ sym e1 ⟩ | |
| 88 | 345 Inverse.to perm Π.⟨$⟩ suc t |
| 346 ≡⟨ cong (λ k → Inverse.to perm Π.⟨$⟩ k ) (sym e0) ⟩ | |
| 347 Inverse.to perm Π.⟨$⟩ ( Inverse.from perm Π.⟨$⟩ suc x ) | |
| 348 ≡⟨ inverseʳ perm ⟩ | |
| 61 | 349 suc x |
| 350 ∎ | |
| 50 | 351 |
| 352 piso→ : (x : Fin n ) → p← ( p→ x ) ≡ x | |
| 88 | 353 piso→ x with perm ⟨$⟩ʳ (suc x) | inspect (_⟨$⟩ʳ_ perm ) (suc x) |
| 354 piso→ x | zero | record { eq = e } = ⊥-elim ( sh1 perm p0=0 {x} e ) | |
| 355 piso→ x | suc t | _ with perm ⟨$⟩ˡ (suc t) | inspect (_⟨$⟩ˡ_ perm ) (suc t) | |
| 356 piso→ x | suc t | _ | zero | record { eq = e } = ⊥-elim ( sh2 perm p0=0 e ) | |
| 61 | 357 piso→ x | suc t | record { eq = e0 } | suc t1 | record { eq = e1 } = begin |
| 358 t1 | |
| 359 ≡⟨ plem2 plem3 ⟩ | |
| 53 | 360 x |
| 61 | 361 ∎ where |
| 362 open ≡-Reasoning | |
| 363 plem2 : suc t1 ≡ suc x → t1 ≡ x | |
| 364 plem2 refl = refl | |
| 365 plem3 : suc t1 ≡ suc x | |
| 366 plem3 = begin | |
| 367 suc t1 | |
| 368 ≡⟨ sym e1 ⟩ | |
| 88 | 369 Inverse.from perm Π.⟨$⟩ suc t |
| 370 ≡⟨ cong (λ k → Inverse.from perm Π.⟨$⟩ k ) (sym e0 ) ⟩ | |
| 371 Inverse.from perm Π.⟨$⟩ ( Inverse.to perm Π.⟨$⟩ suc x ) | |
| 372 ≡⟨ inverseˡ perm ⟩ | |
| 61 | 373 suc x |
| 374 ∎ | |
| 57 | 375 |
| 88 | 376 shrink-iso : { n : ℕ } → {perm : Permutation n n} → shrink (pprep perm) refl =p= perm |
| 377 shrink-iso {n} {perm} = record { peq = λ q → refl } | |
| 378 | |
| 57 | 379 FL→perm : {n : ℕ } → FL n → Permutation n n |
| 380 FL→perm f0 = pid | |
| 381 FL→perm (x :: fl) = pprep (FL→perm fl) ∘ₚ pins ( toℕ≤pred[n] x ) | |
| 382 | |
|
60
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
383 t40 = (# 2) :: ( (# 1) :: (( # 0 ) :: f0 )) |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
384 t4 = FL→perm ((# 2) :: t40 ) |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
385 |
| 61 | 386 -- t1 = plist (shrink (pid {3} ∘ₚ (pins (n≤ 1))) refl) |
|
60
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
387 t2 = plist ((pid {5} ) ∘ₚ transpose (# 2) (# 4)) ∷ plist (pid {5} ∘ₚ reverse ) ∷ [] |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
388 t3 = plist (FL→perm t40) -- ∷ plist (pprep (FL→perm t40)) |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
389 -- ∷ plist ( pprep (FL→perm t40) ∘ₚ pins ( n≤ 0 {3} )) |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
390 -- ∷ plist ( pprep (FL→perm t40 )∘ₚ pins ( n≤ 1 {2} )) |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
391 -- ∷ plist ( pprep (FL→perm t40 )∘ₚ pins ( n≤ 2 {1} )) |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
392 -- ∷ plist ( pprep (FL→perm t40 )∘ₚ pins ( n≤ 3 {0} )) |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
393 ∷ plist ( FL→perm ((# 0) :: t40)) -- (0 ∷ 1 ∷ 2 ∷ []) ∷ |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
394 ∷ plist ( FL→perm ((# 1) :: t40)) -- (0 ∷ 2 ∷ 1 ∷ []) ∷ |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
395 ∷ plist ( FL→perm ((# 2) :: t40)) -- (1 ∷ 0 ∷ 2 ∷ []) ∷ |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
396 ∷ plist ( FL→perm ((# 3) :: t40)) -- (2 ∷ 0 ∷ 1 ∷ []) ∷ |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
397 -- ∷ plist ( FL→perm ((# 3) :: ((# 2) :: ( (# 0) :: (( # 0 ) :: f0 )) ))) -- (1 ∷ 2 ∷ 0 ∷ []) ∷ |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
398 -- ∷ plist ( FL→perm ((# 3) :: ((# 2) :: ( (# 1) :: (( # 0 ) :: f0 )) ))) -- (2 ∷ 1 ∷ 0 ∷ []) ∷ |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
399 -- ∷ plist ( (flip (FL→perm ((# 3) :: ((# 1) :: ( (# 0) :: (( # 0 ) :: f0 )) ))))) |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
400 -- ∷ plist ( (flip (FL→perm ((# 3) :: ((# 1) :: ( (# 0) :: (( # 0 ) :: f0 )) ))) ∘ₚ (FL→perm ((# 3) :: (((# 1) :: ( (# 0) :: (( # 0 ) :: f0 )) )))) )) |
| 57 | 401 ∷ [] |
| 50 | 402 |
| 58 | 403 |
|
49
8b3b95362ca9
remove (fromℕ≤ a<sa) perm is no good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
48
diff
changeset
|
404 perm→FL : {n : ℕ } → Permutation n n → FL n |
|
8b3b95362ca9
remove (fromℕ≤ a<sa) perm is no good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
48
diff
changeset
|
405 perm→FL {zero} perm = f0 |
| 89 | 406 perm→FL {suc n} perm = (perm ⟨$⟩ʳ (# 0)) :: perm→FL (remove (# 0) perm) |
| 407 -- perm→FL {suc n} perm = (perm ⟨$⟩ʳ (# 0)) :: perm→FL (shrink (perm ∘ₚ flip (pins (toℕ≤pred[n] (perm ⟨$⟩ʳ (# 0))))) (p=0 perm) ) | |
|
60
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
408 |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
409 -- t5 = plist t4 ∷ plist ( t4 ∘ₚ flip (pins ( n≤ 3 ) )) |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
410 t5 = plist (t4) ∷ plist (flip t4) |
| 74 | 411 ∷ ( toℕ (t4 ⟨$⟩ˡ fromℕ< a<sa) ∷ [] ) |
| 61 | 412 ∷ ( toℕ (t4 ⟨$⟩ʳ (# 0)) ∷ [] ) |
|
60
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
413 -- ∷ plist ( t4 ∘ₚ flip (pins ( n≤ 1 ) )) |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
414 ∷ plist (remove (# 0) t4 ) |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
415 ∷ plist ( FL→perm t40 ) |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
416 ∷ [] |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
417 |
|
48926e810f5f
perm→FL done. pprep fix.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
59
diff
changeset
|
418 t6 = perm→FL t4 |
|
49
8b3b95362ca9
remove (fromℕ≤ a<sa) perm is no good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
48
diff
changeset
|
419 |
| 63 | 420 postulate |
| 421 FL→iso : {n : ℕ } → (fl : FL n ) → perm→FL ( FL→perm fl ) ≡ fl | |
| 422 -- FL→iso f0 = refl | |
| 423 -- FL→iso (x :: fl) = {!!} -- with FL→iso fl | |
| 61 | 424 -- ... | t = {!!} |
|
49
8b3b95362ca9
remove (fromℕ≤ a<sa) perm is no good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
48
diff
changeset
|
425 |
|
8b3b95362ca9
remove (fromℕ≤ a<sa) perm is no good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
48
diff
changeset
|
426 open _=p=_ |
| 63 | 427 postulate |
| 428 FL←iso : {n : ℕ } → (perm : Permutation n n ) → FL→perm ( perm→FL perm ) =p= perm | |
| 429 -- FL←iso {0} perm = record { peq = λ () } | |
| 430 -- FL←iso {suc n} perm = {!!} | |
|
49
8b3b95362ca9
remove (fromℕ≤ a<sa) perm is no good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
48
diff
changeset
|
431 |
| 66 | 432 lem2 : {i n : ℕ } → i ≤ n → i ≤ suc n |
| 433 lem2 i≤n = ≤-trans i≤n ( refl-≤s ) | |
| 434 | |
|
67
3825082ebdbd
∀-FL : (n : ℕ ) → List (FL (suc n))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
66
diff
changeset
|
435 ∀-FL : (n : ℕ ) → List (FL (suc n)) |
|
3825082ebdbd
∀-FL : (n : ℕ ) → List (FL (suc n))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
66
diff
changeset
|
436 ∀-FL x = fls6 x where |
|
3825082ebdbd
∀-FL : (n : ℕ ) → List (FL (suc n))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
66
diff
changeset
|
437 fls4 : ( i n : ℕ ) → (i<n : i ≤ n ) → FL n → List (FL (suc n)) → List (FL (suc n)) |
|
3825082ebdbd
∀-FL : (n : ℕ ) → List (FL (suc n))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
66
diff
changeset
|
438 fls4 zero n i≤n perm x = (zero :: perm ) ∷ x |
| 74 | 439 fls4 (suc i) n i≤n perm x = fls4 i n (≤-trans refl-≤s i≤n ) perm ((fromℕ< (s≤s i≤n) :: perm ) ∷ x) |
|
67
3825082ebdbd
∀-FL : (n : ℕ ) → List (FL (suc n))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
66
diff
changeset
|
440 fls5 : ( n : ℕ ) → List (FL n) → List (FL (suc n)) → List (FL (suc n)) |
|
3825082ebdbd
∀-FL : (n : ℕ ) → List (FL (suc n))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
66
diff
changeset
|
441 fls5 n [] x = x |
|
3825082ebdbd
∀-FL : (n : ℕ ) → List (FL (suc n))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
66
diff
changeset
|
442 fls5 n (h ∷ x) y = fls5 n x (fls4 n n lem0 h y) |
|
3825082ebdbd
∀-FL : (n : ℕ ) → List (FL (suc n))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
66
diff
changeset
|
443 fls6 : ( n : ℕ ) → List (FL (suc n)) |
|
3825082ebdbd
∀-FL : (n : ℕ ) → List (FL (suc n))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
66
diff
changeset
|
444 fls6 zero = (zero :: f0) ∷ [] |
|
3825082ebdbd
∀-FL : (n : ℕ ) → List (FL (suc n))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
66
diff
changeset
|
445 fls6 (suc n) = fls5 (suc n) (fls6 n) [] |
| 65 | 446 |
| 48 | 447 all-perm : (n : ℕ ) → List (Permutation (suc n) (suc n) ) |
| 448 all-perm n = pls6 n where | |
| 38 | 449 lem1 : {i n : ℕ } → i ≤ n → i < suc n |
| 450 lem1 z≤n = s≤s z≤n | |
| 451 lem1 (s≤s lt) = s≤s (lem1 lt) | |
| 40 | 452 pls4 : ( i n : ℕ ) → (i<n : i ≤ n ) → Permutation n n → List (Permutation (suc n) (suc n)) → List (Permutation (suc n) (suc n)) |
| 48 | 453 pls4 zero n i≤n perm x = (pprep perm ∘ₚ pins i≤n ) ∷ x |
| 454 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 | 455 pls5 : ( n : ℕ ) → List (Permutation n n) → List (Permutation (suc n) (suc n)) → List (Permutation (suc n) (suc n)) |
| 456 pls5 n [] x = x | |
| 457 pls5 n (h ∷ x) y = pls5 n x (pls4 n n lem0 h y) | |
| 458 pls6 : ( n : ℕ ) → List (Permutation (suc n) (suc n)) | |
| 459 pls6 zero = pid ∷ [] | |
| 48 | 460 pls6 (suc n) = pls5 (suc n) (rev (pls6 n) ) [] -- rev to put id first |
| 461 | |
| 462 pls : (n : ℕ ) → List (List ℕ ) | |
| 75 | 463 pls n = Data.List.map plist (all-perm n) |