Mercurial > hg > Members > kono > Proof > automaton
annotate automaton-in-agda/src/finiteSetUtil.agda @ 278:e89957b99662
dup in finiteSet in long list
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 26 Dec 2021 12:38:37 +0900 |
| parents | 8006cbd87b20 |
| children | 681df12f0edc |
| rev | line source |
|---|---|
| 163 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 2 | |
| 3 module finiteSetUtil where | |
| 141 | 4 |
| 5 open import Data.Nat hiding ( _≟_ ) | |
|
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6 open import Data.Fin renaming ( _<_ to _<<_ ; _>_ to _f>_ ; _≟_ to _f≟_ ) hiding (_≤_ ) |
| 141 | 7 open import Data.Fin.Properties |
| 8 open import Data.Empty | |
| 9 open import Relation.Nullary | |
| 10 open import Relation.Binary.Definitions | |
| 11 open import Relation.Binary.PropositionalEquality | |
| 12 open import logic | |
| 13 open import nat | |
| 14 open import finiteSet | |
| 163 | 15 open import fin |
| 141 | 16 open import Data.Nat.Properties as NatP hiding ( _≟_ ) |
| 17 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) | |
| 18 | |
| 163 | 19 record Found ( Q : Set ) (p : Q → Bool ) : Set where |
| 20 field | |
| 21 found-q : Q | |
| 22 found-p : p found-q ≡ true | |
| 23 | |
| 264 | 24 open Bijection |
| 25 | |
| 268 | 26 open import Axiom.Extensionality.Propositional |
| 27 open import Level hiding (suc ; zero) | |
| 28 postulate f-extensionality : { n : Level} → Axiom.Extensionality.Propositional.Extensionality n n -- (Level.suc n) | |
| 29 | |
| 163 | 30 module _ {Q : Set } (F : FiniteSet Q) where |
| 31 open FiniteSet F | |
| 268 | 32 equal?-refl : { x : Q } → equal? x x ≡ true |
| 33 equal?-refl {x} with F←Q x ≟ F←Q x | |
| 34 ... | yes refl = refl | |
| 35 ... | no ne = ⊥-elim (ne refl) | |
| 163 | 36 equal→refl : { x y : Q } → equal? x y ≡ true → x ≡ y |
| 37 equal→refl {q0} {q1} eq with F←Q q0 ≟ F←Q q1 | |
| 38 equal→refl {q0} {q1} refl | yes eq = begin | |
| 39 q0 | |
| 40 ≡⟨ sym ( finiso→ q0) ⟩ | |
| 41 Q←F (F←Q q0) | |
| 42 ≡⟨ cong (λ k → Q←F k ) eq ⟩ | |
| 43 Q←F (F←Q q1) | |
| 264 | 44 ≡⟨ finiso→ q1 ⟩ |
| 163 | 45 q1 |
| 46 ∎ where open ≡-Reasoning | |
| 47 End : (m : ℕ ) → (p : Q → Bool ) → Set | |
| 48 End m p = (i : Fin finite) → m ≤ toℕ i → p (Q←F i ) ≡ false | |
| 49 first-end : ( p : Q → Bool ) → End finite p | |
| 50 first-end p i i>n = ⊥-elim (nat-≤> i>n (fin<n {finite} {i}) ) | |
| 51 next-end : {m : ℕ } → ( p : Q → Bool ) → End (suc m) p | |
| 52 → (m<n : m < finite ) → p (Q←F (fromℕ< m<n )) ≡ false | |
| 53 → End m p | |
| 54 next-end {m} p prev m<n np i m<i with NatP.<-cmp m (toℕ i) | |
| 55 next-end p prev m<n np i m<i | tri< a ¬b ¬c = prev i a | |
| 56 next-end p prev m<n np i m<i | tri> ¬a ¬b c = ⊥-elim ( nat-≤> m<i c ) | |
| 57 next-end {m} p prev m<n np i m<i | tri≈ ¬a b ¬c = subst ( λ k → p (Q←F k) ≡ false) (m<n=i i b m<n ) np where | |
| 58 m<n=i : {n : ℕ } (i : Fin n) {m : ℕ } → m ≡ (toℕ i) → (m<n : m < n ) → fromℕ< m<n ≡ i | |
| 264 | 59 m<n=i i refl m<n = fromℕ<-toℕ i m<n |
| 163 | 60 found : { p : Q → Bool } → (q : Q ) → p q ≡ true → exists p ≡ true |
| 61 found {p} q pt = found1 finite (NatP.≤-refl ) ( first-end p ) where | |
| 62 found1 : (m : ℕ ) (m<n : m Data.Nat.≤ finite ) → ((i : Fin finite) → m ≤ toℕ i → p (Q←F i ) ≡ false ) → exists1 m m<n p ≡ true | |
| 63 found1 0 m<n end = ⊥-elim ( ¬-bool (subst (λ k → k ≡ false ) (cong (λ k → p k) (finiso→ q) ) (end (F←Q q) z≤n )) pt ) | |
| 64 found1 (suc m) m<n end with bool-≡-? (p (Q←F (fromℕ< m<n))) true | |
| 264 | 65 found1 (suc m) m<n end | yes eq = subst (λ k → k \/ exists1 m (<to≤ m<n) p ≡ true ) (sym eq) (bool-or-4 {exists1 m (<to≤ m<n) p} ) |
| 163 | 66 found1 (suc m) m<n end | no np = begin |
| 264 | 67 p (Q←F (fromℕ< m<n)) \/ exists1 m (<to≤ m<n) p |
| 163 | 68 ≡⟨ bool-or-1 (¬-bool-t np ) ⟩ |
| 264 | 69 exists1 m (<to≤ m<n) p |
| 70 ≡⟨ found1 m (<to≤ m<n) (next-end p end m<n (¬-bool-t np )) ⟩ | |
| 163 | 71 true |
| 72 ∎ where open ≡-Reasoning | |
| 268 | 73 not-found : { p : Q → Bool } → ( (q : Q ) → p q ≡ false ) → exists p ≡ false |
| 74 not-found {p} pn = not-found2 finite NatP.≤-refl where | |
| 75 not-found2 : (m : ℕ ) → (m<n : m Data.Nat.≤ finite ) → exists1 m m<n p ≡ false | |
| 76 not-found2 zero _ = refl | |
| 77 not-found2 ( suc m ) m<n with pn (Q←F (fromℕ< {m} {finite} m<n)) | |
| 78 not-found2 (suc m) m<n | eq = begin | |
| 79 p (Q←F (fromℕ< m<n)) \/ exists1 m (<to≤ m<n) p | |
| 80 ≡⟨ bool-or-1 eq ⟩ | |
| 81 exists1 m (<to≤ m<n) p | |
| 82 ≡⟨ not-found2 m (<to≤ m<n) ⟩ | |
| 83 false | |
| 84 ∎ where open ≡-Reasoning | |
| 85 found← : { p : Q → Bool } → exists p ≡ true → Found Q p | |
| 86 found← {p} exst = found2 finite NatP.≤-refl (first-end p ) where | |
| 87 found2 : (m : ℕ ) (m<n : m Data.Nat.≤ finite ) → End m p → Found Q p | |
| 88 found2 0 m<n end = ⊥-elim ( ¬-bool (not-found (λ q → end (F←Q q) z≤n ) ) (subst (λ k → exists k ≡ true) (sym lemma) exst ) ) where | |
| 89 lemma : (λ z → p (Q←F (F←Q z))) ≡ p | |
| 90 lemma = f-extensionality ( λ q → subst (λ k → p k ≡ p q ) (sym (finiso→ q)) refl ) | |
| 91 found2 (suc m) m<n end with bool-≡-? (p (Q←F (fromℕ< m<n))) true | |
| 92 found2 (suc m) m<n end | yes eq = record { found-q = Q←F (fromℕ< m<n) ; found-p = eq } | |
| 93 found2 (suc m) m<n end | no np = | |
| 94 found2 m (<to≤ m<n) (next-end p end m<n (¬-bool-t np )) | |
| 95 not-found← : { p : Q → Bool } → exists p ≡ false → (q : Q ) → p q ≡ false | |
| 96 not-found← {p} np q = ¬-bool-t ( contra-position {_} {_} {_} {exists p ≡ true} (found q) (λ ep → ¬-bool np ep ) ) | |
| 97 | |
| 163 | 98 |
| 99 | |
| 264 | 100 iso-fin : {A B : Set} → FiniteSet A → Bijection A B → FiniteSet B |
| 141 | 101 iso-fin {A} {B} fin iso = record { |
| 264 | 102 Q←F = λ f → fun→ iso ( FiniteSet.Q←F fin f ) |
| 103 ; F←Q = λ b → FiniteSet.F←Q fin (fun← iso b ) | |
| 104 ; finiso→ = finiso→ | |
| 105 ; finiso← = finiso← | |
| 141 | 106 } where |
| 264 | 107 finiso→ : (q : B) → fun→ iso (FiniteSet.Q←F fin (FiniteSet.F←Q fin (Bijection.fun← iso q))) ≡ q |
| 141 | 108 finiso→ q = begin |
| 264 | 109 fun→ iso (FiniteSet.Q←F fin (FiniteSet.F←Q fin (Bijection.fun← iso q))) |
| 110 ≡⟨ cong (λ k → fun→ iso k ) (FiniteSet.finiso→ fin _ ) ⟩ | |
| 111 fun→ iso (Bijection.fun← iso q) | |
| 112 ≡⟨ fiso→ iso _ ⟩ | |
| 141 | 113 q |
| 264 | 114 ∎ where open ≡-Reasoning |
| 115 finiso← : (f : Fin (FiniteSet.finite fin ))→ FiniteSet.F←Q fin (Bijection.fun← iso (Bijection.fun→ iso (FiniteSet.Q←F fin f))) ≡ f | |
| 141 | 116 finiso← f = begin |
| 264 | 117 FiniteSet.F←Q fin (Bijection.fun← iso (Bijection.fun→ iso (FiniteSet.Q←F fin f))) |
| 118 ≡⟨ cong (λ k → FiniteSet.F←Q fin k ) (Bijection.fiso← iso _) ⟩ | |
| 141 | 119 FiniteSet.F←Q fin (FiniteSet.Q←F fin f) |
| 120 ≡⟨ FiniteSet.finiso← fin _ ⟩ | |
| 121 f | |
| 122 ∎ where | |
| 123 open ≡-Reasoning | |
| 124 | |
| 125 data One : Set where | |
| 126 one : One | |
| 127 | |
| 128 fin-∨1 : {B : Set} → (fb : FiniteSet B ) → FiniteSet (One ∨ B) | |
| 129 fin-∨1 {B} fb = record { | |
| 130 Q←F = Q←F | |
| 131 ; F←Q = F←Q | |
| 132 ; finiso→ = finiso→ | |
| 133 ; finiso← = finiso← | |
| 134 } where | |
| 135 b = FiniteSet.finite fb | |
| 136 Q←F : Fin (suc b) → One ∨ B | |
| 137 Q←F zero = case1 one | |
| 138 Q←F (suc f) = case2 (FiniteSet.Q←F fb f) | |
| 139 F←Q : One ∨ B → Fin (suc b) | |
| 140 F←Q (case1 one) = zero | |
| 141 F←Q (case2 f ) = suc (FiniteSet.F←Q fb f) | |
| 142 finiso→ : (q : One ∨ B) → Q←F (F←Q q) ≡ q | |
| 143 finiso→ (case1 one) = refl | |
| 144 finiso→ (case2 b) = cong (λ k → case2 k ) (FiniteSet.finiso→ fb b) | |
| 145 finiso← : (q : Fin (suc b)) → F←Q (Q←F q) ≡ q | |
| 146 finiso← zero = refl | |
| 147 finiso← (suc f) = cong ( λ k → suc k ) (FiniteSet.finiso← fb f) | |
| 148 | |
| 149 | |
| 150 fin-∨2 : {B : Set} → ( a : ℕ ) → FiniteSet B → FiniteSet (Fin a ∨ B) | |
| 151 fin-∨2 {B} zero fb = iso-fin fb iso where | |
| 264 | 152 iso : Bijection B (Fin zero ∨ B) |
| 153 iso = record { | |
| 154 fun← = fun←1 | |
| 155 ; fun→ = λ b → case2 b | |
| 156 ; fiso→ = fiso→1 | |
| 157 ; fiso← = λ _ → refl | |
| 141 | 158 } where |
| 264 | 159 fun←1 : Fin zero ∨ B → B |
| 160 fun←1 (case2 x) = x | |
| 161 fiso→1 : (f : Fin zero ∨ B ) → case2 (fun←1 f) ≡ f | |
| 162 fiso→1 (case2 x) = refl | |
| 141 | 163 fin-∨2 {B} (suc a) fb = iso-fin (fin-∨1 (fin-∨2 a fb) ) iso |
| 164 where | |
| 264 | 165 iso : Bijection (One ∨ (Fin a ∨ B) ) (Fin (suc a) ∨ B) |
| 166 fun← iso (case1 zero) = case1 one | |
| 167 fun← iso (case1 (suc f)) = case2 (case1 f) | |
| 168 fun← iso (case2 b) = case2 (case2 b) | |
| 169 fun→ iso (case1 one) = case1 zero | |
| 170 fun→ iso (case2 (case1 f)) = case1 (suc f) | |
| 171 fun→ iso (case2 (case2 b)) = case2 b | |
| 172 fiso← iso (case1 one) = refl | |
| 173 fiso← iso (case2 (case1 x)) = refl | |
| 174 fiso← iso (case2 (case2 x)) = refl | |
| 175 fiso→ iso (case1 zero) = refl | |
| 176 fiso→ iso (case1 (suc x)) = refl | |
| 177 fiso→ iso (case2 x) = refl | |
| 141 | 178 |
| 179 | |
| 264 | 180 FiniteSet→Fin : {A : Set} → (fin : FiniteSet A ) → Bijection (Fin (FiniteSet.finite fin)) A |
| 181 fun← (FiniteSet→Fin fin) f = FiniteSet.F←Q fin f | |
| 182 fun→ (FiniteSet→Fin fin) f = FiniteSet.Q←F fin f | |
| 183 fiso← (FiniteSet→Fin fin) = FiniteSet.finiso← fin | |
| 184 fiso→ (FiniteSet→Fin fin) = FiniteSet.finiso→ fin | |
| 141 | 185 |
| 186 | |
| 187 fin-∨ : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A ∨ B) | |
| 188 fin-∨ {A} {B} fa fb = iso-fin (fin-∨2 a fb ) iso2 where | |
| 189 a = FiniteSet.finite fa | |
| 190 ia = FiniteSet→Fin fa | |
| 264 | 191 iso2 : Bijection (Fin a ∨ B ) (A ∨ B) |
| 192 fun← iso2 (case1 x) = case1 (fun← ia x ) | |
| 193 fun← iso2 (case2 x) = case2 x | |
| 194 fun→ iso2 (case1 x) = case1 (fun→ ia x ) | |
| 195 fun→ iso2 (case2 x) = case2 x | |
| 196 fiso← iso2 (case1 x) = cong ( λ k → case1 k ) (Bijection.fiso← ia x) | |
| 197 fiso← iso2 (case2 x) = refl | |
| 198 fiso→ iso2 (case1 x) = cong ( λ k → case1 k ) (Bijection.fiso→ ia x) | |
| 199 fiso→ iso2 (case2 x) = refl | |
| 141 | 200 |
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201 open import Data.Product hiding ( map ) |
| 141 | 202 |
| 203 fin-× : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A × B) | |
| 204 fin-× {A} {B} fa fb with FiniteSet→Fin fa | |
| 205 ... | a=f = iso-fin (fin-×-f a ) iso-1 where | |
| 206 a = FiniteSet.finite fa | |
| 207 b = FiniteSet.finite fb | |
| 264 | 208 iso-1 : Bijection (Fin a × B) ( A × B ) |
| 209 fun← iso-1 x = ( FiniteSet.F←Q fa (proj₁ x) , proj₂ x) | |
| 210 fun→ iso-1 x = ( FiniteSet.Q←F fa (proj₁ x) , proj₂ x) | |
| 211 fiso← iso-1 x = lemma where | |
| 141 | 212 lemma : (FiniteSet.F←Q fa (FiniteSet.Q←F fa (proj₁ x)) , proj₂ x) ≡ ( proj₁ x , proj₂ x ) |
| 213 lemma = cong ( λ k → ( k , proj₂ x ) ) (FiniteSet.finiso← fa _ ) | |
| 264 | 214 fiso→ iso-1 x = cong ( λ k → ( k , proj₂ x ) ) (FiniteSet.finiso→ fa _ ) |
| 141 | 215 |
| 264 | 216 iso-2 : {a : ℕ } → Bijection (B ∨ (Fin a × B)) (Fin (suc a) × B) |
| 217 fun← iso-2 (zero , b ) = case1 b | |
| 218 fun← iso-2 (suc fst , b ) = case2 ( fst , b ) | |
| 219 fun→ iso-2 (case1 b) = ( zero , b ) | |
| 220 fun→ iso-2 (case2 (a , b )) = ( suc a , b ) | |
| 221 fiso← iso-2 (case1 x) = refl | |
| 222 fiso← iso-2 (case2 x) = refl | |
| 223 fiso→ iso-2 (zero , b ) = refl | |
| 224 fiso→ iso-2 (suc a , b ) = refl | |
| 141 | 225 |
| 226 fin-×-f : ( a : ℕ ) → FiniteSet ((Fin a) × B) | |
| 227 fin-×-f zero = record { Q←F = λ () ; F←Q = λ () ; finiso→ = λ () ; finiso← = λ () ; finite = 0 } | |
| 228 fin-×-f (suc a) = iso-fin ( fin-∨ fb ( fin-×-f a ) ) iso-2 | |
| 229 | |
| 230 open _∧_ | |
| 231 | |
| 232 fin-∧ : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A ∧ B) | |
| 233 fin-∧ {A} {B} fa fb with FiniteSet→Fin fa -- same thing for our tool | |
| 234 ... | a=f = iso-fin (fin-×-f a ) iso-1 where | |
| 235 a = FiniteSet.finite fa | |
| 236 b = FiniteSet.finite fb | |
| 264 | 237 iso-1 : Bijection (Fin a ∧ B) ( A ∧ B ) |
| 238 fun← iso-1 x = record { proj1 = FiniteSet.F←Q fa (proj1 x) ; proj2 = proj2 x} | |
| 239 fun→ iso-1 x = record { proj1 = FiniteSet.Q←F fa (proj1 x) ; proj2 = proj2 x} | |
| 240 fiso← iso-1 x = lemma where | |
| 141 | 241 lemma : record { proj1 = FiniteSet.F←Q fa (FiniteSet.Q←F fa (proj1 x)) ; proj2 = proj2 x} ≡ record {proj1 = proj1 x ; proj2 = proj2 x } |
| 242 lemma = cong ( λ k → record {proj1 = k ; proj2 = proj2 x } ) (FiniteSet.finiso← fa _ ) | |
| 264 | 243 fiso→ iso-1 x = cong ( λ k → record {proj1 = k ; proj2 = proj2 x } ) (FiniteSet.finiso→ fa _ ) |
| 141 | 244 |
| 264 | 245 iso-2 : {a : ℕ } → Bijection (B ∨ (Fin a ∧ B)) (Fin (suc a) ∧ B) |
| 246 fun← iso-2 (record { proj1 = zero ; proj2 = b }) = case1 b | |
| 247 fun← iso-2 (record { proj1 = suc fst ; proj2 = b }) = case2 ( record { proj1 = fst ; proj2 = b } ) | |
| 248 fun→ iso-2 (case1 b) = record {proj1 = zero ; proj2 = b } | |
| 249 fun→ iso-2 (case2 (record { proj1 = a ; proj2 = b })) = record { proj1 = suc a ; proj2 = b } | |
| 250 fiso← iso-2 (case1 x) = refl | |
| 251 fiso← iso-2 (case2 x) = refl | |
| 252 fiso→ iso-2 (record { proj1 = zero ; proj2 = b }) = refl | |
| 253 fiso→ iso-2 (record { proj1 = suc a ; proj2 = b }) = refl | |
| 141 | 254 |
| 255 fin-×-f : ( a : ℕ ) → FiniteSet ((Fin a) ∧ B) | |
| 256 fin-×-f zero = record { Q←F = λ () ; F←Q = λ () ; finiso→ = λ () ; finiso← = λ () ; finite = 0 } | |
| 257 fin-×-f (suc a) = iso-fin ( fin-∨ fb ( fin-×-f a ) ) iso-2 | |
| 258 | |
| 259 -- import Data.Nat.DivMod | |
| 260 | |
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261 open import Data.Vec hiding ( map ; length ) |
| 141 | 262 import Data.Product |
| 263 | |
| 264 exp2 : (n : ℕ ) → exp 2 (suc n) ≡ exp 2 n Data.Nat.+ exp 2 n | |
| 265 exp2 n = begin | |
| 266 exp 2 (suc n) | |
| 267 ≡⟨⟩ | |
| 268 2 * ( exp 2 n ) | |
| 269 ≡⟨ *-comm 2 (exp 2 n) ⟩ | |
| 270 ( exp 2 n ) * 2 | |
| 271 ≡⟨ *-suc ( exp 2 n ) 1 ⟩ | |
| 272 (exp 2 n ) Data.Nat.+ ( exp 2 n ) * 1 | |
| 273 ≡⟨ cong ( λ k → (exp 2 n ) Data.Nat.+ k ) (proj₂ *-identity (exp 2 n) ) ⟩ | |
| 274 exp 2 n Data.Nat.+ exp 2 n | |
| 275 ∎ where | |
| 276 open ≡-Reasoning | |
| 277 open Data.Product | |
| 278 | |
| 279 cast-iso : {n m : ℕ } → (eq : n ≡ m ) → (f : Fin m ) → cast eq ( cast (sym eq ) f) ≡ f | |
| 280 cast-iso refl zero = refl | |
| 281 cast-iso refl (suc f) = cong ( λ k → suc k ) ( cast-iso refl f ) | |
| 282 | |
| 283 | |
| 284 fin2List : {n : ℕ } → FiniteSet (Vec Bool n) | |
| 285 fin2List {zero} = record { | |
| 286 Q←F = λ _ → Vec.[] | |
| 287 ; F←Q = λ _ → # 0 | |
| 288 ; finiso→ = finiso→ | |
| 289 ; finiso← = finiso← | |
| 290 } where | |
| 291 Q = Vec Bool zero | |
| 292 finiso→ : (q : Q) → [] ≡ q | |
| 293 finiso→ [] = refl | |
| 294 finiso← : (f : Fin (exp 2 zero)) → # 0 ≡ f | |
| 295 finiso← zero = refl | |
| 296 fin2List {suc n} = subst (λ k → FiniteSet (Vec Bool (suc n)) ) (sym (exp2 n)) ( iso-fin (fin-∨ (fin2List ) (fin2List )) iso ) | |
| 297 where | |
| 298 QtoR : Vec Bool (suc n) → Vec Bool n ∨ Vec Bool n | |
| 299 QtoR ( true ∷ x ) = case1 x | |
| 300 QtoR ( false ∷ x ) = case2 x | |
| 301 RtoQ : Vec Bool n ∨ Vec Bool n → Vec Bool (suc n) | |
| 302 RtoQ ( case1 x ) = true ∷ x | |
| 303 RtoQ ( case2 x ) = false ∷ x | |
| 304 isoRQ : (x : Vec Bool (suc n) ) → RtoQ ( QtoR x ) ≡ x | |
| 305 isoRQ (true ∷ _ ) = refl | |
| 306 isoRQ (false ∷ _ ) = refl | |
| 307 isoQR : (x : Vec Bool n ∨ Vec Bool n ) → QtoR ( RtoQ x ) ≡ x | |
| 308 isoQR (case1 x) = refl | |
| 309 isoQR (case2 x) = refl | |
| 264 | 310 iso : Bijection (Vec Bool n ∨ Vec Bool n) (Vec Bool (suc n)) |
| 311 iso = record { fun← = QtoR ; fun→ = RtoQ ; fiso← = isoQR ; fiso→ = isoRQ } | |
| 141 | 312 |
| 313 F2L : {Q : Set } {n : ℕ } → (fin : FiniteSet Q ) → n < suc (FiniteSet.finite fin) → ( (q : Q) → toℕ (FiniteSet.F←Q fin q ) < n → Bool ) → Vec Bool n | |
| 314 F2L {Q} {zero} fin _ Q→B = [] | |
| 315 F2L {Q} {suc n} fin (s≤s n<m) Q→B = Q→B (FiniteSet.Q←F fin (fromℕ< n<m)) lemma6 ∷ F2L {Q} fin (NatP.<-trans n<m a<sa ) qb1 where | |
| 316 lemma6 : toℕ (FiniteSet.F←Q fin (FiniteSet.Q←F fin (fromℕ< n<m))) < suc n | |
| 317 lemma6 = subst (λ k → toℕ k < suc n ) (sym (FiniteSet.finiso← fin _ )) (subst (λ k → k < suc n) (sym (toℕ-fromℕ< n<m )) a<sa ) | |
| 318 qb1 : (q : Q) → toℕ (FiniteSet.F←Q fin q) < n → Bool | |
| 319 qb1 q q<n = Q→B q (NatP.<-trans q<n a<sa) | |
| 320 | |
| 321 List2Func : { Q : Set } → {n : ℕ } → (fin : FiniteSet Q ) → n < suc (FiniteSet.finite fin) → Vec Bool n → Q → Bool | |
| 322 List2Func {Q} {zero} fin (s≤s z≤n) [] q = false | |
| 323 List2Func {Q} {suc n} fin (s≤s n<m) (h ∷ t) q with FiniteSet.F←Q fin q ≟ fromℕ< n<m | |
| 324 ... | yes _ = h | |
| 325 ... | no _ = List2Func {Q} fin (NatP.<-trans n<m a<sa ) t q | |
| 326 | |
| 327 open import Level renaming ( suc to Suc ; zero to Zero) | |
| 328 open import Axiom.Extensionality.Propositional | |
| 268 | 329 -- postulate f-extensionality : { n : Level} → Axiom.Extensionality.Propositional.Extensionality n n |
| 141 | 330 |
| 331 F2L-iso : { Q : Set } → (fin : FiniteSet Q ) → (x : Vec Bool (FiniteSet.finite fin) ) → F2L fin a<sa (λ q _ → List2Func fin a<sa x q ) ≡ x | |
| 332 F2L-iso {Q} fin x = f2l m a<sa x where | |
| 333 m = FiniteSet.finite fin | |
| 334 f2l : (n : ℕ ) → (n<m : n < suc m )→ (x : Vec Bool n ) → F2L fin n<m (λ q q<n → List2Func fin n<m x q ) ≡ x | |
| 335 f2l zero (s≤s z≤n) [] = refl | |
| 163 | 336 f2l (suc n) (s≤s n<m) (h ∷ t ) = lemma1 lemma2 lemma3f where |
| 141 | 337 lemma1 : {n : ℕ } → {h h1 : Bool } → {t t1 : Vec Bool n } → h ≡ h1 → t ≡ t1 → h ∷ t ≡ h1 ∷ t1 |
| 338 lemma1 refl refl = refl | |
| 339 lemma2 : List2Func fin (s≤s n<m) (h ∷ t) (FiniteSet.Q←F fin (fromℕ< n<m)) ≡ h | |
| 340 lemma2 with FiniteSet.F←Q fin (FiniteSet.Q←F fin (fromℕ< n<m)) ≟ fromℕ< n<m | |
| 341 lemma2 | yes p = refl | |
| 342 lemma2 | no ¬p = ⊥-elim ( ¬p (FiniteSet.finiso← fin _) ) | |
| 343 lemma4 : (q : Q ) → toℕ (FiniteSet.F←Q fin q ) < n → List2Func fin (s≤s n<m) (h ∷ t) q ≡ List2Func fin (NatP.<-trans n<m a<sa) t q | |
| 344 lemma4 q _ with FiniteSet.F←Q fin q ≟ fromℕ< n<m | |
| 345 lemma4 q lt | yes p = ⊥-elim ( nat-≡< (toℕ-fromℕ< n<m) (lemma5 n lt (cong (λ k → toℕ k) p))) where | |
| 346 lemma5 : {j k : ℕ } → ( n : ℕ) → suc j ≤ n → j ≡ k → k < n | |
| 347 lemma5 {zero} (suc n) (s≤s z≤n) refl = s≤s z≤n | |
| 348 lemma5 {suc j} (suc n) (s≤s lt) refl = s≤s (lemma5 {j} n lt refl) | |
| 349 lemma4 q _ | no ¬p = refl | |
| 163 | 350 lemma3f : F2L fin (NatP.<-trans n<m a<sa) (λ q q<n → List2Func fin (s≤s n<m) (h ∷ t) q ) ≡ t |
| 351 lemma3f = begin | |
| 141 | 352 F2L fin (NatP.<-trans n<m a<sa) (λ q q<n → List2Func fin (s≤s n<m) (h ∷ t) q ) |
| 353 ≡⟨ cong (λ k → F2L fin (NatP.<-trans n<m a<sa) ( λ q q<n → k q q<n )) | |
| 354 (f-extensionality ( λ q → | |
| 355 (f-extensionality ( λ q<n → lemma4 q q<n )))) ⟩ | |
| 356 F2L fin (NatP.<-trans n<m a<sa) (λ q q<n → List2Func fin (NatP.<-trans n<m a<sa) t q ) | |
| 357 ≡⟨ f2l n (NatP.<-trans n<m a<sa ) t ⟩ | |
| 358 t | |
| 359 ∎ where | |
| 360 open ≡-Reasoning | |
| 361 | |
| 362 | |
| 363 L2F : {Q : Set } {n : ℕ } → (fin : FiniteSet Q ) → n < suc (FiniteSet.finite fin) → Vec Bool n → (q : Q ) → toℕ (FiniteSet.F←Q fin q ) < n → Bool | |
| 364 L2F fin n<m x q q<n = List2Func fin n<m x q | |
| 365 | |
| 366 L2F-iso : { Q : Set } → (fin : FiniteSet Q ) → (f : Q → Bool ) → (q : Q ) → (L2F fin a<sa (F2L fin a<sa (λ q _ → f q) )) q (toℕ<n _) ≡ f q | |
| 367 L2F-iso {Q} fin f q = l2f m a<sa (toℕ<n _) where | |
| 368 m = FiniteSet.finite fin | |
| 163 | 369 lemma11f : {n : ℕ } → (n<m : n < m ) → ¬ ( FiniteSet.F←Q fin q ≡ fromℕ< n<m ) → toℕ (FiniteSet.F←Q fin q) ≤ n → toℕ (FiniteSet.F←Q fin q) < n |
| 370 lemma11f n<m ¬q=n q≤n = lemma13 n<m (contra-position (lemma12 n<m _) ¬q=n ) q≤n where | |
| 141 | 371 lemma13 : {n nq : ℕ } → (n<m : n < m ) → ¬ ( nq ≡ n ) → nq ≤ n → nq < n |
| 372 lemma13 {0} {0} (s≤s z≤n) nt z≤n = ⊥-elim ( nt refl ) | |
| 373 lemma13 {suc _} {0} (s≤s (s≤s n<m)) nt z≤n = s≤s z≤n | |
| 374 lemma13 {suc n} {suc nq} n<m nt (s≤s nq≤n) = s≤s (lemma13 {n} {nq} (NatP.<-trans a<sa n<m ) (λ eq → nt ( cong ( λ k → suc k ) eq )) nq≤n) | |
| 163 | 375 lemma3f : {a b : ℕ } → (lt : a < b ) → fromℕ< (s≤s lt) ≡ suc (fromℕ< lt) |
| 376 lemma3f (s≤s lt) = refl | |
| 377 lemma12f : {n m : ℕ } → (n<m : n < m ) → (f : Fin m ) → toℕ f ≡ n → f ≡ fromℕ< n<m | |
| 378 lemma12f {zero} {suc m} (s≤s z≤n) zero refl = refl | |
| 379 lemma12f {suc n} {suc m} (s≤s n<m) (suc f) refl = subst ( λ k → suc f ≡ k ) (sym (lemma3f n<m) ) ( cong ( λ k → suc k ) ( lemma12f {n} {m} n<m f refl ) ) | |
| 141 | 380 l2f : (n : ℕ ) → (n<m : n < suc m ) → (q<n : toℕ (FiniteSet.F←Q fin q ) < n ) → (L2F fin n<m (F2L fin n<m (λ q _ → f q))) q q<n ≡ f q |
| 381 l2f zero (s≤s z≤n) () | |
| 382 l2f (suc n) (s≤s n<m) (s≤s n<q) with FiniteSet.F←Q fin q ≟ fromℕ< n<m | |
| 383 l2f (suc n) (s≤s n<m) (s≤s n<q) | yes p = begin | |
| 384 f (FiniteSet.Q←F fin (fromℕ< n<m)) | |
| 385 ≡⟨ cong ( λ k → f (FiniteSet.Q←F fin k )) (sym p) ⟩ | |
| 386 f (FiniteSet.Q←F fin ( FiniteSet.F←Q fin q )) | |
| 387 ≡⟨ cong ( λ k → f k ) (FiniteSet.finiso→ fin _ ) ⟩ | |
| 388 f q | |
| 389 ∎ where | |
| 390 open ≡-Reasoning | |
| 163 | 391 l2f (suc n) (s≤s n<m) (s≤s n<q) | no ¬p = l2f n (NatP.<-trans n<m a<sa) (lemma11f n<m ¬p n<q) |
| 141 | 392 |
| 393 fin→ : {A : Set} → FiniteSet A → FiniteSet (A → Bool ) | |
| 394 fin→ {A} fin = iso-fin fin2List iso where | |
| 395 a = FiniteSet.finite fin | |
| 264 | 396 iso : Bijection (Vec Bool a ) (A → Bool) |
| 397 fun← iso x = F2L fin a<sa ( λ q _ → x q ) | |
| 398 fun→ iso x = List2Func fin a<sa x | |
| 399 fiso← iso x = F2L-iso fin x | |
| 400 fiso→ iso x = lemma where | |
| 141 | 401 lemma : List2Func fin a<sa (F2L fin a<sa (λ q _ → x q)) ≡ x |
| 402 lemma = f-extensionality ( λ q → L2F-iso fin x q ) | |
| 403 | |
| 404 | |
| 405 Fin2Finite : ( n : ℕ ) → FiniteSet (Fin n) | |
| 406 Fin2Finite n = record { F←Q = λ x → x ; Q←F = λ x → x ; finiso← = λ q → refl ; finiso→ = λ q → refl } | |
| 407 | |
| 408 data fin-less { n : ℕ } { A : Set } (fa : FiniteSet A ) (n<m : n < FiniteSet.finite fa ) : Set where | |
| 409 elm1 : (elm : A ) → toℕ (FiniteSet.F←Q fa elm ) < n → fin-less fa n<m | |
| 410 | |
| 411 get-elm : { n : ℕ } { A : Set } {fa : FiniteSet A } {n<m : n < FiniteSet.finite fa } → fin-less fa n<m → A | |
| 412 get-elm (elm1 a _ ) = a | |
| 413 | |
| 414 get-< : { n : ℕ } { A : Set } {fa : FiniteSet A } {n<m : n < FiniteSet.finite fa }→ (f : fin-less fa n<m ) → toℕ (FiniteSet.F←Q fa (get-elm f )) < n | |
| 415 get-< (elm1 _ b ) = b | |
| 416 | |
| 417 fin-less-cong : { n : ℕ } { A : Set } (fa : FiniteSet A ) (n<m : n < FiniteSet.finite fa ) | |
| 418 → (x y : fin-less fa n<m ) → get-elm {n} {A} {fa} x ≡ get-elm {n} {A} {fa} y → get-< x ≅ get-< y → x ≡ y | |
| 419 fin-less-cong fa n<m (elm1 elm x) (elm1 elm x) refl HE.refl = refl | |
| 420 | |
| 421 fin-< : {A : Set} → { n : ℕ } → (fa : FiniteSet A ) → (n<m : n < FiniteSet.finite fa ) → FiniteSet (fin-less fa n<m ) | |
| 422 fin-< {A} {n} fa n<m = iso-fin (Fin2Finite n) iso where | |
| 423 m = FiniteSet.finite fa | |
| 264 | 424 iso : Bijection (Fin n) (fin-less fa n<m ) |
| 163 | 425 lemma8f : {i j n : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n } → i<n ≅ j<n |
| 426 lemma8f {zero} {zero} {suc n} refl {s≤s z≤n} {s≤s z≤n} = HE.refl | |
| 427 lemma8f {suc i} {suc i} {suc n} refl {s≤s i<n} {s≤s j<n} = HE.cong (λ k → s≤s k ) ( lemma8f {i} {i} refl ) | |
| 428 lemma10f : {n i j : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n } → fromℕ< i<n ≡ fromℕ< j<n | |
| 429 lemma10f refl = HE.≅-to-≡ (HE.cong (λ k → fromℕ< k ) (lemma8f refl )) | |
| 430 lemma3f : {a b c : ℕ } → { a<b : a < b } { b<c : b < c } { a<c : a < c } → NatP.<-trans a<b b<c ≡ a<c | |
| 431 lemma3f {a} {b} {c} {a<b} {b<c} {a<c} = HE.≅-to-≡ (lemma8f refl) | |
| 432 lemma11f : {n : ℕ } {x : Fin n } → (n<m : n < m ) → toℕ (fromℕ< (NatP.<-trans (toℕ<n x) n<m)) ≡ toℕ x | |
| 433 lemma11f {n} {x} n<m = begin | |
| 141 | 434 toℕ (fromℕ< (NatP.<-trans (toℕ<n x) n<m)) |
| 435 ≡⟨ toℕ-fromℕ< _ ⟩ | |
| 436 toℕ x | |
| 437 ∎ where | |
| 438 open ≡-Reasoning | |
| 264 | 439 fun← iso (elm1 elm x) = fromℕ< x |
| 440 fun→ iso x = elm1 (FiniteSet.Q←F fa (fromℕ< (NatP.<-trans x<n n<m ))) to<n where | |
| 141 | 441 x<n : toℕ x < n |
| 442 x<n = toℕ<n x | |
| 443 to<n : toℕ (FiniteSet.F←Q fa (FiniteSet.Q←F fa (fromℕ< (NatP.<-trans x<n n<m)))) < n | |
| 444 to<n = subst (λ k → toℕ k < n ) (sym (FiniteSet.finiso← fa _ )) (subst (λ k → k < n ) (sym ( toℕ-fromℕ< (NatP.<-trans x<n n<m) )) x<n ) | |
| 264 | 445 fiso← iso x = lemma2 where |
| 141 | 446 lemma2 : fromℕ< (subst (λ k → toℕ k < n) (sym |
| 447 (FiniteSet.finiso← fa (fromℕ< (NatP.<-trans (toℕ<n x) n<m)))) (subst (λ k → k < n) | |
| 448 (sym (toℕ-fromℕ< (NatP.<-trans (toℕ<n x) n<m))) (toℕ<n x))) ≡ x | |
| 449 lemma2 = begin | |
| 450 fromℕ< (subst (λ k → toℕ k < n) (sym | |
| 451 (FiniteSet.finiso← fa (fromℕ< (NatP.<-trans (toℕ<n x) n<m)))) (subst (λ k → k < n) | |
| 452 (sym (toℕ-fromℕ< (NatP.<-trans (toℕ<n x) n<m))) (toℕ<n x))) | |
| 453 ≡⟨⟩ | |
| 454 fromℕ< ( subst (λ k → toℕ ( k ) < n ) (sym (FiniteSet.finiso← fa _ )) lemma6 ) | |
| 455 ≡⟨ lemma10 (cong (λ k → toℕ k) (FiniteSet.finiso← fa _ ) ) ⟩ | |
| 456 fromℕ< lemma6 | |
| 457 ≡⟨ lemma10 (lemma11 n<m ) ⟩ | |
| 458 fromℕ< ( toℕ<n x ) | |
| 459 ≡⟨ fromℕ<-toℕ _ _ ⟩ | |
| 460 x | |
| 461 ∎ where | |
| 462 open ≡-Reasoning | |
| 463 lemma6 : toℕ (fromℕ< (NatP.<-trans (toℕ<n x) n<m)) < n | |
| 464 lemma6 = subst ( λ k → k < n ) (sym (toℕ-fromℕ< (NatP.<-trans (toℕ<n x) n<m))) (toℕ<n x ) | |
| 264 | 465 fiso→ iso (elm1 elm x) = fin-less-cong fa n<m _ _ lemma (lemma8 (cong (λ k → toℕ (FiniteSet.F←Q fa k) ) lemma ) ) where |
| 141 | 466 lemma13 : toℕ (fromℕ< x) ≡ toℕ (FiniteSet.F←Q fa elm) |
| 467 lemma13 = begin | |
| 468 toℕ (fromℕ< x) | |
| 469 ≡⟨ toℕ-fromℕ< _ ⟩ | |
| 470 toℕ (FiniteSet.F←Q fa elm) | |
| 471 ∎ where open ≡-Reasoning | |
| 264 | 472 lemma : FiniteSet.Q←F fa (fromℕ< (NatP.<-trans (toℕ<n (Bijection.fun← iso (elm1 elm x))) n<m)) ≡ elm |
| 141 | 473 lemma = begin |
| 264 | 474 FiniteSet.Q←F fa (fromℕ< (NatP.<-trans (toℕ<n (Bijection.fun← iso (elm1 elm x))) n<m)) |
| 141 | 475 ≡⟨⟩ |
| 476 FiniteSet.Q←F fa (fromℕ< ( NatP.<-trans (toℕ<n ( fromℕ< x ) ) n<m)) | |
| 264 | 477 ≡⟨ cong (λ k → FiniteSet.Q←F fa k) (lemma10 lemma13 ) ⟩ |
| 141 | 478 FiniteSet.Q←F fa (fromℕ< ( NatP.<-trans x n<m)) |
| 264 | 479 ≡⟨ cong (λ k → FiniteSet.Q←F fa (fromℕ< k )) (HE.≅-to-≡ (lemma8 refl)) ⟩ |
| 141 | 480 FiniteSet.Q←F fa (fromℕ< ( toℕ<n (FiniteSet.F←Q fa elm))) |
| 481 ≡⟨ cong (λ k → FiniteSet.Q←F fa k ) ( fromℕ<-toℕ _ _ ) ⟩ | |
| 482 FiniteSet.Q←F fa (FiniteSet.F←Q fa elm ) | |
| 483 ≡⟨ FiniteSet.finiso→ fa _ ⟩ | |
| 484 elm | |
| 485 ∎ where open ≡-Reasoning | |
| 486 | |
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487 open import Data.List |
| 141 | 488 |
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489 open FiniteSet |
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490 |
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491 memberQ : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool |
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492 memberQ {Q} finq q [] = false |
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493 memberQ {Q} finq q (q0 ∷ qs) with equal? finq q q0 |
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494 ... | true = true |
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495 ... | false = memberQ finq q qs |
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496 |
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497 phase2 : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool |
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498 phase2 finq q [] = false |
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499 phase2 finq q (x ∷ qs) with equal? finq q x |
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500 ... | true = true |
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501 ... | false = phase2 finq q qs |
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502 phase1 : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool |
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503 phase1 finq q [] = false |
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504 phase1 finq q (x ∷ qs) with equal? finq q x |
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505 ... | true = phase2 finq q qs |
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506 ... | false = phase1 finq q qs |
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507 |
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508 dup-in-list : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool |
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509 dup-in-list {Q} finq q qs = phase1 finq q qs |
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510 |
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511 |
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512 dup-in-list+1 : { Q : Set } (finq : FiniteSet Q) |
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513 → (q : Q) (qs : List Q ) → dup-in-list finq q qs ≡ true |
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514 → dup-in-list (fin-∨1 finq) (case2 q) (map case2 qs ) ≡ true |
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515 dup-in-list+1 {Q} finq q qs p = 1-phase1 qs p where |
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516 dup04 : {q x : Q} → equal? finq q x ≡ equal? (fin-∨1 finq) (case2 q) (case2 x) |
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517 dup04 {q} {x} with F←Q finq q f≟ F←Q finq x |
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518 ... | yes _ = refl |
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519 ... | no _ = refl |
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520 1-phase2 : (qs : List Q) → phase2 finq q qs ≡ true → phase2 (fin-∨1 finq) (case2 q) (map case2 qs ) ≡ true |
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521 1-phase2 (x ∷ qs ) p with equal? finq q x | equal? (fin-∨1 finq) (case2 q) (case2 x) | dup04 {q} {x} |
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522 ... | true | true | t = refl |
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523 ... | false | false | t = 1-phase2 qs p |
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524 1-phase1 : (qs : List Q) → phase1 finq q qs ≡ true → phase1 (fin-∨1 finq) (case2 q) (map case2 qs ) ≡ true |
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525 1-phase1 (x ∷ qs ) p with equal? finq q x | equal? (fin-∨1 finq) (case2 q) (case2 x) | dup04 {q} {x} |
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526 ... | true | true | t = 1-phase2 qs p |
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527 ... | false | false | t = 1-phase1 qs p |
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528 |
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529 dup-in-list+iso : { Q : Set } (finq : FiniteSet Q) |
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530 → (q : Q) (qs : List Q ) |
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531 → dup-in-list (Fin2Finite (finite finq)) (F←Q finq q) (map (F←Q finq) qs) ≡ true |
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532 → dup-in-list finq q qs ≡ true |
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533 dup-in-list+iso {Q} finq q qs p = i-phase1 qs p where |
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534 dup05 : {q x : Q} → equal? finq q x ≡ equal? (Fin2Finite (finite finq)) (F←Q finq q) (F←Q finq x) |
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535 dup05 {q} {x} with F←Q finq q f≟ F←Q finq x |
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536 ... | yes _ = refl |
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537 ... | no _ = refl |
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538 i-phase2 : (qs : List Q) → phase2 (Fin2Finite (finite finq)) (F←Q finq q) (map (F←Q finq) qs) ≡ true |
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539 → phase2 finq q qs ≡ true |
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540 i-phase2 (x ∷ qs) p with equal? finq q x | equal? (Fin2Finite (finite finq)) (F←Q finq q) (F←Q finq x) | dup05 {q} {x} |
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541 ... | true | true | t2 = refl |
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542 ... | false | false | t2 = i-phase2 qs p |
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543 i-phase1 : (qs : List Q) → dup-in-list (Fin2Finite (finite finq)) (F←Q finq q) (map (F←Q finq) qs) ≡ true |
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544 → phase1 finq q qs ≡ true |
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545 i-phase1 (x ∷ qs) p with equal? finq q x | equal? (Fin2Finite (finite finq)) (F←Q finq q) (F←Q finq x) | dup05 {q} {x} |
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546 ... | true | true | t2 = i-phase2 qs p |
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547 ... | false | false | t2 = i-phase1 qs p |
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548 |
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549 record Dup-in-list {Q : Set } (finq : FiniteSet Q) (qs : List Q) : Set where |
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550 field |
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551 dup : Q |
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552 is-dup : dup-in-list finq dup qs ≡ true |
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553 |
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554 |
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555 dup-in-list>n : {Q : Set } → (finq : FiniteSet Q) → (qs : List Q) → (len> : length qs > finite finq ) → Dup-in-list finq qs |
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556 dup-in-list>n {Q} finq qs lt = record { |
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557 dup = dup-05 |
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558 ; is-dup = dup-in-list+iso finq dup-05 qs dup-06 } where |
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559 LEM-dup : Dup-in-list finq qs ∨ ( ¬ Dup-in-list finq qs ) |
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560 LEM-dup with exists finq ( λ q → dup-in-list finq q qs ) | inspect (exists finq) ( λ q → dup-in-list finq q qs ) |
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561 ... | true | record { eq = eq1 } = case1 ( record { dup = Found.found-q dup-01 ; is-dup = Found.found-p dup-01} ) where |
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562 dup-01 : Found Q ( λ q → dup-in-list finq q qs ) |
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563 dup-01 = found← finq eq1 |
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564 ... | false | record { eq = eq1 } = case2 (λ D → ¬-bool ( not-found← finq eq1 (Dup-in-list.dup D)) (Dup-in-list.is-dup D) ) |
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565 record NList (n : ℕ) : Set where |
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566 field |
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567 ls : List (Fin n) |
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568 ls>n : length ls > n |
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569 dup-02 : (n : ℕ) → (ls : NList n ) → Dup-in-list (Fin2Finite n) (NList.ls ls) |
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570 dup-02 zero ls = {!!} |
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571 dup-02 (suc n) ls = dup-03 where |
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572 n1 : Fin (suc n) |
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573 n1 = fromℕ< refl-≤ |
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574 d-phase2 : (qs : List (Fin (suc n)) ) → NList n ∨ ( phase2 (Fin2Finite (suc n)) n1 qs ≡ true ) |
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575 d-phase2 [] = {!!} |
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576 d-phase2 (x ∷ qs) with equal? (Fin2Finite (suc n)) n1 x |
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577 ... | true = case2 refl |
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578 ... | false with d-phase2 qs |
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579 ... | case1 p = case1 record { ls = {!!} ∷ NList.ls p ; ls>n = {!!} } |
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580 ... | case2 eq = case2 eq |
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581 d-phase1 : (qs : List (Fin (suc n)) ) → NList n ∨ ( phase1 (Fin2Finite (suc n)) n1 qs ≡ true ) |
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582 d-phase1 [] = {!!} |
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583 d-phase1 (x ∷ qs) with equal? (Fin2Finite (suc n)) n1 x |
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584 ... | true with d-phase2 qs |
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585 ... | case1 p = case1 record { ls = {!!} ∷ NList.ls p ; ls>n = {!!} } |
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586 ... | case2 eq = case2 eq |
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587 d-phase1 (x ∷ qs) | false with d-phase1 qs |
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588 ... | case1 p = case1 record { ls = {!!} ∷ NList.ls p ; ls>n = {!!} } |
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589 ... | case2 eq = case2 eq |
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590 dup-03 : Dup-in-list (Fin2Finite (suc n)) (NList.ls ls) |
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591 dup-03 with d-phase1 (NList.ls ls) |
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592 ... | case1 ls1 = record { dup = fin+1 (Dup-in-list.dup dup-04) ; is-dup = dup-07 } where |
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593 dup-04 : Dup-in-list (Fin2Finite n) (NList.ls ls1) |
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594 dup-04 = dup-02 n ls1 |
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595 dup-07 : dup-in-list (Fin2Finite (suc n)) (fin+1 (Dup-in-list.dup dup-04)) (NList.ls ls) ≡ true |
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596 dup-07 = dup-in-list+iso finq {!!} {!!} (dup-in-list+1 {!!} {!!} qs {!!}) |
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597 ... | case2 dup = record { dup = n1 ; is-dup = dup } |
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598 dup-05 : Q |
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599 dup-05 = Q←F finq (Dup-in-list.dup (dup-02 (finite finq) record { ls = map (F←Q finq) qs ; ls>n = {!!} } )) |
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600 dup-06 : dup-in-list (Fin2Finite (finite finq)) (F←Q finq dup-05) (map (F←Q finq) qs) ≡ true |
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601 dup-06 = subst (λ k → dup-in-list (Fin2Finite (finite finq)) k (map (F←Q finq) qs) ≡ true ) |
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602 {!!} (Dup-in-list.is-dup (dup-02 (finite finq) record { ls = map (F←Q finq) qs ; ls>n = {!!} } ) ) |
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603 |