Mercurial > hg > Members > kono > Proof > automaton
annotate automaton-in-agda/src/finiteSetUtil.agda @ 362:6d5344d3be9c
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 19 Jul 2023 07:08:43 +0900 |
| parents | c66d664593e9 |
| children | 21aa222b11c9 |
| rev | line source |
|---|---|
| 163 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 2 | |
| 3 module finiteSetUtil where | |
| 141 | 4 |
| 5 open import Data.Nat hiding ( _≟_ ) | |
| 347 | 6 open import Data.Fin renaming ( _<_ to _<<_ ; _>_ to _f>_ ; _≟_ to _f≟_ ) hiding (_≤_ ; pred ) |
| 7 open import Data.Fin.Properties hiding ( <-trans ; ≤-trans ; ≤-refl ; <-irrelevant ) renaming ( <-cmp to <-fcmp ) | |
| 141 | 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 |
| 337 | 16 open import Data.Nat.Properties as NP hiding ( _≟_ ) |
| 141 | 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 | |
| 318 | 47 eqP : (x y : Q) → Dec ( x ≡ y ) |
| 48 eqP x y with F←Q x ≟ F←Q y | |
| 49 ... | yes eq = yes (subst₂ (λ j k → j ≡ k ) (finiso→ x) (finiso→ y) (cong Q←F eq) ) | |
| 50 ... | no n = no (λ eq → n (cong F←Q eq)) | |
| 163 | 51 End : (m : ℕ ) → (p : Q → Bool ) → Set |
| 52 End m p = (i : Fin finite) → m ≤ toℕ i → p (Q←F i ) ≡ false | |
| 53 first-end : ( p : Q → Bool ) → End finite p | |
| 360 | 54 first-end p i i>n = ⊥-elim (nat-≤> i>n (fin<n {finite} i) ) |
| 163 | 55 next-end : {m : ℕ } → ( p : Q → Bool ) → End (suc m) p |
| 56 → (m<n : m < finite ) → p (Q←F (fromℕ< m<n )) ≡ false | |
| 57 → End m p | |
| 337 | 58 next-end {m} p prev m<n np i m<i with NP.<-cmp m (toℕ i) |
| 163 | 59 next-end p prev m<n np i m<i | tri< a ¬b ¬c = prev i a |
| 60 next-end p prev m<n np i m<i | tri> ¬a ¬b c = ⊥-elim ( nat-≤> m<i c ) | |
| 61 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 | |
| 62 m<n=i : {n : ℕ } (i : Fin n) {m : ℕ } → m ≡ (toℕ i) → (m<n : m < n ) → fromℕ< m<n ≡ i | |
| 264 | 63 m<n=i i refl m<n = fromℕ<-toℕ i m<n |
| 163 | 64 found : { p : Q → Bool } → (q : Q ) → p q ≡ true → exists p ≡ true |
| 337 | 65 found {p} q pt = found1 finite (NP.≤-refl ) ( first-end p ) where |
| 163 | 66 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 |
| 67 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 ) | |
| 68 found1 (suc m) m<n end with bool-≡-? (p (Q←F (fromℕ< m<n))) true | |
| 264 | 69 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 | 70 found1 (suc m) m<n end | no np = begin |
| 264 | 71 p (Q←F (fromℕ< m<n)) \/ exists1 m (<to≤ m<n) p |
| 163 | 72 ≡⟨ bool-or-1 (¬-bool-t np ) ⟩ |
| 264 | 73 exists1 m (<to≤ m<n) p |
| 74 ≡⟨ found1 m (<to≤ m<n) (next-end p end m<n (¬-bool-t np )) ⟩ | |
| 163 | 75 true |
| 76 ∎ where open ≡-Reasoning | |
| 268 | 77 not-found : { p : Q → Bool } → ( (q : Q ) → p q ≡ false ) → exists p ≡ false |
| 337 | 78 not-found {p} pn = not-found2 finite NP.≤-refl where |
| 268 | 79 not-found2 : (m : ℕ ) → (m<n : m Data.Nat.≤ finite ) → exists1 m m<n p ≡ false |
| 80 not-found2 zero _ = refl | |
| 81 not-found2 ( suc m ) m<n with pn (Q←F (fromℕ< {m} {finite} m<n)) | |
| 82 not-found2 (suc m) m<n | eq = begin | |
| 83 p (Q←F (fromℕ< m<n)) \/ exists1 m (<to≤ m<n) p | |
| 84 ≡⟨ bool-or-1 eq ⟩ | |
| 85 exists1 m (<to≤ m<n) p | |
| 86 ≡⟨ not-found2 m (<to≤ m<n) ⟩ | |
| 87 false | |
| 88 ∎ where open ≡-Reasoning | |
| 89 found← : { p : Q → Bool } → exists p ≡ true → Found Q p | |
| 337 | 90 found← {p} exst = found2 finite NP.≤-refl (first-end p ) where |
| 268 | 91 found2 : (m : ℕ ) (m<n : m Data.Nat.≤ finite ) → End m p → Found Q p |
| 92 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 | |
| 93 lemma : (λ z → p (Q←F (F←Q z))) ≡ p | |
| 94 lemma = f-extensionality ( λ q → subst (λ k → p k ≡ p q ) (sym (finiso→ q)) refl ) | |
| 95 found2 (suc m) m<n end with bool-≡-? (p (Q←F (fromℕ< m<n))) true | |
| 96 found2 (suc m) m<n end | yes eq = record { found-q = Q←F (fromℕ< m<n) ; found-p = eq } | |
| 97 found2 (suc m) m<n end | no np = | |
| 98 found2 m (<to≤ m<n) (next-end p end m<n (¬-bool-t np )) | |
| 99 not-found← : { p : Q → Bool } → exists p ≡ false → (q : Q ) → p q ≡ false | |
| 100 not-found← {p} np q = ¬-bool-t ( contra-position {_} {_} {_} {exists p ≡ true} (found q) (λ ep → ¬-bool np ep ) ) | |
| 351 | 101 Q←F-inject : {x y : Fin finite} → Q←F x ≡ Q←F y → x ≡ y |
| 102 Q←F-inject eq = subst₂ (λ j k → j ≡ k ) (finiso← _) (finiso← _) (cong F←Q eq) | |
| 103 F←Q-inject : {x y : Q } → F←Q x ≡ F←Q y → x ≡ y | |
| 104 F←Q-inject eq = subst₂ (λ j k → j ≡ k ) (finiso→ _) (finiso→ _) (cong Q←F eq) | |
| 268 | 105 |
| 163 | 106 |
| 107 | |
| 264 | 108 iso-fin : {A B : Set} → FiniteSet A → Bijection A B → FiniteSet B |
| 141 | 109 iso-fin {A} {B} fin iso = record { |
| 330 | 110 Q←F = λ f → fun→ iso ( FiniteSet.Q←F fin f ) |
| 264 | 111 ; F←Q = λ b → FiniteSet.F←Q fin (fun← iso b ) |
| 112 ; finiso→ = finiso→ | |
| 113 ; finiso← = finiso← | |
| 141 | 114 } where |
| 264 | 115 finiso→ : (q : B) → fun→ iso (FiniteSet.Q←F fin (FiniteSet.F←Q fin (Bijection.fun← iso q))) ≡ q |
| 141 | 116 finiso→ q = begin |
| 264 | 117 fun→ iso (FiniteSet.Q←F fin (FiniteSet.F←Q fin (Bijection.fun← iso q))) |
| 118 ≡⟨ cong (λ k → fun→ iso k ) (FiniteSet.finiso→ fin _ ) ⟩ | |
| 119 fun→ iso (Bijection.fun← iso q) | |
| 120 ≡⟨ fiso→ iso _ ⟩ | |
| 141 | 121 q |
| 264 | 122 ∎ where open ≡-Reasoning |
| 123 finiso← : (f : Fin (FiniteSet.finite fin ))→ FiniteSet.F←Q fin (Bijection.fun← iso (Bijection.fun→ iso (FiniteSet.Q←F fin f))) ≡ f | |
| 141 | 124 finiso← f = begin |
| 264 | 125 FiniteSet.F←Q fin (Bijection.fun← iso (Bijection.fun→ iso (FiniteSet.Q←F fin f))) |
| 126 ≡⟨ cong (λ k → FiniteSet.F←Q fin k ) (Bijection.fiso← iso _) ⟩ | |
| 141 | 127 FiniteSet.F←Q fin (FiniteSet.Q←F fin f) |
| 128 ≡⟨ FiniteSet.finiso← fin _ ⟩ | |
| 129 f | |
| 130 ∎ where | |
| 131 open ≡-Reasoning | |
| 132 | |
| 133 data One : Set where | |
| 134 one : One | |
| 135 | |
| 136 fin-∨1 : {B : Set} → (fb : FiniteSet B ) → FiniteSet (One ∨ B) | |
| 137 fin-∨1 {B} fb = record { | |
| 330 | 138 Q←F = Q←F |
| 139 ; F←Q = F←Q | |
| 140 ; finiso→ = finiso→ | |
| 141 ; finiso← = finiso← | |
| 141 | 142 } where |
| 143 b = FiniteSet.finite fb | |
| 144 Q←F : Fin (suc b) → One ∨ B | |
| 145 Q←F zero = case1 one | |
| 146 Q←F (suc f) = case2 (FiniteSet.Q←F fb f) | |
| 147 F←Q : One ∨ B → Fin (suc b) | |
| 148 F←Q (case1 one) = zero | |
| 149 F←Q (case2 f ) = suc (FiniteSet.F←Q fb f) | |
| 150 finiso→ : (q : One ∨ B) → Q←F (F←Q q) ≡ q | |
| 151 finiso→ (case1 one) = refl | |
| 152 finiso→ (case2 b) = cong (λ k → case2 k ) (FiniteSet.finiso→ fb b) | |
| 153 finiso← : (q : Fin (suc b)) → F←Q (Q←F q) ≡ q | |
| 154 finiso← zero = refl | |
| 155 finiso← (suc f) = cong ( λ k → suc k ) (FiniteSet.finiso← fb f) | |
| 156 | |
| 157 | |
| 158 fin-∨2 : {B : Set} → ( a : ℕ ) → FiniteSet B → FiniteSet (Fin a ∨ B) | |
| 159 fin-∨2 {B} zero fb = iso-fin fb iso where | |
| 264 | 160 iso : Bijection B (Fin zero ∨ B) |
| 161 iso = record { | |
| 162 fun← = fun←1 | |
| 163 ; fun→ = λ b → case2 b | |
| 164 ; fiso→ = fiso→1 | |
| 165 ; fiso← = λ _ → refl | |
| 141 | 166 } where |
| 264 | 167 fun←1 : Fin zero ∨ B → B |
| 168 fun←1 (case2 x) = x | |
| 169 fiso→1 : (f : Fin zero ∨ B ) → case2 (fun←1 f) ≡ f | |
| 170 fiso→1 (case2 x) = refl | |
| 141 | 171 fin-∨2 {B} (suc a) fb = iso-fin (fin-∨1 (fin-∨2 a fb) ) iso |
| 172 where | |
| 264 | 173 iso : Bijection (One ∨ (Fin a ∨ B) ) (Fin (suc a) ∨ B) |
| 174 fun← iso (case1 zero) = case1 one | |
| 175 fun← iso (case1 (suc f)) = case2 (case1 f) | |
| 176 fun← iso (case2 b) = case2 (case2 b) | |
| 177 fun→ iso (case1 one) = case1 zero | |
| 178 fun→ iso (case2 (case1 f)) = case1 (suc f) | |
| 179 fun→ iso (case2 (case2 b)) = case2 b | |
| 180 fiso← iso (case1 one) = refl | |
| 181 fiso← iso (case2 (case1 x)) = refl | |
| 182 fiso← iso (case2 (case2 x)) = refl | |
| 183 fiso→ iso (case1 zero) = refl | |
| 184 fiso→ iso (case1 (suc x)) = refl | |
| 185 fiso→ iso (case2 x) = refl | |
| 141 | 186 |
| 187 | |
| 264 | 188 FiniteSet→Fin : {A : Set} → (fin : FiniteSet A ) → Bijection (Fin (FiniteSet.finite fin)) A |
| 189 fun← (FiniteSet→Fin fin) f = FiniteSet.F←Q fin f | |
| 190 fun→ (FiniteSet→Fin fin) f = FiniteSet.Q←F fin f | |
| 191 fiso← (FiniteSet→Fin fin) = FiniteSet.finiso← fin | |
| 192 fiso→ (FiniteSet→Fin fin) = FiniteSet.finiso→ fin | |
| 141 | 193 |
| 194 | |
| 195 fin-∨ : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A ∨ B) | |
| 196 fin-∨ {A} {B} fa fb = iso-fin (fin-∨2 a fb ) iso2 where | |
| 197 a = FiniteSet.finite fa | |
| 198 ia = FiniteSet→Fin fa | |
| 264 | 199 iso2 : Bijection (Fin a ∨ B ) (A ∨ B) |
| 200 fun← iso2 (case1 x) = case1 (fun← ia x ) | |
| 201 fun← iso2 (case2 x) = case2 x | |
| 202 fun→ iso2 (case1 x) = case1 (fun→ ia x ) | |
| 203 fun→ iso2 (case2 x) = case2 x | |
| 204 fiso← iso2 (case1 x) = cong ( λ k → case1 k ) (Bijection.fiso← ia x) | |
| 205 fiso← iso2 (case2 x) = refl | |
| 206 fiso→ iso2 (case1 x) = cong ( λ k → case1 k ) (Bijection.fiso→ ia x) | |
| 207 fiso→ iso2 (case2 x) = refl | |
| 141 | 208 |
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209 open import Data.Product hiding ( map ) |
| 141 | 210 |
| 211 fin-× : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A × B) | |
| 212 fin-× {A} {B} fa fb with FiniteSet→Fin fa | |
| 213 ... | a=f = iso-fin (fin-×-f a ) iso-1 where | |
| 214 a = FiniteSet.finite fa | |
| 215 b = FiniteSet.finite fb | |
| 264 | 216 iso-1 : Bijection (Fin a × B) ( A × B ) |
| 217 fun← iso-1 x = ( FiniteSet.F←Q fa (proj₁ x) , proj₂ x) | |
| 218 fun→ iso-1 x = ( FiniteSet.Q←F fa (proj₁ x) , proj₂ x) | |
| 219 fiso← iso-1 x = lemma where | |
| 141 | 220 lemma : (FiniteSet.F←Q fa (FiniteSet.Q←F fa (proj₁ x)) , proj₂ x) ≡ ( proj₁ x , proj₂ x ) |
| 221 lemma = cong ( λ k → ( k , proj₂ x ) ) (FiniteSet.finiso← fa _ ) | |
| 264 | 222 fiso→ iso-1 x = cong ( λ k → ( k , proj₂ x ) ) (FiniteSet.finiso→ fa _ ) |
| 141 | 223 |
| 264 | 224 iso-2 : {a : ℕ } → Bijection (B ∨ (Fin a × B)) (Fin (suc a) × B) |
| 225 fun← iso-2 (zero , b ) = case1 b | |
| 226 fun← iso-2 (suc fst , b ) = case2 ( fst , b ) | |
| 227 fun→ iso-2 (case1 b) = ( zero , b ) | |
| 228 fun→ iso-2 (case2 (a , b )) = ( suc a , b ) | |
| 229 fiso← iso-2 (case1 x) = refl | |
| 230 fiso← iso-2 (case2 x) = refl | |
| 231 fiso→ iso-2 (zero , b ) = refl | |
| 232 fiso→ iso-2 (suc a , b ) = refl | |
| 141 | 233 |
| 234 fin-×-f : ( a : ℕ ) → FiniteSet ((Fin a) × B) | |
| 235 fin-×-f zero = record { Q←F = λ () ; F←Q = λ () ; finiso→ = λ () ; finiso← = λ () ; finite = 0 } | |
| 236 fin-×-f (suc a) = iso-fin ( fin-∨ fb ( fin-×-f a ) ) iso-2 | |
| 237 | |
| 238 open _∧_ | |
| 239 | |
| 240 fin-∧ : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A ∧ B) | |
| 241 fin-∧ {A} {B} fa fb with FiniteSet→Fin fa -- same thing for our tool | |
| 242 ... | a=f = iso-fin (fin-×-f a ) iso-1 where | |
| 243 a = FiniteSet.finite fa | |
| 244 b = FiniteSet.finite fb | |
| 264 | 245 iso-1 : Bijection (Fin a ∧ B) ( A ∧ B ) |
| 246 fun← iso-1 x = record { proj1 = FiniteSet.F←Q fa (proj1 x) ; proj2 = proj2 x} | |
| 247 fun→ iso-1 x = record { proj1 = FiniteSet.Q←F fa (proj1 x) ; proj2 = proj2 x} | |
| 248 fiso← iso-1 x = lemma where | |
| 141 | 249 lemma : record { proj1 = FiniteSet.F←Q fa (FiniteSet.Q←F fa (proj1 x)) ; proj2 = proj2 x} ≡ record {proj1 = proj1 x ; proj2 = proj2 x } |
| 250 lemma = cong ( λ k → record {proj1 = k ; proj2 = proj2 x } ) (FiniteSet.finiso← fa _ ) | |
| 264 | 251 fiso→ iso-1 x = cong ( λ k → record {proj1 = k ; proj2 = proj2 x } ) (FiniteSet.finiso→ fa _ ) |
| 141 | 252 |
| 264 | 253 iso-2 : {a : ℕ } → Bijection (B ∨ (Fin a ∧ B)) (Fin (suc a) ∧ B) |
| 254 fun← iso-2 (record { proj1 = zero ; proj2 = b }) = case1 b | |
| 255 fun← iso-2 (record { proj1 = suc fst ; proj2 = b }) = case2 ( record { proj1 = fst ; proj2 = b } ) | |
| 256 fun→ iso-2 (case1 b) = record {proj1 = zero ; proj2 = b } | |
| 257 fun→ iso-2 (case2 (record { proj1 = a ; proj2 = b })) = record { proj1 = suc a ; proj2 = b } | |
| 258 fiso← iso-2 (case1 x) = refl | |
| 259 fiso← iso-2 (case2 x) = refl | |
| 260 fiso→ iso-2 (record { proj1 = zero ; proj2 = b }) = refl | |
| 261 fiso→ iso-2 (record { proj1 = suc a ; proj2 = b }) = refl | |
| 141 | 262 |
| 263 fin-×-f : ( a : ℕ ) → FiniteSet ((Fin a) ∧ B) | |
| 264 fin-×-f zero = record { Q←F = λ () ; F←Q = λ () ; finiso→ = λ () ; finiso← = λ () ; finite = 0 } | |
| 265 fin-×-f (suc a) = iso-fin ( fin-∨ fb ( fin-×-f a ) ) iso-2 | |
| 266 | |
| 267 -- import Data.Nat.DivMod | |
| 268 | |
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269 open import Data.Vec hiding ( map ; length ) |
| 141 | 270 import Data.Product |
| 271 | |
| 272 exp2 : (n : ℕ ) → exp 2 (suc n) ≡ exp 2 n Data.Nat.+ exp 2 n | |
| 273 exp2 n = begin | |
| 274 exp 2 (suc n) | |
| 275 ≡⟨⟩ | |
| 276 2 * ( exp 2 n ) | |
| 277 ≡⟨ *-comm 2 (exp 2 n) ⟩ | |
| 278 ( exp 2 n ) * 2 | |
| 279 ≡⟨ *-suc ( exp 2 n ) 1 ⟩ | |
| 280 (exp 2 n ) Data.Nat.+ ( exp 2 n ) * 1 | |
| 281 ≡⟨ cong ( λ k → (exp 2 n ) Data.Nat.+ k ) (proj₂ *-identity (exp 2 n) ) ⟩ | |
| 282 exp 2 n Data.Nat.+ exp 2 n | |
| 283 ∎ where | |
| 284 open ≡-Reasoning | |
| 285 open Data.Product | |
| 286 | |
| 287 cast-iso : {n m : ℕ } → (eq : n ≡ m ) → (f : Fin m ) → cast eq ( cast (sym eq ) f) ≡ f | |
| 288 cast-iso refl zero = refl | |
| 289 cast-iso refl (suc f) = cong ( λ k → suc k ) ( cast-iso refl f ) | |
| 290 | |
| 291 | |
| 292 fin2List : {n : ℕ } → FiniteSet (Vec Bool n) | |
| 293 fin2List {zero} = record { | |
| 294 Q←F = λ _ → Vec.[] | |
| 295 ; F←Q = λ _ → # 0 | |
| 296 ; finiso→ = finiso→ | |
| 297 ; finiso← = finiso← | |
| 298 } where | |
| 299 Q = Vec Bool zero | |
| 300 finiso→ : (q : Q) → [] ≡ q | |
| 301 finiso→ [] = refl | |
| 302 finiso← : (f : Fin (exp 2 zero)) → # 0 ≡ f | |
| 303 finiso← zero = refl | |
| 304 fin2List {suc n} = subst (λ k → FiniteSet (Vec Bool (suc n)) ) (sym (exp2 n)) ( iso-fin (fin-∨ (fin2List ) (fin2List )) iso ) | |
| 305 where | |
| 306 QtoR : Vec Bool (suc n) → Vec Bool n ∨ Vec Bool n | |
| 307 QtoR ( true ∷ x ) = case1 x | |
| 308 QtoR ( false ∷ x ) = case2 x | |
| 309 RtoQ : Vec Bool n ∨ Vec Bool n → Vec Bool (suc n) | |
| 310 RtoQ ( case1 x ) = true ∷ x | |
| 311 RtoQ ( case2 x ) = false ∷ x | |
| 312 isoRQ : (x : Vec Bool (suc n) ) → RtoQ ( QtoR x ) ≡ x | |
| 313 isoRQ (true ∷ _ ) = refl | |
| 314 isoRQ (false ∷ _ ) = refl | |
| 315 isoQR : (x : Vec Bool n ∨ Vec Bool n ) → QtoR ( RtoQ x ) ≡ x | |
| 316 isoQR (case1 x) = refl | |
| 317 isoQR (case2 x) = refl | |
| 264 | 318 iso : Bijection (Vec Bool n ∨ Vec Bool n) (Vec Bool (suc n)) |
| 319 iso = record { fun← = QtoR ; fun→ = RtoQ ; fiso← = isoQR ; fiso→ = isoRQ } | |
| 141 | 320 |
| 321 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 | |
| 322 F2L {Q} {zero} fin _ Q→B = [] | |
| 337 | 323 F2L {Q} {suc n} fin (s≤s n<m) Q→B = Q→B (FiniteSet.Q←F fin (fromℕ< n<m)) lemma6 ∷ F2L {Q} fin (NP.<-trans n<m a<sa ) qb1 where |
| 141 | 324 lemma6 : toℕ (FiniteSet.F←Q fin (FiniteSet.Q←F fin (fromℕ< n<m))) < suc n |
| 325 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 ) | |
| 326 qb1 : (q : Q) → toℕ (FiniteSet.F←Q fin q) < n → Bool | |
| 337 | 327 qb1 q q<n = Q→B q (NP.<-trans q<n a<sa) |
| 141 | 328 |
| 329 List2Func : { Q : Set } → {n : ℕ } → (fin : FiniteSet Q ) → n < suc (FiniteSet.finite fin) → Vec Bool n → Q → Bool | |
| 330 List2Func {Q} {zero} fin (s≤s z≤n) [] q = false | |
| 331 List2Func {Q} {suc n} fin (s≤s n<m) (h ∷ t) q with FiniteSet.F←Q fin q ≟ fromℕ< n<m | |
| 332 ... | yes _ = h | |
| 337 | 333 ... | no _ = List2Func {Q} fin (NP.<-trans n<m a<sa ) t q |
| 141 | 334 |
| 335 open import Level renaming ( suc to Suc ; zero to Zero) | |
| 336 open import Axiom.Extensionality.Propositional | |
| 268 | 337 -- postulate f-extensionality : { n : Level} → Axiom.Extensionality.Propositional.Extensionality n n |
| 141 | 338 |
| 339 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 | |
| 340 F2L-iso {Q} fin x = f2l m a<sa x where | |
| 341 m = FiniteSet.finite fin | |
| 342 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 | |
| 343 f2l zero (s≤s z≤n) [] = refl | |
| 163 | 344 f2l (suc n) (s≤s n<m) (h ∷ t ) = lemma1 lemma2 lemma3f where |
| 141 | 345 lemma1 : {n : ℕ } → {h h1 : Bool } → {t t1 : Vec Bool n } → h ≡ h1 → t ≡ t1 → h ∷ t ≡ h1 ∷ t1 |
| 346 lemma1 refl refl = refl | |
| 347 lemma2 : List2Func fin (s≤s n<m) (h ∷ t) (FiniteSet.Q←F fin (fromℕ< n<m)) ≡ h | |
| 348 lemma2 with FiniteSet.F←Q fin (FiniteSet.Q←F fin (fromℕ< n<m)) ≟ fromℕ< n<m | |
| 349 lemma2 | yes p = refl | |
| 350 lemma2 | no ¬p = ⊥-elim ( ¬p (FiniteSet.finiso← fin _) ) | |
| 337 | 351 lemma4 : (q : Q ) → toℕ (FiniteSet.F←Q fin q ) < n → List2Func fin (s≤s n<m) (h ∷ t) q ≡ List2Func fin (NP.<-trans n<m a<sa) t q |
| 141 | 352 lemma4 q _ with FiniteSet.F←Q fin q ≟ fromℕ< n<m |
| 353 lemma4 q lt | yes p = ⊥-elim ( nat-≡< (toℕ-fromℕ< n<m) (lemma5 n lt (cong (λ k → toℕ k) p))) where | |
| 354 lemma5 : {j k : ℕ } → ( n : ℕ) → suc j ≤ n → j ≡ k → k < n | |
| 355 lemma5 {zero} (suc n) (s≤s z≤n) refl = s≤s z≤n | |
| 356 lemma5 {suc j} (suc n) (s≤s lt) refl = s≤s (lemma5 {j} n lt refl) | |
| 357 lemma4 q _ | no ¬p = refl | |
| 337 | 358 lemma3f : F2L fin (NP.<-trans n<m a<sa) (λ q q<n → List2Func fin (s≤s n<m) (h ∷ t) q ) ≡ t |
| 163 | 359 lemma3f = begin |
| 337 | 360 F2L fin (NP.<-trans n<m a<sa) (λ q q<n → List2Func fin (s≤s n<m) (h ∷ t) q ) |
| 361 ≡⟨ cong (λ k → F2L fin (NP.<-trans n<m a<sa) ( λ q q<n → k q q<n )) | |
| 141 | 362 (f-extensionality ( λ q → |
| 363 (f-extensionality ( λ q<n → lemma4 q q<n )))) ⟩ | |
| 337 | 364 F2L fin (NP.<-trans n<m a<sa) (λ q q<n → List2Func fin (NP.<-trans n<m a<sa) t q ) |
| 365 ≡⟨ f2l n (NP.<-trans n<m a<sa ) t ⟩ | |
| 141 | 366 t |
| 367 ∎ where | |
| 368 open ≡-Reasoning | |
| 369 | |
| 370 | |
| 371 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 | |
| 372 L2F fin n<m x q q<n = List2Func fin n<m x q | |
| 373 | |
| 374 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 | |
| 375 L2F-iso {Q} fin f q = l2f m a<sa (toℕ<n _) where | |
| 376 m = FiniteSet.finite fin | |
| 163 | 377 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 |
| 378 lemma11f n<m ¬q=n q≤n = lemma13 n<m (contra-position (lemma12 n<m _) ¬q=n ) q≤n where | |
| 141 | 379 lemma13 : {n nq : ℕ } → (n<m : n < m ) → ¬ ( nq ≡ n ) → nq ≤ n → nq < n |
| 380 lemma13 {0} {0} (s≤s z≤n) nt z≤n = ⊥-elim ( nt refl ) | |
| 381 lemma13 {suc _} {0} (s≤s (s≤s n<m)) nt z≤n = s≤s z≤n | |
| 337 | 382 lemma13 {suc n} {suc nq} n<m nt (s≤s nq≤n) = s≤s (lemma13 {n} {nq} (NP.<-trans a<sa n<m ) (λ eq → nt ( cong ( λ k → suc k ) eq )) nq≤n) |
| 163 | 383 lemma3f : {a b : ℕ } → (lt : a < b ) → fromℕ< (s≤s lt) ≡ suc (fromℕ< lt) |
| 384 lemma3f (s≤s lt) = refl | |
| 385 lemma12f : {n m : ℕ } → (n<m : n < m ) → (f : Fin m ) → toℕ f ≡ n → f ≡ fromℕ< n<m | |
| 386 lemma12f {zero} {suc m} (s≤s z≤n) zero refl = refl | |
| 387 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 | 388 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 |
| 389 l2f zero (s≤s z≤n) () | |
| 390 l2f (suc n) (s≤s n<m) (s≤s n<q) with FiniteSet.F←Q fin q ≟ fromℕ< n<m | |
| 391 l2f (suc n) (s≤s n<m) (s≤s n<q) | yes p = begin | |
| 392 f (FiniteSet.Q←F fin (fromℕ< n<m)) | |
| 393 ≡⟨ cong ( λ k → f (FiniteSet.Q←F fin k )) (sym p) ⟩ | |
| 394 f (FiniteSet.Q←F fin ( FiniteSet.F←Q fin q )) | |
| 395 ≡⟨ cong ( λ k → f k ) (FiniteSet.finiso→ fin _ ) ⟩ | |
| 396 f q | |
| 397 ∎ where | |
| 398 open ≡-Reasoning | |
| 337 | 399 l2f (suc n) (s≤s n<m) (s≤s n<q) | no ¬p = l2f n (NP.<-trans n<m a<sa) (lemma11f n<m ¬p n<q) |
| 141 | 400 |
| 401 fin→ : {A : Set} → FiniteSet A → FiniteSet (A → Bool ) | |
| 402 fin→ {A} fin = iso-fin fin2List iso where | |
| 403 a = FiniteSet.finite fin | |
| 264 | 404 iso : Bijection (Vec Bool a ) (A → Bool) |
| 405 fun← iso x = F2L fin a<sa ( λ q _ → x q ) | |
| 406 fun→ iso x = List2Func fin a<sa x | |
| 407 fiso← iso x = F2L-iso fin x | |
| 408 fiso→ iso x = lemma where | |
| 141 | 409 lemma : List2Func fin a<sa (F2L fin a<sa (λ q _ → x q)) ≡ x |
| 410 lemma = f-extensionality ( λ q → L2F-iso fin x q ) | |
| 411 | |
| 412 | |
| 413 Fin2Finite : ( n : ℕ ) → FiniteSet (Fin n) | |
| 414 Fin2Finite n = record { F←Q = λ x → x ; Q←F = λ x → x ; finiso← = λ q → refl ; finiso→ = λ q → refl } | |
| 415 | |
| 416 data fin-less { n : ℕ } { A : Set } (fa : FiniteSet A ) (n<m : n < FiniteSet.finite fa ) : Set where | |
| 417 elm1 : (elm : A ) → toℕ (FiniteSet.F←Q fa elm ) < n → fin-less fa n<m | |
| 418 | |
| 419 get-elm : { n : ℕ } { A : Set } {fa : FiniteSet A } {n<m : n < FiniteSet.finite fa } → fin-less fa n<m → A | |
| 420 get-elm (elm1 a _ ) = a | |
| 421 | |
| 422 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 | |
| 423 get-< (elm1 _ b ) = b | |
| 424 | |
| 425 fin-less-cong : { n : ℕ } { A : Set } (fa : FiniteSet A ) (n<m : n < FiniteSet.finite fa ) | |
| 426 → (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 | |
| 427 fin-less-cong fa n<m (elm1 elm x) (elm1 elm x) refl HE.refl = refl | |
| 428 | |
| 429 fin-< : {A : Set} → { n : ℕ } → (fa : FiniteSet A ) → (n<m : n < FiniteSet.finite fa ) → FiniteSet (fin-less fa n<m ) | |
| 430 fin-< {A} {n} fa n<m = iso-fin (Fin2Finite n) iso where | |
| 431 m = FiniteSet.finite fa | |
| 264 | 432 iso : Bijection (Fin n) (fin-less fa n<m ) |
| 163 | 433 lemma8f : {i j n : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n } → i<n ≅ j<n |
| 434 lemma8f {zero} {zero} {suc n} refl {s≤s z≤n} {s≤s z≤n} = HE.refl | |
| 435 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 ) | |
| 436 lemma10f : {n i j : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n } → fromℕ< i<n ≡ fromℕ< j<n | |
| 437 lemma10f refl = HE.≅-to-≡ (HE.cong (λ k → fromℕ< k ) (lemma8f refl )) | |
| 337 | 438 lemma3f : {a b c : ℕ } → { a<b : a < b } { b<c : b < c } { a<c : a < c } → NP.<-trans a<b b<c ≡ a<c |
| 163 | 439 lemma3f {a} {b} {c} {a<b} {b<c} {a<c} = HE.≅-to-≡ (lemma8f refl) |
| 337 | 440 lemma11f : {n : ℕ } {x : Fin n } → (n<m : n < m ) → toℕ (fromℕ< (NP.<-trans (toℕ<n x) n<m)) ≡ toℕ x |
| 163 | 441 lemma11f {n} {x} n<m = begin |
| 337 | 442 toℕ (fromℕ< (NP.<-trans (toℕ<n x) n<m)) |
| 141 | 443 ≡⟨ toℕ-fromℕ< _ ⟩ |
| 444 toℕ x | |
| 445 ∎ where | |
| 446 open ≡-Reasoning | |
| 264 | 447 fun← iso (elm1 elm x) = fromℕ< x |
| 337 | 448 fun→ iso x = elm1 (FiniteSet.Q←F fa (fromℕ< (NP.<-trans x<n n<m ))) to<n where |
| 141 | 449 x<n : toℕ x < n |
| 450 x<n = toℕ<n x | |
| 337 | 451 to<n : toℕ (FiniteSet.F←Q fa (FiniteSet.Q←F fa (fromℕ< (NP.<-trans x<n n<m)))) < n |
| 452 to<n = subst (λ k → toℕ k < n ) (sym (FiniteSet.finiso← fa _ )) (subst (λ k → k < n ) (sym ( toℕ-fromℕ< (NP.<-trans x<n n<m) )) x<n ) | |
| 264 | 453 fiso← iso x = lemma2 where |
| 141 | 454 lemma2 : fromℕ< (subst (λ k → toℕ k < n) (sym |
| 337 | 455 (FiniteSet.finiso← fa (fromℕ< (NP.<-trans (toℕ<n x) n<m)))) (subst (λ k → k < n) |
| 456 (sym (toℕ-fromℕ< (NP.<-trans (toℕ<n x) n<m))) (toℕ<n x))) ≡ x | |
| 141 | 457 lemma2 = begin |
| 458 fromℕ< (subst (λ k → toℕ k < n) (sym | |
| 337 | 459 (FiniteSet.finiso← fa (fromℕ< (NP.<-trans (toℕ<n x) n<m)))) (subst (λ k → k < n) |
| 460 (sym (toℕ-fromℕ< (NP.<-trans (toℕ<n x) n<m))) (toℕ<n x))) | |
| 141 | 461 ≡⟨⟩ |
| 462 fromℕ< ( subst (λ k → toℕ ( k ) < n ) (sym (FiniteSet.finiso← fa _ )) lemma6 ) | |
| 463 ≡⟨ lemma10 (cong (λ k → toℕ k) (FiniteSet.finiso← fa _ ) ) ⟩ | |
| 464 fromℕ< lemma6 | |
| 465 ≡⟨ lemma10 (lemma11 n<m ) ⟩ | |
| 466 fromℕ< ( toℕ<n x ) | |
| 467 ≡⟨ fromℕ<-toℕ _ _ ⟩ | |
| 468 x | |
| 469 ∎ where | |
| 470 open ≡-Reasoning | |
| 337 | 471 lemma6 : toℕ (fromℕ< (NP.<-trans (toℕ<n x) n<m)) < n |
| 472 lemma6 = subst ( λ k → k < n ) (sym (toℕ-fromℕ< (NP.<-trans (toℕ<n x) n<m))) (toℕ<n x ) | |
| 264 | 473 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 | 474 lemma13 : toℕ (fromℕ< x) ≡ toℕ (FiniteSet.F←Q fa elm) |
| 475 lemma13 = begin | |
| 476 toℕ (fromℕ< x) | |
| 477 ≡⟨ toℕ-fromℕ< _ ⟩ | |
| 478 toℕ (FiniteSet.F←Q fa elm) | |
| 479 ∎ where open ≡-Reasoning | |
| 337 | 480 lemma : FiniteSet.Q←F fa (fromℕ< (NP.<-trans (toℕ<n (Bijection.fun← iso (elm1 elm x))) n<m)) ≡ elm |
| 141 | 481 lemma = begin |
| 337 | 482 FiniteSet.Q←F fa (fromℕ< (NP.<-trans (toℕ<n (Bijection.fun← iso (elm1 elm x))) n<m)) |
| 141 | 483 ≡⟨⟩ |
| 337 | 484 FiniteSet.Q←F fa (fromℕ< ( NP.<-trans (toℕ<n ( fromℕ< x ) ) n<m)) |
| 264 | 485 ≡⟨ cong (λ k → FiniteSet.Q←F fa k) (lemma10 lemma13 ) ⟩ |
| 337 | 486 FiniteSet.Q←F fa (fromℕ< ( NP.<-trans x n<m)) |
| 264 | 487 ≡⟨ cong (λ k → FiniteSet.Q←F fa (fromℕ< k )) (HE.≅-to-≡ (lemma8 refl)) ⟩ |
| 141 | 488 FiniteSet.Q←F fa (fromℕ< ( toℕ<n (FiniteSet.F←Q fa elm))) |
| 489 ≡⟨ cong (λ k → FiniteSet.Q←F fa k ) ( fromℕ<-toℕ _ _ ) ⟩ | |
| 490 FiniteSet.Q←F fa (FiniteSet.F←Q fa elm ) | |
| 491 ≡⟨ FiniteSet.finiso→ fa _ ⟩ | |
| 492 elm | |
| 493 ∎ where open ≡-Reasoning | |
| 494 | |
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495 open import Data.List |
| 141 | 496 |
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497 open FiniteSet |
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498 |
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499 memberQ : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool |
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500 memberQ {Q} finq q [] = false |
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501 memberQ {Q} finq q (q0 ∷ qs) with equal? finq q q0 |
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502 ... | true = true |
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503 ... | false = memberQ finq q qs |
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504 |
| 316 | 505 -- |
| 506 -- there is a duplicate element in finite list | |
| 507 -- | |
| 508 | |
| 330 | 509 -- |
| 510 -- How about this? | |
| 511 -- get list of Q | |
| 512 -- remove one element for each Q from list | |
| 513 -- there must be remaining list > 1 | |
| 514 -- theses are duplicates | |
| 515 -- actualy it is duplicate | |
| 516 | |
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517 phase2 : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool |
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518 phase2 finq q [] = false |
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519 phase2 finq q (x ∷ qs) with equal? finq q x |
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520 ... | true = true |
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521 ... | false = phase2 finq q qs |
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522 phase1 : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool |
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523 phase1 finq q [] = false |
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524 phase1 finq q (x ∷ qs) with equal? finq q x |
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525 ... | true = phase2 finq q qs |
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526 ... | false = phase1 finq q qs |
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527 |
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528 dup-in-list : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool |
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529 dup-in-list {Q} finq q qs = phase1 finq q qs |
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530 |
| 316 | 531 -- |
| 532 -- if length of the list is longer than kinds of a finite set, there is a duplicate | |
| 533 -- prove this based on the theorem on Data.Fin | |
| 534 -- | |
| 535 | |
| 283 | 536 dup-in-list+fin : { Q : Set } (finq : FiniteSet Q) |
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537 → (q : Q) (qs : List Q ) |
| 283 | 538 → fin-dup-in-list (F←Q finq q) (map (F←Q finq) qs) ≡ true |
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539 → dup-in-list finq q qs ≡ true |
| 283 | 540 dup-in-list+fin {Q} finq q qs p = i-phase1 qs p where |
| 541 i-phase2 : (qs : List Q) → fin-phase2 (F←Q finq q) (map (F←Q finq) qs) ≡ true | |
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542 → phase2 finq q qs ≡ true |
| 294 | 543 i-phase2 (x ∷ qs) p with equal? finq q x | inspect (equal? finq q ) x | <-fcmp (F←Q finq q) (F←Q finq x) |
| 544 ... | true | _ | t = refl | |
| 545 ... | false | _ | tri< a ¬b ¬c = i-phase2 qs p | |
| 546 ... | false | record { eq = eq } | tri≈ ¬a b ¬c = ⊥-elim (¬-bool eq | |
| 547 (subst₂ (λ j k → equal? finq j k ≡ true) (finiso→ finq q) (subst (λ k → Q←F finq k ≡ x) (sym b) (finiso→ finq x)) ( equal?-refl finq ))) | |
| 548 ... | false | _ | tri> ¬a ¬b c = i-phase2 qs p | |
| 283 | 549 i-phase1 : (qs : List Q) → fin-phase1 (F←Q finq q) (map (F←Q finq) qs) ≡ true |
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550 → phase1 finq q qs ≡ true |
| 294 | 551 i-phase1 (x ∷ qs) p with equal? finq q x | inspect (equal? finq q ) x | <-fcmp (F←Q finq q) (F←Q finq x) |
| 552 ... | true | record { eq = eq } | tri< a ¬b ¬c = ⊥-elim ( nat-≡< (cong (λ x → toℕ (F←Q finq x)) ( equal→refl finq eq )) a ) | |
| 553 ... | true | _ | tri≈ ¬a b ¬c = i-phase2 qs p | |
| 554 ... | true | record { eq = eq} | tri> ¬a ¬b c = ⊥-elim ( nat-≡< (cong (λ x → toℕ (F←Q finq x)) (sym ( equal→refl finq eq ))) c ) | |
| 555 ... | false | _ | tri< a ¬b ¬c = i-phase1 qs p | |
| 556 ... | false | record {eq = eq} | tri≈ ¬a b ¬c = ⊥-elim (¬-bool eq | |
| 557 (subst₂ (λ j k → equal? finq j k ≡ true) (finiso→ finq q) (subst (λ k → Q←F finq k ≡ x) (sym b) (finiso→ finq x)) ( equal?-refl finq ))) | |
| 558 ... | false | _ | tri> ¬a ¬b c = i-phase1 qs p | |
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559 |
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560 record Dup-in-list {Q : Set } (finq : FiniteSet Q) (qs : List Q) : Set where |
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561 field |
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562 dup : Q |
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563 is-dup : dup-in-list finq dup qs ≡ true |
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565 dup-in-list>n : {Q : Set } → (finq : FiniteSet Q) → (qs : List Q) → (len> : length qs > finite finq ) → Dup-in-list finq qs |
| 283 | 566 dup-in-list>n {Q} finq qs lt = record { dup = Q←F finq (FDup-in-list.dup dl) |
| 567 ; is-dup = dup-in-list+fin finq (Q←F finq (FDup-in-list.dup dl)) qs dl01 } where | |
| 294 | 568 maplen : (qs : List Q) → length (map (F←Q finq) qs) ≡ length qs |
| 569 maplen [] = refl | |
| 570 maplen (x ∷ qs) = cong suc (maplen qs) | |
| 283 | 571 dl : FDup-in-list (finite finq ) (map (F←Q finq) qs) |
| 294 | 572 dl = fin-dup-in-list>n (map (F←Q finq) qs) (subst (λ k → k > finite finq ) (sym (maplen qs)) lt) |
| 283 | 573 dl01 : fin-dup-in-list (F←Q finq (Q←F finq (FDup-in-list.dup dl))) (map (F←Q finq) qs) ≡ true |
| 574 dl01 = subst (λ k → fin-dup-in-list k (map (F←Q finq) qs) ≡ true ) | |
| 294 | 575 (sym (finiso← finq _)) ( FDup-in-list.is-dup dl ) |
| 337 | 576 |
| 577 open import bijection using ( InjectiveF ; Is ) | |
| 578 | |
| 347 | 579 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
| 345 | 580 |
| 337 | 581 inject-fin : {A B : Set} (fa : FiniteSet A ) |
| 582 → (fi : InjectiveF B A) | |
| 583 → (is-B : (a : A ) → Dec (Is B A (InjectiveF.f fi) a) ) | |
| 584 → FiniteSet B | |
| 347 | 585 inject-fin {A} {B} fa fi is-B with finite fa | inspect finite fa |
| 360 | 586 ... | zero | record { eq = eq1 } = record { |
| 587 finite = 0 | |
| 588 ; Q←F = λ () | |
| 589 ; F←Q = λ b → ⊥-elim ( lem00 b) | |
| 590 ; finiso→ = λ b → ⊥-elim ( lem00 b) | |
| 591 ; finiso← = λ () | |
| 592 } where | |
| 593 lem00 : ( b : B) → ⊥ | |
| 594 lem00 b with subst (λ k → Fin k ) eq1 (F←Q fa (InjectiveF.f fi b)) | |
| 595 ... | () | |
| 347 | 596 ... | suc pfa | record { eq = eq1 } = record { |
| 597 finite = maxb | |
| 360 | 598 ; Q←F = λ fb → CountB.b (cb00 _ (fin<n {_} fb)) |
| 347 | 599 ; F←Q = λ b → fromℕ< (cb<mb b) |
| 351 | 600 ; finiso→ = iso1 |
| 601 ; finiso← = iso0 | |
| 337 | 602 } where |
| 347 | 603 f = InjectiveF.f fi |
| 604 pfa<fa : pfa < finite fa | |
| 605 pfa<fa = subst (λ k → pfa < k ) (sym eq1) a<sa | |
| 606 0<fa : 0 < finite fa | |
| 607 0<fa = <-transˡ (s≤s z≤n) pfa<fa | |
| 608 | |
| 609 count-B : ℕ → ℕ | |
| 610 count-B zero with is-B (Q←F fa ( fromℕ< {0} 0<fa )) | |
| 611 ... | yes isb = 1 | |
| 612 ... | no nisb = 0 | |
| 350 | 613 count-B (suc n) with <-cmp (finite fa) (suc n) |
| 347 | 614 ... | tri< a ¬b ¬c = count-B n |
| 615 ... | tri≈ ¬a b ¬c = count-B n | |
| 616 ... | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) | |
| 617 ... | yes isb = suc (count-B n) | |
| 618 ... | no nisb = count-B n | |
| 619 | |
| 620 record CountB (n : ℕ) : Set where | |
| 621 field | |
| 622 b : B | |
| 623 cb : ℕ | |
| 624 b=cn : cb ≡ toℕ (F←Q fa (f b)) | |
| 625 cb=n : count-B cb ≡ suc n | |
| 626 cb-inject : (cb1 : ℕ) → (c1<a : cb1 < finite fa) → Is B A f (Q←F fa (fromℕ< c1<a)) → count-B cb ≡ count-B cb1 → cb ≡ cb1 | |
| 627 | |
| 628 maxb : ℕ | |
| 629 maxb = count-B (finite fa) | |
| 630 | |
| 352 | 631 count-B-mono : {i j : ℕ} → i ≤ j → count-B i ≤ count-B j |
| 632 count-B-mono {i} {j} i≤j with ≤-∨ i≤j | |
| 633 ... | case1 refl = ≤-refl | |
| 634 ... | case2 i<j = lem00 _ _ i<j where | |
| 635 lem00 : (i j : ℕ) → i < j → count-B i ≤ count-B j | |
| 636 lem00 i (suc j) (s≤s i<j) = ≤-trans (count-B-mono i<j) (lem01 j) where | |
| 637 lem01 : (j : ℕ) → count-B j ≤ count-B (suc j) | |
| 638 lem01 zero with <-cmp (finite fa) 1 | |
| 639 lem01 zero | tri< a ¬b ¬c = ≤-refl | |
| 640 lem01 zero | tri≈ ¬a b ¬c = ≤-refl | |
| 641 lem01 zero | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) | is-B (Q←F fa ( fromℕ< {0} 0<fa )) | inspect count-B 0 | |
| 642 ... | yes isb1 | yes isb0 | record { eq = eq0 } = s≤s z≤n | |
| 643 ... | yes isb1 | no nisb0 | record { eq = eq0 } = z≤n | |
| 644 ... | no nisb1 | yes isb0 | record { eq = eq0 } = refl-≤≡ (sym eq0) | |
| 645 ... | no nisb1 | no nisb0 | record { eq = eq0 } = z≤n | |
| 359 | 646 lem01 (suc i) with <-cmp (finite fa) (suc i) | <-cmp (finite fa) (suc (suc i)) | inspect count-B (suc i) | inspect count-B (suc (suc i)) |
| 647 ... | tri< a ¬b ¬c | tri< a₁ ¬b₁ ¬c₁ | record { eq = eq0 } | record { eq = eq1 } = refl-≤≡ (sym eq0) | |
| 648 ... | tri< a ¬b ¬c | tri≈ ¬a b ¬c₁ | _ | _ = ⊥-elim (nat-≡< b (<-trans a a<sa)) | |
| 649 ... | tri< a ¬b ¬c | tri> ¬a ¬b₁ c | _ | _ = ⊥-elim (nat-<> a (<-trans a<sa c) ) | |
| 650 ... | tri≈ ¬a b ¬c | tri< a ¬b ¬c₁ | record { eq = eq0 } | _ = refl-≤≡ (sym eq0) | |
| 651 ... | tri≈ ¬a b ¬c | tri≈ ¬a₁ b₁ ¬c₁ | _ | _ = ⊥-elim (nat-≡< (sym b) (subst (λ k → _ < k ) (sym b₁) a<sa) ) | |
| 652 ... | tri≈ ¬a b ¬c | tri> ¬a₁ ¬b c | _ | _ = ⊥-elim (nat-≡< (sym b) (<-trans a<sa c)) | |
| 653 ... | tri> ¬a ¬b c | tri< a ¬b₁ ¬c | _ | _ = ⊥-elim (nat-≤> a (<-transʳ c a<sa ) ) | |
| 360 | 654 ... | tri> ¬a ¬b c | tri≈ ¬a₁ b ¬c | record { eq = eq0 } | record { eq = eq1 } with is-B (Q←F fa (fromℕ< c)) |
| 655 ... | yes isb = refl-≤≡ (sym eq0) | |
| 656 ... | no nisb = refl-≤≡ (sym eq0) | |
| 657 lem01 (suc i) | tri> ¬a ¬b c | tri> ¬a₁ ¬b₁ c₁ | record { eq = eq0 } | record { eq = eq1 } | |
| 658 with is-B (Q←F fa (fromℕ< c)) | is-B (Q←F fa (fromℕ< c₁)) | |
| 659 ... | yes isb0 | yes isb1 = ≤-trans (refl-≤≡ (sym eq0)) a≤sa | |
| 660 ... | yes isb0 | no nisb1 = refl-≤≡ (sym eq0) | |
| 661 ... | no nisb0 | yes isb1 = ≤-trans (refl-≤≡ (sym eq0)) a≤sa | |
| 662 ... | no nisb0 | no nisb1 = refl-≤≡ (sym eq0) | |
| 352 | 663 |
| 353 | 664 lem31 : (b : B) → 0 < count-B (toℕ (F←Q fa (f b))) |
| 358 | 665 lem31 b = lem32 (toℕ (F←Q fa (f b))) refl where |
| 666 lem32 : (i : ℕ) → toℕ (F←Q fa (f b)) ≡ i → 0 < count-B i | |
| 667 lem32 zero eq with is-B (Q←F fa ( fromℕ< {0} 0<fa )) | |
| 668 ... | yes isb = s≤s z≤n | |
| 354 | 669 ... | no nisb = ⊥-elim ( nisb record { a = b ; fa=c = lem33 } ) where |
| 670 lem33 : f b ≡ Q←F fa ( fromℕ< {0} 0<fa ) | |
| 671 lem33 = begin | |
| 355 | 672 f b ≡⟨ sym (finiso→ fa _) ⟩ |
| 360 | 673 Q←F fa ( F←Q fa (f b)) ≡⟨ sym (cong (λ k → Q←F fa k) ( fromℕ<-toℕ _ (fin<n _))) ⟩ |
| 674 Q←F fa ( fromℕ< (fin<n _) ) ≡⟨ cong (λ k → Q←F fa k) (fromℕ<-cong _ _ eq (fin<n _) 0<fa) ⟩ | |
| 354 | 675 Q←F fa ( fromℕ< {0} 0<fa ) ∎ where |
| 676 open ≡-Reasoning | |
| 358 | 677 lem32 (suc i) eq with <-cmp (finite fa) (suc i) | inspect count-B (suc i) |
| 360 | 678 ... | tri< a ¬b ¬c | _ = ⊥-elim ( nat-≡< eq (<-trans (fin<n _) a) ) |
| 679 ... | tri≈ ¬a eq1 ¬c | _ = ⊥-elim ( nat-≡< eq (subst (λ k → toℕ (F←Q fa (f b)) < k ) eq1 (fin<n _))) | |
| 358 | 680 ... | tri> ¬a ¬b c | record { eq = eq1 } with is-B (Q←F fa (fromℕ< c)) |
| 356 | 681 ... | yes isb = s≤s z≤n |
| 358 | 682 ... | no nisb = ⊥-elim ( nisb record { a = b ; fa=c = lem33 } ) where |
| 683 lem33 : f b ≡ Q←F fa ( fromℕ< c) | |
| 684 lem33 = begin | |
| 685 f b ≡⟨ sym (finiso→ fa _) ⟩ | |
| 360 | 686 Q←F fa ( F←Q fa (f b)) ≡⟨ sym (cong (λ k → Q←F fa k) ( fromℕ<-toℕ _ (fin<n _))) ⟩ |
| 687 Q←F fa ( fromℕ< (fin<n _) ) ≡⟨ cong (λ k → Q←F fa k) (fromℕ<-cong _ _ eq (fin<n _) c ) ⟩ | |
| 358 | 688 Q←F fa ( fromℕ< c ) ∎ where |
| 689 open ≡-Reasoning | |
| 690 | |
| 347 | 691 cb<mb : (b : B) → pred (count-B (toℕ (F←Q fa (f b)))) < maxb |
| 353 | 692 cb<mb b = sx≤y→x<y ( begin |
| 693 suc ( pred (count-B (toℕ (F←Q fa (f b))))) ≡⟨ sucprd (lem31 b) ⟩ | |
| 694 count-B (toℕ (F←Q fa (f b))) ≤⟨ lem02 ⟩ | |
| 695 count-B (finite fa) ∎ ) where | |
| 352 | 696 open ≤-Reasoning |
| 697 lem02 : count-B (toℕ (F←Q fa (f b))) ≤ count-B (finite fa) | |
| 360 | 698 lem02 = count-B-mono (<to≤ (fin<n {_} (F←Q fa (f b)))) |
| 347 | 699 |
| 700 cb00 : (n : ℕ) → n < count-B (finite fa) → CountB n | |
| 348 | 701 cb00 n n<cb = lem09 (finite fa) (count-B (finite fa)) (<-transˡ a<sa n<cb) refl where |
| 341 | 702 |
| 361 | 703 lem06 : (i j : ℕ) → (i<fa : i < finite fa) (j<fa : j < finite fa) |
| 704 → Is B A f (Q←F fa (fromℕ< i<fa)) → Is B A f (Q←F fa (fromℕ< j<fa)) → count-B i ≡ count-B j → i ≡ j | |
| 362 | 705 lem06 i j i<fa j<fa bi bj eq = lem08 where |
| 361 | 706 lem20 : (i j : ℕ) → i < j → (i<fa : i < finite fa) (j<fa : j < finite fa) |
| 707 → Is B A f (Q←F fa (fromℕ< i<fa)) → Is B A f (Q←F fa (fromℕ< j<fa)) → count-B j ≡ count-B i → ⊥ | |
| 362 | 708 lem20 zero (suc j) i<j i<fa j<fa bi bj le with <-cmp (finite fa) (suc j) |
| 709 ... | tri< a ¬b ¬c = ⊥-elim (¬c j<fa) | |
| 710 ... | tri≈ ¬a b ¬c = ⊥-elim (¬c j<fa) | |
| 711 ... | tri> ¬a ¬b c with is-B (Q←F fa ( fromℕ< 0<fa )) | inspect count-B 0 | is-B (Q←F fa (fromℕ< c)) | inspect count-B (suc j) | |
| 712 ... | no nisc | _ | _ | _ = ⊥-elim (nisc record { a = Is.a bi ; fa=c = lem26 } ) where | |
| 713 lem26 : f (Is.a bi) ≡ Q←F fa (fromℕ< 0<fa) | |
| 714 lem26 = trans (Is.fa=c bi) (cong (Q←F fa) (fromℕ<-cong _ _ refl i<fa 0<fa) ) | |
| 715 ... | yes _ | _ | no nisc | _ = ⊥-elim (nisc record { a = Is.a bj ; fa=c = lem26 } ) where | |
| 716 lem26 : f (Is.a bj) ≡ Q←F fa (fromℕ< c) | |
| 717 lem26 = trans (Is.fa=c bj) (cong (Q←F fa) (fromℕ<-cong _ _ refl j<fa c) ) | |
| 718 ... | yes _ | record { eq = eq1 } | yes _ | record { eq = eq2 } = ⊥-elim ( nat-≤> lem25 a<sa) where | |
| 719 lem24 : count-B j ≡ 0 | |
| 720 lem24 = cong pred le | |
| 721 lem25 : 1 ≤ 0 | |
| 722 lem25 = begin | |
| 723 1 ≡⟨ sym eq1 ⟩ | |
| 724 count-B 0 ≤⟨ count-B-mono {0} {j} z≤n ⟩ | |
| 725 count-B j ≡⟨ lem24 ⟩ | |
| 726 0 ∎ where open ≤-Reasoning | |
| 361 | 727 lem20 (suc i) zero () bi bj le |
| 728 lem20 (suc i) (suc j) (s≤s i<j) bi bj le = ? | |
| 362 | 729 |
| 361 | 730 lem08 : i ≡ j |
| 731 lem08 with <-cmp i j | |
| 362 | 732 ... | tri< a ¬b ¬c = ⊥-elim ? -- ( lem20 i j a i<fa j<fa bi bj (sym eq) ) |
| 361 | 733 ... | tri≈ ¬a b ¬c = b |
| 362 | 734 ... | tri> ¬a ¬b c₁ = ⊥-elim ? -- ( lem20 j i c₁ j<fa i<fa bj bi eq ) |
| 361 | 735 |
| 347 | 736 lem09 : (i j : ℕ) → suc n ≤ j → j ≡ count-B i → CountB n |
| 348 | 737 lem09 0 (suc j) (s≤s le) eq with is-B (Q←F fa (fromℕ< {0} 0<fa )) | inspect count-B 0 |
| 738 ... | no nisb | record { eq = eq1 } = ⊥-elim ( nat-≡< (sym eq) (s≤s z≤n) ) | |
| 739 ... | yes isb | record { eq = eq1 } with ≤-∨ (s≤s le) | |
| 349 | 740 ... | case1 eq2 = record { b = Is.a isb ; cb = 0 ; b=cn = lem10 ; cb=n = trans eq1 (sym (trans eq2 eq)) |
| 362 | 741 ; cb-inject = λ cb1 c1<fa b1 eq → lem06 0 cb1 0<fa c1<fa isb b1 eq } where |
| 349 | 742 lem10 : 0 ≡ toℕ (F←Q fa (f (Is.a isb))) |
| 743 lem10 = begin | |
| 744 0 ≡⟨ sym ( toℕ-fromℕ< 0<fa ) ⟩ | |
| 745 toℕ (fromℕ< {0} 0<fa ) ≡⟨ cong toℕ (sym (finiso← fa _)) ⟩ | |
| 746 toℕ (F←Q fa (Q←F fa (fromℕ< {0} 0<fa ))) ≡⟨ cong (λ k → toℕ ((F←Q fa k))) (sym (Is.fa=c isb)) ⟩ | |
| 747 toℕ (F←Q fa (f (Is.a isb))) ∎ where open ≡-Reasoning | |
| 748 ... | case2 (s≤s lt) = ⊥-elim ( nat-≡< (sym eq) (s≤s (<-transʳ z≤n lt) )) | |
| 350 | 749 lem09 (suc i) (suc j) (s≤s le) eq with <-cmp (finite fa) (suc i) | inspect count-B (suc i) |
| 348 | 750 ... | tri< a ¬b ¬c | _ = lem09 i (suc j) (s≤s le) eq |
| 751 ... | tri≈ ¬a b ¬c | _ = lem09 i (suc j) (s≤s le) eq | |
| 752 ... | tri> ¬a ¬b c | record { eq = eq1 } with is-B (Q←F fa (fromℕ< c)) | |
| 753 ... | no nisb = lem09 i (suc j) (s≤s le) eq | |
| 754 ... | yes isb with ≤-∨ (s≤s le) | |
| 350 | 755 ... | case1 eq2 = record { b = Is.a isb ; cb = suc i ; b=cn = lem11 ; cb=n = trans eq1 (sym (trans eq2 eq )) |
| 362 | 756 ; cb-inject = λ cb1 c1<fa b1 eq → lem06 (suc i) cb1 c c1<fa isb b1 eq } where |
| 350 | 757 lem11 : suc i ≡ toℕ (F←Q fa (f (Is.a isb))) |
| 758 lem11 = begin | |
| 759 suc i ≡⟨ sym ( toℕ-fromℕ< c) ⟩ | |
| 760 toℕ (fromℕ< c) ≡⟨ cong toℕ (sym (finiso← fa _)) ⟩ | |
| 349 | 761 toℕ (F←Q fa (Q←F fa (fromℕ< c ))) ≡⟨ cong (λ k → toℕ ((F←Q fa k))) (sym (Is.fa=c isb)) ⟩ |
| 762 toℕ (F←Q fa (f (Is.a isb))) ∎ where open ≡-Reasoning | |
| 348 | 763 ... | case2 (s≤s lt) = lem09 i j lt (cong pred eq) |
| 347 | 764 |
| 360 | 765 iso0 : (x : Fin maxb) → fromℕ< (cb<mb (CountB.b (cb00 (toℕ x) (fin<n _)))) ≡ x |
| 351 | 766 iso0 x = begin |
| 360 | 767 fromℕ< (cb<mb (CountB.b (cb00 (toℕ x) (fin<n _)))) ≡⟨ fromℕ<-cong _ _ ( begin |
| 768 pred (count-B (toℕ (F←Q fa (f (CountB.b (cb00 (toℕ x) (fin<n _))))))) ≡⟨ sym (cong (λ k → pred (count-B k)) (CountB.b=cn CB)) ⟩ | |
| 351 | 769 pred (count-B (CountB.cb CB)) ≡⟨ cong pred (CountB.cb=n CB) ⟩ |
| 770 pred (suc (toℕ x)) ≡⟨ refl ⟩ | |
| 360 | 771 toℕ x ∎ ) (cb<mb (CountB.b CB)) (fin<n _) ⟩ |
| 772 fromℕ< (fin<n {_} x) ≡⟨ fromℕ<-toℕ _ (fin<n {_} x) ⟩ | |
| 351 | 773 x ∎ where |
| 774 open ≡-Reasoning | |
| 360 | 775 CB = cb00 (toℕ x) (fin<n _) |
| 351 | 776 |
| 360 | 777 iso1 : (b : B) → CountB.b (cb00 (toℕ (fromℕ< (cb<mb b))) (fin<n _)) ≡ b |
| 778 iso1 b = begin | |
| 351 | 779 CountB.b CB ≡⟨ InjectiveF.inject fi (F←Q-inject fa (toℕ-injective (begin |
| 780 toℕ (F←Q fa (f (CountB.b CB))) ≡⟨ sym (CountB.b=cn CB) ⟩ | |
| 360 | 781 CountB.cb CB ≡⟨ CountB.cb-inject CB _ (fin<n _) isb lem30 ⟩ |
| 351 | 782 toℕ (F←Q fa (f b)) ∎ ) )) ⟩ |
| 783 b ∎ where | |
| 784 open ≡-Reasoning | |
| 360 | 785 CB = cb00 (toℕ (fromℕ< (cb<mb b))) (fin<n _) |
| 786 isb : Is B A f (Q←F fa (fromℕ< (fin<n {_} (F←Q fa (f b)) ))) | |
| 787 isb = record { a = b ; fa=c = lem33 } where | |
| 788 lem33 : f b ≡ Q←F fa (fromℕ< (fin<n (F←Q fa (f b)))) | |
| 789 lem33 = begin | |
| 790 f b ≡⟨ sym (finiso→ fa _) ⟩ | |
| 791 Q←F fa (F←Q fa (f b)) ≡⟨ cong (Q←F fa) (sym (fromℕ<-toℕ _ (fin<n (F←Q fa (f b))))) ⟩ | |
| 792 Q←F fa (fromℕ< (fin<n (F←Q fa (f b)))) ∎ | |
| 351 | 793 lem30 : count-B (CountB.cb CB) ≡ count-B (toℕ (F←Q fa (InjectiveF.f fi b))) |
| 794 lem30 = begin | |
| 795 count-B (CountB.cb CB) ≡⟨ CountB.cb=n CB ⟩ | |
| 796 suc (toℕ (fromℕ< (cb<mb b))) ≡⟨ cong suc (toℕ-fromℕ< (cb<mb b)) ⟩ | |
| 353 | 797 suc (pred (count-B (toℕ (F←Q fa (f b))))) ≡⟨ sucprd (lem31 b) ⟩ |
| 351 | 798 count-B (toℕ (F←Q fa (f b))) ∎ |
| 799 | |
| 800 | |
| 347 | 801 -- end |