Mercurial > hg > Members > kono > Proof > automaton
annotate automaton-in-agda/src/finiteSetUtil.agda @ 413:ad086c3161d7 default tip
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 18 Jun 2024 14:05:44 +0900 |
| parents | a60132983557 |
| children |
| rev | line source |
|---|---|
| 403 | 1 {-# OPTIONS --cubical-compatible --safe #-} |
| 163 | 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 |
| 163 | 18 record Found ( Q : Set ) (p : Q → Bool ) : Set where |
| 19 field | |
| 20 found-q : Q | |
| 21 found-p : p found-q ≡ true | |
| 22 | |
| 264 | 23 open Bijection |
| 24 | |
| 268 | 25 open import Axiom.Extensionality.Propositional |
| 26 open import Level hiding (suc ; zero) | |
| 27 | |
| 163 | 28 module _ {Q : Set } (F : FiniteSet Q) where |
| 29 open FiniteSet F | |
| 405 | 30 equal?-refl : { x : Q } → equal? x x ≡ true |
| 268 | 31 equal?-refl {x} with F←Q x ≟ F←Q x |
| 403 | 32 ... | yes eq = refl |
| 268 | 33 ... | no ne = ⊥-elim (ne refl) |
| 163 | 34 equal→refl : { x y : Q } → equal? x y ≡ true → x ≡ y |
| 35 equal→refl {q0} {q1} eq with F←Q q0 ≟ F←Q q1 | |
| 36 equal→refl {q0} {q1} refl | yes eq = begin | |
| 37 q0 | |
| 38 ≡⟨ sym ( finiso→ q0) ⟩ | |
| 39 Q←F (F←Q q0) | |
| 40 ≡⟨ cong (λ k → Q←F k ) eq ⟩ | |
| 41 Q←F (F←Q q1) | |
| 264 | 42 ≡⟨ finiso→ q1 ⟩ |
| 163 | 43 q1 |
| 44 ∎ where open ≡-Reasoning | |
| 318 | 45 eqP : (x y : Q) → Dec ( x ≡ y ) |
| 46 eqP x y with F←Q x ≟ F←Q y | |
| 47 ... | yes eq = yes (subst₂ (λ j k → j ≡ k ) (finiso→ x) (finiso→ y) (cong Q←F eq) ) | |
| 48 ... | no n = no (λ eq → n (cong F←Q eq)) | |
| 163 | 49 End : (m : ℕ ) → (p : Q → Bool ) → Set |
| 405 | 50 End m p = (i : Fin finite) → m ≤ toℕ i → p (Q←F i ) ≡ false |
| 163 | 51 first-end : ( p : Q → Bool ) → End finite p |
| 360 | 52 first-end p i i>n = ⊥-elim (nat-≤> i>n (fin<n {finite} i) ) |
| 163 | 53 next-end : {m : ℕ } → ( p : Q → Bool ) → End (suc m) p |
| 54 → (m<n : m < finite ) → p (Q←F (fromℕ< m<n )) ≡ false | |
| 55 → End m p | |
| 405 | 56 next-end {m} p prev m<n np i m<i with NP.<-cmp m (toℕ i) |
| 163 | 57 next-end p prev m<n np i m<i | tri< a ¬b ¬c = prev i a |
| 58 next-end p prev m<n np i m<i | tri> ¬a ¬b c = ⊥-elim ( nat-≤> m<i c ) | |
| 59 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 | |
| 60 m<n=i : {n : ℕ } (i : Fin n) {m : ℕ } → m ≡ (toℕ i) → (m<n : m < n ) → fromℕ< m<n ≡ i | |
| 405 | 61 m<n=i i refl m<n = fromℕ<-toℕ i m<n |
| 163 | 62 found : { p : Q → Bool } → (q : Q ) → p q ≡ true → exists p ≡ true |
| 337 | 63 found {p} q pt = found1 finite (NP.≤-refl ) ( first-end p ) where |
| 163 | 64 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 |
| 65 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 ) | |
| 66 found1 (suc m) m<n end with bool-≡-? (p (Q←F (fromℕ< m<n))) true | |
| 405 | 67 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 | 68 found1 (suc m) m<n end | no np = begin |
| 264 | 69 p (Q←F (fromℕ< m<n)) \/ exists1 m (<to≤ m<n) p |
| 163 | 70 ≡⟨ bool-or-1 (¬-bool-t np ) ⟩ |
| 264 | 71 exists1 m (<to≤ m<n) p |
| 72 ≡⟨ found1 m (<to≤ m<n) (next-end p end m<n (¬-bool-t np )) ⟩ | |
| 163 | 73 true |
| 74 ∎ where open ≡-Reasoning | |
| 268 | 75 not-found : { p : Q → Bool } → ( (q : Q ) → p q ≡ false ) → exists p ≡ false |
| 337 | 76 not-found {p} pn = not-found2 finite NP.≤-refl where |
| 268 | 77 not-found2 : (m : ℕ ) → (m<n : m Data.Nat.≤ finite ) → exists1 m m<n p ≡ false |
| 78 not-found2 zero _ = refl | |
| 79 not-found2 ( suc m ) m<n with pn (Q←F (fromℕ< {m} {finite} m<n)) | |
| 80 not-found2 (suc m) m<n | eq = begin | |
| 405 | 81 p (Q←F (fromℕ< m<n)) \/ exists1 m (<to≤ m<n) p |
| 268 | 82 ≡⟨ bool-or-1 eq ⟩ |
| 405 | 83 exists1 m (<to≤ m<n) p |
| 268 | 84 ≡⟨ not-found2 m (<to≤ m<n) ⟩ |
| 85 false | |
| 86 ∎ where open ≡-Reasoning | |
| 87 found← : { p : Q → Bool } → exists p ≡ true → Found Q p | |
| 337 | 88 found← {p} exst = found2 finite NP.≤-refl (first-end p ) where |
| 268 | 89 found2 : (m : ℕ ) (m<n : m Data.Nat.≤ finite ) → End m p → Found Q p |
| 403 | 90 found2 0 m<n end = ⊥-elim ( ¬-bool f01 exst ) where |
| 91 f01 : exists p ≡ false | |
| 92 f01 = not-found (λ q → subst ( λ k → p k ≡ false ) (finiso→ _) (end (F←Q q) z≤n )) | |
| 268 | 93 found2 (suc m) m<n end with bool-≡-? (p (Q←F (fromℕ< m<n))) true |
| 94 found2 (suc m) m<n end | yes eq = record { found-q = Q←F (fromℕ< m<n) ; found-p = eq } | |
| 405 | 95 found2 (suc m) m<n end | no np = |
| 96 found2 m (<to≤ m<n) (next-end p end m<n (¬-bool-t np )) | |
| 97 not-found← : { p : Q → Bool } → exists p ≡ false → (q : Q ) → p q ≡ false | |
| 268 | 98 not-found← {p} np q = ¬-bool-t ( contra-position {_} {_} {_} {exists p ≡ true} (found q) (λ ep → ¬-bool np ep ) ) |
| 351 | 99 Q←F-inject : {x y : Fin finite} → Q←F x ≡ Q←F y → x ≡ y |
| 100 Q←F-inject eq = subst₂ (λ j k → j ≡ k ) (finiso← _) (finiso← _) (cong F←Q eq) | |
| 101 F←Q-inject : {x y : Q } → F←Q x ≡ F←Q y → x ≡ y | |
| 102 F←Q-inject eq = subst₂ (λ j k → j ≡ k ) (finiso→ _) (finiso→ _) (cong Q←F eq) | |
| 268 | 103 |
| 163 | 104 |
| 105 | |
| 405 | 106 iso-fin : {A B : Set} → FiniteSet A → Bijection A B → FiniteSet B |
| 141 | 107 iso-fin {A} {B} fin iso = record { |
| 330 | 108 Q←F = λ f → fun→ iso ( FiniteSet.Q←F fin f ) |
| 264 | 109 ; F←Q = λ b → FiniteSet.F←Q fin (fun← iso b ) |
| 405 | 110 ; finiso→ = finiso→ |
| 111 ; finiso← = finiso← | |
| 141 | 112 } where |
| 264 | 113 finiso→ : (q : B) → fun→ iso (FiniteSet.Q←F fin (FiniteSet.F←Q fin (Bijection.fun← iso q))) ≡ q |
| 141 | 114 finiso→ q = begin |
| 405 | 115 fun→ iso (FiniteSet.Q←F fin (FiniteSet.F←Q fin (Bijection.fun← iso q))) |
| 264 | 116 ≡⟨ cong (λ k → fun→ iso k ) (FiniteSet.finiso→ fin _ ) ⟩ |
| 117 fun→ iso (Bijection.fun← iso q) | |
| 118 ≡⟨ fiso→ iso _ ⟩ | |
| 141 | 119 q |
| 264 | 120 ∎ where open ≡-Reasoning |
| 121 finiso← : (f : Fin (FiniteSet.finite fin ))→ FiniteSet.F←Q fin (Bijection.fun← iso (Bijection.fun→ iso (FiniteSet.Q←F fin f))) ≡ f | |
| 141 | 122 finiso← f = begin |
| 405 | 123 FiniteSet.F←Q fin (Bijection.fun← iso (Bijection.fun→ iso (FiniteSet.Q←F fin f))) |
| 264 | 124 ≡⟨ cong (λ k → FiniteSet.F←Q fin k ) (Bijection.fiso← iso _) ⟩ |
| 405 | 125 FiniteSet.F←Q fin (FiniteSet.Q←F fin f) |
| 141 | 126 ≡⟨ FiniteSet.finiso← fin _ ⟩ |
| 127 f | |
| 128 ∎ where | |
| 129 open ≡-Reasoning | |
| 130 | |
| 131 data One : Set where | |
| 132 one : One | |
| 133 | |
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134 finOne : FiniteSet One |
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135 finOne = record { finite = 1 ; Q←F = λ _ → one ; F←Q = λ _ → # 0 ; finiso→ = fin00 ; finiso← = fin1≡0 } where |
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136 fin00 : (q : One) → one ≡ q |
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137 fin00 one = refl |
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138 |
| 405 | 139 fin-∨1 : {B : Set} → (fb : FiniteSet B ) → FiniteSet (One ∨ B) |
| 141 | 140 fin-∨1 {B} fb = record { |
| 330 | 141 Q←F = Q←F |
| 142 ; F←Q = F←Q | |
| 143 ; finiso→ = finiso→ | |
| 144 ; finiso← = finiso← | |
| 141 | 145 } where |
| 146 b = FiniteSet.finite fb | |
| 147 Q←F : Fin (suc b) → One ∨ B | |
| 148 Q←F zero = case1 one | |
| 149 Q←F (suc f) = case2 (FiniteSet.Q←F fb f) | |
| 150 F←Q : One ∨ B → Fin (suc b) | |
| 151 F←Q (case1 one) = zero | |
| 405 | 152 F←Q (case2 f ) = suc (FiniteSet.F←Q fb f) |
| 141 | 153 finiso→ : (q : One ∨ B) → Q←F (F←Q q) ≡ q |
| 154 finiso→ (case1 one) = refl | |
| 155 finiso→ (case2 b) = cong (λ k → case2 k ) (FiniteSet.finiso→ fb b) | |
| 156 finiso← : (q : Fin (suc b)) → F←Q (Q←F q) ≡ q | |
| 157 finiso← zero = refl | |
| 158 finiso← (suc f) = cong ( λ k → suc k ) (FiniteSet.finiso← fb f) | |
| 159 | |
| 160 | |
| 405 | 161 fin-∨2 : {B : Set} → ( a : ℕ ) → FiniteSet B → FiniteSet (Fin a ∨ B) |
| 141 | 162 fin-∨2 {B} zero fb = iso-fin fb iso where |
| 264 | 163 iso : Bijection B (Fin zero ∨ B) |
| 164 iso = record { | |
| 165 fun← = fun←1 | |
| 166 ; fun→ = λ b → case2 b | |
| 167 ; fiso→ = fiso→1 | |
| 168 ; fiso← = λ _ → refl | |
| 141 | 169 } where |
| 264 | 170 fun←1 : Fin zero ∨ B → B |
| 405 | 171 fun←1 (case2 x) = x |
| 264 | 172 fiso→1 : (f : Fin zero ∨ B ) → case2 (fun←1 f) ≡ f |
| 173 fiso→1 (case2 x) = refl | |
| 141 | 174 fin-∨2 {B} (suc a) fb = iso-fin (fin-∨1 (fin-∨2 a fb) ) iso |
| 175 where | |
| 264 | 176 iso : Bijection (One ∨ (Fin a ∨ B) ) (Fin (suc a) ∨ B) |
| 177 fun← iso (case1 zero) = case1 one | |
| 178 fun← iso (case1 (suc f)) = case2 (case1 f) | |
| 179 fun← iso (case2 b) = case2 (case2 b) | |
| 180 fun→ iso (case1 one) = case1 zero | |
| 181 fun→ iso (case2 (case1 f)) = case1 (suc f) | |
| 182 fun→ iso (case2 (case2 b)) = case2 b | |
| 183 fiso← iso (case1 one) = refl | |
| 184 fiso← iso (case2 (case1 x)) = refl | |
| 185 fiso← iso (case2 (case2 x)) = refl | |
| 186 fiso→ iso (case1 zero) = refl | |
| 187 fiso→ iso (case1 (suc x)) = refl | |
| 188 fiso→ iso (case2 x) = refl | |
| 141 | 189 |
| 190 | |
| 264 | 191 FiniteSet→Fin : {A : Set} → (fin : FiniteSet A ) → Bijection (Fin (FiniteSet.finite fin)) A |
| 192 fun← (FiniteSet→Fin fin) f = FiniteSet.F←Q fin f | |
| 193 fun→ (FiniteSet→Fin fin) f = FiniteSet.Q←F fin f | |
| 194 fiso← (FiniteSet→Fin fin) = FiniteSet.finiso← fin | |
| 195 fiso→ (FiniteSet→Fin fin) = FiniteSet.finiso→ fin | |
| 405 | 196 |
| 141 | 197 |
| 405 | 198 fin-∨ : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A ∨ B) |
| 141 | 199 fin-∨ {A} {B} fa fb = iso-fin (fin-∨2 a fb ) iso2 where |
| 200 a = FiniteSet.finite fa | |
| 201 ia = FiniteSet→Fin fa | |
| 264 | 202 iso2 : Bijection (Fin a ∨ B ) (A ∨ B) |
| 203 fun← iso2 (case1 x) = case1 (fun← ia x ) | |
| 204 fun← iso2 (case2 x) = case2 x | |
| 205 fun→ iso2 (case1 x) = case1 (fun→ ia x ) | |
| 206 fun→ iso2 (case2 x) = case2 x | |
| 207 fiso← iso2 (case1 x) = cong ( λ k → case1 k ) (Bijection.fiso← ia x) | |
| 208 fiso← iso2 (case2 x) = refl | |
| 209 fiso→ iso2 (case1 x) = cong ( λ k → case1 k ) (Bijection.fiso→ ia x) | |
| 210 fiso→ iso2 (case2 x) = refl | |
| 141 | 211 |
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212 open import Data.Product hiding ( map ) |
| 141 | 213 |
| 405 | 214 fin-× : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A × B) |
| 141 | 215 fin-× {A} {B} fa fb with FiniteSet→Fin fa |
| 216 ... | a=f = iso-fin (fin-×-f a ) iso-1 where | |
| 217 a = FiniteSet.finite fa | |
| 218 b = FiniteSet.finite fb | |
| 264 | 219 iso-1 : Bijection (Fin a × B) ( A × B ) |
| 405 | 220 fun← iso-1 x = ( FiniteSet.F←Q fa (proj₁ x) , proj₂ x) |
| 221 fun→ iso-1 x = ( FiniteSet.Q←F fa (proj₁ x) , proj₂ x) | |
| 264 | 222 fiso← iso-1 x = lemma where |
| 141 | 223 lemma : (FiniteSet.F←Q fa (FiniteSet.Q←F fa (proj₁ x)) , proj₂ x) ≡ ( proj₁ x , proj₂ x ) |
| 224 lemma = cong ( λ k → ( k , proj₂ x ) ) (FiniteSet.finiso← fa _ ) | |
| 264 | 225 fiso→ iso-1 x = cong ( λ k → ( k , proj₂ x ) ) (FiniteSet.finiso→ fa _ ) |
| 141 | 226 |
| 264 | 227 iso-2 : {a : ℕ } → Bijection (B ∨ (Fin a × B)) (Fin (suc a) × B) |
| 228 fun← iso-2 (zero , b ) = case1 b | |
| 229 fun← iso-2 (suc fst , b ) = case2 ( fst , b ) | |
| 230 fun→ iso-2 (case1 b) = ( zero , b ) | |
| 231 fun→ iso-2 (case2 (a , b )) = ( suc a , b ) | |
| 232 fiso← iso-2 (case1 x) = refl | |
| 233 fiso← iso-2 (case2 x) = refl | |
| 234 fiso→ iso-2 (zero , b ) = refl | |
| 235 fiso→ iso-2 (suc a , b ) = refl | |
| 141 | 236 |
| 405 | 237 fin-×-f : ( a : ℕ ) → FiniteSet ((Fin a) × B) |
| 141 | 238 fin-×-f zero = record { Q←F = λ () ; F←Q = λ () ; finiso→ = λ () ; finiso← = λ () ; finite = 0 } |
| 239 fin-×-f (suc a) = iso-fin ( fin-∨ fb ( fin-×-f a ) ) iso-2 | |
| 240 | |
| 241 open _∧_ | |
| 242 | |
| 405 | 243 fin-∧ : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A ∧ B) |
| 141 | 244 fin-∧ {A} {B} fa fb with FiniteSet→Fin fa -- same thing for our tool |
| 245 ... | a=f = iso-fin (fin-×-f a ) iso-1 where | |
| 246 a = FiniteSet.finite fa | |
| 247 b = FiniteSet.finite fb | |
| 264 | 248 iso-1 : Bijection (Fin a ∧ B) ( A ∧ B ) |
| 405 | 249 fun← iso-1 x = record { proj1 = FiniteSet.F←Q fa (proj1 x) ; proj2 = proj2 x} |
| 264 | 250 fun→ iso-1 x = record { proj1 = FiniteSet.Q←F fa (proj1 x) ; proj2 = proj2 x} |
| 251 fiso← iso-1 x = lemma where | |
| 141 | 252 lemma : record { proj1 = FiniteSet.F←Q fa (FiniteSet.Q←F fa (proj1 x)) ; proj2 = proj2 x} ≡ record {proj1 = proj1 x ; proj2 = proj2 x } |
| 253 lemma = cong ( λ k → record {proj1 = k ; proj2 = proj2 x } ) (FiniteSet.finiso← fa _ ) | |
| 264 | 254 fiso→ iso-1 x = cong ( λ k → record {proj1 = k ; proj2 = proj2 x } ) (FiniteSet.finiso→ fa _ ) |
| 141 | 255 |
| 264 | 256 iso-2 : {a : ℕ } → Bijection (B ∨ (Fin a ∧ B)) (Fin (suc a) ∧ B) |
| 257 fun← iso-2 (record { proj1 = zero ; proj2 = b }) = case1 b | |
| 258 fun← iso-2 (record { proj1 = suc fst ; proj2 = b }) = case2 ( record { proj1 = fst ; proj2 = b } ) | |
| 259 fun→ iso-2 (case1 b) = record {proj1 = zero ; proj2 = b } | |
| 260 fun→ iso-2 (case2 (record { proj1 = a ; proj2 = b })) = record { proj1 = suc a ; proj2 = b } | |
| 261 fiso← iso-2 (case1 x) = refl | |
| 262 fiso← iso-2 (case2 x) = refl | |
| 263 fiso→ iso-2 (record { proj1 = zero ; proj2 = b }) = refl | |
| 264 fiso→ iso-2 (record { proj1 = suc a ; proj2 = b }) = refl | |
| 141 | 265 |
| 405 | 266 fin-×-f : ( a : ℕ ) → FiniteSet ((Fin a) ∧ B) |
| 141 | 267 fin-×-f zero = record { Q←F = λ () ; F←Q = λ () ; finiso→ = λ () ; finiso← = λ () ; finite = 0 } |
| 268 fin-×-f (suc a) = iso-fin ( fin-∨ fb ( fin-×-f a ) ) iso-2 | |
| 269 | |
| 270 -- import Data.Nat.DivMod | |
| 271 | |
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272 open import Data.Vec hiding ( map ; length ) |
| 141 | 273 import Data.Product |
| 274 | |
| 275 exp2 : (n : ℕ ) → exp 2 (suc n) ≡ exp 2 n Data.Nat.+ exp 2 n | |
| 276 exp2 n = begin | |
| 277 exp 2 (suc n) | |
| 278 ≡⟨⟩ | |
| 279 2 * ( exp 2 n ) | |
| 280 ≡⟨ *-comm 2 (exp 2 n) ⟩ | |
| 281 ( exp 2 n ) * 2 | |
| 282 ≡⟨ *-suc ( exp 2 n ) 1 ⟩ | |
| 283 (exp 2 n ) Data.Nat.+ ( exp 2 n ) * 1 | |
| 284 ≡⟨ cong ( λ k → (exp 2 n ) Data.Nat.+ k ) (proj₂ *-identity (exp 2 n) ) ⟩ | |
| 285 exp 2 n Data.Nat.+ exp 2 n | |
| 286 ∎ where | |
| 287 open ≡-Reasoning | |
| 288 open Data.Product | |
| 289 | |
| 403 | 290 cast-iso : {n m : ℕ } → (eq : n ≡ m ) → (f : Fin m ) → Data.Fin.cast eq ( Data.Fin.cast (sym eq ) f) ≡ f |
| 141 | 291 cast-iso refl zero = refl |
| 292 cast-iso refl (suc f) = cong ( λ k → suc k ) ( cast-iso refl f ) | |
| 293 | |
| 294 | |
| 405 | 295 fin2List : {n : ℕ } → FiniteSet (Vec Bool n) |
| 141 | 296 fin2List {zero} = record { |
| 297 Q←F = λ _ → Vec.[] | |
| 298 ; F←Q = λ _ → # 0 | |
| 405 | 299 ; finiso→ = finiso→ |
| 300 ; finiso← = finiso← | |
| 141 | 301 } where |
| 302 Q = Vec Bool zero | |
| 303 finiso→ : (q : Q) → [] ≡ q | |
| 304 finiso→ [] = refl | |
| 305 finiso← : (f : Fin (exp 2 zero)) → # 0 ≡ f | |
| 306 finiso← zero = refl | |
| 307 fin2List {suc n} = subst (λ k → FiniteSet (Vec Bool (suc n)) ) (sym (exp2 n)) ( iso-fin (fin-∨ (fin2List ) (fin2List )) iso ) | |
| 308 where | |
| 309 QtoR : Vec Bool (suc n) → Vec Bool n ∨ Vec Bool n | |
| 310 QtoR ( true ∷ x ) = case1 x | |
| 311 QtoR ( false ∷ x ) = case2 x | |
| 405 | 312 RtoQ : Vec Bool n ∨ Vec Bool n → Vec Bool (suc n) |
| 141 | 313 RtoQ ( case1 x ) = true ∷ x |
| 314 RtoQ ( case2 x ) = false ∷ x | |
| 315 isoRQ : (x : Vec Bool (suc n) ) → RtoQ ( QtoR x ) ≡ x | |
| 316 isoRQ (true ∷ _ ) = refl | |
| 317 isoRQ (false ∷ _ ) = refl | |
| 318 isoQR : (x : Vec Bool n ∨ Vec Bool n ) → QtoR ( RtoQ x ) ≡ x | |
| 319 isoQR (case1 x) = refl | |
| 320 isoQR (case2 x) = refl | |
| 264 | 321 iso : Bijection (Vec Bool n ∨ Vec Bool n) (Vec Bool (suc n)) |
| 322 iso = record { fun← = QtoR ; fun→ = RtoQ ; fiso← = isoQR ; fiso→ = isoRQ } | |
| 141 | 323 |
| 324 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 | |
| 325 F2L {Q} {zero} fin _ Q→B = [] | |
| 337 | 326 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 | 327 lemma6 : toℕ (FiniteSet.F←Q fin (FiniteSet.Q←F fin (fromℕ< n<m))) < suc n |
| 328 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 ) | |
| 329 qb1 : (q : Q) → toℕ (FiniteSet.F←Q fin q) < n → Bool | |
| 337 | 330 qb1 q q<n = Q→B q (NP.<-trans q<n a<sa) |
| 141 | 331 |
| 405 | 332 List2Func : { Q : Set } → {n : ℕ } → (fin : FiniteSet Q ) → n < suc (FiniteSet.finite fin) → Vec Bool n → Q → Bool |
| 141 | 333 List2Func {Q} {zero} fin (s≤s z≤n) [] q = false |
| 334 List2Func {Q} {suc n} fin (s≤s n<m) (h ∷ t) q with FiniteSet.F←Q fin q ≟ fromℕ< n<m | |
| 335 ... | yes _ = h | |
| 337 | 336 ... | no _ = List2Func {Q} fin (NP.<-trans n<m a<sa ) t q |
| 141 | 337 |
| 405 | 338 open import Level renaming ( suc to Suc ; zero to Zero) |
| 141 | 339 |
| 340 | |
| 341 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 | |
| 405 | 342 L2F fin n<m x q q<n = List2Func fin n<m x q |
| 141 | 343 |
| 344 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 | |
| 345 L2F-iso {Q} fin f q = l2f m a<sa (toℕ<n _) where | |
| 346 m = FiniteSet.finite fin | |
| 163 | 347 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 |
| 348 lemma11f n<m ¬q=n q≤n = lemma13 n<m (contra-position (lemma12 n<m _) ¬q=n ) q≤n where | |
| 141 | 349 lemma13 : {n nq : ℕ } → (n<m : n < m ) → ¬ ( nq ≡ n ) → nq ≤ n → nq < n |
| 350 lemma13 {0} {0} (s≤s z≤n) nt z≤n = ⊥-elim ( nt refl ) | |
| 351 lemma13 {suc _} {0} (s≤s (s≤s n<m)) nt z≤n = s≤s z≤n | |
| 337 | 352 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 | 353 lemma3f : {a b : ℕ } → (lt : a < b ) → fromℕ< (s≤s lt) ≡ suc (fromℕ< lt) |
| 354 lemma3f (s≤s lt) = refl | |
| 405 | 355 lemma12f : {n m : ℕ } → (n<m : n < m ) → (f : Fin m ) → toℕ f ≡ n → f ≡ fromℕ< n<m |
| 163 | 356 lemma12f {zero} {suc m} (s≤s z≤n) zero refl = refl |
| 357 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 | 358 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 |
| 359 l2f zero (s≤s z≤n) () | |
| 405 | 360 l2f (suc n) (s≤s n<m) (s≤s n<q) with FiniteSet.F←Q fin q ≟ fromℕ< n<m |
| 361 l2f (suc n) (s≤s n<m) (s≤s n<q) | yes p = begin | |
| 362 f (FiniteSet.Q←F fin (fromℕ< n<m)) | |
| 141 | 363 ≡⟨ cong ( λ k → f (FiniteSet.Q←F fin k )) (sym p) ⟩ |
| 364 f (FiniteSet.Q←F fin ( FiniteSet.F←Q fin q )) | |
| 365 ≡⟨ cong ( λ k → f k ) (FiniteSet.finiso→ fin _ ) ⟩ | |
| 405 | 366 f q |
| 141 | 367 ∎ where |
| 368 open ≡-Reasoning | |
| 337 | 369 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 | 370 |
| 405 | 371 Fin2Finite : ( n : ℕ ) → FiniteSet (Fin n) |
| 141 | 372 Fin2Finite n = record { F←Q = λ x → x ; Q←F = λ x → x ; finiso← = λ q → refl ; finiso→ = λ q → refl } |
| 373 | |
| 406 | 374 -- |
| 375 -- fin→ is in finiteFunc.agda | |
| 376 -- we excludeit becauase it uses f-extensionarity | |
| 377 | |
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378 open import Data.List |
| 141 | 379 |
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380 open FiniteSet |
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381 |
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382 memberQ : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool |
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383 memberQ {Q} finq q [] = false |
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384 memberQ {Q} finq q (q0 ∷ qs) with equal? finq q q0 |
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385 ... | true = true |
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386 ... | false = memberQ finq q qs |
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387 |
| 316 | 388 -- |
| 389 -- there is a duplicate element in finite list | |
| 390 -- | |
| 391 | |
| 330 | 392 -- |
| 393 -- How about this? | |
| 394 -- get list of Q | |
| 395 -- remove one element for each Q from list | |
| 396 -- there must be remaining list > 1 | |
| 397 -- theses are duplicates | |
| 398 -- actualy it is duplicate | |
| 399 | |
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400 phase2 : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool |
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401 phase2 finq q [] = false |
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402 phase2 finq q (x ∷ qs) with equal? finq q x |
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403 ... | true = true |
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404 ... | false = phase2 finq q qs |
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405 phase1 : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool |
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406 phase1 finq q [] = false |
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407 phase1 finq q (x ∷ qs) with equal? finq q x |
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408 ... | true = phase2 finq q qs |
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409 ... | false = phase1 finq q qs |
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410 |
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411 dup-in-list : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool |
| 405 | 412 dup-in-list {Q} finq q qs = phase1 finq q qs |
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413 |
| 316 | 414 -- |
| 415 -- if length of the list is longer than kinds of a finite set, there is a duplicate | |
| 416 -- prove this based on the theorem on Data.Fin | |
| 417 -- | |
| 418 | |
| 405 | 419 dup-in-list+fin : { Q : Set } (finq : FiniteSet Q) |
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420 → (q : Q) (qs : List Q ) |
| 283 | 421 → fin-dup-in-list (F←Q finq q) (map (F←Q finq) qs) ≡ true |
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422 → dup-in-list finq q qs ≡ true |
| 283 | 423 dup-in-list+fin {Q} finq q qs p = i-phase1 qs p where |
| 424 i-phase2 : (qs : List Q) → fin-phase2 (F←Q finq q) (map (F←Q finq) qs) ≡ true | |
| 405 | 425 → phase2 finq q qs ≡ true |
| 403 | 426 i-phase2 (x ∷ qs) p with equal? finq q x in eq | <-fcmp (F←Q finq q) (F←Q finq x) |
| 427 ... | true | t = refl | |
| 428 ... | false | tri< a ¬b ¬c = i-phase2 qs p | |
| 429 ... | false | tri≈ ¬a b ¬c = ⊥-elim (¬-bool eq | |
| 294 | 430 (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 ))) |
| 403 | 431 ... | false | tri> ¬a ¬b c = i-phase2 qs p |
| 405 | 432 i-phase1 : (qs : List Q) → fin-phase1 (F←Q finq q) (map (F←Q finq) qs) ≡ true |
| 433 → phase1 finq q qs ≡ true | |
| 403 | 434 i-phase1 (x ∷ qs) p with equal? finq q x in eq | <-fcmp (F←Q finq q) (F←Q finq x) |
| 435 ... | true | tri< a ¬b ¬c = ⊥-elim ( nat-≡< (cong (λ x → toℕ (F←Q finq x)) ( equal→refl finq eq )) a ) | |
| 436 ... | true | tri≈ ¬a b ¬c = i-phase2 qs p | |
| 437 ... | true | tri> ¬a ¬b c = ⊥-elim ( nat-≡< (cong (λ x → toℕ (F←Q finq x)) (sym ( equal→refl finq eq ))) c ) | |
| 438 ... | false | tri< a ¬b ¬c = i-phase1 qs p | |
| 439 ... | false | tri≈ ¬a b ¬c = ⊥-elim (¬-bool eq | |
| 294 | 440 (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 ))) |
| 403 | 441 ... | false | tri> ¬a ¬b c = i-phase1 qs p |
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442 |
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443 record Dup-in-list {Q : Set } (finq : FiniteSet Q) (qs : List Q) : Set where |
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444 field |
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445 dup : Q |
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446 is-dup : dup-in-list finq dup qs ≡ true |
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448 dup-in-list>n : {Q : Set } → (finq : FiniteSet Q) → (qs : List Q) → (len> : length qs > finite finq ) → Dup-in-list finq qs |
| 283 | 449 dup-in-list>n {Q} finq qs lt = record { dup = Q←F finq (FDup-in-list.dup dl) |
| 450 ; is-dup = dup-in-list+fin finq (Q←F finq (FDup-in-list.dup dl)) qs dl01 } where | |
| 294 | 451 maplen : (qs : List Q) → length (map (F←Q finq) qs) ≡ length qs |
| 452 maplen [] = refl | |
| 453 maplen (x ∷ qs) = cong suc (maplen qs) | |
| 283 | 454 dl : FDup-in-list (finite finq ) (map (F←Q finq) qs) |
| 294 | 455 dl = fin-dup-in-list>n (map (F←Q finq) qs) (subst (λ k → k > finite finq ) (sym (maplen qs)) lt) |
| 283 | 456 dl01 : fin-dup-in-list (F←Q finq (Q←F finq (FDup-in-list.dup dl))) (map (F←Q finq) qs) ≡ true |
| 457 dl01 = subst (λ k → fin-dup-in-list k (map (F←Q finq) qs) ≡ true ) | |
| 294 | 458 (sym (finiso← finq _)) ( FDup-in-list.is-dup dl ) |
| 337 | 459 |
| 460 open import bijection using ( InjectiveF ; Is ) | |
| 461 | |
| 405 | 462 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
| 345 | 463 |
| 405 | 464 inject-fin : {A B : Set} (fa : FiniteSet A ) |
| 465 → (fi : InjectiveF B A) | |
| 337 | 466 → (is-B : (a : A ) → Dec (Is B A (InjectiveF.f fi) a) ) |
| 467 → FiniteSet B | |
| 405 | 468 inject-fin {A} {B} fa fi is-B with finite fa in eq1 |
| 403 | 469 ... | zero = record { |
| 360 | 470 finite = 0 |
| 471 ; Q←F = λ () | |
| 472 ; F←Q = λ b → ⊥-elim ( lem00 b) | |
| 473 ; finiso→ = λ b → ⊥-elim ( lem00 b) | |
| 474 ; finiso← = λ () | |
| 475 } where | |
| 476 lem00 : ( b : B) → ⊥ | |
| 477 lem00 b with subst (λ k → Fin k ) eq1 (F←Q fa (InjectiveF.f fi b)) | |
| 478 ... | () | |
| 403 | 479 ... | suc pfa = record { |
| 347 | 480 finite = maxb |
| 360 | 481 ; Q←F = λ fb → CountB.b (cb00 _ (fin<n {_} fb)) |
| 347 | 482 ; F←Q = λ b → fromℕ< (cb<mb b) |
| 351 | 483 ; finiso→ = iso1 |
| 484 ; finiso← = iso0 | |
| 337 | 485 } where |
| 347 | 486 f = InjectiveF.f fi |
| 487 pfa<fa : pfa < finite fa | |
| 488 pfa<fa = subst (λ k → pfa < k ) (sym eq1) a<sa | |
| 489 0<fa : 0 < finite fa | |
| 405 | 490 0<fa = <-transˡ (s≤s z≤n) pfa<fa |
| 347 | 491 |
| 492 count-B : ℕ → ℕ | |
| 493 count-B zero with is-B (Q←F fa ( fromℕ< {0} 0<fa )) | |
| 494 ... | yes isb = 1 | |
| 495 ... | no nisb = 0 | |
| 350 | 496 count-B (suc n) with <-cmp (finite fa) (suc n) |
| 347 | 497 ... | tri< a ¬b ¬c = count-B n |
| 498 ... | tri≈ ¬a b ¬c = count-B n | |
| 499 ... | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) | |
| 500 ... | yes isb = suc (count-B n) | |
| 501 ... | no nisb = count-B n | |
| 502 | |
| 503 record CountB (n : ℕ) : Set where | |
| 504 field | |
| 505 b : B | |
| 506 cb : ℕ | |
| 507 b=cn : cb ≡ toℕ (F←Q fa (f b)) | |
| 508 cb=n : count-B cb ≡ suc n | |
| 509 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 | |
| 510 | |
| 511 maxb : ℕ | |
| 512 maxb = count-B (finite fa) | |
| 513 | |
| 352 | 514 count-B-mono : {i j : ℕ} → i ≤ j → count-B i ≤ count-B j |
| 515 count-B-mono {i} {j} i≤j with ≤-∨ i≤j | |
| 516 ... | case1 refl = ≤-refl | |
| 517 ... | case2 i<j = lem00 _ _ i<j where | |
| 518 lem00 : (i j : ℕ) → i < j → count-B i ≤ count-B j | |
| 519 lem00 i (suc j) (s≤s i<j) = ≤-trans (count-B-mono i<j) (lem01 j) where | |
| 520 lem01 : (j : ℕ) → count-B j ≤ count-B (suc j) | |
| 405 | 521 lem01 zero with <-cmp (finite fa) 1 |
| 352 | 522 lem01 zero | tri< a ¬b ¬c = ≤-refl |
| 523 lem01 zero | tri≈ ¬a b ¬c = ≤-refl | |
| 405 | 524 lem01 zero | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) | is-B (Q←F fa ( fromℕ< {0} 0<fa )) |
| 403 | 525 ... | yes isb1 | yes isb0 = s≤s z≤n |
| 526 ... | yes isb1 | no nisb0 = z≤n | |
| 527 ... | no nisb1 | yes isb0 = refl-≤≡ (sym lem14 ) where | |
| 405 | 528 lem14 : count-B 0 ≡ 1 -- in-equality does not work we have to repeat the proof |
| 403 | 529 lem14 with is-B (Q←F fa ( fromℕ< {0} 0<fa )) |
| 530 ... | yes isb = refl | |
| 531 ... | no ne = ⊥-elim (ne isb0) | |
| 532 ... | no nisb1 | no nisb0 = z≤n | |
| 405 | 533 lem01 (suc i) with <-cmp (finite fa) (suc i) | <-cmp (finite fa) (suc (suc i)) |
| 404 | 534 ... | tri< a ¬b ¬c | tri< a₁ ¬b₁ ¬c₁ = refl-≤≡ (sym lem14) where |
| 535 lem14 : count-B (suc i) ≡ count-B i | |
| 405 | 536 lem14 with <-cmp (finite fa) (suc i) |
| 404 | 537 ... | tri< a ¬b ¬c = refl |
| 538 ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a a ) | |
| 539 ... | tri> ¬a ¬b c = ⊥-elim ( ¬a a ) | |
| 403 | 540 ... | tri< a ¬b ¬c | tri≈ ¬a b ¬c₁ = ⊥-elim (nat-≡< b (<-trans a a<sa)) |
| 541 ... | tri< a ¬b ¬c | tri> ¬a ¬b₁ c = ⊥-elim (nat-<> a (<-trans a<sa c) ) | |
| 405 | 542 ... | tri≈ ¬a b ¬c | tri< a ¬b ¬c₁ = refl-≤≡ (sym lem14 ) where |
| 543 lem14 : count-B (suc i) ≡ count-B i | |
| 544 lem14 with <-cmp (finite fa) (suc i) | |
| 545 ... | tri< a ¬b ¬c = refl | |
| 546 ... | tri≈ ¬a b ¬c = refl | |
| 547 ... | tri> ¬a ¬b c = ⊥-elim ( ¬c c ) | |
| 403 | 548 ... | tri≈ ¬a b ¬c | tri≈ ¬a₁ b₁ ¬c₁ = ⊥-elim (nat-≡< (sym b) (subst (λ k → _ < k ) (sym b₁) a<sa) ) |
| 549 ... | tri≈ ¬a b ¬c | tri> ¬a₁ ¬b c = ⊥-elim (nat-≡< (sym b) (<-trans a<sa c)) | |
| 550 ... | tri> ¬a ¬b c | tri< a ¬b₁ ¬c = ⊥-elim (nat-≤> a (<-transʳ c a<sa ) ) | |
| 405 | 551 ... | tri> ¬a ¬b c | tri≈ ¬a₁ b ¬c with is-B (Q←F fa (fromℕ< c)) |
| 552 ... | yes isb = refl-≤≡ (sym lem14) where | |
| 553 lem14 : count-B (suc i) ≡ suc (count-B i) | |
| 554 lem14 with <-cmp (finite fa) (suc i) | |
| 555 ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) | |
| 556 ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) | |
| 557 ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) | |
| 558 ... | yes isb = refl | |
| 559 ... | no ne = ⊥-elim (ne record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) | |
| 560 ... | no nisb = refl-≤≡ (sym lem14) where | |
| 561 lem14 : count-B (suc i) ≡ count-B i | |
| 562 lem14 with <-cmp (finite fa) (suc i) | |
| 563 ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) | |
| 564 ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) | |
| 565 ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) | |
| 566 ... | yes isb = ⊥-elim (nisb record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) | |
| 567 ... | no ne = refl | |
| 568 lem01 (suc i) | tri> ¬a ¬b c | tri> ¬a₁ ¬b₁ c₁ | |
| 569 with is-B (Q←F fa (fromℕ< c)) | is-B (Q←F fa (fromℕ< c₁)) | |
| 570 ... | yes isb0 | yes isb1 = ≤-trans (refl-≤≡ (sym lem14)) a≤sa where | |
| 571 lem14 : count-B (suc i) ≡ suc (count-B i) | |
| 572 lem14 with <-cmp (finite fa) (suc i) | |
| 573 ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) | |
| 574 ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) | |
| 575 ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) | |
| 576 ... | no ne = ⊥-elim (ne record {a = Is.a isb0 ; fa=c = trans (Is.fa=c isb0) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) | |
| 577 ... | yes isb = refl | |
| 578 ... | yes isb0 | no nisb1 = refl-≤≡ (sym lem14) where | |
| 579 lem14 : count-B (suc i) ≡ suc (count-B i) | |
| 580 lem14 with <-cmp (finite fa) (suc i) | |
| 581 ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) | |
| 582 ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) | |
| 583 ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) | |
| 584 ... | no ne = ⊥-elim (ne record {a = Is.a isb0 ; fa=c = trans (Is.fa=c isb0) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) | |
| 585 ... | yes isb = refl | |
| 586 ... | no nisb0 | yes isb1 = ≤-trans (refl-≤≡ (sym lem14)) a≤sa where | |
| 587 lem14 : count-B (suc i) ≡ count-B i | |
| 588 lem14 with <-cmp (finite fa) (suc i) | |
| 589 ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) | |
| 590 ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) | |
| 591 ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) | |
| 592 ... | no ne = refl | |
| 593 ... | yes isb = ⊥-elim (nisb0 record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) | |
| 594 ... | no nisb0 | no nisb1 = refl-≤≡ (sym lem14) where | |
| 595 lem14 : count-B (suc i) ≡ count-B i | |
| 596 lem14 with <-cmp (finite fa) (suc i) | |
| 597 ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) | |
| 598 ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) | |
| 599 ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) | |
| 600 ... | no ne = refl | |
| 601 ... | yes isb = ⊥-elim (nisb0 record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) | |
| 352 | 602 |
| 353 | 603 lem31 : (b : B) → 0 < count-B (toℕ (F←Q fa (f b))) |
| 358 | 604 lem31 b = lem32 (toℕ (F←Q fa (f b))) refl where |
| 405 | 605 lem32 : (i : ℕ) → toℕ (F←Q fa (f b)) ≡ i → 0 < count-B i |
| 358 | 606 lem32 zero eq with is-B (Q←F fa ( fromℕ< {0} 0<fa )) |
| 405 | 607 ... | yes isb = s≤s z≤n |
| 354 | 608 ... | no nisb = ⊥-elim ( nisb record { a = b ; fa=c = lem33 } ) where |
| 609 lem33 : f b ≡ Q←F fa ( fromℕ< {0} 0<fa ) | |
| 610 lem33 = begin | |
| 355 | 611 f b ≡⟨ sym (finiso→ fa _) ⟩ |
| 360 | 612 Q←F fa ( F←Q fa (f b)) ≡⟨ sym (cong (λ k → Q←F fa k) ( fromℕ<-toℕ _ (fin<n _))) ⟩ |
| 613 Q←F fa ( fromℕ< (fin<n _) ) ≡⟨ cong (λ k → Q←F fa k) (fromℕ<-cong _ _ eq (fin<n _) 0<fa) ⟩ | |
| 354 | 614 Q←F fa ( fromℕ< {0} 0<fa ) ∎ where |
| 615 open ≡-Reasoning | |
| 405 | 616 lem32 (suc i) eq with <-cmp (finite fa) (suc i) |
| 403 | 617 ... | tri< a ¬b ¬c = ⊥-elim ( nat-≡< eq (<-trans (fin<n _) a) ) |
| 618 ... | tri≈ ¬a eq1 ¬c = ⊥-elim ( nat-≡< eq (subst (λ k → toℕ (F←Q fa (f b)) < k ) eq1 (fin<n _))) | |
| 405 | 619 ... | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) |
| 620 ... | yes isb = s≤s z≤n | |
| 358 | 621 ... | no nisb = ⊥-elim ( nisb record { a = b ; fa=c = lem33 } ) where |
| 622 lem33 : f b ≡ Q←F fa ( fromℕ< c) | |
| 623 lem33 = begin | |
| 624 f b ≡⟨ sym (finiso→ fa _) ⟩ | |
| 360 | 625 Q←F fa ( F←Q fa (f b)) ≡⟨ sym (cong (λ k → Q←F fa k) ( fromℕ<-toℕ _ (fin<n _))) ⟩ |
| 626 Q←F fa ( fromℕ< (fin<n _) ) ≡⟨ cong (λ k → Q←F fa k) (fromℕ<-cong _ _ eq (fin<n _) c ) ⟩ | |
| 358 | 627 Q←F fa ( fromℕ< c ) ∎ where |
| 628 open ≡-Reasoning | |
| 405 | 629 |
| 347 | 630 cb<mb : (b : B) → pred (count-B (toℕ (F←Q fa (f b)))) < maxb |
| 353 | 631 cb<mb b = sx≤y→x<y ( begin |
| 405 | 632 suc ( pred (count-B (toℕ (F←Q fa (f b))))) ≡⟨ sucprd (lem31 b) ⟩ |
| 633 count-B (toℕ (F←Q fa (f b))) ≤⟨ lem02 ⟩ | |
| 353 | 634 count-B (finite fa) ∎ ) where |
| 352 | 635 open ≤-Reasoning |
| 636 lem02 : count-B (toℕ (F←Q fa (f b))) ≤ count-B (finite fa) | |
| 405 | 637 lem02 = count-B-mono (<to≤ (fin<n {_} (F←Q fa (f b)))) |
| 347 | 638 |
| 639 cb00 : (n : ℕ) → n < count-B (finite fa) → CountB n | |
| 348 | 640 cb00 n n<cb = lem09 (finite fa) (count-B (finite fa)) (<-transˡ a<sa n<cb) refl where |
| 341 | 641 |
| 405 | 642 lem06 : (i j : ℕ) → (i<fa : i < finite fa) (j<fa : j < finite fa) |
| 361 | 643 → 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 | 644 lem06 i j i<fa j<fa bi bj eq = lem08 where |
| 405 | 645 lem20 : (i j : ℕ) → i < j → (i<fa : i < finite fa) (j<fa : j < finite fa) |
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646 → 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 |
| 405 | 647 lem20 zero (suc j) i<j i<fa j<fa bi bj with <-cmp (finite fa) (suc j) |
| 362 | 648 ... | tri< a ¬b ¬c = ⊥-elim (¬c j<fa) |
| 649 ... | tri≈ ¬a b ¬c = ⊥-elim (¬c j<fa) | |
| 405 | 650 ... | tri> ¬a ¬b c with is-B (Q←F fa ( fromℕ< 0<fa )) | is-B (Q←F fa (fromℕ< c)) |
| 403 | 651 ... | no nisc | _ = ⊥-elim (nisc record { a = Is.a bi ; fa=c = lem26 } ) where |
| 362 | 652 lem26 : f (Is.a bi) ≡ Q←F fa (fromℕ< 0<fa) |
| 653 lem26 = trans (Is.fa=c bi) (cong (Q←F fa) (fromℕ<-cong _ _ refl i<fa 0<fa) ) | |
| 403 | 654 ... | yes _ | no nisc = ⊥-elim (nisc record { a = Is.a bj ; fa=c = lem26 } ) where |
| 362 | 655 lem26 : f (Is.a bj) ≡ Q←F fa (fromℕ< c) |
| 656 lem26 = trans (Is.fa=c bj) (cong (Q←F fa) (fromℕ<-cong _ _ refl j<fa c) ) | |
| 405 | 657 ... | yes isb1 | yes _ = lem25 where |
| 658 lem14 : count-B 0 ≡ 1 | |
| 659 lem14 with is-B (Q←F fa ( fromℕ< 0<fa )) | |
| 660 ... | no ne = ⊥-elim (ne record {a = Is.a isb1 ; fa=c = trans (Is.fa=c isb1) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) | |
| 661 ... | yes isb = refl | |
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662 lem25 : 2 ≤ suc (count-B j) |
| 362 | 663 lem25 = begin |
| 405 | 664 2 ≡⟨ cong suc (sym lem14) ⟩ |
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665 suc (count-B 0) ≤⟨ s≤s (count-B-mono {0} {j} z≤n) ⟩ |
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666 suc (count-B j) ∎ where open ≤-Reasoning |
| 405 | 667 lem20 (suc i) zero () i<fa j<fa bi bj |
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668 lem20 (suc i) (suc j) (s≤s i<j) i<fa j<fa bi bj = lem21 where |
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669 -- |
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670 -- i < suc i ≤ j |
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671 -- cb i < suc (cb i) < cb (suc i) ≤ cb j |
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672 -- |
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673 lem23 : suc (count-B j) ≡ count-B (suc j) |
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674 lem23 with <-cmp (finite fa) (suc j) |
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675 ... | tri< a ¬b ¬c = ⊥-elim (¬c j<fa) |
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676 ... | tri≈ ¬a b ¬c = ⊥-elim (¬c j<fa) |
| 405 | 677 ... | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) |
| 403 | 678 ... | yes _ = refl |
| 679 ... | no nisa = ⊥-elim ( nisa record { a = Is.a bj ; fa=c = lem26 } ) where | |
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680 lem26 : f (Is.a bj) ≡ Q←F fa (fromℕ< c) |
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681 lem26 = trans (Is.fa=c bj) (cong (Q←F fa) (fromℕ<-cong _ _ refl j<fa c) ) |
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682 lem21 : count-B (suc i) < count-B (suc j) |
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683 lem21 = sx≤py→x≤y ( begin |
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684 suc (suc (count-B (suc i))) ≤⟨ s≤s ( s≤s ( count-B-mono i<j )) ⟩ |
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685 suc (suc (count-B j)) ≡⟨ cong suc lem23 ⟩ |
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686 suc (count-B (suc j)) ∎ ) where |
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687 open ≤-Reasoning |
| 362 | 688 |
| 361 | 689 lem08 : i ≡ j |
| 690 lem08 with <-cmp i j | |
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691 ... | tri< a ¬b ¬c = ⊥-elim (nat-≡< eq ( lem20 i j a i<fa j<fa bi bj )) |
| 361 | 692 ... | tri≈ ¬a b ¬c = b |
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693 ... | tri> ¬a ¬b c₁ = ⊥-elim (nat-≡< (sym eq) ( lem20 j i c₁ j<fa i<fa bj bi )) |
| 361 | 694 |
| 347 | 695 lem09 : (i j : ℕ) → suc n ≤ j → j ≡ count-B i → CountB n |
| 405 | 696 lem09 0 (suc j) (s≤s le) eq with is-B (Q←F fa (fromℕ< {0} 0<fa )) |
| 403 | 697 ... | no nisb = ⊥-elim ( nat-≡< (sym eq) (s≤s z≤n) ) |
| 698 ... | yes isb with ≤-∨ (s≤s le) | |
| 405 | 699 ... | case1 eq2 = record { b = Is.a isb ; cb = 0 ; b=cn = lem10 ; cb=n = trans lem14 (sym (trans eq2 eq)) |
| 362 | 700 ; cb-inject = λ cb1 c1<fa b1 eq → lem06 0 cb1 0<fa c1<fa isb b1 eq } where |
| 405 | 701 lem14 : count-B 0 ≡ 1 |
| 702 lem14 with is-B (Q←F fa ( fromℕ< 0<fa )) | |
| 703 ... | no ne = ⊥-elim (ne record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) | |
| 704 ... | yes isb = refl | |
| 349 | 705 lem10 : 0 ≡ toℕ (F←Q fa (f (Is.a isb))) |
| 706 lem10 = begin | |
| 707 0 ≡⟨ sym ( toℕ-fromℕ< 0<fa ) ⟩ | |
| 708 toℕ (fromℕ< {0} 0<fa ) ≡⟨ cong toℕ (sym (finiso← fa _)) ⟩ | |
| 709 toℕ (F←Q fa (Q←F fa (fromℕ< {0} 0<fa ))) ≡⟨ cong (λ k → toℕ ((F←Q fa k))) (sym (Is.fa=c isb)) ⟩ | |
| 710 toℕ (F←Q fa (f (Is.a isb))) ∎ where open ≡-Reasoning | |
| 711 ... | case2 (s≤s lt) = ⊥-elim ( nat-≡< (sym eq) (s≤s (<-transʳ z≤n lt) )) | |
| 405 | 712 lem09 (suc i) (suc j) (s≤s le) eq with <-cmp (finite fa) (suc i) |
| 403 | 713 ... | tri< a ¬b ¬c = lem09 i (suc j) (s≤s le) eq |
| 714 ... | tri≈ ¬a b ¬c = lem09 i (suc j) (s≤s le) eq | |
| 715 ... | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) | |
| 348 | 716 ... | no nisb = lem09 i (suc j) (s≤s le) eq |
| 717 ... | yes isb with ≤-∨ (s≤s le) | |
| 405 | 718 ... | case1 eq2 = record { b = Is.a isb ; cb = suc i ; b=cn = lem11 ; cb=n = trans lem14 (sym (trans eq2 eq )) |
| 362 | 719 ; cb-inject = λ cb1 c1<fa b1 eq → lem06 (suc i) cb1 c c1<fa isb b1 eq } where |
| 405 | 720 lem14 : count-B (suc i) ≡ suc (count-B i) |
| 721 lem14 with <-cmp (finite fa) (suc i) | |
| 722 ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) | |
| 723 ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) | |
| 724 ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) | |
| 725 ... | yes isb = refl | |
| 726 ... | no ne = ⊥-elim (ne record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) | |
| 350 | 727 lem11 : suc i ≡ toℕ (F←Q fa (f (Is.a isb))) |
| 728 lem11 = begin | |
| 729 suc i ≡⟨ sym ( toℕ-fromℕ< c) ⟩ | |
| 730 toℕ (fromℕ< c) ≡⟨ cong toℕ (sym (finiso← fa _)) ⟩ | |
| 349 | 731 toℕ (F←Q fa (Q←F fa (fromℕ< c ))) ≡⟨ cong (λ k → toℕ ((F←Q fa k))) (sym (Is.fa=c isb)) ⟩ |
| 732 toℕ (F←Q fa (f (Is.a isb))) ∎ where open ≡-Reasoning | |
| 348 | 733 ... | case2 (s≤s lt) = lem09 i j lt (cong pred eq) |
| 347 | 734 |
| 405 | 735 iso0 : (x : Fin maxb) → fromℕ< (cb<mb (CountB.b (cb00 (toℕ x) (fin<n _)))) ≡ x |
| 351 | 736 iso0 x = begin |
| 405 | 737 fromℕ< (cb<mb (CountB.b (cb00 (toℕ x) (fin<n _)))) ≡⟨ fromℕ<-cong _ _ ( begin |
| 360 | 738 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 | 739 pred (count-B (CountB.cb CB)) ≡⟨ cong pred (CountB.cb=n CB) ⟩ |
| 740 pred (suc (toℕ x)) ≡⟨ refl ⟩ | |
| 405 | 741 toℕ x ∎ ) (cb<mb (CountB.b CB)) (fin<n _) ⟩ |
| 742 fromℕ< (fin<n {_} x) ≡⟨ fromℕ<-toℕ _ (fin<n {_} x) ⟩ | |
| 351 | 743 x ∎ where |
| 744 open ≡-Reasoning | |
| 360 | 745 CB = cb00 (toℕ x) (fin<n _) |
| 351 | 746 |
| 360 | 747 iso1 : (b : B) → CountB.b (cb00 (toℕ (fromℕ< (cb<mb b))) (fin<n _)) ≡ b |
| 748 iso1 b = begin | |
| 351 | 749 CountB.b CB ≡⟨ InjectiveF.inject fi (F←Q-inject fa (toℕ-injective (begin |
| 750 toℕ (F←Q fa (f (CountB.b CB))) ≡⟨ sym (CountB.b=cn CB) ⟩ | |
| 360 | 751 CountB.cb CB ≡⟨ CountB.cb-inject CB _ (fin<n _) isb lem30 ⟩ |
| 351 | 752 toℕ (F←Q fa (f b)) ∎ ) )) ⟩ |
| 753 b ∎ where | |
| 754 open ≡-Reasoning | |
| 360 | 755 CB = cb00 (toℕ (fromℕ< (cb<mb b))) (fin<n _) |
| 756 isb : Is B A f (Q←F fa (fromℕ< (fin<n {_} (F←Q fa (f b)) ))) | |
| 757 isb = record { a = b ; fa=c = lem33 } where | |
| 405 | 758 lem33 : f b ≡ Q←F fa (fromℕ< (fin<n (F←Q fa (f b)))) |
| 360 | 759 lem33 = begin |
| 760 f b ≡⟨ sym (finiso→ fa _) ⟩ | |
| 761 Q←F fa (F←Q fa (f b)) ≡⟨ cong (Q←F fa) (sym (fromℕ<-toℕ _ (fin<n (F←Q fa (f b))))) ⟩ | |
| 405 | 762 Q←F fa (fromℕ< (fin<n (F←Q fa (f b)))) ∎ |
| 351 | 763 lem30 : count-B (CountB.cb CB) ≡ count-B (toℕ (F←Q fa (InjectiveF.f fi b))) |
| 764 lem30 = begin | |
| 765 count-B (CountB.cb CB) ≡⟨ CountB.cb=n CB ⟩ | |
| 766 suc (toℕ (fromℕ< (cb<mb b))) ≡⟨ cong suc (toℕ-fromℕ< (cb<mb b)) ⟩ | |
| 353 | 767 suc (pred (count-B (toℕ (F←Q fa (f b))))) ≡⟨ sucprd (lem31 b) ⟩ |
| 405 | 768 count-B (toℕ (F←Q fa (f b))) ∎ |
| 351 | 769 |
| 770 | |
| 347 | 771 -- end |