Mercurial > hg > Members > kono > Proof > galois
annotate src/FLutil.agda @ 272:ce372f6347d6
Foundamental definition done
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 24 Jan 2023 19:15:38 +0900 |
| parents | 6d1619d9f880 |
| children | f59a9f4cfd78 |
| rev | line source |
|---|---|
| 153 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 151 | 2 module FLutil where |
| 134 | 3 |
| 4 open import Level hiding ( suc ; zero ) | |
| 5 open import Data.Fin hiding ( _<_ ; _≤_ ; _-_ ; _+_ ; _≟_) | |
| 156 | 6 open import Data.Fin.Properties hiding ( <-trans ; ≤-refl ; ≤-trans ; ≤-irrelevant ; _≟_ ) renaming ( <-cmp to <-fcmp ) |
| 172 | 7 open import Data.Fin.Permutation -- hiding ([_,_]) |
| 134 | 8 open import Data.Nat -- using (ℕ; suc; zero; s≤s ; z≤n ) |
| 173 | 9 open import Data.Nat.Properties as DNP |
| 172 | 10 open import Relation.Binary.PropositionalEquality hiding ( [_] ) |
| 153 | 11 open import Data.List using (List; []; _∷_ ; length ; _++_ ; tail ) renaming (reverse to rev ) |
| 134 | 12 open import Data.Product |
| 13 open import Relation.Nullary | |
| 14 open import Data.Empty | |
| 15 open import Relation.Binary.Core | |
| 172 | 16 open import Relation.Binary.Definitions |
| 137 | 17 open import logic |
| 18 open import nat | |
| 134 | 19 |
| 20 infixr 100 _::_ | |
| 21 | |
| 22 data FL : (n : ℕ )→ Set where | |
| 23 f0 : FL 0 | |
| 24 _::_ : { n : ℕ } → Fin (suc n ) → FL n → FL (suc n) | |
| 25 | |
| 26 data _f<_ : {n : ℕ } (x : FL n ) (y : FL n) → Set where | |
| 27 f<n : {m : ℕ } {xn yn : Fin (suc m) } {xt yt : FL m} → xn Data.Fin.< yn → (xn :: xt) f< ( yn :: yt ) | |
| 28 f<t : {m : ℕ } {xn : Fin (suc m) } {xt yt : FL m} → xt f< yt → (xn :: xt) f< ( xn :: yt ) | |
| 29 | |
| 30 FLeq : {n : ℕ } {xn yn : Fin (suc n)} {x : FL n } {y : FL n} → xn :: x ≡ yn :: y → ( xn ≡ yn ) × (x ≡ y ) | |
| 31 FLeq refl = refl , refl | |
| 32 | |
| 208 | 33 FLpos : {n : ℕ} → FL (suc n) → Fin (suc n) |
| 34 FLpos (x :: _) = x | |
| 35 | |
| 134 | 36 f-<> : {n : ℕ } {x : FL n } {y : FL n} → x f< y → y f< x → ⊥ |
| 37 f-<> (f<n x) (f<n x₁) = nat-<> x x₁ | |
| 38 f-<> (f<n x) (f<t lt2) = nat-≡< refl x | |
| 39 f-<> (f<t lt) (f<n x) = nat-≡< refl x | |
| 40 f-<> (f<t lt) (f<t lt2) = f-<> lt lt2 | |
| 41 | |
| 42 f-≡< : {n : ℕ } {x : FL n } {y : FL n} → x ≡ y → y f< x → ⊥ | |
| 43 f-≡< refl (f<n x) = nat-≡< refl x | |
| 44 f-≡< refl (f<t lt) = f-≡< refl lt | |
| 45 | |
| 46 FLcmp : {n : ℕ } → Trichotomous {Level.zero} {FL n} _≡_ _f<_ | |
| 47 FLcmp f0 f0 = tri≈ (λ ()) refl (λ ()) | |
| 48 FLcmp (xn :: xt) (yn :: yt) with <-fcmp xn yn | |
| 49 ... | tri< a ¬b ¬c = tri< (f<n a) (λ eq → nat-≡< (cong toℕ (proj₁ (FLeq eq)) ) a) (λ lt → f-<> lt (f<n a) ) | |
| 50 ... | tri> ¬a ¬b c = tri> (λ lt → f-<> lt (f<n c) ) (λ eq → nat-≡< (cong toℕ (sym (proj₁ (FLeq eq)) )) c) (f<n c) | |
| 51 ... | tri≈ ¬a refl ¬c with FLcmp xt yt | |
| 52 ... | tri< a ¬b ¬c₁ = tri< (f<t a) (λ eq → ¬b (proj₂ (FLeq eq) )) (λ lt → f-<> lt (f<t a) ) | |
| 53 ... | tri≈ ¬a₁ refl ¬c₁ = tri≈ (λ lt → f-≡< refl lt ) refl (λ lt → f-≡< refl lt ) | |
| 54 ... | tri> ¬a₁ ¬b c = tri> (λ lt → f-<> lt (f<t c) ) (λ eq → ¬b (proj₂ (FLeq eq) )) (f<t c) | |
| 55 | |
| 138 | 56 f<-trans : {n : ℕ } { x y z : FL n } → x f< y → y f< z → x f< z |
| 57 f<-trans {suc n} (f<n x) (f<n x₁) = f<n ( Data.Fin.Properties.<-trans x x₁ ) | |
| 58 f<-trans {suc n} (f<n x) (f<t y<z) = f<n x | |
| 59 f<-trans {suc n} (f<t x<y) (f<n x) = f<n x | |
| 60 f<-trans {suc n} (f<t x<y) (f<t y<z) = f<t (f<-trans x<y y<z) | |
| 61 | |
| 134 | 62 infixr 250 _f<?_ |
| 63 | |
| 64 _f<?_ : {n : ℕ} → (x y : FL n ) → Dec (x f< y ) | |
| 65 x f<? y with FLcmp x y | |
| 66 ... | tri< a ¬b ¬c = yes a | |
| 67 ... | tri≈ ¬a refl ¬c = no ( ¬a ) | |
| 68 ... | tri> ¬a ¬b c = no ( ¬a ) | |
| 69 | |
| 70 _f≤_ : {n : ℕ } (x : FL n ) (y : FL n) → Set | |
| 71 _f≤_ x y = (x ≡ y ) ∨ (x f< y ) | |
| 72 | |
| 73 FL0 : {n : ℕ } → FL n | |
| 74 FL0 {zero} = f0 | |
| 75 FL0 {suc n} = zero :: FL0 | |
| 76 | |
| 77 fmax : { n : ℕ } → FL n | |
| 78 fmax {zero} = f0 | |
| 79 fmax {suc n} = fromℕ< a<sa :: fmax {n} | |
| 80 | |
| 81 fmax< : { n : ℕ } → {x : FL n } → ¬ (fmax f< x ) | |
| 82 fmax< {suc n} {x :: y} (f<n lt) = nat-≤> (fmax1 x) lt where | |
| 83 fmax1 : {n : ℕ } → (x : Fin (suc n)) → toℕ x ≤ toℕ (fromℕ< {n} a<sa) | |
| 84 fmax1 {zero} zero = z≤n | |
| 85 fmax1 {suc n} zero = z≤n | |
| 86 fmax1 {suc n} (suc x) = s≤s (fmax1 x) | |
| 87 fmax< {suc n} {x :: y} (f<t lt) = fmax< {n} {y} lt | |
| 88 | |
| 89 fmax¬ : { n : ℕ } → {x : FL n } → ¬ ( x ≡ fmax ) → x f< fmax | |
| 90 fmax¬ {zero} {f0} ne = ⊥-elim ( ne refl ) | |
| 91 fmax¬ {suc n} {x} ne with FLcmp x fmax | |
| 92 ... | tri< a ¬b ¬c = a | |
| 93 ... | tri≈ ¬a b ¬c = ⊥-elim ( ne b) | |
| 94 ... | tri> ¬a ¬b c = ⊥-elim (fmax< c) | |
| 95 | |
| 235 | 96 x≤fmax : {n : ℕ } → {x : FL n} → x f≤ fmax |
| 97 x≤fmax {n} {x} with FLcmp x fmax | |
| 98 ... | tri< a ¬b ¬c = case2 a | |
| 99 ... | tri≈ ¬a b ¬c = case1 b | |
| 100 ... | tri> ¬a ¬b c = ⊥-elim ( fmax< c ) | |
| 134 | 101 |
| 151 | 102 open import Data.Nat.Properties using ( ≤-trans ; <-trans ) |
| 103 fsuc : { n : ℕ } → (x : FL n ) → x f< fmax → FL n | |
| 104 fsuc {n} (x :: y) (f<n lt) = fromℕ< fsuc1 :: y where | |
| 105 fsuc1 : suc (toℕ x) < n | |
| 106 fsuc1 = Data.Nat.Properties.≤-trans (s≤s lt) ( s≤s ( toℕ≤pred[n] (fromℕ< a<sa)) ) | |
| 107 fsuc (x :: y) (f<t lt) = x :: fsuc y lt | |
| 108 | |
| 174 | 109 open import fin |
| 110 | |
| 151 | 111 flist1 : {n : ℕ } (i : ℕ) → i < suc n → List (FL n) → List (FL n) → List (FL (suc n)) |
| 112 flist1 zero i<n [] _ = [] | |
| 113 flist1 zero i<n (a ∷ x ) z = ( zero :: a ) ∷ flist1 zero i<n x z | |
| 114 flist1 (suc i) (s≤s i<n) [] z = flist1 i (Data.Nat.Properties.<-trans i<n a<sa) z z | |
| 115 flist1 (suc i) i<n (a ∷ x ) z = ((fromℕ< i<n ) :: a ) ∷ flist1 (suc i) i<n x z | |
| 116 | |
| 117 flist : {n : ℕ } → FL n → List (FL n) | |
| 118 flist {zero} f0 = f0 ∷ [] | |
| 119 flist {suc n} (x :: y) = flist1 n a<sa (flist y) (flist y) | |
| 120 | |
| 185 | 121 FL1 : List ℕ → List ℕ |
| 122 FL1 [] = [] | |
| 123 FL1 (x ∷ y) = suc x ∷ FL1 y | |
| 124 | |
| 125 FL→plist : {n : ℕ} → FL n → List ℕ | |
| 126 FL→plist {0} f0 = [] | |
| 127 FL→plist {suc n} (zero :: y) = zero ∷ FL1 (FL→plist y) | |
| 128 FL→plist {suc n} (suc x :: y) with FL→plist y | |
| 129 ... | [] = zero ∷ [] | |
| 130 ... | x1 ∷ t = suc x1 ∷ FL2 x t where | |
| 131 FL2 : {n : ℕ} → Fin n → List ℕ → List ℕ | |
| 132 FL2 zero y = zero ∷ FL1 y | |
| 133 FL2 (suc i) [] = zero ∷ [] | |
| 134 FL2 (suc i) (x ∷ y) = suc x ∷ FL2 i y | |
| 135 | |
| 136 tt0 = (# 2) :: (# 1) :: (# 0) :: zero :: f0 | |
| 137 tt1 = FL→plist tt0 | |
| 138 | |
| 139 open _∧_ | |
| 140 | |
| 141 find-zero : {n i : ℕ} → List ℕ → i < n → Fin n ∧ List ℕ | |
| 142 find-zero [] i<n = record { proj1 = fromℕ< i<n ; proj2 = [] } | |
| 143 find-zero x (s≤s z≤n) = record { proj1 = fromℕ< (s≤s z≤n) ; proj2 = x } | |
| 144 find-zero (zero ∷ y) (s≤s (s≤s i<n)) = record { proj1 = fromℕ< (s≤s (s≤s i<n)) ; proj2 = y } | |
| 145 find-zero (suc x ∷ y) (s≤s (s≤s i<n)) with find-zero y (s≤s i<n) | |
| 146 ... | record { proj1 = i ; proj2 = y1 } = record { proj1 = suc i ; proj2 = suc x ∷ y1 } | |
| 147 | |
| 251 | 148 plist→FL : {n : ℕ} → List ℕ → FL n -- wrong implementation |
| 185 | 149 plist→FL {zero} [] = f0 |
| 150 plist→FL {suc n} [] = zero :: plist→FL {n} [] | |
| 151 plist→FL {zero} x = f0 | |
| 152 plist→FL {suc n} x with find-zero x a<sa | |
| 153 ... | record { proj1 = i ; proj2 = y } = i :: plist→FL y | |
| 154 | |
| 155 tt2 = 2 ∷ 1 ∷ 0 ∷ 3 ∷ [] | |
| 156 tt3 : FL 4 | |
| 157 tt3 = plist→FL tt2 | |
| 251 | 158 tt4 = FL→plist tt3 |
| 159 tt5 = plist→FL {4} (FL→plist tt0) | |
| 160 | |
| 161 -- maybe FL→iso can be easier using this ... | |
| 162 -- FL→plist-iso : {n : ℕ} → (f : FL n ) → plist→FL (FL→plist f ) ≡ f | |
| 163 -- FL→plist-iso = {!!} | |
| 164 -- FL→plist-inject : {n : ℕ} → (f g : FL n ) → FL→plist f ≡ FL→plist g → f ≡ g | |
| 165 -- FL→plist-inject = {!!} | |
| 151 | 166 |
| 134 | 167 open import Relation.Binary as B hiding (Decidable; _⇔_) |
| 168 open import Data.Sum.Base as Sum -- inj₁ | |
| 138 | 169 open import Relation.Nary using (⌊_⌋) |
| 172 | 170 open import Data.List.Fresh hiding ([_]) |
| 134 | 171 |
| 153 | 172 FList : (n : ℕ ) → Set |
| 173 FList n = List# (FL n) ⌊ _f<?_ ⌋ | |
| 134 | 174 |
| 153 | 175 fr1 : FList 3 |
| 134 | 176 fr1 = |
| 177 ((# 0) :: ((# 0) :: ((# 0 ) :: f0))) ∷# | |
| 178 ((# 0) :: ((# 1) :: ((# 0 ) :: f0))) ∷# | |
| 179 ((# 1) :: ((# 0) :: ((# 0 ) :: f0))) ∷# | |
| 180 ((# 2) :: ((# 0) :: ((# 0 ) :: f0))) ∷# | |
| 181 ((# 2) :: ((# 1) :: ((# 0 ) :: f0))) ∷# | |
| 182 [] | |
| 183 | |
| 184 open import Data.Product | |
| 135 | 185 open import Relation.Nullary.Decidable hiding (⌊_⌋) |
| 172 | 186 -- open import Data.Bool hiding (_<_ ; _≤_ ) |
| 135 | 187 open import Data.Unit.Base using (⊤ ; tt) |
| 188 | |
| 138 | 189 -- fresh a [] = ⊤ |
| 190 -- fresh a (x ∷# xs) = R a x × fresh a xs | |
| 191 | |
| 192 -- toWitness | |
| 193 -- ttf< : {n : ℕ } → {x a : FL n } → x f< a → T (isYes (x f<? a)) | |
| 194 -- ttf< {n} {x} {a} x<a with x f<? a | |
| 195 -- ... | yes y = subst (λ k → Data.Bool.T k ) refl tt | |
| 196 -- ... | no nn = ⊥-elim ( nn x<a ) | |
| 135 | 197 |
|
176
cf7622b185a6
∀Flist non terminating
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
175
diff
changeset
|
198 ttf : {n : ℕ } {x a : FL (n)} → x f< a → (y : FList (n)) → fresh (FL (n)) ⌊ _f<?_ ⌋ a y → fresh (FL (n)) ⌊ _f<?_ ⌋ x y |
| 143 | 199 ttf _ [] fr = Level.lift tt |
| 200 ttf {_} {x} {a} lt (cons a₁ y x1) (lift lt1 , x2 ) = (Level.lift (fromWitness (ttf1 lt1 lt ))) , ttf (ttf1 lt1 lt) y x1 where | |
| 141 | 201 ttf1 : True (a f<? a₁) → x f< a → x f< a₁ |
| 202 ttf1 t x<a = f<-trans x<a (toWitness t) | |
| 203 | |
| 151 | 204 -- by https://gist.github.com/aristidb/1684202 |
| 205 | |
| 153 | 206 FLinsert : {n : ℕ } → FL n → FList n → FList n |
| 207 FLfresh : {n : ℕ } → (a x : FL (suc n) ) → (y : FList (suc n) ) → a f< x | |
| 148 | 208 → fresh (FL (suc n)) ⌊ _f<?_ ⌋ a y → fresh (FL (suc n)) ⌊ _f<?_ ⌋ a (FLinsert x y) |
| 138 | 209 FLinsert {zero} f0 y = f0 ∷# [] |
| 210 FLinsert {suc n} x [] = x ∷# [] | |
| 148 | 211 FLinsert {suc n} x (cons a y x₁) with FLcmp x a |
| 212 ... | tri≈ ¬a b ¬c = cons a y x₁ | |
| 213 ... | tri< lt ¬b ¬c = cons x ( cons a y x₁) ( Level.lift (fromWitness lt ) , ttf lt y x₁) | |
| 153 | 214 FLinsert {suc n} x (cons a [] x₁) | tri> ¬a ¬b lt = cons a ( x ∷# [] ) ( Level.lift (fromWitness lt) , Level.lift tt ) |
| 215 FLinsert {suc n} x (cons a y yr) | tri> ¬a ¬b a<x = cons a (FLinsert x y) (FLfresh a x y a<x yr ) | |
| 147 | 216 |
| 150 | 217 FLfresh a x [] a<x (Level.lift tt) = Level.lift (fromWitness a<x) , Level.lift tt |
| 218 FLfresh a x (cons b [] (Level.lift tt)) a<x (Level.lift a<b , a<y) with FLcmp x b | |
| 151 | 219 ... | tri< x<b ¬b ¬c = Level.lift (fromWitness a<x) , Level.lift a<b , Level.lift tt |
| 149 | 220 ... | tri≈ ¬a refl ¬c = Level.lift (fromWitness a<x) , Level.lift tt |
| 151 | 221 ... | tri> ¬a ¬b b<x = Level.lift a<b , Level.lift (fromWitness (f<-trans (toWitness a<b) b<x)) , Level.lift tt |
| 150 | 222 FLfresh a x (cons b y br) a<x (Level.lift a<b , a<y) with FLcmp x b |
| 151 | 223 ... | tri< x<b ¬b ¬c = Level.lift (fromWitness a<x) , Level.lift a<b , ttf (toWitness a<b) y br |
| 224 ... | tri≈ ¬a refl ¬c = Level.lift (fromWitness a<x) , ttf a<x y br | |
| 150 | 225 FLfresh a x (cons b [] br) a<x (Level.lift a<b , a<y) | tri> ¬a ¬b b<x = |
| 151 | 226 Level.lift a<b , Level.lift (fromWitness (f<-trans (toWitness a<b) b<x)) , Level.lift tt |
| 150 | 227 FLfresh a x (cons b (cons a₁ y x₁) br) a<x (Level.lift a<b , a<y) | tri> ¬a ¬b b<x = |
| 151 | 228 Level.lift a<b , FLfresh a x (cons a₁ y x₁) a<x a<y |
| 134 | 229 |
| 138 | 230 fr6 = FLinsert ((# 1) :: ((# 1) :: ((# 0 ) :: f0))) fr1 |
| 151 | 231 |
| 154 | 232 open import Data.List.Fresh.Relation.Unary.Any |
| 156 | 233 open import Data.List.Fresh.Relation.Unary.All |
| 154 | 234 |
| 222 | 235 x∈FLins : {n : ℕ} → (x : FL n ) → (xs : FList n) → Any (x ≡_) (FLinsert x xs) |
| 154 | 236 x∈FLins {zero} f0 [] = here refl |
| 237 x∈FLins {zero} f0 (cons f0 xs x) = here refl | |
| 238 x∈FLins {suc n} x [] = here refl | |
| 239 x∈FLins {suc n} x (cons a xs x₁) with FLcmp x a | |
| 240 ... | tri< x<a ¬b ¬c = here refl | |
| 241 ... | tri≈ ¬a b ¬c = here b | |
| 242 x∈FLins {suc n} x (cons a [] x₁) | tri> ¬a ¬b a<x = there ( here refl ) | |
| 243 x∈FLins {suc n} x (cons a (cons a₁ xs x₂) x₁) | tri> ¬a ¬b a<x = there ( x∈FLins x (cons a₁ xs x₂) ) | |
| 244 | |
| 222 | 245 nextAny : {n : ℕ} → {x h : FL n } → {L : FList n} → {hr : fresh (FL n) ⌊ _f<?_ ⌋ h L } → Any (x ≡_) L → Any (x ≡_) (cons h L hr ) |
| 173 | 246 nextAny (here x₁) = there (here x₁) |
| 247 nextAny (there any) = there (there any) | |
| 248 | |
| 222 | 249 insAny : {n : ℕ} → {x h : FL n } → (xs : FList n) → Any (x ≡_) xs → Any (x ≡_) (FLinsert h xs) |
| 197 | 250 insAny {zero} {f0} {f0} (cons a L xr) (here refl) = here refl |
| 251 insAny {zero} {f0} {f0} (cons a L xr) (there any) = insAny {zero} {f0} {f0} L any | |
| 198 | 252 insAny {suc n} {x} {h} (cons a L xr) any with FLcmp h a |
| 253 ... | tri< x<a ¬b ¬c = there any | |
| 254 ... | tri≈ ¬a b ¬c = any | |
| 255 insAny {suc n} {a} {h} (cons a [] (Level.lift tt)) (here refl) | tri> ¬a ¬b c = here refl | |
| 256 insAny {suc n} {x} {h} (cons a (cons a₁ L x₁) xr) (here refl) | tri> ¬a ¬b c = here refl | |
| 257 insAny {suc n} {x} {h} (cons a (cons a₁ L x₁) xr) (there any) | tri> ¬a ¬b c = there (insAny (cons a₁ L x₁) any) | |
| 197 | 258 |
| 152 | 259 -- FLinsert membership |
| 260 | |
| 261 module FLMB { n : ℕ } where | |
| 262 | |
| 263 FL-Setoid : Setoid Level.zero Level.zero | |
| 264 FL-Setoid = record { Carrier = FL n ; _≈_ = _≡_ ; isEquivalence = record { sym = sym ; refl = refl ; trans = trans }} | |
| 265 | |
| 266 open import Data.List.Fresh.Membership.Setoid FL-Setoid | |
| 267 | |
| 153 | 268 FLinsert-mb : (x : FL n ) → (xs : FList n) → x ∈ FLinsert x xs |
| 154 | 269 FLinsert-mb x xs = x∈FLins {n} x xs |
| 153 | 270 |
| 154 | 271 |