Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate Ordinals.agda @ 342:b1ccdbb14c92
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Mon, 13 Jul 2020 13:55:46 +0900 |
| parents | 27d2933c4bd7 |
| children | 06f10815d0b3 |
| rev | line source |
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| 16 | 1 open import Level |
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2 module Ordinals where |
| 3 | 3 |
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4 open import zf |
| 3 | 5 |
| 23 | 6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
| 75 | 7 open import Data.Empty |
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8 open import Relation.Binary.PropositionalEquality |
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9 open import logic |
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10 open import nat |
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11 open import Data.Unit using ( ⊤ ) |
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12 open import Relation.Nullary |
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13 open import Relation.Binary |
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14 open import Relation.Binary.Core |
| 3 | 15 |
| 320 | 16 record IsOrdinals {n : Level} (ord : Set n) (o∅ : ord ) (osuc : ord → ord ) (_o<_ : ord → ord → Set n) (next : ord → ord ) : Set (suc (suc n)) where |
| 16 | 17 field |
| 221 | 18 Otrans : {x y z : ord } → x o< y → y o< z → x o< z |
| 19 OTri : Trichotomous {n} _≡_ _o<_ | |
| 20 ¬x<0 : { x : ord } → ¬ ( x o< o∅ ) | |
| 21 <-osuc : { x : ord } → x o< osuc x | |
| 22 osuc-≡< : { a x : ord } → x o< osuc a → (x ≡ a ) ∨ (x o< a) | |
| 329 | 23 not-limit : ( x : ord ) → Dec ( ¬ ((y : ord) → ¬ (x ≡ osuc y) )) |
| 339 | 24 next-limit : { y : ord } → (y o< next y ) ∧ ((x : ord) → x o< next y → osuc x o< next y ) ∧ |
| 25 ( (x : ord) → y o< x → x o< next y → ¬ ((z : ord) → ¬ (x ≡ osuc z) )) | |
| 324 | 26 TransFinite : { ψ : ord → Set n } |
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27 → ( (x : ord) → ( (y : ord ) → y o< x → ψ y ) → ψ x ) |
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28 → ∀ (x : ord) → ψ x |
| 330 | 29 TransFinite1 : { ψ : ord → Set (suc n) } |
| 30 → ( (x : ord) → ( (y : ord ) → y o< x → ψ y ) → ψ x ) | |
| 31 → ∀ (x : ord) → ψ x | |
| 16 | 32 |
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33 |
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34 record Ordinals {n : Level} : Set (suc (suc n)) where |
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35 field |
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36 ord : Set n |
| 221 | 37 o∅ : ord |
| 38 osuc : ord → ord | |
| 39 _o<_ : ord → ord → Set n | |
| 320 | 40 next : ord → ord |
| 41 isOrdinal : IsOrdinals ord o∅ osuc _o<_ next | |
| 17 | 42 |
| 221 | 43 module inOrdinal {n : Level} (O : Ordinals {n} ) where |
| 3 | 44 |
| 221 | 45 Ordinal : Set n |
| 46 Ordinal = Ordinals.ord O | |
| 47 | |
| 48 _o<_ : Ordinal → Ordinal → Set n | |
| 49 _o<_ = Ordinals._o<_ O | |
| 218 | 50 |
| 221 | 51 osuc : Ordinal → Ordinal |
| 52 osuc = Ordinals.osuc O | |
| 218 | 53 |
| 221 | 54 o∅ : Ordinal |
| 55 o∅ = Ordinals.o∅ O | |
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56 |
| 320 | 57 next : Ordinal → Ordinal |
| 58 next = Ordinals.next O | |
| 59 | |
| 221 | 60 ¬x<0 = IsOrdinals.¬x<0 (Ordinals.isOrdinal O) |
| 61 osuc-≡< = IsOrdinals.osuc-≡< (Ordinals.isOrdinal O) | |
| 62 <-osuc = IsOrdinals.<-osuc (Ordinals.isOrdinal O) | |
| 235 | 63 TransFinite = IsOrdinals.TransFinite (Ordinals.isOrdinal O) |
| 330 | 64 TransFinite1 = IsOrdinals.TransFinite1 (Ordinals.isOrdinal O) |
| 320 | 65 next-limit = IsOrdinals.next-limit (Ordinals.isOrdinal O) |
| 321 | 66 |
| 221 | 67 o<-dom : { x y : Ordinal } → x o< y → Ordinal |
| 68 o<-dom {x} _ = x | |
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69 |
| 221 | 70 o<-cod : { x y : Ordinal } → x o< y → Ordinal |
| 71 o<-cod {_} {y} _ = y | |
| 147 | 72 |
| 221 | 73 o<-subst : {Z X z x : Ordinal } → Z o< X → Z ≡ z → X ≡ x → z o< x |
| 74 o<-subst df refl refl = df | |
| 94 | 75 |
| 221 | 76 ordtrans : {x y z : Ordinal } → x o< y → y o< z → x o< z |
| 77 ordtrans = IsOrdinals.Otrans (Ordinals.isOrdinal O) | |
| 94 | 78 |
| 221 | 79 trio< : Trichotomous _≡_ _o<_ |
| 80 trio< = IsOrdinals.OTri (Ordinals.isOrdinal O) | |
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81 |
| 221 | 82 o<¬≡ : { ox oy : Ordinal } → ox ≡ oy → ox o< oy → ⊥ |
| 83 o<¬≡ {ox} {oy} eq lt with trio< ox oy | |
| 84 o<¬≡ {ox} {oy} eq lt | tri< a ¬b ¬c = ¬b eq | |
| 85 o<¬≡ {ox} {oy} eq lt | tri≈ ¬a b ¬c = ¬a lt | |
| 86 o<¬≡ {ox} {oy} eq lt | tri> ¬a ¬b c = ¬b eq | |
| 23 | 87 |
| 221 | 88 o<> : {x y : Ordinal } → y o< x → x o< y → ⊥ |
| 89 o<> {ox} {oy} lt tl with trio< ox oy | |
| 90 o<> {ox} {oy} lt tl | tri< a ¬b ¬c = ¬c lt | |
| 91 o<> {ox} {oy} lt tl | tri≈ ¬a b ¬c = ¬a tl | |
| 92 o<> {ox} {oy} lt tl | tri> ¬a ¬b c = ¬a tl | |
| 23 | 93 |
| 221 | 94 osuc-< : { x y : Ordinal } → y o< osuc x → x o< y → ⊥ |
| 95 osuc-< {x} {y} y<ox x<y with osuc-≡< y<ox | |
| 96 osuc-< {x} {y} y<ox x<y | case1 refl = o<¬≡ refl x<y | |
| 97 osuc-< {x} {y} y<ox x<y | case2 y<x = o<> x<y y<x | |
| 180 | 98 |
| 221 | 99 osucc : {ox oy : Ordinal } → oy o< ox → osuc oy o< osuc ox |
| 100 ---- y < osuc y < x < osuc x | |
| 101 ---- y < osuc y = x < osuc x | |
| 102 ---- y < osuc y > x < osuc x -> y = x ∨ x < y → ⊥ | |
| 103 osucc {ox} {oy} oy<ox with trio< (osuc oy) ox | |
| 104 osucc {ox} {oy} oy<ox | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 105 osucc {ox} {oy} oy<ox | tri≈ ¬a refl ¬c = <-osuc | |
| 106 osucc {ox} {oy} oy<ox | tri> ¬a ¬b c with osuc-≡< c | |
| 107 osucc {ox} {oy} oy<ox | tri> ¬a ¬b c | case1 eq = ⊥-elim (o<¬≡ (sym eq) oy<ox) | |
| 108 osucc {ox} {oy} oy<ox | tri> ¬a ¬b c | case2 lt = ⊥-elim (o<> lt oy<ox) | |
| 109 | |
| 338 | 110 osucprev : {ox oy : Ordinal } → osuc oy o< osuc ox → oy o< ox |
| 111 osucprev {ox} {oy} oy<ox with trio< oy ox | |
| 112 osucprev {ox} {oy} oy<ox | tri< a ¬b ¬c = a | |
| 113 osucprev {ox} {oy} oy<ox | tri≈ ¬a b ¬c = ⊥-elim (o<¬≡ (cong (λ k → osuc k) b) oy<ox ) | |
| 114 osucprev {ox} {oy} oy<ox | tri> ¬a ¬b c = ⊥-elim (o<> (osucc c) oy<ox ) | |
| 115 | |
| 221 | 116 open _∧_ |
| 84 | 117 |
| 221 | 118 osuc2 : ( x y : Ordinal ) → ( osuc x o< osuc (osuc y )) ⇔ (x o< osuc y) |
| 119 proj2 (osuc2 x y) lt = osucc lt | |
| 120 proj1 (osuc2 x y) ox<ooy with osuc-≡< ox<ooy | |
| 121 proj1 (osuc2 x y) ox<ooy | case1 ox=oy = o<-subst <-osuc refl ox=oy | |
| 122 proj1 (osuc2 x y) ox<ooy | case2 ox<oy = ordtrans <-osuc ox<oy | |
| 129 | 123 |
| 221 | 124 _o≤_ : Ordinal → Ordinal → Set n |
| 326 | 125 a o≤ b = a o< osuc b -- (a ≡ b) ∨ ( a o< b ) |
| 221 | 126 |
| 129 | 127 |
| 221 | 128 xo<ab : {oa ob : Ordinal } → ( {ox : Ordinal } → ox o< oa → ox o< ob ) → oa o< osuc ob |
| 129 xo<ab {oa} {ob} a→b with trio< oa ob | |
| 130 xo<ab {oa} {ob} a→b | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 131 xo<ab {oa} {ob} a→b | tri≈ ¬a refl ¬c = <-osuc | |
| 132 xo<ab {oa} {ob} a→b | tri> ¬a ¬b c = ⊥-elim ( o<¬≡ refl (a→b c ) ) | |
| 88 | 133 |
| 221 | 134 maxα : Ordinal → Ordinal → Ordinal |
| 135 maxα x y with trio< x y | |
| 136 maxα x y | tri< a ¬b ¬c = y | |
| 137 maxα x y | tri> ¬a ¬b c = x | |
| 138 maxα x y | tri≈ ¬a refl ¬c = x | |
| 84 | 139 |
| 308 | 140 omin : Ordinal → Ordinal → Ordinal |
| 141 omin x y with trio< x y | |
| 142 omin x y | tri< a ¬b ¬c = x | |
| 143 omin x y | tri> ¬a ¬b c = y | |
| 144 omin x y | tri≈ ¬a refl ¬c = x | |
| 88 | 145 |
| 308 | 146 min1 : {x y z : Ordinal } → z o< x → z o< y → z o< omin x y |
| 221 | 147 min1 {x} {y} {z} z<x z<y with trio< x y |
| 148 min1 {x} {y} {z} z<x z<y | tri< a ¬b ¬c = z<x | |
| 149 min1 {x} {y} {z} z<x z<y | tri≈ ¬a refl ¬c = z<x | |
| 150 min1 {x} {y} {z} z<x z<y | tri> ¬a ¬b c = z<y | |
| 84 | 151 |
| 221 | 152 -- |
| 153 -- max ( osuc x , osuc y ) | |
| 154 -- | |
| 155 | |
| 156 omax : ( x y : Ordinal ) → Ordinal | |
| 157 omax x y with trio< x y | |
| 158 omax x y | tri< a ¬b ¬c = osuc y | |
| 159 omax x y | tri> ¬a ¬b c = osuc x | |
| 160 omax x y | tri≈ ¬a refl ¬c = osuc x | |
| 86 | 161 |
| 221 | 162 omax< : ( x y : Ordinal ) → x o< y → osuc y ≡ omax x y |
| 163 omax< x y lt with trio< x y | |
| 164 omax< x y lt | tri< a ¬b ¬c = refl | |
| 165 omax< x y lt | tri≈ ¬a b ¬c = ⊥-elim (¬a lt ) | |
| 166 omax< x y lt | tri> ¬a ¬b c = ⊥-elim (¬a lt ) | |
| 86 | 167 |
| 221 | 168 omax≡ : ( x y : Ordinal ) → x ≡ y → osuc y ≡ omax x y |
| 169 omax≡ x y eq with trio< x y | |
| 170 omax≡ x y eq | tri< a ¬b ¬c = ⊥-elim (¬b eq ) | |
| 171 omax≡ x y eq | tri≈ ¬a refl ¬c = refl | |
| 172 omax≡ x y eq | tri> ¬a ¬b c = ⊥-elim (¬b eq ) | |
| 91 | 173 |
| 221 | 174 omax-x : ( x y : Ordinal ) → x o< omax x y |
| 175 omax-x x y with trio< x y | |
| 176 omax-x x y | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 177 omax-x x y | tri> ¬a ¬b c = <-osuc | |
| 178 omax-x x y | tri≈ ¬a refl ¬c = <-osuc | |
| 16 | 179 |
| 221 | 180 omax-y : ( x y : Ordinal ) → y o< omax x y |
| 181 omax-y x y with trio< x y | |
| 182 omax-y x y | tri< a ¬b ¬c = <-osuc | |
| 183 omax-y x y | tri> ¬a ¬b c = ordtrans c <-osuc | |
| 184 omax-y x y | tri≈ ¬a refl ¬c = <-osuc | |
| 185 | |
| 186 omxx : ( x : Ordinal ) → omax x x ≡ osuc x | |
| 187 omxx x with trio< x x | |
| 188 omxx x | tri< a ¬b ¬c = ⊥-elim (¬b refl ) | |
| 189 omxx x | tri> ¬a ¬b c = ⊥-elim (¬b refl ) | |
| 190 omxx x | tri≈ ¬a refl ¬c = refl | |
| 191 | |
| 192 omxxx : ( x : Ordinal ) → omax x (omax x x ) ≡ osuc (osuc x) | |
| 193 omxxx x = trans ( cong ( λ k → omax x k ) (omxx x )) (sym ( omax< x (osuc x) <-osuc )) | |
| 194 | |
| 195 open _∧_ | |
| 16 | 196 |
| 326 | 197 o≤-refl : { i j : Ordinal } → i ≡ j → i o≤ j |
| 198 o≤-refl {i} {j} eq = subst (λ k → i o< osuc k ) eq <-osuc | |
| 221 | 199 OrdTrans : Transitive _o≤_ |
| 326 | 200 OrdTrans a≤b b≤c with osuc-≡< a≤b | osuc-≡< b≤c |
| 201 OrdTrans a≤b b≤c | case1 refl | case1 refl = <-osuc | |
| 202 OrdTrans a≤b b≤c | case1 refl | case2 a≤c = ordtrans a≤c <-osuc | |
| 203 OrdTrans a≤b b≤c | case2 a≤c | case1 refl = ordtrans a≤c <-osuc | |
| 204 OrdTrans a≤b b≤c | case2 a<b | case2 b<c = ordtrans (ordtrans a<b b<c) <-osuc | |
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205 |
| 221 | 206 OrdPreorder : Preorder n n n |
| 207 OrdPreorder = record { Carrier = Ordinal | |
| 208 ; _≈_ = _≡_ | |
| 209 ; _∼_ = _o≤_ | |
| 210 ; isPreorder = record { | |
| 211 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
| 326 | 212 ; reflexive = o≤-refl |
| 221 | 213 ; trans = OrdTrans |
| 214 } | |
| 215 } | |
| 165 | 216 |
| 258 | 217 FExists : {m l : Level} → ( ψ : Ordinal → Set m ) |
| 221 | 218 → {p : Set l} ( P : { y : Ordinal } → ψ y → ¬ p ) |
| 219 → (exists : ¬ (∀ y → ¬ ( ψ y ) )) | |
| 220 → ¬ p | |
| 258 | 221 FExists {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p ) |
| 221 | 222 |
| 339 | 223 next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z |
| 224 next< {x} {y} {z} x<nz y<nx with trio< y (next z) | |
| 225 next< {x} {y} {z} x<nz y<nx | tri< a ¬b ¬c = a | |
| 226 next< {x} {y} {z} x<nz y<nx | tri≈ ¬a b ¬c = ⊥-elim ((proj2 (proj2 next-limit)) (next z) x<nz (subst (λ k → k o< next x) b y<nx) | |
| 227 (λ w nz=ow → o<¬≡ nz=ow (subst₂ (λ j k → j o< k ) (sym nz=ow) nz=ow (proj1 (proj2 next-limit) w (subst (λ k → w o< k ) (sym nz=ow) <-osuc) )))) | |
| 228 next< {x} {y} {z} x<nz y<nx | tri> ¬a ¬b c = ⊥-elim (proj2 (proj2 next-limit) (next z) x<nz (ordtrans c y<nx ) | |
| 229 (λ w nz=ow → o<¬≡ (sym nz=ow) (proj1 (proj2 next-limit) _ (subst (λ k → w o< k ) (sym nz=ow) <-osuc )))) | |
| 230 | |
| 342 | 231 osuc< : {x y : Ordinal} → osuc x ≡ y → x o< y |
| 232 osuc< {x} {y} refl = <-osuc | |
| 233 | |
| 340 | 234 nexto=n : {x y : Ordinal} → x o< next (osuc y) → x o< next y |
| 235 nexto=n {x} {y} x<noy = next< (proj1 (proj2 next-limit) _ (proj1 next-limit)) x<noy | |
| 236 | |
| 341 | 237 nexto≡ : {x : Ordinal} → next x ≡ next (osuc x) |
| 238 nexto≡ {x} with trio< (next x) (next (osuc x) ) | |
| 342 | 239 -- next x o< next (osuc x ) -> osuc x o< next x o< next (osuc x) -> next x ≡ osuc z -> z o o< next x -> osuc z o< next x -> next x o< next x |
| 240 nexto≡ {x} | tri< a ¬b ¬c = ⊥-elim ((proj2 (proj2 next-limit)) _ (proj1 (proj2 next-limit) _ (proj1 next-limit) ) a | |
| 241 (λ z eq → o<¬≡ (sym eq) ((proj1 (proj2 next-limit)) _ (osuc< (sym eq))))) | |
| 341 | 242 nexto≡ {x} | tri≈ ¬a b ¬c = b |
| 342 | 243 -- next (osuc x) o< next x -> osuc x o< next (osuc x) o< next x -> next (osuc x) ≡ osuc z -> z o o< next (osuc x) ... |
| 341 | 244 nexto≡ {x} | tri> ¬a ¬b c = ⊥-elim ((proj2 (proj2 next-limit)) _ (ordtrans <-osuc (proj1 next-limit)) c |
| 342 | 245 (λ z eq → o<¬≡ (sym eq) ((proj1 (proj2 next-limit)) _ (osuc< (sym eq))))) |
| 341 | 246 |
| 309 | 247 record OrdinalSubset (maxordinal : Ordinal) : Set (suc n) where |
| 248 field | |
| 249 os→ : (x : Ordinal) → x o< maxordinal → Ordinal | |
| 250 os← : Ordinal → Ordinal | |
| 251 os←limit : (x : Ordinal) → os← x o< maxordinal | |
| 252 os-iso← : {x : Ordinal} → os→ (os← x) (os←limit x) ≡ x | |
| 253 os-iso→ : {x : Ordinal} → (lt : x o< maxordinal ) → os← (os→ x lt) ≡ x | |
| 254 |