Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate filter.agda @ 379:7b6592f0851a
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 21 Jul 2020 02:19:07 +0900 |
| parents | 8cade5f660bd |
| children | 0a116938a564 |
| rev | line source |
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| 190 | 1 open import Level |
| 236 | 2 open import Ordinals |
| 3 module filter {n : Level } (O : Ordinals {n}) where | |
| 4 | |
| 190 | 5 open import zf |
| 236 | 6 open import logic |
| 7 import OD | |
| 193 | 8 |
| 363 | 9 open import Relation.Nullary |
| 10 open import Relation.Binary | |
| 11 open import Data.Empty | |
| 190 | 12 open import Relation.Binary |
| 13 open import Relation.Binary.Core | |
| 363 | 14 open import Relation.Binary.PropositionalEquality |
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15 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
| 363 | 16 import BAlgbra |
| 293 | 17 |
| 18 open BAlgbra O | |
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19 |
| 236 | 20 open inOrdinal O |
| 21 open OD O | |
| 22 open OD.OD | |
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23 open ODAxiom odAxiom |
| 190 | 24 |
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25 import ODC |
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26 |
| 236 | 27 open _∧_ |
| 28 open _∨_ | |
| 29 open Bool | |
| 30 | |
| 295 | 31 -- Kunen p.76 and p.53, we use ⊆ |
| 329 | 32 record Filter ( L : HOD ) : Set (suc n) where |
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33 field |
| 329 | 34 filter : HOD |
| 290 | 35 f⊆PL : filter ⊆ Power L |
| 329 | 36 filter1 : { p q : HOD } → q ⊆ L → filter ∋ p → p ⊆ q → filter ∋ q |
| 37 filter2 : { p q : HOD } → filter ∋ p → filter ∋ q → filter ∋ (p ∩ q) | |
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38 |
| 292 | 39 open Filter |
| 40 | |
| 329 | 41 record prime-filter { L : HOD } (P : Filter L) : Set (suc (suc n)) where |
| 295 | 42 field |
| 43 proper : ¬ (filter P ∋ od∅) | |
| 329 | 44 prime : {p q : HOD } → filter P ∋ (p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q ) |
| 292 | 45 |
| 329 | 46 record ultra-filter { L : HOD } (P : Filter L) : Set (suc (suc n)) where |
| 295 | 47 field |
| 48 proper : ¬ (filter P ∋ od∅) | |
| 329 | 49 ultra : {p : HOD } → p ⊆ L → ( filter P ∋ p ) ∨ ( filter P ∋ ( L ＼ p) ) |
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50 |
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51 open _⊆_ |
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52 |
| 329 | 53 trans-⊆ : { A B C : HOD} → A ⊆ B → B ⊆ C → A ⊆ C |
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54 trans-⊆ A⊆B B⊆C = record { incl = λ x → incl B⊆C (incl A⊆B x) } |
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55 |
| 374 | 56 refl-⊆ : {A : HOD} → A ⊆ A |
| 57 refl-⊆ {A} = record { incl = λ x → x } | |
| 58 | |
| 329 | 59 power→⊆ : ( A t : HOD) → Power A ∋ t → t ⊆ A |
| 331 | 60 power→⊆ A t PA∋t = record { incl = λ {x} t∋x → ODC.double-neg-eilm O (t1 t∋x) } where |
| 329 | 61 t1 : {x : HOD } → t ∋ x → ¬ ¬ (A ∋ x) |
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62 t1 = zf.IsZF.power→ isZF A t PA∋t |
| 292 | 63 |
| 329 | 64 ∈-filter : {L p : HOD} → (P : Filter L ) → filter P ∋ p → p ⊆ L |
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65 ∈-filter {L} {p} P lt = power→⊆ L p ( incl (f⊆PL P) lt ) |
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66 |
| 329 | 67 ∪-lemma1 : {L p q : HOD } → (p ∪ q) ⊆ L → p ⊆ L |
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68 ∪-lemma1 {L} {p} {q} lt = record { incl = λ {x} p∋x → incl lt (case1 p∋x) } |
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69 |
| 329 | 70 ∪-lemma2 : {L p q : HOD } → (p ∪ q) ⊆ L → q ⊆ L |
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71 ∪-lemma2 {L} {p} {q} lt = record { incl = λ {x} p∋x → incl lt (case2 p∋x) } |
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72 |
| 329 | 73 q∩q⊆q : {p q : HOD } → (q ∩ p) ⊆ q |
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74 q∩q⊆q = record { incl = λ lt → proj1 lt } |
| 265 | 75 |
| 331 | 76 open HOD |
| 77 | |
| 295 | 78 ----- |
| 79 -- | |
| 80 -- ultra filter is prime | |
| 81 -- | |
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82 |
| 329 | 83 filter-lemma1 : {L : HOD} → (P : Filter L) → ∀ {p q : HOD } → ultra-filter P → prime-filter P |
| 295 | 84 filter-lemma1 {L} P u = record { |
| 85 proper = ultra-filter.proper u | |
| 86 ; prime = lemma3 | |
| 87 } where | |
| 329 | 88 lemma3 : {p q : HOD} → filter P ∋ (p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q ) |
| 295 | 89 lemma3 {p} {q} lt with ultra-filter.ultra u (∪-lemma1 (∈-filter P lt) ) |
| 90 ... | case1 p∈P = case1 p∈P | |
| 91 ... | case2 ¬p∈P = case2 (filter1 P {q ∩ (L ＼ p)} (∪-lemma2 (∈-filter P lt)) lemma7 lemma8) where | |
| 331 | 92 lemma5 : ((p ∪ q ) ∩ (L ＼ p)) =h= (q ∩ (L ＼ p)) |
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93 lemma5 = record { eq→ = λ {x} lt → record { proj1 = lemma4 x lt ; proj2 = proj2 lt } |
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94 ; eq← = λ {x} lt → record { proj1 = case2 (proj1 lt) ; proj2 = proj2 lt } |
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95 } where |
| 331 | 96 lemma4 : (x : Ordinal ) → odef ((p ∪ q) ∩ (L ＼ p)) x → odef q x |
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97 lemma4 x lt with proj1 lt |
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98 lemma4 x lt | case1 px = ⊥-elim ( proj2 (proj2 lt) px ) |
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99 lemma4 x lt | case2 qx = qx |
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100 lemma6 : filter P ∋ ((p ∪ q ) ∩ (L ＼ p)) |
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101 lemma6 = filter2 P lt ¬p∈P |
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102 lemma7 : filter P ∋ (q ∩ (L ＼ p)) |
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103 lemma7 = subst (λ k → filter P ∋ k ) (==→o≡ lemma5 ) lemma6 |
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104 lemma8 : (q ∩ (L ＼ p)) ⊆ q |
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105 lemma8 = q∩q⊆q |
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106 |
| 295 | 107 ----- |
| 108 -- | |
| 109 -- if Filter contains L, prime filter is ultra | |
| 110 -- | |
| 111 | |
| 329 | 112 filter-lemma2 : {L : HOD} → (P : Filter L) → filter P ∋ L → prime-filter P → ultra-filter P |
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113 filter-lemma2 {L} P f∋L prime = record { |
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114 proper = prime-filter.proper prime |
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115 ; ultra = λ {p} p⊆L → prime-filter.prime prime (lemma p p⊆L) |
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116 } where |
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117 open _==_ |
| 331 | 118 p+1-p=1 : {p : HOD} → p ⊆ L → L =h= (p ∪ (L ＼ p)) |
| 119 eq→ (p+1-p=1 {p} p⊆L) {x} lt with ODC.decp O (odef p x) | |
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120 eq→ (p+1-p=1 {p} p⊆L) {x} lt | yes p∋x = case1 p∋x |
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121 eq→ (p+1-p=1 {p} p⊆L) {x} lt | no ¬p = case2 (record { proj1 = lt ; proj2 = ¬p }) |
| 331 | 122 eq← (p+1-p=1 {p} p⊆L) {x} ( case1 p∋x ) = subst (λ k → odef L k ) diso (incl p⊆L ( subst (λ k → odef p k) (sym diso) p∋x )) |
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123 eq← (p+1-p=1 {p} p⊆L) {x} ( case2 ¬p ) = proj1 ¬p |
| 329 | 124 lemma : (p : HOD) → p ⊆ L → filter P ∋ (p ∪ (L ＼ p)) |
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125 lemma p p⊆L = subst (λ k → filter P ∋ k ) (==→o≡ (p+1-p=1 p⊆L)) f∋L |
| 293 | 126 |
| 329 | 127 record Dense (P : HOD ) : Set (suc n) where |
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128 field |
| 329 | 129 dense : HOD |
| 379 | 130 d⊆P : dense ⊆ Power P |
| 329 | 131 dense-f : HOD → HOD |
| 368 | 132 dense-d : { p : HOD} → p ⊆ P → dense ∋ dense-f p |
| 133 dense-p : { p : HOD} → p ⊆ P → p ⊆ (dense-f p) | |
| 266 | 134 |
| 329 | 135 record Ideal ( L : HOD ) : Set (suc n) where |
| 293 | 136 field |
| 329 | 137 ideal : HOD |
| 293 | 138 i⊆PL : ideal ⊆ Power L |
| 329 | 139 ideal1 : { p q : HOD } → q ⊆ L → ideal ∋ p → q ⊆ p → ideal ∋ q |
| 140 ideal2 : { p q : HOD } → ideal ∋ p → ideal ∋ q → ideal ∋ (p ∪ q) | |
| 293 | 141 |
| 142 open Ideal | |
| 143 | |
| 329 | 144 proper-ideal : {L : HOD} → (P : Ideal L ) → {p : HOD} → Set n |
| 293 | 145 proper-ideal {L} P {p} = ideal P ∋ od∅ |
| 146 | |
| 329 | 147 prime-ideal : {L : HOD} → Ideal L → ∀ {p q : HOD } → Set n |
| 293 | 148 prime-ideal {L} P {p} {q} = ideal P ∋ ( p ∩ q) → ( ideal P ∋ p ) ∨ ( ideal P ∋ q ) |
| 149 | |
| 374 | 150 record F-Filter {n : Level} (L : Set n) (PL : (L → Set n) → Set n) ( _⊆_ : L → L → Set n) (_∩_ : L → L → L ) : Set (suc n) where |
| 370 | 151 field |
| 374 | 152 filter : L → Set n |
| 371 | 153 f⊆P : PL filter |
| 374 | 154 filter1 : { p q : L } → PL (λ x → q ⊆ x ) → filter p → p ⊆ q → filter q |
| 371 | 155 filter2 : { p q : L } → filter p → filter q → filter (p ∩ q) |
| 156 | |
| 374 | 157 Filter-is-F : {L : HOD} → (f : Filter L ) → F-Filter HOD (λ p → (x : HOD) → p x → x ⊆ L ) _⊆_ _∩_ |
| 158 Filter-is-F {L} f = record { | |
| 159 filter = λ x → Lift (suc n) ((filter f) ∋ x) | |
| 160 ; f⊆P = λ x f∋x → power→⊆ _ _ (incl ( f⊆PL f ) (lower f∋x )) | |
| 161 ; filter1 = λ {p} {q} q⊆L f∋p p⊆q → lift ( filter1 f (q⊆L q refl-⊆) (lower f∋p) p⊆q) | |
| 162 ; filter2 = λ {p} {q} f∋p f∋q → lift ( filter2 f (lower f∋p) (lower f∋q)) | |
| 163 } | |
| 164 | |
| 379 | 165 record F-Dense {n : Level} (L : Set n) (PL : (L → Set n) → Set n) ( _⊆_ : L → L → Set n) (_∩_ : L → L → L ) : Set (suc n) where |
| 166 field | |
| 167 dense : L → Set n | |
| 168 d⊆P : PL dense | |
| 169 dense-f : L → L | |
| 170 dense-d : { p : L} → PL (λ x → p ⊆ x ) → dense ( dense-f p ) | |
| 171 dense-p : { p : L} → PL (λ x → p ⊆ x ) → p ⊆ (dense-f p) | |
| 172 | |
| 173 Dense-is-F : {L : HOD} → (f : Dense L ) → F-Dense HOD (λ p → (x : HOD) → p x → x ⊆ L ) _⊆_ _∩_ | |
| 174 Dense-is-F {L} f = record { | |
| 175 dense = λ x → Lift (suc n) ((dense f) ∋ x) | |
| 176 ; d⊆P = λ x f∋x → power→⊆ _ _ (incl ( d⊆P f ) (lower f∋x )) | |
| 177 ; dense-f = λ x → dense-f f x | |
| 178 ; dense-d = λ {p} d → lift ( dense-d f (d p refl-⊆ ) ) | |
| 179 ; dense-p = λ {p} d → dense-p f (d p refl-⊆) | |
| 180 } where open Dense | |
| 181 | |
| 182 ------- | |
| 183 -- the set of finite partial functions from ω to 2 | |
| 184 -- | |
| 185 -- | |
| 186 | |
| 187 data Two : Set n where | |
| 188 i0 : Two | |
| 189 i1 : Two | |
| 190 | |
| 191 data Three : Set n where | |
| 192 j0 : Three | |
| 193 j1 : Three | |
| 194 j2 : Three | |
| 195 | |
| 196 open import Data.List hiding (map) | |
| 197 | |
| 198 import OPair | |
| 199 open OPair O | |
| 200 | |
| 201 record PFunc : Set (suc n) where | |
| 202 field | |
| 203 dom : Nat → Set n | |
| 204 map : (x : Nat ) → dom x → Two | |
| 205 meq : {x : Nat } → { p q : dom x } → map x p ≡ map x q | |
| 206 | |
| 207 open PFunc | |
| 208 | |
| 209 data Findp : List Three → (x : Nat) → Set n where | |
| 210 v0 : {n : List Three} → Findp ( j0 ∷ n ) Zero | |
| 211 v1 : {n : List Three} → Findp ( j1 ∷ n ) Zero | |
| 212 vn : {n : List Three} {d : Three} → {x : Nat} → Findp n x → Findp (d ∷ n) (Suc x) | |
| 213 | |
| 214 FPFunc→PFunc : List Three → PFunc | |
| 215 FPFunc→PFunc fp = record { dom = λ x → findp fp x ; map = λ x p → find fp x p ; meq = λ {x} {p} {q} → feq fp } where | |
| 216 findp : List Three → (x : Nat) → Set n | |
| 217 findp n x = Findp n x | |
| 218 find : (n : List Three ) → (x : Nat) → Findp n x → Two | |
| 219 find (j0 ∷ _) 0 v0 = i0 | |
| 220 find (j1 ∷ _) 0 v1 = i1 | |
| 221 find (d ∷ n) (Suc x) (vn {n} {d} {x} p) = find n x p | |
| 222 feq : (n : List Three) → {x : Nat} {p q : Findp n x } → find n x p ≡ find n x q | |
| 223 feq n {0} {v0} {v0} = refl | |
| 224 feq n {0} {v1} {v1} = refl | |
| 225 feq [] {Suc x} {()} | |
| 226 feq (_ ∷ n) {Suc x} {vn p} {vn q} = subst₂ (λ j k → j ≡ k ) {!!} {!!} (feq n {x} {p} {q}) | |
| 227 | |
| 228 record _f⊆_ (f g : PFunc) : Set (suc n) where | |
| 229 field | |
| 230 extend : {x : Nat} → (fr : dom f x ) → dom g x | |
| 231 feq : {x : Nat} → {fr : dom f x } → map f x fr ≡ map g x (extend fr) | |
| 232 | |
| 233 open _f⊆_ | |
| 234 | |
| 374 | 235 min = Data.Nat._⊓_ |
| 236 -- m≤m⊔n = Data.Nat._⊔_ | |
| 237 open import Data.Nat.Properties | |
| 238 | |
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239 _f∩_ : (f g : PFunc) → PFunc |
| 379 | 240 f f∩ g = record { dom = λ x → (dom f x ) ∧ (dom g x ) ∧ ((fr : dom f x ) → (gr : dom g x ) → map f x fr ≡ map g x gr) |
| 241 ; map = λ x p → map f x (proj1 p) ; meq = meq f } | |
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243 _↑_ : (Nat → Two) → Nat → PFunc |
| 379 | 244 f ↑ i = record { dom = λ x → Lift n (x ≤ i) ; map = λ x _ → f x ; meq = λ {x} {p} {q} → refl } |
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246 record Gf (f : Nat → Two) (p : PFunc ) : Set (suc n) where |
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247 field |
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248 gn : Nat |
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249 f<n : (f ↑ gn) f⊆ p |
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250 |
| 379 | 251 record FiniteF (p : PFunc ) : Set (suc n) where |
| 252 field | |
| 253 f-max : Nat | |
| 254 f-func : Nat → Two | |
| 255 f-p⊆f : p f⊆ (f-func ↑ f-max) | |
| 256 f-f⊆p : (f-func ↑ f-max) f⊆ p | |
| 257 | |
| 258 open FiniteF | |
| 259 | |
| 260 Dense-Gf : F-Dense PFunc (λ x → Lift (suc n) One) _f⊆_ _f∩_ | |
| 261 Dense-Gf = record { | |
| 262 dense = λ x → FiniteF x | |
| 263 ; d⊆P = lift OneObj | |
| 264 ; dense-f = λ x → record { dom = {!!} ; map = {!!} } | |
| 265 ; dense-d = λ {p} d → {!!} | |
| 266 ; dense-p = λ {p} d → {!!} | |
| 267 } | |
| 268 | |
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269 open Gf |
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270 |
| 374 | 271 GF : (Nat → Two ) → F-Filter PFunc (λ x → Lift (suc n) One ) _f⊆_ _f∩_ |
| 371 | 272 GF f = record { |
| 372 | 273 filter = λ p → Gf f p |
| 374 | 274 ; f⊆P = lift OneObj |
| 275 ; filter1 = λ {p} {q} _ fp p⊆q → record { gn = gn fp ; f<n = f1 fp p⊆q } | |
| 276 ; filter2 = λ {p} {q} fp fq → record { gn = min (gn fp) (gn fq) ; f<n = f2 fp fq } | |
| 372 | 277 } where |
| 374 | 278 f1 : {p q : PFunc } → (fp : Gf f p ) → ( p⊆q : p f⊆ q ) → (f ↑ gn fp) f⊆ q |
| 379 | 279 f1 {p} {q} fp p⊆q = record { extend = λ {x} x<g → extend p⊆q (extend (f<n fp ) x<g) ; feq = λ {x} {fr} → lemma {x} {fr} } where |
| 280 lemma : {x : Nat} {fr : Lift n (x ≤ gn fp)} → map (f ↑ gn fp) x fr ≡ map q x (extend p⊆q (extend (f<n fp) fr)) | |
| 281 lemma {x} {fr} = begin | |
| 282 map (f ↑ gn fp) x fr | |
| 283 ≡⟨ feq (f<n fp ) ⟩ | |
| 284 map p x (extend (f<n fp) fr) | |
| 285 ≡⟨ feq p⊆q ⟩ | |
| 286 map q x (extend p⊆q (extend (f<n fp) fr)) | |
| 287 ∎ where open ≡-Reasoning | |
| 374 | 288 f2 : {p q : PFunc } → (fp : Gf f p ) → (fq : Gf f q ) → (f ↑ (min (gn fp) (gn fq))) f⊆ (p f∩ q) |
| 379 | 289 f2 {p} {q} fp fq = record { extend = λ {x} x<g → lemma2 x<g ; feq = λ {x} {fr} → lemma3 fr } where |
| 290 fmin = f ↑ (min (gn fp) (gn fq)) | |
| 291 lemma1 : {x : Nat} → ( x<g : Lift n (x ≤ min (gn fp) (gn fq)) ) → (fr : dom p x) (gr : dom q x) → map p x fr ≡ map q x gr | |
| 292 lemma1 {x} x<g fr gr = begin | |
| 293 map p x fr | |
| 294 ≡⟨ meq p ⟩ | |
| 295 map p x (extend (f<n fp) (lift ( ≤-trans (lower x<g) (m⊓n≤m _ _)))) | |
| 296 ≡⟨ sym (feq (f<n fp)) ⟩ | |
| 297 map (f ↑ (min (gn fp) (gn fq))) x x<g | |
| 298 ≡⟨ feq (f<n fq) ⟩ | |
| 299 map q x (extend (f<n fq) (lift ( ≤-trans (lower x<g) (m⊓n≤n _ _)))) | |
| 300 ≡⟨ meq q ⟩ | |
| 301 map q x gr | |
| 302 ∎ where open ≡-Reasoning | |
| 303 lemma2 : {x : Nat} → ( x<g : Lift n (x ≤ min (gn fp) (gn fq)) ) → dom (p f∩ q) x | |
| 304 lemma2 x<g = record { proj1 = extend (f<n fp ) (lift ( ≤-trans (lower x<g) (m⊓n≤m _ _))) ; | |
| 305 proj2 = record {proj1 = extend (f<n fq ) (lift ( ≤-trans (lower x<g) (m⊓n≤n _ _))) ; proj2 = lemma1 x<g }} | |
| 306 f∩→⊆ : (p q : PFunc ) → (p f∩ q ) f⊆ q | |
| 307 f∩→⊆ p q = record { | |
| 308 extend = λ {x} pq → proj1 (proj2 pq) | |
| 309 ; feq = λ {x} {fr} → (proj2 (proj2 fr)) (proj1 fr) (proj1 (proj2 fr)) | |
| 310 } | |
| 311 lemma3 : {x : Nat} → ( fr : Lift n (x ≤ min (gn fp) (gn fq))) → map (f ↑ min (gn fp) (gn fq)) x fr ≡ map (p f∩ q) x (lemma2 fr) | |
| 312 lemma3 {x} fr = | |
| 313 map (f ↑ min (gn fp) (gn fq)) x fr | |
| 314 ≡⟨ feq (f<n fq) ⟩ | |
| 315 map q x (extend (f<n fq) ( lift (≤-trans (lower fr) (m⊓n≤n _ _)) )) | |
| 316 ≡⟨ sym (feq (f∩→⊆ p q ) {x} {lemma2 fr} ) ⟩ | |
| 317 map (p f∩ q) x (lemma2 fr) | |
| 318 ∎ where open ≡-Reasoning | |
| 319 | |
| 370 | 320 |
| 363 | 321 ODSuc : (y : HOD) → infinite ∋ y → HOD |
| 322 ODSuc y lt = Union (y , (y , y)) | |
| 323 | |
| 366 | 324 data Hω2 : (i : Nat) ( x : Ordinal ) → Set n where |
| 325 hφ : Hω2 0 o∅ | |
| 326 h0 : {i : Nat} {x : Ordinal } → Hω2 i x → | |
| 327 Hω2 (Suc i) (od→ord (Union ((< nat→ω i , nat→ω 0 >) , ord→od x ))) | |
| 328 h1 : {i : Nat} {x : Ordinal } → Hω2 i x → | |
| 329 Hω2 (Suc i) (od→ord (Union ((< nat→ω i , nat→ω 1 >) , ord→od x ))) | |
| 330 he : {i : Nat} {x : Ordinal } → Hω2 i x → | |
| 331 Hω2 (Suc i) x | |
| 332 | |
| 333 record Hω2r (x : Ordinal) : Set n where | |
| 334 field | |
| 335 count : Nat | |
| 336 hω2 : Hω2 count x | |
| 337 | |
| 338 open Hω2r | |
| 363 | 339 |
| 340 HODω2 : HOD | |
| 366 | 341 HODω2 = record { od = record { def = λ x → Hω2r x } ; odmax = next o∅ ; <odmax = odmax0 } where |
| 365 | 342 ω<next : {y : Ordinal} → infinite-d y → y o< next o∅ |
| 343 ω<next = ω<next-o∅ ho< | |
| 366 | 344 lemma : {i j : Nat} {x : Ordinal } → od→ord (Union (< nat→ω i , nat→ω j > , ord→od x)) o< next x |
| 345 lemma = {!!} | |
| 346 odmax0 : {y : Ordinal} → Hω2r y → y o< next o∅ | |
| 347 odmax0 {y} r with hω2 r | |
| 348 ... | hφ = x<nx | |
| 349 ... | h0 {i} {x} t = next< (odmax0 record { count = i ; hω2 = t }) (lemma {i} {0} {x}) | |
| 350 ... | h1 {i} {x} t = next< (odmax0 record { count = i ; hω2 = t }) (lemma {i} {1} {x}) | |
| 351 ... | he {i} {x} t = next< (odmax0 record { count = i ; hω2 = t }) x<nx | |
| 363 | 352 |
| 370 | 353 ω→2 : HOD |
| 379 | 354 ω→2 = Replace (Power infinite) (λ p → Replace infinite (λ x → < x , repl p x > )) where |
| 370 | 355 repl : HOD → HOD → HOD |
| 356 repl p x with ODC.∋-p O p x | |
| 357 ... | yes _ = nat→ω 1 | |
| 358 ... | no _ = nat→ω 0 | |
| 368 | 359 |
| 370 | 360 record _↑n (f : HOD) (ω→2∋f : ω→2 ∋ f ) : Set n where |
| 361 -- field | |
| 362 -- n : HOD | |
| 363 -- ? : Select f (λ x f∋x → ω→nat (π1 f∋x) < ω→nat n | |
| 364 | |
| 372 | 365 -- Gf : {f : HOD} → ω→2 ∋ f → HOD |
| 366 -- Gf {f} lt = Select HODω2 (λ x H∋x → {!!} ) | |
| 368 | 367 |
| 368 G : (Nat → Two) → Filter HODω2 | |
| 369 G f = record { | |
| 365 | 370 filter = {!!} |
| 371 ; f⊆PL = {!!} | |
| 372 ; filter1 = {!!} | |
| 373 ; filter2 = {!!} | |
| 374 } where | |
| 375 filter0 : HOD | |
| 376 filter0 = {!!} | |
| 377 f⊆PL1 : filter0 ⊆ Power HODω2 | |
| 378 f⊆PL1 = {!!} | |
| 379 filter11 : { p q : HOD } → q ⊆ HODω2 → filter0 ∋ p → p ⊆ q → filter0 ∋ q | |
| 380 filter11 = {!!} | |
| 381 filter12 : { p q : HOD } → filter0 ∋ p → filter0 ∋ q → filter0 ∋ (p ∩ q) | |
| 382 filter12 = {!!} | |
| 383 | |
| 363 | 384 -- the set of finite partial functions from ω to 2 |
| 385 | |
| 386 Hω2f : Set (suc n) | |
| 387 Hω2f = (Nat → Set n) → Two | |
| 388 | |
| 389 Hω2f→Hω2 : Hω2f → HOD | |
| 390 Hω2f→Hω2 p = record { od = record { def = λ x → (p {!!} ≡ i0 ) ∨ (p {!!} ≡ i1 )}; odmax = {!!} ; <odmax = {!!} } | |
| 391 | |
| 392 |