Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/generic-filter.agda @ 1266:a27f28dbed87
...
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 21 Mar 2023 11:19:15 +0900 |
| parents | 48d37da98331 |
| children | 0d278b809c01 |
| rev | line source |
|---|---|
| 1200 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 1244 | 2 import Level |
| 431 | 3 open import Ordinals |
| 1242 | 4 module generic-filter {n : Level.Level } (O : Ordinals {n}) where |
| 431 | 5 |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
6 -- import filter |
| 431 | 7 open import zf |
| 8 open import logic | |
| 9 -- open import partfunc {n} O | |
| 1244 | 10 import OD |
| 431 | 11 |
| 1244 | 12 open import Relation.Nullary |
| 13 open import Relation.Binary | |
| 14 open import Data.Empty | |
| 431 | 15 open import Relation.Binary |
| 16 open import Relation.Binary.Core | |
| 17 open import Relation.Binary.PropositionalEquality | |
| 1244 | 18 open import Data.Nat |
| 19 import BAlgebra | |
| 431 | 20 |
| 1124 | 21 open BAlgebra O |
| 431 | 22 |
| 23 open inOrdinal O | |
| 24 open OD O | |
| 25 open OD.OD | |
| 26 open ODAxiom odAxiom | |
| 27 import OrdUtil | |
| 28 import ODUtil | |
| 29 open Ordinals.Ordinals O | |
| 30 open Ordinals.IsOrdinals isOrdinal | |
| 31 open Ordinals.IsNext isNext | |
| 32 open OrdUtil O | |
| 33 open ODUtil O | |
| 34 | |
| 35 | |
| 36 import ODC | |
| 37 | |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
38 -- open filter O |
| 431 | 39 |
| 40 open _∧_ | |
| 41 open _∨_ | |
| 42 open Bool | |
| 43 | |
| 44 | |
| 45 open HOD | |
| 46 | |
| 47 ------- | |
| 48 -- the set of finite partial functions from ω to 2 | |
| 49 -- | |
| 50 -- | |
| 51 | |
| 52 open import Data.List hiding (filter) | |
| 1244 | 53 open import Data.Maybe |
| 431 | 54 |
| 1218 | 55 open import ZProduct O |
| 431 | 56 |
| 1242 | 57 record CountableModel : Set (Level.suc (Level.suc n)) where |
| 431 | 58 field |
| 461 | 59 ctl-M : HOD |
| 1242 | 60 ctl→ : ℕ → Ordinal |
| 1244 | 61 ctl<M : (x : ℕ) → odef (ctl-M) (ctl→ x) |
| 1242 | 62 ctl← : (x : Ordinal )→ odef (ctl-M ) x → ℕ |
| 1244 | 63 ctl-iso→ : { x : Ordinal } → (lt : odef (ctl-M) x ) → ctl→ (ctl← x lt ) ≡ x |
| 1248 | 64 TC : {x y : Ordinal} → odef ctl-M x → odef (* x) y → odef ctl-M y |
| 1249 | 65 is-model : (x : HOD) → & x o< & ctl-M → ctl-M ∋ (x ∩ ctl-M) |
| 1174 | 66 -- we have no otherway round |
| 1242 | 67 -- ctl-iso← : { x : ℕ } → ctl← (ctl→ x ) (ctl<M x) ≡ x |
| 446 | 68 -- |
| 69 -- almmost universe | |
| 70 -- find-p contains ∃ x : Ordinal → x o< & M → ∀ r ∈ M → ∈ Ord x | |
| 1244 | 71 -- |
| 436 | 72 |
| 1244 | 73 -- we expect P ∈ * ctl-M ∧ G ⊆ L ⊆ Power P , ¬ G ∈ * ctl-M, |
| 434 | 74 |
| 1244 | 75 open CountableModel |
| 431 | 76 |
| 77 ---- | |
| 78 -- a(n) ∈ M | |
| 1239 | 79 -- ∃ q ∈ L ⊆ Power P → q ∈ a(n) ∧ p(n) ⊆ q |
| 431 | 80 -- |
| 1242 | 81 PGHOD : (i : ℕ) (L : HOD) (C : CountableModel ) → (p : Ordinal) → HOD |
| 457 | 82 PGHOD i L C p = record { od = record { def = λ x → |
| 1239 | 83 odef L x ∧ odef (* (ctl→ C i)) x ∧ ( (y : Ordinal ) → odef (* p) y → odef (* x) y ) } |
| 1244 | 84 ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L (proj1 lt) } |
| 431 | 85 |
| 86 --- | |
| 1239 | 87 -- p(n+1) = if ({q | q ∈ a(n) ∧ p(n) ⊆ q)} != ∅ then q otherwise p(n) |
| 1244 | 88 -- |
| 1242 | 89 find-p : (L : HOD ) (C : CountableModel ) (i : ℕ) → (x : Ordinal) → Ordinal |
| 90 find-p L C zero x = x | |
| 91 find-p L C (suc i) x with is-o∅ ( & ( PGHOD i L C (find-p L C i x)) ) | |
| 457 | 92 ... | yes y = find-p L C i x |
| 93 ... | no not = & (ODC.minimal O ( PGHOD i L C (find-p L C i x)) (λ eq → not (=od∅→≡o∅ eq))) -- axiom of choice | |
| 431 | 94 |
| 95 --- | |
| 1239 | 96 -- G = { r ∈ L ⊆ Power P | ∃ n → r ⊆ p(n) } |
| 431 | 97 -- |
| 457 | 98 record PDN (L p : HOD ) (C : CountableModel ) (x : Ordinal) : Set n where |
| 431 | 99 field |
| 1242 | 100 gr : ℕ |
| 1244 | 101 pn<gr : (y : Ordinal) → odef (* x) y → odef (* (find-p L C gr (& p))) y |
| 457 | 102 x∈PP : odef L x |
| 431 | 103 |
| 104 open PDN | |
| 105 | |
| 106 --- | |
| 107 -- G as a HOD | |
| 108 -- | |
| 457 | 109 PDHOD : (L p : HOD ) (C : CountableModel ) → HOD |
| 110 PDHOD L p C = record { od = record { def = λ x → PDN L p C x } | |
| 1244 | 111 ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L {y} (PDN.x∈PP lt) } |
| 431 | 112 |
| 113 open PDN | |
| 114 | |
| 115 P∅ : {P : HOD} → odef (Power P) o∅ | |
| 116 P∅ {P} = subst (λ k → odef (Power P) k ) ord-od∅ (lemma o∅ o∅≡od∅) where | |
| 117 lemma : (x : Ordinal ) → * x ≡ od∅ → odef (Power P) (& od∅) | |
| 118 lemma x eq = power← P od∅ (λ {x} lt → ⊥-elim (¬x<0 lt )) | |
| 119 x<y→∋ : {x y : Ordinal} → odef (* x) y → * x ∋ * y | |
| 120 x<y→∋ {x} {y} lt = subst (λ k → odef (* x) k ) (sym &iso) lt | |
| 121 | |
| 1242 | 122 gf05 : {a b : HOD} {x : Ordinal } → (odef (a ∪ b) x ) → ¬ odef a x → ¬ odef b x → ⊥ |
| 1244 | 123 gf05 {a} {b} {x} (case1 ax) nax nbx = nax ax |
| 1242 | 124 gf05 {a} {b} {x} (case2 bx) nax nbx = nbx bx |
| 125 | |
| 446 | 126 open import Data.Nat.Properties |
| 1266 | 127 open import nat hiding ( exp ) |
| 433 | 128 |
| 1242 | 129 p-monotonic1 : (L p : HOD ) (C : CountableModel ) → {n : ℕ} → (* (find-p L C n (& p))) ⊆ (* (find-p L C (suc n) (& p))) |
| 1096 | 130 p-monotonic1 L p C {n} {x} with is-o∅ (& (PGHOD n L C (find-p L C n (& p)))) |
| 131 ... | yes y = refl-⊆ {* (find-p L C n (& p))} | |
| 1242 | 132 ... | no not = λ lt → proj2 (proj2 fmin∈PGHOD) _ lt where |
| 447 | 133 fmin : HOD |
| 457 | 134 fmin = ODC.minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) |
| 135 fmin∈PGHOD : PGHOD n L C (find-p L C n (& p)) ∋ fmin | |
| 136 fmin∈PGHOD = ODC.x∋minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) | |
| 438 | 137 |
| 1242 | 138 p-monotonic : (L p : HOD ) (C : CountableModel ) → {n m : ℕ} → n ≤ m → (* (find-p L C n (& p))) ⊆ (* (find-p L C m (& p))) |
| 139 p-monotonic L p C {zero} {zero} n≤m = refl-⊆ {* (find-p L C zero (& p))} | |
| 140 p-monotonic L p C {zero} {suc m} z≤n lt = p-monotonic1 L p C {m} (p-monotonic L p C {zero} {m} z≤n lt ) | |
| 141 p-monotonic L p C {suc n} {suc m} (s≤s n≤m) with <-cmp n m | |
| 1244 | 142 ... | tri< a ¬b ¬c = λ lt → p-monotonic1 L p C {m} (p-monotonic L p C {suc n} {m} a lt) |
| 1096 | 143 ... | tri≈ ¬a refl ¬c = λ x → x |
| 446 | 144 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c ) |
| 438 | 145 |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
146 record Expansion (p : HOD) (dense : HOD) : Set (Level.suc n) where |
| 1254 | 147 field |
|
1259
4993a95d3fc8
... filter definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1258
diff
changeset
|
148 exp : HOD |
| 1265 | 149 D∋exp : dense ∋ exp |
| 150 p⊆exp : p ⊆ exp | |
| 1254 | 151 |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
152 record Dense (L : HOD ) : Set (Level.suc n) where |
| 1239 | 153 field |
| 154 dense : HOD | |
| 155 d⊆P : dense ⊆ L | |
| 1265 | 156 has-exp : {p : HOD} → (Lp : L ∋ p) → Expansion p dense |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
157 |
|
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
158 record Exp2 (I : HOD) ( p q : HOD ) : Set (Level.suc n) where |
| 1265 | 159 field |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
160 exp : HOD |
| 1265 | 161 I∋exp : I ∋ exp |
| 162 p⊆exp : p ⊆ exp | |
| 163 q⊆exp : q ⊆ exp | |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
164 |
|
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
165 record ⊆-Ideal {L P : HOD } (LP : L ⊆ Power P) : Set (Level.suc n) where |
|
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
166 field |
| 1265 | 167 ideal : HOD |
| 168 i⊆L : ideal ⊆ L | |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
169 ideal1 : { p q : HOD } → L ∋ q → ideal ∋ p → q ⊆ p → ideal ∋ q |
|
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
170 exp : { p q : HOD } → ideal ∋ p → ideal ∋ q → Exp2 ideal p q |
| 1239 | 171 |
| 1256 | 172 record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (Level.suc n) where |
| 1255 | 173 field |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
174 genf : ⊆-Ideal {L} {P} LP |
|
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
175 generic : (D : Dense L ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ ⊆-Ideal.ideal genf ) ≡ od∅ ) |
| 1255 | 176 |
| 1266 | 177 ---- |
| 178 -- Generic Filter on L ⊆ Power P from HOD's Countable Ordinal (G ⊆ Power P ≡ G i.e. ℕ → P → Set ) | |
| 179 -- | |
| 180 -- p 0 ≡ ∅ | |
| 181 -- p (suc n) = if ∃ q ∈ M ∧ p n ⊆ q → q (by axiom of choice) ( q = * ( ctl→ n ) ) | |
| 182 --- else p n | |
| 183 | |
| 1244 | 184 P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 |
| 1241 | 185 → (C : CountableModel ) → GenericFilter {L} {P} LP ( ctl-M C ) |
| 1255 | 186 P-GenericFilter P L p0 L⊆PP Lp0 C = record { |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
187 genf = record { ideal = PDHOD L p0 C ; i⊆L = x∈PP ; ideal1 = ideal1 ; exp = λ ip iq → exp1 ip iq } |
| 1256 | 188 ; generic = fdense |
| 431 | 189 } where |
| 1256 | 190 ideal1 : {p q : HOD} → L ∋ q → PDHOD L p0 C ∋ p → q ⊆ p → PDHOD L p0 C ∋ q |
| 1265 | 191 ideal1 {p} {q} Lq record { gr = gr ; pn<gr = pn<gr ; x∈PP = x∈PP } q⊆p = |
| 1256 | 192 record { gr = gr ; pn<gr = λ y qy → pn<gr y (gf00 qy) ; x∈PP = Lq } where |
| 1265 | 193 gf00 : {y : Ordinal } → odef (* (& q)) y → odef (* (& p)) y |
| 1256 | 194 gf00 {y} qy = subst (λ k → odef k y ) (sym *iso) (q⊆p (subst (λ k → odef k y) *iso qy )) |
| 1266 | 195 Lan : (i : ℕ ) → odef L (find-p L C i (& p0)) |
| 196 Lan zero = Lp0 | |
| 197 Lan (suc i) with is-o∅ ( & ( PGHOD i L C (find-p L C i (& p0))) ) | |
| 198 ... | yes y = Lan i | |
| 1265 | 199 ... | no not = proj1 ( ODC.x∋minimal O ( PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq))) |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
200 exp1 : {p q : HOD} → (ip : PDHOD L p0 C ∋ p) → (ip : PDHOD L p0 C ∋ q) → Exp2 (PDHOD L p0 C) p q |
| 1265 | 201 exp1 {p} {q} record { gr = pgr ; pn<gr = ppn ; x∈PP = PPp } |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
202 record { gr = qgr ; pn<gr = qpn ; x∈PP = PPq } = gf01 where |
| 1265 | 203 Pp = record { gr = pgr ; pn<gr = ppn ; x∈PP = PPp } |
| 204 Pq = record { gr = qgr ; pn<gr = qpn ; x∈PP = PPq } | |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
205 gf17 : {q : HOD} → (Pq : PDHOD L p0 C ∋ q ) → PDHOD L p0 C ∋ * (find-p L C (gr Pq) (& p0)) |
| 1265 | 206 gf17 {q} Pq = record { gr = PDN.gr Pq ; pn<gr = λ y qq → subst (λ k → odef (* k) y) &iso qq |
| 1266 | 207 ; x∈PP = subst (λ k → odef L k ) (sym &iso) (Lan (PDN.gr Pq)) } |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
208 gf01 : Exp2 (PDHOD L p0 C) p q |
| 1256 | 209 gf01 with <-cmp pgr qgr |
| 1265 | 210 ... | tri< a ¬b ¬c = record { exp = * (find-p L C (gr Pq) (& p0)) ; I∋exp = gf17 Pq ; p⊆exp = λ px → gf15 _ px |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
211 ; q⊆exp = λ {x} qx → qpn _ (subst (λ k → odef k x) (sym *iso) qx) } where |
| 1256 | 212 gf16 : gr Pp ≤ gr Pq |
| 213 gf16 = <to≤ a | |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
214 gf15 : (y : Ordinal) → odef p y → odef (* (find-p L C (gr Pq) (& p0))) y |
|
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
215 gf15 y xpy = p-monotonic L p0 C gf16 (PDN.pn<gr Pp y (subst (λ k → odef k y) (sym *iso) xpy) ) |
|
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
216 ... | tri≈ ¬a refl ¬c = record { exp = * (find-p L C (gr Pq) (& p0)) ; I∋exp = gf17 Pq |
| 1265 | 217 ; p⊆exp = λ {x} px → ppn _ (subst (λ k → odef k x) (sym *iso) px) |
| 218 ; q⊆exp = λ {x} qx → qpn _ (subst (λ k → odef k x) (sym *iso) qx) } | |
| 219 ... | tri> ¬a ¬b c = record { exp = * (find-p L C (gr Pp) (& p0)) ; I∋exp = gf17 Pp ; q⊆exp = λ qx → gf15 _ qx | |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
220 ; p⊆exp = λ {x} px → ppn _ (subst (λ k → odef k x) (sym *iso) px) } where |
| 1256 | 221 gf16 : gr Pq ≤ gr Pp |
| 222 gf16 = <to≤ c | |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
223 gf15 : (y : Ordinal) → odef q y → odef (* (find-p L C (gr Pp) (& p0))) y |
|
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
224 gf15 y xqy = p-monotonic L p0 C gf16 (PDN.pn<gr Pq y (subst (λ k → odef k y) (sym *iso) xqy) ) |
|
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
225 fdense : (D : Dense L ) → (ctl-M C ) ∋ Dense.dense D → ¬ (Dense.dense D ∩ (PDHOD L p0 C)) ≡ od∅ |
| 1256 | 226 fdense D MD eq0 = ⊥-elim ( ∅< {Dense.dense D ∩ PDHOD L p0 C} fd01 (≡od∅→=od∅ eq0 )) where |
| 227 open Dense | |
| 228 open Expansion | |
| 229 an : ℕ | |
| 1265 | 230 an = ctl← C (& (dense D)) MD |
| 1256 | 231 pn : Ordinal |
| 232 pn = find-p L C an (& p0) | |
| 233 pn+1 : Ordinal | |
| 234 pn+1 = find-p L C (suc an) (& p0) | |
| 1265 | 235 d=an : dense D ≡ * (ctl→ C an) |
| 1256 | 236 d=an = begin dense D ≡⟨ sym *iso ⟩ |
| 237 * ( & (dense D)) ≡⟨ cong (*) (sym (ctl-iso→ C MD )) ⟩ | |
| 238 * (ctl→ C an) ∎ where open ≡-Reasoning | |
| 239 fd07 : odef (dense D) pn+1 | |
| 240 fd07 with is-o∅ ( & ( PGHOD an L C (find-p L C an (& p0))) ) | |
| 241 ... | yes y = ⊥-elim ( ¬x<0 ( _==_.eq→ fd10 fd21 ) ) where | |
| 242 L∋pn : L ∋ * (find-p L C an (& p0)) | |
| 1266 | 243 L∋pn = subst (λ k → odef L k) (sym &iso) (Lan an ) |
|
1259
4993a95d3fc8
... filter definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1258
diff
changeset
|
244 ex = has-exp D L∋pn |
|
4993a95d3fc8
... filter definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1258
diff
changeset
|
245 L∋df : L ∋ ( exp ex ) |
|
4993a95d3fc8
... filter definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1258
diff
changeset
|
246 L∋df = (d⊆P D) (D∋exp ex) |
|
4993a95d3fc8
... filter definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1258
diff
changeset
|
247 pn∋df : (* (ctl→ C an)) ∋ ( exp ex) |
|
4993a95d3fc8
... filter definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1258
diff
changeset
|
248 pn∋df = subst (λ k → odef k (& ( exp ex))) d=an (D∋exp ex ) |
|
4993a95d3fc8
... filter definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1258
diff
changeset
|
249 pn⊆df : (y : Ordinal) → odef (* (find-p L C an (& p0))) y → odef (* (& (exp ex))) y |
|
4993a95d3fc8
... filter definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1258
diff
changeset
|
250 pn⊆df y py = subst (λ k → odef k y ) (sym *iso) (p⊆exp ex py) |
|
4993a95d3fc8
... filter definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1258
diff
changeset
|
251 fd21 : odef (PGHOD an L C (find-p L C an (& p0)) ) (& (exp ex)) |
| 1256 | 252 fd21 = ⟪ L∋df , ⟪ pn∋df , pn⊆df ⟫ ⟫ |
| 253 fd10 : PGHOD an L C (find-p L C an (& p0)) =h= od∅ | |
| 254 fd10 = ≡o∅→=od∅ y | |
| 255 ... | no not = fd27 where | |
| 256 fd29 = ODC.minimal O ( PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
| 257 fd28 : PGHOD an L C (find-p L C an (& p0)) ∋ fd29 | |
| 258 fd28 = ODC.x∋minimal O (PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
| 259 fd27 : odef (dense D) (& fd29) | |
| 1265 | 260 fd27 = subst (λ k → odef k (& fd29)) (sym d=an) (proj1 (proj2 fd28)) |
| 1256 | 261 fd03 : odef (PDHOD L p0 C) pn+1 |
| 1266 | 262 fd03 = record { gr = suc an ; pn<gr = λ y lt → lt ; x∈PP = Lan (suc an)} |
| 1256 | 263 fd01 : (dense D ∩ PDHOD L p0 C) ∋ (* pn+1) |
| 1265 | 264 fd01 = ⟪ subst (λ k → odef (dense D) k ) (sym &iso) fd07 , subst (λ k → odef (PDHOD L p0 C) k) (sym &iso) fd03 ⟫ |
| 448 | 265 |
| 431 | 266 open GenericFilter |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
267 -- open Filter |
| 431 | 268 |
| 1266 | 269 record Incompatible (L p : HOD ) (L∋a : L ∋ p ) : Set (Level.suc (Level.suc n)) where |
| 431 | 270 field |
| 1245 | 271 q r : HOD |
| 272 Lq : L ∋ q | |
| 273 Lr : L ∋ r | |
| 1265 | 274 p⊆q : p ⊆ q |
| 275 p⊆r : p ⊆ r | |
| 1255 | 276 ¬compat : (s : HOD) → L ∋ s → ¬ ( (q ⊆ s) ∧ (r ⊆ s) ) |
| 431 | 277 |
| 1246 | 278 lemma232 : (P L p0 : HOD ) → (LPP : L ⊆ Power P) → (Lp0 : L ∋ p0 ) |
| 1265 | 279 → (C : CountableModel ) |
| 1249 | 280 → ctl-M C ∋ L |
| 1266 | 281 → ( {p : HOD} → (Lp : L ∋ p ) → Incompatible L p Lp ) |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
282 → ¬ ( ctl-M C ∋ ⊆-Ideal.ideal (genf ( P-GenericFilter P L p0 LPP Lp0 C ))) |
| 1266 | 283 lemma232 P L p0 LPP Lp0 C ML NC MF = ¬G∩D=0 D∩G=0 where |
| 1265 | 284 PG = P-GenericFilter P L p0 LPP Lp0 C |
| 1249 | 285 GF = genf PG |
| 1266 | 286 G = ⊆-Ideal.ideal (genf PG) |
| 1245 | 287 M = ctl-M C |
| 1265 | 288 D : HOD |
| 1266 | 289 D = L ＼ G |
|
1261
2fccbe475cf7
lemma232 (Generic filter is not an element of M)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1260
diff
changeset
|
290 D<M : & D o< & (ctl-M C) |
|
2fccbe475cf7
lemma232 (Generic filter is not an element of M)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1260
diff
changeset
|
291 D<M = ordtrans≤-< (⊆→o≤ proj1 ) (odef< ML) |
| 1248 | 292 M∋DM : M ∋ (D ∩ M ) |
|
1261
2fccbe475cf7
lemma232 (Generic filter is not an element of M)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1260
diff
changeset
|
293 M∋DM = is-model C D D<M |
| 1266 | 294 -- G⊆M : G ⊆ M |
|
1261
2fccbe475cf7
lemma232 (Generic filter is not an element of M)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1260
diff
changeset
|
295 -- G⊆M {x} rx = TC C ML (subst (λ k → odef k x) (sym *iso) (⊆-Ideal.i⊆L GF rx)) |
|
2fccbe475cf7
lemma232 (Generic filter is not an element of M)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1260
diff
changeset
|
296 -- D⊆M : D ⊆ M |
|
2fccbe475cf7
lemma232 (Generic filter is not an element of M)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1260
diff
changeset
|
297 -- D⊆M {x} dx = TC C ML (subst (λ k → odef k x) (sym *iso) (proj1 dx)) |
| 1265 | 298 D=D∩M : D ≡ D ∩ M |
|
1261
2fccbe475cf7
lemma232 (Generic filter is not an element of M)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1260
diff
changeset
|
299 D=D∩M = ==→o≡ record { eq→ = ddm00 ; eq← = proj1 } where |
|
2fccbe475cf7
lemma232 (Generic filter is not an element of M)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1260
diff
changeset
|
300 ddm00 : {x : Ordinal} → odef D x → odef (D ∩ M) x |
|
2fccbe475cf7
lemma232 (Generic filter is not an element of M)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1260
diff
changeset
|
301 ddm00 {x} ⟪ Lx , ¬Gx ⟫ = ⟪ ⟪ Lx , ¬Gx ⟫ , TC C ML (subst (λ k → odef k x) (sym *iso) Lx ) ⟫ |
| 1265 | 302 M∋D : M ∋ D |
|
1261
2fccbe475cf7
lemma232 (Generic filter is not an element of M)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1260
diff
changeset
|
303 M∋D = subst (λ k → M ∋ k ) (sym D=D∩M) M∋DM |
| 1246 | 304 D⊆PP : D ⊆ Power P |
| 1265 | 305 D⊆PP {x} ⟪ Lx , ngx ⟫ = LPP Lx |
| 306 DD : Dense L | |
|
1259
4993a95d3fc8
... filter definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1258
diff
changeset
|
307 DD = record { dense = D ; d⊆P = proj1 ; has-exp = exp } where |
| 1265 | 308 exp : {p : HOD} → (Lp : L ∋ p) → Expansion p D |
| 1254 | 309 exp {p} Lp = exp1 where |
| 310 q : HOD | |
| 1266 | 311 q = Incompatible.q (NC Lp) |
| 1254 | 312 r : HOD |
| 1266 | 313 r = Incompatible.r (NC Lp) |
| 1255 | 314 Lq : L ∋ q |
| 1266 | 315 Lq = Incompatible.Lq (NC Lp) |
|
1259
4993a95d3fc8
... filter definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1258
diff
changeset
|
316 Lr : L ∋ r |
| 1266 | 317 Lr = Incompatible.Lr (NC Lp) |
| 1265 | 318 exp1 : Expansion p D |
| 1266 | 319 exp1 with ODC.p∨¬p O (G ∋ q) |
| 320 ... | case2 ngq = record { exp = q ; D∋exp = ⟪ Lq , ngq ⟫ ; p⊆exp = Incompatible.p⊆q (NC Lp)} | |
| 321 ... | case1 gq with ODC.p∨¬p O (G ∋ r) | |
| 322 ... | case2 ngr = record { exp = r ; D∋exp = ⟪ Lr , ngr ⟫ ; p⊆exp = Incompatible.p⊆r (NC Lp)} | |
| 323 ... | case1 gr = ⊥-elim ( Incompatible.¬compat (NC Lp) ex2 Le ⟪ q⊆ex2 , r⊆ex2 ⟫ ) where | |
|
1260
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
324 ex2 = Exp2.exp (⊆-Ideal.exp GF gq gr) |
|
8a8052df5254
generic filter modification
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1259
diff
changeset
|
325 Le = ⊆-Ideal.i⊆L GF (Exp2.I∋exp (⊆-Ideal.exp GF gq gr)) |
| 1265 | 326 q⊆ex2 = Exp2.p⊆exp (⊆-Ideal.exp GF gq gr) |
| 327 r⊆ex2 = Exp2.q⊆exp (⊆-Ideal.exp GF gq gr) | |
| 1266 | 328 ¬G∩D=0 : ¬ ( (D ∩ G ) =h= od∅ ) |
| 329 ¬G∩D=0 eq = generic PG DD M∋D (==→o≡ eq) | |
| 330 D∩G=0 : (D ∩ G ) =h= od∅ -- because D = L ＼ G | |
| 331 D∩G=0 = record { eq→ = λ {x} G∩D → ⊥-elim( proj2 (proj1 G∩D) (proj2 G∩D)) | |
| 332 ; eq← = λ lt → ⊥-elim (¬x<0 lt) } | |
| 431 | 333 |
| 334 -- | |
| 1174 | 335 -- P-Generic Filter defines a countable model D ⊂ C from P |
| 336 -- | |
| 337 | |
| 338 -- | |
| 339 -- in D, we have V ≠ L | |
| 340 -- | |
| 341 | |
| 342 -- | |
| 431 | 343 -- val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > } |
| 344 -- | |
| 436 | 345 |
| 1263 | 346 val< : {x y p : Ordinal} → odef (* x) ( & < * y , * p > ) → y o< x |
| 1264 | 347 val< {x} {y} {p} xyp = osucprev ( begin |
| 348 osuc y ≤⟨ osucc (odef< (subst (λ k → odef (* y , * y) k) &iso (v00 _ _ ) )) ⟩ | |
| 349 & (* y , * y) <⟨ c<→o< (v01 _ _) ⟩ | |
| 350 & < * y , * p > <⟨ odef< xyp ⟩ | |
| 351 & (* x) ≡⟨ &iso ⟩ | |
| 1265 | 352 x ∎ ) where |
| 1264 | 353 v00 : (x y : HOD) → odef (x , y) (& x) |
| 354 v00 _ _ = case1 refl | |
| 1265 | 355 v01 : (x y : HOD) → < x , y > ∋ (x , x) |
| 1264 | 356 v01 _ _ = case1 refl |
| 357 open o≤-Reasoning O | |
| 436 | 358 |
| 1263 | 359 record valS (G : HOD) (x z : Ordinal) (val : (y : Ordinal) → y o< x → HOD): Set n where |
| 436 | 360 field |
| 1263 | 361 y p : Ordinal |
| 362 G∋p : odef G p | |
| 363 is-val : odef (* x) ( & < * y , * p > ) | |
| 1264 | 364 z=valy : z ≡ & (val y (val< is-val)) |
| 365 z<x : z o< x | |
| 436 | 366 |
| 1266 | 367 val : (x : HOD) → (G : HOD) → HOD |
| 437 | 368 val x G = TransFinite {λ x → HOD } ind (& x) where |
| 1266 | 369 ind : (x : Ordinal) → (valy : (y : Ordinal) → y o< x → HOD) → HOD |
| 370 ind x valy = record { od = record { def = λ z → valS G x z valy } ; odmax = x ; <odmax = v02 } where | |
| 371 v02 : {z : Ordinal} → valS G x z valy → z o< x | |
| 1264 | 372 v02 {z} lt = valS.z<x lt |
| 1266 | 373 |
| 374 TCS : (G : HOD) → Set (Level.suc n) | |
| 375 TCS G = {x y : HOD} → G ∋ x → x ∋ y → G ∋ y | |
| 376 | |
| 377 val∋→∋p : {G : HOD} → (TCS G) → {x y : HOD} → {p : HOD} → G ∋ p → ( val x G ∋ val y G ) → x ∋ < y , p > | |
| 378 val∋→∋p = ? | |
| 379 | |
| 380 p∋→val∋ : {G : HOD} → (TCS G) → {x y : HOD} → {p : HOD} → G ∋ p → x ∋ < y , p > → ( val x G ∋ val y G ) | |
| 381 p∋→val∋ {G} TG {x} {y} {p} Gp = subst (λ k → k ∋ < y , p > → ( val k G ∋ val y G ) ) *iso ( | |
| 382 TransFinite0 {λ x → ( * x ) ∋ < y , p > → ( val (* x) G ∋ val y G )} ? (& x) ) where | |
| 383 pind : (x : Ordinal) → ((z : Ordinal) → z o< x → ( * z ) ∋ < y , p > → val (* z) G ∋ val y G) → | |
| 384 (* x ) ∋ < y , p > → val (* x) G ∋ val y G | |
| 385 pind x prev xyp = ? | |
| 386 -- subst (λ k → odef ( TransFinite (generic-filter.ind (* x) G) (& (* x))) (& (val y G))) ? ( | |
| 387 -- record { y = ? ; p = ? ; G∋p = ? ; is-val = ? ; z=valy = ? ; z<x = ? } ) | |
| 388 | |
| 389 | |
| 390 |