Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/generic-filter.agda @ 1260:8a8052df5254
generic filter modification
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 18 Mar 2023 11:30:42 +0900 |
| parents | 4993a95d3fc8 |
| children | 2fccbe475cf7 |
| 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 |
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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 | |
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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 ---- | |
| 1242 | 116 -- Generic Filter on Power P for HOD's Countable Ordinal (G ⊆ Power P ≡ G i.e. ℕ → P → Set ) |
| 431 | 117 -- |
| 118 -- p 0 ≡ ∅ | |
| 434 | 119 -- p (suc n) = if ∃ q ∈ M ∧ p n ⊆ q → q (by axiom of choice) ( q = * ( ctl→ n ) ) |
| 431 | 120 --- else p n |
| 121 | |
| 122 P∅ : {P : HOD} → odef (Power P) o∅ | |
| 123 P∅ {P} = subst (λ k → odef (Power P) k ) ord-od∅ (lemma o∅ o∅≡od∅) where | |
| 124 lemma : (x : Ordinal ) → * x ≡ od∅ → odef (Power P) (& od∅) | |
| 125 lemma x eq = power← P od∅ (λ {x} lt → ⊥-elim (¬x<0 lt )) | |
| 126 x<y→∋ : {x y : Ordinal} → odef (* x) y → * x ∋ * y | |
| 127 x<y→∋ {x} {y} lt = subst (λ k → odef (* x) k ) (sym &iso) lt | |
| 128 | |
| 1242 | 129 gf05 : {a b : HOD} {x : Ordinal } → (odef (a ∪ b) x ) → ¬ odef a x → ¬ odef b x → ⊥ |
| 1244 | 130 gf05 {a} {b} {x} (case1 ax) nax nbx = nax ax |
| 1242 | 131 gf05 {a} {b} {x} (case2 bx) nax nbx = nbx bx |
| 132 | |
| 446 | 133 open import Data.Nat.Properties |
| 134 open import nat | |
| 433 | 135 |
| 1242 | 136 p-monotonic1 : (L p : HOD ) (C : CountableModel ) → {n : ℕ} → (* (find-p L C n (& p))) ⊆ (* (find-p L C (suc n) (& p))) |
| 1096 | 137 p-monotonic1 L p C {n} {x} with is-o∅ (& (PGHOD n L C (find-p L C n (& p)))) |
| 138 ... | yes y = refl-⊆ {* (find-p L C n (& p))} | |
| 1242 | 139 ... | no not = λ lt → proj2 (proj2 fmin∈PGHOD) _ lt where |
| 447 | 140 fmin : HOD |
| 457 | 141 fmin = ODC.minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) |
| 142 fmin∈PGHOD : PGHOD n L C (find-p L C n (& p)) ∋ fmin | |
| 143 fmin∈PGHOD = ODC.x∋minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) | |
| 438 | 144 |
| 1242 | 145 p-monotonic : (L p : HOD ) (C : CountableModel ) → {n m : ℕ} → n ≤ m → (* (find-p L C n (& p))) ⊆ (* (find-p L C m (& p))) |
| 146 p-monotonic L p C {zero} {zero} n≤m = refl-⊆ {* (find-p L C zero (& p))} | |
| 147 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 ) | |
| 148 p-monotonic L p C {suc n} {suc m} (s≤s n≤m) with <-cmp n m | |
| 1244 | 149 ... | tri< a ¬b ¬c = λ lt → p-monotonic1 L p C {m} (p-monotonic L p C {suc n} {m} a lt) |
| 1096 | 150 ... | tri≈ ¬a refl ¬c = λ x → x |
| 446 | 151 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c ) |
| 438 | 152 |
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153 record Expansion (p : HOD) (dense : HOD) : Set (Level.suc n) where |
| 1254 | 154 field |
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155 exp : HOD |
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156 D∋exp : dense ∋ exp |
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157 p⊆exp : p ⊆ exp |
| 1254 | 158 |
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159 record Dense (L : HOD ) : Set (Level.suc n) where |
| 1239 | 160 field |
| 161 dense : HOD | |
| 162 d⊆P : dense ⊆ L | |
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163 has-exp : {p : HOD} → (Lp : L ∋ p) → Expansion p dense |
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164 |
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165 record Exp2 (I : HOD) ( p q : HOD ) : Set (Level.suc n) where |
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166 field |
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167 exp : HOD |
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168 I∋exp : I ∋ exp |
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169 p⊆exp : p ⊆ exp |
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170 q⊆exp : q ⊆ exp |
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171 |
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172 record ⊆-Ideal {L P : HOD } (LP : L ⊆ Power P) : Set (Level.suc n) where |
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173 field |
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174 ideal : HOD |
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175 i⊆L : ideal ⊆ L |
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176 ideal1 : { p q : HOD } → L ∋ q → ideal ∋ p → q ⊆ p → ideal ∋ q |
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177 exp : { p q : HOD } → ideal ∋ p → ideal ∋ q → Exp2 ideal p q |
| 1239 | 178 |
| 1256 | 179 record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (Level.suc n) where |
| 1255 | 180 field |
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181 genf : ⊆-Ideal {L} {P} LP |
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182 generic : (D : Dense L ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ ⊆-Ideal.ideal genf ) ≡ od∅ ) |
| 1255 | 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 { |
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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 |
| 191 ideal1 {p} {q} Lq record { gr = gr ; pn<gr = pn<gr ; x∈PP = x∈PP } q⊆p = | |
| 192 record { gr = gr ; pn<gr = λ y qy → pn<gr y (gf00 qy) ; x∈PP = Lq } where | |
| 193 gf00 : {y : Ordinal } → odef (* (& q)) y → odef (* (& p)) y | |
| 194 gf00 {y} qy = subst (λ k → odef k y ) (sym *iso) (q⊆p (subst (λ k → odef k y) *iso qy )) | |
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195 Lfp : (i : ℕ ) → odef L (find-p L C i (& p0)) |
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196 Lfp zero = Lp0 |
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197 Lfp (suc i) with is-o∅ ( & ( PGHOD i L C (find-p L C i (& p0))) ) |
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198 ... | yes y = Lfp i |
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199 ... | no not = proj1 ( ODC.x∋minimal O ( PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq))) |
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200 exp1 : {p q : HOD} → (ip : PDHOD L p0 C ∋ p) → (ip : PDHOD L p0 C ∋ q) → Exp2 (PDHOD L p0 C) p q |
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201 exp1 {p} {q} record { gr = pgr ; pn<gr = ppn ; x∈PP = PPp } |
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202 record { gr = qgr ; pn<gr = qpn ; x∈PP = PPq } = gf01 where |
| 1256 | 203 Pp = record { gr = pgr ; pn<gr = ppn ; x∈PP = PPp } |
| 204 Pq = record { gr = qgr ; pn<gr = qpn ; x∈PP = PPq } | |
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205 gf17 : {q : HOD} → (Pq : PDHOD L p0 C ∋ q ) → PDHOD L p0 C ∋ * (find-p L C (gr Pq) (& p0)) |
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206 gf17 {q} Pq = record { gr = PDN.gr Pq ; pn<gr = λ y qq → subst (λ k → odef (* k) y) &iso qq |
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207 ; x∈PP = subst (λ k → odef L k ) (sym &iso) (Lfp (PDN.gr Pq)) } |
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208 gf01 : Exp2 (PDHOD L p0 C) p q |
| 1256 | 209 gf01 with <-cmp pgr qgr |
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210 ... | tri< a ¬b ¬c = record { exp = * (find-p L C (gr Pq) (& p0)) ; I∋exp = gf17 Pq ; p⊆exp = λ px → gf15 _ px |
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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 | |
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214 gf15 : (y : Ordinal) → odef p y → odef (* (find-p L C (gr Pq) (& p0))) y |
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215 gf15 y xpy = p-monotonic L p0 C gf16 (PDN.pn<gr Pp y (subst (λ k → odef k y) (sym *iso) xpy) ) |
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216 ... | tri≈ ¬a refl ¬c = record { exp = * (find-p L C (gr Pq) (& p0)) ; I∋exp = gf17 Pq |
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217 ; p⊆exp = λ {x} px → ppn _ (subst (λ k → odef k x) (sym *iso) px) |
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218 ; q⊆exp = λ {x} qx → qpn _ (subst (λ k → odef k x) (sym *iso) qx) } |
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219 ... | tri> ¬a ¬b c = record { exp = * (find-p L C (gr Pp) (& p0)) ; I∋exp = gf17 Pp ; q⊆exp = λ qx → gf15 _ qx |
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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 | |
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223 gf15 : (y : Ordinal) → odef q y → odef (* (find-p L C (gr Pp) (& p0))) y |
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224 gf15 y xqy = p-monotonic L p0 C gf16 (PDN.pn<gr Pq y (subst (λ k → odef k y) (sym *iso) xqy) ) |
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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 fd09 : (i : ℕ ) → odef L (find-p L C i (& p0)) | |
| 230 fd09 zero = Lp0 | |
| 231 fd09 (suc i) with is-o∅ ( & ( PGHOD i L C (find-p L C i (& p0))) ) | |
| 232 ... | yes _ = fd09 i | |
| 233 ... | no not = fd17 where | |
| 234 fd19 = ODC.minimal O ( PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
| 235 fd18 : PGHOD i L C (find-p L C i (& p0)) ∋ fd19 | |
| 236 fd18 = ODC.x∋minimal O (PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
| 237 fd17 : odef L ( & (ODC.minimal O ( PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq))) ) | |
| 238 fd17 = proj1 fd18 | |
| 239 an : ℕ | |
| 240 an = ctl← C (& (dense D)) MD | |
| 241 pn : Ordinal | |
| 242 pn = find-p L C an (& p0) | |
| 243 pn+1 : Ordinal | |
| 244 pn+1 = find-p L C (suc an) (& p0) | |
| 245 d=an : dense D ≡ * (ctl→ C an) | |
| 246 d=an = begin dense D ≡⟨ sym *iso ⟩ | |
| 247 * ( & (dense D)) ≡⟨ cong (*) (sym (ctl-iso→ C MD )) ⟩ | |
| 248 * (ctl→ C an) ∎ where open ≡-Reasoning | |
| 249 fd07 : odef (dense D) pn+1 | |
| 250 fd07 with is-o∅ ( & ( PGHOD an L C (find-p L C an (& p0))) ) | |
| 251 ... | yes y = ⊥-elim ( ¬x<0 ( _==_.eq→ fd10 fd21 ) ) where | |
| 252 L∋pn : L ∋ * (find-p L C an (& p0)) | |
| 253 L∋pn = subst (λ k → odef L k) (sym &iso) (fd09 an ) | |
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254 ex = has-exp D L∋pn |
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255 L∋df : L ∋ ( exp ex ) |
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256 L∋df = (d⊆P D) (D∋exp ex) |
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257 pn∋df : (* (ctl→ C an)) ∋ ( exp ex) |
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258 pn∋df = subst (λ k → odef k (& ( exp ex))) d=an (D∋exp ex ) |
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259 pn⊆df : (y : Ordinal) → odef (* (find-p L C an (& p0))) y → odef (* (& (exp ex))) y |
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260 pn⊆df y py = subst (λ k → odef k y ) (sym *iso) (p⊆exp ex py) |
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261 fd21 : odef (PGHOD an L C (find-p L C an (& p0)) ) (& (exp ex)) |
| 1256 | 262 fd21 = ⟪ L∋df , ⟪ pn∋df , pn⊆df ⟫ ⟫ |
| 263 fd10 : PGHOD an L C (find-p L C an (& p0)) =h= od∅ | |
| 264 fd10 = ≡o∅→=od∅ y | |
| 265 ... | no not = fd27 where | |
| 266 fd29 = ODC.minimal O ( PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
| 267 fd28 : PGHOD an L C (find-p L C an (& p0)) ∋ fd29 | |
| 268 fd28 = ODC.x∋minimal O (PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
| 269 fd27 : odef (dense D) (& fd29) | |
| 270 fd27 = subst (λ k → odef k (& fd29)) (sym d=an) (proj1 (proj2 fd28)) | |
| 271 fd03 : odef (PDHOD L p0 C) pn+1 | |
| 272 fd03 = record { gr = suc an ; pn<gr = λ y lt → lt ; x∈PP = fd09 (suc an)} | |
| 273 fd01 : (dense D ∩ PDHOD L p0 C) ∋ (* pn+1) | |
| 274 fd01 = ⟪ subst (λ k → odef (dense D) k ) (sym &iso) fd07 , subst (λ k → odef (PDHOD L p0 C) k) (sym &iso) fd03 ⟫ | |
| 448 | 275 |
| 431 | 276 open GenericFilter |
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277 -- open Filter |
| 431 | 278 |
| 1245 | 279 record NotCompatible (L p : HOD ) (L∋a : L ∋ p ) : Set (Level.suc (Level.suc n)) where |
| 431 | 280 field |
| 1245 | 281 q r : HOD |
| 282 Lq : L ∋ q | |
| 283 Lr : L ∋ r | |
| 284 p⊆q : p ⊆ q | |
| 285 p⊆r : p ⊆ r | |
| 1255 | 286 ¬compat : (s : HOD) → L ∋ s → ¬ ( (q ⊆ s) ∧ (r ⊆ s) ) |
| 431 | 287 |
| 1246 | 288 lemma232 : (P L p0 : HOD ) → (LPP : L ⊆ Power P) → (Lp0 : L ∋ p0 ) |
| 1245 | 289 → (C : CountableModel ) |
| 1249 | 290 → ctl-M C ∋ L |
| 1245 | 291 → ( {p : HOD} → (Lp : L ∋ p ) → NotCompatible L p Lp ) |
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292 → ¬ ( ctl-M C ∋ ⊆-Ideal.ideal (genf ( P-GenericFilter P L p0 LPP Lp0 C ))) |
| 1255 | 293 lemma232 P L p0 LPP Lp0 C ML NC MF = ¬rgf∩D=0 record { eq→ = λ {x} rgf∩D → ⊥-elim( proj2 (proj1 rgf∩D) (proj2 rgf∩D)) |
| 1245 | 294 ; eq← = λ lt → ⊥-elim (¬x<0 lt) } where |
| 1255 | 295 PG = P-GenericFilter P L p0 LPP Lp0 C |
| 1249 | 296 GF = genf PG |
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297 rgf = ⊆-Ideal.ideal (genf PG) |
| 1245 | 298 M = ctl-M C |
| 299 D : HOD | |
| 1248 | 300 D = L ＼ rgf |
| 301 M∋DM : M ∋ (D ∩ M ) | |
| 1249 | 302 M∋DM = is-model C D ? |
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303 -- M∋G : M ∋ rgf |
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304 -- M∋G = MF |
| 1254 | 305 M∋D : M ∋ D |
| 306 M∋D = ? | |
| 1246 | 307 D⊆PP : D ⊆ Power P |
| 308 D⊆PP {x} ⟪ Lx , ngx ⟫ = LPP Lx | |
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309 DD : Dense L |
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310 DD = record { dense = D ; d⊆P = proj1 ; has-exp = exp } where |
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311 exp : {p : HOD} → (Lp : L ∋ p) → Expansion p D |
| 1254 | 312 exp {p} Lp = exp1 where |
| 313 q : HOD | |
| 314 q = NotCompatible.q (NC Lp) | |
| 315 r : HOD | |
| 316 r = NotCompatible.r (NC Lp) | |
| 1255 | 317 Lq : L ∋ q |
| 318 Lq = NotCompatible.Lq (NC Lp) | |
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319 Lr : L ∋ r |
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320 Lr = NotCompatible.Lr (NC Lp) |
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321 exp1 : Expansion p D |
| 1254 | 322 exp1 with ODC.p∨¬p O (rgf ∋ q) |
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323 ... | case2 ngq = record { exp = q ; D∋exp = ⟪ Lq , ngq ⟫ ; p⊆exp = NotCompatible.p⊆q (NC Lp)} |
| 1254 | 324 ... | case1 gq with ODC.p∨¬p O (rgf ∋ r) |
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325 ... | case2 ngr = record { exp = r ; D∋exp = ⟪ Lr , ngr ⟫ ; p⊆exp = NotCompatible.p⊆r (NC Lp)} |
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326 ... | case1 gr = ⊥-elim ( NotCompatible.¬compat (NC Lp) ex2 Le ⟪ q⊆ex2 , r⊆ex2 ⟫ ) where |
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327 ex2 = Exp2.exp (⊆-Ideal.exp GF gq gr) |
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328 Le = ⊆-Ideal.i⊆L GF (Exp2.I∋exp (⊆-Ideal.exp GF gq gr)) |
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329 q⊆ex2 = Exp2.p⊆exp (⊆-Ideal.exp GF gq gr) |
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330 r⊆ex2 = Exp2.q⊆exp (⊆-Ideal.exp GF gq gr) |
| 1254 | 331 ¬rgf∩D=0 : ¬ ( (D ∩ rgf ) =h= od∅ ) |
| 332 ¬rgf∩D=0 eq = generic PG DD M∋D (==→o≡ eq) | |
| 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 |
| 1242 | 346 record valR (x : HOD) {P L : HOD} {LP : L ⊆ Power P} (C : CountableModel ) (G : GenericFilter {L} {P} LP (ctl-M C) ) : Set (Level.suc n) where |
| 437 | 347 field |
| 348 valx : HOD | |
| 436 | 349 |
| 437 | 350 record valS (ox oy oG : Ordinal) : Set n where |
| 436 | 351 field |
| 437 | 352 op : Ordinal |
| 1244 | 353 p∈G : odef (* oG) op |
| 437 | 354 is-val : odef (* ox) ( & < * oy , * op > ) |
| 436 | 355 |
| 459 | 356 val : (x : HOD) {P L : HOD } {LP : L ⊆ Power P} |
| 1096 | 357 → (G : GenericFilter {L} {P} LP {!!} ) |
| 436 | 358 → HOD |
| 437 | 359 val x G = TransFinite {λ x → HOD } ind (& x) where |
| 360 ind : (x : Ordinal) → ((y : Ordinal) → y o< x → HOD) → HOD | |
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361 ind x valy = record { od = record { def = λ y → valS x y (& (⊆-Ideal.ideal (genf G))) } ; odmax = {!!} ; <odmax = {!!} } |
| 437 | 362 |