Mercurial > hg > Members > kono > Proof > ZF-in-agda
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 01 Jul 2024 23:04:17 +0900 |
| parents | 171c3f3cdc6b |
| children |
| rev | line source |
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| 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 |
| 6 open import logic | |
| 1244 | 7 import OD |
| 431 | 8 |
| 1244 | 9 open import Relation.Nullary |
| 10 open import Relation.Binary | |
| 11 open import Data.Empty | |
| 431 | 12 open import Relation.Binary |
| 13 open import Relation.Binary.Core | |
| 14 open import Relation.Binary.PropositionalEquality | |
| 1244 | 15 open import Data.Nat |
| 16 import BAlgebra | |
| 431 | 17 |
| 1124 | 18 open BAlgebra O |
| 431 | 19 |
| 20 open inOrdinal O | |
| 21 open OD O | |
| 22 open OD.OD | |
| 23 open ODAxiom odAxiom | |
| 24 import OrdUtil | |
| 25 import ODUtil | |
| 26 open Ordinals.Ordinals O | |
| 27 open Ordinals.IsOrdinals isOrdinal | |
| 1300 | 28 -- open Ordinals.IsNext isNext |
| 431 | 29 open OrdUtil O |
| 30 open ODUtil O | |
| 31 | |
| 32 | |
| 33 import ODC | |
| 34 | |
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35 -- open filter O |
| 431 | 36 |
| 37 open _∧_ | |
| 38 open _∨_ | |
| 39 open Bool | |
| 40 | |
| 41 | |
| 42 open HOD | |
| 43 | |
| 44 ------- | |
| 45 -- the set of finite partial functions from ω to 2 | |
| 46 -- | |
| 47 -- | |
| 48 | |
| 49 open import Data.List hiding (filter) | |
| 1244 | 50 open import Data.Maybe |
| 431 | 51 |
| 1218 | 52 open import ZProduct O |
| 431 | 53 |
| 1458 | 54 -- L : definable HOD in Agda |
| 55 -- L Countable | |
| 56 -- Dense in Ordinal | |
| 57 | |
| 58 --- Dense L | |
| 59 -- x : Ord → ∃ l ∈ L → x ⊆ l | |
| 60 -- | |
| 61 -- ω =c= Power ω ∩ L c< Power ω | |
| 62 -- ω c< Power ω ∩ G[L] c< Power ω -- CH counter example | |
| 63 -- Power (G[L]) | |
| 64 -- | |
| 65 | |
| 66 | |
| 1242 | 67 record CountableModel : Set (Level.suc (Level.suc n)) where |
| 431 | 68 field |
| 461 | 69 ctl-M : HOD |
| 1242 | 70 ctl→ : ℕ → Ordinal |
| 1244 | 71 ctl<M : (x : ℕ) → odef (ctl-M) (ctl→ x) |
| 1242 | 72 ctl← : (x : Ordinal )→ odef (ctl-M ) x → ℕ |
| 1244 | 73 ctl-iso→ : { x : Ordinal } → (lt : odef (ctl-M) x ) → ctl→ (ctl← x lt ) ≡ x |
| 1248 | 74 TC : {x y : Ordinal} → odef ctl-M x → odef (* x) y → odef ctl-M y |
| 1249 | 75 is-model : (x : HOD) → & x o< & ctl-M → ctl-M ∋ (x ∩ ctl-M) |
| 1174 | 76 -- we have no otherway round |
| 1242 | 77 -- ctl-iso← : { x : ℕ } → ctl← (ctl→ x ) (ctl<M x) ≡ x |
| 446 | 78 -- |
| 79 -- almmost universe | |
| 80 -- find-p contains ∃ x : Ordinal → x o< & M → ∀ r ∈ M → ∈ Ord x | |
| 1244 | 81 -- |
| 436 | 82 |
| 1244 | 83 -- we expect P ∈ * ctl-M ∧ G ⊆ L ⊆ Power P , ¬ G ∈ * ctl-M, |
| 434 | 84 |
| 1244 | 85 open CountableModel |
| 431 | 86 |
| 87 ---- | |
| 88 -- a(n) ∈ M | |
| 1239 | 89 -- ∃ q ∈ L ⊆ Power P → q ∈ a(n) ∧ p(n) ⊆ q |
| 431 | 90 -- |
| 1242 | 91 PGHOD : (i : ℕ) (L : HOD) (C : CountableModel ) → (p : Ordinal) → HOD |
| 457 | 92 PGHOD i L C p = record { od = record { def = λ x → |
| 1239 | 93 odef L x ∧ odef (* (ctl→ C i)) x ∧ ( (y : Ordinal ) → odef (* p) y → odef (* x) y ) } |
| 1244 | 94 ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L (proj1 lt) } |
| 431 | 95 |
| 96 --- | |
| 1239 | 97 -- p(n+1) = if ({q | q ∈ a(n) ∧ p(n) ⊆ q)} != ∅ then q otherwise p(n) |
| 1244 | 98 -- |
| 1242 | 99 find-p : (L : HOD ) (C : CountableModel ) (i : ℕ) → (x : Ordinal) → Ordinal |
| 100 find-p L C zero x = x | |
| 101 find-p L C (suc i) x with is-o∅ ( & ( PGHOD i L C (find-p L C i x)) ) | |
| 457 | 102 ... | yes y = find-p L C i x |
| 103 ... | 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 | 104 |
| 105 --- | |
| 1239 | 106 -- G = { r ∈ L ⊆ Power P | ∃ n → r ⊆ p(n) } |
| 431 | 107 -- |
| 457 | 108 record PDN (L p : HOD ) (C : CountableModel ) (x : Ordinal) : Set n where |
| 431 | 109 field |
| 1242 | 110 gr : ℕ |
| 1244 | 111 pn<gr : (y : Ordinal) → odef (* x) y → odef (* (find-p L C gr (& p))) y |
| 457 | 112 x∈PP : odef L x |
| 431 | 113 |
| 114 open PDN | |
| 115 | |
| 116 --- | |
| 117 -- G as a HOD | |
| 118 -- | |
| 457 | 119 PDHOD : (L p : HOD ) (C : CountableModel ) → HOD |
| 120 PDHOD L p C = record { od = record { def = λ x → PDN L p C x } | |
| 1244 | 121 ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L {y} (PDN.x∈PP lt) } |
| 431 | 122 |
| 123 open PDN | |
| 124 | |
| 125 P∅ : {P : HOD} → odef (Power P) o∅ | |
| 126 P∅ {P} = subst (λ k → odef (Power P) k ) ord-od∅ (lemma o∅ o∅≡od∅) where | |
| 127 lemma : (x : Ordinal ) → * x ≡ od∅ → odef (Power P) (& od∅) | |
| 128 lemma x eq = power← P od∅ (λ {x} lt → ⊥-elim (¬x<0 lt )) | |
| 129 x<y→∋ : {x y : Ordinal} → odef (* x) y → * x ∋ * y | |
| 130 x<y→∋ {x} {y} lt = subst (λ k → odef (* x) k ) (sym &iso) lt | |
| 131 | |
| 1242 | 132 gf05 : {a b : HOD} {x : Ordinal } → (odef (a ∪ b) x ) → ¬ odef a x → ¬ odef b x → ⊥ |
| 1244 | 133 gf05 {a} {b} {x} (case1 ax) nax nbx = nax ax |
| 1242 | 134 gf05 {a} {b} {x} (case2 bx) nax nbx = nbx bx |
| 135 | |
| 446 | 136 open import Data.Nat.Properties |
| 1266 | 137 open import nat hiding ( exp ) |
| 433 | 138 |
| 1242 | 139 p-monotonic1 : (L p : HOD ) (C : CountableModel ) → {n : ℕ} → (* (find-p L C n (& p))) ⊆ (* (find-p L C (suc n) (& p))) |
| 1096 | 140 p-monotonic1 L p C {n} {x} with is-o∅ (& (PGHOD n L C (find-p L C n (& p)))) |
| 141 ... | yes y = refl-⊆ {* (find-p L C n (& p))} | |
| 1242 | 142 ... | no not = λ lt → proj2 (proj2 fmin∈PGHOD) _ lt where |
| 447 | 143 fmin : HOD |
| 457 | 144 fmin = ODC.minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) |
| 145 fmin∈PGHOD : PGHOD n L C (find-p L C n (& p)) ∋ fmin | |
| 146 fmin∈PGHOD = ODC.x∋minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) | |
| 438 | 147 |
| 1242 | 148 p-monotonic : (L p : HOD ) (C : CountableModel ) → {n m : ℕ} → n ≤ m → (* (find-p L C n (& p))) ⊆ (* (find-p L C m (& p))) |
| 149 p-monotonic L p C {zero} {zero} n≤m = refl-⊆ {* (find-p L C zero (& p))} | |
| 150 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 ) | |
| 151 p-monotonic L p C {suc n} {suc m} (s≤s n≤m) with <-cmp n m | |
| 1244 | 152 ... | tri< a ¬b ¬c = λ lt → p-monotonic1 L p C {m} (p-monotonic L p C {suc n} {m} a lt) |
| 1096 | 153 ... | tri≈ ¬a refl ¬c = λ x → x |
| 446 | 154 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c ) |
| 438 | 155 |
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156 record Expansion (p : HOD) (dense : HOD) : Set (Level.suc n) where |
| 1254 | 157 field |
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158 exp : HOD |
| 1265 | 159 D∋exp : dense ∋ exp |
| 160 p⊆exp : p ⊆ exp | |
| 1254 | 161 |
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162 record Dense (L : HOD ) : Set (Level.suc n) where |
| 1239 | 163 field |
| 164 dense : HOD | |
| 165 d⊆P : dense ⊆ L | |
| 1265 | 166 has-exp : {p : HOD} → (Lp : L ∋ p) → Expansion p dense |
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167 |
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168 record Exp2 (I : HOD) ( p q : HOD ) : Set (Level.suc n) where |
| 1265 | 169 field |
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170 exp : HOD |
| 1265 | 171 I∋exp : I ∋ exp |
| 172 p⊆exp : p ⊆ exp | |
| 173 q⊆exp : q ⊆ exp | |
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174 |
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175 record ⊆-Ideal {L P : HOD } (LP : L ⊆ Power P) : Set (Level.suc n) where |
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176 field |
| 1265 | 177 ideal : HOD |
| 178 i⊆L : ideal ⊆ L | |
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179 ideal1 : { p q : HOD } → L ∋ q → ideal ∋ p → q ⊆ p → ideal ∋ q |
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180 exp : { p q : HOD } → ideal ∋ p → ideal ∋ q → Exp2 ideal p q |
| 1239 | 181 |
| 1256 | 182 record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (Level.suc n) where |
| 1255 | 183 field |
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184 genf : ⊆-Ideal {L} {P} LP |
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185 generic : (D : Dense L ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ ⊆-Ideal.ideal genf ) ≡ od∅ ) |
| 1255 | 186 |
| 1266 | 187 ---- |
| 188 -- Generic Filter on L ⊆ Power P from HOD's Countable Ordinal (G ⊆ Power P ≡ G i.e. ℕ → P → Set ) | |
| 189 -- | |
| 1270 | 190 -- p 0 ≡ p0 |
| 1266 | 191 -- p (suc n) = if ∃ q ∈ M ∧ p n ⊆ q → q (by axiom of choice) ( q = * ( ctl→ n ) ) |
| 192 --- else p n | |
| 193 | |
| 1244 | 194 P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 |
| 1241 | 195 → (C : CountableModel ) → GenericFilter {L} {P} LP ( ctl-M C ) |
| 1255 | 196 P-GenericFilter P L p0 L⊆PP Lp0 C = record { |
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197 genf = record { ideal = PDHOD L p0 C ; i⊆L = x∈PP ; ideal1 = ideal1 ; exp = λ ip iq → exp1 ip iq } |
| 1256 | 198 ; generic = fdense |
| 431 | 199 } where |
| 1256 | 200 ideal1 : {p q : HOD} → L ∋ q → PDHOD L p0 C ∋ p → q ⊆ p → PDHOD L p0 C ∋ q |
| 1265 | 201 ideal1 {p} {q} Lq record { gr = gr ; pn<gr = pn<gr ; x∈PP = x∈PP } q⊆p = |
| 1256 | 202 record { gr = gr ; pn<gr = λ y qy → pn<gr y (gf00 qy) ; x∈PP = Lq } where |
| 1265 | 203 gf00 : {y : Ordinal } → odef (* (& q)) y → odef (* (& p)) y |
| 1256 | 204 gf00 {y} qy = subst (λ k → odef k y ) (sym *iso) (q⊆p (subst (λ k → odef k y) *iso qy )) |
| 1266 | 205 Lan : (i : ℕ ) → odef L (find-p L C i (& p0)) |
| 206 Lan zero = Lp0 | |
| 207 Lan (suc i) with is-o∅ ( & ( PGHOD i L C (find-p L C i (& p0))) ) | |
| 208 ... | yes y = Lan i | |
| 1265 | 209 ... | 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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210 exp1 : {p q : HOD} → (ip : PDHOD L p0 C ∋ p) → (ip : PDHOD L p0 C ∋ q) → Exp2 (PDHOD L p0 C) p q |
| 1265 | 211 exp1 {p} {q} record { gr = pgr ; pn<gr = ppn ; x∈PP = PPp } |
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212 record { gr = qgr ; pn<gr = qpn ; x∈PP = PPq } = gf01 where |
| 1265 | 213 Pp = record { gr = pgr ; pn<gr = ppn ; x∈PP = PPp } |
| 214 Pq = record { gr = qgr ; pn<gr = qpn ; x∈PP = PPq } | |
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215 gf17 : {q : HOD} → (Pq : PDHOD L p0 C ∋ q ) → PDHOD L p0 C ∋ * (find-p L C (gr Pq) (& p0)) |
| 1265 | 216 gf17 {q} Pq = record { gr = PDN.gr Pq ; pn<gr = λ y qq → subst (λ k → odef (* k) y) &iso qq |
| 1266 | 217 ; x∈PP = subst (λ k → odef L k ) (sym &iso) (Lan (PDN.gr Pq)) } |
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218 gf01 : Exp2 (PDHOD L p0 C) p q |
| 1256 | 219 gf01 with <-cmp pgr qgr |
| 1265 | 220 ... | 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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221 ; q⊆exp = λ {x} qx → qpn _ (subst (λ k → odef k x) (sym *iso) qx) } where |
| 1256 | 222 gf16 : gr Pp ≤ gr Pq |
| 223 gf16 = <to≤ a | |
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224 gf15 : (y : Ordinal) → odef p y → odef (* (find-p L C (gr Pq) (& p0))) y |
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225 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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226 ... | tri≈ ¬a refl ¬c = record { exp = * (find-p L C (gr Pq) (& p0)) ; I∋exp = gf17 Pq |
| 1265 | 227 ; p⊆exp = λ {x} px → ppn _ (subst (λ k → odef k x) (sym *iso) px) |
| 228 ; q⊆exp = λ {x} qx → qpn _ (subst (λ k → odef k x) (sym *iso) qx) } | |
| 229 ... | 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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230 ; p⊆exp = λ {x} px → ppn _ (subst (λ k → odef k x) (sym *iso) px) } where |
| 1256 | 231 gf16 : gr Pq ≤ gr Pp |
| 232 gf16 = <to≤ c | |
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233 gf15 : (y : Ordinal) → odef q y → odef (* (find-p L C (gr Pp) (& p0))) y |
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234 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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235 fdense : (D : Dense L ) → (ctl-M C ) ∋ Dense.dense D → ¬ (Dense.dense D ∩ (PDHOD L p0 C)) ≡ od∅ |
| 1256 | 236 fdense D MD eq0 = ⊥-elim ( ∅< {Dense.dense D ∩ PDHOD L p0 C} fd01 (≡od∅→=od∅ eq0 )) where |
| 237 open Dense | |
| 238 open Expansion | |
| 239 an : ℕ | |
| 1265 | 240 an = ctl← C (& (dense D)) MD |
| 1256 | 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) | |
| 1265 | 245 d=an : dense D ≡ * (ctl→ C an) |
| 1256 | 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)) | |
| 1266 | 253 L∋pn = subst (λ k → odef L k) (sym &iso) (Lan 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) | |
| 1265 | 270 fd27 = subst (λ k → odef k (& fd29)) (sym d=an) (proj1 (proj2 fd28)) |
| 1256 | 271 fd03 : odef (PDHOD L p0 C) pn+1 |
| 1266 | 272 fd03 = record { gr = suc an ; pn<gr = λ y lt → lt ; x∈PP = Lan (suc an)} |
| 1256 | 273 fd01 : (dense D ∩ PDHOD L p0 C) ∋ (* pn+1) |
| 1265 | 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 |
| 1266 | 279 record Incompatible (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 | |
| 1265 | 284 p⊆q : p ⊆ q |
| 285 p⊆r : p ⊆ r | |
| 1255 | 286 ¬compat : (s : HOD) → L ∋ s → ¬ ( (q ⊆ s) ∧ (r ⊆ s) ) |
| 431 | 287 |
| 1268 | 288 Incompatible→¬M∋G : (P L p0 : HOD ) → (LPP : L ⊆ Power P) → (Lp0 : L ∋ p0 ) |
| 1265 | 289 → (C : CountableModel ) |
| 1249 | 290 → ctl-M C ∋ L |
| 1266 | 291 → ( {p : HOD} → (Lp : L ∋ p ) → Incompatible L p Lp ) |
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292 → ¬ ( ctl-M C ∋ ⊆-Ideal.ideal (genf ( P-GenericFilter P L p0 LPP Lp0 C ))) |
| 1268 | 293 Incompatible→¬M∋G P L p0 LPP Lp0 C ML NC MF = ¬G∩D=0 D∩G=0 where |
| 1265 | 294 PG = P-GenericFilter P L p0 LPP Lp0 C |
| 1249 | 295 GF = genf PG |
| 1266 | 296 G = ⊆-Ideal.ideal (genf PG) |
| 1245 | 297 M = ctl-M C |
| 1265 | 298 D : HOD |
| 1266 | 299 D = L ＼ G |
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300 D<M : & D o< & (ctl-M C) |
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301 D<M = ordtrans≤-< (⊆→o≤ proj1 ) (odef< ML) |
| 1248 | 302 M∋DM : M ∋ (D ∩ M ) |
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303 M∋DM = is-model C D D<M |
| 1266 | 304 -- G⊆M : G ⊆ M |
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305 -- G⊆M {x} rx = TC C ML (subst (λ k → odef k x) (sym *iso) (⊆-Ideal.i⊆L GF rx)) |
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306 -- D⊆M : D ⊆ M |
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307 -- D⊆M {x} dx = TC C ML (subst (λ k → odef k x) (sym *iso) (proj1 dx)) |
| 1265 | 308 D=D∩M : D ≡ D ∩ M |
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309 D=D∩M = ==→o≡ record { eq→ = ddm00 ; eq← = proj1 } where |
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310 ddm00 : {x : Ordinal} → odef D x → odef (D ∩ M) x |
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311 ddm00 {x} ⟪ Lx , ¬Gx ⟫ = ⟪ ⟪ Lx , ¬Gx ⟫ , TC C ML (subst (λ k → odef k x) (sym *iso) Lx ) ⟫ |
| 1265 | 312 M∋D : M ∋ D |
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313 M∋D = subst (λ k → M ∋ k ) (sym D=D∩M) M∋DM |
| 1246 | 314 D⊆PP : D ⊆ Power P |
| 1265 | 315 D⊆PP {x} ⟪ Lx , ngx ⟫ = LPP Lx |
| 316 DD : Dense L | |
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317 DD = record { dense = D ; d⊆P = proj1 ; has-exp = exp } where |
| 1265 | 318 exp : {p : HOD} → (Lp : L ∋ p) → Expansion p D |
| 1254 | 319 exp {p} Lp = exp1 where |
| 320 q : HOD | |
| 1266 | 321 q = Incompatible.q (NC Lp) |
| 1254 | 322 r : HOD |
| 1266 | 323 r = Incompatible.r (NC Lp) |
| 1255 | 324 Lq : L ∋ q |
| 1266 | 325 Lq = Incompatible.Lq (NC Lp) |
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326 Lr : L ∋ r |
| 1266 | 327 Lr = Incompatible.Lr (NC Lp) |
| 1265 | 328 exp1 : Expansion p D |
| 1266 | 329 exp1 with ODC.p∨¬p O (G ∋ q) |
| 330 ... | case2 ngq = record { exp = q ; D∋exp = ⟪ Lq , ngq ⟫ ; p⊆exp = Incompatible.p⊆q (NC Lp)} | |
| 331 ... | case1 gq with ODC.p∨¬p O (G ∋ r) | |
| 332 ... | case2 ngr = record { exp = r ; D∋exp = ⟪ Lr , ngr ⟫ ; p⊆exp = Incompatible.p⊆r (NC Lp)} | |
| 333 ... | case1 gr = ⊥-elim ( Incompatible.¬compat (NC Lp) ex2 Le ⟪ q⊆ex2 , r⊆ex2 ⟫ ) where | |
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334 ex2 = Exp2.exp (⊆-Ideal.exp GF gq gr) |
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335 Le = ⊆-Ideal.i⊆L GF (Exp2.I∋exp (⊆-Ideal.exp GF gq gr)) |
| 1265 | 336 q⊆ex2 = Exp2.p⊆exp (⊆-Ideal.exp GF gq gr) |
| 337 r⊆ex2 = Exp2.q⊆exp (⊆-Ideal.exp GF gq gr) | |
| 1266 | 338 ¬G∩D=0 : ¬ ( (D ∩ G ) =h= od∅ ) |
| 339 ¬G∩D=0 eq = generic PG DD M∋D (==→o≡ eq) | |
| 340 D∩G=0 : (D ∩ G ) =h= od∅ -- because D = L ＼ G | |
| 341 D∩G=0 = record { eq→ = λ {x} G∩D → ⊥-elim( proj2 (proj1 G∩D) (proj2 G∩D)) | |
| 342 ; eq← = λ lt → ⊥-elim (¬x<0 lt) } | |
| 431 | 343 |
| 344 -- | |
| 1174 | 345 -- P-Generic Filter defines a countable model D ⊂ C from P |
| 346 -- | |
| 347 | |
| 348 -- | |
| 1270 | 349 -- val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > } |
| 350 -- | |
| 351 -- We can define the valuation, but to use this, we need V=L, which makes things complicated. | |
| 1272 | 352 |
| 353 val< : {x y p : Ordinal} → odef (* x) ( & < * y , * p > ) → y o< x | |
| 354 val< {x} {y} {p} xyp = osucprev ( begin | |
| 355 osuc y ≤⟨ osucc (odef< (subst (λ k → odef (* y , * y) k) &iso (v00 _ _ ) )) ⟩ | |
| 356 & (* y , * y) <⟨ c<→o< (v01 _ _) ⟩ | |
| 357 & < * y , * p > <⟨ odef< xyp ⟩ | |
| 358 & (* x) ≡⟨ &iso ⟩ | |
| 359 x ∎ ) where | |
| 360 v00 : (x y : HOD) → odef (x , y) (& x) | |
| 361 v00 _ _ = case1 refl | |
| 362 v01 : (x y : HOD) → < x , y > ∋ (x , x) | |
| 363 v01 _ _ = case1 refl | |
| 364 open o≤-Reasoning O | |
| 365 | |
| 366 record valS (G : HOD) (x z : Ordinal) (val : (y : Ordinal) → y o< x → HOD): Set n where | |
| 367 field | |
| 368 y p : Ordinal | |
| 369 G∋p : odef G p | |
| 370 is-val : odef (* x) ( & < * y , * p > ) | |
| 371 z=valy : z ≡ & (val y (val< is-val)) | |
| 372 z<x : z o< x | |
| 373 | |
| 374 val : (x : HOD) → (G : HOD) → HOD | |
| 375 val x G = TransFinite {λ x → HOD } ind (& x) where | |
| 376 ind : (x : Ordinal) → (valy : (y : Ordinal) → y o< x → HOD) → HOD | |
| 377 ind x valy = record { od = record { def = λ z → valS G x z valy } ; odmax = x ; <odmax = v02 } where | |
| 378 v02 : {z : Ordinal} → valS G x z valy → z o< x | |
| 379 v02 {z} lt = valS.z<x lt | |
| 1269 | 380 |
| 1268 | 381 -- |
| 1270 | 382 -- What we nedd |
| 383 -- M[G] : HOD | |
| 384 -- M ⊆ M[G] | |
| 385 -- M[G] ∋ G | |
| 386 -- M[G] ∋ ∪G | |
| 387 |