Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/generic-filter.agda @ 1284:45cd80181a29
remove import zf
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 20 May 2023 09:48:37 +0900 |
parents | 30540f151ae0 |
children | 47d3cc596d68 |
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 | |
28 open Ordinals.IsNext isNext | |
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 |
1242 | 54 record CountableModel : Set (Level.suc (Level.suc n)) where |
431 | 55 field |
461 | 56 ctl-M : HOD |
1242 | 57 ctl→ : ℕ → Ordinal |
1244 | 58 ctl<M : (x : ℕ) → odef (ctl-M) (ctl→ x) |
1242 | 59 ctl← : (x : Ordinal )→ odef (ctl-M ) x → ℕ |
1244 | 60 ctl-iso→ : { x : Ordinal } → (lt : odef (ctl-M) x ) → ctl→ (ctl← x lt ) ≡ x |
1248 | 61 TC : {x y : Ordinal} → odef ctl-M x → odef (* x) y → odef ctl-M y |
1249 | 62 is-model : (x : HOD) → & x o< & ctl-M → ctl-M ∋ (x ∩ ctl-M) |
1174 | 63 -- we have no otherway round |
1242 | 64 -- ctl-iso← : { x : ℕ } → ctl← (ctl→ x ) (ctl<M x) ≡ x |
446 | 65 -- |
66 -- almmost universe | |
67 -- find-p contains ∃ x : Ordinal → x o< & M → ∀ r ∈ M → ∈ Ord x | |
1244 | 68 -- |
436 | 69 |
1244 | 70 -- we expect P ∈ * ctl-M ∧ G ⊆ L ⊆ Power P , ¬ G ∈ * ctl-M, |
434 | 71 |
1273 | 72 record COD : Set (Level.suc (Level.suc n)) where |
73 field | |
74 CO : Ordinals {n} | |
75 CA : OD.ODAxiom CO | |
76 cod→ : ℕ → Ordinals.Ordinal CO | |
77 cod← : Ordinals.Ordinal CO → ℕ | |
78 cod-iso→ : { x : Ordinals.Ordinal CO } → cod→ (cod← x) ≡ x | |
79 cod-iso← : { x : ℕ } → cod← (cod→ x) ≡ x | |
80 -- Since it is countable, it is HOD. | |
81 | |
1284 | 82 -- CM : COD → CountableModel |
83 -- CM cod = record { | |
84 -- ctl-M = ? | |
85 -- ; ctl→ = λ n → ? | |
86 -- ; ctl<M = λ n → ? | |
87 -- ; ctl← = λ x → ? | |
88 -- ; ctl-iso→ = ? | |
89 -- ; TC = ? | |
90 -- ; is-model = ? | |
91 -- } | |
1273 | 92 |
1244 | 93 open CountableModel |
431 | 94 |
95 ---- | |
96 -- a(n) ∈ M | |
1239 | 97 -- ∃ q ∈ L ⊆ Power P → q ∈ a(n) ∧ p(n) ⊆ q |
431 | 98 -- |
1242 | 99 PGHOD : (i : ℕ) (L : HOD) (C : CountableModel ) → (p : Ordinal) → HOD |
457 | 100 PGHOD i L C p = record { od = record { def = λ x → |
1239 | 101 odef L x ∧ odef (* (ctl→ C i)) x ∧ ( (y : Ordinal ) → odef (* p) y → odef (* x) y ) } |
1244 | 102 ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L (proj1 lt) } |
431 | 103 |
104 --- | |
1239 | 105 -- p(n+1) = if ({q | q ∈ a(n) ∧ p(n) ⊆ q)} != ∅ then q otherwise p(n) |
1244 | 106 -- |
1242 | 107 find-p : (L : HOD ) (C : CountableModel ) (i : ℕ) → (x : Ordinal) → Ordinal |
108 find-p L C zero x = x | |
109 find-p L C (suc i) x with is-o∅ ( & ( PGHOD i L C (find-p L C i x)) ) | |
457 | 110 ... | yes y = find-p L C i x |
111 ... | 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 | 112 |
113 --- | |
1239 | 114 -- G = { r ∈ L ⊆ Power P | ∃ n → r ⊆ p(n) } |
431 | 115 -- |
457 | 116 record PDN (L p : HOD ) (C : CountableModel ) (x : Ordinal) : Set n where |
431 | 117 field |
1242 | 118 gr : ℕ |
1244 | 119 pn<gr : (y : Ordinal) → odef (* x) y → odef (* (find-p L C gr (& p))) y |
457 | 120 x∈PP : odef L x |
431 | 121 |
122 open PDN | |
123 | |
124 --- | |
125 -- G as a HOD | |
126 -- | |
457 | 127 PDHOD : (L p : HOD ) (C : CountableModel ) → HOD |
128 PDHOD L p C = record { od = record { def = λ x → PDN L p C x } | |
1244 | 129 ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L {y} (PDN.x∈PP lt) } |
431 | 130 |
131 open PDN | |
132 | |
133 P∅ : {P : HOD} → odef (Power P) o∅ | |
134 P∅ {P} = subst (λ k → odef (Power P) k ) ord-od∅ (lemma o∅ o∅≡od∅) where | |
135 lemma : (x : Ordinal ) → * x ≡ od∅ → odef (Power P) (& od∅) | |
136 lemma x eq = power← P od∅ (λ {x} lt → ⊥-elim (¬x<0 lt )) | |
137 x<y→∋ : {x y : Ordinal} → odef (* x) y → * x ∋ * y | |
138 x<y→∋ {x} {y} lt = subst (λ k → odef (* x) k ) (sym &iso) lt | |
139 | |
1242 | 140 gf05 : {a b : HOD} {x : Ordinal } → (odef (a ∪ b) x ) → ¬ odef a x → ¬ odef b x → ⊥ |
1244 | 141 gf05 {a} {b} {x} (case1 ax) nax nbx = nax ax |
1242 | 142 gf05 {a} {b} {x} (case2 bx) nax nbx = nbx bx |
143 | |
446 | 144 open import Data.Nat.Properties |
1266 | 145 open import nat hiding ( exp ) |
433 | 146 |
1242 | 147 p-monotonic1 : (L p : HOD ) (C : CountableModel ) → {n : ℕ} → (* (find-p L C n (& p))) ⊆ (* (find-p L C (suc n) (& p))) |
1096 | 148 p-monotonic1 L p C {n} {x} with is-o∅ (& (PGHOD n L C (find-p L C n (& p)))) |
149 ... | yes y = refl-⊆ {* (find-p L C n (& p))} | |
1242 | 150 ... | no not = λ lt → proj2 (proj2 fmin∈PGHOD) _ lt where |
447 | 151 fmin : HOD |
457 | 152 fmin = ODC.minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) |
153 fmin∈PGHOD : PGHOD n L C (find-p L C n (& p)) ∋ fmin | |
154 fmin∈PGHOD = ODC.x∋minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) | |
438 | 155 |
1242 | 156 p-monotonic : (L p : HOD ) (C : CountableModel ) → {n m : ℕ} → n ≤ m → (* (find-p L C n (& p))) ⊆ (* (find-p L C m (& p))) |
157 p-monotonic L p C {zero} {zero} n≤m = refl-⊆ {* (find-p L C zero (& p))} | |
158 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 ) | |
159 p-monotonic L p C {suc n} {suc m} (s≤s n≤m) with <-cmp n m | |
1244 | 160 ... | tri< a ¬b ¬c = λ lt → p-monotonic1 L p C {m} (p-monotonic L p C {suc n} {m} a lt) |
1096 | 161 ... | tri≈ ¬a refl ¬c = λ x → x |
446 | 162 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c ) |
438 | 163 |
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164 record Expansion (p : HOD) (dense : HOD) : Set (Level.suc n) where |
1254 | 165 field |
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166 exp : HOD |
1265 | 167 D∋exp : dense ∋ exp |
168 p⊆exp : p ⊆ exp | |
1254 | 169 |
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170 record Dense (L : HOD ) : Set (Level.suc n) where |
1239 | 171 field |
172 dense : HOD | |
173 d⊆P : dense ⊆ L | |
1265 | 174 has-exp : {p : HOD} → (Lp : L ∋ p) → Expansion p dense |
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175 |
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176 record Exp2 (I : HOD) ( p q : HOD ) : Set (Level.suc n) where |
1265 | 177 field |
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178 exp : HOD |
1265 | 179 I∋exp : I ∋ exp |
180 p⊆exp : p ⊆ exp | |
181 q⊆exp : q ⊆ exp | |
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182 |
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183 record ⊆-Ideal {L P : HOD } (LP : L ⊆ Power P) : Set (Level.suc n) where |
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184 field |
1265 | 185 ideal : HOD |
186 i⊆L : ideal ⊆ L | |
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187 ideal1 : { p q : HOD } → L ∋ q → ideal ∋ p → q ⊆ p → ideal ∋ q |
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188 exp : { p q : HOD } → ideal ∋ p → ideal ∋ q → Exp2 ideal p q |
1239 | 189 |
1256 | 190 record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (Level.suc n) where |
1255 | 191 field |
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192 genf : ⊆-Ideal {L} {P} LP |
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193 generic : (D : Dense L ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ ⊆-Ideal.ideal genf ) ≡ od∅ ) |
1255 | 194 |
1266 | 195 ---- |
196 -- Generic Filter on L ⊆ Power P from HOD's Countable Ordinal (G ⊆ Power P ≡ G i.e. ℕ → P → Set ) | |
197 -- | |
1270 | 198 -- p 0 ≡ p0 |
1266 | 199 -- p (suc n) = if ∃ q ∈ M ∧ p n ⊆ q → q (by axiom of choice) ( q = * ( ctl→ n ) ) |
200 --- else p n | |
201 | |
1244 | 202 P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 |
1241 | 203 → (C : CountableModel ) → GenericFilter {L} {P} LP ( ctl-M C ) |
1255 | 204 P-GenericFilter P L p0 L⊆PP Lp0 C = record { |
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205 genf = record { ideal = PDHOD L p0 C ; i⊆L = x∈PP ; ideal1 = ideal1 ; exp = λ ip iq → exp1 ip iq } |
1256 | 206 ; generic = fdense |
431 | 207 } where |
1256 | 208 ideal1 : {p q : HOD} → L ∋ q → PDHOD L p0 C ∋ p → q ⊆ p → PDHOD L p0 C ∋ q |
1265 | 209 ideal1 {p} {q} Lq record { gr = gr ; pn<gr = pn<gr ; x∈PP = x∈PP } q⊆p = |
1256 | 210 record { gr = gr ; pn<gr = λ y qy → pn<gr y (gf00 qy) ; x∈PP = Lq } where |
1265 | 211 gf00 : {y : Ordinal } → odef (* (& q)) y → odef (* (& p)) y |
1256 | 212 gf00 {y} qy = subst (λ k → odef k y ) (sym *iso) (q⊆p (subst (λ k → odef k y) *iso qy )) |
1266 | 213 Lan : (i : ℕ ) → odef L (find-p L C i (& p0)) |
214 Lan zero = Lp0 | |
215 Lan (suc i) with is-o∅ ( & ( PGHOD i L C (find-p L C i (& p0))) ) | |
216 ... | yes y = Lan i | |
1265 | 217 ... | 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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218 exp1 : {p q : HOD} → (ip : PDHOD L p0 C ∋ p) → (ip : PDHOD L p0 C ∋ q) → Exp2 (PDHOD L p0 C) p q |
1265 | 219 exp1 {p} {q} record { gr = pgr ; pn<gr = ppn ; x∈PP = PPp } |
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220 record { gr = qgr ; pn<gr = qpn ; x∈PP = PPq } = gf01 where |
1265 | 221 Pp = record { gr = pgr ; pn<gr = ppn ; x∈PP = PPp } |
222 Pq = record { gr = qgr ; pn<gr = qpn ; x∈PP = PPq } | |
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223 gf17 : {q : HOD} → (Pq : PDHOD L p0 C ∋ q ) → PDHOD L p0 C ∋ * (find-p L C (gr Pq) (& p0)) |
1265 | 224 gf17 {q} Pq = record { gr = PDN.gr Pq ; pn<gr = λ y qq → subst (λ k → odef (* k) y) &iso qq |
1266 | 225 ; x∈PP = subst (λ k → odef L k ) (sym &iso) (Lan (PDN.gr Pq)) } |
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226 gf01 : Exp2 (PDHOD L p0 C) p q |
1256 | 227 gf01 with <-cmp pgr qgr |
1265 | 228 ... | 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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229 ; q⊆exp = λ {x} qx → qpn _ (subst (λ k → odef k x) (sym *iso) qx) } where |
1256 | 230 gf16 : gr Pp ≤ gr Pq |
231 gf16 = <to≤ a | |
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232 gf15 : (y : Ordinal) → odef p y → odef (* (find-p L C (gr Pq) (& p0))) y |
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233 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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234 ... | tri≈ ¬a refl ¬c = record { exp = * (find-p L C (gr Pq) (& p0)) ; I∋exp = gf17 Pq |
1265 | 235 ; p⊆exp = λ {x} px → ppn _ (subst (λ k → odef k x) (sym *iso) px) |
236 ; q⊆exp = λ {x} qx → qpn _ (subst (λ k → odef k x) (sym *iso) qx) } | |
237 ... | 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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238 ; p⊆exp = λ {x} px → ppn _ (subst (λ k → odef k x) (sym *iso) px) } where |
1256 | 239 gf16 : gr Pq ≤ gr Pp |
240 gf16 = <to≤ c | |
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241 gf15 : (y : Ordinal) → odef q y → odef (* (find-p L C (gr Pp) (& p0))) y |
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242 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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243 fdense : (D : Dense L ) → (ctl-M C ) ∋ Dense.dense D → ¬ (Dense.dense D ∩ (PDHOD L p0 C)) ≡ od∅ |
1256 | 244 fdense D MD eq0 = ⊥-elim ( ∅< {Dense.dense D ∩ PDHOD L p0 C} fd01 (≡od∅→=od∅ eq0 )) where |
245 open Dense | |
246 open Expansion | |
247 an : ℕ | |
1265 | 248 an = ctl← C (& (dense D)) MD |
1256 | 249 pn : Ordinal |
250 pn = find-p L C an (& p0) | |
251 pn+1 : Ordinal | |
252 pn+1 = find-p L C (suc an) (& p0) | |
1265 | 253 d=an : dense D ≡ * (ctl→ C an) |
1256 | 254 d=an = begin dense D ≡⟨ sym *iso ⟩ |
255 * ( & (dense D)) ≡⟨ cong (*) (sym (ctl-iso→ C MD )) ⟩ | |
256 * (ctl→ C an) ∎ where open ≡-Reasoning | |
257 fd07 : odef (dense D) pn+1 | |
258 fd07 with is-o∅ ( & ( PGHOD an L C (find-p L C an (& p0))) ) | |
259 ... | yes y = ⊥-elim ( ¬x<0 ( _==_.eq→ fd10 fd21 ) ) where | |
260 L∋pn : L ∋ * (find-p L C an (& p0)) | |
1266 | 261 L∋pn = subst (λ k → odef L k) (sym &iso) (Lan an ) |
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262 ex = has-exp D L∋pn |
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263 L∋df : L ∋ ( exp ex ) |
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264 L∋df = (d⊆P D) (D∋exp ex) |
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265 pn∋df : (* (ctl→ C an)) ∋ ( exp ex) |
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266 pn∋df = subst (λ k → odef k (& ( exp ex))) d=an (D∋exp ex ) |
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267 pn⊆df : (y : Ordinal) → odef (* (find-p L C an (& p0))) y → odef (* (& (exp ex))) y |
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268 pn⊆df y py = subst (λ k → odef k y ) (sym *iso) (p⊆exp ex py) |
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269 fd21 : odef (PGHOD an L C (find-p L C an (& p0)) ) (& (exp ex)) |
1256 | 270 fd21 = ⟪ L∋df , ⟪ pn∋df , pn⊆df ⟫ ⟫ |
271 fd10 : PGHOD an L C (find-p L C an (& p0)) =h= od∅ | |
272 fd10 = ≡o∅→=od∅ y | |
273 ... | no not = fd27 where | |
274 fd29 = ODC.minimal O ( PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
275 fd28 : PGHOD an L C (find-p L C an (& p0)) ∋ fd29 | |
276 fd28 = ODC.x∋minimal O (PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
277 fd27 : odef (dense D) (& fd29) | |
1265 | 278 fd27 = subst (λ k → odef k (& fd29)) (sym d=an) (proj1 (proj2 fd28)) |
1256 | 279 fd03 : odef (PDHOD L p0 C) pn+1 |
1266 | 280 fd03 = record { gr = suc an ; pn<gr = λ y lt → lt ; x∈PP = Lan (suc an)} |
1256 | 281 fd01 : (dense D ∩ PDHOD L p0 C) ∋ (* pn+1) |
1265 | 282 fd01 = ⟪ subst (λ k → odef (dense D) k ) (sym &iso) fd07 , subst (λ k → odef (PDHOD L p0 C) k) (sym &iso) fd03 ⟫ |
448 | 283 |
431 | 284 open GenericFilter |
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285 -- open Filter |
431 | 286 |
1266 | 287 record Incompatible (L p : HOD ) (L∋a : L ∋ p ) : Set (Level.suc (Level.suc n)) where |
431 | 288 field |
1245 | 289 q r : HOD |
290 Lq : L ∋ q | |
291 Lr : L ∋ r | |
1265 | 292 p⊆q : p ⊆ q |
293 p⊆r : p ⊆ r | |
1255 | 294 ¬compat : (s : HOD) → L ∋ s → ¬ ( (q ⊆ s) ∧ (r ⊆ s) ) |
431 | 295 |
1268 | 296 Incompatible→¬M∋G : (P L p0 : HOD ) → (LPP : L ⊆ Power P) → (Lp0 : L ∋ p0 ) |
1265 | 297 → (C : CountableModel ) |
1249 | 298 → ctl-M C ∋ L |
1266 | 299 → ( {p : HOD} → (Lp : L ∋ p ) → Incompatible L p Lp ) |
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300 → ¬ ( ctl-M C ∋ ⊆-Ideal.ideal (genf ( P-GenericFilter P L p0 LPP Lp0 C ))) |
1268 | 301 Incompatible→¬M∋G P L p0 LPP Lp0 C ML NC MF = ¬G∩D=0 D∩G=0 where |
1265 | 302 PG = P-GenericFilter P L p0 LPP Lp0 C |
1249 | 303 GF = genf PG |
1266 | 304 G = ⊆-Ideal.ideal (genf PG) |
1245 | 305 M = ctl-M C |
1265 | 306 D : HOD |
1266 | 307 D = L \ G |
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308 D<M : & D o< & (ctl-M C) |
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309 D<M = ordtrans≤-< (⊆→o≤ proj1 ) (odef< ML) |
1248 | 310 M∋DM : M ∋ (D ∩ M ) |
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311 M∋DM = is-model C D D<M |
1266 | 312 -- G⊆M : G ⊆ M |
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313 -- G⊆M {x} rx = TC C ML (subst (λ k → odef k x) (sym *iso) (⊆-Ideal.i⊆L GF rx)) |
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314 -- D⊆M : D ⊆ M |
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315 -- D⊆M {x} dx = TC C ML (subst (λ k → odef k x) (sym *iso) (proj1 dx)) |
1265 | 316 D=D∩M : D ≡ D ∩ M |
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317 D=D∩M = ==→o≡ record { eq→ = ddm00 ; eq← = proj1 } where |
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318 ddm00 : {x : Ordinal} → odef D x → odef (D ∩ M) x |
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319 ddm00 {x} ⟪ Lx , ¬Gx ⟫ = ⟪ ⟪ Lx , ¬Gx ⟫ , TC C ML (subst (λ k → odef k x) (sym *iso) Lx ) ⟫ |
1265 | 320 M∋D : M ∋ D |
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321 M∋D = subst (λ k → M ∋ k ) (sym D=D∩M) M∋DM |
1246 | 322 D⊆PP : D ⊆ Power P |
1265 | 323 D⊆PP {x} ⟪ Lx , ngx ⟫ = LPP Lx |
324 DD : Dense L | |
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325 DD = record { dense = D ; d⊆P = proj1 ; has-exp = exp } where |
1265 | 326 exp : {p : HOD} → (Lp : L ∋ p) → Expansion p D |
1254 | 327 exp {p} Lp = exp1 where |
328 q : HOD | |
1266 | 329 q = Incompatible.q (NC Lp) |
1254 | 330 r : HOD |
1266 | 331 r = Incompatible.r (NC Lp) |
1255 | 332 Lq : L ∋ q |
1266 | 333 Lq = Incompatible.Lq (NC Lp) |
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334 Lr : L ∋ r |
1266 | 335 Lr = Incompatible.Lr (NC Lp) |
1265 | 336 exp1 : Expansion p D |
1266 | 337 exp1 with ODC.p∨¬p O (G ∋ q) |
338 ... | case2 ngq = record { exp = q ; D∋exp = ⟪ Lq , ngq ⟫ ; p⊆exp = Incompatible.p⊆q (NC Lp)} | |
339 ... | case1 gq with ODC.p∨¬p O (G ∋ r) | |
340 ... | case2 ngr = record { exp = r ; D∋exp = ⟪ Lr , ngr ⟫ ; p⊆exp = Incompatible.p⊆r (NC Lp)} | |
341 ... | case1 gr = ⊥-elim ( Incompatible.¬compat (NC Lp) ex2 Le ⟪ q⊆ex2 , r⊆ex2 ⟫ ) where | |
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342 ex2 = Exp2.exp (⊆-Ideal.exp GF gq gr) |
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343 Le = ⊆-Ideal.i⊆L GF (Exp2.I∋exp (⊆-Ideal.exp GF gq gr)) |
1265 | 344 q⊆ex2 = Exp2.p⊆exp (⊆-Ideal.exp GF gq gr) |
345 r⊆ex2 = Exp2.q⊆exp (⊆-Ideal.exp GF gq gr) | |
1266 | 346 ¬G∩D=0 : ¬ ( (D ∩ G ) =h= od∅ ) |
347 ¬G∩D=0 eq = generic PG DD M∋D (==→o≡ eq) | |
348 D∩G=0 : (D ∩ G ) =h= od∅ -- because D = L \ G | |
349 D∩G=0 = record { eq→ = λ {x} G∩D → ⊥-elim( proj2 (proj1 G∩D) (proj2 G∩D)) | |
350 ; eq← = λ lt → ⊥-elim (¬x<0 lt) } | |
431 | 351 |
352 -- | |
1174 | 353 -- P-Generic Filter defines a countable model D ⊂ C from P |
354 -- | |
355 | |
356 -- | |
1270 | 357 -- val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > } |
358 -- | |
359 -- We can define the valuation, but to use this, we need V=L, which makes things complicated. | |
1272 | 360 |
361 val< : {x y p : Ordinal} → odef (* x) ( & < * y , * p > ) → y o< x | |
362 val< {x} {y} {p} xyp = osucprev ( begin | |
363 osuc y ≤⟨ osucc (odef< (subst (λ k → odef (* y , * y) k) &iso (v00 _ _ ) )) ⟩ | |
364 & (* y , * y) <⟨ c<→o< (v01 _ _) ⟩ | |
365 & < * y , * p > <⟨ odef< xyp ⟩ | |
366 & (* x) ≡⟨ &iso ⟩ | |
367 x ∎ ) where | |
368 v00 : (x y : HOD) → odef (x , y) (& x) | |
369 v00 _ _ = case1 refl | |
370 v01 : (x y : HOD) → < x , y > ∋ (x , x) | |
371 v01 _ _ = case1 refl | |
372 open o≤-Reasoning O | |
373 | |
374 record valS (G : HOD) (x z : Ordinal) (val : (y : Ordinal) → y o< x → HOD): Set n where | |
375 field | |
376 y p : Ordinal | |
377 G∋p : odef G p | |
378 is-val : odef (* x) ( & < * y , * p > ) | |
379 z=valy : z ≡ & (val y (val< is-val)) | |
380 z<x : z o< x | |
381 | |
382 val : (x : HOD) → (G : HOD) → HOD | |
383 val x G = TransFinite {λ x → HOD } ind (& x) where | |
384 ind : (x : Ordinal) → (valy : (y : Ordinal) → y o< x → HOD) → HOD | |
385 ind x valy = record { od = record { def = λ z → valS G x z valy } ; odmax = x ; <odmax = v02 } where | |
386 v02 : {z : Ordinal} → valS G x z valy → z o< x | |
387 v02 {z} lt = valS.z<x lt | |
1269 | 388 |
1268 | 389 -- |
1270 | 390 -- What we nedd |
391 -- M[G] : HOD | |
392 -- M ⊆ M[G] | |
393 -- M[G] ∋ G | |
394 -- M[G] ∋ ∪G | |
1272 | 395 -- |
396 -- assume countable L as M | |
1270 | 397 -- L is a countable subset of Power ω i.e. Power ω ∩ M |
1272 | 398 -- it defines countable Ordinal |
1268 | 399 -- |
1270 | 400 -- Generic Filter is a countable sequence of element of M |
401 -- Mg is set of all elementns of M which contains an element of G | |
402 -- | |
403 -- Mg : HOD | |
404 -- Mg | |
405 | |
406 record Mg {L P : HOD} (LP : L ⊆ Power P) (M : HOD) (G : GenericFilter {L} {P} LP M) (x : Ordinal) : Set n where | |
407 field | |
408 gi : Ordinal | |
409 G∋gi : odef (⊆-Ideal.ideal (genf G)) gi | |
410 x∋gi : odef (* x) gi | |
1266 | 411 |
1271 | 412 MgH : {L P : HOD} (LP : L ⊆ Power P) (M : HOD) (G : GenericFilter {L} {P} LP M) → HOD |
1273 | 413 MgH {L} {P} LP M G = record { od = record { def = λ x → (x o< & M) ∧ Mg LP M G x } ; odmax = & M ; <odmax = proj1 } |
1271 | 414 |
1272 | 415 MG1 : {L P : HOD} (LP : L ⊆ Power P) (M : HOD) (G : GenericFilter {L} {P} LP M) → HOD |
416 MG1 {L} {P} LP M G = M ∪ Union (MgH LP M G) | |
1271 | 417 |
1266 | 418 TCS : (G : HOD) → Set (Level.suc n) |
419 TCS G = {x y : HOD} → G ∋ x → x ∋ y → G ∋ y | |
420 | |
1268 | 421 GH : (P L p0 : HOD ) → (LPP : L ⊆ Power P) → (Lp0 : L ∋ p0 ) |
422 → (C : CountableModel ) → HOD | |
423 GH P L p0 LPP Lp0 C = ⊆-Ideal.ideal (genf ( P-GenericFilter P L p0 LPP Lp0 C )) | |
1284 | 424 -- |
425 -- module _ {L P : HOD} (LP : L ⊆ Power P) (M : HOD) (GF : GenericFilter {L} {P} LP M) where | |
426 -- | |
427 -- MG = MG1 {L} {P} LP M GF | |
428 -- G = ⊆-Ideal.ideal (genf GF) | |
429 -- | |
430 -- M⊆MG : M ⊆ MG | |
431 -- M⊆MG = case1 | |
432 -- | |
433 -- MG∋G : MG ∋ G | |
434 -- MG∋G = case2 record { owner = ? ; ao = ? ; ox = ? } where | |
435 -- gf00 : ? | |
436 -- gf00 = ? | |
437 -- | |
438 -- MG∋UG : MG ∋ Union G | |
439 -- MG∋UG = case2 record { owner = ? ; ao = ? ; ox = ? } where | |
440 -- gf00 : ? | |
441 -- gf00 = ? | |
442 -- | |
443 -- |