Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/generic-filter.agda @ 1205:83ac320583f8
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author | kono |
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date | Fri, 03 Mar 2023 10:42:58 +0800 |
parents | 42000f20fdbe |
children | 362e43a1477c |
rev | line source |
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1200 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
431 | 2 open import Level |
3 open import Ordinals | |
4 module generic-filter {n : Level } (O : Ordinals {n}) where | |
5 | |
6 import filter | |
7 open import zf | |
8 open import logic | |
9 -- open import partfunc {n} O | |
10 import OD | |
11 | |
12 open import Relation.Nullary | |
13 open import Relation.Binary | |
14 open import Data.Empty | |
15 open import Relation.Binary | |
16 open import Relation.Binary.Core | |
17 open import Relation.Binary.PropositionalEquality | |
18 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) | |
1124 | 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 | |
38 open filter O | |
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) | |
53 open import Data.Maybe | |
54 | |
55 import OPair | |
56 open OPair O | |
57 | |
453
e5f0ac638c01
P should be an order structure not Power Ser
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
452
diff
changeset
|
58 record CountableModel : Set (suc (suc n)) where |
431 | 59 field |
461 | 60 ctl-M : HOD |
434 | 61 ctl→ : Nat → Ordinal |
461 | 62 ctl<M : (x : Nat) → odef (ctl-M) (ctl→ x) |
63 ctl← : (x : Ordinal )→ odef (ctl-M ) x → Nat | |
64 ctl-iso→ : { x : Ordinal } → (lt : odef (ctl-M) x ) → ctl→ (ctl← x lt ) ≡ x | |
1174 | 65 -- we have no otherway round |
66 -- ctl-iso← : { x : Nat } → ctl← (ctl→ x ) (ctl<M x) ≡ x | |
446 | 67 -- |
68 -- almmost universe | |
69 -- find-p contains ∃ x : Ordinal → x o< & M → ∀ r ∈ M → ∈ Ord x | |
70 -- | |
436 | 71 |
457 | 72 -- we expect P ∈ * ctl-M ∧ G ⊆ L ⊆ Power P , ¬ G ∈ * ctl-M, |
434 | 73 |
74 open CountableModel | |
431 | 75 |
76 ---- | |
77 -- a(n) ∈ M | |
457 | 78 -- ∃ q ∈ L ⊆ Power P → q ∈ a(n) ∧ q ⊆ p(n) |
431 | 79 -- |
457 | 80 PGHOD : (i : Nat) (L : HOD) (C : CountableModel ) → (p : Ordinal) → HOD |
81 PGHOD i L C p = record { od = record { def = λ x → | |
82 odef L x ∧ odef (* (ctl→ C i)) x ∧ ( (y : Ordinal ) → odef (* x) y → odef (* p) y ) } | |
83 ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L (proj1 lt) } | |
431 | 84 |
85 --- | |
464 | 86 -- p(n+1) = if ({q | q ∈ a(n) ∧ q ⊆ p(n))} != ∅ then q otherwise p(n) |
446 | 87 -- |
457 | 88 find-p : (L : HOD ) (C : CountableModel ) (i : Nat) → (x : Ordinal) → Ordinal |
89 find-p L C Zero x = x | |
90 find-p L C (Suc i) x with is-o∅ ( & ( PGHOD i L C (find-p L C i x)) ) | |
91 ... | yes y = find-p L C i x | |
92 ... | 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 | 93 |
94 --- | |
457 | 95 -- G = { r ∈ L ⊆ Power P | ∃ n → p(n) ⊆ r } |
431 | 96 -- |
457 | 97 record PDN (L p : HOD ) (C : CountableModel ) (x : Ordinal) : Set n where |
431 | 98 field |
99 gr : Nat | |
457 | 100 pn<gr : (y : Ordinal) → odef (* (find-p L C gr (& p))) y → odef (* x) y |
101 x∈PP : odef L x | |
431 | 102 |
103 open PDN | |
104 | |
105 --- | |
106 -- G as a HOD | |
107 -- | |
457 | 108 PDHOD : (L p : HOD ) (C : CountableModel ) → HOD |
109 PDHOD L p C = record { od = record { def = λ x → PDN L p C x } | |
110 ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L {y} (PDN.x∈PP lt) } | |
431 | 111 |
112 open PDN | |
113 | |
114 ---- | |
115 -- Generic Filter on Power P for HOD's Countable Ordinal (G ⊆ Power P ≡ G i.e. Nat → P → Set ) | |
116 -- | |
117 -- p 0 ≡ ∅ | |
434 | 118 -- p (suc n) = if ∃ q ∈ M ∧ p n ⊆ q → q (by axiom of choice) ( q = * ( ctl→ n ) ) |
431 | 119 --- else p n |
120 | |
121 P∅ : {P : HOD} → odef (Power P) o∅ | |
122 P∅ {P} = subst (λ k → odef (Power P) k ) ord-od∅ (lemma o∅ o∅≡od∅) where | |
123 lemma : (x : Ordinal ) → * x ≡ od∅ → odef (Power P) (& od∅) | |
124 lemma x eq = power← P od∅ (λ {x} lt → ⊥-elim (¬x<0 lt )) | |
125 x<y→∋ : {x y : Ordinal} → odef (* x) y → * x ∋ * y | |
126 x<y→∋ {x} {y} lt = subst (λ k → odef (* x) k ) (sym &iso) lt | |
127 | |
446 | 128 open import Data.Nat.Properties |
129 open import nat | |
433 | 130 |
457 | 131 p-monotonic1 : (L p : HOD ) (C : CountableModel ) → {n : Nat} → (* (find-p L C (Suc n) (& p))) ⊆ (* (find-p L C n (& p))) |
1096 | 132 p-monotonic1 L p C {n} {x} with is-o∅ (& (PGHOD n L C (find-p L C n (& p)))) |
133 ... | yes y = refl-⊆ {* (find-p L C n (& p))} | |
134 ... | no not = λ lt → proj2 (proj2 fmin∈PGHOD) _ lt where | |
447 | 135 fmin : HOD |
457 | 136 fmin = ODC.minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) |
137 fmin∈PGHOD : PGHOD n L C (find-p L C n (& p)) ∋ fmin | |
138 fmin∈PGHOD = ODC.x∋minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) | |
438 | 139 |
457 | 140 p-monotonic : (L p : HOD ) (C : CountableModel ) → {n m : Nat} → n ≤ m → (* (find-p L C m (& p))) ⊆ (* (find-p L C n (& p))) |
1096 | 141 p-monotonic L p C {Zero} {Zero} n≤m = refl-⊆ {* (find-p L C Zero (& p))} |
142 p-monotonic L p C {Zero} {Suc m} z≤n lt = (p-monotonic L p C {Zero} {m} z≤n ) (p-monotonic1 L p C {m} lt ) | |
457 | 143 p-monotonic L p C {Suc n} {Suc m} (s≤s n≤m) with <-cmp n m |
1096 | 144 ... | tri< a ¬b ¬c = λ lt → (p-monotonic L p C {Suc n} {m} a) (p-monotonic1 L p C {m} lt ) |
145 ... | tri≈ ¬a refl ¬c = λ x → x | |
446 | 146 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c ) |
438 | 147 |
1096 | 148 P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → (C : CountableModel ) → GenericFilter {L} {P} LP ( ctl-M C ) |
457 | 149 P-GenericFilter P L p0 L⊆PP Lp0 C = record { |
460 | 150 genf = record { filter = PDHOD L p0 C ; f⊆L = f⊆PL ; filter1 = λ L∋q PD∋p p⊆q → f1 L∋q PD∋p p⊆q ; filter2 = f2 } |
151 ; generic = fdense | |
431 | 152 } where |
461 | 153 f⊆PL : PDHOD L p0 C ⊆ L |
1096 | 154 f⊆PL lt = x∈PP lt |
460 | 155 f1 : {p q : HOD} → L ∋ q → PDHOD L p0 C ∋ p → p ⊆ q → PDHOD L p0 C ∋ q |
156 f1 {p} {q} L∋q PD∋p p⊆q = record { gr = gr PD∋p ; pn<gr = f04 ; x∈PP = L∋q } where | |
157 f04 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (* (& q)) y | |
1096 | 158 f04 y lt1 = subst₂ (λ j k → odef j k ) (sym *iso) &iso (p⊆q (subst₂ (λ j k → odef k j ) (sym &iso) *iso ( pn<gr PD∋p y lt1 ))) |
446 | 159 -- odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (& q)) y |
461 | 160 f2 : {p q : HOD} → PDHOD L p0 C ∋ p → PDHOD L p0 C ∋ q → L ∋ (p ∩ q) → PDHOD L p0 C ∋ (p ∩ q) |
161 f2 {p} {q} PD∋p PD∋q L∋pq with <-cmp (gr PD∋q) (gr PD∋p) | |
162 ... | tri< a ¬b ¬c = record { gr = gr PD∋p ; pn<gr = λ y lt → subst (λ k → odef k y ) (sym *iso) (f3 y lt ) ; x∈PP = L∋pq } where | |
460 | 163 f3 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (p ∩ q) y |
448 | 164 f3 y lt = ⟪ subst (λ k → odef k y) *iso (pn<gr PD∋p y lt) , subst (λ k → odef k y) *iso (pn<gr PD∋q y (f5 lt)) ⟫ where |
460 | 165 f5 : odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (* (find-p L C (gr PD∋q) (& p0))) y |
1096 | 166 f5 lt = subst (λ k → odef (* (find-p L C (gr PD∋q) (& p0))) k ) &iso ( (p-monotonic L p0 C {gr PD∋q} {gr PD∋p} (<to≤ a)) |
460 | 167 (subst (λ k → odef (* (find-p L C (gr PD∋p) (& p0))) k ) (sym &iso) lt) ) |
461 | 168 ... | tri≈ ¬a refl ¬c = record { gr = gr PD∋p ; pn<gr = λ y lt → subst (λ k → odef k y ) (sym *iso) (f4 y lt) ; x∈PP = L∋pq } where |
460 | 169 f4 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (p ∩ q) y |
447 | 170 f4 y lt = ⟪ subst (λ k → odef k y) *iso (pn<gr PD∋p y lt) , subst (λ k → odef k y) *iso (pn<gr PD∋q y lt) ⟫ |
461 | 171 ... | tri> ¬a ¬b c = record { gr = gr PD∋q ; pn<gr = λ y lt → subst (λ k → odef k y ) (sym *iso) (f3 y lt) ; x∈PP = L∋pq } where |
460 | 172 f3 : (y : Ordinal) → odef (* (find-p L C (gr PD∋q) (& p0))) y → odef (p ∩ q) y |
173 f3 y lt = ⟪ subst (λ k → odef k y) *iso (pn<gr PD∋p y (f5 lt)), subst (λ k → odef k y) *iso (pn<gr PD∋q y lt) ⟫ where | |
174 f5 : odef (* (find-p L C (gr PD∋q) (& p0))) y → odef (* (find-p L C (gr PD∋p) (& p0))) y | |
1096 | 175 f5 lt = subst (λ k → odef (* (find-p L C (gr PD∋p) (& p0))) k ) &iso ( (p-monotonic L p0 C {gr PD∋p} {gr PD∋q} (<to≤ c)) |
460 | 176 (subst (λ k → odef (* (find-p L C (gr PD∋q) (& p0))) k ) (sym &iso) lt) ) |
461 | 177 fdense : (D : Dense L⊆PP ) → (ctl-M C ) ∋ Dense.dense D → ¬ (filter.Dense.dense D ∩ PDHOD L p0 C) ≡ od∅ |
178 fdense D MD eq0 = ⊥-elim ( ∅< {Dense.dense D ∩ PDHOD L p0 C} fd01 (≡od∅→=od∅ eq0 )) where | |
448 | 179 open Dense |
462 | 180 fd09 : (i : Nat ) → odef L (find-p L C i (& p0)) |
181 fd09 Zero = Lp0 | |
182 fd09 (Suc i) with is-o∅ ( & ( PGHOD i L C (find-p L C i (& p0))) ) | |
183 ... | yes _ = fd09 i | |
463 | 184 ... | no not = fd17 where |
185 fd19 = ODC.minimal O ( PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
186 fd18 : PGHOD i L C (find-p L C i (& p0)) ∋ fd19 | |
187 fd18 = ODC.x∋minimal O (PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
188 fd17 : odef L ( & (ODC.minimal O ( PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq))) ) | |
189 fd17 = proj1 fd18 | |
461 | 190 an : Nat |
191 an = ctl← C (& (dense D)) MD | |
192 pn : Ordinal | |
193 pn = find-p L C an (& p0) | |
194 pn+1 : Ordinal | |
195 pn+1 = find-p L C (Suc an) (& p0) | |
464 | 196 d=an : dense D ≡ * (ctl→ C an) |
197 d=an = begin dense D ≡⟨ sym *iso ⟩ | |
463 | 198 * ( & (dense D)) ≡⟨ cong (*) (sym (ctl-iso→ C MD )) ⟩ |
199 * (ctl→ C an) ∎ where open ≡-Reasoning | |
461 | 200 fd07 : odef (dense D) pn+1 |
201 fd07 with is-o∅ ( & ( PGHOD an L C (find-p L C an (& p0))) ) | |
462 | 202 ... | yes y = ⊥-elim ( ¬x<0 ( _==_.eq→ fd10 ⟪ fd13 , ⟪ fd14 , fd15 ⟫ ⟫ ) ) where |
203 fd12 : L ∋ * (find-p L C an (& p0)) | |
204 fd12 = subst (λ k → odef L k) (sym &iso) (fd09 an ) | |
205 fd11 : Ordinal | |
206 fd11 = & ( dense-f D fd12 ) | |
207 fd13 : L ∋ ( dense-f D fd12 ) | |
1096 | 208 fd13 = (d⊆P D) ( dense-d D fd12 ) |
462 | 209 fd14 : (* (ctl→ C an)) ∋ ( dense-f D fd12 ) |
464 | 210 fd14 = subst (λ k → odef k (& ( dense-f D fd12 ) )) d=an ( dense-d D fd12 ) |
462 | 211 fd15 : (y : Ordinal) → odef (* (& (dense-f D fd12))) y → odef (* (find-p L C an (& p0))) y |
1096 | 212 fd15 y lt = subst (λ k → odef (* (find-p L C an (& p0))) k ) &iso ( (dense-p D fd12 ) fd16 ) where |
462 | 213 fd16 : odef (dense-f D fd12) (& ( * y)) |
214 fd16 = subst₂ (λ j k → odef j k ) (*iso) (sym &iso) lt | |
215 fd10 : PGHOD an L C (find-p L C an (& p0)) =h= od∅ | |
216 fd10 = ≡o∅→=od∅ y | |
463 | 217 ... | no not = fd27 where |
218 fd29 = ODC.minimal O ( PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
219 fd28 : PGHOD an L C (find-p L C an (& p0)) ∋ fd29 | |
220 fd28 = ODC.x∋minimal O (PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
221 fd27 : odef (dense D) (& fd29) | |
464 | 222 fd27 = subst (λ k → odef k (& fd29)) (sym d=an) (proj1 (proj2 fd28)) |
461 | 223 fd03 : odef (PDHOD L p0 C) pn+1 |
224 fd03 = record { gr = Suc an ; pn<gr = λ y lt → lt ; x∈PP = fd09 (Suc an)} | |
225 fd01 : (dense D ∩ PDHOD L p0 C) ∋ (* pn+1) | |
226 fd01 = ⟪ subst (λ k → odef (dense D) k ) (sym &iso) fd07 , subst (λ k → odef (PDHOD L p0 C) k) (sym &iso) fd03 ⟫ | |
448 | 227 |
431 | 228 open GenericFilter |
229 open Filter | |
230 | |
461 | 231 record NonAtomic (L a : HOD ) (L∋a : L ∋ a ) : Set (suc (suc n)) where |
431 | 232 field |
461 | 233 b : HOD |
234 0<b : ¬ o∅ ≡ & b | |
235 b<a : b ⊆ a | |
431 | 236 |
461 | 237 lemma232 : (P L p : HOD ) (C : CountableModel ) |
238 → (LP : L ⊆ Power P ) → (Lp0 : L ∋ p ) | |
239 → ( {q : HOD} → (Lq : L ∋ q ) → NonAtomic L q Lq ) | |
240 → ¬ ( (ctl-M C) ∋ filter ( genf ( P-GenericFilter P L p LP Lp0 C )) ) | |
1101 | 241 lemma232 P L p C LP Lp0 NA MG = {!!} where |
242 D : HOD -- P - G | |
243 D = ? | |
431 | 244 |
245 -- | |
1174 | 246 -- P-Generic Filter defines a countable model D ⊂ C from P |
247 -- | |
248 | |
249 -- | |
250 -- in D, we have V ≠ L | |
251 -- | |
252 | |
253 -- | |
431 | 254 -- val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > } |
255 -- | |
436 | 256 |
1096 | 257 record valR (x : HOD) {P L : HOD} {LP : L ⊆ Power P} (C : CountableModel ) (G : GenericFilter {L} {P} LP (ctl-M C) ) : Set (suc n) where |
437 | 258 field |
259 valx : HOD | |
436 | 260 |
437 | 261 record valS (ox oy oG : Ordinal) : Set n where |
436 | 262 field |
437 | 263 op : Ordinal |
264 p∈G : odef (* oG) op | |
265 is-val : odef (* ox) ( & < * oy , * op > ) | |
436 | 266 |
459 | 267 val : (x : HOD) {P L : HOD } {LP : L ⊆ Power P} |
1096 | 268 → (G : GenericFilter {L} {P} LP {!!} ) |
436 | 269 → HOD |
437 | 270 val x G = TransFinite {λ x → HOD } ind (& x) where |
271 ind : (x : Ordinal) → ((y : Ordinal) → y o< x → HOD) → HOD | |
439 | 272 ind x valy = record { od = record { def = λ y → valS x y (& (filter (genf G))) } ; odmax = {!!} ; <odmax = {!!} } |
437 | 273 |