Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/generic-filter.agda @ 1240:fbe072447243
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sun, 12 Mar 2023 13:02:09 +0900 |
parents | 5223f0b40d91 |
children | 5f1572d1f19a |
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 | |
1218 | 55 open import ZProduct O |
431 | 56 |
453
e5f0ac638c01
P should be an order structure not Power Ser
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
452
diff
changeset
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57 record CountableModel : Set (suc (suc n)) where |
431 | 58 field |
461 | 59 ctl-M : HOD |
434 | 60 ctl→ : Nat → Ordinal |
461 | 61 ctl<M : (x : Nat) → odef (ctl-M) (ctl→ x) |
62 ctl← : (x : Ordinal )→ odef (ctl-M ) x → Nat | |
63 ctl-iso→ : { x : Ordinal } → (lt : odef (ctl-M) x ) → ctl→ (ctl← x lt ) ≡ x | |
1174 | 64 -- we have no otherway round |
65 -- ctl-iso← : { x : Nat } → ctl← (ctl→ x ) (ctl<M x) ≡ x | |
446 | 66 -- |
67 -- almmost universe | |
68 -- find-p contains ∃ x : Ordinal → x o< & M → ∀ r ∈ M → ∈ Ord x | |
69 -- | |
436 | 70 |
457 | 71 -- we expect P ∈ * ctl-M ∧ G ⊆ L ⊆ Power P , ¬ G ∈ * ctl-M, |
434 | 72 |
73 open CountableModel | |
431 | 74 |
75 ---- | |
76 -- a(n) ∈ M | |
1239 | 77 -- ∃ q ∈ L ⊆ Power P → q ∈ a(n) ∧ p(n) ⊆ q |
431 | 78 -- |
457 | 79 PGHOD : (i : Nat) (L : HOD) (C : CountableModel ) → (p : Ordinal) → HOD |
80 PGHOD i L C p = record { od = record { def = λ x → | |
1239 | 81 odef L x ∧ odef (* (ctl→ C i)) x ∧ ( (y : Ordinal ) → odef (* p) y → odef (* x) y ) } |
457 | 82 ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L (proj1 lt) } |
431 | 83 |
84 --- | |
1239 | 85 -- p(n+1) = if ({q | q ∈ a(n) ∧ p(n) ⊆ q)} != ∅ then q otherwise p(n) |
446 | 86 -- |
457 | 87 find-p : (L : HOD ) (C : CountableModel ) (i : Nat) → (x : Ordinal) → Ordinal |
88 find-p L C Zero x = x | |
89 find-p L C (Suc i) x with is-o∅ ( & ( PGHOD i L C (find-p L C i x)) ) | |
90 ... | yes y = find-p L C i x | |
91 ... | 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 | 92 |
93 --- | |
1239 | 94 -- G = { r ∈ L ⊆ Power P | ∃ n → r ⊆ p(n) } |
431 | 95 -- |
457 | 96 record PDN (L p : HOD ) (C : CountableModel ) (x : Ordinal) : Set n where |
431 | 97 field |
98 gr : Nat | |
1239 | 99 pn<gr : (y : Ordinal) → odef (* x) y → odef (* (find-p L C gr (& p))) y |
457 | 100 x∈PP : odef L x |
431 | 101 |
102 open PDN | |
103 | |
104 --- | |
105 -- G as a HOD | |
106 -- | |
457 | 107 PDHOD : (L p : HOD ) (C : CountableModel ) → HOD |
108 PDHOD L p C = record { od = record { def = λ x → PDN L p C x } | |
109 ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L {y} (PDN.x∈PP lt) } | |
431 | 110 |
111 open PDN | |
112 | |
113 ---- | |
114 -- Generic Filter on Power P for HOD's Countable Ordinal (G ⊆ Power P ≡ G i.e. Nat → P → Set ) | |
115 -- | |
116 -- p 0 ≡ ∅ | |
434 | 117 -- p (suc n) = if ∃ q ∈ M ∧ p n ⊆ q → q (by axiom of choice) ( q = * ( ctl→ n ) ) |
431 | 118 --- else p n |
119 | |
120 P∅ : {P : HOD} → odef (Power P) o∅ | |
121 P∅ {P} = subst (λ k → odef (Power P) k ) ord-od∅ (lemma o∅ o∅≡od∅) where | |
122 lemma : (x : Ordinal ) → * x ≡ od∅ → odef (Power P) (& od∅) | |
123 lemma x eq = power← P od∅ (λ {x} lt → ⊥-elim (¬x<0 lt )) | |
124 x<y→∋ : {x y : Ordinal} → odef (* x) y → * x ∋ * y | |
125 x<y→∋ {x} {y} lt = subst (λ k → odef (* x) k ) (sym &iso) lt | |
126 | |
446 | 127 open import Data.Nat.Properties |
128 open import nat | |
433 | 129 |
457 | 130 p-monotonic1 : (L p : HOD ) (C : CountableModel ) → {n : Nat} → (* (find-p L C (Suc n) (& p))) ⊆ (* (find-p L C n (& p))) |
1096 | 131 p-monotonic1 L p C {n} {x} with is-o∅ (& (PGHOD n L C (find-p L C n (& p)))) |
132 ... | yes y = refl-⊆ {* (find-p L C n (& p))} | |
1239 | 133 ... | no not = ? where -- λ lt → proj2 (proj2 fmin∈PGHOD) _ ? where |
447 | 134 fmin : HOD |
457 | 135 fmin = ODC.minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) |
136 fmin∈PGHOD : PGHOD n L C (find-p L C n (& p)) ∋ fmin | |
137 fmin∈PGHOD = ODC.x∋minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) | |
438 | 138 |
457 | 139 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 | 140 p-monotonic L p C {Zero} {Zero} n≤m = refl-⊆ {* (find-p L C Zero (& p))} |
141 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 | 142 p-monotonic L p C {Suc n} {Suc m} (s≤s n≤m) with <-cmp n m |
1096 | 143 ... | tri< a ¬b ¬c = λ lt → (p-monotonic L p C {Suc n} {m} a) (p-monotonic1 L p C {m} lt ) |
144 ... | tri≈ ¬a refl ¬c = λ x → x | |
446 | 145 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c ) |
438 | 146 |
1239 | 147 record Dense {L P : HOD } (LP : L ⊆ Power P) : Set (suc n) where |
148 field | |
149 dense : HOD | |
150 d⊆P : dense ⊆ L | |
151 dense-f : {p : HOD} → L ∋ p → HOD | |
152 dense-d : { p : HOD} → (lt : L ∋ p) → dense ∋ dense-f lt | |
153 dense-p : { p : HOD} → (lt : L ∋ p) → (dense-f lt) ⊆ p | |
154 | |
155 record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (suc n) where | |
156 field | |
157 genf : Filter {L} {P} LP | |
158 generic : (D : Dense {L} {P} LP ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ Replace (Filter.filter genf) (λ x → P \ x )) ≡ od∅ ) | |
159 rgen : HOD | |
160 rgen = Replace (Filter.filter genf) (λ x → P \ x ) | |
161 | |
1096 | 162 P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → (C : CountableModel ) → GenericFilter {L} {P} LP ( ctl-M C ) |
457 | 163 P-GenericFilter P L p0 L⊆PP Lp0 C = record { |
1239 | 164 genf = record { filter = Replace (PDHOD L p0 C) (λ x → P \ x) ; f⊆L = ? ; filter1 = ? ; filter2 = ? } |
1240 | 165 ; generic = λ D cd → subst (λ k → ¬ (Dense.dense D ∩ k) ≡ od∅ ) (sym gf00) (fdense D cd ) |
431 | 166 } where |
461 | 167 f⊆PL : PDHOD L p0 C ⊆ L |
1096 | 168 f⊆PL lt = x∈PP lt |
460 | 169 f1 : {p q : HOD} → L ∋ q → PDHOD L p0 C ∋ p → p ⊆ q → PDHOD L p0 C ∋ q |
1239 | 170 f1 {p} {q} L∋q PD∋p p⊆q = ? |
171 f2 : {p q : HOD} → ? ∋ p → ? ∋ q → L ∋ (p ∩ q) → ? ∋ (p ∩ q) | |
461 | 172 f2 {p} {q} PD∋p PD∋q L∋pq with <-cmp (gr PD∋q) (gr PD∋p) |
1239 | 173 ... | tri< a ¬b ¬c = ? |
174 ... | tri≈ ¬a eq ¬c = ? | |
175 ... | tri> ¬a ¬b c = ? | |
1240 | 176 gf00 : Replace (Replace (PDHOD L p0 C) (λ x → P \ x)) (_\_ P) ≡ PDHOD L p0 C |
177 gf00 = ? | |
178 fdense : (D : Dense {L} {P} L⊆PP ) → (ctl-M C ) ∋ Dense.dense D → ¬ (Dense.dense D ∩ (PDHOD L p0 C)) ≡ od∅ | |
1239 | 179 fdense D MD eq0 = ? where |
448 | 180 open Dense |
181 | |
431 | 182 open GenericFilter |
183 open Filter | |
184 | |
461 | 185 record NonAtomic (L a : HOD ) (L∋a : L ∋ a ) : Set (suc (suc n)) where |
431 | 186 field |
461 | 187 b : HOD |
188 0<b : ¬ o∅ ≡ & b | |
189 b<a : b ⊆ a | |
431 | 190 |
461 | 191 lemma232 : (P L p : HOD ) (C : CountableModel ) |
192 → (LP : L ⊆ Power P ) → (Lp0 : L ∋ p ) | |
193 → ( {q : HOD} → (Lq : L ∋ q ) → NonAtomic L q Lq ) | |
1239 | 194 → ¬ ( (ctl-M C) ∋ rgen ( P-GenericFilter P L p LP Lp0 C )) |
1101 | 195 lemma232 P L p C LP Lp0 NA MG = {!!} where |
196 D : HOD -- P - G | |
197 D = ? | |
431 | 198 |
199 -- | |
1174 | 200 -- P-Generic Filter defines a countable model D ⊂ C from P |
201 -- | |
202 | |
203 -- | |
204 -- in D, we have V ≠ L | |
205 -- | |
206 | |
207 -- | |
431 | 208 -- val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > } |
209 -- | |
436 | 210 |
1096 | 211 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 | 212 field |
213 valx : HOD | |
436 | 214 |
437 | 215 record valS (ox oy oG : Ordinal) : Set n where |
436 | 216 field |
437 | 217 op : Ordinal |
218 p∈G : odef (* oG) op | |
219 is-val : odef (* ox) ( & < * oy , * op > ) | |
436 | 220 |
459 | 221 val : (x : HOD) {P L : HOD } {LP : L ⊆ Power P} |
1096 | 222 → (G : GenericFilter {L} {P} LP {!!} ) |
436 | 223 → HOD |
437 | 224 val x G = TransFinite {λ x → HOD } ind (& x) where |
225 ind : (x : Ordinal) → ((y : Ordinal) → y o< x → HOD) → HOD | |
439 | 226 ind x valy = record { od = record { def = λ y → valS x y (& (filter (genf G))) } ; odmax = {!!} ; <odmax = {!!} } |
437 | 227 |