Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/generic-filter.agda @ 1218:362e43a1477c
brain damaged fix
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Mon, 06 Mar 2023 10:45:34 +0900 |
parents | 42000f20fdbe |
children | 5223f0b40d91 |
rev | line source |
---|---|
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
|
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 | |
457 | 77 -- ∃ q ∈ L ⊆ Power P → q ∈ a(n) ∧ q ⊆ p(n) |
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 → | |
81 odef L x ∧ odef (* (ctl→ C i)) x ∧ ( (y : Ordinal ) → odef (* x) y → odef (* p) y ) } | |
82 ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L (proj1 lt) } | |
431 | 83 |
84 --- | |
464 | 85 -- p(n+1) = if ({q | q ∈ a(n) ∧ q ⊆ p(n))} != ∅ 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 --- | |
457 | 94 -- G = { r ∈ L ⊆ Power P | ∃ n → p(n) ⊆ r } |
431 | 95 -- |
457 | 96 record PDN (L p : HOD ) (C : CountableModel ) (x : Ordinal) : Set n where |
431 | 97 field |
98 gr : Nat | |
457 | 99 pn<gr : (y : Ordinal) → odef (* (find-p L C gr (& p))) y → odef (* x) y |
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))} | |
133 ... | no not = λ lt → proj2 (proj2 fmin∈PGHOD) _ lt 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 |
1096 | 147 P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → (C : CountableModel ) → GenericFilter {L} {P} LP ( ctl-M C ) |
457 | 148 P-GenericFilter P L p0 L⊆PP Lp0 C = record { |
460 | 149 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 } |
150 ; generic = fdense | |
431 | 151 } where |
461 | 152 f⊆PL : PDHOD L p0 C ⊆ L |
1096 | 153 f⊆PL lt = x∈PP lt |
460 | 154 f1 : {p q : HOD} → L ∋ q → PDHOD L p0 C ∋ p → p ⊆ q → PDHOD L p0 C ∋ q |
155 f1 {p} {q} L∋q PD∋p p⊆q = record { gr = gr PD∋p ; pn<gr = f04 ; x∈PP = L∋q } where | |
156 f04 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (* (& q)) y | |
1096 | 157 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 | 158 -- odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (& q)) y |
461 | 159 f2 : {p q : HOD} → PDHOD L p0 C ∋ p → PDHOD L p0 C ∋ q → L ∋ (p ∩ q) → PDHOD L p0 C ∋ (p ∩ q) |
160 f2 {p} {q} PD∋p PD∋q L∋pq with <-cmp (gr PD∋q) (gr PD∋p) | |
161 ... | 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 | 162 f3 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (p ∩ q) y |
448 | 163 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 | 164 f5 : odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (* (find-p L C (gr PD∋q) (& p0))) y |
1096 | 165 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 | 166 (subst (λ k → odef (* (find-p L C (gr PD∋p) (& p0))) k ) (sym &iso) lt) ) |
461 | 167 ... | 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 | 168 f4 : (y : Ordinal) → odef (* (find-p L C (gr PD∋p) (& p0))) y → odef (p ∩ q) y |
447 | 169 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 | 170 ... | 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 | 171 f3 : (y : Ordinal) → odef (* (find-p L C (gr PD∋q) (& p0))) y → odef (p ∩ q) y |
172 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 | |
173 f5 : odef (* (find-p L C (gr PD∋q) (& p0))) y → odef (* (find-p L C (gr PD∋p) (& p0))) y | |
1096 | 174 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 | 175 (subst (λ k → odef (* (find-p L C (gr PD∋q) (& p0))) k ) (sym &iso) lt) ) |
461 | 176 fdense : (D : Dense L⊆PP ) → (ctl-M C ) ∋ Dense.dense D → ¬ (filter.Dense.dense D ∩ PDHOD L p0 C) ≡ od∅ |
177 fdense D MD eq0 = ⊥-elim ( ∅< {Dense.dense D ∩ PDHOD L p0 C} fd01 (≡od∅→=od∅ eq0 )) where | |
448 | 178 open Dense |
462 | 179 fd09 : (i : Nat ) → odef L (find-p L C i (& p0)) |
180 fd09 Zero = Lp0 | |
181 fd09 (Suc i) with is-o∅ ( & ( PGHOD i L C (find-p L C i (& p0))) ) | |
182 ... | yes _ = fd09 i | |
463 | 183 ... | no not = fd17 where |
184 fd19 = ODC.minimal O ( PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
185 fd18 : PGHOD i L C (find-p L C i (& p0)) ∋ fd19 | |
186 fd18 = ODC.x∋minimal O (PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
187 fd17 : odef L ( & (ODC.minimal O ( PGHOD i L C (find-p L C i (& p0))) (λ eq → not (=od∅→≡o∅ eq))) ) | |
188 fd17 = proj1 fd18 | |
461 | 189 an : Nat |
190 an = ctl← C (& (dense D)) MD | |
191 pn : Ordinal | |
192 pn = find-p L C an (& p0) | |
193 pn+1 : Ordinal | |
194 pn+1 = find-p L C (Suc an) (& p0) | |
464 | 195 d=an : dense D ≡ * (ctl→ C an) |
196 d=an = begin dense D ≡⟨ sym *iso ⟩ | |
463 | 197 * ( & (dense D)) ≡⟨ cong (*) (sym (ctl-iso→ C MD )) ⟩ |
198 * (ctl→ C an) ∎ where open ≡-Reasoning | |
461 | 199 fd07 : odef (dense D) pn+1 |
200 fd07 with is-o∅ ( & ( PGHOD an L C (find-p L C an (& p0))) ) | |
462 | 201 ... | yes y = ⊥-elim ( ¬x<0 ( _==_.eq→ fd10 ⟪ fd13 , ⟪ fd14 , fd15 ⟫ ⟫ ) ) where |
202 fd12 : L ∋ * (find-p L C an (& p0)) | |
203 fd12 = subst (λ k → odef L k) (sym &iso) (fd09 an ) | |
204 fd11 : Ordinal | |
205 fd11 = & ( dense-f D fd12 ) | |
206 fd13 : L ∋ ( dense-f D fd12 ) | |
1096 | 207 fd13 = (d⊆P D) ( dense-d D fd12 ) |
462 | 208 fd14 : (* (ctl→ C an)) ∋ ( dense-f D fd12 ) |
464 | 209 fd14 = subst (λ k → odef k (& ( dense-f D fd12 ) )) d=an ( dense-d D fd12 ) |
462 | 210 fd15 : (y : Ordinal) → odef (* (& (dense-f D fd12))) y → odef (* (find-p L C an (& p0))) y |
1096 | 211 fd15 y lt = subst (λ k → odef (* (find-p L C an (& p0))) k ) &iso ( (dense-p D fd12 ) fd16 ) where |
462 | 212 fd16 : odef (dense-f D fd12) (& ( * y)) |
213 fd16 = subst₂ (λ j k → odef j k ) (*iso) (sym &iso) lt | |
214 fd10 : PGHOD an L C (find-p L C an (& p0)) =h= od∅ | |
215 fd10 = ≡o∅→=od∅ y | |
463 | 216 ... | no not = fd27 where |
217 fd29 = ODC.minimal O ( PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
218 fd28 : PGHOD an L C (find-p L C an (& p0)) ∋ fd29 | |
219 fd28 = ODC.x∋minimal O (PGHOD an L C (find-p L C an (& p0))) (λ eq → not (=od∅→≡o∅ eq)) | |
220 fd27 : odef (dense D) (& fd29) | |
464 | 221 fd27 = subst (λ k → odef k (& fd29)) (sym d=an) (proj1 (proj2 fd28)) |
461 | 222 fd03 : odef (PDHOD L p0 C) pn+1 |
223 fd03 = record { gr = Suc an ; pn<gr = λ y lt → lt ; x∈PP = fd09 (Suc an)} | |
224 fd01 : (dense D ∩ PDHOD L p0 C) ∋ (* pn+1) | |
225 fd01 = ⟪ subst (λ k → odef (dense D) k ) (sym &iso) fd07 , subst (λ k → odef (PDHOD L p0 C) k) (sym &iso) fd03 ⟫ | |
448 | 226 |
431 | 227 open GenericFilter |
228 open Filter | |
229 | |
461 | 230 record NonAtomic (L a : HOD ) (L∋a : L ∋ a ) : Set (suc (suc n)) where |
431 | 231 field |
461 | 232 b : HOD |
233 0<b : ¬ o∅ ≡ & b | |
234 b<a : b ⊆ a | |
431 | 235 |
461 | 236 lemma232 : (P L p : HOD ) (C : CountableModel ) |
237 → (LP : L ⊆ Power P ) → (Lp0 : L ∋ p ) | |
238 → ( {q : HOD} → (Lq : L ∋ q ) → NonAtomic L q Lq ) | |
239 → ¬ ( (ctl-M C) ∋ filter ( genf ( P-GenericFilter P L p LP Lp0 C )) ) | |
1101 | 240 lemma232 P L p C LP Lp0 NA MG = {!!} where |
241 D : HOD -- P - G | |
242 D = ? | |
431 | 243 |
244 -- | |
1174 | 245 -- P-Generic Filter defines a countable model D ⊂ C from P |
246 -- | |
247 | |
248 -- | |
249 -- in D, we have V ≠ L | |
250 -- | |
251 | |
252 -- | |
431 | 253 -- val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > } |
254 -- | |
436 | 255 |
1096 | 256 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 | 257 field |
258 valx : HOD | |
436 | 259 |
437 | 260 record valS (ox oy oG : Ordinal) : Set n where |
436 | 261 field |
437 | 262 op : Ordinal |
263 p∈G : odef (* oG) op | |
264 is-val : odef (* ox) ( & < * oy , * op > ) | |
436 | 265 |
459 | 266 val : (x : HOD) {P L : HOD } {LP : L ⊆ Power P} |
1096 | 267 → (G : GenericFilter {L} {P} LP {!!} ) |
436 | 268 → HOD |
437 | 269 val x G = TransFinite {λ x → HOD } ind (& x) where |
270 ind : (x : Ordinal) → ((y : Ordinal) → y o< x → HOD) → HOD | |
439 | 271 ind x valy = record { od = record { def = λ y → valS x y (& (filter (genf G))) } ; odmax = {!!} ; <odmax = {!!} } |
437 | 272 |