Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/generic-filter.agda @ 1241:5f1572d1f19a
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 13 Mar 2023 01:30:55 +0900 |
| parents | fbe072447243 |
| children | e843ace90b31 |
| 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 | |
| 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 | |
| 1241 | 162 -- ＼-⊆ |
| 163 | |
| 164 P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) | |
| 165 → (C : CountableModel ) → GenericFilter {L} {P} LP ( ctl-M C ) | |
| 166 P-GenericFilter P L p0 L⊆PP Lp0 NEG C = record { | |
| 167 genf = record { filter = Replace (PDHOD L p0 C) (λ x → P ＼ x) ; f⊆L = gf01 ; filter1 = ? ; filter2 = ? } | |
| 1240 | 168 ; generic = λ D cd → subst (λ k → ¬ (Dense.dense D ∩ k) ≡ od∅ ) (sym gf00) (fdense D cd ) |
| 431 | 169 } where |
| 1241 | 170 GP = Replace (PDHOD L p0 C) (λ x → P ＼ x) |
| 461 | 171 f⊆PL : PDHOD L p0 C ⊆ L |
| 1096 | 172 f⊆PL lt = x∈PP lt |
| 1241 | 173 gf01 : Replace (PDHOD L p0 C) (λ x → P ＼ x) ⊆ L |
| 174 gf01 {x} record { z = z ; az = az ; x=ψz = x=ψz } = subst (λ k → odef L k) (sym x=ψz) ( NEG (subst (λ k → odef L k) (sym &iso) (f⊆PL az)) ) | |
| 460 | 175 f1 : {p q : HOD} → L ∋ q → PDHOD L p0 C ∋ p → p ⊆ q → PDHOD L p0 C ∋ q |
| 1239 | 176 f1 {p} {q} L∋q PD∋p p⊆q = ? |
| 1241 | 177 f2 : {p q : HOD} → GP ∋ p → GP ∋ q → L ∋ (p ∩ q) → GP ∋ (p ∩ q) |
| 178 f2 {p} {q} record { z = xp ; az = Pp ; x=ψz = peq } | |
| 179 record { z = xq ; az = Pq ; x=ψz = qeq } L∋pq with <-cmp (gr Pp) (gr Pq) | |
| 1239 | 180 ... | tri< a ¬b ¬c = ? |
| 181 ... | tri≈ ¬a eq ¬c = ? | |
| 182 ... | tri> ¬a ¬b c = ? | |
| 1240 | 183 gf00 : Replace (Replace (PDHOD L p0 C) (λ x → P ＼ x)) (_＼_ P) ≡ PDHOD L p0 C |
| 184 gf00 = ? | |
| 185 fdense : (D : Dense {L} {P} L⊆PP ) → (ctl-M C ) ∋ Dense.dense D → ¬ (Dense.dense D ∩ (PDHOD L p0 C)) ≡ od∅ | |
| 1239 | 186 fdense D MD eq0 = ? where |
| 448 | 187 open Dense |
| 188 | |
| 431 | 189 open GenericFilter |
| 190 open Filter | |
| 191 | |
| 461 | 192 record NonAtomic (L a : HOD ) (L∋a : L ∋ a ) : Set (suc (suc n)) where |
| 431 | 193 field |
| 461 | 194 b : HOD |
| 195 0<b : ¬ o∅ ≡ & b | |
| 196 b<a : b ⊆ a | |
| 431 | 197 |
| 461 | 198 lemma232 : (P L p : HOD ) (C : CountableModel ) |
| 1241 | 199 → (LP : L ⊆ Power P ) → (Lp0 : L ∋ p ) ( NEG : {p : HOD} → L ∋ p → L ∋ ( P ＼ p)) |
| 461 | 200 → ( {q : HOD} → (Lq : L ∋ q ) → NonAtomic L q Lq ) |
| 1241 | 201 → ¬ ( (ctl-M C) ∋ rgen ( P-GenericFilter P L p LP Lp0 NEG C )) |
| 202 lemma232 P L p C LP Lp0 NEG NA MG = {!!} where | |
| 1101 | 203 D : HOD -- P - G |
| 204 D = ? | |
| 431 | 205 |
| 206 -- | |
| 1174 | 207 -- P-Generic Filter defines a countable model D ⊂ C from P |
| 208 -- | |
| 209 | |
| 210 -- | |
| 211 -- in D, we have V ≠ L | |
| 212 -- | |
| 213 | |
| 214 -- | |
| 431 | 215 -- val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > } |
| 216 -- | |
| 436 | 217 |
| 1096 | 218 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 | 219 field |
| 220 valx : HOD | |
| 436 | 221 |
| 437 | 222 record valS (ox oy oG : Ordinal) : Set n where |
| 436 | 223 field |
| 437 | 224 op : Ordinal |
| 225 p∈G : odef (* oG) op | |
| 226 is-val : odef (* ox) ( & < * oy , * op > ) | |
| 436 | 227 |
| 459 | 228 val : (x : HOD) {P L : HOD } {LP : L ⊆ Power P} |
| 1096 | 229 → (G : GenericFilter {L} {P} LP {!!} ) |
| 436 | 230 → HOD |
| 437 | 231 val x G = TransFinite {λ x → HOD } ind (& x) where |
| 232 ind : (x : Ordinal) → ((y : Ordinal) → y o< x → HOD) → HOD | |
| 439 | 233 ind x valy = record { od = record { def = λ y → valS x y (& (filter (genf G))) } ; odmax = {!!} ; <odmax = {!!} } |
| 437 | 234 |