Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/generic-filter.agda @ 453:e5f0ac638c01
P should be an order structure not Power Ser
definition of dense is wrong
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 15 Mar 2022 14:09:20 +0900 |
| parents | 76aba34438f2 |
| children | d5909d3c725a |
| rev | line source |
|---|---|
| 431 | 1 open import Level |
| 2 open import Ordinals | |
| 3 module generic-filter {n : Level } (O : Ordinals {n}) where | |
| 4 | |
| 5 import filter | |
| 6 open import zf | |
| 7 open import logic | |
| 8 -- open import partfunc {n} O | |
| 9 import OD | |
| 10 | |
| 11 open import Relation.Nullary | |
| 12 open import Relation.Binary | |
| 13 open import Data.Empty | |
| 14 open import Relation.Binary | |
| 15 open import Relation.Binary.Core | |
| 16 open import Relation.Binary.PropositionalEquality | |
| 17 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) | |
| 18 import BAlgbra | |
| 19 | |
| 20 open BAlgbra O | |
| 21 | |
| 22 open inOrdinal O | |
| 23 open OD O | |
| 24 open OD.OD | |
| 25 open ODAxiom odAxiom | |
| 26 import OrdUtil | |
| 27 import ODUtil | |
| 28 open Ordinals.Ordinals O | |
| 29 open Ordinals.IsOrdinals isOrdinal | |
| 30 open Ordinals.IsNext isNext | |
| 31 open OrdUtil O | |
| 32 open ODUtil O | |
| 33 | |
| 34 | |
| 35 import ODC | |
| 36 | |
| 37 open filter O | |
| 38 | |
| 39 open _∧_ | |
| 40 open _∨_ | |
| 41 open Bool | |
| 42 | |
| 43 | |
| 44 open HOD | |
| 45 | |
| 46 ------- | |
| 47 -- the set of finite partial functions from ω to 2 | |
| 48 -- | |
| 49 -- | |
| 50 | |
| 51 open import Data.List hiding (filter) | |
| 52 open import Data.Maybe | |
| 53 | |
| 54 import OPair | |
| 55 open OPair O | |
| 56 | |
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453
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P should be an order structure not Power Ser
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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57 record CountableModel : Set (suc (suc n)) where |
| 431 | 58 field |
| 434 | 59 ctl-M : Ordinal |
| 60 ctl→ : Nat → Ordinal | |
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P should be an order structure not Power Ser
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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61 ctl<M : (x : Nat) → odef (* ctl-M) (ctl→ x) |
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P should be an order structure not Power Ser
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parents:
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62 ctl← : (x : Ordinal )→ odef (* ctl-M ) x → Nat |
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parents:
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63 ctl-iso→ : { x : Ordinal } → (lt : odef (* ctl-M) x ) → ctl→ (ctl← x lt ) ≡ x |
| 446 | 64 ctl-iso← : { x : Nat } → ctl← (ctl→ x ) (ctl<M x) ≡ x |
| 65 -- | |
| 66 -- almmost universe | |
| 67 -- find-p contains ∃ x : Ordinal → x o< & M → ∀ r ∈ M → ∈ Ord x | |
| 68 -- | |
| 436 | 69 |
| 446 | 70 -- we expect P ∈ * ctl-M ∧ G ⊆ Power P , ¬ G ∈ * ctl-M, |
| 434 | 71 |
| 72 open CountableModel | |
| 431 | 73 |
| 74 ---- | |
| 75 -- a(n) ∈ M | |
| 76 -- ∃ q ∈ Power P → q ∈ a(n) ∧ p(n) ⊆ q | |
| 77 -- | |
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P should be an order structure not Power Ser
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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78 PGHOD : (i : Nat) (P : HOD) (C : CountableModel ) → (p : Ordinal) → HOD |
| 436 | 79 PGHOD i P C p = record { od = record { def = λ x → |
| 431 | 80 odef (Power P) x ∧ odef (* (ctl→ C i)) x ∧ ( (y : Ordinal ) → odef (* p) y → odef (* x) y ) } |
| 81 ; odmax = odmax (Power P) ; <odmax = λ {y} lt → <odmax (Power P) (proj1 lt) } | |
| 82 | |
| 83 --- | |
| 436 | 84 -- p(n+1) = if (f n) != ∅ then (f n) otherwise p(n) |
| 446 | 85 -- |
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P should be an order structure not Power Ser
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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86 find-p : (P : HOD ) (C : CountableModel ) (i : Nat) → (x : Ordinal) → Ordinal |
| 436 | 87 find-p P C Zero x = x |
| 447 | 88 find-p P C (Suc i) x with is-o∅ ( & ( PGHOD i P C (find-p P C i x)) ) |
| 89 ... | yes y = find-p P C i x | |
| 90 ... | no not = & (ODC.minimal O ( PGHOD i P C (find-p P C i x)) (λ eq → not (=od∅→≡o∅ eq))) -- axiom of choice | |
| 431 | 91 |
| 92 --- | |
| 450 | 93 -- G = { r ∈ Power P | ∃ n → p(n) ⊆ r } |
| 431 | 94 -- |
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P should be an order structure not Power Ser
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
452
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95 record PDN (P p : HOD ) (C : CountableModel ) (x : Ordinal) : Set n where |
| 431 | 96 field |
| 97 gr : Nat | |
| 446 | 98 pn<gr : (y : Ordinal) → odef (* (find-p P C gr (& p))) y → odef (* x) y |
| 431 | 99 x∈PP : odef (Power P) x |
| 100 | |
| 101 open PDN | |
| 102 | |
| 103 --- | |
| 104 -- G as a HOD | |
| 105 -- | |
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P should be an order structure not Power Ser
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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106 PDHOD : (P p : HOD ) (C : CountableModel ) → HOD |
| 436 | 107 PDHOD P p C = record { od = record { def = λ x → PDN P p C x } |
| 431 | 108 ; odmax = odmax (Power P) ; <odmax = λ {y} lt → <odmax (Power P) {y} (PDN.x∈PP lt) } |
| 109 | |
| 110 open PDN | |
| 111 | |
| 112 ---- | |
| 113 -- Generic Filter on Power P for HOD's Countable Ordinal (G ⊆ Power P ≡ G i.e. Nat → P → Set ) | |
| 114 -- | |
| 115 -- p 0 ≡ ∅ | |
| 434 | 116 -- p (suc n) = if ∃ q ∈ M ∧ p n ⊆ q → q (by axiom of choice) ( q = * ( ctl→ n ) ) |
| 431 | 117 --- else p n |
| 118 | |
| 119 P∅ : {P : HOD} → odef (Power P) o∅ | |
| 120 P∅ {P} = subst (λ k → odef (Power P) k ) ord-od∅ (lemma o∅ o∅≡od∅) where | |
| 121 lemma : (x : Ordinal ) → * x ≡ od∅ → odef (Power P) (& od∅) | |
| 122 lemma x eq = power← P od∅ (λ {x} lt → ⊥-elim (¬x<0 lt )) | |
| 123 x<y→∋ : {x y : Ordinal} → odef (* x) y → * x ∋ * y | |
| 124 x<y→∋ {x} {y} lt = subst (λ k → odef (* x) k ) (sym &iso) lt | |
| 125 | |
| 446 | 126 open import Data.Nat.Properties |
| 127 open import nat | |
| 433 | 128 open _⊆_ |
| 129 | |
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P should be an order structure not Power Ser
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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130 p-monotonic1 : (P p : HOD ) (C : CountableModel ) → {n : Nat} → (* (find-p P C n (& p))) ⊆ (* (find-p P C (Suc n) (& p))) |
| 447 | 131 p-monotonic1 P p C {n} with is-o∅ (& (PGHOD n P C (find-p P C n (& p)))) |
| 132 ... | yes y = refl-⊆ | |
| 133 ... | no not = record { incl = λ {x} lt → proj2 (proj2 fmin∈PGHOD) (& x) lt } where | |
| 134 fmin : HOD | |
| 135 fmin = ODC.minimal O (PGHOD n P C (find-p P C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) | |
| 136 fmin∈PGHOD : PGHOD n P C (find-p P C n (& p)) ∋ fmin | |
| 137 fmin∈PGHOD = ODC.x∋minimal O (PGHOD n P C (find-p P C n (& p))) (λ eq → not (=od∅→≡o∅ eq)) | |
| 438 | 138 |
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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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139 p-monotonic : (P p : HOD ) (C : CountableModel ) → {n m : Nat} → n ≤ m → (* (find-p P C n (& p))) ⊆ (* (find-p P C m (& p))) |
| 446 | 140 p-monotonic P p C {Zero} {Zero} n≤m = refl-⊆ |
| 141 p-monotonic P p C {Zero} {Suc m} z≤n = trans-⊆ (p-monotonic P p C {Zero} {m} z≤n ) (p-monotonic1 P p C {m} ) | |
| 142 p-monotonic P p C {Suc n} {Suc m} (s≤s n≤m) with <-cmp n m | |
| 447 | 143 ... | tri< a ¬b ¬c = trans-⊆ (p-monotonic P p C {Suc n} {m} a) (p-monotonic1 P p C {m} ) |
| 446 | 144 ... | tri≈ ¬a refl ¬c = refl-⊆ |
| 145 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c ) | |
| 438 | 146 |
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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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147 P-GenericFilter : (P p0 : HOD ) → Power P ∋ p0 → (C : CountableModel ) → GenericFilter P |
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440
d1c9f5ae5d0a
give up this generic filter definition
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148 P-GenericFilter P p0 Pp0 C = record { |
| 436 | 149 genf = record { filter = PDHOD P p0 C ; f⊆PL = f⊆PL ; filter1 = f1 ; filter2 = f2 } |
| 448 | 150 ; generic = fdense |
| 431 | 151 } where |
| 436 | 152 PGHOD∈PL : (i : Nat) → (x : Ordinal) → PGHOD i P C x ⊆ Power P |
| 434 | 153 PGHOD∈PL i x = record { incl = λ {x} p → proj1 p } |
| 436 | 154 f⊆PL : PDHOD P p0 C ⊆ Power P |
| 446 | 155 f⊆PL = record { incl = λ {x} lt → x∈PP lt } |
| 436 | 156 f1 : {p q : HOD} → q ⊆ P → PDHOD P p0 C ∋ p → p ⊆ q → PDHOD P p0 C ∋ q |
| 446 | 157 f1 {p} {q} q⊆P PD∋p p⊆q = record { gr = gr PD∋p ; pn<gr = f04 ; x∈PP = power← _ _ (incl q⊆P) } where |
| 158 f04 : (y : Ordinal) → odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (& q)) y | |
| 159 f04 y lt1 = subst₂ (λ j k → odef j k ) (sym *iso) &iso (incl p⊆q (subst₂ (λ j k → odef k j ) (sym &iso) *iso ( pn<gr PD∋p y lt1 ))) | |
| 160 -- odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (& q)) y | |
| 436 | 161 f2 : {p q : HOD} → PDHOD P p0 C ∋ p → PDHOD P p0 C ∋ q → PDHOD P p0 C ∋ (p ∩ q) |
| 447 | 162 f2 {p} {q} PD∋p PD∋q with <-cmp (gr PD∋p) (gr PD∋q) |
| 163 ... | 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 = ODC.power-∩ O (x∈PP PD∋p) (x∈PP PD∋q) } where | |
| 164 f3 : (y : Ordinal) → odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (p ∩ q) y | |
| 448 | 165 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 |
| 166 f5 : odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (* (find-p P C (gr PD∋q) (& p0))) y | |
| 167 f5 lt = subst (λ k → odef (* (find-p P C (gr PD∋q) (& p0))) k ) &iso ( incl (p-monotonic P p0 C {gr PD∋p} {gr PD∋q} (<to≤ a)) | |
| 168 (subst (λ k → odef (* (find-p P C (gr PD∋p) (& p0))) k ) (sym &iso) lt) ) | |
| 447 | 169 ... | 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 = ODC.power-∩ O (x∈PP PD∋p) (x∈PP PD∋q) } where |
| 170 f4 : (y : Ordinal) → odef (* (find-p P C (gr PD∋p) (& p0))) y → odef (p ∩ q) y | |
| 171 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) ⟫ | |
| 448 | 172 ... | 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 = ODC.power-∩ O (x∈PP PD∋p) (x∈PP PD∋q) } where |
| 173 f3 : (y : Ordinal) → odef (* (find-p P C (gr PD∋q) (& p0))) y → odef (p ∩ q) y | |
| 174 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 | |
| 175 f5 : odef (* (find-p P C (gr PD∋q) (& p0))) y → odef (* (find-p P C (gr PD∋p) (& p0))) y | |
| 176 f5 lt = subst (λ k → odef (* (find-p P C (gr PD∋p) (& p0))) k ) &iso ( incl (p-monotonic P p0 C {gr PD∋q} {gr PD∋p} (<to≤ c)) | |
| 177 (subst (λ k → odef (* (find-p P C (gr PD∋q) (& p0))) k ) (sym &iso) lt) ) | |
| 178 fdense : (D : Dense P ) → ¬ (filter.Dense.dense D ∩ PDHOD P p0 C) ≡ od∅ | |
| 179 fdense D eq0 = ⊥-elim ( ∅< {Dense.dense D ∩ PDHOD P p0 C} fd01 (≡od∅→=od∅ eq0 )) where | |
| 180 open Dense | |
| 450 | 181 p0⊆P : p0 ⊆ P |
| 182 p0⊆P = ODC.power→⊆ O _ _ Pp0 | |
| 448 | 183 fd : HOD |
| 450 | 184 fd = dense-f D p0⊆P |
| 448 | 185 PP∋D : dense D ⊆ Power P |
| 186 PP∋D = d⊆P D | |
| 449 | 187 fd00 : PDHOD P p0 C ∋ p0 |
| 188 fd00 = record { gr = 0 ; pn<gr = λ y lt → lt ; x∈PP = Pp0 } | |
| 450 | 189 fd02 : dense D ∋ dense-f D p0⊆P |
| 190 fd02 = dense-d D p0⊆P | |
| 191 fd04 : dense-f D p0⊆P ⊆ P | |
| 449 | 192 fd04 = ODC.power→⊆ O _ _ ( incl PP∋D fd02 ) |
| 450 | 193 fd03 : PDHOD P p0 C ∋ dense-f D p0⊆P |
| 194 fd03 = f1 {p0} {dense-f D p0⊆P} fd04 fd00 ( dense-p D (ODC.power→⊆ O _ _ Pp0 ) ) | |
| 448 | 195 fd01 : (dense D ∩ PDHOD P p0 C) ∋ fd |
| 196 fd01 = ⟪ fd02 , fd03 ⟫ | |
| 197 | |
| 431 | 198 open GenericFilter |
| 199 open Filter | |
| 200 | |
| 450 | 201 record Incompatible (P p : HOD ) (PP∋p : p ⊆ P ) : Set (suc (suc n)) where |
| 431 | 202 field |
| 450 | 203 q r : HOD |
| 204 PP∋q : q ⊆ P | |
| 205 PP∋r : r ⊆ P | |
| 206 p⊆q : p ⊆ q | |
| 207 p⊆r : p ⊆ r | |
| 208 incompatible : ∀ ( s : HOD ) → s ⊆ P → (¬ ( q ⊆ s )) ∨ (¬ ( r ⊆ s )) | |
| 431 | 209 |
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453
e5f0ac638c01
P should be an order structure not Power Ser
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
452
diff
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210 lemma725 : (P p : HOD ) (C : CountableModel ) |
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parents:
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211 → (PP∋p : Power P ∋ p ) |
| 450 | 212 → * (ctl-M C) ∋ (Power P ∩ * (ctl-M C)) -- M is a Model of ZF |
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213 → * (ctl-M C) ∋ ( (Power P ∩ * (ctl-M C)) ＼ filter ( genf ( P-GenericFilter P p PP∋p C)) ) -- M ∋ G and M is a Model of ZF |
| 450 | 214 → ((p : HOD) → (PP∋p : p ⊆ P ) → Incompatible P p PP∋p ) |
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215 → ¬ ( * (ctl-M C) ∋ filter ( genf ( P-GenericFilter P p PP∋p C ))) |
| 450 | 216 lemma725 P p C PP∋p M∋PM M∋D I M∋G = D∩G≠∅ D∩G=∅ where |
| 217 G = filter ( genf ( P-GenericFilter P p PP∋p C )) | |
| 218 M = * (ctl-M C) | |
| 219 D : HOD | |
| 220 D = Power P ＼ G | |
| 221 p⊆P : p ⊆ P | |
| 222 p⊆P = ODC.power→⊆ O _ _ PP∋p | |
| 223 df : {x : HOD} → x ⊆ P → HOD | |
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224 df {x} PP∋x with ODC.∋-p O G ( Incompatible.r (I x PP∋x) ) |
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225 ... | yes y = Incompatible.q (I x PP∋x) |
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226 ... | no n = Incompatible.r (I x PP∋x) |
| 452 | 227 df¬⊆P : {x : HOD} → (lt : x ⊆ P) → df lt ⊆ P |
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228 df¬⊆P {x} PP∋x with ODC.∋-p O G ( Incompatible.r (I x PP∋x) ) |
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229 ... | yes _ = Incompatible.PP∋q (I x PP∋x) |
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230 ... | no _ = Incompatible.PP∋r (I x PP∋x) |
| 452 | 231 ¬G∋df : {x : HOD} → (lt : x ⊆ P) → ¬ G ∋ (df lt ) |
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232 ¬G∋df {x} lt with ODC.∋-p O G ( Incompatible.r (I x lt ) ) |
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233 ... | no n = n |
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234 ... | yes y with Incompatible.incompatible (I x lt ) (Incompatible.q (I x lt )) (Incompatible.PP∋q (I x lt )) |
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235 ... | case1 ¬q⊆pn = λ _ → ¬q⊆pn refl-⊆ |
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236 ... | case2 ¬r⊆pn = {!!} |
| 450 | 237 df-d : {x : HOD} → (lt : x ⊆ P) → D ∋ df lt |
| 452 | 238 df-d {x} lt = ⟪ power← P _ (incl (df¬⊆P lt)) , ¬G∋df lt ⟫ |
| 450 | 239 df-p : {x : HOD} → (lt : x ⊆ P) → x ⊆ df lt |
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240 df-p {x} lt with ODC.∋-p O G ( Incompatible.r (I x lt) ) |
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241 ... | yes _ = Incompatible.p⊆q (I x lt) |
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242 ... | no _ = Incompatible.p⊆r (I x lt) |
| 450 | 243 D-Dense : Dense P |
| 244 D-Dense = record { | |
| 245 dense = D | |
| 246 ; d⊆P = record { incl = λ {x} lt → proj1 lt } | |
| 247 ; dense-f = df | |
| 248 ; dense-d = df-d | |
| 249 ; dense-p = df-p | |
| 250 } | |
| 251 D∩G=∅ : ( D ∩ G ) =h= od∅ | |
| 451 | 252 D∩G=∅ = ≡od∅→=od∅ ([a-b]∩b=0 {Power P} {G}) |
| 450 | 253 D∩G≠∅ : ¬ (( D ∩ G ) =h= od∅ ) |
| 254 D∩G≠∅ eq = generic (P-GenericFilter P p PP∋p C) D-Dense ( ==→o≡ eq ) | |
| 431 | 255 |
| 433 | 256 open import PFOD O |
| 257 | |
| 258 -- HODω2 : HOD | |
| 259 -- | |
| 260 -- ω→2 : HOD | |
| 261 -- ω→2 = Power infinite | |
| 262 | |
| 450 | 263 lemma725-1 : (p : HOD) → (PP∋p : p ⊆ HODω2 ) → Incompatible HODω2 p PP∋p |
| 431 | 264 lemma725-1 = {!!} |
| 265 | |
|
453
e5f0ac638c01
P should be an order structure not Power Ser
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
452
diff
changeset
|
266 lemma726 : (C : CountableModel ) |
| 450 | 267 → Union ( Replace' (Power HODω2) (λ p lt → filter ( genf ( P-GenericFilter HODω2 p lt C )))) =h= ω→2 -- HODω2 ∋ p |
| 431 | 268 lemma726 = {!!} |
| 269 | |
| 270 -- | |
| 271 -- val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > } | |
| 272 -- | |
| 436 | 273 |
| 437 | 274 record valR (x : HOD) {P : HOD} (G : GenericFilter P) : Set (suc n) where |
| 275 field | |
| 276 valx : HOD | |
| 436 | 277 |
| 437 | 278 record valS (ox oy oG : Ordinal) : Set n where |
| 436 | 279 field |
| 437 | 280 op : Ordinal |
| 281 p∈G : odef (* oG) op | |
| 282 is-val : odef (* ox) ( & < * oy , * op > ) | |
| 436 | 283 |
| 437 | 284 val : (x : HOD) {P : HOD } |
| 436 | 285 → (G : GenericFilter P) |
| 286 → HOD | |
| 437 | 287 val x G = TransFinite {λ x → HOD } ind (& x) where |
| 288 ind : (x : Ordinal) → ((y : Ordinal) → y o< x → HOD) → HOD | |
| 439 | 289 ind x valy = record { od = record { def = λ y → valS x y (& (filter (genf G))) } ; odmax = {!!} ; <odmax = {!!} } |
| 437 | 290 |
| 436 | 291 |
| 292 -- | |
| 431 | 293 -- W (ω , H ( ω , 2 )) = { p ∈ ( Nat → H (ω , 2) ) | { i ∈ Nat → p i ≠ i1 } is finite } |
| 294 -- | |
| 295 | |
| 296 | |
| 297 |