Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate filter.agda @ 375:8cade5f660bd
Select : (X : HOD ) → ((x : HOD ) → X ∋ x → Set n ) → HOD does not work
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 20 Jul 2020 16:22:44 +0900 |
parents | b265042be254 |
children | 7b6592f0851a |
rev | line source |
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190 | 1 open import Level |
236 | 2 open import Ordinals |
3 module filter {n : Level } (O : Ordinals {n}) where | |
4 | |
190 | 5 open import zf |
236 | 6 open import logic |
7 import OD | |
193 | 8 |
363 | 9 open import Relation.Nullary |
10 open import Relation.Binary | |
11 open import Data.Empty | |
190 | 12 open import Relation.Binary |
13 open import Relation.Binary.Core | |
363 | 14 open import Relation.Binary.PropositionalEquality |
191
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15 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
363 | 16 import BAlgbra |
293 | 17 |
18 open BAlgbra O | |
191
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19 |
236 | 20 open inOrdinal O |
21 open OD O | |
22 open OD.OD | |
277
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23 open ODAxiom odAxiom |
190 | 24 |
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25 import ODC |
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26 |
236 | 27 open _∧_ |
28 open _∨_ | |
29 open Bool | |
30 | |
295 | 31 -- Kunen p.76 and p.53, we use ⊆ |
329 | 32 record Filter ( L : HOD ) : Set (suc n) where |
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33 field |
329 | 34 filter : HOD |
290 | 35 f⊆PL : filter ⊆ Power L |
329 | 36 filter1 : { p q : HOD } → q ⊆ L → filter ∋ p → p ⊆ q → filter ∋ q |
37 filter2 : { p q : HOD } → filter ∋ p → filter ∋ q → filter ∋ (p ∩ q) | |
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38 |
292 | 39 open Filter |
40 | |
329 | 41 record prime-filter { L : HOD } (P : Filter L) : Set (suc (suc n)) where |
295 | 42 field |
43 proper : ¬ (filter P ∋ od∅) | |
329 | 44 prime : {p q : HOD } → filter P ∋ (p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q ) |
292 | 45 |
329 | 46 record ultra-filter { L : HOD } (P : Filter L) : Set (suc (suc n)) where |
295 | 47 field |
48 proper : ¬ (filter P ∋ od∅) | |
329 | 49 ultra : {p : HOD } → p ⊆ L → ( filter P ∋ p ) ∨ ( filter P ∋ ( L \ p) ) |
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50 |
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51 open _⊆_ |
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52 |
329 | 53 trans-⊆ : { A B C : HOD} → A ⊆ B → B ⊆ C → A ⊆ C |
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54 trans-⊆ A⊆B B⊆C = record { incl = λ x → incl B⊆C (incl A⊆B x) } |
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55 |
374 | 56 refl-⊆ : {A : HOD} → A ⊆ A |
57 refl-⊆ {A} = record { incl = λ x → x } | |
58 | |
329 | 59 power→⊆ : ( A t : HOD) → Power A ∋ t → t ⊆ A |
331 | 60 power→⊆ A t PA∋t = record { incl = λ {x} t∋x → ODC.double-neg-eilm O (t1 t∋x) } where |
329 | 61 t1 : {x : HOD } → t ∋ x → ¬ ¬ (A ∋ x) |
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62 t1 = zf.IsZF.power→ isZF A t PA∋t |
292 | 63 |
329 | 64 ∈-filter : {L p : HOD} → (P : Filter L ) → filter P ∋ p → p ⊆ L |
294
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65 ∈-filter {L} {p} P lt = power→⊆ L p ( incl (f⊆PL P) lt ) |
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66 |
329 | 67 ∪-lemma1 : {L p q : HOD } → (p ∪ q) ⊆ L → p ⊆ L |
294
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68 ∪-lemma1 {L} {p} {q} lt = record { incl = λ {x} p∋x → incl lt (case1 p∋x) } |
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69 |
329 | 70 ∪-lemma2 : {L p q : HOD } → (p ∪ q) ⊆ L → q ⊆ L |
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71 ∪-lemma2 {L} {p} {q} lt = record { incl = λ {x} p∋x → incl lt (case2 p∋x) } |
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72 |
329 | 73 q∩q⊆q : {p q : HOD } → (q ∩ p) ⊆ q |
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74 q∩q⊆q = record { incl = λ lt → proj1 lt } |
265 | 75 |
331 | 76 open HOD |
77 | |
295 | 78 ----- |
79 -- | |
80 -- ultra filter is prime | |
81 -- | |
294
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82 |
329 | 83 filter-lemma1 : {L : HOD} → (P : Filter L) → ∀ {p q : HOD } → ultra-filter P → prime-filter P |
295 | 84 filter-lemma1 {L} P u = record { |
85 proper = ultra-filter.proper u | |
86 ; prime = lemma3 | |
87 } where | |
329 | 88 lemma3 : {p q : HOD} → filter P ∋ (p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q ) |
295 | 89 lemma3 {p} {q} lt with ultra-filter.ultra u (∪-lemma1 (∈-filter P lt) ) |
90 ... | case1 p∈P = case1 p∈P | |
91 ... | case2 ¬p∈P = case2 (filter1 P {q ∩ (L \ p)} (∪-lemma2 (∈-filter P lt)) lemma7 lemma8) where | |
331 | 92 lemma5 : ((p ∪ q ) ∩ (L \ p)) =h= (q ∩ (L \ p)) |
294
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93 lemma5 = record { eq→ = λ {x} lt → record { proj1 = lemma4 x lt ; proj2 = proj2 lt } |
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94 ; eq← = λ {x} lt → record { proj1 = case2 (proj1 lt) ; proj2 = proj2 lt } |
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95 } where |
331 | 96 lemma4 : (x : Ordinal ) → odef ((p ∪ q) ∩ (L \ p)) x → odef q x |
294
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97 lemma4 x lt with proj1 lt |
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98 lemma4 x lt | case1 px = ⊥-elim ( proj2 (proj2 lt) px ) |
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99 lemma4 x lt | case2 qx = qx |
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100 lemma6 : filter P ∋ ((p ∪ q ) ∩ (L \ p)) |
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101 lemma6 = filter2 P lt ¬p∈P |
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102 lemma7 : filter P ∋ (q ∩ (L \ p)) |
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103 lemma7 = subst (λ k → filter P ∋ k ) (==→o≡ lemma5 ) lemma6 |
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104 lemma8 : (q ∩ (L \ p)) ⊆ q |
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105 lemma8 = q∩q⊆q |
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106 |
295 | 107 ----- |
108 -- | |
109 -- if Filter contains L, prime filter is ultra | |
110 -- | |
111 | |
329 | 112 filter-lemma2 : {L : HOD} → (P : Filter L) → filter P ∋ L → prime-filter P → ultra-filter P |
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113 filter-lemma2 {L} P f∋L prime = record { |
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114 proper = prime-filter.proper prime |
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115 ; ultra = λ {p} p⊆L → prime-filter.prime prime (lemma p p⊆L) |
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116 } where |
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117 open _==_ |
331 | 118 p+1-p=1 : {p : HOD} → p ⊆ L → L =h= (p ∪ (L \ p)) |
119 eq→ (p+1-p=1 {p} p⊆L) {x} lt with ODC.decp O (odef p x) | |
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120 eq→ (p+1-p=1 {p} p⊆L) {x} lt | yes p∋x = case1 p∋x |
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121 eq→ (p+1-p=1 {p} p⊆L) {x} lt | no ¬p = case2 (record { proj1 = lt ; proj2 = ¬p }) |
331 | 122 eq← (p+1-p=1 {p} p⊆L) {x} ( case1 p∋x ) = subst (λ k → odef L k ) diso (incl p⊆L ( subst (λ k → odef p k) (sym diso) p∋x )) |
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123 eq← (p+1-p=1 {p} p⊆L) {x} ( case2 ¬p ) = proj1 ¬p |
329 | 124 lemma : (p : HOD) → p ⊆ L → filter P ∋ (p ∪ (L \ p)) |
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125 lemma p p⊆L = subst (λ k → filter P ∋ k ) (==→o≡ (p+1-p=1 p⊆L)) f∋L |
293 | 126 |
329 | 127 record Dense (P : HOD ) : Set (suc n) where |
269
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128 field |
329 | 129 dense : HOD |
368 | 130 incl : dense ⊆ Power P |
329 | 131 dense-f : HOD → HOD |
368 | 132 dense-d : { p : HOD} → p ⊆ P → dense ∋ dense-f p |
133 dense-p : { p : HOD} → p ⊆ P → p ⊆ (dense-f p) | |
266 | 134 |
329 | 135 record Ideal ( L : HOD ) : Set (suc n) where |
293 | 136 field |
329 | 137 ideal : HOD |
293 | 138 i⊆PL : ideal ⊆ Power L |
329 | 139 ideal1 : { p q : HOD } → q ⊆ L → ideal ∋ p → q ⊆ p → ideal ∋ q |
140 ideal2 : { p q : HOD } → ideal ∋ p → ideal ∋ q → ideal ∋ (p ∪ q) | |
293 | 141 |
142 open Ideal | |
143 | |
329 | 144 proper-ideal : {L : HOD} → (P : Ideal L ) → {p : HOD} → Set n |
293 | 145 proper-ideal {L} P {p} = ideal P ∋ od∅ |
146 | |
329 | 147 prime-ideal : {L : HOD} → Ideal L → ∀ {p q : HOD } → Set n |
293 | 148 prime-ideal {L} P {p} {q} = ideal P ∋ ( p ∩ q) → ( ideal P ∋ p ) ∨ ( ideal P ∋ q ) |
149 | |
364 | 150 ------- |
363 | 151 -- the set of finite partial functions from ω to 2 |
152 -- | |
153 -- | |
154 | |
155 import OPair | |
156 open OPair O | |
157 | |
370 | 158 data Two : Set n where |
159 i0 : Two | |
160 i1 : Two | |
161 | |
162 record PFunc : Set (suc n) where | |
163 field | |
164 restrict : Nat → Set n | |
165 map : (x : Nat ) → restrict x → Two | |
166 | |
167 open PFunc | |
168 | |
169 record _f⊆_ (f g : PFunc) : Set (suc n) where | |
170 field | |
371 | 171 extend : (x : Nat) → (fr : restrict f x ) → restrict g x |
172 feq : (x : Nat) → (fr : restrict f x ) → map f x fr ≡ map g x (extend x fr) | |
173 | |
372 | 174 open _f⊆_ |
370 | 175 |
374 | 176 record F-Filter {n : Level} (L : Set n) (PL : (L → Set n) → Set n) ( _⊆_ : L → L → Set n) (_∩_ : L → L → L ) : Set (suc n) where |
370 | 177 field |
374 | 178 filter : L → Set n |
371 | 179 f⊆P : PL filter |
374 | 180 filter1 : { p q : L } → PL (λ x → q ⊆ x ) → filter p → p ⊆ q → filter q |
371 | 181 filter2 : { p q : L } → filter p → filter q → filter (p ∩ q) |
182 | |
374 | 183 Filter-is-F : {L : HOD} → (f : Filter L ) → F-Filter HOD (λ p → (x : HOD) → p x → x ⊆ L ) _⊆_ _∩_ |
184 Filter-is-F {L} f = record { | |
185 filter = λ x → Lift (suc n) ((filter f) ∋ x) | |
186 ; f⊆P = λ x f∋x → power→⊆ _ _ (incl ( f⊆PL f ) (lower f∋x )) | |
187 ; filter1 = λ {p} {q} q⊆L f∋p p⊆q → lift ( filter1 f (q⊆L q refl-⊆) (lower f∋p) p⊆q) | |
188 ; filter2 = λ {p} {q} f∋p f∋q → lift ( filter2 f (lower f∋p) (lower f∋q)) | |
189 } | |
190 | |
191 min = Data.Nat._⊓_ | |
192 -- m≤m⊔n = Data.Nat._⊔_ | |
193 open import Data.Nat.Properties | |
194 | |
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195 _f∩_ : (f g : PFunc) → PFunc |
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196 f f∩ g = record { restrict = λ x → (restrict f x ) ∧ (restrict g x ) ∧ ((fr : restrict f x ) → (gr : restrict g x ) → map f x fr ≡ map g x gr) |
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197 ; map = λ x p → map f x (proj1 p) } |
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198 |
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199 _↑_ : (Nat → Two) → Nat → PFunc |
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200 f ↑ i = record { restrict = λ x → Lift n (x ≤ i) ; map = λ x _ → f x } |
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201 |
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202 record Gf (f : Nat → Two) (p : PFunc ) : Set (suc n) where |
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203 field |
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204 gn : Nat |
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205 f<n : (f ↑ gn) f⊆ p |
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206 |
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207 open Gf |
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208 |
374 | 209 GF : (Nat → Two ) → F-Filter PFunc (λ x → Lift (suc n) One ) _f⊆_ _f∩_ |
371 | 210 GF f = record { |
372 | 211 filter = λ p → Gf f p |
374 | 212 ; f⊆P = lift OneObj |
213 ; filter1 = λ {p} {q} _ fp p⊆q → record { gn = gn fp ; f<n = f1 fp p⊆q } | |
214 ; filter2 = λ {p} {q} fp fq → record { gn = min (gn fp) (gn fq) ; f<n = f2 fp fq } | |
372 | 215 } where |
374 | 216 f1 : {p q : PFunc } → (fp : Gf f p ) → ( p⊆q : p f⊆ q ) → (f ↑ gn fp) f⊆ q |
217 f1 {p} {q} fp p⊆q = record { extend = λ x x<g → extend p⊆q x (extend (f<n fp ) x x<g) ; feq = λ x fr → {!!} } where | |
218 f2 : {p q : PFunc } → (fp : Gf f p ) → (fq : Gf f q ) → (f ↑ (min (gn fp) (gn fq))) f⊆ (p f∩ q) | |
219 f2 {p} {q} fp fq = record { extend = λ x x<g → record { | |
220 proj1 = extend (f<n fp ) x (lift ( ≤-trans (lower x<g) (m⊓n≤m _ _))) | |
221 ; proj2 = record {proj1 = extend (f<n fq ) x (lift ( ≤-trans (lower x<g) (m⊓n≤n _ _))) | |
222 ; proj2 = {!!} }} ; | |
223 feq = λ x fr → {!!} } where | |
370 | 224 |
363 | 225 ODSuc : (y : HOD) → infinite ∋ y → HOD |
226 ODSuc y lt = Union (y , (y , y)) | |
227 | |
366 | 228 data Hω2 : (i : Nat) ( x : Ordinal ) → Set n where |
229 hφ : Hω2 0 o∅ | |
230 h0 : {i : Nat} {x : Ordinal } → Hω2 i x → | |
231 Hω2 (Suc i) (od→ord (Union ((< nat→ω i , nat→ω 0 >) , ord→od x ))) | |
232 h1 : {i : Nat} {x : Ordinal } → Hω2 i x → | |
233 Hω2 (Suc i) (od→ord (Union ((< nat→ω i , nat→ω 1 >) , ord→od x ))) | |
234 he : {i : Nat} {x : Ordinal } → Hω2 i x → | |
235 Hω2 (Suc i) x | |
236 | |
237 record Hω2r (x : Ordinal) : Set n where | |
238 field | |
239 count : Nat | |
240 hω2 : Hω2 count x | |
241 | |
242 open Hω2r | |
363 | 243 |
244 HODω2 : HOD | |
366 | 245 HODω2 = record { od = record { def = λ x → Hω2r x } ; odmax = next o∅ ; <odmax = odmax0 } where |
365 | 246 ω<next : {y : Ordinal} → infinite-d y → y o< next o∅ |
247 ω<next = ω<next-o∅ ho< | |
366 | 248 lemma : {i j : Nat} {x : Ordinal } → od→ord (Union (< nat→ω i , nat→ω j > , ord→od x)) o< next x |
249 lemma = {!!} | |
250 odmax0 : {y : Ordinal} → Hω2r y → y o< next o∅ | |
251 odmax0 {y} r with hω2 r | |
252 ... | hφ = x<nx | |
253 ... | h0 {i} {x} t = next< (odmax0 record { count = i ; hω2 = t }) (lemma {i} {0} {x}) | |
254 ... | h1 {i} {x} t = next< (odmax0 record { count = i ; hω2 = t }) (lemma {i} {1} {x}) | |
255 ... | he {i} {x} t = next< (odmax0 record { count = i ; hω2 = t }) x<nx | |
363 | 256 |
370 | 257 ω→2 : HOD |
258 ω→2 = Replace (Power infinite) (λ p p⊆i → Replace infinite (λ x i∋x → < x , repl p x > )) where | |
259 repl : HOD → HOD → HOD | |
260 repl p x with ODC.∋-p O p x | |
261 ... | yes _ = nat→ω 1 | |
262 ... | no _ = nat→ω 0 | |
368 | 263 |
370 | 264 record _↑n (f : HOD) (ω→2∋f : ω→2 ∋ f ) : Set n where |
265 -- field | |
266 -- n : HOD | |
267 -- ? : Select f (λ x f∋x → ω→nat (π1 f∋x) < ω→nat n | |
268 | |
372 | 269 -- Gf : {f : HOD} → ω→2 ∋ f → HOD |
270 -- Gf {f} lt = Select HODω2 (λ x H∋x → {!!} ) | |
368 | 271 |
272 G : (Nat → Two) → Filter HODω2 | |
273 G f = record { | |
365 | 274 filter = {!!} |
275 ; f⊆PL = {!!} | |
276 ; filter1 = {!!} | |
277 ; filter2 = {!!} | |
278 } where | |
279 filter0 : HOD | |
280 filter0 = {!!} | |
281 f⊆PL1 : filter0 ⊆ Power HODω2 | |
282 f⊆PL1 = {!!} | |
283 filter11 : { p q : HOD } → q ⊆ HODω2 → filter0 ∋ p → p ⊆ q → filter0 ∋ q | |
284 filter11 = {!!} | |
285 filter12 : { p q : HOD } → filter0 ∋ p → filter0 ∋ q → filter0 ∋ (p ∩ q) | |
286 filter12 = {!!} | |
287 | |
363 | 288 -- the set of finite partial functions from ω to 2 |
289 | |
290 Hω2f : Set (suc n) | |
291 Hω2f = (Nat → Set n) → Two | |
292 | |
293 Hω2f→Hω2 : Hω2f → HOD | |
294 Hω2f→Hω2 p = record { od = record { def = λ x → (p {!!} ≡ i0 ) ∨ (p {!!} ≡ i1 )}; odmax = {!!} ; <odmax = {!!} } | |
295 | |
296 |