Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate filter.agda @ 387:8b0715e28b33
...
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Sat, 25 Jul 2020 09:09:00 +0900 |
parents | 24a66a246316 |
children | 19687f3304c9 |
rev | line source |
---|---|
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 Data.Empty | |
190 | 11 open import Relation.Binary.Core |
363 | 12 open import Relation.Binary.PropositionalEquality |
13 import BAlgbra | |
293 | 14 |
15 open BAlgbra O | |
191
9eb6a8691f02
choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
190
diff
changeset
|
16 |
236 | 17 open inOrdinal O |
18 open OD O | |
19 open OD.OD | |
277
d9d3654baee1
seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
276
diff
changeset
|
20 open ODAxiom odAxiom |
190 | 21 |
294
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
22 import ODC |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
23 |
236 | 24 open _∧_ |
25 open _∨_ | |
26 open Bool | |
27 | |
295 | 28 -- Kunen p.76 and p.53, we use ⊆ |
329 | 29 record Filter ( L : HOD ) : Set (suc n) where |
191
9eb6a8691f02
choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
190
diff
changeset
|
30 field |
329 | 31 filter : HOD |
290 | 32 f⊆PL : filter ⊆ Power L |
329 | 33 filter1 : { p q : HOD } → q ⊆ L → filter ∋ p → p ⊆ q → filter ∋ q |
34 filter2 : { p q : HOD } → filter ∋ p → filter ∋ q → filter ∋ (p ∩ q) | |
191
9eb6a8691f02
choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
190
diff
changeset
|
35 |
292 | 36 open Filter |
37 | |
329 | 38 record prime-filter { L : HOD } (P : Filter L) : Set (suc (suc n)) where |
295 | 39 field |
40 proper : ¬ (filter P ∋ od∅) | |
329 | 41 prime : {p q : HOD } → filter P ∋ (p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q ) |
292 | 42 |
329 | 43 record ultra-filter { L : HOD } (P : Filter L) : Set (suc (suc n)) where |
295 | 44 field |
45 proper : ¬ (filter P ∋ od∅) | |
329 | 46 ultra : {p : HOD } → p ⊆ L → ( filter P ∋ p ) ∨ ( filter P ∋ ( L \ p) ) |
294
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
47 |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
48 open _⊆_ |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
49 |
329 | 50 ∈-filter : {L p : HOD} → (P : Filter L ) → filter P ∋ p → p ⊆ L |
294
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
51 ∈-filter {L} {p} P lt = power→⊆ L p ( incl (f⊆PL P) lt ) |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
52 |
329 | 53 ∪-lemma1 : {L p q : HOD } → (p ∪ q) ⊆ L → p ⊆ L |
294
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
54 ∪-lemma1 {L} {p} {q} lt = record { incl = λ {x} p∋x → incl lt (case1 p∋x) } |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
55 |
329 | 56 ∪-lemma2 : {L p q : HOD } → (p ∪ q) ⊆ L → q ⊆ L |
294
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
57 ∪-lemma2 {L} {p} {q} lt = record { incl = λ {x} p∋x → incl lt (case2 p∋x) } |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
58 |
329 | 59 q∩q⊆q : {p q : HOD } → (q ∩ p) ⊆ q |
294
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
60 q∩q⊆q = record { incl = λ lt → proj1 lt } |
265 | 61 |
331 | 62 open HOD |
63 | |
295 | 64 ----- |
65 -- | |
66 -- ultra filter is prime | |
67 -- | |
294
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
68 |
329 | 69 filter-lemma1 : {L : HOD} → (P : Filter L) → ∀ {p q : HOD } → ultra-filter P → prime-filter P |
295 | 70 filter-lemma1 {L} P u = record { |
71 proper = ultra-filter.proper u | |
72 ; prime = lemma3 | |
73 } where | |
329 | 74 lemma3 : {p q : HOD} → filter P ∋ (p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q ) |
295 | 75 lemma3 {p} {q} lt with ultra-filter.ultra u (∪-lemma1 (∈-filter P lt) ) |
76 ... | case1 p∈P = case1 p∈P | |
77 ... | case2 ¬p∈P = case2 (filter1 P {q ∩ (L \ p)} (∪-lemma2 (∈-filter P lt)) lemma7 lemma8) where | |
331 | 78 lemma5 : ((p ∪ q ) ∩ (L \ p)) =h= (q ∩ (L \ p)) |
294
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
79 lemma5 = record { eq→ = λ {x} lt → record { proj1 = lemma4 x lt ; proj2 = proj2 lt } |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
80 ; eq← = λ {x} lt → record { proj1 = case2 (proj1 lt) ; proj2 = proj2 lt } |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
81 } where |
331 | 82 lemma4 : (x : Ordinal ) → odef ((p ∪ q) ∩ (L \ p)) x → odef q x |
294
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
83 lemma4 x lt with proj1 lt |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
84 lemma4 x lt | case1 px = ⊥-elim ( proj2 (proj2 lt) px ) |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
85 lemma4 x lt | case2 qx = qx |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
86 lemma6 : filter P ∋ ((p ∪ q ) ∩ (L \ p)) |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
87 lemma6 = filter2 P lt ¬p∈P |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
88 lemma7 : filter P ∋ (q ∩ (L \ p)) |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
89 lemma7 = subst (λ k → filter P ∋ k ) (==→o≡ lemma5 ) lemma6 |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
90 lemma8 : (q ∩ (L \ p)) ⊆ q |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
91 lemma8 = q∩q⊆q |
4340ffcfa83d
ultra-filter P → prime-filter P done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
293
diff
changeset
|
92 |
295 | 93 ----- |
94 -- | |
95 -- if Filter contains L, prime filter is ultra | |
96 -- | |
97 | |
329 | 98 filter-lemma2 : {L : HOD} → (P : Filter L) → filter P ∋ L → prime-filter P → ultra-filter P |
296
42f89e5efb00
if Filter contains L, prime filter is ultra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
295
diff
changeset
|
99 filter-lemma2 {L} P f∋L prime = record { |
42f89e5efb00
if Filter contains L, prime filter is ultra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
295
diff
changeset
|
100 proper = prime-filter.proper prime |
42f89e5efb00
if Filter contains L, prime filter is ultra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
295
diff
changeset
|
101 ; ultra = λ {p} p⊆L → prime-filter.prime prime (lemma p p⊆L) |
42f89e5efb00
if Filter contains L, prime filter is ultra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
295
diff
changeset
|
102 } where |
42f89e5efb00
if Filter contains L, prime filter is ultra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
295
diff
changeset
|
103 open _==_ |
331 | 104 p+1-p=1 : {p : HOD} → p ⊆ L → L =h= (p ∪ (L \ p)) |
105 eq→ (p+1-p=1 {p} p⊆L) {x} lt with ODC.decp O (odef p x) | |
296
42f89e5efb00
if Filter contains L, prime filter is ultra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
295
diff
changeset
|
106 eq→ (p+1-p=1 {p} p⊆L) {x} lt | yes p∋x = case1 p∋x |
42f89e5efb00
if Filter contains L, prime filter is ultra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
295
diff
changeset
|
107 eq→ (p+1-p=1 {p} p⊆L) {x} lt | no ¬p = case2 (record { proj1 = lt ; proj2 = ¬p }) |
331 | 108 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 )) |
296
42f89e5efb00
if Filter contains L, prime filter is ultra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
295
diff
changeset
|
109 eq← (p+1-p=1 {p} p⊆L) {x} ( case2 ¬p ) = proj1 ¬p |
329 | 110 lemma : (p : HOD) → p ⊆ L → filter P ∋ (p ∪ (L \ p)) |
296
42f89e5efb00
if Filter contains L, prime filter is ultra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
295
diff
changeset
|
111 lemma p p⊆L = subst (λ k → filter P ∋ k ) (==→o≡ (p+1-p=1 p⊆L)) f∋L |
293 | 112 |
329 | 113 record Dense (P : HOD ) : Set (suc n) where |
269
30e419a2be24
disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
268
diff
changeset
|
114 field |
329 | 115 dense : HOD |
379 | 116 d⊆P : dense ⊆ Power P |
329 | 117 dense-f : HOD → HOD |
368 | 118 dense-d : { p : HOD} → p ⊆ P → dense ∋ dense-f p |
119 dense-p : { p : HOD} → p ⊆ P → p ⊆ (dense-f p) | |
266 | 120 |
329 | 121 record Ideal ( L : HOD ) : Set (suc n) where |
293 | 122 field |
329 | 123 ideal : HOD |
293 | 124 i⊆PL : ideal ⊆ Power L |
329 | 125 ideal1 : { p q : HOD } → q ⊆ L → ideal ∋ p → q ⊆ p → ideal ∋ q |
126 ideal2 : { p q : HOD } → ideal ∋ p → ideal ∋ q → ideal ∋ (p ∪ q) | |
293 | 127 |
128 open Ideal | |
129 | |
329 | 130 proper-ideal : {L : HOD} → (P : Ideal L ) → {p : HOD} → Set n |
293 | 131 proper-ideal {L} P {p} = ideal P ∋ od∅ |
132 | |
329 | 133 prime-ideal : {L : HOD} → Ideal L → ∀ {p q : HOD } → Set n |
293 | 134 prime-ideal {L} P {p} {q} = ideal P ∋ ( p ∩ q) → ( ideal P ∋ p ) ∨ ( ideal P ∋ q ) |
135 | |
374 | 136 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 | 137 field |
374 | 138 filter : L → Set n |
371 | 139 f⊆P : PL filter |
374 | 140 filter1 : { p q : L } → PL (λ x → q ⊆ x ) → filter p → p ⊆ q → filter q |
371 | 141 filter2 : { p q : L } → filter p → filter q → filter (p ∩ q) |
142 | |
374 | 143 Filter-is-F : {L : HOD} → (f : Filter L ) → F-Filter HOD (λ p → (x : HOD) → p x → x ⊆ L ) _⊆_ _∩_ |
144 Filter-is-F {L} f = record { | |
145 filter = λ x → Lift (suc n) ((filter f) ∋ x) | |
146 ; f⊆P = λ x f∋x → power→⊆ _ _ (incl ( f⊆PL f ) (lower f∋x )) | |
147 ; filter1 = λ {p} {q} q⊆L f∋p p⊆q → lift ( filter1 f (q⊆L q refl-⊆) (lower f∋p) p⊆q) | |
148 ; filter2 = λ {p} {q} f∋p f∋q → lift ( filter2 f (lower f∋p) (lower f∋q)) | |
149 } | |
150 | |
379 | 151 record F-Dense {n : Level} (L : Set n) (PL : (L → Set n) → Set n) ( _⊆_ : L → L → Set n) (_∩_ : L → L → L ) : Set (suc n) where |
152 field | |
153 dense : L → Set n | |
154 d⊆P : PL dense | |
155 dense-f : L → L | |
156 dense-d : { p : L} → PL (λ x → p ⊆ x ) → dense ( dense-f p ) | |
157 dense-p : { p : L} → PL (λ x → p ⊆ x ) → p ⊆ (dense-f p) | |
158 | |
159 Dense-is-F : {L : HOD} → (f : Dense L ) → F-Dense HOD (λ p → (x : HOD) → p x → x ⊆ L ) _⊆_ _∩_ | |
160 Dense-is-F {L} f = record { | |
161 dense = λ x → Lift (suc n) ((dense f) ∋ x) | |
162 ; d⊆P = λ x f∋x → power→⊆ _ _ (incl ( d⊆P f ) (lower f∋x )) | |
163 ; dense-f = λ x → dense-f f x | |
164 ; dense-d = λ {p} d → lift ( dense-d f (d p refl-⊆ ) ) | |
165 ; dense-p = λ {p} d → dense-p f (d p refl-⊆) | |
166 } where open Dense | |
167 | |
386 | 168 |
169 record GenericFilter (P : HOD) : Set (suc n) where | |
170 field | |
171 genf : Filter P | |
172 generic : (D : Dense P ) → ¬ ( (Dense.dense D ∩ Filter.filter genf ) ≡ od∅ ) | |
173 | |
174 record F-GenericFilter {n : Level} (L : Set n) (PL : (L → Set n) → Set n) ( _⊆_ : L → L → Set n) (_∩_ : L → L → L ) : Set (suc n) where | |
175 field | |
176 GFilter : F-Filter L PL _⊆_ _∩_ | |
177 Intersection : (D : F-Dense L PL _⊆_ _∩_ ) → { x : L } → F-Dense.dense D x → L | |
178 Generic : (D : F-Dense L PL _⊆_ _∩_ ) → { x : L } → ( y : F-Dense.dense D x) → F-Filter.filter GFilter (Intersection D y ) | |
179 |