Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate OD.agda @ 364:67580311cc8e
...
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Sat, 18 Jul 2020 11:38:33 +0900 |
parents | aad9249d1e8f |
children | 7f919d6b045b |
rev | line source |
---|---|
364 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
16 | 2 open import Level |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
3 open import Ordinals |
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
4 module OD {n : Level } (O : Ordinals {n} ) where |
3 | 5 |
14
e11e95d5ddee
separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
11
diff
changeset
|
6 open import zf |
23 | 7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
14
e11e95d5ddee
separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
11
diff
changeset
|
8 open import Relation.Binary.PropositionalEquality |
e11e95d5ddee
separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
11
diff
changeset
|
9 open import Data.Nat.Properties |
6 | 10 open import Data.Empty |
11 open import Relation.Nullary | |
12 open import Relation.Binary | |
13 open import Relation.Binary.Core | |
14 | |
213
22d435172d1a
separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
210
diff
changeset
|
15 open import logic |
22d435172d1a
separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
210
diff
changeset
|
16 open import nat |
22d435172d1a
separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
210
diff
changeset
|
17 |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
18 open inOrdinal O |
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
19 |
27 | 20 -- Ordinal Definable Set |
11 | 21 |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
22 record OD : Set (suc n ) where |
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
23 field |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
24 def : (x : Ordinal ) → Set n |
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
25 |
141 | 26 open OD |
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
27 |
120 | 28 open _∧_ |
213
22d435172d1a
separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
210
diff
changeset
|
29 open _∨_ |
22d435172d1a
separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
210
diff
changeset
|
30 open Bool |
44
fcac01485f32
od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
43
diff
changeset
|
31 |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
32 record _==_ ( a b : OD ) : Set n where |
43
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
33 field |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
34 eq→ : ∀ { x : Ordinal } → def a x → def b x |
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
35 eq← : ∀ { x : Ordinal } → def b x → def a x |
43
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
36 |
234
e06b76e5b682
ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
228
diff
changeset
|
37 id : {A : Set n} → A → A |
43
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
38 id x = x |
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
39 |
271 | 40 ==-refl : { x : OD } → x == x |
41 ==-refl {x} = record { eq→ = id ; eq← = id } | |
43
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
42 |
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
43 open _==_ |
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
44 |
271 | 45 ==-trans : { x y z : OD } → x == y → y == z → x == z |
46 ==-trans x=y y=z = record { eq→ = λ {m} t → eq→ y=z (eq→ x=y t) ; eq← = λ {m} t → eq← x=y (eq← y=z t) } | |
43
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
47 |
271 | 48 ==-sym : { x y : OD } → x == y → y == x |
49 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t } | |
50 | |
43
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
51 |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
52 ⇔→== : { x y : OD } → ( {z : Ordinal } → def x z ⇔ def y z) → x == y |
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
53 eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m |
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
54 eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m |
120 | 55 |
277
d9d3654baee1
seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
276
diff
changeset
|
56 -- next assumptions are our axiom |
322 | 57 -- |
58 -- OD is an equation on Ordinals, so it contains Ordinals. If these Ordinals have one-to-one | |
59 -- correspondence to the OD then the OD looks like a ZF Set. | |
60 -- | |
61 -- If all ZF Set have supreme upper bound, the solutions of OD have to be bounded, i.e. | |
62 -- bbounded ODs are ZF Set. Unbounded ODs are classes. | |
63 -- | |
290 | 64 -- In classical Set Theory, HOD is used, as a subset of OD, |
277
d9d3654baee1
seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
276
diff
changeset
|
65 -- HOD = { x | TC x ⊆ OD } |
290 | 66 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x. |
322 | 67 -- This is not possible because we don't have V yet. So we assumes HODs are bounded OD. |
290 | 68 -- |
309 | 69 -- We also assumes HODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks. |
322 | 70 -- There two contraints on the HOD order, one is ∋, the other one is ⊂. |
71 -- ODs have an ovbious maximum, but Ordinals are not, but HOD has no maximum because there is an aribtrary | |
72 -- bound on each HOD. | |
290 | 73 -- |
322 | 74 -- In classical Set Theory, sup is defined by Uion, since we are working on constructive logic, |
290 | 75 -- we need explict assumption on sup. |
309 | 76 -- |
77 -- ==→o≡ is necessary to prove axiom of extensionality. | |
277
d9d3654baee1
seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
276
diff
changeset
|
78 |
303 | 79 data One : Set n where |
80 OneObj : One | |
81 | |
82 -- Ordinals in OD , the maximum | |
83 Ords : OD | |
84 Ords = record { def = λ x → One } | |
85 | |
86 record HOD : Set (suc n) where | |
302
304c271b3d47
HOD and reduction mapping of Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
301
diff
changeset
|
87 field |
303 | 88 od : OD |
304 | 89 odmax : Ordinal |
308 | 90 <odmax : {y : Ordinal} → def od y → y o< odmax |
302
304c271b3d47
HOD and reduction mapping of Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
301
diff
changeset
|
91 |
304c271b3d47
HOD and reduction mapping of Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
301
diff
changeset
|
92 open HOD |
304c271b3d47
HOD and reduction mapping of Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
301
diff
changeset
|
93 |
277
d9d3654baee1
seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
276
diff
changeset
|
94 record ODAxiom : Set (suc n) where |
d9d3654baee1
seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
276
diff
changeset
|
95 field |
304 | 96 -- HOD is isomorphic to Ordinal (by means of Goedel number) |
303 | 97 od→ord : HOD → Ordinal |
98 ord→od : Ordinal → HOD | |
99 c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y | |
335 | 100 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) |
303 | 101 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x |
322 | 102 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x |
335 | 103 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y |
104 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal | |
306 | 105 sup-o< : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal } → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o< sup-o A ψ |
302
304c271b3d47
HOD and reduction mapping of Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
301
diff
changeset
|
106 |
277
d9d3654baee1
seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
276
diff
changeset
|
107 postulate odAxiom : ODAxiom |
d9d3654baee1
seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
276
diff
changeset
|
108 open ODAxiom odAxiom |
d9d3654baee1
seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
276
diff
changeset
|
109 |
363 | 110 -- odmax minimality |
111 -- | |
112 -- since we have ==→o≡ , so odmax have to be unique. We should have odmaxmin in HOD. | |
113 -- We can calculate the minimum using sup but it is tedius. | |
114 -- Only Select has non minimum odmax. | |
115 -- We have the same problem on 'def' itself, but we leave it. | |
116 | |
117 odmaxmin : Set (suc n) | |
118 odmaxmin = (y : HOD) (z : Ordinal) → ((x : Ordinal)→ def (od y) x → x o< z) → odmax y o< osuc z | |
119 | |
344
e0916a632971
possible order restriction makes ω ZFSet
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
343
diff
changeset
|
120 -- possible order restriction |
339 | 121 hod-ord< : {x : HOD } → Set n |
122 hod-ord< {x} = od→ord x o< next (odmax x) | |
123 | |
335 | 124 -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD |
125 ¬OD-order : ( od→ord : OD → Ordinal ) → ( ord→od : Ordinal → OD ) → ( { x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y) → ⊥ | |
126 ¬OD-order od→ord ord→od c<→o< = osuc-< <-osuc (c<→o< {Ords} OneObj ) | |
277
d9d3654baee1
seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
276
diff
changeset
|
127 |
335 | 128 -- Open supreme upper bound leads a contradition, so we use domain restriction on sup |
129 ¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥ | |
130 ¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where | |
131 next-ord : Ordinal → Ordinal | |
132 next-ord x = osuc x | |
301
b012a915bbb5
contradiction found ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
300
diff
changeset
|
133 |
179 | 134 -- Ordinal in OD ( and ZFSet ) Transitive Set |
303 | 135 Ord : ( a : Ordinal ) → HOD |
304 | 136 Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where |
137 lemma : {x : Ordinal} → x o< a → x o< a | |
138 lemma {x} lt = lt | |
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
139 |
303 | 140 od∅ : HOD |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
141 od∅ = Ord o∅ |
40 | 142 |
303 | 143 odef : HOD → Ordinal → Set n |
144 odef A x = def ( od A ) x | |
123 | 145 |
335 | 146 -- If we have reverse of c<→o<, everything becomes Ordinal |
147 o<→c<→HOD=Ord : ( o<→c< : {x y : Ordinal } → x o< y → odef (ord→od y) x ) → {x : HOD } → x ≡ Ord (od→ord x) | |
303 | 148 o<→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where |
149 lemma1 : {y : Ordinal} → odef x y → odef (Ord (od→ord x)) y | |
150 lemma1 {y} lt = subst ( λ k → k o< od→ord x ) diso (c<→o< {ord→od y} {x} (subst (λ k → odef x k ) (sym diso) lt)) | |
151 lemma2 : {y : Ordinal} → odef (Ord (od→ord x)) y → odef x y | |
152 lemma2 {y} lt = subst (λ k → odef k y ) oiso (o<→c< {y} {od→ord x} lt ) | |
95 | 153 |
303 | 154 _∋_ : ( a x : HOD ) → Set n |
155 _∋_ a x = odef a ( od→ord x ) | |
156 | |
157 _c<_ : ( x a : HOD ) → Set n | |
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
158 x c< a = a ∋ x |
103 | 159 |
361 | 160 cseq : HOD → HOD |
308 | 161 cseq x = record { od = record { def = λ y → odef x (osuc y) } ; odmax = osuc (odmax x) ; <odmax = lemma } where |
162 lemma : {y : Ordinal} → def (od x) (osuc y) → y o< osuc (odmax x) | |
163 lemma {y} lt = ordtrans <-osuc (ordtrans (<odmax x lt) <-osuc ) | |
95 | 164 |
303 | 165 odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x |
166 odef-subst df refl refl = df | |
95 | 167 |
361 | 168 otrans : {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y |
187 | 169 otrans x<a y<x = ordtrans y<x x<a |
123 | 170 |
303 | 171 odef→o< : {X : HOD } → {x : Ordinal } → odef X x → x o< od→ord X |
172 odef→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( odef-subst {X} {x} lt (sym oiso) (sym diso) )) diso diso | |
44
fcac01485f32
od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
43
diff
changeset
|
173 |
51 | 174 -- avoiding lv != Zero error |
303 | 175 orefl : { x : HOD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y |
51 | 176 orefl refl = refl |
177 | |
303 | 178 ==-iso : { x y : HOD } → od (ord→od (od→ord x)) == od (ord→od (od→ord y)) → od x == od y |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
179 ==-iso {x} {y} eq = record { |
303 | 180 eq→ = λ d → lemma ( eq→ eq (odef-subst d (sym oiso) refl )) ; |
181 eq← = λ d → lemma ( eq← eq (odef-subst d (sym oiso) refl )) } | |
51 | 182 where |
303 | 183 lemma : {x : HOD } {z : Ordinal } → odef (ord→od (od→ord x)) z → odef x z |
184 lemma {x} {z} d = odef-subst d oiso refl | |
51 | 185 |
303 | 186 =-iso : {x y : HOD } → (od x == od y) ≡ (od (ord→od (od→ord x)) == od y) |
187 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym oiso) | |
57 | 188 |
303 | 189 ord→== : { x y : HOD } → od→ord x ≡ od→ord y → od x == od y |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
190 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where |
303 | 191 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (ord→od ox) == od (ord→od oy) |
271 | 192 lemma ox ox refl = ==-refl |
51 | 193 |
303 | 194 o≡→== : { x y : Ordinal } → x ≡ y → od (ord→od x) == od (ord→od y) |
271 | 195 o≡→== {x} {.x} refl = ==-refl |
51 | 196 |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
197 o∅≡od∅ : ord→od (o∅ ) ≡ od∅ |
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
198 o∅≡od∅ = ==→o≡ lemma where |
303 | 199 lemma0 : {x : Ordinal} → odef (ord→od o∅) x → odef od∅ x |
200 lemma0 {x} lt = o<-subst (c<→o< {ord→od x} {ord→od o∅} (odef-subst {ord→od o∅} {x} lt refl (sym diso)) ) diso diso | |
201 lemma1 : {x : Ordinal} → odef od∅ x → odef (ord→od o∅) x | |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
202 lemma1 {x} lt = ⊥-elim (¬x<0 lt) |
303 | 203 lemma : od (ord→od o∅) == od od∅ |
150 | 204 lemma = record { eq→ = lemma0 ; eq← = lemma1 } |
205 | |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
206 ord-od∅ : od→ord (od∅ ) ≡ o∅ |
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
207 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) |
80 | 208 |
303 | 209 ∅0 : record { def = λ x → Lift n ⊥ } == od od∅ |
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
210 eq→ ∅0 {w} (lift ()) |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
211 eq← ∅0 {w} lt = lift (¬x<0 lt) |
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
212 |
303 | 213 ∅< : { x y : HOD } → odef x (od→ord y ) → ¬ ( od x == od od∅ ) |
271 | 214 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
215 ∅< {x} {y} d eq | lift () |
57 | 216 |
303 | 217 ∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
218 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x ) |
51 | 219 |
303 | 220 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x |
221 odef-iso refl t = t | |
76 | 222 |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
223 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ ) |
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
224 is-o∅ x with trio< x o∅ |
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
225 is-o∅ x | tri< a ¬b ¬c = no ¬b |
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
226 is-o∅ x | tri≈ ¬a b ¬c = yes b |
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
227 is-o∅ x | tri> ¬a ¬b c = no ¬b |
57 | 228 |
335 | 229 -- the pair |
338 | 230 _,_ : HOD → HOD → HOD |
308 | 231 x , y = record { od = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } ; odmax = omax (od→ord x) (od→ord y) ; <odmax = lemma } where |
232 lemma : {t : Ordinal} → (t ≡ od→ord x) ∨ (t ≡ od→ord y) → t o< omax (od→ord x) (od→ord y) | |
233 lemma {t} (case1 refl) = omax-x _ _ | |
234 lemma {t} (case2 refl) = omax-y _ _ | |
235 | |
343 | 236 pair-xx<xy : {x y : HOD} → od→ord (x , x) o< osuc (od→ord (x , y) ) |
237 pair-xx<xy {x} {y} = ⊆→o≤ lemma where | |
238 lemma : {z : Ordinal} → def (od (x , x)) z → def (od (x , y)) z | |
239 lemma {z} (case1 refl) = case1 refl | |
240 lemma {z} (case2 refl) = case1 refl | |
241 | |
339 | 242 -- another form of infinite |
343 | 243 -- pair-ord< : {x : Ordinal } → Set n |
244 pair-ord< : {x : HOD } → ( {y : HOD } → od→ord y o< next (odmax y) ) → od→ord ( x , x ) o< next (od→ord x) | |
245 pair-ord< {x} ho< = subst (λ k → od→ord (x , x) o< k ) lemmab0 lemmab1 where | |
246 lemmab0 : next (odmax (x , x)) ≡ next (od→ord x) | |
247 lemmab0 = trans (cong (λ k → next k) (omxx _)) (sym nexto≡) | |
248 lemmab1 : od→ord (x , x) o< next ( odmax (x , x)) | |
249 lemmab1 = ho< | |
188
1f2c8b094908
axiom of choice → p ∨ ¬ p
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
187
diff
changeset
|
250 |
344
e0916a632971
possible order restriction makes ω ZFSet
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
343
diff
changeset
|
251 pair<y : {x y : HOD } → y ∋ x → od→ord (x , x) o< osuc (od→ord y) |
e0916a632971
possible order restriction makes ω ZFSet
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
343
diff
changeset
|
252 pair<y {x} {y} y∋x = ⊆→o≤ lemma where |
e0916a632971
possible order restriction makes ω ZFSet
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
343
diff
changeset
|
253 lemma : {z : Ordinal} → def (od (x , x)) z → def (od y) z |
e0916a632971
possible order restriction makes ω ZFSet
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
343
diff
changeset
|
254 lemma (case1 refl) = y∋x |
e0916a632971
possible order restriction makes ω ZFSet
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
343
diff
changeset
|
255 lemma (case2 refl) = y∋x |
e0916a632971
possible order restriction makes ω ZFSet
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
343
diff
changeset
|
256 |
361 | 257 -- another possible restriction. We reqest no minimality on odmax, so it may arbitrary larger. |
258 odmax<od→ord : { x y : HOD } → x ∋ y → Set n | |
259 odmax<od→ord {x} {y} x∋y = odmax x o< od→ord x | |
260 | |
79 | 261 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
262 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) |
59
d13d1351a1fa
lemma = cong₂ (λ x not → minimul x not ) oiso { }6
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
58
diff
changeset
|
263 |
318 | 264 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD |
265 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } | |
141 | 266 |
360 | 267 _∩_ : ( A B : HOD ) → HOD |
268 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x } | |
269 ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))} | |
308 | 270 |
303 | 271 record _⊆_ ( A B : HOD ) : Set (suc n) where |
271 | 272 field |
303 | 273 incl : { x : HOD } → A ∋ x → B ∋ x |
271 | 274 |
275 open _⊆_ | |
190 | 276 infixr 220 _⊆_ |
277 | |
335 | 278 od⊆→o≤ : {x y : HOD } → x ⊆ y → od→ord x o< osuc (od→ord y) |
279 od⊆→o≤ {x} {y} lt = ⊆→o≤ {x} {y} (λ {z} x>z → subst (λ k → def (od y) k ) diso (incl lt (subst (λ k → def (od x) k ) (sym diso) x>z ))) | |
280 | |
281 -- if we have od→ord (x , x) ≡ osuc (od→ord x), ⊆→o≤ → c<→o< | |
338 | 282 ⊆→o≤→c<→o< : ({x : HOD} → od→ord (x , x) ≡ osuc (od→ord x) ) |
283 → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) ) | |
284 → {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y | |
285 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (od→ord x) (od→ord y) | |
286 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a | |
287 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (peq {x}) (pair<y (subst (λ k → k ∋ x) (sym ( ==→o≡ {x} {y} (ord→== b))) y∋x ))) | |
288 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c = | |
289 ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where | |
290 lemma : {z : Ordinal} → (z ≡ od→ord x) ∨ (z ≡ od→ord x) → od→ord x ≡ z | |
291 lemma (case1 refl) = refl | |
292 lemma (case2 refl) = refl | |
293 y⊆x,x : {z : Ordinals.ord O} → def (od (x , x)) z → def (od y) z | |
294 y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x | |
295 lemma1 : osuc (od→ord y) o< od→ord (x , x) | |
296 lemma1 = subst (λ k → osuc (od→ord y) o< k ) (sym (peq {x})) (osucc c ) | |
335 | 297 |
360 | 298 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → (A ∩ x ) ∋ y ) ⇔ ( x ⊆ A ) |
271 | 299 subset-lemma {A} {x} = record { |
300 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } | |
301 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } | |
190 | 302 } |
303 | |
312 | 304 power< : {A x : HOD } → x ⊆ A → Ord (osuc (od→ord A)) ∋ x |
305 power< {A} {x} x⊆A = ⊆→o≤ (λ {y} x∋y → subst (λ k → def (od A) k) diso (lemma y x∋y ) ) where | |
306 lemma : (y : Ordinal) → def (od x) y → def (od A) (od→ord (ord→od y)) | |
307 lemma y x∋y = incl x⊆A (subst (λ k → def (od x) k ) (sym diso) x∋y ) | |
308 | |
261
d9d178d1457c
ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
260
diff
changeset
|
309 open import Data.Unit |
d9d178d1457c
ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
260
diff
changeset
|
310 |
324 | 311 ε-induction : { ψ : HOD → Set n} |
303 | 312 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) |
313 → (x : HOD ) → ψ x | |
261
d9d178d1457c
ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
260
diff
changeset
|
314 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where |
d9d178d1457c
ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
260
diff
changeset
|
315 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) |
d9d178d1457c
ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
260
diff
changeset
|
316 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) |
d9d178d1457c
ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
260
diff
changeset
|
317 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) |
d9d178d1457c
ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
260
diff
changeset
|
318 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy |
d9d178d1457c
ε-induction from TransFinite induction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
260
diff
changeset
|
319 |
335 | 320 -- level trick (what'a shame) |
330 | 321 ε-induction1 : { ψ : HOD → Set (suc n)} |
322 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) | |
323 → (x : HOD ) → ψ x | |
324 ε-induction1 {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where | |
325 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) | |
326 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) | |
327 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) | |
328 ε-induction-ord ox {oy} lt = TransFinite1 {λ oy → ψ (ord→od oy)} induction oy | |
329 | |
303 | 330 HOD→ZF : ZF |
331 HOD→ZF = record { | |
332 ZFSet = HOD | |
43
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
333 ; _∋_ = _∋_ |
363 | 334 ; _≈_ = hod→zf._=h=_ |
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
335 ; ∅ = od∅ |
28 | 336 ; _,_ = _,_ |
363 | 337 ; Union = hod→zf.Union |
338 ; Power = hod→zf.Power | |
339 ; Select = hod→zf.Select | |
340 ; Replace = hod→zf.Replace | |
341 ; infinite = hod→zf.infinite | |
342 ; isZF = hod→zf.isZF | |
28 | 343 } where |
363 | 344 module hod→zf where |
303 | 345 ZFSet = HOD -- is less than Ords because of maxod |
346 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD | |
308 | 347 Select X ψ = record { od = record { def = λ x → ( odef X x ∧ ψ ( ord→od x )) } ; odmax = odmax X ; <odmax = λ y → <odmax X (proj1 y) } |
310 | 348 Replace : HOD → (HOD → HOD) → HOD |
318 | 349 Replace X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → od→ord (ψ (ord→od y)))) ∧ def (in-codomain X ψ) x } |
350 ; odmax = rmax ; <odmax = rmax<} where | |
351 rmax : Ordinal | |
352 rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y))) | |
353 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax | |
354 rmax< lt = proj1 lt | |
303 | 355 Union : HOD → HOD |
318 | 356 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) } |
357 ; odmax = osuc (od→ord U) ; <odmax = umax< } where | |
358 umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (ord→od u)) y) → y o< osuc (od→ord U) | |
319 | 359 umax< {y} not = lemma (FExists _ lemma1 not ) where |
360 lemma0 : {x : Ordinal} → def (od (ord→od x)) y → y o< x | |
361 lemma0 {x} x<y = subst₂ (λ j k → j o< k ) diso diso (c<→o< (subst (λ k → def (od (ord→od x)) k) (sym diso) x<y)) | |
362 lemma2 : {x : Ordinal} → def (od U) x → x o< od→ord U | |
363 lemma2 {x} x<U = subst (λ k → k o< od→ord U ) diso (c<→o< (subst (λ k → def (od U) k) (sym diso) x<U)) | |
364 lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (ord→od x)) y → ¬ (od→ord U o< y) | |
365 lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) ) | |
366 lemma : ¬ ((od→ord U) o< y ) → y o< osuc (od→ord U) | |
367 lemma not with trio< y (od→ord U) | |
368 lemma not | tri< a ¬b ¬c = ordtrans a <-osuc | |
369 lemma not | tri≈ ¬a refl ¬c = <-osuc | |
370 lemma not | tri> ¬a ¬b c = ⊥-elim (not c) | |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
371 _∈_ : ( A B : ZFSet ) → Set n |
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
372 A ∈ B = B ∋ A |
312 | 373 |
374 OPwr : (A : HOD ) → HOD | |
360 | 375 OPwr A = Ord ( sup-o (Ord (osuc (od→ord A))) ( λ x A∋x → od→ord ( A ∩ (ord→od x)) ) ) |
312 | 376 |
303 | 377 Power : HOD → HOD |
300
e70980bd80c7
-- the set of finite partial functions from ω to 2
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
291
diff
changeset
|
378 Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x ) |
277
d9d3654baee1
seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
276
diff
changeset
|
379 -- {_} : ZFSet → ZFSet |
335 | 380 -- { x } = ( x , x ) -- better to use (x , x) directly |
109
dab56d357fa3
remove o<→c< and add otrans in OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
103
diff
changeset
|
381 |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
382 data infinite-d : ( x : Ordinal ) → Set n where |
161 | 383 iφ : infinite-d o∅ |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
384 isuc : {x : Ordinal } → infinite-d x → |
161 | 385 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) |
386 | |
328 | 387 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair. |
338 | 388 -- We simply assumes infinite-d y has a maximum. |
328 | 389 -- |
338 | 390 -- This means that many of OD may not be HODs because of the od→ord mapping divergence. |
346 | 391 -- We should have some axioms to prevent this such as od→ord x o< next (odmax x). |
328 | 392 -- |
393 postulate | |
394 ωmax : Ordinal | |
395 <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax | |
396 | |
303 | 397 infinite : HOD |
328 | 398 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax } |
303 | 399 |
339 | 400 infinite' : ({x : HOD} → od→ord x o< next (odmax x)) → HOD |
401 infinite' ho< = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where | |
402 u : (y : Ordinal ) → HOD | |
403 u y = Union (ord→od y , (ord→od y , ord→od y)) | |
364 | 404 -- next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z |
405 lemma8 : {y : Ordinal} → od→ord (ord→od y , ord→od y) o< next (odmax (ord→od y , ord→od y)) | |
406 lemma8 = ho< | |
407 --- (x,y) < next (omax x y) < next (osuc y) = next y | |
408 lemmaa : {x y : HOD} → od→ord x o< od→ord y → od→ord (x , y) o< next (od→ord y) | |
409 lemmaa {x} {y} x<y = subst (λ k → od→ord (x , y) o< k ) (sym nexto≡) (subst (λ k → od→ord (x , y) o< next k ) (sym (omax< _ _ x<y)) ho< ) | |
410 lemma81 : {y : Ordinal} → od→ord (ord→od y , ord→od y) o< next (od→ord (ord→od y)) | |
411 lemma81 {y} = nexto=n (subst (λ k → od→ord (ord→od y , ord→od y) o< k ) (cong (λ k → next k) (omxx _)) lemma8) | |
412 lemma9 : {y : Ordinal} → od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y , ord→od y)) | |
413 lemma9 = lemmaa (c<→o< (case1 refl)) | |
414 lemma71 : {y : Ordinal} → od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y)) | |
415 lemma71 = next< lemma81 lemma9 | |
416 lemma1 : {y : Ordinal} → od→ord (u y) o< next (osuc (od→ord (ord→od y , (ord→od y , ord→od y)))) | |
417 lemma1 = ho< | |
418 --- main recursion | |
339 | 419 lemma : {y : Ordinal} → infinite-d y → y o< next o∅ |
348
08d94fec239c
Limit ordinal and possible OD bound
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
346
diff
changeset
|
420 lemma {o∅} iφ = x<nx |
364 | 421 lemma (isuc {y} x) = next< (lemma x) (next< (subst (λ k → od→ord (ord→od y , (ord→od y , ord→od y)) o< next k) diso lemma71 ) (nexto=n lemma1)) |
363 | 422 |
423 nat→ω : Nat → HOD | |
424 nat→ω Zero = od∅ | |
425 nat→ω (Suc y) = Union (nat→ω y , (nat→ω y , nat→ω y)) | |
426 | |
427 ω→nat : (n : HOD) → infinite ∋ n → Nat | |
428 ω→nat n = lemma where | |
429 lemma : {y : Ordinal} → infinite-d y → Nat | |
430 lemma iφ = Zero | |
431 lemma (isuc lt) = Suc (lemma lt) | |
338 | 432 |
364 | 433 ω∋nat→ω : {n : Nat} → def (od infinite) (od→ord (nat→ω n)) |
434 ω∋nat→ω {Zero} = subst (λ k → def (od infinite) k) {!!} iφ | |
435 ω∋nat→ω {Suc n} = subst (λ k → def (od infinite) k) {!!} (isuc ( ω∋nat→ω {n})) | |
436 | |
303 | 437 _=h=_ : (x y : HOD) → Set n |
438 x =h= y = od x == od y | |
161 | 439 |
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
440 infixr 200 _∈_ |
96 | 441 -- infixr 230 _∩_ _∪_ |
303 | 442 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite |
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
443 isZF = record { |
271 | 444 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } |
247 | 445 ; pair→ = pair→ |
446 ; pair← = pair← | |
72 | 447 ; union→ = union→ |
448 ; union← = union← | |
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
449 ; empty = empty |
129 | 450 ; power→ = power→ |
76 | 451 ; power← = power← |
186 | 452 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} |
274 | 453 ; ε-induction = ε-induction |
78
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
454 ; infinity∅ = infinity∅ |
160 | 455 ; infinity = infinity |
116 | 456 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
135 | 457 ; replacement← = replacement← |
317 | 458 ; replacement→ = λ {ψ} → replacement→ {ψ} |
274 | 459 -- ; choice-func = choice-func |
460 -- ; choice = choice | |
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
461 } where |
129 | 462 |
303 | 463 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y ) |
464 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x )) | |
465 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y )) | |
247 | 466 |
303 | 467 pair← : ( x y t : ZFSet ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t |
468 pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x)) | |
469 pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y)) | |
247 | 470 |
303 | 471 empty : (x : HOD ) → ¬ (od∅ ∋ x) |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
472 empty x = ¬x<0 |
129 | 473 |
271 | 474 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) |
475 o<→c< lt = record { incl = λ z → ordtrans z lt } | |
155 | 476 |
271 | 477 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y |
155 | 478 ⊆→o< {x} {y} lt with trio< x y |
479 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc | |
480 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc | |
271 | 481 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) |
155 | 482 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) |
151 | 483 |
303 | 484 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
157
afc030b7c8d0
explict logical definition of Union failed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
156
diff
changeset
|
485 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx |
303 | 486 ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } )) |
487 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) | |
258 | 488 union← X z UX∋z = FExists _ lemma UX∋z where |
303 | 489 lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z)) |
490 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } | |
144 | 491 |
303 | 492 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y |
144 | 493 ψiso {ψ} t refl = t |
303 | 494 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) |
144 | 495 selection {ψ} {X} {y} = record { |
496 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
497 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
498 } | |
311 | 499 sup-c< : (ψ : HOD → HOD) → {X x : HOD} → X ∋ x → od→ord (ψ x) o< (sup-o X (λ y X∋y → od→ord (ψ (ord→od y)))) |
500 sup-c< ψ {X} {x} lt = subst (λ k → od→ord (ψ k) o< _ ) oiso (sup-o< X lt ) | |
303 | 501 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x |
311 | 502 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {X} {x} lt ; proj2 = lemma } where |
318 | 503 lemma : def (in-codomain X ψ) (od→ord (ψ x)) |
150 | 504 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) |
303 | 505 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y)) |
150 | 506 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where |
303 | 507 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) |
508 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y))) | |
144 | 509 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where |
303 | 510 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y)) |
511 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq ) | |
512 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) ) | |
513 lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 )) | |
144 | 514 |
515 --- | |
516 --- Power Set | |
517 --- | |
303 | 518 --- First consider ordinals in HOD |
100 | 519 --- |
360 | 520 --- A ∩ x = record { def = λ y → odef A y ∧ odef x y } subset of A |
100 | 521 -- |
522 -- | |
303 | 523 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a ) |
142 | 524 ∩-≡ {a} {b} inc = record { |
525 eq→ = λ {x} x<a → record { proj2 = x<a ; | |
303 | 526 proj1 = odef-subst {_} {_} {b} {x} (inc (odef-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; |
142 | 527 eq← = λ {x} x<a∩b → proj2 x<a∩b } |
100 | 528 -- |
258 | 529 -- Transitive Set case |
360 | 530 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is (Ord a) ∩ t =h= t |
531 -- OPwr (Ord a) is a sup of (Ord a) ∩ t, so OPwr (Ord a) ∋ t | |
532 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( A ∩ (ord→od x )) ) ) | |
100 | 533 -- |
303 | 534 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t |
535 ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t} | |
127 | 536 lemma refl (lemma1 lemma-eq )where |
360 | 537 lemma-eq : ((Ord a) ∩ t) =h= t |
97
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
538 eq→ lemma-eq {z} w = proj2 w |
f2b579106770
power set using sup on Def
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
539 eq← lemma-eq {z} w = record { proj2 = w ; |
303 | 540 proj1 = odef-subst {_} {_} {(Ord a)} {z} |
541 ( t→A (odef-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } | |
542 lemma1 : {a : Ordinal } { t : HOD } | |
360 | 543 → (eq : ((Ord a) ∩ t) =h= t) → od→ord ((Ord a) ∩ (ord→od (od→ord t))) ≡ od→ord t |
544 lemma1 {a} {t} eq = subst (λ k → od→ord ((Ord a) ∩ k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq )) | |
312 | 545 lemma2 : (od→ord t) o< (osuc (od→ord (Ord a))) |
546 lemma2 = ⊆→o≤ {t} {Ord a} (λ {x} x<t → subst (λ k → def (od (Ord a)) k) diso (t→A (subst (λ k → def (od t) k) (sym diso) x<t))) | |
360 | 547 lemma : od→ord ((Ord a) ∩ (ord→od (od→ord t)) ) o< sup-o (Ord (osuc (od→ord (Ord a)))) (λ x lt → od→ord ((Ord a) ∩ (ord→od x))) |
311 | 548 lemma = sup-o< _ lemma2 |
129 | 549 |
144 | 550 -- |
303 | 551 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first |
258 | 552 -- then replace of all elements of the Power set by A ∩ y |
144 | 553 -- |
300
e70980bd80c7
-- the set of finite partial functions from ω to 2
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
291
diff
changeset
|
554 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y ) |
166 | 555 |
556 -- we have oly double negation form because of the replacement axiom | |
557 -- | |
303 | 558 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x) |
258 | 559 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where |
142 | 560 a = od→ord A |
303 | 561 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y))) |
317 | 562 lemma2 = replacement→ {λ x → A ∩ x} (OPwr (Ord (od→ord A))) t P∋t |
303 | 563 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x) |
166 | 564 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) |
303 | 565 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y))) |
566 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 )) | |
567 lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x)) | |
166 | 568 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not |
569 | |
303 | 570 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
311 | 571 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where |
142 | 572 a = od→ord A |
303 | 573 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x |
142 | 574 lemma0 {x} t∋x = c<→o< (t→A t∋x) |
300
e70980bd80c7
-- the set of finite partial functions from ω to 2
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
291
diff
changeset
|
575 lemma3 : OPwr (Ord a) ∋ t |
142 | 576 lemma3 = ord-power← a t lemma0 |
152 | 577 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t |
578 lemma4 = let open ≡-Reasoning in begin | |
579 A ∩ ord→od (od→ord t) | |
580 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ | |
581 A ∩ t | |
317 | 582 ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩ |
152 | 583 t |
584 ∎ | |
317 | 585 sup1 : Ordinal |
360 | 586 sup1 = sup-o (Ord (osuc (od→ord (Ord (od→ord A))))) (λ x A∋x → od→ord ((Ord (od→ord A)) ∩ (ord→od x))) |
313 | 587 lemma9 : def (od (Ord (Ordinals.osuc O (od→ord (Ord (od→ord A)))))) (od→ord (Ord (od→ord A))) |
588 lemma9 = <-osuc | |
360 | 589 lemmab : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) o< sup1 |
315 | 590 lemmab = sup-o< (Ord (osuc (od→ord (Ord (od→ord A))))) lemma9 |
591 lemmad : Ord (osuc (od→ord A)) ∋ t | |
317 | 592 lemmad = ⊆→o≤ (λ {x} lt → subst (λ k → def (od A) k ) diso (t→A (subst (λ k → def (od t) k ) (sym diso) lt))) |
360 | 593 lemmac : ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) =h= Ord (od→ord A) |
317 | 594 lemmac = record { eq→ = lemmaf ; eq← = lemmag } where |
360 | 595 lemmaf : {x : Ordinal} → def (od ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A)))))) x → def (od (Ord (od→ord A))) x |
317 | 596 lemmaf {x} lt = proj1 lt |
360 | 597 lemmag : {x : Ordinal} → def (od (Ord (od→ord A))) x → def (od ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A)))))) x |
317 | 598 lemmag {x} lt = record { proj1 = lt ; proj2 = subst (λ k → def (od k) x) (sym oiso) lt } |
360 | 599 lemmae : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A))))) ≡ od→ord (Ord (od→ord A)) |
315 | 600 lemmae = cong (λ k → od→ord k ) ( ==→o≡ lemmac) |
311 | 601 lemma7 : def (od (OPwr (Ord (od→ord A)))) (od→ord t) |
315 | 602 lemma7 with osuc-≡< lemmad |
603 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab ) | |
317 | 604 lemma7 | case1 eq with osuc-≡< (⊆→o≤ {ord→od (od→ord t)} {ord→od (od→ord (Ord (od→ord t)))} (λ {x} lt → lemmah lt )) where |
605 lemmah : {x : Ordinal } → def (od (ord→od (od→ord t))) x → def (od (ord→od (od→ord (Ord (od→ord t))))) x | |
606 lemmah {x} lt = subst (λ k → def (od k) x ) (sym oiso) (subst (λ k → k o< (od→ord t)) | |
607 diso | |
608 (c<→o< (subst₂ (λ j k → def (od j) k) oiso (sym diso) lt ))) | |
609 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where | |
610 lemmai : od→ord (Ord (od→ord A)) ≡ od→ord t | |
611 lemmai = let open ≡-Reasoning in begin | |
612 od→ord (Ord (od→ord A)) | |
613 ≡⟨ sym (cong (λ k → od→ord (Ord k)) eq) ⟩ | |
614 od→ord (Ord (od→ord t)) | |
615 ≡⟨ sym diso ⟩ | |
616 od→ord (ord→od (od→ord (Ord (od→ord t)))) | |
617 ≡⟨ sym eq1 ⟩ | |
618 od→ord (ord→od (od→ord t)) | |
619 ≡⟨ diso ⟩ | |
620 od→ord t | |
621 ∎ | |
622 lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where | |
623 lemmak : od→ord (ord→od (od→ord (Ord (od→ord t)))) ≡ od→ord (Ord (od→ord A)) | |
624 lemmak = let open ≡-Reasoning in begin | |
625 od→ord (ord→od (od→ord (Ord (od→ord t)))) | |
626 ≡⟨ diso ⟩ | |
627 od→ord (Ord (od→ord t)) | |
628 ≡⟨ cong (λ k → od→ord (Ord k)) eq ⟩ | |
629 od→ord (Ord (od→ord A)) | |
630 ∎ | |
631 lemmaj : od→ord t o< od→ord (Ord (od→ord A)) | |
632 lemmaj = subst₂ (λ j k → j o< k ) diso lemmak lt | |
310 | 633 lemma1 : od→ord t o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x))) |
634 lemma1 = subst (λ k → od→ord k o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x)))) | |
311 | 635 lemma4 (sup-o< (OPwr (Ord (od→ord A))) lemma7 ) |
318 | 636 lemma2 : def (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t) |
151 | 637 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where |
638 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) | |
317 | 639 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ {t} {A} t→A ))) |
142 | 640 |
311 | 641 |
271 | 642 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a)) |
643 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where | |
303 | 644 lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y |
271 | 645 lemma lt y<x with osuc-≡< lt |
646 lemma lt y<x | case1 refl = c<→o< y<x | |
647 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a | |
262 | 648 |
276 | 649 continuum-hyphotheis : (a : Ordinal) → Set (suc n) |
650 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) | |
129 | 651 |
303 | 652 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B |
653 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
654 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
186 | 655 |
303 | 656 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) |
186 | 657 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d |
658 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d | |
129 | 659 |
223
2e1f19c949dc
sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
219
diff
changeset
|
660 infinity∅ : infinite ∋ od∅ |
303 | 661 infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where |
161 | 662 lemma : o∅ ≡ od→ord od∅ |
663 lemma = let open ≡-Reasoning in begin | |
664 o∅ | |
665 ≡⟨ sym diso ⟩ | |
666 od→ord ( ord→od o∅ ) | |
667 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ | |
668 od→ord od∅ | |
669 ∎ | |
303 | 670 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) |
671 infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where | |
161 | 672 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) |
673 ≡ od→ord (Union (x , (x , x))) | |
674 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso | |
675 | |
234
e06b76e5b682
ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
228
diff
changeset
|
676 |
303 | 677 Union = ZF.Union HOD→ZF |
678 Power = ZF.Power HOD→ZF | |
679 Select = ZF.Select HOD→ZF | |
680 Replace = ZF.Replace HOD→ZF | |
363 | 681 infinite = ZF.infinite HOD→ZF |
303 | 682 isZF = ZF.isZF HOD→ZF |
363 | 683 |